Mikhail Raikh, raikh@physics.utah.eduCo-authors: R. C. RoundySpin dynamics of a diffusively moving electron in a random hyperfine field

We study the dynamics, 〈Sz(t)〉, of the average spin of electron hopping over sites which host random hyperfinemagnetic fields. If the typical waiting time for a hop is τ and the typical magnetic fields is bs0 , then thetypical spin-precession angle on a given site is δφ ∼ b0τ 1. Then the Markovian theory predicts thatthe spin, initially oriented along the z-axis decays, on average, as 〈Sz(t)〉 = exp(−t/τs), where τs =1/b20τ is the spin-relaxation time. We find that in low dimensions, d = 1, 2, the decay, 〈Sz(t)〉, is non-exponential at all times. The origin of the effect is that for d = 1, 2 a typical random-walk trajectory exhibitsnumerous self-intersections. Multiple visits of the carrier to the same site accelerates the relaxation since thecorresponding partial rotations, δφ, of spin during these visits add up. As a result, the Markovian descriptiondoes not apply. For one-dimensional diffusion of electron over sites, the average, 〈Sz(t)〉, is the universalfunction of t3/2/τ1/2τs, so that the characteristic decay time is τ1/3τ2/3s is much shorter than τs. Moreover,when the random magnetic fields are located in the (x, y) plane, the decay of 〈Sz(t)〉 to zero is preceded by areversal of 〈Sz(t)〉 to the value 〈Sz〉 = −0.16 at intermediate times. We develop an analytical self-consistentdescription of the spin dynamics which explains this reversal. Another consequence of self-intersections of therandom-walk trajectories is that, in all dimensions, the average, 〈Sz(t)〉, becomes sensitive to a weak externalmagnetic field directed along z. Our analytical predictions are complemented by the numerical simulations of〈Sz(t)〉.

Continuum Models and Discrete Systems-13 Thermodynamics, transport theory, electrical properties and statistical mechanics

for continuum and discrete systems

Spin relaxation of diffusive moving carrier in a random hyperfine field

M.E. Raikh Department of Physics

University of Utah

Supported by: MRSEC DMR-1121252

(in collaboration with R.C. Roundy) arXiv:1401.4796

Spin a top on a flat surface, and you will see it's top end slowly revolve about the vertical direction, a process called precession. As the spin of the top slows, you will see this precession get faster and faster. It then begins to bob up and down as it precesses, and finally falls over. Showing that the precession speed gets faster as the spin speed gets slower is a classic problem in mechanics. The process is summarized in the illustration below.

A rapidly spinning top will precess in a direction determined by the torque exerted by its weight. The precession angular velocity is inversely proportional to the spin angular velocity, so that the precession is faster and more pronounced as the top slows down. The direction of the precession torque can be visualized with the help of the right-hand rule.

The Nobel Prize in Physics 2007

Peter Grünberg Albert Fert

"for the discovery of Giant Magnetoresistance"

Input 1 The effect is observed as a significant change in the electrical resistance depending on whether the magnetization of adjacent ferromagnetic layers are in a parallel or an antiparallel alignment. The overall resistance is relatively low for parallel alignment and relatively high for antiparallel alignment. The magnetization direction can be controlled, for example, by applying an external magnetic field. The effect is based on the dependence of electron scattering on the spin orientation.

hard drives MRAM

biosensors

spin-valve Input 2

device efficiency is limited due to spin-orbit coupling in the metallic active layer

Nature 427, 821-824 (26 February 2004)

( )( )s

s

dd

RRR

TMRλλ/exp1

/exp2−−

−=

−=

↑↑

↑↑↑↓

21

21

PPPP

tunnel (or giant) magnetoresistance

spin diffusion length polarizations of the electrodes

nms 45≈λ

d

Input 3

device efficiency is quantified via:

