Theory

Bilayer Graphene Data

The measured voltages were divided by the 10 nA current to convert them to resistances. The resistance is plotted vs. magnetic field strength and back gate voltage is shown in Figure 1. A 1 dimensional cut of the resistance (constant magnetic field, changing back gate voltage) is shown in Figure 2.

In the 2D plot of measured resistance vs. magnetic field and back gate voltage, quantization is visible as plateaus of constant color. KIt is not clear why the red plateau is broken up along the magnetic field axis. One possibility is that it was not a plateau, but just an SDH peak. In this case it might not always appear, since it is not exactly defined like a Hall resistance plateau.

The effect of the two wire measurement is clearly visible in Figure 2. The plateaus are rounded and not well defined. This is because the two wire measurement included both the Hall resistance plateaus and the longitudinal SDH oscillations.

The data taken at field strengths of 8 and -8 Tesla were analyzed. The data was treated like it only contained Hall resistance, since the two components could not be separated. Points were picked by hand from the plateaus. The plateau resistance was taken to be the average of these points, the error was their standard deviation.

The resistance and its error were converted to conductances.

Ideally, these plateau conductances should be fit to Hall conductance equation for bilayer graphene.

To determine N for each plateau, the averaged conductances were divided by the expected slope. Five plateaus were found in the 8T data. The N values were expected to be close to integers, so they could be rounded. Using the unrounded values would have biased the fit towards the value of Planck's constant used in calculating the expected slope. Five plateaus were found for the 8T data. (These combine the equivalent plateaus on the left and right side of Figure 2). their rounded N numbers were 1, 1.7, 2.3 ,3 and 4. This was an unexpected result. In the Integer Quantum Hall Effect, N can only be an integer. There also should not have been two plateaus between N=1 and N=3. It was not clear why there were two non-integer plateaus. One possibility is that impurities in the sample caused the SDH oscillations to behave strangely. Because the SDH oscillations occur at transitions between plateaus, there should have been only 1 SDH peak per plateau, which could not explain the unexpected result. But, it is more likely that something affected the longitudinal resistance, since it is not exactly quantized, and depended on the bulk properties of the sample.

Monolayer Graphene Data

Graduate students in Ke Wang's lab provided raw data from four wire measurements of the Integer Quantum Hall Effect in monolayer graphene. We analyzed this data to demonstrate the potential accuracy of the experiment if the longitudinal and Hall resistances are separated. The measured Hall resistance is shown as a 2D plot in Figure 3, and Figure 4 is a 1D cut at constant magnetic field that shows both the Hall and longitudinal resistance.

The plateaus can be seen as the bands of constant color in Figure 3. The plateaus are well defined compared to the plateaus seen in the two wire measurement data. the plateau resistance were again determined by picking points off of plateaus and averaging them. Data from all magnetic field strengths were used. The error was again the standard deviation. The resistances were converted to conductances and fit to the equation for the Hall conductance if monolayer graphene.

The results of this fit and the residuals are shown below. the reduced chi squared was 1.4.

Using NIST's 2014 values for the electron charge [1], Planck's constant was calculated from the slope to be:

This was 5.3 σ from the accepted value [1] of Planck's constant:

However, the measurement was accurate, as the percent difference was 0.4 %. Since the error was small compared to the measured value in Planck's constant (about 1 part in 1000). This means the difference in σ was sensitive to small systematic errors that were not accounted for. One source of systematic error could have been the assumption that the current through the sample was 10 nA. The voltage drop across the sample and resistor was 1 V. The sample resistance was neglected because it was on the order of 10s of , and the resistor had a resistance of 100 MΩ. Adding 10 kΩ to the 100 MΩ resistance would have changed the total resistance by only 0.01. The current used to calculate the Hall resistance and conductance was thus 10 nA.

However, it is unlikely that the resistor used had a resistance of exactly 100 MΩ . If this was the case, the current would not have been 10 nA. Since this current was used to calculate the Hall conductance from the voltage, the error would have affected the fit. It was not possible to determine the actual resistance, because the raw data came from an experiment done by graduate students in Ke Wang's lab. Because precise analysis was not their goal, they did not record the resistor that they used, or its resistance. To demonstrate how a deviation of less than 1% in the resistance could have caused the disagreement, 3 correction factors were applied to the conductance of the Hall plateaus and the error. The factors changed the conductances to what they would have been if resistances of 100.2 MΩ , 100.5 MΩ , and 101 MΩ were used in the analysis. A linear fit was done on the corrected data. In the table below, the corrected value of Planck's constant and the deviation from the accepted value is given for each resistance.

The results are sensitive to the small deviations introduced by changing the resistance, so the systematic error introduced by assuming the resistance was 100 MΩ could explain the 5.3σ disagreement between the calculated and accepted values of Planck's constant. However, since the actual resistance was not known and could not be measured after the analysis was done, this source of error cannot be empirically confirmed.

Go To Looking Forward

Back To Home

References

[1] NIST. “CODATA Internationally Recommended 2014 Values of the Fundamental Physical Constants.” Fundamental Physical Constants from NIST, 2015, physics.nist.gov/cuu/Constants/index.html.