Direct comparison of fractional and integer quantized Hall resistance

The bridge setup comprised two cryo-magnets hosting the iQH and the fQH resistances to be compared. The iQHR cryostat was a standard LHe bath cryostat with a superconducting magnet operated at 10 T (the center of the filling factor 2 plateau of the iQHE device), the temperature of which was held at 2.2 K by a λ-cooler. The fQHR cryostat was a top-loading dilution refrigerator equipped with a solenoid capable of 18 T at 4.2 K. A bath temperature of 40 mK was typically used, but depending on current level the electron temperature of the fQHE device was higher. From previously measured temperature dependences of Shubnikov–de Haas oscillations of similar samples it was estimated that a current of 1 µA would cause an electron temperature of around 100 mK under these conditions.

The resistance comparison was performed with a CCC bridge [22] which featured, at the core of its feedback loop, a DC SQUID to detect the flux balance of the coils N₁ and N₂, with oppositely flowing currents, all operated in a helium dewar at 4.2 K. The nanovolt detector used in the bridge is described in [23]. A schematic of the cabling of the measurement is shown in figure 2. As it differs from comparing two standard resistors, or a standard resistor with the iQHR, the use of an auxiliary winding for compensating the deviation from a perfect integer ratio is not needed here. With the choice of a number-of-turns ratio N₁/N₂ equal to the 6:1 ratio of fQHR and iQHR, the relative deviation from this ratio is then simply obtained as $\Delta U/\Delta \left({{I}_{i}}{{R}_{i}} \right)$. Here $\Delta U$ is the average bridge voltage difference during synchronous reversals of the currents ${{I}_{1}}$ and ${{I}_{2}}$ through resistor ${{R}_{1}}={{R}^{[1/3]}}$ (fQHR) and resistor ${{R}_{2}}={{R}^{[2]}}$ (iQHR), and $\Delta \left({{I}_{i}}{{R}_{i}} \right)\,(i=1,2)$ represents the voltage drop across each of those. The influence of thermal voltages and their drifts is practically eliminated by the current reversals, which had typical reversal periods of tens of seconds, corresponding to an effective measurement frequency of the order of tens of millihertz. Transient artefacts due to current reversals were eliminated by discarding the first half of the data points of each reversal half-cycle. The influence of a possible leakage resistance (most harmful when in parallel to the high-resistance fQHR arm of the bridge) was reduced by using shielded and guarded cabling. Nevertheless, it was considered as a type-B uncertainty, assuming a worst-case leakage resistance of 50 TΩ. This value derives from the previously determined 10¹⁴ Ω isolation resistance of our standard QHE setup, downscaled to take the longer cable path to the fQHE cryostat into account. The corresponding uncertainty u_L is given in line 5 of table 1.

Table 1. Measurement currents I, measurement durations t for one voltage set $\{a,b,c,d\}$, maximum flux levels seen by the SQUID, estimated type-B uncertainties ${{u}_{B}}$ for the given flux level, and uncertainty ${{u}_{L}}$ due to leakage resistance (lines 1–5). Actual bridge voltage readings a to d and the consistency check value a ‒ c  +  d ‒ b, with type-A uncertainties (67% confidence level) obtained from the statistical distribution of the raw data values, are in lines 6–10. The last two lines list longitudinal resistances ${{R}_{xx}}=(b-c)/2I$ and resistance deviations $\delta {{R}_{xy}}=(a+d)/2I$, both in milliohms. Their uncertainties include ${{u}_{B}}$ and ${{u}_{L}}$ from lines 4 and 5 and were calculated as $\sqrt{{{u}_{A}}^{2}+{{u}_{B}}^{2}/3+{{u}_{L}}^{2}/3}$, with ${{u}_{A}}$ calculated from the $\{a,b,c,d\}$ uncertainties.

