U NIVERSIT E´ C ATHOLIQUE DE L OUVAIN FACULT E´ DES S CIENCES A PPLIQU E´ ES D E´ PARTEMENT DE S CIENCE DES M AT E´ RIAUX ´ D E´ S ET DES P ROC E Unit´e de Physico-chimie et de Physique des Mat´eriaux Bˆatiment Boltzmann Place Croix du Sud 1 B-1348 Louvain-la-Neuve

INVESTIGATION OF LOVE WAVES SENSORS. O PTIMISATION FOR B IOSENSING A PPLICATIONS .

P ROMOTORS : Prof. dr. ir. P. BERTRAND Dr. ir. A. CAMPITELLI

M ASTER T HESIS SUBMITTED

IN

FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

M ATERIALS S CIENCE ENGINEER by Laurent FRANCIS in collaboration with IMEC vzw Interuniversity MicroElectronics Center Kapeldreef 75 B-3001 Leuven

2000–2001

Abstract The current demand for sensors able to determine the presence of specific molecules at low concentration levels, for the early detection of diseases and cancers, is increasing. There is a necessity to obtain sensors that present a large sensitivity to minute amount of analyte molecules supported in liquid media. Love waves biosensors present a cutting-edge technology to achieve the realisation of such sensors. Love waves are acoustic modes guided in a plate deposited on top of a substrate, their characteristics are particularly well suited to biosensing applications: a large sensitivity in comparison to other acoustic waves based devices and a shear-horizontal polarisation of the waves that present a low coupling with the liquid media. The waves are influenced by the changes in the environmental parameters. The biosensor is used as a gravimetric sensor: changes in the mass of a film on top of the structure introduce frequency shifts of the Love wave. These shifts are monitored by an electronic instrumentation and are related to the sensitivity. The theoretical study of the sensitivity is presented for two types of materials deposited on a ST-cut quartz substrate: silicon dioxide and epoxy polymer. The key factor for the study is the influence of the layer thickness. The sensitivity of the device presents a peak for an optimum thickness that depends upon the material selection. The Love waves structure was modelled by an equivalent transmission line network and the parameters of the equivalent transmission lines were extracted. The physical interpretation of the model is easily done with comparison of the structure of the device. The model serves to determine the optimum thickness corresponding to a maximum of sensitivity upon an energy consideration. This whole modelling of the Love waves biosensor was not reported previously in the literature under this form. The results obtained fit well to the experiments. In addition, the model presents the ability to study multilayered structures. In particular, the study of the composite structure based on the integration of both silicon dioxide and epoxy polymer could be achieved. It is believed from the results that such a structure presents a high potential for biosensing application in comparison to device incorporating separately the two types of materials. In the frame of this investigation, Love waves devices were fabricated and characterised. Several designs have been developed and the influence of the mounting procedure is reported. Although only a few experiments could be realised, the results achieved are promising for further developments of this technology.

R´esum´e La demande actuelle pour des capteurs capables de d´etecter la pr´esence de mol´ecules sp´ecifiques a` de faibles niveaux de concentration, pour la d´etection pr´ecoce des maladies et des cancers, est actuellement en augmentation. Il y a une n´ecessit´e a` cr´eer des capteurs qui pr´esentent une forte sensibilit´e a` de tr`es faibles quantit´es de mol´ecules sp´ecifiques support´ees par les milieux liquides. Les capteurs a` ondes de Love pr´esentent une technologie particuli`erement int´eressante a` cet e´ gard. Les ondes de Love sont des modes acoustiques guid´es dans une couche d´epos´ee sur un substrat piezo´electrique, leurs caract´eristiques sont tr`es bien adapt´ees pour leur application comme biocapteurs: une sensibilit´e e´ lev´ee en comparaison des autres dispositifs bas´es sur le principe des ondes acoustiques et une polarisation de l’onde qui pr´esente un faible couplage avec les milieux liquides. Les ondes sont influenc´ees par les modifications survenant dans leur environnement. Le biocapteur est configur´e comme un capteur gravim´etrique: les changements de masse d’un film pr´esent en surface de la structure introduisent des variations dans la fr´equence de l’onde de Love. Ces sauts de fr´equence sont suivis par une instrumentation e´ lectronique et sont reli´es a` la sensibilit´e. L’´etude th´eorique de la sensibilit´e est pr´esent´ee pour deux types de mat´eriaux d´epos´es sur un substrat de quartz de coupe ST : le dioxide de silicium et un polym`ere de type e´ poxy. Le point essentiel de cette e´ tude est l’influence de l’´epaisseur de la couche qui guide l’onde. La sensibilit´e du dispositif pr´esente un pic pour une e´ paisseur optimale qui d´epend du choix des mat´eriaux. La structure supportant les ondes de Love est mod´elis´ee par un r´eseau de lignes de transmission e´ quivalentes. Les param`etres caract´eristiques de ces lignes de transmission ont e´ t´e obtenus. L’interpr´etation physique du mod`ele est facilement compar´ee avec la structure du dispositif. Ce mod`ele est utilis´e pour d´eterminer l’´epaisseur optimale qui correspond a` un maximum de sensibilit´e selon un raisonnement bas´e sur des consid´erations e´ nerg´etiques. Cette mod´elisation compl`ete du biocapteur a` ondes de Love n’a pas e´ t´e report´ee telle quelle dans la litt´erature. Les r´esultats obtenus par cette mod´elisation sont en accord avec les r´esultats exp´erimentaux. De plus, ce mod`ele est adapt´e a` l’´etude des structures multicouches. En particulier, l’´etude des structures composites bas´ees sur l’int´egration conjointe du dioxide de silicium et du polym`ere (´epoxy) dans le mˆeme dispositif a pu eˆ tre entreprise. Il en r´esulte que cette structure pr´esente un potentiel e´ lev´e d’application en tant que biocapteur en comparaison de structures incluant ces deux mat´eriaux s´epar´ement. Dans le cadre de cette e´ tude, des dispositifs a` ondes de Love ont e´ t´e fabriqu´es et caract´eris´es. Diff´erentes architectures ont e´ t´e imagin´ees et l’influence du montage des dispositifs a e´ galement e´ t´e e´ tudi´ee. Malgr´e le fait que seulement peu d’exp´eriences ont pu eˆ tre r´ealis´ees, les r´esultats obtenus sont prometteurs pour le d´eveloppement futur de cette technologie.

Acknowledgements

I would like to thank here all the people that have been involved in the realisation of this master thesis. First of all, my promotor from UCL, Prof. Patrick Bertrand, and my promotor from IMEC, Dr. Andrew Campitelli, for their support. I acknowledge Prof. Vincent Bayot for his presence in the reading committee of my thesis. I wish to thank especially Dr. Pedro Banda for his interest in my progress and good advices he gave me, Dr. Jean-Michel Friedt for his friendly co-operation to this project and also for the technical help with the report, and Filip Frederix for the information about biochemistry. I am really indebted to the technical staff involved in the fabrication of the sensors and to the people who help me in any way for the processing: Ann de Caussemaeker, Brigitte Parmentier, Agnes Verbist, Chikhi Abdelhafid, Bert Du Bois, and Johan Mees. A special thank to Carmen Bartic and Dr. Anne Jourdain for their pleasant mood. Thank also to Wang Wenfei and Kristof Daemen. I wish to thank all my friends, colleagues, and the other thesis students from the MCP/BIO and the MCP/MEMS groups at IMEC for their enthusiasm, their advice and the (technical) conversations. All my acknowledgements to IMEC for the hosting. I am grateful to my parents for their presence and their support during all those years.

Science sans conscience n’est que ruine de l’ˆame. Franc¸ois Rabelais

Table of Contents 1

Introduction

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Acoustic waves biosensors 2.1 Sensor concepts . . . . . . . . . . . . . . . . . 2.2 Biosensors . . . . . . . . . . . . . . . . . . . . 2.2.1 Definition, market and features . . . . . 2.2.2 Acoustic transducers . . . . . . . . . . 2.2.3 Applications . . . . . . . . . . . . . . 2.3 A review of acoustic waves biosensors . . . . . 2.3.1 Bulk acoustic waves . . . . . . . . . . 2.3.2 Surface acoustic waves and similar . . . 2.3.3 Plate modes . . . . . . . . . . . . . . . 2.3.4 Surface skimming bulk waves . . . . . 2.3.5 Acoustic waves biosensors comparison 2.4 Biochemical interface . . . . . . . . . . . . . . 2.4.1 Immobilisation techniques . . . . . . . 2.4.2 Self-assembled monolayers . . . . . . 2.4.3 Mixed self-assembled monolayers . . .

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Love waves biosensors 3.1 A word of History . . . . . . . . . . . . . . . . . . . . 3.2 State-of-the-art . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Love waves biosensor . . . . . . . . . . . 3.2.2 A first experience on Love waves biosensor . . 3.2.3 Theoretical prediction of the sensitivity . . . . 3.2.4 Composite devices and recent developments . . 3.3 Transmission line equivalent model . . . . . . . . . . 3.3.1 Necessity of a modelling . . . . . . . . . . . . 3.3.2 Partial waves propagation . . . . . . . . . . . 3.3.3 Transmission line properties . . . . . . . . . . 3.3.4 Analogy between acoustic and electrical waves 3.3.5 Decomposition of the Love waves device . . . 3.3.6 Expression of the resonance condition . . . . . 3.3.7 Conclusions about the equivalent model . . . . 3.4 Simulations of the Love waves biosensor . . . . . . . . 3.4.1 Single Love plate biosensors . . . . . . . . . . 3.4.2 Composite biosensors . . . . . . . . . . . . .

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TABLE OF CONTENTS

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Instrumentation 4.1 Introduction . . . . . . . . . . . . . . . . . . . 4.2 Interdigital transducer . . . . . . . . . . . . . . 4.2.1 Principle . . . . . . . . . . . . . . . . 4.2.2 Equivalent model . . . . . . . . . . . . 4.2.3 Two ports equivalent network . . . . . 4.2.4 Design and simulation . . . . . . . . . 4.2.5 Second order effects and improvements 4.3 Dual channels oscillator configuration . . . . . 4.3.1 Introduction . . . . . . . . . . . . . . . 4.3.2 Closed loop operation . . . . . . . . . 4.3.3 Open loop operation . . . . . . . . . . 4.3.4 Stabilisation of the basic set-up . . . . 4.3.5 From open loop to closed loop . . . . . 4.4 Temperature control . . . . . . . . . . . . . . .

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Experimentation 5.1 Introduction . . . . . . . . 5.2 Elaboration of the sensors . 5.2.1 Process flow . . . 5.2.2 Devices design . . 5.2.3 Fabrication . . . . 5.3 Experimental results . . . 5.4 Analysis and interpretation

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A Table of Symbols B Acoustic waves propagation in solids B.1 Introduction . . . . . . . . . . . . . . . . . B.2 Propagation of time-harmonic plane waves . B.2.1 Elastic fields in solids . . . . . . . B.2.2 Christoffel equation . . . . . . . . B.2.3 Isotropic solids . . . . . . . . . . . B.2.4 Anisotropic solids . . . . . . . . . B.2.5 Piezoelectric solids . . . . . . . . . B.2.6 Attenuation . . . . . . . . . . . . . B.2.7 Slowness surfaces . . . . . . . . . B.3 Reflection and refraction . . . . . . . . . . B.3.1 Snell’s law . . . . . . . . . . . . . B.3.2 Critical angle . . . . . . . . . . . . B.3.3 Acoustic impedances . . . . . . . . B.4 Energy considerations . . . . . . . . . . . . B.4.1 Acoustic Poynting’s Theorem . . .

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Complex acoustic Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 101 Energy transport of plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

C Materials data

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D Devices layout

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E Simulation codes E.1 Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2 Sensitivity by the theory of perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . E.3 Transmission line model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Bibliography

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List of Figures 1.1

Organisation of the approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

The sensor principle: the sensor transfer a physical signal IN into an other physical signal OUT. The relation between IN and OUT is known to be the transfer function of the sensor. Transfer function of a sensor: the relation between IN and OUT when IN is varying (dynamic conditions) can follow any kind of curve that depends on the properties of the sensor. Variations of IN can be around a certain point, the operating point, where the transfer function can be approximated at the first order by a straight line. The slope of this line is the sensitivity of the sensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of biosensors. Biosensors are made of several elements in order to capture specific biochemical species in an analyte and transform one property of the species into an electrical signal. A whole biosensor system includes also signal processing and data display to the final user. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The acoustic waves family. This family is fairly large and has to be distinguished between bulk acoustic waves (BAW) and surface generated acoustic waves (SGAW). . . . . . . . . A bulk acoustic waves biosensor : the quartz crystal microbalance (QCM) is the typical example of BAW resonating in a thickness shear mode (TSM). . . . . . . . . . . . . . . . Configuration of biosensors. Drawings include the presence of the electrodes for transducing and position of the liquid cell. The liquid cell contains the biochemical species and is a part of a more complex system of liquid delivering around the biosensor. . . . . . . . . Alkanethiols in contact with gold lead to a self-assembled monolayer. . . . . . . . . . . . Target principle of biosensor. The biochemical species present in the analyte can attach to specific probes (bioreceptors) at the surface of the sensor. The bioreceptors are embedded in a self-assembled monolayer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The biological recognition layer based on mixed SAMs and a covalent bound antibody. . .

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A typical Love waves biosensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guideline for the selection of materials as Love plate for quartz based devices. The base 2 logarithmic value of the theoretical sensitivity is expressed as a function of the relative density and the relative acoustic wave velocity in the material in regards to the quartz. The silicon dioxide and the PMMA are reported as examples. . . . . . . . . . . . . . . . . . Experimental measurements of the mass sensitivity of silicon dioxide devices (diamond dots) and comparison with the theoretical prediction (solid line) given by the perturbation theory after Du et al. [DHCD97]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental measurements of single material Love plate (only silicon dioxide or only PMMA) and of composite devices (silicon dioxide and PMMA) after Du et al. [DH98]. The measured (left) mass sensitivity, and (right) insertion loss as a function of the total overlay thickness are represented. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

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Structure of the Love waves device. Axis are set as following: the wave propagates in the direction x, the direction z is aligned to the normal of the layers. At least one layer has to enter the structure of the device, but several layers numbered from 1 to n are allowed to be a part of the structure. On top of the device, the vacuum is usually considered but it could also be a newtonian liquid or a very thin layer (for instance a gold layer with self-assembled monolayer on it). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Love wave results from the interaction in each layer of partial waves propagating in opposite direction along z but with the same component along x. The common component along x gives the Love wave velocity. The partial waves are described by the slowness curves for each layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The partial wave propagates along direction z0 that results of a rotation of the axis around y by an angle θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infinitesimal equivalent circuit of a transmission line. . . . . . . . . . . . . . . . . . . . . The equivalent transmission line model of the propagation of shear wave in different medium: (left) isotropic, lossless medium, (center) isotropic, lossy medium, and (right) piezoelectric, lossless medium. The inductance L has the value ρ and the capacitance C the value c−1 44 . For a viscous medium, the shunt arm is the series combination of the mechanical capacitance with a conductance G that has the value η−1 44 . For a piezoelectric media, a piezoelectric capacitance C piezo is included in series with the mechanical capacih  2 i−1 ex4 2 . . . tance. It results in a stiffened capacitance of value Csti f f ened = c44 + εS sin 2θ xx The equivalence between shear waves in solids and transmission line is applied to the Love waves device. (a) For a proper Love wave velocity, each layer (layer 1,2, . . . ) has a different angle of propagation θi (with i associated to each layer). (b) In the longitudinal direction x, the layers structure is the superposition of the equivalent transmission line of each layer. These lines are only linked by the angles θi and they admit all the same propagation function which corresponds to the Love wave velocity. (c) In the transversal direction z, the device is the series network of the transmission lines associated to each layer. The Love wave corresponds to a resonating scheme in this structure. . . . . . . . . . The equivalent network in the direction z set the resonance condition. The line presents two impedances Z + and Z − at any point z0 of the layered structure. These two impedances are a function of the geometry of the structure, of the materials combination, and of the angles of propagation of the partial waves. . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of the procedure to follow to determine the Love modes in the frame of the equivalent transmission line model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Love wave phase and group velocities as a function of the normalised thickness for (left) silicon dioxide and (right) polymer Love plate. The 0th order and 1st order Love modes are presented. For a given frequency, the Love modes appear at a constant variation of the Love plate thickness that is a function of the materials parameters. . . . . . . . . . . . . . Repartition of the 0th order Love mode energy in a cross-section of the device for (left) silicon dioxide and (right) polymer Love plate as a function of the normalised thickness. The total amount of energy is constant. For increasing values of the thickness, the energy is more embedded in the Love plate and subsequently less energy is present in the substrate.

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3.15 Position of the maximum of energy as a function of the normalised thickness for (left) silicon dioxide and (right) polymer Love plate. The position is computed from the origin located at the interface between the Love plate and the substrate. For low value of thickness, the position of the peak of energy is attached close to the top of the device. For a critical value, the peak shifts towards the interface between the substrate and the Love plate. 3.16 Gradient of the energy in the substrate as a function of the normalised thickness for (left) silicon dioxide and (right) polymer Love plate. The gradient presents a peak for a specific normalised thickness. This peak is sharper for the polymer than for the silicon dioxide. . . 3.17 Composite Love waves device is composed of a polymer plate on top of a silicon dioxide plate. Love wave (left) phase and (right) group velocity as a function of the normalised thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18 Composite device. The position of the maximum of energy in each layer is plotted with respect to its normalised value (ratio position of the maximum in the layer to the thickness of the layer). The position is plotted for (left) silicon dioxide and (right) polymer Love plate. 3.19 Composite device. Repartition of the energy in the (top left) substrate, the (top right) silicon dioxide and the (bottom left) polymer as a function of the normalised thickness. For each layer, the values have been normalised between 0 and 1. The gradient of energy for increasing thickness of the polymer layer is given in the bottom right figure. . . . . . . 3.20 Graphical representation of the Love wave in the real space as a function of the propagation angle. Left: single layer device. Right: double layers device. . . . . . . . . . . . . . . . . 4.1

4.2

4.3 4.4 4.5 4.6 4.7 4.8 4.9

Instrumentation of the Love wave biosensor. Two interdigital transducers and a space between them are the elements of a delay line. The delay line and a RF amplifier constitute a closed-loop oscillator. A device is made of two delay lines (or two channels). An active area is presents at the top of one of them, that is the biosensitive area. The signals of the two delay lines are mixed together and filtered (low-pass) to obtain the synchronous frequency of the biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the delay line: input and output interdigital transducers (IDT) are metal fingers at the surface of a piezoelectric substrate, the spacing between them determines the delay time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Crossed-Field equivalent model of the IDT . . . . . . . . . . . . . . . . . . . . . . The In-line equivalent model of the IDT . . . . . . . . . . . . . . . . . . . . . . . . . . The interdigital transducer is equivalent to a three-ports network . . . . . . . . . . . . . The Love wave device transfer function can be computed by this two-ports equivalent network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transfer function of the device with acoustic velocity of 5060 m/s and vacuum on top of the quartz substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The design can be improved to reduce diffraction from the input IDT and the reflections from the output IDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic circuit (left) of an oscillator whose frequency is set to the resonance frequency of the SGAW device (Φ and G must then verify the condition ∑ Φ = 0 (sum of the phases must be null so that the output signal is equal to the input signal) and ΠG ≥ 1 (energy input must be greater or equal to the energy losses in the circuit) for the oscillator to run). Middle and right: typical frequency responses (amplitude at center, phase at right) of an acoustic wave device (lithium tantalate substrate). . . . . . . . . . . . . . . . . . . . .

39

40

41

41

42 44

. 46

. . . .

47 48 48 48

. 50 . 52 . 54

. 55

vii

LIST OF FIGURES

4.10 Observed frequencies of a dual-lines configuration sensor with lithium tantalate as substrate (frequency of each line, and difference (compensation for external disturbances to the lines). Considering a temperature coefficient of 30 ppm/K for lithium tantalate, a stability of 1 kHz on each individual frequency of 100 MHz means temperature fluctuations of 0.3 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Comparison between the amplitude response of an acoustic wave device as observed by the HP network analyser, and the VCO. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Single loop PLL synthesiser principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Dual loops PLL synthesiser principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Left: short term stability of the VCO (frequency fluctuations within 700 Hz, acquisition time: 1h). Middle: long term (in)stability of the VCO (acquisition during 5h30). The two sharp frequency shifts are due to touching the output cable. Right: sensitivity of the VCO to the output emitter-follower NPN transistor’s polarisation voltage (moved from 12V to 15V, inducing a 30 to 50 kHz shift, certainly due to impedance variation). . . . . . . . . 4.15 Top view of a metal strip as temperature sensitive element . . . . . . . . . . . . . . . . 5.1 5.2 5.3 5.4

5.5 5.6 5.7

5.8

5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21

. 56 . 57 . 58 . 59

. 59 . 61

Process flow for the fabrication of the devices . . . . . . . . . . . . . . . . . . . . . . . . Left: open circuit in the IDEs due to not-well cleaned wafer surface. Right: problem in the CVD silicon dioxide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Left: the resist can stick to the mask (hard-contact photolithography) and damage the pattern of the IDEs. Right: open circuit on the IDEs caused by dust or hair. . . . . . . . . Left: over-etching of the silicon dioxide has also remove the aluminium contact pad, the resist not postbaked can hardly sustain the buffer HF and has cracked on the edges. Right: gold lift-off, edges are critical points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . When everything is running fine, the result looks better. Left: a partial view of the device. Right: IDEs well patterned and not over-etched have the same width and are equally spaced. Test sensor: the quartz wafer is not covered by a Love plate. Left: repartition of the tested devices. Right: theoretical voltage transfer function for this configuration. . . . . . . . . . Silicon dioxide sensor: the quartz wafer is covered with silicon dioxide, thickness 6.8µm. Left: repartition of the tested devices. Right: theoretical voltage transfer function for this configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymer sensor: the quartz wafer is covered with the polymer, thickness 3.3µm. Left: repartition of the tested devices. Right: theoretical voltage transfer function for this configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (Vacuum) Left: device 13, line 1, mounting 1. Right: device 51, line 1, mounting 3. . . . . (Vacuum) Device 14, line 1, mounting 2. Left: in the air. Right: with water on top . . . . . (Vacuum) Device 31, mounting 3. Left: line 1. Right: line 2. . . . . . . . . . . . . . . . . (Vacuum) Device 41, line 2, mounting 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . (Vacuum) Device 61, mounting 1. Left: line 1. Right: line 2. . . . . . . . . . . . . . . . . (SiO2 ) Left: device 13, line 1, mounting 3. Right: device 14, line 2, mounting 4. . . . . . . (SiO2 ) Device 53, line 1, mounting 4. Left: in the air. Right: with water on top. . . . . . . (SiO2 ) Device 54, mounting 1. Left: line 1. Right: line 2. . . . . . . . . . . . . . . . . . . (SiO2 ) Device 31, mounting 4. Left: line 1. Right: line 2. . . . . . . . . . . . . . . . . . . (SiO2 ) Device 41, line 2, mounting 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (SiO2 ) Device 44, line 1, mounting 1. Left: 20 MHz of span. Right: 2 MHz span. . . . . . (Polymer) Device 14, mounting 1. Left: line 1. Right: line 2. . . . . . . . . . . . . . . . . (Polymer) Device 44, mounting 1. Left: line 1. Right: line 2. . . . . . . . . . . . . . . . .

63 65 66

66 67 68

68

69 69 70 70 71 71 72 72 73 73 74 74 75 75

viii

LIST OF FIGURES

5.22 Evolution of the signal in the temperature sensor from room temperature 20 ◦ C to approximately 30 ◦ C for (left) 5 s and (right) 10 s. . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.23 Closed-loop operation of the silicon dioxide device with water. The oscillator was monitored in open air for 900 seconds before getting water. The water was removed from the device after 1600 seconds. The in-box is a close-up view of the first 600 seconds. . . . . . 76 IL −ILmean 5.24 Graphical representation of the ratio IL peak . Left: vacuum devices. Right: CVD peak +ILmean silicon dioxide devices. The abscise is formatted to give Device-Line,Mount. . . . . . . . . 79 B.1 Linear model of solid : atoms with mass M linked by elastic bonds of constant K . . . . B.2 Slowness curves for silicon dioxide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Slowness curves for YZ-plane in LiNbO3 with (solid lines) and without (dashed lines) taking account of piezoelectricity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Slowness curves for ST-cut α − SiO2 without taking account of piezoelectricity. . . . . . B.5 Reflections and refractions at boundary between two isotropic solids. . . . . . . . . . . . B.6 Relation of dispersion: phase velocity V and group velocity Vg are represented . . . . . .

. 86 . 93 . . . .

