Theory of Nonequilibrium Intrinsic Spin Torque in a Single Nanomagnet

PHYSICAL REVIEW B 78, 212405 ͑2008͒

Theory of nonequilibrium intrinsic spin torque in a single nanomagnet

A. Manchon and S. Zhang Department of Physics, University of Arizona, Tucson, Arizona 85721, USA ͑Received 20 October 2008; revised manuscript received 17 November 2008; published 10 December 2008͒ In a single nanomagnet with spin-orbit interactions, the electrical current can generate a nonequilibrium spin density that gives rise to a spin torque on the magnetization. This spin torque does not involve spin transfer mechanism and originates from the band structure itself. We show that this spin torque can be effectively used to switch the direction of the magnetization and the critical switching current density could be as low as 104 –106 A/cm2 for a number of magnetic systems. Several magnetic systems for possible experimental realization are discussed.

DOI: 10.1103/PhysRevB.78.212405 PACS number͑s͒: 75.60.Jk, 75.75.ϩa, 72.25.Ϫb, 72.10.Ϫd

͗ ͘ The manipulation of magnetization in submicronic de- equilibrium, Heff=0 because p is zero. In the presence of ͗ ͘ vices is an important topic related to data storage and mag- the current, p is proportional to the current je and thus the netic memory technology.1 Besides field-induced magnetiza- average magnetic field on the magnetization is proportional ␣ ͑ ϫ ͒ tion switching, which is known to be a size and power to R je zˆ ; it is this current-driven effect that can be used consuming technique,2 a current-driven switching based on to switch the direction of the magnetization. We now carry the spin transfer effect has been realized.3 This mechanism out the rigorous calculation below. requires a noncollinear magnetization structure such as spin The one-electron eigenenergy and wave function are valves, tunnel junctions, and domain walls. The current den- ប2k2 sity observed in transition-metal-based materials is on the Ϯ Ϯ ͉␣ ͑ ϫ ͒ ˆ ͉͑͒ Ek = R k zˆ − JsdM 2 order of 107 A/cm2, which is higher than the current break- 2m 2 down of semiconductor-based transistors. and In this Brief Report, we propose another current-driven ␥ 1 Ϯei k mechanism for the manipulation of the magnetization of a ⌿Ϯ ͩ ͪ ͑ ͒ ͑ ͒ ͑ ͒ k = exp ik · r , 3 single magnetic two-dimensional electron gas 2DEG when ͱ2A 1 a Rashba spin-orbit coupling4 is present. Contrary to the con- ␥ ͑␣ ␪͒/͑␣ ␪͒ ventional spin-valve structures, this spin torque arises from where tan k = Rkx +Jsd sin Rky −Jsd cos and A is the band structure of the nanomagnet without the need for the area of the film. noncollinear ferromagnetic layers and does not involve the To calculate the nonequilibrium spin density and the spin usual spin transfer mechanism. The transport properties with torque in the presence of the current, we use the Boltzmann Rashba interactions in two-dimensional electron gas have equation for the two bands, been widely studied mainly for paramagnetic systems where ␴ f ץ there is no spontaneous ferromagnetic magnetization. Inoue ͩ 0 ͪ ͵ 2 Ј ␴␴Ј͑ ␴ ␴Ј͒ ͑ ͒ eEx − = ͚ d k W fk − f , 4 kkЈ kЈ ץ 5 et al. showed that the presence of an in-plane charge current kx ␴Ј induces an out-of-equilibrium transverse spin accumulation. ␴ This spin accumulation can then exert a torque on neighbor- where Ex is the electric field in the x direction, f0 is the ͑␴ Ϯ͒ ing ferromagnetic contacts.6,7 In contrast, we consider here a equilibrium distribution of the two bands = , and the single ferromagnetic 2DEG sandwiched between two dis- scattering probability is similar materials, so that the electric potential is highly ␴␴Ј ͉͗⌿␴͉ ͑ ͉͒⌿␴Ј͉͘2␦͑ ␴ ␴Ј͒ WkkЈ = ͚ k Vsc r − Ri kЈ Ek − EkЈ asymmetric, leading to the presence of the Rashba interac- i tion at the interfaces or within the ferromagnetic layer. The ␲ Hamiltonian is thus 2 ni ␴ ␴Ј = V2͓1+␴␴Ј cos͑␥ − ␥ ͔͒␦͑E − E ͒, ͑5͒ ប k kЈ k kЈ p2 ␣ R ͑ ϫ ͒ ␴ ␴ ˆ ͑ ͒ H = + p zˆ · − Jsd · M, 1 ͑ ͒ ␦͑ ͒ 2m ប where Vsc r =V r is the impurity scattering potential, Ri is the impurity position, and ni is their concentration. The where we confine the electron motion in two dimensions above definition accounts for both intraband ͑␴=␴Ј͒ and in- ͑x,y͒, Mˆ =cos ␪xˆ +sin ␪yˆ is the unit vector for the direction terband ͑␴=−␴Ј͒ scatterings, so that Eq. ͑4͒ may be solved of the local magnetization, Jsd is the exchange coupling numerically. However in the limiting case considered below ͑ ӷ ӷ␣ ͒ ͑␥ ␥ ͒ ͑␣2 ͒ strength between the conduction electron and the magnetiza- EF Jsd RkF , cos k − kЈ =1+o R , so that interband ␣ ␣ tion, and R is the Rashba parameter. To immediately see the transitions can be neglected to the first order in R. This current-driven effect due to the Rashba term, let us first con- ␴␴Ј ␲ 2 /ប yields to an isotropic scattering probability WkkЈ = niV , ˆ sider a large Jsd such that the conduction-electron spin and M which is equivalent to a constant relaxation-time approxima- are parallel and the Rashba term is equivalent to the Zeeman tion. In this case, the Boltzmann distribution in Eq. ͑4͒ has a ͑␣ /ប͒͑͗ ͘ϫ ͒ energy with the magnetic field Heff=− R p zˆ .In simple solution,

