Transcript Fermi energy

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From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fermi-Dirac Statistics
The Fermi-Dirac function
1
f (E) 
 E  EF 
1  exp 

 kT 
where EF is a constant called the Fermi energy.
f(E) = the probability of finding an electron in a state with energy
E is given
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
The fermi-Dirac f(E) describes the statistics
of electrons in a solid. The electrons
interact with each other and the environment,
obeying the Pauli exclusion principle.
• EF = Fermi level, or Fermi Energy
• At T = 0 K, f(E) = 1 for E < EF and
f(E) = 0 for E > EF
• At T > 0 K, f(E) is non-zero above
EF and less than one below EF
Fig 4.26
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Free Electron Model
Fig 4.27
(a) Above 0K, due to thermal excitation, some of the electrons are at energies above EF.
(b) The density of states, g(E) versus E in the band.
(c) The probability of occupancy of a state at an energy E is f (E).
(d) The product of g(E) f (E) is the number of electrons per unit energy per unit volume,
or the electron concentration per unit energy. The area under the curve on the energy
axis is the concentration of electrons in the band.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Free Electron Model (Cont.)
• In a small energy range, E to (E + dE), there are nE dE
electrons per unit volume
• When we sum all nE dE from the bottom to the top of the
band (E = 0 to E = EF + F), we get the total number of
valence electrons per unit volume, n, in the metal:
• Since f(E) falls sharply when E > EF, we can carry the
integration to E = ∞, rather than (EF + F) because f  0
when E >> EF, so plugging in for g(E) and f(E), we get:
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fermi Energy
If we could integrate previous integral, we would obtain an expression relating n and
EF. However, at T = 0K, EF = EFO and the integrand exists only for E < EFO.
Integrating at 0 K yields:
Fermi energy at T = 0 K
 h  3n 
 
EFO  
 8me   
2
2/3
n is the concentration of conduction electrons (free carrier concentration)
Utilizing various mathematical approximations, it is not too difficult to integrate
previous integral to obtain the Fermi energy at a temperature T:
Fermi energy at T (K)
  2  kT  2 
 
EF (T )  EFO 1  
 12  EFO  
This shows that the Fermi energy is only weakly temperature dependent since EFO >> kT
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Average energy per electron
The Fermi energy has important significance in terms of average energy, Eav, of
conduction electrons in a metal
Average energy per electron at 0 K
Sub g(E)f(E) and integrate
3
Eav (0)  EFO
5
Average energy per electron at T (K)
2
2

