A study of quantum Hall devices with different working magnetic fields for primary resistance metrology

Two kinds of quantum Hall devices with different working magnetic fields are fabricated and compared with the LEP (Laboratoires d’Electronique Philips) device, which is currently used in the primary resistance standard system of the National Institute of Metrology (NIM) of China. The comparison is made by calibrating the same 1 Ω standard resistor made by the National Measurement Laboratory of Australia using a Tinsley 100 Ω resistor as the intermediate standard and a cryogenic current comparator. The calibrated values from the NIM device with high working magnetic field and those from the LEP device agreed to within  −0.69  ×  10−9 with an uncertainty of 4.9  ×  10−9. The values from the NIM device with low working magnetic field and those from BIPM-2 agreed to within 2.5  ×  10−9 with an uncertainty of 7.1  ×  10−9.


Introduction
The quantum Hall effect (QHE) provides a metrological resistance standard in terms of the Planck constant h and elementary charge e [1]. The QHE relies on a 2D electron gas (2DEG), which can be realized in several material systems such as Si-MOSFETs, GaAs/Al x Ga 1−x As heterostructures or graphene. The GaAs/Al x Ga 1−x As-based device is the one most used for resistance standard applications [2]. Practical metrological measurements are performed at the i = 2 plateau, the quantized resistance at which a filling factor of 2 equals half of R K-90 , which is 12 906.4035 Ω [3][4][5][6][7][8][9][10].
In the National Institute of Metrology (NIM) of China, the currently used quantum Hall devices for the resistance standard system (known as BIPM-2) are made by Laboratoires d'Electronique Philips (LEP) and supplied by BIPM (the International Bureau of Weights and Measures) [11,12]. NIM has recently started to develop its own devices for the quantum Hall resistance standard [13][14][15]. In this paper we report the development of two devices for the quantum Hall standard: one has a high working magnetic field above 10 T (NIM-1), and the other has a low working magnetic field below 7 T (NIM-2). Devices operating at a low magnetic field could be used as an economic metrological resistance system, having a small magnet which uses less liquid helium [16]. NIM-1, NIM-2 and BIPM-2 are compared by calibrating the same NML 1 Ω standard resistor using a Tinsley 100 Ω resistor as the intermediate standard and a cryogenic current comparator (CCC) as the method of measurement.
Two kinds of quantum Hall devices with different working magnetic fields are fabricated and compared with the LEP (Laboratoires d'Electronique Philips) device, which is currently used in the primary resistance standard system of the National Institute of Metrology (NIM) of China. The comparison is made by calibrating the same 1 Ω standard resistor made by the National Measurement Laboratory of Australia using a Tinsley 100 Ω resistor as the intermediate standard and a cryogenic current comparator. The calibrated values from the NIM device with high working magnetic field and those from the LEP device agreed to within −0.69 × 10 −9 with an uncertainty of 4.9 × 10 −9 . The values from the NIM device with low working magnetic field and those from BIPM-2 agreed to within 2.5 × 10 −9 with an uncertainty of 7.1 × 10 −9 .
Keywords: quantum Hall effect, resistance standards, heterojunction, cryogenic current comparator (Some figures may appear in colour only in the online journal) Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. of the NIM-1 and NIM-2 wafers are given in table 1. The heterojunction was modulation doped. The central magnetic flux density B 2c for the i = 2 plateau, which is proportional to the carrier concentration of the 2DEG, was tuned by changing the thickness of the spacer layer. Figure 1 shows the structure of the quantum Hall resistance (QHR) device. The 2400 µm × 400 µm Hall bar mesa (pink layer) was defined by a standard photolithography process followed by wet chemical etching using a solution of H 3 PO 4 , H 2 O 2 and deionized water with a volume ratio of 2:8:90. The Au(1072 nm)/Ge(528 nm)/Ni(400 nm) contacts (blue layer) were formed by sequential e-beam deposition and a lift-off process, and then Ohmic contacts were achieved by rapid thermal annealing at 370 °C for 120 s and 430 °C for 50 s in a N 2 /H 2 gas atmosphere. The portion of the contact pads outside the Hall bar was used for Au wire bonding to ensure that the heterojunction was not affected during the wire bonding process. No SiN x protection layer was grown on top of the QH mesa since it would introduce stress on the 2DEG layer and possibly result in changes in the electrical properties. The fabricated devices were mounted on home-made non-magnetic TO-8 ceramic holders. The inset of figure 2 shows a photograph of a prepared QHR device.

