week ending PRL 104, 146802 (2010) 9 APRIL 2010

Inverse Spin-Galvanic Effect in the Interface between a Topological and a Ferromagnet

Ion Garate1,2 and M. Franz1 1Department of Physics and Astronomy, The University of British Columbia, Vancouver, BC V6T 1Z1, Canada 2Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada (Received 2 November 2009; published 9 April 2010) When a ferromagnet is deposited on the surface of a topological insulator the topologically protected surface state develops a gap and becomes a two-dimensional quantum Hall . We demonstrate that the Hall current in such a liquid, induced by an external electric field, can have a dramatic effect on the magnetization dynamics of the ferromagnet by changing the effective anisotropy field. This change is dissipationless and may be substantial even in weakly spin-orbit coupled ferromagnets. We study the possibility of dissipationless current-induced magnetization reversal in monolayer-thin, insulating ferro- magnets with a soft perpendicular anisotropy and discuss possible applications of this effect.

DOI: 10.1103/PhysRevLett.104.146802 PACS numbers: 73.43.f, 75.30.Gw, 75.70.i, 72.15.Gd

Introduction.—Understanding the electric-field control parameters and is not related to Ampere’s law; instead it is of magnetization and harnessing its technological potential the topological counterpart of the inverse spin-galvanic are among the most important objectives of . effect (ISGE) [7], which has only recently been exploited Current-induced spin torques can reverse the magnetiza- in conducting ferromagnets with spin-orbit interaction [8]. tion of conducting ferromagnets (FM) and move magnetic The topological variant of ISGE is unique in that it is domain walls [1]. However, the Joule heating generated by quantized, topologically protected, and occurs in insulating transport currents remains a handicap from a practical ferromagnets without spin-orbit interaction. In ultrathin viewpoint. An electric field can also reorient the magneti- (thickness &1nm) ferromagnetic insulators deposited on zation of insulating compounds with broken inversion a surface of TI (Fig. 1) the topological ISGE produce symmetry via the magnetoelectric coupling [2]. While torques that may be comparable to the coercive field, they overcome the issue with Joule heating, these multi- thus opening an unprecedented avenue for current-induced ferroic materials are fewer and more difficult to engineer control of magnetization without Joule heating. than common metallic ferromagnets. Recently, a novel Functional integral formalism.—We begin by reviewing magnetoelectric effect has been discovered [3] in topologi- the equation of motion for the magnetization M M^ of cal insulators that are coated with ferromagnetic films. a classical ferromagnet (in units of 1=volume). At low en- Topological insulators (TIs) are bulk insulators with an ergies the magnitude M is approximately constant and the anomalous band structure that supports topologically ro- only dynamical variable is the direction ^ ¼ðx;y;zÞ. bust gapless states at the surfaces [4]. These materials are predicted to display a variety of unconventional spintronics effects [5]. One unique feature is the universal quantized topological magnetoelectric effect [3], described by e2 M ¼C E: (1) top 1 2 M C Here top is the induced magnetization, 1 is a half- integer topological invariant that depends solely on the sign of the time-reversal-symmetry-breaking perturbation, E C e2= is the applied electric field, and 1 2 H is the FIG. 1. Corbino-disk-shaped TI coated with an ultrathin ferro- Hall conductance (@ 1 throughout). Unfortunately, the magnet. (a) Top view: In the absence of electric fields, the prospects for manipulating the magnetization of real ferro- magnetization of the ferromagnet points outside the page (dotted magnets via Eq. (1) are limited because below the thresh- circles). When a voltage difference is applied between the inner old Hall current density (jH < 1A=m, see Ref. [6]) the and outer circles, a dissipationless Hall current flows at the B M & 6 interface between the two materials ( arrows). This current topological magnetic field top ¼ 0 top 10 T is * : magnetizes the of the TI (inverse spin-galvanic very small compared to typical coercive fields ( 0 01 T) effect) along the radial direction (dashed arrows), thus exerting a in a ferromagnet. spin torque on the magnetization of the ferromagnet. (b) Cross In this Letter we unveil a qualitatively new contribution sectional view: The shaded region is the TI, whereas the un- H to the topological magnetoelectric effect, which stems shaded region is the ferromagnetic film. FM is the anisotropy from the current-induced spin polarization of the TI sur- field in electric equilibrium. HCS;1 is a topological magnetic field face states. Unlike Eq. (1), this effect depends on material proportional to (and parallel to) the applied electric field.