Ϯ f ץ Ϯ Ϯ f = f − eE ␶ͩ− 0 ͪ, ͑6͒ ץ k 0 x kx where ␶ is the relaxation time ͑eϾ0͒. With the above distri- ␦ bution, the spin density m and the spin current js can be readily evaluated; i.e.,

␦ ͵ 2 ͑ + −͒͑ ␥ ␥ ͒͑͒ m =2 d k fk − fk xˆ cos k − yˆ sin k 7 and

͑ ͒ ͵ 2 ͑ + + − −͒͑ ␥ ␥ ͒ ͑ ͒ FIG. 1. Color online Absolute ratio between the torque and the js = e d k vx fk − vx fk xˆ cos k − yˆ sin k , 8 current density as a function of the Rashba spin-orbit coupling ␣R. For these simulations we took EF =10 meV. ␦ ץ/ Ϯ ץϮ ͑ /ប͒ where vx = 1 Ek kx is the velocity. Once m is ob- ␦ ϫ ˆ ប tained, the spin torque is T=−Jsd m M. eHKMs ӷ␣ j = . ͑14͒ In the limiting case of Jsd RkF, where kF is the Fermi c 2␣ mP wave vector, one can analytically integrate out the momen- R ͑ ͒ ͑ ͒ ␣ / ␣ −11 tum in Eqs. 7 and 8 . Up to the first order in R Jsd, If we take the Rashba constant R =10 eV m for a typical value of a two-dimensional electron gas with a spatial inver- ␣ ␥ ␪ R ͑ 2 ␪ ␪ ␪͒͑͒ sion asymmetry, the spin polarization P=0.5, the saturation ءcos k = − cos + ky sin + kx sin cos 9 magnetization M =104 ͑J/Tm3͒, the effective mass m Jsd s =0.45 of heavy holes in GaMnAs,8 and an anisotropy field of ͑ ͒ ϫ 6 / 2 and 200 Oe , we estimate that jc is about 5 10 A cm , which is comparable to the critical current in spin valves and tun- ␣ ␥ ␪ R ͑ 2 ␪ ␪ ␪͒ ͑ ͒ neling junctions. Depending on the material parameters sin k = sin + kx cos + ky sin cos . 10 ͑ ͒ Jsd Rashba coupling, effective mass, etc. , we estimate that the current threshold can be as low as 104 –106 A/cm2. ͑ ͒ ͑ ͒ ӷ␣ By placing the above expressions into Eqs. 7 and 8 and While the above limiting case Jsd RkF is most interest- by noticing that the terms linear in ky average out after inte- ing and most relevant for observing the spin torque, we can gration, we have also consider a weak ferromagnet where Jsd is smaller than ␣ ͑␥ ␥ ͒ ͑␾ ␾ ͒ ͑ ͓͒ RkF. In this case, cos k − kЈ =cos − Ј +o Jsd with k ␣ m j ͑ ␾ ␾͔͒ ␦m =2 R e ͑cos ␪ sin ␪xˆ − cos2 ␪yˆ͒͑11͒ =k cos ,sin , and interband scattering is allowed at the បe EF zeroth order. Note that, since the Rashba interaction is now dominant, the zeroth order is sufficient to obtain the nonequi- ˆ Ϸ / and js = PjeM, where P Jsd EF is the spin polarization librium torque. In this case, we can use the formalism devel- of the current. Thus, the spin torque is oped by Schliemann and Loss9 to solve Eq. ͑4͒. Provided ͑ ␣ /ប ͒ ␪ ␴␴Ј ␴Ј␴ T=−zˆ 2 RmPje e cos . If one defines a current-driven that WkkЈ =WkЈk , the authors showed that the distribution ␥ ϫ ʈ magnetic field via T=− M Hcd, we have ␶ function can be expressed as a function of longitudinal k ␶Ќ ͑ ͒ ␣ m and transverse k relaxation times see Ref. 9 for details , H =2 R Pj ͑zˆ ϫ jˆ ͒, ͑12͒ Ϯ ץ Ϯ ץ cd ប e e eM Ϯ Ϯ ʈ f Ќ f s f = f − eE ͫ␶ ͩ− 0 ͪ + ␶ ͩ− 0 ͪͬ. ͑15͒ kץ k kץ k 0 x k ˆ x y where Ms is the saturation magnetization and je represents ␶ʈ ␶ ␶Ќ the unit vector in the direction of the current. Interestingly, we find that k = and k =0 at the zeroth The current-driven magnetic field or torque discussed order; then we can apply the relaxation-time approximation above competes with other torques in a ferromagnet. Similar and we find to the current-driven spin torque in spin-valve structures, we ␣ m j can write the dynamic equation of the magnetization by in- ␦m =− R e yˆ ͑16͒ បe E cluding Hcd, F and dM ␣ dM =−␥M ϫ ͑H + H ͒ + M ϫ , ͑13͒ dt eff cd M dt ␣ m J s T =−zˆ R sd j cos ␪. ͑17͒ ប e e EF where Heff is the effective field due to other magnetic inter- ␣ / actions. If we consider a single domain ferromagnet with an For an arbitrary ratio of RkF Jsd we show, in Fig. 1, the anisotropy field HK in yˆ direction, we obtain the critical numerical calculation of the torque efficiency defined as the ͉ ͉/ switching current density by setting Hcd=HK; i.e., ratio between the spin torque and the current density T je

