Download 5. Conductivity of solids

ELEKTRONIKOS PAGRINDAI
1
2008
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
VGTU EF ESK
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ELEKTRONIKOS PAGRINDAI
2
2008
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
Fp = −
1
τr
mn v E
Thus, the total force acting on an electron is given by
1
d vE
FΣ = F + Fr = qE − mn vE = ... = mn a = mn
τr
dt
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ELEKTRONIKOS PAGRINDAI
3
2008
Conduction process in solids
d vE
qE − mn v E = mn
τr
dt
1
... 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
vE =
qτ r
E = µn E
mn
µn =
qτ r
mn
The coefficient µn is called mobility. If E = 1, we have that v E = µ n .
… 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.
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ELEKTRONIKOS PAGRINDAI
4
2008
Conductivity
Let us consider a parallelepiped.
V = vE
N Σ = nV = nv E
j = qnv E
j = qnµ n E = σE
… 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.
σ = qnµ n
σ = q(nµ n + pµ p )
… Conductivity is dependent on charge carriers density and their
mobility.
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ELEKTRONIKOS PAGRINDAI
5
2008
Relaxation time and mobility
Now let us suppose that after some initial application of the field it is suddenly
reduced to zero.
d vE
qE − mn v E = mn
τr
dt
1
 t
v E (t ) = v E (0) exp −
 τr
d v E (t )
1
= − v E (t )
τr
dt



... 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.
If we know relaxation time, we can find the average distance between two
collisions.
λ = vτ r
... Charge carriers mobility:
VGTU EF ESK
kλ = vτ r
µ n ,p =
τ r = kλ v
q kλ
mn , p v
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v ≅ vT
ELEKTRONIKOS PAGRINDAI
6
2008
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.
µ n ,p =
q kλ
mn , p vT
λ ~ 1/ T
µ n , p ~ T −3 / 2
k ≅1
vT ~ T
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ELEKTRONIKOS PAGRINDAI
7
2008
Mobility of charge carriers in semiconductors
µ n , p ~ T −3 / 2
... The carrier mobility in an intrinsic semiconductor
decreases 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.
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.
λ = const k ~ (v T ) 4 ~ T 2
VGTU EF ESK
vT ~ T
µn,p
kλ T 2
~
~
~ T 3/ 2
vT
T
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ELEKTRONIKOS PAGRINDAI
8
2008
Mobility of charge carriers in semiconductors
µn,p
kλ T 2
~
~
~ T 3/ 2
vT
T
... 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.
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ELEKTRONIKOS PAGRINDAI
9
2008
Mobility at normal temperature
Mobility is also dependent upon impurity density.
µ n ,p
q kλ
=
mn , p v
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.
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
1
µ
VGTU EF ESK
≅
1
∑µ
i
i
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ELEKTRONIKOS PAGRINDAI
10
2008
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.
VGTU EF ESK
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ELEKTRONIKOS PAGRINDAI
11
2008
Conductivity of semiconductors
Two types of carriers exist in semiconductors. According to this
σ = q(nµ n + pµ p )
In the intrinsic semiconductor densities of electrons and holes are equal. Then
σ i = qni ( µ n + µ p )
ni ~ T 3 / 2 exp(−∆W / 2kT )
µ n , p ~ T −3 / 2
σ i = σ 0 e − ∆W / 2 kT
ln σ i = ln σ 0 −
∆W
2kT
... 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.
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ELEKTRONIKOS PAGRINDAI
12
2008
Conductivity of semiconductors
The specific conductivity of a doped semiconductor depends on the
majority carrier density and their mobility.
σ n = qnµ n σ p = qpµ 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.
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ELEKTRONIKOS PAGRINDAI
13
2008
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.
j = σE.
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
j = σ (E)E
σ = q(nµ n + pµ p )
... The strong field effects may be caused by carrier mobility or density variation
in the strong field.
VGTU EF ESK
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ELEKTRONIKOS PAGRINDAI
14
2008
Mobility versus strength of electrical field. Carrier velocity saturation
µ n ,p
q kλ
=
mn , p v
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).
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ELEKTRONIKOS PAGRINDAI
15
2008
Mobility in compound semiconductors. The Gunn effect
The energy of a free microparticle is given by
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.
h2 2
h2
p2
Wk =
k =
=
2
2m
2m
2 mλ
In low electric fields, electrons reside in the
lower central valley. Then the electron
mobility in GaAs is high (about
8000 cm2/(V⋅s)).
v E1 = µ n1E
j = qnµ n1E
As the field is increased, electrons transfer
to the upper valley and their mobility is
considerably reduced (to 0,01 m2/(V⋅s))
vE 2 = µ n 2 E
VGTU EF ESK
j = qnµ n 2 E
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2nd
valley
1st
valley
ELEKTRONIKOS PAGRINDAI
16
2008
Mobility in compound semiconductors. The Gunn effect
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:
σ d = d j /d E < 0, ρ d < 0
... 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.
Devices based on the Gunn effect are called either Gunn diodes or
transferred-electron devices (TEDs). They can be used in amplifiers or
oscillators.
VGTU EF ESK
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ELEKTRONIKOS PAGRINDAI
17
2008
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.
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ELEKTRONIKOS PAGRINDAI
18
2008
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.
W > ∆W
ν min = ∆W / h
VGTU EF ESK
λmax = c /ν min = hc / ∆W
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ELEKTRONIKOS PAGRINDAI
19
2008
Photoconductivity
W > ∆W
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.
VGTU EF ESK
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ELEKTRONIKOS PAGRINDAI
20
2008
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
µn =
VGTU EF ESK
q λF k F
mn v F
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21
2008
Conductivity of metals and alloys
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
µn ~ 1 / T
σ ~ 1/ T
ρ = 1/ σ ~ T
ρ ≅ ρ 0 [1 + α (T − T0 )]
... 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.
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.
VGTU EF ESK
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ELEKTRONIKOS PAGRINDAI
22
2008
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.
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ELEKTRONIKOS PAGRINDAI
23
2008
Superconductivity
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.
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.
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ELEKTRONIKOS PAGRINDAI
24
2008
Superconductivity
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
VGTU EF ESK
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ELEKTRONIKOS PAGRINDAI
25
2008
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 nonmagnetic 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 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.
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).
VGTU EF ESK
Substrate
Tin strip
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SiO2
Lead
strip
ELEKTRONIKOS PAGRINDAI
26
2008
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.
VGTU EF ESK
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ELEKTRONIKOS PAGRINDAI
27
2008
Josephson effects
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.
VGTU EF ESK
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ELEKTRONIKOS PAGRINDAI
28
2008
Josephson effects
The dissipation of energy is in the form of electromagnetic waves.
I = I 0 sin ϕ
∆WC = 2qU
∆ν = ∆WC / h = 2qU / h
 2 qU 
 2qU 
I
=
I
sin
t  = I 0 sin(2π f t )

