S9 Band Theory 2011

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SECTION 9: BAND THEORY OF SOLIDS

1. Energy bands: a qualitative introduction 2

2. A one-dimensional model of a solid 3

2.1 Periodic potentials 3

2.2 The displacement operator 4

2.3 Bloch functions 4

2.4 Crystal momentum k 5

2.5 Brillouin Zones 5

2.6 Periodic boundary conditions 6

2.7 The periodic potential 7

2.8 Formation of energy bands 9

3. Electrical Properties of Solids 10

4. Band Theory of Electrical Conductivity 11

4.1 Insulators 11

4.2 Conductors 12

4.3 Intrinsic semiconductors 13

4.4 Extrinsic or impurity semiconductors 15

References

The original version of these notes was based loosely on the textbook by Griffiths and Quantum Physics, by

S Gasiorowitz, published by Wiley (1974). See also Fundamentals of Physics, 6thedition, by C Halliday, R

Resnick and J Walker, published by Wiley. Specific references to these and other books are given as footnotes

to the text.

Original Version: October 2002; Last Major Revision: October 2007

Revision Date: 23 October 2009

Printed:14 June 2010

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1.ENERGY BANDS: A QUALITATIVE INTRODUCTIONWe consider only crystalline solids made up of a regular, repeating pattern of atoms or molecules forming a

rigid three-dimensional lattice structure; this specifically excludes materials such as rubber, wood, plastic and

glass.In a crystalline solid, a few of the loosely bound valence electrons in each atom roam throughout the material

subject to the combined potential of the entire crystal lattice, and not just of the individual atoms. We

introduce a primitive model as an introduction to the band theory of solids, and explore its consequences for

electrical conductivity in solids.

Consider the energy states of a solid consisting of a regular lattice of Nidentical atoms1. The solid can be built

up by bringing together the atoms one by one, as illustrated schematically in Fig. 1.

Figure 1: Formation of energy bands in solids

As we have seen in the discussion of atomic structure, for a single electron, the energy levels will belabelled 1s, 2s, 2p, 3setc. in order of increasing energy, with degeneracies 2, 2, 6, 2 etc.

If we have two atoms sufficiently far apart that the electron wave functions do not overlap significantly,the electron energy spectrum will be unchanged, except that the degeneracy of each level is doubled.

Imagine the two atoms are now brought closer and closer together. The electron wave functions begin tooverlap, starting with the outermost electrons which on average correspond to higher energy. As soon as

the two sets of electron wave functions overlap significantly, in order to satisfy the requirements of the

Pauli principle each energy level splits into two (as discussed in the section on exchange symmetry); thecloser the two atoms, the larger the splitting.

If we have three atoms close together, the energy levels will be split into three; with four atoms, each levelis split into four, and so on. With an extremely large number Nof atoms, interacting to form a solid, each

multi-electron energy level is split into Nextremely close-lying levels, forming an almost continuous band

of electron levels. The bands are normally separated by energy gaps where no electron states exist.

These bands have the following properties:

1The original version of this section was based on section 42-3 of the textbook byHalliday, Resnick and Walker.

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The number of states in each band is equal to Ntimes the number of states in the original electron energylevel from which the band was created. The bands can be labelled with the quantum numbers of the

original single-electron levels.

Electrons in lower energy levels spend most of their time close to the nucleus at the centre of the atom;the wave functions of these electrons therefore do not overlap as much those in higher energy levels andthe splitting of the resulting bands is less. The bands of higher energy are therefore wider than those of

lower energy.

Typically bands are a few eV wide and they are separated by gaps a few eV wide. Higher-lying bands mayactually overlap each other; i.e. the energy gap disappears.

2.A ONE-DIMENSIONAL MODEL OF A SOLID2In a crystalline solid, the atoms are arranged in a regular pattern in a lattice. The potential experienced by

an electron due to the atomic nuclei is therefore periodic in space.

