Mikhail Raikh, raikh@physics.utah.edu
Co-authors: R. C. Roundy
Spin dynamics of a diffusively moving electron in a random hyperfine field
We study the dynamics, hSz (t)i, of the average spin of electron hopping over sites which host random hyperfine
magnetic fields. If the typical waiting time for a hop is τ and the typical magnetic fields is bs0 , then the
typical spin-precession angle on a given site is δφ ∼ b0 τ 1. Then the Markovian theory predicts that
the spin, initially oriented along the z-axis decays, on average, as hSz (t)i = exp(−t/τs ), where τs =
1/b20 τ is the spin-relaxation time. We find that in low dimensions, d = 1, 2, the decay, hSz (t)i, is nonexponential at all times. The origin of the effect is that for d = 1, 2 a typical random-walk trajectory exhibits
numerous self-intersections. Multiple visits of the carrier to the same site accelerates the relaxation since the
corresponding partial rotations, δφ, of spin during these visits add up. As a result, the Markovian description
does not apply. For one-dimensional diffusion of electron over sites, the average, hSz (t)i, is the universal
2/3
function of t3/2 /τ 1/2 τs , so that the characteristic decay time is τ 1/3 τs is much shorter than τs . Moreover,
when the random magnetic fields are located in the (x, y) plane, the decay of hSz (t)i to zero is preceded by a
reversal of hSz (t)i to the value hSz i = −0.16 at intermediate times. We develop an analytical self-consistent
description of the spin dynamics which explains this reversal. Another consequence of self-intersections of the
random-walk trajectories is that, in all dimensions, the average, hSz (t)i, becomes sensitive to a weak external
magnetic field directed along z. Our analytical predictions are complemented by the numerical simulations of
hSz (t)i.
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
(in collaboration with R.C. Roundy)
Department of Physics
University of Utah
Supported by: MRSEC DMR-1121252
arXiv:1401.4796
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.
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.
Input 1
The Nobel Prize in Physics 2007
Peter Grünberg
Albert Fert
"for the discovery of Giant Magnetoresistance"
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.
Input 2
spin-valve
MRAM
hard drives
biosensors
device efficiency is
limited due to spin-orbit
coupling in the metallic active layer
Input 3
Cited by 2911
Nature 427, 821-824 (26 February 2004)
device efficiency is quantified via:
tunnel (or giant) magnetoresistance
R ↑↓ − R ↑↑
2P1 P2 exp(− d / λ s )
TMR =
=
R ↑↑
1 − P1 P2 exp(− d / λ s )
Cited by 847
polarizations of the electrodes spin diffusion length
d
characteristics of the
spin memory loss
λ s ≈ 45 nm
Mechanism of Spin Memory Loss
Precession of Spinning Top
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)]:

 
dSi
= bi × Si
dt
[
S z ( t ) = Sz (0) exp(− t / τ s )
]
typical spin - rotation
angle
δφ ~ b0 τ << 1
typical
on-site field
< Sz ( N) > = Sz (0) exp( − N δφ 2 )
t = Nτ
< Sz ( t ) > = Sz (0) exp( − t / τ s )
typical
waiting time
where
τs =
1
b02 τ
spin relaxation
time
cos (φ1 + φ 2 + ... + φ N ) = Re exp(i φ1 + i φ 2 + ...i φ N ) = exp (i φ )
N
 Nδφ 2 

= exp  −
2 

exponential decay of
S z (t )
Numerical simulation of spin evolution
[
(
)]
(
)
(


  
 
