1
Institute for Materials Research & WPI-AIMR, Tohoku UniversityKavli Institute of NanoScience, TU Delft
Modelling and Simulation:a physicist’s point of view
Part 1
I want you (to talk about Modelling and Simulation)
Uncle Minzhong
What is simulation
History: Horse simulator (Wikipedia)
Simulant (German) = Malingerer
Simulation is the imitation of the operation of a real-world process or system over time.
The act of simulating something requires a model.
Modelling (Wikipedia)
Modelling: A scientific activity, the aim of which is to make a particular part or feature of the world easier to understand, define, quantify, visualize, or simulate by referencing it to existing and usually commonly accepted knowledge ©
P. Rekacewicz
Philosophy
Model Simulation
intuition
experimentsparameters,methods
2
Our model
0i ieA m
Modeling and simulation
All of condensed matter theory is modelling and simulation!
The fundamental–applied divide
acad
emic
indu
stria
l
Contents• Modelling and simulation: a physicists
point of view
Part 1I. M&S of electronic structureII. M&S of magnetizationIII. M&S of spin and charge transportIV. M&S of MRAM elements
Part 2• Case study: heterostructures with magnetic
insulators
I. M&S of electronic structure Many-body problem
• Exact solutions (hydrogen molecule, 1D systems, 2D Ising model)
• Many-body perturbation theory, Green's functions,diagrammatic methods (also for alloys)
• Configuration interaction (chemistry)• Coupled cluster expansions (chemistry)• Quantum Monte-Carlo approaches• Density functional theory• Lattice gauge theory• Quantum tensor network theory• Numerical renormalization group• AdS-CFT (string theory)
…
3
“Mean-field” theory
Models
- Dirac Hamiltonian- Pauli spin (non rel.)- tight-binding models- Hubbard models - spin models
Simulation methods
- Hartree-Fock
- GW-method
- Density functional theory
Local density approximation
Corrections: LDA+U, LDA+C
- Dynamic mean-field theory
Density functional theory
0 0
min
min
HE T E
E E
The ground state energy E [] is a functional of the electron (spin) density r(Hohenberg & Kohn, 1964):
Constrained search:
Kohn-Sham non-interacting auxiliary particles:
2
1
N
ii
r r
minKS H XCE T E E
1 11 , ,KS
N NN r r
2
1i dr r
Kohn-Sham equations
0E N E
r r
Variational principle: ground state energy is minimal for the ground state density. The variation under particle number constraintis stationary:
N dr r
chemical potential
*
*
2 2
0
2
i i i
KS i i ii
KS H xc
E NH
H V Vm
r rr
r r
xc
xc
EV r
r
LDAxc xcE dr r local density approximation
Spin density-functional theory
2occ
ii
r r
,
,KS i i i
KS i i i
H
H
r r
r r
,
,KS i i i
KS i i i
H
H
r r
r r
2 2
,,2 nucS xcK l CV r Vr VHm
r
Spin-polarized Kohn-Sham equations:
starting,
construct KS Hamiltonians
solve KSequations
determine densities converged?
no
yes
Self-consistency cycle:
exchange (-correlation) potentials
Hartree Hamiltonian
Band structure and Fermi surfaces of Co
Zwierzycki et al. (2008)
Baga
yoki
et a
l. (1
983)
DMFT
Lichtenstein et al. (2001)
4
Transition metals and s-d hybridization
10sd sd
sd
E eV t fsE
sp and d- electrons are strongly hybridized and cannot be distinguished on electron transport time scales.
d bandsd bands
EF
pF px
py
Stoner metallic ferromagnet
Halfmetallic ferromagnet
2
2ˆ2
Hm
: exchange(-correlation) potential that reflects the energy splitting betweenmajory and minority spin states.
pF
Spin-dependent densities, Fermi momenta, and mobilities
2
II. M&S of magnetization (dynamics)
© G. SchützSimulated magnetic vortex switching
Magnetization dynamics
Magnetic dipole in a magnetic field
-
B
M L Torqueddt
BMTL
d ddt dt
TM L M B
F M B Force
EnergyU BM
Spin equation of motionHeisenberg spin-operator equation of motion:
, ,i ii
d i iH B sdt
B
s s s
s
,x y zs s i s
ddt
m m B
, ,x y zs s s s s
Spin-vector (Landau-Lifshitz) equation of motion:
m s
5
Magnetism, statics and dynamics
Quantum spin modelsHeisenbergIsingX-Y….
