Variation of Energy Bands with Alloy Composition E 3.0 E.8.6 L 0.3eV Al x GaAs AlAs 1- xas 1.43eV.16eV X k.4 L. X.0 X 1.8 L 1.6 1.4 0 0. 0.4 0.6 X 0.8 1 1
Carriers in intrinsic Semiconductors Ec 4º 1º 5º 6º 7º 8º º 0º 3º 9º K 300º 14º 15º 0º 19º 11º 10º 1º 13º 17º 16º 18º K Eg Ev Electron E Hole Pair
Effective Mass Example: Find the (E,k) relationship for a free electron and relate it to the electron mass. E p mv hk k E 1 mv 1 p m h m k d E dk h m Most energy bands are close to parabolic at their minima (for conduction bands) or maxima (for valence bands).
Intrinsic Material A perfect semiconductor crystal with no impurities or lattice defects is called an Intrinsic semiconductor. In such material there are no charge carriers at 0º K, since the valence band is filled with electrons and the conduction band is empty. 4
Intrinsic Material e - Si E g h + n=p=n i 5
Extrinsic Material In addition to the intrinsic carriers generated thermally, it is possible to create carriers in semiconductors by purposely introducing impurities into the crystal. This process, called doping, is the most common technique for varying the conductivity of semiconductors. When a crystal is doped such that the equilibrium carrier concentrations n 0 and p 0 are different from the intrinsic carrier concentration n i, the material is said to be extrinsic. 6
Extrinsic Material (n-type) V P As Sb 4º 1º 5º 6º 7º 8º º 0º 3º 9º K 50º 14º 15º 0º 19º 11º 10º 1º 13º 17º 16º 18º K Donor Ec Ed Ev
Extrinsic Material (p-type) ш B Al Ga In 4º 1º 5º 6º 7º 8º º 0º 3º 9º K 50º 14º 15º 0º 19º 11º 10º 1º 13º 17º 16º 18º K Acceptor Ec Ea Ev
Extrinsic Material Al h + Si e - Sb
Carriers Concentrations In calculating semiconductor electrical properties and analyzing device behavior, it is often necessary to know the number of charge carriers per cm 3 in the material. The majority carrier concentration is usually obvious in heavily doped material, since one majority carrier is obtained for each impurity atom (for the standard doping impurities). The concentration of minority carriers is not obvious, however, nor is the temperature dependence of the carrier concentration.
Fermi-Dirac distribution function Electrons in solids obey Fermi-Dirac statistics. In the development of this type of statistics: Indistinguishability of the electrons Their wave nature Pauli exclusion principle must be considered. The distribution of electrons over a range of allowed energy levels at thermal equilibrium is f ( E) ( EE f ) kt 1 e 1 k : Boltzmann s constant f(e) : Fermi-Dirac distribution function E f : Fermi level
Effect of temperature on Fermi level f 1 ( E f ) E f E f ) 1 e f(e) 1 11 ( kt 1 1 T=0ºK 1 >0ºK >T 1 1/ E f E 1
Function f(e), the Fermi Dirac distribution function, gives the probability that an available energy state at E will be occupied by an electron at absolute temperature. Put E = E F in f(e) and we get f(e F ) = 1 / Thus an energy state at the Fermi level has a probability of ½ of being occupied by an electron.
Effect of temperature on Fermi level Every available energy state upto E F is filled at 0K.
