Spin Hall Effect and Origins of Nonlocal Resistance in Adatom-Decorated Graphene

Recent experiments reporting an unexpectedly large spin Hall effect (SHE) in graphene decorated with adatoms have raised a fierce controversy. We apply numerically exact Kubo and Landauer-Büttiker formulas to realistic models of gold-decorated disordered graphene (including adatom clustering) to obtain the spin Hall conductivity and spin Hall angle, as well as the nonlocal resistance as a quantity accessible to experiments. Large spin Hall angles of $\sim 0.1$ are obtained at zero temperature, but their dependence on adatom clustering differs from the predictions of semiclassical transport theories. Furthermore, we find multiple background contributions to the nonlocal resistance, some of which are unrelated to the SHE or any other spin-dependent origin, as well as a strong suppression of the SHE at room temperature. This motivates us to design a multiterminal graphene geometry which suppresses these background contributions and could, therefore, quantify the upper limit for spin-current generation in two-dimensional materials.

Figure 1

Schematic view of a six-terminal graphene employed to compute the nonlocal resistance $R_{\mathrm{NL}}=V_{\mathrm{NL}}/I_1$ and the spin Hall angle ${\theta }_{\text{sH}}=I_5^{S_z}/I_1$. For nonlocal transport, the injected transverse charge current between leads 1 and 2 generates the longitudinal spin current $I_5^{S_z}$ in lead 5 as well as the mediative spin current $I_M^{S_z}$, whose conversion into the voltage drop $V_{\mathrm{NL}}=V_3-V_4$ between leads 3 and 4 generates $R_{\mathrm{NL}}$. The dashed region illustrates the sample of size $400\mathrm{nm}\times 400\mathrm{nm}$, with periodic boundary conditions, used for calculations of Kubo conductivities. The enlargement shows carbon atoms (black circles) and Au adatoms (yellow circles).

Figure 2

Spin Hall ${\sigma }_{\text{sH}}$ (main frame) and longitudinal ${\sigma }_{xx}$ (insets) conductivities for the two cases of $n_i=15\%$ Au-adatom distributions: (a) scattered and (b) clustered, where Au islands have varying radius $\in [1,3]\mathrm{nm}$. In both cases, the effect of the presence (red lines, $V_I=0.007{\gamma }_0$) or absence (black lines, $V_I=0$) of the enhanced intrinsic SOC within the hexagons hosting adatoms is also shown. All results are averaged over 400 disorder configurations.

Figure 3

Spin Hall angle ${\theta }_{\text{sH}}={\sigma }_{\text{sH}}/{\sigma }_{xx}$ corresponding to Fig. 2 for scattered (black curve) and clustered (red curve) distributions of Au adatoms, which are illustrated in the insets.

Figure 4

(a) Nonlocal resistances for six-terminal graphene in Fig. 1 with $n_i=15\%$ of scattered Au adatoms, fixed channel width $W=50\mathrm{nm}$, and several channel lengths: $L=10\mathrm{nm}$ (main frame); $L=100\mathrm{nm}$ (left inset); and $L=300\mathrm{nm}$ (right inset). Dotted lines plot $R_{\mathrm{NL}}$ when all SOC terms in Eq. (1) are switched off ($\mathrm{SOC}\equiv 0\iff V_I=V_R=0$). (b) Spin Hall angle, obtained from LB formula calculations, for the same concentration of Au adatoms which are scattered (main frame) or clustered (inset). All curves are averaged over ten disorder configurations.