Plasmons due to the Interplay of Dirac and Schrödinger Fermions

We study the interplay between Dirac and Schrödinger fermions in the polarization properties of a two-dimensional electron gas (2DEG). Specifically, we analyze the low-energy sector of narrow-gap semiconductors described by a two-band Kane model. In the context of quantum spin Hall insulators, particularly, in Hg(Cd)Te quantum wells, this model is named the Bernevig-Hughes-Zhang model. Interestingly, it describes electrons with intermediate properties between Dirac and Schrödinger fermions. We calculate the dynamical dielectric function of such a model at zero temperature within random phase approximation. Surprisingly, plasmon resonances are found in the intrinsic (undoped) limit, whereas they are absent—in that limit—in graphene as well as ordinary 2DEGs. Additionally, we demonstrate that the optical conductivity offers a quantitative way to identify the topological phase of Hg(Cd)Te quantum wells from a bulk measurement.

Figure 1

Dispersion relation and pseudospin properties of the eigenstates of the BHZ model for a NI phase with ${\xi }_M=1/2$ (a) and TI phase with ${\xi }_M=-1/2$ (b). The two bands have been artificially seperated by an additional $2{\epsilon }_{X,\lambda }$ for a better illustration of the pseudospin.

Figure 2

Real (a) and imaginary (b) part of ${\mathrm{\Pi }}^R(X,\mathrm{\Omega })$ of the BHZ model for ${\xi }_M={\xi }_D=0$. In (c) line cuts are drawn of $\mathrm{Im}[{\mathrm{\Pi }}^R]$ for $X=0.5$, with ${\xi }_D=-0.5$, 0. The line cuts in (d) show $\mathrm{Re}[{\mathrm{\Pi }}^R]$ for $X=2$ with ${\xi }_M=-4/9$, 0, and 4/9, with green triangles, blue dots, and black stars, respectively.

Figure 3

(a) Illustration of the damping ratio $\mathrm{\Gamma }/\mathrm{\Omega }$ and (b) the function $1/\mathrm{Re}[g(X,\mathrm{\Omega }-i\mathrm{\Gamma })]$, for ${\xi }_M=-4/9$. (c) Plasmon dispersion together with the loss function $\mathrm{Im}[-1/ϵ]$ for ${\xi }_D=-0.5$ and $\alpha =0.4$. (d) Plasmon frequency and damping at $X=0.5$ [dashed line in (b) and (c)] for ${\xi }_M\in \{-4/9,4/9\}$ and ${\xi }_D=-0.5$ in green triangles, black stars, and blue dots, respectively.

Figure 4

Illustration of $\mathrm{Im}[{\sigma }_0(\mathrm{\Omega })]$ for ${\xi }_M=-4/9$, 0, and 4/9, with dot-dashed, long dashed, and solid lines, respectively. The inset shows the ratio $R=\mathrm{Im}{[{\sigma }_0(\mathrm{\Omega })]}_{M>0}/\mathrm{Im}{[{\sigma }_0(\mathrm{\Omega })]}_{M<0}$ for $|{\xi }_M|=1/6$, 1/3, and 1/2 in dot-dashed, long dashed, and solid lines, respectively.