Transport theory and simulations in hybrid structures:
the need for a bottom-up approach in new semiconductor spintronic systems
JAIRO SINOVA
Texas A&M University
Institute of Physics ASCR
Institute of Physics ASCR
Tomas Jungwirth, Vít Novák, Karel Vyborny, et al
Hitachi Cambridge
Joerg Wünderlich, A. Irvine, et al
Hamburg University
Jan Jacob, Bodo Krause-Kyora, et al
International Symposium High Performance
Computing in Nano-Spintronics
Hamburg, November 30th, 2011
Research fueled by:
Transport theory and simulations in hybrid structures:
the need for a bottom-up approach in new spintronic systems
I. Introduction: using the dual personality of the electron
•Internal coupling of charge and spin: origin and present use
II.Spintronic devices and the need to understand them microscopically
III. Spin injection Hall effect FET: a new paradigm in exploiting SO coupling
•Spin based FET: old and new paradigm in charge-spin transport
II. Spin filters in InAs nano-wires
III. A bottom to top approach: from mesoscopic to microscopic
• Why is it important to approach the problem from the mesoscopic regime
V. Theoretical tools of the mesoscopic world:
• Landauer-Büttiker (coherent regime)
• Non-Equilibrium Green’s Function approach: quantum Boltzmann Eq.
• Beyond coherent transport: the basic master equation of the nonequilibrium density matrix
VI. Computational Challenges of NEGF approach
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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The electron:
the key character with dual personalities
SPIN 1/2
Makes the electron
antisocial: a fermion
CHARGE
Easy to manipulate:
Coulomb interaction
quantum mechanics
E=p2/2m
E→ iħ d/dt
p→ -iħ d/dr
+
special relativity
E2/c2=p2+m2c2
(E=mc2 for p=0)
& spin
= particles/antiparticles
Dirac equation
“Classical” external manipulation of charge & spin
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Using charge and spin in information technology
Using charge to create a field effect transistor:
work horse of information processing
Vg >0
S
gate
insulator
semiconductor
total wf antisymmetric = orbital wf
antisymmetric × spin wf symmetric (aligned)
D
substrate
HIGH tunablity of electronic transport
properties the key to FET success in
processing technology
What about the internal communication
between charge & spin? (spintronics)
Sinova
Using spin: Pauli exclusion principle and
Coulomb repulsion →ferromagnetism
work horse of information storage
• Robust (can be as strong as bonding in solids)
• Strong coupling to magnetic field
(weak fields = anisotropy fields needed
only to reorient macroscopic moment)
e-
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Internal communication between spin and charge:spinorbit coupling interaction
(one of the few echoes of relativistic physics in the solid state)
e-
Classical explanation (in reality it arises from a second order expansion of
Dirac equation around the non-relativistic limit)
• “Impurity” potential
•
V(r)
Produces
an electric field
In the rest frame of an electron
the electric field generates an
Motion of an electron
effective magnetic field
This gives an effective interaction with the electron’s magnetic moment
s
p
V
Beff
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
5
Internal communication between spin and charge:spinorbit coupling interaction
(one of the few echoes of relativistic physics in the solid state)
e-
Classical explanation (in reality it arises from a second order expansion of
Dirac equation around the non-relativistic limit)
• “Impurity” potential
•
V(r)
Produces
an electric field
In the rest frame of an electron
the electric field generates an
Motion of an electron
effective magnetic field
This gives an effective interaction with the electron’s magnetic moment
V
s
p
Beff
Consequence #1
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
6
Internal communication between spin and charge:spinorbit coupling interaction
(one of the few echoes of relativistic physics in the solid state)
e-
Classical explanation (in reality it arises from a second order expansion of
Dirac equation around the non-relativistic limit)
• “Impurity” potential
•
V(r)
Produces
an electric field
In the rest frame of an electron
the electric field generates an
Motion of an electron
effective magnetic field
This gives an effective interaction with the electron’s magnetic moment
V
s
p
Beff
Consequence #2
Mott scattering
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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How spintronics has impacted your life: Metallic spintronics
1992 - dawn of (metallic) spintronics
•Anisotropic magnetoresistance (AMR): In
ferromagnets the current is sensitive to the
relative direction of magnetization and current
direction
magnetization
e-
current
Appreciable sensitivity, simple design, cheap BUT only a 2-8 % effect
Giant magnetoresistance (GMR) read head - 1997
Fert, Grünberg et al. 1998
e-
×
Nobel Price 2007
Fert and Grünberg
High sensitivity, very large effect 30-100%
 and  are almost on and off states:
“1” and “0” & magnetic  memory bit
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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What next? The need for basic research
Industry has been successful in doubling of transistor numbers on a chip
approximately every 18 months (Moore’s law). Although expected to continue for
several decades several major challenges will need to be faced.
