hall effect experiment simulation

Hall-Effect : Determination of carrier concentration

If a metal or a semiconductor carrying a current I is placed in transverse magnetic field B, Potential difference V H is produced in a direction normal to both the magnetic field and current directions. This is known as HALL EFFECT. Let a semiconductor sample of thickness t and area of cross section A, carrying a current I be acted upon by a transverse magnetic field B. The magnetic field tends to deflect charge carriers in the sample towards one of its faces leading to an accumulation of charges there. This in turn produces an electric field E H in a direction which opposes the Lorentz force due to the magnetic field. The electric field builts till it exactly compensated for the effect of magnetic field. The potential difference V H arising due to E H is given by

V H = BI/net

Where n- carrier concentration , e- charge of an electron and t- thickness of the sample. The ration 1/ne is known as the Hall Coefficient (R H )

R H = (V H t)/BI

Knowing the Hall coefficient, the concentration of charge carrier in the sample can be determined by

n= BI/V H et

Due to difference in the velocity of the carriers, a correction factor is applied to the formula for the concentration. So, the final formula is

n=(3⫪/8)(BI/etV H )

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The hall effect in metals and semiconductors.

In this experiment, students learn about the classical Hall effect which is the basis of most sensors used in magnetic field measurements. Thin film samples consisting of a semiconductor (InAs) and two different metals (aluminum and gold) are investigated to determine the sign and density of the charge carriers. The sample is placed in a DC magnetic field, and the transverse (Hall) voltage is measured as a function of the current through the sample. Students observe the inverse relationship between the magnitude of the Hall voltage and charge carrier density, a relationship which accounts for the almost exclusive use of semiconductors as Hall effect/magnetic field sensing devices.

Experiment Information

• Lecture slides on Optical Properties of solids (Olmstead) Gives basic theory behind dielectric constants and conductivity.
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Discussion Questions

• Find some values for mobility in tables that are listed in texts, such as Kittel's or other references, such as the Landolt-Bornstein series of books available in the library. Compare your results with the tabulated values. How close are your results to the ones cited for InAs? Can you explain any differences?
• In the case of the aluminum sample, you will find in such texts as Kittel's "Introduction to Solid State Physics" that the assumed charge carriers are "1-hole per atom". Why should we assume that the charge carrier in aluminum is a hole? This is not predicted by the simple Drude theory of electrons in metals.
• Your results for aluminum are likely to be different from the predicted or tabulated values, which are usually quoted as being measured at low temperature and high magnetic field. See what you can find out about the Hall coefficient for a range of temperatures and field strengths, and discuss how your measurement fits in with these.

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Observing Hall Effect in Semiconductors

Further Readings and References

• Hall effect sensing Honeywell: Sensing and contro, Hall Effect Sensing and Application, .
• Principles of Electronic Materials and Device Principles of Electronic Materials and Devices, McGraw-Hill, S. O. Kasap, 145, (2006).
• Experiments in Modern Physics Experiments in Modern Pysics, Academic Press, Adrian C. Melissinos, 63, (2003).

Pictorial Procedure

1. Provided equipment for this experiment.

2. Connection of electromagnet with power supply.

3. Placement of the Hall probe in between the pole pieces of electromagnet.

4. Side view of printed circuit board (PCB) and Hall probe placed vertically in between the pole pieces of electromagnet.

5. Top view of printed circuit board (PCB) and Hall probe placed vertically in between the pole pieces of electromagnet.

Measurement of Planck's constant using a light bulb

Electron energy loss investigation through the nobel prize winning franck-hertz experiment.

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Hall effect in a two-dimensional disordered Lorentz gas

Benjamin sanvee, jakob schluck, mihai cerchez, dominique mailly, hans w. schumacher, klaus pierz, thomas heinzel, and jürgen horbach, phys. rev. b 108 , 035301 – published 5 july 2023.

• Citing Articles (1)
• INTRODUCTION
• THE DRUDE AND (GENERALIZED) BOLTZMANN…
• LORENTZ GAS MODEL: SIMULATION DETAILS
• EXPERIMENTAL SETUP AND MEASUREMENT…
• RESULTS ON THE MAGNETOTRANSPORT
• SUMMARY AND CONCLUSIONS

Using a combination of experiment and simulation, we study the magnetotransport in a two-dimensional disordered Lorentz gas with circular obstacles. Our focus is on the investigation of the Hall effect at obstacle densities beyond the low-density limit. However, as a reference, we also consider very low obstacle densities. Here, the magnetotransport properties, as obtained from the simulation and the experiment of a pristine sample, can be well described in terms of the Drude-Boltzmann model. For intermediate and high obstacle density, only for very low magnetic fields B , we find a linear dependence of the Hall resistance ϱ x y on B , albeit with a Hall coefficient that does not reflect properly the carrier density . At larger magnetic fields but still below the onset of the Landau quantization as well as the magnetic-field-induced conductor-to-insulator transition, striking nonlinearities of ϱ x y ( B ) due to classical memory effects are observed. Moreover, the scattering time obtained within the Drude-Boltzmann model develops into a phenomenological parameter that decreases with increasing magnetic field.

