Strongly modulated transmission of a spin-split quantum wire with local Rashba interaction

Strongly modulated transmission of a spin-split quantum wire with local Rashba interaction

David Sánchez Departament de Física, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain    Llorenç Serra Departament de Física, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain Institut Mediterrani d’Estudis Avançats IMEDEA (CSIC-UIB), E-07122 Palma de Mallorca, Spain    Mahn-Soo Choi Department of Physics, Korea University, Seoul 136-701 Korea
September 16, 2019

We investigate the transport properties of ballistic quantum wires in the presence of Zeeman spin splittings and a spatially inhomogeneous Rashba interaction. The Zeeman interaction is extended along the wire and produces gaps in the energy spectrum which allow electron propagation only for spinors lying along a certain direction. For spins in the opposite direction the waves are evanescent far away from the Rashba region, which plays the role of the scattering center. The most interesting case occurs when the magnetic field is perpendicular to the Rashba field. Then, the spins of the asymptotic wavefunctions are not eigenfunctions of the Rashba Hamiltonian and the resulting coupling between spins in the Rashba region gives rise to sudden changes of the transmission probability when the Fermi energy is swept along the gap. After briefly examining the energy spectrum and eigenfunctions of a wire with extended Rashba coupling, we analyze the transmission through a region of localized Rashba interaction, in which a double interface separates a region of constant Rashba interaction from wire leads free from spin-orbit coupling. For energies slightly above the propagation threshold, we find the ubiquitous occurrence of transmission zeros (antiresonances) which are analyzed by matching methods in the one-dimensional limit. We find that a a minimal tight-binding model yields analytical transmission lineshapes of Fano antiresonance type. More general angular dependences of the external magnetic field is treated within projected Schrödinger equations with Hamiltonian matrix elements mixing wavefunction components. Finally, we consider a realistic quantum wire where the energy subbands are coupled via the Rashba intersubband coupling term and discuss its effect on the transmission zeros. We find that the antiresonances are robust against intersubband mixing, magnetic field changes, and smooth variations of the wire interfaces, which paves the way for possible applications of spin-split Rashba wires as spintronic current modulators.

71.70.Ej, 72.25.Dc, 73.63.Nm

I Introduction

i.1 Motivation

Since the pioneering Datta-Das proposal of an electronic field-effect transistor in which the current flow is controlled by magnetic means only,dat90 the study of the Rashbaras60 ; byc84 spin-orbit interaction in one-dimensional (1D) and quasi-one-dimensional ballistic channels (quantum wires) has attracted a lot of interest.sat99 ; mor99 ; mir01 ; kis01 ; mol01 ; egu02 ; gov02 ; fev02 ; bul02 ; and03 ; stre03 ; scha04 ; per04 ; nes04 ; cah04 ; wan04 ; she04 ; zha05 ; kno05 ; per05 ; rom05 ; deb05 ; ser05 ; zhan05 ; zhan06 ; rey06 ; san06 ; jeo06 ; sha06 ; zha06 ; per07 ; nev07 Precise tunability of the strength of the Rashba coupling has been also experimentally demonstrated in quantum wells.nit97 ; eng97 ; gru00 Typically, semiconductor quantum wires are built from two-dimensional electron gases formed at the interface of a semiconductor heterostructure when the lateral motion of electrons is restricted by a transversal confinement potential to effective widths of the order of the de Broglie electron wavelength. For very clean quantum wires (e.g., quantum point contacts) transport is ballistic and conductance is quantized to integer values of .wee88 ; wha88

