1. Semiconductor Device Modeling and Characterization – EE5342 Lecture 4 – Spring 2011 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/

2. ©rlc L04 26Jan20112 First Assignment e-mail to listserv@listserv.uta.edu –In the body of the message include subscribe EE5342 This will subscribe you to the EE5342 list. Will receive all EE5342 messages If you have any questions, send to ronc@uta.edu, with EE5342 in subject line.

3. ©rlc L04 26Jan20113 Second Assignment Submit a signed copy of the document that is posted at www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf

4. ©rlc L04 26Jan2011 Semiconductor Electronics - concepts thus far Conduction and Valence states due to symmetry of lattice “Free-elec.” dynamics near band edge Band Gap –direct or indirect –effective mass in curvature Thermal carrier generation Chemical carrier gen (donors/accept) 4

5. ©rlc L04 26Jan2011 Counting carriers - quantum density of states function 1 dim electron wave #s range for n+1 “atoms” is 2  /L < k < 2  /a where a is “interatomic” distance and L = na is the length of the assembly (k = 2  / ) Shorter s, would “oversample” if n increases by 1, dp is h/L Extn 3D: E = p 2 /2m = h 2 k 2 /2m so a vol of p-space of 4  p 2 dp has h 3 /L x L y L z 5

6. ©rlc L04 26Jan2011 QM density of states (cont.) So density of states, g c (E) is (Vol in p-sp)/(Vol per state*V) = 4  p 2 dp/[(h 3 /L x L y L z )*V] Noting that p 2 = 2mE, this becomes g c (E) = {4  2m n *) 3/2 /h 3 }(E-E c ) 1/2 and E - E c = h 2 k 2 /2m n * Similar for the hole states where E v - E = h 2 k 2 /2m p * 6

7. ©rlc L04 26Jan2011 Fermi-Dirac distribution fctn The probability of an electron having an energy, E, is given by the F-D distr f F (E) = {1+exp[(E-E F )/kT]} -1 Note: f F (E F ) = 1/2 E F is the equilibrium energy of the system The sum of the hole probability and the electron probability is 1 7

8. ©rlc L04 26Jan2011 Fermi-Dirac DF (continued) So the probability of a hole having energy E is 1 - f F (E) At T = 0 K, f F (E) becomes a step function and 0 probability of E > E F At T >> 0 K, there is a finite probability of E >> E F 8

9. ©rlc L04 26Jan2011 Maxwell-Boltzman Approximation f F (E) = {1+exp[(E-E F )/kT]} -1 For E - E F > 3 kT, the exp > 20, so within a 5% error, f F (E) ~ exp[-(E-E F )/kT] This is the MB distribution function MB used when E-E F >75 meV (T=300K) For electrons when E c - E F > 75 meV and for holes when E F - E v > 75 meV 9

10. ©rlc L04 26Jan2011 Electron Conc. in the MB approx. Assuming the MB approx., the equilibrium electron concentration is 10

11. ©rlc L04 26Jan2011 Electron and Hole Conc in MB approx Similarly, the equilibrium hole concentration is p o = N v exp[-(E F -E v )/kT] So that n o p o = N c N v exp[-E g /kT] n i 2 = n o p o, N c,v = 2{2  m* n,p kT/h 2 } 3/2 N c = 2.8E19/cm3, N v = 1.04E19/cm3 and n i = 1E10/cm3 11

12. ©rlc L04 26Jan2011 Calculating the equilibrium n o The ideal is to calculate the equilibrium electron concentration n o for the FD distribution, where f F (E) = {1+exp[(E-E F )/kT]} -1 g c (E) = [4  2m n *) 3/2 (E-E c ) 1/2 ]/h 3 12

13. ©rlc L04 26Jan2011 Equilibrium con- centration for n o Earlier quoted the MB approximation n o = N c exp[-(E c - E F )/kT],(=N c exp  F ) The exact solution is n o = 2N c F 1/2 (  F )/  1/2 Where F 1/2 (  F ) is the Fermi integral of order 1/2, and  F = (E F - E c )/kT Error in n o,  is smaller than for the DF:  = 31%, 12%, 5% for -  F = 0, 1, 2 13

14. ©rlc L04 26Jan2011 Equilibrium con- centration for p o Earlier quoted the MB approximation p o = N v exp[-(E F - E v )/kT],(=N v exp  ’ F ) The exact solution is p o = 2N v F 1/2 (  ’ F )/  1/2 Note: F 1/2 (  ) = 0.678, (  1/2 /2) = 0.886 Where F 1/2 (  ’ F ) is the Fermi integral of order 1/2, and  ’ F = (E v - E F )/kT Errors are the same as for p o 14

15. ©rlc L04 26Jan2011 Degenerate and nondegenerate cases Bohr-like doping model assumes no interaction between dopant sites If adjacent dopant atoms are within 2 Bohr radii, then orbits overlap This happens when N d ~ N c (E F ~ E c ), or when N a ~ N v (E F ~ E v ) The degenerate semiconductor is defined by E F ~/> E c or E F ~/< E v 15

16. ©rlc L04 26Jan2011 Donor ionization The density of elec trapped at donors is n d = N d /{1+[exp((E d - E F )/kT)/2]} Similar to FD DF except for factor of 2 due to degeneracy (4 for holes) Furthermore n d = N d - N d +, also For a shallow donor, can have E d -E F >> kT AND E c -E F >> kT: Typically E F -E d ~ 2kT 16

17. ©rlc L04 26Jan2011 Donor ionization (continued) Further, if E d - E F > 2kT, then n d ~ 2N d exp[-(E d -E F )/kT],  < 5% If the above is true, E c - E F > 4kT, so n o ~ N c exp[-(E c -E F )/kT],  < 2% Consequently the fraction of un- ionized donors is n d /n o = 2N d exp[(E c -E d )/kT]/N c = 0.4% for N d (P) = 1e16/cm 3 17

18. ©rlc L04 26Jan201118 Classes of semiconductors Intrinsic: n o = p o = n i, since N a &N d << n i =[N c N v exp(E g /kT)] 1/2,(not easy to get) n-type: n o > p o, since N d > N a p-type: n o < p o, since N d < N a Compensated: n o =p o =n i, w/ N a - = N d + > 0 Note: n-type and p-type are usually partially compensated since there are usually some opposite-type dopants

19. ©rlc L04 26Jan201119 References *Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, 1989. **Semiconductor Physics & Devices, by Donald A. Neamen, 2nd ed., Irwin, Chicago. M&K = Device Electronics for Integrated Circuits, 3rd ed., by Richard S. Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons, New York, 2003. 1 Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986. 2 Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981.