S14HallEffect

Abstract:

We measured the temperature dependence of the conductivity and mobility of a silicon sample doped with boron in the range of 77 K to 200 K. The conductivity was supposed to fall as T^-2.2, and our data showed a curve of T^-1.99±0.01, which is 21 sigma away from the expected value. Our mobility data followed the curve of T^-1.45±0.04 , compared to the expected T^-3/2 , which is 1.25 sigma away.

Theory:

The conductivity and mobility of a doped semiconductor are dependent on the temperature of the sample. This is due to scattering effects in the semiconductor, which are dominant over different temperature ranges. An example plot of the mobility is shown below:

At lower temperatures, impurity scattering is dominant over the mobility, whereas at higher temperatures, the scattering is due to lattice vibrations in the crystal structure of the semiconductor.

The conductivity can be found from the resistivity of the sample, which is found using the Van der Pauw method. Then, by applying a magnetic field to the sample, the mobility can be measured as proportional to the inverse of the Hall Voltage, as measured via the Hall Effect.

By going to liquid nitrogen temperatures, we were able to see the 'peak' in the conductivity and mobility, and then the decay at warmer temperatures. To observe the effects of the impurity scattering, the temperature of the sample has to be lowered further.

Apparatus:

Our sample was contained in the cryostat, which was cooled with liquid nitrogen, and the leads were connected to the sample via indium dotting. The switchbox cycled through the cases of the Van der Pauw method and the Hall effect. A permanent magnet was used to apply the magnetic field when testing the Hall Effect. The sample was first cooled to liquid nitrogen temperatures, and then allowed to warm to room temperature for both the conductivity and mobility measurements.

The data was input into the computer via the switchbox and saved into an Excel file for further processing.

Results:

Our conductivity data is as follows:

This showed a temperature dependence of T^-0.9, which signified an error in our data taking, where we did not switch from a voltage source to a current source, as is required by the Van der Pauw method (which was later confirmed using the mobility results). An attempt to correct for the error gave a temperature dependence of T^-1.99±0.01, which is 21 sigma away from the expected T^-2.2 curve.

Our mobility data:

Our mobility data showed a curve of T^-1.45±0.04, which is 1.25 sigma away from the expected T^-3/2. As stated previously, this data confirmed that our conductivity data had been taken incorrectly, because this data did not show a large systematic error. Our data for the mobility was also left in arbitrary units, as solving for the proportionality constant involved using the conductivity, which had been determined to be incorrectly taken.

Our data was analyzed partially with Excel, and the Van der Pauw equation was solved using Mathematica for each data point. This meant that analyzing the conductivity data took some time, as a large part had to be done in Excel, then by hand, then put into Mathematica. In the future, differently shaped samples that produce "horizontal" and "vertical" resistances (as defined by the Van der Pauw method) that are equal to each other would provide simpler and faster analysis of the conductivity data, as the Van der Pauw equation can be solved algebraically instead of numerically.

Conclusion:

We measured the temperature dependence of the conductivity and mobility of a boron-doped silicon sample. Due to errors measuring our conductivity, the curve our data was supposed to follow was T^-2.2, but our data followed a curve of T^-1.99±0.01, which is 21 sigma away from the expected value. Our mobility data followed the curve of T^-1.45±0.04 , 1.25 sigma away from the expected T^-3/2 . Future improvements in this experiment could include differently shaped and doped samples, ensuring correct measurements of the conductivity data, utilizing the heater via the temperature controller, and cooling to liquid helium temperatures. In particular, choosing a square or clover shaped sample (as outlined in the recommended shapes for the Van der Pauw method) might allow for the use of the algebraic solution to the Van der Pauw equation, which would ease data analysis.

Sources:

Ross, H. and Olson, N. (2013). Temperature Dependence of a Doped Semiconductor and Measuring the Hall Effect. MXP Wiki. Web.

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.

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

Sah, Chih- Tang. (1991). Fundamentals of Solid-state Electronics. Singapore: World Scientific.