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Spin relaxation of “upstream” electrons: beyond the driftdiﬀusion model
Sandipan Pramanik and Supriyo Bandyopadhyay
∗
Department of Electrical and Computer Engineering,Virginia Commonwealth University, Richmond, VA 23284, USA
Marc Cahay
Department of Electrical and Computer Engineering and Computer Science,University of Cincinnati, Cincinnati, OH 45221, USA
Abstract
The classical drift diﬀusion (DD) model of spin transport treats spin relaxation via an empiricalparameter known as the “spin diﬀusion length”. According to this model, the ensemble averagedspin of electrons drifting and diﬀusing in a solid decays exponentially with distance due to spindephasing interactions. The characteristic length scale associated with this decay is the spindiﬀusion length. The DD model also predicts that this length is diﬀerent for “upstream” electronstraveling in a decelerating electric ﬁeld than for “downstream” electrons traveling in an acceleratingﬁeld. However this picture ignores energy quantization in conﬁned systems (e.g. quantum wires)and therefore fails to capture the nontrivial inﬂuence of subband structure on spin relaxation. Herewe highlight this inﬂuence by simulating upstream spin transport in a multisubband quantum wire,in the presence of D’yakonovPerel’ spin relaxation, using a semiclassical model that accounts forthe subband structure rigorously. We ﬁnd that upstream spin transport has a complex dynamicsthat deﬁes the simplistic deﬁnition of a “spin diﬀusion length”. In fact, spin does not decayexponentially or even monotonically with distance, and the drift diﬀusion picture fails to explainthe qualitative behavior, let alone predict quantitative features accurately. Unrelated to spintransport, we also ﬁnd that upstream electrons undergo a “population inversion” as a consequenceof the energy dependence of the density of states in a quasi onedimensional structure.
PACS numbers: 72.25.Dc, 85.75.Hh, 73.21.Hb, 85.35.Ds
∗
Corresponding author. Email: sbandy@vcu.edu
I. INTRODUCTION
Spin transport in semiconductor structures is a subject of much interest from the perspective of both fundamental physics and device applications. A number of diﬀerent formalismshave been used to study this problem, primary among which are a classical drift diﬀusionapproach [1, 2, 3], a kinetic theory approach [4], and a microscopic semiclassical approach[3, 5, 6, 7, 8, 9, 10, 11]. The central result of the drift diﬀusion approach is a diﬀerentialequation that describes the spatial and temporal evolution of carriers with a certain spinpolarization
n
σ
. Ref. [3] derived this equation for a number of special cases starting fromthe Wigner distribution function. In a coordinate system where the xaxis coincides withthe direction of electric ﬁeld driving transport, this equation is of the form:
∂n
σ
∂t
−
D
∂
2
n
σ
∂x
2
−
A
∂n
σ
∂x
+
B
n
σ
= 0 (1)where
D
=
D
0 00
D
00 0
D
,
(2)(3)
D
is the diﬀusion coeﬃcient, and
A
and
B
are dyadics (9component tensors) that dependon
D
, the mobility
µ
and the spin orbit interaction strength in the material.Solutions of Equation (1), with appropriate boundary conditions, predict that the ensemble averaged spin

S

(
x
) =
S
x
2
(
x
) +
S
y
2
(
x
) +
S
z
2
(
x
) should decay exponentiallywith
x
according to:

S

(
x
) =

S

(0)
e
−
x/L
(4)where1
L
=
µE
2
D
+
µE
2
D
2
+
C
2
.
(5)Here
E
is the strength of the driving electric ﬁeld and
C
is a parameter related to the spinorbit interaction strength.The quantity
L
is the characteristic length over which

