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Shubnikov-de Haas (SdH) oscillations and magneto-intersubband oscillations of magnetoresistance of structures with single HgTe quantum wells with a width of (10-18) nm have been experimentally studied. The spectrum of the conduction band in these structures is split by the spin-orbit interaction. This leads to beats of the SdH oscillations and the appearance of low-frequency magneto-intersubband oscillations. The mutual position of the antinodes of the SdH oscillations and the maxima of the magneto-interband oscillations is unusual -- in low magnetic fields it is directly opposite to the predictions of the theory. Measurements in high magnetic fields, in which the relative amplitude of the SdH oscillations becomes greater than 0.2-0.3, show a change in the relative position of the antinodes of the SdH oscillations and the maxima of low-frequency oscillations. Numerical calculations and additional measurements at different temperatures show that the observed effects are due to oscillations of the Fermi level in the magnetic field., Comment: 7 two-column pages, 10 figures

The strictly gauge invariant approach to the construction of the analog of guiding center integrals of motion in spatially homogeneous/inhomogeneous constant magnetic fields is considered. With their help the gauge invariant equations, describing the wave functions of highly degenerate Landau levels in the classical non-relativistic case, are formulated. The proposed gauge-invariant approach was used also for the construction of the equations describing the quasi-relativistic carriers behavior in the homogeneous/inhomogeneous magnetic field in the single layer graphene., Comment: 15 pages

The results of systematic experimental studies of quantum oscillations of resistivity, Hall coefficient and capacitance in GaAs and In$_x$Ga$_{1-x}$As quantum wells (QWs) with a simple electron spectrum and HgTe QWs with a complicated non-parabolic spectrum and strong spin-orbit interaction are reported. A striking result is the ratio of the amplitudes of the resistance and Hall coefficient oscillations. In GaAs QW with a simple spectrum characterized by negligibly small Zeeman and spin-orbit splitting, the ratio of amplitudes is close to that predicted theoretically. In HgTe QWs, this ratio is very different and behaves differently in QWs with normal and inverted electron spectra. In HgTe WQs with a normal spectrum, it tends to a theoretical value with an increase of the filling factor ($N$), while for HgTe QWs with an inverted spectrum, it differs significantly from the theoretical one for all available $N$. It is assumed that such a difference in the ratio of amplitudes in GaAs and HgTe QWs is due not to the peculiarities of the energy spectrum of HgTe, but to the peculiarities of electron scattering due to spin-orbit interaction with the potential of the scatterers. This assumption is justified by analysis of experimental results obtained for a heterostructure with a In$_{0.2}$Ga$_{0.8}$As QW, which spectrum is very close to the GaAs QW spectrum, but characterizes by much stronger spin-orbit splitting value. It has been found that the positions of the resistance and capacitance oscillations, the difference between the phases of the resistance and Hall coefficient oscillations and its $N$ dependence are close to those observed in GaAs QW. At the same time the ratio of the amplitude of the resistance oscillations to the Hall coefficient oscillations and its $N$ dependence differs very strongly and they are close to that observed in HgTe quantum wells., Comment: 11 pages, 14 figures

A magnetotransport and quantum capacitance of the two-dimensional electron gas in HgTe/Cd$_x$Hg$_{1-x}$Te quantum wells of a width ($20.2-46.0$)~nm are experimentally investigated. It is shown that the first energy subband of spatial quantization is split due to the spin-orbit interaction and the split branches are single-spin, therewith the splitting strength increases with the increase of the quantum well width. The electron effective masses in the branches are close to each other within the actual density range. Magneto-intersubband oscillations (MISO) observed in the structures under study exhibit the growing amplitude with the increasing electron density that contradicts to the expected decrease of wave function overlap for the rectangular quantum well. To interpret the data obtained, we have used a self-consistent approach to calculate the electron energy spectrum and the wave function within framework of the \emph{kP}-model. It has been in particular shown that the MISO amplitude increase results from the increasing overlap of the wave functions due to their shift from the gate electrode with the gate voltage increase known as phenomenon of the negative electron polarizability. The results obtained from the transport experiments are supported by quantum capacitance measurements., Comment: 12 pages, 15 figures

