Effective Mass Equation Research Articles

Magnetic-field-dependent optical properties and interdot correlations in coupled quantum dots

We theoretically investigate the properties of neutral and charged excitons in vertically coupled quantum dots, as a function of the in-plane magnetic field. The single-particle states are computed by numerically solving the 3D effective-mass equation, while the neutral- and charged-exciton states are obtained by means of a configuration interaction approach. We show that the field determines an enhancement of the interdot correlations, resulting in unexpected carrier localization. The field effect on the excitonic binding energies is also discussed, and is shown to strongly depend on the charging.

Exciton states and oscillator strength in two vertically coupled InP/InGaP quantumdiscs

Quantum mechanical coupling and strain in two vertically arranged InP/InGaP quantumdots is studied as a function of the size of the dots and the spacer thickness. Thestrain distribution is determined by the continuum mechanical model, while thesingle-band effective-mass equation and the multiband theory are employed to compute the conduction and valence band energy levels,respectively. The exciton states are obtained from an exact diagonalization approach, andwe also compute the oscillator strength for recombination. We found that the light holesare confined by strain to the spacer, which is the reason that the hole statesexhibit coupling at much larger distances as compared with the electrons. At smalld,the doublet structure of the hole energy levels arises as a consequence of the relocation of the lighthole from the matrix to the regions located outside the stack, close to the dot–matrix interface. Whend varies, the exciton ground state exhibits numerous anticrossings with other states,which are related to the changing spatial localization of the hole as a function ofd. The oscillator strength of the exciton recombination is strongly reduced in a certain rangeof spacer thicknesses, which effectively turns a bright exciton state into a darkone. This effect is associated with anticrossings between exciton energy levels.

Effective Mass Approach for n-MOSFETs on Arbitrarily Oriented Wafers

The general theory for quantum simulation of cubic semiconductor n-MOSFETs is presented within the effective mass equation approach. The full three-dimensional transport problem is described in terms of coupled transverse subband modes which arise due to quantum confinement along the body thickness direction. Couplings among the subbands are generated for two reasons: due to spatial variations of the confinement potential along the transport direction, and due to non-alignment of the device coordinate system with the principal axes of the constant energy conduction band ellipsoids. The problem simplifies considerably if the electrostatic potential is separable along transport and confinement directions, and further if the potential variations along the transport direction are slow enough to prevent dipolar coupling (Zener tunneling) between subbands. In this limit, the transport problem can be solved by employing two unitary operators to transform an arbitrarily oriented constant energy ellipsoid into a regular ellipsoid with principal axes along the transport, width and confinement directions of the device.

A modeling of the optical properties of the zinc oxide–zinc magnesium oxide double barrier system

We have modeled the optical properties of the ZnMgO/ZnO/ZnMgO system and correlated them to its transport properties by employing an effective mass equation method. By its construction our method was sensitive to the barrier thickness of the system. From our results we deduce that there are more states in the well than resonant tunneling peaks in the transport data. The “missing” states in the electrical characteristics are not seen because of their much lower localization in the well. However these states are responsible for the UV activity of the system. Reducing the barrier thickness changes dramatically the localization of the states in the well without affecting significantly the energy levels. Hence the electrical properties can be changed without changing significantly the optical properties.

Self-consistent simulations of mesoscopic devices operating under a finite bias

A novel numerical algorithm based on the solution of the two-dimensional effective mass equation for current-carrying scattering states in mesoscopic devices is developed. Using this while allowing for an energy dependent transmission matrix, the total charge density distribution based on all electrons injected into the device is calculated through integration over energy. By coupling this energy-resolved calculation of charge density distribution iteratively with a potential calculation a fully self-consistent calculation, which allows for accurate simulations of mesoscopic devices with arbitrary complex device geometries operating under a finite bias, is achieved. Thus it is possible to self-consistently study space charge effects in mesoscopic devices. The developed method is described and tested on a number of sample geometries.