Cited by 847

Cited by 2911

characteristics of the spin memory loss

Precession of Spinning Top

Mechanism of Spin Memory Loss

Spin Precession in a Magnetic Field

Random Magnetic Fields of Nuclei Surrounding the Sites

How does the average spin polarization evolve with time?

expected result [D’yakonov-Perel’ (1971)]: ( )sz tSt τ/exp)0()( −=zS

1~ 0 <<τδφ b )exp()0()( 2δφNSNS zz −=><τNt =

)/exp()0()( szz tStS τ−=><

ττ 2

0

1bs =where

typical spin - rotation angle

typical typical waiting time

on-site field

spin relaxation time

)(tzSexponential decay of

( ) ( ) ( )

−==++=+++

2expexp...expRe...cos

2

2121δφφφφφφφφ Niiii N

NN

[ ]iii Sbtd

Sd

×=

time in the units

Numerical simulation of spin evolution

)(tzS

unidirectional hops : simple-exponential decay

Decay of is strongly non-exponential !

of

2D hops with random planar fields

( )[ ] ( ) ( )101011 sincos −−−− ⋅+×+⋅−= iiiiiiiiii SnnbSnbSnnSS

ττ

ττ 2

0

1bs =

direction of the hyperfine field on site i

qualitative explanation: multiple visitations of the same site in course of a random walk lead to accelerated spin-relaxation.

for a 3D random walk of N steps only a small portion ~ N-1/2 sites are

In course of random walk 1 2 3 4 5 3 6 the site 3 is visited twice: the corresponding partial spin rotations add up

The number of self-intersections of the random-walk trajectories depends strongly on the dimension:

for a 2D random walk of N steps each site is visited ~ 2 times visited twice

qualitative consideration

if all partial rotations are statistically independent:

1~ 0 <<τδφ b )exp()0()( 2δφNSNS zz −=><τNt =

)/exp()0()( szz tStS τ−=><

1D random walk

on the other hand, after N steps of a 1D random walk,

2/1NN →

δφδφ 2/1N→

−=><

szz

tStSττ 2/1

2/3

exp)0()(

ii. each site is visited ~ N1/2 times

i. N1/2 sites are visited

faster than a simple exponent

at the minimum

Planar hyperfine fields: relaxation proceeds via spin reversal

scaling t3/2 is correct, but the shape is not.

ττ 2

0

1bs =

16.0)( −=tSz

( ) sss bb

ττττ

τ <<== 3/203/13/4

0

1~

StbtdSd

×= )(analytical treatment:

Equation of spin dynamics

( ) )()()(cos1)( 2210

20

120

1

tSttdtdtbtS z

tt

z φφ −−= ∫∫

closed equation for )(tSz

magnitude

formal solution for a given realization: of the field

random in-plane orientation of the 2D field

Without returns, averaging should be performed with the help of the “usual” correlator

( ) ( ) )',(/|'|exp)'()(cos 0 ttCtttt =−−=− τφφ

describing the stay on a given site for a short-time, τ

−=

sz

ttSτ

exp)(0

term-by-term averaging reproduces the standard result

Poisson’s distribution of the waiting times

With returns:

for a diffusively traveling particle, there is a probability that it returns to the same site a long time later!

( ) )',()'(2

1)'()(cos2/

ttCttD

tt D

d

=

−

=−π

φφ

( )

−−=

Dtrr

DttrrP d 2

||exp

21),,(

221

2/21

πReturn: 12 rr

=

−−→

sDD

ttttCttCτ2

|'|exp)',()',(

short-time hops modify the diffusive correlator

Spin-memory is lost between two subsequent

diffusion coefficient

visits to the same site

diffusive correlator

returns

Comparison with

g2 is chosen to be 0.75 instead of 1

∫

−−= −−

s

s

ss

ttt

z egetSτ

ττ

ττ

π

2/

2/

/2/)( u

s

eu2τt

udu

analytical result in 2D

numerics

−−

sD

ttttCτ2

|'|exp)',(averaged with correlator

in the limit t >> the τs diffusive correction behaves as:

