Photo-spintronics of spin-orbit active electric weak links R. I. Shekhter,1 O. Entin-Wohlman,2,3 M. Jonson,1,4, and A. Aharony2,3 ∗ 1Department of Physics, University of Gothenburg, SE-412 96 Go¨teborg, Sweden 2Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel 3Physics Department, Ben Gurion University, Beer Sheva 84105, Israel 4SUPA, Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland, UK (Dated: January 18, 2017) We show that a carbon nanotube can serve as a functional electric weak link performing photo- 7 spintronictransduction. Aspincurrent,facilitatedbystrongspin-orbitinteractionsinthenanotube 1 andnotaccompaniedbyachargecurrent,isinducedinadevicecontainingthenanotubeweaklink 0 by circularly polarized microwaves. Nanomechanical tuning of the photo-spintronic transduction 2 can be achieved due to the sensitivity of the spin-orbit interaction to geometrical deformations of the weak link. n a J INTRODUCTION form a tube. The strain-induced SOI in a simple one- 7 dimensional model of such a nanotube is described by 1 Spintronics is a rapidly developing research area of the Hamiltonian l] modern solid state physics. In contrast to traditional ˆstrain =(cid:126)v kstrainσ nˆ, (1) l electronics,wheretheelectricchargeofelectronsisinfo- Hso F so · a wherev istheFermivelocity,kstrain isaphenomenolog- h cus,thefieldofspintronicsreliesonanotherfundamental F so - propertyofelectrons,viz. theirmagneticmoment,which icalparameterthatgivesthestrengthoftheSOIinunits s of inverse length, σ is a vector whose components are e is associated with their spin degree of freedom. m Issues related to the electrical control of spin currents the Pauli matrices σx,y,z, and nˆ is a unit vector point- ing along the longitudinal axis of the nanotube in the . as well as to spin control of charge currents are at the t direction of electron propagation (nˆ =kˆ). Equation (1), a heart of spintronics research today, both from a funda- m which was used in Ref. 3, is a simplified form of the SOI mental and an applied perspective [1]. Recently it has Hamiltonian previously derived for a realistic model of - been suggested that such controls may be effectively im- d such a nanotube [6]. plementedinnanodevicescontaininganelectricweaklink n The SOI active weak-link device shown in Fig. 1 com- o with strong spin-orbit interactions (SOI) that bridges prises a nanowire that bridges two bulk electronic reser- c bulk electrodes [2–4]. In [2] it was demonstrated that a [ spin-orbit coupling results in a “splitting” of the spin of voirs. The spin-orbit interaction described by Eq. (1) is restricted to the nanowire and has the effect of scat- 1 electrons passing through such a weak link (Rashba spin tering the spins of electrons that pass through the wire. v splitting), which under certain conditions may generate Following the approach developed in Refs. 2–4 we will 5 a spin current. This was shown to occur if an imbal- 6 describe the transfer of electrons through the nanowire- ance of the population of spin states in the electrodes 6 basedweaklinkwiththehelpofaspin-dependenttunnel is established by spin-flip assisted electronic transitions 4 Hamiltonian. Hence,thetotalHamiltonianofthesystem 0 due to the absorption (or emission) of circularly polar- can be written as a sum of three parts, . ized photons created by microwave pumping [5]. The 1 0 SOI-induced spin generation inside the weak link makes ˆ = ˆL+ ˆR+ ˆT , (2) H H H H 7 it a point-like source of a spin current due to a photo- where 