EXAMINING THE CHARGE CARRIERS; THE HALL EFFECT

Transcription

1 MISN EXAMINING THE CHARGE CARRIERS; THE HALL EFFECT l d y b B z I x EXAMINING THE CHARGE CARRIERS; THE HALL EFFECT by Peter Signell 1. Introduction. Why We Study the Hll Effect b. Mutully Perpendiculr I, B, V c. An E Field so Lorentz Force Delineting the Conditions. Mesurement Lyout b. Force Independent of Chrge Sign c. Downwrd Drift Reches Equilibrium The Equilibrium Equtions. Equilibrium E nd v b. Equilibrium Voltge c. Crrier Velocity d. Crrier Density; the Hll Constnt e. Actul Mesurements Acknowledgments A. Hll Constnt for Two Crriers Project PHYSNET Physics Bldg. Michign Stte University Est Lnsing, MI 1

2 ID Sheet: MISN Title: Exmining the Chrge Crriers; The Hll Effect Author: Peter Signell, Dept. of Physics, Mich. Stte Univ Version: 3/1/2000 Evlution: Stge 0 Length: 1 hr; 16 pges Input Skills: 1. Stte the expression for the force (Lorentz Force) on chrged prticle in electric nd mgnetic fields (MISN-0-122). 2. Stte the expression relting set of chrges nd their common velocity to their equivlent vlue s current (MISN-0-123). 3. Vocbulry: conduction in metls (MISN-0-118). Output Skills (Knowledge): K1. Strting from the Lorentz force, derive the equtions which give the drift velocity nd density of the microscopic chrges constituting n electricl current in terms of the observble mcroscopic Hll effect quntities. K2. Outline briefly how one deduces from the Hll effect tht the chrge crriers in copper nd silver re essentilly the vlence electrons of the constituent toms. Output Skills (Rule Appliction): R1. Use the (given) mesured vlue of the Hll constnt to clculte the drift velocity of the chrges in specified current in specified conductor. Externl Resources (Optionl): 1. C. M. Hurd, The Hll Effect in Metls nd Alloys, Plenum Press (1972). For ccess, see this module s Locl Guide. 2. F. Bitter, Currents, Fields, nd Prticles, Wiley (1957). For ccess, see this module s Locl Guide. 3. C. Kittel, Introduction to Solid Stte Physics, John Wiley nd Sons, New York (1986), p For ccess, see this module s Locl Guide. THIS IS A DEVELOPMENTAL-STAGE PUBLICATION OF PROJECT PHYSNET The gol of our project is to ssist network of eductors nd scientists in trnsferring physics from one person to nother. We support mnuscript processing nd distribution, long with communiction nd informtion systems. We lso work with employers to identify bsic scientific skills s well s physics topics tht re needed in science nd technology. A number of our publictions re imed t ssisting users in cquiring such skills. Our publictions re designed: (i) to be updted quickly in response to field tests nd new scientific developments; (ii) to be used in both clssroom nd professionl settings; (iii) to show the prerequisite dependencies existing mong the vrious chunks of physics knowledge nd skill, s guide both to mentl orgniztion nd to use of the mterils; nd (iv) to be dpted quickly to specific user needs rnging from single-skill instruction to complete custom textbooks. New uthors, reviewers nd field testers re welcome. PROJECT STAFF Andrew Schnepp Eugene Kles Peter Signell Webmster Grphics Project Director ADVISORY COMMITTEE D. Aln Bromley Yle University E. Leonrd Jossem The Ohio Stte University A. A. Strssenburg S. U. N. Y., Stony Brook Views expressed in module re those of the module uthor(s) nd re not necessrily those of other project prticipnts. c 2001, Peter Signell for Project PHYSNET, Physics-Astronomy Bldg., Mich. Stte Univ., E. Lnsing, MI 48824; (517) For our liberl use policies see: 3 4

