ELEKTRONIKOS PAGRINDAI 2008 of... · 2009-11-19 · Then the carriers are not in thermal...

CONDUCTIVITY OF SOLIDS

Objectives:

Revelation on what, why and how conductivity of solids and currents in solids

depend.

Content:

1. Conduction process in solids

2. Mobility of charge carriers in semiconductors

3. Conductivity of semiconductors

4. Strong field effects• Carrier velocity saturation

• The Gunn effect

• Strong field effects caused by variation of carrier density

5. Photoconductivity

6. Conductivity of metals

7. Superconductivity

8. Josephson effects

9. Hall effect

10.Diffusion of charge carriers in semiconductors

11.Total current flow in semiconductors

ELEKTRONIKOS PAGRINDAI 2008

VGTU EF ESK [email protected]

1

Conduction process in solids

Let us consider a solid containing conduction electrons.

If the external electrical field is not applied, electrons

move randomly colliding with crystal lattice defects.

After application of the field, electrons began to drift in

the direction opposite to the direction of the field.

The drift velocity is dependent on the strength of the field.

At applied field, force F = qE acts on an electron.

During the collision with the lattice, the electron losses its velocity.

The action of a crystal lattice on a drifting electron can be accounted by a

restraining force. It is given by

EvmF nr

p

1

τ−=

Thus, the total force acting on an electron is given by

t

vmamvmEFFF E

Ed

d...

1q nnn

r

r ===−=+=Σ τ



2

t

vmvmE E

Ed

d1q nn

r

=−τ

... When the external electrical field is applied, the drift velocity increases. At

the same time the restraining force also increases. The drift velocity becomes

constant when the restraining force becomes equal to the accelerating force.

Then

EEm

vE nn

rqµ

τ==

n

rn

q

m

τµ =

The coefficient µn is called mobility. If E = 1, we have that .nE µ=v

… The mobility can be defined as the incremental average electron velocity per

unit of electric field.

… The mobility of charge carriers in a solid is limited by the scattering of the

carriers by the crystal lattice.

Conduction process in solids



3

Conductivity

Let us consider a parallelepiped.

EvV = EnvnVN ==Σ

Envj q= EEnj σµ == nq

… The drift velocity in a solid is proportional to the strength of the electric field.

The last formula can be rearranged to I = U / R (Ohm’s law).

σ is conductivity.

nq µσ n= )(q pn µµσ pn +=

… Conductivity is dependent on charge carriers density and their

mobility.



4

Relaxation time and mobility

Now let us suppose that after some initial application of the field it is suddenly

reduced to zero.

)(1

d

)(d

r

tvt

tvE

E

τ−=

−=

r

exp)0()(τt

vtv EE

... The last equation indicates that the drift velocity relaxes back to zero from the

initial value exponentially. The process during that the system returns to

equilibrium after its excitation is called relaxation.

The time constant τr is the electron relaxation time.

rτλ v= rτλ vk = vk r λτ =

... Charge carriers mobility:v

k

m

λµ

p,np,n

q= Tvv ≅

t

vmvmE E

Ed

d1q nn

r

=−τ

If we know relaxation time, we can find the average distance between two

collisions.



5

Mobility of charge carriers in semiconductors

Charge carriers are microparticles. It is necessary to take into account their dual

nature.

The lengths of de Broglie’s waves corresponding to the free electrons and holes

are much longer than crystal lattice constants. Then free electrons and holes do

not reflect from the nodes of a perfect crystal lattice. They are reflected only by

lattice defects caused by thermal vibrations, impurity atoms, dislocations and

other lattice imperfections.

An intrinsic semiconductor at 0 K is transparent (clear) for electronic waves.

There are no impurities in intrinsic semiconductors. Scattering of carriers is

caused by phonons.

With the increase of temperature thermal vibrations of the crystal lattice arise.

These vibrations cause thermal defects that cause scattering of electrons and

holes.

Tv

k

m

λµ

p,np,n

q=

T/1~λ

TvT ~

2/3p,n ~ −Tµ1≅k



6

2/3p,n ~ −Tµ ... The carrier mobility in an intrinsic semiconductor

decreases with temperature.

