ECE 340 Lecture 9 : Carrier Concentrations and the ...

1 ECE 340 Lecture 9 : Carrier Concentrations and the Temperature dependence Class Outline: Intrinsic Carrier Concentrations Extrinsic Carrier Concentrations Thermal Effects Revisited What is the charge neutrality relationship? How are the Carrier Concentrations determined for extrinsic material? What role does temperature play in Carrier Concentrations in extrinsic material? What is compensated material? Gilbert ECE 340 Lecture 9 09/17/12 Things you should know when you Key Questions Gilbert ECE 340 Lecture 9 09/17/12 Intrinsic Carrier Concentrations We recall that by using the density of states and the Fermi For electrons: For holes.

2 ()() =topcEEcdEEfEgn()()[] =vbottomEEvdEEfEgp1We can simplify these integrals to find closed form 232*232*2222 = = TkmNTkmNbpvbnc230*,319, =mmcmNpnvc3kbT 3kbT ()()TkEEvTkEEcbfvbcfeNpeNn == Gilbert ECE 340 Lecture 9 09/17/12 Intrinsic Carrier Concentrations Is this the best and easiest?? () deFcc +=0212/11()()32232/13232 ==TkEEFbcfcc ()()TkEEcbcfeF = 2/1 But we can use curve fitting (no physics) to cover all of the ranges: + = ++=vvbvccbcfNpNpTkENnNnTkEE81ln81lnJoyce -Dixon Approximation: Gilbert ECE 340 Lecture 9 09/17/12 Intrinsic Carrier Concentrations Recall that we can also find the dependence on For intrinsic semiconductors, we know the following: n = p = ni and Ei = Ef Then the relations for n and p become.

3 TkEiEviTkEEcibvbcieNneNn ==TkEEivTkEEicbvibicenNenN ==By combining equations, we arrive at two very important and useful results: TkEEiTkEEibfibifenpenn ==TkEvcibgeNNn2 = Gilbert ECE 340 Lecture 9 09/17/12 Extrinsic Carrier Concentrations In most situations, we are dealing with extrinsic Ed Ea Ev Ev x Donor Doped Semiconductor Band Diagram: Acceptor Doped Semiconductor Band Diagram: Gilbert ECE 340 Lecture 9 09/17/12 Extrinsic Carrier Concentrations Let s consider a small separate parts of a uniformly doped + + + - Gilbert ECE 340 Lecture 9 09/17/12 Extrinsic Carrier Concentrations The preceding analysis leads to the charge neutrality 0arg3= + = + = + +ADADNNnpqNqNqnqpcmech0= ADNNnpAADDNNNN== + Gilbert ECE 340 Lecture 9 09/17/12 Extrinsic Carrier Concentrations Let s use the charge neutrality nnpi2=02= + ADiNNnnn()

4 022= iADnNNnn212222 + + =iADADnNNNNn212222 + + = Gilbert ECE 340 Lecture 9 09/17/12 Extrinsic Carrier Concentrations The previous equations are generic, we can simplify in 4 NA = ND = 0 n = p = ni DiDNnpNn2= AiANnnNp2= or n = p = ni Gilbert ECE 340 Lecture 9 09/17/12 Extrinsic Carrier Concentrations What about the 4th case? Compensated Semiconductors + + + + + + + - - - - - - - Sum of positive charges (holes and ionized donors) must balance sum of negative charges (electrons and ionized acceptors): !

5 P0 +Nd+ = n0 +Na- Gilbert ECE 340 Lecture 9 09/17/12 Extrinsic Carrier Concentrations Where is the intrinsic level, Ei, really located? In intrinsic material, we know that n=p. Now use the Boltzmann equations: TkEEvTkEEcbifbfieNeN =But, 23** =npcvmmNNSo, ++=** Gilbert ECE 340 Lecture 9 09/17/12 Extrinsic Carrier Concentrations So where does the Fermi level lie for a doped semiconductor?? Solve for Ef Ei: TkEEiTkEEibfibifenpenn == = = iAbfiiDbifnNTkEEnNTkEElnlnND >> NA , ND >> ni NA >> ND , NA >> ni Gilbert ECE 340 Lecture 9 09/17/12 Extrinsic Carrier Concentrations So where does the Fermi level lie for a doped semiconductor?

6 ? Fermi level positioning in Si at 300 K as a function of the doping Gilbert ECE 340 Lecture 9 09/17/12 Thermal Effects Revisited What is the role of temperature??