Journal ArticleDOI

# Rashba spin precession in a magnetic field

, Jun Wang2
11 Feb 2004-Physical Review B (American Physical Society)-Vol. 69, Iss: 8, pp 085304

AbstractSpin precession due to Rashba spin-orbit coupling in a two-dimension electron gas is the basis for the spin field effect transistor, in which the overall perfect spin-polarized current modulation could be acquired. There is a prerequisite, however, that a strong transverse confinement potential should be imposed on the electron gas or the width of the confined quantum well must be narrow. We propose relieving this rather strict limitation by applying an external magnetic field perpendicular to the plane of the electron gas because the effect of the magnetic field on the conductance of the system is equivalent to the enhancement of the lateral confining potential. Our results show that the applied magnetic field has little effect on the spin precession length or period although in this case Rashba spin-orbit coupling could lead to a Zeeman-type spin splitting of the energy band.

Topics: Spintronics (65%), Spin–orbit interaction (65%), Larmor precession (64%), Spin polarization (64%), Spin Hall effect (63%)

### Summary

• Spin precession due to Rashba spin-orbit coupling in a two-dimension electron gas is the basis for the spin field effect transistor, in which the overall perfect spin-polarized current modulation could be acquired.
• There is a prerequisite, however, that a strong transverse confinement potential should be imposed on the electron gas or the width of the confined quantum well must be narrow.
• The authors propose relieving this rather strict limitation by applying an external magnetic field perpendicular to the plane of the electron gas because the effect of the magnetic field on the conductance of the system is equivalent to the enhancement of the lateral confining potential.

