Energy Spectrum of the Valence Band in HgTe Quantum Wells on the Way from a Two- to Three-Dimensional Topological Insulator

The magnetic field and temperature dependences of longitudinal magnetoresistance and the Hall effect have been measured in order to determine the energy spectrum of the valence band in HgTe quantum wells with the width dQW = 20–200 nm. The comparison of hole densities determined from the period of Shubnikov–de Haas oscillations and the Hall effect shows that states at the top of the valence band are doubly degenerate in the entire dQW range, and the cyclotron mass \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{m}_{h}}$$\end{document} determined from the temperature dependence of the amplitude of Shubnikov–de Haas oscillation increases monotonically from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.2{{m}_{0}}$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.3{{m}_{0}}$$\end{document} (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{m}_{0}}$$\end{document} is the mass of the free electron) with increasing hole density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document} from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2 \times {{10}^{{11}}}$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$6 \times {{10}^{{11}}}$$\end{document} cm–2. The determined dependence has been compared to theoretical dependences \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{m}_{h}}(p,{{d}_{{{\text{QW}}}}})$$\end{document} calculated within the four-band kP model. These calculations predict an approximate stepwise increase in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{m}_{h}}$$\end{document} owing to the pairwise merging of side extrema with increasing hole density, which should be observed at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p = (4{-} 4.5) \times {{10}^{{11}}}$$\end{document} and 4 × 1010 cm–2 for dQW = 20 and 200 nm, respectively. The experimental dependences are strongly inconsistent with this prediction. It has been shown that the inclusion of additional factors (electric field in the quantum well, strain) does not remove the contradiction between the experiment and theory. Consequently, it is doubtful that the mentioned kP calculations adequately describe the valence band at all dQW values.


INTRODUCTION
Structure with HgTe quantum wells (QWs) attract great attention for several reasons.First, the QW is formed from a gapless semiconductor, whereas HgCdTe barriers are formed from a semiconductor with normal band ordering. 1Second, the band structures of the parent materials HgTe and HgCdTe are studied in detail and their parameters are well known.Third, the multiband kP method for the calculation of the energy spectrum E(k) in HgTe quantum wells is well developed (see, e.g., [1][2][3][4] and references therein).These calculations show that various energy spectra occur depending on the width of the quantum well dQW from the spectrum similar to the spectrum of a narrow-gap semiconductor at dQW < 6.3 nm to a semimetallic spectrum at dQW > ≈15 nm.Fourth, the theory predicts that the HgTe QW with dQW > 6.5 nm is a two-dimensional topological insulator, where one-dimensional edge states are formed in addition to two-dimensional states.The HgTe QW with dQW > 60-80 nm is a three-dimensional topological insulator, where two-dimensional single-spin surface states are formed with the characteristic localization length in the z direction, which is perpendicular to the plane of the QW, much larger than dQW.Fifth, the technology of growth of HgCdTe/HgTe/HgCdTe structures is well developed [5,6].
All these circumstances seem to allow detailed understanding of (transport, optical, etc.) properties of HgCdTe/HgTe/HgCdTe structures.Theoretical calculations predict that the conduc- metry in the 8.3-nm-wide quantum well with the tion band is quite simple, nearly isotropic and nonpar-parameter g4 = 0.8.This figure demonstrates that abolic.Its spectrum in structures with dQW = 4-80 nm constant-energy contours of two extrema presented in the figure are merged with a decrease in the energy is thoroughly studied by the optical and magnetotransport methods, as well as by the photoelectric method in a wide photon energy range beginning with the terahertz band [8][9][10][11][12].It was shown that the energy spectrum is reasonably described in general within the (i.e., with an increase in the hole density) to 18-18.5 meV, which should lead to the doubling of the cyclotron mass mh caused by the doubling of dS /dE .
The energy spectrum of the valence band is much less studied experimentally [15][16][17].It was shown that four-band kP model, and some discrepancies between the extremum of the valence band at dQW= 5-7 nm, theory and experiment were discussed in [13].
The energy spectrum of the valence band is much when it is located at k = 0, is strongly split owing to the more complex.The theory predicts that the top of the valence band at dQW < 7-7.5 nm is located at k = 0 and has the curvature (mass) close to the curvature of the conduction band.The top of the valence band at dQW > 7-7.5 nm is formed by the four side extrema so that the states at the top of the valence band in symmetric quantum wells have a degree of degeneracy of K = 8 (2 owing to "spin" multiplied by 4, which is the number of side extrema).The anisotropy of these states near extrema is small and the cyclotron mass of holes given by the formula mh = (h 2 /)dS /dE , where S is the area of the constant-energy cross section at the energy E , is (0.2−0.3)m0 at p < (5−9)  10 11 cm -2 .The interface inversion asymmetry in interface inversion asymmetry [15,17].At dQW > 7-7.5 nm, when the top of the valence band is formed by four side extrema, the energy spectrum is experimentally studied much less.In [18], we show that the effective mass in the QW with dQW = 8-20 nm at p = (2−5)  10 11 cm -2 is close to the theoretical value, but the degree of degeneracy is K = 2 rather than 4, as should be the case taking into account the interface inversion asymmetry in the symmetric quantum well.It was shown that the additional asymmetry (e.g., different widths of heterointerfaces or different parameters g4L and g4R characterizing the contribution from the interface inversion asymmetry on the left and right walls, respectively) results in halving of the degree of degeneracy K and, thereby, makes it possi-HgCdTe/HgTe/HgCdTe structures, which is described by a single parameter g4 within the four-ble to remove this discrepancy with theory.
Experimental studies of the cyclotron mass of holes band kP model [14], leads to the "spin" splitting of states at the top of the valence band, so that the degree of degeneracy decreases to 4. Figure 1 presents constant-energy contours of the upper split state calculated taking into account the interface inversion asym-in the QW with dQW = 8-20 nm in the hole density range of p = (2−5)  10 11 cm -2 showed that the effective mass is close to the theoretical value [17].The results from [17] are presented in Fig. 2.  In this work, the effective mass of holes and the degree of degeneracy of the top of the valence band in the QW with dQW = 20-200 nm in the hole density range of (2−6)  10 11 cm -2 are studied experimentally.