1 spintronic effect on the nanometer length scale. Esti- v: mations show, however, that if the SOI is caused by an HˆL(R) = εk(p)c†k(p)σck(p)σ (3) Xi external electric field, as implicitly assumed in [2], that k(cid:88)(p)σ field has to be quite strong for the induced Rashba spin are Hamiltonians that describe the electrons in the left r splitting to be significant. The aim of the present work a andrightleads. Theseelectronsarecharacterizedbythe is to demonstrate that a much stronger photo-spintronic momentumquantumnumberskandp,respectively,and transduction effect can be achieved if a material with an bythespinprojectiononthezˆ axis. Welabelthelatter intrinsic SOI, here assumed to be induced by stresses, is − by σ = 1, so that the spin projections are s = (cid:126)σ/2. used for the weak link. ± ThetunnelHamiltonianisexpressedintermsoftheprob- Thepreciseformofthestress-inducedSOIdependson ability amplitudes [W ] for electron transmission the material used for the electric weak link and the type through the wire, p,k σ,σ(cid:48) of strains involved. In a single-wall carbon nanotube, which will be considered here, the strain can be thought HˆT = (c†pσ(cid:48)[Wp,k]σ(cid:48),σckσ+H.c.) . (4) of as occurring when a graphene ribbon is rolled up to (cid:88)k,p(cid:88)σ,σ(cid:48) 2 These amplitudes have to be calculated taking the spin that the SOI, which is restricted to the weak link, is the dynamics given by Hamiltonian (1) into account. dominant spin relaxation mechanism the SOI becomes a sourceofaspincurrent, whichflowsoutoftheweaklink into the two reservoirs and which counteracts the spin pumping. Referring to Fig. 1, we note that both the ori- " entation of the plane that contains the weak link with respect to the pumped spin orientation and the bending angle of the link can be used to tune the generated spin ! JL JR current. spin spin z The spin bias U can be defined by noting that for y ˆ = 0 the electron spin reservoirs are described by T H x Fermi-Dirac distributions with different chemical potentials, FnaIGno.w1i:reSwchitehmaatsitcropnigctsuprien-oofrbthiteindteevriacceticoonnbsirdidergeeds.twAobbeunlkt fkσ(p) =(cid:104)c†k(p)σck(p)σ(cid:105)HˆT=0 =nF(εk(p)−µσ) , (5) reservoirs. Thebentnanowireismodeledbytwoequal-length where straight wire segments which form angles θ and θ with the xˆ−axis, respectively, while the plane that cont−ains the µσ =µ−σU/2 , σ =±1 , (6) nanowireistiltedbyanangleγawayfromtheyˆ axistowards − and µ is the chemical potential of both leads at equilib- the zˆ axis. The device is irradiated by circularly-polarized micro−waveradiation(wavyarrows)thatcreatesadifferencein rium. The spin current generated by electrons tunneling thepopulationofspin-upandspin-downelectrons(thickver- out of the SOI-active weak link can be obtained as the tical arrows) corresponding to a “spin biasing” of the device. time derivative of the total spin, Sˆ , of the electrons, Spin-flip transitions induced by the spin-orbit interaction in (cid:104) (cid:105) thenanowirecreateaspincurrentJspin =JsLpin+JsRpin travel- J = d(cid:104)Sˆ(cid:105) = (cid:126)σ dNˆσ , (7) ing out from the wire with a magnitude that depends on the spin dt 2 (cid:104) dt (cid:105) angles θ and γ. σ= 1 (cid:88)± where SPIN-BIASED ELECTRIC WEAK LINK Nˆσ =NˆLσ+NˆRσ , NˆL(R)σ = c†k(p)σck(p)σ . (8) k(cid:88)(p) A straightforward calculation of the spin current (7) can Electronic transport through the weak link shown in bedoneusingthetunnelHamiltonian(1)tolowestorder