Local injection of pure spin current generates electric current vortices

Injection of pure spin current and its subsequent manipulation in spintronic devices¹ has been viewed as a milestone in realization of “spin electronics,” where electron spin would be carrying signal on a par with the charge. In the classic experiments of Johnson and Silsbee,² pure spin current was injected into an electrically disconnected device. Since there was no electric current $j$ entering or leaving the device, it was tacitly assumed that $j$ should also be zero everywhere inside it. Johnson and Silsbee found that spin current injection nevertheless generates a voltage V between the ferromagnetic (F) and normal (N) elements (Fig. 1). In a diffusive transport regime, where electron momentum relaxes much faster than its spin, such a voltage can be described in terms of the “Valet-Fert model,”^3–7 as outlined below. Johnson and Silsbee² predicted V to be proportional to the spin accumulation at the F/N boundary and independent of the measuring probe position—as long as the electric current was absent and the F-probe was placed at a point where spin accumulation has relaxed to zero (i.e., further than several spin relaxation lengths λ_s away from the F/N boundary).

This statement is true and transparent in the case of a narrow F/N contact (Fig. 1(a)). However, if the contact is wide enough for the spin accumulation to vary substantially along the F/N interface Fig. 1(b)), then it is not clear which accumulation value should be used in the Johnson-Silsbee formula. This practical issue was investigated, e.g., in Ref. 8, where it was found that V does depend on the probe position, even if the thickness t_F of the F layer exceeds λ_s (Fig. 1).

On the one hand, the emergence of a non-uniform voltage in a system with non-uniform spin accumulation appears to be natural. On the other hand, a potential gradient in the F region with vanishing non-equilibrium spin accumulation can only mean the presence of electric current. How does this correspond to the absence of $j$ in the Johnson-Silsbee picture? Here, we show that even if electric currents do not enter the device,⁹ they are still induced inside it. These internal currents circulate along the closed loops that cross the F/N interface and are maintained by the external source that produces the pure-spin injection. We demonstrate the existence of electric current loops and study their influence on the voltage and spin accumulation distributions. The current loops are akin to the Eddy currents generated by oscillating magnetic fields, except that the present phenomenon occurs in a non-equilibrium steady state. We show that such electric vortices are not limited to spin transport and shall be expected whenever electric current is coupled to another diffusive current by linear relationships with the Onsager cross-coefficients.

We will consider setups with collinear magnetization. As detailed in Ref. 7, in the Valet-Fert model, carrier distributions for spin $α=↑,↓$ are characterized by different electrochemical potentials $μα$⁠. With two conductivities $σ↑,↓$ being different in a ferromagnet, the currents¹² carried by the two spin populations are given by $jα=−(σα/e2)∇μα$⁠. The conservation of electric current (⁠$∂tn+divj=0$⁠) and spontaneous relaxation of spin (⁠$∂tns+divjs=−ns/τs$⁠) yield steady-state equations

where $j=j↑+j↓$ and $js=j↑−j↓$ are electric and spin currents, respectively, n and n_s are the non-equilibrium charge density and spin accumulation, respectively, and τ_s is the spin relaxation time. The average potential $μ=(μ↑+μ↓)/2$ is the quantity measured by an ideal voltmeter, while the spin potential $μs=μ↑−μ↓$ characterizes the non-equilibrium spin accumulation. The currents $j$ and $js$ can be written as

with $σ=σ↑+σ↓$ and the polarization $p=(σ↑−σ↓)/σ$⁠. Note that in Eqs. (2) and (3), spin and charge are coupled by $p≠0$⁠. We will assume σ, p and τ_s to be piecewise constant, undergoing jumps at interfaces between different materials. Within each uniform region, Eqs. (1)–(3) yield

with λ_s being the spin relaxation length.⁷ The interfaces will be assumed transparent (continuity of μ and μ^s) and spin-inactive (continuity of the $j⊥s$ component, normal to the boundary).

First, we show that non-uniform spin accumulation near the F/N boundary inevitably produces electric current, even if the latter is not injected from outside. Eq. (2) implies

Now, p evolves from p = 0 in the normal metal to $p≠0$ in the ferromagnet, thus $∇p≠0$⁠. If $∇μs$ has a component perpendicular to $∇p$⁠, i.e., if spin accumulation varies along the interface between materials with different polarizations p, then $curl(e2j/σ)≠0$ and thus $j≠0$⁠. Since current cannot cross an outer boundary of an electrically disconnected device, it circulates inside forming closed loops. These loops cannot be confined to any of the uniform parts of the device and thus cross the boundaries between them. Indeed, the presence of a current loop in a region of constant σ and p would mean $curl j≠0$⁠, which is impossible due to Eq. (2): in uniform regions, $j$ is the gradient of a function. Thus, the current lines must form a vortex with the core somewhere at the F/N boundary.

