© Copyright National University of Singapore. All Rights Reserved. ENHANCING THERMOELECTRIC EFFICIENCY FOR NANOSTRUCTURES AND QUANTUM DOTS Jian-Sheng Wang Department of Physics, National University of Singapore
Jan 12, 2016
© Copyright National University of Singapore. All Rights Reserved.
ENHANCING THERMOELECTRIC EFFICIENCY FOR NANOSTRUCTURES AND QUANTUM DOTS
Jian-Sheng WangDepartment of Physics, National University of Singapore
© Copyright National University of Singapore. All Rights Reserved. 2
OUTLINE
• Seebeck effect & thermoelectric efficiency• Disordered graphene/graphane• Quantum master equation: electron-
phonon interaction to thermoelectric efficiency in quantum dots
• Enhancing thermoelectric efficiency by time-dependent driven
• Conclusion
1ST CONFERENCE ON CONDENSED MATTER PHYSICS, 15-17 JULY 2015
© Copyright National University of Singapore. All Rights Reserved. 3
SEEBECK EFFECTSeebeck coefficient: how much voltage difference can one generate per temperature difference? S = dV/dT
How efficient is it comparing to Carnot engine, W/Q = 1 – Tc/Th?
Ans: Determined by a material parameter called
ZT = S2T/
: electric conductivity, : thermal conductivity, T: absolute temperature
From “Physics Today,” June 2014, p.14
© Copyright National University of Singapore. All Rights Reserved. 4
ENHANCING ZT BY DISORDERING
Ni, Liang, Wang & Li, Appl. Phys. Lett. 95, 192114 (2009).
ZT at 300K for graphene/graphane calculated using ballistic NEGF formulation for the armchair ribbons with fraction of H-bond disorder, with DFT structure determination.
graphane graphene
© Copyright National University of Singapore. All Rights Reserved. 5
QUANTUM DOT ELECTRON-PHONON INTERACTION
Model:
© Copyright National University of Singapore. All Rights Reserved. 6
QUANTUM MASTER EQUATION APPROACH
• Advantage of NEGF: any strength of system-bath coupling V; disadvantage: difficult to deal with nonlinear systems.
• QME: advantage - center can be any form of Hamiltonian, in particular, nonlinear systems; disadvantage: weak system-bath coupling, small system.
• Can we improve?
WANG, ET AL, FRONT. PHYS. 9, 673 (2014); THINGNA, ET AL, J. CHEM. PHYS. (2014)
© Copyright National University of Singapore. All Rights Reserved.
DYSON EXPANSIONS
7
0 0 0
0
0
( )
0
( )
0
2 42 4 6
( ) Tr ( , ) ( ) ( , )
Tr ( ) ,
( ) Tr ( ) [ ( ), ( )] ,
( )2! 4!
| |,
C
C
H B
V d
B c
V d
H B c
T T T
nm
O t S t O t S t
iT O t e
dO t T O t O t V t e
dt
X X V X V O
X n m
0 1Tr ..B cT d
© Copyright National University of Singapore. All Rights Reserved.
DIVERGENCE
8
1
+
1 1,
is diagonal, 0
if | |
d d
T n T n
d
m nmn
X V X V Vn i n
E EX m n
© Copyright National University of Singapore. All Rights Reserved.
UNIQUE ONE-TO-ONE MAP, 0↔; ORDERED CUMULANTS
9
2
2
2
2 4 42 4 2
0 2
42 3
4
4 42 3 6
2! 4! 2!
[ , ] [ , ]3!
[ , ]2!
( )3! 2!
T
T
T
T T T
X V
T T
T m nmnX V
H X V
T LCL C
X V X V X V
di X V V X V V
dt
E EX V V
I VV VV VV
V p V u
© Copyright National University of Singapore. All Rights Reserved.
ORDER-BY-ORDER SOLUTION
10
(0)
(0)
( 2) ( 2) (0)
2(0)
(0) 2 (2) 4
(0)
(2)
3
( )
0
1[ , ]
...
