Quasiparticle interference in the unconventional metamagnetic compound ${\text{Sr}}_3{\text{Ru}}_2{\text{O}}_7$

Quasiparticle interference (QPI) in spectroscopic imaging scanning tunneling microscopy provides a powerful method to detect orbital band structures and orbital ordering patterns in transition-metal oxides. We use the $T$-matrix formalism to calculate the QPI spectra for the unconventional metamagnetic system of ${\text{Sr}}_3{\text{Ru}}_2{\text{O}}_7$ with a $t_{2g}$-orbital band structure. A detailed tight-binding model is constructed accounting for features such as spin-orbit coupling, bilayer splitting, and the staggered rotation of the RuO octahedra. The band parameters are chosen by fitting the calculated Fermi surfaces with those measured in the angular-resolved photoemission spectroscopy experiment. The calculated quasiparticle interference at zero magnetic field exhibits a hollow squarelike feature arising from the nesting of the quasi-one-dimensional $d_{xz}$ and $d_{yz}$ orbital bands, in agreement with recent measurements by Lee et al. [Nat. Phys. 5, 800 (2009)]. Rotational symmetry breaking in the nematic metamagnetic state also manifests in the quasiparticle interference spectra.

Figure 1

(Color online) The lattice structure in a single layer of ${\text{Sr}}_3{\text{Ru}}_2{\text{O}}_7$. The small yellow circle represents the octahedra oxygen which rotate about $6.8°$ (the angle in the plot is a little exaggerated) with respect to the $z$ axis on the Ru sites. The red curves show the orientations of the Ru $d_{xy}$ orbitals. Because the direction of the rotation is opposite for nearest-neighbor Ru sites, two types of the sublattice are identified as A (blue dot) and B (white dot). The direction of rotation is also opposite from bottom to top layers, leading to the switch of the sublattices A and B in different layers.

Figure 2

(Color online) Hopping processes for (a) $t_1$ and $t_3$ (b) $t_2$. For each $i,j,k$, it can be $\widehat{x},\widehat{y},\widehat{z}$, but $i\ne j\ne k$. (a) The hopping processes described by $t_1$ and $t_3$ are assisted by the $p$ orbital of oxygen. (b) The hopping process described by $t_2$ is through the direct overlap between two identical orbitals on the nearest-neighbor Ru sites without going through the oxygen, thus it is much weaker than $t_1$ and $t_3$.

Figure 3

(Color online) The Wannier wave functions of the $d_{yz}$ and $d_{xz}$ with the lattice distortion. The blue and white dots denote sublattices A (with $x+y$ odd) and B (with $x+y$ even), and the gray dots denote the oxygen. The sign indicates the sign of the wave function in the positive $z$ plane and the wave functions in the negative $z$ plane have opposite signs.

Figure 4

(Color online) The wave functions viewed from the top of the material. The dashed line represents the wave function of the $d$ orbitals on the top layer and the solid line for those on the bottom layer. The smaller figures represent the $p$ orbital of the oxygen between layers with the red lobe having positive sign and the white lobe having negative sign. Note that the signs of the $d$ orbitals indicates those of the wave functions closest to the oxygen. (a) $d_{xz}$ at bottom layer and $d_{yz}$ at top layer and (b) $d_{yz}$ at bottom layer and $d_{xz}$ at top layer.

Figure 5

(Color online) The Fermi surfaces using the bilayer tight-binding model with the parameters: $t_1=0.5$, $t_2=0.05$, $t_3=0.5$, $t_4=0.1$, $t_5=-0.03$, $t_6=0.05$, $t_{\perp }=0.3$, $t_{\text{INT}}=t_{\text{INT}}^{\perp }=0.05$, $\lambda =0.1$, $V_{xy}=0.15$, and $\mu =0.47$ for (a) $V_{\text{bias}}=0$, (b) $V_{\text{bias}}=0.1$, (c) $V_{\text{bias}}=0.2$, and (d) $V_{\text{bias}}=0.3$. The thick dashed lines mark the boundary of half Brillouin zone due to the unit cell doubling induced by the rotation of octahedra oxygen. (a) For $V_{\text{bias}}=0$, the Fermi surfaces of the bonding ($k_z=0$, black solid lines) and the antibonding bands ($k_z=\pi$, red dashed lines) could cross since $k_z$ is a good quantum number. (b) As $V_{\text{bias}}$ is turned on, the crossings of the Fermi surfaces with different $k_z$ are avoided. (c) The optimized Fermi surfaces are obtained with $V_{\text{bias}}=0.2$. Fermi surface sheets of ${\alpha }_1$, ${\alpha }_2$, ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$, and $\beta$ are marked. The ${\gamma }_{1,2}$ sheets have dominant 2D $d_{xy}$ orbital character while the ${\alpha }_{1,2}$ sheets are mostly formed by quasi-1D $d_{yz,xz}$ orbitals. The ${\gamma }_3$ sheets are not seen in the ARPES measurements. (d) For $V_{\text{bias}}=0.3$, the Fermi sheets of ${\gamma }_2$ disappear.

