Spin-orbit coupling effects in one-dimensional ballistic quantum wires
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Spin-orbit coupling effects in one-dimensional ballistic quantum wires
J.E. Birkholz and V. Meden
Institut f¨ur Theoretische Physik, Universit¨at G¨ottingen,Friedrich-Hund-Platz 1, D-37077 G¨ottingen, Germany
We study the spin-dependent electronic transport through a one-dimensional ballistic quantumwire in the presence of Rashba spin-orbit interaction. In particular, we consider the effect of thespin-orbit interaction resulting from the lateral confinement of the two-dimensional electron gasto the one-dimensional wire geometry. We generalize a situation suggested earlier [P. Strˇeda andP. Sˇeba, Phys. Rev. Lett. , 256601 (2003)] which allows for spin-polarized electron transport.As a result of the lateral confinement, the spin is rotated out of the plane of the two-dimensionalsystem. We furthermore investigate the spin-dependent transmission and the polarization of anelectron current at a potential barrier. Finally, we construct a lattice model which shows similarlow-energy physics. In the future, this lattice model will allow us to study how the electron-electroninteraction affects the transport properties of the present setup. PACS numbers: 72.25.Dc, 71.70.Ej, 72.25.Mk
I. INTRODUCTION
Spin-orbit coupling is a relativistic effect of order O ( v /c ), where v is the electron velocity, which followsdirectly from the Dirac equation. It is described by theHamiltonian (for ∇ × E = 0) H SO = − e ~ m c σ · h E × (cid:16) p − ec A (cid:17)i , (1)where the electric field E = −∇ V /e ( e < − eA/c to the canonicalmomentum is abandoned. In order to confine electronsto nanostructure devices, sharp potentials are necessary,which lead to nonnegligible spin-orbit interaction (SOI),especially in systems with structural inversion asymme-try like e.g. semiconductor heterostructures. This effectcan be used to achieve control over the electron spinand leads to spin-dependent transport properties, suchas spin-polarized currents, even in systems without fer-romagnetic leads.The emerging field of spintronics might result in an ex-tensive use of the spin degree of freedom for informationprocessing. In a two-dimensional electron gas (2DEG)obtained by a strong confinement in z -direction, the SOIis usually described by the so-called Rashba term H R = ~ m α z ( σ x p y − σ y p x ) , (2)contributing to the Hamiltonian of the electron system. Here the components of the electron momentum oper-ator are denoted by p i , the Pauli matrices by σ i , and α z ∝ E z is the SOI coupling coefficient set by the con-fining electric field. As discussed by Datta and Das, afurther confinement of the 2DEG to a wire geometry al-lows for a particular control over the spin, if α z or thelength of the wire are varied. This insight led to exten-sive studies on the transport properties of noninteractingelectrons in quasi one-dimensional (1D) quantum wires with SOI. In particular, the effect of sub-band mixing and a magnetic field perpendicular tothe plane of the underlying 2DEG was investigated.A very promising candidate for a system to experimen-tally produce spin-polarized currents using SOI is thesetup suggested by Strˇeda and Sˇeba where the magneticfield points in the wire direction and an additional po-tential step is placed in the quantum wire. It is as-sumed that due to the large energy level spacing onlythe lowest subband of the quantum wire is occupied andsubband mixing can be neglected. Restricting the con-siderations to this subband, one does not have to in-clude explicitly the potential confining the electrons tothe wire. Furthermore, the strong lateral confinementallows to take into account only the momentum in thewire direction, p x = p , p y = p z = 0 in Eq. (2). The en-ergy dispersion of the 1D electron gas ε ( k ) = ~ k / (2 m ),where k = k x , is split by the Rashba term Eq. (2) intotwo branches ε ( s ) ( k ) = ~ ( k + sα z ) / (2 m ) − E α z , with s = ± and E α z = ~ α z / (2 m ). The eigenenergies arefourfold degenerate with two left and two right movingstates. The spin expectation values are h σ y i k,s = s and h σ x i k,s = h σ z i k,s = 0, independent of k . In presence of anexternal magnetic field (parallel to the wire), describedby a Zeeman term H Z = ǫ Z σ x / , (3)an “energy gap” of size ǫ Z opens up at k = 0 [see Fig. 1 a)]and states within this “gap” are only twofold degenerate(one left and one right moving state). A potential stepcan then be used to generate a tunable spin polarization,in mainly the y -direction, of the linear response current.In order to achieve this, the height of the step V > x < x > ± y -direction into the ± x -direction when | k | →
