quantum thermal transport in nanostrucutures
DESCRIPTION
Quantum Thermal Transport in Nanostrucutures. Jian-Sheng Wang Center for Computational Science and Engineering and Department of Physics, NUS; IHPC & SMA. Outline. Nonequilibrium Green’s function (NEGF) approach to thermal transport, ballistic and nonlinear - PowerPoint PPT PresentationTRANSCRIPT
Quantum Thermal Transport in Nanostrucutures
Jian-Sheng WangCenter for Computational Science and Engineering
and Department of Physics, NUS; IHPC & SMA
Suzhou 08
2
Outline
• Nonequilibrium Green’s function (NEGF) approach to thermal transport, ballistic and nonlinear
• QMD – classical molecular dynamics with quantum baths
• Outlook and conclusion
Suzhou 08
3
Approaches to Heat Transport
Molecular dynamics Strong nonlinearity Classical, break down at low temperatures
Green-Kubo formula Both quantum and classical
Linear response regime, apply to junction?
Boltzmann-Peierls equation
Diffusive transport Concept of distribution f(t,x,k) valid at nanoscale?
Landauer formula Ballistic transport T→0, no nonlinear effect
Nonequilibrium Green’s function
A first-principle method Perturbative. A theory valid for all T?
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4
Nonequilibrium Green’s Function Approach
, ,
,
1 1,
2 21
3
T TL LC C C CR Rn
L C R
T T
C C Cn ijk i j k
ijk
H H u V u u V u H
H u u u K u
H T u u u
Left Lead, TL Right Lead, TR
Junction Part
T for matrix transpose
mass m = 1,
ħ = 1
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5
Heat Current
( 0)
1Tr [ ]
2
1Tr [ ] [ ] [ ] [ ]
2
L L
LCCL
r aL L
CL LCL L
I H t
V G d
G G d
V g V
Where G is the Green’s function for the junction part, ΣL is self-energy due to the left lead, and gL is the (surface) Green’s function of the left lead.
Suzhou 08
6
Landauer/Caroli Formula
• In systems without nonlinear interaction the heat current formula reduces to that of Laudauer formula:
0
/( )
1[ ] ,
2
[ ] Tr ,
,
1
1B
L R L R
r aL R
r a
k T
I I d T f f
T G G
i
fe
See, e.g., Mingo & Yang, PRB 68 (2003) 245406; JSW, Wang, & Lü, Eur. Phys. J. B, 62, 381 (2008).
Suzhou 08
7
Contour-Ordered Green’s Functions
( '') ''
0
'
0
( , ') ( ) ( ') ,
( , ') lim ( , ' '),
, , , ,
,
ni H dT
t t
r t a t
G i T u u e
G t t G t i t i
G G G G G G G G
G G G G G G
τ complex plane
See Keldysh, or Meir & Wingreen, or Haug & Jauho
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Adiabatic Switch-on of Interactions
t = 0
t = −
HL+HC+HR
HL+HC+HR +V
HL+HC+HR +V +Hn
gG0
G
Governing Hamiltonians
Green’s functions
Equilibrium at Tα
Nonequilibrium steady state established
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9
Contour-Ordered Dyson Equations
0 1 2 1 1 2 0 2
0 1 2 0 1 1 2 2
†
0 0 2
0 0 0
1
0
( , ') ( , ') ( , ) ( , ) ( , ')
( , ') ( , ') ( , ) ( , ) ( , ')
Solution in frequency domain:
1, 0
( )
,
1,
C C
n
r aC r
r a
r
r rn
r an
G g d d g G
G G d d G G
G Gi I K
G G G
GG
G G G
0( ) ( )r r a an nI G G I G
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10
Feynman Diagrams
Each long line corresponds to a propagator G0; each vertex is associated with the interaction strength Tijk.
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Leading Order Nonlinear Self-Energy
' ' ', 0, 0,
'' '' '', ' 0, 0,
, ''
4
'[ ] 2 [ '] [ ']
2
'2 '' [0] [ ']
2
( )
n jk jlm rsk lr mslmrs
jkl mrs lm rslmrs
ijk
di T T G G
di T T G G
O T
σ = ±1, indices j, k, l, … run over particles
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12
Energy Transmissions
The transmissions in a one-unit-cell carbon nanotube junction of (8,0) at 300 Kelvin. From JSW, J Wang, N Zeng, Phys. Rev. B 74, 033408 (2006).
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Thermal Conductance of Nanotube Junction
Con
d-m
at/0
6050
28
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14
Molecular Dynamics
• Molecular dynamics (MD) for thermal transport– Equilibrium ensemble, using Green-Kubo formula
– Non-equilibrium simulation• Nosé-Hoover heat-bath• Langevin heat-bath
• Disadvantage of classical MD– Purely classical statistics
• Heat capacity is quantum below Debye temperature of 1000 K
• Ballistic transport for small systems is quantum
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15
Quantum Corrections
• Methods due to Wang, Chan, & Ho, PRB 42, 11276 (1990); Lee, Biswas, Soukoulis, et al, PRB 43, 6573 (1991).
