fermi energy

[Type text]

FERMI ENERGY LEVEL CONCEPTS

IES material

2010

Mukund Bihari ASE ,TCS

[email protected] 08882215887

mailto:[email protected]

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)


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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.


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


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

fermi energy

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