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

共Received 11 August 2003; published 11 February 2004兲

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 number共s兲: 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 共spin light-

emitting diode兲, creating spin-based memory device and uti-

lizing the properties of spin coherence for quantum compu-

tation and communication. In the spintronics 共spin-based

electronics兲 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 共SFET兲 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 共RSO兲 coupling

3

in

two-dimensional electron gas 共2DEG兲. 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: 共1兲 long spin-relaxation time

in 2DEG; 共2兲 gate voltage control of RSO coupling, and 共3兲

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

⫽ (

␣

/ប)(

⫻ 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.,

具

n

兩

H

R

兩

n⫹ 1

典

/(

n⫹1

⫺

n

)Ⰶ 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 WⰆប

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

共xy plane, it is assumed that x is the current direction of the

PHYSICAL REVIEW B 69, 085304 共2004兲

0163-1829/2004/69共8兲/085304共5兲/$22.50 ©2004 The American Physical Society69 085304-1

device兲. 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⫽

兺

lm

lm

C

lm

†

C

lm

⫺ t

兺

lm

兵

C

l⫹ 1,m

†

C

lm

⫹ C

l,m⫹ 1,

†

C

lm

⫹ H.c

其

⫺ t

so

兺

lm

⬘

兵

C

l⫹ 1,m,

⬘

†

共

i

y

兲

⬘

C

lm

⫺ C

l,m⫹ 1,

⬘

†

共

i

x

兲

⬘

C

lm

⫹ H.c.

其

, 共1兲

where C

lm

†

(C

lm

) is the creation 共annihilation兲 operator of

electron at site 共lm兲 with spin

and

lm

⫽ 4t is the site-

energy, t⫽ប

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

共

iប

c

l/2t

兲

⫽

共

T

lm⫹ 1,lm

兲

쐓

;T

lm,l⫹ 1,m

⫽

共

T

l⫹ 1m,lm

兲

쐓

⫽ t, 共2兲

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 共electric current direc-

tion兲. In magnetic ﬁeld, RSO coupling Hamiltonian is reex-

pressed as H

R

⫽ (

␣

/ប)

⫻ (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

关

⌫

L

G

r

⬘

⌫

R

⬘

G

a

⬘

兴

, 共3兲

where ⌫

L(R)

⫽ i

关

⌺

L(R)

r

⫺ ⌺

L(R)

a

兴

, ⌺

L(R)

r

⫽ (⌺

L(R)

a

)

쐓

is the self

energy from the left 共right兲 lead, G

r(a)

is the retarded 共ad-

vanced兲 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 共where the spin preces-

sion of incident electron is induced兲 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

បw

c

⫽ 0, there are 6 modes 共subbands兲 at the Fermi energy

(E

F

⫽ 0.4t) contributing to the conductance. When បw

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 共more detailed later兲. The transmitted modes

below E

F

thus becomes less and the conductance is quan-

tized and decreases with B. Basically, when បw

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 共AB兲 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 ប

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 共2004兲

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. 2共a兲, 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 k⫽0 still exists. When the intersubband mixing

from RSO coupling is neglected, the spin-resolved eigenval-

ues are approximately at every subband

⫾

(k)⫽

n

⫹ ប

2

k

2

/2m⫾

␣

k, ⫾ denotes two spin-splitting bands due to

the RSO coupling and not the eigenstates of

z

yet. Once

B⫽0, the Landau levels form in the system and this is the

reason of the platform appearing at the bottom of subbands

共near k⫽ 0) as shown in Fig. 2共b兲. Both the subbands and

their energy difference are enhanced in comparison with Fig.

2共a兲 when the external magnetic ﬁeld increases, moreover,

the gap of intersubbands is basically equivalent to បw

c

.Itis

interesting to note that the RSO coupling will lead to a

Zeeman-type energy-band split under B⫽0. 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

␣

冑

2m/ប)

冑

បw

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. 3共b兲, where

FIG. 2. The subband energy spectrum with

RSO coupling strength t

so

⫽ 0.02; 共a兲 no mag-

netic ﬁeld ប

c

⫽ 0 and 共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. 共a兲ប

c

⫽ 0 and 共b兲ប

c

⫽ 0.40.

RASHBA SPIN PRECESSION IN A MAGNETIC FIELD PHYSICAL REVIEW B 69, 085304 共2004兲

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 B⫽0, 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 共not shown兲

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. We ﬁnd that the

perfect spin modulation of the conductance due to the RSO

coupling would not be destroyed in the large coupling con-

stant region because the RSO coupling between different

transmitted modes is negligible compared with their energy

difference. We conclude that our proposal of applying a mag-

netic ﬁeld to the system can be a proper alternative to the

prerequisite of the narrow width of the quantum wire or the

strong transverse conﬁning potential for the functional opera-

tion of a spin ﬁeld effect transistor.

