L06 31Jan021 Semiconductor Device Modeling and Characterization EE5342, Lecture 6-Spring 2002 Professor Ronald L. Carter

L06 31Jan022 General Instructions: All projects should be submitted on 8.5" x 11" paper with a cover sheet attached, or electronically as a single document file which will print as such. If submitted as a paper project report, it should be stapled only in the upper left-hand corner and no other cover or binder or folder should be used.

L06 31Jan023 Format The cover sheet –your name, –the project title, –the course name and number, and –your address. The report includes – purpose of the project and the theoretical background, –a narrative explaining how you did the project, –answers to all questions asked in the project assignment, and a –list of references used in the order cited in the report (the reference number should appear in the report each time the reference is used).

L06 31Jan024 All figures and tables should be clearly marked with a figure or table number and caption. The caption and labels on the figures should make the information in the figure comprehensible without reading further in the text of the report. Circuits used should be shown in the text. Auxiliary information (such as SPICE data outputs, etc.) should be included in appropriate Appendices at the end of the report. Be sure to describe exactly how all results were obtained, giving enough information for anyone who understands EE 5342 to repeat your work. All work submitted must be original. If derived from another source, a full bibliographical citation must be given. (See all of Notes 5 and 6 in the syllabus.)

L06 31Jan025 The temperature dependence of the mobility of carriers in silicon (the Arora model - see Arora, Hauser and Roulston, Electron and Hole Mobilities in Silicon as a Function of Concentration and Temperature, IEEE Trans. Electron Devices, ED- 29, p. 292, ff., 1982) is quoted by Casey (Devices for Integrated Circuits : Silicon and III-V Compound Semiconductors, by H. Craig Casey, John Wiley, New York, 1999, p. 75) and also quoted by Muller and Kamins (Device Electronics for Integrated Circuits, 2nd ed., by Richard S. Muller and Theodore I. Kamins, John Wiley and Sons, New York, 1986, p. 35).

L06 31Jan026 Question 1: Careful examination of the form of  n (N,T) and  p (N,T) (N = doping concentration, T = temperature) will reveal that Casey and Muller and Kamins do not agree. Resolve the differences and determine the correct equation for the model. This model will be referred to as  n AHR (N,T) and  p AHR (N,T).

L06 31Jan027 Question 2: Determine the values of the model [  n AHR (N,T) and  p AHR (N,T)] for the 3x3 matrix of values of T= 0, 30, and 60C and N=1E15, 3E16, and 1E18 cm -3. Show your results in table format, i.e., one table will be values of  n AHR (N,T) for all nine conditions described in the 3x3 matrix of N,t values, and a similar table will be developed for the  p AHR (N,T) values.

L06 31Jan028 Another model is discussed by Mohammad, Bemis, Carter and Renbeck (Temperature, Electric field and Doping Dependent Mobilities of Electrons and Holes in Semiconductors”, Solid-State Electronics, Vol. 36, No. 12, PP , 1993.) This model will be referred to as  n MBCR (T,E,N) and  p MBCR (T,E,N)

L06 31Jan029 Question 3: Determine the same tables defined in Question 2 for the models  n MBCR (T,E,N),  p MBCR (T,E,N)] for the case where E = 0.

L06 31Jan0210 Question 4: Determine the tables of values for the conditions defined in Question 2 for the relative differences between the models when E = 0. Use the following definitions for the relative differences: rdn  |  n MBCR (T,E,N) -  n AHR (N,T)|   n AHR (N,T) and rdp  |  p MBCR (T,E,N) -  p AHR (N,T)|/   p AHR (N,T)

L06 31Jan0211 Questions 5 and 6 5: Comment on the results of Question 4. What possible reasons can you give for the differences between the two models? 6: Comment on the application of a  n (T,N) and a  p (T,N) model to determine a R(T,N) model for an integrated circuit resistor. For one thing, what additional modeling issues would need to be considered?

L06 31Jan0212 Energy bands for p- and n-type s/c p-type EcEc EvEv E Fi E Fp q  p = kT ln(n i /N a ) EvEv EcEc E Fi E Fn q  n = kT ln(N d /n i ) n-type

L06 31Jan0213 Making contact in a p-n junction Equate the E F in the p- and n-type materials far from the junction E o (the free level), E c, E fi and E v must be continuous N.B.: q  = 4.05 eV (Si), and q  = q   E c - E F EoEo EcEc EfEf E fi EvEv q  (electron affinity) qFqF qq (work function)

L06 31Jan0214 Band diagram for p + -n jctn* at V a = 0 EcEc E fN E fi EvEv EcEc E fP E fi EvEv 0 xnxn x -x p -x pc x nc q  p < 0 q  n > 0 qV bi = q(  n -  p ) *N a > N d -> |  p | >  n p-type for x<0 n-type for x>0

L06 31Jan0215 A total band bending of qV bi = q(  n -  p ) = kT ln(N d N a /n i 2 ) is necessary to set E fP = E fN For -x p < x < 0, E fi - E fP < -q  p, = |q  p | so p < N a = p o, (depleted of maj. carr.) For 0 < x < x n, E fN - E fi < q  n, so n < N d = n o, (depleted of maj. carr.) -x p < x < x n is the Depletion Region Band diagram for p + -n at V a =0 (cont.)

