Nonparabolic Energy Dispersion Research Articles

Revised Fowler–Dubridge model for multiphoton over-barrier electron emission from two-dimensional materials

We revise the Fowler–Dubridge model for multiphoton over-barrier photoemission from two-dimensional (2D) materials to include the effects of reduced dimensionality, non-parabolic, and anisotropic energy dispersion of 2D materials. Two different directions of electron emission are studied, namely, vertical emission from the surface and lateral emission from the edge. Our analytical model reveals a universal temperature scaling of Tβ with β = 1 for the surface emission over a wide class of 2D materials and β = 3/2 for the edge emission from the 2D material with anisotropic parabolic energy dispersion, which are distinct from the traditional scaling of β = 2 originally derived for the traditional bulk materials. Our comparison shows good agreement with two experiments of photoelectron emitted from graphene for both surface and edge emission. Our calculations also show that the photoelectron emission is more pronounced than the coexisting thermionic emission for materials with low temperature and Fermi energy. This model provides helpful guidance in choosing proper combinations of light intensity, temperature range, and type of 2D materials for the design of photoemitters, photodetectors and other optoelectronics.

Time-resolved analysis of dual-gate FETs with non-parabolic energy dispersion for THz applications

The investigation of charge carrier transport in state-of-the-art nanoelectronic devices based on III/V semiconductors proves to be challenging, even more so when the highly non-parabolic energy dispersion exhibited by these materials is taken into account. Unlike the common approach of neglecting this behavior by the use of the parabolic band approximation, a novel combination of a tight-binding approach with a quantum Liouville-type equation is introduced here, where any arbitrary energy dispersion can effectively be included. This leads to a discretization based on the atomic structure without the need for finite difference approximations of the Hamiltonian. Because this allows for the stationary as well as the transient simulation of quantum charge carrier transport, it is well suited for the analysis of ultrathin FETs such as dual-gate FETs when it is combined with a mode-space approach. We demonstrate that the parabolic approximation not only vastly underestimates the current densities when compared to the non-parabolic case but also fails to capture transient effects such as gain compression when amplifier operation is considered.

Efficient strain-modified improved nonparabolic-band energy dispersion model that considers the effect of conduction band nonparabolicity in mid-infrared quantum cascade lasers.

Intersubband polar-optical-phonon (POP) scattering plays an important role in determining the population inversion and optical gain of mid-infrared (mid-IR) quantum cascade lasers (QCLs). In particular, the nonparabolicity of the conduction band (CB) significantly affects the energy dispersion relation and intersubband POP scattering time. However, the currently used parabolic-band (PB) and nonparabolic-band (NPB) energy dispersion models are not appropriate for mid-IR QCLs because they are unsuitable for high electron wave vectors and do not consider the effect of applied strain on the energy dispersion relation of the CB. The eight-band k·p method can provide a relatively accurate nonparabolic energy dispersion relation for high electron wave vectors but has the disadvantages of high computational complexity and spurious solutions to be discarded. Consequently, we propose a strain-modified improved nonparabolic-band (INPB) energy dispersion model that has no spurious solution and acceptable accuracy, compared to the eight-band k·p method. To demonstrate the accuracy and efficiency of our proposed INPB model compared with those of the PB, NPB, and eight-band k·p models, we calculate the energy dispersion relations and intersubband POP scattering times in a strain-compensated QCL with a lasing wavelength of 3.58 µm. Calculation results reveal that our proposed model is almost as accurate as the eight-band k·p model; however, it enables much faster calculations and is free from spurious solutions.

Entropy of electron gas in semiconductor nanowires

The thermodynamic potential and entropy of the electron gas of a semiconductor nanowire are obtained. When calculating the entropy of an electron gas with a nonparabolic energy dispersion, the Nelson method was used. The entropy of the electron gas with the value of the chemical potential of the smaller line, which is equal to or greater than the energy of an arbitrary discrete energy level, is studied. An analytical expression for the entropy is obtained in the case when the chemical potential of the electron gas is equal to the energy of the discrete level. The temperature dependences of the entropy of the electron gas of semiconductor nanowires on the degree of nonparabolicity of the energy dispersion law are found.

