Spin edge states: An exact solution and oscillations of the spin current

Spin edge states: An exact solution and oscillations of the spin current

V. L. Grigoryan,¹A. Matos Abiague,²and S. M. Badalyan^1,2,

*

1Department of Radiophysics, Yerevan State University, 1 A. Manoukian Street, 375025 Yerevan, Armenia

2Department of Physics, University of Regensburg, 93040 Regensburg, Germany

共Received 14 August 2009; revised manuscript received 24 September 2009; published 15 October 2009兲 We study the spin edge states, induced by the combined effect of spin-orbit interaction共SOI兲and hard-wall confining potential, in a two-dimensional electron system, exposed to a perpendicular magnetic field. We find an exact solution of the problem and show that the spin-resolved edge states are separated in space. The SOI-generated rearrangement of the spectrum results in a peaked behavior of the net-spin current versus the Fermi energy. The predicted oscillations of the spin current with a period, determined by the SOI-renormalized cyclotron energy, can serve as an effective tool for controlling the spin motion in spintronic devices.

DOI:10.1103/PhysRevB.80.165320 PACS number共s兲: 73.21.Fg, 73.63.Hs, 72.25.Dc, 72.10.⫺d

I. INTRODUCTION

The principal importance of spin-orbit interaction共SOI兲is in its ability to link the electron charge and spin degrees of freedom, which is fertile for novel physical phenomena.^1–3 Unlike the charge, the electron spin is double valued and identifies two system components, which can be separated as in the spin-Hall effect^4,5 or mixed via the spin-Coulomb drag.^6,7 There are different mechanisms, realizing SOI,¹ and the interplay between them produces another rich arena for study and potential applications in spintronics.^8,9

In two-dimensional electron systems 共2DES兲 of the quantum-Hall-effect geometry, the extended edge state play a central role in understanding of transport phenomena.^10–12 The suppression of backscattering and interedge-state relaxation^13–15 make possible nondissipative transport through edge channels. In the presence of SOI the nonlocal transport through spin-polarized edge channels holds prom- ise of providing even richer phenomenology and greater power of electronic applications. Another strong motivation for investigations of spin edge states is related to the recent experimental realization of the Mach-Zehnder.¹⁶

Recently several theoretical papers have addressed the ef- fect of SOI on the edge states in restricted 2DES 共Refs.

17–21兲 and along magnetic interfaces.^22,23All these works, however, find unlikely an exact analytical solution of the edge state problem and adopt different numerical approaches,18–20,22,23 use a parabolic confining potential,²¹ which has no flat domain, or give an analytical approxima- tion in the limit of strong magnetic fields¹⁷where the effec- tive SOI coupling is small.

Here we present an analytical solution to the spin edge states, induced by the combined effect of SOI and hard-wall confining potential. We derive anexactformula for the elec- tron energy dispersions and calculate their spectral and trans- port properties. We find that due to SOI the spin edge states are resolved not only in the energy but are alsoseparated in space: an effect, which is not captured by the approximate approach, adopted in Ref.17. From the energy spectrum we calculate the electron-group velocity and the average spin components. We find that the magnitude of spin components are not equal in the up and down states. In the quasibulk states the electron spins are mainly aligned along the mag-

netic field while near the hard wall the spins of edge states become mainly perpendicular to their propagation direction.

Using these ingredients we calculate the components of the net-spin current as a function of the Fermi energy and show that the SOI-induced splitting results in the peaked behavior of the spin current. We argue that the predicted oscillations of the spin current with a period, determined by the renor- malized cyclotron frequency, can serve as a new tool for manipulating spin currents in a controllable manner. The de- veloped approach here is equally applicable to the spin edge states along magnetic interfaces in nonhomogeneous mag- netic fields.

