Interplay between altermagnetism and superconductivity in two dimensions: intertwined symmetries and singlet-triplet mixing

Abstract

We study the interplay between altermagnetism and unconventional superconductivity for the case of two-dimensional square- and triangular-lattice systems. Our approach is based on an effective single particle Hamiltonian which mimics the alternating spin splitting characteristic for the $d$-$wave$ and $i$-$wave$ altermagnetic state. By supplementing the model with intersite pairing term we characterize the principal features of the coexistent altermagnetic-superconducting state as well as the possibility of inducing the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. Our calculations show that the subtle interplay between the symmetries of the superconducting and altermagnetic order parameters as well as the shape/size of the Fermi surface lead to various types of anisotropic behaviors of the resultant non-zero momentum pairing, which has not been possible in the originally proposed FFLO state. Moreover, in the considered systems additional pairing symmetries appear leading to an exotic multi-component order parameter with singlet-triplet mixing. To interpret the obtained data we analyze the Cooper pair density in the momentum space and the corresponding Fermi wave vector mismatch resulting from the altermagnetic spin splitting. We discuss our result in the context of possible applications like, e.g., the superconducting diode.

Figures (14)

• Figure 1: The values of the direction dependent $\alpha_{ij}$ factor appearing in Eq. (\ref{['eq:H_t']}) for nearest neighbor hopping on a square lattice which leads to the $d$-$wave$ altermagnetic symmetry (a) and for the third nearest neighbor hopping on a triangular lattice which leads to the $i$-$wave$ altermagnetic symmetry (b). Additionally in (a) and (b) we show the nearest neighbor hoppings marked by black arrows. In (c) and (d) we show the resultant spin split Fermi surfaces corresponding to the situations visualized in (a) and (b), respectively.
• Figure 2: Cooper pair density $\gamma_{\mathbf{kQ_0}}=\gamma_{\mathbf{kQ_0}}^{\bar{\sigma}\sigma}$, for $\mathbf{Q}_0=(0,0)$, $t_{\mathrm{am}}=0$, and for two selected band fillings $n=0.12$ (a) and $n=0.64$ (b) where the $extended$$s$-$wave$ and $d$-$wave$ pairing symmetries are stable, respectively. The white dashed lines correspond to the so-called nodal lines where due to symmetry reasons the SC gap has to be zero. In (c) and (d) we show the free energy of the $extended$$s$-$wave$ (for $n=0.12$) and $d$-$wave$ (for $n=0.64$) superconducting states, respectively, together with the normal state energy, all as functions of $t_{\mathrm{am}}$. The yellow dashed line marks the transition point where SC becomes unstable and $E^{SC}>E^{NS}$.s
• Figure 3: Free energy of the $extended$$s$-$wave$ superconducting state as a function of Cooper pair momentum, obtained for $n$ = 0.12. Three different values of the $d$-$wave$ altermagnetic spin splitting $t_{am}$ have been selected: (a) $t_{am}=0.08|t|$; (b) $t_{am}=0.214|t|$ and (c) $t_{am}=0.216|t|$. All energies are shifted relative to energy for $\mathbf{Q}=(0,0)$-paired state. (d) The $\varepsilon_{F}=\varepsilon_{\pm\mathbf{k}\uparrow}$ and $\varepsilon_{F}=\varepsilon_{\mp\mathbf{k}\downarrow}$ Fermi surfaces with four regions of large Fermi wave vector mismatch marked by red color for the $t_{\mathrm{am}}$ value corresponding to the FF phase formation presented in (c).
• Figure 4: (a) Free energy of the $extended$$s$-$wave$ superconducting state as a function of $Q_d=Q_x=Q_y$, where the Cooper pair momentum has the form $\mathbf{Q}=(Q_x,Q_y)$. Dashed lines represents curves calculated for $t_{am}$ from range $0.214|t|$-$0.216|t|$, with step of $0.0002|t|$. (b) Values of $Q_d$ corresponding to the Cooper pair momentum minimizing free energy, as function of altermagnetic spin splitting amplitude. (c) Free energy of the SC (paired state with $\mathbf{Q}=0$), FF (paired state with $\mathbf{Q}\neq0$), and NS (normal state) as a function of $t_{\mathrm{am}}$. (d) Superconducting gap amplitudes versus $t_{am}$ close to the transition point between the SC and FF states.
• Figure 5: The $\varepsilon_{\mathbf{k}\sigma}$ and $\varepsilon_{\mathbf{-k+Q}\bar{\sigma}}$ Fermi surfaces [(a) and (c)] between which the pairing appears for the case of $d$-$wave$ altermagnetic spin splitting amplitude $t_{\mathrm{am}}=0.216|t|$, for which the FF phase is stable with non-zero centre-of-mass momentum of the Cooper pairs $\mathbf{Q}\approx(0.05,0.05)\;1/a$ (cf. Figs. \ref{['fig:cooper_square_n_0_12']}). The red color marks the regions in which significant Fermi wave vector mismach appears. The pair density $\gamma^{\bar{\sigma}\sigma}_{\mathbf{kQ}}$ for the same model parameters is shown in (b) and (d). Note that the depairing regions for which $\gamma^{\bar{\sigma}\sigma}_{\mathbf{kQ}}=0$ in (b) and (d) correspond to the areas of significant Fermi wave vector mismatch seen in (a) and (c). Due to the non-zero value of $\mathbf{Q}$ pairing takes place at a large extent of the Fermi surface.