1

The Hall Effect

C1

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.

2

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 Handling

X

Hazardous Substances

Electrical X

Other

Lone Working Permitted?

Yes No Permit-to-Work Required?

Yes No

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

1

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.

2

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 𝐼𝑥 = 𝐽𝑥𝑤𝑡 = 𝑛𝑞𝑣𝑥𝑤𝑡, 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, 𝑭 = 𝑞 (𝒗 × 𝑩), 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, 𝑉𝐻 = −𝑤𝐸𝑦 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

𝑉𝐻 = (1

𝑛𝑞)

𝐼𝑥𝐵𝑧

𝑡. (1)

Equation 1 shows that the Hall voltage is proportional to both the applied magnetic

field and current. The quantity 1 𝑛𝑞⁄ = 𝑅𝐻 is the Hall coefficient, which is an intrinsic

property of the material. It is often convenient to consider the sheet carrier density

𝑛𝑠ℎ = 𝑛𝑡 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 𝑣𝑑 = 𝜇𝐸. 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

𝜌 = 1𝑛𝑒𝜇⁄ . (2)

3

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

𝜇 = |𝑅𝐻

𝜌⁄ |. (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.

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 𝑅𝑖𝑗,𝑘𝑙 = 𝑉𝑘𝑙/𝐼𝑖𝑗 , 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

𝑅𝑠ℎ =𝜋

ln (2)

(𝑅𝐴+𝑅𝐵)

2𝑓 (

𝑅𝐴

𝑅𝐵) (4)

where,

𝑅𝐴 = (𝑅12,43 + 𝑅21,34 + 𝑅43,12 + 𝑅34,21)/4

and

𝑅𝐵 = (𝑅14,23 + 𝑅41,32 + 𝑅23,14 + 𝑅32,41)/4

1

2

3

4 4

4

are the two characteristic longitudinal resistances of the sample. 𝑓 (𝑅𝐴

𝑅𝐵) is a function

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

𝜌 = 𝑅𝑠ℎ𝑡. (5)

The Hall resistance is determined by a similar average (𝑅13,24 + 𝑅31,42 + 𝑅24,31 +

𝑅42,13)/4.

NOTE: From the reciprocity theorem, 𝑅𝑖𝑗,𝑘𝑙 = 𝑅𝑘𝑙,𝑖𝑗. 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,

5

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.

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.

300nm SiO2

Si

graphene Vsd

Vgate

6

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

7

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?

8

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

𝑁𝐸𝐹 =𝑉𝑜𝑙𝑡𝑎𝑔𝑒 𝑛𝑜𝑖𝑠𝑒 (𝑉)

𝑅𝑒𝑠𝑝𝑜𝑛𝑠𝑖𝑣𝑡𝑦 (𝑉/𝑇). (6)

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

⟨𝑉𝑡ℎ2 ⟩ = 4𝑘𝐵𝑇𝑅∆𝑓, (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

9

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

10

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 resistance

Resistivity Hall Coefficient

Carrier density

Mobility