II. THEORETICAL CONCEPT

We assume that the 2DES resides in a quantum well, formed in the 共001兲 plane of a zincblende semiconductor heterostructure, and is exposed to a perpendicular homoge- neous magnetic field,Bជ=B₀zˆ. The motion of electrons in the 2DES is confined by an infinite hard-wall potential, V共x兲

=⬁ for x⬍0. Such a system is described by a two- dimensional Hamiltonian of the form

H=H₀+H_SOI+V共x兲, 共1兲

where the Hamiltonian of free particle in a magnetic field is H₀=共␲ជ²/2m^ⴱ兲␶^ˆ and the Rashba SOI Hamiltonian H_SOI

=␣R共␲x␴^ˆy−␲y␴^ˆx兲,²⁴m^ⴱdenotes the electron effective mass,

␲ជ^=pជ⁻共e/c兲Aជ the kinetic momentum with pជ= −iបⵜជ_{. We} choose the Landau gauge so that the components of vector potential areAជ=共0 ,xB₀, 0兲. The unity matrix␶^ˆand the Pauli- spin matrices ␴^ˆ act in the spin space. It is assumed that electrons are confined to the lowest-energy subband in thez direction.

Using the ansatz ⌿共x,y兲=e^ikyy␹k_y共x兲, we can reduce the two-dimensional Schrödinger equationH⌿=E⌿to the one- dimensional problem whereEis the electron total energy and ky the electron momentum inydirection. Then the electron wave function␹k_y共x兲inxdirection should satisfy the follow- ing equation

再 ^冋

^{dxd22+␯+12 −关x−X共k4 y兲兴2}

^册

^␶ˆ

+␥

冋

^{idxd ␴ˆy−x−X共k2 y兲␴ˆx}

册 冎

^{␹ky共x兲= 0, 共2兲}

where the effective potential V_eff共x兲=关x−X共ky兲兴²/4 in x di- rection depends on the wave vectorky. In Eq.共2兲we express the energyE→共␯+ 1/2兲ប␻Bin units of the cyclotron energy, ប␻B⬅បeB0/m^ⴱc, and the lengthx→xlB/

冑

2 in the magnetic length, lB⬅

冑

បc/eB₀. We introduce also the dimensionless SOI coupling constant␥=

冑

2␣R/vBwith the cyclotron veloc- ityvB=ប²/m^ⴱlBand the dimensionless coordinate of the cen- ter of orbital rotationX共k_y兲=

冑

2k_yl_B. Taking into account ex- plicitly that the eigenstates of Eq.共2兲are spinors

␹k_y共x兲=

冏

^␹␹1k2kyy^关x关x⁻−^X共kX共k^y_y^兲兴兲兴

冏

^{, 共3兲}

we write the Schrödinger equation in the following compact form:

冉

^{hh␯− hh+␯}

冊 冏

^␹␹1k2kyy^共共^␰␰^兲兲

冏

^{= 0. 共4兲}

Here we introduce the following operators:

h_␯=

冉

d^d␰²²⁺␯⁺¹ 2−␰²

4

冊

^{, 共5兲}

h_⫾= −␥

冋

^␰2⫿dd␰

册

^{. 共6兲}

The system of equations, obtained from Eq. 共4兲 has to be solved under the boundary conditions ␹k_y共x兲→0 when x

→0 and +⬁. In the absence of SOI, h_⫾= 0, the solution is given in terms of the parabolic cylindrical functions, D_␯共x兲.

In the presence of SOI we search the bulk solution of matrix Eq. 共4兲 as ␹1k_y共␰兲=aD_␮共␰兲 and ␹2k_y共␰兲=bD_␮−1共␰兲 where a andb are thex-independent spinor coefficients and␮ ^{is an} arbitrary index, different from ␯. Making use of the follow- ing recurrent properties of the parabolic cylindrical func- tions:

h_␯D_␮共␰兲=共␯⁻␮兲D_␮共␰兲, 共7兲

h_⫾D_␮共␰兲= −␥

再

^{␮DD␮+1␮−1共␰共␰兲兲}

冎

^{, 共8兲}

we obtain

␮_⫾共␯^,␥兲=␯⁺¹ 2+␥²

2 ⫾

冑

^␯␥2+14共1 +␥²兲², 共9兲 c_⫾共␯^,␥兲= −1

␥

冋

^12+␥22⫾

^冑

^{␯␥2+14共1 +␥2兲2}

册

^共10兲

wherec_⫾=b_⫾/a_⫾. Thus, the two independent bulk solutions of Eq.共4兲are given by the spinor wave functions

