Hall Effect in Semiconductors PH5XX Indian Institute of Technology Gandhinagar Department of Physics 2014
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Obje Objectiv ctive e • To measure the Hall coefficient in both P & N type semiconductors.
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Theory
When a conductor (o ra semiconductor) is placed in a magnetic field, which is perpendicular to the direction of the current, a voltage is induced across the specimen in the direction perpendicular to both the current and the magnetic field. The reason behind the build up of this voltage is because the moving charges are forced to one side of the specimen by the applied magnetic field. The charges accumulate on a face of the specimen until the electric field associated with them is large enough to cancel the force exerted by the magnetic magnetic field. This corresponds corresponds to the steady state condition condition for that value of the applied magnetic field. We are concerned only with the steady state condition. Besides being proportional to the magnetic field, the hall voltage is also proportional the applied current current and depends on the nature and the shape of the sample as well. well. Howeve However, r, if these factors factors are kept constant the Hall voltage is an accurate measure of the magnetic field. The basic physical physical principle underlying underlying the Hall effect is the Lorentz Lorentz force. For an electron electron e moving with a velocity v in the influence of electron field E & & magnetic field the force experienced by = −e +v × the particle is given by F − e[E + E B ]. Where Where v is the particle velocity and e and e = = (1. (1.602 602× × 10−19 C ) this force is termed as Lorentz force. For an n-type, bar-shaped semiconductor such as that shown in Fig.1, the carriers carriers are predominat predominately ely electrons electrons of bulk density density n. We assume that a constant 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 direction toward the negative y-axis, resulting in an excess negative surface electrical charge on this side of the sample. This charge charge results in the Hall voltage, voltage, a poten p otential tial drop across the two sides of the sample. (Note that the force on holes is toward the same side because of their opposite velocity and positive charge.) charge.) This transverse transverse voltage voltage is the Hall voltage voltage V H IB/qnd, H and its magnitude is equal to IB/qnd, where I is the current, B is the magnetic field, d is the sample thickness, and q = q = (1. (1.602 × 10−19 C ) is the element elementary ary charge. charge. In some cases, it is convenien convenientt to use layer layer or sheet density density (ns ( ns = nd) nd) instead instead of bulk density density. One then obtains obtains the equation equation n n s = IB/q = IB/q |V H H |
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Figure 1: Schematic of the Hall effect in a long, thin bar of semiconductor with four ohmic contacts. The direction of the magnetic field B is along the z-axis and the sample has a finite thickness d. The Lorentz force on a charge carrier is = e(E + vxB) F
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The steady state Hall electric field in the Y-direction is given by the condition F y = e(E y − vx Bz ) = 0
(2)
For the given geometry. Thus E y = v x Bz but the conductivity σ = neµ where n is the carrier concentration and µ is the mobility of the charge species, i.e. velocity per unit field. So s = nev e J o r sE x = nevx gives the current density J x along x direction. Substituting vx = ne we have E y = J x Bz is called the Hall coefficient. x
x
x
1 (3) ne The Hall coefficient can take both positive and negative values depending on the type of charge holes and electron respectively. Besides the concentration of the charge carriers the thickness of the sample also affects the hall voltage, lower the carrier concentration and thinner the specimen the higher is the Hall voltage. The external variables magnetic field and the sample current. Hall voltage is given by. RH =
E y (4) w Where w is the distance between Hall voltage probes, i.e. the width of the sample, we have V H =
V H =
RH IBz V H d orRH = d IBz 2
(5)
where d is the thickness of the specimen in the direction of the magnetic field. Solving for B z we get V H d Bz = (6) RH I
Figure 2: Experimental set-up
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Experimental Set-up
The Hall effect probe consists of InAs film with four electrically attached conductive leads. The film is 500 nm thick and is covered with a protective epoxy resin. The film and its conductive leads are mounted on a plastic handle. The electrical probes in the sample are shown in figure (3) Make the connections as shown in figure 4. Switches are provided to revere both the sample current and the magnet field. The Hal voltage developed depends on the angle θ between the magnetic field and the normal to the plane of the film. It is maximum when theta= 180o or0o for a sample current (say 1A) rotate the probe using the protractor arrangement till V H is maximum. Do not exceed the sample current beyond 2.5 A or you will burn the hall probe, Switch off the current when not making measurement
