4. Carrier densities in SC 08 - VGTU Elektronikos fakulteto

CHARGE CARRIERS IN SEMICONDUCTORS

Objectives:

• Discovery of the nature of charge carriers in intrinsic and extrinsic

semiconductors

• Finding on what, how and why densities of charge carriers in semiconductors

depend

• Calculation methods of of charge carrier densities in semiconductors

Content:

Charge carriers in intrinsic semiconductors

Nature of charge carriers

Fermi level in an intrinsic semiconductor

Densities of carriers

Charge carriers in extrinsic semiconductors

n-type semiconductors

p-type semiconductors

Compensation doping

Excess carriers and lifetime

ELEKTRONIKOS PAGRINDAI 2008

VGTU EF ESK [email protected]

1

Nature of charge carriers in intrinsic semiconductors

Carefully refined semiconductors are called intrinsic semiconductors. .

In a silicon crystal each atom is surrounded by four neighbour atoms. At 0 K

all valence electrons take part in covalent bonding and none are free to move

through the crystal.

As the temperature increases, the lattice vibrations arise. Some of the energy

of the lattice vibrations is transferred to the valence electrons. If sufficient

energy is given to an electron, it leaves a bond and becomes free.

At very low temperatures the

valence band is full (filled to

capacity) and the conduction band

is completely empty.

Thus, at a very low temperature

(close to 0 K) the crystal behaves

as an insulator.



2

The jump of an electron from the

valence band to the conduction

band corresponds to the release

of an electron from covalent

bonding. The minimum energy

required for that is equal to the

width of the forbidden band.

If the width of the forbidden band is less, electrons are released at lower

temperature.

When an electron is released, a positively charged vacancy appears. This

vacancy may be considered as a positive hole.

According to the energy band diagram an uncompleted allowed energy level in

the valence band corresponds to a hole.




3

Above occupied levels there are

unoccupied energy levels in the

conduction and valence bands.

The situation is similar to that in

conductors: electrons are able to accept

energy, change their velocity and

participate in conductivity.

… a hole behaves as a positive charge

carrier having a positive charge equal in

magnitude to the electronic charge.

... In the intrinsic semiconductor the number of electrons in the

conduction band equals to the number of holes in the valence band.

Charge carriers appear as a result of charge carrier generation.

Positive holes attract negative electrons. If an electron is drawn into the bond, it

recombines with a hole.

The jump of an electron from the conduction band to the valence band

corresponds to the recombination process.




4

Fermi level in an intrinsic semiconductor

n

pcvF lnk

4

3

2 m

mT

WWW +

+=

If the Fermi level is below the bottom of the conduction

band, it is possible to use the simplified formula

... The Maxwell-Boltzmann distribution function can be used for calculation of the

probability of occupation of energy levels in the conduction band.

The same conclusion can be made for holes in the valence band.

TWWWf

k/)(F

Fe)(−−

≅

The Fermi level in an intrinsic

semiconductor lays at the middle

of the forbidden band.



5

Densities of charge carriers in intrinsic semiconductors

We have derived the expression for density of electrons in a non-generate system.

Now it is necessary to take into account that conduction electrons exist in a crystal.

We must consider the effective mass:

( ) TWWTWWN

Tmn

k/)(c

k/)(

3

2/3n FcFc eeh

kπ22 −−−−==

Acting in a similar way we can find density or holes in the valence band.

( )TWWTWW

NTm

pk/)(

vk/)(

3

2/3p

vFvF eeh

kπ22−−−−

==

TWTWWNNNNnp k/vc

k/)(vc ee vc ∆−−−

==

2i

2iii pnpnnp ===

TWNNnppn k2/vcii e ∆−===

... The densities of electrons and

holes in an intrinsic semiconductor

are strongly dependent on

temperature and gap energy.



6

The number of conduction electrons and holes increases rapidly with the

increase of temperature and decreases with the increase of the gap

energy.

( )Tba

T

WNNpn

11

k2ln2

1lnln vcii −≅−==

∆

k2tan

W∆α ≅

... Using the expressions for the densities of electrons and holes and taking into

account the condition n= p, it is possible to derive the formula for the Fermi

level in an intrinsic semiconductor.

