CH.8 Electrical and Thermal Properties of Materials

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CH.8 Electrical and Thermal Properties of
Materials
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• 8.1 MACROSCOPIC ELECTRICAL PROPERTIES
• 8.1.1 Generalized Ohms law conductivity
• J current density, conductivity,
  electric field.
• N number density of electrons, e charge, v
  average drift velocity
  
• 
  mobility

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• In metals the charge carriers are electrons
  and so we are concerned in this case only with
  the number density, charge and mobility of
  electrons. In semiconductors both electrons and
  holes contribute to the conduction, so that using
  a similar equation for the current density in
  terms of both contributions from holes and
  electrons leads to

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• 8.1.2 Temperature dependence of conductivity in
  metals
• Assume one type of charge carrier for
  simplicity.
• The temperature dependence of the conductivity
  is dependent on the temperature dependence of N
  and µ
• In a metal N is the density of valence
  (conduction) electrons. This has a value of
  typically N 1028m-3 in a metal, and is largely
  temperature independent. Therefore the
  temperature dependence of conductivity should be
  due to a temperature dependence of mobility

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• The mobility of electrons in metals is of the
  order of µ 10-3 10-1 m2 (Vs)-1 and so this
  leads to a conductivity
• of typically 106 108 (ohm m)-1. In fact
  all of the observed temperature dependence of
  in metals arises from the temperature
  dependence of the electron mobility µ which is
  affected by phonon scattering and impurity
  scattering of electrons in the metal

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ELECTRON MOBILITY

• vd drift velocity (average electron velocity)
• µe electron mobility (the frequency of
  scattering events)

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FREE ELECTRON THEORY OF CONDUCTION

• charge? mass m? free electron?
  electric field E? ????.
• electron? ??? force F?
• but, electrons periodically collide with the
  atom in the lattice and loose their kinetic
  energy. ?Scattering

v? ??? linear? ??
v electronic draft velocity
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average
mean free path
quantum theory
thermal velocity of electrons at Fermi level
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METALS RESISTIVITY VS T, IMPURITIES

• assumption scattering simply involves the
  collision of an electron with an ion core.
• Thermal component of ? ?th
• interaction between moving electrons and atomic
  vibration atomic vibration ? with Temp.?
• ? mean free path?

mean free path due to thermal vibration mean free
path due to impurities
Mathiessens rule
of phonon
proportionality constant
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• Impurity component of ? ?I
• at fixed impurity concentration ?I constant
• Impurity? ???? scattering ?? ? lI ?? ? ?I ??

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• 8.1.3 Temperature dependence of conductivity in
  semiconductors
• In intrinsic semiconductors, the number
  density of charge carriers increases with
  temperature according to the equation
• Where Eg is the band gap and the above
  equation assumes that the Fermi level is in the
  middle of the band gap. This equation show that
  there is an increase in the number density of
  conduction electrons with temperature.

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• 8.1.4 Temperature dependence of mobility
• If the mean free time between collisions is
  , the charge e and the mass m, then the
  mobility is given by,
• And it can be seen that it is the temperature
  dependence of which determines the mobility,
  or alternatively we can view this as the
  temperature dependence of the resistive
  coefficient in the equation of motion of the
  electrons. In a metal increases with
  temperature,
• ? reduction in mobility ,decrease in
  conductivity

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• 8.1.5 Different types of mobility
• Four different kinds of mobility of electrons
• Microscopic mobility
• This is defined for a particular electron
  moving with drift velocity v in an electric field
• 2. Conductivity mobility
• This is the macroscopic or average mobility
  which is determined from measurement of
  electrical conductivity
• assuming N and e are both known.

