We obtain analytic formulas for the frequency-dependent spin-Hall
conductivity of a two-dimensional electron gas (2DEG) in the presence of
impurities, linear spin-orbit Rashba interaction, and external magnetic field
perpendicular to the 2DEG. We show how different mechanisms (skew-scattering,
side-jump, and spin precession) can be brought in or out of focus by changing
controllable parameters such as frequency, magnetic field, and temperature. We
find, in particular, that the d.c. spin Hall conductivity vanishes in the
absence of a magnetic field, while a magnetic field restores the
skew-scattering and side-jump contributions proportionally to the ratio of
magnetic and Rashba fields.Comment: Some typos correcte

E. M. Hankiewicz

G. Vignale

M. I. Dyakonov

English

arXiv

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Phase Diagram of the Spin Hall Effect

URL:http://link.aps.org/doi/10.1103/PhysRevLett.100.026602
DOI:10.1103/PhysRevLett.100.026602We obtain analytic formulas for the frequency-dependent spin-Hall conductivity of a two-dimensional electron gas (2DEG) in the presence of impurities, linear spin- orbit Rashba interaction, and external magnetic field perpendicular to the 2DEG. We show how different mechanisms (skew scattering, side jump, and spin precession) can be brought in or out of focus by changing controllable parameters such as frequency, magnetic field, and temperature. We find, in particular, that the dc spin- Hall conductivity vanishes in the absence of a magnetic field, while a magnetic field restores the skew-scattering and side jump contributions proportionally to the ratio of magnetic and Rashba fields.This work was supported by NSF Grant No. DMR-0313681

Hankiewicz, E. M.

