@article{iafrate_sokolov_2021, title={Bloch-electron dynamics in homogeneous electric fields: Application to multiphoton absorption in semiconductors and insulators}, volume={104}, ISSN={["2469-9934"]}, DOI={10.1103/PhysRevA.104.063113}, abstractNote={The theory of Bloch-electron dynamics for carriers in a homogeneous electric field of arbitrary time dependence is developed in consideration of the properties of multiphoton absorption (MPA) in semiconductors and insulators. The general approach is to utilize the accelerated Bloch-state representation (ABR) as a basis thereby treating the electric field exactly; also, the electric field is described in the vector potential gauge. In developing the ABR, the instantaneous eigenstates for the central Hamiltonian are obtained as the accelerated Bloch states and utilized to find the time-dependent solution to the Schr\"odinger equation. In introducing the Wigner-Weisskopf approximation (WWA), the transition probability amplitude between the states of the system is obtained and used in the application to MPA in semiconductors and insulators. The probability amplitude for MPA is derived for a plane-polarized radiation field. The periodicity of the time-dependent electric field serves as a principle time constant in developing the probability amplitude per period. Utilizing the WWA, the general MPA transition probability is derived for transitions between an arbitrary set of valence and conduction bands for a time-varying electric field in the $x$ direction. Within the WWA, the MPA transition probability, equivalent to the well-known Keldysh transition amplitude, is derived and expressed in terms of infinite-variable generalized Bessel functions and modified Bessel functions. The exact MPA result is analyzed in the small ($\ensuremath{\lesssim}{10}^{6}$ V/cm) and large ($\ensuremath{\gtrsim}{10}^{8}$ V/cm) electric field amplitude limits.}, number={6}, journal={PHYSICAL REVIEW A}, author={Iafrate, G. J. and Sokolov, V. N.}, year={2021}, month={Dec} }
 @article{iafrate_sokolov_2020, title={The Bloch Electron Response to Electric Fields: Application to Graphene}, volume={257}, ISSN={["1521-3951"]}, DOI={10.1002/pssb.201900660}, abstractNote={The theory of Bloch electron dynamics for carriers in homogeneous electric fields of arbitrary time dependence is developed in consideration of the electronic transport properties in graphene. The general approach is to use the accelerated Bloch state representation (ABR) as a basis so that the dependence upon the electric field, including Zener tunneling, is treated exactly; also, the electric field is described in the vector potential gauge. Within the ABR, the instantaneous eigenstates for the central Hamiltonian are described and utilized to develop both the time‐dependent wave functions and the single‐particle density matrix pictures with explicit application to intrinsic graphene. The Bloch electron analysis for graphene in a constant electric field reveals the explicit manifestation of electron–hole pair creation, Bloch oscillations, and Wannier–Stark localization. The average velocity and acceleration are established as an explicit function of the electric field, and their behavior is characterized on short and long time scales. Further, it is shown that the average acceleration in a nonvanishing electric field gives rise to an electric field induced dynamical effective mass tensor in the direction of the field.}, number={6}, journal={PHYSICA STATUS SOLIDI B-BASIC SOLID STATE PHYSICS}, author={Iafrate, Gerald and Sokolov, Valeriy}, year={2020}, month={Jun} }
 @article{barry_sokolov_kim_trew_2010, title={Large-Signal Analysis of Terahertz Generation in Submicrometer GaN Diodes}, volume={10}, ISSN={1530-437X 1558-1748}, url={http://dx.doi.org/10.1109/JSEN.2009.2038132}, DOI={10.1109/jsen.2009.2038132}, abstractNote={The conditions for microwave power generation in a submicrometer GaN diode, with a relatively lightly doped active channel, coupled to an external resonant circuit are investigated. Applying a high-field electron transport model based on the local quasi-static approximation, we show that oscillations in group III-nitride diodes can be supported in the terahertz-frequency range near the limited space-charge accumulation regime. The shape of the diode voltage and electronic current waveforms are examined in terms of the circuit parameters and operating frequencies over the bandwidth of active generation. Based on a Fourier series analysis of the diode voltage and current, the generated power and dc-to-RF conversion efficiency at the fundamental and the second or higher order harmonic frequencies are estimated. The calculation results clearly indicate that submicrometer GaN diodes (channel doping of 1 × 1017 cm-3) can achieve large output powers (> 1 W) in the absence of Gunn domain formation, over a wide range of frequencies, near 0.5 THz.}, number={3}, journal={IEEE Sensors Journal}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Barry, E. A. and Sokolov, V. N. and Kim, K. W. and Trew, R. J.}, year={2010}, month={Mar}, pages={765–771} }
