v d = (VII) (II) (IV)

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1 P Pag 1/5 Objcts of th xprints 1. Masuring of th Hall voltag as function of th currnt at a constant agntic fild: dtrination of th dnsity and obility of charg carrirs.. Masuring of th Hall voltag for as function of th agntic fild at a constant currnt: dtrination of th Hall cofficint.. Masuring of th Hall voltag as function of tpratur: invstigation of th transition fro xtrinsic to intrinsic conductivity. Principls Th Hall ffct is an iportant xprintal thod of invstigation to dtrin th icroscopic paratrs of th charg transport in tals or dopd siconductors. To invstigat th Hall ffct in this xprint a rctangular strip of p-dopd graniu is placd in a unifor agntic fild B according Fig. 1. If a currnt I flows through th rctangular shapd sapl an lctrical voltag (Hall voltag) is st up prpndicular to th agntic fild B and th currnt I du to th Hall ffct: U H I B = (I) d R H is th Hall cofficint which dpnds on th atrial and th tpratur. At quilibriu conditions (Fig. 1) for wak agntic filds th Hall cofficint R H can b xprssd as function of th charg dnsity (carrir concntration) and th obility of lctrons and hols: 1 p µ p n µ n (II) 0 (p µ p + n µ n ) 0 = As (lntary charg) p = p E + p S (total dnsity of hols) p E : dnsity of hols (intrinsic conduction) p S : dnsity of hols (hol conduction du to p-doping) n = n E :dnsity of lctrons (intrinsic conduction) µ p : obility of hols µ n obility of lctrons Fro quation (II) follows: Th polarity of prdoinant charg carrirs can b dtrind fro th Hall cofficint R H if th dirctions of th currnt I and agntic fild B ar known. Th thinnr th conducting strip th highr th Hall voltag. Th doping of group III lnts lik.g. B, Al, In or Ga into th crystal lattic of graniu crats positiv chargd hols in th valnc band (Fig. ). Thir activation nrgy E A of about 0.01 V is significantly sallr than th activation nrgy E g (band gap) to gnrat lctrons and hols by thral activation (intrinsic charg carrirs). At roo tpraturs in p-dopd graniu th dnsity of hols p S can prdoinat th dnsity of intrinsic charg carrirs (p E and n E ). In this cas whr th charg transport is prdoinatly du to hols fro th dopants (n = n E = p E 0). Th dnsity of p S can b dtrind by asuring th Hall voltag U H as function of th currnt I. With quation (I) and (II) follows: B I ps (III) 0 d Th obility is a asur of th intraction btwn th charg carrirs and th crystal lattic. Th obility is dfind as (in cas p- dopd graniu it is th obility µ P of th hols cratd by th dopants, i.. accptors): v p µ P = E (IV) v p : drift vlocity E: lctric fild du to th voltag drop Th lctric fild E can b dtrind by th voltag drop U and th lngth w of th p-dopd graniu strip: U E = (V) w Th drift vlocity v p can b dtrind fro th quilibriu condition, whr th Lorntz forc copnsats th lctrical forc which is du to th Hall fild (Fig. 1) 0 v d B = 0 EH (VI) which can b xprssd using th rlation E H = b U H as v d = (VII) b B Substituting quation (V) and (VII) in quation (IV) th obility µ p of hols can b stiatd at roo tpraturs as follows: U H w µ p = (VIII) b B U I + _ U H w _ v - F L n F + F F L Fig. 1: Hall ffct in a rctangular sapl of thicknss d, hight b and lngth w: At quilibriu conditions th Lorntz forc F L acting on th oving charg carrirs is balancd by th. lctrical forc F which is du to th lctric fild of th Hall ffct. Th currnt I in a siconductor crystal is ad up of both hol currnts and lctron currnts (Fig. 1): B I = b d (np µ p + nn µ n ) (IX) Th carrir dnsity dpnds on th dopant concntration and th tpratur. Thr diffrnt rgions can b distinguishd for p-dopd graniu: At vry low tpraturs th xcitation fro lctrons of th valnc band into th accptor lvls is th only sourc of charg carrirs. Th dnsity of hols p S incrass with tpratur. It follows a rgion whr th dnsity p S is indpndnt of tpratur as all accptor lvls ar occupid (xtrinsic conductivity). In this rgi th charg transport du to intrinsic charg carrirs can b nglctd. A furthr incras in tpratur lads to a dirct thral xcitation of lctrons fro th valnc band into th conduction band. Th charg transport incrass du to intrinsic conductivity and finally prdoinats (Fig. ). Ths transition fro pur xtrinsic conduction to a prdoinatly intrinsic conduction can b obsrvd by asuring th Hall voltag U H as function of th tpratur. To dscrib th Hall voltag as function of tpratur U H basd on a sipl thory quation (I) and (II) hav to b xtndd in th following way: v p b d LEYBOLD DIDACTIC GMBH. Lyboldstrass 1. D-5054 Hürth. Phon (0) Fax (0) ail: info@lybold-didactic.d

