Amplification and Relaxation of
Electron Spin Polarization in
Semiconductor Devices
Yuriy V. Pershin and Vladimir Privman
Center for Quantum Device Technology, Clarkson University,
Potsdam, New York 13699-5720, USA
Spin Relaxation of Conduction Electrons
Due to Interaction with Nuclear Spins
Abstract: Relaxation of conduction electron spins in semiconductors owing to
the hyperfine interaction with spin-½ nuclei, in zero applied magnetic field, is
investigated. We calculate the electron spin relaxation time scales, in order to
evaluate the importance of this relaxation mechanism. Master equations for
the electron spin density matrix are derived and solved. Polarized nuclear
spins can be used to polarize the electrons in spintronic devices.
G
σ
G
Ii
G
I i +1
G
I i+2
G
I i +3
G
I i +4
Main assumptions of the model:
ƒ Electrons move classically and interact with nuclear spins located on electron trajectories.
ƒ At each moment of time an electron interact with only one nuclear spin or does not interact at all.
ƒ Change of electron spin state occurs only due to interaction with nuclear spins (zero external magnetic field,
spin-orbit coupling, etc..).
G
G
H = σ ⋅ Ai ( t ) I i
ƒ Hamiltonian of electron-nuclear spin interaction:
.
i
Ai = Ai θ t i + δ ti − θ (ti )
ƒ The constant of interaction is selected in the form:
.
ƒ Low electron density: back reaction of electron on nuclear spins is neglected.
((
∑
)
)
Evolution of the electron density matrix during a single interaction
ρ (t i ) = ρ ei −1 ⊗ ρ i
The initial density matrix of the two-spin system is
The evolution equation :
The resulting electron density matrix:
sin ( 4ai ) x
1
ρ = ρ cos ( 2ai ) + (1 + Pi z ) sin 2 ( 2 ai ) +
( Pi Im ρ10i−1 − Pi y Re ρ10i −1 )
2
2
sin ( 4 ai ) 

 e − i 2 ai

x
y
i −1
sin
2
cos ( 2ai ) 
ρ10i = ρ10i −1  cos 2 ( 2ai ) + iPi z
+
+
− ρ 00
i
P
iP
a
(
)
 ( i
i )
i 
2
2




i
00
i −1
00
ρ ( ti + δ ti ) = e
i
− H i δ ti
=
ρ (ti ) e
ρ ei = trI ρ ( ti + δ ti )
i
H i δ ti
=
.
.
.
i
2
Master equations that connect the electron density
matrix elements before and after the interaction.
Electron density matrix after interaction with many nuclear spins
We found that electron spin relaxes exponentially to the direction of nuclear spin polarization.
Longitudinal and transverse spin relaxation time: T = −
Polarized nuclear spins
a)
b)
∆t
∆t
, T⊥ = −
,
G
2 
2 ln(cos(2a ))

