1. 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

2. First Assignment Send e-mail to ronc@uta.edu
On the subject line, put “5342 ”
In the body of message include
Your address
Your Name as it appears in the UTA Record - no more, no less
Last four digits of your Student ID: _____
The name you would like me to use when speaking to you.
L04 January 27

3. Second Assignment e-mail to listserv@listserv.uta.edu
In the body of the message include subscribe EE5342
This will subscribe you to the EE5342 list. Will receive all EE5342 messages
If you have any questions, send to with EE5342 in subject line.
L04 January 27

4. 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

5. 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

6. 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

7. 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

8. Exp. mobility model function for Si1
Parameter As P B
mmin
mmax
Nref e e e17 a
L04 January 27

9. 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 previous slide). Figure (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

10. 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

11. Carrier mobility functions (ex.)
Let Nd = 1.78E17/cm3 of phosphorous, so mmin = 68.5, mmax = 1414, Nref = 9.20e16 and a = 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 = Thus mn = 189 cm2/V-s
L04 January 27

12. 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

13. 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

14. 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

15. 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

16. 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 (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

17. 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

18. 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

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

20. 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) E9 T E8 T-0.52
Ec (V/cm) T T1.68
b E-2 T T0.17
L04 January 27

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

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

23. 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

24. 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

25. Diffusion of Carriers (cont.)
L04 January 27

26. Current density components
L04 January 27

27. Total current density
L04 January 27

28. 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

29. 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

30. 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

31. 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