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Rashba spin precession in a magnetic ﬁeld
Jun Wang,
1,2
H. B. Sun,
1
and D. Y. Xing
2
1
Center of Quantum Computing Technology and School of Physical Sciences, University of Queensland, Brisbane Qld 4072, Australia
2
National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
Received 11 August 2003; published 11 February 2004
Spin precession due to Rashba spin-orbit coupling in a two-dimension electron gas is the basis for the spin
ﬁeld effect transistor, in which the overall perfect spin-polarized current modulation could be acquired. There
is a prerequisite, however, that a strong transverse conﬁnement potential should be imposed on the electron gas
or the width of the conﬁned quantum well must be narrow. We propose relieving this rather strict limitation by
applying an external magnetic ﬁeld perpendicular to the plane of the electron gas because the effect of the
magnetic ﬁeld on the conductance of the system is equivalent to the enhancement of the lateral conﬁning
potential. Our results show that the applied magnetic ﬁeld has little effect on the spin precession length or
period although in this case Rashba spin-orbit coupling could lead to a Zeeman-type spin splitting of the energy
band.
DOI: 10.1103/PhysRevB.69.085304 PACS numbers: 73.21.b, 71.70.Ej, 73.40.Sx
Spin-polarized electron transport in microstructures has
attracted considerable attention since last decade, fueled by
the possibility of producing efﬁcient photoemitters with a
high degree of polarization of the electron beam spin light-
emitting diode, creating spin-based memory device and uti-
lizing the properties of spin coherence for quantum compu-
tation and communication. In the spintronics spin-based
electronics ﬁeld, both degree of freedom of spin and charge
are exploited, even spin could entirely replace the electric
charge to carry information. This is the basis for a new gen-
eration of electric devices.
1
The spin-polarized ﬁeld effect transistor SFET proposed
by Datta and Das
2
is one of the most attractive spintronic
devices for it may switch faster than the traditional transistor
since it can avoid redistributing charges during operation.
The idea is based on Rashba spin-orbit RSO coupling
3
in
two-dimensional electron gas 2DEG. It results in spin pre-
cession as electrons move along a heterostructure and can be
controlled by an external electric ﬁeld. This novel spintronic
device has three requirements: 1 long spin-relaxation time
in 2DEG; 2 gate voltage control of RSO coupling, and 3
high spin injection efﬁciency. At present, the ﬁrst two con-
ditions have been basically satisﬁed in experiments.
4,5
It ap-
pears to be very difﬁcult, however, to achieve an efﬁcient
injection of spin-polarized carriers from a ferromagnetic
metal into 2DEG, and a great deal of work has been dedi-
cated to this challenge.
6–11
Apart from the three requirements above for an SFET, in
fact, there is another basic limitation to the ultimate imple-
mentation of SFET, i.e., in order to restrict the angular dis-
tribution of electrons in a 2DEG,
2
a strong enough transverse
conﬁning potential must be imposed on the 2DEG or the
width of the conﬁned quantum well must be very narrow.
The RSO interaction in the 2DEG comes from the inversion
asymmetry of the structure and can be expressed as
3
H
R
(
/)(
p), where
is Pauli matrix,
is the RSO cou-
pling constant proportional to the external electric ﬁeld E,
and p is the momentum operator. The term H
R
itself can lift
the degeneracy of spin space but not lead to a Zeeman-type
split of energy band, because the time inversion symmetry of
system remains unchangeable under this RSO interaction.
When an electron propagates, however, the RSO coupling
can result in spin precession of electronic current along its
propagating way due to the interference of two spin-splitting
electronic waves. To ensure the perfect spin modulation of
electric current in SFET, the energy gap between two neigh-
boring subbands due to the lateral conﬁning potential, which
is generally assumed to have reﬂection symmetry, must be
much larger than the intersubband mixing from RSO
coupling,
2,12
i.e.,
n
H
R
n 1
/(
n1
n
) 1 with n being
the index of subband. Therefore, the subband energy disper-
sion from RSO keeps linear k dependence. It has been
argued
12
that in the hard-wall conﬁning potential, the width
of the quantum well must satisfy W
2
/
m with m being
the effective mass of electrons. From this, one can see that
the RSO coupling constant
modulated by an external elec-
tric ﬁeld is strongly limited by the width W of transverse
conﬁned potential well.
In this paper, we propose to employ an external magnetic
ﬁeld to relieve the limitation to a strong transverse conﬁning
potential or the narrow well width. The Landau level will
form in the magnetic ﬁeld so that the energy gap of intersub-
bands could be enlarged and the RSO coupling constant
could be modulated in a larger range, in which the perfect
spin-current modulation could not be destroyed. The mag-
netic ﬁeld effect is equivalent to the enhancement of the
conﬁning potential or the reduction of the effective width of
the quantum wire. However, does it introduce another factor
to destroy the spin precession? To answer this question is
another motive of this paper that investigates the interplay
between the RSO coupling and the external magnetic ﬁeld.
Our numerical results show that the RSO coupling in mag-
netic ﬁeld will lead to spin split of the subband spectrum like
Zeeman effect, while the spin precession from RSO coupling
