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Renormalization of the electron g factor in the degenerate two-dimensional electron gas of ZnSe- and CdTe-based quantum wells
Abstract
The effective electron g factor, geff, is measured in a two-dimensional electron gas (2DEG) in modulation-doped ZnSe- and CdTe-based quantum wells by means of time-resolved pump-probe Kerr rotation. The measurements are performed in magnetic fields applied in the Voigt geometry, i.e., normal to the optical axis parallel to the quantum well plane, in the field range 0.05–6 T at temperatures 1.8–50K. The geff absolute value considerably increases with increasing electron density ne. geff changes in the ZnSe-based QWs from +1.1 to +1.9 in the ne range 3×1010−1.4×1012cm−2 and in the CdTe-based QWs from −1.55 down to −1.76 in the ne range 5×109−3×1011cm−2. The modification of geff reduces with increasing magnetic field, increasing temperature of lattice and 2DEG, the latter achieved by a higher photoexcitation density. A theoretical model is developed that considers the renormalization of the spin-orbit coupling constant of the two-dimensional electrons by the electron-electron interaction and takes into account corrections to the electron-electron interaction in the Hubbard form. The model results are in good agreement with experimental data.
... The second contribution originates from the commutator of λ 2 [k × σ] in Eq. (41) and V in Eq. (31). From Eqs. (31), (43), we see that the ratio between the SOC constant of the velocity operator and the SOC constant of the Hamiltonian equals 1 + λ 2 /λ SOC . In a system described by the Pauli Hamiltonian, λ SOC = λ 2 9 and thus this ratio is 2 (which is a known result [24,[32][33][34]). ...
... Together with Eqs. (41), (42), (43), they constitute the second most important result of this paper. Table 2: SOC constants for a system described by the Pauli Hamiltonian, the Kane model, InSb, and GaAs. ...
... Another effect that might be measurable experimentally is the renormalization of the spin g-factor by the first square bracketed group in Eq. (31). One can expect to observe the density dependence of the g-factor, not related to electron-electron interactions [42,43] or the orbital magnetism [44]. ...
We diagonalize the 8-band Kane Hamiltonian with a proper inclusion of the interband matrix elements of the scalar and vector potentials. This leads, among other results, to a modification of the conventional expression for the spin-orbit coupling (SOC) strength in narrow-gap semiconductors with the zinc blende symmetry. We find that in GaAs, at low temperatures, the correct expression for the SOC strength is actually twice as large as usually considered. In InSb it is 1.76 1.76 times larger. We also provide a proper treatment of the interband matrix elements of the position operator. We show that the velocity operator in a crystal should be defined as a time-derivative of a fictitious position operator rather than the physical one. We compute the expressions for both these position operators projected to the conduction band of the 8-band Kane model. We also derive an expression for the projected velocity operator and demonstrate that the SOC strength in it differs from the SOC strength in the Hamiltonian. The ratio between them is not equal to 1 1 , as it is often assumed for the Rashba model. It does not equal 2 2 either. The correct result for this ratio is given by a rational function of the parameters of the model. This function takes values between 4(23+3\sqrt{2})/73≈ 1.49 4 ( 23 + 3 2 ) / 73 ≈ 1.49 and 2 2 . Our findings modify a vast number of research results obtained using the Rashba model and provide a path for consistent treatment of the latter in future applications.
... In this respect, we point out that SOS in 2D TMDs, ∆(n), may be sensi-tive to the carrier density, n, due to exchange interaction between electrons [69]. This effect is analogous to the exchange enhancement of electron's g-factor established in various 2D systems (such as accumulation layers in semiconductor heterostructures) [70][71][72][73]. Due to such renormalization, the apparent absolute value of SOS, ∆(n * ), at the threshold density, n * , would substantially differ from the single-particle value ∆ 0 (provided by DFT computations [33,74,75]), as illustrated in Fig. 1 for electrons in the spin-split bands shown in the sketch. ...
We show that spin-orbit splitting (SOS) in monolayers of semiconducting transition metal dichalcogenides (TMDs) is substantially enhanced by electron-electron interaction. This effect, similar to the exchange-enhancement of the electron g-factor, is most pronounced for conduction band electrons (in particular, in MoS), and it has a non-monotonic dependence on the carrier sheet density, n. That is, the SOS enhancement is peaked at the onset of filling of the higher-energy spin-split band by electrons, , which also separates the regimes of slow (at ) and fast (for ) spin and valley relaxation of charge carriers. Moreover, this density itself is determined by the enhanced SOS value, making the account of exchange renormalisation important for the analysis of spintronic performance of field-effect transistors based on two-dimensional TMDs.
... In this respect, we point out that SOS in 2D TMDs, (n), may be sensitive to the carrier density n due to an exchange interaction between electrons [68]. This effect is analogous to the exchange enhancement of the electron's g factor established in various 2D systems (such as accumulation layers in semiconductor heterostructures) [69][70][71][72]. Due to such renormalization, the apparent absolute value of SOS, (n * ), at the threshold density n * would substantially differ from the single-particle value 0 (provided by DFT computations [33,73,74]), as illustrated in Fig. 1 for electrons in the spin-split bands shown in the sketch. ...
