Doped semiconductors: donor impurities

A silicon lattice with a single impurity atom (Phosphorus, P) added.

As compared to Si, the Phosphorus has one extra valence electron which, after all bonds are made, has very weak bonding.

Very small energy is required to create a free electron from an impurity atom.

This type of impurity is called donor.

Note, that there is no hole created when a free electron comes from the impurity atom.

Free electron concentration in donor - doped semiconductors

For Si and other semiconductors, the typical doping levels are:

ND = 1015 cm-3 ….1018 cm-3

nD = 1015 cm-3 ….1018 cm-3 (compare to ni = 1.3×1010 cm-3 in intrinsic Si)

nD >> ni

Doping provides a flexible control over semiconductor conductivity.

The vast majority of microelectronic devices are based on doped semiconductors

If the concentration of donor impurity (e.g. Phosphor) in Si is ND, the concentration of free electrons,

n ≈ ND

When donor atoms are introduced into the semiconductor material, they are all ionized.

Each donor atom creates one free electron.

How much would be the resistance of the (1 cm×1cm× 1cm) Si sample doped with donor impurities with concentration 2×1016 cm-3?

Resistance of Donor-Doped Silicon sample

n = 2×1016 cm-3

µn = 1000 cm2/(V ×s)q = 1.6 ×10-19 C

σ = 1.6 ×10-19 C × 2×1016 cm-3 × 1000 cm2/(V ×s)

σ = 3.2 (Ohm × cm)-1

AL

ALR

qn

×==

=

σρ

µσ1

;

ρ = 0.325 Ohm × cm

R = 0.325 (Ohm × cm) ×1 cm /(1cm ×1cm) = 0.325 Ohm

The resistance of a doped Si crystal can be significantly lower than that of intrinsic Si

A silicon lattice with a single impurity atom (Boron, B) added. Boron has only three valence electrons, one electron less than the Si atom.Having only three valence electrons - not enough to fill all four bonds - it creates an excess hole that can be used in conduction.

This type of impurity is called acceptor.

There is no corresponding free electron created from acceptor impurity

Doped semiconductors: acceptor impurities

Hole concentration in acceptor - doped semiconductors

The vast majority of microelectronic devices using hole conductivity, are based on doped semiconductors

In doped semiconductors, the concentration of intrinsic electrons and holes can be neglected as compared to those coming from donor and acceptor impurities.

For Si and other semiconductors, the typical acceptor doping levels are:

NA = 1015 cm-3 ….1018 cm-3

pA = 1015 cm-3 ….1018 cm-3 (compare to ni = 1.3×1010 cm-3 in intrinsic Si);

pA >> ni

If the concentration of acceptor impurity (B atoms) in Si is NA, the hole concentration

pA ≈ NA

Concentration –temperature dependence in doped semiconductors

T

n, cm-3

Typical dependence for n-Si (i.e. donor-doped)(for p-Si (i.e. acceptor doped) the dependences are similar

Impurity electrons

Intrinsic electrons,intrinsic holes

ND

100 K 300 K 400 K200 K

Mobile charge carriers energy

Bound electron

Atom

Ec

Ev

Intrinsic material at low temperature. There are no free electrons or holes – no free carriers. The mobile charge energy does not make sense.

valence band

In semiconductors, the mobile charge carriers are the free electrons and holes

Conductance band energy

Free electron

Atom

When the electron in the valence band acquires sufficient extraenergy, it can be detached from its parent atom and reaches reach the “conductance band”The minimum energy of the conduction band is denoted as EC

conductance band

Hole

Ec

Ev

Energy Band Gap (Eg)

Generally no electron can have the energy between Ec and Ev

The “band-gap” is the energy difference between Ec and Ev:Eg=Ec-Ev

Ec

Ev

Forbidden Energyregion

Band-gap

Ec

Ev

Free electron

Atom

conductance band

Hole

Intrinsic material at high temperature. Temperature generates free electrons and holes in equal concentrations.

The energy of free electrons is close to EC; the energy of holes is close to EV

valence band

Mobile charge carriers energy

Ec

Ev

Average free carrier Energy – Fermi energyconductance

band

The energy of free electrons is close to EC; the energy of holes is equal to EV

valence band

The average energy of all the mobile charges in semiconductor:Eave Average [(Electron Average Energy + Hole Average Energy)] ≈(EC + EV)/2.

The average energy of all the mobile charges in semiconductor is called Fermi energy EF.In intrinsic semiconductor:EF ≈ (EC + EV)/2.

EF

n-type semiconductor

In the n-type material most of the mobile charges are free electrons.Therefore, the average energy of mobile charges is close to EC:

EF ≈ EC

EFn

Phosphorus (P) has 5 outer shell electrons.

Extra free electron

EC

EV

p-type semiconductor

In the p-type material most of the mobile charges are holes.Therefore, the average energy of mobile charges is close to EV:

EF ≈ EV

EFp

EC

EV

Boron (B) has 3 outer shell electrons.

Extra electron vacancy or hole

Carrier Concentration and Fermi level: n-type material

Electron concentration:

ND - Donor atoms concentration

n Dn N≈

Hole concentration in the n-type material:

2i

nn

npn

=

Fermi energy level: F CE E≈

2n n ip n n=

Carrier Concentration and Fermi level: p-type material

Hole concentration:

NA - Acceptor atoms concentration

p Ap N≈

Electron concentration in the p-type material:

2i

pp

nnp

=

Fermi energy level: F VE E≈

2n n ip n n=

Compensation

If both donor and acceptor are added to an intrinsic semiconductor then the semiconductor is said to be compensated

If ND > NA, the free electron concentration:n = ND-NA

If ND < NA, the hole concentration:p = NA-ND

Drift CurrentThe electric current due to electric field is called the

Drift Current.

E,n drift nJ q nEµ=

The electron current density (current per unit area):

µn is the electron mobility and n is the electron concentration.

Jn,drift

Jp,drift

Similarly the hole current density:

pEqJ pdrift,p µ=

µp is the hole mobility and p is the hole concentration.

…cont… Drift Current and conductivity

pEqnEq

JJJ

pn

drift,pdrift,ndrift

µ+µ=

+=

The total (electron + hole) drift current density:

( )drift n pJ q n p Eµ µ= +Conductivity:

( )n pq n pσ µ µ= + driftJ Eσ=

1 1( )n pq n p

ρσ µ µ

= =+

Resistivity:

Diffusion CurrentDiffusion is due to concentration difference between two regions of a semiconductorThe carriers will move from higher concentration region to the lower one.

Con

cent

ratio

n

x

Abr

upt c

once

ntra

tion

chan

ge

Con

cent

ratio

n

x

Gra

dual

con

cent

ratio

n ch

ange

Hole diffusion

Jp,diff

…continued… Diffusion CurrentThe electron diffusion current density: ,n diff n

dnJ qDdx

=

Dn is the diffusion coefficient of electrons

The hole diffusion current density: ,p diff pdpJ qDdx

=−

Ele

ctro

n C

once

ntra

tion

x

Electron diffusion

Jn,diff

Hol

e C

once

ntra

tion

x

Dp is the diffusion coefficient of holes

Total Currents in semiconductors with bothelectric field and concentration gradients

Electron current density

Hole current density:

, ,n n drift n diff n ndnJ J J q nE qDdx

µ= + = +

, ,p p drift p diff p pdpJ J J q pE qDdx

µ= + = −

Total current density: n pJ J J= +

Total current: ( )×An pI J A J J= × = + A is the sample cross-section area

Total electron current

n nI J A= ×

Total hole current

p pI J A= ×