# Hall effect for sensing and materials characterization

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Since the discovery of the Hall effect in 1879 by Edwin Hall, it has been widely applied in measurements, particularly in materials characterization and sensing. Today, it is being used to characterize new materials, enabling the discovery of new phenomena, such as the quantum Hall effect (QHE), spin Hall effect and more recently, topological insulators ^{[1]}. The Hall effect also serves as a platform for many device applications, such as current sensing in the automotive industry, metrology, non-destructive inspection and testing and security screening applications.

The Hall effect appears when a conductor is placed in a magnetic field, as shown in Figure 1. The charge carriers flowing through the conductor are deflected due to the Lorentz force (F_{L}) causing a transverse, or Hall voltage (V_{xy}), between the sides of the material, perpendicular to both the magnetic field and the current (I_{R}.) The Hall voltage is linearly proportional to the magnetic field (B), while the longitudinal voltage (V_{xx}) is magnetic field independent. These are the classical Hall effect signatures for metallic conductors and semiconductors, as illustrated in Figure 2 (a). The resistivity of a bulk conductor is defined as *ρ* = R w t/l, where R = V/I is the resistance, w is the width, t is the thickness and l is the length of the conductor. The resistivities *ρ*_{xx} and *ρ*_{xy} are derived from the classical Drude model in the form:

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 1a describes the transverse Hall resistivity with a linear magnetic field and inverse carrier density dependence, whereas Equation 1b describes the longitudinal resistivity, which is field independent and reflects the transport characteristics of the material in the direction of current. Two main applications of the classical Hall effect are as follows.

### 1. Materials characterization

A material’s conducting properties are determined using Hall effect measurements. From the Hall resistivity, the carrier concentration n and carrier type are determined using Equation 1a because the magnetic field dependence is linear. Longitudinal resistivity measurements are used with Equation 1b to determine conductivity *σ*_{xx} = 1/*ρ*_{xx} and carrier mobility μ = *σ*_{xx}/n e. The Hall bar geometry (as shown in Figure 1) was designed to extract the voltages V_{xx} and V_{xy} as well as I_{R} directly.

### 2. Sensing

When the conducting properties of a material are known, the Hall voltage and its linear proportionality to the perpendicular magnetic field component can be used to build sensitive and accurate magnetic field sensors. In this case, the measured voltage is linearly proportional to the magnetic field and can be extracted from Equation 1a. Hall sensors are widely used in industrial applications as Hall probes, speedometers, Hall switches, proximity and current sensors for magnetic fields spanning from zero to several Tesla. In many of these applications, measurement speed is important, while the signals are often embedded in a large signal and noise background coming from various static and time-varying field noise sources. This sets demanding requirements for the measurement electronics.

AC measurement techniques are particularly well suited for such requirements as they help to avoid certain systematic measurement errors, such as thermal offsets, thermal drift and unwanted components in the background noise spectrum. In addition, a higher signal-to-noise ratio (SNR) is achieved compared to DC techniques, primarily because the background noise falls as 1/f with increasing frequency (f). Higher SNR usually also leads to faster measurements. In addition, AC techniques often come with higher measurement resolution, enabling a larger dynamic range.

## Lock-in amplifiers for accurate and fast measurements

AC measurements are best performed with lock-in amplifiers, as they measure reliably down to a few nanovolts (nV) with adjustable bandwidth, even in the presence of significant background noise. These instruments use phase-sensitive detection to measure the amplitude of a signal and the phase against a reference frequency. Background noise at frequencies other than the reference frequency and outside the measurement bandwidth are rejected and do not impact the measurement.

A typical setup using the Hall bar geometry and two Zurich Instruments MFLI 500 kHz lock-in amplifiers, is shown in Figure 1. The two lock-in amplifiers measure the transverse V_{xy} and V_{xx}, while one of them also provides the current through the sample over a so-called current limiting resistor (R_{L}). This resistor is chosen to be much larger than any of the resistances combined in the circuit, so the current can be assumed constant. Materials characterization measurements are typically performed at frequencies up to a few tens of Hertz, with the two lock-in amplifiers synchronized in terms of measurement frequency and data sampling. The constant current assumption holds in many cases, but in studies where the sample impedance changes dramatically over the course of a measurement, the current needs to be carefully monitored. With its current sensing input, the MFLI has this capability. It measures the current at the same frequency as the voltage input. For a more detailed treatment of lock-in amplifiers and how they work, see Principles of Lock-in Detection^{[3]}.

## Quantum Hall effect in 2DEGs

When the electrons of a 2D electron gas (2DEG) are placed in a magnetic field, the Hall effect exhibits qualitatively new features. The Hall resistivity changes in steps with plateau structures occurring as the magnetic field increases. At the same time, the longitudinal resistivity drops to zero. This is known as the QHE, a quantum mechanical phenomenon where the discrete energy levels, the so-called Landau levels, are formed in the density of states of a 2DEG. The typical behavior is shown in Figure 2 (b). While on a plateau, the electrons follow discrete circular orbits with quantized energy levels and travel macroscopic distances along the edges of a sample without any resistance. This is quite remarkable as the conduction is still carried by electrons and its dissipation-less nature points to the topologically protected edge states. In addition, the electrons moving between the Landau levels can cause observable “quantum oscillations,” known as Shubnikov-de Haas (SdH) oscillations (Figure 3). The integer QHE was discovered by Klaus von Klitzing in 1980^{[4]}, winning him the Nobel Prize in 1985.

