Enhanced Nonlinear Response by Manipulating the Dirac Point at the (111) ${\mathrm{LaTiO}}_3/{\mathrm{SrTiO}}_3$ Interface

Tunable spin-orbit interaction (SOI) is an important feature for future spin-based devices. In the presence of a magnetic field, SOI induces an asymmetry in the energy bands, which can produce nonlinear transport effects ($V\sim I^2$). Here, we focus on such effects to study the role of SOI in the (111) ${\mathrm{LaTiO}}_3/{\mathrm{SrTiO}}_3$ interface. This system is a convenient platform for understanding the role of SOI since it exhibits a single-band Hall response through the entire gate-voltage range studied. We report a pronounced rise in the nonlinear longitudinal resistance at a critical in-plane field $H_{\mathrm{cr}}$. This rise disappears when a small out-of-plane field component is present. We explain these results by considering the location of the Dirac point formed at the crossing of the spin-split energy bands. An in-plane magnetic field pushes this point outside of the Fermi contour, and consequently changes the symmetry of the Fermi contours and intensifies the nonlinear transport. An out-of-plane magnetic field opens a gap at the Dirac point, thereby significantly diminishing the nonlinear effects. We propose that magnetoresistance effects previously reported in interfaces with SOI could be comprehended within our suggested scenario.

Figure 1

Hall and superconductivity measurements. (a) Hall effect for various gate voltages at 10 K, displaying a perfectly linear one-band behavior. (b) Superconducting transitions for various gates voltages. (c),(d) In-plane and out-of-plane superconducting critical fields at various temperatures and gates exhibiting a two-dimensional behavior.

Figure 2

The second harmonic resistance versus the in-plane magnetic field at $T=3\mathrm{K}$ and $R_s=380\mathrm{\Omega }$; $H_{\mathrm{cr}}$ is indicated. Top inset: dependence of $\gamma =(2R_{2\omega }/R_{0}HI)$ on the in-plane angle $\varphi$. In the low-field regime, the effect depends only on the component of the field perpendicular to the current. In the high-field regime, the effect also depends on the in-plane field angle $\varphi$. Bottom inset: sketch of the sample, current, and magnetic field. In all such sketches throughout this Letter, the ${\mathrm{SrTiO}}_3$ is facing up, while the ${\mathrm{LaTiO}}_3$ is facing down.

Figure 3

Effects of the out-of-plane magnetic field. (a),(b) $\gamma$ is shown as a function of the out-of-plane magnetic field angle $\theta$ for total field magnitudes of 9 T (a) and 3 T (b). A sharp increase in $\gamma$ is observed for the high-field regime when the magnetic field is in plane. (c) $\gamma$ is shown versus the in-plane angle $\varphi$ for $H_{\parallel }=7.8\mathrm{T}$ and $H_{\perp }=4.5\mathrm{T}$ (red), $-4.5\mathrm{T}$ (black), and 0 T (blue). (d) Similar magnetic field scheme like (c) for the first harmonic measurements. Note the mirror symmetry breaking for the red and black curves. (e) Sketch of the magnetic field configurations employed in the measurements presented in this figure. Temperatures and sheet resistances for these figures slightly vary due to different cooldowns: (a) $T=3\mathrm{K}$ and $R_s=380\mathrm{\Omega }$; (b) $T=1.8\mathrm{K}$ and $R_s=412\mathrm{\Omega }$; (c),(d) $T=1.9\mathrm{K}$ and $R_s=407\mathrm{\Omega }$.

Figure 4

(a) Second harmonic resistance versus transverse magnetic field at different temperatures. (b) $\gamma$ at 9 T after subtracting the low-field contribution versus temperature. (c),(d) Similar to (a),(b) but as a function of gate voltage represented by the sheet resistance ($R_s$). The effect quickly decays at more negative gate voltages (higher resistances). Temperature-dependent measurements were conducted for the lowest $R_s$; gate-dependent measurements were conducted at 2 K.

Figure 5

Minimal model calculations. (a) Calculated second harmonic resistance as function of transverse magnetic field with no out-of-plane field (top panel) and a small out-of-plane field (bottom panel). (b) Fermi contours for $B_{\parallel }<B_{\mathrm{cr}}$, $B_{\parallel }=B_{\mathrm{cr}}$, $B_{\parallel }>B_{\mathrm{cr}}$. For every scenario, the location of the Dirac point is marked. (c) Second harmonic resistance versus the perpendicular magnetic field with constant high transverse magnetic field. Inset: illustration of band gap opening with perpendicular magnetic field.