semiconductor physics. introduction semiconductors are materials whose electronic properties are...
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Semiconductor
Physics
Introduction
• Semiconductors are materials whose electronic properties are intermediate between those of Metals and Insulators.
• They have conductivities in the range of 10 -4
to 10 +4S/m.
• The interesting feature about semiconductors is that they are bipolar and current is transported by two charge carriers of opposite sign.
• These intermediate properties are determined by
1.Crystal Structure bonding Characteristics. 2.Electronic Energy bands.
• Silicon and Germanium are elemental semiconductors and they have four valence electrons which are distributed among the outermost S and p orbital's.
• These outer most S and p orbital's of Semiconductors involve in Sp3 hybridanisation.
• These Sp3 orbital's form four covalent bonds of equal angular separation leading to a tetrahedral arrangement of atoms in space results tetrahedron shape, resulting crystal structure is known as Diamond cubic crystal structure
Semiconductors are mainly two types
1. Intrinsic (Pure) Semiconductors 2. Extrinsic (Impure) Semiconductors
Intrinsic Semiconductor
• A Semiconductor which does not have any kind of impurities, behaves as an Insulator at 0k and behaves as a Conductor at higher temperature is known as Intrinsic Semiconductor or Pure Semiconductors.
• Germanium and Silicon (4th group elements) are the best examples of intrinsic semiconductors and they possess diamond cubic crystalline structure.
Si
Si
SiSiSi
Valence Cell
Covalent bonds
Intrinsic Semiconductor
E
Ef
Ev
Valence band
Ec
Conduction band
Ec
Electronenergy
Distance
KE ofElectron = E - Ec
KE of Hole = Ev - E
Fermi energy level
Carrier Concentration in Intrinsic Semiconductor
When a suitable form of Energy is supplied to a Semiconductor then electrons take transition from Valence band to Conduction band.
Hence a free electron in Conduction band and simultaneously free hole in Valence band is formed. This phenomenon is known as Electron - Hole pair generation.
In Intrinsic Semiconductor the Number of Conduction electrons will be equal to the Number of Vacant sites or holes in the valence band.
)1......(..........)()(
)()(band theof top
cE
dEEFEzn
EFdEEZdn
Calculation of Density of Electrons
Let ‘dn’ be the Number of Electrons available between energy interval ‘E and E+ dE’ in the Conduction band
Where Z(E) dE is the Density of states in the energy interval E and E + dE and F(E) is the Probability of Electron occupancy.
dEEEmh
dEEZ ce2
1
2
3
3)()2(
4)(
Since the E starts at the bottom of the Conduction band Ec
dEEmh
dEEZ e2
1
2
3
3)2(
4)(
We know that the density of states i.e., the number of energy states per unit volume within the energy interval E and E + dE is given by
dEEmh
dEEZ 2
1
2
3
3)2(
4)(
)exp()(exp)(
)exp(
1)(
res temperatupossible allFor
)exp(1
1)(
kT
EE
kT
EEEF
kT
EEEF
kTEEkT
EEEF
FF
f
F
f
Probability of an Electron occupying an energy state E is given by
)2.....()exp()()exp()2(4
)exp()()2(4
)exp()()2(4
)()(
2
1
2
3
3
2
12
3
3
2
1
2
3
3
band theof top
c
c
c
c
E
cF
e
E
Fce
E
Fce
E
dEkT
EEE
kT
Em
hn
dEkT
EEEEm
hn
dEkT
EEEEm
hn
dEEFEzn
Substitute Z(E) and F(E) values in Equation (1)
)3.....()(exp)()exp()2(4
)(exp)()exp()2(4
)exp()()exp()2(4
0
2
1
2
3
3
0
2
1
2
3
3
0
2
1
2
3
3
dxkT
xx
kT
EEm
hn
dxkT
xEx
kT
Em
hn
dEkT
EEE
kT
Em
hn
dxdE
xEE
xEE
cFe
cFe
cF
e
c
c
To solve equation 2, let us put
)exp()2
(2
}2
){()exp()2(4
2
3
2
2
1
2
3
2
3
3
kT
EE
h
kTmn
kTkT
EEm
hn
cFe
cFe
)3(
2)()exp()(
2
1
2
3
0
2
1
equationinsubstitute
kTdEkT
xxthatknowwe
The above equation represents Number of electrons per unit volume of the Material
Calculation of density of holes
)1......(..........)}(1){(
)}(1{)(
band theof bottom
Ev
dEEFEzp
EFdEEZdp
Let ‘dp’ be the Number of holes or Vacancies in the energy interval ‘E and E + dE’ in the valence band
Where Z(E) dE is the density of states in the energy interval E and E + dE and 1-F(E) is the probability of existence of a hole.
