Normal State ${}^{17}\mathrm{O}$ NMR Studies of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ under Uniaxial Stress

The effects of uniaxial compressive stress on the normal state ${}^{17}\mathrm{O}$ nuclear-magnetic-resonance properties of the unconventional superconductor ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ are reported. The paramagnetic shifts of both planar and apical oxygen sites show pronounced anomalies near the nominal $\mathbf{a}$-axis strain ${ϵ}_{aa}\equiv {ϵ}_v$ that maximizes the superconducting transition temperature $T_c$. The spin susceptibility weakly increases on lowering the temperature below $T\simeq 10\mathrm{K}$, consistent with an enhanced density of states associated with passing the Fermi energy through a van Hove singularity. Although such a Lifshitz transition occurs in the $\gamma$ band formed by the Ru $d_{xy}$ states hybridized with in-plane O $p_{\pi }$ orbitals, the large Hund’s coupling renormalizes the uniform spin susceptibility, which, in turn, affects the hyperfine fields of all nuclei. We estimate this “Stoner” renormalization $S$ by combining the data with first-principles calculations and conclude that this is an important part of the strain effect, with implications for superconductivity.

Popular Summary

After 25 years of study, the unconventional superconductor ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ continues to challenge researchers. While evidence for “in-triplet” pairing—where electrons pair up with their spins pointing in the same direction—emerges in several key measurements, other experimental outcomes are not easily understood in that framework. Resolving this issue is important to condensed-matter physicists because ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ is a notable example of a strongly correlated, quasi-2D system that has few defects or impurities and because the underlying experimental techniques are used throughout the field of superconductivity.

A consequence of the proposed spin-triplet pairing is the expectation that when the material is strained, two different superconducting transition temperatures appear. Here, we use nuclear magnetic resonance to infer the strain-dependent spin susceptibility (a measure of the number of possible states at each energy level, known as the “density of states”), and we identify two physical consequences: an increase in the density of states at the Fermi energy and an associated enhancement in the Stoner factor, which is a signature of intensified ferromagnetic fluctuations that couple paired electrons.

Our study provides two possible explanations for the increase in the superconducting temperature in stressed ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$: While the density-of-states effect is directly more beneficial in the case of spin-singlet pairing, the coincident enhanced Stoner factor could play a role in stabilizing a triplet state. Extending these investigations to the superconducting state will provide a more conclusive answer.

Figure 1

(a) Configurations of planar O(1) and $\mathrm{O}(1^{\prime })$ in the ${\mathrm{RuO}}_2$ plane and apical O(2) in the SrO layer around Ru ion in a unit cell of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. Compressive strain is applied along the $\mathbf{a}$ axis (${ϵ}_{aa}$); magnetic fields are applied orthogonal, $\parallel \mathbf{b},\parallel \mathbf{c}$. Arrows signify the principal axes of Knight-shift tensors. (b) Orbitals forming the $\gamma$ band at the $X$ (left) and $Y$ (right) points in the Brillouin zone. The blue (red) double arrows show positive (negative) signs of orbital overlaps. Note that at the $Y$ point, only $\mathrm{O}(1)p_x$ orbitals participate in the band formation, while $\mathrm{O}(1^{\prime })p_y$ suffers from cancellation of the left and right overlaps. The weak $\mathrm{O}(1)\text{−}\mathrm{O}(1^{\prime })$ overlaps also cancel out, as shown in the figure.

Figure 2

(a) Bands along the $\mathrm{\Gamma }\text{−}X$ and $\mathrm{\Gamma }\text{−}Y$ directions. The partial weights of the $\mathrm{O}(1)p_x$, $\mathrm{O}(1^{\prime })$$p_y$, $\mathrm{O}(1)p_z$, and $\mathrm{O}(1^{\prime })p_z$ orbitals are shown in green, blue, red, and purple, respectively. Other oxygen orbitals have far lesser weight near the Fermi energy. (b) Depictions of the 2D Fermi surfaces, with quasi-2D $\gamma$ ($d_{xy}$) and quasi-1D $\alpha$, $\beta$ ($d_{zx,yz}$) bands.

Figure 3

NMR spectra of the central transitions ($\frac{1}{2}\leftrightarrow -\frac{1}{2}$) of O(1), $\mathrm{O}(1^{\prime })$, and O(2) at various strains for magnetic field along the $\mathbf{b}$ (left) and $\mathbf{c}$ axes (right). The measurements are carried out at fixed temperature ($T=4.3\mathrm{K}$) and field ($B=8.0970\mathrm{T}$) and radio frequency $f_0=46.80\mathrm{MHz}$. The curves are vertically offset for clarity. The dash vertical line corresponds to ${}^{17}\gamma =-5.7719\mathrm{MHz}/\mathrm{T}$ (${\mathrm{D}}_2^{17}\mathrm{O}$) [30] with zero shift.

Figure 4

Measured NMR shifts for $\mathbf{B}\parallel \mathbf{b}$ (a) and for $\mathbf{B}\parallel \mathbf{c}$ (b) at $T=4.3\mathrm{K}$. The solid (open) symbols represent increasing (decreasing) $|{ϵ}_{aa}|$. The error bars are determined by the half-width of the peaks. Similar results are reproduced from several samples; see Fig. S5(a) in the Supplemental Material [32].

Figure 5

(a) Calculated DOS at the critical strain, at which the calculated van Hove singularity is located exactly at the Fermi level. Note the very small width (3 meV) and weight (0.0015 $e^{-}$ per spin channel) of the peak in the DOS. (b) Partial DOS projected onto different O orbitals. The orbitals that are not shown have negligible weight.

Figure 6

Main panel: Temperature dependence of $K_{1\parallel }$ and $K_{1^{\prime }\perp }$ evaluated at the critical strain ${ϵ}_v$, $B=8.0970\mathrm{T}$, and $\mathbf{B}\parallel \mathbf{b}$. Inset: Field dependence of $K_{1\parallel }$ and $K_{1^{\prime }\perp }$ measured at ${ϵ}_v$ and 4.3 K.

Figure 7

(a) Calculated magnetic susceptibility in DFT. ${\chi }_{s0}^{\mathrm{DFT}}\equiv {\mu }_B^{2}N(E_F)$ is the noninteracting susceptibility, ${\chi }_s^{\mathrm{DFT}}$ is obtained by dividing the calculated magnetization by the applied field, $M_s/{\mu }_{B}H$. The average DFT Stoner factor $S={\chi }_s^{\mathrm{DFT}}/{\chi }_{s0}^{\mathrm{DFT}}$ and $S_{\mathrm{RPA}}=1/[1-IN(E_F)]$. Here, $S_{\mathrm{RPA}}$ is normalized to $S$ obtained from the calculated DFT result at zero strain. Its variation with strain is calculated from Eq. (4) and the strain-dependent DFT density of states. (b) Calculated total Knight shifts for $\mathbf{H}\parallel \mathbf{b}$ for the three sites, O(1), $\mathrm{O}(1^{\prime })$, and O(2), as a function of normalized strain. See the text for details.

Figure 8

Calculated Fermi surfaces (nonrelativistic) with no orthorhombic strain (left) and the strain corresponding to the VHS (right). No additional scaling of the Stoner interaction is applied, as opposed to the Knight-shift calculation (Fig. 7 and main text). The surfaces are colored with the calculated exchange splitting in a small uniform external field $H$ normalized to ${\mu }_{B}H=1.6\mathrm{meV}$. Note the different color scales for the two panels.