Impurity density does not depend on temperature, and the mean free path of a

carrier does not depend on temperature. The mean velocity increases with

temperature. The change of the carrier motion direction near an ion is less at

higher velocity. For this reason the number of collisions, after what the electron

losses its velocity in some direction, increases with temperature.

In a heavily doped semiconductor the

impurity scattering predominates in the low

temperature range.

As the temperature is increased the

thermal velocity of charge carriers

increases and the impurity scattering

decreases.

24 ~)(~ Tvk T TvT ~const=λ 2/32

p,n ~~~ TT

T

v

k

T

λµ




7

2/32

p,n ~~~ TT

T

v

k

T

λµ

... In the low temperature range the

mobility in the doped semiconductors

increases with temperature.

In the high temperature range the lattice scattering predominates and mobility in a

doped semiconductor decreases with temperature

In a heavily doped semiconductor the maximum mobility appears in the

middle temperature range (usually at the temperature lower than 300 K).


Mobility is also dependent upon impurity density. At room temperature mobility in

a doped semiconductor is constant for low impurity concentrations because it is

limited by the lattice scattering. For impurity densities greater than 1016 cm-3

mobility decreases as a result of impurity scattering.



8

Mobility at normal temperature

v

k

m

λµ

p,np,n

q=

Electron mobility in silicon is about 3

times greater than that of holes.

Collisions of charge carriers with neutral impurity atoms, dislocations and other

imperfections of lattices also limits the mobility.

It is possible to estimate the influence of various factors using relationship

∑≅i iµµ

11

At room temperature mobility in a doped semiconductor is

constant for low impurity concentrations because it is limited

by the lattice scattering. For impurity densities greater than

1016 cm-3 mobility decreases as a result of impurity

scattering.

Mobility is also dependent upon impurity density.



9

Mobility of charge carriers in semiconductors. Problem

The mobility of electrons in silicon is around 1500 cm2/(V·s).

Estimate the mean free path of an electron. Compare it with the

lattice constant a = 0,543 nm.



10

Conductivity of semiconductors

Two types of carriers exist in semiconductors. According to this

)(q pn µµσ pn +=

In the intrinsic semiconductor densities of electrons and holes are equal. Then

)(q pnii µµσ += n )k2/exp(~ 2/3i TWTn ∆− 2/3

p,n ~ −Tµ

TW k2/0i e ∆σσ −=

T

W

k2lnln 0i

∆σσ −=

... Conductivity of an intrinsic semiconductor is strongly dependent upon

the gap energy and temperature because the density of intrinsic carriers

strongly depends on the gap energy and temperature.



11

The specific conductivity of a doped semiconductor depends on the

majority carrier density and their mobility.

nn q µσ n= pp q µσ p=The conductivity of the doped semiconductor

varies with temperature for two reasons.

One is the dependence of the free charge carrier

density on temperature which is felt either at very

low temperatures (when not all impurity atoms are

ionised) or at very high temperatures (when

thermal generation of intrinsic carriers begins).

The second reason is the dependence of mobility

on temperature. It determines the character of

conductivity in the intermediate range: the

conductivity slowly decreases with increasing

temperature.

At room temperature conductivity depends on the impurity density. If the impurity

density is higher, the density of majority carriers is higher and the conductivity of

the doped semiconductor is also higher.

Conductivity of semiconductors



12

Strong field effects

If the electric field is low, the conductivity of a semiconductor is constant. Then the

current density linearly depends on the strength of the field.

EEj )(σ= )(q pn µµσ pn +=

... The strong field effects may be caused by carrier mobility or density variation

in the strong field.

.Ej σ=

The proportionality between j and E is gradually lost at higher and higher fields. It

is because conductivity becomes dependent on the strength of the field.

This phenomenon is the strong field effect.

In the strong field region



13

Mobility versus strength of electrical field. Carrier velocity saturation

v

k

m

λµ

p,np,n

q=

At small strength of the field the mean charge carrier

velocity equals the mean thermal motion velocity.

The mobility is constant.

If the strength of the field increases, the drift

velocity also increases. This causes an increase of

the mean velocity and decrease of the mobility.

The decrease of the mobility causes the

saturation of the drift velocity.

If the condition that the drift velocity is small is no longer satisfied, we have that

the mean velocity of a charge carrier is greater than it mean thermal velocity.

Then the carriers are not in thermal equilibrium with the crystal lattice.

