Valence band energy spectrum of HgTe quantum wells with an inverted band structure

The energy spectrum of the valence band in $\mathrm{HgTe}/{\mathrm{Cd}}_x{\mathrm{Hg}}_{1-x}\mathrm{Te}$ quantum wells of a width ($8–20$) nm has been studied experimentally by magnetotransport effects and theoretically in the framework of a four-band $kP$ method. Comparison of the Hall density with the density found from a period of the Shubnikov–de Haas (SdH) oscillations clearly shows that the degeneracy of states of the top of the valence band is equal to 2 at the hole density $p<5.5\times {10}^{11}{\mathrm{cm}}^{-2}$. Such degeneracy does not agree with the calculations of the spectrum performed within the framework of the four-band $kP$ method for symmetric quantum wells. These calculations show that the top of the valence band consists of four spin-degenerate extremes located at $k\ne 0$ (valleys) which gives the total degeneracy $K=8$. It is shown that taking into account the “mixing of states” at the interfaces leads to the removal of the spin degeneracy that reduces the degeneracy to $K=4$. Accounting for any additional asymmetry, for example, due to the difference in the mixing parameters at the interfaces, the different broadening of the boundaries of the well, etc., leads to reduction of the valleys degeneracy, making $K=2$. It is noteworthy that for our case twofold degeneracy occurs due to degeneracy of two single-spin valleys. The hole effective mass ($m_h$) determined from analysis of the temperature dependence of the amplitude of the SdH oscillations shows that $m_h$ is equal to $(0.25\pm 0.02)m_0$ and weakly increases with the hole density. Such a value of $m_h$ and its dependence on the hole density are in a good agreement with the calculated effective mass.

Figure 1

Sketch and energy diagram of the structures investigated.

Figure 2

(a) The magnetic field dependencies of ${\rho }_{xy}$ for some $V_g$ values. (b) The gate voltage dependence of the carriers density. The solid circles are the Hall density for holes $p_H=1/e{R}_H(B=1\text{T})$ and for electrons $n_H=-1/e{R}_H(B=0.2\text{T})$. The crosses and empty circles are the hole density found from SdH oscillations for the degeneracy of the Landau levels 8 and 2 (crosses and circles, respectively). The solid line is the carrier density found from the capacitance between the 2D gas area and the gate electrode as $C(V_g^{\text{CNP}}-V_g)/e{S}_g$, where $V_g^{\text{CNP}}$ is the gate voltage corresponding to the charge neutrality point. (c) The density dependence of the hole mobility. $T=1.4$ K.

Figure 3

(a) The magnetic field dependencies of $(d{\rho }_{xx}/dB)/{\rho }_{xx}\left(0.1\text{T}\right)$ at different $V_g, T=1.4$ K. The solid line for $V_g=-5$ V is fit to the Lifshitz-Kosevich formula, Eq. (1). (b) The positions of the maxima of ${\rho }_{xx}$ in coordinates $(B,V_g)$. The dashed lines are provided as a guide for the eye. (c) The points are the positions of the minima of ${\rho }_{xx}$ in inverse magnetic field, $1/B_{\text{min}}$, as a function of the filling factor for the different degeneracy of LL. The solid line is $(e/h)\times (N/p_H)$.

Figure 4

(a) The magnetic field dependencies of $d{\rho }_{xx}/dB$ taken at different temperatures at $p=3.6\times {10}^{11}{\mathrm{cm}}^{-2}$. (b) and (c) The amplitude of the oscillations at different temperatures for $B=2.4$ and 1.9 T, respectively (symbols). The dashed curves are the results of the best fit to Eq. (2) with $m_h/m_0$ as the fitting parameter.

Figure 5

(a) The broadening $\mathrm{\Delta }$ for different temperatures at $p=3.6\times {10}^{11}{\mathrm{cm}}^{-2}$. (b) The values of the hole effective mass found at different hole densities.

Figure 6

The values of hole effective mass measured in all the structures investigated within the whole hole density range. The calculated dependence of $m_h$ with different parameters is presented by the wide semitransparent line (see Sec. 3c).

Figure 7

The isoenergy contours of the top of valence band. The orientation of substrate is (013) and the width of the well is equal to 8.3 nm. We show half of the picture; the second part is a mirror image. The energy is measured from $E(k=0)$. The step is 1 meV. The energies for some contours are shown in the plot. The dashed line shows the direction which passes through the maximum of $E\left(\mathbf{k}\right)$.

Figure 8

(a) The dependencies $E\left(k\right)$ for three directions: $\left[100\right], \left[03\overline{1}\right]$, and the “[diagonal]” one, which passes through the maximum of $E\left(\mathbf{k}\right)$. (b) The dependence of hole density on the Fermi energy. The arrow shows the density of holes at which the two nearby valleys are merged.

Figure 9

(a) The isoenergy contours of the upper split-off state calculated with taking into account IIA. (b) The $E$ versus $k_{\text{diagonal}}$ dependencies for both split-off states. (c) The dependence of hole density in the upper (solid curve) and lower (dashed curve) split-off states on the Fermi energy.

Figure 10

(a) The isoenergy contours of the upper split-off states calculated with the different values of parameter $g_4$ for one and another boundary of the quantum well: $g_4=1$ and 0.8 eV Å. (b) The $E$ versus $k_{\text{diagonal}}$ dependencies for both split-off states of upper valley (solid and dashed lines) and upper states of lower valley (the dotted line). (c) The dependence of hole density in the upper (solid curve) and lower (dashed curve) split-off states on the Fermi energy.

Figure 11

The experimental (symbols) and theoretical (lines) hole density dependence of the hole effective mass in the structure with $d=8.3$ nm. The solid line is calculated when the value of the parameter $g_4$ is the same for both boundaries of the quantum well, $g_4=0.8$ eV Å. The dashed line is the case when the value of $g_4$ is different for different boundaries, $g_4=1$   and 0.8 eV Å.