We show that spin-orbit splitting (SOS) in monolayers of semiconducting transition metal dichalcogenides (TMDs) is substantially enhanced by an electron-electron interaction. This effect, similar to the exchange enhancement of the electron g factor, is most pronounced for conduction band electrons (in particular, in MoS 2 ), and it has a nonmonotonic dependence on the carrier sheet density n . That is, the SOS enhancement is peaked at the onset of filling of the higher-energy spin-split band by electrons, n * , which also separates the regimes of slow (at n < n * ) and fast (for n > n * ) spin and valley relaxation of charge carriers. Moreover, this density itself is determined by the enhanced SOS value, making the account of exchange renormalization important for the analysis of the spintronic performance of field-effect transistors based on two-dimensional TMDs.
Published by the American Physical Society 2024
... Furthermore, the approximate independence of the decay time τ v from B ∥ shows that g factor fluctuations do not play a role. Otherwise, a 1/B ∥ dependence of τ v would be expected 45,46 . Figure 3a shows a summary of all oscillation frequencies ν, extracted by this procedure, versus B ∥ . ...
Layered van-der-Waals materials with hexagonal symmetry offer an extra degree of freedom to their electrons, the so-called valley index or valley pseudospin, which behaves conceptually like the electron spin. Here, we present investigations of excitonic transitions in mono- and multilayer WSe 2 and MoSe 2 materials by time-resolved Faraday ellipticity (TRFE) with in-plane magnetic fields, B ∥ , of up to 9 T. In monolayer samples, the measured TRFE time traces are almost independent of B ∥ , which confirms a close to zero in-plane exciton g factor g ∥ , consistent with first-principles calculations. In contrast, we observe pronounced temporal oscillations in multilayer samples for B ∥ > 0. Our first-principles calculations confirm the presence of a non-zero g ∥ for the multilayer samples. We propose that the oscillatory TRFE signal in the multilayer samples is caused by pseudospin quantum beats of excitons, which is a manifestation of spin- and pseudospin layer locking in the multilayer samples.
... Furthermore, the approximate independence of the decay time τ v from B shows that g factor fluctuations do not play a role. Otherwise, a 1/B dependence of τ v would be expected [40,41]. Figure 3a shows a summary of all oscillation frequencies ν, extracted by this procedure, versus B . ...
Layered van-der-Waals materials with hexagonal symmetry offer an extra degree of freedom to their electrons, the so called valley index or valley pseudospin. This quantity behaves conceptually like the electron spin and the term valleytronics has been coined. In this context, the group of semiconducting transition-metal dichalcogenides (TMDC) are particularly appealing, due to large spin-orbit interactions and a direct bandgap at the K points of the hexagonal Brillouin zone. In this work, we present investigations of excitonic transitions in mono- and multilayer WSe and MoSe materials by time-resolved Faraday ellipticity (TRFE) with in-plane magnetic fields, , of up to 9 T. In monolayer samples, the measured TRFE time traces are almost independent of , which confirms a close to zero in-plane exciton g factor , consistent with first-principles calculations. In stark contrast, we observe pronounced temporal oscillations in multilayer samples for . Remarkably, the extracted in-plane are very close to reported out-of-plane exciton g factors of the materials, namely and for the 1s A excitons in WSe and MoSe multilayers, respectively. Our first-principles calculations nicely confirm the presence of a non-zero for the multilayer samples. We propose that the oscillatory TRFE signal in the multilayer samples is caused by pseudospin quantum beats of excitons, which is a manifestation of spin- and pseudospin layer locking in the multilayer samples. Our results demonstrate ultrafast pseudospin rotations in the GHz- to THz frequency range, which pave the way towards ultrafast pseudospin manipulation in multilayer TMDC samples.
The effective g factor of holes is measured in modulation-doped ZnSe/(Zn,Mg)(S,Se) quantum wells and from surface-state p-doped CdTe/(Cd,Mg)Te quantum wells by time-resolved pump-probe Kerr rotation. The measurements are performed at a temperature of 1.7 K and in magnetic fields up to 5 T applied in the Voigt geometry with orientation perpendicular to the quantum-well growth axis. The absolute value of the in-plane hole g factor increases with growing magnetic field in both studied heterostructures. A theoretical model is developed that considers the influence of magnetic field and interface mixing of heavy-hole and light-hole states on the g factor. The model results are in good agreement with the experimental data.
Time-resolved Kerr-rotation microscopy explores the influence of optical doping on the persistent spin helix in a [001]-grown CdTe quantum well at cryogenic temperatures. Electron spin diffusion dynamics reveal a momentum-dependent effective magnetic field providing SU(2) spin-rotation symmetry, consistent with kinetic theory. The Dresselhaus and Rashba spin-orbit coupling parameters are extracted independently from rotating the spin helix with external magnetic fields applied parallel and perpendicular to the effective magnetic field. Most importantly, a non-uniform spatiotemporal precession pattern is observed. The kinetic theory framework of spin diffusion allows for modeling of this finding by incorporating the photocarrier density into the Rashba () and the Dresselhaus () parameters. Corresponding calculations are further validated by an excitation-density dependent measurement. This work shows universality of the persistent spin helix by its observation in a II-VI compound and the ability to fine-tune it by optical doping.
