Semiconductor Device Modeling and Characterization – EE5342 Lecture 7 – Spring 2011 Professor Ronald L. Carter

©rlc L07-07Feb20112 First Assignment to –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.

©rlc L07-07Feb20113 Second Assignment Submit a signed copy of the document that is posted at

©rlc L07-07Feb20114 Schedule Changes Due to the University Closures last week Plan to meet until noon some days in the next few weeks. This way we will make up the lost time. The first extended class will be Wednesday, February 9. The MT will be postponed until Wednesday, February 16. All other due dates and tests will remain the same.

©rlc L07-07Feb20115 Equipartition theorem The thermodynamic energy per degree of freedom is kT/2 Consequently,

©rlc L07-07Feb20116 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 L07-07Feb20117 v drift [cm/s] vs. E [V/cm] (Sze 2, fig. 29a)

©rlc L07-07Feb20118 Carrier velocity saturation (cont.) At 300K, for electrons,  o = v 1 /E c = 1.53E9(300) /1.01(300) 1.55 = 1504 cm 2 /V-s, the low-field mobility The maximum velocity (300K) is v sat =  o E c = v 1 = 1.53E9 (300) = 1.07E7 cm/s

©rlc L07-07Feb20119 Diffusion of carriers In a gradient of electrons or holes,  p and  n are not zero Diffusion current,  J =  J p +  J n (note D p and D n are diffusion coefficients)

©rlc L07-07Feb 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  J p and the + sign in the definition of  J n

©rlc L07-07Feb Diffusion of Carriers (cont.)

©rlc L07-07Feb Current density components

©rlc L07-07Feb Total current density

©rlc L07-07Feb Doping gradient induced E-field If N = N d -N a = N(x), then so is E f -E fi Define  = (E f -E fi )/q = (kT/q)ln(n o /n i ) For equilibrium, E fi = constant, but for dN/dx not equal to zero, E x = -d  /dx =- [d(E f -E fi )/dx](kT/q) = -(kT/q) d[ln(n o /n i )]/dx = -(kT/q) (1/n o )[dn o /dx] = -(kT/q) (1/N)[dN/dx], N > 0

©rlc L07-07Feb Induced E-field (continued) Let V t = kT/q, then since n o p o = n i 2 gives n o /n i = n i /p o E x = - V t d[ln(n o /n i )]/dx = - V t d[ln(n i /p o )]/dx = - V t d[ln(n i /|N|)]/dx, N = -N a < 0 E x = - V t (-1/p o )dp o /dx = V t (1/p o )dp o /dx = V t (1/N a )dN a /dx

©rlc L07-07Feb The Einstein relationship For E x = - V t (1/n o )dn o /dx, and J n,x = nq  n E x + qD n (dn/dx) = 0 This requires that nq  n [V t (1/n)dn/dx] = qD n (dn/dx) Which is satisfied if

©rlc L07-07Feb Direct carrier gen/recomb gen rec EvEv EcEc EfEf E fi E k EcEc EvEv (Excitation can be by light)

©rlc L07-07Feb Direct gen/rec of excess carriers Generation rates, G n0 = G p0 Recombination rates, R n0 = R p0 In equilibrium: G n0 = G p0 = R n0 = R p0 In non-equilibrium condition: n = n o +  n and p = p o +  p, where n o p o =n i 2 and for  n and  p > 0, the recombination rates increase to R’ n and R’ p

©rlc L07-07Feb Direct rec for low-level injection Define low-level injection as  n =  p < n o, for n-type, and  n =  p < p o, for p-type The recombination rates then are R’ n = R’ p =  n(t)/  n0, for p-type, and R’ n = R’ p =  p(t)/  p0, for n-type Where  n0 and  p0 are the minority- carrier lifetimes

©rlc L07-07Feb Shockley-Read- Hall Recomb EvEv EcEc EfEf E fi E k EcEc EvEv ETET Indirect, like Si, so intermediate state

©rlc L07-07Feb S-R-H trap characteristics 1 The Shockley-Read-Hall Theory requires an intermediate “trap” site in order to conserve both E and p If trap neutral when orbited (filled) by an excess electron - “donor-like” Gives up electron with energy E c - E T “Donor-like” trap which has given up the extra electron is +q and “empty”

©rlc L07-07Feb S-R-H trap char. (cont.) If trap neutral when orbited (filled) by an excess hole - “acceptor-like” Gives up hole with energy E T - E v “Acceptor-like” trap which has given up the extra hole is -q and “empty” Balance of 4 processes of electron capture/emission and hole capture/ emission gives the recomb rates

©rlc L07-07Feb 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, Physics of Semiconductor Devices, Shur, Prentice- Hall, 1990.