Hall effect

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We are discussing physics. remember that!!! I would like you to write me 4 pages talking about those Three articles I have attached. if you think you need other sources please do. what I want is a good paper with a very very great explanation. I do not want to see a collection of an academics words. Try to make your writing as the level as the articles. resistivity, conductivity, hall voltage, directions, and hall effect will be involved here a lot.

1- The subject is related to physics, which is Hall Effect.

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1414 November PROCEEDINGS OF THE I.R.E. Hall Effect* OLOF LINDBERGt Summery--The Hall effect is one of the rich sources of informa tion about the conduction properties of semiconductors. The mobility and carrier concentration can be obtained from the Hall constant in conjunction with the resistivity; this cannot be done with the resistivity alone. The mobility is pertinent to the understanding of transistors since such things a s high-frequency cut-off and the intrinsic current gain of the transistor are related to thie property of germanium. The Hall effect and associated thermomagnetic and galvanomagnetic (Ettingshausen, Nernst, Righi-Leduc, and Ohmic) effects are discussed The elimination of the effect of associated phenomena from the Hall measurement can be achieved in several ways. Some of the methods which are used today in the study of germanium are discussed, and typical apparatus is described. INTRODUCTION HE TRANSISTOR makes use of the special conduction properties of semiconductors to gain the T advantages it has over the vacuum tube. To understand the advantages and limitations of the transistor one must understand these special conduction properties. For example, the high-frequency cut-off of the type A transistor can be predicted from the injected carrier’s transit time which is related to the mobility. The intrinsic current gain in the type A transistor is related to the ratio of the mobility of electrons to the mobility of holes. In dealing with current flow in semiconductors, one must take account of the fact that other carriers than electrons may be present. The nature of these carriers (whether they are electrons or holes), the number of carriers per unit volume, and the ease with which they respond to an applied electric field (mobility) are all important quantities to know. The Hall effect provides a direct measurement of the carrier type and concentration and, in conjunction with the resistivity, yields the mobility. The density of carriers in semiconductors is determined in part by the density of foreign atoms in the material (for example, aluminum in germanium). Thus the Hall measurement is used to determine the density of impurity atoms. This measure of impurity density in high-purity samples is several orders of magnitude more sensitive than the best chemical procedures. Because of this, the measurement of the Hall constant is one of the basic procedures in experimental studies of semiconductors. The Hall effect occurs when a substance carrying a current is subjected to a magnetic field perpendicular to the direction of the current, If the current is flowing in the x-direction and the magnetic field is applied in the s-direction, a potential gradient will appear across the sample in the y-direction. This transverse potential gradient is found to be proportional to the product of the current density in the sample and the applied mag* Decimal classification R282.12. Original manuscript received by the Institute, June 30, 1952. t Westinghouse Research Laboratories, Eaet Pittsburgh, Pa. netic field ; the constant of proportionality is called the “Hall constant.” Mathematically this can be expressed as follows: Grad VII = - RiH. (1) Grad VH-- (the transverse potential gradient) = - EH, where EII is the Hall field. i-the current density. H-the applied magnetic field. R-the Hall constant. If the sample is a rectangular solid of width a and thickness b and if the distribution of current is assumed uniform, (1) can be rewritten in terms of the total current I, the Hall voltage VH, and the dimensions of the sample. - RIB Grad V”H = k - a ab - RIH VII =-, b (2) The Hall effect can be explained on the basis of the particle nature of conduction. The current consists of streams of charged particles drifting under the influence of the electric field. When not under the influence’ of the magnetic field, the current flows longitudinally in the sample. On application of the magnetic field, the current carriers experience a force e/c(νxH) and are swept to the edges of the sample. The charge continues to build up on the edges of the sample until the field due to the nonuniform charge distribution exerts a force equal to the deflecting force of the magnetic field. L Fig. l-Rotation of equipotentials by Hall effect, The Hall effect rotates the equipotentials so that th ey are no longer normal to the current flow. Fig. 1 shows that the equipotentials are no longer normal to the current flow, but have been rotated through an angle 0, called the “Hall angle.” Examination of the vector diagram of the