A major barrier to the development of spin-based electronics is the transition from current-driven spin torque, or magnetic-field-driven magnetization reversal, to a more scalable voltage-driven magnetization reversal. To achieve this, multiferroic materials appear attractive, however the effects in current materials occur at very large voltages or at low temperatures. Here the potential of a new class of hybrid multiferroic materials is described, consisting of a topological insulator adjacent to a magnetic insulator, for which an applied electric field reorients the magnetization. As these materials lack conducting states at the chemical potential in their bulk, no dissipative charge currents flow in the bulk. Surface states at the interface, if present, produce effects similar to surface recombination currents in bipolar devices, but can be passivated using magnetic doping. Even without conducting states at the chemical potential, for a topological insulator there is a finite spin Hall conductivity provided by filled bands below the chemical potential. Spin accumulation at the interface with the magnetic insulator provides a torque on the magnetization. Properly timed voltage pulses can thus reorient the magnetic moment with only the flow of charge current required in the leads to establish the voltage. If the topological insulator is sufficiently thick the resulting low capacitance requires little charge current.

## I. INTRODUCTION

Topological insulators^{1,2} exhibit peculiar bulk transport properties, which can be harnessed to produce unique spin-accumulation properties when a voltage is applied to the material. Although the focus of recent research has been mostly on the surface states that exist at the interface between a topological insulator and a trivial insulator, the bulk transport properties exist independently of that surface, and correspond to coherent transport of spin across the material, in response to a voltage, due to the evolution properties of the entire full band under the effect of an electric field.^{3} The resulting spin accumulation provides a torque at the interface with a magnetic insulator,^{4} which can rapidly reorient the magnetization of that material. In analogy to a multiferroic material, in which an applied electric field can generate a reorientation of the bulk magnetization, for these hybrid topological insulator/magnetic insulator structures an electric field applied to the topological insulator generates a reorientation of the magnetic insulator’s magnetization. Thus this hybrid material can be described as a special type of hybrid multiferroic.

Here the dependence of the switching power on device geometry and material parameters is presented, and is exceptionally low, suggesting that such hybrid multiferroics could form the basis of a new architecture for spin-based memory and logic. The role of surface-passivating topological insulator materials, which would play a similar role to the presence of a barrier for electronic states in quantum wells, is also described. These general considerations of the role of topological insulators in a voltage-driven spintronic architecture should assist in the design of functional, efficient heterostructures relying on voltage-driven magnetization dynamics instead of current-driven spin torque^{5–8} or magnetic-field-driven reversal.^{9,10} With this approach, and by encoding logical information in the magnetic orientation, switching energies less than an attoJoule and switching frequencies of order 100 GHz can be achieved for currently-known material properties.

## II. TOPOLOGICAL INSULATOR SPIN TRANSPORT WITHOUT SURFACE STATES

Topological insulators are distinguished from regular (so-called trivial) insulators in that their electronic ground state possesses a nonzero Berry curvature. The spin Hall conductivity in the clean static limit, evaluated as the linear response of the spin current to an electric field using a Kubo approach, consists of a sum of the Berry curvature:^{11}

where *e* is the electric charge, *ℏ* is Planck’s constant, *V* is the volume of the system, **k** is the crystal momentum, *n* is a band index, and the Berry curvature $\Omega nz(\mathbf{k})$ is

Here the Fermi-Dirac distribution function (*f*_{kn}) ensures that the sum is over filled states, corresponding to all the filled bands at temperature 0K. *u*_{nk} and *u*_{n′k} are Bloch states associated with crystal momentum **k** and band index *n* and *n*′. The spin current and velocity operators, $j^ij$ and $v^i$, are

where $\sigma j$ is the spin operator along direction *j* and $H^$ is the Hamiltonian of the material.

The spin Hall conductivity of Eq. (1), which is the spin current flow in response to an applied electric field, does not vanish even when there are no states at the Fermi energy, as it involves a sum over all the filled states (*f*_{kn}). It thus differs from many longitudinal charge transport calculations in which the current flow depends on the difference between Fermi occupation factors ($f\mathbf{k}n\u2212f\mathbf{k}\u2032n\u2032$) evaluated over the small range of energies accessible due to inelastic scattering (such as within a phonon energy of the chemical potential) or at the same energies for elastic scattering. Thus even if all the single-particle states are much farther than the thermal energy *k*_{B}*T* from the Fermi energy, which is possible in a bulk topological insulator such as Bi_{2}Se_{3} whose band gap exceeds 300 meV, there remains a finite spin Hall conductivity.

