1. EE 5340 Semiconductor Device Theory Lecture 11 – Spring 2011 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc

2. ©rlc L11-24Feb20112 Metal/semiconductor system types n-type semiconductor Schottky diode - blocking for  m >  s contact - conducting for  m <  s p-type semiconductor contact - conducting for  m >  s Schottky diode - blocking for  m <  s

3. ©rlc L11-24Feb20113 Real Schottky band structure 1 Barrier transistion region,  Interface states above  o acc, p neutrl below  o dnr, n neutrl D it  -> oo, q  Bn  = E g -  o Fermi level “pinned” D it  -> 0, q  Bn  =  m -  Goes to “ideal” case

4. ©rlc L11-24Feb20114 Fig 8.4 1 (a) Image charge and electric field at a metal-dielectric interface (b) Distortion of potential barrier at E=0 and (c) E  0

5. ©rlc L11-24Feb20115 Poisson’s Equation The electric field at (x,y,z) is related to the charge density  =q(Nd-Na-p-n) by the Poisson Equation:

6. ©rlc L11-24Feb20116 Poisson’s Equation n = n o +  n, and p = p o +  p, in non-equil For n-type material, N = (N d - N a ) > 0, n o = N, and (N d -N a +p-n)=-  n +  p +n i 2 /N For p-type material, N = (N d - N a ) < 0, p o = -N, and (N d -N a +p-n) =  p-  n-n i 2 /N So neglecting n i 2 /N

7. ©rlc L11-24Feb20117 Ideal metal to n-type barrier diode (  m >  s,V a =0) E Fn EoEo EcEc EvEv E Fi q  s,n qsqs n-type s/c qmqm E Fm metal q  Bn q  bi q’nq’n No disc in E o E x =0 in metal ==> E o flat  Bn =  m -  s = elec mtl to s/c barr  bi =  Bn -  n =  m -  s elect s/c to mtl barr Depl reg 0 x n x nc

8. ©rlc L11-24Feb20118 Depletion Approximation For 0 < x < x n, assume n << n o = N d, so  = q(N d -N a +p-n) = qN d For x n < x < x nc, assume n = n o = N d, so  = q(N d -N a +p-n) = 0 For x = 0 -, there is a pulse of charge balancing the qN d x n in 0 < x < x n

9. ©rlc L11-24Feb20119 Ideal n-type Schottky depletion width (V a =0) xnxn x qN d  Q’ d = qN d x n x  ExEx -E m xnxn (Sheet of negative charge on metal)= -Q’ d

10. ©rlc L11-24Feb201110 Debye length The DA assumes n changes from N d to 0 discontinuously at x n. In the region of x n, Poisson’s eq is  E =  /  --> dE x /dx = q(N d - n), and since E x = -d  /dx, we have -d 2  /dx 2 = q(N d - n)/  to be solved n x xnxn NdNd 0

11. ©rlc L11-24Feb201111 Debye length (cont) Since the level E Fi is a reference for equil, we set  = V t ln(n/n i ) In the region of x n, n = n i exp(  /V t ), so d 2  /dx 2 = -q(N d - n i e  /Vt ), let  =  o +  ’, where  o = V t ln(N d /n i ) soN d - n i e  /Vt = N d [1 - e  /Vt-  o/Vt ], for  -  o =  ’ <<  o, the DE becomes d 2  ’/dx 2 = (q 2 N d /  kT)  ’,  ’ <<  o

12. ©rlc L11-24Feb201112 Debye length (cont) So  ’ =  ’(x n ) exp[+(x-x n )/L D ]+con. and n = N d e  ’/Vt, x ~ x n, where L D is the “Debye length”

13. ©rlc L11-24Feb201113 Debye length (cont) L D estimates the transition length of a step-junction DR. Thus, For V a = 0,  i ~ 1V, V t ~ 25 mV  < 11%  DA assumption OK

14. ©rlc L11-24Feb201114 Effect of V  0 Define an external voltage source, V a, with the +term at the metal contact and the -term at the n-type contact For V a > 0, the V a induced field tends to oppose E x caused by the DR For V a < 0, the V a induced field tends to aid E x due to DR Will consider V a < 0 now

15. ©rlc L11-24Feb201115 Effect of V  0

16. ©rlc L11-24Feb201116 Ideal metal to n-type Schottky (V a > 0) qV a = E fn - E fm Barrier for electrons from sc to m reduced to q(  bi -V a ) q  Bn the same DR decr E Fn EoEo EcEc EvEv E Fi q  s,n qsqs n-type s/c qmqm E Fm metal q  Bn q(  i -V a ) q’nq’n Depl reg

17. ©rlc L11-24Feb201117 Schottky diode capacitance xnxn x qN d -Q-  Q Q’ d = qN d x n x  ExEx -E m xnxn  Q’

18. ©rlc L11-24Feb201118 Schottky Capacitance (continued) The junction has +Q’ n =qN d x n (exposed donors), and Q’ n = - Q’ metal (Coul/cm 2 ), forming a parallel sheet charge capacitor.

19. ©rlc L11-24Feb201119 Schottky Capacitance (continued) This Q ~ (  i -V a ) 1/2 is clearly non- linear, and Q is not zero at V a = 0. Redefining the capacitance,

20. ©rlc L11-24Feb201120 Schottky Capacitance (continued) So this definition of the capacitance gives a parallel plate capacitor with charges  Q’ n and  Q’ p (=-  Q’ n ), separated by, L (=x n ), with an area A and the capacitance is then the ideal parallel plate capacitance. Still non-linear and Q is not zero at V a =0.

