Single-parameter spin-pumping in driven metallic rings with spin-orbit coupling
J. P. Ramos, L. E. F. Foa Torres, P. A. Orellana, V. M. Apel
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Single-parameter spin-pumping in driven metallic rings withspin-orbit coupling
J. P. Ramos, L. E. F. Foa Torres, P. A. Orellana, and V. M. Apel Departamento de F´ısica, Universidad Cat´olica del Norte,Angamos 0610, Casilla 1280, Antofagasta, Chile Instituto de F´ısica Enrique Gaviola (CONICET) and FaMAF,Universidad Nacional de C´ordoba, Ciudad Universitaria 5000, C´ordoba, Argentina Departamento de F´ısica, Universidad Federico Santa Mar´ıa,Avenida Vicu˜na Mackenna 3939, San Joaquin, Santiago, Chile (Dated: May 8, 2019)
Abstract
We consider the generation of a pure spin-current at zero bias voltage with a single time-dependent potential. To such end we study a device made of a mesoscopic ring connected toelectrodes and clarify the interplay between a magnetic flux, spin-orbit coupling and non-adiabaticdriving in the production of a spin and electrical current. By using Floquet theory, we show thatthe generated spin to charge current ratio can be controlled by tuning the spin-orbit coupling. . INTRODUCTION Largely forgotten during the early decades of nanoelectronics, the spin degree of freedom isbecoming ever closer to the center of the research stage . Indeed, generating and detectingspin-currents is now a fascinating field of research with applications in future electronics ,quantum computing and information storage . Among the many ways of harnessing theelectron spin, spin orbit interaction (SOI) in two-dimensional electron gases is a promisingone since the spin transport properties can be controlled simply by applying an electricfield .Most of the proposals aiming at the control of the spin degree of freedom use static elec-tric or magnetic fields . Here we follow a different path and use alternating fields (ac) asin . The time-dependence introduced by the alternating fields provide an avenue forexploring new phenomena including the opening of a laser-induced bandgap or chiraledge states and, more generally, the tuning of its topological properties . Anotherstriking phenomena is the coherent generation of a current at zero bias voltage (termed quantum charge pumping ) and, as shown below, the generation of a pure spin-currentsthrough spin pumping. Quantum pumping is usually achieved by driving a sample connectedto electrodes through ac gate voltages. Within the adiabatic approximation , pumping anon-vanishing charge requires the presence of at least two time-dependent parameters (typ-ically constituted by gate voltages) and has been widely studied in many systems includingpristine and disordered graphene . But beyond the adiabatic approximation single-parameter pumping is also possible as predicted theoretically (similar to the mesoscopicphotovoltaic effect predicted earlier in ) and achieved in careful experiments . Besidesreducing the burden of adding more contacts in a nanoscale sample, a single parametersetup could also prove advantageous (as a compared to a two-parameter one) in reducingcapacitive effects and crosstalk between time-dependent gates.Here we address the effect of spin-orbit coupling and its interplay with a single time-dependent field in the generation of non-adiabatic spin current at zero bias voltage. To suchend we consider a setup as the one represented in Fig.1, where a nanoscale ring is connectedto electrodes and has a quantum dot embeded in one of its arms. The time dependence isintroduced as an alternating gate applied to the quantum dot and does not break neithertime-reversal nor inversion symmetry (parity). Crucial to the generation of pumped current2s the addition of a magnetic flux threading the ring as shown in Fig. (1). The spin-orbitcoupling is introduced as an additional spin-dependent flux. In this paper we show how thissimple setup is able to provide a minimal model where a pure spin-current can be achieved. FIG. 1. Scheme of setup considered in the text, a quantum ring with a magnetic field cross themand a quantum dot embedded in one of its arms, driven by an ac voltage source.
