1. EE 5340 Semiconductor Device Theory Lecture 8 - Fall 2009 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc

2. L 08 Sept 182 Test 1 – Sept. 24, 2009 8 AM Room 108 Nedderman Hall Open book - 1 legal text or ref., only. You may write notes in your book. Calculator allowed A cover sheet will be included with full instructions. See http://www.uta.edu/ronc/5340/tests/ for examples from previous semesters.

3. L 08 Sept 173 Si and Al and model (approx. to scale) q  m,Al ~ 4.1 eV EoEo E Fm E Fp E Fn EoEo EcEc EvEv E Fi q  s,n q  si ~ 4.05 eV EoEo EcEc EvEv E Fi q  s,p metaln-type s/cp-type s/c q  si ~ 4.05 eV

4. L 08 Sept 174 Making contact be- tween metal & s/c Equate the E F in the metal and s/c materials far from the junction E o (the free level), must be continuous across the jctn. 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)

5. L 08 Sept 175 Equilibrium Boundary Conditions w/ contact No discontinuity in the free level, E o at the metal/semiconductor interface. E F,metal = E F,semiconductor to bring the electron populations in the metal and semiconductor to thermal equilibrium. E o - E C = q  semiconductor in all of the s/c. E o - E F,metal = q  metal throughout metal.

6. L 08 Sept 176 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iqi 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  i =  Bn -  n =  m -  s elect s/c to mtl barr Depl reg

7. L 08 Sept 177 Metal to n-type non-rect cont (  m <  s ) E Fn EoEo EcEc EvEv E Fi q  s,n qsqs n-type s/c qmqm E Fm metal q  B,n qnqn No disc in E o E x =0 in metal ==> E o flat  B,n =  m -  s = elec mtl to s/c barr  i =  Bn -  n < 0 Accumulation region Acc reg qiqi

8. L 08 Sept 178 Ideal metal to p-type barrier diode (  m <  s ) E Fp EoEo EcEc EvEv E Fi q  s,p qsqs p-type s/c qmqm E Fm metal q  Bn qiqi q  p <0 No disc in E o E x =0 in metal ==> E o flat  Bn =  m -  s = elec mtl to s/c barr  Bp =  m -  s + E g = hole m to s  i =  Bp -  s,p = hole s/c to mtl barr Depl reg q  Bp qiqi

9. L 08 Sept 179 Metal to p-type non-rect cont (  m >  s ) No disc in E o E x =0 in metal ==> E o flat  B,n =  m -  s,n = elec mtl to s/c barr  Bp =  m -  s + E g = hole m to s Accumulation region E Fi EoEo EcEc EvEv E fP q  s,n qsqs n-type s/c qmqm E Fm metal q  Bn q(  i ) qpqp Accum reg q  Bp qiqi

10. L 08 Sept 1710 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

11. L 08 Sept 1711 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

12. L 08 Sept 1712 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

13. L 08 Sept 1713 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:

14. L 08 Sept 1714 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

15. L 08 Sept 1715 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

16. L 08 Sept 1716 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

17. L 08 Sept 1717 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

18. L 08 Sept 1718 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

19. L 08 Sept 1719 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

20. L 08 Sept 1720 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”

21. L 08 Sept 1721 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

22. L 08 Sept 1722 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

23. L 08 Sept 1723 Effect of V  0

24. L 08 Sept 1724 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

25. L 08 Sept 1825 Schottky diode capacitance xnxn x qN d -Q-  Q Q’ d = qN d x n x  ExEx -E m xnxn  Q’

26. L 08 Sept 1826 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.

27. L 08 Sept 1827 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,

28. L 08 Sept 1828 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.

29. L 08 Sept 1829 Schottky Capacitance (continued) The C-V relationship simplifies to

30. L 08 Sept 1830 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

31. L 08 Sept 1731 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.