Prof.P. Ravindran,folk.uio.no/ravi/cutn/semiphy/12.carrier_const6.pdf · Prof.P. Ravindran, ......

P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI

http://folk.uio.no/ravi/semi2013

Prof.P. Ravindran, Department of Physics, Central University of Tamil

Nadu, India

Carriers Concentration in Semiconductors - VI

1


Heavy Doping

Light doping: impurity

atoms do not interact with

each other impurity level

Heavy doping: perturb the

band structure of the host

crystal reduction of

bandgap

E

r(E)

Ec

Ed

Ev

Eg


Metal-Insulator Transition

Average impurity-impurity distance = Bohr radius

Mott criterionB

3 164dN a

If the number of donor is higher than the Mott’s criterion the semiconductor

to metal transition will takes place.


Semiconductor Equilibrium

Charge carriers

Electrons in conductance

n(E) = gc(E)fF(E)

n(E) - prob. dens. of electrons

gc(E) – density of states of electrons.

fF(E) - Fermi-Dirac prob. function

Holes in valence

p(E) = gV(E)(1 - fF(E))

p(E) - prob. dens. of holes

gv(E) – density of states of holes.

fF(E) - Fermi-Dirac prob. function

Semiconductors at equilibrium the concentration of carriers will not change

with time.



n0 gc (E) fF (E )dE

n0 Nc exp(EC EF )

kT

NC 22mn

*kT

h2

3 / 2

p0 gv (E)(1 fF (E))dE

Nv 22mp

*kT

h2

3 / 2

p0 Nv exp(EF Ev )

kT

Number of Electrons:

Number of

Electrons:



Example

Find the probability that a state in the conduction band is occupied and

calculate the electron concentration in silicon at T = 300K. Assume Fermi

energy is .25 eV below the conduction band. Nc=2.8.1019

Note low probability per state but large number of states implies

reasonable concentration of electrons.

fF (E) exp(Ec EF )

kT

exp(0.25 /.0259) 6.43 105

n0 Nc(E)exp(Ec EF )

kT

(2.8 1019)(6.43 105) 1.8 1015cm3



The Extrinsic Semiconductor

Example

Consider doped silicon at 300K. Assume that the Fermi enery is .25 eV

below the conduction band and .87 eV above the valence band. Calculate the

thermal equilibrium concentration of e’s and holes.

Nc=2.8.1019;Nv=1.04.1019.

31519

0 108.1)0259.0/25.0exp()108.2()(

exp)(

cm

kT

EEENn Fc

c

3419

0 107.2)0259.0/87.0exp()1004.1()(

exp)(

cm

kT

EEENp vF

v

At thermal equilibrium the generation and recombination rate will be same and hence

the electron and hole concentration will not change.


Carrier Concentration in N-type Semiconductor

Consider Nd is the donor Concentration i.e., the numberof donor atoms per unit volume of the material and Ed isthe donor energy level.

At very low temperatures all donor levels are filled withelectrons.

With increase of temperature more and more donoratoms get ionized and the density of electrons in theconduction band increases.


)exp()2

(2 2

3

2 kT

EE

h

kTmn cFe

The density of Ionized donors is given by

)exp()}(1{kT

EENEFN Fd

ddd

At very low temperatures, the number of electrons in the conduction band must be equal

to the number of ionized donors.

)exp()exp()2

(2 2

3

2 kT

EEN

kT

EE

h

kTm Fdd

cFe

Density of electrons in conduction band is given by


Taking logarithm and rearranging we get

2

)(

0.,

)2

(2

log22

)(

)2

(2

log)(2

)2

(2loglog)()(

2

3

2

2

3

2

2

3

2

cdF

e

dcdF

e

dcdF

ed

FdcF

EEE

kat

h

kTm

NkTEEE

h

kTm

NkTEEE

h

kTmN

kT

EE

kT

EE

At 0k Fermi level lies exactly at the middle of the Donor level and the bottom of the Conduction band


Density of electrons in the conduction band

kT

EE

h

kTm

N

kT

EE

h

kTm

N

kT

EE

kT

EE

kT

E

h

kTm

N

kT

EE

kT

EE

kT

E

h

kTm

NkTEE

kT

EE

kT

EE

h

kTmn

cd

e

dcF

e

dcdcF

c

e

dcdcF

c

e

dcd

cF

cFe

2

)(exp

])2

(2[

)()exp(

}

])2

(2[

)(log

2

)(exp{)exp(

}

])2

(2[

)(log

2

)(exp{)exp(

}

}

)2

(2

log22

)({

exp{)exp(

)exp()2

(2

2

1

2

1

2

1

2

3

2

2

1

2

3

2

2

1

2

3

2

2

1

2

3

2

2

3

2


kT

EE

h

kTmNn

kT

EE

h

kTm

N

h

kTmn

kT

EE

h

kTmn

cded

cd

e

de

cFe

2

)(exp)

2()(2

}2

)(exp

])2

(2[

)({)

2(2

)exp()2

(2

4

3

22

1

2

3

2

2

1

2

3

2

2

3

2

2

1

Thus we find that the density of electrons in the conduction band is proportional to the

square root of the donor concentration at moderately low temperatures.


