Supplementary information
L
ħ
T
kB
q
s
cqs
qs
nqs
qs
Dqs
L

V0
s
D,s
M
s
d m
s
pq
x
Lattice thermal conductivity
Planck constant divided by 2
Temperature
Boltzmann constant
Wavevector
Phonon polarization
Group velocity
Phonon frequency
Bose-Einstein equilibrium phonon distribution
function
Total relaxation time
Phonon density of states
Effective diameter of the bulk sample
Impurity scattering parameter
Volume per atom
Grüneisen parameter
Debye temperature
Atomic mass average
Inverse of Knudsen number
Film thickness.
Bulk mean free path
Wavevector dependent specularity parameter
Integration variable
The thermal conductivity simulation of free-standing membranes was carried out, first, by deriving the
lattice thermal conductivity in Si bulk system using the modified Callaway model1,2 under the single
mode relaxation time approximation. Then, once the thermal conductivity for bulk Si is determined, we
introduced the effect of finite size through the so-called Fuchs-Sondheimer boundary corrections3,4.
 bulk 
2
3k BT 2
c
qs2  qs nqs (nqs  1)Dqs dq
2
qs
(1)
s

 film
3 1  pq 
1  e  x
3
5
 1
(x  x )
dx
 bulk
2  s 1
1  pq e  x
s
s
(2)
where pq = exp(42q2) is the fraction of phonons that are specularly reflected by the boundaries5 and s
= dms is the inverse of Knudsen number.
The dispersion relation, qs, was determinate applying the approach suggested by Hopkins et al.6 and
obtained an analytical form from a fourth-order polynomial fit to the experimental data for
direction.
  Aq  Bq 2  Cq3  Dq 4 ,
(3)
with A, B, C and D constants determined via numerical fitting of experimental values. The total bulk, qs,
relaxation time for each polarization, s, is limited by various scattering mechanisms such as: boundary
B,qs, mass defect I,qs and Umklapp phonon-phonon interactions U,qs. This can be obtained via the
Matthiessen rule as
 q,1s   B1,qs   I,1qs   U1,qs ,
where the relaxation times B,qs, I,qs and U,qs are given by7–9
(4)
 B1,qs 
 I,1qs
 U1,qs 
cqs
L
V
 0 3 qs4
4cqs
 s2
qs2 exp   D ,s /(3T )
2
Mcqs D ,s
(5)
(6)
(7)
The parameters main used in the calculations are shown in table 1.
L

L
T
(K)
(K)
 (mm)
2.01
Silicon
5.0
 


Table 1 Silicon parameters used in the calculations.
Material
L
T
Figure 1 (a) Phonon dispersion relation of bulk silicon: experimental results (red dots)
from Ref.10 and fourth-order polynomial fit (black solid line). (b) Temperaturedependence of the lattice thermal conductivity: black solid line calculated temperaturedependence of the lattice thermal conductivity of bulk silicon. Red dots: the
experimental data of silicon obtained from Ref.11
With the thermal conductivity determined we introduce the effect of the finite size using the equation 2.
The surface roughness surface parameter, , was fixed in 0.5 nm taken from the estimated value in ref12
for Si membranes.
References
(1)
Callaway, J. Physical Review 1959, 113, 1046–1051.
(2)
Holland, M. Physical Review 1963, 132, 2461–2471.
(3)
Fuchs, K. Mathematical Proceedings of the Cambridge Philosophical Society 1938, 34, 100–108.
(4)
Sondheimer, E. H. Advances in Physics 1952, 1, 1–42.
(5)
Ziman, J. M. Electrons and Phonons: The Theory of Transport Phenomena in Solids; Oxford
University Press, USA, 1960.
(6)
Hopkins, P. E.; Rakich, P. T.; Olsson, R. H.; El-kady, I. F.; Phinney, L. M. Applied Physics
Letters 2009, 95, 161902.
(7)
Klemens, P. G. Proceedings of the Physical Society. Section A 1955, 68, 1113–1128.
(8)
Slack, G.; Galginaitis, S. Physical Review 1964, 133, A253–A268.
(9)
Casimir, H. B. G. Physica 1938, 5, 495–500.
(10)
Group IV Elements, IV-IV and III-V Compounds. Part a - Lattice Properties; Madelung, O.;
Rössler, U.; Schulz, M., Eds.; Springer-Verlag: Berlin/Heidelberg, 2001; Vol. a.
(11)
C. J. Glassbrenner and Glen A. Slack Phys. Rev. 1964, 134, 1058–1069.
(12)
Cuffe, J.; Ristow, O.; Chávez, E.; Shchepetov, A.; Chapuis, P.-O.; Alzina, F.; Hettich, M.;
Prunnila, M.; Ahopelto, J.; Dekorsy, T.; Sotomayor Torres, C. M. Physical Review Letters 2013,
110, 095503.