f(e) distribution in intrinsic and extrinsic semiconductors f(e c ) f(e c ) E E c E f [1-f(E c )] E v f(e) 1 1/ 0 Intrinsic n-type p-type 15
f(e) distribution in intrinsic and extrinsic semiconductors 16
On semiconductors there are two charge carriers: electrons and holes Electrons on solids obey the Fermi-Dirac statistics. In equilibrium, the electron distribution over the allowed energy level interval obeys f ( E) EE / kt 1 e where E F is called Fermi level For T > 0K the probability to have a state with E=E F occupied, is 1 F f 1 1 ( EF ) EF EF / kt 1 e 11 1
A more detailed review of f(e) indicates that at 0K, the distribution assumes the rectangular form pictured below. That means that at 0K any available energy state from up to E F is filled with electrons and every states over E F are unoccupied f ( E) seee 1 0 seee F F For T> 0K there s a finite probability, f(e), that the states over E F are filled (e.g. T=T 1 ) and a corresponding probability, [1-f(E)], that the states below E F are unoccupied
Electron and hole concentrations at equilibrium To know the concentrations of electrons and holes in a semiconductor, we need to know the densities of available states. e.g. Conc. Of electrons in CB is n f ( E). N( E) de o E c Probability of occupancy Density of energy states (states/cm 3 ) in the energy range de
Thermal equilibrium = No excitations except thermal energy Schematic band diagram, density of states, Fermi Dirac distribution and the carrier concentrations for intrinsic SCs at thermal equilibrium
In n-type semiconductors n o = N c f(e c ) Effective density of states Probability of occupancy n o = N C e -(E C -E )/ KT F (IN CONDUCTION BAND) As E F moves closer to the CB, the electron concentration increases.
Electron and Hole Concentrations at Equilibrium The concentration of electrons in the conduction band is n f ( E) N( E) de 0 E C N(E)dE : is the density of states (states. cm -3 ) in the energy range de. The result of the integration is the same as that obtained if we repres-ent all of the distributed electron states in the conduction band edge E C.
Similarly conc. of holes in the valence band is p o = N v [ 1 f(e v ) ] Probability of finding an empty state at E v. p o = N V e -(E F E V )/KT (In Valence Band) Hole concentration increases as E F moves closer to the valence band. 1 December 015 MEL G631(L1) 3 BITS, Pilani
Schematic band diagram, density of states, Fermi Dirac distribution and the carrier concentrations for n-type SCs at thermal equilibrium
n o = N e -(E C C -E )/ KT F (IN p o = N V e -(E F E V )/KT CONDUCTION BAND) (IN VALENCE BAND) Equations are valid, whether the material is intrinsic or doped, provided thermal equilibrium is maintained. n o p o = [N C e -(E C -E F )/ KT ].(N V e -(E F E V )/KT )
For an intrinsic material, E F lies at some intrinsic level. Hence E F = E i Thus for an intrinsic material, electron and hole concentrations are n i = N C e -(E C -E i ) / KT p i = N V e -(E i E V ) /KT Find product of n i and p i from here :
Since n i = p i We Get n i = (N c N v )1/ e-(eg / KT) This Is Called Mass-action Law -Shows that intrinsic carrier concentration varies with temperature
Electron and Hole Concentrations at Equilibrium N(E)f(E) E Electrons E C E f N(E)[1-f(E)] E V Holes Intrinsic n-type p-type 8
Electron and Hole Concentrations at Equilibrium f ( E C ) 1 e 1 ( EC E F ) kt e ( E C E F ) kt n 0 N C e ( E C E F ) kt N C * mnkt ( h 3 )
Electron and Hole Concentrations at Equilibrium p N [1 ( E 0 V V f )] 1 f ( E V ) 1 1 e 1 ( EV E F ) kt e ( E F E V ) kt p 0 N V e ( E F E V ) kt N V m ( h * p kt 3 ) 30
Calculation of carrier concentration The results is: n const T 3/ exp E c E kt F p const T 3/ exp E F E kt v If there is no doping: n = p = n i it is called intrinsic material E c E F E F E c E F E v E v = E i E F : Fermi-level 31
.:: CALCULATION Consider 1 cm 3 of Silicon. How many atoms does this contain? Solution: The atomic mass of silicon is 8.1 g which contains Avagadro s number of atoms. Avagadro s number N is 6.0 x 10 3 atoms/mol. The density of silicon:.3 x 10 3 kg/m 3 so 1 cm 3 of silicon weighs.3 gram and so contains 6.0 10 8.1 3.3 4.9310 atoms This means that in a piece of silicon just one cubic centimeter in volume, each electron energy-level has split up into 4.93 x 10 smaller levels!