Magnons
Quantum Classical
First principles methods(Car-Parinello)
Classical spin modelsLandau-Lifshitz-BlochLandau-Lifshitz-Gilbert
Spin waves
Landau-Lifshitz-Gilbert equation
eff
d t d tt
dt dt
m mm B
Landau-Lifshitz-Gilbert equation; Gilbert damping constant
m
precession
effB
m
damping (>0)
t
mm
tm
rotation
; ss
MM
Mm M
eff
s
FM
Bmm
Landau-Lifshitz-Gilbert equation
eff
d t d tt
dt dt
m mm B
Landau-Lifshitz-Gilbert equation; Gilbert damping constant
; ss
MM
Mm M
eff
s
FM
Bmm
Micromagnetism (static)
Course grain minimization of magnetostatic energy with
2
2effCF d M M r B M r M r r
Yu et al., PRB 60(1999)7352
.12 2
anisoteff appl dipolar
sr M
M
BB B B
C spin-wave stiffness
0 :M
transverse magnetic head-to-head domain wall
Spin waves or magnons
Plane wave solution of LL equation:
William Fuller Brown, Jr. (1904-1983)
Thermal Fluctuations of a Single-Domain ParticleWilliam Fuller Brown, Jr.Phys. Rev. 130, 1677 –Published 1 June 1963
FATHER OF MICROMAGNETICS
6
Thermal magnetization noise
effM HeffM H
0T 0 cT T
0M T M
eff t hm m H m
0
2' 'Bi j ij
k Tt t t tM
h h
Thermal equilibrium: 0 0( ) 12MB
B eff
kTM T M f d MH
Ω Ω
Fluctuation-dissipation theorem:
Stochastic field:
Fokker-Planck equation for probability distribution function P (m,t )
Magnetic domain walls
B
© T. Schrefl
Moving domain walls by magnetic fields:
LL(Gilbert) vs. LLB(loch)
LLG: .sM const M
LLB: sM TM
eff eff
eff
t
t
ddt
B B b
B
m m m m
m m b
Garanin (1997)
transverse dynamics
longitudinal dynamics+ FD theorem
Atomistic spin simulations for localized spins
© J. BarkerEllis et al., arXiv:1505.07367
Nxi ij i j
j iE J
S S
effi i i i i i S S B S S
eff ii i
i
E t
B b
S
2 B ik l kli
i
k Tb t b t t tM
iS
Parameters
Spin-wave dispersion in the (1, 1, 1) direction of the reciprocal space for LaTiO3 measured by neutron-scattering experiment. The line is the fitting curve by using the Heisenberg model with isotropic J of 15.5 meV on the cubic lattice (Keimer et al., 2000)
Nxi i j
j iJE
S S
micromagnetism
III. Transport
7
Charge and spin transport
- Green function methodsDiagrammatic, Kubo formula, Keldysh
- Scattering theory of transport- Semiclassical methods
Kinetic, Boltzmann, diffusion (Valet-Fert)- Numerical (first principles and model)
Recursive GF, Kwant- Hybrid methods
circuit theory, Continuous-RMT
Regimes dictate the models and methods
Linear vs. non-linear transport Ballistic vs. diffuse transportBulk vs. interfacesZero vs. finite temperatureRelativistic vs. non-relativisticMetals vs. insulators
Ohm’s LawDisorderMultilayersNoiseSpin Hall effectYIG
Qualititative vs. predictive material design
Atomic scale interfaces + disorder Diffusion in bulk 3D metal
