An Automated Potentiometric System For Precision Measurement Of the Quantized Hall Resistance

This paper describes the development of an automated potentiometric measurement system that is used to compare the quantized Hall resistance with that of wire-wound reference resistors having the same nominal value. Conceptual considerations, along with the major practical problems associated with this method, are presented. We then report experimental results which demonstrate that this measurement system is accurate to within a 0.007 ppm one standard deviation uncertainty.


Introduction
The quantization of the Hall resistance (1) (where h is the Planck constant, e the electron charge, and i an integer quantum number) has been demonstrated for such diverse two-dimensional electron systems as those in Si-MOSFETs [1]1 or in heterostructure devices (GaAs/ AIGaAs [2] and more recently InGaAs/InP [3]). This resistance quantization provides a very promising means to About the Authors: Giancarlo Marullo Reedtz is a physicist at the Galileo Ferraris National Electrotechnical Institute, and Marvin E. Cage, who is with the Electricity Division in the NBS Center for Basic Standards, is also a physicist. ' 303 yield not only a constant and reproducible as-maintained unit of resistance but to determine also, by solid state physics techniques, the fine structure constant a [4,5]. Even if the physical phenomenon (as manifested in real samples) is not completely understood, the experimental results from different national laboratories show an agreement between RH values of a few parts in 10' [6] whe~ using different samples and measurement systems.
The precision measurement of RH requires three main steps: l. the metrological characterization of the sample so that a reasonable confidence is reached on the independence of the measured value from the experimental conditions-mainly temperature and current through the sample [7.8]; 2. the comparison of RH with the resistance RR of a highly stable reference resistor; and 3. the calibration of RR in terms of either the as-maintained unit of resistance or in terms of the International System (SI) resistance unit. An automated potentiometric system has been built to compare, as in step 2, the value of RH with a nominally equal reference resistance RR within a one standard deviation (leT) uncertainty of 0.007 ppm (parts per million). This system, which replaces and improves an earlier manually-operated version, is flexible enough to be used also for the precision measurements required in step 1.

Comparison of RH and RR
The conceptual scheme of the measurement circuit is shown in figure I, where the numerical values are specific to the case when R H =6,453.20 n (the i =4 Hall plateau) and the current is 1 = 25 ).I.A. The two resistances to be compared are connected in series, and the high impedance nanovoItmeter D measures the differences between each voltage drop VII or V R and the compensation voltage Vc. The switch SI enables the current direction and the compensation voltage both to be reversed so that the influence of thermal-electric voltages can be reduced; Sz allows for switching between V H and V R • Therefore, SI must not change the resistance of the current circuits too much for the two polarities, I 216 kO and S2 must introduce only a minimum amount of thermal voltages in the voltage circuit.
The measurement sequence needed for a single determination of the ratio RIIIRR should be chosen to reduce the influence of the thermal voltages and the voltage and current drifts, while also trying to minimize the number of switching operations. Using the letters H or R to represent the comparisons between VII or V R and Vc (switch S2), and the symbols + and -for the two current polarities (switch SI), we consider the following sequence of operations: Within each pair of + Imeasurements, the influence of the thermals, if constant, is eliminated by the current reversal. The circuit in figure I is also affected by changes in the thermals, and by drifts in the voltage generators E and E', or in the currents. Linear drifts in the circuit can be represented as changing currents by the relations where D(t) is the reading of the detector at time t, and Rp is the potentiometer resistance as labeled in figure 1. If the measurements are uniformly spaced with time intervals M, then using eqs (3) and (4) and dividing the sum of the RH equations by the sum of the RR equations, we obtain where and t=tl+~M is the mean time of the four measurements.
If the two resistance values are different by a small amount we can expand the denominator in eq (5); neglecting second-order terms: Here the compensation voltage Ve has been written as time independent, which it is in the limits that we require (e.g., Ve need only be constant to I part in 10 3 for DIVe ratios of 10-5 if we want an accuracy of 1 part in 10 8 ). Within these limits the drift of Ve has no influence in eq (7).
We now examine what happens if the switch SI introduces slightly different resistances for the two polarities of the I and I' circuits. Let the currents for the two polarities be I and (-l' + 6./') for the circuit E, and in a similar way I' and (-l' + 6. Subtracting and dividing each pair by two: Adding and reordering: The last (third) term in eq (10) gives a second-order correction if the two resistances to be compared are nearly equal and the variation AI II is small enough. As before, the denominator of the second term need only be known with limited accuracy and can be replaced by the compensation voltage Ve. As a conclusion of this analysis we note that: a) the sequence given by eq (2) eliminates the effects of constant thermals within each pair of current reversal measurements; b) the symmetrical sequence eliminates the effects of linear drifts of thermals and of I and I'; and c) the measurements are not influenced by the slightly different values of I and l' for the two current polarities because each polarity is used to compare both VII and V R with the same compensation voltage Vc.
The measurement sequence actually used is a liltle more complex but fulfills all the requirements outlined above. It can be represented by

