Electric Control of Spin Currents and Spin-Wave Logic
Spin waves in insulating magnets are ideal carriers for spin currents with low energy dissipation. An electric field can modify the dispersion of spin waves, by directly affecting, via spin-orbit coupling, the electrons that mediate the interaction between magnetic ions. Our microscopic calculations based on the super-exchange model indicate that this effect of the electric field is sufficiently large to be used to effectively control spin currents. We apply these findings to the design of a spin-wave interferometric device, which acts as a logic inverter and can be used as a building block for room-temperature, low-dissipation logic circuits.
One of the major challenges of contemporary electronics is to reduce dissipation as the size of devices shrinks to the nanometric scale. In this context, spin-wave spintronics, so called magnonics, with insulating magnets offers interesting possibilities (1); (2). While in metals and semiconductors the spin current is carried by mobile conduction electrons/holes, which inevitably dissipate energy as they move, in a magnetic insulator, such as (YIG), the spin current is carried by a collective motion of magnetic moments – a spin wave – with no charge displaced. The spin current propagating in these insulating material is thus totally free of energy dissipation from Joule heating, and almost free of dissipation from other sources (e.g. electron-magnon scattering): the coherence length can be as large as several centimeters (2). For these reasons, magnetic insulators have attracted considerable attention in recent theoretical (3); (4); (5) and experimental (6); (2) work. For example, Kajiwara et al. (2), have demonstrated injection and extraction of spin waves into and out of a YIG wave guide (8). Kostylev et al. (9), have designed an ingenious scheme of spin-wave logic, based on the interference between spin waves traveling along different arms of a Mach-Zehnder interferometer (a schematic illustration of a Mach-Zehnder spin wave interferometer is shown in Fig. 1).
A crucial element of magnonics (1) is the phase shifter – a device that changes the phase of propagating spin waves. Several mechanisms have been proposed in the past to implement controlled phase shifts on spin waves. The simplest and most direct, is the application of a magnetic field, which shifts the dispersion (10), thus changing the wave vector at constant frequency (9). More sophisticated mechanisms exploited the Berry phase accumulated by spin waves that propagate on a non-collinear magnetic texture (4); (11). In a parallel development, Cao et al. (12) studied the effect on spin waves of an electric field-induced Aharonov-Casher (AC) phase (13). More recently, the influence of electric fields on spin waves has been studied both theoretically (14) and experimentally (15) and a strong shift of spin-wave dispersion induced by an electric field has been reported (15).
In this Letter we directly tackle the problem of controlling the phase of a spin wave (and hence the spin current) by means of an electric field. We will show that the electric field-induced AC phase has important implications for spin wave interferometry, since the effect is much larger than that initially predicted in Ref. (12). Our analysis starts at the microscopic level, with a very simple super-exchange model (16) for the magnetic interaction between two neighboring magnetic ions (e.g. Fe in YIG) in a magnetic insulator. The model is depicted in Fig. 2. There are no itinerant electrons in an insulator, but the virtual hopping of electrons between the -orbitals on the magnetic ions (Fe) and the -orbitals on the the ligand () is sufficient to establish an antiferro-magnetic interaction of the Heisenberg type,
which is responsible for the occurrence of magnetic order in the material. Notice that in this model the physical and orbitals are replaced by doubly degenerate orbitals, which are eigenstates of the -component of the electron spin. Spin-orbit coupling effects are completely neglected up to this point.
We now augment the usual super-exchange model by the inclusion of a spin-orbit (SO) interaction of the form
where is a characteristic length scale that controls the strength of the SO interaction, is the elementary charge and is the Pauli matrix. For electrons in vacuum is the Compton wavelength , but we will see below that, in any realistic model of magnetic insulators, the value of is orders of magnitudes larger (of the order of ). Although there are no itinerant electrons carrying a finite average momentum , we will show below that the effect of the SO interaction on the phases of the virtual hopping translates, at the macroscopic level, into the appearance of a Dzyaloshinskii-Moriya (DM) interaction (17) between the magnetic moments of the ions:
where the vector is given by
with being the distance between the magnetic ions, the unit vector in the direction connecting the ions, and (in vacuum and , is the bare electron mass). As a result of this interaction, the spin waves – obtained by solving the appropriate ferromagnetic Hamiltonian for an infinite chain of identical blocks of magnetic ions (see discussion below) – have their wave vector shifted by , where is the direction of the equilibrium magnetization. This, in principle, gives us a way to control the phase of the spin wave by an electric field. In practice, the feasibility of the proposal depends critically on the strength of the spin-orbit coupling . If we used the value of in vacuum (as was done in Ref. (12)), the effect would be extremely weak, and probably unobservable with realistic electric fields.