94 96 98 103

C.1 Materials matrices for some classes. (a) The matrix is symmetric and has three submatrices (a 6-by-6 matrix for elastic constants, a 3-by-6 matrix for the piezoelectric constants, and a 3-by-3 matrix for the dielectric constants) (b) isotropic and cubic m3m (c) trigonal 32 (d) trigonal 3m. The • and the ◦ are non-zero components. The linked components are identical if the colour is the same and opposite if the colour differ. The × is given by (c11 − c12 )/2 except for the cubic case. . . . . . . . . . . . . . . . . . . . . . . . 105 D.1 D.2 D.3 D.4 D.5 D.6 D.7

Repartition of the devices on the 3” quartz wafer Design of device 11,13, and 14 . . . . . . . . . . Design of device 22, 23, and 24 . . . . . . . . . . Design of device 31, 32, and 33 . . . . . . . . . . Design of device 41, 42, and 44 . . . . . . . . . . Design of device 51, 52, 53, and 54 . . . . . . . . Design of device 61, 62, 63, and 64 . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

106 107 108 109 110 111 112

E.1 Program structure for the sensitivity given by the theory of perturbation . . . . . . . . . . 114 E.2 Program structure for the transmission line model . . . . . . . . . . . . . . . . . . . . . . 117

List of Tables 2.1 2.2 2.3

Some applications of acoustic waves sensors . . . . . . . . . . . . . . . . . . . . . . . . . 8 Biosensors comparison after Du et al. [DHCD97] . . . . . . . . . . . . . . . . . . . . . . 12 Analytical techniques for monolayers characterisation . . . . . . . . . . . . . . . . . . . 16

3.1 3.2

Thickness and properties of various overlayer materials corresponding to optimum sensitivity Comparison between the propagation of a shear wave in an isotropic medium and the propagation of an electrical wave in a transmission line. . . . . . . . . . . . . . . . . . . . Summary of the equivalent transmission line parameters for isotropic solids that have a bulk wave phase velocity V . The impedance Z and the admittance Y are defined for an infinitesimal length. For the line itself, the relevant parameters are the characteristic impedance Zc and the propagation function γ. . . . . . . . . . . . . . . . . . . . . . . . . Materials data for the simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normalised thickness for single layer devices at the optimum of sensitivity and after different approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

3.4 3.5

21 30

30 36 43

4.1 4.2

IDT design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Metals resistivity and first-order temperature coefficients at 20 ◦ C . . . . . . . . . . . . . 61

5.1

Summary of the results for the tested devices . . . . . . . . . . . . . . . . . . . . . . . . 78

B.1 Convention for subscript notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 C.1 density (in kg · m−3 ) and elastic constants (in 1010 N · m−2 ) . . . . . . . . . . . . . . . . . 104 C.2 (piezo)electric constants (in C · m−2 ) and dielectric constants (in 10−11 F · m−1 ) . . . . . . 104

ix

Chapter 1 Introduction The current demand for sensors able to determine the presence of specific molecules at low concentration levels, for the early detection of diseases and cancers, is increasing. There is a necessity to obtain sensors that present a large sensitivity to minute amount of analyte molecules supported in liquid media. A wide variety of transducing systems are suited to this detection and among them acoustic waves based biosensors are a cutting-edge technology particularly interesting for this use. The possibility to achieve small, fast, reliable, low cost biosensors based upon that technology explains the interest they obtain for the past decade. In particular, Love waves biosensors are among them probably the best sensor achieved up to now. This specific biosensor is currently in development and no industrial production of this device has been reported. A good understanding of its characteristics and a complete picture of its ability to become a self-standing biosensor are a priority for its investigation. The design of Love waves biosensor is various; a specific combination of materials has to be selected for an experimental study. This selection, based on the theoretical and experimental approaches reported in the literature, emerge to a necessary optimisation of the sensing properties of the biosensor. This report is dedicated to the investigation of the Love waves sensors and to the optimisation for biosensing applications. The chapter 2 introduces a definition of the biosensors and their market. The efficiency of the Love wave biosensor is presented, its high sensitivity to mass loading makes it clearly adapted to biosensors applications and probably one of the most interesting acoustic waves based device for such application. The selectivity of the device towards biochemical species is realised by an interfacial chemistry. This chemistry based on alkanethiols self-assembled monolayer on gold is briefly presented at the end of the chapter since it constitutes a part of the device. This part is the chemical part of the sensor, for this study only the transducing part is investigated for optimisation. The investigation of the properties of the Love waves and the sensitivity of the device for a mass change of the interfacial layer is presented and detailed in chapter 3. The usual structure of the Love waves device is a material layer deposited on top of a piezoelectric substrate. Love waves are acoustic modes guided in the layer; they present a shear-horizontal polarisation. Their properties of propagation are strongly dependent upon the combination of materials and the thickness of the layer. The literature has reported study of silicon dioxide and polymer as layer on top of a quartz substrate. The integration of both materials leads to a composite device that present interesting properties from a theoretical point of view. The integration of polymers in the device enhances strongly its sensitivity but also increase the acoustic damping. The integration of an epoxy polymer is a part of the investigation led in our study. For this study of the device, an equivalent model was established that presents a high potential to obtain a complete picture of the Love waves biosensor. The sensitivity of the device is qualitatively approached by the variation of the repartition 1

2

CHAPTER 1. INTRODUCTION

of the energy in the structure as a function of the layer thickness. A comparison between the devices is done and some conclusions are expressed for the processing of the sensors. The acoustics waves are launched and monitored in the structure by interdigital electrodes deposited on the surface of the piezoelectric substrate. The instrumentation of the sensor by this means is the object of the chapter 4. A model derived from the literature helps the design procedure for the device fabrication. The device operates as the element of an oscillator configuration. The conditions of existence for the oscillation are presented and the problem of the electronics going around for the monitoring is detailed. The change in frequency of the oscillator is related to the frequency of the device. A method to monitor the temperature drifts in-situ is also presented. In the frame of this investigation, a procedure has been set to fabricate operating devices. A certain number of devices were fabricated and electrically characterised. The experimental results and their analysis are the object of chapter 5. Finally, some conclusions are withdrawn from this investigation and future outlines are set for further developments of the Love waves biosensors in the chapter 6. The approach done for this investigation is summarised in the Figure 1.1. The theory of the acoustic waves propagation serves for the elaboration of a model used for the simulations. These simulations are applied to structure reported in the literature so that an estimation of the efficiency of the model could b established. The results of that combined approach give rise to the characteristics of a set of theoretically optimised Love wave biosensors. A fabrication step has the goal to obtain such devices. Fabricated devices will be electrically characterised under various conditions and in many configurations. The results are analysed to eventually integrate them back into the simulations until a real optimisation will be deduced.

BIOSENSORS

ACOUSTIC WAVES THEORY

STATE-OF-THE-ART

SIMULATIONS

THEORETICALLY OPTIMISED LOVE WAVE BIOSENSOR

FABRICATION

CHARACTERISATION

Figure 1.1: Organisation of the approach

Chapter 2 Acoustic waves biosensors “La vie du pr´esent tisse celle de l’avenir.“ Gustave Le Bon, Hier et Demain

2.1 Sensor concepts This chapter introduces the concepts associated to biosensors. In this introduction, the sensor is represented as a system that transfers any environmental signal into other signal that is more easily processed. The relation between the two signals is given by a key factor in the development of any sensor: the sensitivity. The best sensor is not the sensor that presents the best sensitivity; other factors enter the selection of a sensor. When the environmental parameter that needs to be sensed is related to chemical or biological species, the sensor becomes a biosensor. Biosensors represent a class of sensor, their global market is relatively large and is called to become larger in the future. The choice of a particular transducing system is dictated by some basic necessities; the biosensor has to show in the same time good responsiveness, feasibility to process easily and has to give a sensitivity that relates the mechanisms behind the immobilisation technique needed by (bio)chemical species. Acoustic transducers easily encounter those necessities and acoustic waves based biosensors are given here as illustrative examples, among them the Love modes biosensor. Finally, the significant role played by immobilisation techniques is developed in the last section. A necessary approach in the conception of any sensor is the final goal that needs to be reached by the sensor. A review of all necessary properties would be very extensive and have to be detailed. However, a clear and concise view of the sensor is necessary for the designers to reach objectives within fabrication delays, with fulfilment of the desired properties and quality. A first insight in sensors is given by their fundamental properties. Sensors react to a change of their environment by a change of their own properties [Fra96]. Sensors work in static or dynamic systems. In a static system, an absolute value of the sensed environmental parameter IN gives an output signal OUT. A general representation of sensors is a system that transform a physical signal into an other physical signal1 as represented in Figure 2.1. Under dynamic conditions, the signal IN evolves and the signal OUT follows. The relation between the two signals is the transfer function of the sensor. In some cases, the transfer function is approximated at the first order around an operating point that is given by the mean value of IN as depicted in Figure 2.2. 1a

system that can reciprocally transform the IN signal into the OUT parameter is an actuator

3

4

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

ENVIR. SIGNAL

IN

SENSOR OUTPUT

SENSOR

OUT

Figure 2.1: The sensor principle: the sensor transfer a physical signal IN into an other physical signal OUT. The relation between IN and OUT is known to be the transfer function of the sensor.

Signal OUT Transfer curve

First order approximation of the transfer curve

Sensitivity S

Signal IN Operating point

Figure 2.2: Transfer function of a sensor: the relation between IN and OUT when IN is varying (dynamic conditions) can follow any kind of curve that depends on the properties of the sensor. Variations of IN can be around a certain point, the operating point, where the transfer function can be approximated at the first order by a straight line. The slope of this line is the sensitivity of the sensor.

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

5

The linear relation is described as: OUT = S · IN + OFFSET.

(2.1)

The term S is designated as the sensitivity of the sensor and OFFSET the sensor output at zero input signal. The sensitivity S is thus the relative change in the sensor response to a relative change in a parameter of the sensor environment. Sensitivity is the key factor for the development of any sensor. Usually, the parameter that evolves in the environment changes an intrinsic property of the sensor, and the way that is modified is considered as the sensor’s response. One can see that a larger sensitivity results in a higher output signal. Although a large signal output is usually required for instrumentation around the sensor, the highest sensitivity is not an ultimate goal for two principal reasons. The first one is the influence of the noise on the sensor. A higher sensitivity increases the signal over the minimum detection threshold, giving then an output signal that is easily confused with a real signal, i.e. a change in the environmental parameter although only noise has presented fluctuation. Excessive increase in term of sensitivity does not lead immediately to an increase of sensor’s efficiency since the noise quickly reduces the signal’s quality, reducing in the same time the general performances of the sensor. The second factor is given in term of material limit: highest sensitivities quickly overload the sensor and then produce a bad signal that can even destroy the sensor itself. An optimum of sensitivity need to be reached, which is done by simulations, experiments, trials and errors. It is obvious that the best sensor would produce a transfer function that fits as close as possible a linear curve. For systems with high frequency in the rate of change, a good responsiveness is wanted to follow closely the variations of the environment.

2.2

Biosensors

2.2.1 Definition, market and features Biosensors have been defined by the Union of Pure and Applied Chemistry (IUPAC) Commission as a subgroup of chemical sensors in which a biologically based mechanism is used for detection of the analyte [SR94]. In variety of industrial, medical, health care and consumer applications, small, reliable, disposable and inexpensive biosensors are needed. The global market for biosensors has increased rapidly during the last decade, with an annual rate of 10%. For the year 1996, the global sale reached about $10-15 billion. In the future, it is quite reasonable to think this demand will still increase due to breakthroughs in bio-related areas such as health care, biotechnology, pharmaceutical, . . . Current examples are the growing use of sensors in personal glucose testing for diabetics, cancers and HIV detection. Although the potential market for biosensors is known to be very large, the commercialisation of the numerous biosensors has been slow except rare cases including glucose monitoring, single analyte enzyme electrodes for food industry and biological oxygen demand for water. The limit has been imposed by a lack in the desired performance characteristic of the biosensors in terms of sensitivity, dynamic range, and reproducibility. Anyway, the potential advantages offered by biosensors technology include minimal sample preparation, high speed of analysis and the potential for in situ and flow stream analysis for process control [LL98]. Biological sensing has been demonstrated with potentiometric and amperometric sensors using enzymatic mediation. These sensors measure the electrochemical activity of the enzyme. Potentiometric systems measure the open-circuit voltage, while the amperometric systems measure the short-circuit current. Chemiresistive devices measure a change in electrical conductivity caused by a biochemical reaction.

6

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

Thermal sensors can also be used to detect antibody-antigene and/or enzymatic detection. These devices are based on a change in thermal conductivity between a heater and a heat detector. Optical fibers and optical waveguides have been used in many configurations including phase interferometry, optical absorption, fluorescence and surface plasmon resonance. The technological advances in microelectronics and fibre optics have dramatically increase the number of physical transducers which are available for use in biosensors. In addition, breakthroughs in biochemistry, immunochemistry and biotechnology have made available large amount of well characterised enzymes, antibodies, tissues, living cells and receptors for use in these devices. Biosensor components include a biological recognition element, a physical transducer, an electrical amplifier and a data processing system. Figure 2.3 reports the schematic of biosensors.

Bioreceptors and biofilm

Physical transducer

Electrical signal

Electrical amplifier

Data acquisition and display

Figure 2.3: Schematic of biosensors. Biosensors are made of several elements in order to capture specific biochemical species in an analyte and transform one property of the species into an electrical signal. A whole biosensor system includes also signal processing and data display to the final user. Biological sensing elements may include any biological materials that can selectively recognise an analyte or class of analytes of interest. Biosensors are usually classified by their physical transducers (optical, electrochemical, thermal, acoustic, . . . ). Theoretically, suitable biological species such as enzymes, receptors, immunochemicals, etc., can be interfaced to any of those transducers to measure an appropriate analyte. When considering biosensors, important characteristics for the user need to be taken into account for right selection of the transducer and the biological interface. Some factors need to be considered in charaterising biosensors for potential applications [SR94]: • Is the method have to be reversible, irreversible or regenerable ? • Do reagents other than the analyte of interest have to be added ? • What is the biosensor interface with its environment ? • What are the physical characteristics of the system (i.e. fixed laboratory, semi-portable or portable) ?

2.2.2 Acoustic transducers Acoustic wave based biosensors measure the changes in acoustic wave properties. The changes are due to interactions between the acoustic waves and their environment. Acoustic waves sensors are fabricated with piezoelectric materials in order to make an electro-mechanical transduction to launch and detect acoustic waves. Used materials entering frequently in the fabrication of the sensors include quartz (α − SiO2 ),

7

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

lithium niobate (LiNbO3 ) and lithium tantalate (LiTaO3 ). Piezoelectric materials are very stable, chemically inert and present good mechanical properties and aging characteristics. The biological part of the sensor is realised by adding selective bioreceptors in a biofilm, usually Langmuir-Blodgett films or selfassembled monolayers. Mechanical and/or electrical interaction between the surrounding environment and the bioreceptor will cause a change in the attenuation or in the velocity of the acoustic waves, which is then monitored as the sensor response [AV94]. The acoustic transducers can be distinguished among three groups: • Homogeneous acoustic biosensors: the measurand influences directly the electroacoustic properties of the piezoelectric element. Liquid density and viscosity are sensed. Example: quartz crystal microbalance 2 . • Hybrid acoustic biosensors: the sensors are constituted of a piezoelectric element and a sensing element selective to the measurand of interest. A wide variety of measurands can be detected by changing the properties of the film, those changes are detected by the piezoelectric element. The (thin) film can be biological, metal oxide or polymer and can be used to probe both gaseous and/or liquid media. Examples: mercury detector (thin gold film), water vapour sensor (polyimide film). • Transmission acoustic biosensors: the piezoelectric element is no more involved in sensing but for transmission and/or reception of acoustic waves into/from the environment. The changes in acoustic properties of the media supporting the waves such as density or sound absorption cause change in velocity. Examples: bulk acoustic wave immunosensor, particle size distribution sensor, acoustic emission chemical sensor. Changes that can have influence are numerous: mass, elasticity, viscosity, dielectric (optical), conductivity (electronic, ionic and thermal). The sensitivity is known to be very high. This large sensitivity is due to the mixed character of acoustic waves since they involve both mechanic and electric fields. This is in contrast to many of the other microsensor technologies that usually monitor a change in a single material property. Subsequently, as the response of an acoustic waves biosensor is the combination of all those parameters, the problem of overlapping sensitivities is made more complex and need to be addressed carefully during design. A first expression of that sensitivity can be expressed by the change in wave phase velocity by the mass (mass), electrical (elec), mechanical (mech) and environmental (envir) parameter properties [Cam97]: S'

2.2.3

∆V V

= V1



∂V ∂mass

∂V ∂V ∂V ∆mass + ∂elec ∆elec + ∂mech ∆mech + ∂envir ∆envir



(2.2)

Applications

Some applications of acoustic waves sensors are reported in Table 2.1 [Whi98]. Biosensors find applications in medical diagnostic, environmental monitoring, biotechnology and pharmaceutical industry; the medical area is especially important. The fact that acoustic waves biosensors are able to sense acoustic/mechanical and electric/dielectric properties of the measurand is clearly the unique strength of acoustic sensor technology. A very promising application is immunosensor (they are affinity sensors, they have a high selectivity toward biological species). The market of immunosensors is approximately $1 billion 2 next

section details this specific device

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

8

Characterisation of thin-film materials Film thickness, real-time monitoring of deposition or removal Real-time monitoring of materials modification Corrosion and diffusion Adsorption at surfaces from gases and liquids Surface area and pore size distribution Characterisation of liquid properties Density and viscosity (including cells and macromolecules) Electrochemical studies and sonoelectrochemistry Acoustoelectric interactions to probe solution electrical properties Characterisation of polymers Viscoelastic properties (storage and loss moduli) Temperature transition Diffusion and permeation Chemical and biological sensing Vapour determination (concentration, composition) Biological determination (concentration, identification) Biomedical diagnosis (antigen, DNA, RNA, proteins, bacteria, . . . ) Table 2.1: Some applications of acoustic waves sensors

annually [LL98]. In addition, acoustic waves biosensors can monitor the kinetics of immunological reactions. They were also used for monitoring of DNA hybridisation, detection of urease enzyme, detection of gases or volatile compounds such as H2 S, CO, O2 , NO2 , aliphatic and hydrocarbons. Finally, sensors were also designed for food quality control [LL98].

2.3

A review of acoustic waves biosensors

Acoustic waves can be distinguished between two groups: bulk acoustic waves (BAWs) and surfacegenerated acoustic waves (SGAWs). The acoustic waves family is fairly large and represented in Figure 2.4. The SGAWs travel along or near free surface such as surface acoustic waves (SAW). The Bleustein-Gulyaev is a special kind of SAW. Love waves and pseudo-SAWs are also considered as surfacegenerated acoustic waves although they differ slightly from them by the principle. In this section, we will briefly present the acoustic waves given in Figure 2.4 and look at them as biosensors.

2.3.1

Bulk acoustic waves

Bulk acoustic waves travel in the bulk of a material and present interaction at opposite surfaces. BAW were the first to be developed as acoustic waves based sensors, they were then used for detection of gases and particle matter. The more representative device is the quartz crystal microbalance (QCM) which was firstly developed for monitoring of thin films in microelectronics. As biosensors, these devices are used for antibody-antigene monitoring. Shear waves are used for minimising power dissipation in fluid supporting the biochemical species.

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

9

Figure 2.4: The acoustic waves family. This family is fairly large and has to be distinguished between bulk acoustic waves (BAW) and surface generated acoustic waves (SGAW).

The sensor is based on a resonance principle, a stationary wave is generated and maintained in a plate with thickness multiple of a half-wavelength. This mode is called thickness shear mode (TSM). The bulk acoustic wave phase velocity and the boundary conditions (electrical and mechanical at surfaces) determine the resonance frequencies. Although in principle any kind of piezoelectric material can be employed, the QCM uses the AT-cut of quartz. This cut has been studied for more than 50 years; it provides a shear wave vibration with a resonance vibration that has the advantage to be stable in a large temperature domain. A representation of QCM is given in Figure 2.5 (courtesy of A. Campitelli [Cam97]).

Figure 2.5: A bulk acoustic waves biosensor : the quartz crystal microbalance (QCM) is the typical example of BAW resonating in a thickness shear mode (TSM).

QCM is called “microbalance” due to the frequency decrease when loaded by a mechanical charge at its surfaces. The QCM exhibit best sensitivity for the first modes of resonance and for higher operating

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

10

frequencies. A very sensitive device could then be obtained with a very small thickness but problems for realisation and holding of thinner thickness impose a maximum in term of sensitivity.

2.3.2 Surface acoustic waves and similar Surface acoustic waves (SAW) are known since Lord Rayleigh 3 suggested that they “play an important part in earthquakes, and in the collision of elastic solids.” For that reason, surface acoustic waves are equally named Rayleigh waves. These waves have displacements that decay in an exponential fashion in depth beneath the surface. Almost all of the elastic energy is concentrated within a distance of the order of a wavelength below the free surface. The particle motion on the surface and at each depth is elliptical and entirely contained in the plane perpendicular to the surface. Rayleigh waves know successful applications in the telecommunications with development of filters and delay lines. The deposition of interdigital electrodes at surface of piezoelectric media has widely opened the field of investigations for the gigahertz frequency range. As sensors, they also obtained very enthusiastic investigations and were successfully applied as temperature, pressure, acceleration, (high) voltage, polymer phase transitions and gas concentration sensors but less for biosensing applications. Indeed, the normal component of the Rayleigh waves couple easily with liquid and thus suffer from strong attenuation as they propagate. The literature reports sometime SAW biosensors although the presented devices does not present the SAW properties; actually those sensors support other type of waves (Acoustic Plate Modes or Love Modes) and can then not be considered as pure SAW devices. Bleustein-Gulayev (BG) waves are pure shear horizontal waves propagating in piezoelectric substrates. As any SH waves, BG waves have a high potential for fluid-sensing applications. They have a single electrical component that is perpendicular to the crystal plane. The waves are sensitive to electrical and viscous properties. The BG waves can not propagate in α − SiO2 , lithium niobate, nor in lithium tantalate. Some materials that are candidates to BG are lithium iodate (LiIO3 ), lithium borate (Li2 Bi4 O7 ), bismuth germanate (Bi12 GeO20 ) and KTP (KTiOPO4 ). Up to now, only this latter one has received interest as immunosensor. Pseudo-SAWs propagate along the surface of piezoelectric substrates and attenuate in the direction of propagation. They have a small component of propagation directed to the bulk of the substrate. For that reason, they are also called leaky-SAW. They appear only for certain orientations and have a phase velocity higher than that of the SAW. For some material and cut combination, the attenuation may be very small or almost zero, for instance in the Y −rotated X−propagating cut of lithium niobate and lithium tantalate. Furthermore, the leaky-SAW in those latter materials has almost a shear-horizontal component. The major drawback of the pseudo-SAW is interference from the spurious bulk modes. 3 John

William Strutt (1842-1919), Lord Rayleigh, British physicist. His first paper in 1865 was on Maxwell’s electromagnetic theory. He worked on propagation of sound and, while on an excursion to Egypt taken for health reasons, Strutt wrote Treatise on Sound (1870-1871). In 1879 he wrote a paper on travelling waves, this theory has now developed into the theory of solitons. His theory of scattering (1871) was the first correct explanation of why the sky is blue. In 1873 he succeeded to the title of Baron Rayleigh. From 1879 to 84 he was the second Cavendish professor of experimental physics at Cambridge succeeding Maxwell. Then in 1884 he became secretary of the Royal Society. Rayleigh discovered the inert gas argon in 1895 with Ramsey, work which earned him a Nobel Prize, in 1904. He was awarded the De Morgan Medal of the London Mathematical Society in 1890 and was president of the Royal Society between 1905 and 1908. He became chancellor of Cambridge University in 1908 [source : Encyclopædia Britannica].

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

11

2.3.3 Plate modes Acoustic plate modes (APM) are waves excited in a plate. These modes may have displacements that are either transverse to the propagation direction (shear-horizontal, SH-APM, and shear vertical, SV-APM) or in the propagation direction (longitudinal, L-APM). In terms of liquid-sensing applications, SH-APM are the most attractive since they do not present coupling with the liquid so no energy is radiated in the liquid. Piezoelectric materials that have receive much attention for that application are the ST-cut in quartz and the Z−cut X−propagating cut in lithium niobate. The higher electromechanical coupling constants in lithium niobate than in quartz make lithium niobate more sensitive. The plate can support a large range of acoustic modes, each one with its own frequency. Any perturbation of the top or the bottom boundaries of the piezoelectric plate modify the modes in the plate. In any case, for instrumental reasons, it is more efficient to isolate one mode and then track this mode as the boundary conditions change. Preferentially, this mode should be sufficiently removed from neighbouring modes so that interference can not occur. The APM device use interdigital electrodes (IDEs) in order to launch the acoustic modes, the use of this kind of sensor is such that IDEs are on the back side of the sensor and so are not in contact with the liquid. Lamb waves are considered as the interference between SAW propagating along the parallel faces of a plate. As the thickness of a plate is greater than a few wavelengths, two independents SAW can propagate on each face; as the thickness is reduced to a few wavelengths the two waves interact and result in two Lamb-type plate modes. The even (symmetric) mode behaves like the SAW and the odd (anti-symmetric) mode velocity approaches zero as the plate thickness is decreased; very thin membranes can be fabricated using micromachining techniques. Lamb wave sensors may be used as a fluid phase sensor and has been shown to be extremely sensitive. However, Lamb wave sensors are not really promising for biosensing applications since the device is fragile and difficult to manufacture reproducibly.

2.3.4

Surface skimming bulk waves

Since most SAWs have significant displacement components perpendicular to the crystal surface, they radiate energy into the adjacent liquid, therefore limiting their application as biosensors. Interdigital electrodes are known to excite a spectrum of shear and longitudinal bulk waves that propagate into the piezoelectric medium. Usually, reflections of the bulk waves from the bottom of the sensor occur and interfere with the SAW signal, leading to a further decrease of the sensor response. In some materials, these waves propagate parallel or nearly parallel to the piezoelectric surface and reach the output IDT without reflecting from the bottom. These waves are named surface skimming bulk waves (SSBWs). For sensor application, a reasonable coupling coefficient should be present while the other bulk waves and the SAW should be low to avoid interference. The SSBW can exist in the usual SAW materials such as quartz, lithium niobate and lithium tantalate. For sensor application, the SSBW must be horizontally polarised. If a layer is placed between the input and output transducers, the shear SSBW convert to a waveguide mode called Love mode. This occurs only if the shear acoustic wave velocity is less in the layer than in the substrate. Several modes can exist as function of the layer thickness. For biosensing applications, single mode operation is better since the first mode is more sensitive. Love modes conditions of existence, properties and sensitivity in regards with biosensing applications will be detailed in next chapters.

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

12

2.3.5 Acoustic waves biosensors comparison For biosensors, the specific species that need to be sensed are usually supported in a liquid medium. The presence of the liquid medium makes all the sensors supporting non pure shear waves unavailable for biosensors. Energy losses at the interface between the device supporting the acoustic wave and the liquid supporting the biological species can follow two paths. The first one is a coupling by a non pure shear horizontal component, i.e. normal displacement that generates a compressional wave into the liquid. The second one occurs when the wave phase velocity is greater than the compressional velocity in the liquid, causing generation of a leaky wave into the liquid and thus energy dissipation. The use of Rayleigh devices is directly put away for biosensors; it was seen that the normal component damps the wave too much. In addition, the Rayleigh wave has almost always its velocity higher than the compressional wave in liquid. The needed requirements are encountered by use of devices that present only viscous coupling with the liquid. Effectively, this coupling suffer less attenuation. The QCM, SHSAW (BG and pseudo-SAWs), SH-APM, Lamb modes and Love modes are then suited for biosensors applications. Those devices and the location of the liquid cell is represented in Figure 2.6 (courtesy of A. Campitelli [Cam97]). The usual operating frequency and the measured sensitivity for the known biosensors is reported in Table 2.2 reports [DHCD97]. The Love waves device exhibit the highest sensitivity. Actually, the experimented Love waves device reported in the Table 2.2 was constituted of silicon dioxide on a quartz substrate. We will see further that sensitivity of 380 cm2 · g−1 is the minimum that can be expected from that kind of device. This sensitivity get higher by an appropriated selection of the materials. It is obvious that the Love waves device, which is perfectly suited for liquid (bio)sensing applications, has withdrawn a growing interest. Device SH-APM QCM SAW (Rayleigh) Love

Operating Frequency [MHz] Mass loading sensitivity [cm2 g−1 ] 120-158 10-20 10 23 78 92-115 120 380

Table 2.2: Biosensors comparison after Du et al. [DHCD97]

2.4 2.4.1

Biochemical interface Immobilisation techniques

For biosensors, the development of a biological sensing element depends on a number of issues such as potential user, transduction mechanisms, design and manufacturing for competitive pricing. A successful biosensor must have distinct and compelling cost advantages over other currently available or emerging technologies. These advantages may result from speed, portability, minimal user training requirements. An essential step in the development is the bonding of the biological component to the transduction element. This immobilisation feature dictates the reliability and performance of a biosensor.

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

13

Figure 2.6: Configuration of biosensors. Drawings include the presence of the electrodes for transducing and position of the liquid cell. The liquid cell contains the biochemical species and is a part of a more complex system of liquid delivering around the biosensor.

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

14

When considering this aspect, the choice of a particular technique is ruled by a variety of issues [SR94]: • Nature and type of bio-compound; • Stabilising and mediating chemicals and bio-chemicals; • Transduction surface; • Sensor storage aspects; • Manufacturing process; • Environment in which the sensor will be employed. Obviously, the immobilisation of the analyte on the physical transducer must confine the biologically active material on the sensing element and keep it from leaking out over the lifetime of the biosensor, allow contact to the analyte solution, allow any product to diffuse out of the immobilisation layer, and not denature the biologically active material (bio-compound). For a bio-compound, these include structural features as well as the chemical composition. The transducer surface can be chemically modified or not. Aspects such as how long the sensor has to be stored before use and under what conditions can also affect the selection of a particular immobilisation method. Immobilisation methods for macromolecules or molecular mixtures include a non-covalent adsorption, covalent linkages to a chemically modified physical transducer surface or entrapment and/or covalent linkage to a polymeric membrane.

