ECE 875: Electronic Devices Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University

VM Ayres, ECE875, S14 Chp. 01 Concentrations Degenerate NondegenerateEffect of temperature Contributed by traps Lecture 08, 27 Jan 14 }

VM Ayres, ECE875, S14 Example: Concentration of conduction band electrons for a nondegenerate semiconductor: n: “hot” approximation of Eq’n (16) 3D: Eq’n (14) Three different variables (NEVER ignore this)

VM Ayres, ECE875, S14 N C The effective density of states at the conduction band edge. MCMC Answer: Concentration of conduction band electrons for a nondegenerate semiconductor: n:

VM Ayres, ECE875, S14 Answer: Concentration of conduction band electrons for a nondegenerate semiconductor: n: Nondegenerate: E C is above E F : Sze eq’n (21) Use Appendix G at 300K for N C and n ≈ N D when fully ionised

VM Ayres, ECE875, S14 This part is called N V : the effective density of states at the valence band edge. Typically valence bands are symmetric about  : M V = 1 Lecture 07: Would get a similar result for holes:

VM Ayres, ECE875, S14 Similar result for holes: Concentration of valence band holes for a nondegenerate semiconductor: p: Nondegenerate: E C is above E F : Sze eq’n (23) Use Appendix G at 300K for N V and p ≈ N A when fully ionised

VM Ayres, ECE875, S14 HW03: Pr 1.10: Shown: kinetic energies of e- in minimum energy parabolas: KE  E > E C. Therefore: generic definition of KE as: KE = E - E C

VM Ayres, ECE875, S14 Define: Average Kinetic Energy HW03: Pr 1.10: Single band assumption

VM Ayres, ECE875, S14 “hot” approximation of Eq’n (16) 3D: Eq’n (14) HW03: Pr 1.10: Single band assumption Average Kinetic Energy

VM Ayres, ECE875, S14 “hot” approximation of Eq’n (16) 3D: Eq’n (14) HW03: Pr 1.10: Single band definition Average Kinetic Energy Equation 14:

VM Ayres, ECE875, S14 Considerations:

VM Ayres, ECE875, S14 Therefore: Single band assumption: means:

VM Ayres, ECE875, S14 Start: Average Kinetic Energy Finish: Average Kinetic Energy “hot” approximation of Eq’n (16) 3D: Eq’n (14) Therefore: Use a Single band assumption in HW03: Pr 1.10:

VM Ayres, ECE875, S14 Reference: Some commonly used gamma functions: n is always a positive whole number

VM Ayres, ECE875, S14 - Because nondegenerate: used the Hot limit: = F(E) E C E F E i E V

VM Ayres, ECE875, S14 Consider: as the Hot limit approaches the Cold limit: “within the degenerate limit” E C E F E i E V Use:

VM Ayres, ECE875, S14 Dotted: nondegenerate Will find: useful universal graph: from n: Solid: within the degenerate limit x-axis: how much energy do e-s need: (E F – E C ) versus how much energy can they get: kT y-axis: Fermi-Dirac integral: good for any semiconductor

VM Ayres, ECE875, S14 Concentration of conduction band electrons for a semiconductor within the degenerate limit: n: 3D: Eq’n (14) Three different variables (NEVER ignore this)

VM Ayres, ECE875, S14 Part of strategy: pull all semiconductor-specific info into N C. To get N C :

VM Ayres, ECE875, S14 Next: put the integrand into one single variable:

VM Ayres, ECE875, S14 Therefore have: And have: Next: put the integrand into one single variable:

VM Ayres, ECE875, S14 Remember to also change the limits to  bottom and  top : Change dE: Next: put the integrand into one single variable:

VM Ayres, ECE875, S14 Now have: Next: write “Factor” in terms of N C :

VM Ayres, ECE875, S14 Compare: Write “Factor” in terms of N C :

VM Ayres, ECE875, S14 Write “Factor” in terms of N C :

VM Ayres, ECE875, S14 F 1/2 (  F ) No closed form solution but correctly set up for numerical integration

VM Ayres, ECE875, S14 Note:  F = (E F - E C )/kT is semiconductor-specific F 1/2 (  F ) is semiconductor-specific But: a plot of F 1/2 (  F ) versus  F is universal Could just as easily write this as F 1/2 (x) versus x

VM Ayres, ECE875, S14 Recall: on Slide 5 for a nondegenerate semiconductor: n: “hot” approximation of Eq’n (16) 3D: Eq’n (14) F 1/2 (  F )

VM Ayres, ECE875, S14 Dotted: nondegenerate Useful universal graph: Solid: within the degenerate limit x-axis: how much energy do e-s need: (E F – E C ) versus how much energy can they get: kT y-axis: Fermi-Dirac integral: good for any semiconductor

VM Ayres, ECE875, S14

Shows where hot limit becomes the “within the degenerate limit” ECEC EFEF EiEi EVEV Around -1.0 Starts to diverge -0.35: ECE 874 definition of “within the degenerate limit” Why useful: one reason:

VM Ayres, ECE875, S14 F(  F ) 1/2 integral is universal: can read numerical solution value off this graph for any semiconductor Example: p.18 Sze: What is the concentration n for any semiconductor when E F coincides with E C ? Why useful: another reason:

VM Ayres, ECE875, S14 Answer: Degenerate E F = E C =>  F = 0 Read off the F 1/2 (  F ) integral value at  F = 0 ≈ 0.6 Why useful: another reason: Appendix G

VM Ayres, ECE875, S14 ECEC EFEF EiEi EVEV 0.9 kT Example: What is the concentration of conduction band electrons for degenerately doped GaAs at room temperature 300K when E F – E C = +0.9 kT?

VM Ayres, ECE875, S14 Answer:

VM Ayres, ECE875, S14 For degenerately doped semiconductors (Sze: “degenerate semiconductors”): the relative Fermi level is given by the following approximate expressions:

VM Ayres, ECE875, S14 Compare: Sze eq’ns (21) and (23): for nondegenerate: Compare with degenerate:

VM Ayres, ECE875, S14 Chp. 01 Concentrations Degenerate NondegenerateEffect of temperature Contributed by traps Lecture 08, 27 Jan 14 }

VM Ayres, ECE875, S14 Nondegenerate: will show: this is the Temperature dependence of intrinsic concentrations n i = p i ECE 474

VM Ayres, ECE875, S14 Intrinsic: n = p Intrinsic: E F =E i = E gap /2 Set concentration of e- and holes equal: For nondegenerate: = Correct definition of intrinsic:

VM Ayres, ECE875, S14 E F for n = p is given the special name E i Solve for E F :

VM Ayres, ECE875, S14 Substitute E F = E i into expression for n and p. n and p when E F = E i are given name: intrinsic: n i and p i n i = p i :p i =n i =

VM Ayres, ECE875, S14 Substitute E F = E i into expression for n and p. n and p when E F = E i are given name: intrinsic: n i and p i n i = p i : Units of 4.9 x = ? = cm -3 K -3/2 p i =n i =

VM Ayres, ECE875, S14 Plot: n i versus T: nini Note: temperature is not very low

VM Ayres, ECE875, S When temperature T = high, most electrons in concentration n i come from Si bonds not from dopants Dotted line is same relationship for n i as in the previous picture. However: this is doped Si: < liquid N 2