We study the anomalous Hall conductivity in spin-polarized, asymmetrically
confined two-dimensional electron and hole systems, focusing on skew-scattering
contributions to the transport. We find that the skew scattering, principally
responsible for the extrinsic contribution to the anomalous Hall effect,
vanishes for the two-dimensional electron system if both chiral Rashba subbands
are partially occupied, and vanishes always for the two-dimensional hole gas
studied here, regardless of the band filling. Our prediction can be tested with
the proposed coplanar two-dimensional electron/hole gas device and can be used
as a benchmark to understand the crossover from the intrisic to the extrinsic
anomalous Hall effect.Comment: 4 pages, 2 figures include

Borunda, Mario F.

Jungwirth, T.

Luck, Thomas

MacDonald, A. H.

Nunner, Tamara S.

Sinitsyn, N. A.

Sinova, Jairo

Timm, Carsten

Wunderlich, J.

English

arXiv

Borunda, Mario

L��ck, Thomas

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:cond-mat/0702289v2  [cond-mat.mtrl-sci]  20 Feb 2007Absence of skew scattering in two-dimensional systems:Testing the origins of the anomalous Hall effectMario F. Borunda,1 Tamara S. Nunner,2 Thomas Lu¨ck,2 N. A. Sinitsyn,3, 1 CarstenTimm,4 J. Wunderlich,5 T. Jungwirth,6, 7 A. H. MacDonald,8 and Jairo Sinova11Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA2Institut fu¨r Theoretische Physik, Freie Universita¨t Berlin, Arnimallee 14, 14195 Berlin, Germany3CNLS/CCS-3, Los Alamos National Laboratory, Los Alamos, NM 87545, USA4Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA5Hitachi Cambridge Laboratory, Cambridge CB3 0HE, UK6Institute of Physics ASCR, Cukrovarnicka´ 10, 162 53 Praha 6, Czech Republic7School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK8Department of Physics, University of Texas at Austin, Austin TX 78712-1081, USA(Dated: February 12, 2007)We study the anomalous Hall conductivity in spin-polarized, asymmetrically confined two-dimensional electron and hole systems, focusing on skew-scattering contributions to the transport.We find that the skew scattering, principally responsible for the extrinsic contribution to the anoma-lous Hall effect, vanishes for the two-dimensional electron system if both chiral Rashba subbandsare partially occupied, and vanishes always for the two-dimensional hole gas studied here, regardlessof the band filling. Our prediction can be tested with the proposed coplanar two-dimensional elec-tron/hole gas device and can be used as a benchmark to understand the crossover from the intrisicto the extrinsic anomalous Hall effect.PACS numbers: 72.15.Eb,72.20.Dp,72.25.-bIntroduction.—The observed Hall resistance of a mag-netic film contains the ordinary Hall response to the ex-ternal magnetic field and the anomalous Hall responseto the internal magnetization. Although the anomalousHall effect (AHE) has been used for decades as a ba-sic characterization tool for ferromagnets, its origin isstill being debated, also in the context of a closely re-lated novel phenomenon, the spin Hall effect [1, 2, 3, 4].Three mechanisms giving rise to AHE conductivity havebeen identified: (1) an intrinsic mechanism based solelyon the topological properties of the Bloch states orig-inating from the spin-orbit-coupled electronic structure[5], (2) a skew-scattering mechanism originating from theasymmetry of the scattering rate [6], and (3) a side-jumpcontribution, which semiclassically is viewed as a side-step-type of scattering and contributes to a net currentperpendicular to the initial momentum [7].Recent experimental and theoretical studies of tran-sition-metal ferromagnets