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.