Calibration of scanning thermal microscopes for conductivity measurements Stéphane Lefèvre, Sebastian Volz, Jean-Bernard Saulnier Laboratoire d’Etude Thermique UMR CNRS 6608, BP40109, 86961 Futuroscope Cedex

Abstract Thermal mapping with nanoscale spatial resolution requires the use of contact probe techniques. Among them, the atomic force microscope (AFM) based measurements provide the highest resolution ranging from a few to several dozens of nanometers. We use an AFM probe consisting of a hot platinum wire to measure thermal conductivity. While in contact with the sample surface, the electrical current I injected into the probe is controlled so that electric resistance or wire temperature is maintained by Joule effect. The current variation is directly related to the heat flow lost towards the sample. Traditional calibration is carried out by relating the thermal conductivity of a set of samples to the measured current I. A linear dependence is assumed and a constant tip-sample contact conductance is supposed. However, none of those conditions can reasonably be accepted. We provide analytical and numerical thermal modelling of tip and sample to show the impact of the different heat fluxes on a basic conductivity calibration. A simple expression is finally established to calculate the thermal conductivity from the input current I.

1 Introduction The development of electronic and optical microsystems (nanotransistors, diodes) as well as nanostructures (films, superlattices, nanowires) leads to the renewal of thermal metrologies adapted to sub-microscales. Indeed, to take an example in the heart of the industrial and economic developments current, the life time and the power output of the lasers diode, parameters which represent an essential stake in the development of the information networks, are strongly conditioned by the temperatures and maximum gradients reached within their elementary structures. Nanoscale thermal imaging can not be performed through optical measurements because of diffraction limit, the use of contact probe techniques is therefore clearly inevitable. Among them, the atomic force microscope (AFM) provides the highest resolution ranging from a few to several dozens of nanometers. The probe (Topometrix) consists of a wollaston wire shaped as a tip and etched to uncover a core platinum wire as described in Figure 1.

Figure 1 : Schematic diagram of the thermal Probe

The platinum core is 5µm in diameter and 200µm in length, the tip-sample contact zone is approximated to a disk with radius 50nm. The thin Pt wire with Ag contacts is mounted in a AFM device and used as a thermal element. In the thermal conductivity mode, the probe serves as classical hot wire anemometer: the electrical resistance Rprobe is measured with current-voltage estimation and is used as input signal for a feedback loop to maintain temperature which is linearly dependent to Rprobe. When the probe is brought within close contact to the film surface, the induced change of the dissipate electrical power is supposed proportional to the heat flux toward the sample and consequently, to the sample thermal conductivity (Pilkky, 1996). The heat flux lost in the sample can therefore be estimated at each point on the sample surface. The strategy for calibration simply consists in measuring the heat flux dissipated in reference samples with well known conductivity to fix the parameters of the function relating conductivity and lost electrical power. A linear calibration function is accepted and was even shown in literature. The assumptions of linear dependence and also constant tip-sample contact conductance can however reasonably not be accepted. We provide analytical and numerical thermal modelling of tip and sample to show the impact of the different heat exchanges on a basic conductivity calibration. A simple expression is finally established to calculate the thermal conductivity.

2 Thermal model To simplify the model, a plane symmetry including the tip-sample contact point and crossing the middle of the two probe arms is considered. The resultant system consists in a straight wollaston wire ended by a 100µm long Pt wire. A cylindrical symmetry with axis being the wollaston wire considerably simplifies the model into a 2D representation as illustrated in figure 2.

Figure 2: Probe-sample schematic. Arrows represent exchange with ambient.