3 MISN l d y b B z I x Figure 1. Section of conductor through which is pssing current I. EXAMINING THE CHARGE CARRIERS; THE HALL EFFECT by Peter Signell 1. Introduction 1. Why We Study the Hll Effect. The moving objects constituting n ordinry electricl current re sid by physicists to be negtively chrged electrons, not positive chrges s ssumed in the electricl engineering convention. 1 These electrons drift velocity down household wire re sid by physicists to be typiclly bout 10 feet per hour. How cn 10 ft/hr relly be true when wll switch seems to ctivte lmp cross the room instntly! Here we exmine convincing evidence of the negtive sign of the chrge crriers nd the mgnitude of their drift velocity. Elsewhere we exmine the speed of electricl power trnsmission. 2 The Hll effect is lso interesting for its pplictions: it is used to determine electronic properties of new mterils nd for routine mesurements of unknown mgnetic fields. 1b. Mutully Perpendiculr I, B, V. The term Hll effect refers to specil voltge tht ppers when trnsverse mgnetic field is pplied to n electricl current flowing in mteril (see Fig. 1). This specil voltge, clled the Hll voltge, is t right ngles to both the current nd the mgnetic field. This mens tht ll three quntities involved, current, field, nd voltge, re t right ngles to ech other. In the configurtion shown in Fig. 1 the Hll voltge is mesured long the y-xis, 1 For further informtion see Conductivity nd Resistnce (MISN-0-118) nd Force on Current in Mgnetic Field (MISN-0-123). 2 For further informtion see Signl Velocity in Conductor (MISN-0-150). MISN with the voltmeter s positive led t point b, negtive led t point. Notice the orienttions of the other two quntities. 1c. An E Field so Lorentz Force. The existence of the y-xis voltge in Fig. 1 mens tht there is y-xis electric field, one which, like ll electric fields, exerts y-xis force on chrge crriers like those in the current I. It is not surprising tht such force is exerted on the chrge crriers; it is just the ordinry Lorentz force tht occurs whenever there is mgnetic field t right ngles to the velocity of the chrge. The importnce of the effect is tht the vlue nd sign of the voltge tell us much bout the chrge crriers if we know the mgnetic field. Then, the mesurement hving been mde for prticulr mteril in known mgnetic field, tht mteril s Hll voltge is used industrilly to determine unknown mgnetic fields. 2. Delineting the Conditions 2. Mesurement Lyout. The Hll voltge is mesured cross piece of conducting mteril formed in the shpe of rectngulr br s in Fig. 1. We induce the current I in the x-direction nd pply constnt mgnetic field B in the z-direction. 2b. Force Independent of Chrge Sign. If ech chrge crrier in the current I hs chrge q, then the mgnetic field Lorentz force on the chrge is: 3 F = q vd B, where v D is the velocity of drift in the direction of the current. Show tht if the current consists of positive prticles going to the right (positive x-direction) in Fig. 1 then the force on them is downwrds (negtive y-direction). Then mke the other possible ssumption tht the current to the right consists of negtive prticles going to the left nd show tht such negtive chrge crriers for the sme current would lso experience downwrd mgnetic field Lorentz force. 2c. Downwrd Drift Reches Equilibrium. When the current of Fig. 1 is initited, the Lorentz Force cuses net downwrd migrtion of the chrge-crrier electrons. This downwrd migrtion produces n incresing concentrtion of electrons in the lower prt of the mteril, leving correspondingly incresing concentrtion of positive lttice ions in the upper prt of the mteril. A crrier tht is prt-wy down will thus 3 See Force on Chrge Prticle in Mgnetic Field: The Lorentz Force (MISN ). 5 6

4 MISN experience net upwrd electrosttic field just due to the grdient in the net chrge concentrtion. Downwrd migrtion ceses when the number of lower electrons becomes so lrge tht the upwrd concentrtion grdient force mtches the downwrd Lorentz force. 3. The Equilibrium Equtions 3. Equilibrium E nd v. The condition for equilibrium in downwrd migrtion is tht the net force on crrier is zero. Writing the chrge on the crrier s q, the condition is: F y = 0 = q[e y + ( v D B) y ]. Then ech mobile chrge moves in the x-direction in n electric field whose y-component is: 4 E y = +v D B. (1) Show tht E y is in the upwrd direction for positively chrged crriers, downwrd for negtively chrged crriers, nd tht this checks with the electric field direction being wy from positive chrges nd towrd negtive chrges. 3b. Equilibrium Voltge. The potentil difference between points nd b of Fig. 1 is mesurble with voltmeter, nd is simply relted to E y. Recll tht the voltge of point b with respect to point is the work per unit chrge which you would hve to do in moving chrge from the reference point to point b. Then the Hll voltge is: b V b = E d y = b b ( E y dy) = E y dy = E y l (see Fig. 1), where we hve used the fct tht E y in Eq. (1) hs no dependence on y, hence is constnt in the y-direction. Since E is independent of y we could lso hve obtined V b by: 4 This nlysis hs ssumed tht ll the chrge crriers hve the sme sign. If they don t, then V D in (1) is n verge (see Appendix). MISN Work( b) V b = chrge Applied Force( b) Distnce( b) = chrge Applied Force( b) = Distnce( b) chrge = E y l where we hve used the fct tht the positive downwrd pplied force per unit chrge must exctly cncel the positive upwrd electric field E y. 3c. Crrier Velocity. Combining Eqs. (1) nd (2), we cn eliminte E y in terms of mesurble quntities: (2) v D = V b Bl. (3) Mesurement of the Hll voltge thus gives us direct mesurement of the drift velocity of the chrge crriers. 3d. Crrier Density; the Hll Constnt. We cn lso obtin the sign of the chrge nd the density of the crriers by reclling tht the current is given by: I = Q L v D where (Q/L) is the chrge per unit length long the conductor. reltion cn lso be written s: This I = σ v A v D, (4) where σ v is the crrier chrge per unit volume, nd A is the cross sectionl re of the conductor. Combining Equtions (3) nd (4) nd using A = ld (see Fig. 1): σ v = IB V b d. (5) The inverse of this quntity is clled the Hll constnt nd is written R H or C H. 3e. Actul Mesurements. When one ctully mkes the mesurement on, sy, piece of copper which could hve been used for ordinry house wiring, the Hll voltge turns out to be negtive. This mens tht 7 8