This work was supported by Australian Research Council

under Project LX0347471; D.Y.X. thanks the state key pro-

grams for Basic Research of China for ﬁnancial support un-

der Grant No. 10174011.

1

See, e.g., M. Oestreich, Nature 共London兲 402, 735 共1999兲; S.A.

Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von

Molna

´

r, M.L. Roukes, A.Y. Chtchelkanova, and D.M. Treger,

Science 294, 1488 共2001兲.

2

S. Datta and B. Das, Appl. Phys. Lett. 56, 665 共1990兲.

3

Y.A. Bychkov and E.I. Rashba, J. Phys. C 17, 6039 共1984兲.

4

J.M. Kikkawa and D.D. Awschalom, Phys. Rev. Lett. 80, 4313

共1998兲.

5

J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett.

78, 1335 共1997兲.

6

C.-M. Hu and T. Matsuyama, Phys. Rev. Lett. 87, 066803 共2001兲;

D. Grundler, ibid. 86, 1058 共2001兲.

7

T. Matsuyama, C.-M. Hu, D. Grundler, G. Meier, and U. Merkt,

Phys. Rev. B 65, 155322 共2002兲; P. Mavropoulos, O. Wunnicke,

and Peter H. Dederichs, ibid. 66, 024416 共2002兲.

8

G. Schmidt, D. Ferrand, L.W. Molenkamp, A.T. Filip, and B.J.

van Wees, Phys. Rev. B 62, R4790 共2000兲; E.I. Rashba, ibid. 62,

R16 267 共2000兲; J.D. Albrecht and D.L. Smith, ibid. 66, 113303

共2002兲.

9

A.T. Hanbicki, B.T. Jonker, G. Itskos, G. Kioseoglou, and A.

Petrou, Appl. Phys. Lett. 80, 1240 共2002兲; V.F. Motsuyi, J. De

Boeck, J. Das, W. Van Roy, G. Borghs, E. Goovaerts, and V.I.

Safarov, ibid. 81, 265 共2002兲.

10

R. Fiederling, M. Keim, G. Reuscher, W. Ossau, G. Schmidt, A.

Waag, and L.W. Molenkamp, Nature 共London兲 402, 787 共1999兲;

Y. Ohno, D.K. Young, B. Beschoten, F. Matsukura, H. Ohno,

and D.D. Awschalom, ibid. 402, 790 共1999兲.

11

G. Kirczenow, Phys. Rev. B 63, 054422 共2001兲.

12

F. Mireles and G. Kirczenow, Phys. Rev. B 64, 024426 共2001兲.

13

Z.L. Ji and D.W.L. Sprung, Phys. Rev. B 54, 8044 共1996兲.

14

T.P. Pareek and P. Bruno, Phys. Rev. B 65, 241305 共2002兲; L.W.

Molenkamp, G. Schmidt, and G.E.W. Bauer, ibid. 62, 4790

共2000兲.

15

L.V. Keldysh, Sov. Phys. JETP 20, 1018 共1965兲.

16

P.A. Lee and D.S. Fisher, Phys. Rev. Lett. 47, 882 共1981兲.

17

T. Ando, Phys. Rev. B 44, 8017 共1991兲; M.J. Mclennan, Y. Lee,

and S. Datta, ibid. 43, 13846 共1991兲; H.U. Baranger, D.P. Divin-

cenzo, R.A. Jalaber, and A.D. Stone, ibid. 44, 10637 共1991兲.

FIG. 4. The spin-resolved conductance in different RSO cou-

pling constants t

so

and the lengths of quantum wire N

x

; the solid

line and dash line are same as in Fig. 3. 共a兲 t

so

⫽ 0.02, N

x

⫽ 60;

共b兲 t

so

⫽ 0.04, N

x

⫽ 60; 共c兲 t

so

⫽ 0.02, N

x

⫽ 40, t

so

⫽ 0.04, N

x

⫽ 40.

WANG, SUN, AND XING PHYSICAL REVIEW B 69, 085304 共2004兲

085304-4

18

B.J. van Wees, H. van Houten, C.W.J. Beenakker, J.G. William-

son, L.P. Kouwenhoven, D. van der Marel, and C.T. Foxon,

Phys. Rev. Lett. 60, 848 共1988兲.

19

A.V. Moroz and C.H.W. Barnes, Phys. Rev. B 60, 14272 共1999兲.

20

H. Yoshioka and Y. Nagaoka, J. Phys. Soc. Jpn. 59, 2884 共1990兲.

21

Y. Takagaki and D.K. Ferry, Phys. Rev. B 47, 9913 共1993兲.

22

A.G. Scherbakov, E.N. Bogachek, and U. Landman, Phys. Rev. B

53, 4054 共1996兲.

RASHBA SPIN PRECESSION IN A MAGNETIC FIELD PHYSICAL REVIEW B 69, 085304 共2004兲

085304-5