 @article{sokolov_kochelap_kim_2010, title={Magnetoconcentration effect in intrinsic graphene ribbons}, volume={97}, ISSN={0003-6951 1077-3118}, url={http://dx.doi.org/10.1063/1.3486124}, DOI={10.1063/1.3486124}, abstractNote={We consider transverse redistributions of the electrons and holes in intrinsic graphene ribbons under the influence of crossed electric and magnetic fields, i.e., the magnetoconcentration effect. The electron and hole transport is described from the Boltzmann kinetic equation assuming the local quasiequilibrium distributions of the carriers over energy states. The effective control of the carriers is achieved from deep depletion to accumulation modes depending on the properties of the ribbon edges, provided electron-hole recombination/generation rates at the edges are different from those inside the ribbon. The current-voltage characteristics reflect the behavior of the carrier redistributions across the ribbon. The obtained results suggest that interesting effects such as the population inversion in certain ranges of energies and the induced transparency are possible, which make them attractive for various optoelectronics applications in the terahertz spectral range.}, number={11}, journal={Applied Physics Letters}, publisher={AIP Publishing}, author={Sokolov, V. N. and Kochelap, V. A. and Kim, K. W.}, year={2010}, month={Sep}, pages={112112} }
 @article{wang_zou_araki_luening_kilcoyne_sokolov_ade_rafailovich_2010, title={Probing the Chain and Crystal Lattice Orientation in Polyethylene Thin Films by Near Edge X-ray Absorption Fine Structure (NEXAFS) Spectroscopy}, volume={43}, ISSN={["1520-5835"]}, DOI={10.1021/ma101213h}, abstractNote={The chain and the crystal unit cell orientation of linear low density polyethylene (LLDPE) were measured with near edge X-ray absorption fine structure (NEXAFS) spectroscopy. A strongly attractive substrate, silicon, and a weakly attractive substrate, mica, were used. For a 100 nm thick LLDPE film on the silicon substrate, the crystals exhibit an edge-on lamellar morphology, with the chains predominantly parallel to the substrate, and the orthorhombic unit cell ⟨a, b, c⟩ in the following approximate orientation: b and c are in the film plane with b along the crystal fibril direction and c perpendicular to the fibril direction and a perpendicular to the film plane. On the mica substrates, LLDPE films with thickness below 180 nm completely dewet the surface and form isolated droplets, while a film 366 nm thick crystallizes as spherulites with most of the chains perpendicular to the substrate before annealing and with a twisted lamellar structure after isothermal crystallization at 60 °C. The results demonst...}, number={19}, journal={MACROMOLECULES}, author={Wang, Yantian and Zou, Ying and Araki, Tohru and Luening, Jan and Kilcoyne, A. L. D. and Sokolov, Jonathan and Ade, Harald and Rafailovich, Miriam}, year={2010}, month={Oct}, pages={8153–8161} }
 @article{kong_sokolov_kim_trew_2010, title={Quasi-Coherent Thermal Emission in the Terahertz by Doped Semiconductors}, volume={10}, ISSN={1530-437X 1558-1748}, url={http://dx.doi.org/10.1109/JSEN.2009.2038133}, DOI={10.1109/jsen.2009.2038133}, abstractNote={We investigate thermal emission characteristics mediated by surface plasmon polaritons (SPPs) resonantly excited at a semiconductor-vacuum interface. The characteristic plasma and SPP resonant frequencies in the interval from 0.3 to 10 THz can be controlled with conventional doping densities. All of the cases under consideration (n-doped GaAs, GaN, and Si) demonstrate the spectral energy density in the near field that is several orders of magnitude larger than the blackbody radiation. The strongly resonant SPPs are also shown to enhance drastically the radiative heat transfer between two semi-infinite surfaces separated by nanometric distances. The possibility of extending spatially coherent emission through 1-D binary grating is examined based on a rigorous coupled-wave analysis. Our calculation results clearly indicate that n-doped semiconductors with properly designed surface grating can achieve efficient directional thermal emission in the THz frequency range for potential use in a number of applications including sensing.}, number={3}, journal={IEEE Sensors Journal}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Kong, B. D. and Sokolov, V. N. and Kim, K. W. and Trew, R. J.}, year={2010}, month={Mar}, pages={443–450} }
