SUPPLEMENTARY MATERIAL Lifetimes of Confined Acoustic Phonons in Ultra-Thin Silicon Membranes (Dated: December 7th, 2012) J. Cuffe1†, O. Ristow2, E. Chávez1,3, A. Shchepetov4, P-O. Chapuis1†, F. Alzina1, M. Hettich2, M. Prunnila4, J. Ahopelto4, T. Dekorsy2, C. M. Sotomayor Torres1,3,5.* 1
Catalan Institute of Nanotechnology (ICN2), Campus UAB, 08193 Bellaterra (Barcelona), Spain Department of Phyics and Center of Applied Photonics, Universitaet Konstanz, D-78457 Konstanz, Germany 3 Dept. of Physics, UAB, 08193 Bellaterra (Barcelona), Spain 4 VTT Technical Research Centre of Finland, PO Box 1000, 02044 VTT, Espoo, Finland 5 Institució Catalana de Recerca i Estudis Avançats (ICREA), 08010 Barcelona, Spain
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*Email:
[email protected] † Present address: MIT, 77 Massachusetts Ave. Cambridge, MA 02139, US;
[email protected] †† Present address: Centre de Thermique de Lyon (CETHIL)– CNRS – INSA Lyon, 9, rue de la Physique, Campus La Doua, 69621 Villeurbanne cedex, France
Summary Fourier transform of the time traces................................................................................p2 Sample fabrication and characterization..........................................................................p3 Thickness uniformity and inhomogeneous broadening...................................................p5 Surface roughness as phonon decay mechanism.............................................................p6 Three-phonon interactions...............................................................................................p9
1
FT Power Spectrum (arb. units)
Fourier Transform of the Time Trace
1
D1
142 nm 100 nm 30 nm
D1
D1
0.1
0.01
D2 1E-3
D3
1E-4
D2 1E-5
1E-6 0.02
0.04
0.06 0.08 0.1
0.2
0.3
0.4
Frequency (THz)
Fig. S1: Normalized power spectra of the discrete Fourier Transform of the time traces acquired for the 142 nm (black), 100 nm (red) and 30 nm (blue) membranes. The spectra show that the signals are predominately single frequency (D1), with weak higher order modes observed for thicker membranes.
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Sample fabrication and characterisation The membranes were fabricated on 150 mm silicon-on-insulator (SOI) wafers using Si MEMS processing techniques with areas of about 500 x 500 m2. In this process, the underlying Si substrate and the buried oxide layer are removed through a combination of dry and wet etching techniques to leave a top layer of suspended silicon. The high etch selectivity of the buried oxide with respect to the top SOI layer allows the release of the membrane (Fig. S2(a)). The thickness was obtained from reflectance measurements performed with a FilmTek 2000 spectroscopic reflectometer (Fig. S2(b)). Thickness values were measured at four different points, north, south, east and west of the membranes, approximately 500 m apart. We found that the standard deviation of the thickness measurement was negligible for the purposes of this experiment (Table S.1). The precision or repeatability of these measurements is demonstrated in small standard deviation of the measurements combined with Fig. S2(b), showing that the measurements are sensitive to small changes in thickness. The absolute accuracy of the measurements is model dependent, and is estimated to be better than one nanometer. The total measured thickness includes the native oxide layer.
70
8.6 nm 6.35 nm Calculated
60
R (%)
50 40 30 20 10 0 300
400
500
600
700
800
900
(nm) Figure S.2. Calculated and measured reflectivity spectra as function of wavelength for 6.4 nm (square) and 8.6 nm (circle) Si membranes. The number of experimental data points has been reduced for clarity.
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Table S.1. Average and standard deviation of thickness measurements at four sides of the membranes Thickness Std. Dev. nm
nm
7.7
0.1
8.8
0.3
18.7
0.5
29.3
0.3
49.1
0.2
99.0
1.0
142.3
0.8
194.4
1.4
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Thickness uniformity and inhomogeneous broadening While SOI fabrication processes are highly developed and well controlled, small deviations of the fabricated membranes from the ideal structure must be taken in to account when calculating the phonon lifetime. For example, non-uniformity in thickness over the spot size would lead to an apparent decay of the observed signal, as each thickness would vibrate with a different frequency, and the superposition of these waves would result in dephasing. This phenomenon is known as inhomogeneous broadening. The small deviation in thickness measured across the 500 x 500 m2 membranes as shown in Table S.1 suggests that “large scale” differences in thickness are negligible over the 1.75 m spot size. Note also that inhomogeneous broadening would lead to non-exponential decay of the measured reflectivity signal and to small increase of the signal after a first decay, due to beating between the waves (see Figure S.3b). Indeed, inhomogeneous broadening alone cannot guarantee a real relaxation since it does not lead to loss of energy. While we cannot fully discard the possibility for the additional presence of inhomogeneous broadening in the smallest membranes (d < 10 nm) where the signal-to-noise is low, we have not observed non-exponential decay for the larger membranes.
