EE 5340 Semiconductor Device Theory Lecture 5 - Fall 2009 Professor Ronald L. Carter

L 05 Sept 08 Second Assignment Please print and bring to class a signed copy of the document appearing at 2

L 05 Sept 083 Classes of semiconductors Intrinsic: n o = p o = n i, since N a &N d << n i, n i 2 = N c N v e -Eg/kT, ~1E-13 dopant level ! 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

L 05 Sept 084 n-type equilibrium concentrations N ≡ N d - N a, n type  N > 0 For all N, n o = N/2 + {[N/2] 2 +n i 2 } 1/2 In most cases, N >> n i, so n o = N, and p o = n i 2 /n o = n i 2 /N, (Law of Mass Action is al- ways true in equilibrium)

L 05 Sept 085 p-type equilibrium concentrations N ≡ N d - N a, p type  N < 0 For all N, p o = |N|/2 + {[|N|/2] 2 +n i 2 } 1/2 In most cases, |N| >> n i, so p o = |N|, and n o = n i 2 /p o = n i 2 /|N|, (Law of Mass Action is al- ways true in equilibrium)

L 05 Sept 086 Intrinsic carrier conc. (MB limit) n i 2 = n o p o = N c N v e -Eg/kT N c = 2{2  m* n kT/h 2 } 3/2 N v = 2{2  m* p kT/h 2 } 3/2 E g = 1.17 eV -  T 2 /(T+  )  = 4.73E-4 eV/K  = 636K

L 05 Sept 087 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

L 05 Sept 088 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 )

L 05 Sept 089 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)

L 05 Sept 0810 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.

L 05 Sept 0811 Net intrinsic mobility Considering only lattice scattering

L 05 Sept 0812 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

L 05 Sept 0813 Net extrinsic mobility Considering only lattice and impurity scattering

L 05 Sept 0814 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.)

L 05 Sept 0815 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

L 05 Sept 0816 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].

L 05 Sept 0817 Exp.  (T=300K) model for P, As and B in Si 1

L 05 Sept 0818 Exp. mobility model function for Si 1 ParameterAsPB  min  max N ref 9.68e169.20e162.23e17 

L 05 Sept 0819 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

L 05 Sept 0820 Carrier mobility functions (ex.) Let N d = 1.78E17/cm3 of phosphorous, so  min = 68.5,  max = 1414, N ref = 9.20e16 and  = –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  = –Thus  p = 189 cm2/V-s

L 05 Sept 0821 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 )

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).] L 05 Sept 0822

L 05 Sept 0823 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.)

L 05 Sept 0824 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

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 with the following values of the parameters [3] (see table on next slide). L 04 Sept 0425

L 05 Sept 0826 References 1 Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, –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.