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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
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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.
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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.
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
Semiconductor Equilibrium
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:
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
Semiconductor Equilibrium
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
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
Semiconductor Equilibrium
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.
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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.
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
)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
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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.
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
Concept of a Donor “Adding extra” Electrons
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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.
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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).
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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|/ .
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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.
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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.
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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
P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - VI
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