21. ©rlc L11-24Feb201121 Schottky Capacitance (continued) The C-V relationship simplifies to

22. ©rlc L11-24Feb201122 Schottky Capacitance (continued) If one plots [C j ] -2 vs. V a Slope = -[(C j0 ) 2 V bi ] -1 vertical axis intercept = [C j0 ] -2 horizontal axis intercept =  i C j -2 ii VaVa C j0 -2

23. Diagrams for ideal metal-semiconductor Schottky diodes. Fig. 3.21 in Ref 4. ©rlc L11-24Feb201123

24. ©rlc L11-24Feb2011 Energy bands for p- and n-type s/c p-type EcEc EvEv E Fi E FP q  P = kT ln(n i /N a ) EvEv EcEc E Fi E FN q  n = kT ln(N d /n i ) n-type 24

25. ©rlc L11-24Feb2011 Making contact in a p-n junction Equate the E F in the p- and n-type materials far from the junction E o (the free level), E c, E fi and E v must be continuous N.B.: q  = 4.05 eV (Si), and q  = q   E c - E F EoEo EcEc EFEF E Fi EvEv q  (electron affinity) qFqF qq (work function) 25

26. ©rlc L11-24Feb2011 Band diagram for p + -n jctn* at V a = 0 EcEc E FN E Fi EvEv EcEc E FP E Fi EvEv 0 xnxn x -x p -x pc x nc q  p < 0 q  n > 0 qV bi = q(  n -  p ) *N a > N d -> |  p | >  n p-type for x<0 n-type for x>0 26

27. ©rlc L11-24Feb2011 A total band bending of qV bi = q(  n -  p ) = kT ln(N d N a /n i 2 ) is necessary to set E Fp = E fn For -x p < x < 0, E fi - E FP < -q  p, = |q  p | so p < N a = p o, (depleted of maj. carr.) For 0 < x < x n, E FN - E Fi < q  n, so n < N d = n o, (depleted of maj. carr.) -x p < x < x n is the Depletion Region Band diagram for p + -n at V a =0 (cont.) 27

28. ©rlc L11-24Feb2011 Depletion Approximation Assume p << p o = N a for -x p < x < 0, so  = q(N d -N a +p-n) = -qN a, -x p < x < 0, and p = p o = N a for -x pc < x < -x p, so  = q(N d -N a +p-n) = 0, -x pc < x < -x p Assume n << n o = N d for 0 < x < x n, so  = q(N d -N a +p-n) = qN d, 0 < x < x n, and n = n o = N d for x n < x < x nc, so  = q(N d -N a +p-n) = 0, x n < x < x nc 28

29. ©rlc L11-24Feb2011 Depletion approx. charge distribution xnxn x -x p -x pc x nc  +qN d -qN a +Q n ’=qN d x n Q p ’=-qN a x p Due to Charge neutrality Q p ’ + Q n ’ = 0, => N a x p = N d x n [Coul/cm 2 ] 29

30. ©rlc L11-24Feb2011 Induced E-field in the D.R. The sheet dipole of charge, due to Q p ’ and Q n ’ induces an electric field which must satisfy the conditions Charge neutrality and Gauss’ Law* require thatE x = 0 for -x pc < x < -x p and E x = 0 for -x n < x < x nc ≈0≈0 30

31. ©rlc L11-24Feb2011 Induced E-field in the D.R. xnxn x -x p -x pc x nc O - O - O - O + O + O + Depletion region (DR) p-type CNR ExEx Exposed Donor ions Exposed Acceptor Ions n-type chg neutral reg p-contact N-contact W 0 31

32. ©rlc L11-24Feb2011 Induced E-field in the D.R. (cont.) Poisson’s Equation  E =  / , has the one-dimensional form, dE x /dx =  / , which must be satisfied for  = -qN a, -x p < x < 0, and  = +qN d, 0 < x < x n, with E x = 0 for the remaining range 32

33. ©rlc L11-24Feb2011 Soln to Poisson’s Eq in the D.R. xnxn x -x p -x pc x nc ExEx -E max 33

34. ©rlc L11-24Feb2011 Soln to Poisson’s Eq in the D.R. (cont.) 34

35. ©rlc L11-24Feb2011 Soln to Poisson’s Eq in the D.R. (cont.) 35

36. ©rlc L11-24Feb2011 Comments on the E x and V bi V bi is not measurable externally since E x is zero at both contacts The effect of E x does not extend beyond the depletion region The lever rule [N a x p =N d x n ] was obtained assuming charge neutrality. It could also be obtained by requiring E x (x=0  x  E x (x=0  x)  E max 36

37. ©rlc L11-24Feb2011 Sample calculations V t  25.86 mV at 300K  =  r  o = 11.7*8.85E-14 Fd/cm = 1.035E-12 Fd/cm If N a  5E17/cm 3, and N d  2E15 /cm 3, then for n i  1.4E10/cm 3, then what is V bi = 757 mV 37

38. ©rlc L11-24Feb2011 Sample calculations What are N eff, W ? N eff, = 1.97E15/cm 3 W = 0.707 micron What is x n ? = 0.704 micron What is E max ? 2.14E4 V/cm 38

39. ©rlc L11-24Feb201139 References 1 Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986. See Semiconductor Device Fundamentals, by Pierret, Addison- Wesley, 1996, for another treatment of the  model. 2 Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981. 3 Semiconductor Physics & Devices, 2nd ed., by Neamen, Irwin, Chicago, 1997. 4 Device Electronics for Integrated Circuits, 3/E by Richard S. Muller and Theodore I. Kamins. © 2003 John Wiley & Sons. Inc., New York.

40. ©rlc L11-24Feb201140 References 1 and M&K Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986. See Semiconductor Device Fundamentals, by Pierret, Addison-Wesley, 1996, for another treatment of the  model. 2 Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981. 3 and ** Semiconductor Physics & Devices, 2nd ed., by Neamen, Irwin, Chicago, 1997. Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, 1989.