Thermal ionization energy of hydrogen-like impurities in semiconductor materials

Abstract

In the work the dependence of the thermal ionization energy of hydrogen-like donors and acceptors on their concentration in n- and p-type semiconductors is analyzed analytically and numerically. The impurity concentrations and temperatures at which the semiconductors are on the insulator side of the concentration insulator – metal phase transition (Mott transition) are considered. It is assumed that impurities in the crystal are distributed randomly (according to Poisson), and their energy levels are distributed normally (according to Gauss). In the quasi-classical approximation, it is shown, for the first time, that the decrease in the ionization energy of impurities mainly occurs due to the joint manifestation of two reasons. Firstly, from the excited states of electrically neutral impurities, a quasicontinuous band of allowed energy values is formed for c-band electrons in an n-type crystal (or for v-band holes in a p-type crystal). This reduces the energy required for the thermally activated transition of electron from the donor to the c-band (for the transition of the hole from the acceptor to the v-band). Secondly, from the ground (unexcited) states of impurities a classical impurity band is formed, the width of which at low temperatures is determined only by the concentration of impurity ions. In moderately compensated semiconductors (when the ratio of the concentration of minority impurities to the concentration of majority impurities is less than 50 %) the Fermi level is located closer to the edge of the band of allowed energy values than the middle of the impurity band, that issue reduces thermal ionization energy of impurities from states in the vicinity of the Fermi level (transition of electron from a donor to the c-band, or hole from an acceptor to the v-band). Previously, these two causes of decrease in the thermal ionization energy due to increase in the concentration of impurities were considered separately. The results of calculations according to the proposed formulas are quantitatively agree with the known experimental data for a number of semiconductor materials (germanium, silicon, diamond, gallium arsenide and phosphide, silicon carbide, zinc selenide) with a moderate compensation ratio.

Model of DC Tunneling Conductivity via Hydrogen‐Like Impurities in Heavily Doped Compensated Semiconductors

“…The mobility edge E c ð Þ m (see Figure 2), for the concentration of c-band electrons n ( Kð1 À KÞN d , is [28,29]…”

Maximum hopping direct current conductivity via hydrogen-like impurities in semiconductors

“…The decrease in the thermal ionization energy of the hydrogen-like acceptor due to the confinement of the maximum Bohr radius of the hole orbit on the acceptor is given by the formula [21,22]…”

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

“…Энергия взаимодействия эффективного спинового магнитного момента электрона | g t |µ B с магнитным полем B ± B t учитывается отдельным слагаемым. В итоге для уровней энергии водородоподобного донора [23] при наложении на кристалл внешнего магнитного поля получаем…”

“…Дифференциальная термическая энергия ионизации доноров определяется так (см., напр., [23] и цитируемую там литературу):…”

“…магнитного поля (при B → 0) из (37) получаем энергию ионизации одиночного (уединенного) донора I t (B) →I d = e 2 /8pe r e 0 a d = me 4 /[2(4pe r e 0 ħ) 2 ] для боровского радиуса локализации на нем электрона R t (B) → a d = 4πε r ε 0 ħ 2 /me 2 .Приравнивая выражения (35) и (37) для I t (B), получим уравнение для нахождения радиуса орбиты R t , локализованного на атоме донора электрона в магнитном поле (совпадающее с уравнением (10)):Формула (37) дает энергию ионизации одиночного донора в магнитном поле. Если учесть взаимный «конфайнмент» примесей, т. е. ограничение на максимальный радиус орбиты R t электрона на одном доноре по[23], то (37) принимает вид im = 0,62[N(1 + K )] −1/3 -средний радиус сферической области, приходящейся на каждый атом донорной и акцепторной примесей в кристаллическом полупроводнике. Здесь отметим, что в нулевом магнитном поле радиус орбиты электрона R t = a d , и формула (38) принимает вид E t = I d (1 − a d /R im ).…”