I/µA	±0.0820	±0.1615	±0.3260	±0.6520	±0.9735	±1.3025
t/min	352	352	88	88	88	88
Flux/Φ0	±29.5	±58.2	±117	±235	±351	±469
${{u}_{B}}$/mΩ	2.63	1.33	0.67	0.33	0.22	0.17
${{u}_{L}}$/mΩ	0.12	0.12	0.12	0.12	0.12	0.12
a/nV	−0.70  ±  0.35	−0.65  ±  0.27	−4.21  ±  0.50	−21.09  ±  0.69	−53.97  ±  0.73	−151.54  ±  0.54
b/nV	−0.66  ±  0.38	−0.61  ±  0.31	−2.16  ±  0.53	−7.88  ±  0.83	7.60  ±  0.59	23.09  ±  0.41
c/nV	−0.27  ±  0.22	−0.85  ±  0.28	−4.75  ±  0.45	−26.38  ±  0.81	−97.54  ±  1.04	−266.41  ±  0.45
d/nV	−0.48  ±  0.37	−0.74  ±  0.25	−2.50  ±  0.55	−12.87  ±  0.65	−34.06  ±  0.76	−93.01  ±  0.56
a ‒ c  +  d ‒ b/nV	−0.25  ±  0.67	0.07  ±  0.56	0.19  ±  1.01	0.30  ±  1.50	1.90  ±  1.59	−1.23  ±  0.99
${{\boldsymbol{R}}_{\boldsymbol{xx}}}$/mΩ	−1.19  ±  2.03	0.37  ±  1.01	1.99  ±  0.66	7.09  ±  0.48	27.00  ±  0.34	55.57  ±  0.17
$\boldsymbol{\delta }{{\boldsymbol{R}}_{\boldsymbol{xy}}}$/mΩ	−3.59  ±  2.17	−2.16  ±  0.96	−5.15  ±  0.69	−13.02  ±  0.41	−22.61  ±  0.31	−46.94  ±  0.20

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Direct comparisons have been performed against the middle Hall contact pair of the iQHR, exposed to a field of 10 T, but varying the Hall contact pairs for the fQHR in order to determine longitudinal and Hall resistances. Other variations of experimental parameters include the magnetic field for the fQHR, the settings of the current reversal cycles, and the current bias level (see figure 3) which was varied from 82 nA to 1.3 µA.

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The necessity for low current levels derives from the following consideration. The quasi-particle gap of composite fermions, as determined experimentally in [24, 25], is at filling factor 1/3 approximately 0.7 meV (see figure 3 in [24], obtained with a sample of very similar carrier density to ours), which is 25 times smaller than the Landau gap of GaAs iQHE devices at 10 T. In addition to limiting the typical iQHR operating temperature of 1.4 K to below 50 mK for fQHE devices, this also sets a limit for the Hall electric field which causes a breakdown of quantization when it becomes too large. The Hall field is proportional to current multiplied by resistance, and therefore, due to the resistence being six times higher, an additional 6-fold decrease of current is required.

While at the typical 40 µA currents used with iQHE devices type-A dominated uncertainties of few parts in 10⁹ or lower are routinely achieved with CCC bridges, the sub-microampere current level of the fQHE measurements will not allow such low type-A uncertainties. Also, note that the absolute number of superconducting magnetic flux quanta Φ₀ seen by the SQUID flux balance detector of the CCC bridge becomes as small as  ±30 at the lowest current. In consequence, the 1/f SQUID noise begins to dominate other noise sources. We reduced this effect by employing a new two-stage SQUID with improved noise figure, as described in detail in [26].

Further, in this low-current regime, a usually negligible type-B uncertainty contribution cannot be ignored. It stems from the fact that at low currents and concomitant low flux levels, down-conversion and rectification of high-frequency noise at the non-linear SQUID characteristic becomes more and more significant and can systematically falsify the reading of the bridge. This effect, described in detail in [27], would require very long averaging times to quantify it precisely, with no guarantee that at the actual resistance measurement the same conditions prevail. For our results presented in the next section we estimated as an upper limit for its influence a flux error of  ±1 µΦ₀. This is treated as the limit of a rectangular distribution and is represented as a type-B uncertainty in line 4 of table 1.