S

decays to 1
/e
times its srcinalvalue. Therefore, it is deﬁned as the “spin diﬀusion length”. Equation (5) clearly shows that
spin diﬀusion length depends on the
sign
of the electric ﬁeld
E
. It is smaller for upstreamtransport (when
E
is positive) than for downstream transport (when
E
is negative).This diﬀerence assumes importance in the context of spin injection from a metallic ferromagnet into a semiconducting paramagnet. Ref. [1] pointed out that the spin injectioneﬃciency across the interface between these materials depends on the diﬀerence between thequantities
L
s
/σ
s
and
L
m
/σ
m
where
L
s
is the spin diﬀusion length in the semiconductor,
σ
s
is the conductivity of the semiconductor,
σ
m
is the conductivity of the metallic ferromagnet,and
L
m
is the spin diﬀusion length in the metallic ferromagnet. Generally,
σ
m
>> σ
s
. However, at suﬃciently high retarding electric ﬁeld,
L
s
<< L
m
, so that
L
s
/σ
s
≈
L
m
/σ
m
. Whenthis equality is established, the spin injection eﬃciency is maximized. Thus, ref. [1] claimedthat it is possible to circumvent the infamous “conductivity mismatch” problem [13], whichinhibits eﬃcient spin injection across a metalsemiconductor interface, by applying a highretarding electric ﬁeld in the semiconductor close to the interface. A tunnel barrier betweenthe ferromagnet and semiconductor [14] or a Schottky barrier [15, 16] at the interface doesessentially this and therefore improves spin injection.The result of ref. [1] depends on the validity of the drift diﬀusion model and Equation (4)which predicts an exponential decay of spin polarization in space. Without the exponentialdecay, one cannot even deﬁne a “spin diﬀusion length”
L
. The question then is whetherone expects to see the exponential decay under all circumstances, particularly in quantumconﬁned structures such as quantum wires. The answer to this question is in the
negative
.Equation (1), and similar equations derived within the drift diﬀusion model, do not accountfor energy quantization in quantum conﬁned systems and neglect the inﬂuence of subbandstructure on spin depolarization. This is a serious shortcoming since in a semiconductorquantum wire, the spin orbit interaction strength is
diﬀerent
in diﬀerent subbands. It isthis diﬀerence that results in D’yakonovPerel’ (DP) spin relaxation in quantum wires.Without this diﬀerence, the DP relaxation will be completely absent in quantum wires andthe corresponding spin diﬀusion length will be always inﬁnite [17]. The suband structure istherefore vital to spin relaxation.
II. SEMICLASSICAL MODEL OF SPIN RELAXATION
In this paper, we have studied spin relaxation using a microscopic semiclassical modelthat is derived from the Liouville equation for the spin density matrix [18, 22]. Our modelhas been described in ref. [10] and wil not be repeated here. This model allows us to studyD’yakonovPerel’ spin relaxation taking into account the detailed subband structure in thesystem being studied.In technologically important semiconductors, such as GaAs, spin relaxation is dominatedby the D’yakonovPerel’ (DP) mechanism [12]. This mechanism arises from the Dresselhaus[19] and Rashba [20] spin orbit interactions that act as momentum dependent eﬀective magnetic ﬁelds
B
(
k
). An electron’s spin polarization vector
S
precesses about
B
(
k
) accordingto the equation
d
S
dt
=
Ω
(
k
)
×
S
(6)where Ω(
k
) is the angular frequency of spin precession and is related to
B
(
k
) as
Ω
(
k
) =(
e/m
∗
)
B
(
k
), where
m
∗
is the electron’s eﬀective mass. If the direction of
B(k)
changesrandomly due to carrier scattering which changes
k
, then ensemble averaging over the spinsof a large number of electrons will lead to a decay of the ensemble averaged spin in space andtime. This is the physics of the DP relaxation in bulk and quantum wells. In a quantumwire, the direction of
k
never changes (it is always along the axis of the wire) in spite of scattering. Nevertheless, there can be DP relaxation in a
multisubband
quantum wire, aswe explain in the next paragraphs.We will consider a quantum wire of rectangular crosssection with its axis along the [100]crystallographic orientation (which we label the xaxis), and a symmetry breaking electricﬁeld
E
y
is applied along the yaxis to induce the Rashba interaction (refer Fig. 1). Then,the components of the vector
Ω
(
k
) due to the Dresselhaus and Rashba interactions are givenby
Ω
D
(
k
) = 2
a
42
nπW
y
2
−
mπW
z
2
k
x
ˆ
x
; (
W
z
> W
y
)
Ω
R
(
k
) = 2
a
46
E
y
k
x
ˆ
z
(7)where
a
42
and
a
46
are material constants, (
m,n
) are the transverse subband indices,
k
x
is thewavevector along the axis of the quantum wire, and
W
z
,W
y
are the transverse dimensions
of the quantum wire along the z and ydirections. Therefore,
B
(
k
) =
m
∗
e
[
Ω
D
(
k
) +
Ω
R
(
k
)]= 2
m
∗
a
42
e
−
mπW
z
2
+
nπW
y
2
k
x
ˆ
x
+ 2
m
∗
a
46
e
E
y
k
x
ˆ
z
(8)Thus,
B (k)
lies in the xz plane and subtends an angle
θ
with the wire axis (xaxis)given by
θ
= arctan
a
46
E
y
a
42
mπW
z
+
nπW
y
nπW
y
−
mπW
z
(9)Note from the above that in any given subband in a quantum wire, the
direction
of
B
(
k
)is ﬁxed, irrespective of the magnitude of the wavevector
k
x
, since
θ
is independent of
k
x
. Asa result, there is no DP relaxation in any given subband, even in the presence of scattering.However,
θ
is diﬀerent in diﬀerent subbands because the Dresselhaus interaction is diﬀerent in diﬀerent subbands. Consequently, as electrons transition between subbands becauseof intersubband scattering, the angle
θ
, and therefore the direction of the eﬀective magneticﬁeld
B
(
k
), changes. This causes DP relaxation in a
multisubband
quantum wire. Sincespins precess about diﬀerent axes in diﬀerent subbands, ensemble averaging over electronsin all subbands results in a gradual decay of the net spin polarization. Thus, there is no DPspin relaxation in a quantum wire if a single subband is occupied, but it is present if multiplesubbands are occupied and intersubband scattering occurs. This was shown rigorously inref. [17].The subband structure is therefore critical to DP spin relaxation in a quantum wire. Infact, if a situation arises whereby all electrons transition to a single subband and remainthere, further spin relaxation due to the DP mechanism will cease thereafter. In this case,spin no longer decays, let alone decay exponentially with distance. Hence, spin depolarization (or spin relaxation) cannot be parameterized by a constant spin diﬀusion length.The rest of this paper is organized as follows. In the next section, we describe our modelsystem, followed by results and discussions in section IV. Finally, we conclude in section V.
III. MODEL OF UPSTREAM SPIN TRANSPORT
We consider a non centrosymmetric (e.g. GaAs) quantum wire with axis along [100]crystallographic direction. We choose a three dimensional Cartesian coordinate system with