CHAPTER 4 – THE SEMICONDUCTOR IN EQUILIBRIUMblogs.cae.tntech.edu/bwabegaz42/files/2013/08/Physical... · 2013-09-20 · Equilibrium Distribution of Electrons and Holes The distribution

PHYSICAL ELECTRONICS(ECE3540)

Brook Abegaz, Tennessee Technological University, Fall 2013

Friday, September 20, 2013Tennessee Technological University 1

CHAPTER 4 – THE SEMICONDUCTOR IN EQUILIBRIUM

Chapter 4 – The Semiconductor in Equilibrium Chapter 3: considering a general crystal and

applying to it the concepts of quantum mechanicsin order to determine a few of the characteristics ofelectrons in a single-crystal lattice.

Chapter 4: apply these concepts specifically to asemiconductor material.

Chapter 4: use the density of quantum states in theconduction band and the density of quantum statesin the valence band along with the Fermi-Diracprobability function to determine the concentrationof electrons and holes in the conduction andvalence bands


Equilibrium, or thermal equilibrium, implies that no externalforces such as voltages, electric fields, magnetic fields, ortemperature gradients are acting on the semiconductor.

All properties of the semiconductor will be independent oftime in this case.

An intrinsic semiconductor = a pure crystal with no impurityatoms or defects.

The electrical properties of an intrinsic semiconductor can bealtered in desirable ways by adding controlled amounts ofspecific impurity atoms, called dopant atoms, to the crystal, thuscreating an extrinsic semiconductor.

Adding dopant atoms changes the distribution of electronsamong the available energy states, so the Fermi energybecomes a function of the type and concentration of impurityatoms.


Chapter 4 – The Semiconductor in Equilibrium


Chapter 4 – The Semiconductor in Equilibrium

Fig. 1: Thermal Excitation caused jumping of an electron to CB. There is a corresponding hole created in VB where the electron was located.

Current is the rate at which charge flows. In a semiconductor, two types of charge

carriers, the electron and the hole, cancontribute to a current.

Since the current in a semiconductor isdetermined largely by the number of electronsin the conduction band and the number ofholes in the valence band, an importantcharacteristics is the density of these chargecarriers.


Charge Carriers in Semiconducductors

Equilibrium Distribution of Electrons and Holes

The distribution (with respect to energy) ofelectrons in the conduction band is given by thedensity of allowed quantum states times theprobability that a state is occupied by an electron.

eq. (4.1) where fF(E) is the Fermi-Dirac probability function

and gc(E) is the density of quantum states in theconduction band.

The total electron concentration per unit volume inthe conduction band is then found by integratingEquation (4.1) over the entire conduction-bandenergy.


(E)f (E)g =n(E) Fc


The distribution (with respect to energy) ofholes in the valence bend is the density ofallowed quantum states in the valence handmultiplied by the probability that a state is notoccupied by an electron.

eq. (4.2) The total hole concentration per unit volume is

found by integrating this function over the entire valence-band energy.


(E)] f -(E)[1g = p(E) Fv


An ideal intrinsic semiconductor is a puresemiconductor with no impurity atoms and nolattice defects in the crystal (e.g. pure Silicon).

For an intrinsic semiconductor at T = 0K, allenergy states in the valence band are filled withelectrons and all energy states in the conductionband are empty of electrons.

The Fermi energy must, therefore, besomewhere between Ec and Ev (The Fermi energydoes not need to correspond to an allowedenergy.)


The no and po Equations The equation for the thermal-equilibrium

concentration of electrons may be found byintegrating Equation (4.1) over the conduction bandenergy, as:

eq. (4.3) Applying the Boltzmann approximation to the

Fermi energy calculation, the thermal-equilibriumdensity of electrons in the conduction band is:

eq. (4.4)


dEEfEgn Fc )()(0

dEkT

EEEEhmn F

cE

n

c

])(exp[)()2(43

2/3*

0

The no and po Equations Solving the integral, substitute Nc as the effective density of

states function in the conduction band.eq. (4.5)

If m* = mo, then the value of the effective density ofstates function at T = 300 K is Nc = 2.5 x 1019 cm-3,which is the value of Nc for most semiconductors.

If the effective mass of the electron is larger or smaller thanmo, then the value of the effective density of states functionchanges accordingly, but is still of the same order ofmagnitude.