Vignale, Giovanni, 1957-

University of Missouri: MOspace

Phase Diagram of the Spin Hall EffectE. M. Hankiewicz1,2,* and G. Vignale11Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA2Department of Physics, Fordham University, Bronx, New York 10458, USA(Received 30 July 2007; published 17 January 2008)We obtain analytic formulas for the frequency-dependent spin-Hall conductivity of a two-dimensionalelectron gas (2DEG) in the presence of impurities, linear spin-orbit Rashba interaction, and externalmagnetic field perpendicular to the 2DEG. We show how different mechanisms (skew scattering, sidejump, and spin precession) can be brought in or out of focus by changing controllable parameters such asfrequency, magnetic field, and temperature. We find, in particular, that the dc spin-Hall conductivityvanishes in the absence of a magnetic field, while a magnetic field restores the skew-scattering and sidejump contributions proportionally to the ratio of magnetic and Rashba fields.DOI: 10.1103/PhysRevLett.100.026602 PACS numbers: 72.25.Dc, 72.25.Rb, 73.43.f, 85.75.dThe spin-Hall effect (SHE), i.e., the generation of atransverse spin current in response to a dc electric field[1–6], has attracted much attention recently, particularlyafter a series of experiments [7–10] culminating in theobservation of the SHE at room temperature [10]. On thetheoretical front, however, there remains considerable un-certainty as to the physical origin of the SHE, whichappears to depend on an intricate interplay of three pro-cesses: (i) the skew-scattering (SS) due to spin-orbit inter-action (SOI) between electrons and impurities, (ii) the sidejump (SJ) (due to the noncanonical character of the physi-cal position and velocity variables in the presence of SOIwith impurities), and (iii) the spin precession caused byspin nonconserving terms in the band structure, amongwhich we include the linear Rashba SOI generated by anexternal electric field. To these we may add the influence ofan external magnetic field, which tends to lock the spins ina fixed direction, thus reducing the importance of spinprecession. A first principles theory that includes all ofthese effects on equal footing is very complicated. To ourknowledge, the diagrammatic approach by Tse and DasSarma [11] comes closest to fulfilling the order, and yet itdoes not include magnetic field or frequency. However,these diagrammatic calculations are very difficult to followin detail and do not lead to an intuitive understanding of thestriking nonadditive behavior of impurity and band struc-ture effects.Our goal in this Letter is to present the ‘‘phase diagram’’of the SHE, i.e., to clarify in which range of experimentallycontrollable parameters one should expect the dominanceof each mechanism mentioned above, and how the cross-overs between different regimes occur. We do this for a2DEG with Rashba SOI and a magnetic field perpendicularto the plane. The two parameters that are most easilycontrolled in an experiment are (i) the frequency ! ofthe electric field and (ii) Zeeman energy !0. Ac-cordingly, we plot our ‘‘phase-diagram’’ in the !!0=RkF plane, where  is the electron-impurity scatter-ing time and RkF is the magnitude of the effective mag-netic field due to Rashba SOI for electrons at the Fermiwave vector kF. Throughout the Letter we assume !, 1=,RkF, !0  EF; i.e., all energy scales are smaller than theFermi energy.Our qualitative conclusions are shown in Fig. 1. Incontrast to previous calculations [11], we found that thedc limit (! ! 0) of the spin-Hall conductivity (SHC) iszero in the presence of spin precession and in the absenceof a magnetic field (!0  0). As the magnetic field in-creases, both SS and SJ contributions increase with theratio of the magnetic to the Rashba field (!0=kF), thusrestoring the values they would have had at zero frequencyand zero magnetic field in the absence of spin precession.In high-mobility samples the SS mechanism is the domi-nant mechanism [12,13] overcoming the SJ contribution,which has an opposite sign for attractive impurity poten-tial. However, the SJ mechanism could well dominate inFIG. 1 (color online). Different regimes of spin-Hall effect inthe !!0=RkF plane. D is Drude conductivity; ss is skewscattering. Spin-orbit interactions and mobility are fixed. Seediscussion in the text.PRL 100, 026602 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending18 JANUARY 20080031-9007=08=100(2)=026602(4) 026602-1 © 2008 The American Physical Societylow-mobility samples. As discussed in Ref. [13], the mo-bility can be controlled to some extent by changing thetemperature T, and this allows one in principle to tunebetween SS and SJ contributions. For this reason we usethe label SS/SJ for the left side of diagram, where !  