dEEmh
dEEZ h2
1
2
3
3)2(
4)(
Density of holes in the Valence band is
Since Ev is the energy of the top of the valence band
dEEEmh
dEEZ vh2
1
2
3
3)()2(
4)(
)exp()(1
exp
)}exp(1{1)(1
})exp(1
1{1)(1
1
kT
EEEF
valuesThigherfor
ansionaboveintermsorderhigherneglectkT
EEEF
kT
EEEF
f
f
f
Probability of an Electron occupying an energy state E is given by
)2....()exp()()exp()2(4
)exp()()2(4
)}(1){(
2
1
2
3
3
2
1
2
3
3
band theof bottom
v
v
E
vF
h
E
Fvh
Ev
dEkT
EEE
kT
Em
hp
dEkT
EEEEm
hp
dEEFEzp
Substitute Z(E) and 1 - F(E) values in Equation (1)
0
2
1
2
3
3
0
2
1
2
3
3
2
1
2
3
3
)exp()()exp()2(4
))(exp()()exp()2(4
)exp()()exp()2(4
dxkT
xx
kT
EEm
hp
dxkT
xEx
kT
Em
hp
dEkT
EEE
kT
Em
hp
dxdE
xEE
xEE
Fvh
vFh
E
vF
h
v
v
v
To solve equation 2, let us put
)exp()2
(2
2))(exp()2(
4
2
3
2
2
1
2
3
2
3
3
kT
EE
h
kTmp
kTkT
EEm
hp
Fvh
Fvh
The above equation represents Number of holes per unit volume of the Material
Intrinsic Carrier ConcentrationIn intrinsic Semiconductors n = pHence n = p = n i is called intrinsic Carrier Concentration
)2
exp()()2
(2
)2
exp()()2
(2
)}exp()2
(2)}{exp()2
(2{
4
3
2
3
2
4
3
2
3
2
2
3
22
3
2
2
kT
Emm
h
kTn
kT
EEmm
h
kTn
kT
EE
h
kTm
kT
EE
h
kTmn
npn
npn
ghei
cvhei
FvhcFei
i
i
Fermi level in intrinsic Semiconductors
sidesboth on logarithms taking
)exp()()2
exp(
)exp()2
()exp()2
(
)exp()2
(2)exp()2
(2
pn torssemiconduc intrinsicIn
2
3
2
3
22
3
2
2
3
22
3
2
kT
EE
m
m
kT
E
kT
EE
h
kTm
kT
EE
h
kTm
kT
EE
h
kTm
kT
EE
h
kTm
cv
e
hF
FvhcFe
FvhcFe
E
Ef
Ev
Valence band
Ec
Conduction band
Ec
Electronenergy
Temperature
**eh mm
Thus the Fermi energy level EF is located in the middle of the forbidden band.
)2
(
that know tor wesemiconduc intrinsicIn
)2
()log(4
3
)()log(2
32
2
3
cvF
he
cv
e
hF
cv
e
hF
EEE
mm
EE
m
mkTE
kT
EE
m
m
kT
E
Extrinsic Semiconductors
• The Extrinsic Semiconductors are those in which impurities of large quantity are present. Usually, the impurities can be either 3rd group elements or 5th group elements.
