VOLUME 84, NUMBER 5
P H Y S I C A L R E V I E W L E T T E R S
31 JANUARY 2000
Minimum Thermal Conductivity of Superlattices
M. V. Simkin and G. D. Mahan
Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200
and Solid State Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831
(
Received 23 July 1999)
The phonon thermal conductivity of a multilayer is calculated for transport perpendicular to the layers.
There is a crossover between particle transport for thick layers to wave transport for thin layers. The
calculations show that the conductivity has a minimum value for a layer thickness somewhat smaller
then the mean free path of the phonons.
PACS numbers: 66.70.+f, 68.65.+g
The thermal conductivity is a fundamental transport pa-
rameter [1]. There has been much recent interest in the
thermal conductivity of semiconductor superlattices due to
their possible applications in a variety of devices. Efficient
solid state refrigeration requires a low thermal conductivity
which is often the case experimentally. The effective ther-
mal conductivity of the superlattice is then
2
RSL
L
2L
KSL
.
(1)
L͑1͞K 1 1͞K ͒ 1 2R
1
2
B
This classical prediction is that the thermal conductivity
decreases as the layer thickness L decreases [9].
[2]. Preliminary experimental and theoretical work sug-
gests that the thermal conductivity of superlattices is quite
low, both for transport along the planes [3,4], or perpen-
dicular to the planes [5–8]. The heat is carried by exci-
tations such as phonons and electrons. Most theories use
a Boltzmann equation which treats the excitations as par-
ticles and ignores wave interference [7,9]. These theories
all predict that the thermal conductivity perpendicular to
the layers decreases as the layer spacing is reduced in the
superlattice. The correct description using the Boltzmann
equation would be to use the phonon states of the superlat-
tice as an input to the scattering, but this has not yet been
done by anyone.
We present calculations of the thermal conductivity per-
pendicular to the layers which include the wave interfer-
ence of the superlattice. These calculations, in one, two,
and three dimensions, always predict that the thermal con-
ductivity increases as the layer spacing is reduced in the
superlattice. This behavior is shown to be caused by band
folding in the superlattice. It is a general feature which
should be true in all cases. The particle and wave calcu-
lations are in direct disagreement on the behavior of the
thermal conductivity with decreasing layer spacing. This
disagreement is resolved by calculations which include the
mean free path (mfp) of the phonons. For layers thinner
than the mfp, the wave theory applies. For layers thicker
than the mfp the particle theory applies. The combined
theory predicts a minimum in the thermal conductivity, as
a function of layer spacing. The thickness of the layers for
minimum thermal conductivity depends upon the average
mfp, and is therefore temperature dependent.
The wave theory calculates the actual phonon modes
v ͑k͒ of the superlattice, where l is the band index. They
l
are used to calculate the thermal conductivity from the
usual formula in d dimensions [1],
X Z
d
d k
≠n͑v, T͒
≠T
K͑T͒
h¯ v ͑k͒ jy ͑k͒jᐉ ͑k͒
,
l
z
l
͑2p͒d
l
(2)
where n͑v, T͒ is the Bose-Einstein distribution function.
A rigorous treatment uses Boltzmann theory applied to
the transport in minibands to find the mean free path
ᐉ ͑k͒. At high temperatures, one can approximate n ϳ
l
k T͞ h¯ v ͑k͒, which gives the simpler formula
B
l
X Z
d
d k
K͑T͒ kB
jy ͑k͒jᐉ ͑k͒ .
(3)
z
l
͑2p͒d
l
The above formula is quite general. There are two im-
portant special cases of constant relaxation time ͑K ͒ and
t
constant mfp ͑K ͒
ᐉ
X Z
ddk
͑2p͒d
2
K ͑T͒ k t
t
y ͑k͒ ,
(4)
(5)
B
z
l
X
Z
d
d k
K ͑T͒ k ᐉ
jy ͑k͒j .
ᐉ
B
z
͑
2p͒d
l
Both of these formulas can be related to the distribution
P͑yz͒ of phonon velocities perpendicular to the layers
X Z
d
d k
P͑y ͒
d͑y 2 jy ͑k͒j͒ ,
(6)
(7)
(8)
z
z
z
͑
2p͒d
Z
l
The particle theories use the interface boundary resis-
tance [10] as the important feature of a superlattice. A su-
perlattice with alternating layers has a thermal resistance
for one repeat unit of RSL L ͞K 1 L ͞K 1 2R ,
2
z
K k t dy P͑y ͒y ,
t
B
z
z
Z
1
1
2
2
B
K k ᐉ dy P͑y ͒y .
ᐉ
B
z
z
z
where ͑L , K ͒ are the thickness and thermal conductiv-
j
j
ity of the individual layers, and R is the thermal bound-
Wave interference leads to band folding [11,12]. Band
folding leads to a reduction of the phonon velocities. Both
B
ary resistance. For simplicity assume that L L ϵ L,
1
2
0031-9007͞00͞84(5)͞927(4)$15.00
© 2000 The American Physical Society
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