The presence of impurities or defects in the vicinity of the constriction destroys conductance quantization.chu89 ; bag90 ; fai90 ; tek91 ; gur93 ; noc94 A striking effect arises when the impurity potential is attractive and enables the existence of at least one bound state whose energy is degenerate with the continuum band of propagating states. As a consequence, for energies close to the transition threshold a direct transmission channel can interfere with a wave trajectory that travels across the bound state and this interference is destructive, leading to enhanced backscattering and Fano asymmetric lineshapes.fano ; cer73 ; gor00 ; kob02 Recently, two of ussan06 have demonstrated that a spin-orbit interaction of the Rashba type localized in an infinitely long quantum wire plays a role similar to an attractive potential and pronounced dips are seen in numerical simulations of the conductance curves.she04 ; zhan05 ; san06 ; zha06 It is remarkable that the Rashba interaction provides both the attractive potential that supports bound statesval04 ; cse04 and the mixing term that couples the localized and the propagating states.san06 Interestingly, when charging effects are taken into account, Coulomb blockade resonances can be tuned directly modulating the strength of the Rashba coupling.lop07

A magnetic field applied in the wire plane leads to Zeeman spin splitting of the 1D modes. Evidence of this is shown in the appearance of conductance plateaus at odd multiples of .wha88 In the first plateau the current is fully polarized since only one spin species is allowed to propagate. Quantum states with opposite spin are evanescent asymptotically and do not take part in electron transport unless there exist inhomogeneities that give rise to resonances or Fano-type interferences in which case evanescent states are crucial. Reference ser07, presents a theoretical method to calculate evanescent states in quantum wires with uniform Rashba interaction.

To determine the full transmission pattern of a generic quantum wire, one must first analyze the energy spectrum of the wire. For quantum wires with uniform Rashba interaction in the absence of external magnetic fields, free-electron energy bands are parabolas shifted apart for opposite spin directions.mol01 The splitting size is proportional to the spin-orbit interaction strength and in the quasi-1D case the Rashba interaction produces anticrossings between bands corresponding to opposite spins and adjacent modes.mor99 ; gov02 Moreover, the propagation threshold is shifted, compared to the case with no spin-orbit coupling, down to an energy . In the presence of an in-plane magnetic field, the energy spectrum changes dramatically even for arbitrarily small fields. The field can be either externally applied or originated from stray fields of the ferromagnets coupled to the wire in the Datta-Das setup.dat90 It is shownstre03 ; per04 ; ser05 that the interplay between the magnetic field and the Rashba interaction leads to the openings of gaps in the 1D energy bands at small wavenumbers. In a quasi-1D wire most of the energy dispersions around the gap form energy minima locally in contrast to the maxima encountered in the 1D case.ser05 In those energy windows in which the gap consists of an energy local maximum followed by a local minimum the conductance curves present anomalous steps for chemical potentials within the gap.per04 ; ser05

In short, external magnetic fields lead to the formation of energy gaps in the spectrum while local Rashba interactions produce Fano-type antiresonances due to the formation of quasi-bound states coupled to the channel of direct transmission states. Therefore, we expect a rich interplay between in-plane fields and localized Rashba spin-orbit couplings in the transport properties of a ballistic quantum wire. This paper presents a generic theoretical description of the quantum transmission of an electron subject to Zeeman splittings and spatially modulated Rashba fields.

i.2 Main findings

We find the occurrence of exact tranmission zeros in the conductance curves of a Zeeman-split wire with local Rashba interaction as a function of the Fermi energy. Central to the existence of the transmission zeros are the formation of a Zeeman gap arising from an in-plane magnetic field and the role of the evanescent states within the Rashba region. In fact, the Rashba interaction couples the propagating and evanescent states precisely in the interior of the Rashba region. The transmission antiresonances are almost universal, showing a Fermi energy with vanishingly small transmission at moderately low magnetic fields. This might be relevant for applications since it provides two operation points for working transistors (low and high current states). It is important to stress that these transmission zeros are fundamentally distinct from the suppressed transmission that may take place in a Datta-Das setup due to spin precessiondat90 even in the presence of in-plane magnetic fields.nev07 The antiresonance position can be tuned with a slight change of and are robust against changes of the magnetic field. We only require that the Fermi energy lies within the gap.