Circuit heat generation is one key limiting factor for scaling device speed
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Transport theory and simulations in hybrid structures:
the need for a bottom-up approach in new spintronic systems
I. Introduction: using the dual personality of the electron
•Internal coupling of charge and spin: origin and present use
II.Spintronic devices and the need to understand them microscopically
III. Spin injection Hall effect FET: a new paradigm in exploiting SO coupling
•Spin based FET: old and new paradigm in charge-spin transport
II. Spin filters in InAs nano-wires
III. A bottom to top approach: from mesoscopic to microscopic
• Why is it important to approach the problem from the mesoscopic regime
V. Theoretical tools of the mesoscopic world:
• Landauer-Büttiker (coherent regime)
• Non-Equilibrium Green’s Function approach: quantum Boltzmann Eq.
• Beyond coherent transport: the basic master equation of the nonequilibrium density matrix
VI. Computational Challenges of NEGF approach
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
10
Towards a realistic spin-based non-magnetic FET device
Can we achieve direct spin polarization injection, detection, and manipulation by
electrical means in an all paramagnetic semiconductor system?
Long standing paradigm: Datta-Das FET
Unfortunately it has not worked :
•no reliable detection of spin-polarization in
a diagonal transport configuration
•No long spin-coherence in a Rashba spinorbit coupled system (Dyakonov-Perel
mechanism)
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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New paradigm using SO coupling: SO not so bad for dephasing
Problem: Rashba SO coupling in
the Datta-Das SFET is used for
manipulation of spin (precession)
BUT it dephases the spin too quickly
(DP mechanism).
1) Can we use SO coupling to manipulate spin AND increase spin-coherence?
Use the persistent spin-Helix state and control of SO coupling strength
(Bernevig et al 06, Weber et al 07, Wunderlich et al 09, 10)
• Can we detect the spin in a non-destructive way electrically?
Use AHE to measure injected current polarization at the
nano-scale electrically (Wunderlich, et al 09, 04)
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Spin-dynamics in 2D electron gas with Rashba
and Dresselhauss spin-orbit coupling
1) Can we use SO coupling to manipulate spin AND increase spin-coherence?
a 2DEG is well described by the effective Hamiltonian:
Rashba: from the asymmetry of the
confinement in the z-direction
ky [010]
 > 0, = 0
[110]
kx [100]
_
[110]
Sinova
Dresselhauss: from the broken inversion
symmetry of the material, a bulk property
ky [010]
 = 0, < 0
[110]
kx [100]
_
[110]
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Effects of Rashba and Dresselhaus SO coupling
ky [010]
 > 0, = 0
 = -
[110]
ky [010]
[110]
kx [100]
_
[110]
 = 0, < 0
kx [100]
ky [010]
[110]
_
[110]
kx [100]
_
[110]
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Spin-dynamics in 2D systems with Rashba and Dresselhauss SO coupling
For the same distance traveled along [1-10], the spin precesses by exactly the same angle.
[110]
_
[110]
_
[110]
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Spin-helix state when α ≠ β
In addition we performed MC
Boltzmann transport calculations
incorporating spin physics
For Rashba or Dresselhaus by
themselves NO oscillations are
present; only and over damped
solution exists; i.e. the spin-orbit
coupling destroys the phase
coherence.
There must be TWO competing
spin-orbit interactions for the spin
to survive!!!
Wunderlich, Irvine, Sinova, Jungwirth, et al, Nature Physics 09
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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“Local” magnetization measurement through the
Anomalous Hall Effect
Spin dependent “force” deflects like-spin particles
M⊥
_
__
majority
FSO
FSO
I
minority
ρH=R0B ┴ +4π RsM┴
AHE is does NOT originate from any internal
magnetic field created by M⊥; the field would
have to be of the order of 100T!!!
