**In 1879, Edwin Hall discovered the Hall effect and it has since been widely applied in measurements, particularly in materials characterization and sensing. The discovery of the new phenomena has been enabled only through the characterization of new materials. These discoveries include the quantum Hall effect; spin Hall effect, and topological insulators [1]. The Hall effect can also operate as a platform for several device applications. For example, it can be used in the automotive industry to sense current.**

Figure 1 demonstrates how the Hall effect becomes relevant when a conductor is placed in a magnetic field B. The Lorenz force F_{L} deflects the charge carriers flowing through the conductor resulting in a Hall voltage V_{xy} which runs at a 90-degree angle to the magnetic field and the I_{R} current.

*Figure 1.* *Schematic showing the Hall effect measurement setup in a standard Hall bar geometry. Two lock-in amplifiers are used to measure the voltages V*_{xx} *and V*_{xy} *along and perpendicular to the current flow. The conventional current I*_{R} *flow direction is used in the diagram, so that the motion of electrons is in the opposite direction. The Lorenz force F*_{L} is perpendicular to the magnetic field B and the charge carrier motion.

The longitudinal voltage V_{xx} is independent of the magnetic field but the Hall voltage and the magnetic field have a linearly proportional relationship. Figure 2(a) shows the typical signatures of the Hall effect for metallic conductors and semiconductors.

*Figure 2.* *Illustration of longitudinal and transverse resistivities **ρ*_{xx} *and **ρ*_{xy} *plotted as a function of the magnetic field. (**a**) Classical Hall effect behavior, where **ρ*_{xy} *is co-linear with B, and **ρ*_{xx} *is independent of B. (**b**) Typical signatures of the integer quantum Hall effect. The Hall resistivity **ρ*_{xy} *shows plateaus for a range of magnetic field values, with **ρ*_{xx} *going to zero at the same time.*

The resistivity *ρ *of a bulk conductor can be defined as *ρ *= *R w t/l* where *R=V/I, *the width is *w, *the thickness is *t* and the length of the conductor is *l. *Denoting the density of charge carriers, *n, *the carrier effective mass,* m,* the electron charge, *e,* and the mean free scattering time, *τ,* then the resistivities *ρ*_{xx }and *ρ*_{xy} can be derived using the classical Drude model [2]:

Equation 1(a):

Equation 1(b):

The transverse Hall resistivity with a linear magnetic field and inverse carrier density dependence is given in Equation 1(a) and the longitudinal resistivity in Equation 1(b). There are two main applications for the classical Hall effect:

## Materials Characterization

Materials conducting properties can be determined with Hall effect measurements. The linear magnetic field dependence means that the carrier type and the carrier concentration *n* are determined with Equation 1(a). Conductivity *σ*_{xx}=1/*ρ*_{xx}, carrier mobility *µ*=* σ*_{xx}/*n e* and longitudinal resistivity measurements can be calculated with Equations 1(b). The current *I*_{R} and the voltages *V*_{xx} and *V*_{xy} can be directly extracted with the Hall bar geometry as shown in Figure 1.

## Sensing

Once the conducting properties of a material have been identified, the Hall voltage, with its linear proportionality to the perpendicular magnetic field component, can be employed to build sensitive and accurate magnetic field sensors. In this case, the relationship between the measured voltage and the magnetic field is proportional and can be determined with Equation 1(a).

Industrial applications frequently make use of Hall sensors as the linearity spans from zero to fields of numerous Tesla (T). Such applications include Hall probes, speedometers, Hall switches, and current sensors. It is also important to measure speed with many of these applications but background noise often results in the signals being on top of another big signal, making it difficult to take electronic measurements.

Systematic measurement mistakes such as thermal offsets, thermal drift, and undesired components in the background noise spectrum can occur but AC measurement techniques can help to avoid these.

An elevated signal-to-noise ratio (SNR) is reached with AC techniques as background noise falls 1/*f *with increasing frequency *f. *Quicker measurements are also possible with higher SNR and AC techniques enable a larger dynamic range as they usually come with elevated measurement resolution.

## Lock-In Amplifiers for Accurate and Quick Measurements

Even with elevated background noise lock-in amplifiers can accurately measure down to a few nV with modifiable bandwidth, making them a great tool with which to perform AC measurements. Phase-sensitive detection is used for measuring the amplitude of a signal against a reference frequency. With the exception of background noise at the reference frequency and outside the measurement bandwidth, background noises are omitted and do not interfere with the measurement.

A classic setup is made up of the Hall bar geometry and two Zurich Instruments MFLIs, 500 kHz lock-in amplifiers as shown in Figure 1. Both lock-in amplifiers are operated to measure the transverse Hall voltage *V*_{xy} and longitudinal voltage *V*_{xx} and one of these is used to supply the current over a current-limiting resistor R_{L}.

