Energy quantization at the three-quarter Dirac point in a magnetic field

The quantization of the energy in a magnetic field (Landau quantization) at a three-quarter Dirac point is studied theoretically. The three-quarter Dirac point is realized in the system of massless Dirac fermions with the critically tilted Dirac cone in one direction, where a linear term disappears and a quadratic term ${\alpha }_2{q}_x^2$ with a constant ${\alpha }_2$ plays an important role. The energy is obtained as $E_n\propto {\alpha }_2^{\frac{3}{5}}{\left(nB\right)}^{\frac{4}{5}}$, where $n=1,2,3,\cdots$, by means of numerically solving the differential equation. The same result is obtained analytically by adopting an approximation. The result is consistent with the semiclassical quantization rule studied previously. The existence of the $n=0$ state is studied by introducing the energy gap due to the inversion-symmetry-breaking term, and it is obtained that the $n=0$ state exists in one of a pair of three-quarter Dirac points, depending on the direction of the magnetic field when the energy gap is finite.

Figure 1

Energy as a function of $q_x$ and $q_y$ at $B=0$ at the three-quarter Dirac point. Parameters are $w_x=0.4$, $w_y=1$, $w_{0x}=-w_x$, $w_{0y}=0$, ${\alpha }_2^{\prime }={\alpha }_2^{″}=0.01$.

Figure 2

Wave functions in the three-quarter Dirac point obtained numerically with $E/\left(w_y\sqrt{\hslash eB}\right)=0.3$ (a), 0.31 (b), 0.32 (c), and 0.33 (d). Parameters are $w_x=w_y=1$, $w_{0x}=-1$, $w_{0y}=0$, ${\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y={\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y=0.01$, and $B=1$. The boundary condition at $y=20$ is taken to be Eq. (29).

Figure 3

Wave functions in the three-quarter Dirac point obtained numerically with $E/\left(w_y\sqrt{\hslash eB}\right)=0.3105$ (a), and 0.3108 (b). Parameters are $w_x=w_y=1$, $w_{0x}=-1$, $w_{0y}=0$, ${\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y={\alpha }_2^{″}\sqrt{\hslash eB}/w_y=0.01$, and $B=1$. The boundary condition at $y=20$ is taken to be Eq. (29).

Figure 4

Wave functions in the three-quarter Dirac point obtained numerically with $E/\left(w_y\sqrt{\hslash eB}\right)=0.52906$ (a), 0.529065 (b), and 0.52907 (c). Parameters are $w_x=w_y=1$, $w_{0x}=-1$, $w_{0y}=0$, ${\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y={\alpha }_2^{″}\sqrt{\hslash eB}/w_y=0.01$, and $B=1$. The boundary condition at $y=20$ is taken to be Eq. (29).

Figure 5

Boundary condition $\theta$ at $y=20$ [Eq. (29)] as a function of energy for $w_x=w_y=1$, $w_{0x}=-1$, $w_{0y}=0$, ${\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y={\alpha }_2^{″}\sqrt{\hslash eB}/w_y=0.01$, and $B=1$. (b) and (c) are the close-up of (a) near the energy of the almost-localized states at $y\lesssim 0$.

Figure 6

Green lines are ${\varepsilon }_{{\mathrm{tqD}}_{\pm }}(q_x,q_y=0)/\left(w_y\sqrt{\hslash eB}\right)$ as a function of $y=-q_x/\left(\ell eB\right)=-q_x/\sqrt{\hslash eB}$, i.e. $E/\left(w_y\sqrt{\hslash eB}\right)=({\alpha }_2\sqrt{\hslash eB}/w_y)y^2$ and $E/\left(w_y\sqrt{\hslash eB}\right)=(2w_x/w_y)y+({\tilde{\alpha }}_2\sqrt{\hslash eB}/w_y)y^2$. Classical particles can exist in the cyan-shaded regions. Wave functions of almost-localized state with the quantum number $n=1$, 2, 3, and 4 near the three-quarter Dirac point are plotted as functions of $y$. Zero of the wave functions are shifted to their energies. Wave functions are calculated with the boundary conditions at $y=-15$ as in Sec. 2c. Parameters are $B=1$, $w_x=w_y=1$, $w_{0x}=-1$, $w_{0y}=0$ in (a) and (b), and ${\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y={\alpha }_2^{″}\sqrt{\hslash eB}/w_y=0.005$ (${\alpha }_2\sqrt{\hslash eB}/w_y=0.01$) in (a) and ${\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y={\alpha }_2^{″}\sqrt{\hslash eB}/w_y=0.0025$ (${\alpha }_2\sqrt{\hslash eB}/w_y=0.005$) in (b).