x eV x
eV
x E je x
no quantum interference effectssuch as weak localization:semiclassical diffusion
Ohm’s Law
Current conservation
0 .eEj constx
R
2 resistivitymne
resistanceLR A
A
L
Mesoscopic physics
quantum point contacts
break junctionsquantum wires
Interface conductance
V1
metal 1 metal 2
2 22 1 1
i j FE
LB Sharvinij
e eG G N
h hSharvin conductance
1 2
2 22
1
i j F
ILB
E NILB ij k
ij k
I G V Ve eG t Th h
tijrij'
Landauer-Büttiker formula
V2
8
Metallic interface
kx
ky
U
kx
ky
EFA
AFk B
Fk
BFE
E A B
U EFA
EFB
Specular metallic interface
222 14
AF
AF
ke UGh E
A 222 14
AF
AF
ke UGh E
A
|| || || ||, ,1 for0
k
k
E UT
E U
k k k k
semiclassical approximation:
A interface area
A B
Disorder
U > 0
xx
xx
x
x
x
x
x
x
x
Interface alloying and roughness Bulk disorder scattering
Interfaces
I
/
/ /L R
L R L Rx E j
e x
I IGj
eA
GI interface conductance
L R
RL RI RR equivalent circuit
Spin valve and giant magnetoresistance (GMR)
22
AP P
AP
R RGMRR
R R PR R
Two spin-channel model , ,s
s sx j s
x
Ohm’s Law
spin relaxation sf
j j e n nx
2
2 2sd
x xx
R R R
equivalent circuit
R
R R R R
Rsf Rsf Rsf
sd sfD
spin-flip diffusion
213 FD v
spin diffusion lengthaveraged diffusion constant
9
Nsd sfD
Nsd spin-flip diffusion lengthD diffusion constant sf spin-flip relaxation time
F NSpin accumulation and spin current F|N|F spin valves
L
I
Lsd
F|N|F spin valves
L
I
Lsd
F|N|F spin valves
Lsd
L
I
F|N|F spin valves
Lsd
L
I
F|N|F spin valves
Lsd
L
I
10
F|N|F spin valves: non-collinear
Lsd
L
I
2013 Oliver E. Buckley Condensed Matter Physics Prize
Luc BergerCarnegie Mellon University
John SlonczewskiIBM Research Staff Emeritus
Citation:"For predicting spin-transfer torque and opening the field of current-induced control over magnetic nanostructures."
Electron spin vector operator
2
20 1 0 1 0, ,1 0 0 0 1
e e Bem
ii
g g
σμ s s
s σ
σ
Pauli matrices
Beg
2Bem
Bohr magneton
ˆ 00
1 02 0 1
z e Bz
g BB
BB μ H
0
B sWhen the system oscillates with Larmor frequency 0 .
01 12 2s sm mH
Rotation in quantum mechanics
x
y
z
d
x x ydy y xd
ˆ ,
ˆ1
z
z
R d f f x dx y dyif y x fd L d f
x y
Finite rotation:
limn
nd
ˆ ˆlim 1n
z zn
iR d L
log 1 lim log 1 limn
n nx x x nx
ˆˆ ˆ ˆlog li ˆm z
z z zL
n
i
zR ei iR nd L L
n
( , ,0)x yn
Rotation in quantum mechanics
1,2 2 0x
Rx
y
z
n
cos ,sin ,0With n
1, ,0
R
generates all possible spin states. Example:
ˆR R n
ˆ ˆexpR i n n LRotation operator:
Electron spin: ˆ2
L s σ /2ˆ iR e n σn
Rotation of a state by an angle around an axis with unit vector n:
Spins on the Bloch sphere
1112x
y
0 1 /2/2 1 0
0
0 11,2 ! 21 0
cos sin2 2
sin cos2 2
mmi
me e
m
R
1,0,02x x
s Check:
Spin-up state in x-direction is obtained by rotation about y-axis with .