 @article{sokolov_iafrate_krieger_2009, title={Bloch electron spontaneous emission from a single energy band in a classical ac field}, volume={80}, ISSN={["1098-0121"]}, DOI={10.1103/PhysRevB.80.165328}, abstractNote={A theory for the spontaneous emission of radiation for a Bloch electron in a single superlattice (SL) energy band under the influence of an external, spatially homogeneous, classical ac electric field is presented. The classical external ac electric field is described in the vector-potential gauge. The quantum radiation field is described by the free-space quantized electromagnetic field in the Coulomb gauge. Utilizing the instantaneous eigenstates of the Bloch Hamiltonian as the basis states, the Bloch electron dynamics is described to all orders in the classical ac electric field. It is shown that the spontaneous emission occurs with frequencies equal to integral multiples of the classical ac electric field frequency; this is due to the imposition of temporal periodic motion of the Bloch electrons in the SL miniband from the external periodic ac field. From appropriately derived selection rules for photon frequency and wave-vector transitions, the total spontaneous-emission probability (TSEP) is derived to first-order perturbation theory in the quantized radiation field. A general expression is obtained for the TSEP in terms of arbitrary SL miniband parameters; further, the TSEP is analyzed in detail based on the band model for the nearest-neighbor tight-binding approximation, and results show multiharmonic behavior and ac electric field tuning properties. In the nearest-neighbor tight-binding approximation, specific results for single Bloch electron manifest distinct plateaulike step structure in the analysis of normalized TSEP as a function of the ratio ${\ensuremath{\omega}}_{0}/\ensuremath{\omega}$, where ${\ensuremath{\omega}}_{0}$ is the characteristic frequency, proportional to the ac electric field amplitude, and $\ensuremath{\omega}$ is the ac electric field frequency; the plateau centers of gravity are found to be defined by the Stark delocalization condition established in ac-field transport. Further, the influence of a microcavity waveguide is established and shows enhancement as well as harmonic tuning of the TSEP due to coupling to the microcavity modal environment. Finally, the one-electron TSEP is extended, within the independent electron approximation, so as to include fractional band filling along with a constant-temperature-dependent and electron-density-dependent analysis; from this analysis, TSEP numerical estimates are projected at terahertz external field frequencies for a half-filled GaAs/AlGaAs SL miniband at zero temperature.}, number={16}, journal={PHYSICAL REVIEW B}, author={Sokolov, V. N. and Iafrate, G. J. and Krieger, J. B.}, year={2009}, month={Oct} }
 @article{barry_sokolov_kim_trew_2009, title={Terahertz generation in GaN diodes operating in pulsed regime limited by self-heating}, volume={94}, ISSN={0003-6951 1077-3118}, url={http://dx.doi.org/10.1063/1.3147217}, DOI={10.1063/1.3147217}, abstractNote={The conditions for pulsed regime operation of terahertz power generation in vertical nanoscale GaN-based diodes are investigated via self-consistent simulation of the high-field electron transport in the active channel and thermal transport in the entire device structure. The combined electrothermal model allows for a detailed analysis of the dynamical local distributions of the electric field, drift-velocity, and lattice temperature. We show that stable generation is achievable with a self-heating limited output power of 2.25 W at an operation frequency of 0.71 THz for a pulse width of 3 ns with a few tens of nanosecond duty cycle.}, number={22}, journal={Applied Physics Letters}, publisher={AIP Publishing}, author={Barry, E. A. and Sokolov, V. N. and Kim, K. W. and Trew, R. J.}, year={2009}, month={Jun}, pages={222106} }
