Quantized Hall Resistance Research Articles

The quantized Hall insulator

AbstractThe quantized Hall insulator is characterized by vanishing conductivities and a quantized Hall resistance. For low mobility samples, the quantized Hall insulator is obtained when the magnetic field is increased well above the ν = 1 quantum Hall state. For higher mobility samples, a similar quantization is observed when the magnetic field is increased above the ν = 1/3 fractional quantum Hall states. This quantization, throughout the quantum Hall liquid‐to‐insulator transition, leads to a perfect semicircle relation for the diagonal and Hall conductivities. The measurements were performed in Ge/SiGe quantum Wells and in n‐type InP/InGaAs and GaAs/AlGaAs heterostuctures.

Equivalent electrical circuit representations of AC quantized Hall resistance standards

We use equivalent electrical circuits to analyze the effects of large parasitic impedances existing in all sample probes on four-terminal-pair measurements of the ac quantized Hall resistance RH. The circuit components include the externally measurable parasitic capacitances, inductances, lead resistances, and leakage resistances of ac quantized Hall resistance standards, as well as components that represent the electrical characteristics of the quantum Hall effect device (QHE). Two kinds of electrical circuit connections to the QHE are described and considered: single-series “offset” and quadruple-series. (We eliminated other connections in earlier analyses because they did not provide the desired accuracy with all sample probe leads attached at the device.) Exact, but complicated, algebraic equations are derived for the currents and measured quantized Hall voltages for these two circuits. Only the quadruple-series connection circuit meets our desired goal of measuring RH for both ac and dc currents with a one-standard-deviation uncertainty of 10−8 RH or less during the same cool-down with all leads attached at the device. The single-series “offset” connection circuit meets our other desired goal of also measuring the longitudinal resistance Rx for both ac and dc currents during that same cool-down. We will use these predictions to apply small measurable corrections, and uncertainties of the corrections, to ac measurements of RH in order to realize an intrinsic ac quantized Hall resistance standard of 10−8 RH uncertainty or less.

Quantum Hall effect in a single-mode wire

The quantum Hall effect in a single-mode wire is studied for the first time. It is established that a well-expressed quantum Hall resistance for filling factors i=1 and 2 is observed in magnetic fields in which the magnetic length becomes less than the width of the wire. Breakdown of proportionality with respect to the magnetic field in the arrangement of the Hall quantization plateau and the dissipative conductivity minima is observed.

Exact Quantization of the Even-Denominator Fractional Quantum Hall State atν=5/2Landau Level Filling Factor

We report ultra-low temperature experiments on the obscure fractional quantum Hall effect (FQHE) at Landau level filling factor $\nu$=5/2 in a very high mobility specimen of $\mu=1.7 \times 10^7$ cm$^2$/Vs. We achieve an electron temperature as low as $\sim$ 4~mK, where we observe vanishing $R_{xx}$ and, for the first time, a quantized Hall resistance, $R_{xy}=h/(5/2e^2)$ to within 2 ppm. $R_{xy}$ at the neighboring odd-denominator states $\nu$=7/3 and 8/3 is also quantized. The temperature dependences of the $R_{xx}$-minima at these fractional fillings yield activation energy gaps $\Delta_{5/2}$=0.11K, $\Delta_{7/3}$=0.10K, and $\Delta_{8/3}$=0.055K.

Comments on "Influence of voltage contacts on precision measurements of the quantized Hall resistance: an effect of externally injected current"

T.P. Chen and H.A. Chua (ibid., vol. 47, pp. 592-4, 1998) proposed a model to explain the influence of the voltage contact resistance R, on the value of the Hall resistance R/sub H/ in high-precision measurements. Their model is based on the observation that the offset current of the null detector of the cryogenic bridge generates a voltage in the contact resistance that induces a deviation of the Hall resistance from its ideal value. Using their model, they give a quantitative interpretation of a set of high-precision measurements made at the Swiss Federal Office of Metrology. The main argument of the paper is as follows: the offset current of the null detector I/sub offset/ will develop an error voltage V/sub d/=I/sub offset/R/sub c/, where R/sub c/ is the contact resistance. We think this argument is wrong. In addition to the contact resistance, the Hall resistance is present between the terminals of the voltmeter and, therefore, the error voltage is given by V/sub d/=I/sub offset/(R/sub c/+2R/sub H/). In this case, RH is multiplied by two because we are comparing two Hall resistances.