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sample.JPG
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Experiments to be performed • Measurement of RH and B :- The measured Hall voltage V H is a function of B and I for a given specimen (i.e. given d and RH ). In practice, thermal and misalignment voltages are superimposed on the Hall voltage and some times become comparable. To minimize the error form the thermal and misalignment voltages one has to reverse the direction of sample current and magnetic field respectively and calculate V H using the relation V H =
1 [V H (B, I ) − V (B, −I ) − V H (−B, I ) − V H (−B, −I )] 4
(7)
Keeping I at any convenient value (¡ 2.5 mA) aet the magnet current to a value certain value I 1 and measure V H (B, I ) where B is the magnetic field corresponding to magnet current I 1 . Then reverse the direction of sample current I and measure V H (B, −I ). Repeat the above two observations after reversing the direction of the magnet current and get V H (−B, −I ) and V H (−B, I ). Follow this sequence by varying I 1 from 0 to 1.5 A in steps of 0.1 A. Repeat this procedure for at least three values of the sample current I (1, 105, 2.0 mA) par Plot V H Vs B (use magnet calibration curve). From the slope (Eq 6), knowing I and D, find R H and hence calculate carrier concentration ’n’ (Eq 3). Knowing n one can use the magnet calibration graph to find out any unknown B. • Rotation of the probe The Hall voltage depends on the angle θ between the magnetic field and the normal to plane of the sample. It is maximum when θ is 0o or180o . For a given value of the magnetic field and the sample current, measure V H (by taking all four combinations of B and I) as a function of θ. Plot V H vsθ comment on the shape of the curve.
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Procedure 1. Connect the width wise contacts of the Hall Probe to the terminals marked ’Voltage’ and lengthwise contacts to terminals marked ’Current’. 4
2. Switch ’ON’ the Hall Effect set-up and adjustment current (say few mA). 3. Switch over the display to voltage side. There m ay be some voltage reading even outside the magnetic field. This is due to imperfect alignment of the four contacts of the Hall Probe and is generally known as the ’Zero field Potential’. In case its value is comparable to the Hall Voltage it should be adjusted to a minimum possible (for Hall Probe (Ge) only). In all cases, this error should b e subtracted from the Hall Voltage reading. 4. Now place the probe in the magnetic field as shown in fig. 3 and switch on the electromagnet power supply and adjust the current t o any desired value. Rotate the Hall probe till it become perpendicular to magnetic field. Hall voltage will be maximum in this adjustment. 5. Measure Hall voltage for both the directions of the current and magnetic field (i.e. four observations for a particular value of current and magnetic field). 6. Measure the Hall voltage as a function of current keeping the magnetic field constant. Plot a graph. 7. Measure the Hall voltage as a function of magnet ic field keeping a suitable value of current as constant. Plot graph. 8. Measure the magnetic field by the Gauss meter.
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Questions 1. If you can not get InAs film ass used in this experiment what easily available would you use to demonstrate Hall effect? would you choose a metallic sample? why? 2. Why do you reverse both the sample and magnet current in this experiment? 3. Why is it advantageous to have a thin sample in Hall effect measurement? 4. What information do you get about a solid from Hall effect measurement? applications of Hall effect in instrumentation?
What are the
5. Why the resistance of the sample increases with the increase of magnetic field? 6. Why the Hall voltage should be measured for both the directions of current as well as of magnetic field? 7. A relativistic particle of mass m and charge q with initial velocity vo is moving in the influence & B fields, say (E = E xˆi)&(B = Bz zˆ). What will be the trajectory of the of cross field E particle ? 8. what will be the trajectory of a non relativistic charged particle ?
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References • C. Kittel ”Introduction to solid state Physics” • W.Angrist, Scientific American • E.M.Putley ”The Hal Effect and Related Phenomenon” • Fundamentals of semiconductor Devices, J.Lindmayer and C.Y. Wrigley, Affiliated East-West Press Pvt. Ltd., New Delhi.
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APPENDIX 1. Sample Details (a) Sample : InAs thin film (b) Thickness (z) : 5X 10−2 cm. (c) Resistivity (r) : 10 ohm. cm. or 10 volt coulomb−1 sec cm (d) Conductivity (s) : 0.1 coulomb volt−1 sec−1 cm−1
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