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(a)
(b)
6
R/R (a. u.)
R/R (10-6)
3
0
-3
-6
200
300
400
500
600
0
100
Time (ps)
200
300
400
Time (ps)
Figure S.3. a) Reflectivity signal decay for the 18.7 nm membrane (arbitrary time origin). Blue: experimental signal. Red: damped exponential sine fit. b) Simulation of a reflectivity signal decay linked to inhomogeneous broadening from a Gaussian distribution in thickness with a standard deviation of 0.5 nm with respect to a mean plane.
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Surface roughness as phonon decay mechanism “Small scale” variations in thickness over the size of the spot, i.e. when the typical feature width is not more than approximately 4 time its height, are expected and are considered as surface roughness. We model the effect of the phonon phase difference that occurs as a result of these variations in thickness following the approach of Berman, Foster and Ziman [1,2].
The effect of boundary scattering due to surface roughness may be introduced through a boundary condition on the steady-state Boltzmann transport equation. Intuitively, this boundary condition is that all the phonons which hit the boundary at position rB, will be reflected with the normal component of their velocity vn reversed. Introducing a displaced Bose-Einstein distribution function
, where
is the equilibrium distribution, the boundary
condition can be derived to have the following form, (S.4) In this treatment, the effect of the roughness is described by a single phenomenological parameter, p, which represents the “polish” of the surface, with p = 0 for perfectly rough surfaces and p = 1 for perfectly smooth surfaces. Due to our specific well-defined, two-dimensional geometry, we can derive a phonon wavelength-dependent specularity p() by considering a plane wave normally incident on the boundary. This is a good approximation in our case as the D1q// = 0 mode can be thought of as a standing wave formed from the superposition of two counterpropagating longitudinal plane waves. The change in phase of the wave reflected from the boundary is related to the roughness, i.e. varying thickness, of the membrane, as ,
(S.5)
where x is the direction parallel to the membrane surface, and y(x) is a continuous function representing the deviation of the height of the surface from a reference plane. From this, the wavelength-dependent specularity can be derived by considering the auto-correlation of the phase [2],
(
̅̅̅̅)
, where is the root mean square deviation of
y(x), sometimes known as the asperity, henceforth referred to as the roughness. The physical interpretation of this expression is that shorter wavelengths will feel a greater effect of the surface roughness than longer wavelengths. 6
1.0
0.8
p()
0.6
0.4
= 0.5 nm = 1 nm = 2 nm
0.2
0.0
0
20
40
60
80
100
(nm) Fig. S.4. Wavelength dependent specularity p() as a function of phonon wavelength for roughness values of = 0.5 nm (black), = 1 nm (red), = 2 nm (blue).
With the wavelength-dependent specularity p() we can calculate the effective mean free path in the membrane. After considering multiple reflections from the boundary in series, the mean free path can be written as (S.6) where 0, is the characteristic dimension of the structure, i.e. the thickness of the membrane. As we know the wavelength = 2d or frequency f = vL/(2d) explicitly for the D1q// = 0 mode, we can derive simple expressions for the relaxation time as functions of either the thickness of the membrane or the frequency of the mode in terms of the roughness parameter, , and the velocity, vL: (
)
(
)
oth (
)
oth (
)
(S.7)
7
100 p() p=0 p = 0.5 p = 0.95
Lifetime (ns)
10 1 0.1 0.01 1E-3 Exp. Data
1E-4 10
100
1000
10000
Frequency (GHz) Fig. S.5. Phonon lifetime as a function of frequency for the wavelength dependent specularity model p() with = 0.5 nm (black), as well as for constant specularity values of p = 0 (red), p = 0.5 (blue) and p = 0.95 (pink), compared to experimental data (black squares). When the phonon wavelength is comparable to or smaller than the roughness, the wavelength-dependent model converges to the diffusive limit.
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Three-phonon interactions A crystal with N atoms is often considered to behave as a collection of 3N harmonic oscillators i.e., a system of 3N harmonic phonon modes. While this assumption predicts many important features of acoustic transport and solid state physics in general, it is invalid for real materials at finite temperatures. If atoms in a crystal lattice were governed by a truly harmonic potential, there would be no phonon-phonon interactions, no thermal expansion, and both phonon relaxation times and thermal conductivity would be greatly enhanced. The inclusion of anharmonic terms in the lattice Hamiltonian leads to phonon-phonon interactions, which gain in importance as the temperature increases. Limiting the anharmonicity of the crystal to cubic terms, the relaxation time associated to three-phonon interactions can be treated using the firstorder perturbation theory and four-phonon interactions using the second-order perturbation theory. In general the four-phonon processes are ignored because of their low contribution to thermal conductivity [3].