2.4.2 Self-assembled monolayers The concept of monolayers was introduced in 1917, when Langmuir studied the spreading of amphiphiles on water and realised that this film had the thickness of one molecule. By careful compression of the film a regularly oriented monolayer of molecules could be obtained. Blodgett was able to transfer the monolayer from the air-water interface onto a solid support (LB films) and also repetitive transfer could form multilayers. These films are thermodynamically unstable; under temperature changes or exposure to solvents they degrade. As combinations, self-assembled monolayers (SAM) were found like sulfur-containing absorbates on gold, alkyltrichlorosilanes on glass, and fatty acids on a metal oxide surface. The common point between these systems is the strong interaction between the functional group of the adsorbate and the bare substrate. The process involved in self-assembling of a simple (linear) alkanethiol monolayer on a crystalline gold surface is schematically represented in Figure 2.4.2. Exposition of the gold surface to a relatively low concentration of the alkanethiol in a contacting ambient two interactions result in the spontaneous assembly of a highly ordered film: chemisorption of the thiol head group on the gold (more than 20 kcal/mol) and van der Waals forces between the long alkyl chains (about 0.2 kcal/mol-CH2 unit [RKT+ 92]). The chemisorption results in the high packing of the thiol on the surface and also for the orientation “head down”. The van der Waals forces are responsible for the orientation of the chains. Those forces are maximised with all chains lined up in the same relative orientation. The “tilt angle” between the long axis of the alkanethiol molecules and the substrate normal is in the range of 20◦ − 25◦ for gold surface. This is due to the stronger interaction when the head group occupies three-fold hollow sites on the Au (111) surface. For that orientation, alkyl chains must bend over to optimise the van der Waals interaction between the CH2 groups of adjacent chains. Furthermore, in polycrystalline films of gold obtained by evaporation of sputtering, the (111) orientation is predominant since it has the lowest surface energy.

15

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

S

+

SSSSSSSS Au

Au

Figure 2.7: Alkanethiols in contact with gold lead to a self-assembled monolayer.

In a well-packed monolayer, the total amount of organic material of alkanethiols on gold is approximately 8.3 · 10−10 mol · cm−2 . Combining different analytical methods used in surface characterisation give the complete picture of the SAM. The attachment of the SAM to the substrate and the interactions at the monolayer interface with other molecules are especially relevant and discrimination in the changes of the layer is essential for determining the underneath mechanism of the biochemical part of the biosensor. Table 2.3 give a representation of useful analytical techniques for monolayer characterisation [Fli00, page 7].

2.4.3

Mixed self-assembled monolayers

Self-assembled monolayers are integrated in acoustic wave sensors to monitor mass changes. Figure 2.8 represents symbolically the process involved in mass sensors. Bioreceptors link with the desired chemical species. Since a tailored structure enhances greatly the responsiveness of the sensor and is thus highly desired, self-assembled monolayers have found application in that field. By adding functionality to the alkyl chain, one can obtain a structure that is able to present covalent binding to a specific molecule. Mixed structures, including two kinds of SAM, have been realised (i.e. [FBL00]). The advantage of mixed structure resides in their higher efficiency in comparison with single SAM. A pretty good coverage of the gold surface is necessary to prevent the analyte to attach to the bare gold surface. Attachment of any analyte to the surface spoils the signal and prevents the comprehension of the underneath mechanisms. A total coverage is thus necessary! Hence monolayers present good packing this point is already solved by using them. In second place, bioreceptors have to be attached to the surface with a pretty good control of their concentration and their surface repartition so that only a fixed number of specific analyte can attach to the surface without having interference neither with undesired species, nor with the biochemical area. A unique SAM that presents a covalent bind with bioreceptor will cover the whole area; interactions between adjacent molecules that compose the SAM prevent an efficient repartition of the bioreceptors since those are usually larger than the long tie of SAM molecules. The mixed structure has advantage to solve many of these issues at depends of an increasing complexity in the processing. However, bioreceptors can attach specific molecules of the SAM and are repelled from the surface by the other molecules of the mixed SAM. The statistical repartition of the active bioreceptors and the larger distance between bioreceptors lead to a structure perfectly suited to the biochemical interaction and the physical sensing

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

16

Analytical technique (a) Structural information General Contact Angle Hydrophobicity, order QCM/SAW Changes in mass Optical Infrared spectroscopy Functional groups, molecular orientation UV-vis absorbance Density of adsorbates Fluorescence spectroscopy Density of adsorbates Ellipsometry Layer thickness SPR Layer thickness Vacuum XPS Elemental composition AES Elemental composition SIMS Molecular mass of adsorbate and fragments Microscopy AFM Molecular packing STM Molecular packing Electrochemical Cyclic voltammetry Thickness, order, defects Impedance spectroscopy Thickness, order, defects (a)

Abbreviations: QCM (quartz crystal microbalance), SAW (surface acoustic wave), SPR (surface plasmon resonance), XPS(X-ray photoelectron spectroscopy),

AES (Auger electron spectroscopy), SIMS (secondary ion mass spectrometry), AFM (atomic force microscopy), STM (scanning tunneling microscopy).

Table 2.3: Analytical techniques for monolayers characterisation

Figure 2.8: Target principle of biosensor. The biochemical species present in the analyte can attach to specific probes (bioreceptors) at the surface of the sensor. The bioreceptors are embedded in a self-assembled monolayer.

CHAPTER 2. ACOUSTIC WAVES BIOSENSORS

17

element. An example of mixed self-assembled monolayers is 11-mercapto-1-undecanol in conjunction with 16-mercapto-1-hexadecanoic acid. The acid attaches to a anti-Human Serum Albumin (anti-HSA) and that structure is a biochemical interface. This interface is able to adsorb Human Serum Albumin (see Figure 2.9, courtesy of F. Frederix [FBL00]).

Figure 2.9: The biological recognition layer based on mixed SAMs and a covalent bound antibody.

Chapter 3 Love waves biosensors “All you need is love . . . ” The Beatles

3.1

A word of History

Augustus Edward Hough Love (born April 17, 1863, Westonsuper-Mare, Somerset, England; died June 5, 1940, Oxford) was a British geophysicist and a mathematician graduated from Cambridge. He held the Sedlieian proffessorship of natural philosophy at Oxford University from 1899 to 1949. He discovered a major type of earthquake wave that was subsequently named after him. In his analysis of earthquake waves, Love made the assumption that the Earth consists of concentric layers that differ in density and postulated the occurrence of a seismic wave confined to the surface layer, or crust, of the Earth. This wave would be propagated as a result of the difference in density between the crust and underlying mantle. His prediction was confirmed by recordings of the behaviour of waves in the surface layer of the Earth. He proposed a method based on measurements of Love waves to measure the thickness of the Earth’s crust. From his work on the structure of the Earth Some Problems in Geodynamics, he received the Adams Prize at Cambridge in 1911. In addition to his work on geophysical theory, Love studied elasticity and wrote a two volume work A Treatise on the Mathematical Theory of Elasticity (1892-93). He received many honours, the Royal Society awarded him its Royal Medal in 1909 and its Sylvester Medal in 1937, the London Mathematical Society awarded him its De Morgan Medal in 1926. Finally, a lunar crater hold his name 1 . 1 source:

Encyclopædia Britannica

18

CHAPTER 3. LOVE WAVES BIOSENSORS

19

3.2 State-of-the-art 3.2.1 The Love waves biosensor Love waves are guided acoustic modes in a layer on top of a piezoelectric substrate. The layer is usually named Love plate. A lower velocity of the bulk shear wave in the Love plate than in the substrate is the condition necessary to obtain Love waves. The Love waves present a shear-horizontal polarisation and they can be launched in the structure thanks to an interdigital transducer (IDT Tx). For sensors application, the waves are monitored by a receiving interdigital transducer (IDT Rx). For the specific application of biosensors, an interfacial film has to be put on top of the structure so that a biofilm deposited on top acts as the biochemical part. The mass changes of the biofilm influence the propagation properties of the waves on the path between the two IDTs. This sensitivite path has been named the sweet spot by Harding [Har01]. The sensor acts as a microbalance that receive usually the name of gravimetric sensor. The Figure 3.1 reports schematically the typical structure of the Love waves biosensors and the clue points where something can be studied.

Figure 3.1: A typical Love waves biosensor The interest towards Love wave devices as biosensors has appeared only recently in the early 90’s. This interest is due to the very high sensitivity of the Love waves biosensor in comparison to other kinds of gravimetric acoustic biosensors. Intensive researches have been led around the world to understand the

20

CHAPTER 3. LOVE WAVES BIOSENSORS

physical principles underneath the Love structure, to improve the interfacial chemistry and to process the biosensors. Three main groups are known for their interest in Love waves biosensors: • the Institute of Biotechnology at University of Cambridge, United-Kingdom; • the Electronic Instrumentation Lab at Delft University of Technology, The Netherlands; • the Commonwealth Scientific and Industrial Research Organisation, Australia.

3.2.2

A first experience on Love waves biosensor

The group of Gizeli et al. in Cambridge has studied the selection of the Love plate material as the key factor for an increased mass sensitivity. They realised a test set-up to measure the sensitivity of the device with silica and PMMA as Love plate on a single crystal Y −cut (42.5◦ ) with propagation direction perpendicular to the X− axis. The device operates at 110 MHz. Although the acoustic damping in PMMA is much more higher than in the silica, it was theoretically known that a better sensitivity could be achieved with polymeric materials since they present a better energy entrapment of the waves, making them more sensitive to perturbation on top of the device. The hydrophobic PMMA was directly used as the interfacial layer for the protein absorption. From their measurement, they could bring the experimental proof that PMMA coated devices present a higher mass sensitivity than the one coated with SiO2 ([GSGL91] and [GGLS92]). From the theoretical point of view, it is known that the number of Love modes presents in the structure depends in a unique manner of the Love plate thickness. Each mode appears at a monomode thickness defined by [GSGL92]: h0 =

V VLayer q Substrate 2 2 2 f0 (VSubstrate −VLayer )

(3.1)

where f0 is the operating frequency, VS the substrate bulk shear velocity and VL the Love plate (single layer) bulk shear velocity. For the PMMA coating, the monomode thickness is 5.1 mm. It was also showed that the insertion loss increases with increasing polymer thickness until it reaches a maximum for a thickness of around 1.2 µm where the signal is 8 dB less lossy than in the not coated device.

3.2.3

Theoretical prediction of the sensitivity

The experience related in the previous section proved that the sensitivity of the biosensor is strongly dependent of the overlay materials selection. A more theoretical approach of the device is thus necessary to obtain a simulated estimation of the sensitivity of the device. The mass increase of the top layer is reduced to a change in the characteristic of the device, like the operating frequency or the signal insertion loss. From a theoretical point of view, the sensitivity was modelled as a change in the propagation properties of the shear acoustic wave supported in the Love waves device. The sensitivity is essentially defined by the relative change of oscillation frequency due to the mass loading: Smf =

∆f/f (m/A)

(3.2)

where (m/A) is the mass per surface area. Since the frequency f , the wave velocity V , and the wavenumber along the direction of propagation kx are all linked, others definitions of the sensitivity are allowed that

21

CHAPTER 3. LOVE WAVES BIOSENSORS

involve in the same manner the change ∆V /V or ∆kx /kx . The start point to get this estimation of the sensitivity is based on an expression derived from the perturbation theory [Aul73a, eq. 2.15]. The mass loading is considered as a small perturbation, the theory let the sensitivity to be approximated at the first order by: kx ∼ Sm =

VL |vy |2 ∆kx = kx ∆(ρ p h p ) 4P

(3.3)

where kx is the wavenumber along the direction of propagation, ρ p h p the surface mass density of the perturbing film 2 , VL the Love wave phase velocity, |vy | the amplitude of the particle velocity, and P the time-averaged power in the guiding layer. One can see that the sensitivity so defined is expressed in [m2 /kg]; a more useful representation is [cm2 /g]. A better sensitivity is achieved with a material that present simultaneously a low density and a very low bulk shear wave velocity in comparison to the one of the substrate material. Once a combination of materials has been chosen, the unique parameter that has kx is the thickness h of the Love plate, or equivalently its still an influence on the mass loading sensitivity Sm normalised thickness α ≡ h/λ. This sensitivity presents an optimum for a specific value of α. Table 3.1 reports the optimum value of the thickness for different overlayers on a ST-cut quartz substrate operating at 110 MHz [SGGL93]. Material

Shear acoustic velocity [m · s−1 ] Fused silica 3764 Aluminum 2700 Copper 2325 Zinc 2440 Silver 1610 Gold 1200 Lead 700 PMMA 1100

Density [kg · m−3 ] 2200 3040 8930 7100 10400 19700 11400 1180

Sensitivity [Hz · cm2 · ng−1 ] 15 50 15 16 20 20 120 1500

Thickness [µm] 12 9 7 7 1-2 1-2 1-2 3

Table 3.1: Thickness and properties of various overlayer materials corresponding to optimum sensitivity The perturbation theory gives the relative change of propagation constant kx at a fixed frequency f0 . f The relative change of oscillation frequency Sm is monitored in a delay line or in an oscillation loop kx and S f is direct: [KV92]. The relation between the two sensitivities Sm m Smf

 2 Vg 1 ∆f kx =− Sm ≡ f ∆(ρ p h p ) VL

where Vg is the Love wave group velocity. 2 One

would notice that ρ p h p = m/A.

(3.4)

22

CHAPTER 3. LOVE WAVES BIOSENSORS

Wang et al. [WCJ94] have developed an equation to estimate the sensitivity and the optimum normalised thickness. Their work has been used many times in order to obtain an optimum value for the device in regards with mass loading. Their approach is based on the perturbation theory and under the assumption that the shear wave velocity in the substrate is much higher than in the Love plate. The formula is valid only for single layer Love plate. Subsequently, the analysis of several layers in the Love plate is not possible. From their results, an optimum normalised thickness of 16.9% for silicon dioxide and of 6.4% for PMMA on quartz was found. The corresponding sensitivity was overestimated in comparison to the experimental results but the results were relatively close to the theoretical predictions. From their expression, the Figure 3.2 is set to give an idea of the sensitivity that could be expected for several guiding materials. The relative density r and relative velocity v are the parameters that rule the sensitivity at a given frequency, they correspond respectively to the ratio of the material density to the quartz density and to the velocity in the material to the velocity in the quartz. A higher frequency results in a higher sensitivity since the sensitivity is function of λ−1 .

90

7

70

4.8 4.9

7

2 7.

6. 8 6. 9

60

7. 1

4 7. .5 7 6 7.

7.

3

50

7. 7.

9 8

8

Quartz Love wave device @120MHz − Log2 of sensitivity 3.3. 43 3.7 3.5 3.6 3.8 3.9 4.1 4 4.7 4.2 4.6 4.3 4.4 4.5 5 4.8 5. 5 1 5.3 .2 4.9 4.7 5.4 5.5 5.6 5 5.8 5.7 5.1 5.9 5.2 5.3 6.1 6 5.4 5.5 6.3 6.2 5 . 6.4 6 5.7 5.8 5.9 6.5 6.7 6 6.6 6.1 6.8 6 .2 6.3 6 6.4 7 .9

44.1 4 3.9 44.4 .24.3 .5

6. 5 6. 6

6.

Relative bulk shear wave velocity (s)

80

5 5.355.2.1 5 5..54 5 5.8 .6 5. 5.7 9 6. 6 1 6. 6.3 6 .2 4

4.6

7. 7

7.2

7.1

40

4

8.

8. 2 8. 3

8.

5

7

8.1

30

7.9 8

8.

7.7

9.49.3

9.29.1

30

9

8.7 8.8 40

PMMA 8.2 8.4 8.3 8.68.5 50

8.1 60 Relative density (r)

4.4 4.5

4.6

4.7

4.9

SiO2

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 6.1 6 .2 6.3 6.4

7.8 70

6.5 6.6

6.7

7.1

7.9 8

4.1 4.2 4.3

4.8

7 7.4 .3 7.67.5

9

20

3.8 4

6.8 6.9

7.2

7.8

3.3 3.4 3.5 3.6

3.7

6.5 6.6

6.7

7.4 7.3 7.6 7.5

2.9

3

3.2 3.1

7 7.2

6.8 6.9 7.1

7. 7.3 7.5 4

7.7 80

90

100

Figure 3.2: Guideline for the selection of materials as Love plate for quartz based devices. The base 2 logarithmic value of the theoretical sensitivity is expressed as a function of the relative density and the relative acoustic wave velocity in the material in regards to the quartz. The silicon dioxide and the PMMA are reported as examples. Love waves devices incorporating silicon device have been used as biosensor by Harding et al. to monitor the kinetics of the antibody-antigen binding in aqueous solutions. The detection limit was around 1 nanogram of sheep IgG per milliliter of solution [DHCD97]. Optimisation of the biosensor became

23

CHAPTER 3. LOVE WAVES BIOSENSORS

the main problem for development of reliable and sensitive devices. The experimental study made in this group about the influence of the layer thickness is reported in Figure 3.3. The sensitivity reached a maximum for α ' 14% with a value of 380 cm2 /g. They also studied the influence of the viscosity of √ the liquid on the relative insertion loss and proved experimentally a relation proportional to ηρ, where η is the viscosity of a liquid that presents a density ρ. This relation was already known from the QCM measurements and is typical of acoustic devices. The same relation was theoretically and experimentally studied by Jakoby and Vellekoop [JV98d]. The Love wave sensor can be used as a liquid viscosity sensor.

Figure 3.3: Experimental measurements of the mass sensitivity of silicon dioxide devices (diamond dots) and comparison with the theoretical prediction (solid line) given by the perturbation theory after Du et al. [DHCD97]. The influence of the damping in the chemical layer has been simulated by Jakoby et al. [JV98c] by use of the perturbation theory and experiments have been lead with use of PMMA layers on silicon dioxide devices. They show that the sensitivity is higher with increasing frequency but the damping is also increasing. This study set the maximum thickness of the interfacial layer for an allowed insertion loss. Jakoby et al. have studied the physics of the Love wave device in operation. They improve the device to reduce the sensitivity when used in water. Since water is the usual supporting medium of the biochemical species, the large dielectric constant of the water couples to the interdigital transducer used to launch the wave and then influence the total sensitivity of the device. This reduction is easily obtained by getting a thin metal overlay over the whole area of the device. Actually, a gold layer combines advantage of electrical protection and is a suited material for the biofilm [JV98a]. They also investigate the temperature sensitivity (TCF) of the Love wave sensor that is expressed by:   Vg 1 ∂V 1 ∂L 1 df = − (3.5) TCF = f dT V v ∂T L ∂T | {z } −TCD

CHAPTER 3. LOVE WAVES BIOSENSORS

24

where L is the length of the line. The TCD is the negative temperature coefficient of delay. It was known from this formula that the ST-cut of the quartz presents a zero TCD for Rayleigh waves. For the Love waves in a silicon dioxide layer on quartz, their study brought solutions to reduce the temperature sensitivity. That can be achieved by using a different cut of the quartz substrate 3 and by tuning the thickness of the silicon dioxide layer [JV98b].

3.2.4 Composite devices and recent developments A multilayered structure that includes silicon dioxide and a polymer was evaluated by Du et al. [DH98]. The comparison with single layer device has been experimentally established as a function of the layer thickness. The temperature coefficient was also reduced by the structure and the results is presented in Figure 3.4. A composite structure presents a reasonable insertion loss in comparison to a polymer device and the sensitivity is higher in comparison to a silica device. The influence of the thickness is relatively important on the sensitivity and the insertion loss. The utilised polymer was spin-coated PMMA. A high thickness of the polymer could not be achieved because of instability in the polymer.

Figure 3.4: Experimental measurements of single material Love plate (only silicon dioxide or only PMMA) and of composite devices (silicon dioxide and PMMA) after Du et al. [DH98]. The measured (left) mass sensitivity, and (right) insertion loss as a function of the total overlay thickness are represented. Recently, Harding has studied the influence of a new guiding layer made of silicon-oxy-fluoride [Har01]. The fluorine concentration during the fabrication process allows to tune the acoustic properties of the material. The insertion loss was relatively low (less than 20 dB for most of the devices) and the sensitivity seems promising (about 300 Hz/ng). He concluded with the opportunity to exploit for further works the incorporation of fluorine, chlorine and sulphur that may reduce the acoustic wave velocity and open the way to new Love waves guiding layer materials. 3A

cut with Euler angle (0◦ , 120◦ , 90◦ ).

25

CHAPTER 3. LOVE WAVES BIOSENSORS

3.3 Transmission line equivalent model 3.3.1 Necessity of a modelling The Love waves biosensors presents two parts. The first part is the propagation of the acoustic waves in the structure, and the second part is the sensing characteristics of the structure. The structure of the device can be considered as the stacking of different layers. The layers define areas in the device where the material properties are constants and assumed to be known. Although the perturbation theory is a useful approach to determine, at the first order, the sensitivity of the sensor, it is necessary to compute in a first place the fields in the non perturbed wave. Since this is related to the geometry of the device, a method has to be set to determine exactly the acoustic field for a given structure that could be very simple (only one layer) or more complicated (multilayers). From the non-destructive testing (NDT) field, the propagation of all kinds of acoustic waves has been intensively studied. In particular, methods exist to determine the dispersion curve of Love modes and the influence of the viscosity. Softwares are readily existing since a decade but their application is too large in comparison to the real need encountered for the simulation of the Love waves device (see [ASFJ90]). This kind of software is destined in prior to skilled specialists and has been avoided here for the modelling of the Love waves biosensor. In this part, an equivalent model is established on base of the analogy between acoustic and electrical waves. The acoustic device is entirely described in term of that analogy which is successfully applied to the study of Love waves (bio)sensors.

3.3.2 Partial waves propagation The structure of the Love waves device has been presented in a previous section. For the modelling, the equations that rule the guiding of the wave in the layers have to be developed 4 . The first step is to set a referential that is representative of the problem. The direction of propagation of the wave is set as the x−axis. This axis is parallel to the layers boundaries, the normal to the boundaries is arbitrarily set as the z−axis aligned from the substrate to the top layer. The position of the origin is set at the interface between the substrate and the first Love plate. The n layers that compose the Love plate are numbered from 1 to n from bottom to top. The substrate is assumed to be an infinite half space so that no number is attributed to it. The top half space is also an infinite half space, usually vacuum. In the vacuum, no acoustic wave propagate. The description made here is sketched in Figure 3.5. For the considered problem of shear waves propagating along the direction x, one has to define the initial properties of that propagation. The wave is considered independent of the direction y, i.e. the spatial derivative in that direction is null: ∂• = 0. ∂y

(3.6)

In addition, the wave is also considered to be polarised along this latter direction and so the local particle 4 this

part is using the concepts and the notations introduced in the appendix B.

26

CHAPTER 3. LOVE WAVES BIOSENSORS

z y

vacuum (semi-infinite half-space) x direction of propagation

Love plate

(layer 2) layer 1 substrate (semi-infinite half space)

Figure 3.5: Structure of the Love waves device. Axis are set as following: the wave propagates in the direction x, the direction z is aligned to the normal of the layers. At least one layer has to enter the structure of the device, but several layers numbered from 1 to n are allowed to be a part of the structure. On top of the device, the vacuum is usually considered but it could also be a newtonian liquid or a very thin layer (for instance a gold layer with self-assembled monolayer on it).

displacement u (or equivalently the local particle velocity v) has only y component: u(x, z) = [0 uy (x, z) 0] v(x, z) = [0 vy (x, z) 0] .

(3.7) (3.8)

For waves propagating in a waveguide structure, like the Love plate, an assumption has to be made: the main wave propagating in the x direction results from the interaction of two partial waves propagating in different directions. A second assumption that would simplify the description is to consider all the materials as isotropic. Although it is not really the case for the piezoelectric substrate since this is a crystalline material, the approximation remains valuable if the anisotropy factor of the material is not to high. In each layer, the partial waves are propagating upwards and downwards, they are totally defined in the xz plane. The partial waves present the same component along the direction x and are symmetrical around this axis, the components along z are opposite. This effect results from the reflection of the partial waves at the layer interfaces and is understood by considering the representation of the partial waves in the slowness curves of the materials (see Figure 3.6). A partial wave is considered to propagate along the direction z0 . The direction is related to the axis by the angle θ it forms with one of the in-plane direction (see Figure 3.7). Arbitrarily the z direction is taken as reference: z0 = [sin θ 0 cos θ] .

(3.9)

It is easily seen that in each layer, the angles made by the two partial waves are supplementary, only θ has to be defined since the other one would be (π − θ).

27

CHAPTER 3. LOVE WAVES BIOSENSORS

θ2

Vacuum

θ1

Layer 2 z Layer 1 x

(evanescent field)

Substrate kz ω

θ2 Layer 2

kx ω

θ1 Layer 1

kx ω

Substrate

kx ω

Figure 3.6: The Love wave results from the interaction in each layer of partial waves propagating in opposite direction along z but with the same component along x. The common component along x gives the Love wave velocity. The partial waves are described by the slowness curves for each layer.

28

CHAPTER 3. LOVE WAVES BIOSENSORS

z

direction of propagation of the partial wave z’ θ x

x’

Figure 3.7: The partial wave propagates along direction z0 that results of a rotation of the axis around y by an angle θ.

All those definitions let the translationnal equation of motion (B.3) and the constitutive equation of the solid (B.4) to be written as (in abbreviated subscript notation, see B.1): ∂vy (x, z) ∂T4 (x, z) ∂T6 (x, z) + = ρ ∂z ∂x ∂t ∂vy (x, z) ∂T4 (x, z) = c44 ∂z ∂t ∂vy (x, z) ∂T6 (x, z) c44 = . ∂x ∂t

(3.10) (3.11) (3.12)

The spatial derivatives expressed for z0 are given by ∂ ∂ = sin θ ∂x ∂z0 ∂ ∂ = cos θ. ∂z ∂z0

(3.13) (3.14)

If we define T40 (z0 ) as (T4 (x, z) cos θ + T6 (x, z) sin θ), the relations are finally reduced to ∂vy0 (z0 ) ∂T40 (z0 ) = ρ ∂z0 ∂t 0 ∂vy0 (z ) ∂T40 (z0 ) c44 = , ∂z0 ∂t

(3.15) (3.16)

and obviously the rotation of the axis has not changed the polarisation of the wave so that vy0 (z0 ) = vy (x, z).

3.3.3

Transmission line properties

This section reports the basic relations needed for further modelling of the acoustic wave device. The theory for transmission line has been reported in many books, one can find it in [Vor95]. An infinitesimal length ∂r ( λ) of transmission line is represented by a series impedance Z and a shunt

29

CHAPTER 3. LOVE WAVES BIOSENSORS

admittance Y as shown in Figure 3.8. The voltage V (r) and tension I(r) are linked together by the coupled equations: ∂V (r) = ZI(r) ∂r ∂I(r) − = YV (r). ∂r

−

I(r)

(3.17) (3.18)

Z

V(r)

Y

dr

Figure 3.8: Infinitesimal equivalent circuit of a transmission line. p The transmission line admits a characteristic impedance Z = Z/Y and a propagation function c √ −1 γ = ZY = α + jβ [m ]. The voltage and tension at any point of the line is the summation of two partial values V− and V+ that represent electrical waves propagating in opposite directions: V (r) = V+ e−γ r +V− eγ r Zc I(r) = V+ e−γ r −V− eγ r .

