Heat conduction and phonon localization in disordered

Heat conduction and phonon localization in disordered harmonic
lattices
Anupam Kundu
Abhishek Chaudhuri
Dibyendu Roy
Abhishek Dhar
Joel Lebowitz
Herbert Spohn
Raman Research Institute
NUS, Singapore
February 2010
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July 2010
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Outline
Introduction
Heat transport in disordered harmonic lattices
Results from kinetic theory
Results from localization theory
A heuristic theory
Numerical results
Discussion
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July 2010
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Fourier’s law
For small ∆T = TL − TR and system size L :
Fourier0 s law implies : J ∼ κ
∆T
L
The thermal conductivity κ is an intrinsic material property.
e.g at room temperature:
κ
(RRI)
=
0.025 W /mK air
=
429 W /mK silver
=
2000 W /mK Diamond
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Proof of Fourier’s law
Open problem:
Microscopic derivation/verification of Fourier’s law in a realistic
model with Hamiltonian dynamics.
Review article:
Fourier’s law: A challenge for theorists
(Bonetto, Rey-Bellet, Lebowitz) (2000).
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July 2010
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Results so far
Studies in one and two dimensional systems find that Fourier’s law is not valid.
The thermal conductivity is not an intrinsic material property.
For anharmonic systems without disorder , κ diverges with system size L as:
κ ∼
∼
∼
Lα 1D
0
Lα , log L
0
L
2D
3D
Pinned anharmonic (momentum non-conserving) systems satisfy Fourier’s law.
A.D, Advances in Physics, vol. 57 (2008).
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July 2010
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Disordered Harmonic systems: Results in 1D
Exact expression for nonequilibrium heat current. In classical case:
J
kB ∆T
2π
=
∞
Z
dωT (ω) ,
0
where T (ω) is the phonon transmission function.
Localization implies: T (ω) ∼ e−N/`(ω) with `(ω) ∼ 1/ω 2 for ω → 0 .
<
Hence frequencies ω ∼ N −1/2 “do not see” the randomness and can carry current.
These are ballistic modes.
Hence J ∼
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R N −1/2
0
T (ω)dω.
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Disordered Harmonic systems: 1D
Form of T (ω) (at small ω) depends on boundary conditions.
0.4
1
0.3
T(ω)
T(ω)
0.8
0.2
0.1
0.6
0
0
0.5
0.4
1
ω
1.5
2
0.2
0
0
Fixed BC: T (ω) ∼ ω 2
Free BC: T (ω) ∼ ω 0
0.5
1
ω
1.5
2
J ∼ 1/N 3/2
J ∼ 1/N 1/2
If all sites are pinned then low frequency modes are cut off: hence J ∼ e−N/` .
[Matsuda,Ishii,Rubin,Greer,Casher,Lebowitz,Dhar]
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July 2010
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Higher dimensional systems
What about systems in three dimensions ?
Is Fourier’s law valid ?
Consider dielectric crystals.
Heat conduction is through lattice vibrations .
Harmonic approximation:
solid can be thought of as a gas of phonons with frequencies corresponding to the vibrational
spectrum of ordered harmonic crystal.
Phonons get scattered by:
(i) Impurities
(ii) Other phonons (anharmonicitiy).....Keiji’s Talk.
It is usually expected that these scattering mechanisms should lead to a finite
conductivity.
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July 2010
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Three dimensional systems
Heat conduction in a classical harmonic crystal with isotopic mass
disorder.
Look at L dependence of J in nonequilibrium steady state. (J ∼ 1/L ? )
Look at phonon transmission:
Effect of localization and boundary conditions.
EPL 90, 40001 (2010), PRB 81, 064301 (2010).
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July 2010
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Heat conduction in mass disordered harmonic crystal
H=
X mx
Xk
X ko
q̇x2 +
(qx − qx+ê )2 +
q2
2
2
2 x
x
x
x,ê
qx : scalar displacement,
ko = 0: Unpinned.
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masses mx random.
ko > 0: Pinned.
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Kinetic theory.
In the harmonic approximation the solid can be thought of as a gas of phonons with
frequencies corresponding to the vibrational spectrum.
The phonons do a random walk. A phonon of frequency ω has a mean free path `K (ω).
Thermal conductivity given by:
Z
ωD
dω D(ω) ρ(ω) c(ω, T )
κ=
0
D(ω) = v `K (ω) = phonon diffusivity ,
v = phonon velocity ,
ρ = density of states ,
c = specific heat capacity ,
ωD = Debye frequency .
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Kinetic theory
Rayleigh scattering of phonons gives a mean free path `K (ω) ∼ 1/ω d+1 .
>
For low frequency phonons `K ∼ L , where L is the size of system (ballistic phonons).
ρ(ω) ∼ ω d−1 . Hence
Z
κ
ωD
dω ρ(ω) v `K (ω)
=
1/N
∼
L1/(d+1)
1D : κ ∼ L1/2 ,
2D : κ ∼ L1/3 ,
3D : κ ∼ L1/4 .
Thus kinetic theory predicts a divergence of the conductivity in all dimensions.