1.In the ac regime (!  1) and for low magnetic fieldimpurities become irrelevant, leaving room for the intrinsicSHE with ‘‘universal’’ conductivity e=8 [4]. Finally, inthe ac regime and at high magnetic field the SHC declinesto zero in different manners depending on whether ! orthe magnetic field is kept constant as shown in Fig. 1.In what follows we describe the main points of our newtheoretical approach, which enables us to calculate thespin-Hall conductivity in different regions of the parameterspace and to derive simple analytic formulas describing thecrossovers between different regimes.Our model is defined by the Hamiltonian H^  XNi1p^2i2m V ~^ri  R@p^iyS^ix  p^ixS^iy 1@	p^ixryV ~^ri  rxV ~^rip^iyS^iz !0S^iz: (1)Besides the kinetic energy and the usual electron-impuritypotential V ~^ri, we have two distinct spin-orbit couplings.The 1 coupling between the electrons and the impuritiesconserves the z component of the spin and is responsiblefor SS and the SJ effects. The R coupling—also known asRashba coupling—creates a momentum-dependent mag-netic field in the plane, which breaks the conservation of Szand causes spin precession. This term is responsible, underappropriate conditions, for the intrinsic contribution toSHE. Finally, we have included a magnetic field perpen-dicular to the plane. The Zeeman splitting, !0, and thefrequency ! of the ac electric field (not shown in H^) are thetwo control knobs in terms of which our ‘‘phase diagram’’will be plotted.The skew-scattering effect is easily described in theBoltzmann equation formalism [12], but it is difficult totreat in the diagrammatic approach [11]. On the other hand,spin-precession effects are easily included in the diagram-matic formalism, but are problematic in the Boltzmannequation formalism (the distribution function becomes a2 2 matrix). We get the best of two worlds by combiningthe two approaches in the following manner. First wenotice that the skew-scattering collision term in theBoltzmann equation is formally equivalent to the imposi-tion of a ‘‘spin-electric field’’ Ezy, which accelerates up-spin and down-spin electrons (‘‘up’’ and ‘‘down’’ are de-fined with respect to a z axis) in opposite directions alongthe y axis, perpendicular to the flow of the charge current(x). The problem is now ‘‘reduced’’ to calculating the zspin current jzy which flows along the y axis in response tothe spin-electric field Ezy in the same direction. This can bedone with the help of the standard diagrammatic formal-ism, including both electron-impurity scattering and spinprecession, but not the skew-scattering processes, for theskew scattering has already ‘‘done its job’’ by producingthe spin-electric field Ezy (in other words, we work to firstorder in 1). Thus we have jzyjss  zyyEzy; (2)where zyy is the longitudinal spin conductivity calculatedin the absence of skew-scattering (or side jump) effects. Onthe other hand, Ezy has the well-known expression Ezy  ssjx; (3)where jx is the current density in the x direction and ss m=ne2ss is the skew-scattering resistivity, calculatedfrom the Boltzmann collision integral [12]. An explicitexpression for the skew-scattering rate 1=ss is given inEq. (29) of Ref. [12]: notice that it is proportional to 1 andpositive [14]. jx can be expressed as DEx, where Ex is theelectric field in the x direction and D is the Drude con-ductivity of the electron gas. Therefore, Eq. (2) becomes: jzyjss  zyyssDEx; (4)from which we extract the first important result of thisLetter, namely, the expression for the skew-scattering con-tribution to the SHC SH, SHss  zyyssD: (5)The dependence of this formula on spin-precession rate,frequency of the ac field, and magnetic field will be ob-tained below. In particular, we will show that SHss vanishesat low magnetic field (spin-precession regime), and itrecovers the zero-precession value (2Dss@=e) at highmagnetic field. We will then consider separately sidejump SHsj and purely intrinsic (coming from band struc-ture) SHR contributions to the spin-Hall conductivity.They are most efficiently analyzed in terms of the Kuboformula for the spin-Hall conductivity. The SJ contribu-tion, similar to SS term, vanishes at low magnetic field(spin-precession regime) and recovers its value in highmagnetic fields. The remaining intrinsic contribution iseasily calculated by standard diagrammatics (includingvertex corrections) and leads to the well-known e=8result in the appropriate ac regime.Skew-scattering.