EXPERIMENTAL RESULTS AND DISCUSSION
The studied Hg1−x Cdx Te/HgTe/Hg1−x Cdx Te ( x = 0.6−0.7)structures with quantum wells of the widths dQW = 22, 32, 46, 80, 88, 120, 200 nm were grown by molecular beam epitaxy on the (013) semiinsulating GaAs substrate (in addition, one structure with dQW = 80 nm was grown on the (100) substrate).The measurements were carried out with Hall bars with a channel width of 0.5 mm and potential contacts separated by 0.5 mm.An aluminum gate was deposited after the deposition of a gate dielectric (parylene) on the surface of the bars.The dc measurements were performed in the temperature range of 1.3-4 K in magnetic fields up to 5 T.
Experimental results and their processing are identical for all studied structures.We describe them in detail for structure 180824 with dQW = 32 nm.
The magnetic field dependences of the longitudinal resistance Rxx and the Hall coefficient RH presented in Fig. 3а show that transport involves at least two types of carriers: electrons, which determine the magnetic field dependences of Rxx and RH in low magnetic fields B < 0.3-0.5 T, and holes, which determine the magnetic field dependences of Rxx and RH at magnetic fields B > 0.5 T (similar dependences are observed at all gate voltages Vg < 0).At gate voltages Vg > 0, the hole contribution to the conductivity vanishes and Rxx and RH are determined only by conduction electrons.This occurs because the HgTe quantum well with dQW > 14-15 nm is a semimetal; i.e., the bottom of the conduction band, which is located at the center of the Brillouin zone at k = 0, is below the side extrema in the valence band (see the inset of Fig. 3а).Dependences of the hole and electron densities on Vg are presented in Fig. 3b.The electron density at Vg < 0 was determined from the magnetic field dependences of Rxx and RH in the magnetic field range of B = 0.03-0.6T in the two-carrier conduction model, whereas the electron density at Vg > 0, when RH < 0 and hardly depends on the magnetic field, was determined from the Hall effect at B = 0.03 T as n = (eRH) -1 , where e is the electron charge.
The hole density was determined both from the magnetic field dependences of Rxx and RH in the range dependence of C/Sg, where C is the capacitance between the two-dimensional gas and the gate and Sg is the area of the gate.
For example, we consider oscillations of Rxx(B) in the hole region at Vg = -4 V shown in Fig. 3а.The Fourier spectrum of the oscillatory part Rxx = of B = 0.05-1 T within the two-carrier conduction model and from the frequency F of Shubnikov-de (Rxx − R mon )/R mon of Rxx , where R mon is the mono-Haas oscillations as pSdH = (e/h)FK (Fig. 4а). Figure tonic part of the magnetoresistance, at Vg = -4 V is 3b demonstrates that pSdH at K = 2 coincides within the experimental error with the Hall hole density. 2  Figure 3b also shows that the charge of the quantum well depends linearly on Vg in the entire Vg range, which indicates the absence of missed conduction channels.This conclusion is confirmed by the fact that the slope of the gate-voltage dependence of Q/(eVg) coincides within the error with the gate-voltage 2 This work is focused on the study of the spectrum of the valence band.For this reason, we do not discuss the behavior of R xx and R H in the electron region.The behavior of this structure in the electron region was analyzed in detail in [18].
shown in Fig. 4а.The low-and high-frequency components of the spectrum correspond to the contributions from electrons and holes to oscillations of Rxx(B), respectively.This immediately follows from the temperature dependence of the amplitudes of these components (Fig. 4а).As the temperature increases from 1.32 to 2.4 K, the amplitude of the low-frequency component decreases only by 20% (this is due to a small effective mass of electrons), whereas the amplitude of the high-frequency component decreases by a factor of about 5.
The first conclusion following from Figs. 3b and 4а is that the degree of degeneracy of Landau levels in the Fourier amplitude (arb.units) h valence band is 2.This follows from the fact that the hole density determined from the period of high-frequency oscillations under the assumption that Landau levels are doubly degenerate coincides within the error with the Hall hole density.
To determine the effective mass of holes from the temperature dependence of the amplitude of oscillations, these oscillations were reconstructed by the inverse Fourier transform of the filtered (as shown by the dashed line in Fig. 4а) Fourier spectrum (Fig. 4b).The amplitudes of oscillations in a magnetic field of 1.5 T at several temperatures are presented by circles in Fig. 4с, where the line corresponds to the Lifshitz-Kosevich formula ensuring the best reproduction of the experimental results, which is achieved at mh/m0 = 0.22.To estimate the error, the ratio mh/m0 was determined at different magnetic fields.These results are presented in Fig. 4d.Thus, mh/m0 = 0.22 ± 0.03 at should be pairwise merged, leading to a sharp increase in m h /m0.The dashed rectangle same as in Fig. 2 indicates the m h /m0 region including all experimental results at d QW = 8-20 nm [18].