Spatio-temporal evolution of electron spins in a modulation-doped GaAs single quantum well is investigated using ultrafast two-color Kerr rotation spectroscopy. The evolution is governed by the spin-orbit interaction related to bulk and structural inversion asymmetries which manifests as an effective magnetic field. We directly extract the Dresselhaus and Rashba contributions via the coherent spin precession of the electrons which diffuse and/or drift in the presence of an in-plane electric field. In addition, we explore the effects of altered in-plane electric and magnetic fields on this precession and the corresponding diffusion coefficient. Surprisingly, the spatial spin precession depends strongly on the in-plane electric field, which has not been observed before. These experiments are well supported by corresponding theoretical simulations. Further, we find the Rashba component of the spin-orbit coupling to markedly depend on the electric field applied along the growth direction of the quantum well.
Atheory for the electron (and hole) g factor in multivalley lead-salt IV-VI semiconductor quantum wells (QWs) is presented. An effective Hamiltonian for theQWelectronic states in the presence of an external magnetic field is introduced within the envelope-function approximation, based on the multiband kp Dimmock model for the bulk. The mesoscopic spin-orbit (Rashba-type) and Zeeman interactions are taken into account on an equal footing and the effective g factor in symmetric quantum wells (g*(QW)) is calculated analytically for each nonequivalent conduction-band (and valence-band) valley, and for QWs grown along different crystallographic directions.
We study theoretically the ultrafast spin dynamics of diluted magnetic
semiconductors in the presence of spin-orbit interaction. Our goal is to
explore the interplay or competition between the exchange sd-coupling and the
spin-orbit interaction in both bulk and quantum well systems. For bulk
materials we concentrate on ZnMnSe and take into account the
Dresselhaus interaction, while for quantum wells we examine
HgMnCdTe systems with a strong Rashba coupling. Our
calculations were performed with a recently developed formalism which
incorporates electronic correlations beyond mean-field theory originated from
the exchange sd-coupling. For both bulk and quasi-two-dimensional systems we
find that, by varying the system parameters within realistic ranges, both
interactions can be chosen to play a dominant role or to compete on an equal
footing with each other. The most notable effect of the spin-orbit interaction
in both types of systems is the appearance of strong oscillations where the
exchange sd-coupling by itself only causes an exponential decay of the mean
electronic spin components. The mean-field approximation is also studied and it
is interpreted analytically why it shows a strong suppression of the
spin-orbit-induced dephasing of the spin component parallel to the Mn magnetic
field.
We report on magneto-optical studies of ZnSe/(Zn,Mg)(S,Se) quantum wells with n-type and p-type modulation doping. Negatively and positively charged excitons related to the heavy-hole exciton states are found and identified by their polarization properties. Negatively charged excitons formed with light-hole exciton states are observed. Their binding energy is about 20% less than that related to the heavy-hole exciton. The exciton and trion parameters (radiative and nonradiative dampings) are determined.
Optically detected magnetic resonance is applied to study the fast optical transition processes on a ns time scale with the microwave-induced magnetic transition rate much lower than the optical transition rate. The spin resonance of excess electrons (EESR) is observed with the detection on either the direct exciton or the negatively charged exciton X- emission in type-I CdTe/Cd1-xMgxTe quantum wells with excess electrons of low density. It is found that the electron-spin-dependent and electron-spin-conserved formation and recombination of X- make the optical detection of EESR feasible.
Very high precision measurements of the electron Lande g-factor in GaAs are presented using spin-quantum beat spectroscopy at low excitation densities and temperatures ranging from 2.6 to 300 K. In colligation with available data for the temperature dependent effective mass a temperature dependence of the interband matrix element within a common five level kp-theory can model both parameters consistently. A strong decrease of the interband matrix element with increasing temperature consistently closes a long lasting gap between experiment and theory and substantially improves the modeling of both parameters.
Properties of conduction electrons in GaAs are described theoretically using a five-level k⋅p model, which consistently accounts for inversion asymmetry of the material. The dispersion relation E(k) is computed and it is shown that the conduction band is both nonparabolic and nonspherical. The energy dependence of the electron effective mass, the energy-momentum relation in the forbidden gap, and the spin splitting of the band are calculated. Analytical expressions for the band-edge effective mass, the spin splitting, and the Landé factor g* are presented, taking explicitly into account an interband matrix element of the spin-orbit interaction. A five-level P⋅p theory for the conduction band in the presence of an external magnetic field is developed. Resonant and nonresonant effects due to polar electron–optic-phonon interaction are included in the theory. The spin g value of conduction electrons is calculated as a function of energy and magnetic field. Spin-doublet splitting of the cyclotron resonance and the cyclotron-resonance-mass anisotropy are described. A comparison of the theory with experimental data of various authors is used to determine important band parameters for GaAs. It is shown that away from the band edge the polaron effects in GaAs are comparable to the band-structure effects.