fields in Fig. 2 (opposite page) shows that the Hall angle 6 isdetermined by the following relation: lf52 Lindberg: Hall Effect EH tan 0 = x- E 0 I 3* 1 R=--, for small angles. (3) RiHu Now En = RiH and E, = 2, thus-θ =-= RHa, u i where Q is the conductivity. Fig. 2-Vector diagram for the Hall effect. The Hall angle 8 is the angle of rotation of equipotentials. EH is the Hall field if the carriers are e1ectrons. It is possible to calculate an expression for the Hall constant on the basis of the particle theory. In the derivation we assume that only electrons are present; but if only holes are present, the procedure is analogous. From the discussion above, the condition for the steady state is that the deflecting force of the magnetic field on a current carrier just equals the force exerted by the transverse electric field due to the charge build-up at the edges of the sample. This condition can be met mathematically by setting the y-component of the electric-field force equal in magnitude but opposite in sign to the force experienced by a charge moving in the magnetic field. V.H. eElr = e c ’ (4) i.H* EH - - , the current density i, = ncv= ncc (5) where n the carrier concentration; i, and II, can be replaced by i and H since i, = i, = II, = E?,, = 0. iH EH = - = RiH net 1 R =-* net 1415 (6) The Hall constant derived by this method is valid only insofar as the particle picture of conduction is valid; that is, it applies to simple metals and impure semiconductors, To obtain an expression of more general applicability the mechanism of conduction must be examined more closely on the basis of a Boltzmann distribution of velocity of carriers. The result of such an investigation gives a relation which is generally used in semiconductor work when only one type of carrier is involved.’ 1 W. Shockley, “Electrons and Holes in Semiconductors,” D. Van Nostrand Co., Inc., New York, N. Y., p. 277; 1950. 8 net From the measurement of the Hall constant, the carrier concentration can be determined. In an impuritytype semiconductor the carrier concentration is determined by the density of the dominant impurity. For example, in germanium at room temperature the intrinsic carrier concentration is approximately 5 X 1013 carriers per cubic centimetcr, and impurity densities ten times as great are commonly found: thus it introduces a small error to attribute the entire carrier concentration to impurity atoms. From (6), the carrier concentration is proportional to l/R. This gives a quantitative measure of the impurity density. The value of this procedure can be seen from a calculation of the per cent impurity concentration for an impurity density of 4 X 1016. There are approximately 4 X 1022 atoms of germanium per cubic centimeter; thus concentrations of one part in 10’ are commonly met. To use chemical analysis to determine such impurity concentrations is a hopeless task. The Hall voltage, on the other hand, becomes larger for lower impurity concentration, and therefore more easily measured. The carrier concentration does not give the complete picture of the conduction properties. The conductivity u is related to the carrier concentration, the charge of the electron and the ease with which the carriers move in an electric field (mobility). The latter is defined as the steady-state average velocity of the conducting particle (cm/sec) in unit electric field (1 volt/cm). In germanium the mobility will vary from 1,000 to 3,600 cm/sec per volt/cm. In a semiconductor with both negative and positive (holes) carriers the conductivity must be a function of the concentration and mobility of both holes and electrons. u = nlelbh + PlelcIp hh n-electron concentration (no/cc). p-hole concentration (no/cc). I.c,,-electron mobility (up to 3,600 cm*/volt-see in germanium). pp-hole mobility (up to 1,700 cm*/volt-set in germanium). In the range where the concentrations of holes and electrons are about the same this expression for conductivity must be used. At the operating temperature of a semiconducting device, the concentration of one type carrier is much greater than the concentration of the other so that the conductivity will be dependent on only one term or the other in (8). For p-type sample : u = p e N,,. For n-type sample: u = 7t IeI P,,. Carrier concentration can be determined from the conductivity if the mobility is known or a value is assumed. The Hall effect gives a method for determining the mobility and the carrier concentration for both n-type and p-type semiconductors without assumptions. November PROCEEDINGS OF THE I.R.E. 1416 For an n-type sample the Hall constant is negative since e is negative. - .7r 3 1 R=--. 8 (9) 4elc For a P-type sample the Hall constant is positive because the charge of a hole is positive. 