2 fundamentals of electronic devices

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Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

Fundamentals of Semicondutors

Source: Apple

Ref.: IBM

101110-110-210-310-410-510-610-8Critical

dimension (m)

Ref.: Apple

Introduction to Electronic Devices

(Course Number 300331) Fall 2006

Fundamentals of Semiconductors

Dr. Dietmar Knipp

Assistant Professor of Electrical Engineering

Ref.: Palo Alto Research Center

10-7

Information:

http://www.faculty.iu-

bremen.de/dknipp/


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Introduction to Electronic Devices

2 Fundamentals of Semiconductors

2.1 Semiconductors General Information

2.1.1 General Material Properties2.1.2 Structural Properties of Materials

2.1.2.1 Classification of semiconducting materials2.1.2.2 The unit cell2.1.2.3 Diamond crystal structure2.1.2.4 Crystal Planes and Miller Indices

2.1.3 Basics of Crystal Growth

2.2 Basics of Solid State Physics

2.2.1 The Hydrogen Atom

2.2.2 Energy bands2.2.3 Band structure in Semiconductors2.2.4 Energy-Momentum Diagram2.2.5 Electron energy in a Solid


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2.2.6 Material and Carrier Properties2.2.6.1Carrier Concentration in Semiconductors2.2.6.2 Density of States2.2.6.3 Fermi-Dirac Statistic

2.2.6.4 Fermi Energy in Solids2.2.7 Intrinsic carrier concentration

2.2.8 Donors and Acceptors2.2.9 Electrons and Holes in Semiconductor2.1.10 Compensated Semiconductors2.1.11 Minority and Majority Carriers2.2.12 Degenerated and Non-degenerated Semiconductors

2.2.13 Bulk Potential

References


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2.1 Semiconductors General Information

The purpose of this part of the lecture is to introduce the solid state physicsconcepts, which are needed to understand semiconductor materials andsemiconductor devices. This part of the lecture is kept as comprehensive aspossible.

2.1.1 General Material Properties

Solid-state materials can be grouped in terms of their conducttivity or resistiviy.Accordingly three classes of materials can be difined: Insulators,Semiconductors and conductors. The conductivity of semiconductors isgenerally sensitive to temperature, illumination, radiation, magnetic fields andimpurity atoms.


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Range of electricalconductivities .

Correspondingresistivity:

1=

Ref.: M.S. Sze, Semiconductor Devices

Classification of materials in terms of their conductivity or resititivity.



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Periodic table of semiconductor materials

All materials listed in this periodic table are of interest for electronicapplications. However, silicon (Si) and gallium arsenide (GaAs) are the mostmost important materials. Germanium (Ge) is only of interest for nicheapplications. Silicon has substituted germanium mainly due to the propertiesof silicon oxide.



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Periodic table of semiconductor materials

GaAs is a compound semiconductor, meaning it is an alloy of gallium and

arsenic. GaAs is non-toxic in its solid state phase. GaAs is a III/Vsemiconductor, because it is composed of material out of column III andcolumn V of the periodic table. GaAs can be seen as a alloy of gallium andarsenic. Other important materials out of the group of III/V semiconductors areIndium Phosphide (InP), and Gallium Nitride (GaN).

The electrical and the optical properties of III/V compound materials aredifferent from the properties of silicon. The materials are of main interest forhigh speed electronics, photonics, optical communication and high-end solarcells.


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2.1.2 Structural Properties of Materials

2.1.2.1 Classification of semiconducting materials

In order to build electronic devices we have to understand the electronictransport of charges in the material. However, the electronic properties ofelectronic material highly depend on the strucutral properties of the material.Based on the strucutral propeties of the material different classes of materialscan be distinguished:

Amorphous materials, polycrystalline materials and (mono)crystalline

materials.

The structural order of materials highly depends on the fabrication method andtemperatures. In general, the higher the structural order of the material the betterthe charges can move in the semiconducting material.


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2.1.2.1 Classification of semiconducting materials

Amorphous

materials

Poly crystalline

materials(Mono)Crystalline

materials

No long-range

order

Completely ordered

in segments

Entirely ordered

solid

Ref.: R.F. Pierret, Semiconductor Fundamentals


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2.1.2.2 The unit cell

The periodic arrangement of atoms iscalled lattice!

A unit cell of a material represents theentire lattice. By repeating the unit cellthroughout the crystal, one can generatethe entire lattice.

A unit cell can be characterized by avectorR, where a, b and c are vectorsand m, n and p are integers, so thateach point of a lattice can be found.

R=ma+nb+pc

The vectors a, b, and c are called thelattice constants.

Primitive unit cell.



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2.1.2.2 The unit cell

Different unit cells based on cubic unit cells


Simplecubic unitcell

Body centeredcubic unit cell

(bcc)

Face centeredcubic unit cell

(fcc)


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2.1.2.3 Diamond crystal structure

Silicon and germanium have a diamond

crystal structure.The silicon structure belongs to theclass of face center cubic unit cells. Asilicon unit cell consists of eight siliconatoms.

