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.
Highlights
In order to form highly sensitive photodetectors based on crystalline semiconductors, the hopping conduction via hydrogen-like impurities should be inhibited
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 Kh
In the temperature range T < Th /2, the hopping activation energy e3 behaves as e3 / T3/4 (Mott model)33 or e3 / T1/2 (EfrosShklovskii model).34] This regime is usually called variable range hopping (VRH). This mechanism of conduction is superseded by hops of holes between the nearest neighbor acceptors (NNH) with activation energy e3, which is equal within an order of magnitude to the root-mean-square fluctuation W of their energy levels. [In Ref. 35, a relation of proportionality for boundary temperature between nearest neighbor hopping (NNH) and VRH regimes is obtained based on statistics of pair donors, between which an electron jump takes place, taking into account the D-bot and D-top donor bands in crystalline n-type semiconductors.] Further increase of temperature leads to the emission of the holes from the A-bot band to the A-top band; e2-conduction takes place
Summary
In order to form highly sensitive photodetectors (from infrared to terahertz wavelengths) based on crystalline semiconductors, the hopping conduction via hydrogen-like impurities should be inhibited (see, e.g., Refs. 1 and 2). [In Ref. 35, a relation of proportionality for boundary temperature between NNH and VRH regimes is obtained based on statistics of pair donors, between which an electron jump takes place, taking into account the D-bot and D-top donor bands in crystalline n-type semiconductors.] Further increase of temperature leads to the emission of the holes from the A-bot band to the A-top band; e2-conduction takes place. For temperatures T > Tj, the emission of holes from the A-bot and A-top bands to the t-band occurs as well as the capture of holes from the t-band on the A-bot and A-top bands These processes correspond to the band conduction (BC) with activation energy e1, which is equal within an order of magnitude to the thermal ionization energy of the acceptor.
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