The Hall Effect C1

The Hall Effect

Head of Experiment: Zulfikar Najmudin

The following experiment guide is NOT intended to be a step -by-step manual for the experiment but rather provides an overall introduction to the experiment and outlines the important tasks that need to be performed in order to complete the experiment. Additional sources of documentation may need to be researched and consulted during the experiment as well as for the completion of the report. This additional documentation must be cited in the references of the report.

RISK ASSESSMENT AND STANDARD OPERATING PROCEDURE

1. PERSON CARRYING OUT ASSESSMENT

Name Geoff Green Position Chf Lab Tech Date 18/09/08

2. DESCRIPTION OF ACTIVITY

C1 Hall Effect

3. LOCATION

Campus SK Building Huxley Room 403

4. HAZARD SUMMARY

Accessibility X Mechanical X

Manual X Hazardous Handling Substances

Electrical X Other

Lone Working Yes No Permit-to- Yes No Permitted? Work Required?

5. PROCEDURE PRECAUTIONS

Use of 240v Mains Powered Equipment Isolate Socket using Mains Switch before unplugging or plugging in equipment

Accessibility All bags/coats to be kept out of aisles and walkways.

Use of Electro-magnet See attached Scheme of Work

6. EMERGENCY ACTIONS

All present must be aware of the available escape routes and follow instructions in the event of an evacuation

THE HALL EFFECT

1. Objectives

The goal of this experiment is to make accurate measurements of the charge carrier density and carrier mobility in three different materials where the majority charge carriers are electrons (n-type) or holes (p-type) using the van der Pauw method. The samples provided are n-type Gallium Arsenide (GaAs), p-type GaAs, undoped Indium Antimonide (InSb), and graphene. You will gain insight into basic DC electrical measurements and carrier transport phenomena including, the Hall effect, magnetoresistance and the influence of band structure on the material properties. To accurately make these measurements you must consider the effect of various systematic errors and strategies to remove them through both experimental control and data analysis. 2. Background

2.1 The Hall effect

When a conductor is subject to an applied electric current Jx and magnetic field Bz, a transverse electric field Ey is developed in a direction normal to both Bz and Jx (see Fig. 1).

This principle is known as the Hall effect, the best known of several phenomena in which electrical or thermal currents produce electric fields or temperature gradients in a direction normal both to the magnetic field and to the current. For a detailed description of these effects see Ref. [1].

Fig. 1: The Hall Effect for an ideal conductor of length l, width w and thickness t (l >> w). When subject to a current, Jx, and a magnetic field, Bz, perpendicular to each other, an electric field is set up perpendicular to both, Ey.

In order to get gain some physical insight we can ask what happens to free charges in the material subject to an applied magnetic field. Consider the geometry of a three-dimensional conducting slab shown in Fig. 1. We assume l >> w so that we may ignore the effect of the current supplying contacts on the phenomenon of interest. Charge carriers drift in the applied electric field Ex with an average velocity vx causing a current given by 퐼푥 = 퐽푥푤푡 = 푛푞푣푥푤푡, where q is the charge (q = -e for electrons and q = +e for holes) n is the charge density. When the magnetic field is initially applied, charge carriers experience a Lorentz force, 푭 = 푞 (풗 × 푩), that will deflect them to one side of the slab. After a short period of time, charges will accumulate on the z-x surfaces creating a transverse electric field, Ey. When the system reaches steady-state the force due to the charge accumulation must balance the force due to the magnetic field such that there is no net force on the charge carriers. The resulting potential difference, 푉퐻 = −푤퐸푦 is known as the Hall voltage, which you will measure in this experiment. For the geometry in Fig. 1, and in the simplest case of a single carrier type (e.g. electrons), the Hall voltage can be expressed as

1 퐼 퐵 푉 = ( ) 푥 푧 . (1) 퐻 푛푞 푡

Equation 1 shows that the Hall voltage is proportional to both the applied magnetic 1 field and current. The quantity ⁄푛푞 = 푅퐻 is the Hall coefficient, which is an intrinsic property of the material. It is often convenient to consider the sheet carrier density

푛푠ℎ = 푛푡 and to lump the sample thickness and Hall coefficient together into an effective Hall coefficient expressed in units of Ohms/Tesla that can be directly obtained from the slope of the Hall resistance (VH/Ix) versus Bz. An important property of the Hall coefficient is that it depends on the sign of the charge carrier, making it possible to determine both the charge density and charge carrier type (i.e. electrons or holes) in a material from a straightforward measurement of the Hall voltage. An impressive feat, given the Hall effect was measured a decade before the electron was discovered! 2.2 Carrier mobility

