Quantum dissipative Rashba spin ratchets
Sergey Smirnov, Dario Bercioux, Milena Grifoni, Klaus Richter
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Quantum dissipative Rashba spin ratchets
Sergey Smirnov, Dario Bercioux, , Milena Grifoni, and Klaus Richter Institut f¨ur Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany Physikalisches Institut, Albert-Ludwigs-Universit¨at, D-79104 Freiburg, Germany (Dated: December 1, 2018)We predict the possibility to generate a finite stationary spin current by applying an unbiased acdriving to a quasi-one-dimensional asymmetric periodic structure with Rashba spin-orbit interactionand strong dissipation. We show that under a finite coupling strength between the orbital degreesof freedom the electron dynamics at low temperatures exhibits a pure spin ratchet behavior, i.e. afinite spin current and the absence of charge transport in spatially asymmetric structures. It is alsofound that the equilibrium spin currents are not destroyed by the presence of strong dissipation.
PACS numbers: 03.65.Yz, 72.25.Dc, 73.23.-b, 05.60.Gg
An opportunity to induce a net stationary particlecurrent by unbiased external forces applied to a quan-tum dissipative one-dimensional (1D) periodic structureis provided when the system does not possess a center ofinversion in real space [1]. Then the particle transportoccurs due to the ratchet effect and the device works asa Brownian motor [2]. In the deep quantum regime thecharge ratchet effect can only be achieved when at leastthe two lowest Bloch bands contribute to transport [3].Recently a new research field of condensed matterphysics, spintronics, has emerged. One of its central is-sues is how to generate pure spin currents (SC) in para-magnetic systems due to only spin-orbit interactions andwithout applied magnetic fields. Rashba spin-orbit in-teraction (RSOI) [4] represents one of the possible toolsto reach this goal since the spin-orbit coupling strengthcan be externally controlled by a gate voltage. One wayto get pure SC is due to the intrinsic spin Hall effect[5, 6] expected in a high-mobility two-dimensional semi-conductor systems with RSOI [7]. Such pure SC wereexperimentally detected through the reciprocal spin-Halleffect in Ref. [8]. An alternative is to induce pure SCthrough absorption of polarized light [9]. The generationof pure SC by coherent spin rectifiers [10] has been dis-cussed only recently for a finite size setup with RSOI.However, the presence of dissipation has not been con-sidered up to now.In this letter we address the challenging task of howto implement devices which can work both as Brown-ian charge and spin motors. Here a natural and alsoprinciple question for spintronics arises: Is it possibleto switch a device working as a charge ratchet to a purespin ratchet mode where the charge current (CC) is com-pletely blocked? As mentioned above, when in a dissipa-tive system without RSOI transport is restricted to onlyone Bloch band, the charge ratchet mechanism does notexist [3]. Whether the same effect takes place in a dis-sipative system with RSOI is an open and non-trivialquestion. In fact, the Rashba Hamiltonian is not in-variant under reflection of a transport direction. Thusthe Rashba Hamiltonian itself already has a built-in spa- Q u a s i − D w i r e c o n f i n e m e n t T r a n s v er s e e l ec t r o n s V V − V − V VV A s y m m e t r i c p e r i o d i c g a t e s : U ( x ) A s y m m e t r i c p e r i o d i c g a t e s : U ( x ) z x FIG. 1: (Color online) A schematic picture of the isolatedasymmetric periodic quasi-1D structure described by theHamiltonian (1). In the center of the quasi-1D wire the peri-odic potential is weaker and gets stronger closer to the edges.Thus the electron group velocity is higher in the central regionand tails off away from the center. tial asymmetry which due to the spin-orbit coupling canbe further mixed with the periodic potential symme-try/asymmetry. The presence of dissipation additionallyincreases the complexity of the problem because the in-fluence of a dissipative environment on the orbital motionchanges through RSOI