In the present paper we study magneto-intersubband oscillations (MISO) in HgTe/Hg$_{1-x}$Cd$_x$Te single quantum well with "inverted" and "normal" spectra and in conventional In$_{1-x}$Ga$_x$As/In$_{1-y}$Al$_y$As quantum wells with normal band ordering. For all the cases when two branches of the spectrum arise due to spin-orbit splitting, the mutual arrangement of the antinodes of the Shubnikov-de Haas oscillations and the maxima of MISO occurs opposite to that observed in double quantum wells and in wide quantum wells with two subbands occupied and does not agree with the theoretical predictions. A "toy" model is supposed that explain qualitatively this unusual result., Comment: 8 pages, 8 figures

The results of experimental studies of the Shubnikov-de Haas (SdH) efect in the (013)-HgTe/Hg$_{1-x}$Cd$_x$Te quantum wells (QWs) of electron type of conductivity both with normal and inverted energy spectrum are reported. Comprehensive analysis of the SdH oscillations measured for the different orientations of magnetic field relative to the quantum well plane and crystallographic exes allows us to investigate the anisotropy of the Zeeman effect. For the QWs with inverted spectrum, it has been shown that the ratio of the spin splitting to the orbital one is strongly dependent not only on the orientation of the magnetic field relative to the QW plane but also on the orientation of the in-plane magnetic field component relative to crystallographic axes laying in the QW plane that implies the strong anisotropy of in-plane g-factor. In the QW with normal spectrum, this ratio strongly depends on the angle between the magnetic field and the normal to the QW plane and reveals a very slight anisotropy in the QW plane. To interpret the data, the Landau levels in the tilted magnetic field are calculated within the framework of four-band \emph{kP} model. It is shown that the experimental results can be quantitatively described only with taking into account the interface inversion asymmetry.

The polarizability of electrons occupying the lowest subband of spatial quantization in CdTe/Cd$_x$Hg$_{1-x}$Te/CdTe quantum wells is calculated. It is shown that polarizability in the quantum well without cadmium is negative, i.e., the displacement of an electron in an electric field applied perpendicularly to the quantum well plane is opposite to the force acting on it. The negative polarizability of 2D electrons can reduce the dielectric constant of quantum wells by up to $(10-15)$ percent., Comment: 4 pages, 4 figures

Spin-orbit splitting of conduction band in HgTe quantum wells was studied experimentally. In order to recognize the role of different mechanisms, we carried out detailed measurements of the Shubnikov-de Haas oscillations in gated structures with a quantum well widths from $8$ to $18$ nm over a wide range of electron density. With increasing electron density controlled by the gate voltage, splitting of the maximum of the Fourier spectrum $f_0$ into two components $f_1$ and $f_2$ and the appearance of the low-frequency component $f_3$ was observed. Analysis of these results shows that the components $f_1$ and $f_2$ give the electron densities $n_1$ and $n_2$ in spin-orbit split subbands while the $f_3$ component results from magneto-intersubband oscillations so that $f_3=f_1 - f_2$. Comparison of these data with results of self-consistent calculations carried out within the framework of four-band \emph{kP}-model shows that a main contribution to spin-orbit splitting comes from the Bychkov-Rashba effect. Contribution of the interface inversion asymmetry to the splitting of the conduction band turns out to be four-to-five times less than that for the valence band in the same structures., Comment: 6 pages, 6 figures

The energy spectrum of the conduction band in HgTe/Cd$_x$Hg$_{1-x}$Te quantum wells of a width $d=(4.6-20.2)$ nm has been experimentally studied in a wide range of electron density. For this purpose, the electron density dependence of the effective mass was measured by two methods: by analyzing the temperature dependence of the Shubnikov-de Haas oscillations and by means of the quantum capacitance measurements. There was shown that the effective mass obtained for the structures with $d(7-8)$ nm, the experimental effective mass becomes noticeably less than the calculated ones. This difference increases with the electron density decrease, i.e., with lowering the Fermi energy; the maximal difference between the theory and experiment is achieved at $d = (15-18)$ nm, where the ratio between the calculated and experimental masses reaches the value of two and begins to decrease with a further $d$ increase. We assume that observed behavior of the electron effective mass results from the spectrum renormalization due to electron-electron interaction., Comment: 8 pages, 10 figures

