Semiclassical Rates Research Articles

Modeling Infrared Superlattice Photodetectors: From Nonequilibrium Green’s Functions to Quantum-Corrected Drift Diffusion

Carrier transport in type-II superlattice photodetectors is investigated by means of a rigorous nonequilibrium Green's function model based on a physics-based B\"uttiker-probe formalism. Intraband scattering self-energies (carrier-phonon interactions) are computed in the self-consistent Born approximation, while interband self-energies (Shockley-Read-Hall and optical transitions) are included in terms of semiclassical generation-recombination rates, neglecting interband renormalization effects. Current conservation is achieved with an efficient Newton-Raphson algorithm. While carrier transport in infrared detectors is usually understood in terms of quantities (e.g., mobilities and quasi-Fermi-levels) that are admittedly not germane to nonequilibrium Green's function theory, the proposed model provides a quantum-kinetic description of tunneling, miniband transport, hopping, and carrier extraction within a drift-diffusion-friendly framework. The connection with semiclassical theories allows exploration of the possibilities offered by Poisson-Schr\"odinger or localization landscape drift-diffusion approaches.

Hybrid approach for treating the depolarization of the solar lines of the Ba ii, Ca ii, and Mg ii ions by collisions

ABSTRACT Isotropic collisions between atoms of hydrogen and solar ions emitting polarized light contribute to reducing the observed polarization (depolarization). The aim of this work is to apply a hybrid method in order to provide new collisional depolarization rates of the 2P1/2 and 2P3/2 states of the Mg ii, Ca ii, and Ba ii ions. The hybrid method proposed in this work takes into account the spin effects in the calculation of the interaction potential and in the treatment of the collision dynamics. We detect the region of the interaction potential that is of importance in the determination of the depolarization rates. We conclude that the best strategy is to combine semiclassical and quantum potentials in order to build the so-called hybrid potentials. The dynamics of collisions proposed in this work is based on coupled differential equations that take into account the effects of the spins of the Mg ii, Ca ii, and Ba ii ions and the spin of the hydrogen. Hybrid depolarization rates are then inferred by solving the dynamics of collisions and using hybrid potentials. Comparison with previous quantum and semiclassical rates is presented. Our results should be of use for interpreting solar spectropolarimetric observations and our method can be applied to other ions.

Collisional Depolarization of the Solar Ca, Mg, and Ba Levels

Abstract We study the depolarization of the p-states of the Mg, Ca, and Ba atoms by isotropic collisions with neutral hydrogen. We show that the main source of error in calculating the collisional depolarization is the inaccuracy of the interaction potentials. To better investigate this problem, we determine the region of the interaction potential that is decisive in the calculation of the depolarization rates. We conclude that semiclassical and quantum potentials should be combined in view of obtaining the so-called hybrid potentials, which are the most precise ones. The depolarization rates of the p-states of the Ca i, Mg i, and Ba i are inferred by using hybrid potentials and compared to previously obtained quantum and semiclassical rates. Insights are also given for treating the effects of the spin. Our results should be useful for interpreting the second solar spectrum.

Semiclassical rates for tunnel ionization from power-law potentials induced by a constant or low-frequency electric field

The probabilities of tunnel ionization of particles confined by 1D power-law and logarithmic potentials are calculated for constant and low-frequency electric fields.

Thermodynamically-consistent semi-classical ℓ-changing rates

We compare the results of the semi-classical (SC) and quantum-mechanical (QM) formalisms for angular-momentum changing transitions in Rydberg atom collisions given in a series of papers by Vrinceanu et al, most recently Vrinceanu et al (2012 Astrophys. J. 747 56), with those of the SC formalism using a modified Monte Carlo realization. We find that this revised SC formalism agrees well with the QM results. This provides further evidence that the rates derived from the QM treatment are appropriate to be used when modeling recombination through Rydberg cascades, an important process in understanding the state of material in the early universe. The rates for derived from the QM formalism diverge when integrated to sufficiently large impact parameter, b. Further to the empirical limits to the b integration suggested by Pengelly and Seaton (1964 Mon. Not. R. Astron. Soc. 127 165), we suggest that the fundamental issue causing this divergence in the theory is that it does not fully cater for the finite time taken for such distant collisions to complete.