keeps almost invariable such as its length or period.
The model we adopted is a two-terminal device that a
quasi-one-dimensional quantum wire with RSO coupling is
connected by two ideal leads. This device is subjected to a
magnetic ﬁeld perpendicular to the two-dimensional plane
xy plane, it is assumed that x is the current direction of the
PHYSICAL REVIEW B 69, 085304 2004
0163-1829/2004/698/0853045/$22.50 ©2004 The American Physical Society69 085304-1 device. Considering the real condition in experiment, the magnetic ﬁeld applied to the RSO coupling region is as- sumed inhomogeneous and being tuned adiabatically on and off as in Ref. 13. After discreting procedure, a type of tight- binding Hamiltonian including the RSO coupling on a square lattice is obtained in absence of magnetic ﬁeld, 12 H lm lm C lm C lm t lm C l 1,m C lm C l,m 1, C lm H.c t so lm ␴␴ C l 1,m, i y ␴␴ C lm C l,m 1, i x ␴␴ C lm H.c. , 1 where C lm (C lm ) is the creation annihilation operator of electron at site lm with spin and lm 4t is the site- energy, t 2 /2ma 2 is the hopping energy, a is the lattice constant, and t so /2a is the RSO coupling strength. Here we focus on the case of an impurity-free quantum wire with RSO coupling. The generalization to the case including im- purities is straightforward. 14 In our following calculation, all energy is normalized by the hoping energy t(t 1). When the magnetic ﬁeld B(0,0,1) is introduced, it could be incor- porated into the nearest-neighbor hopping energy by the Peierl’s phase factor such as T lm,lm 1 t exp i c l/2t T lm 1,lm ;T lm,l 1,m T l 1m,lm t, 2 where c eB/mc is the cyclotron frequency. We choose vector potential A(By,0,0) and keep the transitional symme- try of system along the x direction electric current direc- tion. In magnetic ﬁeld, RSO coupling Hamiltonian is reex- pressed as H R ( /) (p eA/c) so that t so has a similar modiﬁcation. The spin-quantum axis is chosen along z direction. The Zeeman effect from the external magnetic ﬁeld is not included here. In the ballistic transport, the conductance of structure is given by Landauer-Buttiker formula G (e 2 /h)T and T is the multichannel transmission coefﬁcient of electron with spin . Based upon the nonequilibrium Green function formalism, 15 the following result for spin-resolved conduc- tance is obtained 14 G ␴␴ e 2 h Tr L G r ␴␴ R G a , 3 where L(R) i L(R) r L(R) a , L(R) r ( L(R) a ) is the self energy from the left right lead, G r(a) is the retarded ad- vanced Green function of the structure, and the lead effect is incorporated into the self energy of green function G r(a) . The trace is over the spatial degrees of freedom. The Green function above is computed by the well-known recursive Green function method 16,17 and the conductance is evaluated at the Fermi energy. Our following discussion is based on the assumption that only spin-up polarized electrons are injected from the left lead into RSO region where the spin preces- sion of incident electron is induced and collected in the right lead. We have chosen the lattice size a much smaller than the Fermi wavelength F ( F 10a) so that our model can simulate a continuum system. The width and the length of the quantum wire are taken as N y a 30a and N x a 60a, respectively. The calculated results of the conductance are plotted in Fig. 1 as a function of the external magnetic ﬁeld in the absence of RSO interaction. It appears that the con- ductance is quantized and decreases with magnetic ﬁeld. At w c 0, there are 6 modes subbands at the Fermi energy (E F 0.4t) contributing to the conductance. When w c 0, all subbands are elevated and the energy gap of intersub- bands increases due to the formation of Landau level as shown in Fig. 2 more detailed later. The transmitted modes below E F thus becomes less and the conductance is quan- tized and decreases with B. Basically, when w c 2E F , there is no transmitted mode contributing to conductance and it decreases to zero. The conductance quantization induced by magnetic ﬁeld resembles that found in quantum point contact, in which the number of modes at E F will change in discrete steps by constricting continuously its width. 18 In other word, the magnetic ﬁeld effect on the conductance of a quantum wire is equivalent to the reduction of its effective width or enhancement of the conﬁning potential because a strong transverse potential will also lead to decrease of the number of modes at E F . 19 Apart from the quantization of conductance in Fig. 1, an- other character, the oscillation of conductance is also found in our results. This oscillation is referred to as the Aharonov- Bohm AB effect 20,21 and originates from the edge state 22 in a magnetic ﬁeld. Due to the multireﬂection of electrons in quantum wire before they escape to the collector, the right- going channel and left-going channel form a loop resulting from the perpendicular magnetic ﬁeld so that the quantum interference will lead to AB effect as that in a mesoscopic ring. The oscillation periodicity is related to the wave vector of the transmitted mode and the length of the multireﬂection region. Consequently, the oscillation becomes apparent just FIG. 1. The conductance of quantum wire as a function of mag- netic ﬁeld c in absence of RSO coupling t so 0. The inset en- larges the points just above the threshold of the second transmitted mode. WANG, SUN, AND XING PHYSICAL REVIEW B 69, 085304 2004 085304-2 above the threshold of every transmitted mode where the wave vector turns out to be smaller. The subband energy spectrum is plotted in Fig. 2 at the presence of RSO coupling. Since the contribution to conduc- tance of those evanescent modes could be neglected, only these modes at E F are shown. When there is no magnetic ﬁeld B 0 in Fig. 2a, the spectra seem to be simple para- bolic due to a weak RSO coupling t so 0.02t used in our calculation. 