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First-principles thermoelectricity in nanostructures Phonon School at IWCE 2012
First-principles quantum transport modeling of thermoelectricity in nanowires
and single-molecule nanojunctions
First-principles quantum transport modeling of thermoelectricity in nanowires
and single-molecule nanojunctions
Branislav
K. NikolićDepartment of Physics and Astronomy, University of Delaware,
Newark, DE 19716, U.S.A.http://wiki.physics.udel.edu/qttg
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
References
http://arxiv.org/abs/1201.1665
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Thermoelectric Phenomena: Fundamentals and Applications
Fundamentals
Applications
electron–hole asymmetry at the Fermi energy generates thermoelectric phenomena
bulk
constrictions and interfaces
Kelvin-Onsager
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Thermoelectric Figure(s) of Merit ZT in the Linear-Response Regime
In the linear-response regime (i.e., close to equilibrium) one operates close to the small voltage V = -
S ΔT which exactly cancels the current induced by the small temperature bias ΔT
As ZT → ∞, the efficiency approaches the ideal Carnot value
η
C
= 1 -
T/(T+ ΔT )
thus, in the linear-response regime ΔT «
T typically investigated for bulk materials, the efficiency stays low η
C
= ΔT /T even if ZT can be made very large
Ultimate pragmatic goal:devices with ZT ≈
2–3 that are stable over a broad temperature range
with low parasitic losses
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Decades of Little Progress in Increasing ZT of Bulk Materials
“phonon glass-electron crystal”
Mahan-Sofo
mechanism
PbTe
doped with TlZT=1.5 at 773 K
Science 321, 554 (2008)
Nature Mater. 8, 83 (2009)
complex bulk materials
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
New Routes for ZT Optimization Brought by Low-Dimensional and Nanoscale
Devices
Electron-phonon coupling
Transmission peaks or nodes Coulomb interaction Nonlinear regime
ACS
Nan
o4,
531
4 (2
010)
PRB 78, 161406(R) (2008) PRB 82, 045412 (2010)
PR B 83, 195415 (2011)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Graphene
as a Building Block of Nanoscale
and Low-Dimensional Devices
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Large-Area Graphene
is not
Suitable for Thermoelectric Applications
Kim Lab, PRL 102, 096807 (2009)Balandin
Lab, New J. Phys. 11, 095012 (2009)
Shi Lab, ACS Nano
5, 321 (2011); Science 328, 213 (2010)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Zigzag and Chiral
GNRs
with Nanopore
Arrays as Potentially High-ZT Thermoelectrics
-1.0 -0.5 0.0 0.50
1
2
3
-1.0 -0.5 0.0 0.502468
1012
-1.0 -0.5 0.0 0.50.0
0.5
1.0
-1.0 -0.5 0.0 0.50
1
2
3 T=77 K T=300 K
T=77 K T=300 K
T=77 K T=300 K
T=77 K T=300 K
(a)
ZT
ZT
(d)(c)
(b)
ZT
Fermi Energy EF (eV)Fermi Energy EF (eV)
Fermi Energy EF (eV) Fermi Energy EF (eV)
Nikolic group, arXiv:1201.1665
20-ZGNR
(8,1)-CGNR
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
NEGF FundamentalsBasic NEGF quantities:
NEGFs
for steady-state transport:
NEGF-based current expression for two-terminal nanostructures:
density of available quantum states: how are those states occupied:
Meir-Wingreen
formulaLandauer-Büttiker-type
formula (phase-coherent transport where Coulomb interaction is treated at
the mean-field level)
NEGF (quantum) vs. Boltzmann (semiclassical) nonequilibrium
statistical mechanics:
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Electronic Thermopower, Conductance and Thermal Conductance via NEGF
Electronic transmission and its integrals:
Electronic conductance, thermopower, and thermal conductance:
Nikolić
group, J. Comp. Electronics
11, 78 (2012)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Third-Nearest-Neighbor π-Orbital Tight-Binding Hamiltonian For Graphene
Nan
oRe
sear
ch 1
, 361
(200
8)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Zigzag GNR: Fundamentals
20 15 10 5 0-3
-2
-1
