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PHYSICAL REVIEW B 69, 121306~R! ~2004!

Effect of spin-orbit interaction and in-plane magnetic field on the conductanceof a quasi-one-dimensional system

Yuriy V. Pershin, James A. Nesteroff, and Vladimir PrivmanCenter for Quantum Device Technology, Department of Physics and Department of Electrical and Computer Engineering,

Clarkson University, Potsdam, New York 13699-5721, USA~Received 6 November 2003; published 22 March 2004!

We study the effect of spin-orbit interaction and in-plane effective magnetic field on the conductance of aquasi-one-dimensional ballistic electron system. The effective magnetic field includes the externally appliedfield, as well as the field due to polarized nuclear spins. The interplay of the spin-orbit interaction with effectivemagnetic field significantly modifies the band structure, producing additional subband extrema and energygaps, introducing the dependence of the subband energies on the field direction. We generalize the Landauerformula at finite temperatures to incorporate these special features of the dispersion relation. The obtainedformula describes the conductance of a ballistic conductor with an arbitrary dispersion relation.

DOI: 10.1103/PhysRevB.69.121306 PACS number~s!: 73.23.Ad, 75.47.Jn, 71.70.Ej

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Recently, there have been numerous studies of the perties of quasi-one-dimensional systems.1–8 The motivationbehind this interest has been the observation of conductquantization. Most quasi-one-dimensional systems, or qutum wires, are created by a split gate technique in a twdimensional electron gas~2DEG!.6,7 When a negative potential is applied to the gates, the electrons are depleunderneath. Thus, a one-dimensional channel is createdtween two reservoirs. For ballistic transport to occur,1 thisconstriction should be less than the electron mean free pand have a width of the order of de Broglie wavelength.6–8

The explanation for conductance quantization is foundusing a noninteracting electron model. With a small biasplied across the channel, the electrons move from one revoir to the other. Due to the transverse confinement inchannel, the electrons are distributed, according toFermi-Dirac distribution, among various subbands inchannel. The calculation of the conductance has been smarized in the Landauer-Bu¨ttiker formalism.6–8 Each of thesubbands contributes 2e2/h to the conductance.

The spin-orbit ~SO! interaction is described by thHamiltonian9–12

HSO5\

4m2c2 ~¹W V3pW !•sW , ~1!

wheresW represents a vector of the Pauli spin matrices,m isthe free electron mass,pW is the momentum operator, and¹W Vis the gradient of a potential, proportional to the electric fieacting on the electron. When dealing with crystals, theretwo main types of spin-orbit interaction. The Dresselhaspin-orbit interaction11 appears as a result of the asymmepresent in certain crystal lattices. The Rashba spin-ointeraction13 arises due to the asymmetry associated withconfinement potential and is of interest because of the abto electrically control the strength of this interaction. Tlatter is utilized, for instance, in the Datta-Das sptransistor.14 The Hamiltonian for the Rashba interactionwritten13 as

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p-

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de-

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HSO5 iaS sx

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]y2sy

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wherea is the coupling constant. In this paper, we limit oconsideration to the systems with only the Rashba intetion. Incorporation of the Dresselhaus interaction into ocalculations is straightforward.

In the simplest case when the external magnetic fieldapplied in the plane of the heterostructure and the spin-ocoupling is neglected, each subband in the channel issplit. The effect this field has on conductance is that esubband now contributese2/h to the conductance, as observed, e.g., in Ref. 3. Another possible source of spsplitting are nuclear spins, which we incorporate into tmodel within an effective-field approximation. We introduc

the total magnetic field asBW 5BW ext1BW N , whereBW ext is the

external magnetic field andBW N is the effective field~for spinmotion only! produced by the polarized nuclear spins via thyperfine interaction. The high level of nuclear spin polization in GaAs, closed to the maximum of 5.3 T,15 has beenachieved experimentally.16,17 It is anticipated that the effecof the nuclear spin polarization on the conductance is simto the effect of in-plane magnetic field.18 The characteristicenergy scale of the interaction between conduction electrand polarized nuclear spins makes it possible to confineelectrons into low-dimensional electron structures usmodulation of the nuclear spin polarization.5,19–22We assume

that BW N is parallel to the external magnetic fieldBW ext.16

Recent studies of 2DEG-based systems23–25have been fo-cused on including the effects of the spin-orbit interactionthe conductance with perfect channel transmissionT(E)51. Other studies8,26–28 have dealt with similar effects inhybrid ferromagnetic-semiconductor systems but with tramission coefficients less than 1. In our calculations wesume perfect transmission through the channel and focuthe effects of the interplay of spin-orbit interaction andin-plane magnetic field on the conductance at finite tempetures. We show that the spin-orbit interaction breaks the s