and of less conventional sys-tems, such as diluted magnetic semiconductors, oxideand spinel ferromagnets, etc., have collected numerousexamples of the intrinsic AHE and of the transition to theextrinsic AHE dominated by disorder scattering [8]. Theunambiguous determination of the origin of the AHE inthese experimental systems is hindered, in part, by theircomplex band structures, which has motivated studies ofsimpler model Hamiltonians, such as the two-dimensional(2D) Rashba and Dirac band models [9, 10, 11, 12, 13].Attempts to describe all the contributions to the AHEwithin the same framework have yielded farraginous re-sults, however. So far a rigorous connection of the moreintuitive semiclassical transport treatment with the moresystematic diagramatic treatment, providing a clear-cutinterpretation of the intrinsic, skew, and side-jump AHEterms, has only been demonstrated for the Dirac Hamil-tonian model [14].In this Letter we calculate the transport coefficientsin these two complementary approaches for asymmetri-cally confined 2D electron and hole gases in the pres-ence of spin-independent disorder, finding perfect agree-ment. The motivation for the study of these systemsis threefold: First, they can be represented by simplespin-orbit-coupled bands, which, similar to the DiracHamiltonian model, allows us to unambiguously identifythe individual AHE contributions. Second, the extrinsicskew-scattering term vanishes for a two-subband occu-pation in the case of the Rashba 2D electron gas andfor any band occupation for the studied 2D hole gas.This provides a clean test of the intrinsic AHE mech-anism and of the transition between the intrinsic andskew-scattering-dominated AHE. Finally, we propose a2D electron gas/2D hole gas coplanar magneto-opticaldevice in which the unique AHE phenomenology foundin our theoretical models can be systematically exploredexperimentally.Model Hamiltonians.—We study the following 2Dmodel Hamiltonians:H =~2k22mσ0+ iαn(σ+kn−−σ−kn+)−hσz +V (r)σ0 (1)with m being the effective in-plane mass, σi the 2 × 2Pauli matrices, k± = kx ± iky, h the exchange field,αn the spin-orbit coupling parameter, and V (r) a spin-independent disorder potential. The exponent n = 1 (3)2describes a 2D electron (hole) gas [15]. The eigenenergiesof the clean system are E± = ~2k2/2m±√h2 + (αnkn)2.The eigenvectors in the clean system take the form|Ψ±k〉 = exp(ik · r)|u±k〉 with k = k (cosφ, sinφ) and|u±k〉 = 1√2λ( ±ie−niφ√λ± h√λ∓ h), (2)where λ =√h2 + (αnkn)2. We now define k±(E) as thewave number for the ± band at a given energy E and de-fine λ± ≡ λ(k±). If E is not specified, it is assumed to bethe Fermi energy. We consider the model of randomly lo-cated δ-function scatterers, V (r) =∑i Vi δ(r−Ri) withRi random and disorder averages satisfying 〈Vi〉dis = 0,〈V 2i 〉dis = V 20 6= 0, and 〈V 3i 〉dis = V 31 6= 0. This modelis different from the standard white-noise disorder with〈|V 0k′k|2〉dis = niV 20 , where ni is the impurity concentra-tion and other correlators are either zero or related to thiscorrelator by Wick’s theorem. The deviation from whitenoise in our model is quantified by V1 6= 0, and is neces-sary to capture part of the skew-scattering contributionto the AHE.Semiclassical approach.—We sketch here the semiclas-sical procedure used in the calculation, for further detailswe refer to Ref. 14. The multi-band Boltzmann equationin a weak electric field E is given by∂fl∂t+ eE · vl dfldǫ= I[f ]coll (3)where l = (k, µ), µ = ± is the subband index, andI[f ]coll = −∑µ′∫d2k′/(2π)2 ωll′ (fl − fl′) is the impu-rity collision integral. The distribution function fl isthe sum of the equilibrium function and a correction,fl = f0l + gl. The scattering rates ωll′ are related to theT-matrix elements through ωll′ = 2π/~ |Tl′l|2δ(ǫl′ − ǫl),where Tl′l = 〈l′|V |ψl〉, and |ψl〉 are eigenstates of thecomplete Hamiltonian, and |l〉 ≡ |Ψµk〉 of the disorder-free Hamiltonian.Skew scattering.