The influence is strongest at the lowest current level, yielding an absolute uncertainty $3{{R}_{K}}\left(1\mu {{\Phi }_{0}}/29.5{{\Phi }_{0}} \right)$ of 2.63 milliohms. This additional uncertainty increases the total uncertainty by 40% at this current (and less at higher currents), which does not severely limit the significance of our results. However, should resistance comparisons at even lower currents be attempted, e.g. when testing the precision of quantization of the recently demonstrated quantized anomalous Hall effect of ferromagnetic 3D topological insulators [28–30], this contribution must be considered.

Because the bridge can be balanced only when it measures a Hall voltage, the longitudinal voltages were obtained as differences of Hall voltages, as is recommended practice in precision resistance measurements (see section 6.2 in [21]). For each single set of measurements at a given magnetic field and current, four Hall voltages $a,b,c,d$ were determined. They are indicated by the 'bow-tie' arrow pattern in figure 4. From these voltages and the known current, longitudinal resistances and differences of resistances to the reference resistance can be calculated, as described in [21] and below, provided the consistency condition a ‒ c  +  d ‒ b  =  0 is fulfilled. However, we did not follow the recommendation in [21] to average all four voltages to obtain a Hall voltage with lower type-A uncertainty. We instead restricted ourselves to calculating $\delta {{R}_{xy}}:={{R}^{[1/3]}}-6{{R}^{[2]}}~~$ as $(a+d)/2I$ and ${{R}_{xx}}$ as $(b-c)/2I$ (referring hereinafter to the fQHE current as I instead of I₁ as in figure 2). In this way, correlations between the $\delta {{R}_{xy}}$ and ${{R}_{xx}}$ data are avoided and statistical subtleties in the subsequent analysis of the $\delta {{R}_{xy}}\left({{R}_{xx}} \right)$ dependence need not be considered.

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Note that the low-resistance but comparatively large Sn-ball contacts cause a small longitudinal contribution to the measured Hall resistance, even for geometrically exactly opposing contacts [31], leading to an apparent linear contribution of ${{R}_{xx}}$ to $\delta {{R}_{xy}}$. Estimated from the contact and Hall bar widths, this alone would contribute approximately 8% of ${{R}_{xx}}$ to the measured $\delta {{R}_{xy}}$ values, as symbolized by the slightly tilted arrows $a$ and $d$.

It is known [32–35] that in iQHE devices thermally activated transport also contributes to the linear correlation between $\delta {{R}_{xy}}$ and ${{R}_{xx}}$, and often dominates it. Such effects are likely to occur in an fQHE device already at the sub-µA current levels used here, due to the smaller energy gap and the fragility of the fractional state. Therefore, one must rely on an extrapolation of the $\delta {{R}_{xy}}\left({{R}_{xx}} \right)$ dependence to zero ${{R}_{xx}}$ for the determination of the 'true' $\delta {{R}_{xy}}$ value. The demonstration that the extrapolated ${{R}_{xx}}\left(I \right)$ becomes zero for zero current is of course a prerequisite for this.

As a preparation step for the comparison we determined the centre of the plateau at filling factor 1/3 from a series of measurements with 1 µA current at six different magnetic fields between 16.1 and 16.4 T. From the six voltage data sets $\{a,b,c,d\}$ longitudinal resistance values were obtained as described above, and the minimum of a parabola fitted to those data determined the magnetic field position of 16.24 T, where the actual comparison was performed. Next, six more data sets $\left\{a,b,c,d \right\}$ were measured at this field with six different current levels ranging from 0.08 to 1.3 µA. The measurement time for one data set was 88 min, except for the two lowest currents, where it was four times longer, nearly 6 h per set.

