Temperature Dependence of a Doped Semiconductor and Measuring the Hall Effect

Hannah Ross and Nicole Olson

University of Minnesota

Methods of Experimental Physics Spring 2013

Introduction

In this experiment we investigate the behavior of an extrinsic semiconductor at different temperatures. The temperature dependence of both the conductivity and the charge carrier mobility is measured between 10 and 300 K. This range was chosen due to time and equipment constraints. We investigate this dependence because temperature is a primary consideration in the use of any material. How a semiconductor will function at any given temperature is vital information for the design of any product using a semiconductor.

Semiconductors have opposite behavior from normal conductors in the fact that conduction decreases with rising temperature, whereas it increases in conductors due to more electrons being freed and raised to the conduction band [1].

Theory

Materials are classified by the size of their energy gap. In semiconductors, this is around 2-3 eV. The term “energy gap” refers to the difference in energy between the top of the valence band the bottom of the conduction band [2].

Semiconductor classification

Since pure semiconductors may not have electrons inside this energy gap, electrons must be supplied with enough energy to completely bridge the gap in order to be promoted from the valence to the conduction band. When atoms of a different material are added to a semiconductor it becomes doped. These different atoms are called impurities because they constitute a small fraction of the material without changing the host structure. If these impurities have a different valence configuration than the host material, states inside the energy gap form in one of two ways: extra filled electron levels are created near the conduction band to create donor energy levels, or extra empty levels are created near the valence band to create acceptor energy levels. This creates n- and p-type semiconductors, respectively. The number of valence electrons in the doping agent determines the type of dominant charge carrier in the semiconductor. For example, assume the semiconductor is silicon. If the dopant has three valence electrons, the impurity will bond with four silicon electrons, causing the impurity atom to become an ion. This ion has taken an electron from another atom, leaving a hole that becomes the dominant charge carrier when the material is doped with a considerable number of these same impurities. This classifies the semiconductor as p-type. In a p-type semiconductor, the addition of energy results in the emptying of filled electron states into the acceptor states and creates holes in the valence band. These holes are then free to move around, and it is this movement of charge that is responsible for conduction. If the dopant has five valence electrons, the semiconductor is an n-type and the charge carriers are the extra electrons that could not bond with the silicon [1].

Conductivity and mobility

The conductivity of a material quantifies how easily the material conducts charge. This conductivity, σ, of a semiconductor is a function of the carrier concentration and the mobility, μ, of the charge carriers. In a doped semiconductor, the expression of conductivity can be reduced to include only the term describing the dominant charge carrier. Both of these quantities are temperature dependent. The mobility, or the freedom of the charge carriers to move, is affected by scattering from both the host lattice and the impurity ions. The effect of lattice scattering decreases at low temperatures due to the decrease in the kinetic energy that slows lattice vibrations. At the same time, this lower energy slows the movement of the charge carriers, giving them more time to interact with the impurity ions. Therefore, the mobility and the conductivity increase with temperature until lattice scattering dominates and decreases these quantities. The temperature dependence of mobility goes as T^3/2 and T^-3/2, when impurity and lattice scattering dominate, respectively [2]. We can find the mobility once we know the conductivity and charge carrier density.

The van der Pauw technique

The Van der Pauw technique allows one to determine the conductivity for an arbitrarily shaped sample and eliminates extraneous effects, such as stray voltages, from a measurement. This technique, illustrated in figure 4 on the following page, involves the use of a slab of material that has four electrical contacts on each corner. While a current is run across contacts A and B the voltage across contacts C and D is measured.

The polarities are then switched and the measurement is taken again. Reversed polarity measurements can vary slightly, but should be on the same order of magnitude. This is repeated for all horizontal and vertical combinations. Using Ohm’s Law, we can find each individual resistance and use the Van der Pauw equation to find the sheet resistance, R_s. The conductivity is then 1/(R_s*t) where t is the thickness of the sample. Using this technique improves the accuracy of the calculation and checks for the repeatability of the measurements [4].

The conductivity results we expect to obtain are shown in Figure 1.

Figure 1: Expected conductivity results

Apparatus

The design of our apparatus is shown in Figure 2.

Figure 2: Experimental apparatus

This experiment will use a boron doped silicon semiconductor, measuring approximately 1 cm². The doping concentration is between 10¹⁵ and 10¹⁶ cm^-3, as stated by the manufacturer. For calculation purposes, we assume the concentration to be 5 x 10¹⁵ cm^-3. The four corners of the sample have undergone an aluminum deposition to create ohmic contacts between the semiconductor and the aluminum. Each of these contacts is connected to the wires of the cryostat via indium dots. To vary the temperature of our sample, a slow flow-through cryostat attached to a programmable temperature controller was used. This controller powers a resistive heater in thermal contact with a copper block to which the sample is attached. The liquid helium was held in a separate Dewar and cycled through the cryostat via a transfer hose. A switchbox consisting of four inputs and four outputs will contain a system of 16 relays. These relays are open circuits that close upon receiving a digital signal from our DAQ card. A 10 μA current is applied across two inputs and the voltage is measured across the remaining two. Each output is connected to one corner of the sample, allowing us to set which contacts the current will be applied to and the voltage measured across. A LabView program controls the entire data acquisition process.

This design was modified after the discovery of one bad contact on our sample, and a two-probe measurement was used to obtain data in the 10-35 K range. The switchbox was removed and a 1 μA current was applied across two contacts while the potential difference was measured across the same contacts. The disadvantages of this method are discussed in the Analysis section.

