FYSP106 / K1 ENERGY GAP OF GERMANIUM
1 Introduction
This measurement is about conductivity of a semiconductor and its dependence on the
temperature. As a quantitative result the energy gap of germanium is measured.
Literature related:
o Harris, Nonclassical Physics, chapters 9.4-9.9
o Young & Freedman, University Physics with Modern Physics, 11th ed., chapters
42.4, 42.6 and 42.7 (10th ed., chapters 44-5, 44-7 and 44-8)
o Smith, Electronics, 3rd ed., chapter 5, Semiconductor Diodes
o Kittel, Introduction to Solid State Physics, chapter 8, Semiconductor Crystals
2
Theoretical background
2.1
Semiconductors
Semiconductors are materials, whose resistivity has a strong dependence on temperature.
At room temperature the resistivity of a semiconductor is around 10 −2 K10 9 Ω cm . At
absolute zero the lattice structures of the most semiconductors are insulators
(resistivity > 1014 Ω cm ). Semiconductors are used in transistors, switches, diodes, photo
cells, sensors, processors, memory circuits etc. applications of electronics technology.
By doping a semiconductor with some other material, the conductivity of the
semiconductor can be adjusted. E.g. silicon is often doped with phosphorus, boron,
antimony or arsenic. On the valence band of silicon (and germanium), there are four
electrons. When a substance with five valence electrons (phosphorus, arsenic, antimony)
is added, in the end there is one extra electron which can carry a current. This is the case
in the n-type semiconductor. If the added substance has three valence electrons (boron),
there are positively charged holes formed in lattice structure, which can carry a current.
This is the p-type semiconductor.
2.2
Temperature dependence of conductivity
At absolute zero all electrons of insulators and semiconductors are on valence band.
Higher conducting band is empty and therefore charges don’t move. Between valence
band and conducting band there is an energy gap (Si: 1.12 eV, Ge: 0.67 eV). To cross this
gap, electrons need energy. This is the reason why the resistance of semiconductors
depends on temperature. Insulators have so broad energy gap (several eV:s) that electrons
can’t get into conducting band at room temperature. Furthermore, the valence band of
insulators is full, so electrons can’t move there either. Conductors have partially filled
valence band and thus electrons can move regardless temperature.
At the temperature scale used in this measurement ( 20 − 150 o C ) the conductivity of a
semiconductor, σ (the inverse number of resistivity) abides relation
σ = σ 0e
(
− E g 2 kT
)
(1)
where E g is the energy gap, k is the Boltzmann constant ( 8.625 *10 −5 eV ) and T is
temperature (in Kelvins).
3
Measurements
The measurement set-up is presented in figure 1. The equipment includes a card, where
necessary connections can be done. The germanium chip is connected in series with DC
source (4.5 V battery) and a resistor. The current shouldn’t exceed the value of 30 mA.
The germanium chip is heated by a coil which is connected to AC or DC source (max
6V). The temperature is measured by using the thermoelement of the measurement board
and thermoelement sensor (Pasco CL-6526). The data is collected by using DataStudio
program.
Figure 1: The measurement devices.
In order to calculate the energy gap of Ge, information of the resistance (R=U/I) and
temperature are needed. Therefore the signal from the temperature sensor and voltages
over the germanium chip and over the resistor should be connected to analogy channels
of the interface of DataStudio. After preparations (calibrating the temperature reading and
calculations of necessary values) two measurements are done. First switch the heating
voltage on and collect data until the rising of temperature slows down. It usually takes
few minutes. The temperature shouldn’t exceed 135o C . The second measurement is done
while the germanium chip cools down. With DataStudio, draw (1/T, ln (1/R)) –graphs
(equation 1) from results. Make a linear fit and from that calculate the size of energy gap.
Workplace instructions
1. Build the set-up as in figure 1.Connect the voltage meters you use to measure
voltages over the Ge-chip and over the resistor. For temperature measurements,
use the thermoelement (glued on the circuit board) and the amplifier (Pasco CL6526). Connect all the sensors to the interface of the DataStudio –program (if you
use Interface 500, the temperature sensor must be connected to either channel B
or C). You can take the heating current from Leybold power source (output 6.3~).
The measurements last approximately half an hour, so there is plenty of time to do
the connections properly.
Figure 1: The measurement set-up
2. When the connections are done, open DataStudio and choose Create Experiment.
An Experiment setup –window is opened where you can choose Voltage
sensors to two analog-channels and a Temperature sensor (K-type) to one
remaining by dragging the icons on the right spot. By double-clicking the channel
icon, a Sensor properties window is opened. From there, following adjustments
are made: Sensitivity: Low and Sample rate: 2 Hz. The channels can be named
by double-clicking the icon in data window.
3. The voltage over the resistor is converted into current by using the Calculate
function (starts by clicking New).
o Write I=x/560 as the equation and choose Data measurement from
Variables menu (choose the channel, where the voltage one in question
is connected).
4. From Displays-window choose Digits and drag the temperature icon from datawindow to the new one. Perform a test measurement (Play) at the room
temperature and check if you get the same result as with the thermometer. Do the
offset correction if necessary. Check also measuring the current and voltage. The
current should no exceed 35 mA.
5. The necessary 1/R and 1/T –values to the graph are found by using the Calculate
function. Because DataStudio has difficulties with exponential fit, you should
make ln (1/R) and then create (1/T, ln (1/R)) graph. This can be done by choosing
Graph from Displays window and dragging ln (1/R) icon to y-axis and 1/T icon
to x-axis from Data window.
6. The first measurement is performed by switching on the heating voltage (6 V~)
and data collecting. The Ge-chip is at its maximum temperature in few minutes,
and the data collecting is stopped. The second measurement is started at the same
time when the heating is switched off. Collect data until the temperature is near
room temperature.
7. To linear fit you can use the Fit function. You can also save the data by choosing
File… Export Data (ln (1/R) vs. 1/T is chosen to data).