The Zeeman splitting and the underlying g factor for conduction-band electrons in GaAs∕AlxGa1−xAs quantum wells have been measured by spin-beat spectroscopy based on a time-resolved Kerr rotation technique. The experimental data are plotted as functions of the lowest band-to-band optical transition energy, i.e., the effective band gap of the quantum wells. The model calculations suggest that in the tracked range of transition energies E from 1.52 to 2.0 eV, the component of electron g factor along the growth axis follows closely the universal dependence g‖(E)≈0.445+3.38(E−1.519)−2.21(E−1.519)2 (with E given in eV), and this universality also embraces AlxGa1−xAs alloys. On the other hand, the in-plane g factor component deviates in a systematic way from this universal curve, with the degree of deviation controlled by the structural anisotropy. The experimental data are in good agreement with the theoretical predictions.
We report a theoretical study of the exchange interaction effects in the electron spin resonance (ESR) in n-type narrow-gap quantum well (QW) heterostructures. Using the Hartree-Fock approximation, based on the eight-band k⋅p Hamiltonian, the many-body correction to the ESR energy is found to be nonzero, providing theoretical evidence of Larmor theorem violation in symmetric narrow-gap QWs. We predict the exchange enhancement of the ESR g-factor and its divergence in low magnetic fields. The 'enhanced' ESR g-factor and quasiparticle g-factor, measured in magnetotransport, coincide at even-valued filling factors of the Landau levels in moderate and quantizing magnetic fields.
The spin coherence of a two-dimensional electron gas (2DEG) at different densities in Cd Te / Cd 0.85 Mg 0.15 Te quantum wells has been examined by the time-resolved Kerr rotation technique using resonant excitation of either trions or excitons. The formation of negatively charged trions, either excited resonantly or via exciton states, causes strong spin polarization of the 2DEG. This effect leads to a long lasting exponential decay in the nanosecond regime. Spin dephasing times T2* of the 2DEG measured as a function of electron density up to 2.4×1011 cm -2 show a nonmonotonic behavior with a maximum at 8×1010 cm -2 .
Using the 'screened' Hartree-Fock approximation based on the eight-band k·p Hamiltonian, we have extended our previous work (Krishtopenko et al 2011 J. Phys.: Condens. Matter 23 385601) on exchange enhancement of the g-factor in narrow-gap quantum well heterostructures by calculating the exchange renormalization of quasiparticle energies, the density of states at the Fermi level and the quasiparticle g-factor for different Landau levels overlapping. We demonstrate that exchange interaction yields more pronounced Zeeman splitting of the density of states at the Fermi level and leads to the appearance of peak-shaped features in the dependence of the Landau level energies on the magnetic field at integer filling factors. We also find that the quasiparticle g-factor does not reach the maximum value at odd filling factors in the presence of large overlapping of spin-split Landau levels. We advance an argument that the behavior of the quasiparticle g-factor in weak magnetic fields is defined by a random potential of impurities in narrow-gap heterostructures.
We report on the study of the exchange enhancement of the g-factor in the two-dimensional (2D) electron gas in n-type narrow-gap semiconductor heterostructures. Our approach is based on the eight-band k⋅p Hamiltonian and takes into account the band nonparabolicity, the lattice deformation, the spin-orbit coupling and the Landau level broadening in the δ-correlated random potential model. Using the 'screened' Hartree-Fock approximation we demonstrate that the exchange g-factor enhancement not only shows maxima at odd values of Landau level filling factors but, due to the conduction band nonparabolicity, persists at even filling factor values as well. The magnitude of the exchange enhancement, the amplitude and the shape of the g-factor oscillations are determined by both the screening of the electron-electron interaction and the Landau level width. The 'enhanced' g-factor values calculated for the 2D electron gas in InAs/AlSb quantum well heterostructures are compared with our earlier experimental data and with those obtained by Mendez et al (1993 Phys. Rev. B 47 13937) in magnetic fields up to 30 T.
Optical control of the spin coherence of quantum well electrons by short laser pulses with circular or linear polarization is studied experimentally and theoretically. For that purpose the coherent electron spin dynamics in a n-doped CdTe/(Cd,Mg)Te quantum well structure was measured by time-resolved pump-probe Kerr rotation, using resonant excitation of the negatively charged exciton (trion) state. The amplitude and phase shifts of the electron spin beat signal in an external magnetic field, that are induced by laser control pulses, depend on the pump-control delay and polarization of the control relative to the pump pulse. Additive and non-additive contributions to pump-induced signal due to the control are isolated experimentally. These contributions can be well described in the framework of a two-level model for the optical excitation of the resident electron to the trion. Comment: 15 pages, 18 figures
CdSe colloidal nanoplatelets are studied by spin-flip Raman scattering in magnetic fields up to 5 T. We find pronounced Raman lines shifted from the excitation laser energy by an electron Zeeman splitting. Their polarization selection rules correspond to those expected for scattering mediated by excitons interacting with resident electrons. Surprisingly, Raman signals shifted by twice the electron Zeeman splitting are also observed. The theoretical analysis and experimental dependencies show that the mechanism responsible for the double flip involves two resident electrons interacting with a photoexcited exciton. Effects related to various orientations of the nanoplatelets in the ensemble and different orientations of the magnetic field are analyzed.
Optically detected magnetic resonance (ODMR) is measured for photoexcited electrons in (In,Al)As/AlAs quantum dots having an indirect band gap and a type-I band alignment. A sharp ODMR resonance corresponding to a g factor of 1.97±0.02 is identified, associated with X-valley electrons. Its variation with optical transition energy allows us to distinguish between the spectral regions of direct and indirect QDs, which is in good agreement with the results on exciton recombination dynamics.
