heat flux - AICES - RWTH Aachen University

RWTH Aachen University, August 25th, 2016 - Lecture 1
Calculating lattice thermal
conductivity
Luciano Colombo
Department of Physics
University of Cagliari, Italy
A synopsis of available theoretical methods
luciano.colombo@unica.it
1
1. Defining the canonical problem
❖ In general, heat, mass, and charge transport phenomena are coupled
Reference (thermodynamics language):
S. Kjelstrup & D. Bedaux “Non-Equilibrium Thermodynamics of Heterogeneous Systems’’ (World Scientific, Singapore, 2008)
Reference (solid-state physics language):
J.M. Ziman “Electrons and Phonons” (Clarendon Press, Oxford, 1960)
❖ Effective heat, mass, and charge transport diffusivity provided by Onsager coefficients
✓ ◆
✓
◆
✓
◆ Jheat : heat flux
d
1
1 dµ
1 dV
µ : chemical potential
Jheat = Lhh
+ Lhm
+ Lhc
dz T
T dz
T dz
V : electrostatic potential
Lhh,hm,hc : Onsager coe↵s.
❖ General theory of Onsager coefficients proves that transport is dominated by the main
coefficients
Lii Ljj Lij Lji 0
i, j : h, m, c
❖ We can therefore focus just on (pure) thermal transport where the usual
1
thermal conductivity is linked to the Onsager coefficient through:
 = 2 Lhh
T
2
❖ Assumptions
1.
2.
3.
4.
5.
6.
Incompressible solid-state medium
Non-polarizable medium
Semiconductor or insulator
Small temperature-gradient
Homogeneous medium
Isotropic medium
no convective modes & no mass transport
no electron contribution to 
linear response regime
constant thermal conductivity
scalar problem
Fourier law
❖ Constitutive equations
J~heat = J~ =
~
rT
Heat diffusion equation
J˙
⇢ Cv @T
1 @T
2
r T+ =
=

 @t
↵ @t
steady-state conditions
Jz =
dT

dz
❖ Most atomistic calculations addressed to calculate thermal conductivity in a steady-state
condition
3
2. Synopsis of theoretical/computational methods
Total-energy methods
Molecular dynamics methods
Equilibrium
Ab initio
Model
potentials
Boltzmann Transport
Equation
Non
equilibrium
Approach to
equilibrium
Green-Kubo
Green-Kubo
4
- setting ∆T
- setting J
AEMD
Part 1: solving Fourier by BTE
❖ Steady-state heat transport condition (one-dimensional case)
Fourier law
1 X
J=
~!q~s vq~s nq~s (T ) =
N⌦
@T

@z
q
~s
@T @nq~s
vq~s
@z @T
BTE
balance equation for the
perturbed phonon population
@nq~s
@t
phonon group velocity
in the direction of heat flux
linearized BTE
✓
non-equilibrium
phonon population
=0
scattering
small perturbation
@nq~s
⇠
@T
(eq)
@nq~s
@T
❖ Standard procedure: single-mode (relaxation time) approximation (known as SMA or RTA)
equilibrium
for the phonon-phonon scattering term
phonon population
✓
individual phonons thermalize independently
@nq~s
@t
=
scattering
nq~s
(eq)
nq~s
⌧q~s
1
phonon population relaxes to equilibrium at rate
⌧q~s
5
❖ Eventually, we get the full ab initio BTE thermal conductivity
h
i
X
~2
(eq)
(eq)
2 2
=
!
v
⌧
n
n
~s q
q
~s q
~s q
~s
q
~s + 1
2
N ⌦kB T
q
~s
linearized & SMA
BTE
consistent with kinetic theory
1 X
=
Cq~s,v vq~2s ⌧q~s
N⌦
q
~s
❖ What is needed to implement the above expression (typically done using DFT-PT)
phonon frequencies
phonon group velocities
phonon equilibrium population
❖
❖
❖
3-phonon scattering rates
❖
2nd-order interatomic force constants
dynamical matrix diagonalization
3rd-order interatomic force constants
account for all scattering processes
Example: scattering rate for this event
1
⌧
(~
q1 s1 ,~
q2 s2 )!~
q 3 s3
V
(3)
⇠
X
~
G
(eq)
(eq)
(eq)
|V (3) (~q1 s1 , ~q2 s2 , ~q3 s3 )|2 nq~1 s1 nq~2 s2 (nq~3 s3 + 1)
(~q1 s1 , ~q2 s2 , ~q3 s3 ) =
✓
~
8N !q~1 s1 !q~2 s2 !q~3 s3
◆ 12
X X
~
r1 ~
r2 ~
r3 ↵
6
↵
~
q
~1 +~
q2 q
~3 ,G
(~q1~r1 , ~q2~r2 , ~q3~r3 )
3rd-order interat.
force consts.