L06 31Jan0216 Depletion Approximation Assume p << p o = N a for -x p < x < 0, so  = q(N d -N a +p-n) = -qN a, -x p < x < 0, and p = p o = N a for -x pc < x < -x p, so  = q(N d -N a +p-n) = 0, -x pc < x < -x p Assume n << n o = N d for 0 < x < x n, so  = q(N d -N a +p-n) = qN d, 0 < x < x n, and n = n o = N d for x n < x < x nc, so  = q(N d -N a +p-n) = 0, x n < x < x nc

L06 31Jan0217 Depletion approx. charge distribution xnxn x -x p -x pc x nc  +qN d -qN a +Q n ’=qN d x n Q p ’=-qN a x p Charge neutrality => Q p ’ + Q n ’ = 0, => N a x p = N d x n [Coul/cm 2 ]

L06 31Jan0218 Induced E-field in the D.R. The sheet dipole of charge, due to Q p ’ and Q n ’ induces an electric field which must satisfy the conditions Charge neutrality and Gauss’ Law* require thatE x = 0 for -x pc < x < -x p and E x = 0 for -x n < x < x nc h 0h 0

L06 31Jan0219 Induced E-field in the D.R. xnxn x -x p -x pc x nc O - O - O - O + O + O + Depletion region (DR) p-type CNR ExEx Exposed Donor ions Exposed Acceptor Ions n-type chg neutral reg p-contact N-contact W 0

L06 31Jan0220 Review of depletion approximation Depletion Approx. p p << p po, -x p < x < 0 n n << n no, 0 < x < x n 0 > E x > -2V bi /W, in DR (-x p < x < x n ) p p =p po =N a & n p =n po = n i 2 /N a, -x pc < x < -x p n n =n no =N d & p n =p no = n i 2 /N d, x n < x < x nc x xnxn x nc -x pc -x p 0 EvEv EcEc qV bi E Fi E Fn E Fp

L06 31Jan0221 Review of D. A. (cont.) x xnxn x nc -x pc -x p ExEx -E max

L06 31Jan0222 Depletion Approxi- mation (Summary) For the step junction defined by doping N a (p-type) for x 0, the depletion width W = {2  (V bi -V a )/qN eff } 1/2, where V bi = V t ln{N a N d /n i 2 }, and N eff =N a N d /(N a +N d ). Since N a x p =N d x n, x n = W/(1 + N d /N a ), and x p = W/(1 + N a /N d ).

L06 31Jan0223 One-sided p+n or n+p jctns If p + n, then N a >> N d, and N a N d /(N a + N d ) = N eff --> N d, and W --> x n, DR is all on lightly d. side If n + p, then N d >> N a, and N a N d /(N a + N d ) = N eff --> N a, and W --> x p, DR is all on lightly d. side The net effect is that N eff --> N -, (- = lightly doped side) and W --> x -

L06 31Jan0224 Debye length The DA assumes n changes from N d to 0 discontinuously at x n, likewise, p changes from N a to 0 discontinuously at -x p. In the region of x n, Poisson’s eq is =* E =  /  --> dE x /dx = q(N d - n), and since E x = -d  /dx, we have -d 2  /dx 2 = q(N d - n)/  to be solved n x xnxn NdNd 0

L06 31Jan0225 Debye length (cont) Since the level E Fi is a reference for equil, we set  = V t ln(n/n i ) In the region of x n, n = n i exp(  /V t ), so d 2  /dx 2 = -q(N d - n i e  /Vt ), let  =  o +  ’, where  o = V t ln(N d /n i ) soN d - n i e  /Vt = N d [1 - e  /Vt-  o/Vt ], for  -  o =  ’ <<  o, the DE becomes d 2  ’/dx 2 = (q 2 N d /  kT)  ’,  ’ <<  o

L06 31Jan0226 Debye length (cont) So  ’ =  ’(x n ) exp[+(x-x n )/L D ]+con. and n = N d e  ’/Vt, x ~ x n, where L D is the “Debye length”

L06 31Jan0227 Debye length (cont) L D estimates the transition length of a step-junction DR (concentrations N a and N d with N eff = N a N d /(N a +N d )). Thus, For V a =0, & 1E13 < N a,N d < 1E19 cm -3 13% DA is OK