Electron & Hole Statisticsin Semiconductors

A “Short Course”. BW, Ch. 6 & S. Ch 3

The following assumes basic knowledge of elementary statistical physics.

• We know thatElectronic Energy Levels in the Bands (Solutions to the Schrödinger Equation in the periodic crystal)

are actually NOT continuous, but are really discrete. We have always treated them as continuous, because there are so many levels & they are so very closely spaced.

• Though we normally treat these levels as if they were continuous, in the next discussion, lets

treat them as discrete for a while • Assume that there are N energy

levels (N >>>1):ε1, ε2, ε3, … εN-1, εN

with degeneracies: g1, g2,…,gN

• Results from quantum statistical physics:

Electrons have the following Fundamental Properties:

• They are indistinguishable• For statistical purposes, they are

Fermions, Spin s = ½

• Indistinguishable Fermions, withSpin s = ½.

• This means that they must obey thePauli Exclusion Principle:

• That is, when doing statistics (counting) for the occupied states:There can be, at most, one e-

occupying a given quantum state (including spin)

• Electrons obey thePauli Exclusion Principle:

• So, when doing statistics for the occupied states:There can be at most, one e-

occupying a given quantum state (including spin)

• Consider the band state (Bloch Function) labeled nk (energy Enk, & wavefunction nk):

Energy level Enk can have2 e- , or 1 e- , or 1 e- , or 0 e- _

Statistical Mechanics Results for Electrons: • Consider a system of n e-, with N Single e- energy

Levels (ε1, ε2, ε3, … εN-1, εN ) with degeneracies (g1, g2,…, gN) at absolute temperature T:

• See any statistical physics book for the proof that the probability that energy level εj (with degeneracy gj) is occupied is:

(<nj/gj ) ≡ (exp[(εj - εF)/kBT] +1)-1

(< ≡ ensemble average, kB ≡ Boltzmann’s constant)• Physical Interpretation: <nj ≡ average

number of e- in energy level εj at temperature TεF ≡ Fermi Energy (or Fermi Level, discussed next)

Fermi-Dirac Distribution

Define:The Fermi-Dirac Distribution Function

(or Fermi distribution)

f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1

Physical Interpretation:The occupation probability for level j is

(<nj/gj ) ≡ f(ε)

• Look at the Fermi Function in more detail. f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1

Physical Interpretation:εF ≡ Fermi Energy ≡ Energy of the

highest occupied level at T = 0.• Consider the limit T 0. It’s easily shown that:

f(ε) 1, ε < εF

f(ε) 0, ε > εFand, for all T

f(ε) = ½, ε = εF

The Fermi Function: f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1

• Limit T 0: f(ε) 1, ε < εF

f(ε) 0, ε > εF

for all T: f(ε) = ½, ε = εF• What is the order of magnitude of εF? Any solid

state physics text discusses a simple calculation of εF. • Typically, it is found, (in temperature units) that

εF 104 K. • Compare with room temperature (T 300K):

kBT (1/40) eV 0.025 eVSo, obviously we always have εF >> kBT

• NOTE! Levels within ~ kBT of εF (in the “tail”,where it differs from a step function) are the ONLY ones which enter conduction (transport) processes!Within that tail, f(ε) ≡ exp[-(ε - ε F)/kBT] ≡ Maxwell-Boltzmann Distribution

Fermi-Dirac Distribution

• Properties of the Free Electron Gas: The Fermi Energy EF & related properties

• Fermi Energy EF Energy of the highest occupied state.

Related Properties• Fermi Velocity vF Velocity of an electron with

energy EF • Fermi Temperature TF Effective temperature of

an electron with energy EF

• Fermi Wavenumber kF Wave number of an electron with energy EF

• Fermi Wavelength λF de Broglie wavelength of an electron with energy

EF

“Free Electrons” in Metals at 0 K

Free Electron Gas:• Fermi Energy EF Energy of highest occupied state.• Fermi Velocity vF Velocity of electron with energy EF

• Fermi Temperature TF Effective temperature of an electron with energy EF

ηe Electron Density in the material• Fermi Wavenumber kF Wave number of an electron

with energy EF: EF = [ħ2(kF)2]/(2m) kF (3π2ηe)⅓

• Fermi Wavelength λF Wavelength of an electron with energy EF : λF (2π/kF) λF [2π/(3π2ηe)⅓]

3

12

322

222

3

322

eF

eF

F

mv

mmkE

B

FF k

ET

EF

EF Fermi EnergyF Work Function

Energy

Band Edge Metal

Vacuum Level

• Sketch of a typical experiment. A sample of metal is “sandwiched” between two larger sized samples of an insulator or semiconductor.