V
Simple electrical measurement
of out of plane magnetization (or
spin polarization ~ n↑-n↓)
InMnAs
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Cartoon of the mechanisms contributing to AHE
Skew scattering
A
~σ~1/ni
Vimp(r) (Δso>ħ/τ)  λ*Vimp(r) (Δso<ħ/τ)
Asymmetric scattering due to the spin-orbit
coupling of the electron or the impurity.
Known as Mott scattering.
Intrinsic deflection B
independent of
impurity density
Electrons deflect to the right or to the left as
they are accelerated by an electric field ONLY
because of the spin-orbit coupling in the
periodic potential (electronics structure)
E
SO coupled quasiparticles
Electrons have an “anomalous” velocity perpendicular to the electric field
related to their Berry’s phase curvature which is nonzero when they have
spin-orbit coupling.
Side jump scattering B
independent of impurity density
Vimp(r) (Δso>ħ/τ)
 λ*Vimp(r) (Δso<ħ/τ)
Electrons deflect first to one side due to the field created by the impurity and deflect back when they leave the impurity
since the field is opposite resulting in a side step. They however come out in a different band so this gives rise to an
anomalous velocity through scattering rates times side jump.
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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AHE contribution to Spin-injection Hall effect in a 2D gas
Two types of contributions:
i)S.O. from band structure interacting with the field (external and internal)
•Bloch electrons interacting with S.O. part of the disorder
Type (i) contribution much smaller in the weak SO coupled regime where the SOcoupled bands are not resolved, dominant contribution from type (ii)
Crepieux et al PRB 01
Nozier et al J. Phys. 79
Lower bound
estimate of skew
scatt. contribution
Wunderlich, Irvine, Sinova, Jungwirth, et al, Nature Physics 09
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Spin-injection Hall device measurements
SIHE ↔ Anomalous Hall
trans. signal
VL
Local Hall voltage changes sign and magnitude along a channel of 6 μm
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Further experimental tests of the observed SIHE
T = 250K
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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SiHE transistor
Vb
Vg
VH
I
x
VH
Vg
Vb
x=1 m
Spin Hall effect transitor:
Wunderlich, Irvine, Sinova, Jungwirth,
et al, Science 2010
Sinova
+
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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SiHE transistor AND gate
Vg2 [V]
+0.1
0
-0.1
0
RH1
[]
12
6
H1
-
Sinova
-
RH2 0
[]
3
6
-
H2
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Next SiHE transistor: all electrical spin FET
Key features:
•Spin injection through thin Fe
contacts into GaAs
•Detection via SHE and Non-Local
Hanle effect
•Spin current control through charge
current bias
K. Olejnik, Wunderlich, et al
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Semiconductor spin filters in InAs nanowires
Jan Jacob’s group
• Splits an unpolarized current into two oppositely spin-polarized currents
A. A. Kiselev and K. W. Kim, Appl. Phys. Lett. 78, 775 (2001); J. Appl. Phys. 94, 4001 (2003).
J. I. Ohe, M. Yamamoto, T. Ohtsuki, and J. Nitta, Phys. Rev. B 72, 041308 (2005).
M. Yamamoto, T. Ohtsuki, and B. Kramer, Phys. Rev. B 72, 115321 (2005).
A. W. Cummings, R. Akis, and D. K. Ferry, Appl. Phys. Lett. 89, 172115 (2006).
M. Yamamoto, K. Dittmer, B. Kramer, and T. Ohtsuki, Physica E 32, 462 (2006).
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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12
Semiconductor spin filters in InAs nanowires
•
In-plane magnetic fields
•
Detect spin-Hall signal in
filter outputs
•
Top-gated spin-filter cascades
•
Single quantum-point contacts
•
Top/Backgate-controlled
spin-orbit coupling
•
Ferromagnet-Semiconductor Spin-Valves
(together with OSU)
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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30
Transport theory and simulations in hybrid structures:
the need for a bottom-up approach in new spintronic systems
I. Introduction: using the dual personality of the electron
•Internal coupling of charge and spin: origin and present use
II.Spintronic devices and the need to understand them microscopically
III. Spin injection Hall effect FET: a new paradigm in exploiting SO coupling
•Spin based FET: old and new paradigm in charge-spin transport
II. Spin filters in InAs nano-wires
III. A bottom to top approach: from mesoscopic to microscopic
• Why is it important to approach the problem from the mesoscopic regime
V. Theoretical tools of the mesoscopic world:
• Landauer-Büttiker (coherent regime)
• Non-Equilibrium Green’s Function approach: quantum Boltzmann Eq.
