**The Hall effect was first discovered by Edwin Hall in 1879. Since then, it has been applied extensively in measurements, in particular, in materials characterization and sensing. More recently, it has been used to characterize new materials which, in turn, empowers the discovery of new phenomena. Examples of these are the quantum Hall effect, spin Hall effect, and topological insulators [1]. **

For several device applications, the Hall effect can operate as a platform. Sensing of currents in the automotive industry, metrology, non-destructive inspection and testing, and security screening applications are all examples of this.

As shown in Figure 1, when a conductor is placed in a magnetic field B the Hall effect becomes relevant. The charge carriers flowing through the conductor are deflected due to the Lorenz force F_{L}. This results in a Hall voltage V_{xy }between the sides of the material, at a 90-degree angle to the magnetic field and the current I_{R}.

*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.There is a linearly proportional relationship between the Hall voltage and the magnetic field. Meanwhile, the longitudinal voltage V_{xx }is independent of the magnetic field. As shown in Figure 2(a) for metallic conductors and semiconductors these are the typical signatures of the Hall effect.

*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.*

If R=V/I denote the resistance*, *the width by *w, *the thickness by *t* and the length of the conductor by *l*, then the resistivity *ρ *of a bulk conductor is described by *ρ *= *R w t/l. *The classical Drude model is used to derive the resistivities *ρ*_{xx} and *ρ*_{xy}:

Equation 1(a):

Equation 1(b):

where *n* is the density of charge carriers, *m* is the carrier effective mass, *e* is the electron charge and *τ *is the mean free scattering time [2]. Equation 1(a) describes the transverse Hall resistivity with a linear magnetic field and inverse carrier density dependence. Equation 1(b) defines the longitudinal resistivity. That is the transport characteristics of the material in the direction of current. This is independent of the field. The classical Hall effect is primarily used for two applications and these are:

## Materials Characterization

Hall effect measurements are used to decide on the conducting properties of a material. As the magnetic field dependence is linear, Equation 1(a) is used to determine the carrier type and the carrier concentration *n*. Equation 1(b) determines conductivity *σ*_{xx}=1/*ρ*_{xx }and carrier mobility *µ*=* σ*_{xx}/*n e* along with longitudinal resistivity measurements. The Hall bar geometry, shown in Figure 1, is designed to extract the current *I*_{R} as well as the voltages *V*_{xx} and *V*_{xy} directly.

## Sensing

Following on from the identification of the conducting properties of a material, sensitive and accurate magnetic field sensors can be built by employing the Hall voltage and its linear proportionality to the perpendicular magnetic field component. Equation 1(a) can be used to determine the proportional relationship between the measured voltage and the magnetic field in this case.

The linearity spans from zero to fields of numerous Tesla (T). This means that Hall sensors are frequently used in industrial applications. Examples include Hall probes, speedometers, Hall switches, proximity and current sensors. In several of these applications, it is also important to measure speed. However, the signals are commonly on top of another big signal such as background noise resulting from various static and time-varying field noise sources. This can make electronic measurement especially demanding.

AC measurement techniques help to avoid certain systematic measurement mistakes, making them particularly appropriate for these demands. Examples of such systematic measurement mistakes are thermal offsets, thermal drift, and undesired components in the background noise spectrum.

In addition, in comparison to DC techniques, an elevated signal-to-noise ratio (SNR) is reached. This is mainly due to the background noise falling as 1/*f *with increasing frequency *f. *Measurements can typically also be taken quicker with higher SNR. Furthermore, a larger dynamic range is allowed with AC techniques as they frequently come with elevated measurement resolution.

## Lock-In Amplifiers for Accurate and Quick Measurements

Lock-in amplifiers accurately measure down to a few nV with modifiable bandwidth even with a great deal of background noise. This makes them the best instruments to perform AC measurements with. They use phase-sensitive detection to measure amplitude of a signal and the phase against a reference frequency. Background noises are omitted and do not interfere with the measurement, except for at the reference frequency and outside the measurement bandwidth.