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• 3. Hall mobility
• is the mobility of charge carriers as
  determined from a Hall effect measurement.
• 4. Drift mobility
• This is determined from measurement of the
  time t required for carriers to travel a distance
  d in the material under the action of an electric
  field

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• 8.2 QUANTUM MECHANICAL DESCRIPTION OF CONDUCTION
  ELECTRON BEHAVIOUR
• In the absence of an electric field, the
  valence electrons in a metal have no net or
  preferential velocity in any direction. If we
  plot the vectors of these electrons in velocity
  space, then for a free-electron metal we obtain a
  velocity sphere, the surface of which corresponds
  to the Fermi velocity. All points inside the
  Fermi sphere are occupied. Integrating over the
  entire sphere we obtain zero drift velocity

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Thermal and Drift Velocities

• The carrier-gas model assumes that the kinetic
  energy of any single carrier is given by
• (3.18)
• The kinetic energy of the carriers is related to
  the crystal temperature T
• (3.19)

Thermal velocity
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Figure 3.5 The concept of drift velocity (a) No
electric field is applied. (b) A small electric
field is applied. (c) A larger electric field is
applied.
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• Fig. 3.5a An electron that returns to the
  initial position after a number of scattering
  events, performing therefore no effective motion
  in any particular direction in the crystal.
• ? no drift current ? drift velocity 0
• Fig. 3.5b If an electric field is applied to
  the electron gas, the electric-field will deviate
  slightly the electron paths between collisions,
  producing an effective shift of the electrons in
  the direction opposite to the direction of the
  electric field.
• ? drift velocity gt 0
• Fig. 3.5c An increase in the electric field
  increases the drift velocity.

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• When an electric field is applied
• The majority of electron velocities cancel,
  but now some are uncompensated and it is these
  electrons which cause the electric current.
• The important result that only certain
  specific electrons which are close to the Fermi
  surface can contribute to the conduction
  mechanism .
• Note that a similar effect was found for heat
  capacity where only those electrons within kBT of
  the Fermi level could contribute to the heat
  capacity

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• 8.2.1 Quantum corrections to the conductivity in
  Ohms law
• The highest energy that electrons can take in
  a metal in its ground state is the Fermi energy
  EF.
• Density of states
• This means that only a small change of energy
  is needed to raise a large number of
  electrons above the Fermi level. The velocity of
  the uncompensated electrons under the action of
  the field is close to the Fermi velocity.

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• Our Ohms law equation of section 8.1.1.
  needs to be slightly modified to take into
  account the fact that not all free electrons
  contribute to the conductivity.
• vF velocity of electrons at the Fermi
  level, N number of displaced electrons

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• 8.2.2 Number of conduction electrons
  contributing to conduction

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• For free electron, we have
  and hence

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• 8.2.3 displacement of the Fermi sphere under the
  action of an electric field
• the mean free time of the electrons
  between collisions.
• With this expression for

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• Only the projections of vF along the
  direction of the electric field , that is
  vFcos?, contribute to the current
• For a spherical Fermi surface there is a slight
  correction which gives.

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• And finally, the conductivity is given by
  , so that
• This quantum mechanical statement of
  conductivity shows that not all conduction
  electrons can contribute to the conductivity, but
  only those close to the Fermi surface. In
  addition, the conductivity is determined by the
  density conduction electron per atom, this
  density is high, leading to high conductivity.

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8.3 DIELECTRIC PROPERTIES capacitance

• A dielectric material is one that is electrically
  insulating (nonmetallic) and exhibits or may be
  made to exhibit an electric dipole structure
  that is, there is a separation of positive and
  negative electrically charged entities on a
  molecular or atomic level.
• As a result of dipole interactions with electric
  fields, dielectric materials are utilized in
  capacitors.
• the capacitance
• the quantity of charge stored on either plate
• the voltage applied across the capacitor

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the area of the plates the distance between
the plates the permittivity of
vacuum 8.8510-12 F/m
the permittivity of the dielectric medium
the dielectric constant (the relative
permittivity)
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• Most electronic applications involve the use
  of alternation electric fields or currents. In
  these cases the atoms in insulators oscillate
  under the action of the applied electric field,
  and these oscillations can be expressed in terms
  of the dielectric constant, e.
• This dielectric constant is actually
  dependent on the frequency of the applied
  electric field. When considering its dependence
  on the frequency of electromagnetic radiation it
  is often represented as e(?).