Spin coherence of resident electrons and holes is measured in ZnSe-based quantum wells by means of time-resolved Kerr rotation technique. At a temperature of 1.8 K spin dephasing time for localized electrons can be as long as 33 ns, and for holes of 0.8 ns. Electron spin precession is clearly observed in a wide temperature range up to 230 K. The electron spin dephasing becomes much shorter of 0.2 ns for the quantum well with a high-density electron gas. Using vector magnet all components of the g-factors tensor are evaluated for the electrons and heavy-holes, revealing strong anisotropy for the heavy-holes.
The g-factor tensors of electron and hole in self-assembled (In,Ga)As/GaAs quantum dots are studied by time-resolved ellipticity measurements in a three dimensional vector magnet system. Both g-factor tensors show considerable deviations from isotropy. These deviations are much more pronounced for the hole than for the electron and are described by different anisotropy factors, which can even have opposite signs.
The theory of the effect of a magnetic field on the optical absorption in semiconductors is developed on the basis of the effective-mass approximation. For simple parabolic conduction and valence bands and a direct transition which is allowed at k=0, absorption peaks occur at energies above the zero-field gap. Since the selection rule for the transition is Δn=0 where n is the magnetic quantum number, the spacing between the peaks is the sum of the cyclotron frequencies for the two bands. For degenerate band edges, the spectrum is more complicated. A detailed treatment of the direct transition in germanium is given in which account is taken of the change in curvature of the bands away from k=0 and the results are in good agreement with the experimental measurements of Zwerdling, Lax, Roth, and Button. The k=0 conduction band mass is found to agree with predictions based on cyclotron resonance in the valence band. In addition, a gyromagnetic ratio for conduction electrons of -2.6 resulted from the calculations. The deviation from g=2.0 is due to spin-orbit interaction. In InSb the effect is much greater, the result being g=-50. These are consistent with experimental results. For bands in which the transition probability vanishes at k=0, absorption peaks will also occur corresponding to Δn=±1 but absorption edges occur for Δn=0. In the case of indirect transitions, the absorption does not exhibit oscillations but consists of a series of "steps" as has been observed in Ge by Zwerdling, Lax, Roth, and Button.
A comparative survey of known experimental and theoretical values of heavy-hole and electron effective mass in GaAs, InAs, and A1As is presented in this work. Recommended room-temperature values of these parameters are given for the above binary solutions and for their ternary compounds.
We calculate, as a function of temperature and density, the electron-electron interaction induced quasiparticle effective-mass renormalization in two-dimensional electron systems within the leading-order dynamically screened Coulomb interaction expansion. We find an unexpected nonanalyticity and nonmonotonicity in the temperature-dependent effective mass with the renormalized mass linearly increasing with temperature at low temperatures for all densities.
The effective mass and the g factor of quasiparticles near the Fermi surface of a two-dimensional electron gas are calculated in the random-phase approximation and in the Hubbard approximation, and are compared with the experimental results for an inversion layer on a (100) surface of silicon.
This review covers results of recent O.D.M.R. measurements of recombination processes in layered semiconductors, II VI compounds, at dep traps and in amorphous compounds, and includes consideration of experimental aspects.
Character tables for the "group of the wave vector" at certain points of symmetry in the Brillouin zone are given. The additional degeneracies due to time reversal symmetry are indicated. The form of energy vs wave vector at these points of symmetry is derived. A possible reason for the complications which may make a simple effective mass concept invalid for some crystals of this type structure will be presented.
The importance of a well-known theorem, originally due to Larmor, is emphasized. It enables a definition of "momentum" and "moment of momentum" for electrons in a magnetic field, hence the possibility of writing the conservation of these quantities when the geometry of the structure is convenient. As typical examples of the method, two special cases are discussed: a plane electron beam and a cylindrical electron beam with longitudinal magnetic field. In both cases it is found that the space-charge density of the beam is entirely controlled by the magnetic field and that the maximum current is obtained for a suitable optimum magnetic field.
The magnitude of the effects of the Coulomb interaction between the electrons in an electron gas at metallic densities is estimated for a variety of phenomena. The calculations are based on an approximation suggested by Hubbard (2) and are presented in the framework of the Landau theory. The effective mass, the Pauli spin susceptibility, and the compressibility are estimated and compared with the experimental results and with previous calculations. The quasi-particle renormalization factor and the interaction energy between two quasi particles on the Fermi surface, i.e., the Landau f(P, P′), are also evaluated in this approximation. The proper vertex part is calculated to lowest order in the dynamically screened interaction.
Different kinds of oscillatory behaviour have been observed in CdTe based modulation doped single quantum wells (MDSQWs). Besides the oscillations of the Landau level half width, the Landau level energy and the electron g-factor, we have found that the g-factor of photo-excited holes oscillates too. This oscillatory property of the g-factor of photo-excited holes in a two-dimensional electron gas (2DEG) is also reflected in the oscillation of the integrated intensity of σ+ and σ− polarised luminescence emission.
