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FERMI ENERGY LEVEL CONCEPTS
IES material
2010
Mukund Bihari ASE ,TCS
[email protected] 08882215887
www.mukundbihari.blogspot.com
FERMI ENERGY
Also known as CHARACTERISTIC ENERGY.
Unit- eV
It is defined as the max. energy possessed by an electron at 0K.
Fermi energy is also defined as max. kinetic energy possessed by
an electron at 0K.
It is also defined as the possessed energy by the fastest moving
electron at 0K.
EF = max. K.E.
EF =0.5m(vmax)2
Max. velocity of electron ,
Vmax = (2Ef/m)1/2
FERMI – DIRAC FUNCTION → f(E)
Also known as FERMI DIRAC PROBABILITY.
In a semiconductor or metal
f(E)=
Fermi-Dirac function indicates the probability of electron existing
has a function of energy.
In formula of f(E) , E = energy possessed by an electron in eV
1
1+e(E-Ef)/kT)
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If T = 0K
We get conditions below
E > EF , f(E) =
f(E) = 0
E < EF , f(E) =
f(E) = 1
If T ≠ 0K
If E = EF ,f(E) =
f(E) = 0.5 = 50%
Fermi level energy is also defined as the probability of
existing is 50 % , if forbidden energy band does not exist.
In metal Fermi –Dirac function f(E) = 1 or 100%
In a semiconductor of an electron existing is given by
f(E) and the probability of hole existing is given by
1-f(E).
1
1+e∞
1
1+e--∞
1
1+e0
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FERMI LEVEL IN INTRINSIC SEMICONDUCTOR
The Intrinsic semiconductor
n=p
Nc e-(Ec-Ef)/kT = Nv e
-(Ef-Ev)/kT
NC/NV = e (Ec+Ev-2Ef)/kT
(Ec+Ev – 2Ef)/kT = ln (Nc/Nv)
EF = (Ec+Ev)/2 – kT/2 ln (Nc/Nv) ---------------[1]
In Intrinsic semiconductor Fermi level depends on Temp
CASE I
Let mn = mp
Then Nc = Nv
Now , EF = (EC + EV)/2
The Fermi level exist at the center of forbidden energy gap
CB
VB
EC
EF
EV
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CASE II
Let T = 0K
Eqn [1] will become
EF = (Ec+Ev)/2
In Intrinsic Semiconductor , Fermi level exist exactly at the center of
forbidden energy band when
If mn = mp
When T = 0K
At T = 0K , electron concentration and hole concentrations is zero and
conductivity is zero and Intrinsic semiconductor at 0K is a perfect
insulator.
At 0K
CB
VB
EC
EF
EF
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CASE III
At T = 300K
EF = (Ec+EV)/2 – kT/2 ln (Nc/NV)
Electron conc. = Hole conc.
Because of electron concentration
and concentration .There will be a
conductivity in intrinsic semiconductor T = 300K
at room temperature.
CASE IV
Position of Fermi level at different
temp. T > 300K
As temp. increases electron concentration
Increases & the hole concentration
Increases & therefore conductivity
Increases.
In intrinsic semiconductor σ ↑ temp. ↑.
VB
CB
EF
Hole
Concentration
Electron
Concentration
CB
VB
T > 300K
T = 300K
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FERMI LEVEL IN n –TYPE SEMICONDUCTOR
n = ND
NC e-(Ec-Ef)/kT = ND
Nc/ND = e(Ec-Ef)/kT
ln(NC/ND) = (Ec – EF)/kT
Ec – EF = kT ln(NC/ND)
EF = Ec – kT ln(NC/ND).
In n-Type semiconductor , Fermi Level depends on temp. and doping
concentration.
Case I
If T = 0K , then EF = EC.
EF coincides with EC
CB
VB
ED
EC = EF
EV
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Electron conc. and holes conc. is zero. The n-type semiconductor at 0 K
is an INSULATOR.
Case II
If T = 300K
EF = EC – kT ln(NC/ND).
In n-type semiconductor , Fermi level exist just below the donor
energy level.
+
EC
ED
EF
VB
CB
At 300K
Electron
concentration
Hole concentration
EC
EV
At 300K
EF ED
VB
CB
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CASE III
EC – EF = kT ln(NC/ND).
Let Temp.↑
Let NC ↑ and NC > ND
EC – EF > 0 » EC > EF
As temp. ↑ , n-type semiconductor , EF moves away from CB or EF
moves towards the center of energy gap. Hence σ ↓ temp. ↑.
At curie temp. the Fermi level exist at the center of energy gap.
Let doping ↑
Let ND and ND > NC
EC - EF < 0
As doping ↑ , n-type semiconductor EF moves into CB or EF is away
from the center of the energy gap.Hence , σ increasing with doping.
In n-type semiconductor as doping increases Fermi-level takes
upward shift.
300k
T > 300K
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In a highly doped n-type semiconductor or highly (doped)
degenerative n –type semiconductor , the Fermi level lies in the
conduction band.
N+ semiconductor at 300K
CASE IV
Shift in the position of EF of N-type semiconductor w.r.t the center of
the enery gap. Or
Shift in the position of EF of N-type semiconductor due to doping is
given by
Hole concentration
Electron concentration
EC
EV Hole
concentration
Electron
concentration
VB
CB
EC
ED
EV
Shift = KT loge(ND/ni) eV
Shift = KT loge(n/ni) eV
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FERMI – LEVEL IN P - TYPE SEMICONDUCTOR
P ≈ NA
NV e (Ef-Ev)/kT = NA
NV/ NA = e (Ef-Ev)/kT
ln(NV/ NA ) = (EF – EV)/kT
In the P – type semiconductor, Fermi-level is a function of temp. and
doping concentration.