0, while h σ z i k,s remains zero. Depending on the chosen param- V E F a) V E F c b) FIG. 1: (Color online) a) A potential step of height V andb) a potential barrier of height V and width 2 x c . The cor-responding dispersions in the different regions are sketched(solid line: s = +, dashed line: s = − ). eters, this leads to a small x -component of the groundstate magnetization, whereas the y - and z -componentsare exactly zero as will be explained below.We here generalize the situation studied in Ref. 9 inseveral ways. We first study how the above scenario ismodified in the presence of an additional Rashba term H ′ R = ~ m α y σ z p x (4)resulting from the confinement of the 2DEG to the wiregeometry, a term which so far was mainly ignored. As wealso focus on the lowest subband and do not study sub-band mixing, the exact shape of the potential confiningthe electrons to the wire is not important. As its maineffect, H ′ R will lead to nonvanishing spin expectation val-ues h σ z i k,s and thus a spin polarization component per-pendicular to the plain of the underlying 2DEG. We also study the transmission current and the spin polarizationat a potential barrier and discuss the interplay of α y and α z . In addition, we present a lattice model which in anappropriate parameter regime shows the same physicsas the continuum model. This model will allow us tostudy the effect of the electron-electron interaction onthe spin polarization in a forthcoming publication us-ing the functional renormalization group method. II. CONTINUUM MODEL
The model we consider is given by the Hamiltonian H = p x m − ~ α z m σ y p x + ~ α y m σ z p x − e ~ mc σ · B . (5)We slightly generalized the situation discussed above andallow for a Zeeman term with a magnetic field B = B (sin θ cos ϕ, sin θ sin ϕ, cos θ ) pointing in arbitrary direc-tion. The normalized eigenstates with quantum numbers k and s = ± are given by the product of a plane wave(in x -direction) and a two-component spinor φ ( s ) k ( x ) = 1 √ π e ikx A ( s ) k B ( s ) k ! . (6)Applying the Hamiltonian Eq. (5) to this ansatz we ob-tain (cid:18) k + 2 α y k + 2 k Z cos θ − ǫ, ikα z + 2 k Z e − iϕ sin θ ikα z + 2 k Z e iϕ sin θ, k − α y k − k Z cos θ − ǫ (cid:19) A ( s ) k B ( s ) k ! = 0 , (7)with ǫ = 2 mE/ ~ , α y = eE y / (4 mc ), α z = eE z / (4 mc ), and k Z = − eB/ (2 ~ c ). Note that α y , α z < ~ / m ) ǫ ( s ) ( k ) = k + 2 s sgn( k − k ) p C ( k ) , (8)with C ( k ) = ( α y + α z ) k + 2 k Z k ( α y cos θ − α z sin θ sin ϕ ) + k Z and k = − k Z ( α y cos θ − α z sin θ sin ϕ ) / ( α y + α z ) beingthe wave number at which the “energy gap” becomes smallest [see Fig. 2]. The corresponding eigenfunctions are φ ( s ) k ( x ) = 1 √ π r (cid:12)(cid:12)(cid:12) a ( s ) k (cid:12)(cid:12)(cid:12) e ikx (cid:18) a ( s ) k (cid:19) , (9)with a ( s ) k = − iα z k − k Z e − iϕ sin θα y k + k Z cos θ − s sgn( k − k ) q ( α y + α z ) k + 2 k Z k ( α y cos θ − α z sin θ sin ϕ ) + k Z (10)and the spin expectation values are given by h σ x + iσ y i k,s = 2 (cid:16) a ( s ) k (cid:17) ∗ (cid:12)(cid:12)(cid:12) a ( s ) k (cid:12)(cid:12)(cid:12) , h σ z i k,s = − (cid:12)(cid:12)(cid:12) a ( s ) k (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) a ( s ) k (cid:12)(cid:12)(cid:12) . (11) As can be seen from Eq. (11), the necessary condition h σ x i k,s + h σ y i k,s + h σ z i k,s = 1 holds for all values of s e n e r gy / E α s = +s = - -3 -2 -1 0 1 2 3 k / α -1-0.500.51 s p i n ( s =+ ) a) < σ x >< σ y >< σ z > e n e r gy / E α s = +s = - -3 -2 -1 0 1 2 3 k / α -1-0.500.51 s p i n ( s =+ ) b) < σ x >< σ y >< σ z > e n e r gy / E α s = +s = - -3 -2 -1 0 1 2 3 k / α -1-0.500.51 s p i n ( s =+ ) c) < σ x >< σ y >< σ z > FIG. 2: (Color online) Dispersion and spin expectation values on the ( s = +)-branch for a magnetic field in a) x - , b) y - andc) z -direction, α y /α = − . α z /α = − . k Z /α = 0 .
