Current Patterns and Orbital Magnetism in Mesoscopic dc Transport

We present ab initio calculations of the local current density $\mathbf{j}(\mathbf{r})$ as it arises in dc-transport measurements. We discover pronounced patterns in the local current density, ring currents (“eddies”), that go along with orbital magnetism. Importantly, the magnitude of the ring currents can exceed the (average) transport current by orders of magnitude. We find associated magnetic fields that exhibit drastic fluctuations with field gradients reaching $1\mathrm{T}{\mathrm{nm}}^{-1}{\mathrm{V}}^{-1}$. The relevance of our observations for spin relaxation in systems with very weak spin-orbit interaction, such as organic semiconductors, is discussed. In such systems, spin relaxation induced by bias driven orbital magnetism competes with relaxation induced by the hyperfine interaction and appears to be of similar strength. We propose a NMR-type experiment in the presence of dc-current flow to observe the spatial fluctuations of the induced magnetic fields.

Figure 1

Local current density per spin (integrated over the out-of-plane direction) in a wide hydrogen-terminated armchair graphene nanoribbon ($41\times 8$) that has been functionalized with an additional 20% hydrogen atoms (additional 66 hydrogen atoms). The current exhibits very strong mesoscopic fluctuations that reflect in a logarithmic color scale covering 4 decades. Some interesting current paths are drawn in the picture for illustration: local current vortices exceeding the spatial average current by orders of magnitude (see dark red regions; the average current is only $\int (d{j}_x(\mathbf{r})/d{V}_{\text{bias}})dz{|}_{\text{avg}}=\mathcal{T}(e^2/h)w^{-1}=10^{-3}\mathrm{a}.\mathrm{u}.$; width $w=5.2\mathrm{nm}$) and a local backflow channel where the current runs against the average current (see central arrow from bottom to top; average current direction: from top to bottom). Plot shows current amplitude (color), current direction (arrows), carbon atoms (gray crosses), and hydrogen atoms (red crosses). Sample magnetization per bias: $d{m}_z/d{V}_{\text{bias}}=-38\mathrm{a}.\mathrm{u}.=-2.8({\mu }_B/\mathrm{V})$. (See Sec. S5 of the Supplemental Material [14] for the detailed atomic structure.)

Figure 2

Magnetic field distribution per spin (in out-of-plane direction) for the ribbon of Fig. 1. The magnetic field strongly varies and changes sign from region to region. The field is plotted in the (averaged) carbon plane ($z=0$), but being divergence free, it hardly changes with $z$ (checked for $z=\pm 1\AA$). The current paths drawn in Fig. 1 are repeated for convenience.

Figure 3

The transmission functions (red, left $y$ axis) belong to the ribbons shown in Fig. 1 (left) and Fig. 4 (right). The transmission functions of the pristine ribbons show sharp steps (orange, left $y$ axis). The blue arrows mark the energy for which the current response is plotted in Figs. 1 and 5. Both ribbons exhibit a finite band gap at $E={ϵ}_F$ with $\mathcal{T}=0$. The sample magnetization per bias $d{m}_z/d{V}_{\text{bias}}$ (perpendicular to the graphene plane, dashed black, right $y$ axis) shows sign changes in the vicinity of antiresonances.

Figure 4

Local current density per spin (left) and induced magnetic field (right) in a narrow armchair ribbon (five carbon atoms wide) with one nitrogen atom replacing a carbon atom (at right edge) The current reveals a massive tendency for (local) vortices that exceeds the spatial average of the current density (through current) by an order of magnitude. The average current is only $\int (d{j}_x(\mathbf{r})/d{V}_{\text{bias}})dz{|}_{\text{avg}}=\mathcal{T}(e^2/h)w^{-1}=1.5\times 10^{-3}\mathrm{a}.\mathrm{u}.$; width $w=6.8\AA$). Average current direction: from top to bottom. Sample magnetization per bias: $d{m}_z/d{V}_{\text{bias}}=2.6\mathrm{a}.\mathrm{u}.=0.19$ ${\mu }_B/\mathrm{V}$.

Figure 5

Transport current and orbital current (magnetization) as calculated for the two-path (toy) model. Currents are proportional to the incoming current $j_{\mathrm{in}}$. Inset: graphical representation of the (tight-binding) two-path model. ($t^{\prime }=t/\sqrt{2}$; ${ϵ}_T=-{ϵ}_B=\frac{1}{2}t$; $m_z=(j_B-j_T)A/2$).