  
Si = Si −1 − ni ni ⋅ Si −1 cos b0 τ + ni × Si −1 sin b0 τ + ni ni ⋅ Si −1
unidirectional hops :
simple-exponential decay
direction of the hyperfine
field on site i
time in the units
of τ = 1
s
b02 τ
2D hops with random planar fields
Decay of
S z (t )
is strongly non-exponential !
)
qualitative explanation:
multiple visitations of the same site in course of a random walk lead to
accelerated spin-relaxation.
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 3D random walk of N steps only a small portion ~ N-1/2 sites are
visited twice
for a 2D random walk of N steps each site is visited ~ 2 times
qualitative consideration
1D random walk
τs =
1
b02 τ
if all partial rotations are statistically independent:
δφ ~ b0 τ << 1
< Sz ( N) > = Sz (0) exp( − N δφ 2 )
t = Nτ
on the other hand, after N steps of a 1D random walk,
i.
N1/2
sites are visited
ii. each site is visited ~ N1/2 times
N→N
1/ 2
δφ → N1/ 2 δφ
< Sz ( t ) > = Sz (0) exp( − t / τ s )
 t 3/ 2 
< Sz ( t ) > = Sz (0) exp  − 1/ 2 
 τ τs 
faster than a simple exponent
~τ =
s
1
2/3
= τ s (b0 τ ) << τ s
4 / 3 1/ 3
b0 τ
at the minimum S z (t ) = −0.16
Planar hyperfine fields:
relaxation proceeds via
spin reversal
scaling t3/2 is correct,
but the shape is not.
Equation of spin dynamics


dS 
= b (t ) × S
dt
analytical treatment:
closed equation for S z (t )
t
t1
0
0
S z (t ) = 1 − b02 ∫ dt1 ∫ dt 2 cos(φ (t1 ) − φ (t 2 ) )S z (t 2 )
formal solution for a given realization:
magnitude
of the field
random in-plane
orientation of the 2D field
Without returns, averaging should be performed with the help of
the “usual” correlator
Poisson’s distribution of the waiting times
cos(φ (t ) − φ (t ' ) ) = exp(− | t − t ' | / τ ) = C0 (t , t ' )
describing the stay on a given site for a short-time, τ
term-by-term averaging
reproduces the standard result
 t
S (t ) = exp − 
 τs 
0
z
With returns:
for a diffusively traveling particle, there is a probability that it returns
to the same site a long time later!
 
P( r1 , r2 , t ) =
returns
 
 | r1 − r2 |2 
1
exp −

d/ 2
2Dt 
(2π Dt )

diffusion
coefficient
 
Return: r2 = r1


1
cos(φ( t ) − φ( t ' ) ) = 

 2π D ( t − t ' ) 
d/ 2
= CD ( t , t ' )
diffusive correlator
short-time hops modify the diffusive correlator
 | t − t '| 
C D (t , t ' ) → C D (t , t ' ) exp −

2
τ
s 

Spin-memory is lost between two subsequent
visits to the same site
analytical result in 2D
 | t − t'| 
averaged with correlator CD ( t , t ' ) exp −

2
τ
s 

t / 2τ
 u
g2 − t / τs s du  t
− t / τs

− e
− u  e
Sz ( t ) = e
∫
u  2τ s
π

τ / 2 τs
Comparison with
numerics
in the limit t >>τs the
diffusive correction behaves as:
−1
−
g2 is chosen to be 0.75
instead of 1
 t 
g2  t 

 exp −

π  2τs 
2
τ
s

 t 
 − 
exp
falls off slower than
 τs 
there should be a sign reversal
With magnetic field, B, along the z-axis
t
t1
0
0
S z (t ) = 1 − b02 ∫ dt1 ∫ dt 2 cos[φ (t1 ) − φ (t 2 ) + B (t1 − t 2 )] S z (t 2 )
Without returns, the decay remains exponential