[ , ] 0
1[ , ] [ , ] [ , ]
3!1
[ , ]2!
d
d f
T
T T Td f
f
Tf f
Td
T T Td d d
Td X V
O
X X X
X V Vi
X V V
X V V X V V X V V
X V V
|1 1 | 0 0 ...
0 | 2 2 | 0
0 0 | 3 3 |
... ...
0 |1 2 | |1 3 | ...
| 2 1| 0 | 2 3 |
| 3 1| | 3 2 | 0
... ... ...
d
f
X
X
© Copyright National University of Singapore. All Rights Reserved. 11
TIME-DEPENDENT DRIVEN
• Nonequilibrium Green’s function (NEGF), we use Jauho, Wingreen, Meir (PRB 1994) theory
• Master equation case - drive change the eigenvalues but not eigenvectors, easily generalize
© Copyright National University of Singapore. All Rights Reserved. 12
ELECTRON CURRENT
ZHOU, ET AL, PHYS REV B 90, 045410 (2015).
Electron current in units of . Background temperature , lead temperature , , . . Chemical potentials set to 0 for all leads.
Notice the current changes direction with gate voltage.
© Copyright National University of Singapore. All Rights Reserved. 13
REDUCTION OF ZT UNDER ELECTRON-PHONON INTERACTION
(a) ZT vs ep interaction strength , for different gate voltage .
(b) back-action (deviation from equilibrium).
© Copyright National University of Singapore. All Rights Reserved. 14
WHY DRIVE THE SYSTEM? CAN HAVE HIGHER EFFICIENCY
• Far from thermal equilibrium
• Break down of Onsager relation (due to break down of time translational invariance)
© Copyright National University of Singapore. All Rights Reserved. 15
HOW TO QUANTIFY EFFICIENCY IN TIME-DEPENDENT SITUATION?
(𝐼 𝑒𝛼(𝑡)𝐼 h𝛼(𝑡))=(𝐿1 1 𝐿12 𝐿𝑒
𝐷[ .]𝐿21 𝐿22 𝐿𝑒
𝐷[ .])(Δ𝛼𝜇𝑒Δ𝛼𝑇𝑇
𝐹 (𝑡′)) ,𝛼=𝐿 ,𝑅
T or
Current I
Do this analysis for each fixed t.
© Copyright National University of Singapore. All Rights Reserved. 16
MEASURING EFFICIENCY
SEE, EG., H J GOLDSMID, “INTRO TO THERMOELECTRICITY.”
𝜂 (𝑡 )=𝐼𝑒
2 𝑅𝐿
det (𝐿) 𝑅𝑀 Δ𝑇𝑇
+𝐿21𝑅𝑀 𝐼𝑒−𝐼𝑒
2 𝑅𝑀
2
Optimal efficiency in steady state by maximizing , one obtain
Work done to load
Entropy flow
Peltier heat Joule heat flow back
© Copyright National University of Singapore. All Rights Reserved. 17
EFFICIENCY ENHANCEMENT BY DRIVEN
ZHOU, ET AL, ARXIV:1505.06132
Ballistic quantum dot:
Under step control of the drive . (a) efficiency, (b) Onsager relation broken-down, (c) entropy flow.
© Copyright National University of Singapore. All Rights Reserved. 18
WITH EP INTERACTION, ARXIV:1505.06132
Wave form of drive.
Entropy transport
Breakdown of Onsager relation
Displacement current
Normalized efficiency
© Copyright National University of Singapore. All Rights Reserved. 19
SUMMARY
• Structure change does bring in higher ZT, e.g., disorder, or go down to 0-dimension (quantum dot)
• Electron-phonon interaction reduces ZT
• Dynamic drive (forcing) improves ZT, up to a factor of 4
© Copyright National University of Singapore. All Rights Reserved. 20
ACKNOWLEDGEMENTS
• Students: Hangbo Zhou, Juzar Thingna, Xiaoxi Ni
• Collaborators: Peter Hänggi, Baowen Li, Albert Liang GC
© Copyright National University of Singapore. All Rights Reserved.
THANK YOU