 @article{kong_sokolov_kim_trew_2008, title={Terahertz emission mediated by surface plasmon polaritons in doped semiconductors with surface grating}, volume={103}, ISSN={0021-8979 1089-7550}, url={http://dx.doi.org/10.1063/1.2840063}, DOI={10.1063/1.2840063}, abstractNote={Doped semiconductors with one-dimensional surface gratings are studied theoretically for application to terahertz emission. The presence of free carriers supports thermally excited surface plasmon polaritons at the semiconductor-vacuum interface whose resonance frequency can be controlled by doping. The calculation based on the fluctuational electrodynamics shows that the near-field spectral energy density of this surface excitation can be many orders of magnitude larger than those in the blackbody radiation. At the same time, a rigorous coupled-wave analysis with a properly designed surface microstructure clearly indicates narrow angular lobes in the directional emissivity at the targeted frequencies, illustrating strong coupling between surface plasmon polaritons and propagating waves. Thus, doped semiconductors with engineered radiation may provide an efficient terahertz source with spatial and spectral control.}, number={5}, journal={Journal of Applied Physics}, publisher={AIP Publishing}, author={Kong, B. D. and Sokolov, V. N. and Kim, K. W. and Trew, R. J.}, year={2008}, month={Mar}, pages={056101} }
 @article{barry_sokolov_kim_trew_2008, title={Terahertz generation in GaN diodes in the limited space-charge accumulation mode}, volume={103}, ISSN={0021-8979 1089-7550}, url={http://dx.doi.org/10.1063/1.2946717}, DOI={10.1063/1.2946717}, abstractNote={The conditions for terahertz power generation are investigated theoretically in a nanoscale GaN-based diode coupled to an external resonant circuit for operation in the limited space-charge accumulation (LSA) mode under the high-field transport regime. The generation criteria are revisited in terms of a phase plane analysis of the diode high-field transport and circuit equations. Based on a Fourier series analysis, the waveforms of the diode voltage and current are examined and the generated power and conversion efficiencies are estimated at the fundamental and lowest harmonic frequencies. The advantages of group-III nitride LSA diodes are elucidated including their ability to simultaneously achieve large output powers (>10 mW) and high dc-to-rf conversion efficiencies (>1%) over a wide range of frequencies near 1 THz.}, number={12}, journal={Journal of Applied Physics}, publisher={AIP Publishing}, author={Barry, E. A. and Sokolov, V. N. and Kim, K. W. and Trew, R. J.}, year={2008}, month={Jun}, pages={126101} }
 @article{sokolov_iafrate_krieger_2007, title={Microcavity enhancement of spontaneous emission for Bloch oscillations}, volume={75}, ISSN={["1098-0121"]}, DOI={10.1103/physrevb.75.045330}, abstractNote={A theory for the spontaneous emission of a Bloch electron traversing a single energy miniband of a superlattice while accelerating under the influence of a constant external electric field and radiating into a microcavity is presented. In the analysis, the quantum electromagnetic radiation field is described by the dominant microcavity ${\mathrm{TE}}_{10}$ rectangular waveguide mode in the Coulomb gauge, and the instantaneous eigenstates of the Bloch Hamiltonian are utilized as the basis states in describing the Bloch electron dynamics to all orders in the constant external electric field. The results show that the spontaneous emission amplitude, when analyzed over many integral multiple values of the Bloch period, gives rise to selection rules for photon emission in both frequency and wave number with preferred transitions at the Wannier-Stark ladder levels. From these selection rules, the total spontaneous emission probability is derived to first-order perturbation theory in the quantized radiation field. It is shown that the power radiated into the dominant ${\mathrm{TE}}_{10}$ waveguide mode can be enhanced by an order of magnitude over the free-space value by tuning the Bloch frequency to align with the waveguide spectral density peak. A general expression for the total spontaneous emission probability is obtained in terms of arbitrary superlattice single band parameters, showing multiharmonic behavior and cavity tuning properties. For $\mathrm{GaAs}$-based superlattices, described in the nearest-neighbor tight-binding approximation, the power radiated into the waveguide from spontaneous emission due to Bloch oscillations in the terahertz frequency range is estimated to be several microwatts.}, number={4}, journal={PHYSICAL REVIEW B}, author={Sokolov, V. N. and Iafrate, G. J. and Krieger, J. B.}, year={2007}, month={Jan} }
 @article{sokolov_kim_kochelap_muntiian_2007, title={Negative small-signal impedance of nanoscale GaN diodes in the terahertz frequency regime}, volume={90}, ISSN={0003-6951 1077-3118}, url={http://dx.doi.org/10.1063/1.2720758}, DOI={10.1063/1.2720758}, abstractNote={A physical mechanism of electrical instability in the terahertz frequency range is explored in a nanoscale two-terminal configuration of nitride-based structures under a high-field transport regime. The investigated instability utilizes the negative differential resistance effect that is known to occur in bulk group-III nitride semiconductors. A distinctive feature of the proposed mechanism for terahertz generation is that the diode steady-state I-V characteristics maintain a positive differential resistance for a stable operation, while the small-signal response provides negative impedance within the desired frequency range. An additional advantage is that the frequency window of negative small-signal impedance can be tuned electrically.}, number={14}, journal={Applied Physics Letters}, publisher={AIP Publishing}, author={Sokolov, V. N. and Kim, K. W. and Kochelap, V. A. and Muntiian, P. M.}, year={2007}, month={Apr}, pages={142117} }
