Dynamic spin-charge coupling: spin Hall magnetoresistance in non-magnetic conductors

TL;DR

This paper explores how spin and charge currents interact dynamically in non-magnetic conductors, enabling electrical detection of electron spin resonance through impedance measurements influenced by spin relaxation and diffusion.

Contribution

It demonstrates that spin dynamics can be directly observed via electrical impedance, revealing a new method to study spin phenomena in non-magnetic materials.

Findings

Electrical impedance reflects spin dynamics

Electron spin resonance can be detected electrically

Spin relaxation and diffusion define key frequency scales

Abstract

The dynamic coupling between spin and charge currents in non-magnetic conductors is considered. As a consequence of this coupling, the spin dynamics is directly reflected in the electrical impedance of the sample, with a relevant frequency scale defined by spin relaxation and spin diffusion. This allows the observation of the electron spin resonance by purely electrical measurements.

Equations

$$\mathbf{q}=\mu n\mathbf{E}+\gamma D\mathrm{rot}\mathbf{P},$$

$$q_{ij}=-D\frac{\partial P_j}{\partial x_i}+\gamma \mu n{ϵ}_{ijk}{E}_k,$$

$$\frac{\partial P_j}{\partial t}+\frac{\partial q_{ij}}{\partial x_i}+(\Omega \times \mathbf{P}{)}_j+\frac{P_j}{{\tau }_s}=0,$$

$$q_{yy}=-D\frac{d{P}_y}{dy},q_{yz}=-D\frac{d{P}_z}{dy}+\gamma \mu nE\cos \omega t.$$

$$\begin{array}[]{l}\displaystyle D\frac{d^{2}P_{y}}{dy^{2}}=\Big{(}-i\omega+\frac{1}{\tau_{s}}\Big{)}P_{y}+\varOmega P_{z}\>,\\ \\ \displaystyle D\frac{d^{2}P_{z}}{dy^{2}}=\Big{(}-i\omega+\frac{1}{\tau_{s}}\Big{)}P_{z}-\varOmega P_{y}\>,\end{array}$$

$${\frac{d{P}_y}{dy}|}_{y=\pm W/2}=0,{\frac{d{P}_z}{dy}|}_{y=\pm W/2}=\gamma \frac{\mu nE}{D}.$$

$$\begin{array}[]{c}\displaystyle P_{y}(y)=-i\gamma\frac{\mu nE}{2D}\,[\,F_{+}(y)-F_{-}(y)\,],\\ \\ \displaystyle P_{z}(y)=\gamma\frac{\mu nE}{2D}\,[\,F_{+}(y)+F_{-}(y)\,],\end{array}$$

$$F_{\pm }(y)=\frac{\sinh ({\lambda }_{\pm }y)}{{\lambda }_{\pm }\cosh ({\lambda }_{\pm }W/2)},$$

$${\lambda }_{\pm }=\frac{\sqrt{1+i(-\omega \pm \Omega ){\tau }_s}}{L_s},$$

$$\Delta j(y)=\gamma^{2}\frac{e\mu nE}{2}\,\Big{[}\,\frac{dF_{+}}{dy}+\frac{dF_{-}}{dy}\,\Big{]}\>.$$

$$\Delta I={\int }_{-W/2}^{W/2}\Delta j(y)dy.$$

$$\frac{\Delta Z}{Z_{0}}=-\gamma^{2}\Big{[}\frac{\tanh(\lambda_{+}W/2)}{\lambda_{+}W}+\frac{\tanh(\lambda_{-}W/2)}{\lambda_{-}W}\Big{]}\>,$$

$$\mathrm{Re}\Delta Z(\omega ,0)=\Delta Z(0,\Omega =-\omega ).$$

$$\frac{\Delta Z}{Z_{0}}=-\gamma^{2}\Big{(}\,1-\frac{1-i\omega\tau_{s}}{24}\frac{W^{2}}{L_{s}^{2}}\,\Big{)}\>.$$

$$\frac{\Delta Z}{Z_0}=-{\gamma }^2\frac{L_s}{W}\sum _{\pm }\frac{1}{\sqrt{1+i(-\omega \pm \Omega ){\tau }_s}},$$

$$\varrho (\omega ,\Omega )=\frac{\mathrm{Re}\Delta Z(\omega ,\Omega )}{\mathrm{Re}\Delta Z(\omega ,\Omega =\omega )}.$$

