1

H63SSD

Solid State Devices

Lecture 2

Energy bands and carrier concentration

2

Fermi level?

Fermi level expresses the probability of an electron occupying a particular level at absolute T according to Fermi-Dirac statistics.

Fermi energy level is defined highest energy level below which all energy levels are filled at 0 K

In a semiconductor the Fermi level is in the middle of the band gap between the valence band and the conduction band.

The probability of electron occupancy is 50%

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Fermi Level

Electrons in solid obey Fermi-Dirac statistics - must consider electron, its wave nature and Pauli exclusion principle

Other statistics: Maxwell-Boltzmann for classical particles, Bose-Einstein for photons

The distribution of electrons over a range of allowed energy levels at thermal equilibrium is given as

The distribution function. f(E) gives us the probability that states with energy E

are occupied

kTEE FeEf

/)(1

1)(

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Where f(E) is the probability of occupancy of an available state with

energy E and EF is the Fermi level energy

For an energy, EF, the occupation probability is

Thus an energy state at the Fermi level has a probability of 1/2 of being

occupied by an electron.

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111)(

1

1)()(

F

kTF

EF

EF

Ef

e

Ef

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Example 1:

Determine the probability that an energy level 2kT above the Fermi level

is occupied by an electron at 300 K.

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Implies that at OK, every available energy state up to EF is filled with

electrons.

At higher T (say, T1 > OK) some probability exists for states above

the Fermi level to be filled and probability

[1 - f(E)] that states below EF are empty.

f(E)

1

1/2

T = 0K

T = T1

With T = 0,

f (E) = 1/(1+0) = 1

when E < EF

f (E) = 1/ (1 +) = 0

when E > EF.

Fermi level

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In doped semiconductors, either p-type or n-type, the Fermi level is

shifted by the impurities, illustrated by their band gaps.

For intrinsic material, concentration of holes in the valence band is equal to the concentration of electrons in the conduction band.

EF lies in the middle of the band gap

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Concentrations of Electron & Holes inside

the energy bands

To determine the electrical behavior of a semiconductor, we need to

know the no. of electrons and holes available for current conduction

The electron density in the CB and similarly the hole density in the VB

can be obtained if N(E) and f(E) are known

N(E) = density of states function - describes the available density of

energy states that may be occupied by an electron

2/123*

3)()2(4)(* c

/

e EEmhEN

* proof in quantum mechanics me* = electron effective mass

Electron & hole density

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f(E) the probability distribution function is given by

EF = Fermi energy, i.e. energy at which f(E) = 1/2 (when E = EF)

k = Boltzmanns constant = 1.38 x 10-23 J/K

T = Temperature in kelvin.

At energy E, the density of electrons in Conduction Band is given by

)exp(1

1)(

kT

EEEf

F

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Number of electrons in the whole Conduction Band of an intrinsic

semiconductor is given by

..

)()(bc

i dEEfENn

dE

kT

EEEEm

hn

Fc

Eei

c

}

)exp(1

1{)()2(4 2/12/3*3

}exp{}2

{2 23

2

*

kT

EE

h

kTmn Fcei

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termed the effective density of states in the conduction band

Where 23

2

*

}2

{2h

kTmN ec

}exp{kT

EENn Fcci

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Similarly, the density of holes in the valence band is given by

23

2

*

23

2

*

..

}2

{2

}exp{

}exp{}2

{2

)](1)[(

h

kTmN

kT

EENp

kT

EE

h

kTmp

dEEfENp

hv

VFvi

VFhi

bv

i

is the effective density of states in the valence band

Where

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c

vvcF

c

vvcF

vFFc

vFv

Fcc

N

NkTEEE

N

NEEE

kTTake

cNv

N

kT

EE

kT

EE

kT

EEN

kT

EEN

ln2

)2

(

ln)}(2{1:log

)}()exp{(

)exp()exp(

For an intrinsic semiconductor, ii pn

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Therefore, Fermi level in the intrinsic semiconductor:

*

*

ln43

2

ln22

e

hvcFi

c

vvcFi

m

mkT

EEE

N

NkTEEE

If me*= mh*, then the intrinsic Fermi level is exactly in the center of the band gap

If me*> mh*, it is slightly below the center of the band gap (towards VB)

If mh*> me*, the intrinsic Fermi level is slightly above the center (towards CB) For Si, Ge and many other semiconductor, the 3rd term is quite small, and EFi is

generally taken to be at the center of the band gap

For InSb, mh* 20me*, EFi shifted toward CB at 300 K

The density of states function is directly related to the carrier effective mass; thus a larger effective mass means a larger density of stales function

The equilibrium electron-hole product

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We know

ni and pi represent the electron and hole concentrations respectively, in an

intrinsic semiconductor. They are usually referred to as the intrinsic

electron concentration and hole concentration.