The results are summarized in table 1, which lists current values and measurement durations in lines 1 and 2. Line 3 lists the flux seen by the SQUID, for the current level and the number of windings given in the caption of figure 2. In lines 4 and 5 the type-B uncertainties described in sections 3.3 and 3.2 are listed. The actual bridge voltage readings a to d and their type-A uncertainties, as obtained from the current reversal cycles, are listed in lines 6–9, with the consistency check value a ‒ c  +  d ‒ b in line 10.

Finally, the last two lines give the longitudinal resistance ${{R}_{xx}}$ and the resistance difference $\delta {{R}_{xy}}$. The uncertainties in lines 6–10 result from the type-A uncertainties of the $\left\{a,b,c,d \right\}$ values only, whereas for the bold values the type-B uncertainties from lines 4 and 5 have been included, assuming a rectangular probability distribution for the type-B components.

Since the longitudinal resistance is zero only at the lowest current levels, where the relative measurement uncertainty is rather high, we use an extrapolation of ${{R}_{xx}}$ and $\delta {{R}_{xy}}$ to zero current to obtain more reliable values for these quantities. The graphical representations of the data in bold in figures 5 and 6 serve to illustrate the extrapolation procedure. For such an extrapolation, it is more important to describe the data by a smooth functional form with the fewest possible fit parameters than to find a representation backed by a physical transport model. We used two methods for the extrapolation. In the first method, the current dependences ${{R}_{xx}}(I)$ and $\delta {{R}_{xy}}(I)$ are each described by a simple power law, and the value of this function at I  =  0 is taken as an estimate of the extrapolated value. This is shown in figure 5. For the second method, the data are plotted as $\delta {{R}_{xy}}({{R}_{xx}})$, and in a similar way an extrapolated value $\delta {{R}_{xy}}({{R}_{xx}}\to 0)$ is determined. This is shown in figure 6.

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There is no theory predicting how $\delta {{R}_{xy}}$ and ${{R}_{xx}}$ should depend on current. Empirically, a simple power law as we use here has been observed before, e.g. in¹ in the case of a GaAs iQHE device. If only thermal effects were responsible for the inter-edge-channel scattering and the concomitant rise of ${{R}_{xx}}$, it would be tempting to assume an ${{I}^{2}}$ behaviour for extrapolating to zero current³. However, an attempt to fit an ${{I}^{2}}$ law to our data fails, as the solid lines in figure 5 show. (Also the data in², when closely analyzed, follow an ${{I}^{3}}$ rather than an ${{I}^{2}}$ law). However, since the Hall electric field, which can also induce scattering, is proportional to current, exponents larger than 2 are quite reasonable. Consequently, we chose to perform a weighted least squares regression analysis of our data with functions of the more general form ${{\alpha }_{j}}+{{\beta }_{j}}{{I}^{{{\gamma }_{j}}}}$, with $j=x,y$ for ${{R}_{xx}}(I)$ and $\delta {{R}_{xy}}(I)$, respectively.

When analyzing the ${{R}_{xx}}$ and $\delta {{R}_{xy}}$ data sets, the resulting ${{\gamma }_{j}}$ exponents were found to be identical within their standard error. Therefore, an additional regression was performed with $\gamma$ restricted to being identical for both data sets (reducing the number of fit parameters from 6 to 5 as a side effect). The resulting curves and their 67% confidence bands are shown in figure 5 as dashed lines and shaded areas, respectively.

The extrapolated value ${{\alpha }_{x}}$ for ${{R}_{xx}}\left(I\to 0 \right)$, together with its 67% and 95% confidence limits, is given in line 2 of table 2. It confirms that at vanishing current the fQHR device is well quantized at filling factor 1/3. The extrapolated value for $\delta {{R}_{xy}}\left({{R}_{xx}}\left(I \right)\to 0 \right)$ and its confidence limits are given in line 3. They were obtained by using the fact that, since ${{R}_{xx}}$ and $\delta {{R}_{xy}}$ are linear in ${{I}^{\gamma }}$, $\delta {{R}_{xy}}\left(I \right)$ can be written as ${{\alpha }_{y}}+{{\beta }_{y}}\left({{R}_{xx}}\left(I \right)-{{\alpha }_{x}} \right)/{{\beta }_{x}}$. This gives, in the limit ${{R}_{xx}}\left(I \right)\to 0$, for $\delta {{R}_{xy}}$ the estimate ${{\alpha }_{y}}-{{\beta }_{y}}{{\alpha }_{x}}/{{\beta }_{x}}$ for $\delta {{R}_{xy}}\left({{R}_{xx}}\left(I \right)\to 0 \right)$. The confidence limits of this aggregate value were calculated from the confidence limits of the individual ${{\alpha }_{j}},{{\beta }_{j}}$ fit results.