—As discussed above, the central rolein calculating SHss is played by the longitudinal spin-channel conductivity zyy. The formal expression for thereal part of zyy is zyy!   4nem2@=mhhS^zp^y; S^zp^yii!; (6)where p^y, S^z are momentum and spin operators for a singleelectron [15], the double bracket is the usual notation forthe spin-current–spin-current response function, and n isthe areal density of the electron gas. Our objective is tocalculate this Kubo formula including elastic electron-impurity scattering in the Born approximation, spin pre-cession due to a Rashba spin-orbit interaction, and a mag-netic field along the z axis, but neglecting spin-orbitinteractions with the impurities. The Feynman diagramsPRL 100, 026602 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending18 JANUARY 2008026602-2for zyy!, to the desired level of accuracy, are shown inFig. 2. The solid lines represent 2 2 Green’s functionsincluding Rashba spin-orbit coupling, magnetic field, and adiagonal disorder self-energy i=2sgn !, where 1= isthe elastic scattering rate: G^p;!  ! p ~hp ~^S i2 sgn !	! p  i2 sgn !2  h2p=4: (7)Here ~hp  Rky;Rkx; !0 combines the Rashba in-plane field (x and y components) and the magnetic fieldalong the z axis in a single effective field. The dashed linesare electron-impurity interactions (averaged over disor-der), and there are spin-current operators pySz at the ver-tices. The empty bubble, calculated by the standardprocedure with RkF, !0, and 1= all much smaller thanthe Fermi energy, is zyy!jbubble ’ @eD1 !21!202122 ; (8)where D  ne2=m is the Drude conductivity and  RkF2 !20qis the spin-precession frequency.Equation (8) suggests that the spin-channel conductivityis similar to the ordinary Drude conductivity, only some-what reduced by precession about the in-plane Rashbafield. The inclusion of vertex corrections changes the resultdrastically. The sum of the ladder diagrams produces a‘‘dressed’’ spin-current vertex ^zy of the form: ^ zy  p^yS^z Rk2F	!0S^x  1 i!S^y2Rk2F 221 i!: (9)Including ^zy in the calculation of the spin-channel con-ductivity we get zyy!  2@Decos21 cos21 cos22  4!22 ; (10)where cos  !0 is the cosine of the angle between theexternal magnetic field and the full effective magnetic field~. While Eq. (10) reduces to the empty-bubble result in thelimit !0  RkF, its most striking feature is that it van-ishes identically (i.e., at all frequencies) in the oppositelimit !0  RkF. In other words, the spin-channel con-ductivity is zero in the absence of an external magneticfield, as long as an infinitesimal spin-precession rate RkFis present. Accordingly, the skew-scattering spin-Hall con-ductivity from Eq. (5) is [16] SHss  2@2Dssecos21 cos21 cos22  4!22 (11)which vanishes in the absence of an external magnetic fieldand recovers the value 2Dss@=e for strong magneticfields. The vanishing of the spin conductivity is a peculiarfeature of the linear Rashba model, in which the spincurrent p^yS^z is proportional to the time derivative of S^y:p^yS^z   _^Sy=R. The expectation value of a time deriva-tive must vanish at zero frequency. This is the same reasonwhich causes the vanishing of the Rashba spin-Hall con-ductivity in the dc limit.Side jump.—In a recent Letter [13] we have shown howto identify the SJ contribution in the Kubo linear responseformalism. We perturb the Hamiltonian (1) with a uniformelectric field of frequency ! in the x direction. After aseries of manipulations, which have been described inRef. [13], we arrive at the following Kubo formula forSHC: SHyx ! neim!p^yS^z;p^xmR@S^y 1ne2im!4hhp^yS^z;ryV ~^rS^zii@2 hhrxV ~^r;p^xii:(12)The first term on the right-hand side produces SS as well aspossible ‘‘intrinsic’’ contributions which are discussed innext section. The second line of this equation, which ex-plicitly shows a dependence on 1, is responsible for the SJcontribution [17], denoted SHsj . We now focus on thiscontribution. Using the Heisenberg equation of motionfor the momentum operator _^pix  rxV ~^ri in zero orderin 1 (because SJ terms are already explicitly linear in 1),we rewrite the SJ contribution as SHsj ! e1n2im!hh _^px; p^xii 4hhp^yS^z; _^pyS^zii@2	: (13)In the first term in the square brackets we apply thestandard rule of integration, which allows us to replace_^px=i! by p^x and rewrite it (its real part) as: 1en2im!hh _^px; p^xii  1ne2m hhp^x; p^xii 1ne21!22 ;(14)where the last equality follows from the well-known formof the current-current response function in a weakly dis-ordered system. The second term in the square brackets ofEq. (13) can be rewritten as follows:  1i!hhp^yS^z; _^pyS^zii   1i!p^yS^z;ddtp^yS^z  p^y _^Sz:(15)Applying again the integration formula, the first term ofEq. (15) yields [18]: 2en1mhhp^yS^z; p^yS^zii  e1n2cos21 cos221 cos22  4!22 :(16)In the absence of spin precession (i.e., for !0  RkF,cos ’ 1) and for zero frequency the contributions (14)and (16) add up to the ‘‘usual’’ SJ conductivity 1ne ofFIG. 