Essay: Low temperature resistance: lab report

Second Year Laboratory Report

Low temperature resistance
School of Physics and Astronomy

Abstract

The current-voltage relationships of constantan, copper, germanium and a silicon diode were investigated in this experiment. Constantan, copper and germanium showed linear relationships between the current and voltage, displaying Ohmic behaviour. The silicon diode showed an exponential (non-linear) relationship between the current and the voltage.
Furthermore, the experiment investigated how the resistance of constantan, copper and germanium varied with temperature. Copper produced a linear relation with the resistance increasing with temperature. The resistance was independent of the temperature for constantan and for germanium, in one region the resistance was increasing with temperature below 0C and in another, resistance decreasing with increasing temperature above 0C.
Finally, the energy gap of the silicon diode was determined by a graph of the voltage against the temperature to be Eg = 1.26, 1.27 and 1.28 eV all with uncertainty ± 0.01 eV. These values are consistent with each other but not with the accepted value of Eg = 1.11 eV.

1. Introduction

German physicist and mathematician Georg Ohm did his work on resistance in 1825 and 1826 and published his findings in 1827. He used a thermocouple as a voltage source and a galvanometer to measure the current through wires of different materials, lengths and diameters. His experiments resulted in Ohm’s law which stated that the currents furnished by different galvanic cells, or combination of cells, are always directly proportional to the e.m.f.’s existing in the circuits in which the currents flow, and inversely proportional to the total resistances of these circuits [1]. The current through a conductor is directly proportional to the voltage across it.
A diode is an electrical component that only allows current to flow in one direction when a potential difference is applied across it. Ideally, there is no resistance in one direction and infinite resistance in the opposite direction. Developed at around the same time in the early 20th century, there are two types of diodes: thermionic diodes, which use thermionic emission in a vacuum tube and solid-state diodes, which use semiconductors. The main use of a diode is to convert alternating current to direct current in a process called rectification.
The aim of this experiment was to observe the current-voltage relationships of different materials and to understand the difference between linear and non-linear relationships. Also, it was to see how varying the temperature affected the resistance of these materials.

2. Theory

In a conductor, the current density J is dependent on the electric field E and the properties of the conductor. For some materials, particularly metals, J is directly proportional to E at a certain temperature; this relationship is known as Ohm’s law which was discovered by Georg Ohm in 1826 [2]. The relationship is given by
(J=σ E#(1) )
where the constant  is the specific conductivity [3]. The reciprocal of the conductivity is the resistivity . If a conducting wire has current I and cross-sectional area A, then
(I=J A#(2) )
and the potential difference V between the ends of the wire is
(V=E L.#(3) )
Rearranging Equation (2) for J to then substitute into Equation (1) and rearranging for E gives
(E=I/(A )=( I)/A#(4) )
where the reciprocal of conductivity has been replaced by the resistivity. Substituting Equation (4) into Equation (3) gives
(V=(ρ L)/A I.#(5) )
When the resistivity is constant, the current is proportional to the potential difference; the ratio between V and I for fixed resistivity is the resistance R.
(R=V/I#(6) )
For a particular conductor at a constant temperature then the resistivity will be constant, and Ohm’s law applies.
Temperature can affect resistance because in conductors there are free electrons and bound electrons; when a potential difference is applied across the conductor, the free electrons are able to travel through the material. As the temperature of the conductor is increased, the bound electrons vibrate more and collisions between these and the free conduction electrons are more likely. When a conduction electron collides with these vibrating bound electrons, they lose some energy, decreasing the flow of electrons (current). Higher temperatures result in more collisions and greater resistance to the current flow.
The diode characteristics are given by the equation
(I=I_0 e^((-E_g)/(k T)) (e^((e V)/(k T))-1)#(7) )
where I0 is a constant, Eg is the energy gap in the material, T is the temperature, e is the elementary charge (e=1.602×〖10〗^(-19) C) and k is the Boltzmann constant (k=1.381×〖10〗^(-23) J K-1). As exp(eV/kT) >> 1, then
(I=I_0 e^((-E_g)/(k T)) e^((e V)/(k T))#(8) )
and this can be rearranged to give
(V=k/e 〖 ln〗⁡(I/I_0 ) T+ E_g/e#(9) )
A voltage-temperature graph with constant current would have an intercept equal to Eg / e; this would give the magnitude of the energy gap in electron volts. The energy gap is an energy range in which an electron state can’t exist; it describes the energy difference between the valance band and the conduction band in a semiconductor. It is the energy needed to promote a bound valence electron to a conduction electron and become a charge carrier.
The apparatus includes the specimen holder as in Figure 1 and the cylinder for the specimen holder as in Figure 2. A selector is used to select the different specimens on the holder and to change the temperature with the heater. The specimen holder sits in the sample space which can be evacuated to produce a vacuum or filled with helium gas. Liquid nitrogen can be added to the cylinder to cool the specimens; they can be cooled down by adding helium gas into the sample space to conduct heat.

