Energy bands semiconductors Dr. Md.Shakowat Zaman Sarker Assistant Professor Dept. of EEE...

Energy bands semiconductors

Dr. Md.Shakowat Zaman SarkerAssistant Professor

Dept. of EEEInternational Islamic University

Chittagong

2

Outlines

Energy bands, Metals, Semiconductor and Insulators, Direct and indirect semiconductor, variation of Energy band with alloy composition, Electrons and Holes, Effective mass, intrinsic and Extrinsic Semiconductors, Electrons and Holes and hole in quantum wells

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3-1-3. Metals, Semiconductors & Insulators

The difference bet-ween insulators and semiconductor mat-erials lies in the size of the band gap Eg, which is much small-er in semiconductors than in insulators.

Insulator Semiconductor

Filled

Filled

Empty

Empty

Eg

Eg

4

3-1-3. Metals, Semiconductors & Insulators

Metal

Filled

Partially Filled

Overlap

In metals the bands either overlap or are only partially filled. Thus electrons and empty energy states

Metal

are intermixed with-in the bands so that electrons can move freely under the infl-uence of an electric field.

3-2. Carriers in Semiconductors

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The semiconductor has filled valance band and empty conduction band at 0K, we must consider the increase in conduction band electrons by thermal excitations across the band gap as temperature is raised. In addition, after electrons are excited to the conduction band, the empty states left in the valance band can contribute to the conduction process.Impurities has an important effect on the energy band structure and on the availability of charge carriers

3-2-1. Electrons and Holes

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As the temperature of a semiconductor is raised from 0K, some electrons in the valance band receive enough thermal energy to be excited across the band gap to the conduction band. The result is a material with some electrons in an otherwise empty conduction band and some unoccupied states in an otherwise filled valance band. An empty state in a valance band is refer to as hole. If the conduction band electron and the hole are created by the excitation of valance band electron to the conduction band, they are called an electron-hole-pair (EHP)

3-2-1. Electrons and Holes

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Hole current is really due to an electron moving in the opposite direction in the valence band.Electron current is an electron moving from state to state in the conduction band.

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3-2. Carriers in Semiconductors

Ec

Ev

Eg

0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK300ºK

15ºK16ºK17ºK18ºK19ºK20ºK

Electron Hole PairE H P

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3.2.2 Effective mass

Electrons in a crystal are not totally free. The periodic crystal affects how electrons

move through the lattice. We use and effective mass to modify the

mass of an electron in the crystal and then use the E+M equations that describe free electrons.

10

k

E

2

2

2

2

2

2

22

2

*

22

1

dkEd

m

mdk

Ed

km

mvE

kmvp


11

The double derivative of E is a constant Not all semiconductors have a perfectly

parabolic band structure The different atomic spacing in each

direction gives rise to different effective masses in different crystal directions. This can be compensated by using an average value of effective mass.


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3.2.2 Effective mass (for density of states calculation)

Ge Si GaAs mn* 0.55 m0 1.1 m0 0.067 m0

mp* 0.37 m0 .56 m0 0.48 m0

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3-2-3. Intrinsic Material

A perfect semiconductor crystal with no

impurities or lattice defects is called an

Intrinsic semiconductor.

In such material there are no charge

carriers at 0ºK, since the valence band is

filled with electrons and the conduction

band is empty.

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3-2-3. Intrinsic Material

SiEgh+

e-

n=p=ni

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3-2-3. Intrinsic Material If we denote the generation rate of EHPs

as and the recombination rate

as equilibrium requires that:

)(Tgi

)( 3scmEHPri

ii gr Each of these rates is temperature

depe-ndent. For example,

increases when the temperature is

raised.

)( 3scmEHPgi

iirri gnpnr 200

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3-2-4. Extrinsic Material

In addition to the intrinsic carriers generated thermally, it is possible to create carriers in semiconductors by purposely introducing impurities into the crystal. This process, called doping, is the most common technique for varying the conductivity of semiconductors.

When a crystal is doped such that the equilibrium carrier concentrations n0 and p0

are different from the intrinsic carrier concentration ni , the material is said to be

extrinsic.