i.3 Outline

The outline of the paper is as follows. Section II is devoted to analyzing the transport properties of a 1D wire, where only one subband is taken into account and Rashba-induced intersubband coupling is then neglected. In Sec. II.1 we discuss the eigenstates and energy spectrum of a 1D wire subject to Rashba interaction and Zeeman spin splittings. In Sec. II.2 we consider a finite Rashba region with constant Rashba strength and a magnetic field pointing in a direction perpendicular to the Rashba field and calculate the transmission within the scattering formalism and numerical matching. A tight binding description of the problem is considered in Sec. II.3. We derive an exact expression for the transmission in the limit of a minimal Rashba region and discuss the Fano form of the lineshape. To end Sec. II we present in Sec. II.4 results for the angular dependence of the magnetic field direction. In Sec. III we examine a quasi-1D wire. The numerical results are in agreement with the 1D case, thus demonstrating that the sharp antiresonances are robust even when intersubband coupling is present like in realistic wires. We also discuss the case of an arbitrary dependence of the Rashba strength with the position and compare our results when the Rashba interaction smoothly increases at the interfaces. Finally, Sec. IV contains the conclusions.

Ii One-Dimensional Wire

We consider a 2D electron gas formed in the - plane due to strong confinement in the direction. As a result of the interfacial electric field, there arises a spin-orbit coupling of the Rashba type with a Hamiltonian given by


where the Rashba strength can be spatially modulated. The limit of a purely 1D system is obtained by further constraining the electron motion along, e.g., the direction. Then, the term in Eq. (1) is neglected and the Rashba interaction plays the effective role of a momentum-dependent magnetic field with a direction along the axis. In the following section we discuss the spectrum and eigenfunctions of a 1D wire when the Rashba strength is uniform and the external magnetic field points either in the - plane (“in-plane” field) or along the vertical direction (“perpendicular” field).

ii.1 Extended Rashba interaction

ii.1.1 In-plane field

We consider an in-plane magnetic field with arbitrary direction, , giving rise to Zeeman interaction.per04 ; ser05 Then, the single-particle Hamiltonian reads,


where is the Zeeman splitting. When we assume a constant Rashba strength from to , the Hamiltonian is diagonalized using the spinor wavefunctions (we take the spin quantization direction along ),


where is the branch-splitting quantum index and is the wavevector associated to free motion along . The spin orientation is determined from


We note that there exists no common spin quantization axis since depends on .ser05 This is due to the existence of a magnetic field since for the spinors lie along (the Rashba axis). The effect is akin to a 2D system with Rashba interaction for which . However, in the 2D case the spin orientation is always perpendicular to whereas in the 1D case only for asymptotically large momenta () the spin is quantized along . A similar effect arises from Rashba intersubband coupling in quasi-1D systems.gov02

Figure 1: Energy spectrum for a 1D quantum wire with uniform Rashba coupling for a Zeeman splitting and different magnetic field angles: (a) , (b) and (c) .

From the Schrödinger equation, , one finds the energy spectrum,


For the case of interest occurs for small Zeeman splittings, , see Fig. 1(a). The lowest branch of the spectrum develops a local maximum around which has important consequences for transport.stre03 ; per04 ; nes04 ; ser05 As a result, there arises a pseudogap between the two spectrum branches. For Fermi energies lying in the pseudogap region, there is only a wavefunction with a given spin direction for each mover (right-moving or left-moving). In addition, there also exist evanescent waves which are crucial when the Rashba interaction is confined to a finite region. Outside the pseudogap region, there are four real wavevectors for a given .

Increasing from to leads to the progressive reduction of the pseudogap size, see Fig. 1(b) and 1(c). For the gap vanishes and the spinors point along the axis since in this case the field axis and the Rashba axis coincide.