The energy spectrum of the valence band in HgTe/Cd$_x$Hg$_{1-x}$Te quantum wells with a width $(8-20)$~nm has been studied experimentally by magnetotransport effects and theoretically in framework $4$-bands $kP$-method. Comparison of the Hall density with the density found from 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 $p< 5.5\times 10^{11}$~cm$^{-2}$. Such degeneracy does not agree with the calculations of the spectrum performed within the framework of the $4$-bands $kP$-method for symmetric quantum wells. These calculations show that the top of the valence band consists of four spin-degenerate extremes located at $k\neq 0$ (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 two-fold 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 show that $m_h$ is equal to $(0.25\pm0.02)\,m_0$ 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., Comment: 8 pages, 11 figures

Energy spectra both of the conduction and valence bands of the HgTe quantum wells with a width close to the Dirac point were studied experimentally. Simultaneous analysis of the Shubnikov-de Haas oscillations and Hall effect over a wide range of electron and hole densities gives surprising result: the top of the valence band is strongly split by spin-orbit interaction while the splitting of the conduction band is absent, within experimental accuracy. Astonishingly, but such a ratio of the splitting values is observed as for structures with normal spectrum so for structures with inverted one. These results do not consistent with the results of kP calculations, in which the smooth electric filed across the quantum well is only reckoned in. It is shown that taking into account the asymmetry of the quantum well interfaces within a tight-binding method gives reasonable agreement with the experimental data., Comment: 10 pages, 16 figures

The results of experimental study of interference induced magnetoconductivity in narrow HgTe quantum wells of hole-type conductivity with a normal energy spectrum are presented. Interpretation of the data is performed with taking into account the strong spin-orbit splitting of the energy spectrum of the two-dimensional hole subband. It is shown that the phase relaxation time found from the analysis of the shape of magnetoconductivity curves for the relatively low conductivity when the Fermi level lies in the monotonic part of the energy spectrum of the valence band behaves itself analogously to that observed in narrow HgTe quantum wells of electron-type conductivity. It increases in magnitude with the increasing conductivity and decreasing temperature following the $1/T$ law. Such a behavior corresponds to the inelasticity of electron-electron interaction as the main mechanism of the phase relaxation and agrees well with the theoretical predictions. For the higher conductivity, despite the fact that the dephasing time remains inversely proportional to the temperature, it strongly decreases with the increasing conductivity. It is presumed that a nonmonotonic character of the hole energy spectrum could be the reason for such a peculiarity. An additional channel of the inelastic interaction between the carriers in the main and secondary maxima occurs when the Fermi level arrives the secondary maxima in the depth of the valence., Comment: 9 pages including 3 pages of Supplemental Material, 9 figures

The results of an experimental study of the energy spectrum of the valence band in a HgTe quantum well of width d<6.3 nm with normal spectrum in the presence of a strong spin-orbit splitting are reported. The analysis of the temperature, magnetic field and gate voltage dependences of the Shubnikov-de Haas oscillations allows us to restore the energy spectrum of the two valence band branches, which are split by the spin-orbit interaction. The comparison with the theoretical calculation shows that a six-band kP theory well describes all the experimental data in the vicinity of the top of the valence band., Comment: 7 pages, 7 figures