Application of fractional Fowler-Nordheim law on field electron emission from tungsten and lanthanum hexaboride nano-scale emitters

ABSTRACT The Murphy-Good Fowler-Nordheim equation was derived for planar and smooth conducting surfaces based on the Sommerfeld free-electron theory. It is applicable to a wide range of micro-emitters. The analyses of experimental field emission data from nano-scaled emitters showed deviation from the theoretical predictions. This has either been noticed in the mismatch of the current–voltage characteristics and/or in the FN-plot behaviour. Such deviation is attributed to the non-planar character of the emitting surface and the shape of the surface potential barrier through which the emitting electrons are tunnelling. Recently, Zubair et al. derived a FN-type equation that takes into account the reduced dimensionality of the emission surface and the non-parabolic energy dispersion [Zubair et al., IEEE Transactions on Electron Devices 65(6) 2018]. In this model, the constants of the conventional FN equation become two variables that depend on the fractional parameter expressing the surface complexity. In the present work, the fractional law is applied to field electron emission from clean tungsten (W) nano-emitter and lanthanum hexaboride (LaB6) nano-protrusions. This has been accomplished in the whole range of the fractional parameter in order to find the value that reproduces a better fit with the experimental data. The application requires determining the geometrical parameter of the emitter and the field emission setup. Results show that the experimental data can easily be reproduced at certain values of the fractional parameter. These ‘certain’ values depend on the geometrical parameters and thus the method used to calculate the electric field value at the emitter surface.

Rotating bi-electron in two-dimensional systems with mexican-hat single-electron energy dispersion

A number of novel two-dimensional materials and nanostructures demonstrate complex single-electron energy dispersion, which is called the mexican-hat dispersion. In this paper, we analyze interaction of a pair of electrons with such an energy dispersion. We show that relative motion of the electron pair is of a very peculiar character. For example, the real space trajectories corresponding to electron-electron scattering can have three reversal points, reversal points at non-zero radial momentum and other unusual features. Despite the repulsive Coulomb interaction, two electrons can be coupled forming a composite quasi-particle – the bi-electron. The bi-electron corresponds to excited states of the two-electron system. Because the bi-electron coupled states exist in continuum of extended (free) states of the electron pair, these states are quasi-resonant and have finite times of life. We found that rotating bi-electron is a long-living composite quasi-particle. The rotating bi-electrons can be in motion. For slowly moving bi-electrons, we have determined the kinetic energy and the effective mass. Due to strongly nonparabolic energy dispersion, the translational motion of the bi-electron is coupled to its internal motion. This results in effective masses dependent on quantum states of the bi-electron. In the paper, properties of the bi-electron have been illustrated for the example of bigraphene in a transverse electric field. We have suggested that investigation of rotating bi-electrons at the mexican-hat single-electron energy dispersion may bring new interesting effects in low-dimensional and low-temperature physics.

Temperature-driven spin-zero effect in TaAs2

The electrical and thermo-electrical transport properties of the semimetal TaAs2 have been measured in a magnetic field applied along the [−2 0 1] direction. The resulting field dependences of the resistivity as well as the Hall, Seebeck, and Nernst coefficients below T ≈ 100 K can be satisfactorily described within the two-band model consisting of electron and hole pockets. At low temperature, all of the measured coefficients exhibit significant contributions from quantum oscillations. The fast Fourier-transform (FFT) of the oscillatory Nernst signal shows two fundamental frequencies, Fα = 105 T and Fβ = 221 T, and the second harmonic of the latter (F2β = 442 T). The ratio between the FFT amplitudes of Fβ and F2β changes with temperature in an unusual way, indicating the observation of a spin-zero effect caused by temperature change. This is likely related to the substantial temperature dependence of the Landé g-factor, which in turn may stem from the non-parabolic energy dispersion or temperature evolution of the spin-orbit coupling.

