Massless Dirac fermions in III-V semiconductor quantum wells

We report on the clear evidence of massless Dirac fermions in two-dimensional system based on III-V semiconductors. Using a gated Hall bar made on a three-layer InAs/GaSb/InAs quantum well, we restore the Landau level fan chart by magnetotransport and unequivocally demonstrate a gapless state in our sample. Measurements of cyclotron resonance at different electron concentrations directly indicate a linear band crossing at the $\mathrm{\Gamma }$ point of the Brillouin zone. Analysis of experimental data within an analytical Dirac-like Hamiltonian allows us not only to determine the velocity ($v_F=1.8\times {10}^5$ m/s) of massless Dirac fermions, but also to demonstrate a significant nonlinear dispersion at high energies.

Figure 1

(a) Subband energies at zero quasimomentum $\mathbf{k}$ in symmetric InAs/GaSb/InAs QWs as a function of InAs layer thickness $d_{\mathrm{InAs}}$. Blue and red curves correspond to the electronlike and holelike states, respectively. The thickness of the GaSb layer $d_{\mathrm{GaSb}}$ equals 12 monolayers (ML). Here, 1 ML corresponds to half of the lattice constant of the bulk material. The zero energy is referenced to the valence-band edge of bulk GaSb. (b) A 3D plot of the Dirac cone describing the energy spectrum at $\mathbf{k}<0.2{\mathrm{nm}}^{-1}$ for $d_{\mathrm{InAs}}=33$ ML and $d_{\mathrm{GaSb}}=12$ ML. (c) LL fan chart for the gapless state. A pair of zero-mode LLs is presented by the red curves. All the calculations are based on an eight-band Kane model [21].

Figure 2

(a) Longitudinal and Hall resistances as a function of magnetic field at $T=1.7$ K and $V_g=0$ V. Electron microscopy image of the Hall bar is shown in the inset. (b) $R_{xx}$ and $R_{xy}$ vs $V_g$ at $T=1.7$ K and $B=6.5$ T. (c) Color map of the longitudinal resistance as a function of $V_g$ and $B$. Black stars indicate $h/e^2$ values and red dots the peak values of $R_{xx}$.

Figure 3

(a) Band dispersion and (b) cyclotron mass $m_c$ in the conduction band as a function of Fermi momentum, $k_F=\sqrt{2\pi n_S}$ for the gapless states with linear (blue curves) and parabolic band touching at the $\mathrm{\Gamma }$ point. Two cases for parabolic touching, when the conduction and valence bands have the same ($\mathcal{A}=0$) and opposite ($\mathcal{A}\ne 0$) parity [36], are represented by dotted black and solid red curves, respectively. The following band parameters were used in the calculation: $C=0$ meV, $A=150\mathrm{meV}\mathrm{nm}$, $\mathcal{A}=35\mathrm{meV}{\mathrm{nm}}^3$, $B=-600\mathrm{meV}{\mathrm{nm}}^2$, and $D=-450\mathrm{meV}{\mathrm{nm}}^2$.

Figure 4

(a), (b) CR spectra at $T=4.2$ K of electrons measured with (a) BWO at 845 GHz and (b) QCL at 3 THz for different values of electron concentration. The symbols mark the positions of CR lines. The numbers over the curves show $n_S$ in ${10}^{11}{\mathrm{cm}}^{-2}$. (c) Cyclotron mass $m_c$ as a function of Fermi momentum $k_F=\sqrt{2\pi n_S}$. The open and solid symbols correspond to the values obtained by BWO and QCL, respectively. The bold blue curve is the fitting to Eq. (2). (d) Comparison of band dispersion calculated within the eight-band Kane model with the one described by a simplified Dirac-like Hamiltonian with the parameters shown in (c). Parameter $C$ is chosen to have a coincidence of the band crossing point in two models.