5. The spin on the ( s = − )-branch points in the opposite direction,i.e. h σ i i k,s = −h σ i i k, − s . The shape of the dispersion and the k -value at which the “energy gap” becomes smallest clearlydepends on the direction of the magnetic field. and k . The existence of the confinement in y -direction(represented by α y ) leads to a rotation of the spin outof the x - y -plain into the z -direction. This indicates thatthe ratio of α y and α z is crucial for the spin direction.The energy dispersion Eq. (8) and the spin expecta-tion values on the ( s = +)-branch are shown in Fig. 2 asa function of k , with k given in units of α = q α y + α z and the energy in units of E α = ~ α / m . For | k | & α the spin expectation values reach their asymptotic, k -independent values. The spin on the ( s = − )-branchpoints in the opposite direction, i.e. h σ i i k,s = −h σ i i k, − s ,and is not shown explicitly here. In combination withthe fact that for B = ( B, , h σ y i k,s and h σ z i k,s aresymmetric with respect to k = 0 on both branches, thisexplains why there is no ground state magnetization inthe y - and z -direction for B being parallel to the wire.However, there is a nonvanishing ground state magne-tization in the x -direction. The “energy gap” is givenby 4 p C ( k ) [see Eq.(8)] and does not necessarily de-crease from its maximum value 4 k Z , if B is tilted against e x as stated in Ref. 9. In units of the Zeeman energy E Z = 2 ~ k Z / m , the size of the “gap” E G for arbitrarymagnetic field B = B (sin θ cos φ, sin θ sin φ, cos θ ) is givenby E G E Z = 1 − ( α y cos θ − α z sin θ sin φ ) α . (12)Therefore, a finite α y term is necessary for opening the“gap” for B || e y . To emphasize this effect, we choosethe parameter set ( α y , α z , k Z ) /α = ( − . , − . , .
5) inFig. 2. In many experimental systems the confining po-tential in the y -direction might be much weaker thanin the z -direction. In this case | α y | ≪ | α z | but sub-band mixing becomes relevant. The latter strongly af-fects the spin-dependent transport properties as e.g. in-vestigated in Ref. 8, and the polarization effects discussedhere can be expected to disappear. To achieve spin po-larization in the present setup a strong confinement inthe y -direction leading to a sizable α y is thus essential.The lower dispersion branch in Fig. 2 has a “W”-likeshape. For B = ( B, , α y + α z > k Z and becomes much more complex for arbitrary magnetic field. We will focus on the situationwhere B = ( B, , t ss ′ (conductance divided by e / ~ )of an electron current at fixed Fermi energy E F passing apotential step in the wire direction [see Fig. 1 a)] are ob-tained by assuming continuity of the wave functions andtheir derivatives at the interface. Here the first index la-bels the branch to the left and the second index labels thebranch to the right of the potential step. It was arguedin Ref. 15 that one has to consider the continuity of thewave function’s flux and not simply its derivative, butin our setup both conditions lead to the same equationsas we consider a homogeneous SOI. The total transmis-sion T is the sum of the four components t ++ , t + − , t − + ,and t −− . To the right of the potential and for momenta | k | & α , one can assign spins with quantum numbers ↑ , ↓ and a properly chosen quantization axis to the branches s = + , − because of the independence of h σ i k,s on k .However, the polarization vector is given by P = t ++ + t − + T h σ i k, + + t + − + t −− T h σ i k, − . (13)Since the potential step geometry was alreadydiscussed, we will only shortly mention the influence ofthe additional term H ′ R , defined in Eq. (4), and discussthe interesting case of a potential barrier [see Fig. 1 b)] inmore detail. The latter can experimentally be achievedby adding gates to the 1D quantum wire.As shown in Fig. 3, the total polarization P = | P | ofthe current passing the potential step is large for ener-gies in the “gap” and increases with α . Similar to thetransmissions t ss ′ , P as well as the parallel polarization P x depend only on V , k Z , and α for B || e x and not on α y and α z independently. The relevant energy scale of thepolarization shown in Fig. 3 is given by E Z , which definesthe size of the “gap” [see Eq. (12)]. Therefore, energiesare given in units of E Z and wave vectors in units of k Z .The same holds for the transmissions and polarizationsshown further down (see Figs. 4 and 5). The parametersin Fig. 3 are V /E Z = 15, and α/k Z = 2, 2 .