(~!q~1 s1 + ~!q~2 s2
~!q~3 s3 )
e↵ (~r1 |~q1 s1 ) e↵ (~r2 |~q2 s2 ) e↵ (~r3 |~q3 s3 )
p
p
p
m1
m2
m3
❖ Implementation of ab initio (linearized) SMA-BTE thermal conductivity
✓
✓
✓
Matthiessen rule
1 X 1
=
⌧
⌧m
m
ideally crystalline systems
or systems containing few defects
systems containing isotopic defects
systems with boundaries
Show-case application:
thermal conductivity in homogeneous SixGe1-x alloys
•
•
•
homogeneous alloy described as
(harmonic VCA + perturbations)
perturbation #1: mass disorder
an harmonic perturbation to VCA according to
standard Tamura theory
perturbation #2: anharmonicity:
✦ 2nd- and 3rd-order FCs calculated for Si, Ge,
and (VCA-)Si0.5Ge0.5
✦ FCs for SixGe1-x computed by quadratic
interpolation
✦ scattering rates calculated by DFT-PT
BTE SMA
J. Garg, N. Bonini, B. Kozinsky, and N. Marzari,
Phys. Rev. Lett. 106, 045901 (2011)
7
❖ Beyond single-mode (relaxation time) approximation
(eq)
@n
@nq~s
q
~s
✓ still valid: small perturbation
⇠
@T
@T
✓
new formulation of the scattering term
(1st-order expansion)
(eq)
nq~s = nq~s
(eq)
kB T @nq~s
fq~s
~!
@z
2
describes the deviation from
the equilibrium phonon population
❖ The (linearized) BTE is now cast in the form
vq~s
(eq)
@nq~s
@T
=
X
(fq~1 s1 + fq~2 s2
q
~2 s2 ,~
q 3 s3
1
+ (fq~1 s1
2
fq~2 s2
1
fq~3 s3 )
⌧
(~
q1 s1,~
q2 s2)!~
q 3 s3
1
fq~3 s3 )
⌧
q
~1 s1!(~
q2 s2,~
q3 s 3 )
for which there exist feasible “exact” solutions
•
•
•
iterative
variational (with trial funct.)
variational (with CG)
M. Omini, A. Sparavigna, Il Nuovo Cimento D 19, 1537 (1997)
R.H.H. Hamilton, J.E. Parrot, Phys. Rev. 117, 1284 (1969)
G. Fugallo, M. Lazzeri, L. Paulatto, F. Mauri, Phys. Rev. B 88, 045430 (2013)
❖ Full ab initio exact thermal conductivity eventually obtained as
h
i
X
~2
(eq)
(eq)
=
!q~s vq~s nq~s nq~s + 1 fq~s
2
N ⌦kB T
q
~s
8
A. Cepellotti et al., Nature Comm. 6, 6400 (2015)
❖ Is it worthy of?
BTE variational CG
G. Fugallo et al., Phys. Rev. B 88, 045430 (2013)
BTE variational CG
Isotopically enriched
diamond
J. Garg, Ph.D. thesis, MIT (2011)
BTE SMA
9
BTE iterative solution
Part 2: solving Fourier by calculating the heat flux
❖ Framework 1 : linear-response theory of transport coefficients
R(t)
lim
=
t!+1 rB
a time-independent perturbation rB
R(t)
a response
in steady-state conditions
1
=
lim
3V kB T t!+1
Z
t
0
⌘ ⇠ lim
t!+1
⌘
Z
t
0
ds h⇠(s)⇠(0)i
the transport coefficient is the auto-correlation
function of suitable physical quantity
~ · J(0)i
~
ds hJ(s)
❖ Framework 2 : direct MD simulation
1. set initial ∆T
2. age the system until
reaching steady-state
3. compute J
hJ(t)i
=
T
h
10
L i
Common issue:
how to compute the heat flux J
❖ Fundamentals
X
d
(tot)
J~ =
~ri Ei
dt
J~ =
i
(tot)
Ei
X d~vi
dt
i
(tot)
Ei
+
X
i
kinetic (convective)
term
negligible for solid-state systems
1
= mi vi2 + Ui
2
J~ '
X
i
~ri
⇣
F~i · ~vi
⌘
(tot)
dEi
~ri
dt
potential
term
+
Problem: how to define Ui
X
i
dUi
~ri
dt
❖ Practical implementations:
classical many-body potentials
Z. Fang et al., Phys. Rev. B0
92, 094301 (2015)
1
X
X 1X
@
U=
Ui =
Uij A
2
i
i
j6=i
@Ui
@Uj
~
Fji =
~rij
~rji
◆
X X ✓ @Uj
J~ '
~rij
· ~vi
@~rji
i
F~ij =
j6=i
general solution valid for
any many-body interaction potential
ab initio total-energy
A. Marcolongo et al., Nature Physics, 10.1038/nphys3509 (2015)
A very recent paper, finally solving the
dilemma
“… in first principles calculations it is impossible
to uniquely decompose the total energy E(tot) into
individual contributions Ei(tot) from each atom… “
We can compute J~ fully ab initio by DFT
11
❖ Computational challenge in using
1
=
lim
3V kB T t!+1
Z
t
0
~ · J(0)i