The results of experimental study of interference induced magnetoconductivity in narrow quantum well HgTe with the normal energy spectrum are presented. Analysis is performed with taking into account the conductivity anisotropy. It is shown that the fitting parameter \tau_\phi corresponding to the phase relaxation time increases in magnitude with the increasing conductivity (\sigma) and decreasing temperature following the 1/T law. Such a behavior is analogous to that observed in usual two-dimensional systems with simple energy spectrum and corresponds to the inelasticity of electron-electron interaction as the main mechanism of the phase relaxation. However, it drastically differs from that observed in the wide HgTe quantum wells with the inverted spectrum, in which \tau_\phi being obtained by the same way is practically independent of \sigma. It is presumed that a different structure of the electron multicomponent wave function for the inverted and normal quantum wells could be reason for such a discrepancy., Comment: 7 pages, 8 figures

The results of experimental study of the magnetoresistivity, the Hall and Shubnikov-de Haas effects for the heterostructure with HgTe quantum well of 20.2 nm width are reported. The measurements were performed on the gated samples over the wide range of electron and hole densities including vicinity of a charge neutrality point. Analyzing the data we conclude that the energy spectrum is drastically different from that calculated in framework of $kP$-model. So, the hole effective mass is equal to approximately $0.2 m_0$ and practically independent of the quasimomentum ($k$) up to $k^2\gtrsim 0.7\times 10^{12}$ cm$^{-2}$, while the theory predicts negative (electron-like) effective mass up to $k^2=6\times 10^{12}$ cm$^{-2}$. The experimental effective mass near k=0, where the hole energy spectrum is electron-like, is close to $-0.005 m_0$, whereas the theoretical value is about $-0.1 m_0$.

The results of experimental study of the magnetoconductivity of 2D electron gas caused by suppression of the interference quantum correction in HgTe single quantum well heterostructure with the inverted energy spectrum are presented. It is shown that only the antilocalization magnetoconductivity is observed at the relatively high conductivity $\sigma>(20-30)G_0$, where $G_0= e^2/2\pi^2\hbar$. The antilocalization correction demonstrates a crossover from $0.5\ln{(\tau_\phi/\tau)}$ to $1.0\ln{(\tau_\phi/\tau)}$ behavior with the increasing conductivity or decreasing temperature (here $\tau_\phi$ and $\tau$ are the phase relaxation and transport relaxation times, respectively). It is interpreted as a result of crossover to the regime when the two chiral branches of the electron energy spectrum contribute to the weak antilocalization independently. At lower conductivity $\sigma<(20-30)G_0$, the magnetoconductivity behaves itself analogously to that in usual 2D systems with the fast spin relaxation: being negative in low magnetic field it becomes positive in higher one. We have found that the temperature dependences of the fitting parameter $\tau_\phi$ corresponding to the phase relaxation time demonstrate reasonable behavior, close to 1/T, over the whole conductivity range from $5G_0$ up to $130G_0$. However, the $\tau_\phi$ value remains practically independent of the conductivity in distinction to the conventional 2D systems with the simple energy spectrum, in which $\tau_\phi$ is enhanced with the conductivity., Comment: 8 pages, 8 figures

The electron-electron interaction quantum correction to the conductivity of the gated single quantum well InP/In$_{0.53}$Ga$_{0.47}$As heterostructures is investigated experimentally. The analysis of the temperature and magnetic field dependences of the conductivity tensor allows us to obtain reliably the diffusion part of the interaction correction for different values of spin relaxation rate, $1/\tau_s$. The surprising result is that the spin relaxation processes do not suppress the interaction correction in the triplet channel and, thus, do not enhance the correction in magnitude contrary to theoretical expectations even in the case of relatively fast spin relaxation, $1/T\tau_s\simeq (20-25)\gg 1$., Comment: 7 pages, 6 figures

The electron-electron interaction quantum correction to the conductivity of the gated double well Al$_x$Ga$_{1-x}$As/GaAs structures is investigated experimentally. The analysis of the temperature and magnetic field dependences of the conductivity tensor allows us to obtain reliably the diffusion part of the interaction correction for the regimes when the structure is balanced and when only one quantum well is occupied. The surprising result is that the interaction correction does not reveal resonant behavior; it is practically the same for both regimes., Comment: 5 pages, 4 figures