Sample Characterization and System Flexibility
Every prospective sample must be characterized to establish what experimental conditions are required in order to ensure that the measured values of Rn have the desired accuracy [7,8]. The following measurements of Rn and of Rx (the sample resistance in the direction of the current) are made before a device is considered for precision RH/RR comparisons: l) RH and Rx as a function of the sample temperature; 2) RH and Rx as a function of the current through the sample; 3) as in items 1) and 2), but for different pairs of RH and Rx contacts and for both magnetic field directions; and 4) RH values for different points along the Hall plateau to verify its flatness.
All these measurements require precision comparisons between RH and RR' For item 2) the compensation voltage for RH measurements must be varied over a wide range. Also, a high impedance current source is required to maintain constant current when the sample is not in a well-quantized regime. The potentiometric comparison circuit of figure 1 has been built in modular form to allow for these different measurement conditions. A functional diagram of the components of this circuit is shown in figure 2, where solid (dashed) lines are for current (voltage) connections. Dashed lines also symbolize output signals that are measured or recorded.
The Potentiometer Box in figure 2 provides the functions needed for the highest precision measurements. The currents I and I' are generated using matched sets of thermally insulted mercury batteries and fixed-value resistors. Comparisons of R 11/ R R can be made at 10 J-LA and 25 J-LA for i = 4 Hall steps and at 25 J-LA for i = 2 steps. These currents are small enough to provide accuracies better than 1 part in 10 8 for the three GaAs samples so far characterized.
An electronic, high impedance, constant current source can replace the mercury battery source Iof the Potentiometer Box for current dependence measurements and for measurements off of the step. The electrical noise, however, is an order of magnitude higher with the constant current source. A high stability solid-state Zener voltage source 306 followed by a Kelvin-Varley resistance divider can substitute for the battery current source and fixed divider. These again are an order of magnitude noisier.
The Sample Patch Box allows for easy connection of the current leads to the source and drain, as well as to any pair of potential pads on the sample, e.g., V H or V x . The voltages read by the detector can either be direct or compensated by Vc.
Each Reference Resistor Box contains an Electro Scientific Industries resistor 2 , composed of series/parallel combinations of wire-wound resistors constructed to have RR values within a few ppm of the value of R H • They are hermetically sealed in a silicone fluid-filled container and placed in a specially constructed temperature-regulated air bath enclosure. The air temperature is controlled to within ±0.OO2·C at a nominal temperature of -28 ·C.
The Calibration Box contains an adjustable voltage source whose polarity can be reversed by the Hewlett Packard 9836 computer. This box, when connected to the detector input, enables the detector-digital voltmeter combination to be calibrated.
Finally, RH and RR can be easily interchanged in the measurement circuit to check for systematic errors. We have found this test to be essential.

The Insulation Resistance
The high values of the resistances to be compared (6,453.20 n or 12,906.40 n for i =4 or i =2 steps, respectively) require an insulation resistance of the circuit of about 10 12 n to achieve an uncertainty of the order of 10-8 • As a rule, polytetrafluoroethylene (PTFE or Teflon) has been used as the insulating material. Great care was taken in cleaning the insulating parts of cables and connectors with alcohol. This provides leakage resistances greater than 10 14 n. . the nanovoItmeter guard circuit. Therefore, the detector input low is always connected to the potential point nearest circuit ground, and the guard is also connected to this ground. (The low potential point, of course, changes sides on the sample if the magnetic field direction is reversed.) The complete potentiometric measurement system has an insulation resistance of 3 X 10 12 n when the humidity is less than 50%. This leakage resis-307 tance is periodically monitored to ensure that dust, lint, hair, and fingerprints have not reduced it.