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• 8.3.1 Polarization
• The polarization can result from the relative
  displacement of the electrons and ionic cores or
  alternatively from the relative displacement of
  positive and negative ionic cores
• The force F on a charge e under the action
  of an electric field is,
• And it is this force which causes
  polarization of a material by displacing the
  positive and negative charges within an atom in
  opposite directions, or by displacing the ionic
  cores within the lattice

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• Electric polarization of the material (P) An
  electric dipole moment per unit volume, which is
  measured in coulomb metres per cubic metre (or
  effectively coulombs per square metre).
• P Np
• Where p is the dipole moment of an individual
  atom and N is the number of atoms per unit
  volume. P can also be defined as the surface
  density of charge which appears on the faces of
  the specimen when placed in a field. The
  polarization can be expressed in terms of the
  electric field by the equation.
• Where e is the permittivity or dielectric
  constant.

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• The dielectric constant is a measure of the
  amount of electric polarization induced by unit
  field strength. A high dielectric constant means
  that a material is easily polarized in an
  electric field. Typical values of the relative
  dielectric permittivity er are in the range 1.0
  10 (dimensionless), although its value can be
  much higher in some special materials, for
  example er is 94 in titanium dioxide(TiO2)

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• 8.3.2 Dielectric field strength
• The dielectric field strength is a measure of
  the largest electric field strength that an
  insulation material can sustain before the
  electrostatic forces holding the atoms in place
  are overcome. Once this happens the material
  suffers electrical breakdown and suddenly becomes
  an electrical conductor. Typical values of the
  dielectric strength are in the range of
  megavolts per metre.
• The breakdown strength often increases with
  frequency, and in particular for most materials
  breakdown is somewhat inhibited above 108Hz

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• 8.3.4 Polymers
• Most polymers are insulators of course, but
  conduction polymers exist which have electrical
  properties resembling those of conventional
  metals or semiconductors. Polyacetylene contains
  a high degree of crystallinity and a relatively
  high conductivity compared with other polymers.
  Trans-polyacetylene has a conductivity that is
  comparable to silicon. The electron band
  structure of this polymer has even been
  calculated and it has been found that when all
  carbon lengths are equal, this material has a
  band structure which is reminiscent of a metal.
  When the carbon bonds alternate in length it is
  found that band gaps appear in the structure.

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• 8.4 OTHER EFFECTS CAUSED BY ELECTRIC FIELDS,
  MAGNETIC FIELDS AND THERMAL GRADIENTS
• 8.4.1 Magnetoresistance
• ?The change in electrical conductivity
  associated with an applied magnetic field.
• ?Hall effect
• Under equilibrium conditions the motion of
  the change carriers is identical in the presence
  or absence of a magnetic field.
• ? magnetoresistance 0
• If not all the charge carriers have the same
  properties, the current flow is disturbed by the
  presence of a magnetic field and some of the
  charge carriers travel a longer distance.

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• ? drift velocity ?
• ? mobility ?
• ?

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8.4.2 Thermoelectric power (Seeback effect)
SEEBACK EFFECT
Metal
Cold
Hot

• Hot region? electrons? cold region? electrons?? ?
  kinetic energy? ???.
• Electrons? hot ? cold? ??? average K.E.? ???.
• ?potential gradient
• ?Seeback voltage

Hot
Cold
Seeback potential thermal driving force
equilibrium
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• If a material is subjected to a temperature
  gradient, the energy of the carriers at the hot
  end is greater than at the cold end and this
  leads to a carrier concentration gradient along
  the material. Displaced charge resulting from
  this concentration gradient generates a
  counteraction electric field until the total
  current becomes zero. The magnitude of this
  electric field in terms of the voltage per degree
  difference is known as the thermoelectric power
  , In a metal,
• EF The Fermi energy

