Density of states measurements for the heavy subband of holes in HgTe quantum wells

A valence band in narrow HgTe quantum wells contains well-conductive Dirac-like light holes at the $\mathrm{\Gamma }$ point and a poorly conductive heavy hole subband located in the local valleys. Here we propose and employ two methods to measure the density of states for these heavy holes. The first method uses a gate-recharging technique to measure thermodynamical entropy per particle. As the Fermi level is tuned with gate voltage from a light to heavy subband, the entropy increases dramatically, and the value of this increase gives an estimate for the density of states. The second method determines the density of states for heavy holes indirectly from the gate voltage dependence of the period of the Shubnikov–de Haas oscillations for light holes. The results obtained by both methods are in reasonable agreement with each other. Our approaches can be applied to measure large effective carrier masses in other two-dimensional gated systems.

Figure 1

(a) QW thickness dependence of the relevant terms in ${\mathrm{Cd}}_{0.7}{\mathrm{Hg}}_{0.3}\text{Te}$/HgTe/${\mathrm{Cd}}_{0.7}{\mathrm{Hg}}_{0.3}\text{Te}$ quantum wells; 3D view (b) and isoenergy plot (c) of the valence band dispersion in 6-nm-thick (0 1 3) ${\mathrm{Cd}}_{0.7}{\mathrm{Hg}}_{0.3}\text{Te}$/HgTe/${\mathrm{Cd}}_{0.7}{\mathrm{Hg}}_{0.3}\text{Te}$ quantum wells (IIA is taken into account, see text). (d) $E\left(k_{33-1}\right)$ dependencies demonstrating two spin-orbit-split branches of the dispersion law. (e) Corresponding density of states plot (solid line), and the effect of disorder on OHS (dotted line).

Figure 2

(a) Gate-insulator-QW structure and schematic for entropy measurements. The ammeter measures recharging femtoampere-range current in response to temperature modulation. (b) Band diagram of the gate-insulator-QW structure, showing holes (orange) in the QW. Dispersion law for the holes is indicated to the right from the band diagram.

Figure 3

(a) and (b) Capacitance of the samples 110623 and 110622 versus carrier density measured at the same measurement frequency as $\partial S/\partial p$ (solid lines, left axes), density of states at the Fermi level, recalculated using the model for $\partial S/\partial p$ fit (dashed lines, right axes). (c) and (d) $\partial S/\partial p$ value for the same samples versus density at 3 K. Lines: Theoretical fits using a smeared steplike density of states model. Hatched areas show the integrated entropy value for $p=3.7\times {10}^{11}{\mathrm{cm}}^{-2}$. Right axes in (c) and (d) show the root-mean-square value of the recharging current. (e) Density of states model and parameters used for fitting of the data in (b) and (d). Gray arrows denote two regions (A, where only light carriers are present, and B, where also heavy subbands emerge).

Figure 4

(a) Carrier density of light subbands (crosses), determined from ShDH oscillations frequencies and their sum (empty circles) versus gate voltage. Solid boxes: Carrier density determined from Hall constant at 0.2 T. Lines show the corresponding linear $p\left(V_g\right)$ fits. (b) Example of SdHO pattern for sample 110623 and their expansion into two frequencies, corresponding to two spin-orbit split dispersion branches. (c) Dingle plots for both branches and effective mass determination.

Figure 5

Gate voltage dependence of the $\partial S/\partial p$ signal for sample 110622, at 3 K (blue squares) and 12 K, with (crosses) and without (triangles) subtraction of linear-in-$V_g$ correction (see text).

Figure 6

(a) Solid lines: Magnetic field dependencies of the resistivity for 110623 sample at 30 (top two lines) and 4.2 K (bottom line). Dashed lines show the two-component fit of the data. (b) Hall coefficient as a function of magnetic field for the same sample and gate voltages as in (a) (solid lines) and the corresponding theoretical fits (dashed lines). The insert shows Hall resistance (solid line, left axis) and Hall coefficient (dotted line, right axis) at low temperature 4.2 K. The values of the temperatures and gate voltages are indicated in the panels. (c) Comparison of the elevated temperature two-liquid fitting parameters from Table $1p_1$ (triangles) and $(p_1+p_2)$ (stars), with carrier densities determined from low temperature Hall coefficient (black boxes) and Shubnikov–de Haas oscillations $p_{\text{SdH}}$ (empty boxes). Low temperature data and extrapolation of total carrier density (dashed line) are taken from Fig. 4.

Figure 7

(a) Example of magnetic field dependence of the magneto-oscillations for sample 110623 close to OHS. Symbols are experimental data, lines are fits. (b) and (c) Two harmonics at different temperatures from 1.4 to 4.2 K. (d) Effective masses determined at different magnetic fields.