The reason why turns out to be much larger than is that the physical -orbitals in the magnetic insulator (as opposed to the model orbitals we have dealt with so far) have strong intrinsic spin-orbit coupling built in. For example, in the model adopted by Katsura et al. (7), the doubly degenerate orbitals of the super-exchange model are actually spin-orbit-entangled states of the form and . In order to determine the value of (or, better, ) in our model, we observe that the DM interaction Eqs. (3,4), and indeed the spin-orbit interaction (2) itself, can be viewed as the interaction of the electric field with an effective electric dipole, i.e. , where the electric dipole is
The effective electric dipole arises from the hybridization of orbitals centered at different atoms. For example, in the model of Ref. (7) the exact single-particle eigenstates are combinations of orbitals on the magnetic ions and orbitals on the oxygen due to the hopping of the electron: these states carry an electric dipole moment where is the hopping coefficient and with being the Bohr radius and () being the atomic number of O (M). Comparing this to our Eq. (5) we arrive at an unambiguous identification of (and hence ) within our model:
Taking YIG as an example (18); (19), with eV, and , we get and . This is indeed several orders of magnitude larger than and opens the way to practical schemes of electric control of the phase of spin waves.
In the remaining part of this paper we supply more theoretical detail on the calculations supporting the above analysis, then work out the dispersion of spin waves in the presence of the electric field, and apply the results of these calculations to the design of a spin-wave interferometer (see Fig.1), to be used as building block of spin-wave logic circuits.
Microscopic analysis – The insulator we are interested in is YIG, whose magnetic order mainly arises from the super-exchange interaction between in octahedral (a) sites and tetrahedral (d) sites. The fact that the numbers of (a) and (d) sites per unit cell are different makes YIG a ferrimagnet. However, the long wave-length spin waves, whose energy is less than about K, can be understood with an effective ferromagnetic exchange coupling between “block spins” , one per unit cell (19). The question is how an electric field affects this block ferromagnet. We start from the super-exchange model shown in Fig. 2, which can be described by the following Hamiltonian
where () is the creation (annihilation) operator of ligand electrons, which can hop forth and back only between oxygen ligand and the metal ions, and are the orbital energies of a metal ion and the oxygen ligand, respectively. The large repulsion energy ( eV) between two electrons on the same metal ion allows for a maximum occupancy of two electrons per ion (the repulsion between the electrons in the oxygen ligand is negligible in comparison).
The fact that () is much smaller that allows us to use perturbation theory. Keeping up to the fourth order of yields the effective interaction between the spins on the magnetic ions:
where corresponds to the energy difference between the and orbitals in the paper by Katsura et al. (7). Setting and dropping the constant term, we obtain the Heisenberg interaction Eq.(1). A positive implies that the interaction between neighboring magnetic ions is antiferromagnetic. However, this antiferromagnetic interaction gives rise to a ferromagnetic interaction between “block spins” in YIG, due to the unequal magnitudes of the anti-parallel magnetic moments in each block.
Let us now include the spin-orbit interaction from Eq. (2). It is easy to see that the inclusion of this interaction is equivalent to the inclusion of a spin-dependent vector potential , which in turn modifies the hopping term by a spin-dependent phase factor, that is
where , provided that the external electric field, the motion of the electron, and the electron spin () are perpendicular to each other. Notice that the phase is proportional to the distance between neighboring sites and independent of the direction of the localized moments: one can therefore switch to the “block spins” description by simply reinterpreting as the distance between neighboring blocks. The resulting spin Hamiltonian takes the form
where , is the effective exchange coupling for the spin blocks and is in the direction perpendicular to and . In addition to the normal Heisenberg term we now also have a DM term (17), whose strength is linear in . An electric-field induced anisotropy is also present, but is an effect of order and has therefore been neglected for weak electric field.