19 The degeneracy of spin space is lifted by the RSO coupling, however, it does not resemble Zeeman effect that leads to split of the energy band, and here the spin de- generacy at k0 still exists. When the intersubband mixing from RSO coupling is neglected, the spin-resolved eigenval- ues are approximately at every subband (k) n 2 k 2 /2m k, denotes two spin-splitting bands due to the RSO coupling and not the eigenstates of z yet. Once B0, the Landau levels form in the system and this is the reason of the platform appearing at the bottom of subbands near k 0) as shown in Fig. 2b. Both the subbands and their energy difference are enhanced in comparison with Fig. 2a when the external magnetic ﬁeld increases, moreover, the gap of intersubbands is basically equivalent to w c .Itis interesting to note that the RSO coupling will lead to a Zeeman-type energy-band split under B0. In an ideal 2DEG under a magnetic ﬁeld, the plane waves of eigenstates have no group velocity, / k 0, and the RSO modiﬁca- tion has no relation with wave vector k. At this moment, the difference of the two spin eigenvalues is ⌬␧ (2 2m/) w c (n 1), here n denotes the Landau level. Thus the RSO spin-splitting strength is related to the energy level index n and the magnetic ﬁeld B. The effect of an external magnetic ﬁeld is equivalent to the enhancement of the transverse conﬁning potential on a quantum wire, which results in an enlargement of the inter- subband energy gap so that the perfect spin-polarized current modulation can be kept in a larger parameter region. In Fig. 3, two spin-splitted conductances are presented as the RSO coupling strength t so varies. At B 0, the conductance modulation is quickly and clearly weakened by the intersub- band mixing that increases with RSO coupling; otherwise, as B increases, the subband mixing from RSO coupling could be neglected compared to the intersubband energy gap, and the perfect spin modulation of conductance would remain in a larger RSO coupling range as shown in Fig. 3b, where FIG. 2. The subband energy spectrum with RSO coupling strength t so 0.02; a no mag- netic ﬁeld c 0 and b兲ប c 0.24. FIG. 3. The spin modulation of conductance vs RSO coupling strength t so ; the solid line and dash line represent G ↑↑ and G ↑↓ , respectively. a兲ប c 0 and b兲ប c 0.40. RASHBA SPIN PRECESSION IN A MAGNETIC FIELD PHYSICAL REVIEW B 69, 085304 2004 085304-3 only one transmitted mode contributing to conductance is chosen to be plotted as a result of the magnetic ﬁeld effect. This oscillation of conductance originates from the interfer- ence of two RSO spin-splitted electronic waves in one sub- band. The oscillation period can be determined by the accu- mulated phase difference 2t so * N x , here N x a is the length of RSO interaction region. For instance, N x 60 is chosen in our calculation and the oscillation period is esti- mated T t s0 0.512. The RSO coupling is the basic principle for the operation of an SFET. One can control the output spin polarization of the SFET by tuning the RSO coupling con- stant via an external electric ﬁeld. In a real device, it is also required that spin dephasing length should be much longer than device size to avoid spin mixing. While in order to avoid the distraction of spin modulation at the collector by the intersubband mixing, an external magnetic ﬁeld may be an alternative as discussed above. Another point to be noted is that even B0, the oscilla- tion period from RSO coupling is still independent of the energy of injected electrons and the magnetic ﬁeld, i.e., two spin-resolved conductances are mainly determined by the RSO coupling constant and the length of RSO region N x a, when the number of mode at E F is ﬁxed. In Fig. 4, we plot the conductances as a function of magnetic ﬁeld at different and N x . It is shown that the conductance G and G keep almost unvaried as the magnetic ﬁeld B varies. This case is similar to that they are independent of the energy of injected electrons at a weak RSO coupling region unless the different transmitted modes at Fermi energy are involved not shown when the energy of injected electrons varies. 12 In summary, we have investigated the ballistic transport of a quasi-one-dimensional quantum wire considering the RSO interaction under an external magnetic ﬁeld. 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When at large distance$r$, this induced electric field$\stackrel{P\vec}{E}$scales as$1∕{r}^{2}$, whereas the$\stackrel{P\vec}{E}$field generated from the linear spin current goes as$1∕{r}^{3}$. 120 citations Journal ArticleDOI Abstract: We investigate the influence of a perpendicular magnetic field on the spectral and spin properties of a ballistic quasi-one-dimensional electron system with Rashba effect. The magnetic field strongly alters the spin-orbit induced modification to the subband structure when the magnetic length becomes comparable to the lateral confinement. A subband-dependent energy splitting at$k=0\$ is found which can be much larger than the Zeeman splitting. This is due to the breaking of a combined spin orbital-parity symmetry.

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##### References
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