The structure can be seen as twointerpenetrating face centered crystalsublattices with one sublattice displacedfrom the other by one quarter of the

distance along the body diagonal of thecube.

Diamond lattice.



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2.1.2.3 Diamond crystal structure

Most of the III/V semiconductors

grow in a zincblende lattice, which isidentical to a diamond lattice exceptthat one of face center cubic cellsublattices has gallium atom and theother arsenic atoms.

Zincblende lattice.



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2.1.2.4 Crystal Planes and Miller Indices

Miller Indicesof someimportantplanes in acubic crystal.

Crystal properties along different planes are different and the electrical,thermal and mechanical properties can be dependent on the crystal

orientation.Indices (Miller indices) were introduced to define various planes in a crystal.



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Example: Determine the crystal plane

The plane has interceptionsat a, 3a and 2a along thethree coordinates. Taking thereciprocals of the intercepts,we get 1, 1/3 and . The

three smallest integers havethe ratio 6, 2, and 3. Thus,the plane is referred to be the(623) plane.



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Conventions how to define Miller indices:

(hkl): For a plane that intercepts the x-axis on the negative side of the originsuch as (100).

[hkl]: For a crystal direction, such as [100] for the x-axis. By definition, the[100]-direction is perpendicular to the (100)-plane, and the [111]-direction isperpendicular to the (111)-plane.

Ref: M. Shur, Introdcution to Electronic Devices


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Conventions how to define Miller indices:

{hkl}: For planes of equivalent symmetry such as {100} for (100), (010), (001),(100), (010) and (001) in cubic symmetry.

Ref.: M. Shur, Introdcution to Electronic Devices


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Simplified schematic drawingof the Czochralski puller.

95% of the material used in semiconductorindustry is crystalline silicon. Beforegrowing the silicon ingots, the material(SiO2, sand) is purified.

The most common growth method is theCzochralski method. The crucible containspoly crystalline material, which is heated by

radio frequency induction up to 1412C.The system is typically filled with an inertgas like argon to prevent contamination ofthe single crystalline ingot.

A silicon rod is used as the seed forthe growth of the silicon crystal.



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Photo of an ingot. Theingot has a diameter of200mm. After pulling thesingle crystalline ingot thematerial is sawed into

wafers of 300-500mthickness.

A more detailed description of the growth of crystalline materials isgiven in chapter 11 of M.S. Szes book Semiconductor devices,

Physics and Technology.



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2.2 Basics of Solid State Physics

To understand the properties of semiconductors it is essential to understandthe properties of their constituent atoms.

Based on Bohrs model the atom consists of a core, which contains basically

the complete mass of the atom. The shell is nearly without a mass. Despite thefact that nearly all the mass is concentrated in the core the diameter of the coreis small with 10-15m in comparison to the diameter of the shell10-10m=0.1nm=1 (ngstrm).

The core consists of neutrons and protons. The core is positively charged. Theshell (electron shell) is negatively charged due to electrons on is orbital. Overallthe atom is not charged or neutral.

The electrons behave like satellites. The electrons circulate around the core ondefined orbitals. The electrons are stabilized on their orbitals due an equilibrium

of centrifugal and Coulomb forces.

We will discuss the consequences of the model based on a hydrogen atom,which is the simplest atom.


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Due to the equilibrium between the centrifugal forces and the electrostaticforces a direct relation exists between the velocity of the electron and theradius to the core. The velocity of each electron is related to radius of the

orbital. As an electron can have different energies, the electron can havedifferent radius to the core of the atom. However, the model has the followingproblems:

Schematic diagram of a hydrogen atom

Based on classical electrodynamics itcan be expected that a charged

particle on a orbital leads to theformation of a magnetic dipole, whichradiates energy. Due to the loss ofenergy the particle would be moreattracted by the core, which leads to a

spiral like projection. Finally, theparticle would fall into the core of theatom.

+r

q2core

q1

v

electron

Centrifugal

force

Electrostatic

force


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To solve this inconsistency Nils Bohr proposed the following postulate: Theenergy levels of an atom and therefore the radius of the orbitals are quantized.The allowed energy levels for a hydrogen atom are given by

where EB is the Bohr energy and n is the principle quantum number. TheBohr energy is given by

where aB is the Bohr radius. q is the charge of the electron, which is theelementary charge and 0 is the permittivity. Electron energies between theseenergy levels En are not allowed.

,.....3,2,12== n

n

EE Bn

B

B

a

qE

0

2

8

= Bohr engery

Hydrogen energy levels


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As the electron energies are quantized the radius of the energy levels arequantized as well. The energy levels for each element are unique.

The formation or the splitting of these energy levels allows the formation of

energy bands. The energies between the defined energy levels are called theforbidden energy bands.

