Quantum magnetic oscillations in Weyl semimetals with tilted nodes

A Weyl semimetal (WSM) is a three-dimensional topological phase of matter where pairs of nondegenerate bands cross at isolated points in the Brillouin zone called Weyl nodes. Near these points the electronic dispersion is gapless and linear. A magnetic field $B$ changes this dispersion into a set of positive and negative energy Landau levels which are dispersive along the direction of the magnetic field only. In this set, the $n=0$ Landau level is special since its dispersion is linear and unidirectional. The presence of this chiral level distinguishes Weyl from Schrödinger fermions. In this paper we study the quantum oscillations of the orbital magnetization and magnetic susceptibility in Weyl semimetals. We generalize earlier works on these de Haas–van Alphen oscillations by considering the effect of a tilt of the Weyl nodes. We study how the fundamental period of the oscillations in the small $B$ limit and the strength of the magnetic field $B_1$ required to reach the quantum limit (i.e., where the Fermi level is lying in the chiral level) are modified by the magnitude and orientation of the tilt vector $\mathbf{t}$. We show that the magnetization from a single node is finite in the $B\to 0$ limit. Its sign depends on the product of the chirality and sign of the tilt component along the magnetic field direction. We also study the magnetic oscillations from a pair of Weyl nodes with opposite chirality and with opposite or identical tilt. Our calculation shows that these two cases lead to a very different behavior of the magnetization in the small and large $B$ limits. We finally consider the effect of an energy shift $\pm {\mathrm{\Delta }}_0$ of a pair of Weyl nodes on the magnetic oscillations. We assume a constant density of carriers so that both nodes share a common Fermi level and the density of carriers is constantly redistributed between the two nodes as the magnetic field is varied. Our calculation can easily be extended to a WSM with an arbitrary number of pairs of Weyl nodes.

Figure 1

Energy in units of $\hslash v_F/\ell$ for the first Landau levels for two nodes with opposite chirality and bias. Parameters are ${\chi }_1=-1,Q_{0,1}\ell =0.5$ for the node on the left and ${\chi }_2=+1,Q_{0,2}\ell =-0.5$ for the node on the right and (a) $t_{1,z}=t_{2,z}=0.4$ and (b) $t_{1,z}=-t_{2,z}=0.4$. The blue lines and the black dots are the chiral level and Dirac point in each node. The separation between the nodes is arbitrary.

Figure 2

Density of states as a function of the energy $e$ for a single node with zero bias, chirality $\chi =-1$, and for $t_z=\pm 0.4$. The black line is the $B=0,t_z=\pm 0.4$ result which does not depend on the sign of $t_z$.

Figure 3

(a) Magnetic susceptibility at zero bias from a single node for different chiralities and sign of the $z$ component of the tilt vector. The green line is for a node where the chiral level has been artificially removed. The large $B$ behavior is shown in the inset. (b) Magnetisation for a node with chirality $-1$ and tilt vector $t_z=0,0.4,-0.4$ showing the different behaviors in the small $B$ limit.

Figure 4

Angular dependence $\mathrm{{\rm Y}}(t,\theta )$ of $1/B_1$ for tilts $t=0$ and $t_z=0.4$ and both chiralities.

Figure 5

The function $P(\mathbf{t},n)$ as a function of $n$ for different values of the chirality and tilt. The full lines gives the limit ${lim}_{n\to \infty }P(\mathbf{t},\chi ,n)$.

Figure 6

Angular dependence of the function $\mathrm{\Gamma }(\mathbf{t},\theta )$ entering in the fundamental period of the magnetic oscillations.

Figure 7

Density of states for the two WSMs for bias $Q_0\ell =0.5$ and tilt $t_z=0.6$.

Figure 8

Quantum oscillations as a function of the inverse magnetic field for a WSM with zero tilt and for different values of the bias ${\mathrm{\Delta }}_0$: (a) Fermi level, (b) node densities, (c) magnetization, and (d) magnetic susceptibility. The dashed lines in (a) are set at the different values of ${\mathrm{\Delta }}_0$. The density is $n_e=2\times {10}^{22}{\mathrm{m}}^{-3}$.

Figure 9

Susceptibility per carrier calculated for WSM1 for dependent and independent nodes assuming the same total density $n_e=2\times {10}^{22}{\mathrm{m}}^{-3}$, bias ${\mathrm{\Delta }}_0=2$ meV, and tilt $t_z=0.5$. The number above or below each peak indicates the Landau level that is crossed by the Fermi level. Black (blue) numbers are for dependent (independent) nodes and $n\left(n^{\prime }\right)$ stand for node 2 (1). Node 1 (2) is shifted upward (downward) in energy (see Table 1).

Figure 10

Effect of a finite tilt on (a) the magnetic susceptibility and (b) magnetization of both WSMs at zero bias$.$

Figure 11

Position of the first peak in $1/B$ of the quantum oscillations for WSM1 and WSM2.