The nonohmic conductivity of 2D hole gas (2DHG) in single $GaAsIn_{0.2}Ga_{0.8}AsGaAs$ quantum well structures within the temperature range of 1.4 - 4.2K, the carrier's densities $p=(1.5-8)\cdot10^{15}m^{-2}$ and a wide range of conductivities $(10^{-4}-100)G_0$ ($G_0=e^2/\pi\,h$) was investigated. It was shown that at conductivity $\sigma>G_0$ the energy relaxation rate $P(T_h,T_L)$ is well described by the conventional theory (P.J. Price J. Appl. Phys. 53, 6863 (1982)), which takes into account scattering on acoustic phonons with both piezoelectric and deformational potential coupling to holes. At the conductivity range $0.01G_0<\sigma

We investigate the gate voltage dependence of capacitance of a system gate - 2D electron gas (C-Vg). The abrupt drop of capacitance at decreasing concentration was found. The possible reasons of this drop, namely inhomogeneity of electron density distribution and serial resistance of 2D electron gas are discussed. Simultaneous analysis of gate voltage dependences of capacitance and resistance has shown that in heavily doped 2D systems the main role in the drop of capacitance at decreasing concentration plays the resistance of 2D gas. It is found that the investigated systems remains homogeneous down to the low temperature conductivity about (10^-2-10^-3)e^2/h., Comment: 5 pages, 5 figures

The interference quantum correction to the conductivity in the gated double quantum well Al$_x$Ga$_{1-x}$As/GaAs/Al$_x$Ga$_{1-x}$As structures is studied experimentally. The consistent analysis of the interference induced positive magnetoconductivity allows us to find the interwell transition time $\tau_{12}$ and the electron dephasing time $\tau_\phi$. It has been obtained that $\tau_{12}^{-1}$ resonantly depends on the difference between the electron densities in the wells as predicted theoretically. The dephasing times have been determined under the conditions when one and both quantum wells are occupied. The surprising result is that the $\tau_\phi$ value in the one well does not depend on the occupation of the other one., Comment: 7 pages, 8 figures

The nonlinear behavior of the Hall resistivity at low magnetic fields in single quantum well GaAs/In$_x$Ga$_{1-x}$As/GaAs heterostructures with degenerated electron gas is studied. It has been found that this anomaly is accompanied by the weaker temperature dependence of the conductivity as compared with that predicted by the first-order theory of the quantum corrections to the conductivity. We show that both effects in strongly disordered systems stem from the second order quantum correction caused by the effect of weak localization on the interaction correction and vice versa. This correction contributes mainly to the diagonal component of the conductivity tensor, it depends on the magnetic field like the weak localization correction and on the temperature like the interaction contribution., Comment: 7 pages, 7 figures

We study the electron-electron interaction contribution to the conductivity of two-dimensional In$_{0.2}$Ga$_{0.8}$As electron systems in the diffusion regime over the wide conductivity range, $\sigma\simeq(1-150) G_0$, where $G_0=e^2/(2\pi^2\hbar)$. We show that the data are well described within the framework of the one-loop approximation of the renormalization group (RG) theory when the conductivity is relatively high, $\sigma \gtrsim 15 G_0$. At lower conductivity, the experimental results are found to be in drastic disagreement with the predictions of this theory. The theory predicts much stronger renormalization of the Landau's Fermi liquid amplitude, which controls the interaction in the triplet channel, than that observed experimentally. A further contradiction is that the experimental value of the interaction contribution does not practically depend on the magnetic field, whereas the RG theory forecasts its strong decrease due to decreasing diagonal component of the conductivity tensor in the growing magnetic field., Comment: 9 pages, 10 figures

Results of experimental study of the weak localization phenomenon in 2D system with artificial inhomogeneity of potential relief are presented. It is shown that the shape of the magnetoconductivity curve is determined by the statistics of closed paths. The area distribution function of closed paths has been obtained using the Fourier transformation of the magnetoconductivity curves taken at different temperatures. The experimental results are found in a qualitative agreement with the results of computer simulation., Comment: 8 pages, 10 figures