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0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK50ºK15ºK16ºK17ºK18ºK19ºK20ºK

Ec

Ev

Ed

Donor

V

P

As

Sb

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0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK50ºK15ºK16ºK17ºK18ºK19ºK20ºK

Ec

Ev

Ea

Acceptor

ш

B

Al

Ga

In

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h+

Al

e- Sb

Si

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We can calculate the binding energy by using the Bohr model results, consider-ing the loosely bound electron as ranging about the tightly bound “core” electrons in a hydrogen-like orbit.

rKnhK

mqE 022

4

4, 1;2

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Example 3-3: Calculate the approximate donor binding energy for Ge(εr=16, mn

*=0.12m0).

22


eVJ

h

qmE

r

n

0064.01002.1

)1063.6()161085.8(8

)106.1)(1011.9(12.0

)(8

21

234212

41931

220

4*

Answer:

Thus the energy to excite the donor electron from n=1 state to the free state (n=∞) is ≈6meV.

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When a ш-V material is doped with Si or Ge, from column IV, these impurities are called amphoteric.

In Si, the intrinsic carrier concentration ni is about 1010cm-3 at

room tempera-ture. If we dope Si with 1015 Sb Atoms/cm3, the conduction electron concentration changes by five order of magnitude.

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3-3. Carriers Concentrations

In calculating semiconductor electrical pro-perties and analyzing device behavior, it is often necessary to know the number of charge carriers per cm3 in the material.

The majority carrier concentration is usually obvious in heavily doped material, since one majority carrier is obtained for each impurity atom (for the standard doping impurities).

The minority carriers concentration is not obvious, however, nor is the temperature dependence of the carrier concentration.

To obtain equations for the carrier concentrations we must investigate the distribution of carriers over the available energy states. The distribution function as given:

Donors and Acceptors Fermi level , Ef Carrier concentration equations Donors and acceptors both present

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3-3-1. The Fermi Level Electrons in solids obey Fermi-Dirac statistics. In the development of this type of statistics:

Indistinguishability of the electrons Their wave nature Pauli exclusion principle

must be considered. The distribution of electrons over a range of

these statistical arguments is that the distrib-ution of electrons over a range of allowed energy levels at thermal equilibrium is

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3-3-1. The Fermi Level

kTfEE

eEf )(

1

1)(

k : Boltzmann’s constant

f(E) : Fermi-Dirac distribution function

Ef : Fermi level

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2

1

11

1

1

1)( )(

kTfEfE

eEf f

An Energy E equal to the fermi level energy EF, the occupation probability is

The Energy state at the fermilevel has a probability of ½ of being occupied by an electron.

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Ef

f(E)

1

1/2

E

T=0ºKT1>0ºKT2>T1

Fermi-Dirac distribution Function

Exponent positive: F(E)=1 for E<FE

Exponent positive: F(E)=0 for E>FE

T=0


In intrinsic material, the concentration of electron in conduction band and concentration of Hole in valance band is equal. The fermi level FE must lie at the middle of band-gap.

In n-type material, higher concentration of electron in the conduction band compare with hole concentration in valance band.

In p-type material, the FE lie near the valance band

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31


Ev

Ec

Ef

E

f(E)01/21

≈≈

f(Ec

)f(Ec

)

[1-f(Ec)]

Intrinsicn-typep-type

Fermi distribution function applied to Semiconductor

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3-3-2. Electron and Hole Concentrations at Equilibrium

EC

EV

Ef

E

Holes

Electrons

Intrinsicn-typep-type

N(E)[1-f(E)]

N(E)f(E)

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CE

dEENEfn )()(0

The concentration of electrons in the conduction band is

N(E)dE : is the density of states (cm-3) in the energy range dE.

O: Electron and Hole concentration symbol in equilibrium condition.

dE: N# of Electron per unit volume in energy range

N(E): Can be calculate by quantum mechanics


The result of the integration is the same as that obtained if we repres-ent all of the distributed electron states in the conduction band edge EC. The conduction band Electron concentration is simply the effective density of states at Ec times of probability of occupancy at Ec