We assume than an electron moves freely in an effective potential containing

the periodic potential due to the positively-charged atomic nuclei; some of the effects of other electrons by treating them as static distributions of negative charge with the

same periodicity as the lattice.

Since we neglect any other interaction between the electrons, we therefore have an independent-particle model

for the electrons, similar to the central-field approximation for an atom:

The wave function is the product of single-electron wave functions (but properly anti-symmetrised as wehave discussed in the case of atoms).

The energy is the sum of the energies of all the electrons.2.1Periodic potentialsWe consider an idealized one-dimensionalmodel of a crystal, in which the crystal lattice comprises a set of N

identical unit cells of length L; this is a simplified version of the model introduced by Kronig and Penney in

1931.

A unit cellis the basic building block of the crystal. Each cell may contain a single atom, or a group ofatoms in a fixed spatial configuration.

The length Lis called the lattice constant. In most crystals (which are obviously three-dimensional objects)this is typically 25 .

As discussed above, we must solve the Schrdinger equation for a single electron

2 2

2( ) ( ) ( ) ( )

2

dH x V x x E x

m dx

= + =

(1)

2

The original version of this section was based on section 5.3.2 of the textbook by Griffith. See also sections 7.4 and 7.5 of QuantumMechanicsby Sara M McMurry, published by Addison-Wesley (1994).

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where the electron potential is obviously the same at equivalent points in each unit cell, i.e.

( ) ( ), 0,1,2,V x nL V x n + = = (2)

2.2The displacement operatorBecause of the symmetry built into the model, it is convenient to define a displacement operator TLthat

produces a translation through distance Lalong the xaxis in the positive xdirection when it acts on any

function of position x. In particular, if (x) is the wave function of a single electron in the solid:

( )

( ) ( )

( ) ( )

L

n

L

T x x L

T x x nL

= +

= + (3)

where nis any integer.

Clearly, TLcommutes with the kinetic energy operator:

( )Ld d d

T dx d x L dx = =+

and, from eq.(2),

( ) ( ) ( ).LT V x V x L V x = + =

Hence, TLcommutes with the complete Hamiltonian and we can look for simultaneous eigenfunctions of the

Hamiltonian and the displacement operator.

2.3Bloch functionsBlochs theorem(Bloch 1928) states that simultaneous eigenfunctions of Hand TLmust have the particular

form3:

( ) ( )ikxk kx e u x = (4)

where the function uk(x) has the same periodicity as the potential, namely

( ) ( ),k ku x L u x + = (5)

and kcan take on any realvalue and is not a function of x. Wave functions satisfying eqs.(4) and (5) are called

Bloch functions.

Since the Hamiltonian has the same translational symmetry as the crystal lattice, we should expect the

electron probability distribution to satisfy

2 2( ) ( )k kx nL x + = ;

obviously, the Bloch functions satisfy this condition provided kis real.

Note that, from the definition of the displacement operator, eq.(3), and the properties of the Bloch functions,

eqs.(4) and (5), we have

( )( ) ( ) ( )

( ).

i k x L

L k k k

ikx ikL

k

T x x L e u x L

e e u x

+= + = +

=

3For a proof of this theorem, see page 199 (1stedition) or page 225 (2ndedition) of the textbook by Griffith.

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In other words

( ) ( ) ( ).ikLL k k k T x x L e x = + = (6)

The Bloch function k(x) is an eigenfunction of the displacement operator with eigenvalue eikL. Since the eigenvalue is not real, the operator is not Hermitian and cannot represent an observable.Because of Blochs theorem we need only solve the Schrdinger equation within a single unit cell, since the

solution everywhere else can be generated by repeated use of eq.(6).

2.4Crystal momentum kThe wave function of a free electron in an infinite solid is given by

( ) ikxkx C e =

where Cis a normalization constant, and the electron has momentum kand energy2 2

2

k

m

.