Full text

1. Introduction. It was predicted nearly half a century ago DP1 ; DP2 that spin-orbit interaction results in the interconnection between electrical and spin currents: an electrical current produces a transverse spin current and vice versa. This leads respectively to the direct and inverse spin Hall effects. Following the proposal in Ref. Averkiev , the inverse spin Hall effect was observed experimentally by Bakun et al. Bakun in 1984, without causing much excitement at that time.

Twenty years later, after the first experimental observations of the (direct) spin Hall effect Kato ; Wunderlich this topic has become a subject of considerable interest with thousands of publications, see for example a review in Ref. DKh .

Because of the interconnection between the spin and charge currents, anything that happens with spins will influence the charge current, i.e. result in corresponding changes of the electrical resistance, which can be measured with a very high precision. An example of this link is provided by the spin Hall magnetoresistance Dyakonov , the reason for which is the depolarization of spins accumulated at the sample boundaries by a transverse magnetic field and the resulting decrease of the driving electric current (for a given voltage) rem . This effect was experimentally demonstrated in platinum by Vélez et al. Velez .

Earlier, a similar effect was discovered and studied by Nakayama et al. Nakayama in layered structures ferromagnet-normal metal. The magnetization in the ferromagnet can be rotated by an applied magnetic field which results in a change in the normal metal resistivity.

In recent years, the ac spin Hall effect in ferromagnet-normal metal structures has also been studied both experimentally Wei ; Hyde ; Weiler ; Hahn and theoretically Chiba ; Chen ; Ulloa . The precession of the magnetization in a ferromagnet leads to a time-varying injection of spin into the normal metal. Due to the inverse spin Hall effect, the resulting spin current in the normal metal generates the ac electric current.

In particular, the observed ac voltage resonantly depends on the Larmor frequency in the ferromagnet and the frequency of the external ac magnetic field, which excites the precession of magnetization. In this way, with the aid of the spin Hall effect in a normal metal, the ferromagnetic resonance was observed by electric measurements.

While these studies are quite important for achieving the ultimate goal of storing and manipulating information by the use of spin Hall effect for switching magnetic domains in magnetics (see the reviews Hoffmann1 ; Hoffmann2 ), the physics of the layered magnetic structures is quite complicated, and this makes the exact theory and the quantitative analysis of experimental data rather difficult.

Here, following Ref. Dyakonov , we develop a much more simple theory of ac electron magnetotransport controlled by the direct and inverse spin Hall effects in non-magnetic materials, semiconductors or metals. The theory is based on the phenomenological transport equations DP1 ; DP2 ; Dyakonov ; DKh describing the interconnection between spin and charge currents. We show that spin resonance in non-magnetic materials can be observed by purely transport measurements.

2. Transport equations. Consider a conductor in an external ac electric field $\bm{E}(t)\sim\cos(\omega t)$ and a magnetic field $\bm{B}$, see Fig. 1. We assume that the ac frequency $\omega$ is much lower than the cyclotron frequency $\omega_{c}$, and that $\omega\tau\ll 1$, where $\tau$ is the momentum relaxation time. However the spin Larmor frequency $\varOmega$ and the spin relaxation time $\tau_{s}\gg\tau$ are such that $\omega\sim\varOmega\sim 1/\tau_{s}$.

In this frequency range, the basic phenomenological equations for the electron flow density $\bm{q}=\bm{j}/e$, the spin current density tensor $q_{ij}$, and the spin density vector $\bm{P}$ are rem1 ; DKh :

[TABLE]

[TABLE]

[TABLE]

where $n$ is the electron density, $\mu$ is the electron mobility, $D$ is the diffusion coefficient, $\gamma\ll 1$ is the dimensionless parameter proportional to the strength of the spin-orbit interaction and describing the interconnection between the particle and the spin currents, $\epsilon_{ijk}$ is the unit antisymmetric tensor, the vector $\bm{\varOmega}$ is directed along the applied magnetic field, $\varOmega$ being the Larmor frequency for electron spins, and $\tau_{s}$ is the spin relaxation time.