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PERSHIN, NESTEROFF, AND PRIVMAN PHYSICAL REVIEW B69, 121306~R! ~2004!

metry of the subbands in certain cases and manifests itseunique features in the conductance dependence on thevoltage.

Let us consider the energy spectrum of a quasi-odimensional system in which we take into account the effeof spin-orbit interaction and an applied in-plane magnefield. The Hamiltonian inside the conductor will then be

H5p2

2m*1U~y!2 iasy

]

]x1

g* mB

2sW •BW . ~3!

Here p is the electron momentum in thex direction,m* isthe electron effective mass,U(y) is the electron confiningpotential in they direction,g* is the effectiveg factor, andmB is the Bohr magneton. The Rashba termHSO defined inEq. ~2!, was reduced for the motion alongx only. We assumethat the total magnetic field experienced by electrons isplane,BW 5Bxx1Byy. It should be emphasized that the manetic field does not enter into Eq.~3! through the vectorpotential in our approximation.

We consider solutions of the Schro¨dinger equation insidethe constriction that are separable, of the form

c5eikxf~y!S w↑w↓

D , ~4!

wheref(y) is the wave function for the transverse mod~due to the confinement potential! and w↑,↓ are the spinorcomponents for spin up and down, respectively. The eigvalue problem can be solved to obtain

E6(n)5

\2k2

2m*1En

tr7@~g* mBB/2!21g* mBakB sinu

1~ak!2#1/2. ~5!

In this expression, the up and down spin states in the eigbasis are denoted by6, u is the angle of the magnetic fielrelative to the electron transport through the wire,B sinurepresents they component of the total magnetic field, anEn

tr is the spectrum of transverse subbands. Assuming pbolic confinement potential in they direction, we haveEn

tr

5\v(n11/2). The energy spectrum corresponding to E~5! is illustrated in Figs. 1~a!–1~c! for various values ofu.Recently, similar energy-spectrum calculations have beenported in Ref. 29.

It is interesting to note some of the properties of thesub-bands. In the case ofu equal to 0, Fig. 1~c!, or p/4, Fig.1~b!, a gap appears atp50 between two spin-split bandswhich is not observed whenu5p/2. The subbands in Figs1~b! and 1~c! also contain local extrema, due to the Rashterm.

Our goal is to calculate the overall influence of the spdependent interactions in the Hamiltonian on the finite teperature conductance. To do this, we make use ofLandauer-Bu¨ttiker formalism. We consider a model ofquantum wire, which consists of two electron reservowith chemical potentialsmL andmR , separated by a conductor. This conductor is assumed to be devoid of scattererthat the transmission coefficientT(E) is unity. If a biaseV

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FIG. 1. Dispersion relations: energyE in units of \v, for dif-ferent values ofu, ~a! u5p/2, ~b! u5p/4, and ~c! u50, as afunction of the momentump, where h is defined as h5(2m* \v)1/2. These plots were obtained using the parameter vuesg* mBB/(2\v)50.1 anda@2m* /(\3v)#1/251.

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RAPID COMMUNICATIONS

EFFECT OF SPIN-ORBIT INTERACTION AND IN- . . . PHYSICAL REVIEW B 69, 121306~R! ~2004!

5mL2mR is applied across the contacts, such thatmL.mR , then the total currentI through the conductor can bwritten as the difference between the currents flowing inforward and reverse directions, see Ref. 7, p. 52,

I 5e

2p (n,s

E2`

`

vns@q~vns! f ~E,mL!

1q~2vns! f ~E,mR!#dk. ~6!

Here f @E(k),mL,R# is the Fermi-Dirac distribution functionfor electrons in the left and right reservoir,vns is the electronvelocity given by\21]Es

(n)/]k, with s56 denoting the spinstate,Es

(n) is given by Eq.~5!, andq(v) is the step function.The summation overn ands includes contributions from althe subbands in the channel.

