Energy
Gap in a Semiconductor

Purpose: The objective of this lab is to determine the forbidden energy
gap of a germanium semiconductor. This will be accomplished by measuring
the resistance of an intrinsic semiconductor sample as a function of temperature.
Theory:

In an atom electrons occupy distinct energy levels.
When atoms join to make a solid, the allowed energy levels are grouped
into bands. The bands are separated by regions of energy levels that
the electrons are forbidden to be in. These regions are called forbidden
Energy gaps or bandgaps. Energy bands and the forbidden energy gap
is illustrated in figure 1. The electrons of the outermost shell
of an atom are the valence electrons. These occupy the valence band.
Any electrons in the conduction band are not attached to any single atom,
but are free to move through the material when driven by an external electric
field.

In a metal such as copper, the valence and conduction
bands overlap as illustrated in figure 2a. There is no forbidden
energy gap and electrons in the topmost levels are free to absorb energy
and move to higher energy levels within the conduction band. Thus
the electrons are free to move under the influence of an electric field
and conduction is possible. These materials are referred to as conductors.

In an insulator such as silicon dioxide (SiO2),
the conduction band is separated from the valence band by a large energy
gap of 9.0 eV. All energy levels in the valance band are occupied
and all the energy levels in the conduction band are empty. It would
take 9.0 eV to move an electron from the valence band to the conduction
band and small electric fields would not be sufficient to provide the energy,
so SiO2 does not conduct electrons and is called an insulator. Notice the
large energy gap shown in figure 2b.

Semiconductors are similiar to the insulators insofar
as they do have an energy gap only the energy gap for a semiconductor is
much smaller ex. Silicon's energy gap is 1.1 eV and Germanium's energy
gap is 0.7 eV at 300 °K. These are pure intrinsic semiconductors.
Observe the energy gap in figure 2 c.

For finite temperatures, a probability exists that
electrons from the top of the valence band in an intrinsic semiconductor
will be thermally excited across the energy gap into the conduction band.
The vacant spaces left by the electrons which have left the valence band
are called holes which also contribute to the conduction because electrons
can easily move into the vacancies. If an electric field is applied,
the electrons flow in one direction and the holes move in the opposite
direction. The holes act as a positive charge (deficiency of negative
charge) so the direction of current (effective positive charge) is in the
same direction. For pure silicon at 300 °K, the number
of electrons residing in the conduction band as a result of thermal excitement
from the valence band is 1.4 x 10^{10} /cm^{3}.

Semiconductors have a conduction band and a valance
band separated by a forbidden region called the energy gap. If the lowest
energy in the conduction band is designated, Ec , and the highest energy
level of the valance band is called, Ev, then the energy gap between them
would be: Eg = Ec - Ev . The conductivity of a material is
directly proportional to the concentration of election in the conduction
band, hence the resistivity, , of a material would be inversely proportional
to the concentration of electrons in the conduction band. The concentration
of electrons increases with temperature. The resistivity decreases with
temperature according to the following equation:

resistivity = Co T^{-3/2}
exp^{(Eg/ 2kT)} where Co is a constant and T
is temperature in degrees Kelvin, Eg is the energy gap, and k is the Boltzman's
constant. Since Eg is small for a semiconductor on the order of 1
eV = 1.6 x 10^{-19 }joules ,

k = 1.38 x 10^{-23} J/mole °K and T is about 350
°K, the resistivity,r , varies almost linearly with (1/k T).
So the resistivity can be described for small temperature range as:

r = C1(T) exp^{(Eg/ 2kT)} where C1 is a slowly varying function
of temperature.

The resistance of a sample of material would thus be:

R(T) = C2(T) exp^{(Eg/ 2kT)} at temperature, T.

At an initial temperature, To, it would be:

R(To) = C2(To) exp^{(Eg/ 2kTo)} where C2(T) =
C2(To).

Dividing the two equations and taking the natural logarithm of both
sides yields:

ln (R_{T}/Ro)
= (Eg/2) (1/kT) - (Eg/2) (1/kTo) where Eg/2 is
the slope of the linear equation

and (Eg/2) (1/k To) is the y-intercept. By plotting
ln (R_{T}/Ro) vs. (1/k T) , the value of Eg can
be found from the slope.

The sample of germanium should be connected
in the circuit as shown in the diagram. Initial measurements of the
room temperature, and the thermocouple's voltage should be made before
the power supply has been turned on.
The voltage across the sample and the current through the sample should
be read and recorded for every 0.2
mv increase in the thermocouple's voltage.

The values should be recorded in Table 1. The thermocouple's
voltage can be converted to temperature by:

T =
V/0.04 + To . The Resistance of the
sample is given by Voltage/Current in the sample. All calculated
values should be recorded in table 1 using Excel. A graph
of the data should be made using MS Excel, use the trendline feature
to obtain the slope. Compare with the actual value of 0.67 eV.

Initial measurements: temperature:
_____ Thermocouple voltage _______ (mv)

Table I:

Current
Voltage Resistance Thermocouple Temperature
1/kT RT/Ro ln (RT/Ro)

(ma)
(mv) (ohms)
(mv)
(°K)