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EEE 3394Electronic MaterialsChris FerekidesFall 2014Week 10++REM: gC(E)d(E)represents the number of available states (cm-3) in the energy interval E+dE. f(E)is the probability a state is occupied by an electron; (1-f(E) holes);

gC(E)f(E)d(E)gives the number of electrons (cm-3) in the interval E+dE;

Therefore the TOTAL number of electrons n (and holes p) in the conduction band (and in the valence band) can be obtained by integrating the relationships:

Equilibrium Carrier ConcentrationsThe integral is called the Fermi-Dirac integral of order (F1/2(C)) and can be found in tables.

and

let

Equilibrium Carrier Concentrations

where NC and NV are the effective densities of states in the conduction and valence bands respectively.

Note: Effective Density of States: if all electron (hole) states were located at EC (EV).

Solving the above integrals yield:

Equilibrium Carrier ConcentrationsEg(E)g(E) (EEc)1/2f(E)EFnE(E) or pE(E)EEForelectronsFor holes[1f(E)]nE(E)pE(E)Area = pArea = ()ndEEEn=EcEvEvEc0Ec+c EFVBCBWhen does n=pthen the Equilibrium Electron and Hole Concentrations can be expressed as:Note:Previous relationships are GENERAL!Yet inconvenient to calculate every time.HOWEVER, if EF is not close to the conduction band or valence band i.e.

When the Fermi Level is too close to the Conduction (or Valence) Band, or lies within the Conduction (or Valence) Band, the semiconductor is said to be Degenerate; i.e. Heavily Doped, with the electron (or hole) concentration being very high.Equilibrium Carrier ConcentrationsthenNote:Ei is the Fermi Level (energy) in an intrinsic semiconductor.In intrinsic semiconductors the electron and hole concentrations are equal.

Intrinsic Fermi Level

solving for NV and NC

finally..EcEvEFiCBVBEFnEcEvEFpEcEvWhat is ni2 and the product np equal to???

also ..

REMEMBER: ALL OF THE ABOVE WERE CALCULATED ASSUMING EQUILIBRIUM CONDITIONS AND NONDEGENERATE SEMICONDUCTORS.np product A uniformly doped semiconductor under equilibrium conditions is charge-neutral; i.e. the NET charge is equal to zero. (WHY?)

Are there any charges inside the semiconductor?Electrons, holes, ionized acceptors, ionized donors.therefore

Note:Number of ionized Donors:ND+Number of ionized Acceptors:NA-At room temperature complete ionization will be assumed. ND+ = ND NA- = NACharge Neutralitytherefore

The above equations are general expressions and can be simplified for most practical applications.Calculating n & p(1)Intrinsic semiconductorsND=NA=0n=p=ni(2)Doped Semiconductors:ND-NA>>ni or NA-ND>>ni

The above is usually true since the controlled doping of semiconductors yields more donors than acceptors, and since typical intentional dopants are at least 1014 (rem: ni=1010 for Si).

ForForandandCalculating n & p(3)Doped with ni>>ND (or ni>>NA)The semiconductor is considered intrinsic. At high enough temperatures all semiconductors are intrinsic.(4)Compensated Semiconductor:When ND or NA are comparable or equal; keep both concentrations and use general expressions.

Calculating n & pWhat happens when ND>>NA & ND>>ni??

What happens when NA>>ND & NA>>ni??

What happens when ni >> NA & ND??

General Expressions for n and pTemperature Dependence of n (or p)From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)

Temperature dependence of n and p

Note:Derivations will be based on equations derived previously that relate n, p, and EF; given one of these three variables one can calculate the others.

(1)Intrinsic semiconductorsND=NA=0n=p=ni

Note:The Fermi Level for intrinsic semiconductors is located near mid gap!

.. also Note:The fermi level is exactly at mid-gap when the effective masses for electrons and holes are equal or when the temperature is zero!Fermi Level (Energy)(2)Doped Semiconductors: (assuming complete ionization and non-degenerate semiconductors)For p-typeUsing

and

For n-typeFermi Level (Energy)Drift:charged particle motion under an applied E-field.

Inside a semiconductor: Under an applied electric field both the electrons and holes experience a force (qE) and move in opposite directions. The carrier motion is interrupted due to collisions with lattice atoms and ionized impurities (scattering).

Note: the carriers are always moving - even in the absence of an electric field!: thermal motion; however, thermal motion is completely random and it averages out to zero. An electron does not necessarily return to its original position, but examining a group of electrons the net effect of thermal motion is zero; on a macroscopic scale thermal motion can be neglected.Drift Velocity:On a macroscopic scale the average carrier motion can be described by a constant velocity: vdwhere is the mobility of the carrier

DriftMOBILITY: is a constant that describes the ease by which carriers can move within a material. It is a property of semiconductors often used to describe the quality of the materials.Typical RT values:for Si with NA=ND=1014/cm3n=1360 cm2/V-sec p=460 cm2/V-secfor uncompensated (high purity) GaAs with ND or NA po

under low level injection the majority carriers are essentially unchanged but the minority carriers increase by several orders of magnitude.Well assume Low Level Injection:n-type material: p