Phonon-induced rotation of the electronic nematic director in superconducting ${\mathrm{Bi}}_2{\mathrm{Se}}_3$

The doped topological insulator $A_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$, with $A=\{\mathrm{Cu},\mathrm{Sr},\mathrm{Nb}\}$, becomes a nematic superconductor below $T_c\approx 3$–4 K. The associated electronic nematic director is described by an angle $\alpha$ and is experimentally manifested in the elliptical shape of the in-plane critical magnetic field $H_{c2}$. Because of the threefold rotational symmetry of the lattice, $\alpha$ is expected to align with one of three high-symmetry directions corresponding to the in-plane nearest-neighbor bonds, consistent with a $Z_3$-Potts nematic transition. Here, we show that the nematic coupling to the acoustic phonons, which makes the nematic correlation length tend to diverge along certain directions only, can fundamentally alter this phenomenology in trigonal lattices. Compared to hexagonal lattices, the former possesses a sixth independent elastic constant $c_{14}$ due to the fact that the in-plane shear strain doublet $({\epsilon }_{xx}-{\epsilon }_{yy},-2{\epsilon }_{xy})$ and the out-of-plane shear strain doublet $(2{\epsilon }_{yz},-2{\epsilon }_{zx})$ transform as the same irreducible representation. We find that, when $c_{14}$ overcomes a threshold value, which is expected to be the case in doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, the nematic director $\alpha$ unlocks from the high-symmetry directions due to the competition between the quadratic phonon-mediated interaction and the cubic nematic anharmonicity. This implies the breaking of the residual in-plane twofold rotational symmetry ($C_{2x}$), resulting in a triclinic phase. We discuss the implications of these findings for the structure of nematic domains, for the shape of the in-plane $H_{c2}$ in $A_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$, and for the presence of nodes inside the superconducting state.

Figure 1

Schematic representation of the electronic nematic director in doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ compounds. Left: The threefold degenerate nematic directors aligned with the high-symmetry directions of the crystal, superimposed with the unit cell of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The latter displays a characteristic A-B-C stacking pattern for which, along a particular path along the $z$ direction, the A, B, C lattice sites are occupied by either $\mathrm{Bi}$ or $\mathrm{Se}$ atoms (see, e.g., Refs. [12, 44]). Right: When the coupling to acoustic phonons overcomes a threshold value, two effects occur: (i) a rotation of the nematic director away from the high-symmetry directions, and (ii) a splitting of the original director into two, which move towards six “antisymmetry directions.” As a result, the number of nonidentical directors doubles from three to six, and the system loses its residual in-plane twofold rotational symmetry $C_{2x}$ inside the nematic phase.

Figure 2

Phase diagram for the nematic director angle ${\alpha }_0$ with respect to the parameters $|{\tilde{c}}_{E3}|\propto |c_{14}|$ and $({\tilde{\kappa }}_1,{\tilde{\kappa }}_2)={\tilde{\kappa }}_0(cos{\varphi }_{\kappa },sin{\varphi }_{\kappa })$. (a) The absolute value ${\tilde{\kappa }}_0$ changes for fixed ${\varphi }_k=\pi /2$ (left horizontal axis) and ${\varphi }_k=0$ (right horizontal axis). (b) The absolute value is fixed at ${\tilde{\kappa }}_0=0.6$ and the relative angle ${\varphi }_{\kappa }\in [0,\pi /2]$ is varied. There are three distinct regimes: (i) For small values of $|c_{E3}|$, and when ${\kappa }_1$ (or ${\kappa }_2$) is much larger than ${\kappa }_2$ (or ${\kappa }_1$), the nematic director is locked at the high-symmetry directions ${\alpha }_s$ (red region). (ii) For large values of $|c_{E3}|$, regardless of ${\tilde{\kappa }}_i$, the nematic director aligns with the “antisymmetry directions” ${\alpha }_{as}$ (blue region). (iii) For intermediate values of $|c_{E3}|$, or when ${\kappa }_1$ and ${\kappa }_2$ are comparable, the nematic director evolves smoothly between ${\alpha }_s<{\alpha }_0<{\alpha }_{as}$ (green region). The full dependence of ${\alpha }_0$ and ${\widehat{\mathit{q}}}_0$ on $|{\tilde{c}}_{E3}|$, for a given maximum of $R({\widehat{\mathit{q}}}_0,{\alpha }_0)$, is shown in Fig. 3 for the $({\tilde{\kappa }}_1,{\tilde{\kappa }}_2)$ values corresponding to the two red dashed lines. For the six parameter values corresponding to the red stars, the function $R({\widehat{\mathit{q}}}_0,{\alpha }_0\left({\widehat{\mathit{q}}}_0\right))$ is plotted in Fig. 4. The horizontal black dotted line denotes the expected value of ${\tilde{c}}_{E3}=-0.265$ for doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ [53]. The topmost light blue horizontal dashed line denotes the limit of structural stability as given by the condition $|c_{E3}{|}_{\max }=\sqrt{c_{E1}{c}_{E2}}$ [see Eq. (10)]. The analytical thresholds stem from the calculations presented in Sec. 3b4.

