California Polytechnic State University -- Solid State Physics Laboratory
Experiment 25: Location of the Fermi Level by Thermoelectric Power
Measurements
Scope
The sign of charge carriers and the location of the Fermi energy within the energy gap
between the valence and conduction bands of an extrinsic semiconductor is determined at several
temperatures by thermoelectric measurements.
Introduction
see Brown 7.3-5
Location of the Fermi Level in a
Semiconductor
In metals, it is common to define the Fermi
energy as the energy of the most energetic
electron at the absolute zero of temperature.
That is, all states with energies lower than EF
are occupied and all states above EF are vacant
at T = 0 K. In insulators and semiconductors,
however, the Fermi energy resides in the
energy gap between the valence and
conduction bands. At T = 0 K, it is still true
that the states with energies lower than EF are
occupied and those of energy higher than EF
are empty. But the exact location of EF in the
energy gap is determined by the Fermi
distribution function,
1
f (E) =
" E ! EF %
1+ exp $
'
# kT &
That is, EF is the energy such that the Fermi
distribution function f(E) correctly predicts the
distribution of electrons among the available
states in the valence and conduction bands.
For intrinsic semiconductors, one expects EF to
reside in the middle of the energy gap--thus
accounting for the equal probabilities of
electrons in the conduction band and holes in
the valence band at a given temperature.
In extrinsic semiconductors, EF resides (at low
temperatures) either near near the conduction
band (n-type) or the valence band (p-type)
depending on the sign and concentration of the
majority carriers and the absolute temperature.
EC
EC
EF
EV
EF
EV
intrinsic.
f(E)
Electrons
EC
EF
EF
EV
n - type
f(E)
Holes
EC
EF
EV
EF
p - type
f(E)
n - type
EC
Conduction Band
EF
EV
Valence Band
Location of the Fermi Level by Thermoelectric Power Measurements
p - type
T
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California Polytechnic State University -- Solid State Physics Laboratory
As the temperature increases, the Fermi energy shifts toward the center of the energy gap. This
behavior is due to the increasing number of intrinsic electron-hole pairs that are thermally
generated at elevated temperatures.
(The limiting case, of course, is when the temperature is sufficiently high that the intrinsic carrier
concentration far exceeds the dopant concentration thus causing the semiconductor to behave
more like an intrinsic semiconductor. The Fermi level would then reside near the center of the
energy gap.)
Any experiment that is sensitive to the location of the Fermi level in the energy gap should
exhibit the shift of EF when performed at different temperatures.
Thermoelectric Power
If two points in a conducting material are maintained at two different temperatures T1 and
T2 , there will be a thermoelectric voltage ΔV between those two points (this is called the
Seebeck effect). The polarity of the thermally generated emf depends on the sign of the majority
carriers and whether T1 or T2 is the higher temperature. The negative of the temperature gradient
of the voltage is defined as the "thermoelectric power" Q. That is, Q =-dV/dT. For small values
of (T2 - T1) the thermoelectric power can be approximated by
$ #V ' $ #V '
dV
Q="
= "&
) = "&
)
% #T ( % T2 " T1 (
dT
The above definition makes Q positive for p-type and negative for n-type semiconductors. The
thermoelectric power of a semiconductor can also be related to the Fermi level location in the
energy gap (see Azároff and Brophy, pp. 233-236).
!
p - type semiconductor: TQpe = (EF - Ev) + 2kT
n - type semiconductor: -TQne = (Ec - EF) + 2kT
where EF is the Fermi level, Ev is the energy of the top of the valence band, Ec is the energy of
the bottom of the conduction band, e is the electronic charge, k is Boltzmann's constant, and T is
the average absolute (Kelvin) temperature of the sample.
Thus the measurement of the thermoelectric power Q of an extrinsic semiconductor can
determine the location of the Fermi level relative to either the valence band in p-type or the
conduction band in n-type.
Procedure
The specimen is a small bar of germanium similar to that used in the Hall effect
experiment. It is either n-type or p-type. You should be able to make that determination from
your experimental results. The ends of the specimen can be maintained at different temperatures
for short periods of time by means of the thermally insulated reservoirs. The temperature of each
end of the specimen is determined by using thermocouples connected to a digital thermometer
and the thermoelectric voltage ΔV of the germanium sample can be measured with a voltmeter.
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Location of the Fermi Level by Thermoelectric Power Measurements
California Polytechnic State University -- Solid State Physics Laboratory
Data should be taken for at least ten different average sample temperatures. Be sure to
allow enough time for the temperature gradient to be established. You might use warm water
baths, cold-water baths, and dry ice-alcohol mixtures. In order for the thermoelectric power to
be approximated by ΔV/ΔT, the temperature difference between the two ends of the specimen
should not exceed about 10 oC. Keep track of which side of the sample is at which
temperature and the sign of the measured voltage.
CAUTION - The specimen is fragile. It is protected by screens as guards, but could still
be easily damaged. Be careful when introducing and pouring off mixtures which contain ice or
dry-ice. Stir very carefully, keeping the stirring rod away from the region of the specimen.
Report
• Determine the majority carrier type for the extrinsic semiconductor. Justify your
determination by explaining the polarity of ΔV, based on transport arguments, of the majority
carriers given a "polarity" of (T2 - T1).
• Determine the location of the Fermi Level for each value of average sample temperature.
Plot your results as a function of T and discuss the results.
• Compare your graph to Fig. 6 on page 208 of the text by Azároff and Brophy (see next page).
Estimate the approximate dopant concentration in your germanium specimen.
References
Azároff and Brophy, Electronic Processes in Materials, pp. 203-208.
Digital
Thermometer
voltmeter
!V
Tc1
Tc2
T1
T2
Ge Specimen
Location of the Fermi Level by Thermoelectric Power Measurements
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California Polytechnic State University -- Solid State Physics Laboratory
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Location of the Fermi Level by Thermoelectric Power Measurements