The diffusion contribution of the hole-hole interaction to the conductivity is analyzed in gated GaAs/In$_x$Ga$_{1-x}$As/GaAs heterostructures. We show that the change of the interaction correction to the conductivity with the decreasing Drude conductivity results both from the compensation of the singlet and triplet channels and from the arising prefactor $\alpha_i<1$ in the conventional expression for the interaction correction., Comment: 6 pages, 5 figures

The temperature and magnetic field dependences of the conductivity of the heterostructures with asymmetric In$_x$Ga$_{1-x}$As quantum well are studied. It is shown that the metallic-like temperature dependence of the conductivity observed in the structures investigated is quantitatively understandable within the whole temperature range, $T=0.4-20$ K. It is caused by the interference quantum correction at fast spin relaxation for 0.4 K$ < T < 1.5$ K. At higher temperatures, 1.5 K$4-6$ K, the metallic-like behavior is determined by the phonon scattering., Comment: 4 pages, 4 figures

Temperature and magnetic field dependences of the conductivity in heavily doped, strongly disordered two-dimensional quantum well structures GaAs/In$_x$Ga$_{1-x}$As/GaAs are investigated within wide conductivity and temperature ranges. Role of the interference in the electron transport is studied in the regimes when the phase breaking length $L_\phi$ crosses over the localization length $\xi\sim l\exp{(\pi k_Fl/2)}$ with lowering temperature, where $k_F$ and $l$ are the Fermi quasimomentum and mean free path, respectively. It has been shown that all the experimental data can be understood within framework of simple model of the conductivity over delocalized states. This model differs from the conventional model of the weak localization developed for $k_Fl\gg 1$ and $L_\phi\ll\xi$ by one point: the value of the quantum interference contribution to the conductivity is restricted not only by the phase breaking length $L_\phi$ but by the localization length $\xi$ as well. We show that just the quantity $(\tau_\phi^\ast)^{-1}=\tau_\phi^{-1}+\tau_\xi^{-1}$ rather than $\tau_\phi^{-1}$, where $\tau_\phi\propto T^{-1}$ is the dephasing time and $\tau_\xi\sim\tau\exp(\pi k_F l)$, is responsible for the temperature and magnetic field dependences of the conductivity over the wide range of temperature and disorder strength down to the conductivity of order $10^{-2} e^2/h$., Comment: 11 pages, 15 figures

Results of numerical simulation of the weak localization in two-dimensional systems in wide range of magnetic filed are presented. Three cases are analyzed: (i) the isotropic scattering and randomly distributed scatterers; (ii) the anisotropic scattering and randomly distributed scatterers; (iii) the isotropic scattering and the correlated distribution of the scatterers. It is shown that the behavior of the conductivity beyond the diffusion regime strongly depends on the scattering anisotropy and correlation in the scatterer distribution., Comment: 11 pages, 6 figures

The results of an experimental study of interaction quantum correction to the conductivity of two-dimensional electron gas in A$_3$B$_5$ semiconductor quantum well heterostructures are presented for a wide range of $T\tau$-parameter ($T\tau\simeq 0.03-0.8$), where $\tau$ is the transport relaxation time. A comprehensive analysis of the magnetic field and temperature dependences of the resistivity and the conductivity tensor components allows us to separate the ballistic and diffusion parts of the correction. It is shown that the ballistic part renormalizes in the main the electron mobility, whereas the diffusion part contributes to the diagonal and does not to the off-diagonal component of the conductivity tensor. We have experimentally found the values of the Fermi-liquid parameters describing the electron-electron contribution to the transport coefficients, which are found in a good agreement with the theoretical results., Comment: 11 pages, 11 figures

The interaction correction to the conductivity of 2D hole gas in strained GaAs/In$_x$Ga$_{1-x}$As/GaAs quantum well structures was studied. It is shown that the Zeeman splitting, spin relaxation and ballistic contribution should be taking into account for reliable determination of the Fermi-liquid constant $F_0^\sigma$. The proper consideration of these effects allows us to describe both th temperature and magnetic field dependences of the conductivity and find the value of $F_0^\sigma$., Comment: 7 pages, 6 figures