Searching for two-dimensional Weyl superconductors in heterostructures

The two-dimensional Weyl superconductor is the most elusive member of a group of materials with Weyl fermions as low-energy excitations. Here, we propose to realize this state in a heterostructure consisting of thin films of half-metal and spin-singlet superconductors. In particular, for the $d$-wave case, a very robust two-dimensional Weyl superconductor ($d$WSC) is realized independently of the orientation of the spontaneous magnetization of the half metal. The quasiparticle spectra of the $d$WSC show interesting evolution with the direction of the magnetization, featured by a series of Lifshitz transitions in the zero-energy contour of the quasiparticle spectrum. In addition, we find a transition between type-I and type-II Weyl nodes. This is an example of a two-dimensional type-II Weyl node in the presence of a superconducting correlation. For a general magnetization orientation of the half metal, the state is a combination of a superconducting component and a normal fluid component and is different from all known forms of pairings. The symmetries and topological properties of the system are analyzed. We also study the phases in the heterostructure with the half metal replaced by a ferromagnetic metal with a partially spin-polarized Fermi surface.

Figure 1

Schematic drawing of a heterostructure consisting of a half-metal (HM) thin film sandwiched between an $s$-wave or $d$-wave spin-singlet superconductor and an insulating substrate, viewed laterally. $m$ and $\theta$ represent the magnitude and direction of the exchange field (magnetization) in the HM, which is assumed to lie on the $xz$ plane.

Figure 2

Bulk [(a), (b), (c), (d), (e)] and edge [(f), (g), (h), (i), (j)] state spectra of the $d$WSC with $\eta \left(\mathbf{k}\right)=cos{k}_x-cos{k}_y$, for a typical set of parameters, $m=t=1$, $\lambda =0.2t$, ${\mathrm{\Delta }}_0=0.5t$, $\mu =-4.6t$. For the bulk system, the dispersions of the $E_{-1}\left(\mathbf{k}\right)$ and $E_1\left(\mathbf{k}\right)$ bands are shown along four lines in the 2D BZ with fixed $k_x$ values at 0 (black lines), $\frac{1}{2}{k}_{x0}$ (blue lines), $k_{x0}$ (red lines), and $\frac{5}{4}{k}_{x0}$ (green lines). $k_{x0}>0$ is defined by Eq. (10). The edge state spectra correspond to a strip with 1500 unit cells along the $x$ direction. $\theta =0$ for (a) and (f), $\theta ={\theta }_{c1}$ for (b) and (g), $\theta ={\theta }_{c2}$ for (c) and (h), $\theta =0.25\pi$ for (d) and (i), $\theta =0.5\pi$ for (e) and (j).

Figure 3

Bulk [(a), (b), (c), (d), (e)] and edge [(f), (g), (h), (i), (j)] state spectra of the $d$WSC with $\eta \left(\mathbf{k}\right)=sin{k}_{x}sin{k}_y$, for a typical set of parameters, $m=t=1$, $\lambda =0.2t$, ${\mathrm{\Delta }}_0=0.5t$, $\mu =-4.6t$. For the bulk system, the dispersions of the $E_{-1}\left(\mathbf{k}\right)$ and $E_1\left(\mathbf{k}\right)$ bands are shown along three lines in the 2D BZ with fixed $k_x$ values at 0 (black lines), $k_{x0}$ (blue lines), and $k_{xc}$ (red lines). The edge state spectra correspond to a strip with 1500 unit cells along the $x$ direction. $\theta =0$ for (a) and (f), $\theta ={\theta }_{c1}^{\prime }$ for (b) and (g), $\theta ={\theta }_{c2}^{\prime }$ for (c) and (h), $\theta =0.25\pi$ for (d) and (i), $\theta =0.5\pi$ for (e) and (j).

Figure 4

The spectral functions for the left [(a), (c), (e), (g), (i)] and right [(b), (d), (f), (h), (j)] edges of a strip of the heterostructure with $\eta \left(\mathbf{k}\right)=cos{k}_x-cos{k}_y$. Five values of $\theta$ are considered, including $\theta =0$ for (a) and (b), $\theta ={\theta }_{c1}$ for (c) and (d), $\theta ={\theta }_{c2}$ for (e) and (f), $\theta =0.25\pi$ for (g) and (h), $\theta =0.5\pi$ for (i) and (j). The parameters are taken as $m=t=1$, $\lambda =0.2t$, ${\mathrm{\Delta }}_0=0.5t$, $\mu =-4.6t$. The darker the color, the larger the spectral weight.

