Carrier concentrations and optical conductivity of a band-inverted semimetal in two and three dimensions

Here we study the single-particle, electronic transport, and optical properties of a gapped system described by a simple two-band Hamiltonian with inverted valence bands. We analyze its properties in the three-dimensional (3D) and the two-dimensional (2D) case. The insulating phase changes into a metallic phase when the band gap is set to zero. The metallic phase in the 3D case is characterized by a nodal surface. This nodal surface is equivalent to a nodal ring in two dimensions. Within a simple theoretical framework, we calculate the density of states, the total and effective charge carrier concentration, the Hall concentration, and the Hall coefficient, for both 2D and 3D cases. The main result is that the three concentrations always differ from one another in the present model. These concentrations can then be used to resolve the nature of the electronic ground state. Similarly, the optical conductivity is calculated and discussed for the insulating phase. We show that there are no optical excitations in the metallic phase. Finally, we compare the calculated optical conductivity with the rule-of-thumb derivation using the joint density of states.

Figure 1

(a) Valence bands Eq. (2.3) of the GSM case in 1D. The inverted conduction band (orange) spans the energy range between ${\omega }_b$ (green) and ${\omega }_t$ (red). The dashed line indicates the Fermi energy ${\omega }_F$. In the NSSM phase, the bands touch on the surface of a circle of radius ${\kappa }_0$, while in the GSM phase ${\kappa }_0$ is the position of minimum (maximum) of the conduction (valence) band. (b) Analogous valence bands in 2D. (c) The Fermi surface of the 3D system in the case of a partially filled conduction band with the Fermi energy in the range ${\omega }_b<{\omega }_F<{\omega }_t$ as depicted by the black level in panel (a).

Figure 2

The density of states Eq. (3.7) of the 3D system derived from the energy dispersion Eq. (2.2). DOS is a function of a dimensionless parameter $\omega$ and is plotted for several values of the gap parameter, ${\omega }_b=\mathrm{\Delta }$. For the case of $\mathrm{\Delta }=0$ (NSSM), the dome in DOS is clearly visible between the energies $(-1,1)$ (black line) with a maximum height of $2N_0^{\left(3\right)}$. For $\mathrm{\Delta }>0$ at the energy $\omega ={\omega }_b$, DOS has a square-root singularity, and the value at $\omega ={\omega }_t$ is given by red circles. For energies $\omega \gg {\omega }_t$, DOS is $\propto \sqrt{\omega }$, just like in the case of the 3D free electron gas.

Figure 3

The total concentration $n$ (blue dashed line), the effective concentration $n_{\alpha \alpha }$ (orange line), and the Hall concentration $n_H$ (green line) are plotted in units of $n_0^{\left(3\right)}$ as functions of Fermi energy ${\omega }_F$, at several values of the gap parameter $\mathrm{\Delta }$. The positions of $\mathrm{\Delta }$ are designated by vertical dotted lines. It is assumed that $m^{*}=m_e$. The red circle is the value of $n\left({\omega }_t\right)$, the green circle is the value of $n_H\left({\omega }_m\right)$, while the violet circles represent the values of $n_{\alpha \alpha }\left({\omega }_t\right)$.

Figure 4

The Hall coefficient $R_H$ for a 3D (green line) and 2D (red dashed line) system as a function of concentration $n$. The concentration is given in units of concentration needed to fill up the dome of the inverted band $n_t$. In 3D $n_t=2^{3/2}{n}_0^{\left(3\right)}$ while in 2D $n_t=2n_0$. Correspondingly, $R_H$ is given in units of ${\left(e{n}_t\right)}^{-1}$. The two regions $n<n_t$ and $n>n_t$ are divided by a vertical dotted line. Below the critical ratio of 1 (orange circle) for both 3D and 2D $R_H\sim n$, while above $R_H\propto 1/n$.