2. Diagrammatic representation of ladder approximationto calculate the spin-channel conductivity.PRL 100, 026602 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending18 JANUARY 2008026602-3Ref. [13]. In the opposite limit of !0  RkF ( cos  0)this term vanishes for the same reason as SHss . The secondterm on the right-hand side of Eq. (15) can be rewritten ashhS^y; p^y ~^p  ~^Sii (we use the equations of motion _^Sz R ~^p  ~^S and p^yS^z   _^Sy=R). In the limit of zero fre-quency and zero magnetic field this reduces tohhp^yS^y; p^yS^yii  @2=4hhp^x; p^xii, which exactly cancelsthe contribution from (14) and causes the SJ contributionto vanish. For general magnetic fields and frequencies thetotal SHsj is finally given by the formula [19] SHsj ! 1ne2cos21!22 2cos21 cos21 cos22  4!22	;(17)which in the dc limit simplifies to SHsj ! 1necos223 cos21 cos2: (18)Summarizing, the side jump conductivity in the dc limit(!  1) grows from zero at !0  0 to the ‘‘full’’ value(1ne) at high magnetic fields, while it tends to zero inhigh frequency limit.Rashba contribution.—Evaluating the first line ofEq. (12) in the absence of skew scattering and side jump,which we have already taken into account, leads to thewell-known result [20,21] SHR !  e8 !2!21=4 , whichvanishes at !  0 and tends to the ‘‘ballistic’’ limit e=8for !  1=. We have found that the behavior of SHR !as a function of external perpendicular magnetic field andfrequency is given by: SHR ! e82Rk2F24!221 cos22  4!22 ; (19)which is a decreasing function of magnetic field.Conclusions.—The results of our analysis are summa-rized in Fig. 3 which shows the full spin-Hall conductivitySH, including Rashba, skew-scattering, and side jumpterms, as a function of two variables, frequency and mag-netic field, for realistic values of the parameters. The figureshows the smooth crossovers between different regimesand should be useful to experimentalists attempting toextricate the various components of this still quite intrigu-ing effect.We thank H. A. Engel for useful discussions. This workwas supported by NSF Grant No. DMR-0313681.*hankiewicz@fordham.edu[1] M. I. Dyakonov and V. I. Perel, Zh. Eksp. Teor. Fiz. 13,657 (1971) [JETP Lett. 13, 467 (1971)].[2] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).[3] S. Zhang, Phys. Rev. Lett. 85, 393 (2000).[4] J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004).[5] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301,1348 (2003).[6] R. Raimondi and P. Schwab, Phys. Rev. B 71, 033311(2005).[7] Y. K. Kato et al., Science 306, 1910 (2004).[8] V. Sih et al., Nature Phys. 1, 31 (2005).[9] J. Wunderlich et al., Phys. Rev. Lett. 94, 047204 (2005).[10] N. P. Stern et al., Phys. Rev. Lett. 97, 126603 (2006).[11] W. K. Tse and S. D. Sarma, Phys. Rev. B 74, 245309(2006).[12] E. M. Hankiewicz and G. Vignale, Phys. Rev. B 73,115339 (2006).[13] E. M. Hankiewicz, G. Vignale, and M. Flatte´, Phys. Rev.Lett. 97, 266601 (2006).[14] Spin-precession corrections to Eq. (3) are of order of2R1 and we omit them as small.[15] Substitution of operator for total system PNi1 p^iySiz bysum of one-electron operators NpySz is justified becausewe omit electron-electron interactions.[16] Actually, this is not the complete formula for the skew-scattering contribution (which is quite complicated), but asimplified version of it, which is valid in the limit ofRkF  1. However, even in the opposite regimeRkF  1 this formula remains qualitatively correct,in the sense that it exhibits the correct scaling behaviorsin the limits of weak and high magnetic field, low and highfrequency. Notice that contributions of order !2=2,which we have omitted, give finite SS at finite frequencyand zero magnetic field, but are negligible in comparisonto the larger intrinsic contribution.[17] We omitted corrections of order 2R1.[18] Equation (16) can be found from Eq. (10) using Kramers-Kro¨nig relation.[19] Again, we present only a simplified version of the com-plete formula, subject to restrictions explained inRef. [16].[20] J. I. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys.Rev. B 70, 041303(R) (2004).[21] E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys.Rev. Lett. 93, 226602 (2004).012345012345−0.8−0.6−0.4−0.200.20.40.60.81  ωτ  ω0/αR kF  σsH[e/8π]FIG. 3 (color online). Behavior of spin-Hall conductivityas a function of ! and !0=RkF and carrier concentrationn2D  2 1012 cm2,   1 m2=Vs, m  0:067me, 1 0:053 nm2, RkF  10 meV. For calculation of skew scattering,we assumed square potential characterized by =ss  0:002like in Ref. [12].PRL 100, 026602 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending18 JANUARY 2008026602-4

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"Phase Diagram" of the Spin Hall Effect

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