(3.19) (3.20)

For a transmission line of length L, the equivalent circuit is a quadrupole whose origin can be set to one end of the line. This origin is also considered as the attach point for any impedance loading. Thanks to this definition, voltage and current are easily computed at any point of the line if they are known at the origin as respectively V (0) and I(0): V (r) = V (0) cosh γ r + Zc I(0) sinh γ r V (0) sinh γ r − I(0) cosh γ r. I(r) = − Zc

(3.21) (3.22)

For a line that only present a partial value V− and where (r ≤ 0), the equations are: V (r) = V (0)eα r cos(ωt − βr) Zc I(r) = V (0)eα r cos(ωt − βr).

(3.23) (3.24)

This case is typical of an infinite half-line. Finally, for a transmission line of length L loaded by an impedance ZL , the equivalent impedance seen at the end of the line is Z(L) = Zc

Zc + ZL coth γ L . ZL + Zc coth γ L

(3.25)

30

CHAPTER 3. LOVE WAVES BIOSENSORS

3.3.4 Analogy between acoustic and electrical waves From the equations (3.15) and (3.16), a strong analogy with the fundamental equations of a transmission line is set. This analogy was known since a long time and has been reported in [Aul73b] for propagation of waves in the bulk of materials 5 . The analogy is given by comparing the mechanical stress T40 to the inverse of the electrical voltage and the local particle velocity vy to the electrical current: T40 (z0 ) ≡ −V (r) Vy (z0 ) ≡ I(r).

(3.26) (3.27)

The Table 3.2 allows the direct interpretation of the equivalence and helps to determine the equivalent parameters. In that table, the fields are assumed to be time-harmonic plane waves so that the time derivative is replaced by a iω factor multiplication of the term. The equivalent parameters are Z = iωρ and Y = iω/c44 ; they correspond respectively to an inductance L (value ρ) and a capacitance C (value c−1 44 ). The equivalent model in transmission line of the isotropic, lossless, medium is sketched in Figure 3.9 and the parameters are summarised in Table 3.3. Acoustic wave propagation ∂T40 (z0 ) ∂z0

c44

=ρ

∂vy (z0 ) ∂z0

=

∂vy (z0 ) ∂t

Electrical wave propagation − ∂V∂r(r) = ZI(r)

= iωρvy (z0 )

∂T40 (z0 ) ∂t

− ∂I(r) ∂r = YV (r)

= iωT40 (z0 )

Table 3.2: Comparison between the propagation of a shear wave in an isotropic medium and the propagation of an electrical wave in a transmission line.

series impedance

Z = iωρ

shunt admittance

Y=

characteristic impedance propagation function

Zc = γ=

p

Z/Y =

√

iω c44

√

ρc44 = ρV

q ZY = iω cρ44 =

iω V

Table 3.3: Summary of the equivalent transmission line parameters for isotropic solids that have a bulk wave phase velocity V . The impedance Z and the admittance Y are defined for an infinitesimal length. For the line itself, the relevant parameters are the characteristic impedance Zc and the propagation function γ. 5 The

analogy has also been partially used to study Love mode transducers [KzWG99].

31

CHAPTER 3. LOVE WAVES BIOSENSORS

Further models have been developed for viscous solids and piezoelectric solids. For viscous solids, the viscosity is represented by a conductance G in series with the shunt capacitance. The value of G is η−1 44 . For piezoelectric solids, the situation is represented by the combination of a capacitance C piezo in series with C, the result is a stiffened capacitance Csti f f ened whose expression depends upon the direction of propagation and is given by:  2  ex4 2 1 = c44 + S sin 2θ . Csti f f ened εxx One has to be careful that the application of the equivalence is not direct for anisotropic solids and requires usually better developments than showed here. These developments would not be reported in this report since they are quite irrelevant for the modelling of the Love waves device.

I

L

I

L

I

V V

C

C

L

V

C Cstiffened C piezo

G dr dr

dr

Figure 3.9: The equivalent transmission line model of the propagation of shear wave in different medium: (left) isotropic, lossless medium, (center) isotropic, lossy medium, and (right) piezoelectric, lossless medium. The inductance L has the value ρ and the capacitance C the value c−1 44 . For a viscous medium, the shunt arm is the series combination of the mechanical capacitance with a conductance G that has the value η−1 44 . For a piezoelectric media, a piezoelectric capacitance C piezo is included in series with the mechanical capacitance. It results in a stiffened capacitance of value Csti f f ened = i−1 h  2 e 2 c44 + εx4 . S sin 2θ xx

3.3.5

Decomposition of the Love waves device

The equivalence has been set in the previous section for the specific propagation of the partial wave. The equivalence with the Love wave device is given by a decomposition of the Love wave that propagates in each layer of the device along the direction x and the direction z. Along the longitudinal direction (x), the propagation is totally free. The ideal case corresponds to the propagation of the wave in an unbound medium. Along the transversal direction (z) the wave is submitted to a condition of resonance: at each boundary, the partial waves are reflected. Incident, transmitted and scattered waves interact together under specific conditions that are directly derived from the transmission line theory. The Figure 3.10 reports the decomposition of the Love wave device into the two equivalent transmission lines. The parameters of each transmission line are linked by the angle of propagation θi (with i different in each layer). The decomposition allows the impedance of the line to be computed for the two main directions. From the expression reported in B.49, the impedance matrices admit only non-zero component for the central

32

CHAPTER 3. LOVE WAVES BIOSENSORS

θ2

Vacuum

θ1

Layer 2

(a)

z Layer 1 x Substrate

(evanescent field)

Layer 2 z Layer 1

(b)

x Substrate

Layer 2 z Layer 1

(c)

x Substrate

Figure 3.10: The equivalence between shear waves in solids and transmission line is applied to the Love waves device. (a) For a proper Love wave velocity, each layer (layer 1,2, . . . ) has a different angle of propagation θi (with i associated to each layer). (b) In the longitudinal direction x, the layers structure is the superposition of the equivalent transmission line of each layer. These lines are only linked by the angles θi and they admit all the same propagation function which corresponds to the Love wave velocity. (c) In the transversal direction z, the device is the series network of the transmission lines associated to each layer. The Love wave corresponds to a resonating scheme in this structure.

33

CHAPTER 3. LOVE WAVES BIOSENSORS

component. The impedance seen in the directions x and z are then: c66 k sin θ ω c44 k cos θ = . ω

(Zxk )22 =

(3.28)

(Zzk )22

(3.29)

The expression taken by the impedances are in agreement with the decomposition approach; the impedances are given by the projection of the impedance experienced by the partial wave propagating along z0 on the p main directions x and z. Since for isotropic solids, c44 = c66 and ω/k = V = c44 /ρ, the relations are simpler: √ (Zxk )22 = ρc44 sin θ √ k (Zz )22 = ρc44 cos θ.

3.3.6

(3.30) (3.31)

Expression of the resonance condition

The guiding of the Love wave in the structure is possible at the condition that the partial waves reflected and transmitted interact in phase to give rise to a coherent interference that held the signal in the structure and embeds the wave in the layers. The partial waves are forced to verify the boundary conditions at each interface, these mechanical conditions are given by the relations (B.39) and are directly solved in the decomposition when the equivalent transmission lines in the direction z constitute a series network. The continuity of the equivalent voltage and current is appropriated to the continuity of the mechanical stress and particle velocity at each interface. From the classical transmission line theory, waves are able to enter a resonating scheme in the structure if the impedances seen from any point of the structure in the two directions are opposite. Let say that the impedance of the line seen at the point z0 when looking to the increasing values of z is Z + (z0 ) and the impedance of the line seen at the same point when looking to the decreasing values of z is Z − (z0 ) (see Figure 3.11). The resonance condition is then expressed as: Z + (z0 ) = −Z − (z0 ).

(3.32)

It should be noticed that this expression is a general relation that may involve complex numbers, no restriction has been made on the nature of the impedances. In particular, the relation may involve complex values that have to be exactly equal. This means that two scalar relations have to be solved simultaneously to observe the resonance of the wave in the structure. The angle that check the resonance condition correspond to one Love mode. The last problem that needs to be solved is the determination of the impedance seen at the point z0 . The geometry of the device has to be taken into account, in particular the thickness of the different layers. The first step is to determine the angle of propagation in all layers and in the substrate. Snell’s law links the angles (see B.40). A particular layer is selected as the reference layer for the angle. It is usually easier to take the layer that presents the higher slowness since the angle of resonance has necessary to be a real number (a complex angle of propagation is related to evanescent fields). This angle is swept between 0 and π/2, the angles in the different layers have to follow the variation of the angle in the reference layer. For a given angle, the acoustic impedance and the propagation function of the layers are different. The impedance seen at the position z0 is directly deduced by the equation (3.25) where the length L of the

34

CHAPTER 3. LOVE WAVES BIOSENSORS

Z+ h2

Layer 2

h1

Layer 1

z0

z x Substrate

ZSubstrate

Figure 3.11: The equivalent network in the direction z set the resonance condition. The line presents two impedances Z + and Z − at any point z0 of the layered structure. These two impedances are a function of the geometry of the structure, of the materials combination, and of the angles of propagation of the partial waves.

transmission line is equivalent to the thickness of the layer. The relation has to be used from the bottom of the device (position 0) to z0 and from the top of the device to z0 . For modelling, the top layer is always considered to be vacuum, this is a specific medium that is modelled by a zero-impedance element (short-circuit).

3.3.7 Conclusions about the equivalent model The conclusions of the equivalent model presented here are the following: • Love waves device can be modelled by equivalent electrical circuits that are transmission lines networks; the Love wave in each layer i results from the interaction of partial waves propagating in two symmetrical directions that make an angle θi (upwards) and π − θi (downwards) with the transversal direction z. The angles are linked together by Snell’s law. • Each layer corresponds to two transmission lines: one in the longitudinal direction x and one in the transversal direction z. The characteristic impedance of each line is given by the projection of the characteristic acoustic impedance of the medium in which the partial wave propagates along the √ considered direction of interest: Zx = Zc sin θi and Zz = Zc cos θi with Zc = ρc44 . The propagation function γ of the transmission line follows the same rule. • For the longitudinal direction, the equivalent transmission lines are independent in each layer in the way that they are not linked together except by Snell’s law. They admit all the same propagation θi iω function that gives the Love wave velocity: γi = γ = iωVsin = VLove . bulk • For the transversal direction, the equivalent transmission lines are connected together in a series network that corresponds to the structure of the device. The boundary conditions at each interface are solved by this configuration. The transversal direction set the resonance condition by the relation Z + (z0 ) = −Z − (z0 ) with z0 any point in the structure (z0 ≥ 0).

35

CHAPTER 3. LOVE WAVES BIOSENSORS

It results from those conclusions that the fundamental property that summarises all the equivalent parameters is the Love wave velocity. For a given structure with a particular geometry and a specific combination of materials, the determination of this velocity is given by solving the resonance condition along the transversal direction z. The resonance condition in itself is given by inspecting the angles θi that satisfies the impedance condition Z + (z0 ) = −Z − (z0 ). Since Snell’s law links all the angles, a reference layer has to be selected for its higher shear bulk wave slowness. For the reference layer, the angle of propagation has to be swept from 0 to π/2 to find all the resonance angles. A special care should be taken to select angles that lead to non-leaky waves in the substrate. In particular, by checking that the angle in the substrate is higher than π/2. Finally, the wave in the substrate is no more resulting of the interaction of two partial waves since the material fill the bottom half-space and is equivalent to a semi-infinite transmission line. The Figure 3.12 is the schematic of the procedure to follow to determine the Love modes. The next section will apply this equivalent model to the Love wave device and a method will be set to estimate the sensitivity of a (bio)sensor.

Reference layer

Angle of propagation in the reference layer Snell’s law Determination of the angles for all layers and substrate

No resonance modification of the angle

Determination of the equivalent parameters

For a given depth, computation of the impedances Resonance One Love mode has been found

Characterisation of the whole structure (directions x and z)

Figure 3.12: Schematic of the procedure to follow to determine the Love modes in the frame of the equivalent transmission line model.

36

CHAPTER 3. LOVE WAVES BIOSENSORS

3.4 Simulations of the Love waves biosensor The modelling of the device with equivalent transmission lines has been implemented to obtain the characteristics of the Love waves. For the biosensing application, only the 0th order mode has been investigated since its sensitivity was known to be the highest in comparison to the other modes. Three different type of configurations have been simulated: • silicon dioxide Love plate; • polymer Love plate; • composite Love plate (silicon dioxide and polymer). The equivalent model allows the simulation of any number of layers, the model has thus a potential to take the interfacial gold layer and the biofilm into account but this is irrelevant since the thickness of these parts is small in comparison to the thickness of the Love plate. The interfacial layer is always ignored for the simulations. For the polymer, the usually studied PMMA was avoided to the profit of epoxy polymers. The reason that explain the large use of PMMA in previous studies is based on the fact that this polymer is easily processed and also quickly removed from the device in order to make a new deposition and study the influence of the PMMA thickness. The poor acoustic quality of the PMMA was not really limiting factor since the testing made were done in a research frame, not for final application. For a final application, one has to consider the longevity of the device and also the acoustic properties of the material that enter the composition of the device. This explains the use of epoxies for the study done here. From a theoretical point of view, the large class of the polymers presents approximately always the same density and acoustic velocity, only the damping factor is reduced by the configuration and the attachment of the chains in the polymer. Every simulations has been done with the materials data reported in the Table 3.4. The values are provided by [WCJ94] except for the epoxy polymer. The viscosity η44 is not taken into consideration for simulations. Material

Density [kg/m3 ] Epoxy 1230 Gold 19300 PMMA 1180 Silicon dioxide 2200 Quartz α 2650

Bulk shear wave velocity [m/s] 1210 1215 1089 2850 5060

Table 3.4: Materials data for the simulations

37

CHAPTER 3. LOVE WAVES BIOSENSORS

3.4.1 Single Love plate biosensors The Love wave phase and group velocity was computed in all structures as a function of the normalised thickness α = h/λ. Since the computation is done for a given frequency, the values of α are post-processed with the knowledge of the Love wave phase velocity thanks to the relation V = λ f (with f the frequency of the wave). The Figure 3.13 reports for the two monolayer devices their velocities. The shape of the Love wave velocity is always the same, it starts from the substrate velocity at each monomode thickness and decreases for increasing thickness of the Love plate until it reachs the velocity of the wave in Love plate material. The monomode thickness is lower for the polymer material; therefore for an identical normalised thickness, the polymer device supports more Love modes than the silica device. For the biosensor, the necessity to achieve a monomode operation introduces a limitation of the thickness of the material. This limitation is frequency dependant since the monomode thickness is not defined in reference to the Love wave but to the phase velocity of the materials. Love waves phase and group velocities for Quartz/SiO2/Vacuum

Love waves phase and group velocities for Quartz/Polymer/Vacuum

5500

5500 0th order 0th order

1st order

5000

1st order

5000 4500

4000

Velocity [m/s]

Velocity [m/s]

4500

4000

3500

3000

2500 3500 2000

1500 3000 1000

2500

0

0.1

0.2

0.3

0.4 0.5 0.6 Normalised thickness

0.7

0.8

0.9

1

500

0

0.1

0.2

0.3

0.4 0.5 0.6 Normalised thickness

0.7

0.8

0.9

1

Figure 3.13: Love wave phase and group velocities as a function of the normalised thickness for (left) silicon dioxide and (right) polymer Love plate. The 0th order and 1st order Love modes are presented. For a given frequency, the Love modes appear at a constant variation of the Love plate thickness that is a function of the materials parameters. The transmission line model is able to determine the amplitude of the stress and the particle velocity at any point of the structure 6 , and thus also to compute the repartition of the energy in the structure thanks to the definition of Poynting’s vector integrated over a cross-section of the device 7 . If we assume that the energy is constant in the system and is totally independent of the geometry of the device, the repartition of the energy between the substrate and the Love plate can be estimated. The Figure 3.14 presents this repartition for the 0th order mode for an increasing normalised thickness. The energy is moving from the substrate to the Love plate as the thickness of this one is increased. It can not be concluded from this 6 At

the condition that the value of the particle displacement and of the stress is known at one point of the structure. Since the top layer is in contact with the vacuum, this corresponds to a free-stress boundary where the stress is always null. The value of the particle displacement is arbitrarily set to the value 1. 7 A physical approach of the energy flux power in the Love waves devices has also been established by [Ses83].

38

CHAPTER 3. LOVE WAVES BIOSENSORS

observation that the energy is totally embedded in the Love plate for an infinite thickness since other Love modes appears as the thickness increases, then the repartition of energy hopes at the apparition of a higher order Love mode. It can be only concluded that in this range of the normalised thickness, the evanescent field in the substrate is reduced and consequently the energy of the wave is more located in the Love plate. The density of energy is directly given by dividing the partial energy in the Love plate by its thickness. Density of energy is meaningless for the substrate since it is an infinite half-space. The density of energy is decreasing gradually for increasing thickness since the total amount of energy put in the system is kept constant. Energy repartition for Quartz/SiO2/Vacuum

Energy repartition for Quartz/Polymer/Vacuum

1

1 Substrate Love plate

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

Partial energy

Partial energy

Love plate Substrate

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0.1

0.2 Normalised thickness

0.3

0.4

0

0

0.02

0.04

0.06 0.08 Normalised thickness

0.1

0.12

0.14

Figure 3.14: Repartition of the 0th order Love mode energy in a cross-section of the device for (left) silicon dioxide and (right) polymer Love plate as a function of the normalised thickness. The total amount of energy is constant. For increasing values of the thickness, the energy is more embedded in the Love plate and subsequently less energy is present in the substrate. The energy in a cross-section of the device presents a peak. The position of this peak has been reported as a function of the normalised thickness for the two types of devices in the Figure 3.15. For an increasing thickness, the peak is firstly very close to the top of the structure. At a certain thickness, the peak drifts away from the surface and throttles to the interface between the substrate and the Love plate. The critical thickness for that happening could not be clearly identified to a particular parameter of the structure. The question of interest is now: how to obtain an estimation of the sensitivity ? An usual procedure is to use the perturbation theory to compute the responsiveness of the system for a mass loading film on top of the structure. The perturbation theory was reported up to now to estimate the theoretical sensitivity of the Love wave device. We are much more interested in our approach to conceive the application of the transmission line equivalence as a complete picture of the biosensor. Therefore, two procedures can be used to estimate the sensitivity of the biosensor in regard to a mass loading. The first procedure is to simulate the perturbation by a new transmission line and compute again the Love wave velocity. Then, starting back from the definition of the sensitivity, estimate the response of the system to the perturbation. This procedure is not the best one since parameters have to be set arbitrarily to the perturbing layer, these parameters are not identical in every situation. In a second place, the sensitivity

39

CHAPTER 3. LOVE WAVES BIOSENSORS Position of the maximum of energy for the 0th order Love wave

Position of the maximum of energy for the 0th order Love wave

0.4

0.14

Layer thickness

0.12

0.3

Normalised distance

Normalised distance

0.1

0.2

Maximum of energy

0.08

0.06

0.04

0.1

0.02

0

0

0.1

0.2 Normalised thickness

0.3

0.4

0

0

0.02

0.04

0.06 0.08 Normalised thickness

0.1

0.12

0.14

Figure 3.15: Position of the maximum of energy as a function of the normalised thickness for (left) silicon dioxide and (right) polymer Love plate. The position is computed from the origin located at the interface between the Love plate and the substrate. For low value of thickness, the position of the peak of energy is attached close to the top of the device. For a critical value, the peak shifts towards the interface between the substrate and the Love plate.

is defined for a thickness of the perturbing layer that goes to 0, thus leading to instabilities in the computation. The second procedure is based on an energetic consideration. It is known that for an increase of the Love plate thickness, more energy is present in the Love plate than in the substrate. It can be approximated that this is similar with the variation of the repartition of the energy in the structure when a mass loading film attach to the surface. Let consider that the film is relatively thin in comparison to the layer thickness. The mass change due to film of thickness h p and density ρ p on an area A is given by ∆m = ρ p h p A. The thickness of the Love plate h required to obtain the same mass change is ∆m = ρhA. The relation between the normalised thickness is obtained by comparing the two expressions and by dividing by the wavelength: ρ p α p = ρα. It is not absurd to assume that the sensitivity of the Love waves sensor will be in correlation with energy transfer from the substrate to the Love plate. Therefore, the sensitivity can be estimated by the energy variation in the substrate to the mass modification described by the equation 8 : Sm ' lim∆α p →0 ∆Esubstrate = ∆α p

ρ p dEsubstrate ρ dα

(3.33)

Since the value of ρ p depends of the nature of the mass loading, we first consider that the film will have the same nature as the Love plate material so that the ratio ρ p /ρ vanishes from the equation (3.33) 9 . 8 We

have to admit here that this approach is not based on a pure mathematical approach. No real link with a mathematical expression of the sensitivity is rigorously done here for the moment. 9 This assumption is not valid for the study of silicon dioxide based biosensors since the actual density of the biofilm is closer to the density of water than the density of the SiO2 . It can be seen that the value of the sensitivity is overestimated but the position of the maximum as a function of the layer thickness are not modified and that is the most important from a practical point of view.

40

CHAPTER 3. LOVE WAVES BIOSENSORS

The equation (3.33) has been used to draw the Figure 3.16 where the estimation of the sensitivity of the biosensor has been computed for the two types of sensors as a function of the normalised thickness. Although the formula is not directly linked to a quantified value of the sensitivity, a calibration can be operated on reported devices from experiments and litterature. Some observations can be withdrawn from the comparison of the gradient of energy. A special care has to be driven to the axis in the Figure 3.16. The silica device presents a maximum of sensitivity for α∼ = 13% with a gradient of −0.016. The polymer device presents a sharp peak of sensitivity for α ∼ = 6.8% with a gradient of −0.148. It is easily concluded from the comparison that the polymer is a better material to integrate in the biosensor than the silicon dioxide. Nevertheless the low acoustic quality of polymer has not been taken into account for the simulation of the sensor and is responsible of a smoothing of the sensitivity peak. An other conclusion concerns the influence of the technical processing on the resulting sensitivity of the sensor. Let say that the real curve of sensitivity fits perfectly to the theoretical one, due to the sharp shape of the curve for the polymer device the window of the polymer thickness is also reduced strongly around the optimum value. This narrow window of processing, in comparison to the one of the silica device, is something relatively hard to achieve during a typical deposition step. This effect could even easily lead to the fabrication of a wide variety of sensitive devices, making then them useless for further applications. Gradient of the energy in the substrate for Quartz/Polymer/Vacuum 0

−0.002

−0.02

−0.004

−0.04

−0.006

−0.06

Gradient of energy

Gradient of energy

Gradient of the energy in the substrate for Quartz/SiO2/Vacuum 0

−0.008

−0.08

−0.01

−0.1

−0.012

−0.12

−0.014

−0.14

−0.016

0

0.1

0.2 Normalised thickness

0.3

0.4

−0.16

0

0.02

0.04

0.06 0.08 Normalised thickness

0.1

0.12

0.14

Figure 3.16: Gradient of the energy in the substrate as a function of the normalised thickness for (left) silicon dioxide and (right) polymer Love plate. The gradient presents a peak for a specific normalised thickness. This peak is sharper for the polymer than for the silicon dioxide.

3.4.2 Composite biosensors The previous section showed that polymer (epoxy) devices present a larger sensitivity in comparison to the silica device. One can imagine to integrate both of them in the Love plate. The composite device that results from this integration should present a layered structure where the polymer is deposited on top of the silicon dioxide in order to obtain Love modes. The figures reported in this section are established on the same approach that has been done for the monolayered devices but now with two degrees of variation

41

CHAPTER 3. LOVE WAVES BIOSENSORS

since the thickness of the two layers are independent and can be tuned separately. The estimation of the sensitivity by the formula (3.33) is reported in Figure 3.19. The estimation of the sensitivity presents a deep valley for a device that incorporate only polymer. The drawing in the bottom right of Figure 3.19 let compare directly the sensitivity for a monolayer sensor and for a composite sensor. It shows that the silica device is effectively much less sensitive than the polymer device and the positions of the peaks for each separate device are exactly at the same position as computed previously. Love wave phase velocity for Quartz/SiO2/Polymer/Vacuum

Love wave group velocity for Quartz/SiO2/Polymer/Vacuum 5000

5000 0.8

0.8

4000

0.6

0.5

3500

0.4 3000 0.3 2500 0.2

0.7 Normalised thickness of the silicon dioxide

Normalised thickness of the silicon dioxide

4500

4500

0.7

4000 0.6 3500 0.5 3000 0.4 2500 0.3 2000 0.2 1500

2000 0.1

0.1

1000

1500 0.05

0.1

0.15 0.2 0.25 Normalised thickness of the polymer

0.3

0.35

0.05

0.1

0.15 0.2 0.25 Normalised thickness of the polymer

0.3

0.35

Figure 3.17: Composite Love waves device is composed of a polymer plate on top of a silicon dioxide plate. Love wave (left) phase and (right) group velocity as a function of the normalised thickness.

Position of the maximum of energy in the SiO2 for Quartz/SiO2/Polymer/Vacuum

Position of the maximum of energy in the Polymer for Quartz/SiO2/Polymer/Vacuum

0.8

0.8

0.9 0.8

0.6

0.7

0.5

0.6 0.5

0.4 0.4 0.3 0.3 0.2 0.2 0.1

0.1

0.05

0.1

0.15 0.2 0.25 Normalised thickness of the polymer

0.3

0.35

0

0.95 0.7 Normalised thickness of the silicon dioxide

Normalised thickness of the silicon dioxide

0.7

0.9

0.6

0.85

0.5

0.8 0.75

0.4

0.7 0.3 0.65 0.2 0.6 0.1

0.55

0.05

0.1

0.15 0.2 0.25 Normalised thickness of the polymer

0.3

0.35

Figure 3.18: Composite device. The position of the maximum of energy in each layer is plotted with respect to its normalised value (ratio position of the maximum in the layer to the thickness of the layer). The position is plotted for (left) silicon dioxide and (right) polymer Love plate.

42

CHAPTER 3. LOVE WAVES BIOSENSORS

Energy repartition in the Quartz for Quartz/SiO2/Polymer/Vacuum

Energy repartition in the SiO2 for Quartz/SiO2/Polymer/Vacuum

0.8

0.8 0.8

0.9 0.8

0.6

0.7

0.5

0.6 0.5

0.4 0.4 0.3 0.3 0.2 0.2 0.1

0.7 Normalised thickness of the silicon dioxide

Normalised thickness of the silicon dioxide

0.7

0.1

0.05

0.1

0.15 0.2 0.25 Normalised thickness of the polymer

0.3

0.7 0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.35

0.05

0.1

0.15 0.2 0.25 Normalised thickness of the polymer

0.3

0.35

Energy repartition in the Polymer for Quartz/SiO2/Polymer/Vacuum

Gradient of energy in the substrate for Quartz/SiO2/Polymer/Vacuum 0.8 0.9

−0.02

0.8

0.7

0.6

0.6

0.5

0.5 0.4 0.4 0.3 0.3 0.2 0.2

0.7 Normalised thickness of the silicon dioxide

Normalised thickness of the silicon dioxide

0.7

0.8

−0.04

0.6

0.5

−0.06

0.4

−0.08

0.3 −0.1 0.2 −0.12 0.1

0.1

0.1

−0.14 0.05

0.1

0.15 0.2 0.25 Normalised thickness of the polymer

0.3

0.35

0.05

0.1

0.15 0.2 0.25 Normalised thickness of the polymer

0.3

0.35

Figure 3.19: Composite device. Repartition of the energy in the (top left) substrate, the (top right) silicon dioxide and the (bottom left) polymer as a function of the normalised thickness. For each layer, the values have been normalised between 0 and 1. The gradient of energy for increasing thickness of the polymer layer is given in the bottom right figure.