—Skew scattering appears in theBoltzmann equation through the asymmetric part of thescattering rate, i.e., ωll′ 6= ωl′l [6]. The scattering rates tosecond and third order in disorder strength are given byωll′ = ω(2)ll′ +ω(3)ll′ +· · · , where ω(2)ll′ = 2π/~ 〈|Vll′ |2〉disδ(ǫl−ǫl′) is symmetric. Here Vl′l = 〈l′|V |l〉. We break up thethird-order contribution into symmetric and antisymmet-ric parts. We ignore the first, since only the second givesrise to skew scattering. This antisymmetric term is givenby [14]ω(3a)ll′ = −(2π)2~∑l′′δ(ǫl−ǫl′′)Im〈Vll′Vl′l′′Vl′′l〉disδ(ǫl−ǫl′).(4)The solution of the Boltzmann equation (3) is found byfirst looking at the deviation of the distribution functionfrom equilibrium [14],gl = −∂f0µ∂ǫeE|vµ|(Aµ cosφ+Bµ sinφ). (5)Assuming that the transverse conductivity is muchsmaller than the longitudinal one (Aµ ≫ Bµ) and sub-stituting Eq. (5) into Eq. (3) one finds Aµ = τ‖µ andBµ = (τ‖µ)2/τ⊥µ , where1τ‖µ=∑µ′∫d2k′(2π)2ωll′[1− |vl′ ||vl| cos(φ− φ′)], (6)1τ⊥µ=∑µ′∫d2k′(2π)2ωll′|vl′ ||vl| sin(φ− φ′). (7)For symmetric Fermi surfaces, the skew-scattering con-tribution to the conductivity tensor at zero temperaturecan now be expressed using the scattering times,σxx =e24π~∑µτ‖µvF,µkµ, σskewxy =e24π~∑µ(τ‖µ)2τ⊥µvF,µkµ.(8)The calculation of (τ‖µ)−1 and (τ⊥µ )−1 uses the matrixelements of Eq. (4). To simplify the notation we define〈µµ′, µ′µ′′, µ′′µ〉 ≡ Im∫ 2pi0dφ′′〈uµk|uµ′k′〉〈uµ′k′|uµ′′k′′〉〈uµ′′k′′|uµk〉,(9)where all momenta are taken on the Fermi surface. Notethat in Eq. (9) the magnitude of k′′ can be different fromthat of k′ or k since the Fermi momenta of different bandsdo not coincide.The matrix elements appearing in Eq. (9) can be cal-culated directly from the basis functions, yielding〈µµ′, µ′µ′′, µ′′µ〉 = −hπα2nknµknµ′2λµλµ′λµ′′sin(nφ− nφ′), (10)from which we obtainω(3a)ll′ = −1~niV31 δ(ǫl− ǫl′)∑µ′′νµ′′ 〈µµ′, µ′µ′′, µ′′µ〉, (11)where ν± is related to the density of states of each bandat the Fermi energy, (ν±)−1 = ~2/m ± n(αnkn−1± )2/λ±.The symmetric part of the scattering rates to secondorder in the disorder potential is given by ω(2)ll′ =2π/~niV20 |〈uµk |uµ′k′ 〉|2δ(ǫl − ǫl′). The relaxation times arefound by inserting this into Eq. (6) and Eq. (11) intoEq. (7). For n = 1, i.e., for the 2D electron gas, therelaxation rates are then1τ‖µ=1~niV20[νµλµ(h2λµ+α21k2µ4λ++α21k2−4λ−)+νµ¯2(1− h2λ−λ+)], (12)1τ⊥µ= −niV31 hα21νµ8~λµ(k2µλµ− k2µ¯λµ¯)(νµλµ− νµ¯λµ¯), (13)3jy x y xj j jFIG. 1: (color online). Diagramatic representation of theskew-scattering contribution to σyx. Both current vertices,denoted by squares, are renormalized by ladder vertex cor-rections.where µ¯ ≡ −µ. If both subbands are occupied, the lastfactor in Eq. (13) vanishes and there is no skew-scatteringcontribution. If only the majority subband is occupied(EF < h), (τ⊥µ )−1 is non-zero and skew scattering con-tributes. For the skew-scattering Hall conductivity andthe longitudinal conductivity we obtain in this caseσxx =e2π~niV 20(λ−k−ν−)213h2 + λ2−, (14)σskewxy = −e2V 312π~niV 40hλ−α21k4−ν−(3h2 + λ2−)2. (15)If n = 3, i.e., for the 2D hole gas, we obtain1τ‖µ=1~niV20[νµ2λ2µ(λ2µ + h2) +νµ¯(λ−λ+ − h2)2λ−λ+],(16)1τ⊥µ= 0 (17)and skew scattering vanishes irrespective of band filling.Microscopic approach.