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Figure 3—12 Energy band model and chemical bond model of dopants in semiconductors: (a) donation of electrons from donor level to conduction band; (b) acceptance of valence band electrons by an acceptor level, and the resulting creation of holes; (c) donor and acceptor atoms in the covalent bonding model of a Si crystal.

Calculation of the Donor Binding Energy We wish to calculate the energy required to excite the 5th electron of a donor atom into the conduction band. This is called the Donor Binding Energy. assume that the donor impurity (As in Si for example) has its four tightly bound electrons that form covalent bonds with neighbouring crystal lattice atoms. the extra (5th) electron is only loosely bound to the donor impurity. Use the Bohr model to represent the tightly bound “core” electrons in a hydrogen-like orbit Read Section 3.2.4 for more information about extrinsic semiconductors. Particularly the compound semiconductors (Group III-V compounds). In an n-type material, the conduction band electrons outnumber the holes in the valence band by many order of magnitude. the holes are minority carriers, and the electrons are majority carriers. In a p-type material: the electrons are minority carriers, and the holes are majority carriers.

The donor binding energy for GaAs—an example From Bohr model, the ground state energy of an “extra” electron of the donor is (3-8)

5.2 meV = Ec - Ed Compare with the room temperature (300K) thermal energy E=kT≈26meV All donor electrons are freed to the conduction band (ionized) Compare with the intrinsic carrier concentration in GaAs (ni=1.1 x 106 /cm3) We will have an increase in conduction electron concentration by a factor of 1010 if we dope GaAs with 1016 S atoms/cm3

3.2.5 Electrons and Holes in Quantum Wells (Read this Section on your own.) Figure 3—13 Energy band discontinuities for a thin layer of GaAs sandwiched between layers of wider band gap AIGaAs. In this case, the GaAs region is so thin that quantum states are formed in the valence and conduction bands. Electrons in the GaAs conduction band reside on “particle in a potential well” states such a E1 shown here, rather than in the usual conduction band states. Holes in the quantum well occupy similar discrete states, such a Eh .

3.3 Carrier Concentrations - We need to know the Concentration (number per unit volume) of charge carriers - calculating the Electrical Properties of semiconductors - investigating the behaviour of semiconductor Devices. - Majority Carrier concentration: - typically one majority carrier per impurity atom -Minority Carrier concentration: - ??? The Carrier Concentration are Temperature Dependent. We must investigate the Distribution of Charge Carriers over the Available States. Electrons in solids obey Fermi-Dirac Statistics.

3.3 Carrier Concentrations 3.3.1 The Fermi Function and the Fermi Level How will electrons distribute over a range of allowed energy levels at thermal equilibrium? Assume: Each allowed state has a maximum of one electron (Pauli principle) The probability of occupancy of each allowed (degenerate) quantum state is the same All electrons are indistinguishable Fermi-Dirac distribution function EF is called the Fermi Level k is the Boltzmann’s constant (3-10) EF is a very important quantity in semiconductor physics

f (EF) = ½  an energy state at the Fermi level has a probability of ½ of being occupied by an electron At T = 0K  f (E) = 1 for E < EF f (E) = 0 for E > EF At T > 0K  f (E) > 0 for E > EF the probability of an electron with E above EF is nonzero 1-f (E) >0 for E < EF the probability of a state below EF being empty The Fermi function is symmetrical about EF for all temperatures

Other distribution functions? Bose-Einstein distribution function for photons Maxwell-Boltzmann distribution—classic limit

Applying of FD Distribution to Semiconductors The distribution function has values within the band gap between EV and EC, but there are no energy states available, and no electron occupancy results from f (E) in this range. For intrinsic material, the Fermi level lies at the middle of the band gap

- for n-type material, the Fermi level lies close to the conduction band; - the value (EC – EF ) indicates how strongly n-type the material is - for p-type material, the Fermi level lies close to the valence band; - the value (EF – EV ) indicates how strongly p-type the material is - the position of EF is commonly included in band diagrams

3.3.2 Electron Concentrations in the Conduction Band at Equilibrium (3-12) → N(E) is the density of states (cm-3) in the energy range dE (see appendix IV) → N(E)  E½ NC is effective density of state in the conduction band (3-13) for (Ec - EF ) >> kT From Appendix IV (3-15)

Hole Concentration in the Valence Band at Equilibrium (3-17) Nv is the effective density of states in the valence band (3-19) mn* and mp* are density of states effective masses

Electron and Hole Concentrations at Thermal Equilibrium (3-21) (In intrinsic materials, EF is written as Ei) (3-23) (3-24) (3-25)