II. HAMILTONIAN MODEL AND ITS SOLUTION THROUGH FLOQUET THE-ORY
Let us start our discussion by presenting our model Hamiltonian for the situation repre-sented in Fig.(1). The total Hamiltonian H ( ⊔ ) is written as: H ( t ) = H C + H QD ( t ) + H T , (1)where H C represents the left and right contacts and the lower arm of the ring (representedby site j = 0 in the notation below), H QD ( t ) the quantum dot in the upper arm of thering (which for simplicity is taken to be a single level), and H T the tunneling Hamiltonianbetween the quantum-ring and the contacts, which are given by H C = ∞ X j = −∞ ,σ ( ε j c † j,σ c j,σ + γc † j,σ c j +1 ,σ ) + h.c. , (2) H QD ( t ) = ε d ( t ) X σ d † σ d σ (3) H T = X σ ( V σL c †− ,σ d σ + V R c † ,σ d σ ) + h.c. . (4)The time dependence is introduced as a modulation of the energy levels of the quantumdot. For a single-level quantum-dot, this is is achieved through ε ( t ) = ε + v cos(Ω t ). We3onsider a magnetic and electric fields in the system, their contributions to the Hamiltonianare embedded in the hopping matrix elements V σL = V exp[ i π ( φ AB + σφ SO ) /φ ], where φ AB and φ SO are the phases due to the Aharonov-Bohm effect and spin-orbit interactionrespectively, σ is the spin index ( σ = ↑ , ↓ or σ = 1 , −
1) and φ is the flux quantum.Since we are interested in a single-parameter pumping configuration as in , thecalculation of the electrical response requires going beyond the adiabatic theory. Floquettheory offers a suitable framework . Here we use it in combination with Green’s func-tions, then we have a Floquet-Green function denoted by G F defined from the Floquet’sHamiltonian H F as G F = [ E I − H F ] − .If the spin-orbit coupling does not couple different spin channels, as in our case, the dccomponent of the current is given by:¯ I σ = 1 τ Z τ dtI σ ( t ) (5)¯ I σ = eh × (6) X n Z h T ( n )( R,σ ) , ( L,σ ) ( ε ) f L ( ε ) − T ( n )( L,σ ) , ( R,σ ) ( ε ) f R ( ε ) i dε, where T ( n )( R,σ ) , ( L,σ ) ( ε ) is the probability for an electron on the left ( L ) with spin σ and energy ε to be transmitted to the right ( R ) reservoir while exchanging n photons and τ = 2 π/ Ω .These probabilities are weighted by the usual Fermi-Dirac distribution functions f R ( L ) foreach electrode and are given, in terms of Floquet-Green function, by T ( n )( R,σ ) , ( L,σ ) ( ε ) = 4Γ R ( ε + n ~ Ω ) | G ( n ) LR,σ ( ε ) | Γ L ( ε ) , (7)where the probability in opposite direction is described exchanging the index L with R , andΓ L ( R ) is the matrix coupling with left (right) electrode, defined as the imaginary part of theelectrode’s self energies, i. e. Γ L ( R ) = − Im(Σ L ( R ) ).The associated spin-current is ¯ I s = ¯ I ↑ − ¯ I ↓ while the charge current is ¯ I = ¯ I ↑ + ¯ I ↓ . III. RESULTS AND DISCUSSION
Using the model introduced before we now turn to our results for the pumped electric andspin currents. To start with we consider the system in the absence of spin-orbit coupling.4e consider the leads in thermodynamic equilibrium ( i. e. f L ( ε ) = f R ( ε ) = f ( ε )) as a semi-infinite 1d system with nearest neighbor coupling γ , which is used as energy parameter. Theac field frequency is set to Ω such that ~ Ω = γ/
5, and the field magnitude is v = 0 . γ/e .The hopping between the contacts and QD is V = γ/ FIG. 2. (Color online) (a) Transmission probability from left to right as a function of the appliedmagnetic flux and the Fermi energy (for vanishing spin-orbit interaction). (b) Same as (a) for thepumped current. Note the emergence of local maxima/minima close to the parameters where atransmission zero is observed.