~
ds hJ(s)
Y. He et al., Phys. Chem. Chem. Phys. 14, 16209 (2012)
Need long simulations
Need large simulation cells
c-Si sample containing 4096 atoms
c-Si sample @ 300K
12
hJ(t)i
❖ Computational challenge in using the direct method  =
h LT i
Generating a thermal gradient by Langevin thermostats in a 500x3x3 c-Si box - THICK terminal layers
1000
900
local temparature at each slab [K]
800
1. set initial ∆T
2. age the system until reaching
steady-state
3. compute J
700
600
500
400
300
200
100
0
0
200000
400000
600000
timestep [3fs]
800000
1e+06
C. Melis et al., Eur. J. Phys. B 87, 96 (2014)
13
Part 3: Solving Fourier without calculating the heat flux
Key idea: set J and then compute corresponding ∆T in steady-state conditions
PBCs along the heat transport direction
❖ Solution #1 : rescale kinetic energy
P.K. Schelling et al., Phys. Rev. B 65, 144306 (2002)
1. kinetic energy of atoms in the “hot” and “cold”
resevoir is rescaled by the same amount ±∆ekin
for a time ∆t
ekin
2. resulting heat flux: J =
2A t
❖ Solution #2 : swap atomic velocities
F. Müller-Plathe, J. Chem. Phys. 106, 6081 (1997)
1. select the coldest/hottest atom in the“hot”/“cold”
resevoir and swap their velocity vector ~vc $ ~vh
2. repeat swapping for a time ∆t
X 1 v2 v2
c
3. resulting heat flux: J =
m h
2 2A t
#swaps
14
where Jin,out
1 @Win,out
=
A
@t
% error
No PBCs along the heat transport direction
8000
4000
0
-4000
-8000
-12000
0
✓
✓
15
0.2
0.4
0.6
0.8
1
1.2
2
1
0
-1
time average
-2
0
0.2
200
150
100
50
0
-50
-100
0.4
0.6
time (ns)
0.8
1
1.2
1
1.2
time average
0
✓
injected
extracted
time (ns)
heat flux (meV fs-1nm-2)
1. couple left/right ends to a hot/cold thermostat
2. reach steady-state and keep the system there
1
3. resulting heat flux:
J = (Jin + Jout )
2
exchanged energy (eV)
❖ Solution #3 : calculate “thermostatting work” instead of heat flux
0.2
0.4
0.6
time (ns)
0.8
all atoms are used to define L
accurate definition of steady-state
no PBCs needed: allow for interface calcns.
Part 4: Solving Fourier with no heat flux …nor long simulations
C. Melis et al., Eur. J. Phys. B 87, 96 (2014)
Approach to equilibrium MD (AEMD)
1. define an initial non-equilibrium (periodic) condition
2. age the system to reaching equilibrium by a NVE MD
run
3. evaluate on-the-fly the difference between average
temperature
4. formal solution of the heat equation provides
X
↵2n ̄t
TA (t) =
Cn e
n
where
̄ = /⇢cv
thermal diffusivity
↵n = 2⇡n/Lz
Cn = 8(Thot
Tcold )[cos(↵n Lz /2)
1]2
Just fit ∆TA(t) on the calculated hTleft (t)i
16
hTright (t)i
Merits & Limitations
Merits
-
BTE
-
Green-Kubo
-
-
MD
-
no heat flux
-
AEMD
-
Limitations
robust theoretical foundation
can be exploited fully ab initio
available in SMA or “exact” formulation
full quantum theory
robust theoretical foundation
one calculation allows for the full thermal
conductivity tensor
available ab initio as well as by empirical
potentials
any structure (crystalline or not)
does not require the implementation of the
heat flux
implementation straightforward
already available in most MD codes
any structure (crystalline or not)
does not require the implementation of the
heat flux
(almost) non need of coding
comparatively (much) shorter simulations
any structure (crystalline or not)
17
-
implementation not straightforward
computationally heavy
only valid for crystalline systems
-
does require the implementation of the heat
flux
long simulations needed
-
long simulations needed
-
still need to be tested more extensively,
meaning: the method is still in its infancy;
generalizations are needed for non-trivial
configurations
-