Resistively detected NMR as a probe of the topological nature of conducting edge/surface states

Electron spins in edge or surface modes of topological insulators (TIs) with strong spin-orbit coupling cannot be directly manipulated with microwaves due to the locking of electron spin to its momentum. We show by contrast that a resistively detected nuclear magnetic resonance (RDNMR) based technique can be used to probe the helical nature of surface conducting states. In such experiments, one applies a radio frequency (RF) field to reorient nuclear spins that then couple to electronic spins by the hyperfine interaction. The spin of the boundary electrons can thereby be modulated, resulting in changes in conductance at nuclear resonance frequencies. Here, we demonstrate that the conductivity is sensitive to the direction of the applied magnetic field with respect to the helicity of the electrons. This dependence of the RDNMR signal on angle probes the nature of the conductive edge or surface states. In the case of 3D TI in the quantum Hall regime, we establish that the dominant mechanism responsible for the conductance change in a RDNMR experiment is based on the Overhauser field effect. Our findings indicate that the same physics underlying the use of RDNMR to probe TI states also enables us to use RF control of nuclear spins to coherently manipulate topologically protected states, which could be useful for a new generation of devices.

Figure 1

Schematic of RDNMR for a 2-D TI. When a RF field is applied to depolarize/flip the nuclei, the nuclei continuously back-scatter the helical Dirac fermions through the hyperfine interaction. The static weak magnetic field is applied along the ${\widehat{z}}^{\prime }$ direction.

Figure 2

(a) The change of conductance at the NMR resonance frequency vs $V/k_{B}T$ at different angles $\theta$ in a two-terminal experiment, for isotropic hyperfine coupling $|A_z|=|A_{xy}|$. (b) The change of conductance at the NMR resonance frequency versus angle $\theta$ for different anisotropic hyperfine coupling $|A_z/A_{xy}|$. The solid line represents $|A_z/A_{xy}|=1$, the dashed line represents $|A_z/A_{xy}|=4$, and the dotted line is for $|A_z/A_{xy}|=0.25$. The blue lines show the 2D TI while the red lines represent the case when there is one spin-degenerate conducting channel on the edge, as depicted in the insets of (b). In the insets, the black lines represent the edges of the sample while blue and red lines show the counterpropagating conducting channels. ${\alpha }_h={\pi }^2{N}_n{\left|A_{xy}\right|}^2/h^2{v}_F^2$, ${\alpha }_{nh}={\pi }^2{N}_n{J}^2{\left|A_{xy}\right|}^2/16$. Parameters: $I=9/2$, $\mathrm{\Delta }/k_{B}T=0.05$.

Figure 3

(a) Schematic of the setup in our model. The arrows denote the directions of the magnetic field $B$, current $I$, and the nuclear polarization $P$, respectively. The double arrows on the magnetic field $B$ indicate that it can be aligned along either direction. In this case $P$ is nonvanishing only on top and bottom surfaces. (b) The surface spectrum of a rectangular wire under the quantization condition. (c) The schematic of the distribution of Dirac fermion when biased, where solid blue lines denote the occupied states while dashed blue lines represent the unoccupied ones; the full circle is the Fermi surface at equilibrium while the two hemi-circles denote the Fermi surface of the reservoirs when bias is present.

Figure 4

Schematic for a 3D TI in a strong magnetic field. Landau levels are formed on the top and bottom surface with filling factor ($\frac{1}{2},-\frac{1}{2}$) as an example. Due to the thermal activation, there is nonzero carrier density contributing to conductivity on the top and bottom surface.