The Hall bar geometry and two Zurich Instruments MFLIs, 500 kHz lock-in amplifiers make a classic setup as shown in Figure 1. The transverse Hall voltage *V*_{xy} and longitudinal voltage *V*_{xx} are measured by operating both lock-in amplifiers. The current is supplied through the sample by one of the two amplifiers over a current limiting resistor RL.

It is assumed that the current is constant as the selected R_{L} is usually much bigger than any of the combined resistances in the circuit. Materials are usually characterized using frequencies of up to a few tens of Hz. The two lock-in amplifiers are then coordinated in terms of measurement frequency and data sampling.

The assumption of constant current holds true in most cases. However, the current will require careful monitoring in experimental situations where the sample impedance drastically changes over the course of measurements. MFLI is able to carry out such monitoring with its current sensing input. It does this by measuring the current at the same frequency as the voltage input. For more details on lock-in amplifiers and how they function, please see reference [3].

## Quantum Hall Effects in 2DEGs

When the electrons of a two-dimensional electron gas (2DEG) are put in a magnetic field, new features are demonstrated with the Hall effect. As the magnetic field increases, the Hall resistivity alters in steps with the formation of plateau structures. At the same time, the longitudinal resistivity drops to zero.

This quantum mechanical phenomenon is known as the quantum Hall Effect (QHE). Within the density of states of a 2DEG, the discrete energy levels, called Landau levels, are developed. The typical behavior is shown in Figure 2(b). There are distinct circular orbits with quantized energy levels, which the electrons on a plateau follow. They then move macroscopic distances tracking the edges of a sample with no resistance. The conduction remains carried by the electrons and its dissipation-less nature points to the topologically protected edge states.

In addition to this, as shown in Figure 3, Shubnikov-de Haas (SdH) oscillations can occur whereby the electrons move from one Landau level to another causing visible “quantum oscillations”. The integer QHE was discovered by Klaus von Klitzing in 1980 [4]. This earned him the Nobel prize in 1985.

*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.*

Equation 2 gives the Hall resistivity for a specific plateau:

Equation 2:

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

Other forms of Hall effect phenomena have been found in addition to the QHE in a 2DEG, including the fractional Hall effect and spin Hall effect. Research experiments have also found that it is possible to exhibit a QHE at zero external magnetic field with topological insulators as thin films through a “quantum anomalous Hall effect” [5].

## Hall Effect Case Study

The QHE was measured at the ETH Zurich using two MFLIs, where 2DEG was GaAs/Al_{0.3}Ga_{0.7}. Samples were characterized with the intention of studying light-matter coupling [6]. The density of electrons was 3x10^{11} cm^{-2} and the electron mobility µ was 3x10^{6 }cm^{2} /V s. The dimensions of the Hall bar were 166.5 µm length and 39 µm width.

An example of measurements performed at 100 mK and at a frequency of 14 Hz is shown in Figure 3. A RMS voltage of 200 mV across the 10 MΩ current limiting resister was placed in series with a Hall bar to attain a current of 20 nA. A home-built preamplifier was used to achieve signal amplification of 1,000 before the lock-in input. A time constant of 100 ms was used to carry out the lock-in measurements.

Figure 3 shows the longitudinal and transverse resistivities *ρ*_{xx} and *ρ*_{xy}. These are given as a function of the magnetic field B, which was found using two lock-in amplifiers. The black trace shows the SdH oscillations in *ρ*_{xx} and the integer quantum Hall effect plateaus in *ρ*_{xy} are shown in red. It is not displayed here but at around 13 T the first plateau develops. A filling factor as high as 300 was seen, with low temperatures and high sample mobility. This caused the early onset of SdH oscillations, which is reflected in the high quality of the sample. As shown in Figure 3(b), the spin splitting of the Landau levels caused structures in *ρ*_{xx} to develop above 0.4 T.

