Mikhail Raikh, [email protected]: 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
Top Related