5, 3, 5. Theenergy offset is chosen such that E F /E Z = 0 correspondsto the middle of the “gap”. The parallel polarization P x gives the main contribution to the total polarization as po l a r i za ti on α /k Z =2.0 α /k Z =2.5 α /k Z =3.0 α/ k Z =5.0 -6 -4 -2 0 2 4 energy / E Z P x / P FIG. 3: (Color online) Polarization of the transmission cur-rent at a potential step as a function of the Fermi energy for V /E Z = 15 and α/k Z = 2, 2 .
5, 3, 5. The total polarization P is sizable for energies in the “gap” (indicated by the ar-rows). In this regime it is mostly carried by P y and P z . Thepolarization becomes negligible for energies outside the “gap”where P x dominates. the energy departs from the “gap”, P x /P →
1. How-ever, in this region the total polarization is negligibleand within the “gap”, the parallel component plays aninferior role. The ratio of the two perpendicular polar-izations is given by | P z /P y | = α y /α z . Therefore, theorthogonal polarization P ⊥ = (0 , P y , P z ) can be rotatedwithin the y - z -plane by adjusting α y and α z .We next study the transmission current at a potentialbarrier of height V and width 2 x c [see Fig. 1 b)]. Thissituation might be more realistic than a simple poten-tial step if one thinks of further structuring by applyinggates to the quantum wire. Fig. 4 shows the four com-ponents of the transmission as a function of E F /E Z for α/k Z = 2, 2 .
5, 3, V /E Z = 15, and k Z x c = 1. Again, theSOI affects the transmissions t ss ′ only via α . Interest-ingly and in contrast to the potential step, the s -flippingtransmissions are degenerate, t + − = t − + . This can beunderstood, if one considers the possible s -flips at thetwo interfaces leading to an overall s -flip. Labeling theleft interface (1) and the right (2), one simply has to takethe sum of the products of transmissions at each interfaceand obtains t + − = t ++ (1) t + − (2) + t + − (1) t −− (2) ,t − + = t −− (1) t − + (2) + t − + (1) t ++ (2) . (14)An analysis of the potential step problem shows thatthe s -conserving transmissions t ++ and t −− are indepen-dent of the sign of V and the s -flipping transmissionsjust swap, i.e. t + − (1) = t − + (2) and t − + (1) = t + − (2).This leads to exactly the same values of t + − and t − + in Eq. (14). The exponential suppression of t ++ (1) and t − + (1) for energies within the “gap” does not affect thisbehavior. The s -conserving transmissions t ++ and t −− show an oscillatory behavior, which is well known from t r a n s m i ss i on t ss ’ α /k Z =2.0 α /k Z =2.5 α /k Z =3.0 -5 0 5 10 15 energy / E Ζ t ++ t +- t -- t -+ FIG. 4: (Color online) Partial transmissions at a potentialbarrier as a function of the energy for V /E Z = 15 and α/k Z = 2, 2 .