31 1 R I---* 8 pet (10) The Hall-angle formula differs for holes and electrons since mobility is different for holes and electrons. H/h lt-type: e = c W HP%! p-type: e = c (12) effect, the Nernst cffcct, and the Righi-Leduc effect. These effects will give rise to a temperature gradient or a potential gradient when either an electric current or a thermal current is subjected to a magnetic field perpendicular to the direction of current flow. A tcmperature gradient as well as a potential gradient will cause an error in the Hall-effect mcasurcmcnt. In Fig. 4, junction A is at temperature T2 and junction B is at temperature T1, which is less than 1’2. Since the probe material is in general not the same as the sample material, the probes and the sample form a thermocouple and produce a voltage dependent in sign and magnitude on the materials of the probes and the sample. Solving (9) and (10) for the carrier concentrations and (11) and (12) for the mobilities, the following expressions are obtained: 31 -1 3* 1 P =-8 Ret f$3-8 Ret ‘!I f In measurements of the Hall voltage, certain associated effects give rise to potentials which must bc corrected in order to avoid error in the measured value. The largest of these effects is the potential which appears because of the experimental difficulty in aligning the measuring probes on the same equipotential plane. , 1 Fig. 4--Ettingshauscn effect. The Ettinghousen effect c a u s e s the edge at probe A to be at temperature 1’1 grcatcr than 7’t the tcmperature of the ot her edge. In the Ettinghousen effect a p e r m a n t l y maintained temperature grndicnt will appear if an electric current is subjected to a mngnctic field perpindicular to its direction of flow. The temperaturc gradient is found to be proportional to the product of the current density and the magnetic field.2 ∆T - = PIN a PIN A I ix--. ab (13) ∆T = _ I b tB Fig. 3-Source of fR drop error. It is difficult to align the probes A and B so that no voltage will be measured in the absence of a magnetic field. If Fig. 3, A and B are probes for measuring the Hall potential. With no field applied, the equipotentials are ideally planes perpendicular to the lines of current flow If probes ;Q and B are not exactly on the same equipotential, a potential will be measured between them, giving a constant error to the Hall voltage measured. This voltage can easily be of the order of magnitude of the Hall voltage itself. This effect is sometimes referred to as the ‘IR drop.” The IR drop is dependent only on the current and the conductivity of the sample, not being affected by a reversal of magnetic field. In addition to the IR drop, there are three thermomagnetic or galvanomagnetic effects, the Ettingshausen Ai”- the cliffcrcnce of tcmpcraturc bctwccn the edges of the sample. i-current density. I-total current. II-magnetic field perpendicular to the tlircction of current flow. P-Ettingshauscn coefficient. ’ u-width of sample. b-thickness of sample. As in the case of the Hall effect, uniform current distribution and a rectangular solid sample have been assumed. The Nernst effect and the Righi-Lcduc effect are similar to the Hall effect and the Ettingshauscn effect except that they arc produced by a thermal current and a perpendicular magnetic field rather than an elec* P. W. BridEman, “The Thermodynamics of Electrical Phenomena in Metals,” Macmillan Co., New York, N. Y., pp. 135-138; 1934. . . 1 \ I. 1 ’ , !952 Lindberg: Hall Efecl 1417 The Righi-Leduc effect produces a transverse temperature gradient when a longitudinal thermal current flows in a magnetic field. As in the Ettingshausen effect a transverse thermal current will flow from the cold edge to the hot edge while the temperature gradient is being built-up. The thcrmomotive force set up by this transverse thermal current (Righi-Leduc effect) is in opposition to the longitudinal thermal current, and is &-Nernst transverse potential gradient. therefore able to extract energy from the thermal curw-thermal current density. rent and maintain itself. I\‘--thermal conductivity of the sample. As has been explained above, the IX drop, the @-Ncrnst coefficient. Ettingshausen effect, the Ncrnst effect, and the RighiThe Righi-Leduc effect produces a temperature Leduc effect all result in potentials at the probes gradient in the y-direction when a thermal current flows measuring the Hall potential. Consequently, to measure in the x-direction, and a magnetic field is applied in the Hall constant, these sources of error must be eliminated z-direction.* or minimized. The most common method of making Hall WU measurements utilizes dc magnetic fields and dc curAT (15) rents. The experimental setup is illustrated in Fig. 4. a = 57 The probes A and B for measuring the Hall voltage are AT-difference in the temperature between the placed in contact with the sample as shown. The curedges of the sample. rent leads are soldered to the ends of the sample. A dc S-Righi-Lcduc coefficient magnetic field is applied in the e-direction and a curU, w, M, I El = vlf + ‘VE + VN + VRL + VIR, and the voltage due to the Ettingshausen temperature VN + VJZL - VI’m, t+IJ’I, -I) & = - 1//I - T/R + gradient, with the result that the Nernst and Righi(-H, -I) Ea = Vm + Vs - VN - VRL - VIR, Leduc and the IR drop have been eliminated as a source of error and only one measurement need be made. (-H, +I> 6 = - I/‘lf - VE - VN - VRZ, + VIR, The Ettingshausen effect has associated a certain which yield time delay which is required for the temperature gradient to become established. This is accounted for by the El - El + Es - EC . vrr + vs = (17) exponential in (18). For