Nonequilibrium
Nonequilibrium Green
Green Function
Function Algorithms
Algorithms
for
for Time-Dependent
Time-Dependent and
and Steady-State
Steady-State
Problems
Problems in
in Spintronics
Spintronics and
and Nanoelectronics
Nanoelectronics
Branislav K. Nikolić
Department of Physics and Astronomy, University of Delaware,
Newark, DE 19716, U.S.A.
https://wiki.physics.udel.edu/qttg
Physics at the Falls, Buffalo 2013
NEGF algorithms
Collaborators
Experiment:
Prof. John Q. Xiao
Dr. Takahiro Moriyama
Physics at the Falls, Buffalo 2013
Theory & Computation:
Farzad Mahfouzi
Dr. Son-Hsien Chen
Dr. Kamal K. Saha
Dr. Denis A. Areshkin
Prof. Ching-Ray Chang
Prof. Naoto Nagaosa
NEGF algorithms
What is the Key Challenge for
Nanoelectronics?
key parameters: ON/OFF ratio,
transconductance, mobility,
subthreshold swing
Based on fundamental considerations alone:
Pentium 2000, 50W/cm2; ~2025, 40MW/cm2
Physics at the Falls, Buffalo 2013
NEGF algorithms
Overcoming the Fundamental Obstacles
Posed by Energy Dissipation
Energy Dissipation ~ QΔV
Q is limited by noise (“drivability”)
ΔV is limited by Boltzmann (~kBT/e ln10 = 60 mV/dec)
SHORT TERM: squeeze mileage out of CMOS paradigm (e.g., materials allowing
faster electrons)
LONG TERM: “reinvent the transistor”
Reduce Q:
all spin logic
exciton condensate
in bilayer graphene
Mottronics
Physics at the Falls, Buffalo 2013
Reduce ∆V:
Quantum interference
Alternate state variables
(such as pseudospin or valley in
single and bilayer graphene)
Operate near a phase transition
point or out of equilibrium (such
as TunnelFETs, NEMFETs, …)
NEGF algorithms
Proposals for Current Switching Using
Quantum Phase in Graphene-Based Devices
Quantum-interference-controlled
nanotransistors [PRL 105, 236803 (2010)]
Interlayer current in AGNR-AGNR crossbar (or
twisted graphene bilayer) [unpublished]
“Air gap”
third gate
electrode
Physics at the Falls, Buffalo 2013
NEGF algorithms
Nanofabrication Efforts Toward
Multiterminal Graphene-Based Nanoelectronics
van der Zant Lab, Nano Lett. 11, 4607 (2011)
Dai Lab, Nano Res. 3, 387 (2010)
depositing
molecules inside a
few-layer
graphene nanogap
(of the size 1-2
nm) formed by
feedback
controlled
electroburning
Physics at the Falls, Buffalo 2013
NEGF algorithms
Steady-State Quantum Transport via 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
for phase-coherent transport
when inelastic (e-ph, e-e, e-spin)
processes are absent
Physics at the Falls, Buffalo 2013
NEGF algorithms
Rocha, Ph.D. thesis
NEGF
Physics at the Falls, Buffalo 2013
Trans
commercial
DFT
open source
First-Principles Quantum Transport Modeling
via NEGF+DFT
NEGF algorithms
Challenges for NEGF-DFT
in Applications to Realistic Device Modeling
Often the system to be
investigated is large
(in excess of 10,000 atoms)
The electrostatic
potential may be
determined by a
dynamical environment
(for instance a solvent)
Nano Lett. 12, 50 (2012)
Many-body effects
may dominate the
current response
Physics at the Falls, Buffalo 2013
NEGF algorithms
NEGF-DFT Applied to >10,000 Atoms Systems:
Recursive Algorithms and Adaptive Integration
+iη
NEW:
OLD:
Z
Z
1 +∞
1 +∞
r
ρ̂ = −
dE ImĜ (E)f (E−eVR )+
dE Ĝr (E)·Γ̂L (E−eVL )·Ĝa (E) [f (E − eVL ) − f (E − eVR )]
π −∞
2π −∞
Construct the layer
retarded Green functions
needed for charge density
using recursive algorithms
with O(N) complexity
or adaptive integration of spikes
PRB 81, 155450 (2010)
Physics at the Falls, Buffalo 2013
NEGF algorithms
Performing Adaptive Integration of NEGF
Expressions Along the Real Energy Axis
Physics at the Falls, Buffalo 2013
NEGF algorithms
Gate Voltage Effect in All Carbon-Hydrogen
GNRFET Composed of ~7000 Atoms