0
1
2
3
Nz=20
Ener
gy
Conductance (2e2/h)
E=0.42
zigzag
-1 0 1-3
-2
-1
0
1
2
3
kx (1/a)
Areshkin
& White, Nano
Lett. 7, 3253 (2007)
1 2 3 4 5 6 7 8 9 100.0
0.4
0.8
1.2
1.6
-3 -2 -1 0 1 2 30.0
0.4
0.8
1.2
1.6
Loca
l DO
S at
EF=0
.01
Transverse Lattice Site
DO
S
Fermi Energy EF
Son, Cohen, and Louie, PRL 97, 216803 (2006)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
First-Principles Quantum Transport Modeling Charge, Heat and Spin Transport: NEGF+DFT
Trans
DFT
NEGF
NANODCALop
en s
ourc
eco
mm
erci
al
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
How to Apply NEGF-DFT to Devices Containing Thousands of Atoms
Main Obstacles:
Computational complexity O(N3) of matrix inversion to get the retarded GF and hard-to-converge real-axis integration of spiky NEGF expressions to get the density matrix
OLD
:NEW
:Construct the layer retarded Green functions needed for charge density using recursive algorithms
with O(N) complexity
Nikolić
group, PRB 81, 155450 (2010)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Gate Voltage Effect in All Carbon-Hydrogen GNRFET Composed of ~7000 Atoms
Zero Gate Voltage Gate Voltage -3 V
self
-con
sist
ent
non-
self
-con
sist
ent
Nikolić
group, PRB 81, 155450 (2010)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
NEGF-DFT For Multiterminal
DevicesSa
haet
al.,
J. C
hem
. Phy
s. 1
31, 1
6410
5 (2
009)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Phonon Thermal Conductance via NEGF Coupled to Minimal Force Constant 4NNN Model
Phonon conductance:
Empirical 4NNN force constant matrix:
PRB 78, 045410 (2008)
Why no phonon-phonon scattering? at 300 K [APL 98, 141919 (2011)]
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Phonon Thermal Conductance via NEGF Coupled to Brenner Empirical Potential or DFT
First-principles brute force method to obtain the force constant matrix (GPAW):we displace each atom I by QIα
in the direction α={x,y,z} to get the forces FIα,Jβ
on atom J
I in direction β for intra-atomic elements
impose momentum conservation
Brenner empirical interatomic
potential for hydrocarbon systems (GULP or GPAW):The Brenner EIPs
are short range, so they cannot accurately fit the graphene
dispersion over the entire BZ. However, the thermal transport depends more sensitively on the
accuracy of acoustic phonon frequencies near the zone center where the longitudinal-
and transverse-acoustic (LA and TA) velocities and the quadratic curvature of the out-of-
plane acoustic (ZA) branch are determined. Conversely, only weak thermal excitation of
the optical phonons and acoustic phonons near the BZ boundary occurs around room
temperature because of the large Debye temperature (~ 2000 K) of graphene.
PRB
81�
, 205
441
(201
0)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Which Method Should You Use: Minimal 4NNNFC vs. Brenner EIP vs. DFT
0 100 200 3000.0
0.5
1.0
1.5
2.0
2.5
3.0
8-ZGNR + nanopore
uniform 8-ZGNR
DFT (GPAW with DZP basis) Brenner EIP 4NNNFC
8-ZGNR
ph (n
W/K
)
Temperature(K)
ZGNR|18-annulene|ZGNR
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Coupled Electron-Phonon Transport via NEGFPhonon drag:
arises due to interchange of
momentum between acoustic phonons and
electronsElectron drag:
phonons are dragged by
electrons from low into high T region
Three-
and four-
phonon many-body
interactions
PRB 74, 125402 (2006)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Electron and Phonon Transport in ZGNRs
and CGNRs
with Nanopores
Nik
olic
gro
up, a
rXiv
:120
1.166
5
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Graphene
Nanoribbons: FabricationScience 319, 1229 (2008): Chemical
Derivation
Nature Nanotech. 3, 397 (2008): STM Nanolithography
Nature 458, 872 (2009): SWCNT Unzipping
Nature Nanotech. 5, 190 (2010): Graphene
nanomesh
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Band vs
Transport Gaps in GNRs
with Rough Edges
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Can We Control Formation of GNR Edges?