the spin dynamics.In this work we focus on the moderate-to-strong dissi-pation case and address how to implement a device whichunder influence of unbiased external ac-driving yields afinite stationary spin current and at the same time blocksthe directed stationary charge transport. To concretizeour idea of a Brownian spin motor we consider a dis-sipative periodic system with RSOI and show that thespin-orbit interaction alone is not enough to produce SC:The system must additionally lack the spatial symmetryand its orbital degrees of freedom must be coupled.The full Hamiltonian of our problem is ˆ H full ( t ) =ˆ H + ˆ H ext ( t ) + ˆ H bath , where ˆ H is the Hamiltonian of theisolated periodic system, ˆ H ext ( t ) describes an externaldriving, and ˆ H bath is responsible for dissipative processes.The isolated quasi-1D periodic system is formed in atwo-dimensional electron gas (2DEG) with RSOI using aperiodic potential along the x -axis and a harmonic con-finement along the z -axis:ˆ H = ~ ˆ k m + mω ˆ z − ~ k so m (cid:0) ˆ σ x ˆ k z − ˆ σ z ˆ k x (cid:1) + U γ (ˆ x, ˆ z ) , (1)where U γ (ˆ x, ˆ z ) = U (ˆ x )(1 + γ ˆ z /L ), ˆ k is related to themomentum operator as ˆ p = ~ ˆ k , ω is the harmonic con-finement strength, k so the spin-orbit coupling strength, U (ˆ x ) the periodic potential with the period L , and γ > E ( t ) ≡ E ( t )ˆ e x .It can be experimentally implemented using for examplelinearly polarized light. This yields ˆ H ext = eE ( t )ˆ x , where e is the elementary charge. We use the time dependence eE ( t ) ≡ F cos(Ω( t − t )), which is unbiased.The system is also coupled to a thermal bath. We as-sume the transverse confinement to be strong enough sothat the probabilities of the direct bath-excited transi-tions between the transverse modes are negligibly small.Thus the environment couples to the electronic degreesof freedom only through ˆ x . Furthermore, in the spiritof the Caldeira and Leggett model [11], we consider aharmonic bath with bilinear system-bath coupling.The dynamical quantities of interest are the ratchetcharge and spin currents J C,S ( t ) given as the statisticalaverage of the longitudinal charge and spin current op-erators, J C,S ( t ) ≡ Tr[ ˆ J C,S ˆ ρ ( t )], where ˆ ρ ( t ) is the reducedstatistical operator of the system, that is the full one withthe bath degrees of freedom traced out. The CC operatoris ˆ J C ( t ) = − ed ˆ x/dt and for the SC operator we use thedefinition suggested in Ref. [12], ˆ J S ( t ) = d (cid:0) ˆ σ z ˆ x (cid:1) /dt .It is convenient to calculate the traces using the basiswhich diagonalizes both ˆ x and ˆ σ z , because this requiresto determine only the diagonal elements of the reduceddensity matrix. As shown in Ref. [13], for a periodicsystem with RSOI the energy spectrum can be derivedfrom the corresponding truly 1D problem without RSOI.This leads to so-called Bloch sub-bands. The 2DEG is as-sumed to be sufficiently dilute to neglect the Pauli exclu-sion principle in the temperature range of our problem.The upper limit of this temperature range is consideredto be low enough so that only the lowest Bloch sub-bandsare populated. The basis which diagonalizes ˆ x and ˆ σ z becomes in this case discrete. The total number of theBloch sub-bands is equal to the product of the number, N B , of the lowest Bloch bands from the correspondingtruly 1D problem without RSOI, the number, N t , of thelowest transverse modes and the number of spin states.In this work we shall use the model with N B = 1, N t = 2.The total number of the Bloch sub-bands in our problemis thus equal to four. Using N B = 1 we also assume thatthe external field is weak enough and does not excite electrons to higher Bloch bands. The representation interms of the eigen-states of ˆ x for a model with discrete x -values is called discrete variable representation (DVR)[3, 14]. Let us call σ -DVR the representation in whichboth the coordinate and spin operators are diagonal. De-noting the σ -DVR basis states as {| α i} and eigen-valuesof ˆ x and ˆ σ z in a state | α i by x α and σ α , respectively, theCC and SC are rewritten as J C ( t ) = − e P α x α ˙ P α ( t ) and J S ( t ) = P α σ α x α ˙ P α ( t ), where P α ( t ) ≡ h α | ˆ ρ ( t ) | α i is thepopulation of the σ -DVR state | α i at time t .We are interested in the long time limit ¯ J ∞ C,S of thecurrents ¯ J C,S ( t ), averaged over the driving period 2 π/ Ω.The advantage of working in the σ -DVR basis is thatreal-time path integral techniques can be used to traceout exactly the bath degrees of freedom [15, 16]. More-over, at driving frequencies larger than the ones char-acterizing the internal dynamics of the quasi-1D systemcoupled to the bath, the averaged populations ¯ P α ( t ) canbe found from the master equation,˙¯ P α ( t ) = X β, ( β = α ) ¯Γ αβ ¯ P β ( t ) − X β, ( β = α ) ¯Γ βα ¯ P α ( t ) , (2)valid at long times. In Eq. (2) ¯Γ αβ is an averaged tran-sition rate from the state | β i to the state | α i .The first task is thus to identify the σ -DVR basis. Theeigen-states | l, k B , j, σ i of ˆ σ z were found in [13] for thecase γ = 0. The results obtained in [13] are straight-forwardly generalized to our model since for N t = 2 theoperator ˆ z (and any even power of ˆ z ) is effectively di-agonal. The quantum numbers l , k B , j , σ stand for theBloch band index, quasi-momentum, transverse mode in-dex and z -projection of the spin, respectively. As men-tioned above l = 1, j = 0 ,
1. One further finds h l ′ , k ′ B , j ′ , σ ′ | ˆ x | l, k B , j, σ i == δ j ′ ,j δ σ ′ ,σ j h l ′ , k ′ B + σk so | ˆ x | l, k B + σk so i j , (3)where the index j under the bra- and ket-symbols in-dicates that the corresponding electronic states are ob-tained using the periodic potential U γ,j ( x ) ≡ U ( x )[1 + γ ~ ( j + 1 / /mω L ]. For a fixed value of j the di-agonal blocks in Eq. (3) are unitary equivalent andthus the eigen-values of ˆ x do not depend on σ . Theeigen-values of the matrix j h l ′ , k ′ B | ˆ x | l, k B i j are analyti-cally found and have the form x ζ,m,j = mL + d ζ,j , where m = 0 , ± , ± . . . , ζ = 1 , , . . . , N B and the eigen-values d ζ,j are distributed within one elementary cell. Thus onecan label the eigen-states of ˆ x as | ζ, m, j, σ i . The corre-sponding eigen-values are x ζ,m,j,σ = x ζ,m,j . We see thatthe σ -DVR basis states | α i introduced above are just the | ζ, m, j, σ i states, that is {| α i} ≡ {| ζ, m, j, σ i} .To calculate CC and SC we use the tight-bindingapproximation assuming that the matrix elements h ζ ′ , m ′ , j ′ , σ ′ | ˆ H | ζ, m, j, σ i with | m ′ − m | > | m, ξ i ≡ | ζ = 1 , m, ξ i where { ξ } = { ( j, σ ) } and ξ = 1 ⇔ (0 , ξ = 2 ⇔ (0 , − ξ = 3 ⇔ (1 , ξ = 4 ⇔ (1 , − m ′ ,mξ ′ ,ξ ≡ h m ′ , ξ ′ | ˆ H | m, ξ i ( m ′ = m and/or ξ ′ = ξ ) and on-site energies ε ξ ≡ h m, ξ | ˆ H | m, ξ i .Due to the harmonic confinement and RSOI the sys-tem is split into two channels: one with ξ = 1 , ξ = 2 ,
3. The two channels are indepen-dent of each other, that is, transitions between them areforbidden. This picture is general and valid for an ar-bitrary number of the transverse modes. For clarity webelow only consider the channel with ξ = 1 ,
4. Two in-dependent channels were also found for a different typeof confinement in Ref. [17].Assuming that the hopping matrix elements are smallenough we can use the second-order approximation [3]for the averaged transition rates in Eq. (2). We have¯Γ m ′ ,mξ ′ ,ξ = | ∆ m ′ ,mξ ′ ,ξ | ~ Z ∞−∞ dτ exp (cid:20) − ( x m,ξ − x m ′ ,ξ ′ ) ~ Q [ τ,J ( ω )]++ i ε ξ − ε ξ ′ ~ τ (cid:21) J (cid:20) F ( x m,ξ − x m ′ ,ξ ′ ) ~ Ω sin (cid:18) Ω τ (cid:19)(cid:21) , (4)where x m,ξ ≡ x ζ =1 ,m,ξ = mL + d ξ with d ξ ≡ d ,j . InEq. (4) J ( x ) denotes the zero-order Bessel function and Q [ τ, J ( ω )] is the twice integrated bath correlation func-tion being a function of time τ and a functional of thebath spectral density J ( ω ) [3, 16]. The dependence of thetransition rates on the orbit-orbit coupling γ comes fromtwo sources. The first one is the Bloch amplitudes andthe second is the difference ∆ d ≡ d , − d , . In a tight-binding model the periodic potential is strong and thus∆ d can be made less than all the relevant length scales,∆ d/l r ≪