The second material property of interest, the carrier mobility (µ), relates the carrier drift velocity to the applied electric field by 푣푑 = 휇퐸. The mobility describes how ‘easily’ the charge carriers respond to an applied electric field and captures all the momentum-scattering processes present in the material. The carrier mobility and carrier density determine the resistivity,  (=1/σ), an intrinsic material property independent of sample geometry. At zero applied magnetic field, the resistivity of an n-type material is given by

1 휌 = ⁄푛푒휇 . (2)

Note that the sample resistance, R, is related to the resistivity through the usual expression R=l/wt. Combining Eq. 1 and Eq. 2 we see that the carrier mobility can be determined from separate measurements of the resistivity and Hall coefficient through the expression

푅 휇 = | 퐻⁄휌|. (3)

2.3 The van der Pauw method

The most common approach to measuring the transport properties of a material is by four-probe resistivity measurements based on the van der Pauw method [4]. You will notice that each sample provided has four electrical contacts located at the periphery as illustrated in Fig. 2. Note that your samples have been encapsulated in silicone for protection.

2 3 1

Fig. 2. Square four-probe array of contacts on sample surface.

Consider a square sample with four contacts positioned as shown in Fig. 2. The resistance is defined as 푅푖푗,푘푙 = 푉푘푙/퐼푖푗 , where Iij denotes a positive DC current injected from contact i to contact j, and Vkl the DC voltage measured between k and l. Note that there are two distinct measurement configurations; longitudinal and transverse. Using conformal mapping van der Pauw showed that the sheet resistance Rsh can be accurately measured in arbitrary shaped samples provided that (i) the contacts are sufficiently small, (ii) are located at the periphery of the sample, and (iii) the sample is homogeneous with uniform thickness. The general solution for the sheet resistance is

휋 (푅퐴+푅퐵) 푅퐴 푅푠ℎ = 푓 ( ) (4) ln (2) 2 푅퐵 where,

푅퐴 = (푅12,43 + 푅21,34 + 푅43,12 + 푅34,21)/4 and

푅퐵 = (푅14,23 + 푅41,32 + 푅23,14 + 푅32,41)/4

푅 are the two characteristic longitudinal resistances of the sample. 푓 ( 퐴) is a function 푅퐵 of the ratio RA/RB, ranging between 1 and 0 that corrects for sample/contact asymmetry and can be obtained graphically (see Appendix). The resistivity can then be calculated from the sheet resistance if the sample thickness is known, through

휌 = 푅푠ℎ푡. (5)

The Hall resistance is determined by a similar average (푅13,24 + 푅31,42 + 푅24,31 +

푅42,13)/4.

NOTE: From the reciprocity theorem, 푅푖푗,푘푙 = 푅푘푙,푖푗. The permutations of resistance measurements in Eq. 4 serve the purpose of consistency checks.

Questions to consider

(These are provided for your own curiosity but will help in your discussion of the results)

 The linear relationship between the Hall voltage and the applied magnetic field makes Hall effect devices ideally suited to magnetic sensing applications. However, not all materials are suitable for such applications. The most obvious distinction is between metals and semiconductors (why?).  Eqs.1-5 are applicable to a three-dimensional slab of material. How are these expressions modified for a two-dimensional sheet of charge carriers?  In most materials, carrier-phonon scattering limits the mobility at room temperature. Why is phonon scattering important at high temperatures?  What is the significance of sheet resistance?  What is the advantage of four-probe measurements over two-probe measurements? 3. Experiment

In this experiment, you will determine and compare the carrier density and carrier mobility for three different technologically important materials; GaAs, InSb and graphene. GaAs and InSb are model examples of ‘traditional’ bulk semiconductors exhibiting direct band gaps of 1.5eV and 0.17eV at 300K, respectively. InSb has the lightest electron effective mass and largest electron mobility at room temperature of any known semiconductor (e.g. mInSb* = 0.013m0 compared to mGaAs*=0.063m0, where m0 is the free electron mass). Graphene is the monolayer derivative of graphite, constructed from a single atomic layer of carbon atoms arranged in a honeycomb lattice. It was discovered in 2004 as a new phase of crystalline matter by A.K. Geim and K.S. Novoselov (who were awarded the Noble prize for Physics in 2010 as a result), and is one of a new class of truly two-dimensional and flexible materials that have recently emerged. Graphene has a unique band structure described by a linear (rather than parabolic) energy dispersion with zero band gap,

4 which leads to remarkable electronic and optical properties. The 2D nature of graphene makes it very sensitive to electric fields allowing the Fermi level to be continuously tuned via the electric field effect. The Fermi level can be pulled up into the conduction band to make it n-type or pulled down into the valance band to make it p-type by applying a voltage between the graphene and a suitable gate electrode.

