1. EE 5340 Semiconductor Device Theory Lecture 6 - Fall 2010 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc

2. L06 10Sep10 Second Assignment Please print and bring to class a signed copy of the document appearing at http://www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf 2

3. L06 10Sep103 Net intrinsic mobility Considering only lattice scattering

4. L06 10Sep104 Lattice mobility The  lattice is the lattice scattering mobility due to thermal vibrations Simple theory gives  lattice ~ T -3/2 Experimentally  n,lattice ~ T -n where n = 2.42 for electrons and 2.2 for holes Consequently, the model equation is  lattice (T) =  lattice (300)(T/300) -n

5. L06 10Sep105 Net extrinsic mobility Considering only lattice and impurity scattering

6. L06 10Sep106 Net silicon extr resistivity (cont.) Since  = (nq  n + pq  p ) -1, and  n >  p, (  = q  /m*) we have  p >  n Note that since 1.6(high conc.) <  p /  n < 3(low conc.), so 1.6(high conc.) <  n /  p < 3(low conc.)

7. L06 10Sep107 Ionized impurity mobility function The  impur is the scattering mobility due to ionized impurities Simple theory gives  impur ~ T 3/2 /N impur Consequently, the model equation is  impur (T) =  impur (300)(T/300) 3/2

8. L06 10Sep108 Figure 1.17 (p. 32 in M&K 1 ) Low-field mobility in silicon as a function of temperature for electrons (a), and for holes (b). The solid lines represent the theoretical predictions for pure lattice scattering [5].

9. Figure 1.16 (p. 31 M&K) Electron and hole mobilities in silicon at 300 K as functions of the total dopant concentration. The values plotted are the results of curve fitting measurements from several sources. The mobility curves can be generated using Equation 1.2.10 with the following values of the parameters [3] (see table on next slide). L06 10Sep109

10. 10 Exp. mobility model function for Si 1 ParameterAsPB  min 52.268.544.9  max 14171414470.5 N ref 9.68e169.20e162.23e17  0.6800.7110.719

11. L06 10Sep1011 Carrier mobility functions (cont.) The parameter  max models 1/  lattice the thermal collision rate The parameters  min, N ref and  model 1/  impur the impurity collision rate The function is approximately of the ideal theoretical form: 1/  total = 1/  thermal + 1/  impurity

12. L06 10Sep1012 Carrier mobility functions (ex.) Let N d = 1.78E17/cm3 of phosphorous, so  min = 68.5,  max = 1414, N ref = 9.20e16 and  = 0.711. –Thus  n = 586 cm2/V-s Let N a = 5.62E17/cm3 of boron, so  min = 44.9,  max = 470.5, N ref = 9.68e16 and  = 0.680. –Thus  p = 189 cm2/V-s

13. L06 10Sep1013 Drift Current The drift current density (amp/cm 2 ) is given by the point form of Ohm Law J = (nq  n +pq  p )(E x i+ E y j+ E z k), so J = (  n +  p )E =  E, where  = nq  n +pq  p defines the conductivity The net current is

14. L06 10Sep1014 Drift current resistance Given: a semiconductor resistor with length, l, and cross-section, A. What is the resistance? As stated previously, the conductivity,  = nq  n + pq  p So the resistivity,  = 1/  = 1/(nq  n + pq  p )

15. L06 10Sep1015 Drift current resistance (cont.) Consequently, since R =  l/A R = (nq  n + pq  p ) -1 (l/A) For n >> p, (an n-type extrinsic s/c) R = l/(nq  n A) For p >> n, (a p-type extrinsic s/c) R = l/(pq  p A)

16. L06 10Sep1016 Drift current resistance (cont.) Note: for an extrinsic semiconductor and multiple scattering mechanisms, since R = l/(nq  n A) or l/(pq  p A), and (  n or p total ) -1 =   i -1, then R total =  R i (series Rs) The individual scattering mechanisms are: Lattice, ionized impurity, etc.

17. L06 10Sep1017 Net silicon (ex- trinsic) resistivity Since  =  -1 = (nq  n + pq  p ) -1 The net conductivity can be obtained by using the model equation for the mobilities as functions of doping concentrations. The model function gives agreement with the measured  (N impur )

18. L06 10Sep1018 Net silicon extr resistivity (cont.) Since  = (nq  n + pq  p ) -1, and  n >  p, (  = q  /m*) we have  p >  n, for the same N I Note that since 1.6(high conc.) <  p /  n < 3(low conc.), so 1.6(high conc.) <  n /  p < 3(low conc.)