3
5  kT  

 
Eav (T )  EFO 1 
5
 12  EFO  
Since EFO >> kT, the second term in the square brackets is much smaller than unity,
and Eav(T) shows only a very weak temperature dependence
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fermi Energy Significance: Metal-Metal Contacts
Fig 4.28
(a) Electrons are more energetic in
Mo, so they tunnel to the surface
of Pt.
(b) Equilibrium is reached when the
Fermi levels are lined up. When
two metals are brought together,
there is a contact potential V.
• Electron transfer from Mo to
Pt continue until the “contact
potential” is large enough to
prevent any more electron
transfer
• Therefore, equilibrium is
reached, and Fermi levels
align
• The contact potential (or
contact voltage, V) is
determined by the difference
in the work functions:
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fermi Energy Significance: Metal-Metal Contacts
Fig 4.28
• It should be noted that away
from the junction, the e- still
requires energy > F to become
free
• This means that the vacuum
level has a step, F at the
junction
• Since we must do work
equivalent to F to get a free
electron from the Mo surface to
the Pt surface, this represents
a voltage:
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
More on Contact Potential
• However, the 2nd law of
thermodynamics, the contact
voltage cannot do work
• In other words, it cannot
drive current in an external
circuit.
• This is illustrated in Fig. 4.29
Fig 4.29
There is no current when a closed
circuit is formed by two different
metals, even though there is a
contact potential at each contact.
The contact potentials oppose each
other.
• Here, if we close the “circuit”,
we create another junction
(contact) with a contact
voltage that is equal and
opposite to the other junction
• Therefore, net voltage is
zero, and no current flows
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fermi Energy Significance
For a given metal the Fermi energy represents the
free energy per electron called the electrochemical
potential, m. The Fermi energy is a measure of the
potential of an electron to do electrical work (eV) or
nonmechanical work, through chemical or physical
processes.
• In general, when two metals brought together, the Fermi level in each will be
different with respect to vacuum
• This difference means a difference in chemical potential m, which in turn
means the system will do external work, which is not possible
• Instead, we have the immediate transfer of electrons as previously described
such that m = 0.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Seebeck Effect
Fig 4.30
The Seebeck effect. A temperature gradient
along a conductor gives rise to a potential
difference.
• Consider an Al rod that is heated on
one end and cooled on the other
• “Hot” region electrons have more
energy than “cold” region electrons
• Therefore, “hot” electrons have
greater velocities, and consequently,
a net diffusion of electrons from hot
to cold occurs
• This leaves behind positive ions in
the hot region and electron “pile up”
in the cold region
• Eventually, this process stops once
the electric field develops between
the positive ions and the piled up
electrons
• A voltage therefore exists between
the hot and cold ends, with the hot
end at positive potential
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Seebeck Effect
Seebeck effect (thermoelectric power)
is the built-in potential difference V across a material due to
a temperature difference T across it.
V
S
T
Sign of S
is the potential of the cold side with respect to the hot side; negative if
electrons have accumulated in the cold side.
• For copper, this intuitive explanation fails to explain why e- actually diffuse from
the cold to the hot region, giving rise to a positive Seebeck coefficients; thus, the
polarity of voltage is actually reversed for copper
• The net diffusion process depends on how the mean free path, l, and the mean free
time (due to scattering from lattice vibrations) change with the electron energy,
which can be quite complicated
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Seebeck Effect (cont.)
Fig 4.31
• Consider two neighboring
regions H (hot) and C
(cold) with widths
corresponding to the
mean free paths  and '
in H and C.
• Half the electrons in H
would be moving in the
+x direction and the other
half in the –x direction.
• Half of the electrons in H
therefore cross into C, and
half in C cross into H.
• Ex: Suppose that e- concentration in H and C are
approximately the same
• Then, the number of e- crossing from H to C is ½ n ,
and C to H is ½ n ’
• Thus, net diffusion from H to C is  ½ n( - ’)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Seebeck Effect (cont.)
• Suppose that the scattering of e- is such that  increases
strongly with e- energy
– Then, e- in H have longer mean free path  > ’
– So, net migration is from H to C, and S is negative, as in
Aluminum
• Opposite occurs for copper since  decreases strongly with
the energy, so cold region has longer mean free path ’ > 
This qualitative description does not account for n, which is not the same (diffusion
changes n), and mean scattering time change with electron energy is neglected.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Seebeck coefficient for metals
V
S
T
S 
 k T
2
2
3eEFO
x
Mott and Jones thermoelectric power equation
x = a numerical constant that takes into account how various charge
transport parameters, such as the mean free path , depend on the electron
energy. (x values are tabulated in Table 4.3)
• A proper explanation of the Seebeck coefficient has to consider how electrons
around EF – which contribute to electrical conduction – are scattered by lattice
vibrations, impurities, and crystal defects.
• This scattering process controls the mean free path, and hence, the Seebeck coeff.
• Also, the scattering e- need empty states, thereby requiring that we consider how
the density of states changes with energy as well
• The Seebeck coefficient for many metals is given by the Mott and Jones eqn where
x is a numerical constant that takes into account how various charge transport
parameters depend on the electron energy
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
S 
 k T
2
2
3eEFO
x
kT/EFO demonstrates that only those electrons
about a kT around EFO are involved in the
transport and scattering processes
Fig 4.32
(a) If Al wires are used to measure the Seebeck voltage across the Al rod, then the net emf is zero.
(b) The Al and Ni have different Seebeck coefficients. There is therefore a net emf in the Al-Ni
Circuit between the hot and cold ends that can be measured.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Thermocouple
We can only measure differences between thermoelectric powers of
materials.
When two different metals A and B are connected to make a thermocouple,
then the net EMF is the voltage difference between the two elements.
VAB   S A  S B dT   S AB dT
T
T
To
To
https://www.youtube.com/watch?v=LYxcE3kPN20
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Thermocouple Equation
VAB  aT  b(T )
2
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Thermocouples are widely used to measure the
temperature.
LEFT: A thermocouple pair embedded in a
stainless steel sheath-probe. The thermocouple
junction inside the probe is in thermal contact
with the probe tip, and, electrically insulated
from the probe metal.
|SOURCE: Courtesy of Omega
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Thermionic Emission and Vacuum Tube Devices
(a) Thermionic electron
emission in a vacuum
tube.
(b) Current-voltage
characteristics of a
vacuum diode.
Fig 4.34
• Metal cathode is heated; e- obtains enough energy to escape the metal
and become free
• Process is self-limiting: emitted electrons leave a net positive charge
behind, which pulls them back
• Therefore, the “lost” e- need to be replenished and also need to collect the
emitted one – done conveniently in a vacuum tube arrangement in a closed
circuit (vacuum ensures no collisions with air molecules)
• Current increases with the anode voltage until all the emitted electrons are
collected by the anode and the current saturates
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Fig 4.35
Fermi-Dirac function f(E) and the energy density of electrons n(E) (electrons per unit energy
and per unit volume) at three different temperatures. The electron concentration extends more
and more to higher energies as the temperature increases. Electrons with energies in excess of
EF+F can leave the metal (thermionic emission)
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Thermionic Emission
Richardson-Dushman thermionic emission equation
 F
J  BoT exp  

 kT 
2
Bo=4emek2/h3 = 120×106 A m-2 K-2
Richardson-Dushman constant
 F
J  BeT exp  

 kT 
2
where Be = effective emission constant
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Schottky Effect and Field Emission
(a) PE of the electron near the surface
of a conductor.
(b) Electron PE due to an applied
field, that is, between cathode and
anode.
(c) The overall PE is the sum.
Fig 4.36
• Positive voltage is applied to the anode with
respect to the cathode
• Electric field at the cathode helps the thermionic
emission process by lowering the PE barrier F.
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Schottky effect
When a positive voltage is applied to the anode with
respect to the cathode, the electric field at the cathode
helps the thermionic emission process by lowering the
PE barrier F by an amount bSE1/2. The current density
in field assisted thermionic emission is
Metal’s work function
Schottky coefficient
 F  bSE
J  BeT exp  
kT

2
1/ 2



https://www.youtube.com/watch?v=gIEJP4F6Oig
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Field assisted emission is field assisted tunneling from the cathode
(a) Field emission is the tunneling of
an electron at an energy EF through
narrow PE barrier induced by a large
applied field.
(b) For simplicity, we take the barrier
to be rectangular.
(c) A sharp point cathode has the
maximum field at the tip where the
field emission of electrons occurs.
Fig 4.37
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)
Field-assisted Tunneling
Field-assisted tunneling probability
Effective work function due to the Schottky effect
 22me F eff
p  exp 
eE


1/ 2
F


Field-assisted tunneling: the Fowler-Nordheim equation
Constants
J field emission
 Ec 
 CE exp   
 E 
2
Applied field at the cathode
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)