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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)

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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

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The end