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the Mn-doped dilute magnetic semiconductors ͑DMSs͒ in the quantum wells. Recently, 2DEG-DMSs were grown by Bove et al.12 and Teran et al.13 using GaMnAs and CdMnTe quantum well, respectively. Although Rashba spin-orbit coupling has not been inves- tigated in magnetic 2DEG yet, a number of theoretical and experimental studies have underlined the seminal role of interfacial discontinuity of the potential gradient14 in the am- plitude of the Rashba term. Whereas the confined 2DEG conduction electrons are almost insensitive to any 15 FIG. 2. Schematics of a memory bit based on Rashba-induced perpendicularly applied electric field, the interfacial dis- magnetization switching. The 2DEG-DMS is asymmetrically em- continuity of the potential gradient is usually very strong and 14 bedded between two insulators and the writing current is injected in gives rise to a sizable contribution. Consequently, the po- the plane of the 2DEG. The readout is via the magnetic tunnel tential discontinuity responsible for the Rashba spin-orbit in- ͑ junction based on the tunnel magnetoresistance. teraction should arise in every 2DEG conventional semiconductors, DMS, and metals͒ presenting a sufficiently high potential gradient and especially a discontinuity of the band ␣ as a function of R for different values of Jsd, maintaining structure at the interface. ӷ ␣ EF Jsd, RkF. This numerical evaluation has been obtained Besides conventional semiconductor 2DEG, Rashba-in- using the Schliemann and Loss formalism.9 Note that in the duced spin splitting at metallic surfaces has also been inten- case of weak Rashba spin-orbit coupling, the torque arises sively studied both theoretically and experimentally. Ag/ from the small misalignment of the electron spin from the Au͑111͒͑Refs. 16 and 17͒ interface exhibits Rashba interac- ␣ Ϸ ϫ −12 ϫ −11 ͑ ͒ local magnetization, whereas in the case of strong Rashba tion of R 4 10 –3 10 eV m, while Bi/Ag 111 coupling, it stems from the anisotropic electron velocity. ͑Ref. 18͒ shows Rashba coupling even much stronger than in ␣ Ϸ ϫ −10 It is interesting to compare the present spin torque with semiconductors of up to R 3 10 eV m. Magnetic Gd/ the conventional spin transfer torque.3 The spin transfer GdO ͑Ref. 19͒ interface also displays a sizable Rashba inter- ͑␣ Ϸ ϫ −11 torque comes from the absorption of the transverse spin cur- action, depending on the spin projection R 2.5 10 for ␣ Ϸ ϫ −11 ͒ rent by the magnetization and thus it requires a noncollinear minority spins and R 1.5 10 for majority spins . magnetization configuration in the direction of the spin cur- Other systems, e.g., oxide/ferromagnetic interfaces, might rent. For the spin-valve structure, the current must be applied possess an important spin-orbit interaction, as shown by tun- perpendicular to the layers and the magnetization of the two neling anisotropic magnetoresistance ͑TAMR͒ studies in Fe/ magnetic layers must not be collinear. Furthermore, the spin MgO/Fe junctions20,21 as well as by the analysis of magnetic / 22 transfer torque involves energy transfer since it competes anisotropy at AlOx Co interface. Recent investigations on with the damping torque. The present spin torque is created electronic reconstruction-induced quasi-2DEG at oxide by the intrinsic spin-orbit coupling in the nonequilibrium interfaces23,24 exhibit a sharp potential gradient25 that may be condition and thus it does not involve the transfer of the asymmetrically designed in order to obtain a large Rashba