• Based on the impurities present in the Extrinsic Semiconductors, they are classified into two categories.
1. N-type semiconductors 2. P-type semiconductors
When any pentavalent element such as Phosphorous, Arsenic or Antimony is added to the intrinsic Semiconductor , four electrons are involved in covalent bonding with four neighboring pure Semiconductor atoms.
The fifth electron is weakly bound to the parent atom. And even for lesser thermal energy it is released Leaving the parent atom positively ionized.
N - type Semiconductors
N-type Semiconductor
Si
Si
SiPSi
Free electron
Impure atom (Donor)
The Intrinsic Semiconductors doped with pentavalent impurities are called N-type Semiconductors. The energy level of fifth electron is called donor level. The donor level is close to the bottom of the conduction band most of the donor level electrons are excited in to the conduction band at room temperature and become the Majority charge carriers.
Hence in N-type Semiconductors electrons are Majority carriers and holes are Minority carriers.
E Ed
Ev
Valence band
Ec
Conduction band
Ec
Electronenergy
Distance
Donor levelsEg
Carrier Concentration in N-type Semiconductor
• Consider Nd is the donor Concentration i.e., the number of donor atoms per unit volume of the material and Ed is the donor energy level.
• At very low temperatures all donor levels are filled with electrons.
• With increase of temperature more and more donor atoms get ionized and the density of electrons in the conduction 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 Fdddd
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
kath
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
hkTm
N
kT
EE
hkTm
N
kT
EE
kT
EE
kT
E
hkTm
N
kT
EE
kT
EE
kT
E
hkTm
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
hkTm
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
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 breading of covalent bonds and the material tends to behave in intrinsic manner.
The Fermi level gradually moves towards the intrinsic Fermi level Ei.
P-type semiconductors
• When a trivalent elements such as Al, Ga or Indium have three electrons in their outer most orbits , added to the intrinsic semiconductor all the three electrons of Indium are engaged in covalent bonding with the three neighboring Si atoms.
• Indium needs one more electron to complete its bond. this electron maybe supplied by Silicon , there by creating a vacant electron site or hole on the semiconductor atom.
• Indium accepts one extra electron, the energy level of this impurity atom is called acceptor level and this acceptor level lies just above the valence band.
• These type of trivalent impurities are called acceptor impurities and the semiconductors doped the acceptor impurities are called P-type semiconductors.
Si
Si
SiInSi
HoleCo-Valent bonds
Impure atom (acceptor)
E
Ea
Ev
Valence band
Ec
Conduction band
Ec
Electronenergy
temperature
Acceptor levels
Eg
• Even at relatively low temperatures, these acceptor atoms get ionized taking electrons from valence band and thus giving rise to holes in valence band for conduction.
• Due to ionization of acceptor atoms only holes
and no electrons are created.
• Thus holes are more in number than electrons and hence holes are majority carriers and electros are minority carriers in P-type semiconductors.
• Equation of continuity:
• As we have seen already, when a bar of n-type germanium is illuminated on its one face, excess charge carriers are generated at the exposed surface.
• These charge carriers diffuse through out the material. Hence the carrier concentration in the body of the sample is a function of both time and distance.
• Let us now derive the differential equation which governs this fundamental relationship.
• Let us consider the infinitesimal volume element of area A and length dx as shown in figure.
• If tp is the mean lifetime of the holes, the holes lost per sec per unit volume by recombination is p/tp .
• The rate of loss of charge within the volume under consideration
pt
peAdx
If g is the thermal rte of generation of hole-electron pairs per unit volume, rate of increase of charge wthin the volume under consideration
eAdxg
• If i is the current entering the volume at x and i + di the current leaving the volume at x + dx, then decrease of charge per second from the volume under consideration = di
• Because of the above stated three effects the hole density changes with time.