t 
S z ( t ) = Sz (0) exp −

(
B
)
τ
s


1 + B2 τ 2
τs =
b02 τ
external field slows
down the decay for
Bτ ≥ 1
b02 τ d / 2 t 2−d / 2 Fd (Bt )
=−
(2π) d / 2
B = 10 τ s−1
B
magnetic field restores
simple-exponential decay
first correction due to returns
π
κ1 (s) =  
 2s 
1/ 2
[Fd (s) − Fd (0)]s >>1 = κ d (s) − cos2 s
s
κ 2 (s) = − ln s
Sensitivity to a weak magnetic fields B~1/τs << 1/τ
analytical treatment in 1D
b02
+
b04
b04
b04
only diffusive correlators are relevant
+
∞
S z ( t ) = ∑ ( −1) n b02 nμ ( n ) ( t )
n=0
t
μ ( n) ( t ) =
t
∫ dt ∫ dt
1
0
t
2
0
infinite series summation is required in 1D
t
CD ( t 1 , t 2 ) ∫ dt 3 ∫ dt 4 CD ( t 3 , t 4 )...
0
0
recurrent relation:
dμ ( n+1)
=
dt
t
( n)
μ
dt
C
t
t
(
,
)
( t1 )
∫ 1 D 1
0
leads to an integral equation
d Sz
dt
b02 τ1/ 2
=−
(2π )1/ 2
dt ′
∫0 (t − t′)1/ 2 Sz (t′)
t
selection of the
higher –order terms
u = b02 τ1/ 2 t 3 / 2
with a new variable
d Sz
du
=−
u
4
9(2 π )
1/ 2
1/ 3
u
∫ u (u
0
1/ 3
1
du1
2/3
)
2 / 3 1/ 2
1
−u
Sz (u1 )
for planar hyperfine fields
for spherically-symmetric fields
-0.28
d Sz
du
=−
4
9(2 π )
1/ 2
u1/ 3
[
(
u
du1 Sz (u1 ) exp − u
0
1/ 3
1
∫
u
(u
2/3
2/3
−u
)
2 / 3 1/ 2
1
−u
)
2 / 3 3/ 2
1
]
With magnetic field, B, along the z-axis
the decay of
b02 τ1/ 2
1 − Sz ( t ) =
(2π )1/ 2
Sz (t ) is accompanied by oscillations
cos B (t1 − t 2 ) b02 τ1/ 2 t 3 / 2
∫0 dt1 ∫0 dt 2 (t1 − t 2 )1/ 2 = (2π )1/ 2
t
planar random field
t1
B
= 0, 1, 2, 3, 4
4 / 3 1/ 3
b0 τ
 π 1/ 2 cos Bt 
 −

2 
2
Bt
(
)
Bt 


spherical random field
Alternative (quantum) description
β( t )


2

spinor χ (t ) = 
probability
that
spin
points
up
|
β
(
t
)
|
2 
1
|
β
(
t
)
|
−


time evolution of spin : S(t ) = 1 − 2 β(t )
α=
Ωt
2
2
evolution matrix
bx − iby

bz

 cos α − i  sin α − i
sin α

| b|
| b|

ˆ
U b,t = 
b + ib
 − i x  y sin α cos α + i bz sin α

| b|
| b|

( )
spin-flip amplitude







( )

ˆ b, τ χ (t )
χ (t + τ ) = U
( ) ( ) ( )



ˆ b ,τ U
ˆ b ,τ U
ˆ b , τ ....
Coherent spin evolution is described by the product: U
1
2
3
effective field
Conclusion:
interference: amplitude of spin flip after visiting three sites with
orientations of hyperfine fields χ1, χ2, χ3
 u
M = (0 1)
iχ 3
ive
−

M
M
2
if the
χ1 , χ 2
2
χ1 , χ 2 , χ 3
− iveiχ 3  u

u  − iveiχ 2
= v 6 + 3u 4 v 2 ≈ 3v 2
= v 6 + 5u 4 v 2 ≈ 5v 2
− iveiχ 2  u

u  − iveiχ1
1 − Sz
1 − Sz
− iveiχ1  1 
 
u  0 
correlated
=
uncorrelated
acceleration of spin relaxation
sites 1 and 3 are the same χ1 = χ3
5
3