A compact model for III–V nanowire electrostatics including band non-parabolicity

The III–V materials have a highly non-parabolic band structure that significantly affects the MOS transistor electrostatics. The compact models used to simulate circuits involving III–V MOS transistors must account for this band structure non-parabolicity for accurate results. In this work, we propose a modification to the energy dispersion relation to include the band structure non-parabolicity in a way suitable for compact models. Unlike the available non-parabolic energy dispersion relation, the one proposed here is simple and includes the non-parabolicity in both confinement and transport directions. The proposed dispersion relation is then used to model the electrostatics of III–V nanowire transistors. The proposed model is scalable to a higher number of sub-bands and computationally efficient for circuit simulators. The model is also validated with the data from a 2D Poisson–Schrodinger solver for a wide range of nanowire dimensions, III–V channel materials, and found to be in excellent agreement with the simulation data.

High Resolution Optical Spectroscopy of Free Exciton and Electronic Band Structure near the Fundamental Gap in 4H SiC

We use thick, relatively high purity 4H SiC boule material to measure the wavelength modulated absorption spectrum with improved wavelength resolution and sensitivity with respect to previous work. We observe several small 0.6 ± 0.1 meV splittings, which we attribute to electron mass anisotropy and electron-hole exchange interaction. In addition, we identify several features in the absorption spectrum as signatures of nonparabolicity in the free exciton dispersion relations, the primary origin of which is likely the nonparabolic energy dispersion of the valence bands, as revealed by published band structure calculations based on density functional theory.

Fractional Fowler–Nordheim Law for Field Emission From Rough Surface With Nonparabolic Energy Dispersion

The theories of field electron emission from perfectly planar and smooth canonical surfaces are well understood, but they are not suitable for describing emission from rough, irregular surfaces arising in modern nanoscale electron sources. Moreover, the existing models rely on Sommerfeld's free-electron theory for the description of electronic distribution which is not a valid assumption for modern materials with nonparabolic energy dispersion. In this paper, we derive analytically a generalized Fowler-Nordheim (FN) type equation that takes into account the reduced space-dimensionality seen by the quantum mechanically tunneling electron at a rough, irregular emission surface. We also consider the effects of non-parabolic energy dispersion on field-emission from narrow-gap semiconductors and few-layer graphene using Kane's band model. The traditional FN equation is shown to be a limiting case of our model in the limit of a perfectly flat surface of a material with parabolic dispersion. The fractional-dimension parameter used in this model can be experimentally calculated from appropriate current-voltage data plot. By applying this model to experimental data, the standard field-emission parameters can be deduced with better accuracy than by using the conventional FN equation.

Current-Temperature Scaling for a Schottky Interface with Nonparabolic Energy Dispersion

In this paper, we study the Schottky transport in narrow-gap semiconductor and few-layer graphene in which the energy dispersions are highly non-parabolic. We propose that the contrasting current-temperature scaling relation of $J\propto T^2$ in the conventional Schottky interface and $J\propto T^3$ in graphene-based Schottky interface can be reconciled under Kane's $\mathbf{k} \cdot \mathbf{p}$ non-parabolic band model for narrow-gap semiconductor. Our new model suggests a more general form of $J\propto \left(T^2 + \gamma k_BT^3 \right)$, where the non-parabolicty parameter, $\gamma$, provides a smooth transition from $T^2$ to $T^3$ scaling. For few-layer graphene, it is found that $N$-layers graphene with $ABC$-stacking follows $J\propto T^{2/N+1}$ while $ABA$-stacking follows a universal form of $J\propto T^3$ regardless of the number of layers. Intriguingly, the Richardson constant extracted from the Arrhenius plot using an incorrect scaling relation disagrees with the actual value by two orders of magnitude, suggesting that correct models must be used in order to extract important properties for many novel Schottky devices.