CASE I
At T = 0K
EF = Ev
At 0K , electron concentration
and hole concentration are zero
& therefore conductivity is zero
And p-type semiconductor will
work as INSULATOR.
EF = EV + kT ln(NV/ NA )
VB
CB
EC
EV = EF
At 0K
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CASE II
T = 300K
It means that semiconductor
Fermi-level exist just above
the acceptor energy level.
CASE III
EF - EV = kT ln(NV/ NA )
As Temp ↑
Let NV ↑ & NV > NA
EF - EV > 0 »
In p-type semiconductor as Temp. ↑
EF moves away from VB or EF moves
towards the center of the energy gap.
Hence σ ↓ with temp.↑
CB
VB Hole concentration
Electron
concentration
Hole conc. > electron conc.
EF > EV
T > 300K
T = 300K
CB
VB
EC
EA
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As doping ↑
In p-type semiconductor at curie temp. Fermi-Level exists at the center
of energy gap
i.e. NA ↑ & let NA > NV
EF - EV < 0 »
As doping ↑ , in p-type semiconductor , EF moves into the VB or EF
will be shifting away from the center of energy gap.
Hence , in p-type semiconductor σ↑ with doping ↑.
In a highly doped p-type semiconductor or highly degenerative p-type
semiconductor Fermi-Level exist in the valency band.
At 300K
P+ semiconductor at 300K
EF < EV
CB
VB
EA
EV Hole concentration
electron concentration
EC
EV
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In p-type semiconductor as doping increases Fermi-Level takes
downward shift.
CASE IV
Shift in the position of EF p-type semiconductor due to doping or shift
in the position of EF of p-type semiconductor w.r.t. EF of intrinsic
semiconductor is given by
Shift = kT ln(NA / ni) eV
Shift = kT ln(p / ni) eV
shift
VB
CB
EA
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When doping is suddenly introduced in a semiconductor :
In an INTRINSIC semiconductor , at first the conductivity
decreases and thereafter the conductivity increases with doping.
In beginning , conductivity falls because when doping is
introduced few charge carrier are created and therefore the mean
path of electrons and holes get reduced .Therefore the
conductivity decreases .
And when semiconductor enters into steady state the conductivity
σ increases(↑) with the doping(↑).
When ND and NA doping are simultaneously introduced into the semiconductor :
In an INTRINSIC semiconductor , if donor and acceptor impurities are simultaneously introduced then
i. ND > NA------semiconductor turns n-type
ii. NA > ND -----semiconductor turns p-type
iii. NA = ND-----semiconductor remains intrinsic
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LOW LEVEL INJECTION
If the concentration of minority carriers is negligible when compared to the concentration of majority carrier , the semiconductor is under low
level injection.
When minority carrier are introduced
into the semiconductor , these will be
moving from higher concentration to
lower concentration and this minority
carrier flow due to diffusion .
Under low level injection hole drift
current is negligible when compared to hole diffusion current. Hence ,
under low level injection minority carrier current s only due to
diffusion.
WHEN LIGHT FALL ON A SEMICONDUCTOR
When light falls on a semiconductor minority carrier are generated .
The photon energy will ionize the covalent bond and equal no. of electrons and holes are generated.Therefore under steady state condition
∆n = ∆p
The minority carrier so created moves from higher concentration to
lower concentration and therefore this minority carrier moves under
diffusion.
n-type
n≈1016 cm-3
n >> ni
Minority carrier concentration
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When light falls on n-type semiconductor there are two components of
current:
1. Hole drift current
2. Hole diffusion current(dominating over hole drift current)
The generation rate of minority carriers in n-type semiconductor is
Unit : e-h pair/cm3/sec
When light is focused on the n-type semiconductor , the rate of
generation minority carrier is given by following :
p0 denotes hole concentration under thermal equilibrium in n-type semiconductor.
(p-p0)
(p-p0)e-x/lp
Lp
Light is
Light is turned ON turned OFF
dp excess hole generated
dx Minority carrier (hole) life time
=
Excess hole concentration
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The length of diffusion of the hole is defined as the distance into the semiconductor in which the injected concentration falls to 1/e of its peak value.
If the distance x is slightly greater than Lp .The excess hole concentration will be reduce to zero.
PHOTOCONDUCTORS
They are also called PHOTORESISTORS.
When light falls on a semiconductor , its conductivity increases or its resistivity decreases. When light falls on a semiconductor, the photon energy will ionize the covalent bond.Therefore σ ↑ as or ρ ↓.
The property due to which σ increases with the light is called PHOTOCONDUCTIVE EFFECT.
The property due to which the resistivity(ρ) of the material decreases with light is called PHOTORESISTIVE EFFECT.
Photoconductive effect is also called photoresistive effect.
If electrons are excited from valance band to conduction band , it is INTRINSIC EXCITATION.
Minimum photon energy required for intrinsic excitation = Energy Gap.
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According to photon energy equation
For intrinsic excitation
hμ = EC2
A photon energy can excite an electron from donor energy level into
conduction band in the n-type semiconductor OR photon energy can
excite from valance band into acceptor energy level in the p-type
semiconductor .This phenomenon is known as EXTRINSIC EXCITING.
For extrinsic excitation the minimum photon energy required is
0.01 eV → Ge
0.05 eV → Si
n-type p-type
EC2 =hμ = hc/λ
CB
VB
EC2
CB CB
VB VB
ED
EA
EC
EV
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WAVELENGTH OF RADIATING LIGHT
The response curve of response
Ge and Si material
When compared to Si , Ge Si
is more sensitive to light. Ge
wavelength of visible light λ
is in range of 0.38 μm to 0.76 μm.
λ = 1.24/EC2 μm
1.2 1.7