−

−

−

ss

ttgττπ 2

exp2

1

2

falls off slower than

−

s

tτ

exp

there should be a sign reversal

B

With magnetic field, B, along the z-axis

τττ 2

0

221bB

s+

=

−=

)(exp)0()(

BtSt

sz τzS

[ ] )()()()(cos1)( 221210

20

120

1

tSttBttdtdtbtS z

tt

z −+−−= ∫∫ φφ

Without returns, the decay remains exponential external field slows down the decay for

1≥τB

110 −= sB τfirst correction due to returns

2/

2/22/20

)2()(

dd

dd BtFtbπ

τ −

−=

magnetic field restores simple-exponential decay

[ ] 21cos)()0()(s

ssFsF dsdd −=− >> κ

Sensitivity to a weak magnetic fields B~1/τs << 1/τ

2/1

1 2)(

=

ss πκ

ss ln)(2 −=κ

analytical treatment in 1D

only diffusive correlators are relevant

)()1()( )(20

0tbt nn

n

n μ∑∞

=

−=zS

)(),( 1)(

10

1

)1(

tttCdtdt

d nD

tn

μμ∫=

+

)...,(),()( 430 0

43210 0

21)( ttCdtdtttCdtdtt D

t t

D

t tn ∫ ∫∫ ∫=μ

( ) ( )( )tS

tttdb

dtd

z

t

′′−′

−= ∫0

2/12/1

2/120

2πτzS

++

infinite series summation is required in 1D

recurrent relation:

leads to an integral equation

20b 4

0b40b 4

0b

selection of the higher –order terms

( ) ( ) ( )10

2/13/21

3/23/11

13/12/129

4 uSuuu

ududu

dz

u

∫ −−=

πzS

2/32/120 tbu τ=

( )( ) ( )[ ]

( )∫ −

−−−=

uz

uuu

uuuSududu

d

02/13/2

13/23/1

1

2/33/21

3/211

3/12/1

exp29

4π

zS

with a new variable

for planar hyperfine fields

-0.28 for spherically-symmetric fields

With magnetic field, B, along the z-axis the decay of is accompanied by oscillations )(tSz

planar random field spherical random field 4,3,2,1,03/13/40

=τb

B

( )( )

( ) ( ) ( )

−

=

−−

=− ∫∫ 2

2/1

2/1

2/32/120

2/121

21

02

012/1

2/120 cos

22cos

2)(1

1

BtBt

Bttb

ttttBdtdtbtS

tt

zπ

πτ

πτ

Alternative (quantum) description

( )

−

= 2|)(|1)(t

tt

ββ

χspinor 2|)(| tβprobability that spin points up

( ) ( ) 221 ttS β−=time evolution of spin :

( )

++

−

−−−

=ααα

ααα

sin||

cossin||

sin||

sin||

cosˆ

bbi

bibb

i

bibb

ibbi

bzyx

yxz

t,U

2tΩ

=α

spin-flip amplitude

evolution matrix

( ) ( ) ( )tbUt χττχ ,ˆ =+

Coherent spin evolution is described by the product: ( ) ( ) ( )....,ˆ,ˆ,ˆ321 τττ bUbUbU

Conclusion: effective field

interference: amplitude of spin flip after visiting three sites with orientations of hyperfine fields χ1, χ2, χ3

if the sites 1 and 3 are the same χ1 = χ3

acceleration of spin relaxation

2246

,,

2 33321

vvuvM ≈+=χχχ

2246

,

2 5521

vvuvM ≈+=χχ

( )

−−

−−

−−

=01

101

1

2

2

3

3

uiveiveu

uiveiveu

uiveiveuM i

i

i

i

i

i

χ

χ

χ

χ

χ

χ

35

11

=−

−

eduncorrelatz

correlatedz

SS