—Within the diagramatic Kuboformalism the skew-scattering contribution to the off-diagonal conductivity is obtained from the expressionσI(a)xy =e2~2πV∑kTr[vxGRk (EF )vyGAk (EF )], (18)where the bare velocity vertex factors in the linear-in-kRashba model are given byvx =~kxmσ0 − α1~σy, vy =~kymσ0 +α1~σx. (19)As shown in a previous study [14], the skew-scatteringcontribution proportional to V 31 /(niV40 ) corresponds tothe diagrams shown in Fig. 1, where the current ver-tices jx, jy on both sides are the bare velocities vx, vyrenormalized by ladder vertex corrections. Only theskew-scattering diagrams with a single third-order ver-tex, shown in Fig. 1, contribute to order V 31 /(niV40 ). Allother terms from a ladder-type summation of third-ordervertices are smaller because they are either not of the or-der 1/ni or of higher order in V1/V0. The sum of theskew-scattering vertices (i.e., the bold/red part of Fig. 1)givesi4niV31 h(ν−λ−− ν+λ+)(σ0 ⊗ σz − σz ⊗ σ0) . (20)In the linear Rashba model we find ν+/λ+ = ν−/λ−,implying that skew scattering vanishes if both subbandsare occupied. In the case that only one subband is occu-pied the evaluation of Fig. 1 to order V 31 /(niV40 ) yieldsexactly the same expression for σskewxy as in the semiclas-sical Eq. (15). The only effect of the ladder vertex cor-rections is to renormalize each bare velocity by a factorof 2(h2 + λ2−)/(3h2 + λ2−) which reduces to a factor of 1in the limit of small α1kF and to a factor of 2 in the limitof small h.For the 2D hole-gas model Hamiltonian (1) with n = 3the bare velocity vertex factors arevx,y =~kx,ymσ0− 6α3~kxkyσy,x± 3α3~(k2x−k2y)σx,y. (21)Here the vertex corrections disappear because integralsof the type∑k GRk vx,yGAk = 0 vanish. This implies theabsence of skew scattering for any subband filling [16],consistent with the semiclassical result. We note that thesame consistency between semiclassical and microscopicquantum theory calculations for the studied 2D models isalso obtained for the intrinsic and side-jump terms simi-lar to the results in the graphene model [14]; the longerdetails of those calculations will be shown elsewhere andare in general agreement with Ref. 12.The abscence of the skew scattering is akin but notequivalent to the results of spin-Hall-effect calculationsin 2D systems [17]. For the Rashba 2D electron gas thedisappearance of the DC spin Hall conductivity is guar-anteed by sum rules that relate the spin current to thedynamics of the induced spin polarization [18, 19]. In thecase of a charge current no similar sum rule is known. Aswe have shown, the skew-scattering contribution in factbecomes finite when the minority band is depleated. Thevanishing of the Hall conductivity in the Rashba 2D elec-tron gas for EF > h is attributed to the simplicity of theHamiltonian. In particular the relation ν+/λ+ = ν−/λ−does not hold generally beyond the case of the linear-in-k Rashba coupling. The abscence of skew scattering inthe 2D hole system has a different origin: Due to thecubic dependence of spin-orbit coupling on momentum,the matrix elements, Eq. (10), in the antisymmetric partof the collision term behave like sin(3φ− 3φ′). Togetherwith the sin(φ− φ′) dependence of the velocity factor inEq. (7), this makes the integral over k′ vanish.Our results predict that the AHE in 2D electron andhole systems can be dominated by contributions indepen-dent of the impurity concentration, for which the anoma-lous Hall resistance is ∝ σ−2xx . We also predict that inthe Rashba 2D electron gas with only one subband oc-cupied the extrinsic skew-scattering contribution, lead-ing to anomalous Hall resistance proportional to σ−1xx , isnon-zero. Note that this term has not been identified inprevious works that considered only white-noise disorder[9, 10, 11, 12, 13]. Since its corresponding