Compass-like manipulation of electronic nematicity in Sr3Ru2O7

Abstract

Electronic nematicity has been found in a wide range of strongly correlated electron materials, resulting in the electronic states having-4.5pc]Please note that the spelling of the following author name(s) in the manuscript differs from the spelling provided in the article metadata: Izidor Benedičič. The spelling provided in the manuscript has been retained; please confirm. a symmetry that is lower than that of the crystal that hosts them. One of the most astonishing examples is [Formula: see text], in which a small in-plane component of a magnetic field induces significant resistivity anisotropy. The direction of this anisotropy follows the direction of the in-plane field. The microscopic origin of this field-induced nematicity has been a long-standing puzzle, with recent experiments suggesting a field-induced spin density wave driving the anisotropy. Here, we report spectroscopic imaging of a field-controlled anisotropy of the electronic structure at the surface of [Formula: see text]. We track the electronic structure as a function of the direction of the field, revealing a continuous change with the angle. This continuous evolution suggests a mechanism based on spin-orbit coupling resulting in compass-like control of the electronic bands. The anisotropy of the electronic structure persists to temperatures about an order of magnitude higher compared to the bulk, demonstrating novel routes to stabilize such phases over a wider temperature range.

Keywords: magnetism; nematicity; scanning tunneling microscopy; spin–orbit coupling; strongly correlated electron materials.

Fig. 1.

Electronic nematicity in ${\text{Sr}}_3{\text{Ru}}_2{\text{O}}_7$. (A) Close to the metamagnetic critical point at ${\mu }_{0}H\approx 7.8\mathrm{T}$, applying a field with a small in-plane component $H_{\mathit{ab}}\Vert [100]$ (or, equivalently, [010]) induces a large anisotropy in the resistivity. Figure adapted from ref. . (B) Crystal structure of ${\text{Sr}}_3{\text{Ru}}_2{\text{O}}_7$ with the tetragonal unit cell (black lines) in side view (Blue spheres: Sr, black: Ru, red: O). Due to the octahedral rotations, the true unit cell is orthorhombic (blue dotted lines). (C) Topographic STM image showing the Sr square lattice and a twofold symmetric quasiparticle interference pattern around a defect. Top Left inset shows an enlarged image of the topography with the top view of the crystal structure superimposed ($V_{\mathrm{s}}=10\mathrm{mV}$, $I_{\mathrm{s}}=300\mathrm{pA}$). Bottom Left inset: Fourier transformation of a spectroscopic map acquired in zero magnetic field across similar defects in a larger area at $V=-0.3\mathrm{mV}$ ($T=80\mathrm{mK}$), showing the $C_4$ symmetry breaking.

Fig. 2.

Field-control of nematicity. (A–E) Topographic images around defects, A, at zero field with definition of the nematic order parameter $\mathrm{\Psi }$, and, B–E, in in-plane magnetic field $|{\mu }_0\mathbf{H}|=5\mathrm{T}$ for field directions as indicated by the red arrow ($V_{\mathrm{s}}=15\mathrm{mV}$, $I_{\mathrm{s}}=300\mathrm{pA}$, $T=4.2\mathrm{K}$). (F) sketch of the crystal structure and definition of angle $\phi$ in which ${\mu }_0\mathbf{H}$ is applied relative to the crystallographic axes. (G and H) Dependence of nematic order parameter $\mathrm{\Psi }$ (extracted from the white rectangle in A) on the field angle $\phi$ as defined in F. $\mathrm{\Psi }$ shows sign reversal for $\phi =0^{°}$, ${90}^{°}$, ${180}^{°}$, and ${270}^{°}$, with maxima in $|\mathrm{\Psi }|$ when the field is along [110] and [$\overline{1}$10], respectively ($\phi ={45}^{°}$, ${135}^{°}$, ${225}^{°}$, and ${315}^{°}$). (H) shows the same data as in G in a polar plot. The solid line shows a sine function as a guide to the eye. (I) colour plot of the real part of the Fourier transformation Re[$\tilde{z}(\mathbf{q})$] of the topography. Red (blue) symbols indicate the same (opposite) phase relative to the topography at zero field. The Fourier peaks associated with a checkerboard modulation at ${\mathbf{q}}_{\mathrm{ckb}}=(\pm 1/2,\pm 1/2)$ is indicated by a black circle. (J) the intensity of the Fourier peak at ${\mathbf{q}}_{\mathrm{ckb}}$ (black circle in I) as a function of in-plane field direction $\phi$. The checkerboard contrast is maximal for field along [100] or [010], showing a contrast reversal between the two directions. (K) same data as in J shown in a polar plot. Lines in G, H, and K are drawn as guides for the eye.

Fig. 3.

Relation of band structure and modulated density of states. Using a tight-binding description (Eq. 1), modeling the electronic structure of the top surface layer by a single layer of ${\text{Sr}}_2{\text{RuO}}_4$ including a magnetization $\mathbf{M}$ and spin-orbit coupling reproduces the key findings of Fig. 2. To model the influence of the magnetic field on the electronic structure, we assume that the magnetization $\mathbf{M}$ points in the direction of the applied field (red arrow), (A). As a consequence of spin–orbit coupling, the band structure obtained from the minimal model (for details see main text and Materials and Methods), (B), depends on the in-plane direction of the magnetization. For direct comparison with the experimental data, we calculate the real-space continuum local density of states (cLDOS), (C), for the in-plane magnetisation directions shown in A at energy $E_1$ indicated by the orange horizontal line in B, showing symmetry breaking QPI patterns, with the symmetry breaking most obvious for $\phi ={45}^{°}$ and ${135}^{°}$, as also seen in Fig. 2 C and E. (D) cLDOS for magnetization directions indicated in A at energy $E_2$, green line in B, showing the appearance of a checkerboard for $\phi =0^{°}$ and ${90}^{°}$ as in Fig. 2 B and D.

Fig. 4.

Quasiparticle interference imaging. (A–D) Fourier transform of $\mathrm{d}lnI/\mathrm{d}lnV$ maps at different directions of the in-plane field $\mathbf{H}$ along [$\overline{1}$10] (A), [010] (B), [110] (C) and [100] (D), shown at $V=-0.8\mathrm{mV}$ ($V_{\mathrm{s}}=10\mathrm{mV}$, $I_{\mathrm{s}}=100\mathrm{pA}$, $T=4.2\mathrm{K}$, ${\mu }_{0}H=5\mathrm{T}$, in-plane field direction is indicated by an arrow in each panel). (E) Difference of Fourier transforms of spectroscopic maps in A and C with magnetic field ${\mu }_{0}H=5\mathrm{T}$ applied in the orthorhombic directions ($\mathbf{H}\Vert$ [$\overline{1}$10] and $\mathbf{H}\Vert$[110]). (F) Same as E but for B and D with magnetic fields applied in the tetragonal directions ($\mathbf{H}\Vert$[010] and $\mathbf{H}\Vert$[100]).