The first term in Eq. (1) is the usual Drude contribution, while the second term expresses the interconnection (caused by spin-orbit interaction) between particle current and the spin current caused by the inhomogeneity of the spin density.

Eq. (2) describes two contributions to the spin current density $q_{ij}$: the first one is due to diffusion of spin-polarized electrons and the second one is due to the transformation of particle current into spin current.

Eq. (3) is the continuity equation for the spin density, taking into account spin diffusion, rotation of spin in magnetic field, and spin relaxation.

In the geometry of Fig. 1, both the particle and the spin flows depend on the $y$ coordinate only. We consider that the ac electric field, as well as the magnetic field, are directed along the x direction.

As it follows from Eq. (2), the two nonzero components of the spin current density tensor are:

[TABLE]

The absence of spin currents across the sample boudaries is described by the boundary conditions: $q_{yy}=0$ and $q_{yz}=0$ at $y=\pm W/2$.

3. Solution of the transport equations. The nonzero components of the spin density vector $\bm{P}$ can be written in a complex form: $P_{y}(y,t)=[P_{y}(y)e^{-i\omega t}+c.c.]/2$, and similarly for $P_{z}(y,t)$. Then, from Eqs. (3) and (4) we obtain a system of coupled equations for $P_{y}(y)$ and $P_{z}(y)$:

[TABLE]

with the boundary conditions:

[TABLE]

The solution of Eqs. (5) with the boundary conditions (6) yields the spin density profile:

[TABLE]

where

[TABLE]

[TABLE]

and $L_{s}=\sqrt{D\tau_{s}}$ is the spin diffusion length.

Thus for narrow samples, $|\lambda_{\pm}|W\ll 1$, the spin density $\bm{P}$ depends linearly on the coordinate $y$, while for wide samples, $|\lambda_{\pm}|W\gg 1$, the spin density $\bm{P}$ is concentrated near the sample edges. In the last case, spin density exhibits spin resonance at $\omega=\pm\varOmega$ provided that $\omega\tau_{s}\gtrsim 1$. The signs $\pm$ correspond to the contribution to $\bm{P}$ from the components of $\bm{E}(t)$ with the right and the left circular polarizations, respectively.

The current density $j_{x}=eq_{x}$ can now be calculated using Eq. (1): $j_{x}=e\mu nE\cos\omega t+\Delta j(y,t)$, where the first term is the normal Drude contribution (in the assumed limit $\omega\tau\ll 1)$, while the second term is a correction which is of second order in the spin-orbit interaction: $\Delta j(y,t)=[\Delta j(y)e^{-i\omega t}+c.c.]/2$, where

[TABLE]

For wide samples, $|\lambda_{\pm}|W\gg 1$, this correction to the ac current density, like the spin density, is concentrated near the sample edges.

The spin-orbit correction $\Delta I$ to the main Drude part, $I_{0}=e\mu nEW$, of the total current can be calculated from Eq. (10):

[TABLE]

Thus we obtain the final result for the correction to the sample impedance, $\Delta Z=\Delta Z(\omega,\varOmega)$, caused by spin-orbit interaction:

[TABLE]

where $Z_{0}=L/(e\mu nW)$ and $\lambda_{\pm}$ are defined by Eq. (9).

4. Results and discussion. We now analyze our results given by Eq. (12) for some special cases.

*(i) Low frequencies: $\omega\tau_{s}\ll 1$. * The results coincide with those of Ref. Dyakonov for the dc spin Hall magnetoresistance.