However, in a crystal,

it is easily shown that k(x) is not an eigenfunction of the momentum operator x dp idx

=

so that k(x) does not represent an electron state of well-defined linear momentum;

nonetheless, the quantity kis called the crystal momentum or the quasi-momentum of the electron; the effect of the crystal structure on the free electrons is incorporated in the periodic function uk(x).

In the limit that the effects of the crystal structure become weak,

the function uk(x) approaches a constant C, the Bloch function approaches the free-electron wave function Ceikx, and the quantity kbecomes the true momentum of an electron.

2.5Brillouin ZonesConsider two Bloch functions with different crystal momenta kand kwhere

2, 0, 1, 2,k k n n

L

= + = (7)

Now it follows from eqs.(6) and (7) that

2( ) ( )

( )

( )

L

ik L

L k k

i nLikL

k

ikL

k

T x e x

e e x

e x

==

=

since2

1Li nLe

= for ninteger. In other words:

The wave function k(x) is an eigenvector of TLwith the same eigenvalue as k(x). In fact, the two wave functions k(x) and k(x) describe4the same physical state of crystal momentum k. Thus we need consider only values of kwithin a single range of values of width 2/L.4See page 165 of the book by McMurry.

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The different ranges of kof width 2/Lare called Brillouin Zones.

We usually need consider only the first zone, conventionally defined by

.kL L

+

2.6Periodic boundary conditionsIn order to solve the Schrdinger equation for an electron, we must apply boundary conditions on the wave

function at either end of the one-dimensional solid:

1212

x NL

x NL

=

= + (8)

where N is the number of unit cells in the solid (and is extremely large) and we have taken x= 0 as the centre

of the crystal.

The wave function of an electron deep inside a sample of macroscopic size will not depend critically on the

precise form of the boundary conditions. Therefore we choose the conditions to simplify the solution of the

problem, specifically:

1 1.

2 2k kNL NL

= +

But from eq.(6)

( ) ( ),ikNLk kx NL e x + =

leading to

1 1.

2 2ikNL

k kNL e NL + =

Hence to satisfy the boundary conditions eq.(8) we require that eikNL= 1,

which means that

2integer.k

NL

=

Thus, the spectrum of kis discrete. Note that:

This condition implies that within each Brillouin zone of width 2/Lthere are exactly Npossible valuesof k.

Since Nis extremely large, the separation between allowed values of kis extremely small. Thus kiseffectively continuous in the range /Lk+/L.

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2.7The periodic potentialThe qualitative behaviour of a solid does not depend on the details of the periodic potential5. For simplicity

we use a one-dimensional Dirac comb a string of delta-function potential wells (or spikes if is negative):

/2

/2( ) ( ).

N

p NV x x pL

+

== In this equation, is a constant giving the strength of the interaction, and the potential wells are at positions

x= pL, p= integer.

The potential is shown schematically in the figure below6.

Figure 2: The Dirac comb a periodic potential

We now solve the Schrdinger equation (1) for this potential. In the region 0 < x< Lthe potential is

V(r) = 0, and the Schrdinger equation becomes

2 2

2( ) ( ).

2

dx E x

m dx

=

The general solution is

( ) sin cos , 0x A x B x x L = + < < (9)

with

2 2

.2

Em

=

As discussed above, we can generate from this the solution for the range L< x< 0 by using eq.(6):

( ) ( ).ikLk kx L e x + =

This gives

( ) ( )[ ]( ) sin cos , 0.ikLx e A x L B x L L x = + + + < < (10)

We now apply the continuity conditions at the matching point x= 0.

The wave function (x) is continuous at x= 0. We get directly from eqs.(9) and (10)[ ]sin cos .ikLB e A L B L = + (11)

5In their original treatment, Kronig and Penney used finite-depth square well potentials (for further discussion, see pages 98-101 of

the book by Gasiorowicz). In the limit that the wells become infinitely deep and infinitely narrow, this is essentially equivalent to the

present treatment which is based on Griffiths.6Based on Figure 5.5 of the book by Griffiths.