0031-9007=10=104(14)=146802(4) 146802-1 Ó 2010 The American Physical Society week ending PRL 104, 146802 (2010) PHYSICAL REVIEW LETTERS 9 APRIL 2010 ^ Z The time dependence of may be determined using the S ¼ d2xdt ½@ H (7) functional integral approach [9], which is built on the TI 0 partition function is the action for the surface states. is a fermionic spinor, Z @ @ eA S ^ 0 ¼ t 0, is the chemical potential (located in the Z Z D ^ x;t e FM½: ¼ 0 ð Þ (2) A gap), and 0 is the electrostatic potential. After rotating the spins by an angle =2 around z^, Eq. (7) may be rewritten as Z is the partition function corresponding to the equilib- R 0 S ¼ d2xdtc ½@ H~ c with ^ ^ S S E TI 0 rium magnetic configuration ¼ eq. FM ¼ B is ~ x y z the action for small (quadratic) spin fluctuations, where H ¼ vF ðx eaxÞþvF ðy eayÞz ; (8) R _ SB ¼ M dxdt^ ð^ ^ Þ is the Berry phase and R eq where c is the rotated fermion field. In this new basis, a E½^ ¼ dxdt^ 1^ is the micromagnetic energy func- =ðevFÞð^ z^Þ appears as an additional contribution to tional. is the spin-spin response function. The semiclas- the effective vector potential. z acts as a mass term. sical equation of motion can be derived from These massive Dirac fermions may be integratedR out in the S =^ ¼ 0, S ^ FM Z D ^ x;t e eff ½ standard manner [11], whereby ¼ ð Þ . _ 1 E ^ S S ^ ¼ ^ : (3) To second order in the effective action is eff ’ FM þ eq M S S ^ CS þ EB, where A gradient expansion [10]of yields the venerable e2 Z 2 S ¼ C d xdt A@A; (9) Landau-Lifshitz-Gilbert-Slonczewski equation for magne- CS 2 1 tization dynamics in the presence of damping and transport A~ A ;A a ;A a currents, ¼ð 0 x þ x y þ yÞ is the effective vector poten- tial and ¼ t; x; y. The Chern-Simons (CS) action (9) ^_ ^ H ^ ^_ v ^ ¼ eq eq s r arises in (2 þ 1)-dimensional systems with broken time reversal symmetry and nontrivial topology. The topology ^ vs r^ þ: (4) eq of the band structure is encoded in the Thouless, Kohmoto, C H is an effective magnetic field (in energy units) that Nightingale, den Nijs (TKNN) [12] invariant 1. For fer- includes the anisotropy field, the exchange field, as well mions described by a single Dirac Hamiltonian (8), we as external magnetic fields. H determines the easy axis have [13] along which the magnetization of a single-domain ferro- 1 magnet points in equilibrium. vs is the adiabatic spin trans- C ¼ sgnðzÞ: (10) 1 2 fer velocity and is proportional to the transport current. S and characterize dissipative processes in which energy is EB is quadratic in spatial and temporal derivatives of transferred from magnetic to nonmagnetic (e.g., lattice) A and encodes the ordinary dielectric or diamagnetic degrees of freedom. response of the gapped surface state. Herein we focus on S A ^ Topological effective magnetic field.—We now address CS, which is first order in the derivatives of (and ), the magnetization dynamics of an insulating ferromagnet ^ and thus outweighs SEB in the description of ðx;tÞ at long sitting on top of a TI. The low-energy effective length and time scales. It also produces the effective mag- Hamiltonian for the surface states of the TI is [3,4] netic field that underlies the inverse spin-galvanic effect which is central to this study. H ¼ vF ð z^Þ ^ ; (5) The semiclassical magnetization dynamics follows from v i i x; y; z S = ^ where F is the Fermi velocity, ( 2f g) are Pauli eff ¼ 0, matrices denoting real spin of the surface states,  ¼ ^_ ^ H H ; ireA, A is the electromagnetic vector potential, z^ ¼ eq ð FM þ CSÞþ (11) is the unit vector normal to the interface between the TI and H S = M ^ the ferromagnet, and is the exchange coupling between where FM ¼ FM ð Þ is the effective magnetic the surface states and the local moments of the ferromagnet field that collects the anisotropy or exchange fields of the ( > 0 for ferromagnetic coupling). We consider a ferro- isolated ferromagnet