Analytical Study of Electronic Structure in Archimedean Type-Spiral Zig-Zag Graphene Nanoscroll

The semiconducting electronic properties of graphene nanoscroll (GNS) are very much related to its geometric structure. The aim of this study is to construct a GNS energy dispersion model within low-energy transport of 1 eV in identifying its electronic properties and carrier statistics. Non-parabolic energy dispersion is used to incorporate the Archimedean type-spiral model, and the band gap is assessed based on chirality and geometry effects. The energy band within low-energy transport indicates that GNS can achieve a quantum conductance limit of ~6.45 kΩ for ballistic transport. On the other hand, the numbers for three minimum sub-bands are attained based on non-parabolic energy dispersion, and the semi-metallic zig-zag GNS is found at chirality (3j + 1, 0). This work consistently predicts the semiconducting properties of the tight-binding model from previous work. The GNS overlapping region strongly affects its electronic properties. Constantly increasing the length of the overlapping region decreases the band gap exponentially, whilst semimetallic GNS forms when the overlap reaches a certain limit. The carrier density with temperature dependence is subsequently assessed at the intrinsic level, and found that the number of carriers in GNS shows a higher rate of increment (exponentially) compared to carbon nanotubes (CNT), in accordance to their diameter. The results are very useful in giving an intuitive understanding on GNS carrier statistics as subject to geometry changes.

The anisotropic energy spectrum dependence of the optical conductivity in bilayer graphene

In this paper, the longitudinal optical conductivity in bilayer graphene was calculated analytically and numerically. In addition to the quadratic terms in the effective-mass approximation Hamiltonian, the linear term, which relates to the indirect interlayer coupling, was included. The nonparabolic energy dispersion was obtained. Two intra- and inter-band transition channels for optical transition via carriers absorbing the optical energy are observed. The inter-band transition offers the main contribution and is almost a constant when the optical energy is larger than two times the Fermi energy. In the presence of the complex energy and pseudospin angle relationship, doing the numerical integration to the wavevector k, the contribution of the intra-band optical transition to the optical conductivity (σxxintra(ω)) is strengthened in the low optical energy region, while the analytical results with parabolic energy curves contribute less to σxxintra(ω). In addition, the optical conductivity also depends on the electron density (or gate voltage) and the broadening width.

Lower limits to metal-semiconductor contact resistance: Theoretical models and experimental data

We calculate the minimum feasible contact resistivity to n-type and p-type In0.53Ga0.47As, InAs, GaAs, GaSb, InP, and InSb. The calculations consider image force lowering and assume either parabolic or non-parabolic energy dispersion in the semiconductor; their results are compared with recent experimental data. Among significant results, the measured contact resistivity to n-In0.53Ga0.47As at a carrier concentration of 5 × 1019 cm−3 is only 2.3:1 higher than that calculated assuming a 0.2 eV barrier potential, and the measured contact resistivity is only 9.0:1 larger than the Landauer quantum conductivity limit at this carrier concentration. These results indicate that, with the surface preparation procedures presently employed, surface contamination does not markedly increase the interface resistance, and that the transmission coefficient for carriers crossing the interface exceeds 10%.

Photomixing in topological insulator HgTe/CdTe quantum wells in terahertz regime

We reveal that topological insulators (TI) HgTe/CdTe quantum well have a strong nonlinear optical property in the three-photon mixing. While the gapless surface state in TI can exhibit strong nonlinear effect due to the linear energy dispersion, the nonparabolic energy dispersion of the bulk state is responsible for the photo mixing effect reported here. To produce response at terahertz frequency regime from femtosecond electrical fields, the mixing efficiency is around 10−4 comparable to that of nonlinear semiconductor crystals. The optimal temperature for this nonlinear effect is around 100 K. The results suggest a potential application of TI in terahertz photonics.

Seebeck Coefficient in Bulk and Quantum Wells of Non-Parabolic Kane-Type Materials

We present a simplified and quantitative analysis of the Seebeck coefficient in degenerate bulk and quantum well materials whose conduction band electrons obey Kane's non-parabolic energy dispersion relation. We use k.p formalism to include the effect of the overlap function due to the band non-parabolicity in the Seebeck coefficient. We also address the key issues and the conditions in which the Seebeck coefficient in quantum wells should exhibit oscillatory dependency with the film thickness under the acoustic phonon and ionized impurity scattering. The effect of screening length in degenerate bulk and quantum wells has also been generalized for the determination of ionization scattering. The well-known expressions of the Seebeck coefficient in non-degenerate wide band gap materials for both bulk and quantum wells has been obtained as a special case and this provides an indirect proof of our generalized theoretical analysis.