Eq. (2) can also be interpreted as follows. Electric current is driven by two forces: one is the conventional electrochemical potential gradient and the other is an effective electromotive force (EMF) $E=−p(σ/e2)∇μs/2$ due to the non-equilibrium spin accumulation μ^s.⁷ Both μ^s and its gradient decay away from the spin current injection point; thus, an EMF region appears around it (Fig. 2(a)), producing the current loops.

The generation of electric current vortices is not limited to spintronics. Consider coupled electric and heat transport

where j_e is the electric current, $q$ is the heat flux, $ϕ$ is the electric potential, κ is the thermal conductivity, and S and Π are Seebeck and Peltier coefficients, respectively. The Eqs. (6) imply that a temperature gradient satisfying $∇S×∇T≠0$ produces current loops at the interface between materials with different Seebeck coefficients – similarly to how the Eq. (5) follows from the Eq. (2).

Now, we choose a symmetric device in Fig. 2(b) as a simple setting to demonstrate the loop current generation in a specific geometry. As the thickness t_N of the normal metal film decreases, we expect the spin accumulation to become ever more uniform across the N-film. Then, the solution for a realistic device with pure spin current injected from the side as in Fig. 2(a) will be the same as for injection from below, as in Fig. 2(b). In the latter case, the electric current bursts into the ferromagnet like water from a fountain and flows back through the normal film: we will call it a “spin fountain” device.

We place the origin at the spin injection point and direct the axes as shown in Fig. 1. All quantities are assumed to be z-independent. For brevity, we introduce notations $λn ≡ λs(N), λf ≡ λs(F)$⁠, $σN ≡ σ(N), σF ≡ σ(F)$⁠, and $p ≡ p (F)$⁠. For the reasons explained above, we assume $tN/λn≪1$⁠, while $tF/λf$ can take any value. We switch to a “mixed potential” $M=μ+pμs/2$⁠, whereby the bulk equations decouple

The price to pay for this simplification is the change of the boundary conditions. While μ^s remains continuous, M experiences a jump $MF−MN=(p/2)μs$ at the F/N interface. Expressions for the currents now read

In a thin normal film, we approximate $MN(x,y), μNs(x,y)$ by their averages over the film thickness $MN(x)$ and $μNs(x)$⁠, for which we derive effective equations

where $R=σN/σF, λmix2(p)=(1−p2)λn2/R$⁠, and s is a rescaled total injected spin current.

In the ferromagnet, we seek the solutions in the form

With $q2(k)=λf−2+k2$⁠, $μFs$ and M_F automatically satisfy Eq. (7) and the boundary conditions $j⊥=0, j⊥s=0$ at the top surface y = t_F of the ferromagnet.

In the normal film, $μNs(x)=μFs(x,0)$ and M_N is found from the boundary condition on its jump

Substituting the Fourier expansions into Eq. (10), we find the coefficients

with

As per Eq. (13), at p = 0, the electric current vanishes.

The potentials are computed from Eqs. (11) by numerical integration. The magnitude of electric current is proportional to the injected spin current s. To compare with experiment, we rescale s so that spin accumulation $μ0s$ at the injection point has the largest feasible value, estimated^13,14 as $μ0s∼1$ mV. The electric current can be found from (8) using the parameters, typical of a Py/Cu device:⁸ $λn=350$ nm, $λf=4.3$ nm, t_N = 2 nm, p = 0.7, R = 6.6, and $σ(Cu)=48×106$ (Ω m)⁻¹.

A typical contour plot of M(x, y) for $tF→∞$ is shown in Figs. 3(a) and 3(b). The electric current is normal to the M = const lines (8) and forms a fountain-like pattern sketched in Fig. 2(b). Figs. 3(c) and 3(d) give the current components at the F/N interface.

Induced electric current significantly alters the voltage measured in a Johnson-Silsbee experiment and makes it dependent on the position of the voltmeter probe. Let us assume that the F-probe is attached at the top of the F-layer, right above the injection point. For an extended F-electrode, the easiest way to attach an N-probe is at $xN→∞$ (Fig. 4(b)). Then the measured voltage $V1(tF)=μ(0,tF)−μ(xN,0)→μ(0,tF)$⁠. The plot of $V1(tF)$ is given in Fig. 4(a). If the N-probe is attached close to the spin-injection point, the voltage changes to V₂. Both V₁ and V₂ significantly differ from the voltage that would develop in the absence of electric current (solid line in Fig. 4(a)). Voltage dependencies on the F-electrode thickness, even for $tF≫λf$ is another visible change. The j = 0 situation emerges either in a narrow F-electrode, or, more generally, in devices where spin is injected uniformly across the F/N interface: according to Eq. (5), when $∇μs$ is normal to the boundary, the reason for current generation vanishes together with $curl j$⁠. Uniform spin injection generates voltage that approaches the Johnson and Silsbee result $VJS=(p/2)μ0s=0.35$ mV for $y≫λf$⁠.