Densities of charge carriers in intrinsic semiconductors

TWNNnppn k2/vcii e ∆−===



7

Densities of charge carriers in intrinsic semiconductors. Problems

1. Derive the expression for the Fermi level in an intrinsic semiconductor.

2. Find what part of germanium and silicon valence electrons is in the

conduction band at temperature 300 K.

3. Find the ratio of carrier densities in germanium and silicon at room

temperature (T = 300 K).

4. Find how many times carrier density in the intrinsic germanium increases, if

the temperature increases from 20 to 1000C. Repeat the calculations for the

intrinsic silicon. Comment on the results.



8

Extrinsic (doped) semiconductors

The intrinsic carrier densities are very small and depend strongly on

temperature. In order to fabricate devices such as diodes or transistors, it is

necessary to increase the free electron or hole population. This is done

intentionally doping the semiconductor, i. e. adding specific impurities in

controlled amounts.

Doped semiconductor are called extrinsic semiconductors.

... The percentage of impurity in non-degenerate semiconductors must be

small (for example, about 10-5 % in the substrates for integrated circuits). Then

impurity atoms are isolated from each other by semiconductor atoms.

In order to have necessary conductivity type, donor and acceptor impurities are

used.



9

Charge carriers in the semiconductors doped with donor impurities

The elements from the column V of the periodic table, e. g. phosphorus (P),

arsenic (As) and antimony (Sb) are added to an intrinsic elemental

semiconductor to modify the semiconductor into an n-type semiconductor.

If the fifth electron gets such energy, it leaves the

impurity atom and becomes free.

The impurity atom becomes ionized.

... An electron and a positive ion appear as a result

of ionization of a donor impurity atom.

Let us consider silicon doped with phosphorus.

Each pentavalent atom occupies a site usually occupied by a silicon atom. Four

valence electrons are in covalent bonds, the fifth electron rotates round the

positively charged impurity atom.

The binding energy of this extra electron is small. In the case of phosphorus in

silicon it is only about 0.044 eV.



10

The free electron is in the conduction band. So

donor energy levels are in the forbidden band near

the bottom of the conduction band

At room temperature each donor impurity atom

donates an additional charge carrier – a negative

conduction electron. According to this

semiconductors that contain donor impurities are

called donor-type, electronic or n-type

semiconductors.

Because usually impurity density is small, donor

levels are isolated and are shown by the dashed line.




11

Charge carrier densities and Fermi level in extrinsic semiconductors strongly

depend on temperature and impurity density.

At 0 K all allowed energy levels in the valence band are filled by electrons. All

donor levels are filled by unbound electrons. The conduction band is free. So

charge carriers do not exist, and the semiconductor behaves as an insulator.

At 0 K the Fermi level is between the donor levels and the bottom of the

conduction band.

If the temperature increases, unbound

electrons obtain energy and jump to the

conduction band. So, in the low

temperature or impurity ionisation

range, density of conduction electrons

increases.




12

At some temperature, that is sufficiently lower than 300 K, all impurity atoms

become ionised and the density of conduction electrons becomes equal to the

donor density Nd.

If the temperature further increases, in the wide

range the density of conduction electrons

remains constant. This temperature range is

called the extrinsic range.

dn Nn ≅ iin

in pp

n

np <<=

... In the n-type semiconductor electrons are the majority carriers and

holes are the minority carriers.




13

In the extrinsic range

( )[ ] dFccn k/exp NTWWNn =−−=

d

ccF lnk

N

NTWW −=

... The Fermi level in the extrinsic range falls as the temperature

increases.

As the density of the intrinsic charge carriers increases with temperature, at

some temperature that is sufficiently higher than 300 K it becomes equal to Nd.

At higher temperatures thermally exited intrinsic carriers predominate. Then the

semiconductor obtains the properties of an intrinsic semiconductor.

In the high temperature or intrinsic range the Fermi level approaches the mid-

gap position.