Calculations along these lines^{3} are bulk calculations and one might be concerned about the relative contribution of surface states. As described in Ref. 3 the situation is likely to be similar to that known from the quantum Hall effect; current understanding is that the relative contribution of bulk currents and edge currents can vary substantially as the magnetic field is varied across a single quantum Hall plateau^{12} even though the Hall conductivity is unchanged. By analogy, the spin Hall conductivity can be calculated for convenience under conditions where the edge current contribution vanishes, but the value obtained will be the appropriate one for the sum of edge and bulk currents in a system where the spin Hall potential drop is distributed nonuniversally between the bulk and edge regions. The fundamental connection between the nature of the quantum Hall state, for which the Berry curvature of the spin-up and spin-down bands are the same (and thus it possesses chiral edge states), and the topological insulator, for which the Berry curvature of a band and its partner band with time-reversed quantum numbers have the same magnitude but opposite sign (and thus it possesses helical edge states), is exploited here. For the quantum Hall system the charge Hall conductivity does not vanish but the spin Hall conductivity does, whereas in the topological insulator the charge Hall conductivity vanishes but the spin Hall conductivity does not, even though (in both cases) there are no extended single-particle states near the chemical potential in the bulk of the material.

In order to consider the accumulation of spin at the edges of the topological insulator, resulting from this bulk spin Hall conductivity, a convenient geometry is the Corbino geometry, corresponding to a ring of topological insulator material with an inside edge and an outside edge, as shown in Fig. 1. Here any surface states cannot connect the inside surface with the outside surface, so a spin accumulation can be considered at the two edges. A magnetic flux Φ pierces the inner ring, such that no magnetic field is applied to the topological insulator itself. A changing magnetic flux through the inner ring generates an electrochemical potential $\u222b\mathbf{E}(\mathbf{r})\u22c5\theta ^d\theta \u221dd\Phi /dt$, and so a linearly-varying-with-time magnetic flux generates a constant electromotive force around the ring. In response, the topological insulator carries a finite spin Hall current $\sigma SHCE$, even though there are no extended states in the bulk of the ring at energies near the chemical potential (which could carry the spin via single-particle excitations^{3}).

The material can carry spin current through the insulating region, even without individual carriers, through the collective response of the band to the electric field. This phenomenon is similar to that seen in the integer quantum Hall system, in which charge is transported across a region without single-particle states at the chemical potential, through collective evolution of the wave functions of the band in response to the electric field. In the topological insulator, in response to the electric field the wave function of each filled state in the band shifts slightly in the transverse direction. The spin-up states shift in the opposite direction from the spin-down states. In order for this to occur for filled states located near an edge the effect of this motion is to deplete carriers of one spin direction and to accumulate carriers of the other spin direction at the edge, leading to oppositely oriented spin accumulation at each edge. It is a challenge to visualize this phenomenon, however as a heuristic picture one can consider two electron waves that are time-reversed states. These two waves have opposite spin character and oppositely-oriented crystal momentum **k**. The effect of the electric field is to provide a time-dependent additional phase proportional to the electric field that is equal and opposite for these two waves. A time-dependent additional phase will cause the wave to appear to move to the inside edge of the ring or the outside edge of the ring, depending on its sign. Thus, for a time-dependent phase that is equal and opposite for the two time-reversed states, a spin current is produced.

Fig. 1 shows schematically the effective motion of the spins towards the inside edge and the outside edge. It also shows the spin-orbit correlation of filled states in the bulk region of the topological insulator. From Eq. (2) one can see that the response to an electric field involves the velocity operator in the $x^$ direction and in the $y^$ direction, where the $x^$ direction is defined to be in the direction of the electric field, and $y^$ to be towards the inner edge of the ring. Fig. 1 shows the orbital character of the state associated with spin up and with spin down. From Fig. 1 the opposite orbital character of state associated with spin down from that associated with spin up produces a opposite direction of flow for spin down versus spin up, corresponding to a finite spin Hall conductivity.