Contact Resistances, Thermal·Electric Voltages. and Drifts
The cables consist of sets of twisted pairs of PTFE-insulated, silvered, stranded copper wires enclosed within two separate coaxial shields.
British Post Office connectors (model numbers OFF 66148 and 66149) are used whenever possible because they have small and reproducible contact resistances, high leakage resistances, and are completely shielded.
Fixed connections are made by a low melting point Sn-Pb alloy solder, rather than a low thermoelectric power Sn-Cd alloy, because of better mechanical strength and more reliable electrical contacts. Tightly-packed thermal insulation is used to minimize temperature gradients.
The contact resistances of the Leeds & Northrup rotary switches are less than 0.7 mO and have a reproducibility of 0.05 mO. Differences in resistance of the two current circuits, due to slightly different lengths of the wires through the switches for each polarity, are less than 4 mO. That value is less than 2x 10-8 of the resistance of each circuit. As shown in section 2, this has no detectable influence on the result when using the sequence of measurements shown in expression (11). These switches exhibit less than 20 nV thermals 10 seconds after repeated switching.
The drifts of the I and /' sources are of the order of 0.2 ppm or less during the 12 minute time intervals required to carry out the measurement sequence of expression (11). This small drift rate is achieved by using careful thermal insulation of the Potentiometer Box, and by not opening the loads during current reversals. (Momentary double loads in the make-before-break switching are better than open-circuiting for the stability of the mercury batteries.)

The Automated Switching
Stepping motors have been used to rotate the switches SI and S2' They are Clifton Precision model 23-SHAB-49BU motors with a minimum rotation of 0.9° /step and a 0.45 N·m torque. The motors must overcome a significant frictional force. As a result. they occasionally can miss steps. After several misses the switches may open-circuit. To prevent this we realign the switches by deliberately forcing the motors to rotate the switches against stops.
The molors are controlled by a driver (a Clifton Precision model DPB05) which supplies the clock pulses that control the steps, and other TTL signals which determine the direction of rotation, the working mode (D.9°/step or l.So/step), and the currcnt-off-after-switching capability. The last feature is useful to reduce the noise in the system during measurements and to allow for manual switch operation.

4.4 Noise Problems
Great care has been used in shielding the system. Two independent levels of electrostatic shielding have been provided for all the boxes and cables within the instrument. In addition, the electronics rack for the instrument, aluminum shelves for the reference resistors, metal room walls, and copper pipe for the signal cable to the cryostat give a third, nearly complete, shielding level that is electrically insulated from the other two shields. The only exceptions are: the reference resistor boxes which have only two levels of shielding, and the cryostat and quantum Hall sample holder with just one independent shield.
To avoid current loops, all cables have their coaxial shields broken on one side. Shielding connections among different chassis boxes are made by dedicated cables. Only one connection exists between different shielding levels, between the instrument rack and laboratory ground, and between the first shield and the measurement circuit. This last connection can be opened for the leakage resistance tests.
The largest amount of electrical noise in the system comes from the nanovoltmeter detector, which has 150 nV peak-to-peak noise within a 0.5 Hz bandwidth. This noise (obtained with the input short-circuited and with the instrument powered by batteries) corresponds to about 38 nV at the lu level, to be compared with 7 nV (10') Johnson noise of a 6,453.20 0 resistor at room temperature within the same bandwidth.
The nanovolt amplifier noise is about 25 X 10-8 of the voltage to be compared for a current of 25 p.A on the j =4 step. Therefore, long integration times must be used on the detector output to obtain uncertainties less than 1 X 10-8 for the standard deviation of the mean of a RHIRR comparison.
For each operation of the sequence given in expression (11), the computer takes 60 readings in 30 s time intervals from a digital voltmeter at the output of the detector. A wait time of 15 s is introduced after switching. It takes 12 minutes to complete one sequence and thus yield a single value of the ratio RH/RR' The standard deviation) of such a sequence is typically 0.016 ppm. The value is the same whether the reference resistor is compared with a quantum Hall resistor or with another reference resistor. Seven sequences (84 minutes of measurements) are usually sufficient to obtain a standard deviation of the mean (random uncertainty) of 0.006 ppm.
Yfhroughout. all uncertainties are meant to correspond to one standard deviation estimates.