The unit of the energy is usually given in electronvolt (eV). The quantity eV(electron volt) is an energy unit corresponding to the energy gained by an

electron when its potential is increased by 1V (1eV=1.6*10-19

AVs=1.6*10-19

J).The Bohr radius is given by

where h is the Planck constant and me is the mass of the electron.

2

20

qm

ha

eB =

Bohr radius


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Bohr's atom model can be combined with Einstein's photon theory (2. BohrsPostulate). The energy difference between two energy levels n and m is given by

where En corresponds to the higher energy level. The transition from a higher toa lower energy level leads to an energy loss. The energy can be released in the

form of a photon, where f is the frequency of the emitted light. The frequency andthe corresponding wavelength of the light is given by

2222

0

4

,

11

8 nmh

mqf emn =

mn

mnf

c

,

, =

Frequency of the emitted light.

Wavelength of the emitted light.

mnfhEE mnmn >= , Photon energy


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2.2.2 Energy Bands

Moving from a single atom to a solid.

For an isolated atom, the electrons have discrete energy levels. As a number ofp isolated atoms are brought together to form a solid, the orbitals of the outer

electrons overlap and interact with each other. This interaction includesattraction and repulsion forces between the atoms. The forces between theatoms cause a shift of the energy levels. Instead of forming a single levels, as itis the case for a single atom, p energy levels are formed. These energy levelsare closely spaced. When p is large the different levels essentially form a

continuous band. The levels and therefore the bands can extend over severaleV depending on the interatomic or molecular spacing.

Schematic illustration of thesplitting of the degeneratedstates into a continuous bandof allowed states.



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2.2.3 Band structure in SemiconductorsEnergy Band in semiconductors

The 4 remaining valence band electrons are bound weakly and can beinvolved in chemical reactions. Therefore, we can concentrate on the outer

shell (n=3 level). The n=3 level consists of a 3s (n=3 and l=0) and a 3p (n=3and l=1) subshells. The subshell 3s has two allowed quantum states per atomand both states are filled with an electron (at 0 Kelvin). The subshell 3p has 6allowed states and 2 of the states are filled with the remaining electrons.


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Schematic diagram of the formation of theenergy bands in silicon as a function of thelattice spacing

Schematic diagram of theformation of the energy

bands in silicon as theinteratomic distancedecreases and the 3s and3p subshells overlap. At atemperature of absolute

zero, the electronsoccupy the lowest energystates, so that all states inthe lower band (valenceband) will be full and all

states in the upper band(conduction band) areempty.



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The bottom of the conduction band is called Ec and the top of the valenceband is called Ev. The energy difference between the bottom of the

conduction band and the top of the valence band is called bandgap energyEg. The bandgap energy Eg=(Ec- Ev) between the bottom of the conductionband and the top of the valence band is the width of the forbidden energygap. Eg is the energy required to break a bond in the semiconductor to free anelectron to the conduction band and leave a hole in the valence band.

A deficiency of an electron in the valence band is considered to be a hole.The deficiency in the valence band maybe be filled by a neighboring electron,which results in an shift of the deficiency location. A hole is positively charged.Both the electron and the hole contribute to the current flow.


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2.2.4 Energy Momentum DiagramEnergy-band diagram for Silicon and Gallium Arsenide

If an electron is excited to the conduction band it can move freely in the crystal,since the electron can be treated like a particle in free space. The propagation

of the free electron can be described by the wave function, which is the solutionof the Schrdinger equation. The wave function for a free electron is given by

where k is the wave vector, which is given by

P is the momentum of the electron. Due to this expression the electron energycan be given as a function of the wave factor. We speak about the k-spacerepresentation. The energy bands can now be determined as a function of thek-vector.

( ) ( )ikxBikxA += expexp 11 Wave function

2h

pk= Wave vector


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2.2.4 Energy Momentum Diagram

Electron energy in free space

2

2vm

Ee

n =Energy of a free electron

vmp e=Momentum of a free electron

Energy momentum diagram for afree electron

En: Energy of a free electron

me: mass of a free electron

v: velocity of the electron e

en

m

pvmE

22

22

==


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We can rewrite the equation so that the wave vector is expressed in terms ofthe momentum of the electron.


Electron energy in free space

vm

h

e

= DeBroglie equation

2

hkp =

Dualism of waves and matter forelectromagentic waves.

k: wave vector

2h

pk=

Wave vector


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2.2.4 Energy Momentum DiagramEnergy-band diagram for for Silicon and Gallium Arsenide

Ref.: M.S. Sze,Semiconductor

Devices

Silicon GaAs

Indirect semiconductor Direct semiconductor


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Electron energy in a Solid

For a solid the electron energy near the conduction band minimum can beapproximated by a parabolic function similar to an electron in free space.