1, where l r = min[ L, p ~ /mω , ~ Ω /F ]. Hencethe main effect of the orbit-orbit coupling on ¯Γ m ′ ,mξ ′ ,ξ comesonly through the Bloch amplitudes, and we neglect termsof order O (∆ d/l r ).We then arrive at the main results of our work, theabsence of the charge transport, ¯ J ∞ C = 0, and the ex-pression for the non-equilibrium spin current (NESC),¯ J ∞ n-e,S ≡ ¯ J ∞ S − ¯ J ∞ e,S :¯ J ∞ n-e,S = − L (cid:18) I I I + I − I (0)14 I (0)41 I (0)14 + I (0)41 (cid:19) k ~ ω m ×× X k B ,k ′ B sin[( k B − k ′ B ) L ]Im[ F k B ,k ′ B ] , (5)where I ξ ′ ,ξ , I (0) ξ ′ ,ξ are the integrals from (4) with and with-out driving, F = 0 and F = 0, respectively, and F k B ,k ′ B ≡ u DVR γ, ,k B + k so ( d , ) u DVR γ, ,k ′ B − k so ( d , ) ×× [ u DVR γ, ,k B − k so ( d , ) u DVR γ, ,k ′ B + k so ( d , )] ∗ , (6)where u DVR γ,j ;1 ,k B ( d ,j ) is the DVR Bloch amplitude of thefirst band for electrons in the periodic potential U γ,j ( x ). Amplitude of the driving force, F [ h¯ ω /L ] -25-20-15-10-50 S p i n c u rr e n t J n - e , S ∞ [ - L ω ] η=0.5η=0.25η=0.75 -1 -0.5 0 0.5 1 x/L -4048 U ( x ) [ h ¯ ω ] Asymmetric periodic potential U (x)
FIG. 2: (Color online) Non-equilibrium spin current, ¯ J ∞ n-e,S ,as a function of the amplitude, F , of the driving force fordifferent values of the viscosity coefficient η . Temperature k Boltz. T = 0 .
5, spin-orbit coupling strength k so L = π/
2, orbit-orbit coupling strength γ = 0 .
1, driving frequency Ω = 1. Theinset displays the shape of the periodic potential.
In Eq. (5) we have eliminated from ¯ J ∞ S the equilib-rium spin current (ESC), ¯ J ∞ e,S , following Ref. [18]. Thefact that the ESC turns out to be finite shows that thedefinition of SC suggested in Ref. [12] does not automati-cally eliminate the presence of ESC. However, as pointedout in Ref. [12], this current really vanishes in insula-tors. This can be seen from Eq. (5). When the potentialis strong, electrons are localized, the dependence of thefunction F k B ,k ′ B on the quasi-momentum disappears, andas a result both ESC and NESC are equal to zero. Thisreasonable result is ensured by the spin current defini-tion taking proper care of the spin torque. It is inter-esting to note that ESCs are present even in a systemwith strong dissipation. As recently proposed in Ref.[19], ESCs can effectively be measured using a Rashbamedium deposited on a flexible substrate playing a roleof a mechanical cantilever.We can determine the conditions under which the SCis finite. First of all from Eq. (5) it follows that the spin-orbit coupling must be finite, i.e. k so = 0. Further, fromEq. (6) one observes, that when γ = 0, the Bloch ampli-tudes do not depend on j , u DVR γ =0 ,j ;1 ,k B ( d ,j ) ≡ u DVR1 ,k B ( d ),and since [ u DVR1 ,k B ( d )] ∗ = u DVR1 , − k B ( d ) (time-reversal sym-metry), the function F k B ,k ′ B becomes even with respect toits arguments. Then from Eq. (5) one gets zero SC. Thusthe second condition is the presence of the orbit-orbitcoupling. Finally, since for a symmetric periodic poten-tial the Bloch amplitudes are real functions, we concludethat the function F k B ,k ′ B is also real in this case, that isIm[ F k B ,k ′ B ] = 0. As a result the third condition is thepresence of spatial asymmetry.Below we present corresponding numerical results.All energies and frequencies are given in units of ~ ω and ω , respectively. The parameters are taken foran InGaAs/InP quantum wire: ~ ω = 0 . α ≡ ~ k so /m = 9 . · − eV · m; m = 0 . m , respectively. Spin-orbit coupling strength, k so L/ π -2-1.5-1-0.500.511.52 S p i n c u rr e n t J n - e , S ∞ [ L ω ] γ = 0.25 γ = 0.1 γ = 0.5 γ = 0.75 γ = 1.0 FIG. 3: (Color online) Non-equilibrium spin current, ¯ J ∞ n-e,S , asa function of the spin-orbit coupling strength, k so , for differentvalues of the orbit-orbit coupling strength, γ . The drivingamplitude and viscosity coefficient are F = 2 ~ ω /L , η = 0 . For k so L = π/ L = 0 . µ m.The dependence of the NESC on the amplitudeof the external driving is shown in Fig. 2 for theasymmetric periodic potential (see inset) U ( x ) = P n =0 V n cos(2 πnx/L − φ n ) with V = 4, V = − V , V = 3 . φ = φ = 0 . φ = 1 .