conduction-electron spin to the magnetization. Then, this effective interaction. torque exists for a uniformly magnetized layer with the cur- These studies are of great interest for our topic since me- rent in the plane of the layer and acts as an effective field; tallic devices seem easier to control than DMS. Although no reducing the magnitude of the anisotropy field to increase the transition-metals 2DEG have been grown yet, Bihlmayer et torque efficiency will limit the thermal stability of the mag- al.26 and Henk et al.27 showed that the Rashba coupling ex- netic layer. Since the torque does not compete with the tends considerably away from the interface ͓up to 5 ML for damping, it cannot excite current-driven steady magnetiza- Ag/Au͑111͔͒,26 suggesting that Rashba interaction can be tion precessions, in contrast with the spin transfer torques. maintained in 2DEG over a few nanometers. Then, even if We now consider the possible realization of our predicted the electron density is higher in metallic 2DEG than in semi- effects in several systems. The essential materials issue is to conductor 2DEG, the higher Rashba interaction should lead find a ferromagnet with a sizable Rashba interaction. First to a switching current density lower than that determined by note that Rashba spin-orbit coupling arises from a structure the Slowczewski torque. inversion asymmetry that restricts the effect to 2DEG. How- Finally, we propose a way to probe and use this current- ever, a similar torque should exist in three-dimensional lay- driven magnetization switching. Figure 2 shows the schemat- ers showing bulk inversion asymmetry leading to a ics of a memory bit based on Rashba-induced magnetization Dresselhaus10 spin-orbit coupling. Most of the studies on the switching. This device is composed of a conventional mag- Rashba interaction have been carried out for nonmagnetic netic tunnel junction whose free layer is a magnetic 2DEG two-dimensional semiconductors. The spatial inversion inserted between two thin insulators. The writing current is asymmetry leads to a net potential gradient in the growth injected in the plane of the free layer and can switch the direction ͑zˆ͒, which generates the Rashba Hamiltonian due to magnetization orientation in either direction depending on ϰ͑ ϫٌ ͒ ␴ϰ͑ ϫ ͒ ␴ spin-orbit coupling HR p V · p zˆ · . For ex- the sign of V+ −V−. The information contained in the free / ample, InGaAs In0.77Ga0.23As quantum well could produce a layer magnetization can then be read by applying a bias volt- −11 ͑ ͒ Rashba parameter as large as 10 eV m Ref. 11 and one age Vg −V− and measuring the corresponding resistance would expect that the Rashba parameter should be similar for across the bottom insulator using the usual tunneling magne-

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/ toresistance effect. In this device, the Rashba-induced mag- coupling. Systems such as MgO/Fe, AlOx Co, or even Au/Fe netization switching can be controlled by both the perpen- should provide interesting preliminary results. 14 ជ ͑ ͒ dicular applied electric ﬁeld E through Vg and the in- We thank Giovanni Vignale at the University of Missouri ͑ ͒ plane ﬂowing current density through V+ −V− . As discussed for helpful discussions. This work was supported by Seagate earlier, the interface between the magnetic 2DEG and the Technologies, NSF ͑Contract No. DMR-0704182͒, and DOE tunnel barrier should have an important interfacial spin-orbit ͑Contract No. DE-FG02-06ER46307͒.

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