The Hawking cascades of gravitons from higher-dimensional Schwarzschild black holes

It has recently been shown that the Hawking evaporation process of $(3+1)$-dimensional Schwarzschild black holes is characterized by the dimensionless ratio $\eta\equiv\tau_{\text{gap}}/\tau_{\text{emission}}\gg1$, where $\tau_{\text{gap}}$ is the characteristic time gap between the emissions of successive Hawking quanta and $\tau_{\text{emission}}$ is the characteristic timescale required for an individual Hawking quantum to be emitted from the Schwarzschild black hole. This strong inequality implies that the Hawking cascade of gravitons from a $(3+1)$-dimensional Schwarzschild black hole is extremely {\it sparse}. In the present paper we explore the semi-classical Hawking evaporation rates of {\it higher}-dimensional Schwarzschild black holes. We find that the dimensionless ratio $\eta(D)\equiv{{\tau_{\text{gap}}}/{\tau_{\text{emission}}}}$, which characterizes the Hawking emission of gravitons from the $(D+1)$-dimensional Schwarzschild black holes, is a {\it decreasing} function of the spacetime dimension. In particular, we show that higher-dimensional Schwarzschild black holes with $D\gtrsim 10$ are characterized by the relation $\eta(D)<1$. Our results thus imply that, contrary to the $(3+1)$-dimensional case, the characteristic Hawking cascades of gravitons from these higher-dimensional black holes have a {\it continuous} character.

Hybrid Boltzmann–Gross-Pitaevskii theory of Bose-Einstein condensation and superfluidity in open driven-dissipative systems

We derive a theoretical model which describes Bose-Einstein condensation in an open driven-dissipative system. It includes external pumping of a thermal reservoir, finite lifetime of the condensed particles, and energy relaxation. The coupling between the reservoir and the condensate is described with semiclassical Boltzmann rates. This results in a dissipative term in the Gross-Pitaevskii equation for the condensate, which is proportional to the energy of the elementary excitations of the system. We analyze the main properties of a condensate described by this hybrid Boltzmann--Gross-Pitaevskii model, namely, dispersion of the elementary excitations, bogolon distribution function, first-order coherence, dynamic and energetic stability, and drag force created by a disorder potential. We find that the dispersion of the elementary excitations of a condensed state fulfills the Landau criterion of superfluidity. The condensate is dynamically and energetically stable as longs as it moves at a velocity smaller than the speed of excitations. First-order spatial coherence of the condensate is found to decay exponentially in one dimension and with a power law in two dimensions, similarly with the case of conservative systems. The coherence lengths are found to be longer due to the finite lifetime of the condensate excitations. We compare these properties with those of a condensate described by the popular ``diffusive'' models in which the dissipative term is proportional to the local condensate density. In the latter, the dispersion of excitations is diffusive which as soon as the condensate is put into motion implies finite mechanical friction and can lead to an energetic instability.

A database for V–V state-to-state rate constants in N2–N2 and N2–CO collisions in a wide temperature range: dynamical calculations and analytical approximations

Extensive semiclassical calculations have been performed regarding the generation of accurate collision data for single- and multi-quantum vibrational state-selected V–V exchanges in N2(v)–N2(u) and N2(v)–CO(u) collisions over a wide range of vibrational quantum numbers (v, u) and gas temperatures. This highly accurate database is also used to test the applicability of simplified analytical approximations frequently used in vibrational kinetic modeling to calculate large sets of state-to-state V–V rate constants in diatom–diatom collisions. The semiclassical calculations clearly show that the long-range attractive branch of the interaction potential cannot be neglected, even for the N2–N2 system. Therefore, new analytical approximations for V–V exchanges for the two collision systems are proposed which agree very well with the semiclassical rates. The ab initio data, together with that obtained from the newly proposed analytical formula, constitute an accurate database that can be used with confidence in the vibrational kinetic modeling of N2 and CO-based gases and plasmas over a wide temperature range, from thermal to hyper-thermal gas temperatures.