As number of electrons (CB) is same as number of holes (VB) in an

intrinsic semiconductor,

ni = pi,

The equilibrium electron (no) concentration can be written as

Likewise, the equilibrium hole (po) concentration is

}exp{kT

EENn Fcco

}exp{kT

EENp VFvo

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Then the product

Where Eg is the band gap. For a given semiconductor material at

a constant temperature, the value of ni is constant, and is

independent of Fermi energy.

kT

EE

kT

EENNn vFiFicvci

(exp

(exp2

kT

EENNn vcvci

)(exp2

kT

ENN

g

vc exp

2

ioo npn

kT

ENNn

g

vci2

exp

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Equilibrium electron-hole Concentration

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Consider a homogeneous semiconductor to which Nd donors/ cm-3 and

Na acceptors/ cm-3 have been added.

The charge neutrality condition is expressed by equating the density of negative charges to the density of positive charges. We then have,

Nd+ + p0 = Na

- + n0 -Neutrality condition (after doping)

np = ni2

For n-type semiconductor, n = p + Nd (ND)

For p-type semiconductor, p = n + Na (NA)

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For p-type material

a

ii

N

n

p

n

i

aa

n

nnp

NNnp

22

2

For n-type material

d

ii

N

n

n

n

i

dd

p

nnp

NnNpn

22

2

;

Remember

The maximum useful temperature for a

semiconductor

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Remember

}exp{kT

EENp VFvo

)exp(

)exp(

max

2

2

2

kT

ENNn

kT

EENNn

nnp

g

vci

vcvci

i

gvc EEE

Ec

Ev

Eg

}exp{kT

EENn Fcco

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When the working temp. of an intrinsic semiconductor is increased, the

no. of thermally generated carrier pairs increases to the point where they

equal or exceed those contributed by the impurities.

Semiconductor loses its extrinsic nature and becomes intrinsic

Therefore Tmax is the absolute maximum for any semiconductor device

and should stay well below it.

})(

ln{221max

d

vc

g

N

NNk

ET

For n-type semiconductor

where ni = Nd

})(

ln{221max

in

NNk

ET

vc

g

Variation of carrier concentration with

temperature

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100 400 800

Ra

tio

of ca

rrie

r co

ncn

.

to d

on

or

co

ncn

., n

/Nd

T (K)

T from 0 - 150K (freeze out region)

At T 0 K, thermal energy is not sufficient to ionize any donor atoms and excite electrons from VB to CB. As T is increased, some donor atoms

become ionized, while there is still not sufficient energy to excite electrons

from VB to CB.

Extrinsic region

full ionization

1.0

0.5

1.5

Intrinsic region

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Temp. 150 K to 400K (extrinsic region)

All donor atoms are ionized so that n Nd- ionization region This

concentration of Nd remains constant up to room temperature, at which

Nd >>ni.

Temp. > 400 K (intrinsic region)

The intrinsic carrier concentration begins to increase rapidly and

eventually reaches a value greater than Nd. The material becomes

intrinsic again and n >>Nd. In this region, labelled as intrinsic region,

both, concentration of electrons and holes is increased.

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Example 2:

Calculate the thermal equilibrium electron concentration in silicon at 300 K,

assuming the Fermi energy is 0.25 eV below the conduction band.

Given: Boltzmanns Constant, k = 1.38 x 10-23 J K-1 = 8.62 x 10-5 eV K-1

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Example 3:

If a Si sample is doped with 1012 boron atoms per cm3, what is the carrier

concentration in the Si sample at 300K?

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Text Books and Reference Books

Text Books:

1. Solid State Electronic Devices (6th edition), Ben G.

Streetman & Sanjay Kumar Banerjee, Pearson

Prentice Hall, 2006. ISBN: 0132017202

2. Semiconductor Devices, S.M. Sze, Wiley Inter-

Science, ISBN: 0471056618.

Reference Books:

1. Semiconductor Physics and devices (2nd edition),

Donald A. Neamen, McGraw-Hill.

2. Semiconductor device fundamentals, Robert F.

Pierret, Addison Wesley