Figure 6

(Color online) The analysis of the Fermi surface formation for $V_{\text{bias}}=0$. Two copies of the Fermi surfaces of ${\text{Sr}}_2{\text{RuO}}_4$ are labeled as ${\alpha }^e$, ${\beta }^e$, ${\gamma }^e$ for bonding and ${\alpha }^o$, ${\beta }^o$, ${\gamma }^o$ for antibonding bands. The backfolding of the Brillouin zone from the corners produces identical partners for each band appearing at positions connected by the wave vector $\vec{Q}=\left(\pi ,\pi \right)$ (the dotted arrow), leading to the Fermi surfaces plotted in Fig. 5a.

Figure 7

(Color online) QPI imaging at zero sample bias voltage $\left(E=0\right)$ contributed from scatterings (a) within all $t_{2g}$ bands, (b) within 2D band $d_{xy}$, and (c) within two quasi-1D bands $d_{yz}$ and $d_{xz}$. The scattering potential is introduced only for top and bottom layers with $V_0=1.0$, reflecting the fact that the impurities are usually in the top layer. Only the LDOS on the top layer are calculated. (b) The strong features are due to the scatterings between the small hole pockets ${\gamma }_2$ and the parts of ${\gamma }_1$ marked by the solid lines in Fig. 8. A representative strongest wave vector ${\vec{q}}_1$ is also indicated. (c) The strongest wave vectors ${\vec{q}}_{2-4}$ can be understood by scatterings indicated in Fig. 8. The stripelike features enclosed by the ovals (both black and yellow) result from the flat parts of the ${\alpha }_{1,2}$ bands, which are the signatures of the quasi-1D bands.

Figure 8

(Color online) The scattering processes related to strongest features in Fig. 7. The scatterings within and between the parts of Fermi surfaces marked by the red solid lines, which are mostly from ${\gamma }_1$ and ${\gamma }_2$ pockets, contribute the dominant features in the QPI image of 2D band $d_{xy}$ shown in Fig. 7b. A representative strongest wave vector ${\vec{q}}_1$ shown in Fig. 7b is plotted. As for the QPI image of quasi-1D bands $d_{yz,xz}$ shown in Fig. 7c, the dominant scatterings related to strongest wave vectors ${\vec{q}}_{2-4}$ occur mostly within ${\alpha }_1$ band, as indicated by the arrows.

Figure 9

(Color online) Schematic illustration of the wave-function overlap related to the tunneling matrix element for STM tip. The tunneling of electrons from the STM tip to the $d$ orbitals of the Ru atoms must go through the oxygen atoms (white dots). (a) For $d_{iz}$ orbitals $\left(i=x,y\right)$, the tunneling matrix element is large with the help of the $p_i$ orbital of the oxygen atoms. (b) For $d_{xy}$, all $p$ orbitals of the oxygen atoms have zero wave function overlaps with it, leading to much weaker tunneling matrix element compared to $d_{yz,xz}$ orbital.

Figure 10

(Color online) QPI imaging at (a) $E=0$, (b) $E=-0.03$, (c) $E=-0.06$, and (d) $E=-0.1$. The main features are similar because the Fermi surfaces of ${\alpha }_{1,2}$ are relatively insensitive to $E$ throughout this energy range.

Figure 11

(Color online) QPI imaging evaluated from Eq. (42) for (a) $x=0.25$, (b) $x=0.5$, (c) $x=0.75$, and (d) $x=1$. The QPI imaging in (a) fits the experimental result the best.

Figure 12

(Color online) QPI imaging with nematic order. $N/t_1=0.1$ and ${\mu }_{B}B/t_1=0.06$ is chosen. The breaking of the $C_4$ symmetry to $C_2$ symmetry is clearly seen.