The band structure of our system exhibits a numberlocal extrema which have to be taken into account in ordecalculate the conductance using Eq.~6!. For example, if weconsider a single subband with an arbitrary number of loextrema then the calculation of the conductance can becomplished by splitting up the integral in Eq.~6! betweenextremal points in the subband. This gives the relation

I n,s5e

h F EE0(n,s)

f ~E,mR!dE1EE0

(n,s)

E1(n,s)

f ~E,mL!dE

1EE1

(n,s)

E2(n,s)

f ~E,mR!dE1¯G , ~7!

whereI n,s is the contribution from the subband labeled bynand s, while Ei

(n,s) is the i th energy extremum in that subband. If we assume that the applied bias is small, then weTaylor-expand the integrals in Eq.~7! in terms of eV. Bytaking only the first-order terms, summing over differesub-bands and using the relation for conductanceG5I /V,we obtain the result

G5e2

h (n,s

(i

b i(n,s) f ~Ei

(n,s)!. ~8!

Here, the sum is calculated over the extremal points of allsubbands, andb i

(n,s) is either11 for a minimum or21 fora maximum. So, we see that maximum points in a subbactually reduce the conductance.

Experimentally, the conductance plateaus are observechanging the potential applied to the gates. This changpotential can be viewed as a shift of the chemical potenThe calculated values forG, using Eq.~8!, and the corre-sponding temperature dependences, are shown in Fig.functions of the chemical potentialm. In Fig. 2~a!, we seethat asm increases, the zero-temperature conductancecreases in steps ofe2/h. This is half of the increment thawould be obtained without the Zeeman splitting of the eergy subbands.8 As the temperature is increased we observsmearing of the conductance plateaus, since electrons cing in from the reservoirs no longer have a sharp stepenergy distribution at the chemical potential.

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When the applied magnetic field has a nonvanishingxcomponent, new effects occur in the conductance plateespecially at low temperatures. In Figs. 2~b! and 2~c!, weillustrate the conductance plateaus foru5p/4 andu50, re-spectively. In the case ofu5p/4, we can see from Fig. 1~b!

FIG. 2. ~Color online! Conductance as a function of the chemcal potential at different temperaturesT ~in units of \v/kB), fordifferent directions of the magnetic field:~a! u5p/2, ~b! u5p/4,and ~c! u50.

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RAPID COMMUNICATIONS

PERSHIN, NESTEROFF, AND PRIVMAN PHYSICAL REVIEW B69, 121306~R! ~2004!

that the energy subbands are split, so as the chemical potial increases, the first contribution to the conductance wbe from the lowest minimum in Fig. 1~b!, and the secondcontribution occurs when the chemical potential passessecond minimum of the same subband. Asm further in-creases and passes the local maximum, the conductanccreases bye2/h. By this mechanism, peaks of the condutance are formed, as demonstrated in Figs. 2~b!, 2~c!. Wenote that temperature-smeared curves look very similathose in the much investigated 0.7 anomaly phenomen3

though additional investigations would be needed befspeculating that the present mechanism could be an altetive to other explanations of this effect offered in the literture.

It should be emphasized that the external magnetic fielnot necessary to observe the conductance peculiaritiescussed in this paper. Similar effects can be obtained duinteractions that mix spin states. The simplest examplepolarized nuclear spins. The nuclear spin polarization isnamic because of the spin diffusion and relaxation proces

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Since the electron equilibration time scales are much shothan the time scales of dynamics of the nuclear spin systthe adiabatic approximation can be applied.5 Within this ap-proximation, the effective magnetic field due to the polariznuclear spins can be considered quasistatic and relationssented above can be used to describe the conductance.

In summary, this paper has focused on the interplayvarious in-plane magnetic field components, Rashba sorbit interaction, and finite temperature in their effects onconductance. We have found that the angle of the magnfield with respect to the conductance channel has a sigcant effect on the conductance. The variation of the anglethe field generates gaps in the energy subband structurecontrol the pattern of the conductance variation as the checal potentials are varied by applied gate voltages.

We gratefully acknowledge helpful discussions with DS. N. Shevchenko. This research was supported by the~Grant No. DMR-0121146!, and by the NSA and ARDA~un-der the ARO Contract No. DAAD 19-02-1-0035!.

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