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)(0 CC EfNn

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kTFECE

kTFECE

ee

Ef C

)(

)(

1

1)(

kTFECE

eNn C

)(

0

23

) 2

(22

*

h

kTmN nC

Fermi function can be simplified as

Concentration of Electron in conduction band

The Effective density of states Nc

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)](1[0 VV EfNp

kTVEFE

kTFEVE

ee

Ef V

)(

)(

1

11)(1

kTVEFE

eNp V

)(

0

23

) 2

(22

*

h

kTmN pV

Similar argument, the concentration of holes


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The electron and hole concentrations are valid whether the material is intrinsic or doped, provided thermal equilibrium is maintained. Thus for intrinsic material, lies at some intrinsic level Ei near the middle of band gap, and the intrinsic electron and hole concentration are:

kTiEcE

eNn Ci

)(

kTvEiE

eNp Vi

)(


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The product of no and po at equilibrium is a constant for a particular material and temperature, even is doping is varied:

kTgE

kTvEcE

eNNeNNpn vcvc

)(

00

kTgE

eNNpn vcii


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The intrinsic electron and hole concentration are equal, ni=pi; thus the intrinsic concentration is

kTgE

eNNn vci2

The product of electron and hole concentration

200 inpn

kTFEiE

enp i

)(

0

kTiEFE

enn i

)(

0

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Example 3-4: A Si sample is doped with 1017 As Atom/cm3. What is the equilibrium hole concentra-tion p0 at 300°K? Where is EF relative to Ei?

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3317

20

0

2

0 1025.210

1025.2

cmn

np i

Answer: Since Nd»ni, we can approximate

n0=Nd and

kTiEFE

enn i

)(

0

eVn

nkTEE

iiF 407.0

105.1

10ln0259.0ln

10

170

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Answer (Continue) :

Ev

Ec

EF

Ei1.1eV0.407eV

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4.3 Carrier lifetime and photo-conductivity

Direct recombination of Electrons and hole

Electron drops from conduction band to the valence band and recombines with a hole without any change in momentum (E vs K) .

The energy difference is used up in an emitted photon.

This process occurs at a certain rate in the form of how long does a free electron or hole remain free before it recombines (n or p)

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Direct recombination of Electrons and hole n or p are dependant on doping level,

crystal quality and temperature. Indirect recombination; Trapping

The probability of a direct recombination is small in Si and Ge.

A trapping level is needed. No photons generated just phonons (lattice vibrations)

Minority carrier lifetime dominates recombination process.

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The Fermi level (EF) is only meaningful at thermal equilibrium.

Under excitation we use the quasi Fermi level to denote excess hole and electron concentrations.

oppKTFE

i

opnkTEF

i

gppppenp

gnnnnenn

pi

in

,,

,,

0/)(

0/)(

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4.4 Diffusion of carriers

Diffusion process The random motion of similar particles from a

volume with high particle density to volumes with lower particle density

A gradient in the doping level will cause electron or hole flow, which causes an electric field to build up until the force from the gradient equals the force of the electric field.

no current will flow at equilibrium

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4.4Diffusion of carriers

Diffusion process t is the mean free time that 1/2 of the particle will enter the

next dx segment. l is the mean free path of a particle between collisions.

dx

xdpqD

dx

xdpDqdiffJp

dx

xdpD

dx

xdp

t

lx

dx

xdnqD

dx

xdnDqdiffJn

dx

xdnD

dx

xdn

t

lx

pppp

nnnn

)()()(.)(,

)()(

2)(

)()()(.)(,

)()(

2)(

2

2

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4.4 Diffusion and drift of carriers

Drift diffusion equations The hole drift and diffusion current densities are in the same direction. The electron drift and diffusion current densities are in the opposite direction.

)()()( xJxJxJ pn

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4.4 Diffusion and drift of carriers

Drift diffusion equations Minority current flow is primarily diffusion. Majority current flow is primarily drift.

An applied electric field will cause a positive slope in E i (Ev and Ec as well) This can be used to derive the Einstein relation.

q

kTD

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Continuity equation Rate of hole build up = increase of

hole concentration in the volume - the recombination rate

n

p

n

x

Jn

qt

n

p

x

Jp

qt

p

1

1

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Diffusion length Lp is the average distance a hole will

move before recombining. Ln is the average distance an electron

will move before recombining.

ppp

nnn

DL

DL

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References:

Solid State Electronic Devices Ben G. Streetman, third edition

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Energy bands semiconductors Dr. Md.Shakowat Zaman Sarker Assistant Professor Dept. of EEE...

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