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The derivative of (x) is notcontinuous at x= 0, since it is a general rule that the derivative is continuousexceptat points at which the potential is infinite. Nonetheless, we can easily find7an expression for the

magnitude of the discontinuity at x= 0.

To achieve this, we integrate the Schrdinger equation from x= 0 to x= 0 +and take the limit 0.

2 2

2( ) ( ) ( ) ( ) .

2d x

dx V x x dx E x dx m dx

+ + +

+ =

The first integral gives

2

20 0

( )lim lim .

x x

d x d d d dx

dx dx dx dx

+

=+ =

=

The third integral tends to zero as 0 since the wave function is continuous at x= 0:

0lim ( ) 0.x dx

+

=

So we get

2

( ) ( ) ( ) ( )2

dV x x dx x x dx

m dx

+ +

= = +

or, rearranging and using the definition of the delta function to evaluate the integral:

2

2(0).

d m

dx

= (12)

From eq.(9) we get

[ ]

cos sin

in the limit 0

x

dA B

dx

A

=+

=

(13)

and from eq.(10)

( ) ( )[ ]

[ ]

cos sin

cos sin in the limit 0

ikL

x

ikL

de A L B L

dx

e A L B L

=

= + +

(14)

From eq.(9) we also find

(0) .B = (15)

We substitute eqs.(13), (14) and (15) into eq.(12) and use eq.(11) to give, after much rearrangement,

2cos cos sin .mkL L L

=

(16)

This is the fundamental result from which all else follows. For other choices of potential, the equation willbe more complicated, but the qualitative features remain unchanged.

7This derivation is based on page 55 (1stedition) or page 72 (2ndedition) of the textbook by Griffith.

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2.8Formation of energy bandsConsider the form of eq.(16):

The left-hand side of the equation is a function of the crystal momentum k, whereas the right-hand sideis a function of , or electron energy 2 2 /2E m= .

The left-hand side is always in the range 1 to +1, whatever the crystal momentum. In fact, as ktakeson all its allowed values in the first Brillouin zone, coskL increases from 1 to +1 and back to 1.

The right hand side of the equation clearly has a magnitude greater than +1 or less than 1 for certainranges of values of . Thus the equation has no solution for some values of , that is for some values of the

electron energy 2 2 /2E m= . We see that some electron energies are allowed, and some areforbidden.

This last point becomes clearer if we rewrite the right-hand side of the equation (16) as

sin( ) cos

zf z z

z=

where z= Land 2/mL = . The figure below shows a graph of the functionf (z) against zfor a fixed value

of = 5.0.

It shows the formation of energy gaps, which represent ranges of forbidden electron energies (where+1

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The resulting electron energy level diagram is shown schematically in the next figure8.

There are Nqfree electrons in a solid, where Nis the number of atoms and qis the number of freeelectrons per atoms, q= 1, 2, 3,

At sufficiently low temperatures these will fill up the lowest Nq/2 electronic states (the factor of 1/2arises because each energy state can accommodate two electrons with different spin orientations).

Since there are Nelectron states in each band, if qis even the electrons will completely fill the lowestq/2 bands; if qis odd, the last occupied band will be half full.

Figure 4: Schematic diagram showing energy bands and gaps

In a real three-dimensional solid with more realistic potentials, the band structure is more complex and these

detailed conclusions may not be valid. However, the qualitative discussion will remain unaltered (see thedescription of conductors, insulators and semiconductors below).

3.ELECTRICAL PROPERTIES OF SOLIDS9Solids are classified electrically according to three basic properties, as described in the following table.

Property Symbol Unit Description

Resistivity .m Resistivity is defined by E J=

Eis the field at some point in the material

J

is the current density at that point

Temperature coefficient of resistivity K1 Defined by1d

dT

=

Number density of charge carriers n m3 Measured using e.g. Hall effect

8

Based on Figure 5.7 of the book by Griffiths.9The original version of this section was taken from section 42-2 of the textbook by Halliday, Resnick and Walker.

