Physics 445LW Modern Physics Laboratory Hall Effect

Physics 445LW Modern Physics Laboratory Hall Effect Introduction If an electric current flows through a conductor in a magnetic field, the magnetic f...
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Physics 445LW Modern Physics Laboratory Hall Effect

Introduction If an electric current flows through a conductor in a magnetic field, the magnetic field exerts a transverse force on the moving charge carriers. This force tends to push the carriers to one side of the conductor. The Hall effect is used to measure charge carrier concentration (n), charge mobility (µ) as well as to determine the sign of the charge carriers. In these experiments you will investigate this phenomenon by studying a sample of germanium. Theory1 Evolution of Resistance Concepts Electrical characterization of materials evolved in three levels of understanding. In the early 1800s, the resistance R and conductance G were treated as measurable physical quantities obtainable from two-terminal I-V measurements (i.e., current I, voltage V). Later, it became obvious that the resistance alone was not comprehensive enough since different sample shapes gave different resistance values. This led to the understanding (second level) that an intrinsic material property like resistivity (or conductivity) is required that is not influenced by the particular geometry of the sample. For the first time, this allowed scientists to quantify the current-carrying capability of the material and carry out meaningful comparisons between different samples. By the early 1900s, it was realized that resistivity was not a fundamental material parameter, since different materials can have the same resistivity. Also, a given material might exhibit different values of resistivity, depending upon how it was synthesized. This is especially true for semiconductors, where resistivity alone could not explain all observations. Theories of electrical conduction were constructed with varying degrees of success, but until the advent of quantum mechanics, no generally acceptable solution to the problem of electrical transport was developed. This led to the definitions of carrier density n and mobility µ (third level of understanding), which are capable of dealing with even the most complex electrical measurements today. The Hall Effect and the Lorentz Force The basic physical principle underlying the Hall effect is the Lorentz force. When an electron moves in an applied magnetic field, it experiences a force acting normal to both directions and moves in response to this force. For an n-type, bar-shaped semiconductor such as shown in Fig. 1, the carriers are predominately electrons of bulk density n. UMKC, Department of Physics

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Figure 1. Standard geometry for a Hall effect measurement.

We assume, see Figure 1, that a constant current I flows along the x-axis from left to right in the presence of a z-directed magnetic field. Electrons subject to the Lorentz force initially drift away from the current line toward the negative y-axis, resulting in an excess surface electrical charge on the sides of the sample. This charge results in a voltage across the two sides of the sample, which is called the Hall voltage, VH. (Note that the force on holes is toward the same side because they would move in the opposite direction, and due to their positive charge.) The magnitude of the Hall voltage is equal to IB/qnd, where I is the current, B is the magnetic field, d is the sample thickness, and q (1.602 x 10-19 C) is the elementary charge. In some cases, it is convenient to use layer or sheet density (ns = nd) instead of bulk density. One then obtains the equation

n s = IB /qV H

(1)

Thus, by measuring the Hall voltage VH and from the known values of I, B, and q, one can determine the sheet density ns of charge carriers in semiconductors. If the measurement apparatus is set up€as described later in Section III, the Hall voltage is negative for n-type semiconductors and positive for p-type semiconductors. The sheet resistance RS of the semiconductor can be conveniently determined by use of the van der Pauw resistivity measurement technique. Since sheet resistance involves both sheet density and mobility, one can determine the Hall mobility from the equation

µ = V H /RS IB = 1/(qn S RS )

(2)

If the conducting layer thickness d is known, one can determine the bulk resistivity (ρ = RSd) and the bulk density (n = nS/d).

€ UMKC, Department of Physics

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Generally, the Hall angle is defined as the angle between the current density vector and the local electric-field vector in the presence of a perpendicular magnetic field. Since it is independent of a particular sample geometry and even of material inhomogeneities, the Hall angle represents a fundamental intrinsic physical quantity in galvanomagnetic transport.

Formulae Needed Hall coefficient

RH = VHw/BI m3C-1 where VH = Hall voltage in volts, w = width of the sample in m, B = magnetic flux in Tesla

Concentration of charge carriers per unit volume

n = 1/eRH

Resistivity of the sample

r = Vlwd/Il Vl = voltage between two points l cm apart on one face of sample, d = thickness of sample

Mobility

µ = RH/r m2V-1s-1

Hall angle

φH = tan-1(µB)

For our sample

w = 4 mm l = 6 mm d = 0.5 mm B = ______ Gauss = ______ x10-4 Tesla

where e = 1.6 x 10-19 C

Experimental Apparatus and Procedures2 1. Set the Gauss meter to X1. 2. Switch on the Gauss meter and carefully adjust it to zero.

Figure 1. Gauss meter UMKC, Department of Physics

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DO NOT SWITCH ON ELECTROMAGNET (discharge tube power supply) AT THIS TIME. 3. Set multimeter 1 to direct current amps. 4. Switch on the constant current source. 5. Adjust the constant current source so that multimeter 1 shows 5mA. This occurs at about 1 volt. 6. Keep magnetic field at zero as measured by Gauss meter.

Figure 2. Constant current power supply DO NOT SWITCH ON ELECTROMAGNET (discharge tube power supply) AT THIS TIME. 7. Press and hold the range switch on multimeter 2 while turning selector dial to mV. If the meter shows AC, press the yellow switch. 8. If the meter dows not read zero mV, ask the lab assistant to set the zero adjust. DO NOT SWITCH ON ELECTROMAGNET (discharge tube power supply) AT THIS TIME.

Figure 3. Discharge tube power supply

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Figrue 4. Magnet and sample holder 9. Adjust the DC volts knob on the constant current power supply downward until multimeter 1 reads zero current. 10. The voltage monitor selector should be set to 80 V. 11. Rotate all the selector knobs on the electromagnet power supply counter clockwise to the minimum. 12. Turn on the electromagnet power supply. 13. Rotate the red selector knob very slowly clockwise until 17 volts shows. 14. Rotate the black DC amps selector knob until the meter reads 3.5 amps. 15. Set the Gauss meter selector switch to X10. 16. Slowly increase the DC amps selector knob until the Gauss meter reads between 153 and 157 Gauss. 17. Do not change the current in the electromagnet during the experiment. In other words, keep the magnetic field constant. 18. Vary the current on the constant current power supply in small increments. Note the current I(mA) on multimeter 1 and the Hall voltage (mV) on multimeter 2. Record these values. Take at least seven readings. 19. Reverse the direction of the magnetic field by interchanging the "+" and "-" connections of the coils, that is, by interchanging the red and black wires to the coils of the electromagnet. Now note the Hall voltages for the same values of current as in step 18. Note: use only the magnitude and not the sign of the Hall voltage. Remember that all procedures, settings, and readings that you take must be recorded in your lab notebook. Results Include the following items in your lab report: • Hall coefficient • Sign of Hall coefficient and type of material • Concentration of charge carriers per unit volume • Resistivity of the material • Mobility • Hall angle • A graph of Hall voltage vs current

[1] http://www.eeel.nist.gov/812/effe.htm [2] Cenco Physics Instruction Manual: Hall Effect WLS1800-24

UMKC, Department of Physics

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