Table 2. Extrapolated values of ${{R}_{xx}}$ and $\delta {{R}_{xy}}$ for the limit of vanishing current, assuming power law current dependencies $\alpha +\beta {{I}^{\gamma }}$, are given in lines 2 and 3. The extrapolated value of $\delta {{R}_{xy}}$ for the limit of vanishing ${{R}_{xx}}$, obtained by directly analysing the data set $\delta {{R}_{xy}}\left({{R}_{xx}} \right)$, is given in line 4.

	Value	67% confidence	95% confidence
${{R}_{xx}}\left(I\to 0 \right)$	−0.2 mΩ	±1.7 mΩ	±3.9 mΩ
$\delta {{R}_{xy}}\left({{R}_{xx}}\left(I \right)\to 0 \right)$	−4.2 mΩ	±2.0 mΩ	±4.5 mΩ
$\delta {{R}_{xy}}\left({{R}_{xx}}\to 0 \right)$	−4.1 mΩ	±1.9 mΩ	±4.9 mΩ

In the second method of extrapolation, a regression analysis is performed directly on the $\delta {{R}_{xy}}({{R}_{xx}})$ data shown in figure 6, taking errors in both the x- and y-axes into account. The data show a linear trend, as is quite commonly observed in iQHR measurements [21], and as is compatible with the result from method 1. We used York's method [36] for the weighted linear least squares fit. Regarding the confidence estimates, York's algorithm evaluates them at the least squares-adjusted points rather than at the observed points, thereby producing the same estimates as a maximum likelihood approach. The resulting regression curve and its 67% confidence band are shown in the figure, and the obtained values for the axis intercept and its 67% and 95% confidence limits are given in line 4 of table 2.

Converting the numbers with the 95% uncertainty estimate from method 2 from milliohms to relative values, we get as the result of our analysis:

$$\begin{align} \displaystyle {{R}^{\left[ 1/3 \right]}}/6{{R}^{[2]}}=1-\left(5.3~\pm 6.3 \right)\cdot {{10}^{-8}}.\nonumber \end{align}$$

The universality test presented in this paper can be formally qualified as 'passed', since the expected value of zero is just covered by the 95% confidence limit.

However, we point out that in both methods 1 and 2 the goodness-of-fit parameter $\chi _{{\rm reduced}}^{2}$ in the numerical regressions was around 9, and not of order unity, as would be expected for correctly chosen models and realistic uncertainty estimates of the measured data³ [36]. The scatter of our data is obviously larger than the assigned uncertainties, indicating that it is our uncertainty estimate which is not realistic, although all relevant components have been included to the best of our knowledge.

Algorithms performing weighted linear least squares fits of data with errors take care of this by scaling the confidence limits by $\sqrt{\chi _{{\rm reduced}}^{2}}$ (which is equivalent to scale the uncertainties of the individual data points by the same factor). Known as the Birge ratio method (see e.g. [37]), this is basically an uncertainty estimation based on the experimentally found deviation of the data from the linear adjustment. We believe that the application of this method does not invalidate the result of our regression analysis, but in similar future studies advanced regression methods should be applied, when appropriate tools become more widely available. Such methods will likely be based on Bayesian inference, e.g. as described in [38] for the case of linear regression of data with negligible uncertainties in $x$.