## III. SWITCHING A MAGNET WITH SPIN CURRENT FROM A TOPOLOGICAL INSULATOR

An analysis of the time and energy required to switch a magnet with spin current begins by assuming a source of spin current from a large spin-orbit-coupling (SOC) material, which could be a topological insulator but does not need to be. Figure 2 shows a configuration for voltage-controlled switching of a magnetic layer using the spin Hall effect in a SOC material. In the absence of an applied voltage there are no net currents and thus no dissipation. Once a voltage, *V*_{switch}, is applied across the SOC material, a spin current flows transverse to the resulting electric field. The spin current’s spin polarization orients transverse to both the electric field and direction of spin current, as indicated in Fig. 2(b). The major sources of dissipation emerge immediately: one is the longitudinal charge current *J*_{L} in the bulk of the large SOC material, another the spin recombination current (spin-flip current) *J*_{R} that occurs at the edge of the magnetic material, and finally the spin diffusion current *J*_{D} from the spin polarized region at the interface towards the bulk of the SOC material. The advantages of a topological insulator are immediately visible. The spin Hall conductivity for a topological insulator can be as large (or larger) than that of a large SOC metal,^{3} but the longitudinal charge current will vanish for the topological insulator. If the material has a large SOC but is a trivial, non-topological insulator, then the spin Hall conductivity vanishes. Thus a topological insulator provides the advantages of a large spin Hall conductivity, but without the longitudinal dissipation of a metallic material.

The spin current generates a chemical potential difference between spins oriented out of the plane of the figure and those oriented into, which generates a torque on the spins in the magnetic material through an effective exchange field. This torque forces the magnetization of the magnetic material to precess, which it does to point, in Fig. 2(c), in the opposite direction of Fig. 2(a). To evaluate the fundamental limits on this switching behavior, and identify the key features of the hybrid multiferroic required to approach those limits, an estimate of the response of this structure when *J*_{D} and *J*_{R} can be reduced to a negligible value is required. A calculation of the dynamics of the switching process requires the following information: (1) the rate at which spins accumulate at the interface with the magnet, which is provided by the spin Hall conductivity $\sigma SHC$, and is $\sigma SHCVswitch/L$, and (2) the dependence on the density of interfacial spins of the exchange field **H**_{ex} from those spins acting on the magnetic material.

The spin Hall conductivity available from large SOC materials such as topological insulators^{8} exceeds 1000 $(\u210f/e)\Omega \u22121cm\u22121$. The areal number density *n*_{spins} of polarized spins thus accumulates rapidly at the interface. The rate of accumulation of spins at the interface depends on the value of the electric field, and the spin Hall conductivity, so

For a typically large value of the spin Hall conductivity associated with a topological insulator, 1000 $(\u210f/e)\Omega \u22121cm\u22121$, and for a voltage drop across $L\u223c10$ nm (*L* chosen large enough so there is negligible charge leakage current),

The exchange field that results from the *n*_{spins} polarized spins will be estimated for yttrium iron garnet (YIG), which is a ferrimagnet composed of a low density of Fe^{3+} spins arranged in octahedral and tetrahedral sites. The approximate exchange field is $\mathbf{H}ex\u223c104\mathbf{M}$, where **M** is the volume magnetization of spins (measured^{13} to be 140 Gauss). For YIG there are ∼20 Fe spins per nm^{3}, so the exchange field should be $\u223c5\u2009Tesla\u2009nspins$, where *n*_{spins} is the spin density per cubic nanometer. To determine the effect of the nonequilibrium spins, the spins acting on a YIG spin at the edge are assumed to be those within an area of (0.5 nm)^{2} and are distributed in a region approximately 0.5 nm from the surface of the magnet. In this estimate the nonequilibrium spins act only on the top 0.5 nm region of the magnet. This distance scale is similar to the length scale of interactions within YIG, and also of the interface states within the topological insulator. From this estimate

where *n*_{spins} is the areal density of polarized spins at the interface. For a thin magnetic layer $tM\u223c10$ nm, the speed of the switching associated with precession of the magnetic moment will be

Assuming that the switching time is approximately equally divided between the exchange field acting and the polarized spins accumulating, the switching rate depends on the switching voltage as

The energy required to perform this switch depends on the charging energy for the capacitor associated with the drop of the voltage *V*_{switch} across the distance *L*. Thus

For a volume 10 nm on a side (*L* = *W* = 10 nm), $\u03f5r\u223c10$, and *V*_{switch} = 50 mV, for a switching frequency of 100 GHz, the energy required to switch is 10^{−21} J, corresponding to an energy-delay product of $3\xd710\u221232$ Js. The switching voltage is set to 50 mV so that it is lower than the gap opened at the edges due to magnetic doping, which has been reported to have been demonstrated^{14} to be as large as 50 meV. Further improvements in the speed and energy required for switching could be achieved by replacing YIG with a room-temperature ferromagnetic material with an even lower spin density.