␹k⫾_y共␰兲=a_⫾

冏

^c⫾^DD^␮_{␮⫾共␯,␥兲−1}^{⫾共␯,␥兲共␰兲}共␰兲

冏

^{. 共11兲}

The normalization of the wave functions兰d␰␹k†_y共␰兲␹k_y共␰兲= 1 gives the amplitude of eigenstates

a_⫾=

再 ^冕

^{d␰关兩D␮⫾共␰兲兩2+兩c⫾兩2兩D␮⫾−1共␰兲兩2兴}

冎

^{−1/2. 共12兲}

It is easy to see that in the limit of vanishing SOI,␥→0, we have a₊→0 and b₋→0 and recover the usual edge states, which are doubly degenerated with respect to the spin

␹k+_y共x兲 ⬃D_␯共x兲

冏

¹⁰

冏

^{and ␹k−y共x兲 ⬃D␯共x兲}

冏

⁰¹

冏

^{. 共13兲}

On the other hand the solution 共11兲for sufficiently large values ofX共ky兲describes quasibulk Landau states so that the index ␮_⫾共␯^,␥兲 differs only exponentially from the Landau index l= 0 , 1 , 2. . . and the parabolic cylindric functions are given by their asymptotics Hermite polynomials, Dl共␰兲

= 2^−l/2exp共−␰²/4兲H_l共

冑

_␰/2兲. In the limit ofX共k_y兲→⬁ taking

␮_⫾共␯^,␥兲=l in Eqs.共9兲and共10兲, one can exactly reproduce the spectrum and the wave functions of the bulk dispersion- less Landau levels, renormalized by the SOI for l= 1 , 2. . . 共Ref. 24–28兲

E_l^⫾共␥兲=

冉

^l⫾

^冑

^14+l␥2

冊

^{ប␻B, 共14兲}

c_⫾共␯^,␥兲= −1

␥

冉

^21⫿

^冑

^14+l␥2

冊

^{. 共15兲}

As usual thel= 0 Landau level remains not perturbed by the spin-orbit coupling. In order to obtain the spectrum of the spin edge states we require vanishing of the electron wave function Eq. 共11兲at x= 0. The energy,E, obtained from the vanishing conditions for both spinor components at x= 0, should be the same. As seen, however, the different spinor components of the bulk solution Eq. 共11兲 are given by the parabolic cylindric functions with different indices, which makes impossible their vanishing simultaneously. To satisfy the boundary conditions, we construct a linear combination of the two independent bulk solutions as

␺k_y共␰兲=␣␹k+_y共␰兲+␤␹k−_y共␰兲 共16兲 and choose the coefficients␣ ^and␤ so that the components of the new spinor wave function ␺k_y共␰兲vanish at x= 0. The eigenvalue problem for␣^and␤has a solution if the respec- tive determinant vanishes atx= 0. This leads to the following exactdispersion equation:

c₋D_␮

+关−X共k_y兲兴D_␮

−−1关−X共k_y兲兴

=c₊D_␮

−关−X共k_y兲兴D_␮

+−1关−X共k_y兲兴 共17兲 for the spin edge states with the wave functions

␺k_y共␰兲=␣

冏

^c+DD␮+^␮−1^+共共^␰␰^兲兲⁻−^rDrc₋^␮D^−共_␮^␰兲

−−1共␰兲

冏

^{. 共18兲}

Here r=D_␮

+关−X共ky兲兴/D_␮

−关−X共ky兲兴 and ␣ is obtained from the normalization of the wave functions. Recall that the de- pendence on energyE=共␯+ 1/2兲ប␻Bmanifests itself via the functions␮_⫾共␯^,␥兲andc_⫾共␯^,␥兲, given by Eqs.共9兲and共10兲.

