CSE 477. VLSI Systems Design - GUC · Dr. Hassan Mostafa ىفطصم نسح .د [email protected] . ELCT 503: Semiconductors German University in Cairo (GUC) ... C V pn N N e g

ELCT 503: Semiconductors German University in Cairo (GUC)

ELCT503

Semiconductors Fall 2014

Lecture 02: Intrinsic Semiconductors (Cont.)

Dr. Hassan Mostafa

حسن مصطفى. د

[email protected]


Contents

Semiconductors

Crystal structure

Band diagram

Intrinsic carrier concentration

Influence of temperature


Electrons and holes in si

Eg = 1.12 eV

Conduction band

Valence band

electron

hole

Prohibited band

EC

EV

E


INTRINSIC CARRIER CONCENTRATION

Intrinsic = no impurities (Pure Silicon)

Fermi distribution function F(E):

The probability that an electron occupies an electronic state with energy E

Probability that a hole occupies an electronic state with energy E = 1-f(E)

1

1)(

(

kTEE F

EF)/

e

EF Fermi level

k Boltzmann’s constant

T temperature in kelvins


INTRINSIC CARRIER CONCENTRATION

Electron density n : the number of electrons per unit volume (Units: cm-3)

n(E) = the electron density at energy level E

N(E) = density of states = number of allowed energy states per energy

range per unit volume

Number of electrons in the conduction band:

This looks like you sum the product of each energy level by the

probability of finding electrons in that energy level

CE

f(E)N(E)dEn


Density of energy states

2123

32

4 /

c

/

e )E(E)m(h

πN(E)

2123

32

4 /

v

/

h E)(E)m(h

πN(E)

in the conduction band:

in the valence band:

where: h – Planc’s constant, [h] = J·s=kg ·m2 · s-1

me – effective mass of electron

mh – effective mass of hole


Density of energy states

There are a large number of allowed states in the conduction band. However, for an intrinsic semiconductor, the probability of an electron occupying one of these states is small. Thus, there will not be many electrons in the conduction band.

There also are a large number of allowed states in the valence band. the probability of an electron occupying one of these states in the valence band is nearly unity. Thus, most of these energy states are occupied by electrons.

Note that there will be only a few unoccupied electron states in the valence band, that is, holes, in the valence band.


Carrier concentration

CE

i f(E)N(E)dEnpn

CB

VB

E

N(E)

EG

CB

VB

E

n(E),p(E)

EG

density of states carrier

concentration

CB

VB

E

F(E)

EG

0 1 0.5

EC

EV

EFi

Fermi distribution

function

n(E)

p(E)


Electrons in conduction band

Total number of electrons in the conduction band:

C

FC

E

)/kTE(E

CeNf(E)N(E)dEn

Nc – Effective density of states in the conduction band

23

2

212

/

nC

h

kTmπN

23

2

22

/

nC

h

kTmπN

For silicon: For gallium arsenide:

k – Boltzman’s constant, [k] = J· K-1


Holes in valence band

Total number of holes in the valence band:

V

VF

E

)/kTE(E

V eNN(E)dEf(E)p 1

NV – Effective density of states in the valence band

23

2

22

/

hV

h

kTmπN


Fermi level in intrinsic semiconductor

C

VVCiFi

N

NkTEEEE ln

22

from:

)/kTE(E

VVFeNp

)/kTE(E

CFCeNn

and:

inpn Hence:

For an intrinsic semiconductor (thermal equilibrium), the number of

electrons per unit volume in the conduction band is equal to the

number of holes per unit volume in the valence band, that is, n = p =

ni, where ni is the intrinsic carrier density.


Fermi level in intrinsic semiconductor

C

VVCiFi

N

NkTEEEE ln

22

At room temperature, the second term is much smaller than the

bandgap. Hence, the intrinsic Fermi level E, of an intrinsic

semiconductor generally lies very close to the middle of the bandgap.


Intrinsic carrier density

/kTE

VCgeNNpn

2

innp

inpn

kT/ENNn gVCi 2exp


Influence of temperature on band gap

1.125

1.424

GaAs

Si

K636KeV1073.4eV170.1

21-4

g

T

TE

K204KeV10405.5eV519.1

21-4

g

T

TE


Influence of temperature on ni

300K

1000K 250K

1010

GaAs

Si


Summary

/kTE

VCgeNNpn

2

innp

222

g

C

g

VVC

iFi

EE

EE

EEEE

V

VF

E

)/kTE(E

V eNN(E)dEf(E)p 1

C

FC

E

)/kTE(E

CeNf(E)N(E)dEn


Short Break


ELCT503

Semiconductors Fall 2014

Lecture 03: Extrinsic Semiconductors

Dr. Hassan Mostafa

حسن مصطفى. د

[email protected]


content

n-type semiconductors

p-type semiconductors

Mass action law & Charge neutrality

Majority and minority carriers

Influence of temperature on carrier concentration





+5 - -

+4

+4 +4

- -

- -

+4

- - +4

- - +4

- -

+4 - -

+4 - -

- -

- -

- -

- - -

-

-

-

-

-

- -

-

-

- -

CB

VB

x

EG

E

EC

EV

ED

ED donor energy level

Small binding energy to donor atom



EC – ED = Ionization energy


Silicon and other semiconductors

Period II III IV V VI

2 B C N O

3 Mg Al Si P S

4 Zn Ga Ge As Se

5 Cd In Sn Sb Te

6 Hg Pb Bi

donors: pentavalent elements from group V (P,As,Sb,Bi)