3. Experimental approach

First, the sample space, as seen in Figure 2, was evacuated. The specimen selector was set to the Type T thermocouple; this is used to measure the temperature T of the specimens. By placing a second thermocouple in some ice, a reference temperature is provided and the voltage V of the thermocouple, measured by the digital voltmeter (DVM), in the specimen holder (Figure 1) corresponds to the temperature of the specimens by a conversion table.
With the thermocouple measuring room temperature, which was taken as 20C, the first specimen (constantan) was selected with the switch on the selector. The resistance R was set to 0.1 k to obtain the voltage across the constantan for a range of currents I from 0.2 mA, increasing in 0.2 mA increments, to 3.0 mA. Also, 10 k was used to check the relationship between the voltage and current with smaller increases of the current.
This was repeated for the next specimens: copper, germanium and a silicon diode. For germanium the range of currents used was from 0.1 mA to 1.4 mA with 0.1 mA increments and much smaller currents (0.01, 0.02, 0.05, 0.1 mA) were used in addition for the silicon diode to observe the relationship at these smaller currents.
Graphs of the voltage against the current were then produced to check for any relation between the two, the resistance was calculated for constantan, copper and germanium.
Using the selector, the heater in the specimen holder was used to increase the temperature of the specimens. By then, there was a vacuum in the cylinder so the specimens would remain at the same temperature whilst taking measurements. The heater was switched on and using the thermocouple to measure the temperature, T = 40C was reached. The same method used at room temperature was then repeated for all the specimens. Graphs were plotted.
The specimens were also heated to 60 and 80C and the same procedure for the previous two temperatures was repeated. The voltage and current were also measured for germanium for every temperature between 44 and 55C as the resistance of germanium varies a lot around 50C.
Liquid nitrogen was then poured into the cylinder and helium was added into the sample space. The specimens cooled to -175C and the voltage measured for the same range of currents as the previous temperatures. By removing some of the helium from the sample space and using the heater, the temperature of the specimens was increased to -125, -75 and -25C, with the same method repeated for each.
This resulted in voltage-current graphs for all four of the specimens at each temperature. For silicon diode, a graph of V was plotted against ln(I).

4. Results

Graphs of the voltage against the current were produced for each specimen at each temperature. Constantan, copper and germanium produced linear graphs – showing Ohmic behaviour; LSFR [4] was used to calculate the gradient of the graphs and from that, the resistance of each specimen for each temperature with an uncertainty. The graphs for the silicon diode were non-linear; further graphs of V against ln(I) were linear, describing an exponential relationship.
For constantan, copper and germanium, graphs of these calculated resistances were plotted against the temperature; Figure 3 shows this.
Figure 3 shows that for copper, there is a linear relationship between the resistance and the temperature; the resistance increases with temperature. For constantan, Figure 3 indicates that the resistance is independent of the temperature (within the temperature range) as the points are a horizontal line. Germanium’s graph shows two linear relationships. The first is for temperatures below 0C where an increase in the temperature, increases the resistance slightly. The second is for above 0C where the resistance decreases as temperature increases.
Figure 4 shows the variation of resistance to temperature for just germanium; it shows more clearly than Figure 3 that at around 50C, the resistance changes considerably. Also, as the temperature was increased from 44 to 55C, the resistance increases which doesn’t fit the overall trend for that region.
For the silicon diode, the voltage measured at fixed currents (I = 0.1, 0.4, 0.8, 1.2, 2.0 and 3.0 mA) was plotted against the temperature as shown in Figure 5. LSFR was then used to calculate the intercept; using Equation 9, this intercept is the magnitude of the energy gap Eg with uncertainties of silicon. For each of the fixed currents, the energy gaps are tabulated in Table 1. The values of voltages at 333 K (60C) don’t quite fit the overall trend; this may be due to a systematic error as this has occurred for currents only at 60C. This may have been due to not accurately reaching this temperature. Subsequently, the data at 60C was not included when LSFR was used to determine the intercept and the energy gap.
I / mA Eg / eV Error / eV
0.1 1.28 0.01
0.4 1.28 0.01
0.8 1.27 0.01
1.2 1.27 0.01
2.0 1.26 0.01
3.0 1.26 0.01
The values for the energy gap are all consistent with each other, however, the values are not consistent with the accepted value of 1.11 eV (at 300 K) [5]; this may have been due to not accurately reaching the stated temperatures or random fluctuations when taking the voltage measurement. Taking an average of these did not seem appropriate as the energy gap values appear to decrease with increasing current.

5. Conclusions

From this experiment, copper, constantan and germanium were shown to be Ohmic conductors as their voltage-current graphs were linear. Also, copper has been shown to have a linear relationship between temperature and resistance where the resistance increases as the temperature increases. Constantan’s data showed that the resistance of constantan is independent of the temperature and germanium displayed two linear relationships, one where the resistance increased with temperature below 0C and another where resistance decreased with increasing temperature above 0C. The data for germanium also indicated that the resistance is lower at around 50C and doesn’t follow the trend for that region.
Voltage-current graphs were also produced for the silicon diode showing a non-linear, exponential relationship. The graphs of voltage against temperature with points at constant current had intercepts of the energy gap of silicon. The values of the energy gap were Eg = 1.26, 1.27 and 1.28 eV all with an uncertainty of ± 0.01 eV; these values are consistent with each other but not with the accepted value of 1.11 eV.

References

[1] Millikan, Robert A. and Bishop, E. S., Elements of Electricity: A Practical Discussion of the Fundamental Laws and Phenomena of Electricity and Their Practical Applications in the Business and Industrial World, American Technical Society, 1917, page 54.
[2] Young, Hugh D. and Freedman, Roger A., University Physics, Pearson, 14th Edition, 2016, page 844.
[3] Bleaney, B. I. and Bleaney, B., Electricity and Magnetism, Oxford University Press, Second Edition, 1965, page 64.
[4] Python least squares fit script, LSFR.py available from Blackboard.
[5] Semiconductor Band Gaps, http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/Semgap.html Carl R. Nave, accessed 30/01/19.