The dispersion relation Eq. 共17兲is quadratic with respect to the parabolic cylindric functions, therefore for a given band index, n, it has two solutions, Esn共ky兲, where s=↑ and ↓ corresponds to the spin edge states with the two spinor wave functions

␺k↑_y,↓共␰兲=␺k_y共␰兲兩E=E_↑,↓n共k_y兲. 共19兲

III. ENERGY SPECTRUM OF SPIN EDGE STATES AND SPIN CURRENT

Further we carry out the actual calculations of the spec- trum of spin edge states, the average components of spins, and the spin-current components, carried by the skipping or- bits along the edges of 2DES. In the presence of a perpen- dicular magnetic field, the efficiency of SOI is determined by the dimensionless coupling constant ␥, which is inversely proportional to

冑

B₀. Therefore, the SOI effect is significant in weak magnetic fields unless temperature fluctuations smear magnetic-quantization effects. In such fields the Zee- man effect is small²¹ and we do not consider it here. We carry out the actual calculations forB₀ corresponding to the cyclotron splitting about 5 K. In GaAs with the electron ef- fective mass m^ⴱ= 0.067m₀ such a cyclotron splitting is achieved for B₀⬇0.25 T and taking the Rashba coupling

␣R⬇4.72 meV Å, we have␥= 0.03. For such a coupling the spin-orbit effects in the edge state spectrum should be hardly visible. The situation is favorable in InAs where the Rashba coupling is larger,␣R⬇112.49 meV Å.¹Despite the smaller effective mass, m^ⴱ= 0.026m₀, in weak fields about B₀

= 0.1 T, we have␥= 0.45. As we see below such a coupling results in essential modifications in the spectrum and trans- port of spin edge state, measurable in experiment.

In Fig. 1共a兲 we plot the energy spectrum of spin edge states, Esn共ky兲, as a function of momentum ky, which we obtain by solving the dispersion Eq.共17兲. It is seen that for a given quantum number n there are two spin-resolved mag- netic edge states, E_↓n共ky兲 and E_↑n共ky兲. Both branches show monotonic behavior in the whole range of kyvariation. For large positive values of ky, the energy of spin edge states is given by the spin-split quasibulk Landau levels, renormal- ized by SOI, while at negative values of k_y the spectrum describes the current-carrying edge channels. The spin split- ting of edge states increases with the main quantum number n. At the stronger effective SOI coupling the spectrum shows well-pronounced anticrossings. The development of the anti- crossings can be traced clearly in Fig.1共b兲where we calcu- late the energy of spin edge states versus ␥ or, what is the same, versus 1/

冑

B₀. These anticrossings in the energy spec- trum result in additional structures of the spin current versus the Fermi energy. In Fig.1共c兲we plot the average transverse position of the spin edge states from the boundary of 2DES as a function of their center of orbital motion, defined as

⌬x_sn共k_y兲=

冕

0

⬁

dxx兩␺k_y关x−X共k_y兲兴兩E=E2 _sn共k_y兲. 共20兲

It is seen that except for large positive values of ky, the position of skipping orbits takesspin-resolvedvalues so that the up- and down-spin-edge states are separated in space.

Notice this effect of the SOI-induced spatial separation is missed in the approximation, adopted in Ref.17. The differ- ences in the probability density for different spins and wave vectors of the first two bands are clearly shown in the insets.

As seen from the lower inset, the probability density for different bands and spins differs even at large positive values of ky. In this limit, however, irrespective the quantum num- bern and the spin orientation, the particle average thickness is the same and varies linearly with its center of orbital mo- tion. This is because in the quasibulk Landau states far from the interface, electrons oscillate symmetrically with respect to the guiding center,X共ky兲, independent of the spin and band index.

From the obtained spectrum we calculate the correspond- ing group velocities along y direction, vsn共ky兲 FIG. 1. 共Color online兲The energy spectrum of spin edge states as a function共a兲of the momentumk_yfor␥= 0.3 and共b兲of the effective SOI coupling␥共or that is the same as a function ofB^−1/2兲forX共k_y兲= 3. The dashed and solid curves correspond to the up and down spins.

共c兲The particle average position as a function ofk_yfor␥= 0.3. The probability density in the respective symbol positions is shown in the insets.

=⳵^Esn共ky兲/⳵បky共vx⬅0 along xdirection兲as well as the av- erage spin components alongx,zdirections

S_sn^x,z共ky兲=ប 2

冕

0

⬁

dx␺k_y

†共x兲␴^ˆx,z␺k_y共x兲兩E=E_sn共k_y兲. 共21兲

Because the transverse wave functions are real, the y com- ponent of the spin vanishes identically, S_sn^y共ky兲⬅0. In Fig.

2共a兲 we plot S_sn^x,z共ky兲as a function of X共ky兲for the first two bands. At large positive values ofky when electrons are far from the hard wall, the spins are mainly aligned alongzaxes.