9. The gap betweenthe Bloch bands with l = 1 and l = 2 is ∆ E ≈ . F L, ~ Ω < ∆ E that is the numerical resultsare consistent with the theoretical model assumptions.As an example we have used an Ohmic bath with thespectral density J ( ω ) = ηω exp( − ω/ω c ), where the vis-cosity coefficient (in units of mω ) is η = 0 . , . , . ω c = 10. As it can be seen,the NESC has an oscillating nature. However, the oscil-lation amplitude goes down when the driving increases.Physically such behavior can be attributed to an effectiverenormalization of the band structure in a high-frequencyelectric field [15]. The group velocity decreases in a non-monotonous way which due to RSOI slows down the spinkinetics. For increasing values of η the dissipation in-duced decoherence in the system gets more pronounced.The system becomes more classical and thus the tunnel-ing processes become less intensive. This leads to thespin current reduction which one observes in Fig. 2.In Fig. 3 the NESC is plotted versus k so L while γ playsthe role of a parameter. The oscillations of the NESChave minima located at nG/ n = 0 , , , . . . , and G is the reciprocal lattice vector. Physically this reflectsthe fact that for those values of k so the Rashba split be-comes minimal due to the periodicity of the energy spec-trum in the k -space. The magnitude of these oscillationsdecreases with decreasing orbit-orbit coupling, and thecurrent vanishes for γ = 0.In summary, we have studied stationary quantumtransport in a driven dissipative periodic quasi-one-dimensional system with Rashba spin-orbit interactionand orbit-orbit coupling. The spin ratchet effect has beeninvestigated and an analytical expression for the spin cur- rent has been derived and analyzed. This analysis hasrevealed that for the case of moderate-to-strong dissipa-tion the necessary conditions for non-vanishing spin cur-rents are the spatial asymmetry of the periodic potentialas well as a finite strength of the spin-orbit interactionand orbit-orbit coupling. It has been demonstrated thatin a dissipative system equilibrium spin currents can ex-ist. Our numerical calculations have shown characteristicoscillations of the spin current as a function of the am-plitude of the driving force and the spin-orbit couplingstrength. Finally, we note, that since the spin current hasthe in-plane polarization, it can be efficiently measuredby a magneto-optic Kerr microscope using the cleavededge technology as suggested recently in Ref. [20].We thank J. Peguiron for useful discussions. Supportfrom the DFG under the program SFB 689 is acknowl-edged. [1] P. Reimann, M. Grifoni, and P. H¨anggi, Phys. Rev. Lett. , 10 (1997).[2] R. D. Astumian and P. H¨anggi, Phys. Today , 33(2002).[3] M. Grifoni, M. S. Ferreira, J. Peguiron, and J. B. Majer,Phys. Rev. Lett. , 146801 (2002).[4] E. Rashba, Fiz. Tverd. Tela (Leningrad) , 1224 (1960).[5] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science ,1348 (2003).[6] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung-wirth, and A. H. MacDonald, Phys. Rev. Lett. , 126603(2004).[7] For an experimental indication see J. Wunderlich,B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev.Lett. , 047204 (2005).[8] S. O. Valenzuela and M. Tinkham, Nature (London) ,176 (2006).[9] B. Zhou and S.-Q. Shen, Phys. Rev. B , 045339 (2007).[10] M. Scheid, A. Pfund, D. Bercioux, and K. Richter, Phys.Rev. B , 195303 (2007).[11] A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. ,211 (1981).[12] J. Shi, P. Zhang, D. Xiao, and Q. Niu, Phys. Rev. Lett. , 076604 (2006).[13] S. Smirnov, D. Bercioux, and M. Grifoni, Europhys. Lett. , 27003 (2007), arXiv:0705.3830v2.[14] D. O. Harris, G. G. Engerholm, and W. D. Gwinn, J.Chem. Phys. , 1515 (1965).[15] M. Grifoni and P. H¨anggi, Phys. Rep. , 229 (1998).[16] U. Weiss, Quantum Dissipative Systems (World Scien-tific, Singapore, 1999), 2nd ed.[17] C. A. Perroni, D. Bercioux, V. M. Ramaglia, andV. Cataudella, J. Phys. Condens. Matter , 186227(2007).[18] E. I. Rashba, Phys. Rev. B , 241315(R) (2003).[19] E. B. Sonin, Phys. Rev. Lett. , 266602 (2007).[20] P. Kotissek, M. Bailleul, M. Sperl, A. Spitzer, D. Schuh,W. Wegscheider, C. H. Back, and G. Bayreuther, Nat.Phys.3