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Semiconductor Device Theory I: Equilibrium Distribution & Carrier Concentration, Study notes of Physics of semiconductor devices

Arizona State University (ASU) - Tempe Physics of semiconductor devices

An overview of the topics covered in eee 531: semiconductor device theory i at arizona state university. The instructor is dragica vasileska. The definition and calculation of the equilibrium distribution function, the fermi-dirac distribution function, and the maxwell-boltzmann distribution function. It also covers the calculation of electron and hole concentrations in a semiconductor using the effective density of states in the conduction and valence bands.

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EEE 531: Semiconductor Device Theory I

Instructor: Dragica Vasileska

Department of Electrical Engineering

Arizona State University

Topics covered:

•Equilibrium distribution function

•Calculation of n and p

•Fermi-Dirac integral

•Donors and Acceptors

•Fermi level calculation

•Impurity bands

Discover Study notes of Physics of semiconductor devices Arizona State University (ASU) - Tempe

Partial preview of the text

Download Semiconductor Device Theory I: Equilibrium Distribution & Carrier Concentration and more Study notes Physics of semiconductor devices in PDF only on Docsity!

EEE 531: Semiconductor Device Theory I

Instructor: Dragica Vasileska

Department of Electrical Engineering

Arizona State University

Topics covered:

Equilibrium distribution function
Calculation of n and p
Fermi-Dirac integral
Donors and Acceptors
Fermi level calculation
Impurity bands

The probability of a state with energy E being occupied

with spin 1/2 electrons, for which the Pauli exclusion

principle is valid, is given by the following function:

Equilibrium distribution function:

k T

E E

f E

1 exp

Fermi-Dirac distribution functi-

on, valid in thermal equilibrium.

In non-equilibrium conditions,

one actually has to solve for the

distribution function.

fn(E)

T=0 K

T>0 K

EF=Fermi level (electrochemical potential)

Calculation of n and p :

The electron concentration in a semiconductor is calculated

according to:

With the introduction of the following variables:
we arrive at the following expression for n:

( )

E E dE

n g E f E dE

k T

E E E

c n

F c

−

∞

∫ − 













= ∫

1 exp

(^21)

( ) ( )

3 / 2

2 h

k T

E E

k T

E E

F c

c − η =

ε = ,

3 / 2

1 / 2 1 / 2

 π



= η = h

m k T N k T

E E

n N F N F

dn B c B

F c c c

Effective DOS

in the C.B.

Similarly, for holes one gets:
Values for Nc and Nv of some typical semiconductors at

T=300 K:

3 / 2

1 / 2 1 / 2

 π

= 

= η = h

m k T N k T

E E

p N F N F

dp B v B

v F v v

Effective DOS

in the V.B.

Material Nc [cm

] Nv [cm - ]

Si 2.8x

19 1.04x

Ge 1.04x

19 6.0x

GaAs 4.7x

17 7.0x

Degenerate vs non-degenerate semiconductors

When EF is in the energy gap, and is separated by more

than several thermal energies (kBT) from the edges of either

the C.B. or the V.B., the semiconductor (SC) is called non-

degenerate:

In the opposite limit, when EF enters either the conduction

or the valence bands, a SC is called degenerate:

η F (η) ≈e 1 / 2

k T

E E

p N

k T

E E

n N

v F v

F c c

exp

3 / 2 1 / 2 3

4 ( ) η π

F η ≈

E (^) c − E F

E E k T degenerate

E E k T non degenerate

c F B

c F B − < ⇒

E (^) F − E v

E E k T degenerate

E E k T non degenerate

F v B

F v B − < ⇒

A sketch of the temperature dependence of the intrinsic

carrier concentration is given below:

room T => ni small to the doping level
high T => ni large due to the thermal generation of the

carriers

ln(ni)

1/T

GaAs

Slope ~ Eg

From the condition that for intrinsic SC:

n=ni and p=ni,

one easily arrives for the expression for the intrinsic Fermi

level:

ln 4

dp B

c v i m

m k T

E E

E

The Fermi level Ei of an intrinsic SC generally lies

very close to the middle of the bandgap.