An assumption of constant current is made, as the selected RL is typically a lot larger than any of the combined resistances in the circuit. Frequencies of up to a few tens of Hz are used to characterize materials and measurement frequency and data sampling are coordinated through the two lock-in amplifiers.

While constant current can be assumed in the majority of cases, in experimental situations where the sample impedance drastically changes over the course of measurements, the current will require careful monitoring. MFLI with its current sensing input is able to carry out such monitoring by recording the current at the same frequency as the voltage input. Reference [3] provides more details on lock-in amplifiers and how they function.

## Quantum Hall Effects in 2DEGs

The Hall effect demonstrates new features when the electrons of a two-dimensional electron gas (2DEG) are put in a magnetic field. Known as the quantum Hall Effect (QHE), an increase in the magnetic field leads the Hall resistivity to alter in steps with the formation of plateau structures while the longitudinal resistivity drops to zero.

Landau levels are developed within the density of states of a 2DEG, as shown in Figure 2(b). The electrons on a plateau follow distinct circular orbits with quantized energy levels. The edges of a sample are then tracked moving macroscopic distances with no resistance. The electrons carry the conduction and the dissipation-less nature point to the topologically protected edge states.

Shubnikov-de Haas (SdH) oscillations, shown in Figure 3, can occur when electrons move along Landau levels, causing visible “quantum oscillations”. QHE was first discovered in 1980 by Klaus von Klitzing, earning him the Nobel prize in 1985. [4].

*Figure 3.* *(**a***)** Plot of longitudinal and transverse resistivity *ρ*_{xx} *and **ρ*_{xy} *as a function of the magnetic field B, using two lock-in amplifiers. Note that **ρ*_{xy} *changes sign when the field direction is inverted. ***(***b**)** A zoom into the measurement data from ***(***a**)** shows several higher order Hall plateaus at negative fields with the prominent signatures of spin splitting between them.*

The Hall resistivity for a specific plateau can be written as:

Equation 2:

where *v *is the filling factor, and h/e^{2} is the von Klitzing constant, which is equivalent to R_{K}=25813.807557(18)Ω as *ρ *= *R w t/l. *The R_{K }is dependent on the Planck constant h and electron charge e, but not on the material properties, temperature fluctuations, or crystal impurities.

The fractional Hall effect and spin Hall effect are further forms of Hall effect phenomena that have been found in addition to the QHE in a 2DEG. Research experiments have also demonstrated the possibility of a “quantum anomalous Hall effect” - a QHE at zero external magnetic field with topological insulators as thin films [5].

## Hall Effect Case Study

Two MFLIs, where 2DEG GaAs/Al_{0.3}Ga_{0.7}, were used to measure the QHE at the ETH Zurich. The intent was to study light-matter coupling and so samples were characterized accordingly [6]. The density of electrons was 3x10^{11} cm^{-2}, the electron mobility µ was 3x10^{6} cm^{2} /V s and the dimensions of the Hall bar were 166.5 µm length and 39 µm width.

Figure 3 gives an example of measurements performed at 100 mK and at a frequency of 14 Hz. A Hall bar was positioned in series with a RMS voltage of 200 mV across the 10 MΩ current-limiting resister to attain a current of 20 nA. A signal amplification of 1,000 before the lock-in input was achieved using a home-built preamplifier and the lock-in measurements were carried out with a time constant of 100 ms.

Figure 3 shows the longitudinal and transverse resistivities *ρ*_{xx }and *ρ*_{xy }respectively, given as a function of the magnetic field B. This was identified with two lock-in amplifiers. The SdH oscillations in *ρ*_{xx} are represented with the black trace and the integer quantum Hall effect plateaus in *ρ*_{xy }are represented with the red. Although it is not shown in the Figure, at around 13 T the first plateau develops. Low temperatures and high sample mobility were observed with a filling factor as high as 300. This led to an early onset of SdH oscillations, as emulated in the high quality of the sample. Figure 3(b) demonstrates how structures in *ρ*_{xx} developed above 0.4 T due to the spin splitting of the Landau levels.

## Practical Uses of QHE and Related Phenomena

The material type, scattering and temperature are all separate from the resistance of the quantum Hall states. This means that 2DEG materials are employed as resistance standards. QHE was previously only observed at low temperatures but in 2007 graphene measurements at a magnetic field of 20 T showed the QHE at room temperature [7]. This formed a potential basis with which to develop new resistance standards. Even more doors were opened with the discovery that the QHE in topological insulators lacked the presence of a magnetic field. These new materials that conduct in protected states have potential to be used in fast electronics and quantum computation [1].

