**In 1879, Edwin Hall discovered what has become known as the ‘Hall effect’ which has since been widely applied in measurements – especially in the sensing and characterization of materials. In the modern day, this effect is enabling new phenomena to be discovered, as it is being used to characterize new materials. These phenomena include the spin Hall effect, the quantum Hall effect, and, quite recently, topological insulators.[1]**

The Hall effect also constitutes a platform for a variety of device applications, including non-destructive testing and inspection, metrology, security screening applications, and current sensing in the automotive industry.

As demonstrated in Figure 1, the Hall effect appears when a conductor is placed in a magnetic field B. As a result of the Lorentz Force F_{L }causing a transverse (or ‘Hall’ voltage V_{xy}) which is perpendicular to both the current I_{R} and the magnetic field between the material’s sides, the charge carriers which flow through the conductor are deflected. The longitudinal voltage V_{xx} is independent of the magnetic field, while the Hall voltage is linearly proportional to the magnetic field.

These represent the traditional signatures of the Hall effect for metallic semiconductors and conductors, as Figure 2 (a) shows. A bulk conductor’s resistivity is defined as ρ = Rwt/I where w is width, R=V/I represents the resistance, l is the conductor’s length, and t represents thickness. The resistivities ρ_{xx} and ρ_{xy} are derived from the traditional Drude model in the following form:

where m is the carrier of effective mass, n is the density of charge carriers, e is the electron charge, and τ is the mean free scattering time [2].

The transverse Hall resistivity with an inverse carrier density dependence and linear magnetic field is described in Equation 1a, while the longitudinal resistivity which is field independent and reflective of the transport characteristics of the material in the direction of current is described in Equation 1b.

Two primary applications of the traditional Hall effect are as follows:

### 1. Materials Characterization

Hall effect measurements are used to determine a material’s conducting properties. From the Hall resistivity, the carrier type and carrier concentration n are determined with the use of Equation 1a, as the magnetic field dependence is linear. With Equation 1b, longitudinal resistivity measurements are used in order to determine the conductivity σ_{xx} = 1/ρ_{xx} and carrier mobility μ = σ_{xx} /n e. Depicted in Figure 1, the Hall bar geometry was designed so that it could directly extract voltages V_{xx }and V_{xy} as well as the current I_{R}.

### 2. Sensing

The Hall voltage and its linear proportionality to the perpendicular magnetic field component can be used to build accurate and sensitive magnetic field sensors, when a material’s conducting properties are known. The measured voltage is proportional to the magnetic field in this case, and can consequently be extracted from Equation 1a. As the linearity extends from fields of several Tesla (T) down to zero, Halls are used widely in industrial applications as speedometers, proximity and current sensors, Hall switches, and Hall probes.

As the signals are frequently on top of other large signals and background noise emanating from various static and time varying field noise sources, the measurement speed is also important in several of these applications. Demanding requirements are consequently established for the measurement electronics.

For these requirements, AC measurement techniques are especially suited. This is because they are capable of helping to avoid some systematic measurement errors, which include thermal drift, thermal offsets, and unwanted components in the background noise spectrum.

Additionally, as compared with DC techniques, a higher signal-to-noise ratio (SNR) is attained. This is mainly because the background noise falls as 1/f as the frequency f increases. Quicker measurements normally result from a higher SNR. Furthermore, a larger dynamic range is often enabled by AC techniques, as they come with a higher measurement resolution.

## Lock-In Amplifiers for Accurate and Fast Measurements

Lock-in amplifiers are best for the performance of AC measurements, as they are capable of producing reliable measurements down to a few nV with adjustable bandwidth – even in the presence of a great deal of background noise.

Phase-sensitive detection is used by these instruments, in order to measure the amplitude of a signal and the phase against a reference frequency. Background noise at frequencies outside of the reference frequency and beyond the measurement bandwidth are overwhelmingly rejected, and consequently have no impact upon the measurement.

Figure 1 depicts the conventional setup, using two Zurich instruments MFLIs, 500 kHz lock-in amplifiers, and the Hall bar geometry. The transverse Hall voltage V_{xy} and the longitudinal voltage V_{xx} are measured by the two lock-in amplifiers. One of these amplifiers also provides the current through the sample via what is known as a ‘current limiting resistor’ R_{L}. The resistor which is usually picked tends to be significantly larger than any of the resistances in the circuit combined. The current can consequently be assumed to be constant.

