Critical temperature $T_c$ and Pauli limited critical field of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$: Uniaxial strain dependence

Variations of critical temperature $T_c$ and in-plane critical field $H_{c2}$ of ${\text{Sr}}_2\text{Ru}{\text{O}}_4$ under uniaxial stress were recently reported. We compare the strain dependence of $T_c$ and $H_{c2}$ in various pairing channels ($d$ wave, extended $s$ wave, and $p$-wave) with the experimental observations by studying a three-band tight-binding model that includes effects of spin-orbit and Zeeman couplings and a separable pairing interaction. Our study helps narrow down the possibility of pairing channels. The importance of the multiband nature of ${\text{Sr}}_2\text{Ru}{\text{O}}_4$ is also highlighted.

Figure 1

Fermi surfaces of the three-band tight-binding Hamiltonian in Eq. (2). Left: Zero strain $\tau =1$, where the system has tetragonal symmetry. Right: Van Hove strain around $\tau =1.055$, where the $\gamma$ band touches the boundary of the first Brillouin zone.

Figure 2

Critical temperature $T_c$ and Pauli limited critical field $H_c$ for $d$-wave and $p$-wave pairing on the single-orbital model as a function of uniaxial strain (hopping ratio $\tau$). Both quantities have been normalized to unity at zero strain ($\tau =1$). Left: For $d$-wave pairing, the peak position for the two quantities is at the Van Hove strain. $H_{c2}$ clearly has a weaker enhancement than $T_c$ when approaching the Van Hove strain, which is inconsistent with experimental observation. Right: For $p$-wave pairing, $H_{c2}$ and $T_c$ are insensitive to the Van Hove singularity at $\tau =1.077$ since we have assumed that the BCS interaction is strain independent and the $p$-wave gap function (10) vanishes at $(k_x,k_y)=(0,\pi )$. The ratio $H_{c2}/T_c$ decreases when approaching the Van Hove strain, which is inconsistent with experimental observation.

Figure 3

Results for a mixture of extended-$s$-wave and $d$-wave pairing channels in a single-orbital model. A particular set of interaction strengths $(g_d,g_s)$ has been chosen. Left: Pairing magnitude at $T=H=0$ for each pairing channel as a function of strain. Right: Critical temperature and critical field as a function of strain. The enhancement in $H_{c2}$ is weaker than that in $T_c$, which is inconsistent with the experimental observation.

Figure 4

Critical temperature $T_c$ and Pauli limited critical field $H_c$ for $d$-wave and $p$-wave pairing on a three-band model as a function of uniaxial strain (hopping ratio $\tau$). Both quantities have been normalized to unity at zero strain ($\tau =1$). Left: For $d$-wave pairing, the peak position for the two quantities is at the Van Hove strain. $H_{c2}$ clearly has a stronger enhancement than $T_c$ when approaching the Van Hove strain, which is consistent with experimental observation. Right: For $p$-wave pairing, $H_{c2}$ and $T_c$ are not sensitive to the Van Hove singularity since we have assumed that the BCS interaction is strain independent and the $p$-wave gap function vanishes at the Van Hove singularity $(k_x,k_y)=(0,\pi )$. The ratio $H_{c2}/T_c$ does not change when approaching the Van Hove strain, which is inconsistent with experimental observation.

Figure 5

Results for a mixture of extended-$s$-wave and $d$-wave pairing channels in the three-band model. A particular set of interaction strengths $(g_d,g_s)$ has been chosen. Left: Pairing magnitude at $T=H=0$ for each pairing channel as a function of strain. Right: Critical temperature and critical field as a function of strain. The enhancement in $H_{c2}$ is stronger than that in $T_c$, which is consistent with the experimental observation.