 @article{sokolov_kong_kim_trew_2007, title={Quasimonochromatic emission spectra in the near field by polar semiconductor thermal sources}, volume={90}, ISSN={0003-6951 1077-3118}, url={http://dx.doi.org/10.1063/1.2713166}, DOI={10.1063/1.2713166}, abstractNote={The near-field spectra of the electromagnetic field emitted from a planar surface are theoretically investigated for a number of polar semiconductor and dielectric materials that support surface phonon polaritons. All of the studied materials, InP, GaAs, GaN, SiC, and α-Al2O3 (sapphire), exhibit quasimonochromatic thermal emission symbolized by strong peaks of evanescent modes at well-defined frequencies in the near field that correspond to the appropriate peaks in the density of states for surface phonon polaritons. It is also found that the materials with lower polariton frequencies (e.g., InP and GaAs) generally demonstrate a higher peak spectral energy density compared to those with higher frequencies (e.g., SiC). This trend is maintained over the entire range of temperature (300–600K) and the distance from the surface (⩽10μm) considered in the calculation. Thus, the results clearly indicate that among the studied materials InP and GaAs are the best candidates to provide the quasicoherent thermal emission for potential use as a nanoscale thermal source. The energy density stored in the evanescent peaks, when close to the surface, is estimated to be many orders of magnitude larger than that in the blackbody radiation.}, number={11}, journal={Applied Physics Letters}, publisher={AIP Publishing}, author={Sokolov, V. N. and Kong, B. D. and Kim, K. W. and Trew, R. J.}, year={2007}, month={Mar}, pages={113106} }
 @article{sokolov_zhou_iafrate_krieger_2006, title={Spontaneous emission of Bloch oscillation radiation from a single energy band}, volume={73}, ISSN={["2469-9969"]}, DOI={10.1103/physrevb.73.205304}, abstractNote={Abstract : In this study, we explored the possibility of enhanced spontaneous emission of radiation beyond the free space value by analyzing a superlattice structure placed in a microcavity whose resonant modes were tuned to the Bloch frequency. In particular, we considered the spontaneous emission of Bloch radiation into the rectangular waveguide dominant mode. In the analysis, the quantum radiation field was described by the waveguide quantized electromagnetic field in the Coulomb gauge, and the instantaneous eigenstates of the Bloch Hamiltonian were used as basis states to analyze the Bloch dynamics to all orders in the constant external electric field. The results predict that the spontaneous emission occurs with frequencies equal to integral mutltiples of the Bloch frequency without any ad hoc assumptions made concerning the existence of Wannier-Stark ladder levels; such quantization effects arise from a natural consequence of the implicit quantum selection rules.}, number={20}, journal={PHYSICAL REVIEW B}, author={Sokolov, V. N. and Zhou, L. and Iafrate, G. J. and Krieger, J. B.}, year={2006}, month={May} }
 @article{fedorov_sokolov_kim_zavada_2005, title={Coulombic effects of electron-hole plasma in nitride-based nanostructures}, volume={98}, ISSN={0021-8979 1089-7550}, url={http://dx.doi.org/10.1063/1.2060937}, DOI={10.1063/1.2060937}, abstractNote={We study the Coulombic effects of electron-hole plasma on the ground and first excited energy levels and carrier wave functions in a GaN∕AlGaN quantum well (QW) structure. The coupled Schrödinger equations for electron and hole states are solved self-consistently in the Hartree-Fock approximation along with the Poisson equation. As expected, the decreasing QW width and increasing plasma density diminish the charge separation in the QW induced by the internal field, leading to the relatively reduced contribution of the Hartree interaction to the energy-level shift. In contrast, the calculation also demonstrates that the contribution of many-body effects becomes more pronounced. The resulting competition between the many-body and Hartree contributions causes a nonmonotonous dependence of the electron and hole energies on the plasma density at a given field. These findings are applied to explore the potential bistable behavior in the QW electroabsorption under near-band-edge photoexcitation.}, number={6}, journal={Journal of Applied Physics}, publisher={AIP Publishing}, author={Fedorov, I. A. and Sokolov, V. N. and Kim, K. W. and Zavada, J. M.}, year={2005}, month={Sep}, pages={063711} }
