Semiconductor ConductivityCh. 1, S

•It is well-known that in semiconductors, there are

Two charge carriers! Electrons e- & Holes e+

What is a hole?We’ll use a qualitative definition for now!

A quantitative definition will come later!

• Holes are often treated as “positively charged electrons”.

How is this possible?

Are holes really particles?We’ll eventually answer both of these questions as the course proceeds.

A Qualitative Picture of Holes (from Seeger’s book)

An idealized, 2 dimensional, “diamond” lattice for e- & e+ conduction

“Thought Experiment”# 1• Add an extra e- (“conduction electron”) & apply an electric

field E (the material is n-type: negative charge carriers)

E Field Direction e- Motion Direction

(“almost free”)

e-

• Remove an e- leaving, a “hole” e+ & apply an electric field E. (the material is p-type: positive charge carriers)

e- Motion Directione+ Motion Direction

e+

“Thought Experiment”# 2

E Field Direction

Crude Analogy: CO2 Bubbles in Beer!

g (gravity)

GlassBeer

Bubbles

Bubble Motion

• We could develop a formal theory of bubble motion in the earth’s gravitational field. Since the bubbles move vertically upward, in this theory, the

Bubbles would need “negative mass”!

Thermal Pair Generation & Annihilation

• Now: A classical Treatment. Simple, classical, statistical analysis. Later: Quantum Treatment

• Define: Eg Binding energy of a valence electron.

(In the Band Picture: This is the band gap energy).

• Apply an energy Eg to an atom

(from thermal or other excitation).

• An e- is promoted out of a valence level (band) into a conduction level (band). Leaves a hole (e+) behind.

• Later: e- - e+ pair recombine, releasing energy Eg

(in terms of heat, lattice vibrations, …)

• Schematically:

e- + e+ Eg

This chemical “reaction” can go both ways.

As the temperature T increases, more e- - e+ pairs are generated & the electrical

conductivity increases & the

conductivity σ increases with increasing T.

e-, e+ Pair GenerationRecombination

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

p concentration (cm-3) of e+

• Can show (& we will): np = CT3 exp[- Eg /(kBT)] (C = material dependent constant)

From the “Law of mass action” from statistical physics• In a pure material: n = p ni (np = ni

2)ni “Intrinsic carrier concentration”

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

Also: Band Gaps are (slightly) T dependent!

• It can be shown that:

Eg(T) = Eg(0) - αT

Si : α = 2.8 x 10-4 eV/K

Ge : α = 3.9 x 10-4 eV/K

But this doesn’t affect the T dependence of ni!

ni2 = CT3exp[- Eg(T)/(kBT)]

= Cexp(α/kB)T3exp[- Eg(0)/(kBT)]

= BT3exp[- Eg(0)/(kBT)]

where B = Cexp(α/kB) is a new constant prefactor

Intrinsic Concentration vs. T Measurements/Predictions

Note the different scales on the right & left figures!

Doped Materials: Materials with Impurities! 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 : (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- is really not free, but is loosely bound with energy

ΔED << Eg

The 5th e- moves when an E field is applied!It becomes a conduction e-

• Let: D any donor, DX neutral donorD+ ionized donor (e- to the conduction band)

• Consider the chemical “reaction”: e- + D+ DX + ΔED

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

We’ll show later howto calculate this!

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

(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 ionizeddonors & no intrinsic carriers

High T Low T

n vs. 1/T

n vs. 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 : (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)

Doping in Compound Semiconductors

• This is MUCH more complicated!• Semiconductor compound constituents can act as

donors and / or acceptors!• Example: CdS, with a S vacancy

(One S-2 “ion” is missing)• The excess Cd+2 “ion” will be neutralized by 2conduction e-. So, Cd+2 acts as a doubleacceptor, even though it is not an impurity!

CdS with S vacancies is a p-type material,even with no doping with impurities!