EE 5340 Semiconductor Device Theory Lecture 06 – Spring 2011 Professor Ronald L. Carter

©rlc L06-10Feb20112 Review the Following R. L. Carter’s web page: – EE 5340 web page and syllabus. (Refresh all EE 5340 pages when downloading to assure the latest version.) All links at: – University and College Ethics Policies – Makeup lecture at noon Friday (1/28) in 108 Nedderman Hall. This will be available on the web.

©rlc L06-10Feb20113 First Assignment Send to –On the subject line, put “5340 ” –In the body of message include address: ______________________ Your Name*: _______________________ Last four digits of your Student ID: _____ * Your name as it appears in the UTA Record - no more, no less

©rlc L06-10Feb20114 Second Assignment Submit a signed copy of the document posted at

©rlc L06-10Feb20115 Schedule Changes Due to University Weather Closings Make-up class will be held Friday, February 11 at 12 noon in 108 Nedderman Hall. Additional changes will be announced as necessary. Syllabus and lecture dates postings have been updated. Project Assignment has been posted in the initial version.

©rlc L06-10Feb20116 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 L06-10Feb20117 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 L06-10Feb20118 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 L06-10Feb20119 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 L06-10Feb Net intrinsic mobility Considering only lattice scattering

©rlc L06-10Feb 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 L06-10Feb Net extrinsic mobility Considering only lattice and impurity scattering

©rlc L06-10Feb 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 L06-10Feb 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 L06-10Feb Figure 1.17 (p. 32 in M&K 1 ) Low-field mobility in silicon as a function of temperature for electrons (a), and for holes (b). The solid lines represent the theoretical predictions for pure lattice scattering [5].

©rlc L06-10Feb Exp.  (T=300K) model for P, As and B in Si 1

©rlc L06-10Feb Exp. mobility model function for Si 1 ParameterAsPB  min  max N ref 9.68e169.20e162.23e17 

©rlc L06-10Feb 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 L06-10Feb 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  p = 189 cm2/V-s

©rlc L06-10Feb 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 )

Figure 1.15 (p. 29) M&K 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).] ©rlc L06-10Feb201121

©rlc L06-10Feb Net silicon extr resistivity (cont.) Since  = (nq  n + pq  p ) -1, and  n >  p, (  = q  /m*) we have  p >  n, for the same N I Note that since 1.6(high conc.) <  p /  n < 3(low conc.), so 1.6(high conc.) <  n /  p < 3(low conc.)

©rlc L06-10Feb 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

Figure 1.16 (p. 31 M&K) 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 next slide). ©rlc L06-10Feb201124

©rlc L06-10Feb 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 L06-10Feb Equipartition theorem The thermodynamic energy per degree of freedom is kT/2 Consequently,

©rlc L06-10Feb 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 L06-10Feb Carrier velocity 2 carrier velocity vs E for Si, Ge, and GaAs (after Sze 2 )

©rlc L06-10Feb 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 L06-10Feb References M&K and 1 Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, –See Semiconductor Device Fundamen- tals, by Pierret, Addison-Wesley, 1996, for another treatment of the  model. 2 Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981.

©rlc L06-10Feb 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, 2003.