We note in passing that when is constant (or, more generically, an even function of ), there exists a symmetry property of the Hamiltonian given by Eq. (2). Let us concentrate on the case . Thus, is invariant under the transformation


namely, the rotation by around axis in the spin space followed by the parity operator , which yields inversion in the direction. The additional factor of is to ensure that . Similar symmetry properties have been discussed in Refs.bul99, ; kis01, ; deb05, which find that the spin parity, i.e., the combination of parity and a Pauli matrix, is a constant of motion for . Here, commutes with even for nonzero fields when . Hence, one can find a common basis of eigenstates for and . The wave functions given by Eq. (3) are not eigenstates of . In fact, since when . Therefore, one could construct states with definite parities with regard to from even () and odd () combinations of , i.e., .

ii.1.2 Perpendicular field

For fields pointing along , the Zeeman term in Eq. (2) is expressed as . The spectrum is identical to the case, Fig. 1(a). Therefore, for the sake of the present discussion, the cases parallel to and parallel to are equivalent in the 1D case since both are perpendicular to the Rashba field direction (along ). Of course, in the quasi-1D case one should also take into account orbital effects but in this section we can neglect this. Let us focus on the branch and energies within the pseudogap. We will find that strong transmission changes take place in that region.

For there are two propagating solutions


with wavevector [see Fig. 1(a)]


where we have defined


As discussed above, the angle depends on the wavevector:


Further, the pseudogap region admits two more solutions which are evanescent waves. For (i.e., ) we find,


For one must make the replacements and . Moreover,




In Fig. 1(a) we illustrate the “dispersion” relation for the evanescent states and the location of . The evanescent states make sense only for nonvanishing as can be readily seen by setting in Eqs. (8) and (14). becomes and becomes pure imaginary, . This corresponds to four propagating solutions (two left-moving and two right-moving) with a definite spin direction.mol01

ii.2 Local Rashba interaction

We now consider a double interface at and between a normal conduction band ( and ) and a region of localized Rashba interaction extending from to , see Fig. 2. Then, for and elsewhere. Our basic goal is to find the transmission through the Rashba region when the magnetic field is present all along the wire, producing a Zeeman gap. For convenience, we take the direction along since the solutions for are simpler to write down.

Figure 2: Schematic representation of a local Rashba interaction in a 1D quantum wire.

We are interested in energies inside the spin pseudogap, , for which a spin-down (spin-up) electron wave is propagating (evanescent). In the scattering problem, an electron with spin down is injected from and reflected with a certain probability . Since the spin quantization axis in the region depends on the wavevector, we must also take into account the spin-up evanescent waves at and . As a consequence, the scattering wave function for reads,


whereas for one should make the replacements and in the expressions for the evanescent states of the Rashba region. In Eq. (16) the wavevector is written in terms of the Fermi wavevector and defined as before. We note that can be pure imaginary for , though the physical wavevector, , is always a quantity manifestly positive and real. The evanescent wave is described with an exponentially decreasing wave with range where , thereby the probability of finding a spin-up on the left or right sides is nonzero. At small the propagating states with wavevector has its spin approximately along [ in Eq. (3)].

We numerically find the coefficients , , , , , , and from matching equations. At the interfaces the wavefunction must be continuous, e.g., . Moreover, the flux, given by the velocity operator , must also be continuous, i.e., . On the normal sides, is trivially given by while in the spin-orbit region one hasmol01


Importantly, the flux associated to Eq. (7), which is proportional to


is positive (negative) for ().

The transmission and the reflection are simply given by and , respectively. Since we are interested in the relative influence of and while keeping the Rashba region length constant, in the numerical results we give the energies as a function of the energy unit . A characteristic transmission curve is plotted in Fig. 3 for whereas is slightly varied. We make the important observation that there arises an exact transmission zero near when . On increasing , the resonance position shifts to lower energies and, at the same time, the resonance broadening is enhanced. This dependence with will become clear later, when we discuss the tight-binding model.

Figure 3: (Color online) Transmission through a Rashba region for and different spin-orbit intensities: (full line), 6.64 (dashed), 6.88 (dot-dashed), 7.12 (dotted).