The g factor of quasiparticle states near the Fermi energy in a two-dimensional electron gas with Coulomb interactions is calculated in a static approximation. It is argued that this quadiparticle g factor, different from the free-electron g=2.0023, is the relevant quantity for spin splittings in experiments such as those on de Haas-van Alphen or magnetoconductivity oscillations, and, in particular, in experiments on electrons in silicon surface inversion layers. The g factor observed in conduction-electron spin resonance, on the other hand, should be the bulk value of 1.99.
n-type inverted silicon surfaces were studied to examine properties of a two-dimensional electron gas as well as those properties of silicon itself. The effect of a magnetic field tilted up to +/-90° from the normal to the (100) Si surfaces was measured between 1.3 and 4.2°K and up to 90 kOe. The electric field associated with the n-type inversion layer quantizes the surface electrons and they behave as a two-dimensional electron gas. It was found that the oscillatory magnetoconductance, due to magnetic quantization into Landau levels, depends solely upon the normal component of the magnetic field with respect to the surface. The observed splitting due to spin appears to be determined by the total magnetic field. The parallel component of the magnetic field was found to have a negligible effect on the surface quantization. The cyclotron mass, measured from the temperature dependence of the oscillation amplitude, is equal to the transverse mass of the electron in silicon and is independent of the tilt angle. The spin splitting can be quantitatively identified at certain angles where the Landau splitting is twice that of the spin splitting. The Landé g factor determined in this way was found to be substantially larger than 2 and decreases with the increasing surface electron density. The g factor has values between 3.25 and 2.47 for a surface electron density ranging from 1×1012 to 6×1012 cm-2.
Landau's theory of Fermi liquid is applied to obtain the quasi-particle parameters of the interacting two-dimensional electrons in the surface inversion layers of silicon. The calculated g-factor exhibits a good agreement with the experimental results of Fang and Stiles over a wide range of electron concentration and supports on a firmer ground Janak's original suggestion that the deviation of the g-factor from the bulk value is caused by the exchange interaction among surface electrons. Similar effects on the effective mass are briefly discussed.
We report resonant spin-flip Raman scattering measurements in CdTe/Cd1-xMgxTe single quantum wells with x=0.15, 0.26, and 0.5 as well as in Cd1-xMgxTe alloys with x=0, 0.15, and 0.5. Effective g factors of electrons and heavy holes are measured as a function of the quantum well width (18-100 Å) and the angle of the magnetic field with respect to the growth axis. The electron g factors in quantum wells are between the values for bulk CdTe (ge=-1.644+/-0.005) and Cd0.85Mg0.15Te (ge=-0.802+/-0.005). We show that the change in the energy gap induced by the quantum confinement of the carriers provides the dominant contribution to the electron g factor in quantum wells. A large g factor anisotropy for conduction electrons is observed. It is shown that this anisotropy depends on the splitting between light- and heavy-hole states. The experimentally determined g factors are in good agreement with theoretical predictions.
A model is developed which allows one to easily calculate correlation effects of interacting electrons. Upon considering a particular electron one replaces the excitation spectrum of all other electrons by a single mode ω(q), varying between the plasma frequency for small q and ℏq2/2m for large q. The coupling strength between the electron and the plasma modes is found by imposing the f sum rule. ω(q) is determined by requiring the model to have a correct dielectric response. The exchange and correlation contributions to E(k) have nearly opposite k dependence. However, there is a residual oscillation near kF which causes the effective mass m* to be less than unity, even though the mean mass (between k=0 and kF) is greater than unity. A specific local approximation to the exchange and correlation potential Axc=-2.07(na03)0.3Ry, analogous to Slater's n1/3 exchange potential, is accurate over 3 orders of magnitude in density. The (bare) momentum distribution n(k), and the fraction ζ of electrons excited above kF, are calculated as a function of density. For Li and Na, excluding band-structure effects, ζ=0.11 and 0.14, respectively.
The magnitude and sign of the effective magnetic splitting factor g* for conduction electrons in GaAs/AlxGa1-xAs quantum wells have been determined as a function of well width down to 5 nm. The experimental method is based on combined measurements of the decay time of photoluminescence and of the suppression of its circular polarization under polarized optical pumping in a magnetic field perpendicular to the growth axis (Hanle effect). Measurements as a function of hole sheet density in the wells reveal a transition from excitonic behavior with very small apparent g value for low density, to larger absolute values characteristic of free electrons at higher densities. For 20-nm wells g* for electrons is close to the bulk value (-0.44), and increases for narrower wells passing through zero for well width close to 5.5 nm. A theoretical analysis based on three-band k⋅p theory, including allowance for conduction-band nonparabolicity and for wave-function penetration into the barriers, gives a reasonable representation of the data, leading to the conclusion that g* in quantum wells has a value close to that of electrons in the bulk at the confinement energy above the band minimum.
The oscillator strength of negatively charged excitons (trions) in ZnSe/(Zn,Mg)(S,Se) quantum-well structures with n-type modulation doping is studied by means of reflectivity as a function of electron concentration, temperature, and external magnetic fields. The trion oscillator strength is found to increase linearly with increasing electron concentration up to 6×1010cm-2. The trion nonradiative damping shows no dependence on the electron concentration or external magnetic field. An optical method, based on the analysis of the polarization of the trion line, is proposed to investigate two-dimensional electron gases of low density from 109 up to 1011cm-2.