In spite of the presence of the noncollinear DM term, the ferromagnetic configuration is still the ground state of (10). To show this, we make , where is small deviation perpendicular to . Then the variation of the DM term up to the second order of is
where is defined by Eq.(4) with replaced by . Since , we see that , which means up to the first order of . Hence, the ground state is still ferromagnetic. However, the DM term will definitely modify the spin-wave frequency, which involves a correction to the ground state energy at the second order in . Further, we can clearly see that it is only the component of the parallel to (i.e. to the direction of the equilibrium magnetization) that plays a role in the modification.
Spin wave dispersion – We now proceed to solve the dispersion of the spin waves in the presence of the DM interaction derived above. We consider the ring geometry illustrated in Fig.1: the electric field perpendicular to the ring produces a DM vector directed along the -axis. This will affect the dispersion of spin waves if and only if the equilibrium magnetization has a non-vanishing component along the axis. In a flat ring, such as the one shown in Fig.1, the shape anisotropy(21) where is the unit vector along the ring – outweighs other forms of anisotropy, causing the equilibrium magnetization to lie along the ring, in which case the electric field has no influence. As a result, a magnetic field along the axis (Zeeman coupling ) is necessary for us to observe the impact of the DM term on spin waves propagating in the ring. Now, however, the orientation of the equilibrium magnetization is no longer constant in absolute space (even though it is constant relative to the ring). This causes an additional geometric phase () to appear, as shown in Ref. (4), where is the radius of the ring. Putting everything together, i.e., DM interaction, geometric phase, Zeeman coupling and shape anisotropy, we arrive at the following equation of motion:
The large magnitude of the “block spin” of YIG (S=14.3) allows us to use the semiclassical spin-wave approach to get the dispersion relation:
where , , and , which is determined by minimizing the total Hamiltonian in the limit of .
As shown in Fig. 3, one can tune the dispersion by adjusting the electric and the magnetic fields. Just as a magnetic field shifts the spin wave dispersion vertically by increasing or decreasing the frequency at fixed , the electric field shifts the dispersion horizontally by increasing or decreasing the wave vector at fixed frequency.
Spin wave interferometer – Now we are ready to design our spin-wave interferometric device. An insulating ring encircles a metal electrode to which a voltage can be applied. The radial electric field acting upon the electrons in the ring is .
In Fig. 3 (b) we plot the transmission of a spin wave sent through this Mach-Zender interferometer, as a function of and B. The effect of is to change the equilibrium orientation of the magnetization. The white regions in the figure are regions of constructive interference, separated by regions of destructive interference. We see that very modest changes of potentials and magnetic fields, of the order of 1 V and 0.01 T respectively, switch the response of the interferometer from high to low. It is then clear how the device can be used as a logic inverter: the logic input being the voltage on the central electrode, and the logic output the intensity of the spin wave, as measured by an inductive coupler. Advantages of this design are that it would operate at room temperature and GHz frequencies, with very little dissipation, and can be made small by using exchange spin waves – the only type we are really considering here, since magnetostatic spin waves have much longer wavelengths and are hardly affected by the AC phase. Once a logic inverter is available, we can follow Kostylev et al. (9) in constructing more complicated architectures, which implement the NAND, the NOR, and all of classical logic.
It is worth noting that a traveling spin wave is itself a source of electric field: where is the velocity of the spin wave, is the amplitude of the magnetization oscillation and is the vacuum permeability. In our device, the resulting electric field is of the order of 1 V/m, which is negligible in comparison to the control field V/m. Even smaller is the electric field associated with mesoscopic equilibrium spin currents in the ring ( V/m). (3)
In conclusion, we have proposed an energy-efficient way to control the spin current propagating in an insulating magnet by means of an electric field. This possibility arises from the strong coupling that exists between the electric field and the spins of the electrons that mediate the interaction between magnetic ions. The strength of this coupling has been theoretically estimated from microscopic parameters, such as electron hopping coefficient, distance between neighboring magnetic sites, etc. as shown in Eq.(6). Or, it could be indirectly determined from measurements of physical effects that are sensitive to it, e.g. the spin wave spin Hall effect proposed by Meier and Loss (5). Finally, we have applied our theory to an insulating magnetic ring inteferometer, which can be used to implement a voltage-controlled spin-wave-based NOT gate.
We acknowledge support from ARO Grant No. W911NF-08-1-0317 and thank Michael Flatté for helpful discussions.
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