CHAPTER 3. LOVE WAVES BIOSENSORS

43

3.5 Comparison of the devices The observation of the repartition of energy in the structure, or the position of the maximum of energy, is interesting but is not a valuable key to the estimation of the sensitivity of the biosensor. The main parameter under study is the normalised thickness of the layers. The study of the repartition of the energy in the structure has clearly show that this energy is more embedded in the Love plate, and less presents in the substrate, for an increase of the normalised thickness. The sensitivity of the device can be estimated by the gradient of the energy in the substrate in regards to a change in the thickness of the top layer. This approach let consider a qualitative comparison between the devices but not a quantitative value of their efficiency. In addition, one has to remember that the materials have been supposed to present an infinite acoustic quality factor. This assumption is far from reality in the case of polymers, instead among them epoxies are believed to present a better acoustic quality value. For polymers, the damping of the wave in the structure would also be responsible of a high insertion loss since the signal is damped as it travels the structure. For the optimum of the sensitivity, the thickness are compared with regard of different approaches in the Table 3.5. The results from the model developed here fit well to the experimental results, but this is not the proof of the validity of the model for general cases. The study of the energy repartition has also clearly showed that the energy is less present in the substrate as the thickness of the layer increases. From a practical point of view, the energy in the substrate is converted back to an electrical signal; a lower value of energy means a higher insertion loss and could also be responsible of a lower mass loading sensitivity. In addition, the estimation of the sensitivity let appear a more rapid fall-off for large values of thickness, that correspond better to the experimental results than the perturbation theory as presented in Figure 3.3. Approach Theory of perturbation Experimental Transmission line model

Silicon dioxide Polymer 16.9% 6.4% (PMMA) 14% (never optimum) 13% 6.8% (Epoxy polymer)

Table 3.5: Normalised thickness for single layer devices at the optimum of sensitivity and after different approaches. Polymers are more interesting from the point of view of the sensitivity. The main problem with polymers is located in their high acoustic damping and the difficulty to achieve a high resolution of their thickness during the processing. In comparison, the silicon dioxide presents a lower damping for the considered frequencies, but the sensitivity of the silica device is low. A compromise between the two approaches is obtained by the composite structure that would attenuate the combined disadvantages of the two separate types of monolayer devices. The theoretical approach and the simulation could only yield an approximation of the best device but the acoustic damping of the polymer has to be taken into account otherwise the best device is the polymer device, that does not fit to the reality. Experiments on composite devices have to be led to check the validity of the theoretical approach and to manage the realisation of a device that would present a higher sensitivity than never showed up to now. It is believed that the real parameters of the materials would open a window on efficient high sensitivity devices on the estimated sensitivity reported in the Figure 3.19.

44

CHAPTER 3. LOVE WAVES BIOSENSORS z

z (1)

(1) (2) (3) (2’)

(2) (3) VLove plate

(4) x Vsubstrate

(5) V2 V1

x Vsubstrate

Figure 3.20: Graphical representation of the Love wave in the real space as a function of the propagation angle. Left: single layer device. Right: double layers device. A synthetic view of the propagation properties can also be deduced from the definition of the Love wave and Snell’s law. This view is represented on the Figure 3.20. In this figure, the bulk shear wave velocity of each layer, including the substrate, are represented in the xz plane 10 . If the materials are lossless, the relation that gives VLove is reduced to VLove = sinViθi . This relation is interpreted as the length of the 11 appears in the relation and indicate that the arrows with a dashed line in the left figure. The critical angle   Vi angle of propagation is lying between θi,critical = arcsin Vsubstrate (position 1) and π/2 (position 3). The angles before describe leaky waves in the substrate and all other angles (position 2) are guided Love waves. It might be interesting to consider now the case of the composite device, represented on the right in the figure. Two layers, with velocity V1 and V2 , are superimposed. They admit separately two different angles of propagation. Since the Love wave velocity is the same in all layers, the angle of propagation in each layer is directly found by drawing arrows 2 and 2’. At position 3, the angle reach π/2 in the layer 1 and consequently all waves with angle in layer 2 between 3 and 5 (for instance, the position 4) present an evanescent behaviour in the layer 1. One can see that happens when the Love wave reach the velocity in the layer 1. Such waves present probably a lower sensitivity since the coupling of the waves between the different layers becomes less efficient. We must also say that the relation between the thickness of the layers and the Love wave is hardly achieved by the representation in Figure 3.20. Hence, the sensitivity could not be expressed by a simple mean from these drawings.

10 This 11 see

is exactly the same procedure used to draw slowness curves as a function of the direction of propagation equation B.45

Chapter 4 Instrumentation 4.1 Introduction The Love waves are generated in the device by the intermediate of interdigital transducers. The variation in frequency of the device is monitored as the sensitivity of the device. The Figure 4.1 represents the schematic of the device in a usual dual channels closed loop operation. Two identical channels form the device; one serves as reference and the other presents the biochemical sensing part. They operate both as oscillator at a different frequency since the waves are submitted to different conditions. The electrical signal is coupled to a frequency mixer. If the sensing channel is oscillating at frequency f and the reference channel at frequency f0 , the mixed signal corresponds to: 1 1 (4.1) cos(2π f t) × cos(2π f0t) = cos (2π( f + f0 )t) + cos (2π( f − f0 )t) . 2 2 The result is filtered by a low-pass filter that eliminates the ( f + f0 ) component of the signal. From this operation, only the differential frequency between the two devices ∆ f = ( f − f0 ) is monitored. The sensitivity of the device is related to the ratio ∆ f / f . In this section, the properties of interdigital transducer are firstly presented. The expected voltage transfer function is deduced and design parameters are selected. A second part is dedicated to the electrical monitoring of the acoustic devices that has also been successfully applied to testing devices. Finally, temperature monitoring is also presented since the temperature changes introduce shifts in the signals.

4.2 Interdigital transducer 4.2.1

Principle

Generation of ultrasonic surface waves is easily obtained by deposition of a thin metal film, usually aluminium, on the highly polished surface of a piezoelectric substrate. The film is patterned by the conventional method of photolithography in order to generate interdigital electrodes (IDEs). The set formed by the substrate and the IDEs is called interdigital transducer (IDT). Interdigital transducers have been investigated in detail since White and Voltmer suggested that technology in 1965. This part of the report is dedicated to their basic principle for the presented application of the development of a biosensor. The theory of IDTs comes from Auld [Aul73a, chapter 10], Campbell [Cam89, chapter 4], Royer and Dieulesaint [RD99, chapter 7], and Smith [Smi81].

45

46

CHAPTER 4. INSTRUMENTATION

Love wave device

RF Amplifier A Active area

Directional coupler

IDT Active lowpass filter Mixer Directional coupler A RF Amplifier

Figure 4.1: Instrumentation of the Love wave biosensor. Two interdigital transducers and a space between them are the elements of a delay line. The delay line and a RF amplifier constitute a closedloop oscillator. A device is made of two delay lines (or two channels). An active area is presents at the top of one of them, that is the biosensitive area. The signals of the two delay lines are mixed together and filtered (low-pass) to obtain the synchronous frequency of the biosensor.

Once the IDT is connected to an electrical source, it produces an acoustic field in the substrate. The converse piezoelectric effect is used to detect the acoustic field by use of an other IDT located on the propagation path of the acoustic wave. The whole structure, emitter and receiver IDTs, is the basic electrical setup of any surface generated acoustic wave device. The Figure 4.2 is the schematic drawing of the structure viewed from the top. When the electrode fingers periodicity is equal to the acoustic wavelength, the transduction is the most efficient since the wave generated by the IDEs add in phase. The whole structure formed by the input transducer, intertransducers delay path and output transducer is an analog filter. The design procedure of this filter is easier to manage with knowledge of the transfer function H( f ), which is the product of the two IDTs transfer functions Hin ( f ) and Hout ( f ) and a factor that depends on the transducer midpoint separation length d H( f ) =

Vout = Hin ( f )Hout ( f )e(− jkx d) Vin

(4.2)

at the condition that no damping occurs on the delay path. The factor kx is the wavenumber in the direction of propagation and is given by kx = 2π/λ = 2π f /Vl with Vl the Love wave velocity. The separation between the two IDTs introduces a separation between them in the design procedure. Although they can be different, and so many different filter shapes can exist, we will focus here to symmetrical structures where both input and output IDTs are shaped in the same way and then present the same transfer function. For this condition of symmetry, the output IDT transfer function is directly given by the conjugate response of the input IDT Hout ( f ) = Hin∗ ( f ).

47

CHAPTER 4. INSTRUMENTATION

IDT midpoint-to-midpoint distance d

Delay path

Acoustic aperture W

INPUT IDT

OUTPUT IDT

Figure 4.2: Structure of the delay line: input and output interdigital transducers (IDT) are metal fingers at the surface of a piezoelectric substrate, the spacing between them determines the delay time

4.2.2

Equivalent model

The modelling of the filter pass-band could not be based on the usual L-C method for linear filters modelling. Actually, the L-C model refers to analyse the pole-zero of the filter design, this relates to energy storage in the electric and magnetic fields of the reactive components. For a Love wave device, as for any other kind of acoustic waves based device, the filter process with energy transfer between input and output IDTs. Consequently, the pole-zero analysis would contain only zeros, thus making this impossible to generate correctly this kind of filter. Specific models have been developed around the acoustic waves devices: the Delta Function Model, the Crossed-Field model and also the Impulse Response model. The Crossed-Field model is derived from the Mason equivalent model of piezoelectric bulk waves transducers. The electric field distribution is approximated as being normal to the piezoelectric surface as shown in Figure 4.3. Although the model is known to fit better to experiments for high-K 2 piezoelectric substrates, such as lithium niobate, an alternative model known as the in-line model is better for materials such as quartz (see Figure 4.4). Anyway, the Crossed-Field model is applied successfully for the IDT design and simulation of IDT on quartz substrate. In the model, each IDT is represented by a three-port network as sketched in Figure 4.5. There are two acoustic ports (ports 1 and 2) and one electrical port (port 3). The acoustic ports have to be represented by their electrical equivalent by introducing the same parameters used for simulation in chapter 3: the tensile force in the direction of propagation is transformed into electrical voltage −V and local velocity is transformed into electrical current I. This equivalence allows to interpret the acoustic characteristic impedance of the device as its equivalent transmission line characteristic impedance Zc . This definition introduces the electrical characteristic

48

CHAPTER 4. INSTRUMENTATION

-

+

-

Crossed E-field

Figure 4.3: The Crossed-Field equivalent model of the IDT

+

-

In-line E-field

Figure 4.4: The In-line equivalent model of the IDT

Electrical voltage V

Love wave 1

Electrical port

3

Love wave 2

3 Acoustic port

1

2

Acoustic port

Figure 4.5: The interdigital transducer is equivalent to a three-ports network

49

CHAPTER 4. INSTRUMENTATION

admittance G0 = 1/Zc , defined by G0 = K 2Cs f0

(4.3)

where K 2 is the electromechanical coupling constant for the considered direction of propagation, f0 the synchronous frequency and Cs the static capacitance of one periodic section. For identical fingers, the static capacitance is expressed by Cs = C0W, where W is the acoustic aperture (finger apodization overlap) and C0 the unit static capacitance. The static capacitance itself depends on the capacitance structure. For equally spaced fingers that have also the same width as the space between them, the capacitance C0 is expressed as 1 : C0 = ε p + εl

(4.4)

where ε p = (ε11 ε22 )1/2 is the surface electrical dielectric constant of the piezoelectric substrate, and εl is the electrical dielectric constant of the dielectric material superimposed to the substrate. For a ST-cut quartz substrate, the value ε p is equal to 3.92 · 10−11 F/m. The value of εl depends on the material used as Love plate is ranging between 3.5 · 10−11 F/m for polymers and 3.9 · 10−11 F/m for amorphous SiO2 . Within the three-ports model of the IDT, the admittance matrix that relates voltages and currents is given, after simplifications due to reciprocities and symmetries, by   Y11 Y12 Y13 Y =  Y12 Y11 −Y13  (4.5) Y13 −Y13 Y33 and the constitutive elements of the matrix are: Y11 Y12 Y13 Y33

= = = =

− jG0 cot(Nθ) jG0 csc(Nθ) − jG0 tan(θ/4) jωNCs + 4 jNG0 tan(θ/4),

(4.6) (4.7) (4.8) (4.9)

where N is the number of interdigital periods, and θ = 2π( f / f0 ) the electrical transit angle (in radians) through one period. At the synchronous frequency, the electrical transit angle is 2π and therefore the term in tan(θ/4) “blow up”. However, the impedance and transfer functions remain finite and have to be developed in series near the synchronous frequency. The input admittance can then be expressed as: Y33 ( f ) = j2π f NCs + Ga ( f )

(4.10)

and the term Ga is related to the radiation conductance, also expressed as: Ga ( f ) ' 8N 2 G0 1 The


sin x 2 x

(4.11)

expression showed here was developed for structure including vacuum. The estimation of the capacitance supposes that the one of the Love plate may directly replace the dielectric constant of the vacuum. Actually, the expression of the capacitance depends on the thickness of the materials and has to be estimated with help of electromagnetic fields simulator.

50

CHAPTER 4. INSTRUMENTATION

where x = Nπ( f − f0 )/ f0 . From this last expression, it can be clearly seen that the radiation conductance at the synchronous frequency is Ga ( f0 ) = 8N 2 G0 = 8K 2 f0Cs N 2 . The needed number of fingers is less for materials that present a high electromechanical coupling constant. The constant relates the conversion of mechanical energy into electrical energy and inversely. When the coupling constant is high, the number of fingers can be lowered. It should be also noticed that the value of 1/Ga ( f ) is well below 50 ohms, it is thus necessary to introduce a matching network between the source/the load and the transducers.

4.2.3 Two ports equivalent network

INPUT IDT Go

(1)

OUTPUT IDT (2)

(1)

(3) Ia

Vin

(2)

Go

(3)

Va

Ib

Vb

CT

CT Rs

Rl Vl

Figure 4.6: The Love wave device transfer function can be computed by this two-ports equivalent network The model of each IDT can be now integrated to shape the final device as represented in Figure 4.6. This equivalent two-ports network helps to compute the voltage transfer curve, if no loss in the delay line is still assumed. From the classical circuit theory, the admittance parameters of the equivalent model can be obtained. The current-voltage relations are Ia = yaaVa + yabVb Ib = ybaVa + ybbVb

(4.12) (4.13)

and input voltage Vin is driven by a source that presents an impedance Rl so that Vin = Va + Ia Rs , while load voltage across load impedance Rl is Vl = Vb . The resolution of the equation yields the voltage transfer function H( f ): H( f ) = VVinl =

yab Rl (1+yaa Rs )(1+ybb Rl )−y2ab Rs Rl

(4.14)

51

CHAPTER 4. INSTRUMENTATION

The admittance values yaa and ybb are identical since the device is considered to be symmetric, and these values are given by the expression 4.10. The transfer admittance yab is equal to yba . This term involves the interaction between the two IDTs and is given by h i 2( f − f0 ) 2d
sin x 2 jπ 1− f0 − λ 2 yab = 8N G0 x e

(4.15)

If the assumption of no loss in the delay line is drop then a damping factor has to be introduced in the equation (4.15). The damping factor increases the insertion loss of the entire delay line. The damping factor is not easily evaluated and need to be approached by the experiment.

4.2.4

Design and simulation

The design procedure of the IDTs has been explained in the introduction of this section. The design parameters are tightly bound to the specificity of the device itself, and to the physical parameters since the velocity of the Love wave and the dielectric constant of the Love plate enter the whole response of the device. For instrumentation, the operating frequency of the device, its insertion loss, the number of electrical modes, . . . have to be taken into account. At some point of the design procedure of the sensor, one has to focus to the transducing part and set some of the allowed values, then simulate the estimated response of the device. If this response seems suited to the needs of the device, then the design of the IDTs is no more of interest. At the opposite, an unusual response may be due to a design problem and in this case the design procedure has to be made again to check for the wrong points. Parameter IDE periodicity IDTs midpoint-to-midpoint distance number of interdigital periods acoustic aperture

Symbol λ d N W

Value 40 µm 7 mm 100 3 mm

Table 4.1: IDT design parameters For the Love wave device developed, the parameters have been arbitrarily set to the values reported in Table 4.1. The simulated transfer function for this device, without Love plate on the top, is given in Figure 4.7. The dielectric constant of the top layer is the vacuum’s one and the Love wave velocity is reduced to the SSBW velocity that is 5060 m/s. In the figure, the insertion loss and the phase angle are presented around the synchronous frequency of 126.5 MHz; the straight line represents the 3 dB level. The minimum insertion loss of the structure is −25.4 dB and the 3 dB bandwidth is 0.8 MHz. The presence of a Love plate changes the dielectric constant and the acoustic wave velocity. Resulting variation in the voltage transfer curve will be presented in chapter 7 for an easier comparison with the experimental curves. The curves presented in Figure 4.7 is only an illustration of the expected shape of the transfer function of the designed device.

52

CHAPTER 4. INSTRUMENTATION

|H| −20

Insertion loss [dB]

−40 −60 −80 −100 −120 −140

1.2

1.25

1.3 Frequency [Hertz]

1.35 8

x 10

Phase H

Phase angle [rad]

4

2

0

−2

−4

1.2

1.25

1.3 Frequency [Hertz]

1.35 8

x 10

Figure 4.7: Transfer function of the device with acoustic velocity of 5060 m/s and vacuum on top of the quartz substrate

53

CHAPTER 4. INSTRUMENTATION

4.2.5 Second order effects and improvements The second order effects are all phenomena that make the real device to deviate from the theoretical one, as known and reported in the literature. Some of these effects can found easy solutions to implement and that often need to be applied; sometimes these solutions are difficult to implement and would just be reported here as illustrative examples. The diffraction is depending in a complex manner of elastic, dielectric and piezoelectric anisotropy inherent to the presence of the piezoelectric substrate. The wave front diffraction is characterised as a function of the distance of propagation, the phase velocity, the propagation of direction and the acoustic aperture (i.e. the length of the fingers that form the IDEs). Usually a large acoustic aperture can attenuate the diffraction effect. The electrodes influence the propagation. Their edges introduce a discontinuity in term of topography and also in term of acoustic impedance since they show different elastic constants and density. In addition, they also introduce an electric boundary condition since the tangential electric field cancels at their edges. These conditions are respectively known as mass loading and electrical loading. The loads introduce a small reflection at each edge of each electrode. In device where there are numerous pairs of fingers, the signal can be largely spoiled because reflections can add in phase. Hopefully, this problem is easily solved by double electrodes where each electrode finger is split in two identical electrodes. The improvement of the design is sketched in Figure 4.8. The same polarity electrodes have to be spaced of one-quarter wavelength and each approximatively one-eigth wavelength wide. With this configuration, reflections interact destructively and then largely cancel near the synchronous frequency. In addition, the transfer function of the double electrodes transducer is virtually identical to that one of the single electrodes transducer. The disadvantage of this method is the requirement of getting double resolution during electrodes fabrication. The IDT itself generates other acoustic modes than only the wanted surface acoustic modes; two separate and different type of waves are generated in addition to the SAW. The shallow propagating bulk wave and the plate modes. Both of them are undesired modes of propagation that can have influence of the transfer function of the device since they interact with the receiver IDT and create unwanted modes called spurious modes. It might be possible to get rid of them, especially of their reflection from the bottom of the device, by a roughening of the back surface. Bulk waves are always generated by the IDT since the structure is equivalent to a radiating antenna. In the piezoelectric medium, there are preferential directions where the emitted bulk waves interact in phase and reach a maximum strength. The angles corresponding to these directions are given by fP =

Vb cos φ

(4.16)

where P is the fingers periodicity and Vb the bulk waves phase velocity (longitudinal and/or shear) for propagation angles 0 ≤ φ ≤ ±90◦ . In addition to the presence of those unwanted acoustic modes, the multireflections of the waves on the device faces generate many unwanted interferences. For the high frequencies and the near distance separation of IDTs, an electromagnetic feedthrough (cross talk) appears between the two IDTs, due to the direct coupling between input and output electrical signal. The interaction give rise to periodic ripples of amplitude and phase in the filter passband at the ripple frequency fr = 1/τ, where τ = d/V . This is also a source of interferences in the final device, except the presence of other capacitance effects due to packaging and bonding wires, and we will see that could

54

CHAPTER 4. INSTRUMENTATION

SIMPLE IDE

DOUBLE IDE

TO

SIMPLE IDT design

IMPROVED IDE design

Figure 4.8: The design can be improved to reduce diffraction from the input IDT and the reflections from the output IDT

reduce further the instrumentation operation of Love wave devices by changing the electrical condition of oscillation. The incidence of the acoustic wave with IDT connected to a finite electrical impedance source or load introduces reflection of the wave. This effect is partly due to the mass/electrical loading and partly due to regeneration due to the interaction between the different fields (acoustic and electric) and the presence of the finite electrical impedance. The reflection give rise to the triple transit echo (TTE). The echo is located in the filter passband at the ripple frequency fT T E = 1/2τ.

4.3

Dual channels oscillator configuration

4.3.1 Introduction Two complementary informations are gathered from acoustic wave devices: amplitude v.s frequency and phase v.s frequency. This section relates the electronic developments required for monitoring these three quantities (frequency, phase and amplitude), and the various kinds of behaviour monitoring techniques. The operating frequency of acoustic wave devices is ranging between 5 MHz and 150 MHz ; the electronics developed are mainly of the RF type, although these techniques can easily be extended to lower frequency ranges using op-amp based circuits.

55

CHAPTER 4. INSTRUMENTATION

4.3.2 Closed loop operation A sine wave voltage in an electronic circuit is subject to two effects: amplitude damping and phase shift. These two effects are dependant upon the excitation frequency. If the electronic circuit can be tuned so that the output signal is of the same amplitude than the input signal, and has a zero phase in respect to the incoming signal, the sine wave generator can be removed and the loop closed upon itself: a self oscillating phase-locked loop has been obtained 2 . The basic elements of a phase locked loop are (Figure 4.9): • an amplifier so that the output signal can be of amplitude equal or greater to that of the incoming signal (during the open loop tuning phase); • a phase shifter in order to be able to compensate for any phase shift in the circuit and hence be able to get the zero-phase condition ; • a band-pass filter to make sure the phase locked loop (PLL) will oscillate around the wanted mode (and not on an overtone). The acoustic wave sensor is inserted in this loop to monitor changes of its behaviour under changes of external physical conditions. The typical devices we are concerned here have many possible oscillation mode within a restricted frequency range (100-105 MHz): a fine tuning of the gain and band-pass filter is necessary to decide on which mode to oscillate. -15

200 "am1.dat"

Φ

G

ΣΦ=0 ΠG >=1 BPF

AW line

"ph1.dat"

-20

150

-25

100

-30

50

-35

0

-40

-50

-45

-100

-50

-55 8e+07

-150

8.5e+07

9e+07

9.5e+07

1e+08

1.05e+08

1.1e+08

1.15e+08

1.2e+08

-200 8e+07

8.5e+07

9e+07

9.5e+07

1e+08

1.05e+08

1.1e+08

1.15e+08

1.2e+08

Figure 4.9: Schematic circuit (left) of an oscillator whose frequency is set to the resonance frequency of the SGAW device (Φ and G must then verify the condition ∑ Φ = 0 (sum of the phases must be null so that the output signal is equal to the input signal) and ΠG ≥ 1 (energy input must be greater or equal to the energy losses in the circuit) for the oscillator to run). Middle and right: typical frequency responses (amplitude at center, phase at right) of an acoustic wave device (lithium tantalate substrate). The advantage of this closed loop operation is that a very fast acquisition rate can be used for monitoring both amplitude and frequency and observing fast physical phenomena occurring on the surface of the sensor.

2 Such

a phase-locked loop will always start oscillating when switched on because in order to obtain an output amplitude equal to the input amplitude, an amplifier has to be included in the loop to compensate for the energy losses. No amplifier is perfect and noise-free: any initial fluctuation in the loop will grow to full oscillations when power is switched on.

56

CHAPTER 4. INSTRUMENTATION

However, there are some disadvantages: • as mentioned previously, a multi-mode acoustic wave device can oscillate on several close (in terms of frequencies) modes which display very similar amplitude and phase conditions: selection of one of the modes is difficult to achieve; • although shear waves are supposed not to be damped by liquids, a damping of 3 to 6 dB is experimentally observed when a liquid is deposited on the sensing surface. Hence, choosing a fixed gain for the PLL is very difficult, and work with higher gains than theoretically necessary (for an output amplitude equal to the input amplitude during open loop tuning) is necessary, which brings problems of non-linear behaviours of the amplifier (as the output amplitude is greater than the input amplitude, the signal will increase until non-linearities or saturation of the amplifier stop any further increase) and the inability to select which mode to oscillate on by fine-tuning the gain of the loop (so that only one of the modes verifies the gain ≥ 1 condition). 98700 98600 98500 98400 98300 98200 98100 98000 97900 97800 97700

freq1-1.031E8

0

1000

2000

3000

4000

5000

6000

7000

58100 58000 57900 57800 57700 57600 57500 57400 57300 57200 57100

8000

9000

10000

freq2-1.029E8

0 241000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

freq1-freq2

240900 240800 240700 240600 240500 240400 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Figure 4.10: Observed frequencies of a dual-lines configuration sensor with lithium tantalate as substrate (frequency of each line, and difference (compensation for external disturbances to the lines). Considering a temperature coefficient of 30 ppm/K for lithium tantalate, a stability of 1 kHz on each individual frequency of 100 MHz means temperature fluctuations of 0.3 K. Due to these difficulties, open loop monitoring should be firstly investigated. A reliable method to come back to closed loop operation is presented at the end of this section with help of a slightly more complex system (helped by digital electronics and micro-controller based feedback).

4.3.3

Open loop operation

Open loop operation aims at monitoring the amplitude and frequency response of the acoustic wave device under test at various frequencies. As a full spectrum is obtained during each acquisition, many more informations than simply the resonance frequency can be derived. However, a precise frequency sweeping

57

CHAPTER 4. INSTRUMENTATION

circuit must be realised (and characterised), and amplitude and phase detection methods must be developed. The first and easiest method that can be imagined is simply sweeping a voltage-controlled oscillator’s (VCO) frequency, and after compensating the great amplitude losses introduced by the sensor (-20 to -30 dB typically), monitor the output phase and amplitude. The output of the VCO was monitored: it is only stable within 500 Hz during a few minutes, and within 2 kHz over a few hours. Such a stability is not enough for our application: at least 100 Hz stability is wanted (and hence frequency resolution). Although this basis scheme set the system on the right tracks (and is good enough for realising a network analyser in the 60 to 120 MHz range with 2 kHz stability), it needs refinements for the specific application of the Love wave biosensor.