Analyzing the effects of capacitances-to-shield in sample probes on AC quantized Hall resistance measurements

We analyze the effects of the large capacitances-to-shields existing in all sample probes on measurements of the ac quantized Hall resistance RH. The object of this analysis is to investigate how these capacitances affect the observed frequency dependence of RH. Our goal is to see if there is some way to eliminate or minimize this significant frequency dependence, and thereby realize an intrinsic ac quantized Hall resistance standard. Equivalent electrical circuits are used in this analysis, with circuit components consisting of: capacitances and leakage resistances to the sample probe shields; inductances and resistances of the sample probe leads; quantized Hall resistances, longitudinal resistances, and voltage generators within the quantum Hall effect device; and multiple connections to the device. We derive exact algebraic equations for the measured RH values expressed in terms of the circuit components. Only two circuits (with single-series “offset” and quadruple-series connections) appear to meet our desired goals of measuring both RH and the longitudinal resistance Rx in the same cool-down for both ac and dc currents with a one-standard-deviation uncertainty of 10−8 RH or less. These two circuits will be further considered in a future paper in which the effects of wire-to-wire capacitances are also included in the analysis.

Influence of infrared illumination on the accuracy of the quantized hall resistance

The influence of infrared radiation on the accuracy of the quantized Hall resistance devices used in metrology is investigated. Samples exhibiting anomalous quantized Hall resistances due to large contact resistances resulting from a partially depleted electron gas can be brought back to a normal behavior by illumination with short pulses of infrared light. In addition, the critical current in the QHE regime can be considerably increased by the same illumination method.

A first attempt to realize (multiple-QHE devices)-series array resistance standards

Several chips composed of ten quantum Hall effect (QHE) devices placed in series by triple connections have been fabricated. The results obtained on one of these first arrays of quantized Hall resistances (QHR) open the possibility to realize novel QHR standards. The QHR on the i=2 plateau, 10/spl times/R/sub H/(i=2), was compared to R/sub H/(i=2) from a standard type QHE sample acting as a reference. The comparison results agreed to within parts in 10/sup 9/ for the measurements of a 1000 /spl Omega/ standard resistance. This work followed preliminary tests on the multiple connection techniques. In particular, the QHR on the i=2 plateau of a special QHE sample allowing the auto-series configuration with a double connection was found equal to 2/spl times/R/sub H/(i=2) within the total standard uncertainty of 4.4 parts in 10/sup 9/.

Fabrication of quantum Hall devices for low magnetic fields

The quantized Hall resistance of a two-dimensional electron gas (2DEG) in AlGaAs/GaAs heterostructures is used for the realization of the unit of resistance. This effect is more and more widely used. For possible application even in magnets with magnetic flux densities as low as 6 T, such devices have been produced with carrier concentrations in a range from 2.7/spl times/10/sup 15/ m/sup -2/ to 5.4/spl times/10/sup 15/ m/sup -2/ and their metrological quality has been checked. We find that even in low magnetic fields the quantized Hall resistance can be reproduced with highest accuracy.

A cheaper, simpler quantized Hall resistance standard

We have developed an accurate quantized Hall resistance (QHR) standard which is both cheaper and simpler to operate than most of those previously realized. In this paper, we outline the characteristics of the samples used, describe and discuss the design of the dewar and magnet system, and present field sweep data and accurate measurements carried out using a room temperature dc current comparator bridge.

Comparison of capacitance with AC quantized Hall resistance

A quadrature bridge operating at a frequency of 1.233 kHz has been developed to compare 10 nF capacitances to the quantized Hall resistance on the i=2 plateau in order to realize the farad in terms of R/sub K/. Portable precision ceramic capacitance transfer standards of remarkable performance have been developed for this purpose.

Measurement of the ac quantized Hall resistance

The flatness and frequency dependence of the i=2 and four quantized Hall resistance plateaus have been investigated. It was observed that for frequencies up to 6.4 kHz, a flat region exists at the center of the plateaus where the relative resistance is constant to /spl plusmn/2/spl times/10/sup -8/, and its relative value has a linear frequency dependence of (0.119/spl plusmn/0.001) (/spl mu//spl Omega///spl Omega/)/kHz and (0.107/spl plusmn/0.007) (/spl mu//spl Omega///spl Omega/)/kHz for i=2 and 4, respectively.

Improved AC quantized Hall measurements

AC quantized Hall resistance (QHR) measurements have been performed with an improved probe and sample mounting system. We describe investigations of the structure of the R/sub H/ plateau for a range of sample currents and temperatures, and for different wiring configurations. We have observed R/sub xx/ minima smaller than 0.02/spl times/10/sup -6/ of R/sub H/, an order of magnitude smaller than most published values.

Dependence of contact resistance on current for ohmic contacts to quantized Hall resistors

The dependence of contact resistance on current has been measured for a large number of ohmic contacts to quantized Hall resistors under quantum Hall effect conditions. Five different functional forms of current dependence are observed at low currents. The trend from best to worst quality can be correlated with the density of defects in the contact, regardless of the physical cause of the defects. The consequences of different types of contact resistance current dependence on the metrological use of samples are discussed.