ln cos(2a ) cos 2 (2a ) + sin 2 (2a ) P 


Aδ t
where
.
ai = i i
=
Evolution of the electron spin
density matrix due to interaction
with completely polarized nuclear
spins in a) (-z)-direction. and b)
(+x)-direction. Initial electron spin
polarization is in (+z)-direction.
ai=0.01.
Unpolarized nuclear spins
a)
b)
Evolution of the electron spin
density matrix elements caused
by interaction with unpolarized
nuclear spins. Unpolarized
nuclear spins were modeled by
zero polarization vector (dotted
and dashed lines), and by unit
polarization vector directed
randomly (noisy data lines), with
a) ai=0.01 and b) ai=0.003.
Reference: Yu. V. Pershin and V. Privman, Nano Lett., 3, 695 (2003).
Focusing of Spin Polarization in Semiconductors
by Inhomogeneous Doping
Abstract: We study the evolution and distribution of non-equilibrium
electron spin polarization in n-type semiconductors within the twocomponent drift-diffusion model in an applied electric field. Propagation of
spin-polarized electrons through a boundary between two semiconductor
regions with different doping levels (n/n+ junction) is considered. We
assume that inhomogeneous spin polarization is created locally and driven
through the boundary by the electric field. We show that an initially created
narrow region of spin polarization can be further compressed and amplified
near the boundary. Since the boundary involves variation of doping but no
real interface between two semiconductor materials, no significant spinpolarization loss is expected. The proposed mechanism will be therefore
useful in designing new spintronic devices.
Physical Model
∂n↑(↓ )
(
)
G
G
e
e
= div j↑(↓ ) +
n↓(↑ ) − n↑(↓ ) + S ↑(↓ ) (r , t )
2τ sf
G
G ∂t
j↑(↓ ) = σ ↑(↓ ) E + eD∇n↑(↓ )
G
e
(N i − n )
div E =
εε 0
The equation for electric field profile:
The equation for spin polarization density:
Injection of spin-polarized electrons
in a system with two levels of
doping.
Two-component drift-diffusion model.
Here: n= n↑ +n↓ is the electron density;
P = n↑ – n↓ is the spin polarization density;
S describes the source of spin polarization.
j0
e
∂ 2 E e ∂E e 2 N i
+
E
−
E
=
−
+
∇N i
εε 0 D εε 0
∂x 2 kT ∂x kTεε 0
G
G
G
∂P
eE
e∇E
P
= D∆P + D
∇P + D
P−
+ F (r , t )
∂t
k BT
k BT
τ sf
Results:
Electric field profile near the
Dynamics of propagation through
Distribution of the spin polarization
boundary, N2/N1=5.
the boundary of spin-polarized
density created by a point source
electrons injected at τ=0, for
N2/N1=10. The blue curve denotes
located at x=-10. Spin accumulation
effect near the boundary becomes
the electric field. The other curves
more pronounced with increased N2.
show the distribution of the spin
polarization density at different
Conclusions:
times.
ƒ Propagation of spin-polarized electrons through n/n+ junction studied within drift-diffusion model.
ƒ Spatial distribution of the electron spin polarization is calculated for different doping levels.
ƒ We found that electron spin polarization can be enhanced at the boundary between two semiconductors if
electrons drift from low-doped to high-doped region.
ƒ This mechanism of electrons spin amplification can be useful in designing new spintronic devices.
Reference: Yu. V. Pershin and V. Privman, Phys. Rev. Lett., 90, 256602 (2003).
Spin Relaxation of Electrons in 2DEG with
Antidot Lattice
Abstract: We study the effect of Sinai billiard lattice (periodic lattice of
r
a
disks in 2DEG) on electron spin relaxation due to D’yakonov-Perel’
relaxation mechanism. Using a Monte Carlo simulation scheme we
show that in such system electron spin relaxation can be efficiently
suppressed. We found that in a certain regime the electron spin
relaxation time increases exponentially with the disk radius. If the radius
to the lattice period aspect ratio is fixed, a power-law dependence of the
electron spin relaxation time on the lattice period is obtained.
Two-dimensional lattice of antidots.
Physical model:
ƒ Electron spin relaxation is of D’yakonov-Perel’ type.
ƒ Electron spin relaxation time is calculated using a Monte Carlo
algorithm with the following main assumptions:
ƒ the spatial motion of electrons is considered semiclassically;
ƒ all electrons have the same velocity;
ƒ scattering is considered to be elastic and isotropic;
ƒ reflecting and diffusive boundary conditions are assumed;
ƒ Electron spin relaxation time is calculated as a function of a and r.
An example of electron trajectory
when antidots almost are tangent to
each other.
Results
Electron spin relaxation time as a
Relaxation time at fixed r/a as a
Relaxation time at fixed r as a
function of antidot radius for
function of the distance between
function of the distance between
different distances between the
antidotes.
antidotes.
antidots, ηLp=0.2. The straight lines
are fitting functions.
Conclusions:
ƒ Electron spin relaxation time was calculated using a Monte Carlo simulation program.
ƒ It was found that DP spin relaxation mechanism could be suppressed by antidot lattice.
ƒ Spin relaxation time increases exponentially with increase of antidot radius r.
ƒ Scaling of parameters leads to an unusual change of relaxation time.
ƒ Diffusive scattering on the disks leads to an additional increase of electron spin relaxation time.
Reference: Yu. V. Pershin and V. Privman, preprint.
Acknowledgments
This research was supported by the National Science Foundation, grants DMR-0121146 and
ECS-0102500, and by the National Security Agency and Advanced Research and Development
Activity under Army Research Office contract DAAD 19-02-1-0035.