The transmission for and various magnetic fields is shown in Fig. 4. The transmission curves are reminiscent of the Fano lineshapes. An interesting question is thus whether the transmission behavior is indeed related to a Fano-type interference effect. While in strict one-dimensional systems the interference giving rise to Fano lineshapes is not possible due to the existence of one channel only, in this case and due to the Zeeman splitting there exist two modes, namely, the propagating mode (spin down) and the evanescent (spin up) mode. Both modes become coupled locally within the Rashba region. As a result, the effect is due to a subtle combination of spin-orbit interaction and Zeeman splitting which leads to destructive interference in the Rashba region. A simplified model, discussed below, will shed light on this. For the moment, we note that Fig. 4 shows that it is sufficient to have a rather small Zeeman gap above which the antiresonance develops. For we find .

Figure 4: Transmission versus Fermi energy for and different Zeeman splittings: (a), 0.128 (b) and 0.064 (c).

The energy and length scales we considered above are within the scope of present techniques. E.g., for a Rashba region of size  m the value corresponds to  meV nm, which is accessible in an InAs wire.nit97 The Zeeman energy used in Fig. 3 corresponds to a magnetic field  mT in the same material and in Fig. 4(c) is only 60T. Notably, the effect scales with . Therefore, a smaller would require a larger wire for the antiresonance to be observable.

ii.3 Tight-binding model

The continuum model discussed above leads to remarkable predictions for the conductance of a spin-split quantum wire with a local Rashba interaction but to gain further insight it would be highly desirable to have a simplified model capable of yielding closed analytical formulas for the dip position and shape. In this subsection we consider a discretized version of the Hamiltonian [Eq. (2)]. The infinite 1D wire is modeled with a linear chain of sites. Thus, we obtain the following tight-binding Hamiltonian,and89


where the summations over and are carried out on an infinitely extended 1D wire and the last summation is over the sites of the Rashba region. In this equation, couples nearest neighbors, is the Rashba interaction strength which couples electronic states with opposite spin directions along , and is the lattice parameter. The on-site energies are given by [ for ]. The Hamiltonian is equivalent, in the limit , to whose transport properties have been analyzed above. Here, in order to obtain simplified expressions we consider a localized Rashba interaction restricted only to two sites, 0 and 1 (see Fig. 5). We note that this is the minimal model that characterizes the transport properties of a 1D wire with a Rashba region.

Figure 5: In the minimal tight-binding model, the system consists of a linear chain of coupled sites with a localized Rashba interaction at sites and .

For and the energy band spectrum is given by the well known expression . In the presence of a magnetic field, the spectrum becomes spin split. We now focus on an energy range close to the band bottom, . Then, the eigenfunctions corresponding to spin () are propagating (evanescent) waves. Since we intend to solve the scattering problem of an -electron wave impinging onto the Rashba region, we now introduce the wave amplitudes for , and for , with and the transmission and reflection probability amplitudes. The wavenumber is related to the total energy by means of For electrons with spin , their energy outside the Rashba region falls below the band bottom. As a result, we take the wave amplitudes given by for and for , where Substituting the total wavefunction into the Schrödinger equation and projecting over the sites of the Rashba region , we find the transmission,


which allows to determine the exact condition for the occurrence of zeros in the transmission function :


This expression has a very appealing form. For a given value of the antiresonance energy lies to the left of , as shown in the numerical results of Figs. 3 and 5. Moreover, it predicts that the dip shifts to lower energies as the Rashba interaction strength increases. This is reproduced in Fig. 6 where we plot a characteristic as a function of energy for . In addition, the dip broadens as increases, in excellent agreement with the numerical results of the continuum model, see Fig. 3. Equation (21) also explains why the critical Zeeman splitting below which the dip disappears is so small since the antiresonance is observable only if lies above the band bottom, i.e., . It then follows that where is a slowly increasing function of for . As a result, , which is at least two orders of magnitude smaller than the bandwidth. Incidentally, we also find an upper bound of the Rashba strength, , above which the dip vanishes. For , we obtain . This is an interesting feature for applications since very strong Rashba couplings are not necessary to generate the antiresonance.