0 network analyzer (200 ms/800 pts) -10 -20 -30 -40 -50 -60 6e+007 3500

7e+007

8e+007

9e+007

1e+008

3500

1.2e+008

1.3e+008

VCO (10 min/65536 pts)

VCO : zoom 91-109.5 MHz

3000

1.1e+008

3000

2500

2500 2000

2000 1500

1500

1000 30000

35000

40000

45000

50000

1000 0

10000

20000

30000

40000

50000

60000

70000

Figure 4.11: Comparison between the amplitude response of an acoustic wave device as observed by the HP network analyser, and the VCO. Even though the VCO is not stable enough for monitoring fine properties changes, amplitude/phase at a given frequency or resonance frequency, of the acoustic wave device under external physical parameter changes, it still allows the measurement of a rough response of the acoustic wave device with frequency. The chart of Figure 4.11 was obtained by sending the output of the VCO, after going through a testing acoustic line, to a +25 dB amplifier (MAR-6), an operational amplifier mounted as a follower, an amplitude detector and finally a 12 bits AD574 A/D converter. The result is compared to that of a commercial HP network analyser.

4.3.4

Stabilisation of the basic set-up

Monitoring the VCO frequency and compensating for drifts is necessary in order to achieve the required variable frequency stability. Fixed frequency quartz crystals have much better stabilities than VCOs (usually within a few hertz for resonators oscillating at a few MHz), the output of the VCO can be compared

58

CHAPTER 4. INSTRUMENTATION

with the signal obtained from a fixed crystal oscillator. Because the VCO aims at having a variable output frequency fo , the only way to compare its signal with that of a fixed oscillator fr is by finding integers N and M so that fo /N= fr /M. Because the VCO should not start oscillating at very low frequencies but only above 60 MHz, a mixer has to be included before the divider by N and thus subtract a starting-frequency fm to fo so that the condition comes back to ( fo − fm )/N = fr /M (Figure 4.12). The requirements of the monitoring system will now define fr and fm : fm is the the minimum frequency at which the VCO will oscillate. For the acoustic device developed here, fm must be a little bit N above 60 MHz, and in practice a fm above 90 MHz is needed. Because fo = fm + M fr , fr /M is in fact the step size between to values of fo . If a 100 Hz step sizes is wanted, then fr /M = 100 Hz. An additional constraint on the choice of fr /M is that not only is this quantity the step size between two increases of fo , but it also gives an idea of the feedback loop action speed of the PLL over the VCO frequency. For instance, an ideally fr /M = 1 Hz could be asked, but the PLL will only be able to correct the VCO frequency each second or so. That means the system should hold on a few seconds after each frequency change of the VCO for its output signal to stabilise (i.e. for the PLL to stabilise). A compromise must be chosen between VCO steps size (which needs to be as small as possible) and feedback loop bandwidth (inverse of its reaction time), which should be as high as possible. Until here, development of a single-PLL synthesiser was showed. However, because the VCO does not start oscillating a very low frequencies, but only works above 60 MHz, fm the minimum frequency at which the VCO will oscillate was introduced.

fm fo fr divide by M

P.D

divide fo − fm by N LPF

VCO

Figure 4.12: Single loop PLL synthesiser principle The question is : where does fm come from ? A fixed quartz oscillator can hardly generate such a high frequency and a stability of fm has to be garanteed. The solution is simply by generating fm from a VCO included in a second PLL loop. Another advantage of generating fm in a second PLL is linked to digital electronics dividers. The frequency range fo − fm hqs to be divided by N to reach the fr /M range. The freauency fo can sweep from 60 to 120 MHz so that fo − fm can go up to 120 − 60 = 60 MHz. On the other hand, fr /M is in the 100 Hz range: the divider by N should thus be able to reach N=

60.106 = 600000 > 219 , 100

so a 20 bits (at least) divider is required. Such a divider is not commercially available. Furthermore, this

59

CHAPTER 4. INSTRUMENTATION

20 bits divider must be able to work at frequencies up to 60 MHz. If fm may vary by larger steps (compared to fr /M) rather than having a fixed frequency, these 20 bits can be split in 16 + 4 bits (i.e. one 16 bits divider, commercially available, and one 4 bit divider, also commercially available and able to work at very high frequencies). This result can be obtained by adding a second closed loop with varying fm as depicted in Figure 4.13. LPF divide P.D by M 0 (M 0 < M)

divide by N 0

VCO fm

fm0 fr

divide by M

fo

divide fo − fm by N LPF

P.D

VCO

Figure 4.13: Dual loops PLL synthesiser principle

9.0372e+007

9.0275e+007

9.038e+007

"vco12.dat"

"vco14.dat"

"vco13.dat"

9.0274e+007

9.036e+007

9.03715e+007 9.0273e+007

9.034e+007 9.0371e+007 9.0272e+007 9.032e+007 9.03705e+007

9.0271e+007 9.03e+007 9.027e+007

9.037e+007 9.028e+007 9.0269e+007 9.03695e+007

9.026e+007

9.0268e+007

9.0369e+007 19000

9.0267e+007 19500

20000

20500

21000

21500

22000

22500

23000

23500

24000

9.024e+007 0

5000

10000

15000

20000

25000

30000

0

1000

2000

3000

4000

5000

6000

7000

8000

Figure 4.14: Left: short term stability of the VCO (frequency fluctuations within 700 Hz, acquisition time: 1h). Middle: long term (in)stability of the VCO (acquisition during 5h30). The two sharp frequency shifts are due to touching the output cable. Right: sensitivity of the VCO to the output emitter-follower NPN transistor’s polarisation voltage (moved from 12V to 15V, inducing a 30 to 50 kHz shift, certainly due to impedance variation).

4.3.5 From open loop to closed loop Once the VCO is stabilised source, the closed loop monitoring of a property of the acoustic wave delay line can be considered again(zero phase, maximum amplitude) by programming a feedback (software PLL) on the microcontroller/computer controlling the counter in the feedback loop of the VCO. The limitation here is the settling time of the feedback loop stabilising the VCO: a compromise must be chosen between the resolution of the frequency steps and the settling time of the PLL controlling the VCO’s frequency. This latter problem does not arise when monitoring a given parameter (amplitude or phase) at a given frequency: the VCO always stays set around the same frequency and the PLL stabilising is never disturbed. This solution gives rise to a fully operational closed loop operation for the delay line.

60

CHAPTER 4. INSTRUMENTATION

4.4 Temperature control The generation of acoustic wave in the substrate gives rise to an elevation of temperature in the device. Although this effect is relatively small, it has influence on the behaviour of the sensor. The drift in temperature with time has to be taken into account for further development of the sensor, corrections to bring to the measured values will be deduced from the temperature drift. It should be also noticed that the dual channels operation reduce the common rise in temperature of the two channels. In the design procedure and for the instrumentation, a temperature sensitive element should be introduced into the device. This temperature sensor should present several characteristics that made it suited for long-term development of the whole biosensor: • present on the device after fabrication procedure of the biosensor; • low acoustic wave propagation perturbation; • sensing properties easy to handle; • avoid crossed sensitivity (only sensitive to the temperature). Actually, a simple metal resistor made with the same metal as for the IDEs is enough if well designed. The temperature sensitive element would then be present at the interface between the substrate and the Love plate, for measure in-situ of the temperature modification. The sensitivity in temperature of a metal strip is given by the variation in its value with temperature. The relative variation in resistance is linked to the relative variation of the metal resistivity: ∆R ∆ρ = . R ρ

(4.17)

The absolute variation of the resistivity ∆ρ is proportional to the temperature by the first-order temperature coefficient. Furthermore, the resistance R is given by the well-known relation R=

ρL A

(4.18)

where L is the metal strip length and A its cross-section area. By introducing R from equation (4.18) into the equation (4.17), the sensitivity with temperature of the resistor comes out ∆R =

L ∆ρ A

(4.19)

The sensitivity of the resistor is characterised by its geometry (ratio L/A) and the metal intrinsic property (by the temperature coefficient). The first order temperature coefficients for several metals that could enter the fabrication of the biosensor are reported 3 in Table 4.2. Since this last parameter is almost the same for the presented materials in the table, the factor of influence is only the geometry. One can see that a higher sensitivity would be obtained by the higher ratio L/A, so by a very thin and long metal strip. The design of a resistor is something quite easy to manage. The metal strip is structured as a zig-zag to increase its length to spatial occupation ratio as shown in Figure 4.15. 3 source:

Handbook of Chemistry and Physics, 56th edition. CRC Press, Ohio, 1975-1976.

61

CHAPTER 4. INSTRUMENTATION Metal Resistivity [µOhm · cm] Aluminum 2.6548 Copper 1.6730 Gold 2.35

  Temperature coefficient K −1 0.00429 0.0068 0.004

Table 4.2: Metals resistivity and first-order temperature coefficients at 20 ◦ C

Figure 4.15: Top view of a metal strip as temperature sensitive element

For instrumentation, the resistor is the fourth element of a balanced Wheatstone bridge. The highest sensitivity of the bridge itself is obtained by getting all resistors with the same value equals to the value of the temperature sensitive resistor. Furthermore, the other resistors have to be chosen with a very small temperature coefficient in order to avoid interference from environmental temperature changes.

Chapter 5 Experimentation “L’imagination est plus importante que le savoir.” Albert Einstein

5.1 Introduction The physical realisation of Love waves devices is presented in this chapter. The process flow relative to the fabrication of three types of devices including silicon dioxide and epoxy materials, and the design of the masks, is introduced. Pictures taken during the processing are shown to specify the critical parts of the fabrication process. The knowledge of those parts is necessary in order to obtain an ameliorated process flow that would lead to reproducible, free of defects, devices. For the devices efficiently fabricated and mounted, the experimental curves are reported for their characterisation in the air. Some of them were characterised with water. A single channel oscillator has been performed on a selected silica device. This device was monitored with liquid on the sensing area and has showed a really nice frequency response, good stability and low noise level. Some conclusions are withdrawn in regard to future biosensor developments.

5.2 Elaboration of the sensors 5.2.1 Process flow Three types of devices have to be considered in agreement with the theoretical approach done in previous chapters: silicon device devices, epoxy devices, and composite silica-epoxy on 3 inches ST-cut quartz wafer. Although the structure of the device is relatively simple, the processing could include many critical parts that have to be achieved carefully. The processing of polymer as photoresist has been reported intensively and is a usual part of any microelectronics processing 1 . Effectively, the photo-induced polymerisation of polymers is the basic principle at the origin of their use as resist material. The polymers can be either positive or negative as their UV-lighting result respectively in their polymerisation or in their decomposition. This principle is used in order to select the nature of the masks (dark or light field) used for the patterning as a function of the nature of the polymer and of the processing steps. 1 [SM89]

could serve as reference about the use of polymers in microelectronics.

62

63

CHAPTER 5. EXPERIMENTATION

Clean substrate

Metallisation

Photolithography

Photolithography

Metallisation

Etching

Lift-off

Formation of SiO 2 layer

Negative polymer spin coating

Photolithography

Etching

Metallisation

Photolithography

Photolithography

Metallisation

Etching

Lift-off

Wafer dicing

Mounting and electrical testing

Biochemical deposition

Figure 5.1: Process flow for the fabrication of the devices

CHAPTER 5. EXPERIMENTATION

64

In the processing developed for the fabrication of the devices, the epoxy is always a positive resist. In the case of a composite device, the epoxy serves equally as Love plate material and as resist before the silicon dioxide etching. The process flow of the devices is summarised in Figure 5.1.

5.2.2

Devices design

The design of the mask has been done in order to obtain a wide characterisation of design influence on the device. A maximum of three levels is used in the fabrication, they are reported as the deposition of the IDT metal layer (metal 1), the opening of the layers for the contact pads and the top interfacial gold layer (metal 2). Six different types of testing structure have been set, their design is reported schematically in the appendix D. Since negative and positive resists are used, each mask level was done in dark and light fields 2 . Each device included two channels. The major differences between the different design are explained here after. For the level metal 1, a large metal strip was linking the center part between the two channels. For the same level, a large metal area is drawn between the IDTs in order to estimate the influence of the metal on the wave propagation. For the level metal 2, the difference resides mainly in the metal area. This is done to observe the influence of the covering of the IDTs by the metal. It was known from the literature that a larger area increases the insertion loss of the device but cancels the influence of liquids with high dielectric constant on the propagation of the acoustic wave. A total of 20 devices could fit to the 3 inches wafer. Each device has dimension 15 × 12 mm. The repartition of the devices and their numbering 3 for identification is reported in Figure D.1. On one device, the upper channel is identified as line 1 and the second channel is line 2. It should be noticed that a careful interpretation would be done once the results would be known since a wide variety of parameters are varying in the same time. The testing design made here is not optimised from a design-of-experiments point of view.

5.2.3 Fabrication This section is devoted to the description of some results of the fabrication. The process flow is something quite ideal, the reality could be very different and some critical points of the fabrication process have to be put in evidence. Those points can happen at any place of the process and have to be distinguished among two classes: • badly designed process flow; • misuse of the procedures. A badly designed process flow is related to bad choices in the procedures for the fabrication. A particular attention should be driven to the compatibility of the successive steps of the process flow. The possibility of getting a cross-contamination of the etching bath with different type of metal is one of the most important of them. A badly designed process flow adds unnecessary steps and/or can lead to a device different than the expected one. For the specific process flow developed here, only the processing of gold could lead to problem since this metal is considered as a contaminant material. Its processing should always be the last step of the processing. 2 The 3 The

I.

design of the alignment crosses and the quality testing structures is a separate problem not reported in this report. numbering is formatted as IJ, I identifies one of the six design and the J identifies the position of the device of design

CHAPTER 5. EXPERIMENTATION

65

The design is not the only factor, the procedures themselves are critical points. Since the process flow is linear, successive steps of fabrication rely on the quality of the previous one. Problems that occur during one step may destroy completely the whole wafer that has then to be rejected ! It is even worse when the procedures are not perfectly suited to the needs of a specific process flow or if the parameters of a standard procedure have to be changed to fit the desired process flow. It might be necessary in those latter cases to make first several tests in order to obtain the procedure parameters. This particular problem is typical of the optimisation of polymer processing. Effectively, this processing is dependant of a large number of factors such as the UV-lighting time, the postbaking time and temperature. In particular, for the Love wave devices where it was clearly shown that the layer thickness has to be tuned with precision; processing characterisation of the epoxy is lacking to this study. From the effectively performed processing, several critical points were discovered and are briefly summarised here after. They are partially base on the pictures taken during the processing and reported in Figures 5.2 to 5.5. These points to check or to improve are the following: • Cleaning of the substrate. Although residues from the cleaning are scarce, their influence on the following processing steps is dramatic and results generally in the discarding of the processed wafer. • IDE patterning and etching has to be controlled in order to avoid over- or under-etched IDEs. • Wet etching of the CVD silicon dioxide is something really dependant of the quality of the deposited film. The etching rate may vary resulting in over-etching. The difficult part of the processing with silicon dioxide is that it could not be differentiated easily from the quartz substrate. In addition, the aluminium is also quickly etched, the result of a too long etch is dramatic. • Gold lift-off over large area could present problem at the edges.

Figure 5.2: Left: open circuit in the IDEs due to not-well cleaned wafer surface. Right: problem in the CVD silicon dioxide.

CHAPTER 5. EXPERIMENTATION

66

Figure 5.3: Left: the resist can stick to the mask (hard-contact photolithography) and damage the pattern of the IDEs. Right: open circuit on the IDEs caused by dust or hair.

Figure 5.4: Left: over-etching of the silicon dioxide has also remove the aluminium contact pad, the resist not postbaked can hardly sustain the buffer HF and has cracked on the edges. Right: gold lift-off, edges are critical points.

CHAPTER 5. EXPERIMENTATION

67

Figure 5.5: When everything is running fine, the result looks better. Left: a partial view of the device. Right: IDEs well patterned and not over-etched have the same width and are equally spaced.

5.3

Experimental results

Three wafers have been successfully processed and electrically characterised. One with no Love plate (vacuum), this one served as reference. One with CVD silicon dioxide as Love plate, with a thickness of 6.8 ± 0.5 µm (α = 17 ± 1.3%). One with epoxy polymer with thickness 3.3 ± 0.1 µm (α = 8.3 ± 0.3%). The devices on the wafer were diced and mounted. Four different kinds of mounting have been envisaged, numbered here from 1 to 4. The choice of the mounting procedure is a relatively important parameter since the quality of the electrical bounding can change radically the aspect of the signal at the operating frequency of the acoustic devices (from 100 MHz to 125 Mhz). The devices have been electrically tested in open air. They operated in open loop and a network analyser measured their transfer function. Figures 5.6,5.7, and 5.8 give the repartition of the tested devices and the mounting for respectively the vacuum, silica and epoxy wafers. Near the wafer repartition, the simulation of the voltage transfer function expected from the devices has been included. Figures 5.9 to 5.13 are the experimental curves for the vacuum devices. Figures 5.14 to 5.19 are the experimental curves for the vacuum devices. Figures 5.20 and 5.21 are the experimental curves for the vacuum devices. On the Figures 5.10 and 5.15, a drop of water was put on top of the device in order to estimate the influence of the liquid loading on the device. The temperature sensitive element of the device was monitored. Its change is reported in Figure 5.22 for a pulse of 10 ◦ C applied during 5 and 10 seconds. One of the device (silica device number 53, line 1, mounting type 4) has been selected and mounted in the oscillator configuration. Its frequency change to a water loading has been monitored during one half-hour. The Figure 5.23 reports the frequency monitored on a frequency counter.

68

CHAPTER 5. EXPERIMENTATION

|H| −40

52 51

23

13

24

−60

11 14

MOUNTING 1

Insertion loss [dB]

22

53 54

MOUNTING 2

−80 −100 −120 −140 −160 1.2

33

31

42

62

61

63

64

41 44

1.22

1.23

1.24 1.25 1.26 Frequency [Hertz]

1.27

1.28

1.29

1.3 8

x 10

Phase H

MOUNTING 3

4

Phase angle [rad]

32

1.21

MOUNTING 4

2

0

−2

−4 1.2

FLAT

1.21

1.22

1.23

1.24 1.25 1.26 Frequency [Hertz]

1.27

1.28

1.29

1.3 8

x 10

Figure 5.6: Test sensor: the quartz wafer is not covered by a Love plate. Left: repartition of the tested devices. Right: theoretical voltage transfer function for this configuration.

|H| −40

52 51

23

13

24

−50

11 14

MOUNTING 1 MOUNTING 2

Insertion loss [dB]

22

53 54

−60 −70 −80 −90 −100

33

31

42

62

61

63

64

FLAT

41 44

MOUNTING 3 MOUNTING 4

1.01

1.02

1.03

1.04 1.05 1.06 Frequency [Hertz]

1.07

1.08

1.09

1.1 8

x 10

Phase H 4

Phase angle [rad]

32

1

2

0

−2

−4

1

1.01

1.02

1.03

1.04 1.05 1.06 Frequency [Hertz]

1.07

1.08

1.09

1.1 8

x 10

Figure 5.7: Silicon dioxide sensor: the quartz wafer is covered with silicon dioxide, thickness 6.8µm. Left: repartition of the tested devices. Right: theoretical voltage transfer function for this configuration.

69

CHAPTER 5. EXPERIMENTATION

|H| −40

52 51

23

13

24

−50

11 14

MOUNTING 1 MOUNTING 2

Insertion loss [dB]

22

53 54

−60 −70 −80 −90 −100

33

31

42

62

61

63

64

FLAT

41 44

MOUNTING 4

1.01

1.02

1.03

1.04 1.05 1.06 Frequency [Hertz]

1.07

1.08

1.09

1.1 8

x 10

Phase H

MOUNTING 3

4

Phase angle [rad]

32

1

2

0

−2

−4

1

1.01

1.02

1.03

1.04 1.05 1.06 Frequency [Hertz]

1.07

1.08

1.09

1.1 8

x 10

Figure 5.8: Polymer sensor: the quartz wafer is covered with the polymer, thickness 3.3µm. Left: repartition of the tested devices. Right: theoretical voltage transfer function for this configuration.

Figure 5.9: (Vacuum) Left: device 13, line 1, mounting 1. Right: device 51, line 1, mounting 3.

CHAPTER 5. EXPERIMENTATION

Figure 5.10: (Vacuum) Device 14, line 1, mounting 2. Left: in the air. Right: with water on top

Figure 5.11: (Vacuum) Device 31, mounting 3. Left: line 1. Right: line 2.

70

CHAPTER 5. EXPERIMENTATION

Figure 5.12: (Vacuum) Device 41, line 2, mounting 4.

Figure 5.13: (Vacuum) Device 61, mounting 1. Left: line 1. Right: line 2.

71

CHAPTER 5. EXPERIMENTATION

Figure 5.14: (SiO2 ) Left: device 13, line 1, mounting 3. Right: device 14, line 2, mounting 4.

Figure 5.15: (SiO2 ) Device 53, line 1, mounting 4. Left: in the air. Right: with water on top.

72

CHAPTER 5. EXPERIMENTATION

Figure 5.16: (SiO2 ) Device 54, mounting 1. Left: line 1. Right: line 2.

Figure 5.17: (SiO2 ) Device 31, mounting 4. Left: line 1. Right: line 2.

73

CHAPTER 5. EXPERIMENTATION

Figure 5.18: (SiO2 ) Device 41, line 2, mounting 2.

Figure 5.19: (SiO2 ) Device 44, line 1, mounting 1. Left: 20 MHz of span. Right: 2 MHz span.

74

CHAPTER 5. EXPERIMENTATION

Figure 5.20: (Polymer) Device 14, mounting 1. Left: line 1. Right: line 2.

Figure 5.21: (Polymer) Device 44, mounting 1. Left: line 1. Right: line 2.

75

76

CHAPTER 5. EXPERIMENTATION

Figure 5.22: Evolution of the signal in the temperature sensor from room temperature 20 ◦ C to approximately 30 ◦ C for (left) 5 s and (right) 10 s.

frequency [MHz] 104.650000

104.640000

104.630000

104.643700

104.643650

104.620000 104.643600

104.610000 104.643550

104.600000 104.643500

104.590000

104.643450 0

100

200

300

400

500

600

104.580000

104.570000 0

200

400

600

800

1000

1200

1400

1600

1800

2000

time [s] Figure 5.23: Closed-loop operation of the silicon dioxide device with water. The oscillator was monitored in open air for 900 seconds before getting water. The water was removed from the device after 1600 seconds. The in-box is a close-up view of the first 600 seconds.

CHAPTER 5. EXPERIMENTATION

77

5.4 Analysis and interpretation The first observation relating to the curves is their general shape. Although they are not perfectly fitted to the simulations, the peaks are almost present at the right positions. The IDTs development has probably to be optimised but at this moment they present a reasonable possibility for an experimental investigation of the device efficiency. Some ripples occurs in the passband of the device, they could be due to the length of the delay line which is 7 mm center-to-center. The results obtained with the epoxy are not looking like guided acoustic modes, they present almost the same shape as the vacuum devices except that their insertion loss is relatively important. Actually, no real explanation could be obtained for the polymer based sensor since the polymer thickness is relatively important. It might be that the resonance conditions are not achieved in the material or the damping along the delay line itself is too important so that no real signal reach the output transducer. For this latter case, the electromagnetic feedthrough and the wave propagating in the substrate are responsible of the monitored curves. It is obvious that more experiments have to be done on the epoxy device in order to make them acting as really guided Love waves sensors. The interpretation about these devices could not be processed any longer due to a lack of information at this moment. A prime step for next development is the monitoring of the devices for lower, not optimised, thickness of the epoxy layer. This requires in a first part to obtain an extended and valuable characterisation of the polymer processing; something that was not achieved during this study. For the other devices, the Table 5.1 summarises some important values off the experimental results. The table reports the value of the frequency resonance peak and its insertion loss (IL peak ), the mean value of the insertion loss (ILmean ), and the bandwidth corresponding to a 2π shift of the phase around the peak. An estimation of the quality of one device and its mounting is given by the number reported in the column R. The ratio R corresponds to 4 : IL peak − ILmean . R= IL peak + ILmean A high R is an indicator of a better contrast between the mean level of the insertion loss and its peak. A high R is also the indicator of a low minimum insertion loss, which is usually preferable for the electronic instrumentation since it reduces the necessity of high gain amplifiers. The ratio is sketched in Figure 5.24 for a visual comparison. The analysis on the only criteria of R is not sufficient but its value gives a reasonable idea of the efficiency of the devices and the mounting. Due to both a high range of devices and mounting and a low number of experiments, a real comparison between the devices and the mounting is really hard to achieve. However, the results for the vacuum devices should not be really considered with that much attention since they were there only to detect problems during fabrication and mounting. One has to focus on the results for the silicon dioxide devices. For those devices, a clear separation can be done between devices 53 and treatment of the error for the ratio is explained here after. The relative error on the measure of X is represented by ρX . A ∂R B A−B The ratio results from the formula R = A+B . The relative error on R is thus given by |ρR | = ∂R · · |ρ | + ∂B · R · |ρB |. A R ∂A This expression developed gives : 2B A(A + B) −2A B(A + B) · |ρB |. |ρR | = · · |ρA | + · (A + B)2 A − B (A + B)2 A − B 4 The

Obviously, the absolute error is given by RρR .

78

CHAPTER 5. EXPERIMENTATION

Material

Device

Vacuum

31 31 14 41 61 61 13 51 14

Line Mount

1 2 1 2 2 1 1 1 1’

3 3 2 4 1 1 1 3 2

average variance SiO2

53 14 53 13 31 54 54 31 44 41

125.3 125.3 125.1 125.3 125.1 125.1 125.3 125.4 125.1

IL Phase Mean IL [dB] [MHz] [dB] −3 −2 (±5 · 10 ) (±5 · 10 ) (±2)

R [%]

-19.40 -19.07 -31.71 -25.40 -15.80 -16.20 -35.33 -35.86 -37.37

0.74 0.74 0.74 0.68 0.74 0.74 0.74 0.74

-35 -34 -53 -40 -23 -22 -45 -45 -39

29 ± 5 28 ± 5 25 ± 4 22 ± 5 19 ± 8 15 ± 9 12 ± 4 11 ± 4 2±5

-20.22 -20.27 -27.38 -27.40 -26.50 -27.03 -30.75 -27.43 -24.04 -12.01

0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47

-50 -50 -50 -42 -40 -40 -43 -36 -28 -13

42 ± 3 42 ± 3 29 ± 4 21 ± 5 20 ± 5 19 ± 5 17 ± 5 14 ± 5 8±7 4 ± 15

125.22 0.016 1 2 1’ 1 1 2 1 2 1 2

average variance

Peak [MHz] (±5 · 10−2 )

4 4 4 3 4 1 1 4 1 2

104.6 105.35 104.6 103.65 105.4 102.9 102.7 105.45 104.26 104.10 104.30 0.978

Table 5.1: Summary of the results for the tested devices

79

CHAPTER 5. EXPERIMENTATION Ratio for silicon dioxide devices 50

45

45

40

40

35

35

30

30 Ratio [%]

Ratio [%]

Ratio for testing devices (vacuum) 50

25

25

20

20

15

15

10

10

5

5

0

31−1,3

31−2,3

14−1,2

41−2,4

61−2,1 Device

61−1,1

13−1,1

51−1,3

0

14−1’,2

IL

53−1,4 14−2,4 53−1’,4 13−1,3 31−1,4 54−2,1 54−1,1 31−2,4 44−1,1 41−2,2 Device

−IL

mean Figure 5.24: Graphical representation of the ratio IL peak . Left: vacuum devices. Right: CVD peak +ILmean silicon dioxide devices. The abscise is formatted to give Device-Line,Mount.