For a pinned crystal it predicts a finite conductivity.
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July 2010
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Problems with kinetic theory
Is a perturbative result valid in the limit of weak disorder. Rayleigh scattering result for `K (ω)
is obtained for single scattering of sound waves.
Effect of Anderson localization is not accounted for since this requires one to consider
multiple scattering of phonons.
The kinetic theory result can also be derived directly from the Green-Kubo formula and `K (ω)
can be computed using normal mode eigenvalues and eigenstates.
Green-Kubo probably correct for normal transport.
Probably WRONG for anomalous transport.
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July 2010
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Anderson localization
Consider harmonic Hamiltonian:
H=
X1
X
qx φxx0 qx0 .
mx q̇x2 +
2
0
x
x,x
x denotes point on a d-dimensional hypercube of size N d .
φ - Force-matrix .
Equation for q th normnal mode, q = 1, 2, ..., N d . [Closed system]
mx Ω2q aq (x) =
X
φx,x0 aq (x0 ) .
x0
Normal mode frequency: Ωq .
Normal mode wave-function: aq (x).
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Character of normal modes
1
0.5
0
-0.5
-1
60
Extended periodic mode
(Ballistic)
60
Extended random mode
(Diffusive)
60
Localized mode
(Non-conducting)
40
20
20
40
60
5
0
-5
40
20
20
40
60
10
7.5
5
2.5
0
40
20
20
40
60
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July 2010
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Predictions from Renormalization group theory
S. John, H. Sompolinsky, and M.J. Stephen
For unpinned system:
In 1D all modes with ω > 1/N 1/2 are localized.
In 2D all modes with ω > [log(N)]−1/2 are localized.
In 3D there is a finite band of frequencies of non-localized states between
0 to ωcL , where ωcL is independent of system-size.
System size dependence of current ??
Nature of non-localized states ??
Ballistic OR Diffusive OR .....
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July 2010
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Nature of phonon modes
Localization theory tells us that low frequency modes are extended.
At low frequencies the effect of disorder is weak and so we expect kinetic theory to be accurate.
Setting `K (ω) = N gives us the frequency scale ωcK below which modes are ballistic. Since
`K (ω) = 1/ω d+1 this gives:
1D:
ωcK ∼ N −1/2
2D:
ωcK ∼ N −1/3
3D:
ωcK ∼ N −1/4
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July 2010
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Nature of phonon modes
Kinetic theory and localization theory gives us two frequency cut-offs:
ωcK
and
ωcL :
ω
Ballistic
Localized
Diffusive
ωLc
ωK
c
1D :
ωcK ∼ ωcL ∼ N −1/2
2D :
ωcK ∼ N −1/3 ,
ωcL ∼ [ln N]−1/2
3D :
ωcK ∼ N −1/4 ,
ωcL ∼ N 0
Try to estimate the N dependence of J.
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July 2010
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Heuristic theory
Nonequilibrium heat current is given by the Landauer formula:
J=
∆T
2π
∞
Z
dω T (ω) ,
0
where T (ω) is the transmission coefficient of phonons.
Phonons can be classified as:
Ballistic: Plane wave like states with T (ω) independent of system size.
Diffusive: Extended disordered states with T (ω) decaying as 1/N.
Localized states: T (ω) ∼ e−N/`L .
Find contribution to J of the different types of modes and estimate the asymptotic size
dependence.
Ballistic contribution to current depends on boundary conditions.
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July 2010
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Boundary conditions
Free BC: T (ω) ∼ ω d−1
ω→0
Fixed BC: T (ω) ∼ ω d+1
ω→0
Pinned system: No low frequency
ballistic modes
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Predictions of heuristic theory
1D unpinned fixed BC: J ∼ N −3/2
1D unpinned free BC : J ∼ N −1/2
1D pinned:
J ∼ exp(−bN)
These have been proved rigorously.
3D unpinned fixed BC: J ∼ N −1
3D unpinned free BC : J ∼ N −3/4
3D pinned:
J ∼ N −1
2D unpinned fixed BC: J ∼ N −1 (ln N)−1/2
2D unpinned free BC : J ∼ N −2/3
2D pinned:
J ∼ exp(−bN)
We now check these predictions numerically.
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July 2010
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Disordered harmonic systems
Equations of motion:
mx q̈x = −
X
(qx − qx−^e ) − ko qx {+fx from heat baths }
e
^
{mx } chosen from a random distribution.
fx is force from heat baths and acts only on boundary particles. We choose white noise Langevin
baths:
fx = −γ q̇x + (2γT )1/2 ηx (t).
hηx (t)ηx0 (t 0 ) = δ(t − t 0 )δx,x0
Periodic boundary conditions in transverse directions.