The carriers with energies greater than 3kT/2 are called hot carriers (hot

electrons and hot holes).



14

Mobility in compound semiconductors. The Gunn effect

The energy of a free microparticle is given bym

p

mk

mW

22

h

2

2

2

22

2

k ===λ

h

In the case of conduction electrons we must

consider the effective mass.

The effective mass is dependant on energy.

For this reason the graph W(k) becomes

complicate. There are two local minima or two

valleys in the conduction band of GaAs.

In low electric fields, electrons reside in the

lower central valley. Then the electron

mobility in GaAs is high (about

8000 cm2/(V⋅s)). EvE 1n1 µ= Enj n1q µ=

As the field is increased, electrons transfer

to the upper valley and their mobility is

considerably reduced (to 0,01 m2/(V⋅s))

EvE 2n2 µ= Enj 2nq µ=



15

1st

valley

2nd

valley

The reduction of the mobility causes a

reduction of the drift velocity and current

density.

If the current density decreases with E, the

differential conductivity is negative:

... The reduction of mobility leads to the

differential negative resistance effect. It was

suggested by Ridley and Watkins in 1961 and

discovered by J. B. Gunn in 1963 after whom

the effect is named.

Mobility in compound semiconductors. The Gunn effect

0,0/dd dd <<= ρσ Ej

Devices based on the Gunn effect are called either Gunn diodes or

transferred-electron devices (TEDs). They can be used in amplifiers or

oscillators.



16

Strong field effects caused by variation of carrier density

The sufficient increase in the conductivity of semiconductors may appear due to

the ionization of semiconductor atoms.

As a result of semiconductor atoms ionization, densities of charge carriers

increase, current strength increases rapidly with increase of applied voltage and

the strength of the field, and breakdown in semiconductor begins.

Electrons may be liberated when host atoms are struck by other free electrons

accelerated by the strong electric field. This mechanism is called avalanche

breakdown.

In the very strong electric field electrons are stripped from their host atoms by the

strength of the electric field. This mechanism is called Zener breakdown.



17

Photoconductivity

Light with the wavelength less than some critical value can cause a change in

conductivity of a semiconductor. Such optically induced conductivity is called

photoconductivity. It is possible due to the internal photoelectric effect.

If photon energy exceeds the forbidden energy gap, the photon can release an

electron from a covalent bond. In that way electron-hole pairs can be generated

by light. The generated excess carriers cause intrinsic photoconductivity.

h/min W∆ν = W∆νλ /hc/c minmax ==

W > ∆W



18

Photoconductivity

Besides intrinsic photoconductivity extrinsic photoconductivity is possible. In this

instance the carrier is generated from either a donor or acceptor level. Clearly,

necessary for generation energy in these cases is always less than half the

semiconductor band-gap energy. Therefore extrinsic photoconductivity is possible

in the range of longer wavelengths.

The extrinsic photoconductivity is possible only in the low temperature or

ionisation range because impurity atoms are ionised at higher temperatures.

Photoconductivity is dependant on the wavelength.

If the wavelength decreases, photoconductivity increase, becomes maximal, and

after that decreases.

Light can also cause change of mobility of charge carriers.

W > ∆W



19

Conductivity of metals

... The conductivity of a metal depends on the density of free electrons and their

mobility.

When atoms form a metal, the valence electrons become free. Thus the density

of free electrons in a metal depends upon the density of metal atoms and

the valence of the element. The density of the free electrons in metals is about

1022–1023 cm–3.

Conductivity of metals is dependant on electron mobility.

The conductivity of metals is high. Therefore the strength of the electric field in

metals usually is low.

In the low field only those electrons near to the Fermi level can change their

energy and participate in the conduction process.

According to this the mobility of electrons in conductors is given by

F

FF

nn

q

v

k

m

λµ =



20

The Fermi energy depends upon the density of free electrons. Because the density

of free electrons in a metal does not vary with temperature, the Fermi energy is

almost constant. Then the mean velocity of electrons is also almost constant.

Usually electrons in a metal are scattered by phonons. Then

T/1~nµ T/1~σ T~/1 σρ = )](1[ 00 TT −+≅ αρρ

... The resistivity of a metal is proportional to temperature.

... The resistivity of a metal is the sum of a thermal part and a residual resistivity.