The thermal-equilibrium electron concentration in theconduction band is:

eq. (4.6)


2/32

*

)2(2h

kTmN nc

])(exp[0 kTEENn Fc

c

The no and po Equations The thermal-equilibrium concentration of holes in

the valence band is found by integrating Equation(4.2) over the valence band energy as:

eq. (4.7)

eq. (4.8)

Defining the effective density of states function in thevalence band:

eq. (4.9) The thermal-equilibrium concentration of holes in

the valence band is then:eq. (4.10)


])(exp[0 kTEENp vF

v

2/32

*

)2

(2h

kTmN p

v

dEEfEgp Fv )](1)[(0

)exp(1

1)(1

kTEEEf

FF

The no and po Equations The effective density of states functions, Nc and Nv are

constant for a given semiconductor material at a fixedtemperature.

The values of the density of states function and of theeffective masses for Silicon, Gallium Arsenide, andGermanium are:

The thermal equilibrium concentrations of electrons inthe conduction band and of holes in the valence bandare directly related to the effective density of statesconstants and to the Fermi energy level.


Table 4.1 Effective Density of States Function and Effective Mass Values

Nc(cm‐3) Nv(cm‐3) mn*/m0 mp

*/m0

Silicon 2.8*1019 1.04*1019 1.08 0.56

Gallium Arsenide 4.7*1017 7.0*1018 0.067 0.48

Germanium 1.04*1019 6.0*1018 0.55 0.37

The Intrinsic Carrier Concentration For an intrinsic semiconductor, the

concentration of electrons in the conductionband (n) is equal to the concentration of holesin the valence band (p).

These parameters are usually referred to as theintrinsic electron concentration (ni) and intrinsichole concentration (pi).

The Fermi energy level for the intrinsicsemiconductor is called the intrinsic Fermienergy, or Ef = Efi.


The Intrinsic Carrier Concentration For an intrinsic semiconductor:

eq. (4.11)

eq. (4.12)

eq. (4.13)

eq. (4.14)


])(exp[0 kTEENnn Fic

ci

])(exp[0 kTEENnpp vFi

vii

])(exp[.])(exp[2

kTEE

kTEENNn vFiFic

vci

])(

exp[])(exp[2

kTE

NNkT

EENNn gvc

vcvci

The Intrinsic Carrier Concentration For a given semiconductor material at a constant

temperature, the value of ni is a constant, andindependent of the Fermi energy.


Table 4.2 Commonly Accepted Values of ni at T=300K

ni (cm-3)

Silicon 1.5*1010

Gallium Arsenide 1.8*106

Germanium 2.4*1013

Fig. 2: Intrinsic carrier concentration niwith respect to change in To .

Application of Intrinsic Semiconductors High Electron Mobility Transistor High resistivity substrate for RF circuits Amorphous‐Si Solar Cells


Fig. 3: Structure of a Solar Cell.

The nopo Product Using the general expressions for no and po:

eq. (4.15) which is simplified as:

eq. (4.16) Thus, for a semiconductor in thermal

equilibrium, the Mass Action Law states:eq. (4.17)


])(exp[])(exp[00 kTEE

kTEENNpn vFFc

vc

])(

exp[00 kTE

NNpn gvc

200 inpn

Exercise1. Assume the Fermi energy is 0.25eV below the

conduction band. The value of Nc for Silicon at T= 300 K is Nc = 2.8 x 1019 cm-3. Calculate theprobability that a state in the conduction band isoccupied by an electron and calculate the thermalequilibrium electron concentration in silicon at T=300 K.

2. Assume that the Fermi energy is 0.27eV above thevalence band energy. The value of Nv for Siliconat T = 300K is Nv = 1.04 x 1019 cm-3. Calculatethe thermal equilibrium hole concentration insilicon at T = 400 K.


Solution1. The probability that an energy state at E = Ec

is occupied by an electron is given by :

The electron concentration is given by:


)](exp[)exp(1

1)(kT

EE

kTEEEf Fc

FccF

510*43.6)]0259.0

25.0(exp[)( cF Ef

)0259.0

25.0exp()10*8.2(])(exp[ 190

kTEENn Fc

c

3150 10*8.1 cmn

Solution2. The parameter values at T = 400 K are found

as:

The hole concentration is given by:


3192/319 10*60.1)300400)(10*04.1( cmNv

eVkT 03453.0)300400)(0259.0(

)03453.0

27.0exp()10*60.1(])(exp[ 190

kTEENp vF

v

3150 10*43.6 cmp

Exercise3. Calculate the intrinsic carrier concentration in

Gallium Arsenide at T = 300 K and at T =450K. The values of Nc and Nv at 300K forGallium Arsenide are 4.7 x 1017 cm-3 and 7.0 x1018 cm-3, respectively. Both Nc and Nv vary asT3/2. Assume the band gap energy of GalliumArsenide is 1.42eV and does not vary withtemperature over this range. The value of kTat 450 K is:


eVkT 03885.0)300450)(0259.0(

Solution3. For T= 300K:

For T = 450K:

Note: The intrinsic carrier concentration increased by over 4 orders of magnitude as the temperature increased by 150oC.