The graphene samples provided have been synthesised by chemical techniques (chemical vapour deposition) and are supported by a Si/SiO2 substrate. They are not expected to exhibit the properties of ‘ideal’ defect free graphene. For more information see Ref. [5]. The heavily doped Si layer substrate can be used as a back gate electrode and the SiO2 layer as the insulating layer (see Fig. 4). You will be able to measure the transport properties of the graphene sample as a function of gate voltage.

Vsd graphene

300nm SiO2

Si Vgate

Fig.4. Schematic layer structure of a graphene sample on SiO2/Si substrate with wiring configuration for gated measurements. 4. Equipment

To make measurements of the Hall coefficients and resistivities you have been provided with a set of electromagnets, power supplies, digital multimeters, and a magnetic field probe (that happens to also use the Hall effect for its measurement).

The multimeters, power supplies and magnetic field probes are standard lab equipment and manuals for them are available. Datasheets for the samples and electromagnets are also available.

The electromagnets have been modified to be cooled using fans. These should be connected to their own power supply set to voltage of 12V, just keep it on while running current thought the magnets.

All voltages should be measured with the high input impedance and high-resolution digital voltmeter provided.

IMPORTANT: You can run a current up to 10A through the electromagnets using the power source supplied, but only a current of up to 4mA should be passed through the samples. If you run any more you run the risk of damaging them. There is a battery powered current supply available to allow accurate application of these low currents.

Before trying to measure current with a multimeter, make sure that it can handle the whole current range before connecting to it, otherwise you could damage the equipment.

Questions to consider

(These are provided for your own curiosity but will help in your discussion of the results)

 In the introduction we assumed the magnetic field is uniform across the sample. How uniform is the magnetic field produced by the electromagnets? What effect will this have on your measurements?  What effect will the positioning of the sample within the field have on your measurements? How can you keep this consistent?  Often Iron core electromagnets exhibit hysteresis. Do the electromagnets provided show this effect, and if so, is it going to affect your results? Check the response of the electromagnet to applied current.  You are supplied with a constant current source. Is the output delivering the current it says?  What happens to the samples at high currents (this is a thought experiment!)? 5. Procedure

The first thing to do when you arrive in the lab is to familiarise yourself with the equipment provided. Plan the experiment before you begin taking measurements, considering what variables you need to measure, vary and record. This will save you time in the future. Do not try to analyse the results as you go.

Handle the samples with care. Make sure that all the contacts on the device are all working before you begin (this can be done using a multimeter). Inform the demonstrator if a sample is not working. It is often useful to draw a diagram of the sample with reference to measurement configurations. Make sure the power supply output is off when connecting up Hall samples (sudden voltage spikes can damage samples).

Use the van der Pauw method outlined in Section 2.3 to measure the resistivity and Hall effect in the samples provided and calculate the carrier density and mobility and their uncertainties. Table A1 in the appendix can be used if desired to collate your data.

It is advised to begin with resistivity measurements as these do not require magnetic field. For the graphene samples, it is advised to start with zero gate voltage and to investigate the gate voltage dependence of the properties as additional work if you have time. For gated measurements you will need an additional voltage source - please refer to Fig. 4 and check with a demonstrator that you have wired the sample

6 up correctly. You will need to apply large voltages to observe an effect (-30V to 30V) which if applied incorrectly can damage the samples.

The potential probes on the samples are always slightly misaligned (not symmetric) leading to an additional voltage measured on top of the Hall voltage associated with the finite material resistance. However, the permutations described above removes this voltage (how?). Additional sources of error are discussed in the next section.

As discussed in Section 2.1, the Hall voltage is proportional to magnetic field and current. Rather than trying to map out the full 2D parameter space it is sufficient to hold one of these quantities constant while varying the other. If the relationship in Eq. 1 holds, VH will be linear in the varied quantity and RH can be obtained from the gradient of the line. Consider carefully whether you want to vary the current or the magnetic field. What effects might varying each one have? Whichever you choose to keep constant, you will want to repeat the measurements at a few values of this to make sure there is no systematic shift that would invalidate Eq. 1. Make sure you record data for positive and negative magnetic fields.