Results

The temperature dependence of the mobility was determined to follow a power law of μ_p=(8.6x10⁷)T^-2.03 in the range of 201 to 296K, as shown in figure 6. This is 18.7 sigma above the expected power law of μ_p = (2.5x10⁸)T^-2.3. The reduced chi squared for this fit is 0.00052. In the same temperature range, the conductivity of our sample was found to follow σ =(6.9x10⁴)T^-2.03. This is 50.7 sigma below the expected power law of σ =3.6x10⁶ T^-2.72. The reduced chi squared for this fit is 0.0012.

The above results were found using the van der Pauw technique. After abandoning this four-probe method and switching to a two-probe, we obtained low temperature results for mobility and conductivity. At low temperatures the conductivity was found to follow an exponential law of σ =0.11e^0.12(T). This is 110.7 sigma away from our expected result of σ =(2x10^-10)e^0.45(T). The reduced chi squared value for this fit is 0.008. Mobility was found to follow a power law of μ_p = (5.1x10^-7)T^{2.52 ± 0.24} and is 4.3 sigma below the expected value. This fit has a reduced chi squared value of 2.15.

Analysis

The van der Pauw equation cannot be solved analytically. Therefore, a Taylor expansion to the second order was preformed to obtain uncertainties on the sheet resistance calculations that could then be further propagated through calculations of conductivity and mobility.

Much of our data was highly inaccurate, and was discarded after the discovery of one bad contact on our sample. The voltage for each reversed polarity measurement using the van der Pauw technique should be on the same order of magnitude. At very low temperatures, these voltages were up to three orders of magnitude higher than expected. Shortly after this data was taken, it was noticed that we were not creating a complete vacuum in our cryostat, and that the tubing was cold to the touch. Replacing an old o-ring that was too large for the seal, we were able to fully evacuate our cryostat, but our data was still not what we expected to see. Data was then taken in a higher temperature range, from 201 to 296 K. As evidenced in figures 5 and 6 above, as the temperature decreased our measurements started to deviate from the theoretical results. The equipment for this experiment was tested at room temperature before any data was acquired and gave results close to what was expected. Therefore, the low temperatures had to be causing a problem that interfered with our measurements. To isolate the problem, the sample was cooled and the equipment tested again. Removing the switchbox from the set up, a voltage was measured across the same contacts to which a current was applied. Every voltage measured when contact “A” was used was orders of magnitude different than expected. This led us to conclude that something was wrong with this contact and was responsible for the large deviations between what we expected and what we saw. Due to time constraints, there was no opportunity to attempt to fix this contact and continue taking data, or to determine the exact cause of the failure.

A possible reason could be the formation of silicon oxide on the surface of the sample between the hydrofluoric acid treatment to remove oxide impurities and the aluminum deposition that formed our ohmic contacts. Seeing as only one of our contacts was bad enough to bar us from obtaining reasonable data using the van der Pauw method, it is more likely a different cause was responsible, such as an issue with our wires. We had trouble getting the wires from the cryostat to stick to the indium dots on the first sample we tried to mount, so the cryostat was re-wired with significantly thinner wires. It is possible that one wire was precariously connected and failed at low temperatures. Time permitting, this could have been tested by detaching our sample, rotating it 90 degrees, re-attaching the wires, and cooling it down again to see which contact fails: contact “A” or the one attached to the wire the contact “A” was previously connected to. Alternatively, our bad contact could have simply been due to an incomplete attachment of the wire to the indium that separated at cold temperatures.

The two-probe technique adopted for our low temperature data is a less accurate method than the van der Pauw technique because only one edge of the sample is measured. There are no averaging or reversed polarity measurements to increase the accuracy of the calculated resistance.

The mobility of the charge carriers freezes out at low temperatures due to small kinetic energies, as was expected. The standard deviation of this result is so small compared to the other fits due to the comparatively high uncertainty in the exponent. Our high standard deviation values reflect the fact that there was a major problem with our equipment. The reduced chi squared values on the order of 10^-2 and lower tells us that this data was not fit well. Conductivity and mobility in a semiconductor do not follow one law over a 300 K temperature range, as can be seen in figure 3. The poor fits could be attributed to the fact that we have tried to fit over a temperature range that may have been too large to fit with only one law. The expected raw data was pulled from the graph in figure 3 and fit to either a power or an exponential law, which was then taken as our expected results. Since our fits are poor and we are far from our expected results, we conclude only that the temperature affects the conductivity and mobility of a semiconductor, and that it does not follow a set dependence over any given range of non-negligible size.

Conclusion

The deviations of our results from what was expected are due to the systematic error of the increasing malfunction of one of our contacts as temperature decreased. To minimize the time consumed by this error in future repetitions of this experiment, all equipment should be tested for functionality at both extrema of the desired temperature range before any data acquisition is begun. In the future, this experiment could be expanded to incorporate an applied magnetic field to investigate the Hall effect.

References

[1] van Zeghbroeck, B. (2011). Principles of semiconductor devices. Retrieved from http://ecee.colorado.edu/~bart/book/book/title.htm

[2] Callister, W. (2010). Materials Science and Engineering: An Introduction. John Wiley and Sons, Inc.

[3] Morin, F., & Maita, J. (1954). Electrical properties of silicon containing arsenic and boron. Physical Review, 96(1).

[4] van der Pauw, L. (1958). A method of measuring specific resistivity and hall effect of discs of arbitrary shape. Phillips Research Reports, 13(1), 1-9.