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The Hall Effect C1 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. 1 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 2 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. 1 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) 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 44 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 3 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.
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