5, 3. po l a r i za ti on α /k Z =2.0 α /k Z =2.5 α /k Z =3.0 -5 0 5 10 15 20 energy / E Z P x / P FIG. 5: (Color online) Polarization of the transmission cur-rent at a potential barrier as a function of the energy for thesame parameters as in Fig. 4. The polarization is sizable forenergies well beyond the “gap” (indicated by the arrows) andshows oscillatory behavior. The x -component P x is only rel-evant in regimes where the total polarization is small. scattering off a potential step at vanishing SOI. How-ever, especially for low energies, the amplitude stronglydepends on α . The s -flipping transmissions t + − and t − + oscillate as well. The second peak of t ++ , which lies inthe “energy gap”, is suppressed compared to t −− , sinceright-moving ( s = +)-waves are exponentially dampedin the barrier region and therefore, as shown in Ref. 9, t −− is the dominant component at each interface in thisenergy range.Fig. 5 shows P and P x /P for the same parameters as inFig. 4. and α/k Z = 2, 2 .
5, 3. Similarly to the potentialstep case, P = | P | and P x only depend on α and not on α y and α z independently. Surprisingly, the polarizationnow has a sizable value in an energy interval much biggerthan the “gap”, which just goes from − E Z to E Z (see thearrows in Fig. 5). This behavior must be contrasted tothe polarization in the case of a potential step as shown inFig. 3 and first introduced in Ref. 9. It can be traced backto the energy dependence of t + − and t − + shown in Fig. 4.Both have finite weight well beyond the “energy gap”.This might be due to interference effects of transmittedand reflected waves in the barrier region. III. LATTICE MODEL
In a next step, we are aiming at constructing a tight-binding lattice model which in appropriate parameterregimes shows similar physics as our continuum model.This will put us in a position to study the effect ofelectron-electron interaction neglected so far using thefunctional renormalization group method. In 1D wiresthe two-particle interaction is known to strongly alterthe low-energy physics of many-body systems leading toso called Luttinger liquid behavior. It can be expectedthat the interplay of the SOI effects discussed above andcorrelation effects leads to interesting physics. The SOIcan be modeled by spin-flip hopping terms with ampli-tude α y and α z in a usual tight-binding model. We start with a representation of the Hamiltonian interms of Wannier states | j, σ i with j ∈ Z labeling thelattice site and σ = ↑ , ↓ labeling the spin. The spin quan-tization is chosen along the z -direction. With c † j,σ beingthe creation operator of an electron at site j with spin σ ,the lattice model Hamiltonian for an arbitrary magneticfield B = B (sin θ cos ϕ, sin θ sin ϕ, cos θ ) can be writtenas H = H + H pot + H R + H Z , (15)with the free part H = ǫ X j,σ c † j,σ c j,σ − t X j,σ (cid:16) c † j +1 ,σ c j,σ + c † j,σ c j +1 ,σ (cid:17) , (16)containing the on-site energy and the conventional (spin-conserving) hopping, external potential (due to e.g. nano-device structuring) H pot = X j,σ V j,σ c † j,σ c j,σ , (17)the spin-flip (Rashba) hopping terms H R = − α z X j,σ,σ ′ (cid:16) c † j +1 ,σ ( iσ y ) σ,σ ′ c j,σ ′ + H.c. (cid:17) (18)+ α y X j,σ,σ ′ (cid:16) c † j +1 ,σ ( iσ z ) σ,σ ′ c j,σ ′ + H.c. (cid:17) , and the Zeeman term H Z = 2 k Z X j,σ,σ ′ c † j,σ h ( σ x ) σ,σ ′ sin θ cos ϕ (19)+ ( σ y ) σ,σ ′ sin θ sin ϕ + ( σ z ) σ,σ ′ cos θ i c j,σ ′ . We show the analogy to the continuum case suppressing H pot and take as an ansatz for the corresponding eigen-states | k, s i = X j,σ a sσ ( k ) e ikj | j, σ i . (20)This leads to the eigenenergies E ( s ) ( k ) = ǫ − t cos k + 2 s sgn( k − k ) p D ( k ) , (21)with k = arcsin (cid:2) − k Z ( α y cos θ − α z sin θ sin ϕ ) / ( α y + α z ) (cid:3) (22)and D ( k ) = ( α y + α z ) sin k + k Z (23)+2 k Z sin k ( α y cos θ − α z sin θ sin ϕ ) . Eq. (21) has almost the same form as the continuum ver-sion Eq. (8). In fact, choosing the on-site