This is because in the quasibulk Landau states electrons have no preferential direction in the共x,y兲plane of their cyclotron rotation. In the opposite limit of negativeky, the edge chan- nels are formed and the spins are mainly aligned in xdirec- tion, perpendicular to theydirection of electron propagation.

Notice that due to the spin splitting the absolute values of the average spin components do not equal in the up and down states and this asymmetry becomes stronger with the band indexn.

In Figs.2共b兲and2共c兲we calculate thexandzcomponents of the net-spin current alongydirection, defined as

J_y^x,z共EF兲=

兺

s,n

S_sn^x,z共ky兲vsn共ky兲兩E_sn共k_y兲=E_F. 共22兲

We use this definition of the spin current, which is widely accepted and intuitive from the physical point of view. No- tice that according to some recent suggestions,^29,30 the stan- dard definition of spin current共which is the one we use here兲 is a proper definition and there is no need for other definitions.^31–33 It is seen thatJ^x_yexhibits regular peaks as a function of the Fermi energy. Such an oscillatory behavior is due to the SOI-induced splitting in the edge state spectrum.

For a given quantum numbern the splitting always creates a narrow energy region near the peak position关see the inset in Fig.1共b兲兴where only the ↓ spin states contribute positively to the spin current. With an increase in the Fermi energy the

↑ spin edge state starts to contribute negatively at the posi- tion of ↑ spin Landau level. In this region the exponential increase in the velocity³⁴ of the ↑ spin edge state with the energy results in a sharp peak of the spin current. With a further increase in the Fermi energy the ↓ and ↑ spin edge

states of the next n+ 1 band start to contribute similarly but with stronger amplitudes because of the spin-splitting en- hancement with n. Thus the peaks of the spin current are imposed against the monotonic background and have a pe- riod, determined by the cyclotron energy. The latter is renor- malized in the presence of SOI, as seen in Fig. 1共a兲 in the limit ofky→⬁.

As seen in Fig.2共c兲thez-spin currentJ_y^z changes its sign, in addition to its peaked behavior: due to an interplay be- tween the average spin S_sn^z 共k_y兲 and the velocityv_sn共k_y兲, the spin current is negative near the Landau levels and positive between them. In this case, at high energies corresponding to large negative values ofkyin each bandn关cf. Fig.1共a兲兴, the large values ofvsn共ky兲 are compensated by the small values of S_sn^z共ky兲 关cf. Fig. 2共a兲兴. Therefore the overall monotonic increase inJ_y^z withEF becomes less pronounced.

IV. SUMMARY

In conclusion, we present an exact analytical solution to the spin edge states, induced by the combined effect of the Rashba SOI and of the hard-wall confining potential. The exact solution of the problem allows its deeper intuitive un- derstanding and can be a strong input in studying the spin transport through edge channels. We find that due to SOI the spin edge states are resolved not only in the energy but are also separated in space. In the bulk of sample the electron spin is mainly aligned along the magnetic field while near the hard wall the spin becomes mainly perpendicular to the edge state propagation direction. The magnitude of spin components is asymmetric in the up and down states. We show that the spin-current components exhibit oscillations versus the Fermi energy. The predicted oscillations, with a period determined by the renormalized cyclotron energy, can serve as an effective tool to control the spin motion in spin- tronic devices.

ACKNOWLEDGMENTS

We thank J. Fabian and G. Vignale for useful discussions and acknowledge support from the Volkswagen Foundation, EU under Grant No. PIIF-GA-2009-235394, SFB under Grant No. 689, and ANSEF under Grant No. PS-1576.

(a) (b) (c)

FIG. 2.共Color online兲 共a兲Thexandzcomponents of average spins in units ofបas a function ofX共k_y兲for the first two bandsn= 1 , 2 and for␥= 0.3.共b兲Thexcomponent and共c兲thezcomponent of the net-spin current共solid curve兲versus the Fermi energy. The dashed and dotted curves plot the separate contributions to the spin current made by the spin-up and spin-down states together for each bandn. Insets show the absolute value of the separate up- and down-spin-current contributions.

*samvel.badalyan@physik.uni-regensburg.de

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