Dynamics of quantum tunneling: Effects on the rate and transition path of OH on Cu(110)

Recent low-temperature scanning-tunneling microscopy experiments [T. Kumagai et al., Phys. Rev. B 79, 035423 (2009)] observed the possible quantum tunneling of hydroxyl groups between two equivalent adsorption configurations on Cu(110). Here we analyze the quantum nuclear tunneling dynamics of hydroxyl on Cu(110) using density-functional theory based techniques. We calculate classical, semiclassical, and quantum-mechanical transition rates for the flipping of OH between two degenerate energy minima. The classical transition rate is essentially zero at the temperatures used in experiment and the tunneling rate along the minimum-energy path is also much too low compared to experimental observations. When tunneling is taken into account along a direct path connecting the initial and final states with only a minimum amount of the oxygen movement the transition rate obtained is in much better agreement with experiment, suggesting quantum tunneling effects cause a deviation of the reaction coordinate from the classical transition path.

Radiative cascade of highly excited hydrogen atoms in strong magnetic fields

We have studied the radiative decay of atomic hydrogen in strong magnetic fields of up to $4\phantom{\rule{0.3em}{0ex}}\mathrm{T}$. We have followed the radiative cascade from completely $l,m$ mixed distributions of highly excited states as well as from distributions that involve highly excited states with $\ensuremath{\mid}m\ensuremath{\mid}\ensuremath{\sim}n$. We have found that the time it takes to populate the ground state is not affected by the magnetic field for the initial states with $n\ensuremath{\lesssim}20$. For higher $n$ manifolds, the electrons in the most negative $m$ states are substantially slowed down by the magnetic field resulting in a much longer lifetime. We show that less than 10% of the antihydrogen atoms with $n\ensuremath{\sim}35$ generated in antihydrogen experiments at $4\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ will decay to their ground states before they hit the wall of the vacuum container unless they are trapped. We have also found that the decay time is mainly determined by the fraction of atoms that were initially in highest negative $m$ states due to the fact that only $\ensuremath{\mid}\ensuremath{\Delta}m\ensuremath{\mid}+\ensuremath{\mid}\ensuremath{\Delta}\ensuremath{\pi}\ensuremath{\mid}=1$ transitions are allowed in the magnetic field. We give a semiclassical method for calculating the decay rates for circular states and show that when the initial states have high-$\ensuremath{\mid}m\ensuremath{\mid}$, semiclassical rates agree with the full quantum mechanical rates within a couple of percent for states with effective $n\ensuremath{\gtrsim}20$.

Universal low-energy dynamics for rotating black holes

Fundamental string theory has been used to show that low energy excitations of certain black holes are described by a two dimensional conformal field theory. This picture has been found to be extremely robust. In this paper it is argued that many essential features of the low energy effective theory can be inferred directly from a semiclassical analysis of the general Kerr-Newman solution of supersymmetric four-dimensional Einstein-Maxwell gravity, without using string theory. We consider the absorption and emission of scalars with orbital angular momentum, which provide a sensitive probe of the black hole. We find that the semiclassical emission rates -including superradiant emission and greybody factors - for such scalars agree in striking detail with those computed in the effective conformal field theory, in both four and five dimensions. Also the value of the quantum mass gap to the lowest-lying excitation of a charge-$Q$ black hole, $E_{gap}=1/8Q^3$ in Planck units, can be derived without knowledge of fundamental string theory.

High-field mobility in an assembly of conjugated polymer segments.

We present a calculation of the hopping rate between the finite chains of a conjugated polymer. It is based on the optimization of the semiclassical hopping rates with respect to the possible widths and possible positions of the large polaron during the jump. For sufficient long chains and high enough fields, we arrive at an expression for the mobility which resembles a Fowler-Nordheim dependence with the field F, namely, exp-(${\mathit{F}}_{0}$/F). In this regime the mobility is weakly temperature dependent. \textcopyright{} 1996 The American Physical Society.