2 P Pag /5 Apparatus E A CB VB A E E g Fig. : Siplifid diagra of xtrinsic (lft) and intrinsic conduction (right) undr influnc of an lctric fild E: Incorporating of dopants (accptors A) into th crystal lattic crats positiv charg carrirs calld hols in th valnc band (VB). With incrasing tpratur th thral nrgy of valnc lctrons incrass allowing th to brach th nrgy gap E g into th conduction band (CB) laving a vacancy calld hol in th VB. It is assud that th obility of lctrons and hols ar diffrnt. Introducing th ratio of th obility µ n k = (X) µ p quation (II) can b rwrittn as follows: 1 p n k (XI) 0 (p + n k) For undopd siconductors th tpratur dpndncy of th charg carrirs can b assud as k B T n = n0 k B T p = p0 (XII) k B = J/K: Boltzann constant Th product of th dnsitis p and n is tpratur dpndnt: n p = ne (pe + ps ) = η (XIII) whr th ffctiv stat dnsity η is approxiatd as k T N0 B η (XIV) In th xtrinsic conductivity rgi th dnsity p S of hols can b dtrind according quation (III). For th intrinsic charg carrirs p E = n E which lads to a quadratic quation for p E with th solution: PS PS pe = + + η (XV) 4 With quations (XI) and (XV) togthr with th rlations p = p E + p S and n = n E th tpratur dpndncy of Hall voltag U H can b siulatd. Using for E g = 0.7 V th rsult of xprint P as stiat valu for th siulation only two unknown paratrs N 0 and k ar lft. 1 Bas unit for Hall ffct G p-dopd G plug-in board Tangntial B-prob B-box Multicor cabl, 6-pol Snsor CASSY CASSY Lab AC/DC Powr Supply 0 to 15 V DC powr supply U-cor with yok Pair of bord pol pics Coil with 50 turns Stand rod, 5 c Lybold Multi clap Stand bas, V-shap, 0 c Pair of cabls, 1, rd and blu additionally rquird: PC with Windows 95/98/NT or highr Stup Mounting and conncting th plug-in board: Nots: Th p-dopd G crystal is xtrly fragil: Handl th plug-in board carfully and do not subjct it to chanical shocks or loads. Du to its high spcific rsistanc, th p-dopd G crystal wars up vn if only th cross-currnt is applid: Do not xcd th axiu cross-currnt I = A. Turn th control knob for th cross-currnt on th bas unit for Hall ffct to th lft stop. - Insrt th plug-in board with th p-dopd G crystal into th DIN sockt on th bas unit for Hall ffct until th pins ngag in th hols. - Carfully insrt th plug-in board with DIN plug into th DIN sockt on Insrt th bas unit with rod into th hol of th U-cor all th way to th stop; ak sur that th plug-in board is satd paralll to th U-cor (s instruction sht bas unit Hall ffct ). - Carfully attach th pair of bord pol pics with additional pol pic, and slid th additional pol pic as far as th spacrs of th plug-in boards (ak sur that th plug-in board is not bnt). - Turn th currnt liitr of th currnt-controlld powr supply to th lft stop, and connct th powr supply. Masuring th agntic fild: - Th Axial B-prob is fixd by th Stand rod to th V-shapd Stand bas. - Bfor th asuring th agntic induction of th fild B plac th B-prob carfully in th gap (s instruction sht bas unit Hall ffct ) aftr th apparatus is adjustd. - For th asurnt connct B-prob to th Snsor CASSY using th B-box. Copnsation of th Hall voltag: - Bfor prforing a asurnt with a constant currnt I th Hall voltag hav to b copnsatd for B = 0 T: - 1. For asuring th currnt I connct th cabls to th Input A of th Snsor CASSY (Fig., s also instruction sht bas unit Hall ffct ). LEYBOLD DIDACTIC GMBH. Lyboldstrass 1. D-5054 Hürth. Phon (0) Fax (0) ail: info@lybold-didactic.d

P7..1.4 Pag /5. Masuring th Hall voltag as function of agntic fild - First copnsat th Hall voltag (s abov). - St th currnt I to a dsird valu.