1 0,22 0 1
R
11
m
Normal / 2
HMFSlonczewski torque and Andreev reflection
/ 2 / 2
transverse spin current = torquelongitudinal spin currentm
/ 2 / 2
2 2
NeV V
N = number of incoming channels
=m
Spin-mixing conductance
=m
2
ˆ ˆ e Nx z V Ve h m
2 2
Sharvin2 21
FEe eG Nh h
k
ˆ ˆ Rex z V V Ge m
Scattering theory:
2 2*
1FEe e NG r r
h h
k k
k
2 2*1
FEe e NG r rh h
k k
k
z
x
k incoming wave vector
Spin mixing conductance:
Switching by spin transfer Switching by spin transfer
Current-induced magnetization reversal
-1.0 -0.5 0.0 0.5
3.50
3.55
3.60
3.65
Current (mA)
dV
/dI (
Oh
ms)
Electrical currents can controllably reverse the magnetization in small (< 200 nm) magnetic devices (Slonczewski, Berger):
Parallel (P)
Antiparallel (AP)
Cu
Co
Kiselev et al. (2003)
Pt
Ip,c
Positive Ip
Ohm’s and Kirchhoff’s Laws
2 22
nmnm
e eG g t
h h
2 22
nmnm
e eG g t
h h Landauer formula
V
Ic V1 V2 V3,1 2cI
1 2 1 2,1 2c VI VG conductance
Charge current:
1 2G
, 0c j ij
I
,3 2cI
charge conservation
12
Spin currents
,
,
2Re 2Im
2
N Ns s s
s
G G
e
I m V m V m
Iin-plane out-of-plane
spin accumulation in Nspin accumulation in F magnetization direction
mNsVFsV
N F
NsV F
sV
transversespin currents
collinear/longitudinal
, ,, ,c s sI I I
Spin currents
*nm nmnm
g N r r *nm nm
nmg N r r
complex spin-mixing conductancefor transverse spin current (torque + exchange field)
2 2s ss nm nm
nm nmg t N r
2 2s ss nm nm
nm nmg t N r
s=, spin-dependent Landauer conductances for charge andcollinear spin current
transversespin currents
collinear/longitudinal
0 , ,, ,s sI I I
spin accumulation in Nspin accumulation in F magnetization direction
mNsVFsV
N F
NsV F
sV
Pauli matrix notation
*ˆ ˆ ˆ1c s
X XX XX X
X σ
ˆ ˆ ˆ1N N Nc sV V V σ
1 0 0 1 0 1 0ˆ ˆ1, , , ,0 1 1 0 0 0 1
ii
σ *ˆ G GG G G
N F
NsV F
sV
ˆ ˆ ˆ1F F Fc sV V V m σ
ˆ ˆ ˆ1c sI I I σ
Circuit theory (Brataas et al., 2000)
V 0
1m 3m
1G 2G 3G 4G
1V 2V 3V
2s
1 ,1 ,1 1
2 ,2 ,2 2
ˆ ˆ ˆ1ˆ ˆ ˆ1
c s
c s
V V VV V V
m σ
s σ 1 2 3 2ˆ ˆ 0I I
1 2I 3 2I
spin & charge conservation:
Angular magnetoresistance
Exp: S. Urazhdin et al. (2005)
Py (12 nm)
CuPy (6 nm)
AF
CuPy
Py(1.5 nm)
AF
AF
15 1 2Re 0.5 10 mG
III. M&S of MRAM elements
13
Spin currents cause magnetization motion(spin transfer torque, Slonczewski, 1996).
Spin torque and spin pumping
Magnetization motion causes spin currents (spin pumping, Tserkovnyak, 2002).
Onsager reciprocals(Brataas et al., 2011)
g
Dynamics of bilayers
Landau-Lifshitz-Gilbert equation with additional torque term:
m
NsV
,c sI Im
spin pumping Slonczewski torque
0
0
4
4 4
bias pu
Ns
mps s
s
r r
s s
Mg g
t
M M
t
t
m mm B m m m
mm B m m
I
mV
I
ReIm 0
rg gg
FsV
NsV
Macrospin simulation of nanopillarBorlenghi(2011)
Continuous random matrix theory [Waintal c.s.]
Spin oscillators
Kyun
g-Jin
Lee
et a
l. (2
013)
Exchange-only theory is complete
SpinFlow3D (Thierry Valet)
- integration of current dynamics with micromagnetics (including Oersted fields)
Additional topics and challenges- Current induces torques in magnetization textures
(domain wall motion, emf due texture dynamics,topological Hall effect)
- Thermally induced spin currents (spin-dependent Seebeck effect, spin Seebeck effect)
- Spin Hall and related effects- Spin orbit torques (field-like vs. damping-like)- Magnetic insulators (YIG)- Antiferromagnets- Skyrmions