Dai Lab, Nature Phys. 7, 616 (2011)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Thermoelectricity in Single-Molecule Nanojunctions
(see mini-review arXiv:1111.0106)
Cuniberti
group, PRB 81, 235406 (2010)
Maj
umda
rLa
b, S
cien
ce 3
15, 2
568
(200
7)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Toward Metal-Free Molecular Electronics
Nature. Mater. 5, 63 (2006)
Science 311, 356 -
359 (2006)
Nikolić
group, PRL 100, 236803 (2010)
Control of the contact structure between an organic molecule and
a metal electrode (usually gold) is difficult because bonding to metal atoms, although potentially strong, is not strongly directional, leading to poor reproducibility
of most metal-molecule-metal junctions.
Our junctions with strong molecule-electrode
coupling evade problems due to the lack of
derivative discontinuity
in continuous local and semi-
local DFT approximations (LDA and GGA) as a major
source of error
in calculating the I-V
characteristics
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
ZGNR|molecule|ZGNR
Thermoelectric Devices Based on Evanescent Mode Transport
ZGNR|C10|ZGNR
ZGNR|18-annulene|ZGNR
-2 -1 0 1 20.0
0.5
1.0
-2 -1 0 1 2-1000
-500
0
500
1000
-2 -1 0 1 20.0
0.5
1.0
-2 0 2-1500
-1000
-500
0
500
1000
T=77 K T=300 K
E - EF (eV)
LUM
O
HO
MO
LUM
O(a)
E - EF (eV)
S (
V/K
)
Tel(E
)
(d)(c)
(b)
S (
V/K
)
T el(E
) HO
MO
T=77 K T=300 K
E - EF (eV) E-EF (eV)
-2 -1 0 1 20.0
0.2
0.4
0.6
0.8
0 100 200 3000
1
2
3
-2 -1 0 1 20.0
0.2
0.4
0.6
0.8
0 100 200 3000
1
2
3 T=77 K T=300 K
E-EF=0 E-EF=-0.02 eV E-EF=-0.04 eV
Temperature (K)
T=77 K T=300 K
(a)
E - EF (eV)
ZT
ZT(d)(c)
(b)
ZT
ZT
E-EF=0 E-EF=-0.02 eV E-EF=-0.04 eV
E - EF (eV) Temperature (K)
Nikolić
group, PRB
84, 041412(R) (2011) + J. Comp. Electronics 11, 78 (2012)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Three-Terminal Single-Molecule Nanojunction Thermoelectrics
Nikolić
group, PRB
84, 041412(R) (2011)
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Fabrication of Single-Molecule Nanojunctions with Graphene
Electrodes
van der
Zant
Lab, Nano
Lett. 11, 4607 (2011)
depositing molecules inside a few-layer graphene
nanogap
(of the size 1-2 nm) formed by
feedback controlled electroburning
Gatable
I-V characteristics at room temperature
First-principles thermoelectricity in nanostructuresPhonon School at IWCE 2012
Conclusions in Pictures Empirical versus first-principles
phonon transport modeling: Edge currents and nanopores
in GNR thermoelectrics:
Evanescent mode transport in single-
molecule nanojunctions
to optimize power factor:
0 100 200 3000.0
0.5
1.0
1.5
2.0
2.5
3.0
8-ZGNR + nanopore
uniform 8-ZGNR
DFT (GPAW with DZP basis) Brenner EIP 4NNNFC
8-ZGNR
ph (n
W/K
)
Temperature(K)
-2 -1 0 1 20.0
0.5
1.0
-2 -1 0 1 2-1000
-500
0
500
1000
-2 -1 0 1 20.0
0.5
1.0
-2 0 2-1500
-1000
-500
0
500
1000
T=77 K T=300 K
E - EF (eV)
LUM
O
HO
MO
LUM
O(a)
E - EF (eV)
S (
V/K
)
Tel(E
)
(d)(c)
(b)
S (
V/K
)
T el(E
) HO
MO
T=77 K T=300 K
E - EF (eV) E-EF (eV)
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