However, the electron energy of an electron in a solid is quite different from theenergy of an electron in free space. The energy of an electron can be given by:

where mn is the effective mass of the electron. The effective mass can becalculated by:

Energy of a electron in the

conduction band

Effective mass of an electronpE

mn

n 22

1

=

( ) nCn mkh

EkE += 2

22

8

I t d ti t El t i D i F ll 2006 D Di t K i


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2.2.4 Energy Momentum DiagramElectron energy in a Solid

Narrowing the parabola,corresponds to a larger secondderivative, the smaller the

effective mass.

Energy-momentum relation-shipof a special semi-conductor with

an electron effective mass ofmn=0.25m0 in the conductionband and a hole effective massof mp=m0. The actual energy-momentum relationship (also

called energy-band diagram) forsilicon and gallium arsenide aremuch more complex.


Introduction to Electronic Devices Fall 2006 Dr Dietmar Knipp


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2.2.4 Energy Momentum DiagramElectron energy in a Solid

The actual energy-momentum relationship (also called energy-band diagram) forsilicon and gallium arsenide are quite different from the energy momentum

diagram of a free electron. Nevertheless, the general features like the bandgapbetween the bottom of the conduction band and the top of the valence band canbe observed. Second, the minimum and the maximum of the conduction andvalence band are parabolic. For silicon the maximum of the valence band occursfor p=0, but minimum of the conduction band is shifted to p=pc. Therefore, in

silicon in addition to the energy Eg, which is necessary to excite an electron anmomentum pc is necessary. For GaAs the maximum in the valence band and theminimum in the conduction band occur at the same momentum (p=0).

Gallium arsenide is called a direct semiconductor, because it does not require

a change in momentum for an electron transition from the valence band to theconduction band.

Silicon is called an indirect semiconductor, because a change of themomentum is required in a transition.



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2.2.5 Electron energy in a Solid

With the gained knowledge we can schematically explain the enormousdifferences in conductivity of insulators, semiconductors and conductors interms of energy bands.

Metals or conductors are characterized by a very low resistivity. Dependingon the material two different schematic energy band diagrams exist.

The conduction band is either partially filled (e.g.for Cu) or the valance band and the conductionband overlap (e.g. Zn, Pb).

Electrons are free to move with only a small

applied electric fields.

Energy Band diagramin a conductor Ref.: M.S. Sze, Semiconductor Devices



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2.2.5 Electron energy in a Solid

For an insulatorthe valence electrons are strongly bonded to the neighboringatoms. This bonds are difficult to break and consequently there are no freeelectrons, which can participate in an current flow.

Insulators are characterized by a large bandgap. All energy levels in thevalance band are occupied, whereas all energy levels in the conduction bandare empty.

Thermal energy or an applied

electrical field is not sufficient to raisethe uppermost electron in thevalence band to the conductionband.

One of the best insulators is siliconoxide.


Energy Band diagramin an insulator



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t oduct o to ect o c e ces, a 006, et a pp



2.2.5 Electron energy in a SolidMaterials with an bandgap of 0.6eV to 4.0eV are considered to besemiconductors (room temperature). Most of the materials have bandgapsbetween 1.0eV and 2.0eV (room temperature).Silicon has a bandgap of 1.12eV, Gallium arsenide has a bandgap of 1.42eV.Therefore, the conductivity of a (intrinsic) semiconductors is low at roomtemperature. The thermal activation energy is not high enough to excite anelectron from the valence band to the conduction band.

At room temperature the thermalactivation energy is a fraction of thebandgap,

Ethermal=kT=0.0256eV=25.6meV,

so that a small number of electronsget thermally excited, which contributeto a moderate current flow forlow/moderate electric field levels.

Energy Band diagramin a semiconductor.



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pp


2.2.6 Material and Carrier PropertiesIntrinsic and extrinsic Semiconductors

The material is considered to be an intrinsic semiconductorif the materialscontains a relatively small amount of impurities.

The material is considered to be an extrinsic semiconductorif the materialscontains a relatively large amount of impurities.

Semiconductors in Thermal Equilibrium

In the following it is assumed that the semiconductor is an intrinsicsemiconductor. Influences of impurities on the semiconductor properties areneglected. Further, it is assumed that the semiconductor is in thermalequilibrium, which means that the semiconductor is not exposed to additionalexcitements like light, pressure or electric field. The semiconductor material iskept constant temperature throughout the entire sample (no temperaturegradient exists in the semiconductor material).



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41Fundamentals of Semicondutors

2.2.6.1Carrier Concentration in Semiconductors

In the following the carrier concentration in the conduction and the valence bandwill be calculated. The carrier concentration is given by:

where n and p are the electron and hole concentration [1/cm3] (Number ofelectrons and holes per unit volume. Ne(E) and N h(E) are Density of States

(Allowed energy states per energy range and per unit volume). Fe(E) and Fh(E)are the Fermi-Dirac distributions for electrons and holes. The Fermi-Dirac

distribution is a probability function, which indicates whether a state is occupiedby an electron or a hole.