Figure 2a shows a contour plot of the transmission probability as a function of the Fermilevel position and the magnetic flux. There we can observe the presence of a region wherethe transmission is very close to zero (close to the intersection of the dashed lines). This isdue to a destructive quantum interference known as Fano resonance or antiresonance (fora recent review ). Interestingly, the pumped current shown in Fig. 2b achieves a maximum5ntensity whenever the parameters are tuned close to the transmission zero. Besides, we cansee that the sign of the observed maxima is reversed when traversing the transmission zero.We note that a single-time dependent harmonic potential does not break time reversalsymmetry (being defined as the existence of a time t such that the Hamiltonian whichis a function of the time t satisfies H ( t + t ) = H ( t − t )). It is the magnetic field thatbreaks TRS and allows for pumping to occur. Note, however that this is true only whenevermagnetic flux is different from the half integer multiples of the flux quantum. For a magneticflux of π for example, the Hamiltonian does not change upon time-reversal (the phase inthe hopping term V L changes from exp( iπ ) to exp( − iπ ) and therefore there is no pumpedcurrent as observed in Fig. 2-b. On the other hand one should note that the magneticfield alone would not produce pumping. This is the point where the time-dependent fieldenters into the game. Its role in this setup is to provide for additional effective channelsfor transport, thereby circumventing the constraint of phase-rigidity and allowing for thedirectional asymmetry in the transmission probabilities.The addition of spin-orbit interaction breaks the spin degeneracy and at zero bias bothcharge and spin currents are generated. By examining Fig. (2) one can imagine that the spin-orbit phase may be used to tune the working point of our pump for each spin independently .Indeed, the term φ AB + σφ SO enters as an effective spin-dependent flux φ eff σ . In particular,we could choose this spin-orbit phase so that it cancels out for one spin direction (leadingto a vanishing pumped charge for this spin) and adds up for the other, or in such a way asto cancel the charge current while summing up towards the spin current.Figures 3 (a) and (b) show the charge (solid lines) and spin (dashed lines) currents asa function of the Fermi energy for different values of the spin-orbit and Aharanov-Bohmphases, while Figures 3 (c) and (d) show the pumped current for each spin, spin up (blacksolid lines) and down (red dashed lines). As anticipated, the parameters can be chosen sothat the currents for each spin direction have opposite signs (Fig. 3 (d)), thereby leading toa pure spin-current as on Fig.3 (b). In this situation, the charge current cancels out whetherthe spin-current is maximal.A point that needs to be emphasized in this proposal is that the pumped current is in-trinsically non-adiabatic (this contrasts for example with Ref. using a similar setup butwith two time-dependent parameters). An adiabatic calculation would actually give a van-ishing response. Going beyond this adiabatic (low-frequency) limit is therefore mandatory6 FIG. 3. (Color online) a-b Pumped charge ( ¯ I , dashed line) and spin currents ( ¯ I s , solid line), dashedand solid for different values of the applied static flux and spin-orbit interaction: (a) φ AB = 0 . φ , φ SO = 0 . φ and b) φ AB = 0 . φ , φ SO = 0 . φ . One can see that in (b) the charge current vanishesbut the spin current is enhanced. The spin-resolved contributions to the current for the same casesare shown in (c) and (d). justifying the use of Floquet theory. On the other hand, the pumped currents in thiscase emerges as an interplay between photon-assisted processes and the interference in theAharanov-Bohm ring . A similar setup but without contacts to electrodes were consideredin . The spin-orbit coupling allows to obtain spin polarized pumped currents and the keyrole of the time-dependent field is to provide for additional paths for interference breakingphase-rigidity , although it does not break time-reversal symmetry.7he setup discussed here can be realized by using the present technologies. A quantumdot inserted in a mesoscopic ring has been fabricated by several laboratories in the lastdecades . A particularly interesting case would be an InGAs quantum-dot inserted ina mesoscopic quantum-ring since
InGAs has a strong spin-orbit coupling and this couplingcan be controlled by an electric field . IV. FINAL REMARKS
In summary, we study quantum spin-pumping with a single parameter in a configurationwhere the effect of the time-dependent field is reduced to the essential one: providing foradditional channels for transport. The spin-orbit interaction breaks the spin degeneracy andwe exploit it to generate a pure pumping spin-current through the independent tuning ofthe phases dues to Aharonov-Bohm effect and spin-orbit coupling.
V. ACKNOWLEDGMENTS
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