Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/ EE5342 – Semiconductor Device Modeling and Characterization Lecture 04-Spring 2010 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/

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Drift Current The drift current density (A/cm2) is given by the point form of Ohm’s Law J = (nqmn+pqmp)(Exi+ Eyj+ Ezk), so J = (sn + sp)E = sE, where = nqmn+pqmp defines the conductivity & 1/s = r defines the resistivity The net current is L04 January 27

Drift current resistance Given: a semiconductor resistor with length, l, and cross-section, A. What is the resistance? As stated previously, the conductivity, s = nqmn + pqmp So the resistivity, r = 1/s = 1/(nqmn + pqmp) L04 January 27

Drift current resistance (cont.) Consequently, since R = rl/A R = (nqmn + pqmp)-1(l/A) For n >> p, (an n-type extrinsic s/c) R = l/(nqmnA) For p >> n, (a p-type extrinsic s/c) R = l/(pqmpA) L04 January 27

Drift current resistance (cont.) Note: for an extrinsic semiconductor and multiple scattering mechanisms, since R = l/(nqmnA) or l/(pqmpA), and (mn or p total)-1 = S mi-1, then Rtotal = S Ri (series Rs) The individual scattering mechanisms are: Lattice, ionized impurity, etc. L04 January 27

Exp. mobility model function for Si1 Parameter As P B mmin 52.2 68.5 44.9 mmax 1417 1414 470.5 Nref 9.68e16 9.20e16 2.23e17 a 0.680 0.711 0.719 L04 January 27

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 previous slide). Figure 1.16 (p. 31) Device Electronics for Integrated Circuits, 3/E by Richard S. Muller and Theodore I. Kamins Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. L04 January 27

Carrier mobility functions (cont.) The parameter mmax models 1/tlattice the thermal collision rate The parameters mmin, Nref and a model 1/timpur the impurity collision rate The function is approximately of the ideal theoretical form: 1/mtotal = 1/mthermal + 1/mimpurity L04 January 27

Carrier mobility functions (ex.) Let Nd = 1.78E17/cm3 of phosphorous, so mmin = 68.5, mmax = 1414, Nref = 9.20e16 and a = 0.711. Thus mn = 586 cm2/V-s Let Na = 5.62E17/cm3 of boron, so mmin = 44.9, mmax = 470.5, Nref = 9.68e16 and a = 0.680. Thus mn = 189 cm2/V-s L04 January 27

Lattice mobility The mlattice is the lattice scattering mobility due to thermal vibrations Simple theory gives mlattice ~ T-3/2 Experimentally mn,lattice ~ T-n where n = 2.42 for electrons and 2.2 for holes Consequently, the model equation is mlattice(T) = mlattice(300)(T/300)-n L04 January 27

Ionized impurity mobility function The mimpur is the scattering mobility due to ionized impurities Simple theory gives mimpur ~ T3/2/Nimpur Consequently, the model equation is mimpur(T) = mimpur(300)(T/300)3/2 L04 January 27

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 m(Nd,Na,T) Resistivity models developed for extrinsic and compensated materials L04 January 27

Net silicon (ex- trinsic) resistivity Since r = s-1 = (nqmn + pqmp)-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 s(Nimpur) L04 January 27

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).] Figure 1.15 (p. 29) Device Electronics for Integrated Circuits, 3/E by Richard S. Muller and Theodore I. Kamins Copyright © 2003 John Wiley & Sons. Inc. All rights reserved. L04 January 27

Net silicon extr resistivity (cont.) Since r = (nqmn + pqmp)-1, and mn > mp, (m = qt/m*) we have rp > rn Note that since 1.6(high conc.) < rp/rn < 3(low conc.), so 1.6(high conc.) < mn/mp < 3(low conc.) L04 January 27

Net silicon (com- pensated) res. For an n-type (n >> p) compensated semiconductor, r = (nqmn)-1 But now n = N = Nd - Na, and the mobility must be considered to be determined by the total ionized impurity scattering Nd + Na = NI Consequently, a good estimate is r = (nqmn)-1 = [Nqmn(NI)]-1 L04 January 27

Equipartition theorem The thermodynamic energy per degree of freedom is kT/2 Consequently, L04 January 27

Carrier velocity saturation1 The mobility relationship v = mE is limited to “low” fields v < vth = (3kT/m*)1/2 defines “low” v = moE[1+(E/Ec)b]-1/b, mo = v1/Ec for Si parameter electrons holes v1 (cm/s) 1.53E9 T-0.87 1.62E8 T-0.52 Ec (V/cm) 1.01 T1.55 1.24 T1.68 b 2.57E-2 T0.66 0.46 T0.17 L04 January 27

vdrift [cm/s] vs. E [V/cm] (Sze2, fig. 29a) L04 January 27

Carrier velocity saturation (cont.) At 300K, for electrons, mo = v1/Ec = 1.53E9(300)-0.87/1.01(300)1.55 = 1504 cm2/V-s, the low-field mobility The maximum velocity (300K) is vsat = moEc = v1 = 1.53E9 (300)-0.87 = 1.07E7 cm/s L04 January 27

Diffusion of carriers In a gradient of electrons or holes, p and n are not zero Diffusion current,`J =`Jp +`Jn (note Dp and Dn are diffusion coefficients) L04 January 27

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`Jp and the + sign in the definition of`Jn L04 January 27

Diffusion of Carriers (cont.) L04 January 27

Current density components L04 January 27

Total current density L04 January 27

Doping gradient induced E-field If N = Nd-Na = N(x), then so is Ef-Efi Define f = (Ef-Efi)/q = (kT/q)ln(no/ni) For equilibrium, Efi = constant, but for dN/dx not equal to zero, Ex = -df/dx =- [d(Ef-Efi)/dx](kT/q) = -(kT/q) d[ln(no/ni)]/dx = -(kT/q) (1/no)[dno/dx] = -(kT/q) (1/N)[dN/dx], N > 0 L04 January 27

Induced E-field (continued) Let Vt = kT/q, then since nopo = ni2 gives no/ni = ni/po Ex = - Vt d[ln(no/ni)]/dx = - Vt d[ln(ni/po)]/dx = - Vt d[ln(ni/|N|)]/dx, N = -Na < 0 Ex = - Vt (-1/po)dpo/dx = Vt(1/po)dpo/dx = Vt(1/Na)dNa/dx L04 January 27

The Einstein relationship For Ex = - Vt (1/no)dno/dx, and Jn,x = nqmnEx + qDn(dn/dx) = 0 This requires that nqmn[Vt (1/n)dn/dx] = qDn(dn/dx) Which is satisfied if L04 January 27

References *Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, 1989. **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. 1Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986. 2Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981. L04 January 27