Variation of Fermi level with temperature (n-type)

To start with ,with increase of temperature EF increases

slightly.

As the temperature is increased more and more donor atoms

are ionized.

Further increase in temperature results in generation of

Electron - hole pairs due to breaking of covalent bonds and

the material tends to behave in intrinsic manner.

The Fermi level gradually moves towards the intrinsic Fermi

level Ei.

P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI14

Extrinsic Material

We can calculate the binding energy by

using the Bohr model results, considering

the loosely bound electron as ranging about

the tightly bound “core” electrons in a

hydrogen-like orbit.

rKnhK

mqE 022

4

4, 1;2


Concept of a Donor “Adding extra” Electrons


Binding energies in Si: 0.03 ~ 0.06 eV

Binding energies in Ge: ~ 0.01 eV

Binding Energies of Impurity

Hydrogen Like Impurity Potential (Binding Energies)

Effective mass should be used to account the influence of

the periodic potential of crystal.

Relative dielectric constant of the semiconductor should

be used (instead of the free space permittivity).

: Electrons in donor atoms

: Holes in acceptor atoms


Extrinsic Material

Calculate the approximate donor binding

energy for Ge(εr=16, mn*=0.12m0).

eVJ

h

qmE

r

n

0064.01002.1

)1063.6()161085.8(8

)106.1)(1011.9(12.0

)(8

21

234212

41931

22

0

4*

Answer:

Thus the energy to excite the donor electron from n=1 state to the free

state (n=∞) is ≈6meV.


Electron-Hole Recombination

The equilibrium carrier concentrations are denoted with n0

and p0.

The total electron and hole concentrations can be different

from n0 and p0 .

The differences are called the excess carrier concentrations

n’ and p’.

'0 nnn '0 ppp


Charge Neutrality

Charge neutrality is satisfied at equilibrium (n’= p’=

0).

When a non-zero n’ is present, an equal p’ may be

assumed to be present to maintain charge equality and

vice-versa.

If charge neutrality is not satisfied, the net charge will

attract or repel the (majority) carriers through the drift

current until neutrality is restored.

'p'n


Generation and Recombination life time of Charges

In an intrinsic semiconductor the number of holes is equal to the number of free electrons. Thermal agitation, however, continues to generate new hole-electron pairs per unit volume per second, while other hole-electron pairs disappear as a result of recombination.

On an average, a hole (an electron) will exist for ζp (ζ n) seconds before recombination. This time is called the mean lifetime of the hole (electron).

Generation and recombination processes act to change the carrier concentrations, and thereby indirectly affect current flow


Recombination Lifetime

Assume light generates n’ and p’. If the light is

suddenly turned off, n’ and p’ decay with time until

they become zero.

The process of decay is called recombination.

The time constant of decay is the recombination time

or carrier lifetime, .

Recombination is nature’s way of restoring

equilibrium (n’= p’= 0).


Modern Semiconductor Devices for Integrated Circuits (C.

Hu)

is the recombination lifetime.

n’ and p’ are the excess carrier concentrations.

n = n0+ n’

p = p0+ p’

Charge neutrality requires n’= p’.

rate of recombination = n’/ = p’/

Thermal Generation

If n’ is negative, there are fewer electrons than the equilibrium value.

As a result, there is a net rate of thermal generation at the rate of |n|/ .


Quasi-equilibrium and Quasi-Fermi Levels

• Whenever n’ = p’ 0, np ni2. We would like to

preserve and use the simple relations:

• But these equations lead to np = ni2. The solution to this

problem is to introduce two quasi-Fermi levels Efn and Efp

such that

kTEE

cfceNn

/)(

kTEE

vvfeNp

/)(

kTEE

cfnceNn

/)(

kTEE

vvfpeNp

/)(

Even when electrons and holes are not at equilibrium, within

each group the carriers can be at equilibrium. Electrons are

closely linked to other electrons but only loosely to holes.


EXAMPLE: Quasi-Fermi Levels and Low-Level Injection

Consider a Si sample with Nd=1017cm-3 and n’=p’=1015cm-3.

(a) Find Ef .

n = Nd = 1017 cm-3 = Ncexp[–(Ec– Ef)/kT]

Ec– Ef = 0.15 eV. (Ef is below Ec by 0.15 eV.)

Note: n’ and p’ are much less than the majority carrier

concentration. This condition is called low-level

injection.


Now assume n = p = 1015 cm-3.

(b) Find Efn and Efp .

n = 1.011017cm-3 =

Ec–Efn = kT ln(Nc/1.011017cm-3)

= 26 meV ln(2.81019cm-3/1.011017cm-3)

= 0.15 eV

Efn is nearly identical to Ef because n n0 .

kTEE

cfnceN

/)(

EXAMPLE: Quasi-Fermi Levels and Low-Level Injection


EXAMPLE: Quasi-Fermi Levels

p = 1015 cm-3 =

Efp–Ev = kT ln(Nv/1015cm-3)

= 26 meV ln(1.041019cm-3/1015cm-3)

= 0.24 eV

kTEE

vvfpeN

/)(

Ec

Ev

Efp

Ef Efn

Prof.P. Ravindran,folk.uio.no/ravi/cutn/semiphy/12.carrier_const6.pdf · Prof.P. Ravindran, ......

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