3 P Pag /5. Masuring th Hall voltag as function of agntic fild - First copnsat th Hall voltag (s abov). - St th currnt I to a dsird valu. - Masur th Hall voltag U H (Input B on Snsor CASSY) as function of th agntic fild B (Input A on Snsor CASSY). - Aftr conncting th cabls st th paratrs with. - For asuring us th button or F9 in anual asuring od. - Saf your asurnt.. Masuring th Hall voltag as function of tpratur - First copnsat th Hall voltag U H (s abov) and st th currnt I to a dsird valu. - St th agntic fild B to a dsird valu (s abov). - Masur th Hall voltag U H (Input B on Snsor CASSY) as function of th Tpratur ϑ (Input A on Snsor CASSY, s abov). Fig. : Exprintal stup (wiring diagra) for asuring th Hall voltag as function of th currnt I. -. For asuring th Hall voltag U H connct th cabls to th Input B of th Snsor CASSY (Fig. s also instruction sht bas unit Hall ffct ). -. St th cross-currnt I to th axiu valu (s instruction anual for p-dopd G crystal ), switch on th copnsation and zro th Hall voltag U H using th copnsation knob. Masuring xapls 1. Masuring th Hall voltag as function of currnt Masuring th voltag drop: - For asuring th voltag drop U connct th cabls to th Input B of Snsor CASSY (s instruction sht bas unit Hall ffct asur th conductivity as function of tpratur). - Connct th cabls to th Input A of th Snsor CASSY to asur th currnt I (s instruction sht bas unit Hall ffct ). - St th currnt I to th axiu valu and asur th voltag drop U. Masuring th tpratur: - For asuring th tpratur ϑ connct th output signal of th hatr to Input A of th Snsor CASSY (s instruction sht bas unit Hall ffct and Physics Laflts P ) Carrying out th xprint 1. Masuring th Hall voltag as function of currnt - First copnsat th Hall voltag (s abov). - St th agntic fild B to a dsird valu and asur th agntic flux dnsity B (s abov). - St th currnt to th axiu valu and asur th voltag drop U. - Masur th Hall voltag U H (Input B on Snsor CASSY) as function of th currnt I (Input A on Snsor CASSY). - Aftr conncting th cabls st th paratrs with. - For asuring us th button or F9 in anual asuring od. Fig. 4: Hall voltag U H as function of th currnt I for diffrnt agntic filds. Th straight lins corrspond to a fit according quation (I). currnt: I = 0 A voltag drop: U = 1,4 V. - Saf your asurnt. LEYBOLD DIDACTIC GMBH. Lyboldstrass 1. D-5054 Hürth. Phon (0) Fax (0) ail: info@lybold-didactic.d

P7..1.4 Pag 4/5. Masuring th Hall voltag as function of agntic fild With th xprintal rsults at roo tpratur U = 1.4 V B = 0.

p = = b B 1 s Fig. 5: Hall voltag U H as function of th agntic fild B for I = 0 A.. Th straight lin with slop A corrsponds to a fit according quation (I).