Nonlinear cyclotron resonance of a massless quasiparticle in graphene

We consider the classical motion of a massless quasiparticle in a magnetic field and under a weak electromagnetic radiation with the frequency $\ensuremath{\omega}$. Due to the nonparabolic linear energy dispersion, the particle responds not only at the frequency $\ensuremath{\omega}$ but generates a broad frequency spectrum around it. The linewidth of the cyclotron resonance turns out to be very broad even in a perfectly pure material which allows one to explain recent experimental data in graphene.

Ab initioand molecular dynamics predictions for electron and phonon transport in bismuth telluride

Phonon and electron transport in ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$ has been investigated using a multiscale approach, combining the first-principles calculations, molecular dynamics (MD) simulations, and Boltzmann transport equations (BTEs). Good agreements are found with the available experimental results. The MD simulations along with the Green-Kubo autocorrelation decay method are used to calculate the lattice thermal conductivity in both the in-plane and cross-plane directions, where the required classical interatomic potentials for ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$ are developed on the basis of first-principles calculations and experimental results. In the decomposition of the lattice thermal conductivity, the contributions from the short-range acoustic and optical phonons are found to be temperature independent and direction independent, while the long-range acoustic phonons dominate the phonon transport with a strong temperature and direction dependence (represented by a modified Slack relation). The sum of the short-range acoustic and optical phonon contribution is about $0.2\phantom{\rule{0.3em}{0ex}}\mathrm{W}∕\mathrm{m}\phantom{\rule{0.2em}{0ex}}\mathrm{K}$ and signifies the limit when the long-range transport is suppressed by nanostructure engineering. The electrical transport is calculated using the full-band structure from the linearized augmented plane-wave method, BTE, and the energy-dependent relaxation-time models with the nonparabolic Kane energy dispersion. Temperature dependence of the energy gap is found to be important for the prediction of electrical transport in the intrinsic regime. Appropriate modeling of relaxation times is also essential for the calculation of electric and thermal transport, especially in the intrinsic regime. The maximum of the Seebeck coefficient appears when the chemical potential approaches the band edge and can be estimated by a simple expression containing the band gap. The scatterings by the acoustic, optical, and polar-optical phonons dominate the electrical conductivity and electric thermal conductivity.

Nonvanishing spin Hall conductivity in two-dimensional Rashba electron systems with nonparabolic Kane dispersion

We investigate the spin Hall effect in a two-dimensional (2D) Rashba spin-orbit coupled electron gas with nonparabolic Kane energy dispersion using the Green's function approach under self-consistent Born approximation. Nonzero spin Hall current is obtained for this homogenous system and the spin Hall conductivity is found to depend not only on the degree of the band nonparabolicity but also on the detailed properties of the scattering. With the same nonparabolic coefficient and electron density pertinent to a typical InAs 2D system, the surviving total spin Hall conductivity ${\ensuremath{\sigma}}_{\mathrm{sH}}$ appears to be an order of magnitude larger in the case of remote impurity scattering than that of the short-range scattering for Rashba coupling parameter $\ensuremath{\alpha}$ up to $6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}\phantom{\rule{0.3em}{0ex}}\mathrm{eV}\phantom{\rule{0.2em}{0ex}}\mathrm{m}$.

Quantum diffusion and lack of universal one-parameter scaling in one-dimensional disordered lattices with long-range coupling

It is generally believed that all eigenstates in one-dimensional disordered lattices are localized provided disorder is uncorrelated. We show that this statement fails for a one-dimensional Anderson model with a special type of long-range inter-site interaction, resulting in a specific, non-parabolic quasi-particle energy dispersion. Remarkably, the states appearing to be delocalized belong to the tail of the band.