conductiv-ity contribution is inversely proportional to the impurity4FIG. 2: (color online). Top panel: Top-view schematics ofthe Hall bar with coplanar 2D hole and electron gases. Spin-polarized carriers are generated by shining circularly polar-ized light on the p-n junction. Center bottom panel: Crosssection of the heterostructure containing p-type and n-typeAlGaAs/GaAs single junctions. The left band diagram cor-responds to the unetched part of the wafer with the 2D holegas, the right band diagram shows the 2D electron gas in theetched section of the wafer.concentration, the skew-scattering mechanism can dom-inate in clean samples.Proposed experimental setup.—The unique phenome-nology of the AHE in the studied 2D systems, in partic-ular the sudden disappearance of skew scattering whenthe Fermi level crosses the depletion point of the minority2D Rashba band, represents an opportunity for a cleantest of the presence of intrinsic and extrinsic sources ofthe AHE and of the transition between these two regimes.In the absence of 2D ferromagnetic system with Rashbalike spin-orbit interatciton, we proposed an experimentalsetup for this test as shown in Fig. 2. The device is basedon a AlGaAs/GaAs heterostructure containing a copla-nar 2D hole gas/2D electron gas p-n junction. The crosssection of the heterostructure and corresponding banddiagrams are shown in the lower panels of Fig. 2 (formore details see Ref. 3). Under a forward bias the junc-tion was successfully utilized as a light-emitting-diodespin detector for the spin Hall effect [3]. Here we pro-pose to operate the junction in the reverse-bias mode,while shining monochromatic, circularly polarized lightof tuneable wavelength on the p-n junction. The pho-togenerated spin-polarized holes and electrons will prop-agate in opposite directions through the respective 2Dhole and electron channels. The longitudinal voltage andthe generated anomalous Hall voltage can be detected bythe successive sets of Hall probes, as shown in the upperpanel of Fig. 2. For the 2D electron gas the macroscopicspin diffusion length allows to use standard lithographyfor defining the Hall probes. Surface or back gates inclose proximity to the 2D electron system can be usedto modify the effective 2D confinements, carrier density,and spin-orbit coupling in order to control the transi-tion between the intrinsic and extrinsic AHE regimes.The exploration of the AHE in the 2D hole gas is morechallenging due to the expected sub-micron spin diffu-sion length in this system but may still be feasible in theproposed experimental setup.Fruitful discussions with S. Onoda and N. Nagaosa aregratefully acknowledged. This work was supported byONR under Grant No. onr-n000140610122, by the NSFunder Grants No. DMR-0547875 and No. PHY99-07949,by the SRC-NRI (SWAN), by EU Grant IST-015728,by EPSRC Grant GR/S81407/01, by GACR and AVCRGrants 202/05/0575, FON/06/E002, AV0Z1010052, andLC510, by the DOE under Grant No. DE-AC52-06NA25396, and by the Univ. of Kansas General Re-search Fund allocation No. 2302015. Jairo Sinova is aCottrell Scholar of Research Corporation.[1] M. I. Dyakonov and V. I. 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Physical Review Letters

Absence of Skew Scattering in Two-Dimensional Systems: Testing the Origins of the Anomalous Hall Effect

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Absence of skew scattering in two-dimensional systems: Testing the
  origins of the anomalous Hall effect

Absence of skew scattering in two-dimensional systems: Testing the origins of the anomalous Hall effect

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