Nanocrystal molecules and chains

The electron energy structure of linear artificial molecules and one-dimensional chains formed of spherical semiconductor nanocrystals is investigated with and without an applied magnetic field. Both uniform and multilayer nanocrystals are studied. The calculations are performed within the effective mass model by numerically integrating the effective mass equation on a two-dimensional cylindrical grid. Some calculations are, for comparison, performed also in the tight-binding approach. Density contours are presented to illustrate the transformation of states in systems of strongly interacting coupled quantum dots. Strong interaction between the quantum-dot–quantum-well structures in a chain of nanocrystals can lead to the formation of a very narrow ground-state miniband, well separated from the excited levels with the energies almost independent of the magnetic field.

Quantum-well infrared photodetector structure synthesis: methodology and experimental verification

A numerical method for global optimization of quantum-well infrared photodetector (QWIP) performance parameters is presented and experimentally verified. The single-band effective-mass Schroedinger equation is solved by employing the argument principle method (APM) to extract both the bound and quasibound eigen-energies of the quantum heterostructure. APM is combined with a simulated annealing algorithm to determine a set of device design parameters such as potential barrier height V/sub i/, layer thickness d/sub i/, number of material layers N, total device length, applied bias V/sub Bias/ etc., for which the QWIP performance is within a predetermined convergence criterion. The method presented incorporates the effect of energy-dependent effective mass of electrons in nonparabolic conduction bands. The present model can handle many optimization parameters and can incorporate fabrication constraints to achieve physically realizable devices. In addition, the method is not limited to the optimization of absorption structures, and can be used for other intersubband devices such as electron-wave Fabry-Perot filters and quantum-cascade lasers. The strength and versatility of the present method are demonstrated by the design of a bicolor equal-absorption-peak QWIP structure, and experimental verification of the zero-bias absorption spectrum is presented.

Hartree and Exchange Effects in the Calculation of Hole Levels in p-Type Delta-Doped Diamond Systems

Γ-point hole states in boron δ-doped diamond quantum wells are calculated with the use of a local density Thomas–Fermi–Dirac approximation for the potential energy function and the solution of an effective mass equation. The approach includes Hartree and exchange contributions in the hole gas. The calculation assumes a two-independent (hh + lh) hole band model, and different sets of Luttinger parameters for diamond, reported in the literature, are used. It is shown that the hole spectrum is highly sensitive to the values of the valence effective masses. The results for the hole energy states would help to clarify the actual set of valence band parameters if measurements of optical absorption involving transitions from hole energy states in diamond systems with delta doping are performed.

Spatially varying effective mass effects on energy level populations in semiconductor quantum devices

For the analysis and design of semiconductor intersubband devices, accurate values for the Fermi energy and the subband electron population are needed. The effect of the position-dependent electron effective massm* (z) is commonly neglected in the determination of these two intersubband device parameters. This approach is nearly valid for single-well devices in the GaAs/Al xGa1−x As material system. However, in multiple-coupled-well devices, the variable nature of the effective mass must be taken into account. In material systems other than the ones based on GaAs, failure to include the position-dependence of the electron mass may give rise to significant errors in the values of the Fermi energy and other device parameters such as the intersubband absorption coefficient. In this paper, the effects ofm* (z) on the Fermi level and the intersubband charge distribution are explored and quantified.Theoretical formulation for the intersubband Fermi energy and the subband electron distribution, with the inclusion of position-dependent electron mass, is presented. The eigenenergies of the intersubband structures are obtained by solving the single-band effective-mass Schroedinger equation using the argument principle method. The electron distribution and the Fermi energy are calculated using both the approximate method ( m0*) and the rigorous formulation [ m*(z)], and the relative differences in the corresponding values are presented. It is demonstrated that these differences are small in the GaAs/Al xGa1−x As material system, but can become very significant in other materials. In addition, the variable nature of carrier effective mass plays an important role in other types of devices such as interband quantum well photodetectors and lasers that employ optical transitions between the valence and the conduction bands. Electronic devices such as the resonant tunneling diode are also affected by the position-dependence of carrier mass and thus the results are applicable to both optoelectronic and electronic quantum devices.