• Compute an equivalent “quantum” temperature by
• Scale the thermal conductivity by
1 1
exp( / ) 2BB Q
Nk Tk T
MDQ
dT
dT
(1)
(2)
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16
Quantum Heat-Bath & MD
• Consider a junction system with left and right harmonic leads at equilibrium temperatures TL & TR, the Heisenberg equations of motion are
• The equations for leads can be solved, given
,
,
L LCL L C
C CL CRC L R
R RCR R C
u K u V u
u F V u V u
u K u V u
0
2 20
2 2
( ) ( ) ( ') ( ') ',
where
( ) 0, ( ) ( )
tLC
L L L C
L LL L
u t u t g t t V u t dt
d dK u t K g t t I
dt dt
(3)
(4)
1
2
j
u
u
u
u
Suzhou 08
17
Quantum Langevin Equation for Center
• Eliminating the lead variables, we get
where retarded self-energy and “random noise” terms are given as
( ') ( ') 't
CC C L Ru F t t u t dt
0
, ,
, ,
C CL R
CL R
V g V
V u
(5)
(6)
Suzhou 08
18
Properties of Quantum Noise
† 0 0
†
†
( ) 0,
( ) ( ') ( ) ( ')
( ') ( '),
( ') ( ) ( '),
( ') ( ) [ ] 2 ( )Im [ ]
CL T LCL L L L
CL LCL L
T
L L L
T i tL L L L
t
t t V u t u t V
V i g t t V i t t
t t i t t
t t e dt i f
For NEGF notations, see JSW, Wang, & Lü, Eur. Phys. J. B, 62, 381 (2008).
(7)
(8)
(9)
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19
Quasi-Classical Approximation, Schmid (1982)
• Replace operators uC & by ordinary numbers
• Using the quantum correlation, iħ∑> or iħ∑< or their linear combination, for the correlation matrix of .
• Since the approximation ignores the non-commutative nature of , quasi-classical approximation is to assume, ∑> = ∑<.
• For linear systems, quasi-classical approximation turns out exact! See, e.g., Dhar & Roy, J. Stat. Phys. 125, 805 (2006).
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Implementation
• Generate noise using fast Fourier transform
• Solve the differential equation using velocity Verlet• Perform the integration using a simple rectangular
rule• Compute energy current by
2 /1( ) i lk M
kk
t lh ehM
( ') ( ') 't
TLL C L c L
dHI u t t u t dt
dt
(10)
(11)
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21
Comparison of QMD with NEGF
QMD ballistic
QMD nonlinear
Three-atom junction with cubic nonlinearity (FPU-). From JSW, Wang, Zeng, PRB 74, 033408 (2006) & JSW, Wang, Lü, Eur. Phys. J. B, 62, 381 (2008).
kL=1.56 kC=1.38, t=1.8 kR=1.44
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22
Equilibrium Simulation1D linear chain (red lines exact, open circles QMD) and nonlinear quartic onsite (crosses, QMD) of 128 atoms. From Eur. Phys. J. B, 62, 381 (2008).
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23
From Ballistic to Diffusive Transport
1D chain with quartic onsite nonlinearity (Φ4 model). The numbers indicate the length of the chains. From JSW, PRL 99, 160601 (2007).
NEGF, N=4 & 32
4
16
64
256
1024
4096
Classical, ħ 0
,S
J TL
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24
FPU- model
} relaxing
rate NEGF
It is not clear whether QMD over or under estimates the nonlinear effect. From Xu, JSW, Duan, Gu, & Li, arXiv:0807.3819.
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25
Electron Transport & Phonons
• For electrons in the tight-binding form interacting with phonons, the quantum Langevin equations are
†
( ') ( ') ' ,
( ') ( ') '
tk
k
t
ic Hc t t c t dt M u c
u Ku t t u t dt c Mc
† †
†
( ) ( ') ( '), ( ') ( ) ( '),
( ) ( ') ( '), ( ') ( ) ( ')
T
TT
t t i t t t t i t t
t t i t t t t i t t
(12)
(13)
(14)
(15)
Suzhou 08
26
Ballistic to Diffusive
Electronic conductance vs center junction size L. Electron-phonon interaction strength is m=0.1 eV. From Lü & JSW, arXiv:0803.0368.
Suzhou 08
27
Conclusion & Outlook
• NEGF is elegant and efficient for ballistic transport. More work need to be done for nonlinear interactions
• QMD for phonons is correct in the ballistic limit and high-temperature classical limit. Much large systems can be simulated (comparing to NEGF)
• Still need to demonstrate its usefulness for more realistic systems, e.g., nanotubes
Suzhou 08
28
Collaborators
• Baowen Li• Pawel Keblinski• Jian Wang• Jingtao Lü• Nan Zeng
• Lifa Zhang• Xiaoxi Ni• Eduardo Cuansing• Jinwu Jiang
• Saikong Chin• Chee Kwan Gan• Jinghua Lan• Yong Xu
Suzhou 08
29
Definitions of Phonon Green’s functions
( , ') ( ') [ ( ), ( ')] ,
( , ') ( ' ) [ ( ), ( ')] ,
( , ') ( ) ( ') ,
( , ') ( ') ( ) ,
( , ') ( ') ( , ') ( ' ) ( , ')
rjk j k
ajk j k
jk j k
jk k j
t
D t t i t t u t u t
D t t i t t u t u t
D t t i u t u t
D t t i u t u t
D t t t t D t t t t D t t
Suzhou 08
30
Relation among Green’s functions
†
/( )
,
,
, ( )
1( )
1B
r a r a
t t r t
t t r a r a
k T
D D D D D D
D D D D D D D
D D D D D f D D
fe
Suzhou 08
31
From self-energy to Green’s functions, to heat current
2
1,
( ) ,
Tr2
r
C r rn
r an
r aL L L
Gi K
G G G
dI G G