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From measurements of resistivity at room temperature, it is found that some materials effectively do not

conduct electricity at all (or rather, there is no measurable current) these are called insulators. For example

24(diamond) 10 (copper).

Materials that are not insulators can be divided into two major categories conductors(or metals) and

semiconductors. Semiconductors have considerably higher resistivity than conductors. The temperature coefficient of resistivity of conductors is positive (i.e. their resistivity increases with

temperature). Semiconductors have a temperature coefficient of resistivity that is both larger and

negative (i.e. their resistivity decreases with temperature).

Semiconductors have considerably fewer change carriers per unit volume.These features are illustrated in the table below for copper, a metal, and silicon, a semiconductor.

Property Copper Silicon

(.m) 82 10 33 10

(K1) 34 10+ 370 10

n(m3) 289 10 161 10

All values are appropriate at room temperature (300 K).

4.BAND THEORY OF ELECTRICAL CONDUCTIVITYWe now apply the band theory of solids to explain their electrical properties10. As discussed for the simple

one-dimensional model described above:

There are Nqfree electrons in a solid, where Nis the number of atoms in the solid and qis the numberof free electrons per atoms.

At low temperatures these will fill up the lowest Nq/2 electronic states, with the factor of 1/2 arisingbecause each energy state can accommodate two electrons with different spin orientations.

4.1InsulatorsIn some materials there are sufficient electrons to just fill the energy levels up to the top of one band; this band

is called the valence band, and an example of such a material is diamond.

In order for a current to flow when an electric field is applied to a solid, free electrons must be able to gain

kinetic energy by absorbing energy from the field. The flow of electrons, in a direction opposite to the field,

constitutes an electrical current.

10See sections 42-4 to 42-7 of the book by Halliday, Resnick and Walker.

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Figure 5: Energy bands for an insulator

In a material such as diamond, none of the electrons in the valence band can absorb energy to move to ahigher energy level within the same band, since all states in the band are already filled. In addition, for

normal fields an electron at the top of the valence band cannot absorb sufficient energy to jump from the

valence band to the next band, called the conduction band, since the gap is too large. Materials such as

diamond are therefore electrical insulators.

If an extremely large field is applied, some electrons may gain sufficient energy to jump the gap and acurrent will flow; i.e. the insulator breaks down. Such fields will in any case usually disrupt the electronic

structure of the crystal.

The magnitude of the energy gap between the valence and conduction bands also affects the optical properties

of a material. Photons in the visible spectrum have wave lengths in the range 400 to 700 nm, corresponding

to photon energies of 3.1 to 1.8 eV.

Photons in the visible spectrum have insufficient energy to raise electrons from the valence to conductionband. Insulators such as diamond are therefore transparent to visible light, since the photons cannot be

absorbed within the material.

More generally, the absorption of radiation can be used to determine the magnitude of the gap energy ofa material.

4.2ConductorsIn some materials, for example metals, the band of highest energy is only partially filled, as in the diagram be-

low11. The energy of the highest-lying filled electron state is called the Fermi energy; this lies within the

conduction band for a conductor.

11

In materials of high valency such as aluminium, the filled valence band and partially-filled conduction band overlap, leading to apartially filled combined band; all electrons in the combined band can contribute to an electric current.

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Figure 6: Energy bands for a conductor.

Electrons just below the Fermi energy can easily be excited to vacant levels just above the Fermi level,since the spacing of energy levels within a band is extremely small.

Thus when an electric field is applied to the material these electron can absorb energy and form a current.Their motion creates vacancies in the previously filled levels, and these are filled by electrons initially in

lower levels. In this way all the electrons in the conduction band can contribute to the current. The

material is therefore a conductor.

Photons in the visible spectrum can cause excitation of electrons within the conduction band. Suchphotons are readily absorbed by electrons in the conduction band and the material is therefore opaque to

visible light.