Measurement
The BIPM-2, NIM-1 and NIM-2 devices were then characterized in the NIM QH resistance standard system which consists of an Oxford magnet (16 T), a high-accuracy current source, an EM N11 voltage meter and the CCC. The devices were slowly cooled to 1.5 K. Three important device parameters, namely central magnetic flux density B 2c , residual longitudinal resistance R xx at B 2c , and contact resistance R c at the i = 2 plateau, were measured.
The longitudinal resistance R xx is equal to V xx /I 15 , where V xx can be V 24 or V 86 obtained from the EM N11 voltage meter, and I 15 is the current flowing through the device. The central magnetic flux density B 2c at the i = 2 plateau is determined by sweeping the magnetic flux density B and measuring the longitudinal resistance R xx ; the typical bias current I 15 used in this process is 10 µA. Once B 2c is determined, we set the magnetic field to B 2c and measure the residual longitudinal resistance at B 2c . In this process I 15 is set to 38 µA. The contact resistance R c is measured using a three-terminal technique. The width of the i = 2 plateau ΔB 2 , mobility µ and carrier concentration n of the devices were calculated from the R xx -B curves. The characterized parameters of the three devices are listed in table 2. Figure 2 shows the R xx of the devices BIPM-2, NIM-1 and NIM-2 as a function of B at 1.5 K with a 10 µA bias current. The B 2c of NIM-1, which is 10.24 T, is close to that of BIPM-2 due to the similar carrier concentration. The magnetic field width at the i = 2 plateau ΔB 2 is about 2 T for BIPM-2 and NIM-1. The mobility of NIM-1 is lower than that of BIPM-2. NIM-1 is designed to replace BIPM-2 as the standard device for the NIM QHR standard system. NIM-2 is designed specifically for a low magnetic field resistance standard system. The B 2c of NIM-2 is only 6.88 T and the corresponding carrier concentration is as low as 3.32 × 10 15 m −2 . ΔB 2 is only about 0.89 T for NIM-2 because of the closer Landau levels due to the low carrier concentration; the mobility is 33.2 T −1 .   For all three devices, the longitudinal resistance R xx at B 2c is less than 0.63 mΩ (V xx < 0.02 µV) which is close to the measurement limit of the EM N11 voltage meter, as affected by the lab conditions. R c is measured using a three-wire method at B 2c , so both the wire resistance and R xx are included. The R c values of the three devices are all <2 Ω, which is small enough for metrological measurements [17,18]. R xx and R c show that the NIM devices have been completely quantized and can be used as the standard devices in the QHR standard system.
We have compared the devices BIPM-2, NIM-1 and NIM-2 by undertaking a calibration measurement of an NML 1 Ω standard resistor. Figure 3 shows the calibration procedure. First we used these three QH devices to calibrate a Tinsley 100 Ω transfer resistor using a CCC with a winding ratio of 4001:31 and a 0.5 V bridge voltage. The QH voltage V xy was achieved from voltage pads 3 and 7, and I 15 = 38 µA. Then we used the calibrated Tinsley 100 Ω resistor to calibrate the NML 1 Ω standard resistor using the CCC with a winding ratio of 400:4 and a 0.05 V bridge voltage. The calibration values of the NML 1 Ω standard resistor by BIPM-2, NIM-1 and NIM-2 are labeled as R BIPM and R NIM (R NIM-1 , R NIM-2 ). Both the Tinsley 100 Ω and 1 Ω resistors were kept in a 293 K oil bath.

Conclusions
Single quantum Hall devices with high (above 10 T, NIM-1) and low (below 7 T, NIM-2) working magnetic flux density were fabricated and characterized. The two devices were compared with the LEP-made device (BIPM-2) by calibrating the same NML 1 Ω standard resistor. The relative differences between the two calibrated values given by NIM-1 and BIPM-2, and NIM-2 and BIPM-2, are −0.69 parts in 10 9 and 2.5 parts in 10 9 respectively. The calibration results show that NIM-made devices can be used as standard devices for the quantum Hall resistance standard system.