System Performance
This measurement system has been in operation since August, 1986 to compare the value of RH for the i =4 plateau of a GaAsl AIGaAs heterostructure with that of two nominally equal 6,453.20 n reference resistors, and to also intercompare the two reference resistors. Table 1 summarizes the estimated uncertainties assigned to quantum Hall sample-reference resistor intercomparisons for a 25 /-LA current. The standard deviation of the mean (the random or type A measurement uncertainty) is, as noted above, typically ±0.OO6 ppm for a sample-resistor comparison. The position of the sample and resistor are then interchanged in the measurement system and the values are remeasured. The Hall resistance value assigned is the unweighted mean of these two comparisons, and the random uncertainty of the 14 pooled measurements is usually ±0.004 ppm. Pooling all of the measurements is viewed as justified since usually the difference in the values obtained in the two positions is well within the scatter of the data.
In addition to the random uncertainties, there are systematic, or type B, uncertainties associated with systematic corrections and other effects. One such correction is due to a measurement system offset, or interchange, error in which the value of the Hall resistance depends on whether it is measured in the RH position or in the RR position of the measurement circuit. The largest interchange correction to date has been +0.013 ppm, but, as noted above, is usually much less (i.e., indiscernible; the correction is based on the assumption that the mean of the two values obtained is the correct value). It is not completely understood why the interchange correction is less of a problem in this measurement system than it is in the old version of the potentiometric system [9] and in the automated bridge system [9,10). Perhaps it is due to a higher leakage resistance and to better electrical shielding. The estimated type B uncertainty associated with the interchange correction is assumed to be the same as the random uncertainty: ±0.004 ppm.
There is an uncertainty in calibrating the gain of the electronic detector-digital voltmeter pair. The pair can appear to vary by a few tenths of a percent over the input voltage range if the DVM has a dead-band at zero volts [9). The dead-band of our Hewlett Packard 3457A Digital Multimeter is small enough for this effect to be negligible. There remains, however, the problem of the stability of the detector-digital voltmeter gain. The day-to-day gain varies by -0.1 % when the room temperature is controlled to ± I ·C. This instability contributes an estimated ±0.OO3 ppm type B uncertainty to the sample-resistor measurements because our reference resistors are about 2-3 ppm smaller than the quantum Hall resistance.
The 3 X 10 12 n measurement system leakage resistance contributes an estimated ±0.OO2 ppm type B uncertainty.
No temperature-dependent [7] or current-dependent [8] quantized Hall resistance (QHR) corrections could be detected for the Hall-probe set used on this sample. The respective estimated type D uncertainties for the dependence of the QHR on temperature and current are ±0.OO2 ppm and ±O.OOI ppm.
The total root-sum-square uncertainty is typically ±0.OO7 ppm. This uncertainty is only one third that of the old potentiometric measurement system [9] and the bridge system [9.10). The smaller uncertainty owes to: a) the ability to directly interchange the sample and the reference resistor positions in the measurement system, rather than having to substitute another reference resistor for the sample in order to determine the interchange error; b) a smaller interchange error; c) the fact that there is only one uncorrelated uncertainty of the detector-digital voltmeter gain. rather than the three correlated uncertainties in the bridge l'.Ystern; and d) a larger leakage resistance. 309

Results
The United States legal unit of resistance O,"\ is being monitored via the quantum Hall effect [9], a .. outlined in steps 1 to 3 of section 1. Compari\Ons between R" and RR are made as part of this ongoing procedure. Figure 3  . The August 1986 data shows the improved measurement accuracy of the new automated potentiometric system over that of the old, manually-operated potentiometric measurement system and the automated bridge system. This new measurement system has proved to be quite reliable. very flexible, and easy to use.