(ii) Zero magnetic field, $B=0$. In Fig. 2 we plot the ratio $\varrho(\omega)=\mathrm{Re}\Delta Z(\omega,0)/\Delta Z_{0}$ of the real part of the spin-orbit correction $\Delta Z$ [Eq. (12)] in the absence of magnetic field to its value at zero frequency, $\Delta Z_{0}=\Delta Z(0,0)$. It is seen that $\varrho(\omega)$ has a quasi-universal behavior as a function of the parameter $\omega\tau^{*}$. Here $1/\tau^{*}=1/\tau_{s}+1/\tau_{d}$ is the effective total relaxation rate which is the sum of the bulk spin relaxation rate $1/\tau_{s}$ and the diffusion rate for space inhomogeneity in the spin distribution $1/\tau_{s}=4D/W^{2}$.

Thus there are the two relaxation processes: the bulk spin relaxation with the rate $1/\tau_{s}$ and the decay of spin inhomogeniety due to diffusion of spin-polarized electrons. The quasi-universal results in Fig. 2 are quite similar to those obtained in Ref. Dyakonov for the dc spin Hall magnetoresistance.

Indeed, from Eq. (12) one can obtain the relation between the corrections to the ac impedance in zero magnetic field and to the dc magnetoresistance:

[TABLE]

Here $\Delta Z(0,\varOmega)$ is, in fact, the spin Hall magnetoresistance calculated in Ref. Dyakonov and denoted therein as $\Delta R(\varOmega)$.

*(iii) High frequencies: $\omega\tau_{s}\gg 1$. * For narrow samples ($W\ll L_{s}/\sqrt{\omega\tau_{s}}$) the correction $\Delta Z$ depends neither on frequency, nor on magnetic field in the main order by the parameter $W/L_{s}\ll 1$ (at not too high magnetic fields when $\varOmega\lesssim\omega$). With the small correction on the order of $(W/L_{s})^{2}$ included, we obtain:

[TABLE]

For wide samples ($W\gg L_{s}$) Eq. (12) leads to the formula:

[TABLE]

displaying spin resonance at $\omega=\pm\varOmega$.

The general formula (12) is needed for medium sample widths ($L_{s}/\sqrt{\omega\tau_{s}}\ll W\ll L_{s}$). In this case Eq. (12) describes the crossover between the resonant dependence of $\Delta Z$ on $\varOmega$ for wide samples [Eq. (15)] and the non-resonant dependence of $\Delta Z$ on $\varOmega$ for narrow samples [Eq. (14)].

In Fig. 3 we plot the ratio $\zeta=\mathrm{Re}\,\Delta Z/(-\gamma^{2}Z_{0})$ of the real part of the spin-orbit correction $\Delta Z$ [Eq. (12)] to its value for narrow samples [Eq. (14)] as a function of $\varOmega$ at a fixed $\omega$ for different sample widths $W$. The transition from the non-resonant to the resonant behavior of $\zeta(\varOmega)$ with the increase of $W$ is clearly seen. Eq. (15) and Fig. 3 show that the wider is the sample, the smaller are both the amplitude and the width of the resonance peak.

It is interesting to study the behavior of the normalized ac magnetoresistance

[TABLE]

An analysis similar to that performed above for the dependence $\varrho(\omega)$ in zero magnetic field, shows that $\varrho(\omega,\varOmega)$ at a fixed $\omega$ has a quasi-universal behavior as a function of the parameter $(\varOmega-\omega)\tau^{*}$ at $|\varOmega|>\omega$, similar to the behavior of $\varrho(\omega)$ displayed at Fig. 2. However, the dependencies of $\varrho$ on $\Omega$ at $|\varOmega|<\omega$, as it is seen from Fig. 3, are qualitatively different for wide and for narrow samples.

4. Conclusion. We have shown that the combination of the direct and inverse spin Hall effects in nonmagnetic metals and semiconductors offers an interesting possibility to study high-frequency spin phenomena, including spin resonance, by purely electrical measurements. The corresponding corrections to the sample impedance are of second order in the spin-orbit coupling parameter, $\gamma$.

In the absence of external magnetic field, the frequency dependence of the electrical impedance is defined by the sum of the bulk spin relaxation rate and the spin diffusion rate. The interplay between the two corresponding relaxation times defines also the width and the amplitude of the electrically measured spin resonance.