It is seen that the effective hole mass mh/m0 at hole densities below 3  10 11 cm -2 is in good agreement both with the results for dQW = 8-20 nm (Fig. 2) and with the theoretical dependence.However, a sharp increase in mh/m0 caused by the pairwise merging of side extrema is not observed.Dependences of mh/m0 on the hole density calculated with several dQW values in the range of 20-200 nm are presented in Fig. 6 together with experimental results for mh/m0 obtained only in structures with dQW = 200 nm.The experimental mh/m0 values in structures with dQW = 22, 32, 46, 60, 88, and 120 nm lie in the dashed rectangle (experimental values are not presented because they are too numerous and it will be very difficult to understand to which structures different symbols belong).
It is seen that the hole density at which a jump in mh/m0 occurs because of the pairwise merging of side extrema should decrease strongly with increasing dQW and this jump at dQW = 200 nm should be observed at p = 0.4  10 11 cm -2 .However, experimental mh/m0 It could be thought that mh/m0 strongly depends on the orientation of the QW.We tested this assumption for the QW with dQW = 80 nm, which was deposited on two substrates with the (013) and (100) orientations.Experimental mh/m0 values at different hole densities presented in the inset of Fig. 6 show that mh( p)/m0 is 20 nm (Fig. 2) and at d QW = 22, 32, 46, 60, 88, and 120 nm.The inset shows the experimental m h /m0 values for two structures with d QW = 80 nm on the (013) and (100) substrates.
independent of the orientation within the experimental error.
Thus, the degree of degeneracy of states near the top of the valence band is 2 in the entire range dQW =  eracy in this range should be 2 and mh/m0 ≈ 0.17-0.18.8-200 nm at hole densities of (1.5−5.5) 10 11 cm -2 .The effective mass of holes mh/m0 at all dQW values At 5  10 10  p  1.2  10 11 cm -2 , the upper state, where the effective hole mass becomes mh/m0 ≈ 0.36, increases monotonically with the hole density from and the lower state with the effective hole mass 0.2 ± 0.03 to 0.3 ± 0.03.This behavior drastically difmh/m0 ≈ 0.18 are filled.At p  2  10 11 cm -2 , both fers from the theoretical spectrum calculated within the four-band kP model, which predicts a stepwise (by a factor of about 2) increase in mh/m0 at the hole density of (4−4.5) 10 11 cm -2 in the 20-nm-wide QW and at the hole density of 0.4  10 11 cm -2 in the 200-nmwide QW.
What is a reason for such discrepancy?1.The hole density in the experiment was changed by varying the gate voltage, i.e., in the presence of the electric field Ez in the quantum well, whereas the calculation was performed for the "empty" spectrum.More accurate self-consistent calculations require the z distribution of the charge of holes; i.e., it is necessary to know wavefunctions at energies below the Fermi energy at all kx and ky values.This problem seems too difficult and, to estimate the effect of the electric field in the quantum well, we consider Ez = const.The dependences of mh/m0 on the hole density calculated at Ez = 5 × 10 2 V/cm presented in Fig. 7 show that doubly degenerate states at the top of the valence band are split and the mass jump is shifted toward lower hole states with close effective hole masses mh/m0 ≈ 0.35-0.38 are filled.Thus, the theory predicts that both the degree of degeneracy and the effective hole mass in the structure with dQW = 80 nm in the presence of the electric field Ez should change with an increase in the hole density.However, at p  2  10 11 cm -2 , as well as in the absence of the field, the degree of degeneracy should be 2 and mh/m0 ≈ 0.4-0.45.This behavior of mh/m0 is not observed: mh/m0 remains in the interval of 0.2-0.3 in the entire hole density range.Thus, discrepancy between the theory and experiment cannot be explained by the fact that the calculations presented in Fig. 6 are not self-consistent.
2. All calculations were performed under the assumption that deformation in the quantum well remains the same as in narrow wells.However, it can be partially removed in wide wells.To estimate the effect of this factor, we calculated the dependence mh( p)/m0 at two values of deformation-induced addition Hp to the Hamiltonian: Hp corresponding to the densities in one branch and toward higher hole densities in the other branch.In a field of 5 × 10 2 V/cm at complete deformation (narrow wells) and 0.5Hp (Fig. 8).p  2.5  10 10 cm -2 , only one upper state is filled, so that the degree of degeneracy in this range should be 1 and mh/m0 ≈ 0.17.At 2.5  10 10  p  5  10 10 cm -2 , both the upper and lower states are filled (marked as upper and lower in Fig. 7), so that the degree of degen-It is seen that the hole density at which the jump in mh/m0 should be observed hardly depends on deformation.
Thus, reasons for the drastic discrepancy between experimental and theoretical dependences mh( p)/m0 lower remain unclear.Consequently, it is doubtful that the kP calculations adequately describe the valence band at all dQW values.Direct experimental evidence of the existence of four fairly high side extrema is absent.The possibility of achieving agreement between the experiment and