*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 IR 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.

Normally, materials characterization measurements are performed at frequencies reaching just a few tens of Hertz. With the two lock-in amplifiers synchronized in terms of data sampling and measurement frequency, While the constant-current assumption is true in a lot of cases, in studies in which the sample impedance significantly changes over the course of the entire measurement, the current must be carefully monitored.

The MFLI has this capability with its current sensing input. It measures the current and the voltage input at the same frequency. See [3] for a more in-depth analysis of lock-in amplifiers and the manner in which they operate.

## Quantum Hall Effect in 2DEGs

The Hall effect exhibits qualitatively new features when the electrons of a two-dimensional electron gas (2DEG) are placed in a magnetic field. Plateau structures occur as the magnetic field increases, and the Hall resistivity changes in steps. Simultaneously, the longitudinal resistivity decreases to zero.

The above is called the Quantum Hall Effect (QHE). This represents a quantum mechanical phenomenon during which the Landau levels (the discrete energy levels) are formed in the density of states of a 2DEG. Figure 2 (b) illustrates the typical behavior which is exhibited.

**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 electrons conform to discrete circular orbits with quantized energy levels while on a plateau, traveling macroscopic distances along the edges of a sample entirely without resistance. As the conduction is still carried by electrons, this phenomenon is quite remarkable. Its propensity not to dissipate hints at the topologically protected edge states.

Furthermore, observable “quantum oscillations” can be caused by the electrons moving between the Landau levels. These are called Shubnikov-de Haas (SdH) oscillations, and are depicted in Figure 3. In 1980, Klaus von Klitzing discovered the integer QHE,[4] for which he was awarded the Nobel Prize in 1985.

*Figure 3.** (a) Plot of longitudinal and transverse resistivity xx and xy as a function of the magnetic field B, using two lock-in amplifiers. Note that 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 is given by:

In this equation, ν is an integer called the filling factor and h/e^{2} is a fundamental constant known as the von Klitzing constant, which is equal to RK = 25812.807557(18) Ω or the quantum of resistance.

It is worth noting that the resistance and resistivity in 2DEGs have the same unit Ω since ρ = R w/l. The R_{k }is not dependent upon temperature fluctuations, crystal impurities, or the material properties, as it is only dependent on electron charge e and the Planck constant h.

Different examples of the Hall effect phenomena have been discovered, further to the QHE in a 2DEG. These include the spin Hall effect and the fractional Hall effect. More recent studies have also indicated the existence of a “quantum anomalous Hall effect” whereby thin films of topological insulators can exhibit a QHE at zero external magnetic field [5].

## Hall Effect Case Study

Two MFLIs were used to measure the QHE at the ETH Zurich, where 2DEG GaAs/Al0.3Ga0.7As samples were characterized in order to study lightmatter coupling [6]. The electron density n was 3 x 10^{11}cm^{−2 }and the electron mobility μ was 3 x 10^{6} cm^{2}/Vs. The Hall bar dimensions were 39 μm wide and 166.5 μm long. Figure 3 illustrates an example of measurements which were performed at 100 mK at a frequency of 14 Hz.

An RMS voltage of 200 mV was applied across the 10 MΩ current limiting resistor (placed in series with a Hall bar) in order to obtain a current of 20 nA. A home-built preamiplifier with a signal amplification of 1000 was used to amplify the signal before the lock-in input. A constant of 100 ms was used to perform the lock-in measurements.

A plot of transverse and longitudinal resistivities ρ_{xy} and ρ_{xx} as a function of the magnetic field B are shown in Figure 3 (a), which were obtained with two lock-in amplifiers. The integer quantum Hall effect plateaus in ρ_{xy }are shown in red, while the ShH oscillations in ρ_{xx }are shown in black. The first plateau is not shown, but was formed at about 13 T.

The filling factor of up to 300 was observed in this measurement. The early onset of SdH oscillations was caused by the high mobility of the sample and low temperatures, reflecting its high quality. Further, structures in ρ_{xx} which formed above 0.4 T were due to spin-splitting of the Landau levels (as shown in Figure 3 (b).

## Practical Uses of QHE and Related Phenomena

Due to the quantum Hall states’ resistance being independent of material type, scattering and temperature, 2DEG materials are used as resistance standards.