From the recent optical conduction-electron spin-resonance (CESR) measurements of the g factors g* in III-V compounds and the known effective masses m*, in the framework of the k⃗·p⃗ perturbation theory, we determine an experimental value of the interband matrix element P2=(2/m0)|〈S|px|X〉|2 coupling the conduction band and the upper valence bands. P2 ranges from 21 ± 1.5 eV in InP to 29 ± 1 eV in GaAs. This unexpected strong variation can be justified by a crude tight-binding calculation, evidencing the combined influence of ionicity and cell dimension. We show that g* can be calculated with a precision of 10% in a three-band calculation, whereas a multiband approximation is required for m*. The good agreement between our CESR measurements of the g factors in Ga1-xInxAs and Ga1-xAlxAs and the calculated values by k⃗·p⃗ theory shows the correctness of this theory in alloys. Moreover, it is possible to obtain a satisfactory fit of the effective-mass data previously unexplained within simple k⃗·p⃗ theory by using a multiband model and correct values of P2. The modifications to k⃗·p⃗ theory involving random potentials and strains are then not necessary at the precision of the experimental data available up to now.
The gyromagnetic ratios (g values) of electrons in ZnSe quantum wells with ZnxMg1-xSySe1-y barriers have been determined by resonance spin-flip Raman scattering as functions of well width and of the direction of the applied magnetic field. These heterostructures provide an excellent test of k⋅p perturbation theories since there is both a large penetration of the electron wave function into the barrier and a considerable difference in the light-hole and heavy-hole confinement energies in the well. The former leads to exposure of the electron to a region in which the spin-orbit coupling is significantly reduced, while the latter introduces a marked anisotropy in the g tensor. The behavior of the g tensor as the well width is decreased is described very well by simple analytic expressions obtained from three-band k⋅p theory, excellent agreement with experiment being obtained with a 10% conduction band offset ratio.
We study theoretically the renormalization of the spin-orbit-coupling constant of two-dimensional electrons by electron-electron interactions. We demonstrate that, similarly to the g factor, the renormalization corresponds to the enhancement, although the magnitude of the enhancement is weaker than that for the g factor. For high-electron concentrations (small interaction parameter rs) the enhancement factor is evaluated analytically within the static random phase approximation. For large rs∼10, we use an approximate expression for effective electron-electron interaction, which takes into account the local field factor, and calculate the enhancement numerically. We also study the interplay between the interaction-enhanced Zeeman splitting and interaction-enhanced spin-orbit coupling.
Spin-flip Raman scattering spectroscopy has been applied to the study of the wide band-gap semiconductor materials ZnSxSe1-x and Zn1-xMgxSe in order to determine the dependence on the composition, x, of the gyromagnetic ratio of electrons in the Γ6 conduction band and, thereby, to obtain a better understanding of the parameters underlying the band structure of these materials. The measured values are discussed in terms of the k⋅p perturbation theory for the band structure near the direct band gap, at different levels of approximation, and it is found that the observed dependence on composition can be reproduced well by the use of suitable interpolation schemes between the binary end members of the range of materials. Preliminary results for the related quaternary material Zn1-xMgxSySe1-y are discussed within the same model.
The influence of the electron-electron interaction on the energy of electrons in the lowest and first-excited sub-bands of an inversion layer has been evaluated. When these corrections are included very good agreement with experimental results is obtained. A bound inter-sub-band exciton is predicted.
We report on the design and growth by molecular beam epitaxy of two types of graded Cd1−xMnxTe/Cd1−yMgyTe quantum well structures having a precisely controlled spatial profile of either the quantum well width and/or of the number of donors in one of the directions perpendicular to the growth axis. The existence of a spatially varying quantum well width was demonstrated by different wavelengths of the photoluminesce emitted from different regions of the sample. The presence of two-dimensional electron gas with a spatially varying concentration produced by graded modulation doping was evidenced by magneto-optical studies which revealed signatures of either excitons, or negatively charged exciton–electron complexes (X−), or Fermi-edge singularity, all in a single sample grown in one molecular beam epitaxy process. Such structures may be very useful for tunable wavelength radiation sources as well as in detailed studies of various physical characteristics of the quantum wells. © 1998 American Institute of Physics.
We determine a dynamical field correction of the two-dimensional
electron gas, taking into account exchange and self-energy contributions
to the random-phase approximation (RPA). Physical properties like
correlation energy, electron self-energy, effective mass, quasiparticle
renormalization factor, and momentum distribution are computed. With
respect to the RPA, we find a substantial reduction of the correlation
energy, whereas the electron effective mass is much less affected.
We have investigated g-factor oscillations of electrons as well as of photoexcited holes in a quasi-two dimensional electron gas (2DEG) of CdTe/(CdMg)Te modulation-doped single quantum wells by means of photoluminescence (PL) and photoluminescence excitation (PLE) spectroscopy. The oscillation amplitude is found to be related to Landau level numbers and filling factors as, predicted by the theory. The exchange constants ANN, of the g-factor enlargement have been obtained experimentally for the first time. The observed enlargement of the g factor in CdTe/(CdMnMgTe) heterostructures, where the sign of the intrinsic g factor is reversed by the sp-d exchange interaction, confirms the results obtained for nonmagnetic CdTe/(CdMg)Te heterostructures.
