First-principles
First-principles quantum
quantum transport
transport
modeling
modeling of
of thermoelectricity
thermoelectricity in
in nanowires
nanowires
and
and single-molecule
single-molecule nanojunctions
nanojunctions
Branislav K. Nikolić
Department of Physics and Astronomy, University of Delaware,
Newark, DE 19716, U.S.A.
http://wiki.physics.udel.edu/qttg
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
References
http://arxiv.org/abs/1201.1665
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Thermoelectric Phenomena:
Fundamentals and Applications
Fundamentals
bulk
electron–hole asymmetry at
the Fermi energy generates
thermoelectric phenomena
constrictions and interfaces
Applications
Kelvin-Onsager
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
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
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Decades of Little Progress
in Increasing ZT of Bulk Materials
Nature Mater. 8, 83 (2009)
complex bulk materials
“phonon glass-electron crystal”
Mahan-Sofo mechanism
PbTe doped with Tl
ZT=1.5 at 773 K
Science 321, 554 (2008)
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
New Routes for ZT Optimization Brought by
Low-Dimensional and Nanoscale Devices
Phonon School at IWCE 2012
Coulomb interaction
Nonlinear regime
PRB 78, 161406(R) (2008)
PRB 82, 045412 (2010)
Electron-phonon coupling
PR B 83, 195415 (2011)
ACS Nano 4, 5314 (2010)
Transmission peaks or nodes
First-principles thermoelectricity in nanostructures
Graphene as a Building Block of
Nanoscale and Low-Dimensional Devices
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Large-Area Graphene is not Suitable for
Thermoelectric Applications
Balandin Lab, New J. Phys. 11, 095012 (2009)
Kim Lab, PRL 102, 096807 (2009)
Shi Lab, ACS Nano 5, 321 (2011); Science 328, 213 (2010)
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Zigzag and Chiral GNRs with Nanopore Arrays
as Potentially High-ZT Thermoelectrics
20-ZGNR
12
(a)
0.5
(b)
T=77 K
T=300 K
8
ZT
ZT
10
T=77 K
T=300 K
1.0
6
4
2
0.0
-1.0
(8,1)-CGNR
-0.5
0.0
0
-1.0
0.5
Fermi Energy EF (eV)
3
(c)
T=77 K
T=300 K
2
0.0
0.5
(d)
T=77 K
T=300 K
2

ZT
3
-0.5
Fermi Energy EF (eV)
1
1
0
-1.0
-0.5
0.0
0.5
Fermi Energy EF (eV)
0
-1.0
-0.5
0.0
0.5
Fermi Energy EF (eV)
Nikolic group, arXiv:1201.1665
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
NEGF Fundamentals
Basic NEGF quantities:
density of available quantum states:
how are those states occupied:
NEGFs for steady-state transport:
NEGF (quantum) vs. Boltzmann (semiclassical) nonequilibrium statistical mechanics:
NEGF-based current expression for two-terminal nanostructures:
Meir-Wingreen formula
Landauer-Büttiker-type formula
(phase-coherent transport where
Coulomb interaction is treated at
the mean-field level)
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Electronic Thermopower, Conductance and
Thermal Conductance via NEGF
Electronic conductance, thermopower, and thermal conductance:
Phonon School at IWCE 2012
Nikolić group, J. Comp. Electronics 11, 78 (2012)
Electronic transmission and its integrals:
First-principles thermoelectricity in nanostructures
Nano Research 1, 361 (2008)
Third-Nearest-Neighbor π-Orbital
Tight-Binding Hamiltonian For Graphene
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Zigzag GNR: Fundamentals
Areshkin & White,
Nano Lett. 7, 3253 (2007)
3
3
Energy
1
0
1
E=0.42
0
1.6
1.2
0.8
-1
-1
-2
-2
Nz=20
-3
20 15 10
5
2
0 -1
Conductance (2e /h)
0
kx (1/a)
1
-3
1.6
1.2
DOS
2
zigzag
Local DOS at EF=0.01
2
0.8
0.4
0.4
0.0
-3
0.0
1
2
-2
-1
0
1
2
3
Fermi Energy EF
3
4
5
6
7
8
9
Transverse Lattice Site
10
Son, Cohen, and Louie,
PRL 97, 216803 (2006)
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First-principles thermoelectricity in nanostructures
NEGF
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commercial
DFT
open source
First-Principles Quantum Transport Modeling
Charge, Heat and Spin Transport: NEGF+DFT
Trans
NANODCAL
First-principles thermoelectricity in nanostructures
How to Apply NEGF-DFT to Devices
Containing Thousands of Atoms
OLD:
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
Construct the layer
retarded Green functions
needed for charge density
using recursive algorithms
with O(N) complexity
NEW:
Nikolić group, PRB 81, 155450 (2010)
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Gate Voltage Effect in All Carbon-Hydrogen
GNRFET Composed of ~7000 Atoms

non-self-consistent
Gate Voltage -3 V
self-consistent
Zero Gate Voltage
Nikolić group, PRB 81, 155450 (2010)
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Saha et al., J. Chem. Phys. 131, 164105 (2009)
NEGF-DFT For Multiterminal Devices
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First-principles thermoelectricity in nanostructures
Phonon Thermal Conductance via NEGF Coupled
to Minimal Force Constant 4NNN Model
Phonon conductance:
Why no phonon-phonon scattering?
at 300 K
[APL 98, 141919 (2011)]
Empirical 4NNN force constant matrix:
PRB 78, 045410 (2008)
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First-principles thermoelectricity in nanostructures
Phonon Thermal Conductance via NEGF Coupled
to Brenner Empirical Potential or DFT
PRB 81, 205441 (2010)
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-ofplane 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.
