ECE 875: Electronic Devices

ECE 875:Electronic Devices

Prof. Virginia AyresElectrical & Computer EngineeringMichigan State [email protected]

VM Ayres, ECE875, S14

Chp. 01

ConcentrationsDegenerateNondegenerate Effect of temperatureContributed by traps

Lecture 08, 27 Jan 14

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Example:Concentration of conduction band electrons for a nondegenerate semiconductor: n:

EE

E

C

C

dEEFENn

“hot” approximation of Eq’n (16)

3D: Eq’n (14)

Three different variables

(NEVER ignore this)


NC

The effective density of states at the conduction band edge.

MC

32

2

3

2

Cde Mm

dem

Answer:Concentration of conduction band electrons for a nondegenerate semiconductor: n:


Answer:Concentration of conduction band electrons for a nondegenerate semiconductor: n:

n

NkTEE

kT

EENn

CFC

FCC

ln

exp

Nondegenerate: EC is above EF:

kT

EENn CF

C

exp

Sze eq’n (21)

Use Appendix G at 300K for NC and n ≈ ND when fully ionised


This part is called NV: the effective density of states at the valence band edge.

Typically valence bands are symmetric about : MV = 1

Lecture 07: Would get a similar result for holes:


Similar result for holes:Concentration of valence band holes for a nondegenerate semiconductor: p:

p

NkTEE

kT

EENp

VVF

VFV

ln

exp

Nondegenerate: EC is above EF:

kT

EENp FV

V

exp

Sze eq’n (23)

Use Appendix G at 300K for NV and p ≈ NA when fully ionised


HW03: Pr 1.10:

Shown: kinetic energies of e- in minimum energy parabolas: KE E > EC.

Therefore:generic definition of KE as:

KE = E - EC


Define: Average Kinetic Energy

EE

E

EE

E

C

C

C

C

C

Edn

EdnEE )(

HW03: Pr 1.10:

Single band assumption


EE

E

EE

E

C

C

C

C

C

dEEFEN

dEEFENEE

“hot” approximation of Eq’n (16)3D: Eq’n (14)

HW03: Pr 1.10:

Single band assumption

Average Kinetic Energy


EE

E

EE

E

C

C

C

C

C

dEEFEN

dEEFENEE


HW03: Pr 1.10:

Single band definition

Average Kinetic Energy

2/1

32

2

3

2)( C

Cde EEMm

EN Equation 14:


2/1

32

2

3

2)( C

Cde EEMm

EN

Considerations:


Therefore: Single band assumption: means:


Start: Average Kinetic Energy

Finish: Average Kinetic Energy

EE

E

EE

E

C

C

C

C

C

dEEFEN

dEEFENEE


kT2

3

Therefore: Use a Single band assumption in HW03: Pr 1.10:


Reference:http://en.wikipedia.org/wiki/Gamma_function#Integration_problems

Some commonly used gamma functions:

n is always a positive whole number


-

Because nondegenerate: used the Hot limit:

= F(E)

EC

EF

Ei

EV


Consider: as the Hot limit approaches the Cold limit:“within the degenerate limit” ECEF

Ei

EV

Use:


Dotted: nondegenerate

Will find: useful universal graph: from n:

Solid: within the degenerate limit

x-axis: how much energy do e-s need: (EF – EC) versus how much energy can they get: kT

y-axis:Fermi-Dirac integral: good for any semiconductor


Concentration of conduction band electrons for a semiconductor within the degenerate limit: n:

EE

E

C

C

dEEFENn

3D: Eq’n (14)

Three different variables

(NEVER ignore this)


Part of strategy: pull all semiconductor-specific info into NC. To get NC:


Next: put the integrand into one single variable:


Therefore have:

And have:



Remember to also change the limits to bottom and top:

Change dE:



Now have:

Next: write “Factor” in terms of NC:


Compare:

Write “Factor” in terms of NC:


Write “Factor” in terms of NC:


F1/2(F)

No closed form solution but correctly set up for numerical integration


Note:

F = (EF - EC)/kT is semiconductor-specific

F1/2(F) is semiconductor-specific

But: a plot of F1/2(F) versus F is universal

Could just as easily write this as F1/2(x) versus x


Recall: on Slide 5 for a nondegenerate semiconductor: n:

EE

E

C

C

dEEFENn

“hot” approximation of Eq’n (16)

3D: Eq’n (14)

FCCF

C

FCC

NkT

EENn

kT

EENn

exp2

2exp

2

2

exp

F1/2(F)


Dotted: nondegenerate

Useful universal graph:

Solid: within the degenerate limit

x-axis: how much energy do e-s need: (EF – EC) versus how much energy can they get: kT

y-axis:Fermi-Dirac integral: good for any semiconductor


Shows where hot limit becomes the “within the degenerate limit”

EC

EF

Ei

EV

Around -1.0Starts to diverge

-0.35: ECE 874 definition of “within the degenerate limit”

Why useful: one reason:


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 EF coincides with EC?

Why useful: another reason:


Answer:DegenerateEF = EC => F = 0Read off the F1/2(F) integral value at F = 0≈ 0.6

Why useful: another reason:

Appendix G


EC

EF

Ei

EV

0.9 kT

Example:What is the concentration of conduction band electrons for degenerately doped GaAs at room temperature 300K when EF – EC = +0.9 kT?


Answer:


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


Compare: Sze eq’ns (21) and (23): for nondegenerate:

Compare with degenerate:


Chp. 01

ConcentrationsDegenerateNondegenerate Effect of temperatureContributed by traps

Lecture 08, 27 Jan 14

}


Nondegenerate: will show: this is the Temperature dependence of intrinsic concentrations ni = pi

ECE 474


Intrinsic: n = p

Intrinsic: EF =Ei = Egap/2

Set concentration of e- and holes equal: For nondegenerate:

=

Correct definition of intrinsic:


EF for n = p is given the special name Ei

Solve for EF:


Substitute EF = Ei into expression for n and p.

n and p when EF = Ei are given name: intrinsic: ni and pi

ni = pi:pi =ni =


Substitute EF = Ei into expression for n and p.

n and p when EF = Ei are given name: intrinsic: ni and pi

ni = pi:

Units of 4.9 x 1015 = ? = cm-3 K-3/2

pi =ni =


Plot: ni versus T:

ni

Note: temperature is not very low

106

1018


1013

1017

When temperature T = high, most electrons in concentration ni come from Si bonds not from dopants

Dotted line is same relationship for ni as in the previous picture.However: this is doped Si:

< liquid N2

ECE 875: Electronic Devices

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