Singlet and triplet states of negatively charged excitons (trions) in ZnSe/(Zn,Be,Mg)Se quantum wells have been studied by means of photoluminescence in pulsed magnetic fields up to 50 T. Singlet state binding energies, measured for different well widths ranging from 48 to 190 Å, show a monotonic increase with growing magnetic fields with a tendency to saturation. This behavior is in qualitative agreement with results of model calculations. Quantitatively, the binding energy of singlet states is underestimated by about 50%. The triplet state assigned to the “dark” triplet becomes detectable in magnetic fields above 15 T. A crossover of the triplet and singlet states is expected in magnetic fields of about 50 T.
Optical properties of a two-dimensional electron gas in ZnSe/(Zn,Be,Mg)Se quantum well structures have been examined by means Optical properties of a two-dimensional electron gas in ZnSe/(Zn,Be,Mg)Se quantum well structures have been examined by means
of photoluminescence and reflectivity techniques in external magnetic fields up to 50 T. For the studied structures, the Fermi of photoluminescence and reflectivity techniques in external magnetic fields up to 50 T. For the studied structures, the Fermi
energy of the two-dimensional electron gas falls in the range between the trion binding energy and the exciton binding energy, energy of the two-dimensional electron gas falls in the range between the trion binding energy and the exciton binding energy,
which keeps the dominating role of Coulombic interactions between electrons and photoexcited holes. Characteristic peculiarities which keeps the dominating role of Coulombic interactions between electrons and photoexcited holes. Characteristic peculiarities
of the optical spectra are discussed. of the optical spectra are discussed.
The dielectric constants of GaAs, CdTe, and ZnSe and their temperature dependences were found from low‐frequency capacitance measurements. From 100 to 300 °K the dielectric constants vary linearly with temperature. No electric field dependence was found up to 104 V/cm, nor frequency dependence between 20 Hz and 1 MHz. The dielectric constants extrapolated linearly to 0 °K are 12.35±0.09, 10.31±0.08, and 8.80±0.07 for GaAs, CdTe, and ZnSe, respectively. The temperature coefficients λ (≡ϵ (0)-1 dϵ/dt) are 2.01×10-4/°K, 2.27×10-4/°K, and 1.71×10-4/°K, respectively, with an accuracy of ±0.02×10-4/°K.
The k · p method is used for calculating the band parameters of semiconductors with germanium, zincblende, and wurtzite structure at k = 0 and at the L and X points of the zincblende structure. At k = 0, the interaction between the highest valence band and the two lowest conduction bands is considered. The energy gaps between these states are obtained from reflectivity measurements. The wave functions of a polar material are obtained from the wave functions of the isoelectronic non-polar material by the application of an antisymmetric perturbing potential. The matrix elements of p between two given states of the non-polar material are assumed to be the same for all materials. The band parameters of the wurtzite materials are obtained by the application of a small hexagonal crystal-field to the corresponding zincblende-type compound. It is possible by this method to account for the existing experimental data and to predict the band parameters of many materials for which they have not been measured. It is suggested that the absolute valence band maximum in several II–VI compounds may not be at the center of the Brillouin zone.
We present a microscopic theory of the effects of the electron-electron interaction on the effective mass and the anomalous Landé g factor in an inversion layer. We find that the inclusion of previously neglected many-body effects, associated with charge- and spin-fluctuation-induced vertex corrections, is crucial. The present approach is based on a new self-consistent determination of the many-body local fields. Our theory has no free parameters and the results are in good agreement with the established experimental findings.
We have carried out an extensive investigation of the frequency-dependent quasiparticle Coulomb self-energy of a two-dimensional electron gas for various values of the electronic density. Our analysis reveals a richly structured self-energy, a feature that we attribute to the existence in two dimensions of a low-lying collective-excitation spectrum. From the self-energy we have extracted and studied the effective mass, the spectral density, the wave-function renormalization factor, and the momentum-space occupation number. Our approach is based on Hedin's GW approximation and the effective interaction due to Kukkonen and Overhauser. One of the purposes of the present work is to investigate for the first time the many-body effects associated with charge- and spin-fluctuation-induced vertex corrections, which we have included by a suitable choice of the Hubbard-type many-body local fields appearing in the effective interaction. The present study shows that the unjustified neglect of these effects can lead to a seriously inaccurate estimate of various quantities of interest. The validity of the familiar on-shell approximation is also discussed. For simplicity, we have carried out our calculations at zero temperature and have taken advantage of the plasma-pole scheme.
The temperature and density dependence of spin quantum beats of electrons is measured by time-resolved photoluminescence spectroscopy and yields the electron Landé g factor in bulk GaAs, InP, and CdTe. In GaAs the g factor increases linearly from -0.44 at 4 K to -0.30 at 280 K; in InP the g factor is 1.20 at 4 K, exhibiting a very small temperature dependence up to 160 K, and in CdTe the g factor follows between T=4 K and 240 K the empirical equation g=-1.653+4×10-4 T+2.8×10-6 T2. In GaAs we demonstrate the suppression of spin quantum beats due to Fermi blocking in a degenerate electron gas and measure an increase of the GaAs g factor from -0.44 at densities below 1×1016 cm-3 to -0.33 at 1017 cm-3.