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
Phonon School at IWCE 2012
for intra-atomic elements
impose momentum conservation
First-principles thermoelectricity in nanostructures
Which Method Should You Use:
Minimal 4NNNFC vs. Brenner EIP vs. DFT
ZGNR|18-annulene|ZGNR
8-ZGNR
3.0
ph (nW/K)
2.5
2.0
DFT (GPAW with DZP basis)
Brenner EIP
4NNNFC
uniform 8-ZGNR
1.5
1.0
0.5
0.0
0
8-ZGNR + nanopore
100
200
300
Temperature(K)
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Coupled Electron-Phonon Transport via NEGF
Phonon drag:
arises due to
interchange of
momentum between
acoustic phonons and
electrons
Electron drag:
phonons are
dragged by
electrons from low
into high T region
Three- and fourphonon many-body
interactions
PRB 74, 125402 (2006)
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Nikolic group, arXiv:1201.1665
Electron and Phonon Transport
in ZGNRs and CGNRs with Nanopores
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Graphene Nanoribbons: Fabrication
Science 319, 1229 (2008): Chemical Derivation
Nature 458, 872 (2009): SWCNT Unzipping
Nature Nanotech. 5, 190 (2010): Graphene nanomesh
Nature Nanotech. 3, 397 (2008): STM Nanolithography
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Band vs Transport Gaps
in GNRs with Rough Edges
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Can We Control Formation of GNR Edges?
Dai Lab,
Nature Phys. 7, 616 (2011)
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
Thermoelectricity in Single-Molecule
Nanojunctions (see mini-review arXiv:1111.0106)
Majumdar Lab, Science 315, 2568 (2007)
Cuniberti group, PRB 81, 235406 (2010)
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First-principles thermoelectricity in nanostructures
Toward Metal-Free Molecular Electronics
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.
Science 311, 356 - 359 (2006)
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Our junctions with strong
molecule-electrode
coupling evade problems
due to the lack of
derivative discontinuity in
continuous local and semilocal DFT approximations
(LDA and GGA) as a major
source of error in
calculating the I-V
characteristics
Nikolić group, PRL 100, 236803 (2010)
Nature. Mater. 5, 63 (2006)
First-principles thermoelectricity in nanostructures
1000
S (V/K)
(a)
Tel(E)
1.0
LUMO
ZGNR|18-annulene|ZGNR
HOMO
ZGNR|molecule|ZGNR Thermoelectric Devices
Based on Evanescent Mode Transport
0.5
(b)
T=77 K
T=300 K
500
0
-500
0.0
-2
-1
0
1
2
1000
(c)
S (V/K)
HOMO
Tel(E)
1.0
LUMO
E - EF (eV)
-1000
-2
0.5
-1
0
1
2
E - EF (eV)
(d)
500
0
-500
-1000
T=77 K
T=300 K
2
-1500
0.0
-2
Nikolić group, PRB 84, 041412(R) (2011)
+ J. Comp. Electronics 11, 78 (2012)
0.8
-1
0
(a)
2
-2
3
T=77 K
T=300 K
0
E-EF (eV)
0.4
(b)
E-EF=0
2
E-EF=-0.02 eV
ZT
0.6
ZT
1
E - EF (eV)
E-EF=-0.04 eV
1
0.2
0.0
-2
-1
0
1
0
0
2
E - EF (eV)
0.8
(c)
200
300
3
T=77 K
T=300 K
0.6
(d)
E-EF=0
2
E-EF=-0.02 eV
ZT
ZT
100
Temperature (K)
0.4
E-EF=-0.04 eV
1
0.2
ZGNR|C10|ZGNR
Phonon School at IWCE 2012
0.0
-2
-1
0
1
E - EF (eV)
2
0
0
100
200
300
Temperature (K)
First-principles thermoelectricity in nanostructures
Three-Terminal Single-Molecule Nanojunction
Thermoelectrics
Nikolić group, PRB 84, 041412(R) (2011)
Phonon School at IWCE 2012
First-principles thermoelectricity in nanostructures
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
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First-principles thermoelectricity in nanostructures
Conclusions in Pictures
Empirical versus first-principles
phonon transport modeling:
8-ZGNR
3.0
ph (nW/K)
2.5
2.0
DFT (GPAW with DZP basis)
Brenner EIP
4NNNFC
Evanescent mode
transport in singlemolecule nanojunctions to
optimize power factor:
Edge currents and
nanopores in GNR
thermoelectrics:
uniform 8-ZGNR
1.5
1.0
0.5
0.0
0
8-ZGNR + nanopore
100
200
300
1000
S (V/K)
LUMO
(a)
Tel(E)
1.0
HOMO
Temperature(K)
0.5
(b)
T=77 K
T=300 K
500
0
-500
0.0
-2
-1
0
1
2
-1000
-2
1000
(c)
S (V/K)
LUMO
Tel(E)
1.0
HOMO
E - EF (eV)
0.5
-1
0
1
2
E - EF (eV)
(d)
500
0
-500
-1000
-1500
0.0
-2
Phonon School at IWCE 2012
-1
0
E - EF (eV)
1
2
-2
0
E-EF (eV)
T=77 K
T=300 K
2
First-principles thermoelectricity in nanostructures