and z magnet with perpendicular anisotropy (eq ¼ ^) so that in 1 S H H ¼ CS ¼ E þ ðz ^_ Þ equilibrium a gap opens in the energy spectrum of the CS M M ev ev ^ 2D ^ 2D F F surface states. The partition function for this composite system is (12) Z Z is an additional (topological) contribution to the magnetic S ^ S ; ; ^ Z Z D ^ x;t e FM½ D2 x;t e TI½ ; ¼ 0 ð Þ ð Þ (6) field that results from the exchange coupling between the M ferromagnet and the TI. 2D is the areal magnetization at S = H where FM is the ferromagnetic action discussed above and the interface (in units of 1 area). CS depends on material 146802-2 week ending PRL 104, 146802 (2010) PHYSICAL REVIEW LETTERS 9 APRIL 2010

(ii) reverses sign when z !z and vanishes when z ¼ 0, and (iii) exerts a dissipationless torque provided that the ferromagnet is insulating. H =M =ev 2z @ ^ CS;2 ð H 2DÞð FÞ ^ t is associated with the change in the spin response function under a magnetic field [Fig. 2(b)]: X X f f ij i j k;n kþq;n0 ðqÞ¼ 0 0 ; (16) FIG. 2. Feynman diagrams for (a) the electric-field-induced M;B A n;n n ;n E E ! k n;n0 kþq;n0 k;n þ magnetization (inverse spin-galvanic effect), (b) the xy compo- nent of the spin-spin response function, (c) the xy component of where q ¼ð!; qÞ is the energy-momentum of the the spin-spin response function in the presence of an electric i n; k i n0; k q q ij and n;n0 ¼h j j þ i.At ¼ 0 we get M;B ¼ current (it yields the adiabatic spin transfer torque vs r^ ). The 2 ij xy yx xx ð=evFÞ ði!ÞH , where ¼ ¼ 1 and ¼ solid straight lines are propagators for massive Dirac quasipar- yy Hi =M ij ticles (quasiholes). The solid wavy lines are that couple ¼ 0. Thus, CS;2 ¼ð 2DÞ M;Bj simply increases to the spin operator, and the dashed straight lines are photons that (if > 0) or decreases (if < 0) the Berry phase of the couple to the velocity operator. isolated ferromagnet [15], thereby renormalizing the pa- rameters entering Eq. (4). v M When the magnetization of the ferromagnet is uniform, parameters ( F, , 2D) and is proportional to the Hall Eq. (12) captures the entire current-induced spin torque for ¼ C e2= conductivity H 1 2 . Because the exchange cou- weak electric fields. In the presence of inhomogeneous pling between the surface states and the localized moments H magnetic textures, one must add the ordinary spin transfer of the ferromagnet is local in space, the influence of CS torque. The microscopic theory for vs r^ amounts to weakens as the thickness of the ferromagnetic film xy increases. evaluating the change of the spin-spin response function [10] under an electric field [Fig. 2(c)]. Starting from H ; =ðevFM ÞHE can be interpreted as an xy CS 1 2D ðqÞ, perturbing the matrix elements of the spin opera- electric-field induced change of magnetic anisotropy. The M;B E underlying cause of this effect is that the electric field spin tors to first order in [16] and expanding the resulting q polarizes the surface states along a direction (E=E) which expression to first order in , we find (numerically) that v q E q E q is misaligned with the equilibrium easy axis (z^). We illus- s / zð x x y yÞ. Furthermore, for realistic pa- H trate this point by computing the magnetization induced by rameters the torque exerted by CS is found to dominate v q q 1 a static and uniform electric field: over s by an ample margin even when j jnm (note that H does not vary as ^ is slightly tilted away Mi ij Ej; CS E 2D M;E (13) from z^). Current-induced magnetization switching.