Anomalous Phase Shift of Quantum Oscillations in 3D Topological Semimetals

Berry phase physics is closely related to a number of topological states of matter. Recently discovered topological semimetals are believed to host a nontrivial $\pi$ Berry phase to induce a phase shift of $\pm 1/8$ in the quantum oscillation ($+$ for hole and $-$ for electron carriers). We theoretically study the Shubnikov–de Haas oscillation of Weyl and Dirac semimetals, taking into account their topological nature and inter-Landau band scattering. For a Weyl semimetal with broken time-reversal symmetry, the phase shift is found to change nonmonotonically and go beyond known values of $\pm 1/8$ and $\pm 5/8$, as a function of the Fermi energy. For a Dirac semimetal or paramagnetic Weyl semimetal, time-reversal symmetry leads to a discrete phase shift of $\pm 1/8$ or $\pm 5/8$. Different from the previous works, we find that the topological band inversion can lead to beating patterns in the absence of Zeeman splitting. We also find the resistivity peaks should be assigned integers in the Landau index plot. Our findings may account for recent experiments in ${\mathrm{Cd}}_2{\mathrm{As}}_3$ and should be helpful for exploring the Berry phase in various 3D systems.

Figure 1

(a) The conduction and valence bands of the Weyl semimetal as a function of $k_z$ at $k_x=k_y=0$. [(b) and (c)] In the $z$-direction magnetic field, the Landau bands when $n=5$ and 5.5 [see dash lines in (d)]. (d) An example of the numerically calculated resistivities ${\rho }_{xx}$ and ${\rho }_{zz}$ as functions of $1/B$. Inset: The Landau index plot and linear fitting (line) using $n=F/B+\varphi$ to the peaks in ${\rho }_{xx}$ and ${\rho }_{zz}$. In this case, $F=3.927\pm 0.003$ and $\varphi =-1.052\pm 0.007$. The parameters are $k_w=0.1{\mathrm{nm}}^{-1}$, $A=0.5\mathrm{eV}\mathrm{nm}$, $M=5\mathrm{eV}$ ${\mathrm{nm}}^2$, and the Fermi energy $E_F=0.055\mathrm{eV}$.

Figure 2

For the Weyl semimetal with broken time-reversal symmetry. (a) The frequency $F$ obtained numerically (scatters) and analytically (solid curves) vs the Fermi energy $E_F$ for (a) different $A$ at a fixed $M$; and (b) for different $M$ at a fixed $A$. (c) The phase shift $\varphi$ vs $E_F$ for different $E_A=A{k}_w$ and a fixed $E_M=M{k}_w^2=0.05\mathrm{eV}$. The curves break because $F$ and $\varphi$ cannot be fitted when beating patterns form. The insets indicate the location of Fermi energy. The vertical dashed lines mark the Lifshitz point. $k_w=0.1{\mathrm{nm}}^{-1}$ throughout the work.

Figure 3

For the Dirac semimetal or Weyl semimetal with time-reversal symmetry. (a) $d{\sigma }_{zz}/d(1/B)$ as a function of $1/B$ for the Dirac semimetal and its Weyl components [$\mathcal{H}(\mathbf{k})$ and ${\mathcal{H}}^{*}(-\mathbf{k})$]. The parameters: $E_F=0.0954\mathrm{eV}$, $k_w=0.1$/nm, $A=2\mathrm{eV}\mathrm{nm}$, and $M=5\mathrm{eV}{\mathrm{nm}}^2$, so that $\min (E_A,E_M)<E_F<\max (E_A,E_M)$. (b) Each panel is the same as Fig. 2 but for the Dirac semimetal. The data break because $\varphi$ cannot be fitted when beating patterns form (highlighted area).