19. Figure 1.15 (p. 29) M&K Dopant density versus resistivity at 23°C (296 K) for silicon doped with phosphorus and with boron. The curves can be used with little error to represent conditions at 300 K. [W. R. Thurber, R. L. Mattis, and Y. M. Liu, National Bureau of Standards Special Publication 400–64, 42 (May 1981).] L06 10Sep1019

20. L 06 Sept 1020 Net silicon (com- pensated) res. For an n-type (n >> p) compensated semiconductor,  = (nq  n ) -1 But now n = N  N d - N a, and the mobility must be considered to be determined by the total ionized impurity scattering N d + N a  N I Consequently, a good estimate is  = (nq  n ) -1 = [Nq  n (N I )] -1

21. Figure 1.16 (p. 31 M&K) Electron and hole mobilities in silicon at 300 K as functions of the total dopant concentration. The values plotted are the results of curve fitting measurements from several sources. The mobility curves can be generated using Equation 1.2.10 with the following values of the parameters [3] (see table on next slide). L 06 Sept 1021

22. L 06 Sept 1022 Approximate  func- tion for extrinsic, compensated n-Si 1 Param.AsP  min 52.268.5  max 14171414 N ref 9.68e169.20e16  0.6800.711  N d >  N a  n-type n o =  N d -  N a = N  = n o q  n N I =  N d +  N a o N As > N P  As param o N P > N As  P param p o = n i 2 /n o

23. L 06 Sept 1023 Approximate  func- tion for extrinsic, compensated p-Si 1 ParameterB  min 44.9  max 470.5 N ref 2.23e17  0.719  N a >  N d  p-type p o =  N a -  N d = |N|  = p o q  p N I =  N d +  N a o N a = N B  B par n o = n i 2 /p o

24. L 06 Sept 1024 Summary The concept of mobility introduced as a response function to the electric field in establishing a drift current Resistivity and conductivity defined  (N d,N a,T) model equation developed Resistivity models developed for extrinsic and compensated materials

25. L06 10Sep1025 Equipartition theorem The thermodynamic energy per degree of freedom is kT/2 Consequently,

26. L06 10Sep1026 Carrier velocity saturation 1 The mobility relationship v =  E is limited to “low” fields v < v th = (3kT/m*) 1/2 defines “low” v =  o E[1+(E/E c )  ] -1/ ,  o = v 1 /E c for Si parameter electrons holes v 1 (cm/s) 1.53E9 T -0.87 1.62E8 T -0.52 E c (V/cm) 1.01 T 1.55 1.24 T 1.68  2.57E-2 T 0.66 0.46 T 0.17

27. L06 10Sep1027 Carrier velocity 2 carrier velocity vs E for Si, Ge, and GaAs (after Sze 2 )

28. L06 10Sep1028 Carrier velocity saturation (cont.) At 300K, for electrons,  o = v 1 /E c = 1.53E9(300) -0.87 /1.01(300) 1.55 = 1504 cm 2 /V-s, the low-field mobility The maximum velocity (300K) is v sat =  o E c = v 1 = 1.53E9 (300) -0.87 = 1.07E7 cm/s

29. L06 10Sep1029 Diffusion of Carriers (cont.)

30. L06 10Sep1030 Diffusion of carriers In a gradient of electrons or holes,  p and  n are not zero Diffusion current,  J =  J p +  J n (note D p and D n are diffusion coefficients)

31. L06 10Sep1031 Diffusion of carriers (cont.) Note (  p) x has the magnitude of dp/dx and points in the direction of increasing p (uphill) The diffusion current points in the direction of decreasing p or n (downhill) and hence the - sign in the definition of  J p and the + sign in the definition of  J n

32. L06 10Sep1032 Current density components

33. L06 10Sep1033 Total current density

34. L06 10Sep1034 Doping gradient induced E-field If N = N d -N a = N(x), then so is E f -E fi Define  = (E f -E fi )/q = (kT/q)ln(n o /n i ) For equilibrium, E fi = constant, but for dN/dx not equal to zero, E x = -d  /dx =- [d(E f -E fi )/dx](kT/q) = -(kT/q) d[ln(n o /n i )]/dx = -(kT/q) (1/n o )[dn o /dx] = -(kT/q) (1/N)[dN/dx], N > 0

35. L06 10Sep1035 Induced E-field (continued) Let V t = kT/q, then since n o p o = n i 2 gives n o /n i = n i /p o E x = - V t d[ln(n o /n i )]/dx = - V t d[ln(n i /p o )]/dx = - V t d[ln(n i /|N|)]/dx, N = -N a < 0 E x = - V t (-1/p o )dp o /dx = V t (1/p o )dp o /dx = V t (1/N a )dN a /dx

36. L06 10Sep1036 The Einstein relationship For E x = - V t (1/n o )dn o /dx, and J n,x = nq  n E x + qD n (dn/dx) = 0 This requires that nq  n [V t (1/n)dn/dx] = qD n (dn/dx) Which is satisfied if

37. L06 10Sep10 References *Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, 1989. ** and 3 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. 37

38. L06 10Sep1038 References M&K and 1 Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986. –See Semiconductor Device Fundamen- tals, by Pierret, Addison-Wesley, 1996, for another treatment of the  model. 2 Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981.