Fig.1canbeinducedinanumberofdifferentways. The in perturbation theory [2]. Then, one finds for the spin standard method would be to apply a voltage bias V conductanceG (definedinanalogywiththeelectrical across the link, so that the chemical potentials for elec- spin conductance G) the relation tronsintheleftandrightreservoirsareshiftedbyasmall amounteV withrespecttoeachother. Theresultofsuch G =J /U for U 0 . (9) charge biasing is that excess charge of opposite polarity spin spin → is accumulated on either side of the link, which leads to an electrical current through the link in a direction that RASHBA SPIN SPLITTING AS A SOURCE OF counters the bias-induced charge imbalance. SPIN GENERATION IN SOI ACTIVE WEAK Anothermethod,whichwillbeinfocushereandwhich LINK is illustrated in Fig. 1, is to arrange for the chemical potential to be different for the two possible projections of In order to calculate the spin current given by Eq. (7) the electron spin along a certain axis. Such spin biasing one needs to evaluate the electronic transmission ampli- can be achieved by illuminating the entire device with tudes[W ] ,whichappearinthetunnelHamiltonian p,k σ,σ(cid:48) circularly-polarized microwave radiation of a frequency (4), and specifically their dependence on the spin-orbit thatenableselectronspin-flipassistedphotonabsorption. interaction as given by Eq. (1). In our simple model the This photonic pumping of the electronic spin creates an weak link consists of two straight parts of equal length imbalance between the number of electrons with oppo- R = R =d/2joinedbyabend. Neglectingthemo- L R | | | | site spin projections on an axis defined by the direction mentum (but not the spin) dependence, the probability of the radiation and leads to a spin-dependent shift in amplitude for an electron of energy E to pass from, say, the chemical potentials for electrons with opposite spin the left to the right lead can be written as a product of projections. The magnitude of the shift, which extends five factors, throughout the device, depends on the intensity of the W =T G(R ;E) G(R ;E)T . (10) radiation and on the spin relaxation rate. If we assume R R T L L 3 HereW isa2 2matrixinspinspace,T istheproba- and L(R) × bilityamplitudetotunnelfromthewiretotheleft(right) leadandT isthetransfermatrixthroughthebendinthe B=sin(α)cos(α)(nˆR+nˆL)+sin2(α)nˆR×nˆL wire. InRef. 4theGreen’sfunctionG(RL(R);E)forthe =xˆsin(2α)cos(θ) yˆsin2(α)sin(2θ)sin(γ) straight segments of the wire, in which the SOI interac- − +zˆsin2(α)sin(2θ)cos(γ) . (20) tion takes place, was evaluated for a Hamiltonian of the form Hence, the probability for a SOI-induced spin-flip tran- ˆ = (cid:126)2k2 +Q(k) σ. (11) sition is H 2m · ∗ w 2 = B 2+ B 2 AcomparisonwithEq.(1)showsthatinthepresentcase ↑↓ ≡|W↑↓| | x| | y| =sin2(2α)cos2(θ)+sin4(α)sin2(2θ)sin2(γ) . (21) Q(k)=(cid:126)v kstrainnˆ. Hence, from Eqs. (A12) and (A13) F so of Ref. 4 we conclude that The spin conductance can now be expressed in terms of the spin-flip probability w as G(R ;E)=G (R ;E) L(R) 0 L(R) ↑↓ [cos(α) isin(α)nˆ σ] , (12) G × − L(R)· Gspin = e2/(cid:126)w↑↓(θ,γ). (22) wherewehaveusedtheshort-handnotationkstraind/2 so ≡ α, which is a measure of the strength of the SOI, and Equations (21) and (22) represent the main results of where the paper. The strength of the SOI, characterized by the dimensionless parameter α = kstraind/2, determines so G0(RL(R);E)=iπ(m∗/(cid:126)2k0)exp[ik0|RL(R)|] , (13) theamountofphoto-spintronictransductionthatcanbe