Figure 6: Transmission versus Fermi energy for and different spin-orbit strengths. All energies are given in units of .

The considerations made above suggest that the antiresonance has a Fano lineshape but it is hard to demonstrate with controlled approximations that Eq. (20) has exactly this form. Perhaps it is more instructive to consider a closely related model, in which the spin flip interaction due to the Rashba coupling is restricted to a single point, see Fig. 7. Then, the Hamiltonian reads


As before, the spin-up and spin-down energy bands are shifted by a Zeeman splitting. The coupling between spins at the central site may represent the action of external magnetic field pointing in a direction perpendicular to the spin quantization axis or other source of spin flipping. Spin flip interactions in quantum dots have recently received much attention.sf1 ; sf2 ; sf3 ; sf4

We take the wave function ansatz for , and for for the propagating states and for and for for the evanescent states. From the tight-binding equations, we obtain the transmission amplitude,


We can define the broadening which measures the coupling strength between spin-up and spin-down states and is proportional to the density of states (per unit length) for electrons with spin down. It reads,


Using this result and the expressions for and written above, we find the expression for the transmission probability,


valid for energies around the point . We note that Eq. (25) has the desired Fano form. In the conventional Fano effect, the coupling takes place between a bound state immersed in a continuum band and the propagating states.fano Here, the role of the bound state is played by the evanescent modes which, due to the spin flip interaction, are coupled to the propagating states with opposite spins.

Figure 7: Sketch of the system considered in the discussion for a point-like spin-flip interaction between propagating (spin down) and evanescent (spin down) states along a one-dimensional site lattice.

ii.4 Angular dependence

We now discuss the angular dependence of the field direction. In the strict 1D case, only one mode is needed. Thus, we expand the wavefunction in the two-spinor basis which, in the case of an in-plane field , takes the form:




the spinors in the direction.

Substituting in the Schrödinger equation with the Hamiltonian given by Eq. (2), we obtain a pair of coupled equations for and :


where primes indicate and we recall . We use the following gauge transformation,


where , in order to eliminate the first derivatives in Eqs. (28,29), which are transformed into


where the elements of the Hamiltonian matrix are


with . It is clear that for the amplitudes and decouple and for energies inside the gap the transmission is given simply by the propagating state . Then, propagating and evanescent states become decoupled and no dip is expected. Only for angles away from does the problem become nontrivial since couples with the evanescent amplitude . Numerical results shown in Fig. 8 confirm this expectation.

Figure 8: (Color online) Transmission as a function of Fermi energy varying the orientation of the in-plane magnetic field. The azimuthal angle for each curve is given in the legend. We take and .

Iii Quasi-One-Dimensional Wire

The preceding section has shown that the Fano resonance phenomenon manifests itself in 1D quantum wires with Zeeman splitting and a local Rashba interaction. Real wires, of course, have always some small extension in the lateral direction. In this section we analyze the influence of this extra dimension by considering a quasi-1D wire, including the direction. This lateral -confinement is usually weaker than the vertical -confinement. Thus, we neglect the contribution of the -confining electric field to the Rashba strength. For simplicity, we consider a transverse potential of parabolic type, .

Figure 9: (color online) Transmission as a function of the Fermi energy for a quasi 1D wire with transverse parabolic confinement characterized by . A Rashba region of length and a Zeeman energy with the magnetic field along the wire () have been used. The legend gives the numerical values of for the different curves.