14

... In the impurity ionisation (low temperature) range the density of electrons increases with

temperature.

In the extrinsic range (at middle temperatures) the density of electrons is almost constant.

At last in the intrinsic range (at high temperatures) the intrinsic carriers predominate and their

density increases with temperature.

Plotting the variable 1/T along the x-axis and lnn and lnp along the y-axis, we obtain curves

that may be approximated by the segments of a straight line.

( ) 2/1dF =Wf 2/dNn ≅( )dc

dcs

/2lnk NN

WWT

−≅

di Nn =)/ln(k 2

dvc

iNNN

∆WT =




15

... In the vicinity of room temperature the conduction electron density in the n-

type semiconductor is approximately constant and equals to the density of donor

impurity atoms. The hole density is much lower than the intrinsic density and is

strongly dependent upon temperature.

At normal temperature carrier densities and Fermi level

depend on impurity density.

... In the n-type semiconductor the Fermi level is always over the middle of the

forbidden band.

If the donor density is higher, the Fermi level is higher, closer to the bottom of

the conduction band; the electron density is higher, the hole density is lower.

If the donor concentration is lower, the Fermi level is lower, closer to the middle

of the forbidden band.


dn Nn ≅ iin

in pp

n

np <<=



16

1. How many charges and how many charge carriers appear as a result of

ionization of semiconductor atom?


donor atom ionization?

3. A germanium specimen is doped with phosphorus. Its density is 1016 cm–3.

Find densities of electrons and holes and position of the Fermi level at

temperature 300 K.

4. Silicon is doped with phosphorus. The Fermi level is in the forbidden band

at the distance of 0,044 eV from the bottom of the conduction band.

Impurity density is 1016 cm–3. Find the position of the Fermi level at

temperature 300 K.

Charge carriers in the semiconductors doped with donor impurities.

Problems



17

Semiconductors with majority charge carrier density of holes are called p-type.

They can be produced adding impurities from column III of the periodic table, e. g.

boron (B), aluminium (Al), gallium (Ga) or indium (In) to intrinsic silicon or

germanium.

These p-type impurities are characterized by three valence electrons in their outer

shell.

Let us consider silicon doped with

boron.

The vacancy thus created by the

impurity is not a hole, since it is

bound to the atom.

At some temperature above 0 K the electron from a bond of a neighbouring parent

atom can fill the vacant site leaving a hole.

So dopants from column III accept electrons to create holes for conduction.

Therefore they are called acceptors.

Charge carriers in the semiconductors doped with acceptor impurities



18

The energy required by a valence electron to fill the vacancy created by an

impurity atom and thus to create a hole is of the similar magnitude to the

ionisation energy of a donor atom. So acceptors create discrete acceptor levels

just above the top of the valence band.

At 0 K acceptor levels are free. Electrons occupy the valence band of the

semiconductor. There are no charge carriers. Semiconductor has properties of

insulator.

If the temperature increases, electrons jump from the valence band to the

acceptor levels leaving holes in the valence band. So in the impurity ionisation

range the density of holes increases with temperature.




19

In the extrinsic range :

Density of electrons is small and increases with temperature:

ap Np ≅

a

2i

p

2i

pN

n

p

nn ≅=

As the temperature in the extrinsic range is

raised, the Fermi level increases.

a

vvF lnk

N

NTWW =−

At high temperatures the material becomes intrinsic and the Fermi level

approaches midway between the conduction and valence bands.


... Positive holes are the majority carriers and electrons are the minority carriers

in the p-type semiconductor.



20

ap Np ≅a

2i

p

2i

pN

n

p

nn ≅=

In the extrinsic range:

a

vvF lnk

N

NTWW =−


1 – low temperature, impurity ionization range;

2 – middle temperature, extrinsic conductivity range;

3 – high temperature, intrinsic conductivity range



21

At normal temperature, Fermi level and carrier densities

depend on the impurity density.

ap Np ≅ a2ip / Nnn ≅

... The Fermi level below the middle of the forbidden band is the

characteristic feature of the band model of the p-type semiconductor.


