Efficiency of the superconducting diode effect of pair-density-wave states in two-dimensional $d$-wave altermagnets

Abstract

We systematically study the efficiency of the intrinsic superconducting diode effect of several pair-density-wave states that can emerge in two-dimensional $d$-wave metallic altermagnets. To this end, we investigate several scenarios using an effective minimal microscopic model and Ginzburg-Landau analysis in order to derive the corresponding pairing phase diagrams. In addition, we examine also whether the presence of a Rashba spin-orbit coupling and an applied external magnetic field are beneficial to this effect in these systems. As a consequence, our results add further support to the fact that altermagnetic materials indeed provide a good platform for the pursuit of finite-momentum superconductivity, which can lead to an optimization of the diode efficiency in some physically interesting situations. The latter phenomenon has been recently proposed to be key in improving the applicability of new energy-efficient quantum electronic devices.

Figures (6)

• Figure 1: Spin texture of the Fermi surfaces (FS) displayed by the present model described by Eqs. \ref{['FreeH']} to \ref{['MagneticH']} with (a) $t_{{AM}}=0.7$, $\alpha_{{R}}=\mathbf{B}=0$ and (b) $t_{{AM}}=0.7$, $\alpha_{{R}}=0.6$ and $\mathbf{B}=0.7 \mathbf{\hat{x}}$. As can be seen here, the finite in-plane magnetic field shifts the centers of the inner and outer FS in opposite directions by the vectors $\pm\mathbf{k_0} =\pm \boldsymbol{\hat{z}}\times \mathbf{B}/v_F$, such that the distance between the two centers is given by $2|\mathbf{k_0}|$. The color of the FS represents the magnitude of the spin polarization $\langle \sigma_z\rangle$.
• Figure 2: Second-order coefficient $\mathit{a}(\mathbf{q})$ from Eq. \ref{['free-energy']} in the case of an $s$-wave PDW as a function of the momentum $\mathbf{q}$ for (a) $t_{AM} = 0.2$ and $\alpha_R = \mathbf{B}=0$ and for (b) $t_{AM} = \alpha_R = 0.2$ and $\mathbf{B}\approx 0.9 B_{\text{P}} \mathbf{\hat{x}}$. The black dots indicate the global minima of the free energy, identifying $\mathbf{q}_0$. In (a), we have $\mathbf{q}_0=(\pm0.15,\pm0.15)$, while in (b) we have only one minimum located at the incommensurate momentum $\mathbf{q}_0=(0.09,0)$ that is proportional to $\mathbf{B}$.
• Figure 3: Phase diagram and diode efficiency as a function of the altermagnetic splitting $t_{\text{AM}}$ in the pure altermagnetic case, i.e., $\alpha_{R}=\mathbf{B} = 0$. The ranges of $t_{AM}$ with different colors correspond to different $d_{x^2-y^2}$-wave PDW phases, which are stabilized at low temperatures in the present model. The label "N" stands for the normal phase. Note that the SDE efficiency is finite only within the FF$^*$ phase.
• Figure 4: Magnitude of the in-plane Pauli magnetic field along the $x$ direction of the lattice (i.e., $\mathbf{B}_{\text{P}} = B_{\text{P}} \mathbf{\hat{x}}$) normalized by its value $B_{\text{P},0}$ at zero AM splitting, as a function of the RSOC $\alpha_{{R}}$ for different values of $t_{AM}$. Note that, for $\alpha_{{R}}=0$, the Pauli field becomes suppressed as a consequence of the paramagnetic limiting, while the parameter $\alpha_{{R}}$ increases the Pauli field, even for large AM splitting.
• Figure 5: Phase diagram and diode efficiency as a function of $t_{{AM}}$ for the $d_{x^2-y^2}$-wave PDW scenario for $\alpha_{{R}}=0.2$ and $\mathbf{B} = 0.9 B_{\text{P}} \mathbf{\hat{x}}$. The label "N" stands for the normal phase.