Tunneling and time-dependent magnetic phase transitions in a bilayer electron system

In this work, we have numerically integrated in space and time the effective-mass nonlinear Schrodinger equation for an electron wave packet in a bilayer electron system. Considering a time-dependent magnetic phase transition from the usual unpolarized ground state to spin-polarized one, we have calculated the tunneling dynamics between the two quantum wells when an external bias is applied. Due to the nonlinear effective-mass equation, it is found that the charge dynamically trapped in both wells produces a reaction field that modifies the system resonant condition. In this way, we have shown the possibility of having another kind of tunneling oscillations between both quantum wells after an optical excitation of the sample.

Discontinuities in the effective-mass equation

The validity of the Heisenberg uncertainty principle for the effective-mass equation is analytically verified for the first time. The procedure also provides an unambiguous demonstration of the need for a distributional derivative in the solution of the one-dimensional rectangular quantum-well problem for Hamiltonians with a piecewise-constant mass function .

Mechanism for carrier velocity saturation in pure organic crystals

We present a possible mechanism for the saturation of the velocity of photoinjected charge carriers in pure organic crystals, the analysis having been motivated by observations in pentacene. The mechanism is based on the strong wave vector dependence of the damping coefficient appearing in the appropriate effective mass equation. Our primary goal is to explore this mechanism in general terms rather than to apply it to a specific system such as pentacene. Our analysis is based on a Fokker-Planck equation treatment. We indicate how the damping coeffcient is related to relaxation times in usual Boltzmann equations and how it can be computed from Fermi golden rule transition rates.

Semiconductor intersubband laser/detector performance optimization using a simulated annealing algorithm

A numerical method for global optimization of semiconductor intersubband laser/detector performance parameters is presented. The single-band effective-mass Schroedinger equation is solved by employing the argument principle method (APM) to extract both the bound (B) and quasibound (QB) eigen-energies of the quantum heterostructure. APM is combined with a simulated annealing (SA) algorithm to determine a set of device design parameters such as potential barrier height Vi, layer thickness di, applied biasVBias , for which the intersubband device performance is within a predetermined convergence criterion. The method presented incorporates the energy-dependent effective mass of electrons in nonparabolic conduction bands. The performance of the method is evaluated for the design of an asymmetric Fabry–Perot electron-wave interference filter (laser structure) and a dual-band quantum well infrared photodetector (QWIP). Results with and without nonparabolic effects are presented. In addition, results from the present method are compared to results obtained via the optimization technique based on super-symmetric quantum mechanics (SUSYQM) for the case of an optically-pumped quantum cascade (QC) laser. The present method is shown to improve the device performance beyond that obtained via SUSYQM optimization. Further, the present model can handle many optimization parameters and can incorporate fabrication constraints to achieve physically realizable devices.

Tunneling escape process from a spin-polarized two-dimensional electron system

In this work, we have numerically integrated in space and time the effective-mass nonlinear Schrodinger equation for an electron wave packet in a double barrier heterostructure. Considering both polarized and unpolarized magnetic phases, we have studied the tunneling escape process from the two-dimensional electron gas. Due to the nonlinear effective-mass equation, it is found that the charge trapped dynamically in the quantum well produces a reaction field, which modifies the tunneling escape process in the quantum well. At different electronic sheet densities, we have shown the possibility of having magnetic phase-dependent tunneling rates.

Analytic methods for field induced tunneling in quantum wells with arbitrary potential profiles

Electric field induced tunneling is studied in three different types of quantum wells by solving time-independent effective mass equation in analytic methods based on three different Airy function approaches. Comparison of different Airy function methods indicates that they are identical and connected to each other by the Breit-Wigner formula.

Observation of confined states in anIn0.53Ga0.47As/In0.52Al0.48Asmulti-quantum wells structure by quantum confined Stark effects

Photocurrent spectra in an In0.53Ga0.47As /In0.52Al0.48As multi-quantum wells structure containing 9.4 nm wide wells were measured at room temperature in electric fields. The exciton peaks of ground-state transitions shifted fairly in 167 kV cm−1as the quantum confined Stark effect. Stark shifts were calculated by using the Runge–Kutta method using the effective mass equation with our experimental band parameters. Our parameters are the hole effective masses and valence band offset derived from saturation of a highest eigen energy, electron effective mass depending on energies and the conduction band offset derived from observed quantum number. It was possible to sufficiently use our experimental band parameters for the calculation of the Stark shift in the electric field.