14 and the other devices. Their ratio is almost the double than for the other devices. In addition, their transfer curve is in agreement with the estimation of efficiency given by R and they look perfectly suited for the oscillator configuration. It explains why the device 53 was selected for the monitoring represented in the Figure 5.23. The monitored device 53 achieved at room temperature a stability of almost 150 Hz. At the water loading, its oscillation has continued and drifted of approximately 70 kHz. This is a 670 ppm frequency shift, really promising for its testing and characterisation with biochemical species. An immediate but still future work with this device is to obtain the characterisation for a mass loading, so that its sensitivity could be determined. A thin metal layer could be deposited at its top. The metal density and the thickness deposited can serve for an estimation of the mass loading of the sensor. We hope to obtain a value reasonably close to 270 cm2 /g for this specific device. The monitoring of the temperature sensitive element has proved that the temperature observation for drift compensation could be achieved. The application of this factor would enter in a final application for the devices, when a sensitivity curve would be calibrated. This could be the part of a future work. The ratio R is a valid criteria to select devices for further investigation. When R is higher than 33%, the signal presents a peak well discriminated from the mean level. In addition to this selection, the transfer curve must be inspected to identify spurious peaks at proximity of the central peak. These others peaks, if they present an insertion loss close to the central peak, may generate mode hopping in the instrumentation and consequently spoil the estimation of the efficiency of the device. Finally, the conclusion that can be withdrawn from the experimental part has two main points. The first point is related to the design of the devices. Designs 3,4, and 6 present a relative low insertion loss but they have a large number of spurious modes that make them not suited for instrumentation. These designs have a common point: the large metal part that links the IDEs. This part is probably responsible of a coupling between the two dual lines and also between the input and output transducers of a same line.

CHAPTER 5. EXPERIMENTATION

80

The best design is achieved with design 1 and 5. The influence of the large metallic area between the IDTs in design 1 could not be put in evidence at this time. Therefore, the design could absolutely avoid any coupling of the IDEs by a conductive media. A second point is related to the mounting procedure for testing and instrumentation. The mounting 4 has achieved the best results in comparison to the other mounting and could then be used in priority for further developments.

Chapter 6 Conclusions and future outlines “ Tout ira mieux demain, voil`a mon esp´erance. Tout est bien aujourd’hui, voil`a l’illusion.” Voltaire Biosensors are marvellous sensors ! They result of a multidisciplinary adventure where physics, biochemistry, surface chemistry, materials science and electronics meet together. The need for fast, reliable, easy, and sensitive measurements of environmental values is increasing and would encounter a still increasing demand in the future. This demand could be encountered by a rapidly evolving technology: acoustic waves biosensors. Due to the nature of the acoustic wave, acoustic waves biosensors present sensitivity to many parameters. In this approach, a specific acoustic waves biosensor has been investigated for its application as gravimetric sensor. Although the sensor is able to sense liquid properties, the main objective with biosensor is to detect the variations in the mass of a biofilm presents on the surface of the sensor. This mass change is related to the absorption in surface of specific genes or antigenes. The interfacial chemistry was briefly presented. It is the part of biochemical developments of the sensor and it determines the high selectivity of the biosensor to specific (bio)chemical species. The approach done here is the optimisation of the physical parameters of the sensor. The sensor investigated supported Love modes. The Love modes are resonating waves in a layered structure. They present a shear-horizontal polarisation that makes them suited to liquid supported (bio)chemical detection. An equivalent model was developed and successfully helped to determine the characteristics of the acoustic waves. The equivalent model is based on the transmission line theory in reason of the analogy between the acoustic and the electrical waves. This model, used here in its simplest form, has a high potential to determine the wave properties as they propagate in viscous and/or piezoelectric media. An in-depth development of this model would also be able to compute directly the influences of the design (materials and thickness) in an electrical simulator. This could be the part of a further development of the modelling of the Love waves biosensor. The sensitivity of the sensor was identified to the variations of the energy repartition for a mass loading on top of the device. In that way, the approach was different from the perturbation theory but the results obtained were in agreement with the experimental data reported in the literature. The influence of the thickness of the layer was put in evidence: a maximum of sensitivity is achieved for an optimum thickness that depends upon the materials combination. 81

CHAPTER 6. CONCLUSIONS AND FUTURE OUTLINES

82

The literature reported the experiments on devices that included silicon dioxide or/and a polymer (PMMA) on a ST-cut quartz substrate. The influence of the normalised thickness (α =h/λ) on two main parameters of the device were studied and reported: the sensitivity and the insertion loss. It was known that a silicon dioxide based device with a normalised thickness of approximately 14% presents a reasonable sensitivity roughly equals to 380 cm2 /g and a low insertion loss, and the PMMA coated device a higher sensitivity but also a higher insertion loss due to the poor acoustic quality factor of the polymer. A composite structure including both SiO2 and PMMA was also studied since this kind of device present an acceptable insertion loss for a higher sensitivity. In this study, the main objective was to determine the conditions necessary to obtain an optimised biosensor. The integration of an epoxy polymer instead of PMMA could theoretically reduce the insertion loss of the device and still present a high sensitivity. Thank to the equivalent model, an estimation of the device integrating the epoxy polymer was established. The polymer-based device is estimated to be almost 10 times more sensitive than the silicon dioxide based device, with a lower material thickness (h/λ ' 6.8%). Nevertheless, this estimation does not take into account the acoustic damping in the polymer and consequently the maximum expected sensitivity could be lower. The composite structure has also been theoretically investigated, experiments have to be done in order to conclude to the validity of the approach. It was also concluded from the shape of the sensitivity curves as a function of the layer thickness that the processing of the polymer-based devices will require a strong resolution on the thickness during the processing since the sensitivity presents a narrow peak around the optimum thickness. This condition implies a processing of the polymer that is relatively insensitive to the usual variations that occur during a normal fabrication procedure. From a practical point of view, silicon dioxide devices with a normalised thickness of 17 ± 1.3 % and epoxy devices with a normalised thickness of 8.3 ± 0.3 % were successfully fabricated and electrically characterised. Several test structures were designed and masks were fabricated in order to obtain the devices. Four mounting procedures have been considered for comparison. The epoxy based devices could not be tested any longer since their response was not corresponding to a Love mode but looked more like simple mass loaded devices without any layer (blank structure). The high thickness of the processed layer is probably the main cause of this effect. For the silicon dioxide device, one of them could be selected for its nice characteristics. It was monitored in a closed-loop operation under water loading and could prove that the acoustic wave based technology could be investigated further with the design and the mounting it presented. Hopeless, the sensitivity of the device was not studied due to a lack of time, and the application as biosensor was not clearly demonstrated. This would be the part of any future work on this subject. In regard to this specific structure, the next step is to evaluate in practice the composite device. It is strongly believed that this kind of device would maximise the sensitivity with an insertion loss compliant with the electronic instrumentation around. In addition to this study, other materials for getting a better control on the sensor processing could replace the materials themselves. In addition, new processing procedures have also to be investigated. Materials with a stronger piezoelectric coefficient like lithium niobate would probably open the way to new designs instead of its high temperature coefficient. Also, the use of piezoelectric materials directly as Love mode guiding layer would modify the properties of the biosensor.

Appendix A Table of Symbols Symbol

Signification

[Units]

a c cE D E e f fc i j K k k l M S T t u V Vg Vl Vs Y Z Zc

inter-atomic distance elastic stiffness matrix elastic stiffness matrix at constant electric field electric displacement electric field piezoelectric strain matrix wave frequency wave cut-off frequency √ imaginary number √−1 (mathematical notation) imaginary number −1 (electrical notation) atomic bounds rigidity constant wavevector wavenumber direction of propagation atom mass strain field stress field time displacement field wave phase velocity wave group velocity longitudinal wave phase velocity shear wave phase velocity equivalent transmission line admittance equivalent transmission line impedance equivalent transmission line characteristic impedance

[m] [N/m2 ] [N/m2 ] [C/m2 ] [V/m] [C/m] [Hz] [Hz]

83

[N/m] [1/m] [1/m] [kg] [N/m2 ] [s] [m] [m/s] [m/s] [m/s] [m/s] [(m · s)/kg] [kg/(m3 · s)] [kg/(m2 · s)]

84

APPENDIX A. TABLE OF SYMBOLS Symbol Signification α εS Γ Γ γ λ Φ ρ ω

normalised thickness electric permitivity matrix at constant mechanical strain Christoffel matrix stiffened Christoffel matrix equivalent transmission line propagation function wavelength electric potential density angular frequency

[Units]

[F/m]

[1/m] [m] [V] [kg/m3 ] [Hz]

Appendix B Acoustic waves propagation in solids “Le silence est la plus grande pers´ecution.” Blaise Pascal, Pens´ees

B.1 Introduction This appendix presents the propagation of acoustic waves in solids, it is the theoretical support for the modelling 1 . Waves propagation is ruled by Christoffel’s equation that is derived from Newton’s dynamic equation and from the constitutive relations for solids. The propagation of time-harmonic plane waves is investigated since their comportment is well suited to waves that can be generated in solids. These waves present three orthogonal displacement components, also called polarisations. Each polarisation has its own phase velocity. Polarisations are largely influenced by the materials characteristics. For that reasons, propagation in isotropic, anisotropic and piezoelectric solids will be detailed and also attenuation effect in viscous solids. An intuitive graphic representation of the inverse of velocity as a function of direction of propagation is given. This representation will help further comprehension of the mechanisms involved in the propagation of the waves. In bounded media, reflection and refraction occur at boundaries. Snell’s law rules the relation between incident and scattered waves. The link between the waves is given by introducing the definition of the acoustic impedance. The impedance gives a direct relation between stresses and local particle velocity for a given direction of propagation of the waves. The definition of that impedance will be the starting point for model and study of resonant structures. Entrapment of the waves in a multi-reflective structure is an acoustic waveguide that support resonant schemes. Finally, power flow of the waves is considered. The well-known Poynting’s Theorem will found a direct equivalent in acoustic. In the study of sensors, energy repartition and influence of environmental parameters play an important role. Although any wave can propagate in a media, propagation of time-harmonic plane waves make the study easier. The mechanical fields that are generated and that propagate under harmonic solicitations at angular frequency ω with wavevector k are described as functions of the position r and the time t by functions proportional to ei(ωt−kl·r) . This describes the well-known time-harmonic plane waves. The angular frequency ω and the wavenumbers ki (i = 1, 2, 3) are related by a function called relation of dispersion. Actually, the onliest knowledge of that relation is sufficient to describe the propagation of harmonic-time 1 The

sources for this appendix were mainly found in [Aul73b], [Pai99], and [RD96]

85

86

APPENDIX B. ACOUSTIC WAVES PROPAGATION IN SOLIDS

plane waves.

atom mass M

atomic bond rigidity K

interatomic distance a

Figure B.1: Linear model of solid : atoms with mass M linked by elastic bonds of constant K A preliminary note before entering the subject is necessary. A simple model of solid is a linear arrangement of atoms with mass M, separated by a distance a and connected together by elastic bonds of constant K (see Figure B.1). The relation of dispersion for this model is r K ak ω=2 sin (B.1) M 2 and it presents for considerably low values of the wavenumber (ka x1 is e−α(x1 −x2 ) . The exact expression of attenuation factor α is obtained by replacing the assumed solution (B.33) in the equation of motion. The obtained solution has often a complex shape that can be reduced to a simpler form by stating the low-loss approximation for shear waves in isotropic media: 

ωη44 c44



1

(B.34)

This statement is valid for most solids with a viscosity coefficient sufficiently small even at frequencies as high as 1 GHz. The attenuation per wavelength αλ = 2πα k is function of the acoustic quality factor (or acoustic Q) c44 . Q= ωη44 The attenuation is known to be proportional to the square of the frequency. At high frequencies, high quality materials have to be used to avoid large acoustic damping.

B.2.7

Slowness surfaces

For a given angular frequency ω, the wavenumber k is a function of the direction l called the wave vector surface in the k−space. This surface is the slowness surface because of the relation k/ω = 1/V . Solutions to the Christoffel equation usually require help of computational methods except for some cases (propagation along symmetry axes, isotropic media, . . . ). We illustrate here after the slowness surface for some materials including amorphous silicon dioxide (SiO2 , isotropic), quartz ( α − SiO2 ) and lithium niobate (LiNbO3 ). Selection of these materials is due to their illustrative example but, and also, because they play an important role in acoustic sensors. Amorphous silicon dioxide is an isotropic material. As considered before, this kind of material supports pure modes: a longitudinal wave with phase velocity Vl and two shear waves with phase velocity Vs . Reporting the materials data from appendix C for silicon dioxide, the slowness surfaces in the k−space are represented in Figure B.2, for a plane cut the slowness surface takes the name of slowness curve. The figure shows two curves, due to the degeneration of the shear waves. The outer curve is the slowest velocity and is related to the shear waves. The inner surface is the longitudinal wave, faster than the two others. Numbering of the k direction is purely arbitrary and any direction can be chosen. In the whole k−space, slowness surfaces are spheres; whatever the cut and the direction of propagation, slownesses remain the same. Anisotropic and piezoelectric cases are showed simultaneously here after. The slowness curves are strongly dependent on the chosen direction of propagation. We consider first lithium niobate, which belongs to the 3m crystallographic class. Slowness curves in the plane YZ are shown in Figure B.3. They are drawn with (solid line) and without (dashed line) taking piezoelectricity of this compound into account. From Figure B.3, three observations can be withdrawn: • There are three curves that present some symmetry around the central point but these curves are no more circles like in the isotropic case; • The dashed and the solid curves for the pure shear wave are superimposed in the Figure. This wave has polarisation along direction X ([100]).

93

APPENDIX B. ACOUSTIC WAVES PROPAGATION IN SOLIDS

ky ω shear

2e−4 s/m longitudinal 1e−4 s/m

kx ω

Figure B.2: Slowness curves for silicon dioxide.

94

APPENDIX B. ACOUSTIC WAVES PROPAGATION IN SOLIDS

[001] shear, polarisation [100]

quasi−longitudinal

quasi−shear 0.0002 s/m

0.0001 s/m

[010]

Figure B.3: Slowness curves for YZ-plane in LiNbO3 with (solid lines) and without (dashed lines) taking account of piezoelectricity.

APPENDIX B. ACOUSTIC WAVES PROPAGATION IN SOLIDS

95

• Along some directions, slowness for transversal waves crosses but those directions change when considering or not piezoelectricity. One can deduce from those observations that, for a given direction and a specific polarisation, an electrical field either accompanies or not the acoustic wave and this modify the phase velocity of the wave. This effect is quantified by the following relation taken from the expression (B.29): ! [e l ][l e ] K j j i iL cEKL 1 + E (B.35) (cKL )(li εSij l j ) where the fractional change in the elastic constant is given by coefficients called electromechanical coupling constants KKL : (KKL (l))2 =

[eK j l j ][li eiL ] (cEKL )(li εSij l j )

(B.36)

The very expression of KKL depends on the considered wave and direction. The constant determines the maximum variation in phase velocity by Ingebritsen’s relation: ∆V V

∼ =

K2 2

(B.37)

The α − SiO2 belongs to crystallographic class 32. The slowness curves are presented for propagation of waves in the plane corresponding to the ST-cut. This cut contains the X axis and the normal is inclined of 47.5◦ with respect to the Z axis. Slowness curves are presented in Figure B.4, the coupling constants in α − SiO2 are very small so that only not piezoelectric slowness curves are presented in the figure. The two piezoelectric materials presented enter in sensors as transducing elements. A large value of the coupling constant indicates a strong effect between electric and acoustic fields. Electrodes plated to the surface of the material can do excitation of acoustic waves and the coupling constants are related to the conversion of electric energy to acoustic energy. Efficiency of the acoustic transducer is really dependent on this factor. We will discuss of the importance of coupling constants latter, when considering design of the transducer. The dependence of the coupling constants with direction of propagation is used to launch preferentially one kind of wave instead of one another. For instance, in the case of lithium niobate: the coupling constant along direction Y + 163◦ in the YZ−plane is null. Shear waves are generated in lithium niobate crystal with cut Y + 163◦ and faces covered with metallic electrodes. Although longitudinal waves are still allowed to propagate, electric sources can not induce their existence. For the ST-cut α − SiO2 , the coupling constants for both quasi-longitudinal and quasi-shear waves are null for direction orthogonal to the X axis and only of K 2 = 0.11% for the pure shear wave. That direction can be used to launch and propagate shear waves that have a polarisation parallel to the cut plane, they are called shear-horizontal waves (or SH-waves). Anyway, it will be too restrictive to talk only about lithium niobate and quartz for generation of acoustic waves. Other materials are under investigation for that goal and receive much attention like berlinite (AlPO4 ), gallium phosphate (GaPO4 ), aluminium nitride (AlN), gallium arsenide (GaAs), . . . In addition, the two materials showed here are also under investigations for finding new cuts [dCdAF98]. These cuts depend obviously of the final application they want to receive.

96

APPENDIX B. ACOUSTIC WAVES PROPAGATION IN SOLIDS

quasi−shear quasi−longitudinal shear 0.0003 s/m

0.0002 s/m

0.0001 s/m [100]

Figure B.4: Slowness curves for ST-cut α − SiO2 without taking account of piezoelectricity.

APPENDIX B. ACOUSTIC WAVES PROPAGATION IN SOLIDS

B.3

Reflection and refraction

B.3.1

Snell’s law

97

At boundaries, acoustic waves experience a change in propagation properties. They can propagate faster or slower, being partially reflected. We assume a plane boundary between two solids, with normal n; displacements and traction forces have to be continuous across the boundary (quoted values are relative to the second medium): u = u0 T · n = T0 · n

(B.38) (B.39)

and as plane wave fields are described by wave functions e−ik·r = e−i(kx x+ky y+kz z) incident and scattered waves must all have the same component of k tangential to the boundary. The remaining component of the wavevector is determined by a relation called acoustic Snell’s law, which is the acoustic pendant of the well known Snell’s law in electromagnetism. For isotropic solids, the acoustic Snell’s law takes a very simple form that can be compared to the electromagnetic Snell’s law: ω Vl

sin θl = Vωs sin θs = Vω0 sin θl 0 = Vω0 sin θs 0 l

s

(B.40)

where θ is the angle between the normal to the boundary plane and the direction of propagation. For anisotropic solids, a simple analytical expression of Snell’law is often difficult to obtain and numerical methods have to be used. The slowness surface give a useful representation to find directly both directions and number of reflected and refracted waves by a simple geometrical construction. Figure B.5 shows schematically that construction for two isotropic materials with different slowness surfaces. The axis on this figure are components tangential (kk /ω) and perpendicular (k⊥ /ω) to the boundary plane. The plane perpendicular to the boundary plane contains an incident shear wave. This wave arrives at the interface from the solid represented below on the figure. Conservation of the tangential component of k give rise to two reflected waves (both shear waves) in the first solid and three refracted waves (one longitudinal and two shear waves) in the second solid, which has phase velocities slower than in the first solid. All the waves are contained in the same plane perpendicular to the interface plane. Extension of this effect to anisotropic solids is direct: up to three reflected and three refracted waves might arise from boundaries and the direction of each one is geometrically deduced from the slowness surfaces or from analytical solution to the boundary value problem. In piezoelectric solids, the problem is more difficult since two additional boundary conditions are necessary for the electrical values. These two additional conditions can be expressed with use of the quasistatic approximation: Φ = Φ0 D · n = D0 · n

(B.41) (B.42)

Those conditions give rise to up to five reflected and five refracted waves (three acoustic waves and two electromagnetic waves). Solutions can be deduced geometrically by adding electromagnetic slowness curves to the acoustic slowness curves.

98

APPENDIX B. ACOUSTIC WAVES PROPAGATION IN SOLIDS

Solid 2

k8/w 2 refracted shear waves

1 refracted longitudinal wave

k///w

Solid 1

k///w 1 incident shear wave

2 reflected shear waves

Figure B.5: Reflections and refractions at boundary between two isotropic solids.

APPENDIX B. ACOUSTIC WAVES PROPAGATION IN SOLIDS

B.3.2

99

Critical angle

The geometrical constructions in Figure B.5 do not always give real intersections with the slowness surface for all of the scattered waves. Physical significance is deduced from the dispersion relation for transmitted wave for isotropic solids: 2

2

kk 0 + k⊥ 0 =

 ω 2

(B.43)

V0

In this equation, the tangential component kk is greater than (ω/V 0 ) leading to a pure imaginary perpendicular component k⊥ : 

2

k⊥ = ±i kk −

 ω 2 1/2 V0

,

(B.44)

the sign is chosen to satisfy the physical requirement of getting a signal that reduces to zero as distance trends to infinite. From this consideration, one can see that the wave is totally reflected; the refracted wave is evanescent. For isotropic solids and from Snell’s law (B.40), total reflection occurs for incidence angles greater than a critical angle defined by: θcritical = arcsin

B.3.3

V V0




(B.45)

Acoustic impedances

Previous sections have shown a relation between stress fields T and particle displacement fields u, or equivalently between stress fields and particle velocities 2 v = ∂u ∂t . For a given direction of propagation, each plane wave solution has three stress components and three velocity components. If we consider, for a direction kl = k, the following definition (Tk )i = Tik with i = x, y, z, the relation between the two fields is given by: −Tik = (Zak )i j v j

(B.46)

and the 3 by 3 matrix (Zak ) is called the acoustic impedance matrix. Acoustic impedance coefficients have dimension [kg · m−2 · s−1 ] that is called Rayleigh (Rayl.) in acoustic. As values of acoustic impedances are usually large, the use of MRayl. = 106 · Rayl. is relatively spread. For a given direction, there are three different acoustic impedance matrices, corresponding to one (quasi)longitudinal and two (quasi)shear waves. Elements of matrix (B.46) are found from the acoustic field equations. Explicit calculation of this matrix is given here after. For further resolution of scattering at boundaries, the relation (B.46) is generalized to measurement of acoustic impedance in direction n = [nx ny nz ] for plane wave propagating in direction k: −Tin = 2 the

factor ω has to be introduced in the relations

niK cKL kL j vj ω

(B.47)

100

APPENDIX B. ACOUSTIC WAVES PROPAGATION IN SOLIDS

where niK



 nx 0 0 0 nz ny =  0 ny 0 nz 0 nx  0 0 nz ny nx 0

(B.48)

and kL j = klL j is defined by (B.10). Elements of the acoustic matrix for that direction are: (Zak )i j =

niK cKL kL j ω

(B.49)

and one can see that impedance matrix elements can be obtained by a slight modification of the Christoffel matrix in (B.9).

B.4 Energy considerations B.4.1

Acoustic Poynting’s Theorem

Wave propagation requires for complete description the formulation of an energy conservation relation. In electromagnetism, this relation is derived from Maxwell’s equations and is known as Poynting’s Theorem. For acoustic fields, using the strong analogy with electromagnetic fields, a similar relation is found. The scalar product of the translationnal equation of motion (B.3) with −v is taken and added to the double scalar product of (B.2) (differentiated with respect to t) with −T. The result can be expressed as: −∇ · (v · T) = −ρv ·

∂S ∂v −T : +v·F ∂t ∂t

(B.50)

and then integration is carried out over a volume V and the divergence theorem is used to introduce an integral over the bounding surface S: ∂v (−v · T) · n dS = − ρv · dV − ∂t S V

I

Z

∂S T: dV + ∂t V

Z

Z

V

v · F dV.

(B.51)

We assume first that surface S is stress-free so that the surface integral in (B.51) goes to zero. Only R volume integration terms remain, in which the term V v · F dV = Ps is the power supplied to the volume V by the sources. For a system without loss, the expression ∂v ρv · dV + ∂t V

Z

Z

V

T:

∂S dV ∂t

(B.52)

is the rate of change of of stored energy in the system, ∂U/∂t. The first term in (B.52) can be identified as the rate of change of kinetic energy density,   ∂uv ∂v ∂ 1 2 = ρv · = ρv , (B.53) ∂t ∂t ∂t 2 and the other term is the rateof change  of the elastic energy stored energy density, us . For a viscously R ∂S damped medium, the term V T : ∂t dV contains both stored energy and power loss term. With all those considerations, the acoustic Poynting’s Theorem can be written for the general case as H

S (−v · T) · n

dS + ∂U ∂t + Pd = Ps

(B.54)

APPENDIX B. ACOUSTIC WAVES PROPAGATION IN SOLIDS

101

where U=

Z

V

(uv + us ) dV

(B.55)

is the total kinetic and elastic stored energy in V . The surface integral is the total power flow outward through closed surface S. The integrand P · n = −v · T · n is the power flow density in the direction n and we can thereby define an acoustic Poynting vector P = −v · T

(B.56)

that has dimension [W · m−2 ].

B.4.2

Complex acoustic Poynting’s Theorem

Poynting’s Theorem (B.54) relates to instantaneous power flow and energy storage. For time-harmonic fields, Poynting’s Theorem is found to be ! I Z Z ∗ (−v∗ · T) T : S∗ | v |2 v ·F · n dS = iω −ρ dV + dV (B.57) 2 2 2 2 S V V In this equation, we can replace terms by definitions for commodity. The complex power supplied by the source: Z

V

v∗ · F dV = Ps 2

(B.58)

and the peak kinetic stored energy in the system: | v |2 ρ dV = (Uv ) peak . 2 V

Z

(B.59)

The peak strain energy (Us ) peak and the time average power loss due to viscous damping (Pd )AV can be defined by:  Z Z  T : S∗ S : c : S∗ S : η : S∗ iω dV = iω + iω dV = iω(Us ) peak − (Pd )AV . (B.60) 2 2 2 V V Finally, the complex acoustic Poynting’s Theorem is H (−v∗ ·T) S

2

· n dS − iω{(Us ) peak − (Uv ) peak } + (Pd )AV = Ps

(B.61)

and the complex acoustic Poynting vector is defined as P=

−v∗ ·T 2

(B.62)

APPENDIX B. ACOUSTIC WAVES PROPAGATION IN SOLIDS

B.4.3

102

Energy transport of plane waves

For the specific case of time-harmonic plane waves, the expression of Poynting’s Theorem takes a simpler form. The relation of dispersion for those waves implies interchange between kinetic and potential density of energy; equivalent peaks are exactly balanced and the relation (B.61) reduces to a simpler power balance: (−v∗ · T) · n dS + (Pd )AV = Ps 2 S

I

(B.63)

and when there is no attenuation, the complex acoustic Poynting vector is a real number that represents the average power flow density PAV . From this latter condition, the equation that can be obtained is rather interesting: H

S PAV

· n dS = Ps

(B.64)

If the energy velocity is the vector defined by the ratio of the average power flow to the peak (kinetic or potential) energy PAV · n Ve = , UAV and an other relation that can be derived is k · Ve = V.

