VOLUME 87, NUMBER 7 PHYSICAL REVIEW LETTERS 13 AUGUST 2001
Thermal Insulating Behavior in Crystals at High Frequencies
Sebastian G. Volz
Laboratoire d’Etudes Thermiques, UMR 6608 CNRS, Site du Futuroscope, Boîte Postale 109,
86960 Futuroscope Cedex, France
(Received 1 February 2001; published 30 July 2001)
When solving heat-conduction problems with periodic temperature perturbations, the thermal conduc-
tivity is assumed to remain frequency independent. We, however, show by using the molecular dynamics
technique and the fluctuation-dissipation theorem a decrease of the effective thermal conductivity of
2 orders of magnitude when the excitation frequency approaches or exceeds the reverse of the phonon
mean relaxation time. Most of the dielectric and semiconductor materials have to be considered as
strongly insulating in those conditions. The comparison between molecular-dynamics simulations per-
formed in Si crystals and theoretical predictions reveals a clear agreement.
DOI: 10.1103/PhysRevLett.87.074301 PACS numbers: 44.10.+i
Clock frequencies of present microprocessors based
on silicon technology reach the gigahertz 10
9
Hz,
and roadmaps suggest that devices with clock rates of
100 GHz will be produced in the next few years. Under-
standing heat transfer mechanisms at high frequencies,
when local nonequilibrium regimes appear, can therefore
not be dismissed. Short time heat conduction in solids has
generated much interest for four decades, and extended
reviews can be found in the literature [1]. The basic
modeling consists in adding an inertia term in the Fourier
law linking the temperature gradient
!
gradT and the heat
flux q [2,3]:
t
≠ q
≠t
1 q 2l
!
gradT , (1)
where t denotes the heat flux relaxation time, and l is the
thermal conductivity. The Boltzmann transport equation
written in terms of the phonon occupation number con-
firms the intuitive derivation of Eq. (1) and leads to the
equality between t and the phonon mean relaxation time
[4]. Equation (1) is coherent with the fact that phonons,
i.e., heat carriers in solids, rarely scatter together in time
intervals smaller than t. Combining Eq. (1) with the en-
ergy balance equation allows one to derive the well-known
hyperbolic heat conduction equation:
t
≠
2
T
≠t
2
1
≠T
≠t
aDT , (2)
where a represents the thermal diffusivity. The right-
hand side term, however, implies a diffusive heat transport
which cannot appear in the involved spatial scales. Heat
flux or temperature fields calculation actually requires the
solving of the Boltzmann equation that can be reformu-
lated as a radiative transport equation [2,3]. Note that a
simpler approach was recently proposed based on the de-
composition of this last equation into ballistic and diffusive
equations [5].
We report a direct way to derive the dissipated heat flux
when a high frequency AC temperature perturbation is gen-
erated in bulk materials. We predict a decrease of 2 orders
of magnitude for this quantity when the excitation fre-
quency approaches or exceeds the reverse of the phonon
mean relaxation time. We then prove the relevance of this
result based on molecular dynamics (MD) simulations per-
formed in Si crystals. Results show accordance between
predicted and computed heat fluxes concerning frequency
and temperature dependence.
The fluctuation-dissipation theorem derived in the frame
of the linear response theory allows one to express the heat
flux q created by an AC temperature disturbance of fre-
quency v, denoted
!
gradTe
i vt
, in terms of the equilibrium
fluctuations of the heat flux q
0
t [6,7]:
qv
V
3k
B
T
2
Z `
0
q
0
0q
0
t e
i vt
dt k
!
gradT k . (3)
Here, q
0
0q
0
t
1
V
R
V
q
0
0, r q
0
t , r r
0
dV is the
heat flux autocorrelation function, and T
0
is the equi-
librium temperature. Equation (1) directly leads to
q
0
0q
0
t q
0
0
2
e
2t t
, but this relation also relies
on a deeper physical basis, proving that the autocorrelation
of the phonon occupation number follows the same expo-
nential law [8]. Inserting the time explicit autocorrelation
function in Eq. (3) allows one to identify the heat flux
as follows:
qv
k
!
gradT k
V
3k
B
T
2
Z `
0
q
0
0q
0
t e
evt
dt
V q
0
0
2
3k
B
T
2
Z `
0
e
i vt 2t t
dt . (4)
We distinguish two cases knowing that the phonon mean
relaxation time t is of the order of 10–100 ps in solid
crystals at room temperature.
(i) When v ø 1t 0.1 0.01 GHz, the exponential
term e
i vt
can be factorized since it varies very slowly in
the interval 0 3t where the heat flux autocorrelation con-
tribution remains non-negligible. This is the most general
case leading to Fourier’s law jqvjk
!
gradT k l, where
the bulk thermal conductivity is defined by the Green
Kubo formula:
074301-1 0031-9007 01 87(7) 074301(4)$15.00 © 2001 The American Physical Society 074301-1