Spin current tends to decay exponentially with the distance from the injection point. For example, for a non-magnetic top layer, in the present case of $RtN≫λf$⁠, we find $ak≈1/(f(0)+λn2k2)$⁠. Hence, along the interface, the spin potential falls off as $μs(x,0)≈μ0s exp(−x/λ‖)$⁠. The decay length $λ‖=λn/f(0)$ is bound as per $λf<λ‖<λn$⁠. Physically, this means that spins would propagate through a detached normal metal film up to a length of about λ_n, but the spin current leakage into the overlayer shortens their reach.

For a magnetic top layer (⁠$p≠0$⁠), a log plot of $μs(x,0)$ is shown in Fig. 5: the $μs(x,0)$ decays exponentially. However, the decay length crosses over from $λ‖$ at $x≪tF$ to a longer length λ_c at $x≫tF$⁠. In the thick-film limit, $RtN/tF≪1$⁠, we find

For the parameters above, this yields $λc≈84$ nm against the numerically found $λc≈83$ nm.

The λ_c grows with t_F and, at $tF→∞$⁠, the $μs(x,0)$ decays non-exponentially for $x≫λ‖$⁠. In this limit, $tanh(tFk)→sgn(k)$ is non-analytic at k = 0 and the expressions for F(k) and H(k) read

The singularity shows itself as a $|k|3$ term in the expansion of a_k. Using the stationary phase method, we find an asymptotic expression

with $C=s(p2/(1−p2))(6R2/πf2(0))λmix2tN$⁠. Thus, for infinite t_F, the spin accumulation ultimately decays as a power-law, i.e., very slowly (the blue line in Fig. 5).

Our findings mean that a ferromagnetic overlayer makes the injected spin current propagate further along the normal film. This conclusion sounds pronouncedly counter-intuitive: after all, ferromagnetic layer is known to be a spin sink, so one would expect that it could only lower the spin propagation length. The seeming paradox is resolved as follows. As we know, the electric current loops cross the F/N boundary. Upon such crossing, a non-equilibrium spin density is inevitably produced,³ so μ^s cannot decay independently of $j$⁠. Ultimately, the conservation of $j$⁠, expressed by Eq. (1), causes a long-range propagation of both charge and spin. The current-assisted propagation of spin also explains the role of the F-layer thickness. The long-range pattern of $j$ is limited by the outer boundaries of the device. Finite t_F is equivalent to “covering the fountain by a lid,” deflecting $j$ down to the normal film within a distance of the order of t_F. Beyond this distance, the power-law decay of spin accumulation reverts to the exponential form.

In conclusion, we have shown that the gradient of spin accumulation along an F/N interface produces the closed electric current loops. The first consequence of this is a significant reduction of the Johnson-Silsbee voltage, which means that the interpretation of some non-local resistance experiments with wide F-electrodes may need to be revisited. For example, in Refs. 15–18, the current polarization of permalloy was deduced to be $p ≲ 0.3$⁠, which is significantly smaller than $p≈0.7$ inferred from the giant magnetoresistance measurements.^8,19–21 Such a seeming reduction of p may arise due to loop currents; this can be verified by voltage measurements on a series of devices with varying thickness or width of the F-electrodes.

Alternatively, the loop currents will manifest themselves by non-zero voltage drop along the normal film. In the absence of electric current, such a voltage must vanish—but not if $j≠0$⁠. In particular, if one probe is connected at $xN→∞$ and another at x_N = 0, the voltmeter will read off the voltage difference $V2−V1$⁠, as shown in Fig. 4.

Another consequence of loop currents is the long-range propagation of spin accumulation along the F/N interface. This effect can be measured by an additional F-electrode positioned downstream of the wide F-electrode. The non-local voltage on the former will reflect the enhanced propagation of spins brought about by the latter. Notice that the signal to be expected in such an experiment is small: as shown in Fig. 5, spin accumulation drops by the orders of magnitude before the long-range propagation regime becomes sufficiently pronounced.

More generally, electric current vortices at the interface between two materials shall be expected whenever electric current is coupled to another driven diffusive current by linear relationships with the material-dependent Onsager cross-coefficients. For example, coupling with heat flow may induce electric current loops in the spin-caloritronic devices with a temperature gradient along the interface between detector ferromagnet and a normal wire.^22–24 

Ya.B. was supported by the NSF Grant No. DMR-0847159. He is grateful to the Laboratoire de Physique Théorique, Toulouse, for the hospitality and to CNRS for funding the visits.