Figure 5

The spectral functions for the left [(a), (c), (e), (g), (i)] and right [(b), (d), (f), (h), (j)] edges of a strip of the heterostructure with $\eta \left(\mathbf{k}\right)=sin{k}_{x}sin{k}_y$. Five values of $\theta$ are considered, including $\theta =0$ for (a) and (b), $\theta ={\theta }_{c1}^{\prime }$ for (c) and (d), $\theta ={\theta }_{c2}^{\prime }$ for (e) and (f), $\theta =0.25\pi$ for (g) and (h), $\theta =0.5\pi$ for (i) and (j). The parameters are taken as $m=t=1$, $\lambda =0.2t$, ${\mathrm{\Delta }}_0=0.5t$, $\mu =-4.6t$. The darker the color, the larger the spectral weight.

Figure 6

Density of states (DOS) on the two edges of a strip for $\eta \left(\mathbf{k}\right)=cos{k}_x-cos{k}_y$ [(a), (b), (c), (d), (e)] and for $\eta \left(\mathbf{k}\right)=sin{k}_{x}sin{k}_y$ [(f), (g), (h), (i), (j)]. Results for the left and right edges are assigned respectively with a label of $n_x=1$ and $n_x=N_x$. In the calculation, the width of the strip ($N_x$) is assumed as infinite so that there is no finite-size effect. Parameters are taken as $m=t=1$, $\lambda =0.2t$, ${\mathrm{\Delta }}_0=0.5t$, $\mu =-4.6t$. The angles used are (a) $\theta =0$, (b) $\theta ={\theta }_{c1}$, (c) $\theta ={\theta }_{c2}$, (d) $\theta =\frac{\pi }{4}$, (e) $\theta =\frac{\pi }{2}$, (f) $\theta =0$, (g) $\theta ={\theta }_{c1}^{\prime }$, (h) $\theta ={\theta }_{c2}^{\prime }$, (i) $\theta =\frac{\pi }{4}$, and (j) $\theta =\frac{\pi }{2}$. See main text for the definitions of the angles.

Figure 7

The $E=0$ equal-energy contours for the bulk quasiparticle bands. $\eta \left(\mathbf{k}\right)=cos{k}_x-cos{k}_y$ for [(a), (b), (c), (d), (e)] and $\eta \left(\mathbf{k}\right)=sin{k}_{x}sin{k}_y$ for [(f), (g), (h), (i), (j)]. Parameters are taken as $m=t=1$, $\lambda =0.2t$, ${\mathrm{\Delta }}_0=0.5t$, $\mu =-4.6t$. The angles used are (a) $\theta =0$, (b) $\theta ={\theta }_{c1}$, (c) $\theta ={\theta }_{c2}$, (d) $\theta =\frac{\pi }{4}$, (e) $\theta =\frac{\pi }{2}$, (f) $\theta =0$, (g) $\theta ={\theta }_{c1}^{\prime }$, (h) $\theta ={\theta }_{c2}^{\prime }$, (i) $\theta =\frac{\pi }{4}$, and (j) $\theta =\frac{\pi }{2}$. See main text for the definitions of the angles. The numberings of the quasiparticle bands to which the pockets belong are indicated in the figures.

Figure 8

Nonzero Zak phases for the two low-energy quasiparticle bands $E_{\pm 1}\left(\mathbf{k}\right)$, at $\theta =\pi /2$. $\eta \left(\mathbf{k}\right)=cos{k}_x-cos{k}_y$ for (a), and $\eta \left(\mathbf{k}\right)=sin{k}_{x}sin{k}_y$ for (b). The parameters are the same as those used in Sec. 3.

Figure 9

Spectral functions for the two edges of a strip of the system described by Eq. (3), with $\eta \left(\mathbf{k}\right)=1$. (a) $\theta =\pi /2$ which corresponds to an $s$WSC; results for the two edges are the same. (b) and (c) are spectral functions on the left ($n_x=1$) and right ($n_x=N_x$) edges of the strip for $\theta =0$. The parameters used are $t=1$, $m=t$, $\lambda =0.2t$, ${\mathrm{\Delta }}_0=0.2t$, $\mu =-2.6t$. The darker the color, the larger the spectral weight.

Figure 10

Spectral functions for the left [(a), (c)] and right [(b), (d)] edges of a strip of the heterostructure described by Eq. (3), with $\eta \left(\mathbf{k}\right)=cos{k}_x-cos{k}_y$ [(a), (b)] and $\eta \left(\mathbf{k}\right)=sin{k}_{x}sin{k}_y$ [(c), (d)]. The angle is taken as $\theta =0$. The parameters are chosen as $t=1$, $m=t$, $\lambda =0.2t$, ${\mathrm{\Delta }}_0=0.5t$, $\mu =-2.6t$. Darker color means larger spectral function.