Transition temperature from band to hopping direct current conduction in crystalline semiconductors with hydrogen-like impurities: Heat versus Coulomb attraction

Abstract

For nondegenerate bulk semiconductors, we have used the virial theorem to derive an expression for the temperature Tj of the transition from the regime of “free” motion of electrons in the c-band (or holes in the υ-band) to their hopping motion between donors (or acceptors). Distribution of impurities over the crystal was assumed to be of the Poisson type, while distribution of their energy levels was assumed to be of the Gaussian type. Our conception of the virial theorem implementation is that the transition from the band-like conduction to hopping conduction occurs when the average kinetic energy of an electron in the c-band (hole in the υ-band) is equal to the half of the absolute value of the average energy of the Coulomb interaction of an electron (hole) with the nearest neighbor ionized donor (acceptor). Calculations of Tj according to our model agree with experimental data for crystals of Ge, Si, diamond, etc. up to the concentrations of a hydrogen-like impurity, at which the phase insulator-metal transition (Mott transition) occurs. Under the temperature Th ≈ Tj /3, when the nearest neighbor hopping conduction via impurity atoms dominates, we obtained expressions for the electrostatic field screening length Λh in the Debye-Hückel approximation, taking into account a nonzero width of the impurity energy band. It is shown that the measurements of quasistatic capacitance of the semiconductor in a metal-insulator-semiconductor structure in the regime of the flat bands at the temperature Th allow to determine the concentration of doping impurity or its compensation ratio by knowing Λh.

Thermal ionization energy of hydrogen-like impurities in semiconductor materials

“…В работе [19] исходя из теоремы вириала определена характерная температура T j перехода стационарной зонной электропроводности по состояниям c-зоны (σ n ) к стационарной прыжковой электропроводности по донорам (σ h ). Значение температуры T j , при которой σ n = σ h , в пределе низкой концентрации электронов в c-зоне ( ) , ,…”

Drift-diffusion model of hole migration in diamond crystals via states of valence and acceptor bands

“…It is important for the wide bandgap semiconductors, like boron-doped crystalline diamond, for which T j is in the region from liquid nitrogen to room temperatures and higher. For the narrow bandgap semiconductors, like boron-doped crystalline silicon, temperature T j is usually in the region from liquid helium to liquid hydrogen temperatures [4]. Such a difference in temperatures T j between boron-doped diamond and boron-doped silicon is connected with that the thermal energy of ionization by electrically neutral boron atoms in diamond is eight times greater than the one in silicon.…”

“…(The relatively high compensation ratio at this boron concentration can be attributed to the compensation of boron atoms by hydrogen; see also [64].)4. Numerical calculation of the ionization equilibrium in p-Dia:B crystals at the temperature T jIn[4,13,30] the characteristic temperature T j of the transition from the HC regime to the BC regime of hole migration is determined on the basis of the virial theorem. According to these references, the value of temperature T j , where p h s s = (figure 1), is found from the equation…”

Localization by an external magnetic field of electrons on the ions of hydrogen-like donors in non-degenerate semiconductors

“…где M -магнитный момент электрона, связанный с его орбитальным движением относительно иона донора. Тогда полная энергия E системы с учетом (25) есть (27) дает две проекции магнитных моментов:…”

“…Схема расщепления уровней энергии электрона c-зоны и электрона на доноре под действием магнитного поля (a); зависимости энергии электрона c-зоны E n (левая шкала) и локализованного состояния электрона на доноре E t (правая шкала) от магнитной индукции B для кристалла n-GaAs с концентрацией доноров N = N 0 + N +1 = 3•10 14 см −3 и степенью их компенсации K = 0,03: кривые 1 и 2 соответствуют ориентации спина по (↑) и против (↓) направления индукции внешнего поля (b) Scheme for splitting energy levels of the c-band electron and the electron on the donor under the influence of a magnetic field (a); dependences of the c-band electron energy E n (left scale) and the energy of the localized state of the electron on the donor E t (right scale) on the magnetic induction B for an n-GaAs crystal with a donor concentration of N = N 0 + N +1 = 3•10 14 cm −3 and a compensation ratio K = 0.03: curves 1 and 2 correspond to the spin orientation along (↑) and against (↓) the direction of the external field induction (b)кристалла n-GaAs (кривая 1 -спин электрона направлен по вектору индукции B, кривая 2 -спин направлен противоположно B). В отсутствие магнитного поля температура T j , ниже которой темп захвата электронов из c-зоны на водородоподобные доноры превалирует над темпом теплового выброса электронов с доноров в c-зону, для кристаллического полупроводника n-типа при n << K(1 − K )N имеет следующий вид[27]:…”