The ambient temperature T=20°C was imposed on the rear and lateral sample faces as well as on the rear wollaston wire face. The white arrows in figure 2 represent heat exchanges with ambient through coupled convection and

radiative modes. Heat transfer coefficients were estimated based on classical correlation including angle influence, since system sizes are larger than air mean free path. Thermal properties of silver λAg= 420W.m-1.K-1 and platinum λPt=30 W.m-1.K-1 are the reference data. Sample thermal conductivity λs and contact conductance G are input parameters. The finite volume elements method was carried out based on rings elements with square sections of 25 µm2 (sample) or 6.25 µm2 (tip and wollaston). The sample is a 20x20 elements distribution, the uncovered Pt wire consists in a 2x80 mesh and the silver coating is modeled by 14x41 elements. A total of 1136 thermal nodes is computed. We consider that the Pt diameter is larger than heat carrier mean free path in platinum, and that the heated contact zone remains larger than the sample heat carriers mean free path. In those conditions, the assumption of diffusive heat transfer can be suggested. Consequently, the heat conduction equation was integrated on each volume element by a simple matrix inversion procedure.

3 Results and discussion 3.1 Heat flux balance on thermal probe The finite volume element calculation was validate by checking heat flux balance as reported in Table 1. The sample is germanium (λs=60W/mK) and the contact conductance is set to a value proposed in the literature G=10-6 W/K (Majumdar, 1999).

Heat Flux to Ag Heat Flux to Sample Convective Heat Flux Radiative Heat Flux Joule Heat Flux Net Heat Flux from probe

Heat Flux (mW) 2.10 0.949 0.13 2.149E-4 3.18 1.74E-4

Table 1: Heat fluxes lost by the Pt wire in mW, input current I=0.05A, G=10-6 W/K, λs=60W/mK. The average tip temperature value was found to be 112°C.

Table 1 also indicates that the radiative heat flux is negligible and convective heat flux represents only 4% of the input Joule power. The major portion of the joule flux goes to the silver coating (66%).

3.2 Heat flux to sample dependence to sample thermal conductivity Heat flux to sample Qs dependence to thermal conductivity and thermal conductance is reported in figure 3. This graph clearly shows that (i) a linear dependence between Qs and λ is not acceptable and (ii) the probe sensitivity appears as limited to a certain range of thermal conductivity starting from 0.1 W/m.K to a few dozens of W/m.K. As expected, the larger the contact conductance the better the sensitivity.

Figure 3: Heat fluxes to sample (squares) and silver coating (diamond) as a function of sample thermal conductivity and tip-sample contact conductance G.

3.3 Analytical modeling A thermal model allows to retrieve the flux dependence to thermal conductivity. The classical heat conduction equation written for the Pt wire including the Joule source term leads to the local probe temperature θp=Tp-Ta:

−ρ I 2 2 x + Ax + B θ p ( x) = 2λ S 2

(1)

d 2 ( rθ ) bθ = 0 , θ (r ) = 0 2 dr r

(2)

where ρ is the probe electrical resistivity, λ and S representing the probe thermal conductivity and section. The quantity I denotes the input current and x the spatial parameter. A and B are two constants to determine. Assuming that heat diffusion length in the sample is longer than contact radius b, the heat conduction equation written in terms of the spherical coordinate r reads:

where θ0 is the temperature imposed by the tip in the hemisphere of radius b centered in the contact point. The heat flux lost to sample Qs can be expressed as a function of θp:

Qs = −λ S

dθ p   ρI 2L  = − λ + A S  − 2 dx  x = L  λS 

(3)

Let us write heat flux continuity at the tip-sample interface:

Qs = G (θ p ( L ) − θ 0 )

Qs = −λe 2π b 2

dθ   = λeπ bθ 0 dr r =b

(4) (5)

and the temperature at the sample side:

−ρI 2 2 L + AL + B 2λ S 2

θ p ( x = L) =

(6)

The heat flux lost in the silver coating QAg, can be formulated as:

QAg = GAgθ p ( 0 )

QAg = λ S

(7)

dθ p   = λ SA dx  x = 0

(8)

where the thermal conductance GAg includes all conductive phenomena in the silver coating. Finally, the constant B is simply: θ P ( x = 0) = B (9) The seven equations (3)-(9) lead to the determination of the seven unknown A,B,θp(L), θp(0), θ0,Qs,QAg. The heat flux Qs solution can then be expressed as follows:

Qs =

λS .G. ( L.GAg + 2λ S ) .π b.ρ .I 02 .L

S . ( 2 L.GAg .G.λS .π .b + 2λ.S .G.λS π .b + λ .S .GAg .G + 2S .GAg .λS .π .b.λ )

Knowing that the Joule power is PJ =

variables

:

(10)

ρ LI 02 λS and identifying the probe conductance GPt = , the parametric S L

form:

Qs =

AλS B + λS

(11)

leads to the identification of the parameters a and b:

A= B=

(G

+ 2.GPt )

PJ

G 2 ( GAg + GPt ) G + GPt .GAg Ag

G.GAg GPt

2π b G ( GAg + GPt ) + GPt GAg 

(12)

(13)

Considering that GAg>>GPt, a and b expressions turns into:

PJ

G 2 G + GPt GPt G πb B= G + GPt A=

(14)

(15)

Relations (14) and (15) belie predictions of previous analysis including hypothesis of isothermal tip. This assumption was in fact denied by previous works (Gomes, 1999). The parametric form of Equation (11) is retrieved with a good degree of accuracy from the numerical model as proven in Table 2 where both analytical and numerical identification of Joule power and probe conductance are reported.

PJ/2 (W) GPt (W.K-1) GPt/πb (W.K-1.m-1)

Analytical Model 1,591.10-3 5,89.10-6 1,87

FVE Method 1,5604.10-3 6,33.10-6 0,700

% 2% 8% 91%

Table 2: comparison of numerical and analytical values Joule power and probe conductance.

The discrepancy between both approaches concerning GPt/πb values is explained by the fact that the radial size of sample volume elements are 5µm. This means that Qs can not be applied on a disk of radius b=50nm. To have same results in the both methods, it is necessary to add at least 10000 elements in the FVE Model, which cost too much CPU time and memory space.

4 Conclusion We present the thermal modeling of an AFM thermal probe in contact with a sample. Conventional linear fitting between heat flux lost from tip to sample and sample thermal conductivity appears as non-consistent. A parametric form relating thermal conductivity to input current I is derived and convergence between numerical and analytical models is finally shown. The obtained relation appears as more complex than previous ones assuming an isothermal probe. Our modeling clearly shows a very limited sensitivity in a thermal conductivity range of 0,1 to 60 W.m-1.K-1. DC approaches seem however more promising in terms of thermal conductivity sensitivity as well as on contact conductance determination (Fiege, 1999). We emphasize that this approach is fully classical, i.e. heat transfer is considered as diffusive. This hypothesis does not hold when the contact radius b=50nm becomes smaller than phonon mean free path in dielectric samples such as crystals, and electrons mean free path in metals (the phonon mean free path is about 1nm for amorphous materials and reach 100nm for diamond). The determination of contact conductance G remains a significant issue and depends on sample and contact force. G is also a function of the sample topography which makes a precise quantitative analysis difficult. References Fiege G.B.M., Altes A., Heiderhoff R., Balk L.J., Quantitative thermal conductivity measurements with nanometer resolution, Journal of Physics D (Applied-Physics), vol.32, pp.13-17, 1999. Gomes S., Trannoy N., Grossel P., DC thermal microscopy: Study of the thermal exchange between a probe and a sample, Measurement Science&Technology,vol.10, pp. 805-811, 1999. Majumdar A., Scanning thermal microscopy, Annual Review of Materials Science 29:505-85, University of California, Berkeley, 1999. Pilkki R. J., Moyer P. J. and West P. E., Japan Journal of Applied Physics, vol. 33, p. 3785, 1996. Ruiz F., Sun W.D., Pollak F.H., Venkatraman C., Determination of the thermal conductivity of diamond-like nanocomposite films using a scanning thermal microscope, Applied Physics Letters, vol.73, pp. 1802-1804, 1998.