0
ϕ = 2π∆νt = 2π
t
 h 
 h 
2q
f =
U ... Frequency of oscillation is proportional to the voltage drop on
the junction.
h
If the voltage change is 1 mV, the frequency change is 483,6 MHz.
The field of electronics holds great promise for practical applications of superconductors. 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.
VGTU EF ESK
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ELEKTRONIKOS PAGRINDAI
29
2008
Josephson effects
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.
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.
VGTU EF ESK
http://www.lne.eu/en/r_and_d/electrical_metrol
ogy/josephson-effect-ej.asp
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ELEKTRONIKOS PAGRINDAI
30
2008
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 F L = −q [v E × B]
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 FE = qE E = U H / d
In equilibrium
UH =
FL = FE, qv E B = qE
1
jBd = RH jBd
qn
RH = 1 / qn
E = vE B
σ = qnµ n
j = qnv E
RH σ = µ n
... Simultaneous measurement of σ and RH can lead to the
experimental value for the carrier drift mobility.
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ELEKTRONIKOS PAGRINDAI
31
2008
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
grad N =
VGTU EF ESK
dN
dx
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ELEKTRONIKOS PAGRINDAI
32
2008
Diffusion of charge
Krūvininkų
carriers
difuzija
in semiconductors
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.
FΣ = − A
dp
∆x
dx
FD = − kT
VGTU EF ESK
1 dN
N dx
NV = NA∆x
FD =
FΣ
1 dp
=−
NV
N dx
p = N kT
... The diffusion force is proportional to
temperature and concentration gradient.
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33
2008
Diffusion of charge carriers in semiconductors
... 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.
FE = qE
v En
τr
τr
qτ r
= µn E =
E=
qE =
FE
mn
mn
mn
In the case of diffusion force:
vDn =
τr
mn
FD = −
τr
mn
kT
1 dn
1 dn
= − Dn
n dx
n dx
τr
qτ r kT
kT
Dn =
kT =
= µn
mn
mn q
q
The diffusion of electrons causes electron current to flow.
jDn = −qnvDn = qDn
VGTU EF ESK
dn
dx
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ELEKTRONIKOS PAGRINDAI
34
2008
Diffusion of charge carriers in semiconductors
Considering holes, we have :
vDp = − Dp
1 dp
p dx
Dp =
τr
mp
kT = µ p
kT
q
jDp = qpvDp = −qDp
dp
dx
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.
Dn
Dp
kT
=
=
µn µp
q
This equation is known as Einstein’s relation.
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
35
2008
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.
jn = jnE + jnD = qnµ n E + qDn
dn
dx
jp = jpE + jpD = qpµ p E − qDp
dp
dx
j = jn + jp = jnE + jnD + jpE + jpD .
... The total current flow in a semiconductor is the sum of electronic diffusion,
electronic drift, hole diffusion and hole drift currents.
VGTU EF ESK
[email protected]
ELEKTRONIKOS PAGRINDAI
36
2008
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.
VGTU EF ESK
[email protected]