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• In a metal is typically a few microvolts
  per degree Kelvin
• In a semiconductor, for an n type material
• Here A is a constant which depends on the
  specific scattering mechanism, A 2 for lattice
  scattering and A 4 for charged impurity
  scattering

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• The Seeback effect is utilized in the
  thermocouple which is used for measuring
  temperature. The thermoelectric power is
  determined from the open circuit electric field
• caused by a temperature gradient dT/dx

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THOMSON EFFECT

• Thomson effect the effect of heat production
  and/or absorption when an electric current flows
  in a temperature gradient
• Electrons move from cold to hot end ? electrons
  absorb heat from hot region ? thermoelectric
  cooling
• Electron moves from hot to cold end ? electrons
  give up heat to the rod

hot
cold
hot
cold
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8.4.3 PELTIER EFFECT

• ? The evolution or absorption of heat that is
  created when an electric current flows across a
  junction between two dissimilar materials.
• Metal-Semiconductor Ohmic contact

Q
electron flow
metal n-type
electron flow
Q
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• When a current flows in a material, a
  temperature gradient is developed. This of course
  is the inverse of the Seeback effect and is used
  in some cases for temperature control. The
  Peltier coefficient is simply the ratio of
  the electrical current density J to the thermal
  current density JQ

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• 8.4.4 Nernst effect
• When a magnetic field is applied at right
  angles to a temperature gradient, the diffusing
  charge carriers are deflected in the same way as
  when the magnetic field is applied at right
  angles to a conventional electric current, the
  result is a Nernst voltage. However, since charge
  carriers of both signs diffuse in the same
  direction the polarity of the Nernst voltage is
  not dependent on the sign of the charge carrier.

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• 8.4.5 Ettingshausen effect
• In the Hall effect, the application of a
  magnetic field normal to the passage of an
  electric current leads not only to a transverse
  voltage but also to a transverse temperature
  gradient. The appearance of this temperature
  gradient is known as the Ettingshausen effect.
  This arises because charge carriers with
  different energies (velocities) are deflected
  differently by the magnetic field. This is a
  small effect which adds to the Hall voltage

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• 8.5 THERMAL PROPERTIES OF MATERIALS
• 8.5.1 Thermal conductivity
• In the case of metals, the thermal conduction
  mechanism is similar to the electrical conduction
  mechanism and proceeds via the free electrons
  which migrate throughout the material. In
  semiconductors, conduction can take place by the
  electrons which are thermally stimulated into the
  conduction band.
• In insulators another mechanism must be
  involved and in this case the thermal conduction
  is due to phonons which are created at the hot
  part of a solid and destroyed at the cold part.
  These phonons provide the mechanism by which
  energy is transferred through the material.
• Thermal conductivity electron phonon

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• In metals the phonon contribution to thermal
  conductivity is also present, but the electronic
  contribution is so much greater that in these
  cases the phonon contribution is neglected.
• Units W -1m -1k -1

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• 8.5.2 Mechanism of thermal conduction
• If we begin from the assumption that thermal
  conduction can arise from both the motion of free
  electrons and phonons, we can derive a theory of
  the thermal conductivity. Again as in electrical
  conductivity, only those electrons close to the
  Fermi surface can contribute to the thermal
  conductivity.

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0
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Neglect Too small
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• 8.5.3 Thermal conductivity of metals
• The number of participating electrons N is
  determined by the population density at the Fermi
  energy N(EF). To a first approximation, this is
  about 1 of the number of free electrons per unit
  volume.

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• Ein the heat energy flowing in per unit time
  per unit area at the left end
• Eout the heat energy flowing out per unit time
  per unit area at the right end.