—As ex- where H plained above, CS;1 modifies the anisotropy field of X X e f f 0 the ferromagnet in the presence of a Hall current jH ¼ ij 1 i vj k;n k;n M;E ¼ lim n;n0 n0;n (14) z E !! i! A E 0 E ! H ^ : 0 k n;n0 k;n k;n þ K H z z j ; is the linear magnetoelectric response function [Fig. 2(a)]. an ¼ z ^ þ ^ H (17) 0 M2D evFM2D n; n are the band indices of the surface states, Ek;n are the f i band energies, k;n are the Fermi distributions, n;n0 ¼ where K is the anisotropy energy per unit area for the hn; kjijn0; ki, and A is the area of the interface. From magnetic ultrathin film in electric equilibrium. When E ¼ Eq. (5), the velocity operator is related to the spin operator 0 the magnetization of the ferromagnet points along z^. via v ¼ @H =@k ¼vF z^, which allows us to use the After turning on the electric field, the magnetization begins H TKNN formula for conductivity [12] and write to precess around an and (assisted by the damping) equilibrates along the modified easy axis. For instance, in ij H ¼ ij; M;E ev (15) a Corbino disk geometry depicted in Fig. 1 the electric field F produces a crown-shaped magnetization. Provided that where ij is the Kronecker delta and we have used the fact quantum coherence is preserved, this configuration hosts Hi [17] a circulating spin-current proportional to MðÞ that the longitudinal conductivity is zero. Hence, CS;1 ¼ i M j z O E2 ð=M ÞEM . This result is reminiscent of the current- ð þ Þ/zð H ^Þþ ð Þ, which is radially po- 2D 2D induced effective field in single-domain metallic ferromag- larized and persistent (dissipationless). is the azimuthal nets that belong to the gyrotropic crystal class [14]. Some angle around the disk. ^ significant differences between Ref. [14] and the present If jH * evFK=, reaches the interface (z ¼ 0)in H work are that CS;1 (i) does not depend on the strength of the course of the precession. At that moment, according to C @ ^ spin-orbit interactions in the ferromagnet or at the interface Eq. (10), 1 ¼ 0 and hence t ¼ 0; yet this is an un- [Eq. (5) involves vF rather than a ‘‘spin-orbit velocity’’], stable fixed point and an infinitesimal in-plane magnetic 146802-3 week ending PRL 104, 146802 (2010) PHYSICAL REVIEW LETTERS 9 APRIL 2010

field suffices to kick the magnetization towards z < 0. magnetic textures and to switch the magnetization of a Once this occurs the electric field may be turned off and the ferromagnet by 180 without Joule heating. magnetization will equilibrate towards z^. Thus a 180 We thank I. Affleck, J. Folk, A. H. MacDonald, and G. magnetization switching may be completed by combining Sawatzky for helpful comments and questions. This re- a dissipationless Hall charge current with a very small search has been supported by NSERC and CIfAR. I. G. magnetic field. Nevertheless, achieving jH * evFK= in thanks CIfAR for financial support. real materials presents challenges. First, jH (E) cannot be larger than 1A=m (0:5mV=nm) because otherwise the dissipationless quantum will break down [6]. Second, we require relatively small coercive fields: [1] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008); G. S. D. Beach et al., ibid. 320, 1272 (2008). H ¼ K=M & : coer 2D 0 02 T. While such a soft perpendicu- [2] D. Khomskii, Physics 2, 20 (2009). lar anisotropy is inadequate for the magnetic recording [3] X.-L. Qi et al., Phys. Rev. B 78, 195424 (2008). industry, it may find applications in magnetic random [4] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, access memories and magnetic field sensors [18]. Third, 106803 (2007); J. E. Moore and L. Balents, Phys. Rev. B the thickness of the ferromagnet needs to be comparable to 75, 121306(R) (2007); H. Zhang et al., Nature Phys. 5, the penetration depth of the Dirac fermions into the ferro- 438 (2009); D. Hsieh et al., Nature (London) 452, 970 magnetic insulator (&1nm). While ultrathin films are (2008); Y.L. Chen et al., Science 325, 178 (2009). commonplace in metallic ferromagnets [19], insulating [5] X.-L. Qi, T. Hughes, and S.