• Beyond coherent transport: the basic master equation of the nonequilibrium density matrix
VI. Computational Challenges of NEGF approach
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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quantitative modeling of spin-current based devices from bottom up
Semiclassical: Boltzmann-Monte-Carlo (2010) ✓
scale ~ 1μ
Connection to effective Hamiltonians
and boundary conditions of different
phenomena
Mesoscopic: Non-Equil. Green’s Function (2007) ✓
scale ~ 1nm
Microscopic: Ab-initio (tight-binding)
Interface effects of spin-injection
and bulk spin coherence
scale ~ 1Å
Sinova
Molecular level
modeling
COMPETING LENGTH SCALES
OF THE SAME ORDER
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Semiclassical bulk transport theory: Boltzmann “diagonal” transport
Electrons are treated as Newtonian pinballs, but
having an effective mass and g-factor determined by
the band structure and undergo random quantum
scattering
Dynamics of the occupation number
• Very successful for charge transport and
micro-device modeling
• Key feature: l.h.s. classical – r.h.s.
quantum collision term
• Main shortcoming: Does not treat
systematically inter-band coherence or
spin transport
• Applicable in the diffusive metallic regime
ASSUMES COLLISIONS ARE
INSTANTANEOUS AND UNEVENTFUL
SinovaFormalism
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
NEGF
29
Semiclassical approach to bulk Hall transport:
breakdown of collision assumption
Modified
Boltzmann
Equation:
Coordinate shift
(Side jump):
Golden Rule:
Berry curvature:
velocity:
current:
Sinitsyn et al PRB 06
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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What is what in the modified Boltzmann treatment
E
a) Intrinsic deflection
b) Side jump
scattering
c) Skew scattering
SinovaFormalism
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
NEGF
31
1.
2.
3.
Quantum mesoscopic transport
When the mean free path is of the
same order as the system size then
quantum interference effects appear
In the presence of SO coupling bands
are mixed: occupation number is not
the only thing needed to describe
transport
Need to explore the non-equilibrium
dynamics of the whole density matrix
Nikolic et al 04
NEGF formalism is in a sense the
quantum Boltzmann Equation: both the
collisions AND dynamics between
collisions are treated quantum
mechanically
SinovaFormalism
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
NEGF
32
WHAT CAN WE ADDRESS FROM THIS MICROSCOPIC
(1) What is the effect of the nature of the scattering on the induced spin-currents
and spin coherence in general?
(2) What is the nature of the spin-current?
(3) Can these induced currents lead to strong enough spin-accumulation?
(4) Does coherent transport matter?
(5) What is the loss of spin-coherence in relation to scattering?
(6) Is the effect more readily controlled at the mesoscopic scales?
(7) What is the dissipative ratio in spintronic devices analogous to current
devices?
Advantages of NEGF approach:
•No assumptions on spin-currents
•Deals with intrinsic and extrinsic on an equal footing
•Multi-band treatment comes out naturally
•No assumed length scales
•Inelastic processes can be incorporated
Limitations: system sizes and extrapolation to the bulk regime
SinovaFormalism
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
NEGF
33
Mesoscopic transport via Landauer-Büttiker Formalism
(Coherent limit of the NEGF formalism)
Uncorrelated electrons injected from
left lead.
Single channel conductance:
T
transmission
probability
Sharvin conductance
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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1.
Landauer-Büttiker Procedure in a Nutshell (real space)
Calculate the leads self-energies (f-numberelt
by the sample):
Usually known or easily
calculable
2.
Compute the sample retarded GF:
3.
Obtain transmission coefficients:
4. Calculate currents with Büttiker formula:
See, e.g., Electronic Transport in Mesos. Systems by Datta
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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How to actually compute things: Tight-Binding Approximation
Tight-binding lattice
Continuous 2DEG
Artificial length scale
Effective mass Hamiltonian
Tight-binding Hamiltonian
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Example: Quantum Point Contact Conductance
Experimental evidence of
conductance quantization:
Conductance of a ballistic 2DEG
QPC:
Van Wees et al, PRL 60, 848 (1988)
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Limitations of Landauer-Büttiker
•Limited to relatively small systems.
•Must rely on finite size scaling to reach bulk values
•Limited to coherent transport
NEGF formalism advantages
•Expresses physical quantities via correlation functions.