Figure 5

The real part of the optical conductivity calculated from the two-band model (2.1) in units of ${\sigma }_0^{\left(3\right)}$. $\mathrm{Re}\sigma \left(\mathrm{\Omega }\right)$ is plotted for several values of the parameter $\mathrm{\Delta }$ as a function of a dimensionless parameter $\mathrm{\Omega }=\mathcal{E}/\left(2A\right)$. The solid lines depict the intrinsic case $[{\omega }_F=0\to \mathcal{F}\left(\mathrm{\Omega }\right)=1]$ while the dashed red line gives a doped case with ${\omega }_F=5\to \mathcal{F}\left(\mathrm{\Omega }\right)=\mathrm{\Theta }(\mathrm{\Omega }-{\omega }_F)$. All the curves represent the situation of zero interband relaxation as given by Eq. (5.7). For the specific case of $\mathrm{\Delta }=2$, a finite interband relaxation in Eq. (5.1) was used (green line). Orange circles represent $\mathrm{Re}\sigma \left({\mathrm{\Omega }}_t\right)$.

Figure 6

The DOS of the 2D system described by the energy dispersion ${\varepsilon }_{\mathbf{k}}^{c,v}=\pm \sqrt{{(A-B{k}^2)}^2+C^2}$ [Eq. (2.2)] as a function of $\omega$, plotted for several values of the parameter $\mathrm{\Delta }$. For the metallic NSSM case $(\mathrm{\Delta }=0)$, DOS has a steplike shape (black line), while for the gapped GSM case $(\mathrm{\Delta }>0)$ the divergences appear at ${\omega }_b=\mathrm{\Delta }$. The step feature remains visible for ${\omega }_b<1$. The red circles indicate $N\left({\omega }_t\right)$. In the high-$\omega$ limit, DOS becomes constant.

Figure 7

(a) The total concentrations $n$ Eq. (6.2) (blue dashed line), effective concentration $n_{\alpha \alpha }$ Eq. (6.3) (orange line), and the Hall concentration $n_H$ Eq. (6.4) (green line), as a function of Fermi energy ${\omega }_F$ in units of $n_0$ for the case $m_e=m^{*}$. The concentrations are plotted for several values of the ${\omega }_b=\mathrm{\Delta }$ whose positions are denoted by the vertical dotted lines. Particularly interesting is the NSSM case $({\omega }_b=0)$, where the difference between the concentrations is the most profound. All three concentrations have a kink at the ${\omega }_t$. For $n$ and $n_H$ the value at ${\omega }_t$ is $2n_0$ as indicated by the red dot. (b) $n$ (blue dashed line), $n_{\alpha \alpha }$ from Eq. (6.6) (orange line), and $n_H$ from Eq. (6.5) (green line) are shown as a function of $n$ for several values of $\mathrm{\Delta }$, for the case when $m_e=m^{*}$. The arrow indicates the direction of increasing $\mathrm{\Delta }$ in $n_{\alpha \alpha }$. $n$ and $n_H$ do not depend explicitly on ${\omega }_b$ while $n_{\alpha \alpha }$ does. Vertical dotted line designates $2n_0$, which is the concentration needed to fill the system to the top of the dome in Fig. 1.

Figure 8

Real part of the optical conductivity of the 2D system described by the two-band model (2.1) in units of ${\sigma }_0$. $\mathrm{Re}\sigma \left(\mathrm{\Omega }\right)$ is plotted for several values of parameter $\mathrm{\Delta }$ as a function of $\mathrm{\Omega }=\mathcal{E}/\left(2A\right)$ for the intrinsic case $\left[\mathcal{F}\right(\mathrm{\Omega })=1]$. The green curve (with a finite $\mathrm{\Gamma }$) is calculated using Eq. (5.1) with $\mathrm{\Gamma }=0.05$ eV, while the rest are given by Eq. (6.9). The steplike feature in the DOS is not visible in the conductivity. $\mathrm{Re}\sigma \left({\mathrm{\Omega }}_t\right)$ is depicted by orange circles. The circles approach $2{\sigma }_0$ as $\mathrm{\Delta }\to \infty$ (red circle).