Electron concentration

Hole concentration

( ) ( ) =topC

botC

E

E

ee dEEFENn

( ) ( ) =

topV

botV

E

Ehh dEEFENp



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2.2.6.1Carrier Concentration in Semiconductors

In the first step the product of the Density of States Ne(E), Nh(E) and the Fermi-Dirac Distribution Fe(E), Fh(E) is calculated. The product states whether thestates in the conduction and the valence band are occupied by free electrons

and holes. The product corresponds to a carrier density for a given energy. Inorder to determine the overall carrier concentration the integral over all energies(conduction and the valence band) has to be determined.



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2.2.6.1 Carrier Concentration in SemiconductorsSchematic Band Diagram, Density of States, Fermi-Dirac Distribution andCarrier Concentration of an intrinsic semiconductor in thermal equilibrium

SchematicBand Diagram Density ofStates Fermi-DiracDistribution Electron andhole Density




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2.2.6.2 Density of States

The density of states can be calculated by the Schrdinger equation.However, the derivation of the density of state function will not bediscussed here. Further information is given by M.S Sze, Semiconductor

Devices, Appendix H.

The Density of States is determined by a single material parameter, which

is the effective mass of the electron or the hole. Therefore, the density ofstates for electrons and holes are very often different.

( ) ( )ceC EEmh

EN = 33

24

Density of states for electrons

( ) ( )EEmh

EN VhV =3

32

4 Density of states for holes



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2.2.6.3 Fermi-Dirac Statistic

The Fermi-Dirac statics describes the probability that an electronic state for agiven energy E is occupied by an electron. The Fermi-Dirac Statistic is symmetricaround the Fermi energy EF. The Fermi energy can be defined as the energy atwhich the Fermi-Dirac distribution is equal to . In general, the Fermi-Diracstatistic is strongly temperature dependent. With decreasing temperature the

k: Boltzmann constant,

T: temperature in Kelvin,EF: Fermi energy


transition gets sharper. Itmeans that in practical terms anelectronic state is very likely to

be occupied by an electron ifthe energy of the electron is afew kT higher than the Fermienergy. Consequently it is veryunlikely that an electronic state

is occupied by an electron if theenergy is a few kT below thanthe Fermi energy.



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( )( )kTEE

EFF

e +=

exp1

1

( ) ( )

( )kTEE

EFEF

F

eh

+=

==

exp1

1

1

0

0.5

1.0

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

Fh(h)

Fe(E)

Energy E-EF

[eV]

Ferm

iDiracDistribu

tionF(E)

Fermi energy for electrons

Fermi energy for holes

2.2.6.3 Fermi-Dirac StatisticSo far the Fermi-Dirac distribution was only introduced for electrons. TheFermi-Dirac distribution for holes is given by:



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2.2.6.3 Fermi-Dirac StatisticThermal equilibrium

A semiconduting material is in thermal equilibrium, if the temperature at eachposition of the crystal is the same, the overall current through the material is 0,

and the solid state is not illuminated. Furthermore, we assume that no chemicalreaction is taking part.

As a consequence the Fermi energy throughout the material is constant.

( ) .,, constzyxEE FF == Thermal equilibrium


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2.2.6.4 Fermi Energy in Solids

How can we apply now the concept of the Fermi level do different materials likeconductors, insulators and semiconductors?

In the case of a conductor the Fermi level is in the conduction band. Therefore,the conduction band is always occupied with electrons.The situation is quite different for insulators and semiconductors. In the case of asemiconductor it is assumed that the material is an intrinsic semiconductor. As aconsequence the Fermi level is (approximately) in the middle of the bandgap.

However, the bandgap of an insulator is much larger than the bandgap of asemiconductor.The bandgap for a semiconductor is in the range of 0.6eV to 4eV, whereas thebandgap of an insulator is larger than 5.0eV. For example silicon oxide, which isthe insulator in microelectronics, has a bandgap of 9.0eV. As a consequence it is

very difficult to overcome such a high energy barrier.



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2.2.6.5 Boltzmann distributionTo calculate the carrier concentration for electrons and holes the Fermi-Integralhas to be solved.

However, the Fermi integral cannot be solved analytically. Therefore, anapproximation is used to determine the carrier densities. The approximation is

called the Boltzmann distribution.

( )( )kTEENn FCC exp

Electron concentration, Boltzmann distribution

( )( )kTEENn FCC = 21

kTEEfor FC 2


( )( )kTEENp VFV = 21 Hole concentration

( )( )kTEENp VFV exp

Hole concentration, Boltzmann distribution

kTEEfor VF 2



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3

222

=

h

kTmN hV

Effective Density of Statesin the valence band

3

222

=

h

kTmN eC

Effective Density of Statesin the conduction band

2.2.6.5 Boltzmann distributionInstead of using the energy dependent Density of States a new parameter isintroduced, which is the effective Density of States. The effective Density ofStates is again defined for electron and holes. The effective Density of States isindependent of the energy. Therefore, the effective Density of States is a pure

material parameter.



