Quantum oscillations without magnetic field

When the magnetic field $B$ is applied to a metal, nearly all observable quantities exhibit oscillations periodic in $1/B$. Such quantum oscillations reflect the fundamental reorganization of electron states into Landau levels as a canonical response of the metal to the applied magnetic field. We predict here that, remarkably, in the recently discovered Dirac and Weyl semimetals, quantum oscillations can occur in the complete absence of magnetic field. These zero-field quantum oscillations are driven by elastic strain which, in the space of the low-energy Dirac fermions, acts as a chiral gauge potential. We propose an experimental setup in which the strain in a thin film (or nanowire) can generate a pseudomagnetic field $b$ as large as 15 T and demonstrate the resulting de Haas–van Alphen and Shubnikov–de Haas oscillations periodic in $1/b$.

Figure 1

Proposed setup for strain-induced quantum oscillation observation in Dirac and Weyl semimetals. (a) The bent film is analogous, in terms of its low-energy properties, to an unstrained film subject to magnetic field $B$. (b) Detail of the atomic displacements in the bent film. Displacements have been exaggerated for clarity.

Figure 2

Schematically shown low-energy electron excitation spectrum in Dirac and Weyl semimetals. (a) In a Dirac semimetal the bands are doubly degenerate due to the spin degree of freedom, while in a Weyl semimetal they are nondegenerate. (b) Contours of constant energy for $k_y=0$. For magnetic field $\mathit{B}\parallel \widehat{y}$ these correspond to the extremal orbits [20] that give rise to QOs periodic in $1/B$.

Figure 3

Numerical results for the ${\mathrm{Cd}}_3{\mathrm{As}}_2$ lattice Hamiltonian. We model the Hamiltonian (2) in the presence of magnetic field $\mathit{B}=\widehat{y}B$ and strain-induced pseudomagnetic field $\mathit{b}=\widehat{y}b$. In all panels films of thickness 500 lattice points are studied with parameters appropriate for ${\mathrm{Cd}}_3{\mathrm{As}}_2$. The $p-h$ asymmetry terms ${\epsilon }_{\mathit{k}}$ are neglected for simplicity, which makes contributions from the two spin sectors identical. (a) Band structure and density of states (DOS) for zero field and zero strain. The inset shows the first Brillouin zone. (b) Band structure and normalized DOS for $B=1.5\mathrm{T}$. Red crosses indicate the peak positions expected on the basis of the Lifshitz-Onsager quantization condition [20]. (c) Band structure and DOS for $b=1.5\mathrm{T}$. The thin black line shows the expected bulk DOS for ideal Weyl dispersion computed from Eq. (8).

Figure 4

Strain-induced quantum oscillations. The top panel shows oscillations in DOS at energy 10 meV as a function of inverse strain strength expressed as $1/b$. For comparison, ordinary magnetic oscillations are displayed, as well as the result of the bulk continuum theory Eq. (8). Crosses indicate expected peak positions based on the Lifshitz-Onsager theory. The bottom panel shows oscillations in conductivity ${\sigma }_{yy}$ assuming a Fermi energy $E_F=10\mathrm{meV}$. To simulate the effect of disorder, all data are broadened by convolving in energy with a Lorentzian with width $\delta =0.25\mathrm{meV}$. The same geometry and parameters are used as in Fig. 3.