5 MISN the voltmeter will only give positive reding if the leds re reversed so tht the (+) one is t point. The inescpble conclusion is tht these crriers hve negtive chrge. The numericl vlue obtined in Eq. (5) turns out to be very close to the density of vlence-electron-chrge for both of the common good conductors, silver nd copper. Tht vlue, the vlence-electron chrge per unit volume, cn be obtined by dividing the metl s mss density by its mss per tom to get the number of toms per unit volume nd then multiplying tht vlue by the vlence-electron chrge on ech tom. 5 MISN As n exmple, R H is positive for iron, showing dominnce of holes over electrons. Acknowledgments Michel Hrrison provided helpful discussion of the Hll effect. Willim Lne, Stephen Smith nd their students, especilly Jim Peterson, provided much vluble feedbck on erlier versions. Preprtion of this module ws supported in prt by the Ntionl Science Foundtion, Division of Science Eduction Development nd Reserch, through Grnt #SED to Michign Stte University. A. Hll Constnt for Two Crriers (for those interested) When current consists of electrons (negtive chrge) nd oppositely-moving electron holes (positive chrge) s in iron, mgnesium nd semiconductors, the Hll constnt is given by: 6 R H = p nb2 e (p + nb) 2 where p nd n re the number densities of positive nd negtive crriers, respectively, e is the mgnitude of the electronic chrge, nd b is the mobility rtio, (τ m m p )/(τ p m n ), where the τ s re the crriers menfree-times between collisions nd the m s re their (positive) msses. The bove formul for R H is sid to be in the drift velocity pproximtion. 5 See Currents, Fields, nd Prticles, F. Bitter, John Wiley nd Sons, New York (1957), Tble 6.1, p.231, nd The Hll Effect in Metls nd Alloys, C. M. Hurd, Plenum Press, New York (1972), Fig. 1.1 on pge 7, nd Tble 7.6 on pges For ccess, see this module s Locl Guide. 6 For further informtion see Introduction to Solid Stte Physics, C. Kittel, John Wiley nd Sons, New York (1986), p For ccess, see this module s Locl Guide. 9 10

6 MISN LG-1 MISN PS-1 LOCAL GUIDE The redings for this unit re on reserve for you in the Physics-Astronomy Librry, Room 230 in the Physics-Astronomy Building. Ask for them s The redings for CBI Unit 149. Do not sk for them by book title. PROBLEM SUPPLEMENT Note: Problem 3 lso occurs in this module s Model Exm. 1. A Hll probe uses the Hll Effect to mesure mgnetic field strength. If the probe is copper, hs thickness of 0.1 mm, n R H vlue of R H = m 3 /(A s), nd cn mesure potentil of 0.1 mv, wht current is needed to mesure field of 0.2 Tesl? Would you wnt to use copper for Hll probe? 2. Using R H = m 3 /(A s), wht is the chrge crrier drift velocity for 2.0 A current in copper conductor whose cross-sectionl re is (1 unitcm 2 )? Why does the result imply the chrge crriers re negtive? 3. In Hurd s book, pges 278-9, the experimentl vlues of R H for copper cover the rnge: m 3 /(A s) R H m 3 /(A s) Clculte the drift velocity (in feet per hour) for 5 mp current in No. 18 copper wire (cross sectionl re 2 mm 2 ). Note: m/s = ft/hr. Brief Answers: 1. R H = V bd IB so: I = V bd RB = (10 4 V)(10 4 m) ( m 3 /(A s))( T) = 103 A. No: vstly too much current. 2. v D = V b Bl = R HIB/d = R HI Bl ld = ( m 3 /(A s))(2.0 A) (10 2 m)(10 2 = 10 6 m/s. m) Here v D is negtive so it is opposite to the positive chrge direction, so these chrges re negtive

7 MISN PS-2 MISN ME-1 3. Combine the equtions bove to find v D in terms of R H, I nd A. Then: 2.3 ft/hr v D 1.4 ft/hr. MODEL EXAM 1. l I d b Strting from the Lorentz force, derive the Hll effect equtions: v D = V b Bl σ v = IB V b d 2. See Output Skill K2 in this module s ID Sheet. 3. In Hurd s book, pges 278-9, the experimentl vlues of R H for copper cover the rnge: m 3 /A s R H m 3 /A s Clculte the drift velocity (in feet per hour) for 5 mp current in No. 18 copper wire (cross sectionl re 2 mm 2 ). Note: m/s = ft/hr. B Brief Answers: 1. See this module s text. 2. See this module s text. 3. See this module s Problem Supplement, problem

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