4 P Pag 4/5. Masuring th Hall voltag as function of agntic fild With th xprintal rsults at roo tpratur U = 1.4 V B = 0.5 T U H = 7 V and th dinsions of th p-dopd graniu strip b = 10 w = 0 th drift vlocity v p (quation (VII)) and th obility µ p (quation (VIII)) of th charg carrirs in th xtrinsic rgion can b stiatd: v p = = b B 1 s Fig. 5: Hall voltag U H as function of th agntic fild B for I = 0 A.. Th straight lin with slop A corrsponds to a fit according quation (I).. Masuring of th Hall voltag as function of tpratur w c µ p = = 940 b B U Vs. Masuring of th Hall voltag as function of agntic fild As can b sn fro th linar rgrssion of a straight lin through th origin th Hall voltag U H is proportional to th agntic fild B: U H ~ B. Togthr with th rsult of part 1., i.. U H ~ I, th following rlation is found: Fig. 6: Hall voltag U H as function of tpratur T for I = 0 A and diffrnt agntic filds B. Evaluation and rsults 1. Masuring of th Hall voltag as function of currnt For th asurnt with. g. B = 0,5 T and I = 0 A in Fig. 4 th slop B V A = =.1 d A is obtaind by th fitting a straight lin through th origin (right ous click in th diagra and fit function ). With th linar rgrssion rsult and quation (III) th dnsity p S of hols in th xtrinsic conducting rgi can dtrind as follows: d = B = 0.5 T B 1 1 ps = = d A U H ~ I B. Thus th thortically drivd forula (quation (I)) for th Hall voltag U H of a strip-shapd conductor of thicknss d is confird. For th fit of a straight lin to th xprintal data of Fig. 5 th Hall cofficint R H is obtaind as follows: d = I = 0 A A = V/T (slop of Fig. 5) A d = = I As A coparison with. g. th Hall cofficint of th tallic conductor silvr (R H = C -1 xprint P7..1.1) shows that th Hall cofficint is about 10 7 largr for siconductors.. Masuring of th Hall voltag as function of tpratur Using quations (XI) and (XV) togthr with th rlations p = p E + p S and n = n E th Hall voltag U H can b xprssd as follows: U H = ((A+(Sqr(A^/4+B^*Exp(-C*105.9/x))-A/)*(1-D^))/ ((A+(Sqr(A^/4+B^*Exp(-C*105.9/x))-A/)*(1+D))^) *7.49*10^) (XVI) Using quation (XVI) th tpratur bhavior of th Hall voltag U H can b siulatd with th following fit paratrs (For prforing a Fit with CASSY Lab us ky Alt F): A = B = N 0 = C = E g = 0.74 V D = µ n /µ p =1.81 Th rsult of th fit is shown in Fig. 7. LEYBOLD DIDACTIC GMBH. Lyboldstrass 1. D-5054 Hürth. Phon (0) Fax (0) ail: info@lybold-didactic.d

P7..1.4 Pag 5/5 Fig. 7: Fit according quation (XVI) to th xprintal data of Fig 6 for B = 0.5 T and I = 0 A.

5 P Pag 5/5 Fig. 7: Fit according quation (XVI) to th xprintal data of Fig 6 for B = 0.5 T and I = 0 A. Th tpratur dpndncy of th Hall voltag U H probs th transition fro a charg transport du to hols to a bipolar charg transport of lctrons and hols. At roo tpraturs th obsrvd bhavior of U H is du to hols cratd by th accptor atos in th graniu lattic. Incrasing th tpratur, th charg transport is or and or du to thrally activatd lctrons and vacancis lft in th valnc band. Whn th nubr of th fastr lctrons xcds th nubr of hols th Hall voltag U H bcos ngativ. According quation (II) a sign chang of U H taks plac whn p µ p = n µ n. Th ngativ tpratur rang of th Hall voltag is dtrind by th lctrons. Thir drift vlocity and thus thir obility is largr as th drift vlocity and obility of th hols, rspctivly. µ n µ p At high tpraturs th charg dnsity of hols and lctrons ar approxiatly th sa. Th Hall voltag U H approachs finally zro du to th qual but opposit lctrical filds of th lctrons and hols (Fig. 7). For that rason no Hall Effct can b obsrvd in pur siconductors (intrinsic charg carrirs only). Th siplifid odl nglcts corrctions of th quantu thory, i.. band structur and ffctiv ass. Espcially, th ffctiv stat dnsity N 0 is not constant as assud in quation (XIV). N 0 has to b rplacd by th product of th ffctiv stat dnsitis of th conduction band N C and valnc band N V : N0 = NC NV T (XII) Supplntary inforation Th Hall ffct was discovrd in Although th Hall ffct is prsnt in all conducting atrials it raind a laboratory curiosity until th latr half of 000 cntury. With th advnt of siconductor tchnology and dvlopnt of various III- and V-copounds it has bco possibl to produc Hall voltags svral ordrs of agnitud largr than with arlir atrials. In tchnical applications th Hall ffct of siconductors is spcially usd in agntic asurnt probs. LEYBOLD DIDACTIC GMBH. Lyboldstrass 1. D-5054 Hürth. Phon (0) Fax (0) ail: info@lybold-didactic.d

v d = (VII) (II) (IV)

v d = (VII) (II) (IV) P7..1. Pag 1/5 Objcts of th xprints 1. Masuring of th Hall voltag as function of th currnt at a constant agntic fild: dtrination of th dnsity and obility of charg carrirs.. Masuring of th Hall voltag for