•Studies non-coherent transport.
•Can include the effect of interactions and dissipation
gradually
•Can make clearer connection to the bulk transport theories
However, it still has same size limitation as LB
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Nonequilibrium Green Function Formalism
Classical transport:
distribution function
Quantum transport:
density matrix
Key correlation function
Generalizes the density matrix.
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Boltzmann vs. NEGF Functions
(but not literally; r.h.s. knows about interband coherence)
Electron distribution function
Hole distribution function
Scattering functions
Do not take into account phase
correlations.
Sinova
Electron correlation function
Hole correlation function
Scattering functions
Include phase correlations.
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Connection to semiclassical Boltzmann when looking in real space
Using the relation shown and G in real space
SinovaFormalism
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
NEGF
41
NEGF Algorithm - Interactions
1. Compute self-energies
only first iteration
2. Calculate the retarded GF
3. Find the correlation function
4. Calculate interaction self-energy
5. Check for convergence
If not
6. Compute physical quantities
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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NEGF Algorithm – Molecular Transport
The sample Hamiltonian depends on
charge density:
4. Compute the charge density for
biased sample:
1. Calculate self-energies of the
electrodes:
5. Check for convergence:
2. Calculate the retarded GF:
no
6. Compute physical quantities:
3. Compute the correlation function:
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Nonequilibrium Spin Hall Accumulation in mesoscopic
Rashba 2DEG: non-linear transport
eV=0
+eV/2
-eV/2
Spin density (NEGF):
PRL 95, 046601 (2005)
SinovaFormalism
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
NEGF
44
Transport theory and simulations in hybrid structures:
the need for a bottom-up approach in new spintronic systems
I. Introduction: using the dual personality of the electron
•Internal coupling of charge and spin: origin and present use
II.Spintronic devices and the need to understand them microscopically
III. Spin injection Hall effect FET: a new paradigm in exploiting SO coupling
•Spin based FET: old and new paradigm in charge-spin transport
II. Spin filters in InAs nano-wires
III. A bottom to top approach: from mesoscopic to microscopic
• Why is it important to approach the problem from the mesoscopic regime
V. Theoretical tools of the mesoscopic world:
• Landauer-Büttiker (coherent regime)
• Non-Equilibrium Green’s Function approach: quantum Boltzmann Eq.
• Beyond coherent transport: the basic master equation of the nonequilibrium density matrix
VI. Computational Challenges of NEGF approach
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Usuki-transfer matrix method with GPUs (two terminal)
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Key computational challenges
The key issue with NEGF for realistic size microscopic
calculations is how to best deal with large matrices (both
multiplication and inversion)
•In order to obtain bulk realistic values one needs to do finite size
scaling of more than three small sizes
•Can one dynamically deal with sparsity?
•Not all eginevectors are created equal - how to select?
•2-terminal vs. multi-terminal?
•Conductance (two terminal) vs. G< (much more computationally
expensive but contains everything)
Sinova
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Principal Outside Collaborators
Jan Jacob
Tomas Jungwirth Joerg Wunderlich Allan MacDonald U. Hamburg
Texas A&M U. Cambridge-Hitachi U of Texas
Inst. of Phys. ASCR
U. of Nottingham
Sinova
Laurens Molenkamp Bryan Gallagher
U. of Nottingham
Würzburg
Gerrit Bauer
TU Delft
and many others
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
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Sinova’s group
Xin Liu
Texas A&M U.
Vivek Amin
Texas A&M U.
Erin Vehstedt
Texas A&M U.
H. Gao
Texas A&M U.
Jacob Gyles
Texas A&M U.
Oleg Tretiakov
(main PI Abanov)
Texas A&M U.
Previous members
Liviu Zarbo
Alexey Kovalev
Ewelina Hankiewicz
Nikolai Sinitsyn
Texas A&M U. (PD-Texas A&M Univ.) PD-Texas A&M Univ. PD-Texas A&M
Now:
Now:
Now: W2-Professor at
Now:
PD- ASCR
PD-UC Riverside
Würzburg University
Staff LANL
49
International Symposium High Performance Computing in Nano-Spintronics, Hamburg
Xiong-Jun Liu
Texas A&M U.
Now:
PD- U. Maryland
Sinova
Mario Borunda
Texas A&M Univ.
Now:
PD-Harvard Univ.
What next?
Sinova
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Inversion of large sparse matrices
Sinova
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