Figure 3

The soft momentum-space direction ${\widehat{\mathit{q}}}_0$ (parametrized by the polar angle ${\theta }_0$ and the azimuthal angle ${\phi }_0$) and the nematic director angle ${\alpha }_0$ associated with a particular maximum of $R({\widehat{\mathit{q}}}_0,{\alpha }_0)$ are plotted as a function of $|{\tilde{c}}_{E3}|$ for the two indicated $({\tilde{\kappa }}_1,{\tilde{\kappa }}_2)$ values, which correspond to the red dashed lines in Fig. 2.

Figure 4

The function $R(\widehat{\mathit{q}},{\alpha }_0\left(\widehat{\mathit{q}}\right))$ is plotted as a function of the momentum-space polar angle $\theta$ and azimuthal angle $\phi$. Because of the relationship in Eq. (34), we consider only the ranges $\phi \in [0,\pi )$ and $\theta \in [0,\pi ]$. The extremal director angle value ${\alpha }_0\left(\widehat{\mathit{q}}\right)\in [-\frac{\pi }{3},\frac{\pi }{3}]$ is shown at every maximum; note that the other nematic director angles outside of this interval can be obtained from Eq. (35). The value of $|{\tilde{c}}_{E3}|$ increases upon moving from the left to the right panels, encompassing the three regions of the phase diagram of Fig. 2, as indicated by the red stars. The top (bottom) panels correspond to a nonzero ${\kappa }_1$ (${\kappa }_2$). The black arrows indicate the maxima previously presented in Fig. 3.

Figure 5

The renormalized mass (37) as a function of the momentum directions $\widehat{\mathit{q}}=\widehat{\mathit{q}}(\phi ,\theta )$ with the respective maximized director angle ${\alpha }_0\left(\widehat{\mathit{q}}\right)$ inserted according to Eq. (43). The renormalized mass attains its maximum value $M[\widehat{\mathit{q}},{\alpha }_0\left(\widehat{\mathit{q}}\right)]=M_{\mathrm{stat}}$ for the 12 directions ${\widehat{\mathit{q}}}_{1,2}$ [see Eqs. (44) and (45)], indicated by the black dots. The displayed features do not depend on the chosen parameters; for this plot, we set $({\tilde{\kappa }}_1,{\tilde{\kappa }}_2)=(2.7,0.9)$ and use the elastic constant values for ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The static mass $M_{\mathrm{stat}}$ is defined in Eq. (41).

Figure 6

The ratio of the renormalized mass (37) and the static mass (41) as a function of $c_{E3}$ for the two maxima of the function $R(\widehat{\mathit{q}},{\alpha }_0)$ presented in Fig. 3. The renormalized mass is maximum, i.e., $M[{\widehat{\mathit{q}}}_0,{\alpha }_0]=M_{\mathrm{stat}}$, for $c_{E3}=0$, and above the upper threshold value beyond which the acoustic phonon contribution is dominant over the anharmonic contribution.

Figure 7

The analytical function $R\left({\alpha }_0\right)$, given by Eq. (82), for four distinct values of the parameter $\widehat{R}=R_1/2R_2$. Due to condition (31), relevant for $\widehat{R}\ge 0$, the solution is restricted to either the blue or the gray regions. The evolution of the maximum is indicated by the black arrows as $\widehat{R}$ is decreased—or, likewise, $|c_{E3}|$ is increased. The two threshold values are given by $\widehat{R}=1$ and $\widehat{R}=0$.

Figure 8

The nematic director ${\alpha }_0$ that maximizes Eq. (82) as a function of $\widehat{R}=R_1/2R_2$. The trends displayed here coincide with those obtained from the numerical solutions depicted in Fig. 3.

Figure 9

Shape of the in-plane upper critical field $H_{c2}$ superimposed to the schematic distortion of the unit cell. (a) Without nematic order, the upper critical field is sixfold symmetric and respects all of the trigonal point-group symmetries. (b) With nematic order and $\alpha ={\alpha }_s$, the shape of $H_{c2}$ is a twofold symmetric ellipse, and the accompanying lattice distortion is monoclinic. (c) With a rotated nematic director ($\alpha \ne {\alpha }_s$), the residual in-plane twofold symmetry of $H_{c2}$ is lost, and the accompanying lattice distortion is triclinic. The main axis of $H_{c2}$ rotates by $-\alpha /2$ (red line), and the ellipse is deformed, as highlighted in the inset. The inset shows the behavior of the functions $H_{c2}({\phi }_{\mathrm{m}}\pm \delta \phi )$, which correspond to a clockwise and a counterclockwise sweeping of the $H_{c2}$ curve starting from the angle ${\phi }_{\mathrm{m}}$ where $H_{c2}$ is maximum. If a residual twofold symmetry was present, the two curves would overlap.

Figure 10

In the case $\alpha ={\alpha }_s$, the system is expected to form one majority nematic domain, e.g., $\alpha =0$ that is accompanied by minority domains randomly composed of the remaining two states, $\alpha \in \{\frac{2\pi }{3},\frac{4\pi }{3}\}$. In the case $\alpha \ne {\alpha }_s$, there are six nonequivalent nematic directors. The system establishes one majority domain, e.g., $\alpha =\delta$ with $0<\delta <\pi /3$. Then, the surface energy $\sigma (\delta ,-\delta )$ between adjacent domains, $\alpha =\delta$ and $\alpha =-\delta$, is expected to be smaller than the surface energy between other domains, e.g., $\sigma (\delta ,\frac{2\pi }{3}+\delta )$. As a result, the minority domains are expected to be dominated by the $\alpha =-\delta$ domains.