The additional spatial dimension is relevant now because the transverse momentum explicitly appears in the Rashba spin-orbit interaction as shown by Eq. (1). It is also worth stressing that the new term in Eq. (1), proportional to , precludes the use of the analytical solution discussed in Sec. II.1 for a wire with an extended Rashba interaction. In fact, it is well known that it causes the formation of textured spin states lacking well defined spin quantization axis even for a fixed value of the wavenumber .gov02 ; ser05 The Rashba coupling is assumed to be non zero only in a region of length , where it takes the value , as in Sec. II.2. We also include the Zeeman coupling as in Sec. II.1 of a magnetic field oriented along a certain azimuthal angle . The full Hamiltonian thus reads


A natural unit system for the present quasi-1D model is set by the wire transverse potential, with energy unit (oscillator energy) and length unit (oscillator length). In what follows the numerical values for the Rashba region length , spin-orbit intensity and Zeeman energy will be given in these oscillator units. In order to obtain the transmission of the system modeled by Eq. (37) we have used the quantum transmitting boundary algorithm as in Ref. san06, . The Schrödinger equation is discretized on a uniform grid using finite differences for the derivatives and imposing scattering boundary conditions. The reader is referred to Refs. san06, ; qtbm, for additional details of the method.

Figure 10: (Color online) Transmission versus Fermi energy varying the orientation of the in-plane magnetic field. The azimuthal angle for each curve is given in the legend. We use a Rashba region of length , Zeeman energy and spin-orbit coupling strength .

The existence of the Fano lineshapes in a quasi-1D wire, with a transmission zero at a given energy, is clearly shown in Fig. 9. This result proves that the physical effect elucidated with the tight binding model of Sec. II.3 is robust and persists in more realistic models. There is also a nice qualitative agreement with the 1D results of Fig. 3. In all three cases (tight-binding, 1D and quasi-1D) increasing the value of leads to a shift towards lower energies of the transmission zero, and to an important broadening of the transmission dip. These are very appealing features related to practical applications in spintronic devices, since they could allow to control the transmission by tuning ; the device operation would not be very sensitive to small changes in due to the broadness of the dip.

The scales used in Fig. 9 are of the same order as in Fig. 3. E.g., for a confinement strength  meV in an InAs wire, we obtain  m,  meV ( mT) and  meV nm.

When the magnetic field is oriented along the wire as in Fig. 9, the interference leading to the Fano profiles in the transmission is maximal. On the contrary, for transverse orientation it completely disappears (see Fig. 10). This behavior is in agreement with the analysis of Sec. II.4 in the 1D case, where it was shown that the mixings and vanish for .

The evolution with the Zeeman field intensity for the quasi-1D case is shown in Fig. 11. The behavior is again qualitatively similar to the 1D case of Fig. 4, with the dip evolving towards smaller energies when decreasing the value of . We also notice that, as predicted by the tight binding result, even for quite small Zeeman energies there is a dip in the transmission.

Figure 11: (Color online) Transmission as a function of the Fermi energy for a quasi 1D wire with a Rashba region of and a magnetic field along . The different panels correspond to the given Zeeman energies (in units of ). In each panel solid and dashed lines correspond, respectively, to a value of of 0.45 and 0.40, respectively.

Thus far we have considered abrupt interfaces between the normal sides and the Rashba region. Using the quasi-1D grid calculation we can also address the influence of the smoothness in the transition of the Rashba coupling strength from zero to the finite value , which is closer to reality. We model each interface using a Fermi function with a diffusivity ,


Figure 12 shows the results for different values of . The transmission curve coincides with the abrupt interface limit when while for increasing the interface becomes smoother but the transmission zero remains visible and the dip position does not change much. This is a crucial observation—the transmission zeros we find are roughly independent of the precise profile of the Rashba strength. This robustness has an obvious importance for potential spintronic applications.