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2.2.7 Intrinsic carrier concentrationWe already distinguished between intrinsic and extrinsic semiconductors. Thematerial is considered to be an intrinsicsemiconductor, if the material containsa relatively small amount of impurities. Under such conditions the number ofelectrons per volume in the conduction band is equal to the number of holes

per volume in the valence band. Therefore, an intrinsic carriers concentration nican be defined.

Electron, hole and intrinsic carrier concentration. Ref.: M.S. Sze, Semiconductor Devices

inpn ==

Intrinsic carrierconcentration



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2.2.7 Intrinsic carrier concentrationBased on the intrinsic carrier concentration an intrinisc energy can bedetermined. For an intrinsic semiconductor in thermal equlibrium the intriniscenergy is equal to the Fermi energy.

The electron and hole concentration is given by

So that we can derive the following expression for the intrinsic energy.

++=

C

VCVi

N

NkTEEE ln

22Intrinsic energy

( ) iiF EnpnE ===

( )( )kTEENp VFV exp ( )( )kTEENn FCC exp

( )( ) ( )( )kTEENkTEEN iCCViV = expexp



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2.2.7 Intrinsic carrier concentrationThe intrinsic energy is again a pure material parameter. The intrinsic energy isnot affected by light exposure or pressure. The intrinsic energy is constant for asemiconductor even if the material is not in thermal equilibrium anymore (e.g. avoltage is applied to the sample).

At room temperature the second term is much smaller than the first term.Therefore, the intrinsic energy is very close to the middle of the bandgap(EC-EV)/2=Eg/2. For silicon the intrinsic energy deviates from the middle of thebandgap by Ei-(EC+EV)/2-kT/2=-13meV. The intrinsic energy is shifted towardsthe valence band. For Gallium Arsenide the situation is opposite and theintrinsic energy is slightly shifted towards the conduction band: Ei-(EC+EV)/23kT/2=39meV



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2.2.7 Intrinsic carrier concentrationBased on n=p=ni the intrinsic carrierconcentration can be expressed in terms of theeffective density of states for the electrons andholes.

So that the intrinsic concentrationresults to the following expression:

=

kT

ENNn

g

VCi2

expIntrinsic carrierconcentration


=kT

EEnN iCiC exp

= kTEEnN ViiV exp

Intrinsic carrier concentrationfor silicon and GaAs.



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2.2.7 Intrinsic carrier concentrationIn the next step the expression for the carrier concentration (electrons) can bemodified by describing the effective density of states as a function of theintrinsic carrier concentration. As a result a expression for the carrierconcentration can be derived which does not require knowledge of the effective

density of states for the material.

=

kT

EEnn iFi exp

=

kT

EEnp Fii exp

( )( )kTEENn FCC exp

= kTEE

nN

iC

iC exp

kTEEfor FC 2


Hole concentration



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2.2.8 Donor and Acceptors

When a semiconductor is doped, the semiconductor becomes extrinsic andimpurity levels are introduced. In the following the influence of acceptors anddonors on the material properties will be discussed. We will focus here on thedoping of silicon.

If we introduce donors like arsenic andphosphorus in a silicon single crystal a siliconatom is replaced by an donor atom with fivevalence electrons. The arsenic or phosphorusatoms form covalent bonds with its neighboringsilicon atoms. The 5th electron has a low bindingenergy to become a conducting electron. Thearsenic or phosphorus atom is called a donor andthe silicon becomes n-type because of theaddition of the negative charge carrier.Schematic silicon

lattice for n-typedoping with donoratoms (arsenic orphosphorus).




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If we introduce acceptors like boron in the siliconlattice a silicon atom is replaced by a boron atomwith three valence electrons. Additional electronsare accepted to form four covalent bonds. Theboron atom is considered as an acceptor and thesilicon becomes p-type because of the addition ofthe positive charge carrier.

Schematic siliconlattice for p-typedoping with donoratoms (boron).




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2.2.8 Donor and AcceptorsPeriodic table of semiconductor materials

Elements out of column III and column V of the perodic table are ofparticualr interest to intentionally dope silicon. Elements out of column IIIform acceptor states, whereas elements from column V tend to form donorstates.




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2.2.8 Donor and AcceptorsThe introduction of donors like arsenic in the silicon lattice leads to theformation of energy levels very close to the bottom of the conduction band. Atroom temperature the thermal energy kT is high enough to thermally excite theexcess electron to the conduction band. As a consequence positively charged

localized states are left in the material and free and mobile electrons arecreated in the conduction band. A donor state is neutral when it is occupied byan electron and becomes positively charged if the state donates its electron tothe conduction band. Under such conditions the energy level of the donors isvery close to the conduction band.