## Practical Uses of QHE and Related Phenomena

The resistance of the quantum Hall states is separate from material type, scattering and temperature meaning that 2DEG materials are employed as resistance standards. Until recently, QHE was only seen at low temperatures.

A potential basis to develop new resistance standards was discovered in 2007 when graphene measurements at a magnetic field of 20 T showed the QHE at room temperature [7]. Further new possibilities were demonstrated with the discovery of the QHE in topological insulators lacking the presence of a magnetic field. Fast electronics and quantum computation could potentially make use of such new materials that conduct in protected states [1].

## Advantages of Using the MFLI for Hall Effect Measurements

### High Sensitivity to Extract the Smallest Signals

The detection of small signals commonly masked by noise are compromised by measurements of the Hall effect. In Hall Effect measurements, the usual resistance is around 100 Ω or less. This changes into voltages of a few µV for V_{xx} and hundreds of µV for V_{xy} when combined with a current of around 20 nA.

To increase the SNR through amplification and filter broadband noise, it is of benefit to employ a preamplifier. To make room for the full magnetic field sweep, there is demand for the high dynamic range of the MFLI inputs from both voltage measurements. To resolve small features, this is crucial.

### Effective Noise Rejection to Maximize SNR

To attain high SNR, there needs to be an effective rejection of noise. Eight order filters within the MFLI can reject noise as much as a million times larger than the measured signal. This means that they can deliver sufficient margin to maximize measurement speed and accuracy.

The inputs of the MFLI have the lowest power dissipation available, making it the best solution available for conducting measurements at low temperatures [8]. In such scenarios, the lock-in input noise can influence the electronic temperature of the sample and further increase the overall noise.

Quicker measurements can be taken due to the effective noise rejection of the lock-in. This is because it is possible to shorten the filter time constants and reduce the full characterization measurement time by a factor of 10.

### High Accuracy: Dedicated Current Sensing

The current I_{R} must be measured in scenarios where the constant current I_{R} assumption does not hold to maintain accurate resistance measurements and to prevent the systematic error of up to 10%. The current can be measured simultaneously with the Hall voltage by utilizing only one lock-in amplifier unit with the accessibility of the MF-MD option. Minimized setup complexity and maximized measurement fidelity are the benefit of this.

### Efficient Work-Flows: LabOne Software Included

Developed for efficient workflows, the Zurich Instruments control software, LabOne, is also provided with the MFLI. First results are delivered quickly to the user with the instinctive operator interface and a collection of features and tools provide users with high confidence in the collected data. For example, data can be directly recorded at the Signal Inputs with the Scope and, at the same time, the demodulator outputs are capable of being visualized in the time (frequency) domain by utilizing the Plotter (Spectrum Analyzer).

If the measurements require the use of more than one instrument, then the multi-device synchronization (MDS) feature is available. The reference clocks of all synchronized instruments are retained by MDS. This also aligns the timestamps of the recorded data. One session of the LabOne operator interface is sufficient to carry out the measurements.

LabOne also offers APIs for LabVIEW^{®}, MATLAB^{®}, Python, .NET and C should the MFLI require integration into a pre-existing measurement setting or should measurements require automation.

## Conclusion

Within research, SI unit redefinition and industrial applications of the Hall effect measurements are universal. AC techniques benefit the majority of the measurements. Lock-in amplifiers are the primary tool for ensuring high accuracy and SNR through optimum noise rejection.

The latest hardware and software technologies have been used in the development of the MFLI Lock-in Amplifier of Zurich Instruments, combining the benefits of easy use with high-performance digital signal processing.

From basic measurements that detect a Hall voltage at a defined frequency to more complicated setups that demand the use of multiple instruments, MFLI is the best tool available. It can adapt to changing demands, for example, by upgrading the frequency range from DC to 500 kHz to DC to 5 MHz or by adding three additional demodulators to analyze the voltage and current inputs simultaneously.

## 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.