Ab initio spin-flip conductance of hydrogenated graphene nanoribbons: Spin-orbit interaction and scattering with local impurity spins

We calculate the spin-dependent zero-bias conductance $G_{\sigma {\sigma }^{\prime }}$ in armchair graphene nanoribbons with hydrogen adsorbates employing a DFT-based ab initio transport formalism including spin-orbit interaction. We find that the spin-flip conductance $G_{\sigma \overline{\sigma }}$ can reach the same order of magnitude as the spin-conserving one $G_{\sigma \sigma }$, due to exchange-mediated spin scattering. In contrast, the genuine spin-orbit interaction appears to play a secondary role, only.

Figure 1

(a) Structure of the clean AGNR11 device. In the blue marked device region, SOI is present. (b) Between the two leads of (a), an infinitesimal voltage $dV$ is applied. As response, an electronic current $dI$ flows, which is split into four components $d{I}_{\sigma {\sigma }^{\prime }}$. They account for an electronic current being injected with spin $\sigma$ and measured after passing the device region with spin ${\sigma }^{\prime }$. The spin-dependent conductance $G_{\sigma {\sigma }^{\prime }}\left(E\right)$ at a given chemical potential $E$ is then defined as a ratio of $d{I}_{\sigma {\sigma }^{\prime }}$ and $dV$. The reference energy $E_{\text{F}}$ is the chemical potential of the isolated, charge-neutral device. (c) Corresponding conductance of the pristine ribbon (a) according to Eq. (2). The spin quantization axis is chosen in positive $z$ direction. The spin-flip conductance $G_{\sigma \overline{\sigma }}$ is nonvanishing due to SOI. The spin-conserving conductance does not vanish completely inside the band gap (but is reduced by seven orders of magnitude) due to the limited length of the lead system in $x$ direction. This is a methodological artifact that reflects a small numerical difference between the exact lead-induced self-energy and our approximation scheme (decimation technique [47]).

Figure 2

(a) Structure of the AGNR11 device with one single hydrogen adsorbate. (b) Finite-size ribbon for the underlying DFT calculation including SOI. All atoms in the blue box belong to the device region. The red surface denotes the isosurface of the spin density $\mathbf{S}\left(\mathbf{r}\right)$ calculated by the DFT including SOI where we observe the famous zigzag edge magnetism [14, 15, 16, 17]. The total spin is calculated as $\langle \mathbf{S}\rangle =\int \mathbf{S}\left(\mathbf{r}\right)d^3\mathbf{r}$. (c)–(f) Conductance of the device (a) with/without SOI and quantization axis (QA) according to the inset: (c) Collinear case [average magnetization of the sample and spin of incoming electrons are (very nearly) aligned] with SOI, (d) noncollinear case (spin polarization axis of incoming electrons and average magnetization axis of local impurity spins are perpendicular) with SOI, (e) collinear case without SOI, and (f) noncollinear case without SOI. The reference energy $E_{\text{F}}$ is the chemical potential of the isolated, charge-neutral lead. The average spin-flip conductance $G_{\sigma \overline{\sigma },\text{avg}}$ is computed as arithmetic mean of $G_{\uparrow \downarrow }\left(E\right)$ and $G_{\downarrow \uparrow }\left(E\right)$ in the energy interval $[-1.5\text{eV},1.5\text{eV}]$. $G_{\text{clean}}\left(E\right)$ denotes the total conductance of the pristine ribbon.

Figure 3

(a) Finite-size ribbon with two hydrogen adatoms for the underlying DFT calculation including SOI. The atomic structure is relaxed including the six surrounding carbon atoms. All atoms in the blue box belong to the device region. (b) Spin-dependent conductance for $z$-polarized electrons and (c) for $x$-polarized electrons. All symbols are defined in the caption of Fig. 2.

Figure 4

(a) Finite-size ribbon with 12 hydrogen adatoms for the underlying DFT calculation including SOI. The total number of unpaired electrons is $N_{\text{Spin}}=2/\hslash \left|\langle \mathbf{S}\rangle \right|=1.75$. All atoms in the blue box belong to the device region. (b) Spin-dependent conductance for electrons polarized along ${\mathbf{e}}_z$ and (c) for $x$ polarization. All symbols are defined in the caption of Fig. 2.

Figure 5

Local current density response (integrated over the out-of-plane direction; normalized to the conductance per width) in the hydrogenated AGNR11 shown in Fig. 4 (blue box only). The current density exhibits strong mesoscopic fluctuations that reflect in a logarithmic color scale covering three decades. Some interesting current paths are drawn into the picture for illustration: Local current vortices exceeding the spatial average current by one order of magnitude (see dark red regions). Plot shows current amplitude (color), current direction (arrows), carbon atoms (gray crosses), and hydrogen atoms (red crosses).

Figure 6

(a) Input geometry for the SOI-DFT calculation of an AGNR11 with a single adsorbed hydrogen atom. The computed total magnetic moment of the ribbon is $\langle \mathbf{S}\rangle ={(-0.71,-0.48,-0.97)}^T\hslash$, so that there are 2.7 unpaired electrons in the ribbon. For the transport calculation, we partition the ribbon into a device region (blue box) and two contact regions (magenta boxes). (b) Identical geometry and identical DFT calculation as in (a), but different partitioning in device and contact regions. (c) and (d) Result for the spin-dependent conductance according to Eq. (A3) for partitioning (a) [shown in (c)] and for partitioning (b) [shown in (d)] for $z$ quantization so that ${\mathbf{e}}_{\langle \mathbf{S}\rangle }\ne {\mathbf{e}}_z$ enabling the efficient exchange-mediated spin-flip mechanism.

Figure 7

(a) and (b) Tight-binding model to rationalize the conductance curves of the AGNR with a single hydrogen adsorbate including two conducting bands, a localized state with spin $\uparrow$ and $\downarrow$, and a SOI between both localized states. The matrix elements between the quantum wire and the localized states are (a) $t_{\text{LC}}$ in the case of incoming electrons being spin polarized along $\uparrow /\downarrow$ and (b) $t_{\text{LC}}/\sqrt{2}$ for $+/-$ polarization. Here one hopping term is negative, see Eq. (B6). (c) and (d) Spin-dependent conductance for model parameters for the AGNR with hydrogen adatom and (c) $\uparrow /\downarrow$ and (d) $+/-$ quantization of incoming electrons. The SOI strength $\lambda$ is estimated by the SOI strength of 2.5 meV for a hydrogen adatom on graphene [59] in relation to the graphene hopping $t=2.8$ eV.

Figure 8

Average spin-flip conductance $G_{\sigma \overline{\sigma },\text{avg}}$ {arithmetic mean of $G_{\sigma \overline{\sigma }}\left(E\right)$ and $G_{\overline{\sigma }\sigma }\left(E\right)$ in the energy interval $[-2t,2t]$} for the model of Fig. 7 as function of the spacing $\mathrm{\Delta }={\varepsilon }_{\uparrow }-{\varepsilon }_{\downarrow }$ between the two localized levels with opposite spin. The polarization in (a) is $\uparrow /\downarrow$ [model Fig. 7 (a)] and in (b) $+/-$ [model Fig. 7 (b)]. The SOI parameter $\lambda$ and the hopping $t_{\text{LC}}$ are chosen as in Figs. 7 and 7.