The Fermi level Ei of an intrinsic SC generally lies

very close to the middle of the bandgap.

•The simplest calculation of impurity energy levels is based

on the hydrogen-atom-like model:

•Based upon the values of Ec-Ed and Ea-Ev one can distin-

guish between: - shallow impurities

deep-level impurities

•The amount of ionized dopants is given by:

m q EH 13. 2 32

2 2 0

4 0 = π ε

E E eV Si

m m

E

a v

c d

Si d (^0). 05 for

025 for

2 0 − ≈

ε ε

− +

k T

E E

N

k T

E E

N

F d

d d

a F

a a

1 2 exp

1 4 exp

=> Ionization energy

Fermi level calculation

The Fermi level adjusts itself to ensure charge neutrality in

the semiconductor. The charge neutrality condition,

combined with the mass-action law, valid for degenerate

SC as well, leads to:

− +

2 i

a d

np n

n N p N

=> mass-action law

− + − +

− + −

(^22)

1 2

(^22)

1 2

a d a d i

d a d a i

p N N N N n

n N N N N n

=> charge-neutrality

High-T limit:

•For p-type SC and high temperatures, the Fermi level is

given by:

•Temperature variation of EF:

v F v B N

N

E E k T ln



 





 

 = − d

c F c B N

N E E k T ln

Energy

n-type

p-type

•Temperature variation of the electron concentration n:

1/T

n/Nd

intrinsic extrinsic

Freeze-out

•To understand what happens, recall the hydrogen atom:

E gc(E)

High doping Nd

The discrete donor level broadens into a

band which overlaps with the conduction

band, leading to the reduction of the

band-gap => band-gap narrowing effect

One atom Two atoms Lattice

Energy level Anti-bonding

Bonding

C.B.

V.B.

Semiconductor Device Theory I: Equilibrium Distribution & Carrier Concentration, Study notes of Physics of semiconductor devices

Related documents

Partial preview of the text

Download Semiconductor Device Theory I: Equilibrium Distribution & Carrier Concentration and more Study notes Physics of semiconductor devices in PDF only on Docsity!

EEE 531: Semiconductor Device Theory I

Instructor: Dragica Vasileska

Department of Electrical Engineering

Arizona State University

Topics covered:

with spin 1/2 electrons, for which the Pauli exclusion

principle is valid, is given by the following function:

Equilibrium distribution function:

k T

E E

f E

1 exp

Calculation of n and p :

according to:

E E

E E

 π

E E

Effective DOS

in the C.B.

T=300 K:

E E

Effective DOS

in the V.B.

Degenerate vs non-degenerate semiconductors

than several thermal energies (kBT) from the edges of either

the C.B. or the V.B., the semiconductor (SC) is called non-

degenerate:

or the valence bands, a SC is called degenerate:

k T

E E

p N

k T

E E

n N

exp

exp

carrier concentration is given below:

carriers

n=ni and p=ni,

one easily arrives for the expression for the intrinsic Fermi

level:

E E

E

The Fermi level Ei of an intrinsic SC generally lies

very close to the middle of the bandgap.

The Fermi level Ei of an intrinsic SC generally lies

very close to the middle of the bandgap.

•The simplest calculation of impurity energy levels is based

on the hydrogen-atom-like model:

•Based upon the values of Ec-Ed and Ea-Ev one can distin-

guish between: - shallow impurities

•The amount of ionized dopants is given by:

E

ε ε

E E

N

N

E E

N

N

Fermi level calculation

the semiconductor. The charge neutrality condition,

combined with the mass-action law, valid for degenerate

SC as well, leads to:

High-T limit:

•For p-type SC and high temperatures, the Fermi level is

given by:

•Temperature variation of EF:

N

•Temperature variation of the electron concentration n:

•To understand what happens, recall the hydrogen atom:

The discrete donor level broadens into a

band which overlaps with the conduction

band, leading to the reduction of the

band-gap => band-gap narrowing effect