Reflection symmetry and quantized Hall resistivity near the quantum Hall transition

We present a direct numerical evidence for reflection symmetry of longitudinal resistivity $\rho_{xx}$ and quantized Hall resistivity $\rho_{xy}$ near the transition between $\nu=1$ quantum Hall state and insulator, in accord with the recent experiments. Our results show that a universal scaling behavior of conductances, $\sigma_{xx}$ and $\sigma_{xy}$, in the transition regime decide the reflection symmetry of $\rho_{xx}$ and quantization of $\rho_{xy}$, independent of particle-hole symmetry. We also find that in insulating phase away from the transition region $\rho_{xy}$ deviates from the quantization and diverges with $\rho_{xx}$.

Collapse of quantized Hall resistance at high Hall electric fields

Collapse of quantized Hall resistance (QHR) – steep deviation of Hall resistance from the quantized value – at high Hall electric fields has been observed in butterfly-type Hall bars made from GaAs/AlGaAs heterostructures. The collapse is not caused by breakdown of quantum Hall effect (QHE) – abrupt increase in dissipation in the QHE state. A phenomeno-logical model for the collapse of the QHR and the breakdown of the QHE is discussed.

Calculating the Effects of Longitudinal Resistance in Multi-Series-Connected Quantum Hall Effect Devices.

Many ac quantized Hall resistance experiments have measured significant values of ac longitudinal resistances under temperature and magnetic field conditions in which the dc longitudinal resistance values were negligible. We investigate the effect of non-vanishing ac longitudinal resistances on measurements of the quantized Hall resistances by analyzing equivalent circuits of quantized Hall effect resistors. These circuits are based on ones reported previously for dc quantized Hall resistors, but use additional resistors to represent longitudinal resistances. For simplification, no capacitances or inductances are included in the circuits. The analysis is performed for many combinations of multi-series connections to quantum Hall effect devices. The exact algebraic solutions for the quantized Hall resistances under these conditions of finite ac longitudinal resistances provide corrections to the measured quantized Hall resistances, but these corrections do not account for the frequency dependences of the ac quantized Hall resistances reported in the literature.

A Problem in AC Quantized Hall Resistance Measurements and a Proposed Solution.

In all experiments reported to date the measured values of the ac quantized Hall resistances RH varied with the frequency of the applied current, and differed significantly from the dc values of RH, making it difficult to use the ac quantum Hall effect as an absolute impedance standard. We analyze the effects due to the large capacitances-to-shields existing in the sample probes on measurements of RH to see if this is the source of the problem. Equivalent electrical circuits are utilized; they contain capacitances and leakage resistances to the sample probe shields, longitudinal resistances within the quantized Hall effect devices, and multiple connections to the devices. The algebraic solutions for the RH values in these circuits reveal large out-of-phase contributions to the quantized Hall voltages VH that would make it difficult to do accurate measurements with high precision ac bridges. These large out-of-phase contributions could introduce the linear frequency dependences observed in previous RH measurements. We predict, however, that quadruple-series connections to the quantum Hall devices yield only small out-of-phase contributions to VH which may allow accurate measurements of the quantity RH − Rx, where Rx is the longitudinal resistance along the device.

Electrical resistance standards and the quantum Hall effect

This review of electrical resistance standards begins with a description of classical standard resistors and their limitations. Methods of comparing resistance are described; these include bridges based on cryogenic current comparators capable of achieving statistical uncertainties approaching one part in 1010 in the measurement of resistance ratios. Such reproducibility is nearly two orders of magnitude smaller than the overall uncertainty of the most accurate determinations of the ohm from its SI definition via the calculable capacitor. The quantum Hall effect can provide an invariable reference standard of resistance linked to the fundamental physical constants. Many factors, however, limit the accuracy of practical realizations of quantized Hall resistance standards. Ultimately, the accuracy of a specific realization must be confirmed by comparison with similar standards; methods for doing this and the resulting agreement are presented. The ac techniques used in the determinations of the SI ohm by means of the calculable capacitor are now being applied to accurately link the quantized Hall resistance to the impedance of standard capacitors and thereby to provide a new reference standard of capacitance.

Collapse of Quantized Hall Resistance and Breakdown of IQHE in GaAs/AlGaAs Heterostructures

Collapse of quantized Hall resistance (QHR), not resulting from breakdown of the dissipationless current flow, is reported in the integer quantum Hall effect (IQHE) state with i =4 of GaAs/Al 0.3 Ga 0.7 As butterfly-type Hall bars at high Hall electric fields. The Hall electric field F H at the collapse of QHR is about 20% lower than the F H for the breakdown of the dissipationless current flow. The nature of difference between the effect of the F H on the collapse of QHR and the effect of the F H on the breakdown is discussed.