non-self-consistent
Gate Voltage -3 V
self-consistent
Zero Gate Voltage
PRB 81, 155450 (2010)
Physics at the Falls, Buffalo 2013
NEGF algorithms
Nonequilibrium Phase Transition in
Magnetically Ordered ZGNRs at Low T
PRB 79, 205430 (2009)
GPAW developers, PRB 85, 155140 (2012)
Physics at the Falls, Buffalo 2013
NEGF algorithms
J. Chem. Phys. 131, 164105 (2009)
NEGF-DFT For Multiterminal Devices
Physics at the Falls, Buffalo 2013
NEGF algorithms
PRL 100, 236803 (2010)
Source Drain Transmission in the Presence of
the Third Electrode (“Büttiker voltage probe”)
-1
10
I2(
0.2
0.0
-2
10
-0.2
-1
0
Bias Voltage Vds(V)
0.0
-3
10
Physics at the Falls, Buffalo 2013
1
Current I3(A
Current I2(A
0.4
-1.0
-0.5
0.0
0.5
Gate Voltage Vgs(V)
1.0
NEGF algorithms
Interlayer Current in AGNR-AGNR Crossbar
Physics at the Falls, Buffalo 2013
Interlayer Current (A)
Nikolić group, unpublished
Lake group, PRB 86, 045418 (2012)
2
-V/2
0.2
V/2
-V/2
0.02
0.0
V/2
0.2
0.4
0.6
0.8
Bias Voltage (V)
NEGF algorithms
What is Spin Pumping by
Precessing Magnetization?
Science 283, 1905 (1999)
Experimental manifestations: Enhanced Gilbert
damping in F|N Multilayers
cryogenic temperatures and undesired
bias voltage due to stray capacitances
Physics at the Falls, Buffalo 2013
f : microwave frequency
γ : gyromagnetic ratio
NEGF algorithms
Direct Electrical Detection of
Enhanced Pumping Voltage in MTJs
F
|
I
|
F
F |I |N
Xiao Lab at CSB, PRL 100,
067602 (2008)
Physics at the Falls, Buffalo 2013
NEGF algorithms
Exact Rotating Frame Approach to
Spin Pumping in the Absence of Spin Flips
ω
Δ
PRB 79, 054424 (2009)

Ub
...

t
Lead
Physics at the Falls, Buffalo 2013
t
t
Sample
t
t
...
Lead
NEGF algorithms
Spin and Charge Pumping
From a Single Precessing Spin
ω
PRB 79, 054424 (2009)
Δ

Ub

...
t
t
Lead
t
t
Sample
...
t
Lead
Spin current pumping from
a single precesing spin
Charge current pumping from
a single precessing spin
-3
-8
10
Spin Current (e/h)
4
Charge Current (e/h)
10
Sz,left
Jrr'
3
2
1
0
0
Sz,right
Jrr'
1
2
3
4
Barrier Height t
Physics at the Falls, Buffalo 2013
5
6
1
right
Jrr'
0
left
Jrr'
-1
0
1
2
3
4
Barrier Height t
5
6
NEGF algorithms
Interfacial Rashba SO Coupling Effects
in MTJs: STT and TAMR
PRB 85, 054406 (2012)
Physics at the Falls, Buffalo 2013
NEGF algorithms
Time-Dependent Quantum Transport via NEGF:
Fundamentals
Basic NEGF quantities:
density of available quantum states:
how are those states occupied:
Equations of motion:
Time-dependent spin-resolved charge current:
Physics at the Falls, Buffalo 2013
NEGF algorithms
Solving NEGF Equations for
Harmonically Time-Dependent Problems
Exact rotating frame approach valid Continued fraction solution to Floquet-NEGF valid in
in the absence of spin flips:
the presence of spin flips but does not conserve current:
N|F|I|F|N
1.2
0.3
0.8
0.2
F|I|F
0.4
0.1
0.0
0
30
60
90
120
150
Precession Cone Angle  deg)
Physics at the Falls, Buffalo 2013
0.0
180
Exact rotating frame solution vs.
perturbative (V→0) continued fractions
n=±1 tested in the absence of spin flips
DC Voltage Vpump (Ñ/e)
DC Voltage Vpump (Ñ/e)
Finite-size vs. infinite clean F layers → exact approach explains discrepancy
between PRB 79, 054424 (2009) and PRB 77, 180407(R) (2008)
-3
x10
0.4
dF=50|dI=5|dF=50
0.00
-0.05
-0.10
-0.15
0
30
60
90
120
150
Precession Cone Angle (deg)
180
NEGF algorithms
Effect of Interfacial Rashba SO Coupling
on Pumping in semi-MTJs
Charge conserving solution to Floquet NEGF
-2
-2
0.8 x10
(a)

x10
ë
1.0
Vpump(Ñ/e)
Vpump(Ñ/e)
0.6
0.4
0.2
5
RSO=0.5
0
-5
-10
0
F|I|N
30
60
Nph=1
Rashba SOC RSO/
-2
(c)
x10
Nph=1
0.6
Nph
0.4
(Vpump-Vpump )/Vpump (%)
10
Vpump(Ñ/e)
B
PR
85
06
4
54
0
,
2)
1
0
(2
0.2
90 120 150 180
Precession Cone Angle deg)
Physics at the Falls, Buffalo 2013
0.8
0.6