(B.65)

Usually, the energy velocity is equal to the group velocity defined by Vg =

∂ω . ∂k

The energy velocity direction, or the group velocity direction, is the normal to slowness surface at the point given by the intersection between the wavevector and the slowness surface. With that point of view, pure modes are the directions where energy velocity direction is aligned with direction of propagation. When the medium is dispersive, the phase velocity and the group velocity are different. The relation between them can be deduced by differentiating the phase velocity with regard to wavenumber: Vg = V + k · dV dk

(B.66)

This relation gives the influence of the dispersion on the group velocity. Figure B.6 gives the representation of the phase and the group velocity as a function of the dispersion relation. For a given wavenumber, the phase velocity is the slope between the origin (wavenumber null) and the considered point; the group velocity is the slope of the line tangent at the considered point.

103

APPENDIX B. ACOUSTIC WAVES PROPAGATION IN SOLIDS

Angular frequency ω [Hz]

Relation of dispersion

Group velocity Vg

Phase velocity V

Wavenumber k [1/m]

Figure B.6: Relation of dispersion: phase velocity V and group velocity Vg are represented

Appendix C Materials data The data reported here come from [Aul73b, pages 129, 145, and 146]. Material Class Density c11 Aluminum m3m 2702 10.73 Gold m3m 19300 19.25 Silica isotropic 2203 7.85 Lithium niobate 3m 4700 20.3 Lithium tantalate 3m 7450 23.3 Quartz α 32 2648 8.67

c12 6.08 16.30 1.61 5.3 4.7 0.70

c13

c33

7.5 24.5 8.0 27.5 1.19 10.72

c44 2.83 4.24 3.12 6.0 9.4 5.79

c14

0.9 -1.1 -1.79

Table C.1: density (in kg · m−3 ) and elastic constants (in 1010 N · m−2 ) Material Class Lithium niobate 3m Lithium tantalate 3m Quartz α 32

e11 0 0 0.171

e14 e15 0 3.7 0 2.6 -0.0406 0

e22 2.5 1.6 0

e31 0.2 '0 0

e33 1.3 1.9 0

εS11 38.9 36.3 3.92

εS33 25.7 38.1 4.10

Table C.2: (piezo)electric constants (in C · m−2 ) and dielectric constants (in 10−11 F · m−1 )

104

105

APPENDIX C. MATERIALS DATA

Elastic

Piezo-

constants

electric constants

Piezoelectric

Dielectric

constants

constants

(a)

(b)

(c)

(d)

Figure C.1: Materials matrices for some classes. (a) The matrix is symmetric and has three submatrices (a 6-by-6 matrix for elastic constants, a 3-by-6 matrix for the piezoelectric constants, and a 3-by-3 matrix for the dielectric constants) (b) isotropic and cubic m3m (c) trigonal 32 (d) trigonal 3m. The • and the ◦ are non-zero components. The linked components are identical if the colour is the same and opposite if the colour differ. The × is given by (c11 − c12 )/2 except for the cubic case.

Appendix D Devices layout

52 51 22

53 54

11

23

24

13

14

32

31

42

41

33

62

61

63

64

44

FLAT Figure D.1: Repartition of the devices on the 3” quartz wafer

106

107

APPENDIX D. DEVICES LAYOUT

METAL 1

TEMP. SENSOR

IDT 1

IDT 1

LEFT

RIGHT

IDT 2

IDT 2

LEFT

RIGHT

METAL 2

Figure D.2: Design of device 11,13, and 14

108

APPENDIX D. DEVICES LAYOUT

METAL 1

TEMP. SENSOR

IDT 1

IDT 1

LEFT

RIGHT

IDT 2

IDT 2

LEFT

RIGHT

METAL 2

Figure D.3: Design of device 22, 23, and 24

109

APPENDIX D. DEVICES LAYOUT

METAL 1

TEMP. SENSOR

IDT 1

IDT 1

LEFT

RIGHT

IDT 2

IDT 2

LEFT

RIGHT

METAL 2

Figure D.4: Design of device 31, 32, and 33

110

APPENDIX D. DEVICES LAYOUT

METAL 1

TEMP. SENSOR

IDT 1

IDT 1

LEFT

RIGHT

IDT 2

IDT 2

LEFT

RIGHT

METAL 2

Figure D.5: Design of device 41, 42, and 44

111

APPENDIX D. DEVICES LAYOUT

METAL 1

TEMP. SENSOR

IDT 1

IDT 1

LEFT

RIGHT

IDT 2

IDT 2

LEFT

RIGHT

METAL 2

Figure D.6: Design of device 51, 52, 53, and 54

112

APPENDIX D. DEVICES LAYOUT

METAL 1

TEMP. SENSOR

IDT 1

IDT 1

LEFT

RIGHT

IDT 2

IDT 2

LEFT

RIGHT

METAL 2

Figure D.7: Design of device 61, 62, 63, and 64

Appendix E Simulation codes E.1

Computational methods

The simulations are an important part of this report. The software used for the implementation of the models was MATLAB. The version 5.3.1 of this program has been used to perform the computations. Two programs have been developed. The first one for the approach of the optimum sensitivity by the results of Wang et al. [WCJ94], this is based on the theory of perturbation. The second one is the implementation of the equivalent transmission line of the Love waves device developed here. Their respective architecture is given in the Figure E.1 and the Figure E.2. A code of colour is used to manage the relations between the MATLAB functions. During the implementation of the programs, a particular care was attributed to the convergence of the simulation and also to their efficiency. The functions are well documented. The user interface was reduced to the minimum to avoid complications during the implementation.

113

114

APPENDIX E. SIMULATION CODES

Khl_

total

K

khn

beta

mass_sense

eta_

Figure E.1: Program structure for the sensitivity given by the theory of perturbation

E.2 Sensitivity by the theory of perturbation beta.m function [beta1, beta2] = beta(V, Vs1, Vs2) %[beta1, beta2] = beta(V, Vs1, Vs2) % %IN V Love mode velocity % Vs1 velocity of shear wave in the substrate % Vs2 velocity of shear wave in the Love plate %

%OUT beta1 (dimensionnal) damping factor in the substrate % beta2 (dimensionnal) damping factor in the Love plate % % octob. 2000 - L. Francis (MCP/BIO) beta1 = (1-(Vˆ2/Vs1ˆ2))ˆ(0.5); beta2 = ((Vˆ2/Vs2ˆ2)-1)ˆ(0.5);

eta .m function eta_(beta1, beta2) % eta_(beta1, beta2) % % Small utility that displays displacement in the

% Love plate normalized to the displacement at the % surface (placed at origin) % %octob. 2000 - IMEC - L. Francis (MCP/BIO)

115

APPENDIX E. SIMULATION CODES

for i = 1:101 depth(i) = (i-1)/100; eta(i) = cos(beta2*depth(i)); end

plot(depth,eta) disp(’Pause ...’) pause close

K.m function Kres = K(ratio, rho1, rho2, beta1h, beta2h) % Kres = K(ratio, rho1, rho2, beta1h, beta2h) % %K compute the K factor (sensitivity) of a % Love mode device % %IN ratio lambda/h % rho1 specific mass of the substrate % rho2 specific mass of the Love plate % beta1h normalized damping factor of

% the wave in the substrate % beta2h normalized damping factor of % the wave in the Love plate %OUT Kres sensitivity factor % % octob. 2000 - L. Francis (MCP/BIO) Kint = ratio*(1+((cos(beta2h)*sin(beta2h))/(beta2h))+... ( (rho1*((cos(beta2h))ˆ2))/(rho2*beta1h))); Kres = 1/Kint;

Khl .m function [maximumsense, velocity, alpha, beta1, beta2, ... Velocity, Kres]=Khl_(Vs1, Vs2, rho1, rho2, mode, graph_) %[maximumsense, velocity, alpha, beta1, beta2, ... % Velocity, Kres]=Khl_(Vs1, Vs2, rho1, rho2, mode, graph_) % %Khl_ computes and find the maximum sensitivity factor K % %IN Vs1 velocity of shear wave in the isotropic % substrate % Vs2 velocity of shear wave in the isotropic % Love plate % rho1 specific mass of the substrate % rho2 specific mass of the Love plate % mode Love mode % graph_ boolean, graphical version (1) or not (0). % %OUT maximumsense K sensitivity factor maximal value % velocity Love mode velocity for the max. K % alpha normalized thickness (ratio h/lambda) % at max. K % beta1 adimensionnal damping factor in the % substrate at max. K % beta2 adimensionnal damping factor in the % Love plate at max. K % Velocity all values of Love mode velocity % Kres all values of K factor for each value % of Velocity. % % octob. 2000 - L. Francis (MCP/BIO)

[Velocity, beta1h, beta2h, Omegahn, Lambdah] = ... total(Vs1, Vs2, rho1, rho2, mode, graph_, 500); for i=1:length(Velocity) ratio = 1/Lambdah(i); Kres(i) = K(ratio, rho1, rho2, beta1h(i), beta2h(i)); end [maximumsense, index] = max(Kres); maximumsense disp(’Velocity :’) disp(Velocity(index)) velocity = Velocity(index); disp(’Lambda/h :’) disp(Lambdah(index)) alpha = 1/Lambdah(index); beta1 = beta1h(index); beta2 = beta2h(index); if (graph_) plot(Velocity,Kres) title(’Sensitivity : K factor’) pause close plot3(Velocity,Lambdah, Kres) pause close end

khn.m function beta2h = khn(mu1, mu2, beta1, beta2, n) %beta2h = khn(mu1, mu2, beta1, beta2, n) % %IN mu1 shear modulus of the substrate % mu2 shear modulus of the Love plate % beta1 damping factor in the substrate

% beta2 damping factor in the Love plate % %OUT beta2h normalized damping factor in the Love plate % % octob. 2000 - L. Francis (MCP/BIO) beta2h = atan((mu1*beta1)/(mu2*beta2)) + (n*pi);

mass sense.m function out = mass_sense(K_factor, rho2, alpha, V, omega) %out = mass_sense(K_factor, rho2, alpha, V, omega) % %mass_sense give the mass sensitivity of the Love mode device % %IN K_factor K coefficient % rho2 specific mass of the Love plate % alpha normalized thickness (h/lambda) % V Love mode velocity % omega operating angular frequency of the device

%OUT out sensitivity (m2/kg) for the % operating frequency % % Displays also the value of lambda and h % %octob. 2000 - IMEC - L. Francis (MCP/BIO) f = omega/(2*pi); lambda = V/f out = -K_factor/(rho2*lambda); h = alpha*lambda

116

APPENDIX E. SIMULATION CODES

total.m function [Velocity, beta1h, beta2h , Omegahn, ... Lambdah] = total(Vs1, Vs2, rho1, rho2, n, ... graph_, steps) %[Velocity, beta1h, beta2h , Omegahn, Lambdah] = ... % total(Vs1, Vs2, rho1, rho2, n, graph_, steps) % %IN Vs1 velocity of shear mode in the substrate % (semi-infinite layer) % scalar % Vs2 velocity of shear mode in the Love % plate (finite thickness layer) % scalar % rho1 specific mass of the substrate % scalar % rho2 specific mass of the Love plate % scalar % n Love mode % scalar % graph_ allow graphical version (1) or not (0) % boolean % steps number of steps to compute Velocity % (default = 100) % scalar % %OUT Velocity Love velocity % vector of dimension steps % beta1h adimensionnal damping factor of the % wave in the substrate % vector of dimension steps % beta2h adimensionnal damping factor of the % Love wave % vector of dimension steps % Omegahn adimensionnal angular frequency of % the Love mode % vector of dimension steps % Lambdah adimensionnal ratio lambda/h, inverse % of the normalized thickness % vector of dimension steps % %total displays the damping factor and the lambda/h % % octob. 2000 - L. Francis (MCP/BIO) if (Vs1 < Vs2)

error(’wrong velocities’) end if (nargin == 6) steps = 100 end mu1 = Vs1*Vs1*rho1; mu2 = Vs2*Vs2*rho2; Vstep = (Vs1-Vs2)/(steps-1); for i=1:steps Velocity(i) = Vs2+(i-1)*Vstep; [beta1(i), beta2(i)] = beta(Velocity(i), Vs1, Vs2); beta2h(i) = khn(mu1, mu2, beta1(i), beta2(i), n); Omegahn(i)= Velocity(i)*beta2h(i)/... ((((Velocity(i))ˆ2)/(Vs2ˆ2)-1)ˆ(0.5)); beta1h(i) = (Omegahn(i)/Velocity(i))*... ((1-(((Velocity(i))ˆ2)/Vs1ˆ2))ˆ(0.5)); Lambdah(i)= (2*pi*Velocity(i))/Omegahn(i); end if (graph_) plot(Velocity,beta1h,’b:’, Velocity, beta2h,’r’) legend(’Beta1*h’, ’Beta2*h’); title(’Damping factors’); pause close plot(Velocity, Lambdah) title(’Relation Lambda/Velocity’); xlabel(’Velocity’); ylabel(’Lambda/h’); hold on maximum = 35; plot([Vs2 Vs2],[0 maximum], ’:’) plot([Vs1 Vs1],[0 maximum], ’:’) pause close end

117

APPENDIX E. SIMULATION CODES

findVlove

root2

findResonRoots

find_V_Z

constAngles

fctZero

paramX

findRefLayer

paramZ

Snell

findZero

seenImpedance

valInterval

Zin

Figure E.2: Program structure for the transmission line model

E.3

Transmission line model

constAngles.m function angles = constAngles(ref_layer, Vs, theta) %angles = constAngles(ref_layer, Vs, theta); % %IN ref_layer scalar % reference layer % theta complex % value of angle in the reference layer % Vs vector (1*n) of complex % shear bulk wave phase velocity for layer 1 to n % %OUT angles vector (1*n) of complex % angles in each layer % %L. FRANCIS (MCP/BIO) %February, 2001. %-------------------------------------------------------------%BEGIN %number of layers n = length(Vs); %Protection if (n 1) for posit = (ref_layer-1):-1:1 Vs_l(1) = Vs(posit+1); Vs_l(2) = Vs(posit); angles(posit) = Snell(Vs_l, angles(posit+1)); end end %From reference layer to bottom of the structure if (ref_layer < n) for posit = (ref_layer+1):n Vs_l(1) = Vs(posit-1); Vs_l(2) = Vs(posit); angles(posit) = Snell(Vs_l, angles(posit-1)); end end %-------------------------------------------------------------%Normalization of angles angles = asin(sin(angles)); %END %--------------------------------------------------------------

118

APPENDIX E. SIMULATION CODES

doubler.m %THIS IS A SCRIPT %remove old data clear Vlove2 clear h_val clear resonTheta2 clear whu

%Quartz/SiO2/polymer/Vacuum %Polymer velocity is 1210 m/s and density is 1231 kg/m3 rho = [rho(1) rho(2) 1231 rho(3)] comp = [comp(1) comp(2) 1231*1210*1210 comp(3)] %Data initialization Vlove = [] resonTheta = []

%Begin a new set of data %eps is the interfacial layer thickness % h is the intermediate layer thickness %eps = 4.2:0.1:6 eps = 5.1 eps = eps*1.0e-6 h = [9.3e-6 1.0e-5 8]

%Let it go ! for counter = 1:length(eps) %Display passed time and time remained until end disp(’Already computed % : ’) disp(100*counter/length(eps))

%Load data for compliances and density load dataformodes

%Value of thickness of the polymer film whu = eps(counter) [h_val, Vlove2(:,:,counter), resonTheta2(:,:,counter)] = ... findVlove(comp, whu, h, omega, 1, rho, 0.5);

%Operating angular frequency set to 2pi*120 MHz omega = 2*pi*1.2e8; %Modification of constants data :

%Now spare data in a temporary file save TEMP_FILE Vlove2 resonTheta2 whu end

enertot.m %THIS IS A SCRIPT %Clear old data clear energy clear angles clear depthvalreact clear gamma_X clear gamma_Z clear reactsub clear valreact clear Zc_X clear Zc_Z start = now; %Set operating frequency to 120 MHz omega = 2*pi*120.0e6 %Data will spare time d_begin = 10; d_step = 10; d_end = 1000; for count = d_begin:d_step:d_end tot = count/100; %While loop %--- initialization procedure crit = 1; reso = 5; %core of the loop while crit [angles, depthvalreact(count), energy(count,:), ...

gamma_X(count,:), gamma_Z(count,:), ... reactsub(count,:), valreact(count), ... Zc_X(count,:), Zc_Z(count,:)] = ... root2(comp, h_val(count), omega, rho, ... resonTheta(count,1), 0, reso); %Stop criteria ok_tmp = reactsub(count,1)/reactsub(count,2) if ok_tmp < 0.1 crit = 0 else reso = 2*reso if reso >= 40 crit = 0 disp(’Warning info : never reach your stop criteria’) end end end %Save information save nrj angles depthvalreact energy gamma_X save nrj reactsub valreact Zc_X Zc_Z -APPEND %Time remaining before end stop = now; delta = stop-start; endtime = start + (100*delta/tot); stopit = datestr(now,0); endit = datestr(endtime,0); disp(stopit) disp(endit) end

enertot2.m %THIS IS A SCRIPT %Clear old data clear energy clear angles clear depthvalreact clear gamma_X clear gamma_Z clear reactsub clear valreact clear Zc_X clear Zc_Z %Initializations energy =[] angles =[] depthvalreact =[] gamma_X =[] gamma_Z=[]

reactsub valreact Zc_X=[] Zc_Z=[]

=[] =[]

start = now; %Operating frequency set to 120 MHz omega = 2*pi*120.0e6 %Layers thickness %silicium dioxide and polymer sio2 = 1.0e-7:1.0e-7:10.0e-6; pol = 1.0e-7:1.0e-7:6.0e-6; length(sio2) length(pol) comp rho input(’Is this ok ?’)

gamma_Z

119

APPENDIX E. SIMULATION CODES

for count1 = 1:size(Vlove,2) for count2 = 1:size(Vlove,1) tot = 100*(count2+(count1-1)*size(Vlove,1))/(size(Vlove,1)*... size(Vlove,2)); [angles(count2,:,count1), depthvalreact(count2,:,count1), ... energy(count2,:,count1), gamma_X(count2,:,count1), ... gamma_Z(count2,:,count1), reactsub(count2,:,count1), ... valreact(count2,:,count1), Zc_X(count2,:,count1), ... Zc_Z(count2,:,count1)] = root2(comp, ... [sio2(count2) pol(count1)], omega, rho, ... resonTheta(count2,count1), 0, 5);

save nrj angles depthvalreact energy gamma_X gamma_Z reactsub save nrj valreact Zc_X Zc_Z count1 count2 -APPEND stop = now; delta = stop-start; endtime = start + (100*delta/tot); endit = datestr(endtime,0); clc count1 count2 disp(endit) end end

%Save information

fctZero.m function fZero = fctZero(Zminus, Zplus) %fZero = fctZero(Zminus, Zplus) % %IN Zminus impedance seen in the minus direction

% Zplus impedance seen in the plus direction % %OUT fZero impedance function to find zeros fZero = Zminus + Zplus;

findRefLayer.m function [ref_layer, Vs_ref] = findRefLayer(Vs) %[ref_layer, Vs_ref] = findRefLayer(Vs) % %IN Vs vector (1*n) % shear bulk wave phase velocity in % layer 1 to n % %OUT ref_layer positive integer % index of the maximum value of slowness % Vs_ref real % "Vs" in the reference layer % % L. FRANCIS (MCP/BIO) %February, 2001. %---------------------------------------------------------%BEGIN

%(Rejection of vacuum layer as valid layer) [Vs_ref, ref_layer] = min(Vs); Vs_dummy = max(Vs); n = length(Vs); count = 1; while (Vs_ref==0) count = count+1; if (count == n) error(’Error found in findRefLayer - program aborted’) end Vs(ref_layer) = Vs_dummy; [Vs_ref, ref_layer] = min(Vs); end %END %----------------------------------------------------------

findResonRoots.m function [veloNext, theta_sol] = ... findResonRoots(comp, h, omega, veloPrev, Vs, Zs, debug, steps) %function [veloNext, theta_sol] = ... % findResonRoots(comp, h, omega, veloPrev, Vs, Zs, debug, steps) % %IN comp vector (1*n) % shear stiffness constant in layer 1 to n % h vector (1*(n-2)) % thicknesses of layer 2 to (n-1), the other % layers are assumed semi-infinite % omega complex scalar % operating angular frequency % veloPrev vector (1*2) % range of velocity [minimum, maximum] in which % solutions have to be found % Vs vector (1*n) % bulk shear velocity in layer 1 to n % Zs vector (1*n) % bulk acoustic impedance in layer 1 to n % debug boolean, optional, default = false % debuging option, true = debug, false = not debug % steps scalar, optional, default = 1000 % number of points for computation % %OUT veloNext matrix (s*2) % range of velocity where there are solutions % if there are s solution, the matrix has s lines % for each solution, the first column is % the minimum velocity and the second one the % maximum velocity. The given data % should always have one solution otherwise % the program is aborted. % theta_sol matrix (s*2) % corresponding angle of resonance in

% the layer of reference (intern) % %L. FRANCIS (MCP/BIO) %February, 2001. %-----------------------------------------------------------%Pre-processing --------------------------------------------if (nargin == 6) debug = 0; steps = 1000; end %Pre-processing end------------------------------------------

%Initialization---------------------------------------------r = size(veloPrev,1); [ref_layer, Vs_ref] = findRefLayer(Vs); veloNext = []; theta_sol = []; %Initialization end-----------------------------------------%Main loop -------------------------------------------------for counter = 1:r veloInt = valInterval(real(veloPrev(counter,1)), ... real(veloPrev(counter,2)), steps); veloInt = veloInt.*(1-0.0*i); theta = asin(Vs_ref./veloInt); for counter2 = 1:length(veloInt); angles(counter2,:) = constAngles(ref_layer, Vs, ... theta(counter2)); end [gamma_Z, Zc_Z] = paramZ(angles, comp, omega, Zs); [position, Zminus, Zplus] = ...

120

APPENDIX E. SIMULATION CODES

seenImpedance(gamma_Z, h, Zc_Z); %Function to Zero determination fZero = fctZero(Zminus, Zplus); out = findZero(imag(fZero),1); %Debug option --------------------------------------------if debug close plot(real(veloInt), -imag(Zminus),’b’) hold on plot(real(veloInt), -real(Zminus),’bo-’) plot(real(veloInt), imag(Zplus),’r’) plot(real(veloInt), real(Zplus),’ro-’) plot(real(veloInt), imag(fZero),’g’) plot(real(veloInt), real(fZero),’go-’) legend(’-imag Z-’,’-real Z-’,’imag Z+’,’real Z+’, ... ’imag fZero’,’real fZero’) disp(’Paused during display ...’)

pause end %Debug option end ------------------------------------------

%Protection------------------------------------------------if isempty(out) error(’Error 2 found in findResonRoots - program aborted’) end %Protection end---------------------------------------------

%Post-processing ------------------------------------------veloNext = [veloNext; veloInt(out)] ; theta_sol = [theta_sol; theta(out)]; %Post-processing end---------------------------------------end %Main loop end------------------------------------------------%--------------------------------------------------------------

findVlove.m function [h_val, Vlove, resonTheta]= ... findVlove(comp, eps, h, omega, pos, rho, Vtol) % [h_val, Vlove, resonTheta]= ... % findVlove(comp, eps, h, omega, pos, rho, Vtol) % %IN comp vector (1*n) % shear stiffness constants for layer 1 to n % eps vector (1*(n-3)) % fixed thickness values % h evoluting thickness % can be either simple scalar or [hmin hmax steps] % omega operating frequency % pos scalar % position of the evoluting thickness film % numbering begin at 1 for the film % on top of the substrate and move upwards % rho vector (1*n) % density of layer 1 to n % Vtol scalar % tolerance value on velocity % %OUT h_val vector (1*steps) % values of the evoluting thickness % Vlove vector (1*steps) % values of the Love wave phase velocity % resonTheta vector (1*steps) % corresponding angle of resonance % %The program evaluates and display when it will finish % %L. FRANCIS (MCP/BIO) %February, 2001. %BEGIN----------------------------------------------------%Position settings ---------------------------------------if (length(h) == 3) h_val = valInterval(h(1), h(2), h(3)) elseif (length(h) == 1) h_val = h else error(’Error found ind findVlove - program aborted’) end %Position settings end------------------------------------%Initialization ------------------------------------------[Vs, Zs] = find_V_Z(comp, rho); Vlove = []; n = length(Vs); Vs_int = sort(Vs); Vs_min_pos = 1; Vs_max_pos = n; Vs_min = Vs_int(Vs_min_pos); Vs_max = Vs_int(n); %Initialization end---------------------------------------%Iterative loop to avoid vacuum layers -------------------while (Vs_min == 0) disp(’One layer is vacuum’) Vs_min_pos = Vs_min_pos + 1; if (Vs_min_pos >= length(Vs_int)) error(’Error 2 found in findVlove - program aborted’) else Vs_min = Vs_int(Vs_min_pos); end end

%Iterative loop to avoid vacuum layers end----------------start = now; endtime = datestr(now,0); %Main loop -----------------------------------------------for count = 1:length(h_val) clc disp(’COMPUTATION STATISTICS :’) disp(’------------------------’) disp(’% already done : ’) percent = count/length(h_val); disp(100*percent) disp(’end time : ’) disp(endtime) disp(’------------------------’) disp(’ ’) veloPrev = [Vs_min Vs_max]; veloNext = []; veloMean = []; interf = []; eps_pos = 1; for count2 = 1:(length(eps)+1) if (count2 == pos) interf = [interf h_val(count)]; else interf = [interf eps(eps_pos)]; eps_pos = eps_pos + 1; end end disp(’Interface : ’) disp(interf) %While loop, initialization ---------------------------[veloNext, theta_sol] = ... findResonRoots(comp, interf, omega, veloPrev, Vs, Zs); absi = []; cont = (abs(veloNext(:,1)-veloNext(:,2))) > Vtol; %While loop while (cont) disp(’...’) veloPrev = veloNext; [veloNext, theta_sol] = ... findResonRoots(comp, interf, omega, veloPrev, Vs, Zs); absi = abs(veloNext(:,1)-veloNext(:,2)); absi2 = absi > Vtol; cont = sum(absi > Vtol); end %While loop end-----------------------------------------%Post-processing ---------------------------------------veloMean = mean(veloNext,2)’; Vlove(count,1:length(veloMean)) = veloMean; [ref_layer, Vs_ref] = findRefLayer(Vs); resonTheta(count, 1:length(veloMean)) = ... asin(Vs_ref./veloMean); %Post-processing end------------------------------------timing = now; elapsedtime = now - start; totaltime = elapsedtime/percent; endit = start+totaltime;

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APPENDIX E. SIMULATION CODES

endtime = datestr(endit,0);

%END--------------------------------------------------------

end %Main loop end----------------------------------------------

findZero.m function out = findZero(in, option) %out = findZero(in,option); % %IN in vector (1*x) of real % sampled values % option number for indicated options % no number entered (default) = default value returned % option = 1 : only decreasing curve roots % %OUT out vector (r*2) of real % r interval range index were "in" change of % sign or reach a value of 0. % %L. FRANCIS (MCP/BIO) %February, 2001. %----------------------------------------------------------%BEGIN %Intern tolerance : number below that number are %considered to be 0 interntol = 1.0e-10; %----------------------------------------------------------%Sampling and scaling of data for count = 1:length(in) if (abs(in(count))