The phonon-phonon scattering rates can be studied by treating the anharmonicity as a perturbation and applying Fermi’s golden rule,
Wqq'q ''
2
f H pp i E f Ei 2
(S.8)
where Hpp is the anharmonic potential, i and f correspond to initial and final states for the phonon distributions and the delta Dirac function ensures that energy is conserved. Assuming the singlemode relaxation time approximation [4] and a linear Boltzmann transport equation, the total relaxation rate is given by 1 qp
pp' p '' 2 qq' q' ' n n 1 q q ' q '',G q ' p ' q '' p '' Aqq'q '' 3 4 N 0 q ' p ',q '' p '' v p v p 'v p '' nqp 1
( qpq'p' q''p'' )
1 nq ' p ' nq '' p '' ( qpq'p' q''p'' ) 2 nqp
(S.9)
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Where p is the mode polarization, q is the mode wavevector, v is the mode velocity and n is the Bose2
pp' p '' Einstein distribution function. The mode-dependent three-phonon scattering strengths, Aqq , are ' q ''
difficult to calculate quantitatively, can be approximated as [5]:
2
Aqqpp'q' p''''
4 2 2 2 2 2 v p v p ' v p '' v2
(S.10)
with as the mode-average Grüneisen’s parameter, v the phonon average group velocity given by:
v n v n
qp qp
qp
(S.11)
qp
qp
To obtain the relaxation time of three-phonon processes it is necessary have full knowledge of the dispersion relation inside of the first Brillouin zone. The dispersive nature of the phonon branches at large wavevectors complicates the calculation significantly, and the study of threephonon processes can become intractable at various temperature ranges. This motivates the use of a much simpler model based on the Debye approximation. The Debye model uses a linear dispersion relation for longitudinal and transverse polarizations. In this model it is possible to change the sum in q’ by integral in all space of q’ and replacing the sum of q’’ using the Kronecker delta function. N 0
8 d q' 3
3
q' p'
p'
(S.12)
Despite the fact that the out-of-plane acoustic spectrum in a membrane is discrete, we approximate the density of states in the membranes to be Debye-like. This continuum approach is valid when the frequency separation between the modes, frequency of the highest occupied mode,
, is small in comparison to the
, at room temperature, so that many modes are
occupied [6]. This condition is fulfilled for membranes greater than ~30 nm at room 10
temperature [7]. As the D1q// = 0 mode is purely longitudinal, we can represent it as a point on the longitudinal branch of a linear dispersion relation with total wavevector
√
and a corresponding frequency of
vL
(S.13)
d
where d represents the thickness of the membrane. Any mode with a non-zero parallel component of the wavevector would have a mixed polarization due to the coupling caused by the boundaries. After some algebraic manipulation, the expression for the relaxation time for Normal processes under the Debye approximation is given by [8]:
1 qp
2 n n( x'' ) 1 2 q' p' v p ' x' (Cx Dx' ) dx' nqp 1 4vs2 p ', p '' q D5 S p
2
n n( x'' ) 1 2 2 q' p' x' (Cx Dx' ) dx' 2 nq ' p '
(S.14)
where C v p / v p '' , D v p ' / v p '' , x q / qD , x' q' / qD , x Cx Dx' and n( x ) is the BoseEinstein distribution function evaluated in x . As the wavevector of the interacting phonon is small, Normal processes are dominant over Umklapp processes. The energy and momentum conservation rules for Normal processes impose certain restriction on the integration of the x’. The areas of integration in the space x x' for Normal processes are summarized as follows:
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Fig. S.6. Areas of Integration in the x-x’ plane for Normal processes.
As the first-order dilatational mode in the membrane is a longitudinal mode, the allowed interactions are therefore of the form: L L L , L T L , L L L , L T T ,
L L T and L T L .
References [1]
R. Berman, E. L. Foster, and J. M. Ziman, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 231, 130 (1955).
[2]
J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Oxford University Press, USA, 1960).
[3]
D. Ecsedy and P. Klemens, Physical Review B 15, 5957 (1977).
[4]
J. Callaway, Physical Review 113, 1046 (1959).
[5]
A. AlShaikhi and G. Srivastava, Physical Review B 76, (2007).
[6]
K. Johnson, M. Wybourne, and N. Perrin, Physical Review B 50, 2035 (1994).
[7] [8]
. ü, ournal of pplied hysics 104, 54314 (2008). G. P. Srivastava, The Physics of Phonons (Taylor & Francis, 1990).
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