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Register a new userSpin-orbit splitting of the conduction band in HgTe quantum wells: role of different mechanisms
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Added by
Alexander Germanenko
Publication date
2018
fields
Physics
and research's language is
English
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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.
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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.
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 $d<d_c$, where $d_csimeq6.3$ nm is a critical width of quantum well corresponding to the Dirac-like energy spectrum, is close to the calculated values over the whole electron density range; with increasing width, at $d>(7-8)$ 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.
The Zeeman splitting of the conduction band in the HgTe quantum wells both with normal and inverted spectrum has been studied experimentally in a wide electron density range. The simultaneous analysis of the SdH oscillations in low magnetic fields at different tilt angles and of the shape of the oscillations in moderate magnetic fields gives a possibility to find the ratio of the Zeeman splitting to the orbital one and anisotropy of g-factor. It is shown that the ratios of the Zeeman splitting to the orbital one are close to each other for both types of structures, with a normal and inverted spectrum and they are close enough to the values calculated within kP method. In contrast, the values of g-factor anisotropy in the structures with normal and inverted spectra is strongly different and for both cases differs significantly from the calculated ones. We believe that such disagreement with calculations is a result of the interface inversion asymmetry in the HgTe quantum well, which is not taken into account in the kP calculations.
Using $vec{k}$$cdot$$vec{p}$ theory, we derive an effective four band model describing the physics of the typical two-dimensional topological insulator (HgTe/CdTe quantum well) in the presence of out-of-plane in z-direction inversion breaking and in-plane confining potentials. We find that up to third order in perturbation theory, only the inversion breaking potential generates new elements to the four band Hamiltonian that are off-diagonal in spin space. When this new effective Hamiltonian is folded into an effective two band model for the conduction (electron) or valence (heavy hole) bands, two competing terms appear: (1) a Rashba spin-orbit interaction originating from inversion breaking potential in z-direction and (2) an in-plane Pauli term as a consequence of the in-plane confining potential. Spin transport in the conduction band is further analysed within the Landauer-Buttiker formalism. We find that for asymmetrically doped HgTe quantum wells, the behaviour of the spin-Hall conductance is dominated by the Rashba term.
We report on beating appearance in Shubnikov-de Haas oscillations in conduction band of 18-22nm HgTe quantum wells under applied top-gate voltage. Analysis of the beatings reveals two electron concentrations at the Fermi level arising due to Rashba-like spin splitting of the first conduction subband H1. The difference dN_s in two concentrations as a function of the gate voltage is qualitatively explained by a proposed toy electrostatic model involving the surface states localized at quantum well interfaces. Experimental values of dN_s are also in a good quantitative agreement with self-consistent calculations of Poisson and Schrodinger equations with eight-band kp Hamiltonian. Our results clearly demonstrate that the large spin splitting of the first conduction subband is caused by surface nature of $H1$ states hybridized with the heavy-hole band.
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