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• Z the number of electrons per unit time per
  unit area impinging on the end face
• The number density of free electrons (N) is
  similar to the number contributing to the thermal
  heat capacity
• since in both
  cases the electrons must be able to absorb heat
  energy, Substituting z,

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• And from

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• The thermal conductivity increases with mean
  free path
• Number of electrons per unit volume at the
  Fermi surface NF, and velocity of electrons at
  the Fermi surface vF, Remembering that
  , and that
• The thermal conductivity increases with T and
  and decreases with m

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• 8.5.4 Thermal conductivity of insulators
• The thermal conductivity K is related to the
  heat capacity by the expression

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• 8.6 OTHER THERMAL PROPERTIES
• 8.6.1 Thermoluminescence
• Thermoluminescence is the emission of
  electromagnetic radiation, in the visible
  spectrum, when certain materials are heated.
  These materials must be either insulators or
  semiconductors, and they must have a large number
  of electrons trapped in impurity states in the
  band gap.

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• 8.6.2 Mechanism of thermoluminescence

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• The requirements for a material to be able to
  exhibit thermoluminescence are
• Presence of a band gap
• Presence of impurity energy states in the band
  gap
• Long lifetime of electrons in traps
• Material must have been subjected to radiation to
  excite electrons from valence band before
  becoming trapped
• Material must not have been inadvertently heated,
  which could empty electrons from traps.

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• 8.6.3 Theory of thermoluminescence
• Electrons are thermally stimulated from the
  traps into the conduction band ? fall back into
  the valence band, emitting a photon

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• If we have an electron located in a trapped
  state at an energy below the conduction
  band, then the probability of the electron being
  thermally stimulated into the conduction band in
  unit time is given by the Arrhenius equation,
• Where s is a constant, with dimensions time-1
  and typically of magnitude 1011 - 1017s-1. this
  means that there is a time frame associated with
  the occupancy of the electron trap once the
  electron is there

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• 8.6.4 Occupation and vacation of trapped states
  by electrons
• The probability of filling any state in the
  band gap will also be dependent on time. If dN/dt
  is the rate of stimulation of electrons from
  traps into the conduction band, then
• N the number of electrons in traps , p the
  probability of escape in unit time.

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• This is the Randall-Wilkins equation which
  describes the number of electrons remaining in
  traps as a function of both time t and
  temperature T.

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• 8.6.5 Lifetime of electrons in traps
• The lifetime of occupancy of an electron
  state is inversely proportional to the
  probability p of a transition in unit time.
• Lifetime as a function of temperature T
• Raising the temperature T ? decreases the
  expected lifetime of the electrons in the traps.
• More thermal energy increases the probability
  of the electron escaping by thermal stimulation

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• From the exponential decay equation N
  N0exp(-pt) it is possible to define a half-life
  for the occupancy of the electron traps. Simply,
  when the number of traps remaining occupied has
  declined to half, N N0/2, we have the half-life
  of the occupancy

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• 8.6.6 Intensity of light emitted during
  thermoluminescence
• The intensity of light emitted during
  thermoluminescence is dependent on the rate of
  emptying of the electron traps dN/dt. If we
  assume that every electron from a trap enters the
  bottom of the conduction band and then
  instantaneously falls back to the top of the
  valence band with emission of a photon of energy
  equal to the band gap energy, then light of a
  single frequency (Eg/h) will be emitted. The
  intensity of the light will be equal to the rate
  of emptying of electron traps,

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• 8.6.7 Emission of light on heating
• Suppose then the temperature of the specimen
  is raised at a constant rate,
• Then the fractional change in occupancy (dN/N) is

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• This emission assumes a single type of trap
  at an energy
• below the conduction band, a constant
  rate of change of temperature and a constant
  value of s for all traps of the given type

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• 8.6.8 Location of the peaks in thermoluminescent
  intensity
• When intensity of emission I is measured as a
  function of temperature T as the temperature is
  swept at a fixed rate, peaks in the intensity
  will correspond to the depth of electron traps
  below the conduction band.
• An empirical relationship has been given
  between the depth in electron volts (eV)
  and the peak temperature T in Kelvin (K) by
  Urbach

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