 @article{sokolov_kim_kochelap_woolard_2005, title={Terahertz generation in submicron GaN diodes within the limited space-charge accumulation regime}, volume={98}, ISSN={0021-8979 1089-7550}, url={http://dx.doi.org/10.1063/1.2060956}, DOI={10.1063/1.2060956}, abstractNote={The conditions for microwave power generation with hot-electron transport are investigated in a submicron GaN diode when it operates in the limited space-charge accumulation (LSA) mode. Applying a transport model based on the local quasistatic approximation, the analysis shows that the nitride diodes can support the LSA mode of oscillation in the terahertz-frequency range. For a 100nm n-GaN diode with a cross section of 500μm2 and the electron density of 1×1017cm−3, the generated microwave power is estimated to be as high as ≈0.6W with the corresponding dc-to-rf conversion efficiency of ≈9% and the negative differential resistance of ≈−1.3Ω; which thus provides an efficient mechanism to achieve very high-frequency microwave generation in the nitrides.}, number={6}, journal={Journal of Applied Physics}, publisher={AIP Publishing}, author={Sokolov, V. N. and Kim, K. W. and Kochelap, V. A. and Woolard, D. L.}, year={2005}, month={Sep}, pages={064507} }
 @article{sokolov_kim_kochelap_woolard_2004, title={High-frequency small-signal conductivity of hot electrons in nitride semiconductors}, volume={84}, ISSN={0003-6951 1077-3118}, url={http://dx.doi.org/10.1063/1.1738518}, DOI={10.1063/1.1738518}, abstractNote={We investigate the small-signal conductivity of the electrons in group-III nitrides under a high steady-state electric field. The resulting calculation indicates a frequency dependence of the conductivity that is drastically different from that given by the Drude formula. A large and very fast response of the hot electrons in the nitrides is revealed. The complex conductivity is found to be finite up to the frequency of about 10 THz. For the fields above the threshold corresponding to the peak drift velocity, the calculation also predicts a frequency interval with a negative conductivity. A detailed analysis is provided on the field and frequency dependence of the dynamic conductivity at the high electric fields.}, number={18}, journal={Applied Physics Letters}, publisher={AIP Publishing}, author={Sokolov, V. N. and Kim, K. W. and Kochelap, V. A. and Woolard, D. L.}, year={2004}, month={May}, pages={3630–3632} }
 @article{sokolov_kim_kochelap_woolard_2004, title={Phase-plane analysis and classification of transient regimes for high-field electron transport in nitride semiconductors}, volume={96}, ISSN={0021-8979 1089-7550}, url={http://dx.doi.org/10.1063/1.1808900}, DOI={10.1063/1.1808900}, abstractNote={We present a detailed theoretical analysis of steady-state, transient time-dependent, and spatially dependent electron transport in the group-III nitrides at high and ultrahigh electric fields. To develop an analytical model, we derive time-dependent differential equations describing the hot-electron rates of momentum and energy relaxation in electron–polar-optical-phonon scattering and analyze them by employing phase-plane analysis. From the structure of the phase-plane partitioning based on the phase trajectories, the transient regimes are investigated and classified depending on various initial conditions. We have studied different subpicosecond regimes and found a considerable velocity overshoot effect. One of our findings is that when the velocity reaches the maximum, the electron temperature is of a moderate magnitude but increases considerably in the subsequent stage. Dynamic regimes with high electron temperature overshoot have been revealed. For the dominant electron–polar-optical-phonon scattering, the observed overshoot can be treated as a rudiment of the runaway effect typical for that mechanism. In nanoscale nitride diodes with space-charge limited transport, the transient processes are extended to sufficiently larger distances, the overshoot is weaker and the electron heating in the region of the peak velocity is greater than that found for time-dependent problem with a constant electric field throughout a homogeneous sample.}, number={11}, journal={Journal of Applied Physics}, publisher={AIP Publishing}, author={Sokolov, V. N. and Kim, K. W. and Kochelap, V. A. and Woolard, D. L.}, year={2004}, month={Dec}, pages={6492–6503} }