Semiconductors Fermi-Dirac distribution

Intrinsic Semiconductors. For semiconductors in thermal equiHbrium, (Ai( )), the average number of electrons occupying a state with energy E is governed by the Fermi-Dirac distribution. Because, by the Pauli exclusion principle, at most one electron (fermion) can occupy a state, this average number is also the probabiHty, P E), that this state is occupied (see Fig. 2c). In equation 2, K... 

Noise. So fat, as indicated at the beginning of this section on semiconductor statistics, equihbtium statistics have been considered. Actually, there ate fluctuations about equihbtium values, AN = N— < N >. For electrons, the mean-square fluctuation is given by < ANf >=< N > 1- ) where (Ai(D)) is the Fermi-Dirac distribution. This mean-square fluctuation has a maximum of one-fourth when E = E-. These statistical fluctuations act as electrical noise and limit minimum signal levels. 

The distribution of the electrons among the allowed energy states in the semiconductor crystal at thermal equilibrium is described by the Fermi-Dirac distribution function." It is denoted by ME), which has the form... 

Here o is electrical conductivity, u is thermopower, k is thermal conductivity, t is energy of carrier, p is chemical potential, e is bare charge of electron, and f (e) is Fermi-Dirac distribution function. In deriving eq.(2) we treat the lattice thermal conductivity as a constant. Following we consider the n-type semiconductors, then the change of differential conductivity can be given by ... 

On this basis, solids can be divided into insulators, in which the highest occupied band (the valence band) is completely hlled, while the lowest unoccupied band (the conduction band) is completely empty and metals present a partly empty and partly hlled band (the conduction band). Semiconductors are a particular case of insulators where the energy gap between the top of the valence band and the bottom of the conduction band is small enough that, at nonzero temperature, the smoothing out of the Fermi-Dirac distribution causes an appreciable number of states at top of the valence band to be empty and an equivalent number of states at bottom of the conduction band to be hlled. Note that the conductivity in semiconductors is highly temperature dependent. 

Gas adsorption is a chemical interaction between the gas molecules and the semiconductor surface. This interaction is accompanied by charge exchange creating acceptor or donor like band gap level, whose occupation probability is given by the Fermi-Dirac distribution function (Kireev 1978). Its conduction behavior as acceptor or donor will depend on the type of the adsorbed molecule. 

When an oxygen molecule is approaching the surface of an n-type semiconducting oxide, the probability of trapping a conduction electron from the semiconductor is the occupation probability of the band gap acceptor level Ea=%-W, where % is the electron affinity of the gas molecule and is the semiconductor work function. The probability is given by the Fermi-Dirac distribution function (Kireev 1978) ... 

A space-charge region is also formed, and the bands bent, when a potential is apphed to the electrode. As above, the band edges remain pinned at the electrode/solution interface, which arises because the potential drop between the bulk semiconductor and the solution is essentially entirely across the space-charge region rather than at the semiconductor interface. As a consequence, the intrinsic electron transfer rate constant is independent of applied potential. Nevertheless the current (and hence the effective rate constant) does depend on the apphed potential because the concentration of electrons (the majority carriers) at the electrode surface relative to its bulk concentration has a Boltzmann dependence on the energy difference between the band edge and the interior of the electrode. (The Fermi Dirac distribution reduces to a Boltzmann distribution when E > Fp-)... 

The Fermi-Dirac distribution as a function of temperature in a semiconductor. 

For typical semiconductors the bandgap is between 1 and 3 eV. The electronic population of the energy bands is determined by the Fermi-Dirac distribution function, i.e. 

In this expression Ep is the Fenni-energy level, which determines the population statistics. In a non-degenerate semiconductor - Ep and Ep - E are much larger than kT and the Fermi-Dirac distribution can be approximated by the Boltzmann distribution functions for the conduction and valence bands, i.e. 

The Fermi-Dirac and Maxwell-Boltzmann statistical distribution functions are widely used in semiconductor physics, with the latter commonly used as an approximation to the former. The point of this problem is to make you familiar with these distribution functions their forms, their temperature dependencies, and under what conditions they become interchangeable. Throughout this problem, use the energy of silicon s valence band (Evb) as the zero of your energy scale. 

Statistical thermodynamics takes into account the microscopic structure of matter and how it is constituted. According to - Boltzmann the entropy can be correlated with the microscopic disorder of the system. Beside the Boltzmann distribution formula (- Debye-Htickel equation) other statistics (e.g., - Fermi-Dirac statistics) are of importance in electrochemistry (- semiconductors, -> BCS theory). 

The density of states is useful to our discussion because it allows the calculation of an extremely important numerical quantity, the effective density of states in the conduction band of the semiconductor, N. As a first approximation, can be considered to be the number of electronic states in the conduction band of the semiconductor that are available to accept electrons from a donor. is more formally defined as the number of electronic states, N E), for aU energy E within 3kT of the conduction band edge [ (i cb — E) < 3kT]. Given the density of states per unit energy (equation 1), and using Fermi-Dirac statistics to describe the distribution of electrons as a function of energy, it can be shown that ... 

We now describe the behavior of charge carriers in an intrinsic semiconductor (i.e., pure) at equilibrium. The electrical properties of any extended solid depend on the position of the Fermi level, defined as the highest occupied state at T = 0 K. An alternative definition, stemming from the Fermi-Dirac statistics that govern the distribution of electrons, the Fermi level is the energy at which the probability of finding an electron is If the Fermi level falls within a band, the band is partially filled and the material behaves as a conductor. As shown in Fig. 3, the valence and conduction band edges of an intrinsic semiconductor straddle the Fermi level. At T = 0 K, no conduction is possible since all of the states in the valence band are completely filled with electrons while aU of the states in the conduction band are empty. 

In the subsequent chapters in which we will be investigating the thermal, electrical, optical, and magnetic properties of materials, it will be necessary to be able to determine the energy distribution of electrons, holes, photons, and phonons. To do this, we need to introduce some quantum statistical mechanical concepts in order to develop the distribution fimc-tions needed for this purpose. We will develop the Bose-Einstein (B-E) distribution function that applies to all particles except electrons and holes (and other fermions) that obey the Pauli exclusion principle and show how this function becomes the Maxwell-Boltzmann (M-B) distribution in the classical limit. Also, we will show how the Planck distribution results by relaxing the requirement that particles be conserved. Next we develop the Fermi-Dirac (F-D) distribution that applies to electrons and holes and becomes the basis for imderstanding semiconductors and photonic systems. 

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