Measurements on graphene a ta a magnetic field of 20 T taken in 2007 revealed the QHE at room temperature [7], offering the possible basis of newly developed standards of resistance. Another set of possibilities was opened up by the discovery of the QHE in topological insulators without the presence of a magnetic field. The possible uses of these new materials which conduct in protected states are for quantum computation and fast electronics [1].

## Advantages of Using the MFLI for Hall Effect Measurements

### 1. High Sensitivity: Extract the Smallest Signals

Hall effect measurements operate by detecting small signals which are frequently buried in the noise. These Hall effect measurements have a typical resistance of approximately 100 Ω or less. When combined with a current of approximately 20 nA, this translates into voltages of just a few μV for V_{xx} and hundreds of μV for V_{xy}.

It is certainly beneficial to use a preamplifier, as it increases the SNR by filtering the broadband noise and via amplification. The high dynamic range of the MFLI inputs are required for both voltage measurements, in order to accommodate the entire magnetic field sweep. This is also vital for resolving small features.

### 2. Effective Noise Rejection: Maximize SNR

In order to achieve high SNR, effective noise rejection is essential. The MFLI has a great enough margin to optimize both measurement accuracy and speed, as it is equipped with 8 order filters which can reject noise up to a million times greater than the measured signal.

Furthermore, if measurements are undertaken at low temperatures, samples’ electronic temperatures are affected by the noise originating from the lock-in input, which adds to the overall noise. In this regard, because its inputs have the lowest power dissipation on the market, the MFLI provides the best commercial solution which is currently available [8].

The measurement is effectively sped up by the lock-in's efficient noise rejection, as the filter time constants can be shortened. The full characterization measurement time – which can last from between a day to a week in some cases - can be reduced by as much as a factor of 10.

### 3. High Accuracy: Dedicated Current Sensing

In certain cases, the constant-current I_{R} assumption does not hold true. It is vital in these cases that the current I_{R} is measured in order to achieve extremely accurate resistance measurements, as well as to avoid the systematic error of as much as 10%. Courtesy of the availability of the MFMD option, the current can be measured simultaneously with Hall voltage with the use of only one lock-in amplifier unit. The user can benefit from an increased fidelity of measurement and a reduction in setup complexity.

### 4. Efficient Work-flows: LabOne Software Included

Designed for efficient workflows, LabOne is the Zurich Instruments control software which is provided with the MFLI. The user is quickly taken to first results courtesy of the intuitive user interface, while confidence in the acquired data is enhanced by a wide variety of tools and features.

As an example, as data are being directly recorded at the Signal Inputs, the Scope and the demodulator outputs can be visualized in the time (frequency) domain with the use of the Plotter (Spectrum Analyzer).

If the use of more than one instrument is required for these measurements, the multi-device synchronization (MDS) feature will be of use. MDS keeps the time stamps of all the recorded data aligned, as well as making sure all of the instruments’ reference clocks are synchronized. It is still possible to carry out the measurements within just one session of the LabOne user interface.

LabOne provides APIs for LabView, Python, .Net, C., and MATLAB, in case measurements require automation or the MFLI needs integrating into an existing measurement environment.

## Conclusion

In SI unit redefinition, industrial applications, and fundamental research today, Hall effect measurements have become ubiquitous. The majority of the measurements benefit from using AC techniques where lock-in amplifiers are the main tool for guaranteeing precise accuracy and SNR via optimum noise rejection.

The Zurich Instruments MFLI Lock-in Amplifier has been built with the latest software and hardware technologies, combining ease of use with the benefits of high-performance digital signal processing. The MFLI is the ideal tool to use, whether for simple measurements – such as detecting a Hall voltage at a defined frequency – or for more complicated setups which require the use of multiple instruments.

The MFLI can adapt to constantly changing requirements. As an example, it can upgrade the frequency range from DC-500 kHz to DC-5 MHz, or to add 3 additional demodulators in order to analyze the current inputs and voltage simultaneously.

## Acknowledgements

Zurich is grateful to Gian L. Paravicini-Bagliani from the Quantum Optoelectronics Group at ETH Zurich for sharing and discussing the Quantum Hall effect data.

## 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 adn 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. Roomtemperature 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.

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