24

The periodic table of the elements:

Concept of a hole:

In an intrinsic (pure) semiconductor, for every electron that is excited into the conduction band,

there is a vacant (or “missing”) electron at the top of the conduction band:

Conduction band (empty initially)

Eg

Valence band (filled initially)

We have: an electron in the conduction band + absence of an electron in the valence band.

This vacancy behaves and moves (e.g., when an electric field is applied) like a positive charge of

the same mass as an electron in the solid.

INTRINSIC CONDUCTIVITY

This is for highly pure materials (negligible effect of any impurities).

Conductivity occurs by the mechanism just discussed: Electrons are thermally excited (at a

temperature T > 0 Kelvin) from the filled valence band (leaving holes) to the conduction band.

Then there are 2 types of charges to contribute to σ (the electrons and the holes).

The electrons move in an applied field E in the same way as “free” electrons in a metal through

collisions with the lattice and acquire a drift velocity in the direction opposite to the field.

The holes acquire a drift velocity in the same direction as the field, as follows:

25

E

The previous expression that σ = n |e| µe (as used for metals) now generalizes to

σ = n |e| µe + p |e| µh

where p is the number of holes per unit volume and µh is the hole mobility. The two terms are

additive.

In the present case we have n = p (we call this value the intrinsic carrier concentration ni), so

σ = ni |e| (µe + µh)

It turns out that we always have µe > µh (electrons are easier to move than holes).

How big is the ni term?

Example. Calculate ni for intrinsic GaAs given σ = 1 × 10−6

(Ω-m) −1

, µe = 0.85 m2/V-s and µh

= 0.04 m2/V-s (all values at room temperature).

Answer. ni = 6

12

19

107.0 10

| | ( ) (1.6 10 )(0.85 0.04)e he

σ

µ µ

−

−= = ×

+ × + m−3

Electron bonding model

of conduction in

intrinsic Si:

(a) before excitation,

(b) just after excitation,

(c) at a later time

26

Variation of ni with temperature: At T = 0 Kelvin there would be no thermal effects to excite

electrons from the valence band to the conduction band (ni → 0). When T > 0 there is a thermal

Boltzmann factor like exp(−Eg/kBT) for the probability of electron excitation this will increase

sharply with T.

A detailed calculation shows that ni is proportional to exp(−Eg/2kBT), so roughly σ is

proportional to the same factors (since mobilities do not usually change much with T).

EXTRINSIC CONDUCTIVITY

− This is applicable when there are selected impurities in the semiconductor with

concentrations comparable to (or larger than) the calculated ni.

There are 2 cases of interest, called n-type and p-type.

1) n-type extrinsic semiconductors

As an example, consider Si. Each atom has 4 valence electrons, each of which bonds covalently

with one electron of the 4 neighbouring Si atoms (as in previous figure).

Suppose now that an impurity atom of valence 5 is added as a substitutional impurity (some

possibilities are the Group VA elements P, As or Sb).

→ The extra non-bonding electron is left over and is only weakly attached. Thus it

can move if an electric field is applied, and a current will flow.

→ Its energy level typically comes inside the band gap, just below the conduction

band; it is called a donor impurity level.

(a)

(b) (c)

E

In terms of the energy levels and the bands, the description is

Extrinsic n-type conduction model: (a) an impurity atom having 5 valence electrons (such as P)

may substitute for a Si atom. This results in an extra bonding

electron, which is weakly bound to the impurity atom and

orbits it;

(b) thermal excitation to form a free electron in the conduction

band;

(c) motion of this free electron in response to an applied

electric field.

27

Notice that the process does not cause any holes to be formed in the valence band.

∴ n >> p in this case (and hence the name n-type)

For the conductivity, σ ≅ n |e| µe , and the electrons are called the majority carriers. Also the

temperature dependence should be different from the intrinsic case because the excitation energy

is much smaller (see later).

2) p-type extrinsic semiconductors

We again use Si as an example, but in this case we add a substitutional impurity that has valence

3 from group IIIA (e.g. Al, B, or Ga). This produces the opposite effect to the previous case:

there is now a deficiency of one electron in one of the covalent bonds

→ This is like having one hole that is only loosely attached.

→ It’s energy level again comes within the band gap, but it is now just above the

valence band. It is called an acceptor impurity level.

(a) An impurity atom having 3 valence electrons (such as B) may substitute for a Si atom. This results in a

deficiency of 1 valence electron (or equivalently the presence of a hole) associated with the impurity atom. (b) The

motion of this hole in response to an electric field.

(a) Electron energy band

scheme for a donor impurity

level located within the band

gap, just below the bottom of

the conduction band.

(b) Excitation from a donor

state to produce a free electron

in the conduction band.

Extrinsic p-type

conduction model:

(a) Electron energy band

scheme for an acceptor

impurity level located

within the band gap, just

above the top of the

valence band.

(b) Excitation of an

electron into the acceptor

level to leave a free hole

in the valence band.

28

This process leaves holes at the top of the valence band, but it does not form electrons in the

conduction band.

∴ p >> n in this case (and hence the name p-type)

We have σ ≅ p |e| µh , and the holes are called the majority carriers.

The process of intentionally adding the impurities to form n-type or p-type semiconductors as an

alloy is called doping.

Temperature dependence of σ in semiconductors

— this comes from the T-dependence of the carrier concentrations (n and/or p) and

the T-dependence of the mobilities (µe and/or µh)

— the behaviour for the intrinsic and extrinsic situations can be discussed as follows.

Intrinsic carrier concentration ni

(log scale) vs. temperature T for Si

and Ge.

Electron concentration n vs. T for

n-type Si doped with 1021

m-3

of a

donor impurity. The freeze-out,

extrinsic and intrinsic regimes are

shown.

29

To deduce the conductivity σ, we also need to know something about the T-dependence and

impurity dependence of the electron and hole mobilities.

Dependence of (a) the electron mobility and (b) the hole mobility (log scales) vs. T (log scales)

for Si with various impurity concentrations.

Dependence of the room-

temperature electron and

hole mobilities (log scale)

vs. impurity concentration

(log scale) for Si.