• Received 3 April 2023
• Revised 23 June 2023
• Accepted 23 June 2023

DOI: https://doi.org/10.1103/PhysRevB.108.035301

©2023 American Physical Society

Physics Subject Headings (PhySH)

• Research Areas
• Physical Systems

Authors & Affiliations

• 1 Institut für Theoretische Physik II, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
• 2 Institut für Experimentelle Physik der kondensierten Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
• 3 II. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany
• 4 Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau, France
• 5 Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany

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Vol. 108, Iss. 3 — 15 July 2023

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Snapshots of trajectories from the simulation at B ̃ = 0.05 (red lines) and B ̃ = 0.15 (blue lines) for the densities ρ = 0.1963 (upper-left panel), ρ = 0.3927 (upper-right panel), ρ = 0.5890 (lower-left panel), and ρ = 0.7853 (lower-right panel).

Scanning electron microscope picture of a section of the sample with ρ = 0.1963 . The bright area corresponds to the Hall bar and a voltage probe containing the two-dimensional electron gas, which is removed by an etch step in the dark regions.

(a) Hall resistance ϱ x y , as obtained from the experiment, as a function of the magnetic field B in unit of Tesla (lower axis of abscissae) and the reduced magnetic field B ̃ (upper axis of abscissae) at different temperatures T for the array with ρ = 0.589 . (b) Derivative d ϱ x y / d B ( B ) corresponding to the data in (a). The horizontal black line at 2438  Ω /T corresponds to the value of the Drude-Hall coefficient R DH . The vertical black line marks B ̃ = 1.0 . Adapted from Ref. [ 41 ].

Location of the SdH minima, B j − 1 , as a function of the index j for the pristine sample and the samples at the different obstacle densities (here, j = 1 just corresponds to the first minimum that can be identified). Note that the data are shifted on the ordinate with increasing value of ρ in steps of 1  T − 1 . The inset shows the electron density n e as function of ρ , resulting from the slope of B j − 1 .

(a) Conductivity σ x x as a function of B ̃ for the densities ρ = 0.0012 , 0.012, and 0.1963. The results of the simulation are compared to the predictions of the DB theory, Eq. ( 2 ), and the theory of Bobylev et al. [ 2 ], Eq. ( 7 ). The inset shows the corresponding data for the magnetoresistance ϱ x x . (b) Same as in (a), but now for the magnetoconductivity σ x y and the Hall resistance ϱ x y in the inset.

(a) Conductivity σ x x as a function of B for the experimental pristine sample in comparison to the DB theory, Eq. ( 2 ), and the theory of Bobylev et al. [ 2 ], Eq. ( 7 ). In the latter theoretical expressions, the ratio R / ρ = 55.5   µ m is used to obtain the best match with the experimental data (see text). The inset shows the corresponding data for the magnetoresistance ϱ x x . (b) Same as in (a), but now for the magnetoconductivity σ x y and the Hall resistance ϱ x y in the inset.

Comparison between experiment (solid lines) and simulation (circles) for the reduced conductivity σ x x / σ x x ( B = 0 ) as a function of B ̃ for ρ = 0.1963 , 0.3927, 0.5890, and 0.7853. Also included are the data of the pristine sample and the simulation at ρ = 0.012 , both in comparison to DB theory (dashed black lines).

Hall conductivity σ x y (circles) and ω cy τ σ x x (solid lines) as a function of B ̃ for different densities for (a) the experiment and (b) the simulation. The values of τ are indicated in both panels.

Reduced diffusive time τ / τ Dr and reduced conductivity σ x x ( B ̃ = 0 ) / σ 0 as a function of ρ for (a) the experiment and (b) the simulation. The insets show Δ = σ x x ( B ̃ = 0 ) τ Dr / ( τ σ 0 ) as a function of ρ .

Magnetoresistance ϱ x x ( B ̃ ) at different densities, as obtained from (a) the experiment and (b) the simulation. Solid lines correspond to the data while the dashed lines are calculated via Eq. ( 16 ).

Hall resistance ϱ x y ( B ̃ ) , as obtained from the experiment, at the densities (a)  ρ = 0.1963 , (b)  ρ = 0.3927 , (c)  ρ = 0.5890 , and (d)  ρ = 0.7853 . The solid lines represent the data, while the blue dashed lines are calculated via Eq. ( 17 ). The black dotted lines represent the expected behavior at low magnetic fields, ϱ x y = R ̃ H B ̃ with the Hall coefficient R ̃ H given by Eq. ( 21 ). The insets display ϱ x y / B ̃ . At sufficiently low magnetic field, this quantity is supposed to approach Hall coefficient R ̃ H (the corresponding constant is shown as a dotted horizontal line in these plots).

Same as Fig.  11 but now for the simulation.

Reduced Hall coefficient R ̃ H as a function of density for experiment and simulation. Also displayed is the coefficient R ̃ DH = B 0 / ( n e e ) , as expected from Drude theory, taking the estimates of n e , as obtained from the analysis of the SdH oscillations (see inset of Fig.  4 ).

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