F

• Using typical numbers in the formulas for several metals & calculating gives the table below:

312

322

222

3

322

eF

eF

F

mv

mmkE

B

FF k

ET

Element Electron Density, e [1028 m-3]

Fermi Energy EF [eV]

Fermi Temperature TF [104 K]

Fermi Wavelength F [Å]

Fermi Velocity vF [106 m/s]

Work Function [eV]

Cu 8.47 7.00 8.16 4.65 1.57 4.44 Au 5.90 5.53 6.42 5.22 1.40 4.3 Fe 17.0 11.1 13.0 2.67 1.98 4.31 Al 18.1 11.7 13.6 3.59 2.03 4.25

TkEE

Ef

B

Fexp1

1Fermi-Dirac Distribution

0

1

EFElectron Energy

Occ

upat

ion

Prob

abili

ty

Work Function F

Increasing T

T = 0 KkBT

dEEDEEfVE

dEEDEfVN

ee

e

ee

0

0;

Number and Energy Densities

Density of States De(E) Number of electron states available between energies E & E+dE. For 3D spherical bands only, it’s easily shown that:

2222mEmEDe

Number Density:

Energy Density:

T Dependences of e- & e+ Concentrations • n concentration (cm-3) of e-

• p concentration (cm-3) of e+

• Using earlier results & making the Maxwell-Boltzmann approximation to the Fermi Function for energies near EF, it can be shown that

np = CT3 exp[- Eg /(kBT)](C = material dependent constant)

• For all temperatures, it is always true that np = CT3 exp[- Eg /(kBT)]

(C = material dependent constant)• In a pure material: n = p ni (np = ni

2)ni “Intrinsic carrier concentration”. So,

ni = C1/2T3/2exp[- Eg /(2kBT)]At T = 300K

Si : Eg= 1.2 eV, ni =~ 1.5 x 1010 cm-3

Ge : Eg = 0.67 eV, ni =~ 3.0 x 1013 cm-3

Intrinsic Concentration vs. T Measurements/Predictions

Note the different scales on the right & left figures!

Doped Materials: Materials with Impurities!

As already discussed, these are more interesting & useful!• Consider an idealized carbon (diamond) lattice(we could do the following for any Group IV material).

C : (Group IV) valence = 4• Replace one C with a phosphorous P.

P : (Group V) valence = 54 e- go to the 4 bonds

5th e- ~ is “almost free” to move in the lattice (goes to the conduction band; is weakly bound).

• P donates 1 e- to the material P is a DONOR (D) impurity

Doped Materials• The 5th e- isn’t really free, but is loosely

bound with energy ΔED << Eg

(Earlier, we outlined how to calculate ΔED!)• The 5th e- moves when an E field is applied!

It becomes a conduction e-

If there are enough of these, a current is created

Doped Materials• Let: D any donor, DX neutral donor• D+ ionized donor (e- to conduction band) • Consider the chemical “reaction”:

e- + D+ DX + ΔED

• As T increases, this “reaction” goes to the left.But, it works both directions

• Consider very high T All donors are ionized n = ND concentration of donor atoms

(a constant, independent of T)• It is still true that

np = ni2 = CT3 exp[- Eg /(kBT)]

p = (CT3/ND)exp[- Eg /(kBT)] “Minority Carrier Concentration”

• All donors are ionized The minority carrier concentration is T dependent.

• At still higher T, n >>> ND, n ~ ni

The range of T where n = ND The “Extrinsic” Conduction Region.

lllll

Almost no ionized donors & no intrinsic carriers

High T Low T

n vs. 1/T

n vs. T

Low T High T

• Again, consider an idealized C (diamond) lattice.(or any Group IV material).C : (Group IV) valence = 4

• Replace one C with a boron B.B : (Group III) valence = 3

• B needs one e- to bond to 4 neighbors.• B can capture e- from a C

e+ moves to C (a mobile hole is created)

• B accepts 1 e- from the material B is an ACCEPTOR (A) impurity

• The hole e+ is really not free. It is loosely bound by energy ΔEA << Eg

Δ EA = Energy released when B captures e- e+ moves when an E field is applied!

• NA Acceptor Concentration• Let A any acceptor, AX neutral acceptor

A- ionized acceptor (e+ in the valence band) • Chemical “reaction”:

e++A- AX + ΔEA As T increases, this “reaction” goes to the left.

But, it works both directionsJust switch n & p in the previous discussion!

Terminology “Compensated Material” ND = NA

“n-Type Material” ND > NA(n dominates p: n > p)

“p-Type Material” NA > ND(p dominates n: p > n)