Spin Hall effect by surface roughness

The spin Hall and its inverse effects, driven by the spin orbit interaction, provide an interconversion mechanism between spin and charge currents. Since the spin Hall effect generates and manipulates spin current electrically, to achieve a large effect is becoming an important topic in both academia and industries. So far, materials with heavy elements carrying a strong spin orbit interaction, provide the only option. We propose here a new mechanism, using the surface roughness in ultrathin films, to enhance the spin Hall effect without heavy elements. Our analysis based on Cu and Al thin films suggests that surface roughness is capable of driving a spin Hall angle that is comparable to that in bulk Au. We also demonstrate that the spin Hall effect induced by surface roughness subscribes only to the side-jump contribution but not the skew scattering. The paradigm proposed in this paper provides the second, not if only, alternative to generate a sizable spin Hall effect.

Figure 1

A metallic thin film with rough surfaces, the film thickness at $\rho =(x,y)$ is $d\left(\rho \right)$ with $\langle d\left(\rho \right)\rangle =d$.

Figure 2

Longitudinal conductivity for Cu film, $k_F=1.36\times {10}^{10}/\mathrm{m},a=3.61\text{Å}$, and ${\sigma }_0=5.88\times {10}^7\mathrm{S}/\mathrm{m}({\tau }_0\sim 24$ fs). (Left) $\sigma$ as function of film thickness $n_c\simeq k_{F}d/\pi$ for three cases of roughness $\delta =0,a,10a$. (Right) $\sigma$ as function of the surface roughness $\delta /a$ for three cases of film thickness $n_c=10,100,1000$. In the shaded area, the surface is too rough ($\delta /d>0.2$) and the perturbation assumption fails.

Figure 3

Spin Hall conductivity ${\sigma }^{\mathrm{sH}}$ for Cu film with the same parameter as in Fig. 2 and $\overline{\eta }=0.5$. (Left) ${\sigma }^{\mathrm{sH}}$ as function of surface roughness $\delta /a$. (Right) Spin Hall resistivity ${\rho }^{\mathrm{sH}}={\sigma }^{\mathrm{sH}}/{\sigma }^2$ as a function of resistivity $\rho =1/\sigma$ (by varying surface roughness $\delta$) for $n_c=1000$. The linear fit in the surface roughness dominating region (blue squares) has ${\rho }^{\mathrm{sH}}\propto {\rho }^{2.1}$, consistent with the side jump mechanism.

Figure 4

The spin Hall angle (in percent) for Cu film, with the same parameter as in Fig. 2 and $\overline{\eta }=0.5$, (left) as a function of film thickness $n_c\simeq k_{F}d/\pi$. Inset figure shows the ${\theta }^{\mathrm{sH}}$ dependence on the bulk relaxation time ${\tau }_0$ for $n_c=100$ and $\delta =5a$ with the dashed lines given by the limiting values in Eq. (20) (right) as a function of the surface roughness $\delta /a$. The black dot in all plots corresponds to the same point with $n_c=100$ ($d\simeq 23$ nm), $\delta =5a\simeq 1.8$ nm, ${\tau }_0\sim 24$ fs for Cu with conductivity ${\sigma }_0=5.88\times {10}^7\mathrm{S}/\mathrm{m}$, and has spin Hall angle ${\theta }^{\mathrm{sH}}\simeq 0.35\%$.