## Advantages of Using the MFLI for Hall Effect Measurements

### High Sensitivity to Extract the Smallest Signals

It is difficult to detect small signals frequently masked by noise with measurements of the Hall effect as the usual resistance is normally around 100 Ω or less. When combined with a current of around 20 nA, the resistance changes to voltages of a few µV for V_{xx} and hundreds of µV for V_{xy}.

A preamplifier is beneficial to increase the SNR through amplification and filter broadband noise. To resolve small features, it is crucial that space is made for the full magnetic field sweep. This means there is demand for the high dynamic range of the MFLI inputs from both voltage measurements.

### Effective Noise Rejection to Maximize SNR

Within the MFLI, eight order filters can reject noise up to a million times larger than the measured signal, providing an effective noise rejection and allowing for a high SNR to be realized. They can deliver a sufficient margin to maximize measurement speed and accuracy.

When conducting measurements at low temperatures, the best solution available is the MFLI due to its inputs having the lowest power dissipation available [8]. In these situations, the electronic temperature of the sample can be influenced by the lock-in input noise, which may also increase the overall noise.

Effective noise rejection of the lock-in facilitates faster measurements, as the filter time constants can be shortened and the full characterization measurement time reduced by up to ten fold.

### High Accuracy: Dedicated Current Sensing

The current I_{R} needs to be measured in scenarios where the assumption of constant current I_{R} is invalid to maintain accurate resistance measurements and prevent the systematic error of up to 10%. The current can be measured along with the Hall voltage by only using one lock-in amplifier unit with the option for accessibility of the MF-MD. The benefits of this are minimized setup complexity and maximized measurement fidelity.

### Efficient Work-Flows: LabOne Software Included

LabOne is a Zurich Instruments control software that was designed for efficient workflows and also came included with the MFLI. The user receives first results quickly with the instinctive operator interface. Users can also have high confidence in the collected data with a collection of features and tools that are also provided. One example is the direct recording of data at the Signal Inputs with the Scope while also visualizing the demodulator outputs in the time (frequency) domain by utilizing the Plotter (Spectrum Analyzer).

The multi-device synchronization (MDS) feature is available when measurements require the use of more than one instrument. MDS retains the reference clocks of all synchronized instruments and also coordinates the time stamps of the recorded data. The measurements can be carried out in just one session of the LabOne operator interface.

If automation is required or if the MFLI needs to be integrated into a pre-existing measurement setting then LabOne also offers APIs for LabVIEW^{®}, MATLAB^{®}, Python, .NET and C.

## Conclusion

Industrial applications of the Hall effect measurements and SI unit redefinition are universal within the research. AC techniques are beneficial for most measurements. Optimum noise rejection can be achieved with lock-in amplifiers to ensure high accuracy and SNR.

The development of the MFLI Lock-in Amplifier of Zurich Instruments makes use of the latest hardware and software technologies to combine the benefits of easy use with high-performance digital signal processing.

MFLI is the best tool available for basic measurements to detect a Hall voltage at a defined frequency as well as more complicated setups, which demand the use of multiple instruments. It is adaptable to changing demands such as frequency range upgrades from DC to 500 kHz to DC to 5 MHz or the addition of three additional demodulators to simultaneously analyze the voltage and current inputs.

## References

[1] Davide Castelvecchi. The strange topology that is reshaping physics. Nature, 547:272–274, 2017.

[2] N. W Ashcroft and N.D. Mermin. Solid state physics. 1976.

[3] Zurich Instruments AG. Principles of Lock-in Detection, 2017. White Paper.

[4] K. v. Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys. Rev. Lett., 45:494–497, Aug 1980.

[5] A. J Bestwick, E. J. Fox, X. Kou, L. Pan, K. L. Wang, and D. Goldhaber-Gordon. Precise quantization of the anomalous hall effect near zero magnetic field. Phys. Rev. Lett., 114:187201, May 2015.

[6] G. L. Paravicini-Bagliani, F. Appugliese, E. Richter, F. Valmorra, J. Keller, M. Beck, N. Bartolo, C. Rössler and T. Ihn, K. Ensslin, C. Ciuti, G. Scalari, and J. Faist. Magneto-transport controlled by landau polariton states. arXiv:1805.00846, 2018.

[7] K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim. Room temperature quantum hall effect in graphene. Science, 315(5817):1379–1379, 2007.

[8] Zurich Instruments AG. Power Dissipation at Input Connectors of Lock-in Amplifiers, 2017. Technical Note.

This information has been sourced, reviewed and adapted from materials provided by Zurich Instruments AG.

For more information on this source, please visit Zurich Instruments AG.