A simple and efficient model for quantization effects of hole inversion layers in MOS devices

A simple and efficient model is introduced to study the hole quantization effects in the inversion layer of p-MOS devices. It is based on a six-band effective mass equation and a zigzag potential. The strong mixing between the heavy, light, and split-off hole bands is emphasized and quantitatively assessed. All subband dispersions are found to be anisotropic, far from parabolic, and electric field dependent. In addition, there are camel-back structures at some subband minima. The density of states (DOS) profiles show strong deviations from the step-like functions and one or two peaks may appear at the subband minima of the DOS, corresponding to the camel-back band structures. The traditional one-band effective mass approximation (EMA) using effective mass values extracted from bulk Si underestimates and oversimplifies the subband DOS. We justify this model by applying it to the capacitance of hole inversion layer and the threshold voltage shifts due to quantum mechanical (QM) effect. The model simulation shows good agreement with experimental results, demonstrating the accuracy of this model. The model and the characterization of the band structure of Si valence hole quantization lay the ground work in routine simulation of deep submicrometer MOS devices.

Biaxial Strain Effects on Optical Characteristics of Quantum Well Lasers

The effects of biaxial strain produced by the lattice mismatch of constituent materials on the optical properties of strainedIn1−xGaxAsyP1−y/In1−xGaxAsquantum well lasers are investigated.The optical gain and refractive index change of a biaxially stressed quantum well lasers are studied theoretically using the multiband effective mass equation (i.e.,k→⋅p→method), deformation potential theory and Fermi-Golden rule, band mixing effect is retained in the calculations. It is found that the biaxial strain would change the subband structures and optical properties of quantum well lasers, we found the gain of TE mode increases with increasing compressive strain, while the gain of TM mode increases with increasing tensile strain, these will be benefited for reducing the threshold current depending on whether the quantum well lasers are operating in TE or TM mode. On the other hand, the refractive index change in the active region near the TE(TM) mode peak gain becomes more negative when a biaxial compressive(tensile) strain is applied, it leads to the conclusion that the strain weakens the optical confinement, the temperature dependence of gain also becomes stronger when there is strain.Finally, we also found the minimum peak gain occurs when a small tensile strain is applied, but no strain.

Possibility of multiple tunnelling current peaks in a coupled quantum well system

In this work, we have numerically integrated in space and time the effective-mass nonlinear Schrödinger equation for an electron wave packet in a bilayer electron system. Considering both Hartree and exchange-correlation potentials, we have calculated the tunnelling rates between the two quantum wells when an external bias is applied in the double quantum well system. Due to the nonlinear effective-mass equation, it is found that the charge dynamically trapped in both wells produces a reaction field which modifies the system resonant condition. At different electronic sheet densities, we have shown the possibility of having multiple resonant tunnelling peaks in a bilayer electron system.

Energy band and acceptor binding energy of GaN and Al xGa 1− xN

The hole effective-mass Hamiltonian for the semiconductors of wurtzite structure is established, and the effective-mass parameters of GaN and Al x Ga 1− x N are given. Besides the asymmetry in the z and x, y directions, the linear term of the momentum operator in the Hamiltonian is essential in determining the valence band structure, which is different from that of the zinc-blende structure. The binding energies of acceptor states are calculated by solving strictly the effective-mass equations. The binding energies of donor and acceptor for wurtzite GaN are 20 and 131, 97 meV, respectively, which are inconsistent with the recent experimental results. It is proposed that there are two kinds of acceptors in wurtzite GaN. One kind is the general acceptor such as C, substituting N, which satisfies the effective-mass theory, and the other includes Mg, Zn, Cd etc., the binding energy of which deviates from that given by the effective-mass theory. Experimentally, wurtzite GaN was grown by the MBE method, and the PL spectra were measured. Three main peaks are assigned to the DA transitions from the two kinds of acceptor. Some of the transitions were identified as coming from the cubic phase of GaN, which appears randomly within the predominantly hexagonal material. The binding energy of acceptor in AlN is about 239, 158 meV, that in Al x Ga 1− x N alloys ( x≈0.2) is 147, 111 meV, close to that in GaN.