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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