4.3Intrinsic semiconductorsSome materials have a band structure similar to insulators except that the gap between the valence and

conduction bands is significantly smaller. These materials are called semiconductors. For example:

Material Energy gap

0 K 300 K

silicon 1.17 eV 1.11 eV

germanium 0.74 eV 0.67 eVgallium arsenide 1.52 eV 1.43 eV

At low temperatures, these materials are insulators for the reasons discussed above.

However, because of the smaller energy gap, at room temperature the thermal energy of the electrons is

sufficient to raise some of them from near the top of the valence band into the conduction band. This leaves

vacant electron energy levels, or holes, near the top of the valence band. A hole, corresponding to the absence

of an electron, acts just like a positively-charged electron.

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Figure 7: Energy bands for an intrinsic semiconductor at room temperature.

If an electric field is applied to the material:

The electrons in the conduction band because of thermal excitation absorb energy, being excited to vacanthigher energy states within the same band. The absorbed energy becomes kinetic energy, and the electrons

move in a direction opposite to that of the field (creating a current parallel to the field).

Electrons just below the levels left vacant in the valence band by thermal excitation absorb sufficientenergy from the field to fill these vacancies, thereby creating further vacancies. In other words, the holes

created in the valence band move in response to the applied field. The motion of the holes creates a current

in the same direction as the field. The motions of the electrons in the conduction band and holes in the valence band contribute equallyto

a measurable current, since the number of holes in the valence band equals the number of electrons in the

conduction band.

To see why electrons can be excited thermally from the valence band to the conduction band in silicon but not

in diamond, we consider the probability of an electron having sufficient energy at temperature Tto jump the

energy gap Egapbetween the two bands; this can be estimated as follows.

The mean kinetic energy of the electrons at temperature Tis approximately kTwhere k= 8.62 105 eV/K

is Boltzmanns constant. The probability of a single electron having energy Egapis given approximately by the

Boltzmann distribution law as exp gapE

kT .

The table below compares this probability at room temperature for the insulator diamond with some common

semi-conducting materials. These values are very approximate because of the extreme sensitivity of the result

to the magnitude of the energy gap.

In a macroscopic sample of silicon or germanium, with of the order of Avogadros number of free electrons, a

reasonable number will be excited to the conduction band, but for diamond the number effectively zero.

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Material Energy gap Probability n(in cm3)

Diamond 5.5 eV 3 1093

Silicon 1.1 eV 3 1019 1.4 1010

Germanium 0.7 eV 6 1012 2.5 1013

Gallium arsenide 1.4 eV 1 1024 9 106

The approximate density of electrons in the conduction band at a temperature of 300 K is shown in the table

above12. This density rises steeply with temperature since the probability of thermal excitation from the

valence band increase sharply with temperature. (Note the implication that even an insulator can become

conducting at sufficiently high temperature, providing it does not melt!)

For comparison, in metals there are approximately 1022 1023electrons per cm3in the conduction band.

4.4Extrinsic or impurity semiconductorsAll modern electronic devices involve extrinsic or impurity semiconductors, rather than the intrinsic semi-

conductors described above. These are formed by dopinga semiconducting material with atoms with a

different value of the quantity q. For example:

Material Configuration No. of valence electrons

silicon 2 2(3 ) (3 )s p 4

antimony 2 3(5 ) (5 )s p 5

The effect of the antimony atoms can be understood by considering the bonds13that act between atoms in the

crystal lattice.

Figure 8: Covalent bonds in silicon Figure 9: Bonds with antimony atom replacing silicon

In a sample of silicon, the bond between two adjacent silicon atoms is made from an electron donated by each

of the two atoms. As shown in the first diagram, a silicon atom contributes one electron to the formation of

12These values are taken fromMicroelectronic Devicesby K Leaver, 2ndedition, published by Imperial College Press (1997)13

The figures, which are flattened two-dimensional representations of three-dimensional objects, are based on those in the book byLeaver.

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the bond with each of its four neighbouring atoms; therefore for each atom four electrons take part in the

formation of bonds.