• Increase in the number of charges per second within the volume
pdt
dpeAdx
Increase = generation - loss
dIt
peAdxeAdxg
dt
dpeAdx
pp
Since the hole current is the sum of the diffusion current and the drift current
EApedx
dpAeDI hp
Where E is the electric field intensity within the volume. when no externalfield is applied, under thermal equilibrium condition, the hole density attains a constant value .0p
.conditions mequilibriuunder
ion recombinat todue loss of therate toequal is holes
of generation of rate that theindicatesequation this
0dt
dp and 0di conditions eunder thes
0
pt
pg
dx
pEd
x
pD
t
pp
dt
dp
eqcombain
hpp
s
)()(
.5&4,3...,.
2
20
This is called equation of conservation of charge or the continuity equation.
x
En
x
nD
t
nn
t
n
x
Ep
x
pD
t
pp
t
p
x
pE
x
pD
t
pp
t
p
pe
pn
e
ppp
nh
np
p
nnn
hpp
)()(
material type-p in the electrons gconsiderin are weif
)()(
material type-n in the holes gconsiderin are weif
)()(
used. be should sderivative partial
x,andboth t offunction a is p if
2
20
2
20
2
20
Direct band gap and indirect band gap semiconductors:
• We known that the energy spectrum of an electron moving in the presence of periodic potential field is divided into allowed and forbidden zones.
• In crystals the inter atomic distances and the internal potential energy distribution vary with direction of the crystal. Hence the E-k relationship and hence energy band formation depends on the orientation of the electron wave vector to the crystallographic axes.
• In few crystals like gallium arsenide, the maximum of the valence band occurs at the same value of k as the minimum of the conduction band as shown in below. this is called direct band gap semiconductor.
Valence band
Conduction band
gE
k
E
k
E
gE
Valence band
Conduction band
• In few semiconductors like silicon the maximum of the valence band does not always occur at the same k value as the minimum of the conduction band as shown in figure. This we call indirect band gap semiconductor.
• In direct band gap semiconductors the direction of motion of an electron during a transition across the energy gap remains unchanged.
• Hence the efficiency of transition of charge carriers across the band gap is more in direct band gap than in indirect band gap semiconductors.
Hall effect
When a magnetic field is applied perpendicular to a current carrying conductor or semiconductor, voltage is developed across the specimen in a direction perpendicular to both the current and the magnetic field. This phenomenon is called the Hall effect and voltage so developed is called the Hall voltage.
Let us consider, a thin rectangular slab carrying current (i) in the x-direction.If we place it in a magnetic field B which is in the y-direction.Potential difference Vpq will develop between the faces p and q which are perpendicular to the z-direction.
i
B
P
Q
X
Y
Z
+ + ++ +VH
++++++++ +++++++
++ + +
+
-
P – type semiconductor
i
B
X
Y
Z
VH
+
-
__ _
__ _
_
__ __
_ _
_
_
_
_ _ P
Q
N – type semiconductor
Magnetic deflecting force
citydrift velo is vWhere
)(
)(
d
BvE
qEBvq
dH
Hd
Hall eclectic deflecting force
HqEF
When an equilibrium is reached, the magnetic deflecting force on the charge carriers are balanced by the electric forces due to electric Field.
)( BvqF d
ne
Jvd
The relation between current density and drift velocity is
Where n is the number of charge carriers per unit volume.
JB
E
netcoefficienHallR
JBRE
JBne
E
Bne
JE
BvE
HH
HH
H
H
dH
1),.(
)1
(
)(
)(
If VH be the Hall voltage in equilibrium ,the Hall electric field.
IB
tVR
Bt
IRV
JBdRVdt
IJ
d
V
JBR
JB
ER
d
VE
HH
HH
HH
HH
HH
HH
)(
density current anddt issection cross itsThen
sample, theof thickness theis t If
1
slab. theof width theis d Where
• Since all the three quantities EH , J and B are measurable, the Hall coefficient RH and hence the carrier density can be found out.
• Generally for N-type material since the Hall field is developed in negative direction compared to the field developed for a P-type material, negative sign is used while denoting hall coefficient RH.