Figure 7

Wave function ${\mathrm{\Psi }}_1\left(y\right)$ in the three-quarter Dirac point obtained numerically by using the boundary condition at $y=-10$. Parameters are $w_x=w_y=1$, $w_{0x}=-1$, $w_{0y}=0$, ${\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y={\alpha }_2^{″}\sqrt{\hslash eB}/w_y=0.01$, and $B=1$. We take several values of $E$, and we find that an eigenvalue for the almost-localized eigenstates at $y<0$ exists in the region $0.310<E/\left(w_y\sqrt{\hslash eB}\right)<0.311$.

Figure 8

Wave functions of almost-localized eigenstates at $y<0$ with quantum number (a) $n=0$, (b) $n=1$, (c) $n=2$, (d) $n=3$, (e) $n=4$, (f) $n=5$, and (g) $n=6$ in the three-quarter Dirac point obtained numerically with the boundary condition at $y=-10$. Parameters are $w_x=w_y=1$, $w_{0x}=-1$, $w_{0y}=0$, ${\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y={\alpha }_2^{″}\sqrt{\hslash eB}/w_y=0.01$, and $B=1$.

Figure 9

Energy as a function of the quantum number $n$ for the three-quarter Dirac point. Parameters are $w_x=w_y=1$, $w_{0x}=-1$, $w_{0y}=0$, and $B=1$. We take two choices of parameters giving the same ${\alpha }_2$, ${\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y={\alpha }_2^{″}\sqrt{\hslash eB}/w_y=0.01$ and ${\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y=0.02$, ${\alpha }_2^{″}=0$. The obtained values of the energy is well fitted by the red broken line ($E\propto 0.3n^{\frac{4}{5}}$).

Figure 10

Energy with the quantum number $n=3$ as a function of ${\alpha }_2$ for the three-quarter Dirac point. Parameters are $w_x=w_y=1$, $w_{0x}=-1$, $w_{0y}=0$, ${\alpha }_2^{″}=0$, and $B=1$.

Figure 11

Dimensionless energy [$E/\left(w_y\sqrt{\hslash e{B}_0}\right)$ with $B_0=1$] as a function of magnetic field $B$ at the three quarter Dirac point. It is well fitted as a function of $B^{\frac{4}{5}}$.

Figure 12

Schematic plot of the areas of the Fermi surface and the density of states as a function of energy for the three-quarter Dirac fermion, massless 2D Dirac fermion, 2D free fermion, and 2D semi Dirac fermion [12, 13]. The density of states are scaled to be 1 at $\epsilon =1$.

Figure 13

Energy at $B=0$ as a function of $q_x$ and $q_y$ with parameters $w_x=0.4$, $w_y=1$, ${\alpha }_2^{\prime }={\alpha }_2^{″}=0.01$, $w_{0x}=-w_x$, $w_{0y}=0$, and $\mathrm{\Delta }=\pm 0.3$.

Figure 14

Boundary condition $\theta$ at $y=y_{+}=20$ [Eq. (29)], which makes $|{\mathrm{\Psi }}_{1,2}\left(y\right)|\to 0$ at $y\to -\infty$, as a function of the energy. We take parameters $w_x=w_y=1$, ${\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y={\alpha }_2^{″}\sqrt{\hslash eB}/w_y=0.01$, $\mathrm{\Delta }=0.3,0.2,...,-0.3$, $-0.4$, and $B=1$.

Figure 15

Wave functions for (a) $\mathrm{\Delta }=-0.1$, $E/\left(w_y\sqrt{\hslash eB}\right)=0.06$, (b) $\mathrm{\Delta }=-0.3$, $E/\left(w_y\sqrt{\hslash eB}\right)=0.10$, (c) $\mathrm{\Delta }=0.1$, $E/\left(w_y\sqrt{\hslash eB}\right)=0.06$, and (d) $\mathrm{\Delta }=0.3$, $E/\left(w_y\sqrt{\hslash eB}\right)=0.10$, Other parameters are $w_x=w_y=1$, ${\alpha }_2^{\prime }\sqrt{\hslash eB}/w_y={\alpha }_2^{″}\sqrt{\hslash eB}/w_y=0.01$, and $B=1$. Boundary conditions at $y=20$ ($\theta$) are taken as in the case of $\mathrm{\Delta }=0$ in Sec. 2b. The wave functions have large amplitudes in $y\lesssim 0$ region, when $\mathrm{\Delta }<0$ [(a) and (b)], while no peaks are seen in $y\lesssim 0$ region, when $\mathrm{\Delta }>0$ [(c) and (d)].