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.
 energy levels of semiconductor
 

    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.
energy levels for dopped semiconductor
    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 1010  /cm3.
    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 (RT/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 (RT/Ro)  vs.   (1/k T) , the value of Eg can be found from the slope.

Apparatus:
Energy Gap Apparatus
 
 

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)