Highly efficient superconducting diode effect in unconventional $p$-wave magnets

Abstract

We investigate the emergence of superconducting phases, both with zero and finite Cooper-pair center of mass momenta, in recently proposed unconventional $p$-wave magnets. As a consequence, we find that, while these magnetic phases are in principle compatible with a conventional pairing state at zero field, a Fulde-Ferrell phase can generally be promoted as the leading instability under the application of a finite magnetic field. Interestingly, by calculating the efficiency of the superconducting diode effect of this finite momentum pairing state via a Ginzburg-Landau theory, we uncover that a high efficiency can be obtained in these systems for experimentally relevant spin splittings. Therefore, our prediction reveals that the experimental discovery of these new materials represents a promising platform for the construction of energy-efficient logic circuits that can potentially be used, e.g., in the fields of classical and quantum computing.

Figures (8)

• Figure 1: Schematic representation of the spin splitting of the Fermi surfaces in a UPM, for (a) zero magnetic field and (b) an out-of-plane magnetic field given by $\mathbf{B}=B_z \boldsymbol{\hat{z}}$. The spin polarization of the Fermi surfaces displayed by the present model only assumes one of the values $\langle \sigma_z\rangle=\pm \hbar/2$, indicated, respectively, by the colors red and blue in the FS. Note that the Fermi surfaces are always displaced from the center of the zone for a finite $\lambda$, indicated by the wavevector $\mathbf{k}_0 = 2 \frac{m \lambda}{\hbar^2} \boldsymbol{\hat{x}}$. The nodes of the FS are present up to a critical out-of-plane field given by $B_c = \pm \abs{\lambda} \sqrt{\frac{2m\mu}{\hbar^2}}$, where $\mu$ is the chemical potential. Also, we point out that any infinitesimal in-plane magnetic field leads instead to nodeless Fermi surfaces.
• Figure 2: Second-order GL coefficient [Eq. \ref{['FreeEnergy']}] for the $s$-wave FF state in the present model. Here, we choose $\lambda/T_{c,0}=1$, $B/T_{c,0}=1$, and $T=0.1T_{c,0}$. The symbol "$\boldsymbol{\times}$" marks the global minimum of the GL free energy, which yields $\mathbf{Q}\approx -0.7 \mathbf{\hat{x}}$ in this case. The "half-moon" pattern of this plot stems from both time-reversal and inversion symmetry breakings.
• Figure 3: (a) Schematic pairing phase diagram of $B_z$ vs. $\lambda$ in the present model. The white region denotes the UPM phase, the red region stands for the spin-singlet $s$-wave FF phase and the blue region corresponds to the spin-singlet $s$-wave zero-momentum SC phase. (b) Efficiency $\eta$ of the SDE [Eq. \ref{['efficiency']}] for the FF phase with modulation along the direction of the minimizing wave vector $\mathbf{Q}$, as a function of the UPM spin splitting $\lambda$ for a finite magnetic field such that $B_z/T^{(s)}_{c,0} =1$. We also show the behavior of $\eta$ in the model for different temperatures $T$. (c) Supercurrent as a function of the momentum $q_x$ for different values of the spin splitting around the maximum of the efficiency, indicated by three circles shown in (b), at $T=0.01T^{(s)}_{c,0}$.
• Figure S1: Second-order coefficient of the GL free energy for the singlet [(a)-(d)] and triplet [(e)] pairing states with fixed temperature $T=0.1T^{(s)}_{c,0}$ and splitting $\lambda/T^{(s)}_{c,0}=1$. Upper panels (a) and (b) are for $s$-wave states with $B/T^{(s)}_{c,0}=0$ and $B/T^{(s)}_{c,0}=1$, respectively. In the latter case, note the emergence of a single nonzero minimum in the right panel, given by $\mathbf{Q}=-0.7 \widehat{\mathbf{x}}$. Bottom panels (c) and (d) are for $d$-wave states with $B/T^{(s)}_{c,0}=0$ and $B/T^{(s)}_{c,0}=1$, respectively. In the latter case, note the emergence of a single nonzero minimum in the right panel, given by $\mathbf{Q}=-0.7 \widehat{\mathbf{x}}$, so the only possible modulation is the FF one. Lower panel (e) is for helical $p$-wave state, showing the emergence of two minima $\pm \mathbf{Q}$, related by a $C_{2z}$ rotation.
• Figure S2: Fourth-order Feynman diagrams for (a) spin-singlet FF, (b) spin-triplet FF and (c) spin-triplet LO order. The Feynman rules for the vertices are extracted from Eq. \ref{['GeneralFourthOrder']} as: $\bullet \equiv d_0(\mathbf{k}) \sigma_y$ for incoming external legs, $\bullet \equiv [d_0(\mathbf{k}) \sigma_y]^{\dagger}$ for outgoing external legs, $\blacksquare \equiv \mathbf{d}(\mathbf{k}) \cdot \boldsymbol{\sigma} \sigma_y$ for incoming external legs, $\blacksquare \equiv [\mathbf{d}(\mathbf{k}) \cdot \boldsymbol{\sigma} \sigma_y]^\dagger$ for outgoing external legs, $\square \equiv \mathbf{d}(\mathbf{k}+\mathbf{Q}) \cdot \boldsymbol{\sigma} \sigma_y$ for incoming external legs and $\square \equiv [\mathbf{d}(\mathbf{k}+\mathbf{Q}) \cdot \boldsymbol{\sigma} \sigma_y]^\dagger$ for outgoing external legs. Since this is a pairing channel, the following similarity transformation $G^{(h)}(\mathbf{k},i\omega_n) = - [G^{(p)}(-\mathbf{k},-i\omega_n)]^{T}$ was used to derive these diagrams.