Questions to consider

(These are provided for your own curiosity but will help in your discussion of the results)

 Are the electrical contacts to the samples Ohmic? How can you check?  What is a sensible current to use for the measurements? It will not necessarily be the same for each sample (why?). The current should be Ohmic unless you intend to study the properties of ‘hot carriers’.  What happens to the resistivity of the samples in a magnetic field?  The InSb samples are undoped i.e. intrinsic such that n=p. Is the Hall response what you would expect? Why does undoped InSb exhibit n-type properties?  How do your values of mobility compare to those in the literature? Can you comment on any discrepancies? 6. Related effects and systematic errors

Along with the Hall effect, there are three related thermomagnetic effects that can contribute to a transverse voltage: the Nernst effect, the Ettingshausen effect and the Righi-LeDuc effect. These effects arise from an interplay between applied electrical currents, magnetic fields and temperature gradients. Luckily, such a contribution is small, but sadly not negligible. For a good introduction of how these effects are going to have an impact on your measurements, Ref. [2] is recommended. Consider the symmetry of these effects with respect to current and magnetic field and how you may eliminate them. Can you eliminate all of them? Can you use any of the data you plan to take to estimate the coefficients associated with these effects?

Further to these thermomagnetic effects there other ways heating can affect your results: For example, the Hall coefficient and mobility are temperature dependant (particularly in intrinsic materials). The effect is small, but could affect your results and if possible excessive heating during measurements should be avoided.

If the balance of the Hall force and Lorentz force were achieved for all carriers (i.e. if all carriers have the same drift velocity), all carriers would move through the sample unperturbed by the magnetic field. This implies there would be no change in sample resistance when magnetic field is applied. In the case of metals, this is often true. However, in most semiconductors the change in resistance, known as magnetoresistance, can be quite large. For more information see Ref. [3]. Magnetoresistance can occur as a result of sample geometry (through either complete or partial shorting of the Hall field Ey) or physical (intrinsic) origins.

If two or more distinct carrier species are present, Eq. 1 is no longer valid and magnetoresistance increases. Think carefully about whether Eq. 1 is valid for all the samples? If it is not, what is the impact on your determination of carrier density?

The accuracy of the van der Pauw method depends on how well the sample geometry meets the requirements set out in van der Pauw’s theory. The average diameter of the contacts (d) and the sample thickness must be much smaller than the separation of the contacts (L). Relative errors in RH and  are of the order d/L. Can you estimate the errors? (see Ref. [4] for more details).

The Hall effect and magnetoresistance are the basis for most magnetic field sensors used today. Signal and noise are of primary concern to the performance of any sensor. The figure of merit for a magnetic field sensor is the noise-equivalent field (typically given in units of Tesla/Hz) i.e. the minimum magnetic field required to yield a signal-to-noise ratio of 1. The NEF is calculated from

푉표푙푡푎𝑔푒 푛표푖푠푒 (푉) 푁퐸퐹 = . (6) 푅푒푠푝표푛푠푖푣푡푦 (푉/푇)

The most significant contribution to noise is thermal noise (also known as Johnson or Nyquist noise) associated with the random motion of charge carriers in the conductor. The mean square voltage noise is

2 〈푉푡ℎ〉 = 4푘퐵푇푅∆푓, (7) where kB is the Boltzmann constant, T the temperature, R the device resistance and Δf the bandwidth of the noise (Hz). For more information see Ref. [3]. As an extension it is possible to quantitatively discuss the sensitivity of the samples you have measured. Which samples would make the best sensor? Why? What are the merits of using the Hall effect and magnetoresistance for sensors? Ask your demonstrator for guidance. 7. Report

The goal of this experiment is to measure the Hall coefficient and resistivity for each of the samples in order to determine the carrier type, density and mobililty. The values of these properties are very interesting when discussed in the right context. However, the methods used to obtain these values, the handling of both statistical and systematic errors and your data analysis are equally interesting. Emphasis is placed on quantitative discussion and thoughtful analysis/interpretation of results. References

[1] The Hall Effect and Related Phenomena, Putley; Butterworths (1960). pp. 30, 77, 99.

[2] Hall Effect, Olaf Lindberg, 1952, PROCEEDINGS OF THE I.R.E.

[3] Hall effect devices 2nd Ed, R.S. Popovic, IoP publishing.

[4] A method of measuring specific resistivity and Hall effect of discs of arbitrary shape, L.J. van der Pauw, Philips Research Reports, Vol 13 (1958).

[5] A.K. Geim, K.S. Novoselov, The rise of graphene, Nature materials, 6, 187 (2007).

Last revised: Adam Gilbertson 23 September 2014. Appendix

Fig. A1. Van der Pauw correction factor F. Dependence on the ratio RA/RB (taken from Ref. [4]).

Table A1. Material properties input table.

Sample Thickness Sheet Resistivity Hall Carrier Mobility resistance Coefficient density