If acceptor density is higher, the Fermi level is lower,

closer to the top of the valence band; the hole density

is higher, the electron density is lower.

If the acceptor density is lower, the Fermi level is

higher, closer to the middle of the forbidden band.

a

vvF lnk

N

NTWW =−



22


acceptor ionization?

2. Silicon plate is doped by boron. Its density is 1016 cm–3. Find carrier

densities and Fermi level at temperature 300 K.

Charge carriers in the semiconductors doped with acceptor impurities.

Problems



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Compensation doping

When both acceptor and donor impurities are added simultaneously to an

intrinsic semiconductor, the compensation takes place.

At higher donor density the crystal is n-type

semiconductor since n-type impurity predominates.

The free carriers supplied by the less concentrated

dopant recombine with an equal number of carriers

of the opposite type. So some of the donor states

are cancelled by acceptor states..

The process is called compensation.

'' addn NNNn −≅= d2in / Nnp =

'' daap NNNp −≅= a2ip / Nnn =

At higher donor impurity density when ,'' AD NN >>

If

,'' DA NN >>If



24

Compensation doping is widely used in manufacturing of semiconductor devices

and integrated circuits.

The compensation is possible, if impurity density is not very high. Then the

distance between impurity atoms is relatively great and impurity atoms cannot

interact.

If impurity density increases, the distance between impurity atoms decreases.

When impurity density in silicon becomes approximately 1019 cm-3, degeneration

of the semiconductor arises. Then impurity levels split and allowed bands

appear.

p-type material can be converted to the n-type and vice-versa, by the addition of

excess dopant atoms of the appropriate type.

If , the intrinsic charge carriers predominate. The semiconductor

has properties of the intrinsic compensated or near-fully compensated material.

The Fermi level lies near the middle of the forbidden gap.

ADi NNn −>

Compensation doping



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In the degenerate n-type semiconductor, conduction

and donor bands overlap and form the hybrid

conduction band. Energy levels at the bottom of the

hybrid conduction band are occupied by electrons.

The Fermi level is above the bottom of the

conduction band.

In the degenerate p-type semiconductor, we have

overlapping of the valence and acceptor bands.

Then energy levels at the top of the hybrid valence

band are not occupied by electrons. The Fermi level

is below the top of the valence band

W

Wv

Wc

WF

W

Wv

Wc

WF

Compensation doping

In both cases there are unoccupied allowed energy

levels over the levels that are filled by electrons. The

situation is very similar to that in conductors.

Therefore conduction is possible even at 0 K in

degenerate semiconductors and they are sometimes

called semimetals.



26

Germanium is doped with phosphorus and boron. Their densities are

1017 cm–3 and 1016 cm-3, respectively. Find carrier densities and position of

the Fermi level at 300 K.

Compensation doping. Problem



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Under thermal equilibrium the generation rate and the recombination rate are

equal and the carrier densities and remain constant.

The equilibrium may be disturbed by light or carrier injection.

Excess electrons and holes are always equal in number:

nnn ∆+= 0 ppp ∆+= 0 .pn ∆=∆ .2inpn >

If the external excitation (activation) stops, the density of the excess carriers

reduces exponentially:

)/exp()/exp()()( 00 τ−∆=τ−∆=∆=∆ tptntptn

The recombination of the excess carriers

may be radiative or nonradiative, direct

or indirect, band-to-band or through

recombination centres and traps.



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.

,

0n0

0n0

ppp

nnn

∆∆+=

+=

τ

∆∆ n

t

n

t

n−== ...

d

)d(

d

d

τ

∆∆ p

t

p

t

p−== ...

d

)d(

d

d

In practical device design small lifetime may be desirable for high-speed

switching applications.

Nanosecond switching speed is realised using gold doping.

)/exp()/exp()()( 00 τ−∆=τ−∆=∆=∆ tptntptn

The lifetime τ represents the average time a carrier remains free before it

recombines. During the lifetime the number of excess carriers reduces e times.

The recombination rate of the excess carriers is dependent on the lifetime:




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4. Carrier densities in SC 08 - VGTU Elektronikos fakulteto

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