-C. Zhang, Nature Phys. 4, ferromagnets such as EuO or EuS present additional ex- 273 (2008); T. Yokoyama, Y. Tanaka, and N. Nagaosa, perimental difficulties (but see Ref. [20] for recent Phys. Rev. B 81, 121401(R) (2010); J. Gao et al., arXiv:0909.0378. progress). Alternatively, one could electrically manipulate [6] G. Nachtwei, Physica (Amsterdam) 4E, 79 (1999);V. the spin textures caused by magnetic impurities placed on Singh and M. M. Deshmukh, Phys. Rev. B 80, 081404 JM j the surface of the TI [21,22]. Using ¼ 2D, H ¼ (R) (2009). = v 5 = H : 1A m, F ¼ 5 10 m s, and coer ¼ 0 01 T, we esti- [7] E. L. Ivchenko and S. Ganichev, in Spin Physics in mate J * 50 meV nm2 as the condition for magnetization , edited by M. I. Dyakonov (Springer, switching. Hence J=a2 * 0:2eV, where a ’ 0:5nmis a New York, 2008). typical lattice constant for the topological insulator. J=a2 ’ [8] A. Chernyshov et al., Nature Phys. 5, 656 (2009). 0:2eVis an a priori reasonable value [21] for the exchange [9] See, e.g., A. Auerbach, Interacting Electrons and integral between the localized moments of the ferromag- Quantum Magnetism (Springer, New York, 1994). netic insulator and the surface states of the TI. For stronger [10] H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn. 75, perpendicular anisotropies (say, H * 0:05 T) the ex- 113706 (2006); R. A. Duine et al., Phys. Rev. B 75, coer 214420 (2007); I. Garate et al., ibid. 79, 104416 (2009). change integral would need to be of the order of a few [11] X.-G. Wen, Quantum Field Theory of Many-Body Systems eV, and at such strong coupling the surface states of the TI (Oxford University Press, Oxford, England, 2004). would be altered in a way not captured by Eq. (5). From the [12] D. J. Thouless et al., Phys. Rev. Lett. 49, 405 (1982). ! H =@ precession frequency prec ’ B an ’ 1 GHz we infer [13] See, e.g., G. Rosenberg et al., Phys. Rev. B 79, 205102 switching times of the order of a nanosecond. (2009). There has been some recent work along the lines of the [14] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 above discussion, albeit in topologically trivial materials (2008); Phys. Rev. B 79, 094422 (2009); I. Garate and [23]. There are two salient differences between Ref. [23] A. H. MacDonald, Phys. Rev. B 80, 134403 (2009); and the present work. (i) The microscopic origin of the K. M. D. Hals, A. Brataas, and Y. Tserkovnyak, arXiv:0905.4170. change in magnetic anisotropy: in our case it is the current- [15] J.Fernandez-Rossier et al., Phys.Rev.B69,174412 (2004). induced spin-polarization of massive Dirac fermions (the [16] The influence of E on the Fermi factors can be neglected topological inverse spin-galvanic effect), whereas Ref. [23] because the chemical potential of the surface states lies in concentrates on the electrical manipulation of the atomic the energy gap. positions and distortions of the charge distribution. [17] F. Schuetz, M. Kollar, and P. Kopietz, Phys. Rev. Lett. 91, (ii) Symmetry of the anisotropy mechanism: in our case 017205 (2003). it is odd under time reversal (because jH is odd), whereas [18] Y. Ding, J. H. Judy, and J.-P. Wang, J. Appl. Phys. 97, in Ref. [23] it is even under time reversal (because E and 10J117 (2005); H. Meng and J.-P. Wang, Appl. Phys. Lett. charge density are even). 88, 172506 (2006); R. Sayed Hasan et al., New J. Phys. 9, Conclusions.—When a ferromagnetic film with perpen- 364 (2007). [19] C. A. F. Vaz, J. A. C. Bland, and G. Lauhoff, Rep. Prog. dicular anisotropy is placed on top of a topological insu- Phys. 71, 056501 (2008). lator, a quantum Hall current induces a spin torque which [20] T. S. Santos et al., Phys. Rev. Lett. 101, 147201 (2008). substantially modifies the magnetic easy axis. The origin of [21] Q. Liu et al., Phys. Rev. Lett. 102, 156603 (2009). this new torque can be traced to a topological variant of the [22] R. R. Biswas and A. V. Balatsky, arXiv:0910.4604. inverse spin-galvanic effect. In Corbino disk geometries [23] J. Stohr et al., Appl. Phys. Lett. 94, 072504 (2009);S.J. this effect may be exploited to generate crown-shaped Gamble et al., Phys. Rev. Lett. 102, 217201 (2009).

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