→ release of electrons → n-type semiconductor

The most famous are P (Phosphorus) and As (Arsenic)


ionization of donors

complete ionization

of donors:

ionization energy of impurities

DNn

S.M.Sze


Again: Intrinsic carrier concentrations

CB

VB

E

N(E)

CB

VB

E

F(E) 0 0.5 1

E

CB

VB

n(E), p(E)

Fermi distribution

function density of states carrier

concentration

EG EG


Extrinsic carrier concentrations (n-type)

CB

VB

E

N(E)

CB

VB

E

F(E) 0 0.5 1

E

CB

VB

n(E), p(E)

Fermi distribution


concentration

EG EG

EC

CE

f(E)N(E)dEn


Ascending fermi level

DCFC NNkTEE ln

DNn

)/kTE(E

CFCeNn

Valence band

Conduction band

EC

EV

Efi = Ei

ED

EFn



Electron concentration in doped semiconductor

)/kTE(ENn FCC exp

)/kTE(Enn iFi exp

in

)/kTE(E)/kTE(EN FiiCC expexp

)/kTEEE(EN FiiCC exp

Valence band

Conduction band

EC

EV

Efi = Ei

ED

EFn

DCFC NNkTEE ln



Short Break



+3 -

+4

+4 +4

- -

- -

+4

- - +4

- - +4

- -

+4 - -

+4 - -

- -

- -

- -

- - -

-

-

-

-

-

- -

-

-

- -

CB

VB

E

x

EG

EC

EV EA

EA acceptor energy level



EA – EV = Ionization energy


Silicon and other semiconductors

Period II III IV V VI

2 B C N O

3 Mg Al Si P S

4 Zn Ga Ge As Se

5 Cd In Sn Sb Te

6 Hg Pb Bi

acceptors: trivalent elements from group III (B, Al, Ga, In)

→capture of electron→hole remains → p-type semiconductor

The most famous is B (Boron)


ionization of impurities

ionization energy of impurities

ANp

complete ionization

of acceptors:

S.M.Sze


Intrinsic carrier concentrations

Fermi distribution


concentration

CB

VB

E

N(E) 0.5

CB

VB

E

EV

0 1

CB

VB

E

EG

n(E), p(E) F(E)

EG


Extrinsic carrier concentrations (p-type)

Fermi distribution


concentration

CB

VB

E

N(E) 0.5

CB

VB

E

0 1

CB

VB

E

n(E), p(E) F(E)

VE

N(E)dEf(E)p0

1

EG EG


Descending fermi level

AVVF NNkTEE ln

)/kTE(E

VVFeNp

ANp

Valence band

Conduction band

EC

EV

Efi = Ei

EA

EFp



Hole concentration in doped semiconductor

)/kTE(ENp VFV exp

)/kTE(Enp Fii exp

in

)/kTEEE(EN ViiFV exp

)/kTE(E)/kTE(EN iFViV expexp

Valence band

Conduction band

EC

EV

Efi = Ei

EA

EFp


AVVF NNkTEE ln


Mass action law

)/kTE(En)/kTE(Enpn iFiFii expexp

2

inpn

This equation holds for both intrinsic and extrinsic semiconductors


charge neutrality

DA NpNn

In general both acceptors and donors can be present

positive ions negative ions

charge neutrality = zero net charge density


majority and minority carriers

electrons holes

n-doped

(ND > NA)

MAJORITY

carrier

MINORITY

carrier

p-doped

(ND < NA)

MINORITY

carrier

MAJORITY

carrier


Majority and minority carrier (n-type)

majority electrons:

nin nnp /2

22

42

1iADADn nNNNNn

DADnAD NNNnNN

minority holes:

2

inn nnp DnAn NpNn from: and


Majority and minority carrier (p-type)

majority holes:

pip pnn /2

22

42

1iDADAp nNNNNp

ADApDA NNNpNN

minority electrons:

2

ipp nnp DpAp NpNn from: and


Non-degenerate semiconductor

CFCD EENNn

VFVA EENNp

and

DCFC NNkTEE ln

AVVF NNkTEE ln

The approximations used for f(E) can not be used for degenerate semiconductors

(such as heavily doped semiconductors for which NC < n or NV < p and

Fermi levels lie in the conduction band (n-type) and in the valence band (p-type)),

All the above formulas does not work and the integral should be done numerically


Influence of temperature on carrier concentration


electron temperature vs. temperature

S.M.Sze

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CSE 477. VLSI Systems Design - GUC · Dr. Hassan Mostafa ىفطصم نسح .د [email protected] . ELCT 503: Semiconductors German University in Cairo (GUC) ... C V pn N N e g