Fig. 1 .
Fig. 1. (Color online) Constant-energy contours of the upper split state calculated taking into account the interface inversion asymmetry.Constant-energy contours of the lower split state are similar and located approximately 6 meV below in energy.The calculations were performed for the 8.3-nm-wide quantum well on the (013) substrate at the parameter g4 = 0.8.One half of the picture is shown; the second half is a mirror image.The energy is measured from E(k = 0) with a step of 1 meV.Some energies are indicated near the corresponding contours.The dashed line indicates the direction passing through the maximum of E(k).

Fig. 2 .
Fig. 2. (Color online) Cyclotron mass of holes versus the hole density in structures with d QW = 8.3-20 nm (from [18]).The black and red solid lines are calculated for d QW = 8.3 and 20 nm at g4L = 0.8 and g4R = 1, respectively.The dashed rectangle marks the m h /m0 region including all experimental results. g

Fig. 3 .
Fig. 3. (Color online) (а) Magnetic field dependences of R xx and R H .The inset shows the sketch of E c (k) and E v (k) at d QW > 14 nm.(b) Hole, p, and electron, n, densities and the charge number of the quantum well Q/e = pn versus the gate voltage V g determined in the two-carrier conduction model at the gate voltage V g < 0 and the density p determined from the period of Shubnikov-de Haas oscillations under the assumption of double degeneracy of Landau levels.The inset is the mobility of holes versus the hole density.

Fig. 4 .
Fig. 4. (Color online) (а) Fourier spectrum of oscillations of R xx presented in Fig. 3а; the long-dashed line shows the filter for the separation of contribution from holes to oscillations of R xx .(b) Shubnikov-de Haas oscillations of holes found after the filtration of the Fourier spectrum, as shown in panel (a).(с) (Circles) Temperature dependence of the amplitude of Shubnikov-de Haas oscillations in a field of 1.5 T and (line) the Lifshitz-Kosevich formula with m h /m0 = 0.22.(d) Magnetic field dependence of m h /m 0 .

Fig. 5 .
Fig. 5. (Circles) Hole mass versus the hole density at d QW = 32 nm and (line) the calculated dependence.The

Fig. 6 .
Fig. 6. (Color online) (Lines) Calculated density dependences of the hole mass m h ( p)/m 0 at the indicated d QW values and (circles) experimental values for the largest quantum well with the width d QW = 200 nm.The dashed rectangle same as in Figs. 2 and 5 indicates the m h /m 0 region including all experimental results at d QW = 8-

Fig. 7 .
Fig. 7. (Color online) Hole mass m h /m0 versus the hole density in the upper and lower branches of the spectrum split by the electric field E z .The inset shows the hole mass m h /m0 versus the energy measured from the top of the valence band at different electric field E z .

Fig. 8 .
Fig. 8. (Color online) Density dependences of the hole mass m h ( p)/m0 at two additions Hp and 0.5Hp to the Hamiltonian describing the contribution from deformation.