61218172 10*09.5)0259.0

42.1exp()10*0.7)(10*7.4(

cmni

3610*26.2 cmni

621318172 10*48.1)03885.0

42.1exp()300450)(10*0.7)(10*7.4(

cmni

31010*85.3 cmni

Dopant Atoms and Energy Levels The real power of semiconductors is realized by

adding small, controlled amounts of a specificdopant, or impurity atoms.

Adding a group V element, such as Phosphorus,as a substitution impurity in single-crystallineSilicon lattice, four of the valence electrons willcontribute to the covalent bonding with theSilicon atoms, leaving the fifth more looselybound to the Phosphorus atom.

The fifth valence electron is called a donorelectron.


Dopant Atoms and Energy Levels If a small amount of energy, such as thermal energy,

is added to the donor electron, it can be elevated into the conduction band, leaving behind a positively charged donor ion.

The electron in the conduction band can now move through the crystal generating a current, while the positively charged ion is fixed in the crystal.

This type of impurity atom donates an electron to the conduction band so is called a donor impurity atom.

The donor impurity atoms add electrons to the conduction band without creating holes in the valence band thus resulting an n-type semiconductor (n for the negatively charged electron).


Dopant Atoms and Energy Levels


Fig. 4: Silicon lattice doped with a donor impurity (P).

Fig. 5: Donor Electron Energy Level

Dopant Atoms and Energy Levels Adding a group III element, such as Boron, as a

substitution impurity to Silicon. The group IIIelement’s valence electrons are all taken up in thecovalent bonding.

The hole can move through the crystal generating acurrent, while the negatively charged Boron atom isfixed in the crystal. The group III atom accepts anelectron from the valence band and so is referred toas an accepter impurity atom.

The acceptor atom can generate holes in the valencehand without generating electrons in the conductionband. This type of semiconductor material isreferred to as a p-type material (p for the positivelycharged hole).


Dopant Atoms and Energy Levels


Fig. 6: Acceptor Impurity, Boron Impurity Atom in a Silicon Lattice

Fig. 7: Acceptor Electron Energy Level

The Extrinsic Semiconductor An extrinsic semiconductor is a semiconductor in

which controlled amounts of specific dopant orimpurity atoms have been added so that thermal-equilibrium electron and hole concentrations aredifferent from the intrinsic carrier concentration.

Only one carrier type dominates in an extrinsicsemiconductor.

The thermal equilibrium electron and holeconcentrations can be found as:

(eq. 4.18, 4.19)


])(exp[0 kTEEnn FiF

i

])(exp[0 kTEEpp FiF

i

The Extrinsic Semiconductor Also, the Mass Action Law (eq. 4.17) applies as:

eq. (4.17)

If the impurity concentration increases, the distance betweenthe impurity atoms decreases and a point will be reached whendonor electrons, for example, will begin to interact with eachother.

When this occurs, the single discrete donor energy will split intoa band of energies.

As the donor concentration further increases, the band of donorstates widens and may overlap the bottom of the conductionband.

This overlap occurs when the donor concentration becomescomparable with the effective density of states. When theconcentration of electrons in the conduction band exceeds


200 inpn

Degenerate and Nondegenerate Semiconductors If the number of impurity atoms is small, they would be

spread far enough apart so that there is no interactionbetween donor electrons, for example, in an n-typematerial.

Such impurities introduce discrete, non-interacting donorenergy states in the n-type semiconductor and discrete,non-interacting acceptor states in the p-typesemiconductor.

These types of semiconductors are referred to as non-degenetate semiconductors.

If the impurity concentration increases, the distancebetween the impurity atoms decreases and at some point,donor or acceptor states will begin to interact with eachother. When this occurs, the single discrete donor oracceptor energy will split into a band of energies.


Degenerate and Nondegenerate Semiconductors When the concentration of electrons in the

conduction band exceeds the density of states Nc ,the Fermi energy lies in the conduction band.

This type of semiconductor is called a degenerate n-type semiconductor.

As the acceptor doping concentration increases in ap-type semiconductor, the discrete acceptor energystates will split into a band of energy and mayoverlap the top of the valence band.

The Fermi energy will lie in the valence band whenthe concentration of holes exceeds the density ofstates Nv .

This type semiconductor is called a degenerate p-type semiconductor.


Degenerate and Nondegenerate Semiconductors


Nd > Nc Na > Nv

Fig. 8: (a) Degenerate n-type semiconductor and (b) Degenerate p-type semiconductor

Effect of Temperature on Ionization of Dopants


Low Temperature Moderate Temperature High Temperature

Fig. 9: Electron Concentration vs. Temperature in Extrinsic Semiconductor

Charge Neutrality This charge-neutrality condition is used determine the

thermal-equilibrium electron and hole concentrations asa function of the impurity doping concentration.

