1. Semiconductor Device Modeling and Characterization – EE5342 Lecture 3 – Spring 2011 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/

2. ©rlc L03 24Jan20112 Web Pages *Bring the following to the first class R. L. Carter’s web page –www.uta.edu/ronc/ EE 5342 web page and syllabus –http://www.uta.edu/ronc/5342/syllabus.htm University and College Ethics Policies www.uta.edu/studentaffairs/conduct/ www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf

3. ©rlc L03 24Jan20113 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.

4. ©rlc L03 24Jan20114 Second 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.

5. ©rlc L03 24Jan20115 Schrodinger Equation Separation of variables gives  (x,t) =  (x)  (t) The time-independent part of the Schrodinger equation for a single particle with KE = E and PE = V.

6. ©rlc L03 24Jan20116 K-P Potential Function*

7. ©rlc L03 24Jan20117 K-P Static Wavefunctions Inside the ions, 0 < x < a  (x) = A exp(j  x) + B exp (-j  x)  = [8  2 mE/h] 1/2 Between ions region, a < x < (a + b) = L  (x) = C exp(  x) + D exp (-  x)  = [8  2 m(V o -E)/h 2 ] 1/2

8. ©rlc L03 24Jan20118 K-P Impulse Solution Limiting case of V o -> inf. and b -> 0, while  2 b = 2P/a is finite In this way  2 b 2 = 2Pb/a < 1, giving sinh(  b) ~  b and cosh(  b) ~ 1 The solution is expressed by P sin(  a)/(  a) + cos(  a) = cos(ka) Allowed values of LHS bounded by +1 k = free electron wave # = 2  /

9. ©rlc L03 24Jan20119 K-P Solutions* P sin(  a)/(  a) + cos(  a) vs.  a x x

10. ©rlc L03 24Jan201110 K-P E(k) Relationship*

11. ©rlc L03 24Jan201111 Analogy: a nearly-free X electron model Solutions can be displaced by ka = 2n  Allowed and forbidden energies Infinite well approximation by replacing the free electron mass with an “effective” mass (noting E = p 2 /2m = h 2 k 2 /2m) of

12. ©rlc L03 24Jan201112 Silicon Band Structure** Indirect Bandgap Curvature (hence m*) is function of direction and band. [100] is x-dir, [111] is cube diagonal E g = 1.17-  T 2 /(T+  )  = 4.73E-4 eV/K  = 636K

13. ©rlc L03 24Jan201113 Generalizations and Conclusions The symm. of the crystal struct. gives “allowed” and “forbidden” energies (sim to pass- and stop-band) The curvature at band-edge (where k = (n+1)  ) gives an “effective” mass.

14. ©rlc L03 24Jan201114 Silicon Covalent Bond (2D Repr) Each Si atom has 4 nearest neighbors Si atom: 4 valence elec and 4+ ion core 8 bond sites / atom All bond sites filled Bonding electrons shared 50/50 _ = Bonding electron

15. ©rlc L03 24Jan201115 Si Energy Band Structure at 0 K Every valence site is occupied by an electron No electrons allowed in band gap No electrons with enough energy to populate the conduction band

16. ©rlc L03 24Jan201116 Si Bond Model Above Zero Kelvin Enough therm energy ~kT(k=8.62E-5eV/K) to break some bonds Free electron and broken bond separate One electron for every “hole” (absent electron of broken bond)

17. ©rlc L03 24Jan201117 Band Model for thermal carriers Thermal energy ~kT generates electron-hole pairs At 300K Eg(Si) = 1.124 eV >> kT = 25.86 meV, Nc = 2.8E19/cm3 > Nv = 1.04E19/cm3 >> ni = 1.45E10/cm3

18. ©rlc L03 24Jan201118 Donor: cond. electr. due to phosphorous P atom: 5 valence elec and 5+ ion core 5th valence electr has no avail bond Each extra free el, -q, has one +q ion # P atoms = # free elect, so neutral H atom-like orbits

19. ©rlc L03 24Jan201119 Bohr model H atom- like orbits at donor Electron (-q) rev. around proton (+q) Coulomb force, F=q 2 /4  Si  o,q=1.6E-19 Coul,  Si =11.7,  o =8.854E-14 Fd/cm Quantization L = mvr = nh/2  E n = -(Z 2 m*q 4 )/[8(  o  Si ) 2 h 2 n 2 ] ~-40meV r n = [n 2 (  o  Si )h 2 ]/[Z  m*q 2 ] ~ 2 nm for Z=1, m*~m o /2, n=1, ground state

20. ©rlc L03 24Jan201120 Band Model for donor electrons Ionization energy of donor E i = E c -E d ~ 40 meV Since E c -E d ~ kT, all donors are ionized, so N D ~ n Electron “freeze- out” when kT is too small

21. ©rlc L03 24Jan201121 Acceptor: Hole due to boron B atom: 3 valence elec and 3+ ion core 4th bond site has no avail el (=> hole) Each hole, adds --q, has one -q ion #B atoms = #holes, so neutral H atom-like orbits

22. ©rlc L03 24Jan201122 Hole orbits and acceptor states Similar to free electrons and donor sites, there are hole orbits at acceptor sites The ionization energy of these states is E A - E V ~ 40 meV, so N A ~ p and there is a hole “freeze-out” at low temperatures

23. ©rlc L03 24Jan201123 Impurity Levels in Si: E G = 1,124 meV Phosphorous, P: E C - E D = 44 meV Arsenic, As:E C - E D = 49 meV Boron, B: E A - E V = 45 meV Aluminum, Al: E A - E V = 57 meV Gallium, Ga: E A - E V = 65meV Gold, Au: E A - E V = 584 meV E C - E D = 774 meV

24. ©rlc L03 24Jan2011 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) 24

25. ©rlc L03 24Jan201125 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.