The residual resistivity at low temperatures is

caused by lattice defects (impurity scattering).

Increasing impurity concentration in a metal

causes the increase of the residual resistivity

and resistivity.

Conductivity of metals and alloys

Lattice defects caused by other atoms are particularly large in disordered alloys

such as nichrome. Hence such alloys are used when high resistance combined

with a low temperature coefficient of resistance are required.



21

Superconductivity

The residual electrical resistance of many materials drops abruptly to an

unmeasurably small value when the material is cooled below a sharply defined

transition temperature. This phenomenon was discovered by H. Hamerlingh

Onnes in Leiden in 1911 and was called superconductivity. An electric current

induced in a superconducting lead ring can persist (without any battery) for

several years without significant decay.

The modern theory of superconductivity was

created only in 1957. This theory was created

by J. Bardeen, L. Cooper and J. Schrieffer and

is called BSC theory.

According to the theory free electrons form electron pairs called Cooper pairs at

the transition temperature. We can imagine that Cooper pairs appear in this way.

A negative electron attracts positive ions. When they approach the electron a

positive charge appears. This attracts the other electron having different spin

quantum number.

When two electrons having different spin quantum numbers form a Cooper pair

the spin quantum number becomes 0. So Cooper pairs have properties of

bosons.



22

The superconducting material exhibits perfect diamagnetism in weak magnetic flux

densities (the flux inside the material is zero). If the value of the applied flux density

rises to a value greater than a critical value, superconductivity is destroyed. The

value of the transition flux density is a function of the temperature of the material

and its nature. A superconducting current itself can produce the magnetic flux

density greater than the critical value; therefore there is an upper limit of the current

density that may be sustained by the material in the superconducting state.

When two electrons having different spin quantum numbers

form a Cooper pair, their energy decreases. As a

consequence a forbidden energy gap appears in the energy

diagram of a superconductor.

The width of the gap is some milielectronvolts. The energy

levels below the gap are occupied by electrons and Cooper

pairs. The levels above the gap are unoccupied..

At low temperatures near to 0 K the lattice vibrations are not intense. The energy of

the vibrations is not enough to break Cooper pairs. Therefore at low temperatures

Cooper pairs move in the crystal without any collisions and scattering. Due to the

long free path of charge carriers conductivity is also great (infinite) and a

superconductor allows electricity to pass freely, without resistance.

Superconductivity



23

Until 1986, physicists had believed that BCS theory forbade superconductivity at

temperatures above about 30 K. In that year, Bednorz and Müller discovered

superconductivity in a lanthanum-based cuprate perovskite material, which had a

transition temperature of 35 K (Nobel Prize in Physics, 1987). It was shortly found

by Paul C. W. Chu of the University of Houston and M.K. Wu at the University of

Alabama in Huntsville [1] that replacing the lanthanum with yttrium, i.e. making

YBCO, raised the critical temperature to 92 K, which was important because liquid

nitrogen could then be used as a refrigerant (at atmospheric pressure, the boiling

point of nitrogen is 77 K.) This is important commercially because liquid nitrogen

can be produced cheaply on-site with no raw materials, and is not prone to some

of the problems (solid air plugs, etc) of helium in piping. Many other cuprate

superconductors have since been discovered, and the theory of superconductivity

in these materials is one of the major outstanding challenges of theoretical

condensed matter physics.

As of March 2007, the current world record of superconductivity is held by a

ceramic superconductor consisting of thallium, mercury, copper, barium, calcium,

strontium and oxygen (Tc=138 K). Also a patent has been applied for a material

which becomes superconductive at an even higher temperature (up to 150 K).[2]

http://en.wikipedia.org/wiki/Superconductivity

Superconductivity



24

Superconductivity

Superconductors are used to make some of the most powerful electromagnets

known (superconducting magnets)… They can also be used for magnetic separation,

where weakly magnetic particles are extracted from a background of less or non-

magnetic particles, as in the pigment industries.

Superconductors have also been used to make digital circuits (e.g. based on the

Rapid Single Flux Quantum technology) and microwave filters for mobile phone base

stations.