### Hall effect case study

The QHE was measured using two MFLIs at the ETH Zurich, where 2DEG GaAs/Al_{0.3}Ga_{0.7}As samples were characterized for the purpose of a study of light-matter coupling^{[5]}. Figure 3 (a) shows a plot of longitudinal and transverse resistivities, *ρ*_{xx} and *ρ*_{xy}, as a function of the magnetic field B, obtained using two lock-in amplifiers. The black trace shows SdH oscillations in *ρ*_{xx} and the integer QHE plateaus in *ρ*_{xy} are displayed in red. In this experiment, an early onset of SdH oscillations was observed as a result of the low temperatures and the high mobility of the sample, reflecting its high quality. Also, structures in *ρ*_{xx} forming above 0.4 T are due to spin splitting of the Landau levels (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. Until recently, the QHE was observed only at low temperatures. In 2007, measurements on graphene at a magnetic field of 20 T revealed the QHE at room temperature^{[6]}, and the potential basis to develop new resistance standards. The discovery of the QHE in topological insulators without the presence of a magnetic field opened another set of new possibilities. These new materials that conduct in protected states could be used for fast electronics and quantum computation^{[1]}.

## Advantages of using the MFLI for Hall effect measurements

### 1. High sensitivity: Extract the smallest signals

Hall effect measurements involve detection of small signals often buried in the noise. The typical resistance in Hall effect measurements is about 100 Ω or less. Combined with a current of about 20 nA, this converts into voltages of a few μV for V_{xx} and hundreds of μV for V_{xy}. The use of a preamplifier is certainly beneficial to increase the SNR through amplification and filtering of the broadband noise. Both voltage measurements require the high dynamic range of the MFLI inputs to accommodate the full magnetic field sweep. This is also essential to resolve small features.

### 2. Effective noise rejection: Maximize SNR

Effective noise rejection is crucial for achieving high SNR. The MFLI is equipped with eight order filters capable of rejecting noise up to a million times bigger than the measured signal, providing enough margin to optimize for both measurement speed and accuracy. In addition, low-temperature measurements are susceptible to noise originating from the lock-in input, which affects a sample’s electronic temperature and adds to overall noise. In this respect, the MFLI provides the best commercial solution on the market, as its inputs have the lowest power dissipation available^{[7]}. The efficient noise rejection of the lock-in effectively speeds up the measurement since the filter time constants can be shortened and the full characterization measurement time, which in some cases lasts between a day or a week, can be reduced by a factor of up to 10.

### 3. High accuracy: Dedicated current sensing

In cases where the constant current assumption does not hold, it is important to measure the current to achieve highly accurate resistance measurements and avoid a systematic error of up to 10%. With the MF-MD option available, the current can be simultaneously measured with Hall voltage using just one lock-in amplifier unit. The user benefits from reduced setup complexity and increased measurement fidelity.

### 4. Efficient work-flows: LabOne software included

LabOne, Zurich Instruments’ control software provided with the MFLI, is designed for efficient workflows. The intuitive user interface enables the user to obtain results quickly, while a wealth of features and tools help to build high confidence in the acquired data. For instance, while data are directly recorded at the signal inputs with the scope, the demodulator outputs can be visualized in the time (frequency) domain using the plotter (spectrum analyzer).

For measurements requiring the use of more than one instrument, the multi-device synchronization (MDS) feature is valuable. MDS keeps the reference clocks of all instruments synchronized and the time stamps of the recorded data aligned. The measurements can still be carried out within a single session of the LabOne user interface.

For automated measurements or MFLI integration into an existing measurement environment, LabOne provides APIs for LabVIEW®, MATLAB®, Python, .NET and C.

## Conclusion

Today, Hall effect measurements are ubiquitous in fundamental research, SI unit redefinition and industrial applications. Most of these measurements benefit from AC techniques where lock-in amplifiers are the prime tool guaranteeing high accuracy and SNR through optimum noise rejection.

Zurich Instruments’ MFLI Lock-in Amplifier is built on the latest hardware and software technologies, combining the benefits of high performance digital signal processing with ease of use. From simple measurements detecting a Hall voltage at a defined frequency to more complicated setups requiring the use of multiple instruments, the MFLI is the right tool to use. It can adapt to changing requirements, for instance by upgrading the frequency range from DC to 500 kHz to DC to 5 MHz or adding another three demodulators to analyze the voltage and current inputs simultaneously.

Please contact Zurich Instruments to discuss any Hall measurement requirements (info@zhinst.com or +41 44 5150 410). For more information on the MFLI Lock-in Amplifier, visit the Zurich Instruments website.

## 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] 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.

[6] 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.

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