The electrons that form the bonds are the (3s) and (3p) electrons, the valence electrons listed in the table

above. These electrons constitute the valence band of the silicon sample. If an electron is freed from its bond

so that it is able to move through the lattice, we say that it has been raised from the valence band to theconduction band. The minimum energy required to remove an electron from a bond is the gap energyEgap.

If an antimony atom replaces a silicon atom in a crystal, four of its five valence electrons take part in bonds

with neighbouring silicon atoms, and the remaining electron is very loosely bound to the antimony ion core,

as shown in the second diagram. This creates an additional electron state that is situated in the energy gap;

since the energy needed to free this loosely-bound electron is extremely small, the additional energy level is

only just below the conduction band.

Each antimony atom that replaces a silicon atom creates such an additional level. At very low temperatures

these levels will all be filled, but only minor thermal agitation is necessary to ionize the impurity atoms.

Therefore, even at relatively low temperatures, electrons will be excited from the additional levels to the

conduction band. The antimony atoms are called donor atoms,since they donate electrons to the conduction

band, and the additional levels are called a donor levels.

Figure 10: Energy bands for an n-type semiconductor (at very low temperature)

At room temperature, virtually all the electrons from the donor atoms are in the conduction band. Whenan electric field is applied, these electrons are able to absorb energy and form a current. This is in addition

to the current that in any case would be present since silicon is an intrinsic semiconductor. The majority

carriersare now electrons and the holes in the valence band are the minority carriers.

Antimony is referred to as a donor-type impurity, and the doped material is an n-type semiconductor(since the conduction is mainly due to negatively-charged electrons).

A different type of impurity semiconductor is created by doping silicon with atoms that have one fewer valence

electrons, such as indium.

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Material Configuration No. of valence electrons

silicon 2 2(3 ) (3 )s p 4

indium 2(5 ) (5 )s p 3

An indium atom introduced into a sample of silicon can form a covalent bond with only three of the four

neighbouring silicon atom, leaving an unsaturated bond with only one electron and a hole where the second

electron should be. However, it requires only a small amount of energy to remove an electron from a nearby

silicon-silicon bond to fill this hole. This of course creates a hole in the silicon-silicon bond, but this will be

filled by an electron from another bond, thereby creating a new hole. In this way the hole moves through the

lattice.

Figure 11: Bonds with indium atom replacing silicon

The indium atom is called an acceptor atomsince it readily accepts an electron from another bond, i.e. from

the valence band of silicon. The electron occupies an energy level just above the top of the valence band, which

is called an acceptor level. At temperatures close to absolute zero such levels are empty, but only slight

thermal agitation is necessary to move an electron from the valence band into an acceptor level. At room

temperature, all acceptor levels will be occupied.

Under thermal agitation, a silicon electron is excited from the top of the valence band into the acceptorlevel, leaving an unfilled level or hole in the valence band.

When an electric field is applied, the holes in the valence band drift in the direction of the field, creatinga positive current.

The number of holes in the valence band is very much larger than the number of electrons that exist inthe conduction band at room temperature in an intrinsic semiconductor. The majority carriers in this case

are therefore holes and the minority carriers are electrons. Indium is said to be an acceptor impurity and

the material is called a p-type semiconductor.

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Figure 12: Energy bands for a p-type semi-conductor at very low temperature

The table below summarizes the properties of semiconductors created by doping silicon with phosphorus and

aluminium.

Property Type of semiconductor

n type p type

Matrix material silicon silicon

Matrix energy gap 1.14 eV 1.14 eV

Dopant phosphorus aluminium

Type of dopant donor acceptor

Dopant valency 5 3

Dopant energy gap 0.045 eV 0.067 eV

Majority carriers electrons holes

Minority carriers holes electrons

Replacing 1 in 5 million Si atoms with P atoms increases the density of charge carriers by a factor of about1 million, to approximately 1016cm-3. The conductivity is roughly proportional to the density of dopant

atoms.

Typically, 1 silicon atom in 105 1010would be replaced by dopant atoms in a commercially producedsemiconducting material.