Superconductors are used to build Josephson junctions which are the building blocks

of SQUIDs (superconducting quantum interference devices), the most sensitive

magnetometers known…

The presence of a current in one of the strips

changes the superconductivity of the other

element and hence switches it off or on. It is

important that the switching process is very

fast (it lasts only some picoseconds).

SubstrateSiO2

Tin strip Lead

strip

The cryogenic switching devices called cryotrons have very simple construction. A

thin film cryotron consists of two insulated crossing strips made of superconductive

materials with different critical field curves such as tin and lead.



25

Josephson effects

Two superconductive layers with a

thin dielectric layer between them

(Fig 5.12(a)) form a Josephson

structure or Josephson junction.

The terms are named eponymously

after British physicist Brian David

Josephson, who predicted the

existence of the effect in 1962.

The superconducting current can flow across the

junction in the absence of an applied voltage. This

is the direct-current Josephson effect.

The direct-current Josephson effect occurs due to

the tunnelling current. The Cooper pairs can

penetrate through the thin potential barrier without

change in their energy. Then there is no voltage

drop across the junction.



26

The current-voltage

characteristic of the

Josephson junction

If the current exceeds critical value, the voltage

drop, corresponding to the forbidden energy gap,

arises.

… When a small direct voltage is applied, the

alternating current Josephson effect occurs.

The superconducting current across the junction

becomes an alternating current and the junction

radiates electromagnetic waves.

Cooper pairs penetrating the dielectric layer occur

over the gap. After that they jump to the energy

level below the gap. The dissipation of energy is in

the form of electromagnetic waves.

Josephson effects



27

The dissipation of energy is in the form of electromagnetic waves.

ϕsin0II = UW q2C =∆ h/q2h/C U∆W∆ ==ν

tU

t

==h

q2π2π2 ν∆ϕ )π2sin(

q2sin 00 tfIt

UII =

=h

Ufh

q2=

If the voltage change is 1 mV, the frequency change is 483,6 MHz.

... Frequency of oscillation is proportional to the voltage drop on

the junction.

Josephson effects

The field of electronics holds great promise for practical applications of supercon-

ductors. The use of new superconductive films may result in more densely packed

chips which could transmit information more rapidly by several orders of magnitude.

Superconducting electronics have achieved impressive accomplishments in the field

of digital electronics. Logic delays of 13 picoseconds and switching times of 9

picoseconds have been experimentally demonstrated. Through the use of basic

Josephson junctions scientists are able to make very sensitive microwave detectors,

magnetometers, SQUIDs and very stable voltage sources.

The Josephson junction quantum computer was demonstrated in April 1999 by

NEC Fundamental Research Laboratories in Tsukuba, Japan.



28

Conversely, the stability of voltage references

based on the Josephson effect depends solely

on frequency stability, which can easily reach

10-12. For this reason, National Metrology

Laboratories started using the AC Josephson

effect as a representation of the volt and

adopted KJ, KJ-90= 483 597,9 GHz/V as a true

value for the Josephson constant. This value

was accepted by international agreement at the

18th General Conference on Weights and

Measures and came into application on 1st

January 1990. http://www.lne.eu/en/r_and_d/electrical_metrol

ogy/josephson-effect-ej.asp

The definition of the volt in the International System of Units (SI) is as follows:

"The volt is the electromotive force between two points of a conductor carrying a

current of 1 ampere when the power dissipated between the two points is 1 watt"

[3]. Realization of the volt in the SI system rests on experiments comparing an

electrostatic force with a mechanical force, but the uncertainties obtained by this

method are much too great to meet the requirements of modern instrumentation.

Josephson effects



29

Hall effect

... Let us consider that a current is passed through a

semiconductor and a magnetic field is applied at a right

angle to the direction of the current flow. Then an

electric field is induced in the direction mutually

perpendicular to I and B. This phenomenon is known as

the Hall effect, discovered in 1879 by E. H. Hall.

The electrons flowing with some drift velocity experience

the Lorentz force ][qL BvF E ×−=

It tends to drive electrons towards the right face D of the

bar. The electrons moving to the right leave positive ions

and as a consequence the electric field arises in the bar.

This produces the electrical force dUE /H=EFE q=

In equilibrium EBvFF E qq E,L == BvE E= Envj q=

jBdRjBdn

U HHq

1== nR q/1H = nq µσ n= nH µσ =R

... Simultaneous measurement of σ and RH can lead to the

experimental value for the carrier drift mobility.