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July 2010
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Measured quantities
In the nonequilibrium steady state we measure the energy current and temperature profile. These
are given by the following time-averages:
J
=
X
q1,x0 q̇2,x0
x0
Tx
=
X
2
mx,x0 q̇x,x
0
x0
We use two approaches:
1. Numerical evaluation of phonon transmission and use the Landauer formula to find J.
2. Nonequilibrium Simulations.
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July 2010
23 / 33
Current in terms of phonon transmission
J=
∆T
2π
∞
Z
dωT (ω) ,
0
where
T (ω) = 4γ 2 ω 2 |G1N |2
G
=
and
=
[−ω 2 M + Φ − ΣL − ΣR ]−1
0
a1 − iγω −1
0
...
−1
a2
−I
0 ...
B
B
...
...
...
...
B
@
0
... 0 −1
aN−1
0
...
0
−1
0
0
...
−1
aN − iγω
1−1
C
C
C
A
,
al = 2 + ko − ml ω 2 .
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July 2010
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Product of random matrices
Transmission function T (ω) can be expressed in terms of a product of random
matrices.
µ
G1N
=
where T̂l
=
(1
µ
− iγ) T̂N T̂N−1 ...T̂1
¶
2 − ml ω 2 −1
1
0
1
iγ
¶
Generalization to higher dimensions (N d lattice):
T (ω) can be expressed in terms of a product of
2N d−1 × 2N d−1 random matrices (Anupam Kundu).
We implement this numerically for d = 2, 3.
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July 2010
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Numerical and simulation results
Open system. We look at:
• Disorder averaged transmission: T (ω) =
R
• Current: = J ∼ 0∞ dω T (ω).
[T (ω)]
N d−1
Closed system without heat baths.
For the normal modes find:
• Density of states: ρ(ω) =
1
Nd
P
q
δ(ω − Ωq )
• Inverse participation ratio: P −1 =
(
P 4
a
P x 2x 2 .
x ax )
• Diffusivity D(ω) from Green-Kubo.
IPR is large (∼ 1) for localized states, small (∼ N −d ) for extended states.
Binary mass distribution: m1 = 1 + ∆, m2 = 1 − ∆
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(∆ = 0.8.
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J-versus-N plots: 3D
Free BC
Fixed BC
Pinned
0.01
-0.75
J
N
-1
0.001
N
8
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16
N
32
64
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J-versus-N plots: 2D
0.1
Free BC
Fixed BC
J
0.01
-0.6
N
0.001
0.0001
0.001
1e-06
4
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-3.7
1e-05
N
4
16
16
-0.75
N
64
64
N
256
1024
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Comparision with numerical results
3D
Unpinned fixed BC:
Heuristic prediction
J ∼ N −1
Unpinned free BC:
J ∼ N −3/4
Pinned:
J ∼ N −1
(Numerical result)
[∼ N −0.75 ]
[∼ N −0.75 ]
[∼ N −1 ]
2D
Unpinned fixed BC:
Heuristic prediction
[Numerical result]
J ∼ N −1 (ln N)−1/2
[∼ N −0.75 ]
Unpinned free BC:
J ∼ N −2/3
Pinned:
J ∼ exp(−bN)
[ ∼ N −0.6 ]
[∼ exp(−bN) ]
Look at Phonon transmission functions.
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July 2010
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Phonon transmission: pinned 3D
N T(ω)
0.32
(i)
N=8
N=16
N=32
0.24
0.16
0.08
0
2
2.4 2.8 3.2
8
8.4 8.8 9.2 9.6
10 10.4
ω
ρ(ω)
1.5
(ii)
Disordered
Ordered
1
0.5
0
2
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2.4 2.8 3.2
ω
8
8.4 8.8 9.2 9.6 10 10.4
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Transmission: Unpinned 3D
0.08 (i)
T(ω)
N=8
N=16
N=32
0.04
0
0
1
2
3
4
5
6
(ii)
N T(ω)
0.8
8
0.4
0
0
1
2
ρ(ω)
1
0.8 (iii)
0.6
0.4
0.2
0
1
0
2
1000
(iv)
N=8
3
4
ω
5
6
7
8
Ordered Lattice
Disordered Lattice
3
4
5
6
7
8
3
4
ω
5
6
7
8
N=16
100
NP
3 -1
7
N=8
N=16
N=32
10
1
0
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1
2
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Effective phonon mean free path: 3D
`eff = NT (ω)/ω d−1
10
−4
ω
leff(ω)
1
Fixed: N=16
N=32
Free: N=16
N=32
0.1
10
1
0.1
0.01
Fixed
Free
0.01
0.1
0.1
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ω
1
1
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Conclusions
Unpinned case: Results sensitive to boundary conditions. Heuristic arguments suggest that
Fourier’s law valid for generic boundary conditions (fixed).
Numerical verification presumably requires larger system sizes.
Pinned system: First numerical verification of Fourier’s law in a 3D system.
Pinned system: Direct verification of a 2D heat insulator.
Landauer and Green-Kubo formulas.
N T (ω)/2π ? = ? D(ω) ρ(ω) (Proof)
Our results suggest this is true for normal transport but not true for anomalous tranport.
Future questions
Exact results.
Improving numerical methods and study larger system sizes.
What happens in real systems ? Effect of anharmonicity, other degrees of freedom.
(RRI)
July 2010
33 / 33

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