Pair breaking in semiclassical singlet small-bipolaron hopping.

The Holstein-Hubbard model is used to study high-temperature (T\ensuremath{\gtrsim}${\mathit{T}}_{\mathrm{phonon}}$/3) hopping transport when it is energetically favorable for carriers to pair as singlet small bipolarons. The semiclassical rates for one-electron transfers involving small polarons are found to be much greater than those that only involve small bipolarons. In particular, the most rapid one-electron-transfer processes have a small-polaron hopping to a vacant site and one of a small bipolaron's two carriers jumping onto the site of an adjacent small polaron. As a result, even when most carriers form small bipolarons rather than small polarons, one-electron transfers involving small polarons always dominate the dc conductivity. The energy to thermally generate small polarons from small bipolarons thus contributes to the conductivity's activation energy. This pair-breaking energy also manifests itself in a thermally activated contribution to the system's paramagnetic susceptibility. In addition, the carriers' Seebeck coefficient garners a contribution with a temperature dependence that is characterized by the pair-breaking energy. \textcopyright{} 1996 The American Physical Society.

Fusion in flight from the molecular continuum of dtμ

Fusion reaction rates for the muonic molecular system dtμ are calculated in the scattering continuum above the first break-up threshold (tμ) 1s+d, and for a few of the resonant states there embedded. In order to analyze the fusion in flight reaction (tμ) + d, the molecular wave function is obtained within (a) semiclassical, (b) adiabatic, and (c) nonadiabatic descriptions, in the energy range 0.002–10 keV. The semiclassical rates are higher than the corresponding adiabatic fusion rates, but converge towards the latter for high collision energies. In the nonadiabatic treatment, the qualitatively new structure has been obtained. The fusion rate shows an oscillatory behaviour, associated with the presence of a series of virtual states and variations of the overbarrier transmission in the tμ+d potential scattering. Furthermore, the nonadiabatic fusion rates are substantially enhanced at resonant energies near the tμ 2s threshold.

Comment on ‘‘Non-Rice–Ramsperger–Kassel–Marcus dynamics and the statistics of reaction rates in chaotic systems’’ [J. Chem. Phys. 9 8, 7898 (1993)

Results from chaos theory on the scarring of wave functions are applied to the problem of analyzing semiclassical reaction rates. Specifically, we show that uniform support of an eigenstate over the energy surface yields a Rice–Ramsperger–Kassel–Marcus (RRKM) estimate of the rate, while scarring of an eigenstate results in corrections to the RRKM rate due to periodic trajectories which recross the transition state. Non-RRKM behavior is also demonstrated for strongly averaged rates where recrossing effects are negligible.

Intracollisional field effect in the high-field regime of the quantum transport equation using Monte Carlo simulations.

We have studied the importance of the ``intracollisional field effect'' in the quantum-mechanical transport equation derived by Khan, Davies, and Wilkins [Phys. Rev. B 36, 2578 (1987)] via Monte Carlo simulations. This equation is similar to other quantum transport equations in the literature [Barker and Ferry, Phys. Rev. Lett. 42, 1779 (1979)] and contains an electric-field-induced ``broadening'' (denoted by \ensuremath{\surd}\ensuremath{\Vert}\ensuremath{\alpha}\ensuremath{\Vert} ) of the energy-conserving \ensuremath{\delta} function in semiclassical transition rates which is proportional to \ensuremath{\surd}\ensuremath{\Vert}scrE\ensuremath{\cdot}q\ensuremath{\Vert} , where scrE is the electric field and q is the change in electron momentum caused by scattering. We use a single parabolic band with parameters of the central valley of GaAs. The electrons are assumed to scatter with polar optical and acoustic phonons with the scattering parameters chosen to simulate GaAs. Our Monte Carlo analysis of the problem shows that intracollisional field effect is not important. This implies that in the field-dominant regime, i.e., \ensuremath{\surd}\ensuremath{\Vert}\ensuremath{\alpha}\ensuremath{\Vert} &gt;\ensuremath{\Elzxh}/${\ensuremath{\tau}}_{\mathrm{sc}}$, the results obtained from the quantum transport equation reduce to those obtained from the semiclassical Boltzmann equation for a single-band semiconductor of infinite width, where ${\ensuremath{\tau}}_{\mathrm{sc}{}^{\mathrm{\ensuremath{-}}1}}$ is the electron scattering-out rate.