30

Diffusion of charge carriers in semiconductors

It is also possible for current to flow in a semiconductor even in the absence of

the field. It can flow due to a carrier concentration gradient in the crystal. The

diffusion current can flow as a result of non-uniform densities of either electrons

or holes.

In neutral gas the flow or diffusion of particles occurs in the direction from

the high density (high pressure) to the low density (low pressure) region.

The particle flow depends on concentration gradient. It the instance when

density depends on one coordinate, concentration gradient is given by

x

NN

d

dgrad =



31

Krūvininkų difuzija

xx

pAF ∆d

dΣ −= xNANV ∆=

x

p

NNV

FF

d

d1ΣD −== TNp k=

x

N

NTF

d

d1kD −= ... The diffusion force is proportional to

temperature and concentration gradient.


The existence of a gradient implies that if an imaginary surface (for example,

indicated by dashed line) is drawn, the density of particles on one side of the

surface is greater than the density on the other side. The particles are in a random

motion. Accordingly particles move back and forth across the surface. Then in a

given time interval more particles cross the surface from the side of greater density

than in the reverse direction. Thus, the flow or diffusion of particles occurs.



32

... As a result of diffusion the density of neutral particles becomes uniform.

We will see that electrical field can appear in semiconductors as a result of

charge carriers diffusion.

... The diffusion force acts in a way entirely analogous to that in which the force

due to an electric field acts on electrons in a solid. The flow of particles is limited

by collisions.

EFE q= EE Fm

Em

Em

Evn

r

n

r

n

rnn q

q τττµ ====

x

n

nD

x

n

nT

mF

mv

d

d1

d

d1k n

n

rD

n

rDn −=−==

ττ

q

k

q

kqk n

n

r

n

rn

TT

mT

mD µ

ττ===

x

nDnvj

d

dqq nDnDn =−=

The diffusion of electrons causes electron current to flow.

In the case of diffusion force:




33

x

p

pDv

d

d1pDp −=

q

kk p

p

rp

TT

mD µ

τ==

x

pDpvj

d

dqq pDpDp −==

Considering holes, we have :

Notice that the electron diffusion current is in the same direction as the positive

gradient. The hole current flows in the opposite direction with respect to the

positive gradient.

… At any given temperature the diffusion coefficient and the mobility of carriers in

a given material are not independent of each other.

q

k

p

p

n

n TDD==

µµ

This equation is known as Einstein’s relation.




34

Total current flow in semiconductors

... The current flow in a semiconductor is due to the motion of the charge

carriers under the influence of applied fields or concentration gradients. It is

quite possible to have these two effects occurring simultaneously and the net

current flow is then the sum of drift and diffusion currents.

x

nDEnjjj E

d

dqq nnnDnn +=+= µ

x

pDEpjjj E

d

dqq pppDpp −=+= µ

.pDpnDnpn jjjjjjj EE +++=+=

... The total current flow in a semiconductor is the sum of electronic diffusion,

electronic drift, hole diffusion and hole drift currents.



35

Conductivity and current in solids. Problems

1. The mobility of electrons in silicon is around 1500 cm2/(V⋅s) at T = 300 K.

Estimate the mean free path of an electron. Compare it with the lattice

constant a = 0.543 nm.

2. Estimate the conductivity and resistivity of intrinsic silicon at T = 300 K.

3. Find the ratio of the values of electrical conductivity of intrinsic silicon at 40

and 200C. Assume that the forbidden energy gap in silicon is about 1.1 eV.

Comment on the result.

4. Estimate the ratio of the values of electrical conductivity of extrinsic silicon at

40 and 200C.

5. Estimate the critical strength of the electric field in silicon assuming that the

electron drift velocity in the critical field is equal to the mean-square thermal

velocity. Assume that electron mobility in silicon is 0.13 m2/(V·s) at 300 K.

6. The visible light spectrum wavelengths are between 380 and 780 nm.

Estimate the maximum gap energy of material suitable for visible light

detector.

7. The resistivity of a doped silicon crystal is 9.27⋅10-3 m and the Hall coefficient is 3.84⋅10-4 m3/C. Assuming that conduction is determined by a single type of

charge carriers, calculate the density and mobility of the carriers.



36

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