Schematic energy band

representation of asemiconductor with donor ions.EV

EC

Donor levels

Energy

ED

Distance



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With increasing donor concentration the Fermi level will shift closer to thebottom of the conduction band. Therefore, the energy difference between theFermi level and the conduction band (EC-EF) gets smaller with increasing donorconcentration.

M.S. Sze, Semiconductor Devices

SchematicBand Diagram

Density ofStates

Fermi-DiracDistribution

Electron andhole Density



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An analog behavior is observed for increasing acceptor concentration. Thehigher the acceptor concentration the closer the Fermi level will move to thevalence band. At room temperature the thermal activation is already highenough to active an hole from the valence band. As a consequence the

acceptor ions get negative and holes are created in the valence band. Anacceptor is negatively charged when it is occupied it is occupied by andelectron and becomes neutral after accepting an electron from the valenceband.

Schematic energy bandrepresentation of asemiconductor withacceptor ions.EV

EC

Acceptor levelsEnergy

EA

Distance



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2.2.9 Electrons and Holes in Semiconductor

The product of the electron and hole concentration is equal to the square ofthe intrinsic carrier concentration if the semiconductor is in thermalequilibrium. In this case it does not matter, whether the semiconductor is an

intrinsic semiconductor or an extrinsic semiconductor. In the second case thesemiconductor is doped by acceptors or donors.

If the semiconductor is intrinsic the following relationship applies

Doping of a semiconductor leads to the following relationship

2innp =innp ==

Intrinsic semiconductorin thermal equilibrium

ii nnnpnp ,,Extrinsic semiconductorin thermal equilibrium

and

2innp =and



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If a semiconductor samples is uniformly doped (no internal electric field) andno electric field is applied (external electric field) the semiconductor is neutral.In this case charge neutrality applies. To preserve charge neutrality, the totalnegative charges (electrons and ionized acceptors) must equal the totalpositive charges (holes and ionized donors).

If we assume that the material is only doped by donors so that NA=0 theequation is simplified to n=p+ND. Therefore, the semiconductor is an n-typesemiconductor. The hole concentration can now be calculated by

where the index n indicates that we deal with a n-type semiconductor.

DA NpNn +=+

nin nnp2=

Charge neutrality

Hole concentration for an

n-type semiconductor



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The following expression for the electron concentration can be derived:

In most of the cases we can assume that the Donor concentration is higherthan the intrinsic carrier concentration so that the expression is reduced to

If the electron concentration is approximately equal to the Donorconcentration complete ionization can be assumed. Complete ionization isobserved for (shallow) donors and acceptors, which means that theintroduced impurities form defect levels very close to the bands.

( )22 4

2

1

iDDn

nNNn ++=Electron concentration for

an n-type semiconductor

Dn Nn Complete ionization for ann-type semiconductor



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Consequently we get the following term for the hole concentration :

So that the Fermi level can be calculated by using the Boltzmann distribution

The analog behavior can be observed for a p-type doped semiconductor. Ifwe assume that donor concentration is N

D=0 we get the following expression

for the holes: p=n+NA. The electron concentration can be described by

Din Nnp2=

D

C

CF N

NkTEE ln

Fermi level for an n-type

semiconductor

pip pnn2= Electron concentration foran p-type semiconductor



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Subsequently the following expression is obtained for the hole concentration :

If we again assume that the defect levels are very close to the band (valenceband) most of the acceptors will be ionized so that

So that the Fermi level can be calculated by using the Boltzmann distribution

( )22 4

2

1

iAApnNNp ++=

Hole centration for an p-

type semiconductor

Ap Np Complete ionization for anp-type semiconductor

+

A

VVF

N

NkTEE ln

Fermi level for an p-typesemiconductor



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Various impurities in silicon and gallium arsenide


Measured ionizationengeries for variousimpurities in silicon andGaAs.

Si

GaAs



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Influence of the Doping Concentration on the Fermi Level

The energetic position of the Fermi level depends on the concentration of thedopants and the temperature. With increasing temperature the Fermidistribution is getting broader so that the Fermi level is closer to the intrinsic

energy level. With increasing doped concentration the Fermi level shiftscloser to the bands (conduction and valence band). This behavior is similarfor all semiconductor materials.


Influence of thetemperature and thedoping concentration onthe Fermi level in silicon.


2 2 10 C t d S i d t


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2.2.10 Compensated Semiconductor

So far either n-type or p-type semiconductors were considered in thediscussion. However, every often in microelectronics the material is doped bydonors and acceptors. For example a p-type wafer is doped with arsenic (n-typeregion) so that a pn-junction is formed. In this case the semiconductor iscompensated. In order to preserve charge neutrality both dopant concentrationshave to be considered.

However, in most of the cases the concentration of one dopant species is muchhigher than the concentration of the other species so that the semiconductorproperties are determined by the higher dopant concentration.