example, in bismuth the time 4 required for the temperature gradient to reach its maxiThis method has been outlined in detail because it is mum value has been measured by one observer to be a standard method of making Hall measurements in about one minute. (This corresponds to w~=O.l.) This the presence of the associated thermomagnetic and makes it apparent that an ac Ettingshauscn effect galvanomagnetic effects. The method is not completely would not be detectable if the period were sufficiently satisfactory since it doesn’t separate the Ettingshausen short compared to the response time of the effect. This effect from the Hall effect and since a series of four is the basis of a measurement method which will sepameasurements requires sufficient time for possible rate the Hall effect and the Ettingshausen effect. If the changes to occur in the experimental conditions, espe- field and the current are of different frequencies, then cially if measurements are being made as a function of all the resulting signals on the Hall probes will be ac temperature. I=I’ sin od H-H’ sin wJ 4 The measurements must be made rapidly so that the part of the temperature gradient dbe to the Peltier effect does not have time to reverse after the current is reversed. RIH VII - - = y [cos (w, - w)t - cos (WI + w&l b I \ Lindberg: EZaEl Efect 1922 1419 I=lma Ve = D b = 1 mm, As in the previous case, b is sample thickness and oo is a constant of the material. D is a constant to account for the thermoelectric effect, and depends on the probe material and the sample material. VIR = A sin wrl. VII = B sin WZ! VRL = C sin or!. If the difference frequency is high compared to WO, selecting either the sum or the difference frequency with a suitable filter, the measurement will give the Hall voltage directly. This method gives the Hall voltage free from all associated first-order effects, and requires but a single measurement to do so. The Hall constant is The primes indicate peak values and b is the sample thickness. There are other possible methods for making Hall measurements: these utilize various combinations of ac and dc for the held and the sample current. The methods described above are typical and an analysis of any of the other methods would be similar. Fig. S-Typical dc Hall effect apparatus. This is the most common and the simplest method of measuring Hall effect. Fig. 5 illustrates schematically the apparatus for making dc Hall measurements. The dc electromagnet produces a field up to 5,000 gauss in the air gap. Usually a field of 1,000 gauss provides a Hall voltage large enough to be readily detected. The dc power supply, usually a battery, allows a sample current of 1 or 2 ma. The current should be small to produce as little I2R heating of the sample as possible and to keep the Peltier heating at the junctions small, but it must be large enough to produce a detectable Hall voltage. A Hall voltage of about 100 pv is desirable if it is to be measured with a potentiometer. If the results of a Hall measurement are as follows, vg = 100 pv H = 1,000 gauss - lO+ volts 10-l cm - Vnb R =I - x 10’ s (lo*) lo” gauss lOma amp HZ R - - 1 , 0 0 0 cm* coulomb ’ The factor lOa converts voltage and current from electrostatic units to practical units. Equation (7) determines the carrier concentration; 3* 1.17 n=-= 1@X1.6X10-‘D 8Rce = 7 X 101scarriers/cc. In (7) the electronic charge is in electrostatic units: in the expression used above, the charge is in the practical system of units (coulombs). The carriers arc electrons since the Hall constant is negative. -WI Fig. 6-Typical ac Hall effect apparatus. Fig. 6 shows an apparatus for making ac Hall measurements. The electromagnet is supplied by a singlephase, 60-cycle power source, and produces a magnetic field in the gap up to about 5,000 gauss. The 70-cycle generator produces a current of 1 or 2 ma in the sample. The Hall-voltage component (the difference frequency) is amplified by a selective amplifier and the output fed into a measuring device, such as a recording potcntiometer. The system is required to be stable so that accurate calibration can be obtained. This method is especially adapted to making measurements as a function of temperature where a continuous record of the variation of Hall voltage is desired. The measurement of the Hall effect may be made in a variety of ways. The ac Hall apparatus is probably the best type for rapid determinations. The measurement is extremely useful in the determination of impurity concentration as such a deduction requires no assumptions as to other properties of the material. BI B L I O G R A P H Y 1. H. B. Callen, “Application of Onsafer’s reciprocal to thermomagnetic and galvanomagnetrc effects, Phys. RI%, vol. 73, p. 1349; 1948. 2. H. E. Callen, #Note on the adiabatic thermomagnetic effects,” Phys. Rev., vol. 85, p. 16; 1952. and Thermomagnetic Ef3. L. L. Campbell, “Galvanomagnetic fects,” Langmans, Green and Co., New York, N. Y.: 1923. 4. A. Sommerfeld and N. H. Frank, “The statistrcal theory of thermoelectric, galvanomagnetic, and thermomagnetic phenomena in metals,’ Ecu. Mod. PlryS., Vol. 3, p. 1; 1931.
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