Time-dependent transition rates for a multilevel quantum system stochastically coupled to a thermal bath.

Two different approaches (quantum and semiclassical) used to calculate time-dependent transition rates for a multilevel molecular system stochastically interacting with a thermal bath are presented. While the former is based on the use of a relationship between the real and the imaginary parts of the time autocorrelation function (TAF) associated with the interaction Hamiltonian, the latter relies on the replacement of a quantum TAF by the corresponding anticommutator. Two well-known examples for a stochastic interaction Hamiltonian, an Ornstein-Uhlenbeck process and a white-noise process, have been considered. In both cases, explicit expressions for the transition rates in terms of a reduced set of parameters characterizing the bath are obtained. In the Gaussian regime both quantum and semiclassical transition rates show a time-damped oscillatory behavior towards the same steady value. In the white-noise (\ensuremath{\delta}-correlated) regime, while the quantum transition rates are time dependent, the semiclassical rates are not. The high- and the low-temperature limits for the quantum and the semiclassical transition rates are also reported. The semiclassical results reduce to the quantum results only in the high-temperature limit.

Quantum transport equations for high electric fields.

We have studied quantum-mechanical transport equations for nondegenerate electrons in semiconductors in high electric fields. Our calculations use Kadanoff and Baym's formalism based on Green functions, treat only fields that are constant in space and time, and are restricted to weak scattering. First, we derive an approximate solution to the equation of motion of the retarded Green function in an electric field, with a careful check of its validity. This is used in deducing the conditions under which the quantum-mechanical transport equation reduces to the Boltzmann equation. We find, in agreement with some previous studies, that the electric field causes a ``broadening'' of the \ensuremath{\delta} function in semiclassical transition rates, a result of the ``intracollisional field effect.'' The Boltzmann equation fails when this broadening exceeds some characteristic energy scale (usually ${k}_{B}$T), which occurs at fields of a few MV ${m}^{\mathrm{\ensuremath{-}}1}$ in conventional semiconductors. These results are strongly dependent on the ansatz used to reduce the Green function to a distribution function. The scattering-out term is usually much less sensitive to the electric field than the scattering-in term. We exploit this to construct an integral transport equation, valid in high electric fields, which differs from the Boltzmann equation only in having a broadened function replacing the \ensuremath{\delta} function in the scattering-in rates. It should be possible to solve this equation using standard numerical techniques and gain quantitative information on the intracollisional field effect.

Quantum corrections to the Boltzmann equation for transport in semiconductors in high electric fields.

We have investigated quantum corrections to the Boltzmann equation for transport in high, spatially uniform electric fields, with weak scattering. Parabolic bands show no effects for constant fields up to ${10}^{7}$ V/cm; nonparabolic bands or time-varying fields give quantum corrections by dis- torting the energy-conserving \ensuremath{\delta} function in semiclassical transition rates. We have found that in practical cases the scale of this is less than that from collisional broadening, almost invariably neglected. These results extend greatly the known range of validity of the Boltzmann equation in semiconductors.

A proton magnetic resonance study of the motion of NH+4 in dilute solid solutions of NH4Cl in RbCl and KCl

The temperature dependence of the proton spin–lattice relaxation times of dilute solid solutions of about 3 mol % NH4Cl in RbCl and KCl are studied in the range 5–300 K. The potential barriers hindering the NH+4 reorientations are found to be extremely small (∼150 cal/mol). The relaxation rates are found to depart significantly from the usual semiclassical values: at 6 K the semiclassical relaxation rates are some 40 times greater than experimental ones. This departure from the semiclassical theory is partly explained by including the effects of symmetry and the quantization of the molecular rotational states.