DA NpNn +=+ Charge neutrality




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[ ]DniD

iDDn

NnnN

nNNn

>>

++= 22 42

1 [ ]

DiniD

iDD

in

NnpnN

nNN

np

2

22

2

42

1

>>

++=

Majority carriers(n-type semiconductor)

Minority carriers(n-type semiconductor)

[ ]

ApiA

iAAp

NpnN

nNNp

>>

++= 22 42

1 [ ]

AipiA

iAA

ip

NnnnN

nNN

nn

2

22

2

42

1

>>

++=

Majority carriers(p-type semiconductor)

Minority carriers(p-type semiconductor)

AD NN >

DA NN >

Assumption:

(p-typesemiconductor)

Assumption:(n-type

semiconductor)


2 2 11 Mi it d M j it C i


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2.2.11 Minority and Majority Carriers

As complete ionization can assumed for typical dopants like arsenic or boron theconcentration of free carriers is more or less controlled by the dopantconcentration. If for example silicon is doped by arsenic the concentration ofelectrons in the conduction band is much higher than the concentration of holes

in the valence band. In this case the electrons in the conduction band aremajority carriers and the holes in the valence band are minority carriers. Asthe name implies, the electrons represent the majority of carriers and the holesrepresent the minority of carriers. The analog behavior is observed for borondoped material. Here the concentration of holes in the valance is much higher

than the concentration of electrons in the conduction band. Consequently theholes are the majority carriers, whereas the electrons are the minority carriers.

Electrons are majority and holes are minority carriers in n-type materials!

Holes are majority and electrons are minority carriers in p-type materials!

For bipolar electronic devices like diodes (e.g. solar cells, LED) or bipolartransistors the electronic transport is controlled by the minority carriers, becausethe electronic transport is limited by the number or the lifetime of minoritycarriers.


2 2 12 Degenerated and Non degenerated Semiconductors


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2.2.12 Degenerated and Non-degenerated Semiconductors

For most of the electronic devices the electron and hole concentration ismuch lower than the effective density of states in the conduction or thevalence band. The Fermi level is at least 3kT above the valence band or 3kTbelow the conduction band. In such a case we speak about a non-

degenerated semiconductor.For very high levels of doping the concentration of dopants gets higher thanthe effective density of states in the valence or the conduction band. In sucha case the semiconductor is degenerated and the Fermi levels shifts into theconduction or the valence band. Under such conditions the equations whichwere derived here does not apply any more.

However, the fabrication of degenerated semiconducting materials can benecessary. For example the fabrication of laser diodes require populationinversion, which can only achieve if the semiconductor is degenerated.



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2.2.13 Bulk potentialIn following we will introduce the bulk potential. The bulk potential is animportant parameter if it comes to the explanation of bipolar devices likediodes or bipolar transistors. The bulk potential is directly related to the Fermilevel in a material. Therefore, the position of the Fermi level can be expressedby the bulk potential or vice versa.

The electron and the hole concentration of an intrinsic semiconductor can beexpressed in terms of the intrinsic carrier concentration.

Instead of using the energy difference between the intrinsic energy level andthe Fermi level the term can be substituted by the bulk potential.

=

kTEEnn iFi exp = kT

EEnp Fii exp

Bulk potential( )Fib EEq=

1


2.2.13 Bulk potential


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The bulk potential is a measure of the energy difference between the intrinsicenergy level and the Fermi level. Bulk implies that this parameter is related tothe bulk/volume properties of a semiconductor. The complementary termwould be the surfec potential, which corresponds to the potential at the surface

of a semiconductor. The term surface potential will be introduced in chapter 6,Furthermore, the Boltzmann equation can be simplified by using thetemperature voltage

so that electron and hole concentration results to

Therefore, the bulk potential is directly related with the carrier concentration.

( )thbi Vnn exp=

( )thbi Vnp = exp


Hole concentration

Temperature voltageqkTVth

=


2 2 13 Bulk potential


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In order to directly relate the bulk potential with the material properties wehave to rewrite the equation. For an n-type semiconductor the bulk potentialresults to

In most of the cases the Donor concentration is large than the intrinsic carrierconcentration so that:

Accordingly we can derive an expression for an p-type semiconductor.

( ) ++=22 4

21ln iDD

i

thbn nNNn

V

Bulk potential for an n-type semiconductor

0ln

i

Dthbn

n

NV



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References

Michael Shur, Introduction to Electronic Devices, John Wiley & Sons;(January 1996). (Price: US$100)

Simon M. Sze, Semiconductor Devices, Physics and Technology, JohnWiley & Sons; 2nd Edition (2001). (Price: US$115)

R.F. Pierret, G.W. Neudeck, Modular Series on Solid State Devices,Volumes in the Series: Semicondcutor Fundamentals, The pn junctiondiode, The bipolar junction transistor, Field effect devices, (Price: US$25per book)

2 fundamentals of electronic devices

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