Figure 12: (Color online) Transmission versus Fermi energy for and . Each curve corresponds to a different diffusivity , as given in the legend in units of , for a Rashba region of length

Iv Conclusions

We have performed a theoretical analysis of the transport properties of a ballistic quantum wire with spatially inhomogeneous Rashba interaction in the presence of an external magnetic field giving rise to Zeeman spin splitting. When the Rashba coupling dominates the magnetic field an energy pseudogap develops in the wire spectrum. We find abrupt transmission lineshapes when the Fermi energy lies within the pseudogap. The lineshapes are narrow and asymmetric and the transmission reaches zero for energies near the gap closing. We have discussed a minimal tight-binding model that reproduces the essential features of the resonances, yielding analytical expressions for the lineshape dependence on Fermi energy, Rashba intensity and Zeeman splitting. Qualitatively, the evanescent band plays the role of a quasi-bound state that the confined Rashba interaction couples to the propagating states. The evanescent waves are not true bound states but when the Fermi energy approaches the evanescent band bottom electrons scattering off the Rashba region become strongly affected, leading to perfect reflection. Numerical results in realistic quantum wires agree with the purely 1D case. Finally, we have analyzed the behavior of the resonances when the angular orientation of the magnetic field is changed and the interfaces become smoother.

The system studied here could work as a current modulator device. For slight variations of the Fermi energy, which can be externally controlled, we have shown that the transmission changes dramatically between two limit values (1 and 0) across the antiresonance. Our proposal has a number of differences compared to the Datta-Das spin transistor.dat90 First, the latter device modulates the current independently of the energy of the injected electrons since the phase difference that governs the spin precession is independent of the wavevector. Then, small changes of or strongly affects the working points of the transistor whereas in our case these points are not very sensitive to small variations of the external parameters such as the external magnetic field, the Rashba strength or the interface diffusivity. Moreover, a 100 current modulation is hard to achieve in the Datta-Das transistor, especially when intersubband coupling is taken into account, whereas in our case the modulation is rather abrupt and is preserved even when adjacent subbands are coupled, an effect which is unavoidable in real quantum wires. We have reported results for the lowest spin-split subband but have checked that similar pronounced dips are seen in higher subbands. Our results differ also with those of Ref. san06, since in that case the antiresonances reached zero only after a fine tuning of the parameters. Here, our only requirement is that the Fermi energy should lie within the spectrum pseudogap.

As far as the discussion in 1D wires is concerned, the field directions orthogonal to the Rashba field (along , according to the parameterization of the Hamiltonian we have employed) are equivalent. However, in the quasi-1D case one of these directions (the one perpendicular to both the Rashba field and the electron propagation) induce orbital effects. Here, we have restricted ourselves to fields giving rise only to Zeeman splittings since the study of orbital effects in inhomogeneous systems requires knowledge of the evanescent states when the magnetic field is applied perpendicular to the wire plane. This is not a trivial task and it seems to be promising avenue of future research. Reference deb05, finds important changes in the spectrum structure of a quantum wire with uniform Rashba interaction and perpendicular magnetic fields. However, our sharp antiresonances show up even in the presence of rather small Zeeman splittings. Therefore, we expect that the dips should be still visible even when orbital effects are taken into account provided the magnetic field length is much larger than the confinement length.

In our discussion, we have neglected electron-electron interaction effects, which may lead to Luttinger liquid effects in 1D ballistic wires when the interactions are screened like in a wire with electric-field induced spin-orbit interactions.hau01 ; gri05 When Zeeman splittings are present lee05 ; dev05 the transmission seems to be altered by electron-electron interactions, although these works neglect the intersubband coupling term of the Rashba interaction. In fact, for ballistic wires without Rashba coupling but multiple populated subbands a simple mean-field approach demonstratesbut03 that Coulomb interactions are crucial to understand rectification effects observed in nanojunction rectifiers. On the other hand, single-particle effects are shownlop07 to lead to Coulomb blockade antiresonances of the Fano form. Hence, further work is needed to clarify the influence of Coulomb interactions in the conductance of a quantum wire with Zeeman splitting and a localized Rashba interaction.


This work was supported by the Spanish MEC Grant No. FIS2005-02796 and the “Ramón y Cajal” program.


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