To evaluate the challenge of reaching this limit, the spin dissipation resulting from *J*_{D} and *J*_{R} would be required as well. The diffusion current *J*_{D} will be determined by the hopping transport from one localized state to another perpendicular to the interface of the topological insulator, and can be in principle reduced to nearly zero, should the density of localized states in the bulk of the material vanish. The recombination current *J*_{R} depends on the spin relaxation time of the states at the edge of the material, should they become spin polarized. For the magnetic insulator the spin relaxation of states near the edge is likely negligible, but in the topological insulator the large spin-orbit interaction may provide an efficient pathway towards spin relaxation. Some general observations may be made about the values of *J*_{D} and *J*_{R}. Both should be proportional to *n*_{spins} at the interface, as both the gradient of the density and the recombination current will be proportional to *n*_{spins}. Thus low-moment magnetic insulators, such as YIG, are good choices as they require smaller values of *n*_{spins} to switch.

If the spin relaxation time is $\tau s$, then the rate of decay of polarized spins would be

For a *V*_{switch} = 50 mV, and $\tau s=10\u221212$ s (which is an extremely short spin relaxation time), this loss of spins is an order of magnitude smaller than the accumulation rate in Eq. (5). Thus the principal effect will be to reduce the switching rate by about 10%.

The diagram in Fig. 2(b) cannot be the steady-state configuration in the absence of the dissipative currents *J*_{R} and *J*_{D}, as *J*_{SPIN} will continue to accumulate an unbalanced spin population at the interface. The steady-state configuration can be determined by considering the effective electromotive force on the unbalanced spin population. The increased spin density for out-of-plane spins generates a diffusion current away from the interface, and the decreased spin density for into-the-plane spins produces a diffusion current towards the interface. When this diffusion spin current is sufficiently large to cancel the spin current produced by the spin Hall effect then the steady-state condition has been reached, with a steady-state spin polarization at the interface *n*_{spins,max}. The value of *n*_{spins,max} is also of value when *J*_{D} and *J*_{R} are non-zero; the effective spin current producing the spin polarization is then reduced to *J*_{SPIN}-*J*_{R}, and *n*_{spins,max} is reached when *J*_{SPIN} = *J*_{R} + *J*_{D}.

Surface recombination currents in bipolar devices short the minority carrier currents, reducing minority carrier accumulation, whereas the surface states at the topological insulator edge short the spin currents, reducing spin accumulation. Thus surface states at the interface between the topological insulator and the magnetic insulator, if present (i.e. if the magnetic insulator is topologically trivial), act similar to surface recombination currents in bipolar devices,^{15} but can be passivated using magnetic doping,^{14,16–19} which opens up a gap in the single-particle states. Even if a full gap is not opened up, so long as the states remaining are localized rather than extended, the reduction of the surface accumulation would be minimal, and the effect would be limited to an increase in spin relaxation.

Note that it is also possible for the magnetic insulator to have non-trivial topological character. When this occurs the edge states at the boundaries may not need to be passivated through magnetic doping. For such a configuration, as shown in Fig. 3, the spin current passes through the magnetic insulator and reorients it. Spin current then is sourced and sunk at the left and right edges of the heterostructure, either generated from surface recombination spin currents or, in the absence of spin recombination, generated through the accumulation and depletion of localized polarized states. Other potential sources of dissipation include that associated with precession in the magnetic insulator, but for such an estimate a detailed simulation of the Landau-Lifshitz-Gilbert equations is required.

## IV. CONCLUSION

Here is described a hybrid multiferroic that incorporates a topological insulator and a magnetic insulator. In response to an electric field the spin accumulation at the interface between the two causes the magnetic insulator’s magnetization to reorient. Switching times and energies associated with this hybrid material are very small, and suggest its utility in novel spin-based architectures for memory and logic.

## ACKNOWLEDGMENTS

This work was supported in part by C-SPIN, one of six centers of STARnet, a Semiconductor Research Corporation program, sponsored by MARCO and DARPA.

## REFERENCES

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