ë
single Rashba SOC interface
two Rashba SOC interfaces
0.4
0.2
0.0
F|I|N
0.0
0.0
(b)
0.0
F|I|F
0.2
0.4
0.6
Rashba SOC RSO/
10 (d)
8
RSO=0.5
6
4

ë
2
0
0
F|I|N
2
4
6
8
10
Number of Photons
NEGF algorithms
What is Spin-Transfer Torque?
Physics at the Falls, Buffalo 2013
NEGF algorithms
Nonequilibrium Born-Oppenheimer-Type
Formula for STT in the Presence of SOC
NEGF approach to STT in conventional MTJs with
zero SOC [Butler group, PRL 97, 237205 (2006)]:
NEGF approach to STT in the presence of SOC
[PRL 109, 166602 (2012)]:
Physics at the Falls, Buffalo 2013
NEBO approach to current-induced forces
[von Oppen group, PRL 107, 036804 (2011)]:
NEGF algorithms
Gauge-Invariant Nonequilibrium Density Matrix
Properly Removes Equilibrium Contributions
Density matrix often split into “equilibrium” + “nonequilibrium” contributions
for purely computational purposes (like in NEGF+DFT):
Z
Z
1 +∞
1 +∞
r
ρ̂ = −
dE ImĜ (E)f (E−eVR )+
dE Ĝr (E)·Γ̂L (E−eVL )·Ĝa (E) [f (E − eVL ) − f (E − eVR )]
π −∞
2π −∞
The proper gauge-invariant nonequilibrium density matrix is defined by:
ρ̂neq
·
¸
Z
1 +∞
r
= ρ̂ − ρ̂eq = ρ̂ +
dE Im Ĝ (E) f (E)
π −∞
First two terms below remove any equilibrium contribution to physical quantity
whose non-zero value is compatible with time-reversal invariance (arXiv:1305.3810):
EF
Z
⎡
⎛
⎞
⎤
∂ Σ̂L
∂ Σ̂R ⎠ r ⎦
eVR
1
Ĝ0 f (E)
dE Im ⎣Ĝr0 ⎝eU − eVL
Im [Gr0 (EF )] −
− eVR
π
π −∞
∂E
∂E
eVb r
Ĝ0 (EF ) · Γ̂L (EF ) · Ĝa0 (EF ),
+
2π
ρ̂neq = −
Physics at the Falls, Buffalo 2013
NEGF algorithms
STT in TI-based Vertical Heterostructures
-3
Torque (eV b ias/Ñ )
T7
T^
0.0
-0.2
-0.4
30
60
Torque (eVbias /Ñ)
-5
0.5
Pz
out
0.4
0.3
in
0.2
N
x10
0.2
T7
2
0.0
T^
0
-0.4
0
N
-2
30
60
90 120 150 180
1.2
T7
0.8
T^
0.4
0.0
0
30 60
90 120 150 180
-3
-3
0.5
x10
F|I|F
N|I|F
0.4
0.3
(d)
0.2
0.1
0.0
0
N|TI|F
30 60
90 120 150 180
Angle  deg)
0.1
0.0
0
x10
(c) N|I|F
-0.2
out
TI
0.4
90 120 150 180
2
0
Conductance (e /h Ñ)
0.4
0.2
x10
1.6 (b) F |I|F
PRL 109, 166602 (2012)
Torque (eVb ias /Ñ)
-3
x10
0.6 (a) N|TI|F
2
4
6
8
Thickness of TI Layer dTI
Physics at the Falls, Buffalo 2013
10
PRB 71, 195328 (2005)
NEGF algorithms
PRB 77, 014440 (2008)
Inelastic Electron-Magnon Scattering in MTJs
60
Hartree
TMR (%)
Fock
45
40
35
-0.2
40
Left Current
Right Current
Physics at the Falls, Buffalo 2013
I/I 0
20
10
0
1
2
3
4
5
6
7
Self-Consistent Loop Steps
d2I/dV2 (arb. u.)
5
Ralph Lab:
TMR of FeCoB/MgO/FeCoB
increases by 50% upon cooling to 10 K
in-plane and perpendicular torque
both decrease substantially
bias-dependent structure in the
asymmetry of the in-plane torque
Interacting
50
Polarization bubble
(nonequilibrium backaction)
30
Non-Interacting
55
10-2
-0.1
0.0
0.1
Bias Voltage (V)
0.2
Non-Interacting
Interacting
4
3
2
1
0
-1
-0.2
parallel magnetizations
-0.1
0.0
0.1
Bias Voltage (V)
0.2
NEGF algorithms
Summary and Open Questions in Pictures
NEGF+DFT algorithms for large number of
atoms and multiterminal devices
Charge conserving solution to Floquet
NEGF [PRB 85, 054406 (2012) ]
Open questions:
magnon-assisted
spin torque
Hartree
Spin-transfer torque driven by interfacial
SOC + gauge-invariant density matrix
[PRL 109, 166602 (2012) and
arXiv:1305.3810]
Fock
Polarization bubble
(nonequilibrium backaction)
Physics at the Falls, Buffalo 2013
NEGF algorithms