achievedbythestudiedRashbaspin-splitterdevice. The with k0 = (2m∗E/(cid:126)2)1/2, is the propagator on the left sensitivity of the effect to the geometry of the experi- (right) segment in the absence of SOI. It follows that we mental set-up opens the possibility for tuning the device can factor out the dependence on the SOI and write the nanomechanically, by varying the angles γ and θ. The amplitude given by Eq. (10) as dependenceofthespinconductanceontheseexperimen- tallyaccessibledeviceparametersisillustratedinFigs.2 W =W , (14) 0W and 3. The photo-spintronic transduction occurs even for a where straight wire (i.e., when θ = 0). However, wire defor- W =T G (R ;E) G (R ;E)T (15) mation provides a tool for a nanomechanical control of 0 R 0 | R| T 0 | L| L thegeneratedspincurrent. Dependingonthestrengthof is the SOI-independent part and SOI coupling α, both a monotonic and a non-monotonic dependence of the dimensionless spin-conductance w =[cos(α) isin(α)nˆ σ][cos(α) isin(α)nˆ σ] ↑↓ W − R· − L· on mechanical deformations can be achieved. (16) Figure 2 illustrates that the dimensionless conductance, w , is an oscillatory function of the angle γ. contains the effect of the SOI. ↑↓ As is clear from Eq. (21), the maximal [minimal] value, In our geometry the zˆ axis is the spin quantization − wα (θ) [wα (θ)], of w =w (γ) is achieved when the axis, and the bent wire lies in a plane that (i) contains max min ↑↓ ↑↓ normaltotheplanecontainingthenanowireweaklinkis the xˆ axis and (ii) is rotated by an angle γ away from − perpendicular[parallel]tothespinquantizationaxis,i.e. theyˆ axistowardsthezˆ axis. Intheplaneofthewire, − − when γ = π/2 [γ = 0 or π] and therefore sin2(γ) = 1 the left (right) straight leg of the wire forms an angle θ [sin2(γ)=0±], independent of the values of α and θ. (-θ) with the xˆ axis. In other words, − ForastrongenoughSOI,i.e., whensin2(α) 1/2, the ≥ nˆ =cos(θ)xˆ+sin(θ)[cos(γ)yˆ+sin(γ)zˆ] , largest possible value, wα (θ)=1, is reached when the L max bending angle is θ = θ , where cos2(θ ) = 1/[2sin2(α)], nˆ =cos(θ)xˆ sin(θ)[cos(γ)yˆ+sin(γ)zˆ] , (17) α α R − asillustratedinFig. 3. Ontheotherhand,forsin2(α)< whichmeansthatwecanwriteEq.(16)inamatrixform, 1/2theeffectissmallersincethenwα (θ)ateo9⇡tih6a(hhr⌧81,195418(2010).ahna11T⇡ieha(gol,aiae1Oho1Ps↵l[11Tα↓(eµEssgolf,.ai4nemaxhllVa0P2sidd)m(ndreµybo.n,4(lengthscaleshouldenableopportunitiesfornovelspin-21.nmE)llVwa7MFie1m(nrurn6rrlummybinFig.3b.nAe✓yanthex-axisand(ii)isrotatedbyanangle�fromthey-ex7eoo9d2se6luammoowhnA,3ypsmevsM2α.fo0v4ehhef32bn.ps.mny.[6]M.S.RudnerandE.I.Rashba,Phys.Rev.B81,125426(aRa2sM,encasfv040e-0ebn�.irwa(wherecos(✓)=1/[2sin(↵)].Asmallere↵ect,wnn.R7yInourgeometrythez-axisisthespinquantization,na.ca,sha)aahowstrongaphoto-spintronice↵ectcanberealizedin[7]F.Kuemmeth,S.Ilani,D.C.Ralph,andP.L.McEuen,0-0d,ananniird,tt.10y.inia.oc,hhaazpen[9↵anangf1.2,eLiyi(i1a(θi↵aakhEsanopen)nnS6Sq5t,a1.2,,d,ily5γo((iakocmmesanltronicdevices.]s)nnES)S624.,22n,g3,1Bpsa.hlwP5=/,uordctotaxistowardsthez-axis.Intheplaneofthewire,theleftksmenrc2,,2nct,.d3,1)sa.t"#)(2010);K.FlensbergandC.M.Marcus,Phys.Rev.B.ariu,utre.c,rfiuikae.inrht)0,,t)c.d7/,55hBr2oaRdiotan.Nature452,448(2008).u/⇥inbeeDsatfP)s0un↵,obtainsintheweakcouplinglimit,↵1,assketched)cu;7tDaxis,andthebentwireliesinaplanethat(i)containsn.BrNuoRapractice,whichinturndependsonthestrengthofSOIrhio;)anaueh⇥rbme1stc2gP0aasoHulDttenvltiuatu445rhiMM;teug8

Photo-spintronics of spin-orbit active electric weak links

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