energy ǫ = 2 t ,which corresponds just to an overall energy shift, andsubstituting cos k by 1 − k / k by k , which isvalid for sufficiently small | k | , we get exactly the sameform. Note however that, in contrast to the continuumcase, α y , α z and k Z now have the unit of energy, butsince c ( s ) k , defined in Eq. (24) is dimensionless, all formu-las remain valid. We choose for the eigenstates Eq. (20) a s ↓ ( k ) = 1 and obtain a s ↑ ( k ) = c ( s ) k with c ( s ) k = − iα z sin k − k Z e − iϕ sin θα y sin k + k Z cos θ − s sgn( k − k ) q ( α y + α z ) sin k + 2 k Z sin k ( α y cos θ − α z sin θ sin ϕ ) + k Z , (24) e n e r gy / E α s = +s = - -3 -2 -1 0 1 2 3 k / α -1-0.500.51 s p i n ( s =+ ) a) < σ x >< σ y >< σ z > e n e r gy / E α s = +s = - -3 -2 -1 0 1 2 3 k / α -1-0.500.51 s p i n ( s =+ ) b) < σ x >< σ y >< σ z > e n e r gy / E α s = +s = - -3 -2 -1 0 1 2 3 k / α -1-0.500.51 s p i n ( s =+ ) c) < σ x >< σ y >< σ z > FIG. 6: (Color online) Lattice dispersion and spin expectation values on the ( s = +)-branch for a magnetic field in a) x - , b) y -and c) z -direction for t/α = 1, α y /α = − . α z /α = − . k Z /α = 0 .
5. The spin on the ( s = − )-branch points in the oppositedirection, i.e. h σ i i k,s = −h σ i i k, − s . For | k − k | < π/ and the spin expectation values have exactly the contin-uum form h σ x + iσ y i k,s = 2 (cid:16) c ( s ) k (cid:17) ∗ (cid:12)(cid:12)(cid:12) c ( s ) k (cid:12)(cid:12)(cid:12) , h σ z i k,s = − (cid:12)(cid:12)(cid:12) c ( s ) k (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) c ( s ) k (cid:12)(cid:12)(cid:12) . (25)The energy dispersions and the spin expectation valuesfor magnetic fields in x -, y -, and z -direction are shownin Fig. 6. Besides the cosine-like structure, which be-comes especially relevant near the band edges, the dis-persion and spin expectation values have the same shapeas in the continuum model. A direct comparison of Fig. 6and Fig. 2 shows that our lattice model reproduces thelow energy physics, i.e. for | k − k | < π/
2, observed inthe continuum. As above we only show the spin ex-pectation values on the ( s = +)-branch. The spin onthe ( s = − )-branch points in the opposite direction,i.e. h σ i i k,s = −h σ i i k, − s . The direct relation between thedispersion and the spin expectation values for energiesof the order of the “gap” is the essential feature leadingto the remarkable scattering properties of the continuummodel (and eventually a spin polarized conductance) atsteps and barriers. One can thus expect similar trans-port characteristics to be realized in the lattice model. Adetailed discussion of this and in particular the effect ofthe electron-electron interaction on transport will be thetopic of an upcoming publication. IV. CONCLUSIONS
We have investigated the dispersion and spin expec-tation values of a 1D electron system with SOI as well as arbitrary magnetic field, and have shown that an ad-ditional SOI term resulting from the lateral confinementof a 2DEG to a 1D wire geometry leads to a rotationof the spin out of the 2D plane. For the case of a mag-netic field parallel to the quantum wire, the transmissionand polarization of a linear response current at a poten-tial step as well as at a potential barrier were studied.For the latter, we observed an extended energy range,where significant spin polarization can be achieved. Weshowed that this spin polarization can be rotated out ofthe plane of the 2DEG arbitrarily by adjusting the SOIconstants α y and α z . The potential barrier describes asetup which can experimentally be achieved by addingfurther gates to the wire geometry. We then constructeda lattice model which shows the same low energy physicsas the continuum model. This lattice model now enablesus to investigate the interplay of SOI and Coulomb inter-action in quantum wires with potential steps and barriersusing the functional renormalization group method. Acknowledgments
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