A compensated semiconductor is one that contains bothdonor and acceptor impurity atoms in the same region.

A compensated semiconductor can be formed, forexample, by diffusing acceptor impurities into an n-typematerial, or by diffusing donor impurities into a p-typematerial.

An n-type compensated semiconductor occurs when Nd> Na, and a p-type compensated semiconductor occurswhen Na > Nd.

If Na = Nd, we have a completely compensatedsemiconductor that has the characteristics of an intrinsicmaterial.


Charge Neutrality The charge neutrality condition is expressed by

equating the density of charges to the density ofpositive charges.

eq. (4.20, 4.21) where no and po are the thermal-equilibrium

concentrations of electrons and holes in the conductionband and valence band, respectively.

The parameter nd is the concentration of electrons inthe donor energy states, so Nd

+ = Nd – nd is theconcentration of positively charged donor states.Similarly, pa is the concentration of holes in theacceptor states, so Na

- = Na – pa is the concentrationof negatively charged acceptor states.


da NpNn 00

)()( 00 ddaa nNppNn

Charge Neutrality If we assume complete ionization, nd and pa are

both zero, Using po = ni

2/no , eq. (4.22)

The electron concentration no can be determinedusing the quadratic formula:

eq. (4.23)

eq. (4.24) is used to calculate the electron concentration in

an n-type semiconductor, or when Nd > Na.eq. (4.25)


da NpNn 00

do

ia N

nnNn

2

0

220 )

2(

2)(

iadad nNNNNn

0)( 20

20 iad nnNNn

220 )

2(

2)(

idada nNNNNp

Exercise4. Consider an n-type Silicon semiconductor at T

= 300K in which Nd = 1016 cm-1 and Na = 0.The intrinsic carrier concentration is assumedto be ni = 1.5 x 1010 cm-3. Determine thethermal equilibrium electron and holeconcentrations for the given dopingconcentration.


Solution4. Using the formula:

Using Mass Action Law,


31621021616

0 10)10*5.1()2

010(2

)010(

cmn

220 )

2(

2)(

iadad nNNNNn

200 inpn

3416

210

0

2

0 10*25.2)10*1()10*5.1( cm

nnp i

Position of Fermi Energy Level It is possible to determine the position of the Fermi energy level

as a function of the doping concentrations and as a function oftemperature.

eq. (4.26)

eq. (4.27)

For an n-type semiconductor in which Nd >> ni, then no ≈ Nd

eq. (4.28)

As the donor concentration increases, the Fermi level moves closer to the conduction band.


])(exp[0 kTEENn Fc

c

)ln()(0n

NkTEE cFc

)ln()(d

cFc N

NkTEE

Position of Fermi Energy Level Also using the formula:

eq. (4.29)

eq. (4.30) Similar formulas apply for a p-type semiconductor

eq. (4.31)

eq. (4.32)

eq. (4.33) As the acceptor concentration increases, the Fermi

level moves closer to the valence band.Friday, September 20, 2013Tennessee Technological University 40

)ln()(0p

NkTEE vvF

)ln()(a

vvF N

NkTEE

])(exp[0 kTEEnn FiF

i

)ln()(i

oFiF n

nkTEE

)ln()(i

oFFi n

pkTEE

Position of Fermi Energy Level


])(exp[0 kTEEn Fc

cN

)ln()(0n

NkTEE cFc

])(exp[0 kTEEn FiF

in

)ln()(i

oFiF n

nkTEE

Fig. 10: Position of Fermi Energy Level

Summary Position of Fermi level for an (a) n-type and (b) p-type semiconductor.

Variation of EF with doping concentration


)ln(d

ccF N

NkTEE

)ln(a

vvF N

NkTEE

Fig. 10: Position of Intrinsic Fermi level EFi and variation of EF with doping concentration.

Picture Credits Semiconductor Physics and Devices, Donald Neaman, 4th

Edition, McGraw Hill Publications. The Semiconductor in Equilibrium, Department of

Microelectronics and Computer Engineering, DelftUniversity of Technology, Sep. 2008.

Semiconductor Picture, Forbes Megazine:http://www.forbes.com/sites/jimhandy/2011/10/31/why-

read-a-semiconductor-blog/


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CHAPTER 4 – THE SEMICONDUCTOR IN EQUILIBRIUMblogs.cae.tntech.edu/bwabegaz42/files/2013/08/Physical... · 2013-09-20 · Equilibrium Distribution of Electrons and Holes The distribution