Semiconductor Device Modeling and Characterization – EE5342 Lecture 6 – Spring 2011 Professor Ronald L. Carter

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©rlc L06-31Jan20114 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

©rlc L06-31Jan20115 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 )

©rlc L06-31Jan20116 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)

©rlc L06-31Jan20117 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.

©rlc L06-31Jan20118 Exp. mobility model function for Si 1 ParameterAsPB  min  max N ref 9.68e169.20e162.23e17 

©rlc L06-31Jan20119 Exp. mobility model for P, As and B in Si

©rlc L06-31Jan 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

©rlc L06-31Jan 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  n = 189 cm2/V-s

©rlc L06-31Jan 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

©rlc L06-31Jan 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

©rlc L06-31Jan Mobility Summary The concept of mobility introduced as a response function to the electric field in establishing a drift current Resistivity and conductivity defined Model equation def for  (N d,N a,T) Resistivity models developed for extrinsic and compensated materials

©rlc L06-31Jan 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 )

©rlc L06-31Jan Net silicon extr resistivity (cont.)

©rlc L06-31Jan 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.)

©rlc L06-31Jan 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

©rlc L06-31Jan Equipartition theorem The thermodynamic energy per degree of freedom is kT/2 Consequently,

©rlc L06-31Jan 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 E8 T E c (V/cm) 1.01 T T 1.68  2.57E-2 T T 0.17

©rlc L06-31Jan v drift [cm/s] vs. E [V/cm] (Sze 2, fig. 29a)

©rlc L06-31Jan References *Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, **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, Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981.