PHYS-102
Honor's Lab
The Hall Effect – Finding the Hall Coefficient ( R H ) 1.) Objectives The objectives of this experiment are to demonstrate the the effects of a magnetic fied !"# on a c$rrent carr%ing cond$ctor !semicond$ctor&meta# (f a c$rrent carr%ing cond$ctor is paced in a magnetic fied oriented oriented perpendic$ar perpendic$ar to the direction of the c$rrent) c$rrent) a votage votage is deveoped deveoped across the cond$ctor cond$ctor in a direction perpendic$ar to both the the magnetic fied and the direction direction of the c$rrent This effect is *no+n as the Ha ,ffect ,ffect This effect effect is $sef$ in determining determining the nat$re nat$re of charge carriers carriers ! ie) +hether a semicond$cto semicond$ctorr is of n-t%pe or p-t%pe#) p-t%pe#) the n$mber densit% densit% of charge carriers) carriers) and the mobiit% mobiit% of charge carriers carriers Ha obiit% obiit% is an expression expression of the extent to +hich the Ha effect ta*es pace pace in a semicond$cto semicond$ctorr materia materia .or a given magnetic fied intensit% intensit% and c$rrent c$rrent va$e) the votage generated b% the Ha effect is greater +hen the Ha obiit% is higher The Ha obiit% is given b% the prod$ct of the Ha constant and the cond$ctivit% for a given materia (n genera) the greater the carrier ca rrier mobiit% in a semicond$ctor) the greater the Ha mobiit% (n this ab %o$ +i/ 1 St$d% St$d% the the Ha Ha ,ffe ,ffect ct and and deter determin mine/ e/ ● Ha otage V H Ha oefficient R H 1 etermine etermine the t%pe of majorit majorit% % charge carriers carriers ! ie ie determine determine +ether +ether the semicond$cto semicond$ctorr sampe is of the n-t%pe or p-t%pe # 2 ete eterm rmin inee the the char charge ge carri carrier er densi densit% t% ! or carri carrier er conce concent ntra rati tion on per $nit vo$me vo$me # in the the semicond$ctor cr%sta 3 ete eterm rmin inee the the Ha Ha 4ng 4ngee H ●
The Ha effect has man% interestin interesting g appications appications in science and engineering engineering (n materia materia science the Ha Ha ,ffe ,ffect ct can can be $sed $sed to esta estab bis ish h pot potss of majo majori rit% t% carr carrie ierr conce concent ntra rati tion on and mobi mobii it% t% vs vs temperat$re This is an important too for characteri5ing semicond$ctor devices 4s +i be seen in this ab) the Ha ,ffect is aso the basis basis for integrated circ$it devices +hich meas$re magnetic fieds The 6a$ss eter eter $sed in this this ab $ses a Ha Ha ,ffect Transd Transd$cer $cer There are are a arge n$mber of integra integrated ted Ha circ$its in $se toda% an% of these circ$its are $sed as contactess s+itches and mechanica proximit% sensors
2.) Theor 4 charged partice in a magnetic magnetic fied experiences experiences a force that is proportiona proportiona to the magnit$de magnit$de of the charge of the partice) to the veocit% of the partice) and to the magnit$de of the magnetic fied s$rro$nding the charged partice The direction of the force is perpendic$ar to both the direction of the magnetic fied s$rro$nding the partice and to the direction of the veocit% of the charged partice) and foo+s the 78ight Hand 8$e9 as sho+n in .ig$re :ne
B=q v X B F 1
[ 1]
Fig!re One" The force on a charged partice in a magnetic fied
4s has been sho+n in ect$re) that a c$rrent carr%ing +ire ocated in a magnetic fied +i experience a force d$e to the charge carriers moving thro$gh the +ire at a drift veocit%) vd This force is is transferred to the +ire b% the forces that bind the charge carriers to the +ire The force on c$rrent ) and carr%ing +ire is proportiona to the magnit$de of the c$rrent I ) the ength of the +ire L This force can be $sed to provide a the magnit$de if the magnit$de of the magnetic fied B tor;$e on a c$rrent carr%ing oop of +ire) +hich can then be $sed to create an eectric motor
F B = I L X B = N I A X B wire
[ 2] [3]
This force can aso be $sed to investigate properties of materias onsider a c$rrent carr%ing cond$ctor in an externa magnetic fied) as sho+n in .ig$re T+o Since the charge carriers experience a force d$e to the externa magnetic fied) the charge carriers +i be acceerated to+ards one side of the c$rrent carr%ing cond$ctor) b$t +i be trapped inside the cond$ctor
Fig!re T#o" The force on a positive% charged partice in a magnetic fied
(n .ig$re T+o) a sampe of a cond$ctor +ith a +idth) w ) carries a c$rrent ( and is paced in an externa magnetic fied 4 positive charge carrier) a proton) traves in the same direction as the c$rrent +ith a drift veocit%) vd .oo+ing the right hand r$e) the positive charge carrier +i experience a 2
force to+ards the top of the sampe) b$t is trapped in the sampe The force experienced b% the charge d$e to the magnetic fied is e;$a to/
B= q v X B = e vd X B F o ∣ F B∣= e∣v d ∣∣ B∣ sin=0 =e∣v d ∣∣ B∣
[ <] [ >]
Fig!re Three" The force on a negative% charged partice in a magnetic fied
(n .ig$re Three) a sampe of a cond$ctor carries a c$rrent ( and is paced in an externa magnetic fied 4 negative charge carrier) an eectron) traves in the opposite direction as the c$rrent +ith a drift veocit%) vd .oo+ing the right hand r$e) the negative charge carrier +i experience a force to+ards the top of the sampe) b$t is trapped in the sampe The force experienced b% the charge d$e to the magnetic fied is e;$a to/
B= q = −e vd X B F v X B
∣ F B∣=e∣v d ∣∣ B∣
[? ] [ @]
4s the charges b$id $p on the top and the bottom of the sampe) and eectric fied evoves onsidering the motion of a positive charge carriers) there is no+ a force d$e to the eectric fied The net force) sho+n in .ig$re .o$r) *no+ as the Lorent5 .orce) +i be/
q v X B =−e E e v X B F net =q E
3
[ A]
Fig!re Fo!r" The forces on a positive% charged partice in a magnetic fied and in an ind$ced eectrica fied The net force is *no+ as the Lorent5 .orce
(nitia%) the eectric fied is non-existent 4s positive charges move to the top of the sampe and negative charges move to the bottom of the sampe) the eectric fied gro+s as the eectric force become greater ,vent$a% a stead% state evoves and the force d$e to the eectric fied and the force d$e to the magnetic fied become e;$a The t+o forces are e;$a and opposite and the net force approaches 5ero
q v X B =−e E e F net =q E v d X B
∣ F net ∣=−e E e v d B sin =0o=0 e E =e v d B E = B v d
[ =]
4t this point the eectric fied +i e;$a E = B v d as sho+n in e;$ation = 4n eectric potentia difference across the sampe +i res$t from the separation of the positive% charged top s$rface and the negative% charged bottom s$rface This eectric potentia difference if *no+n as the Ha ,ffect otage an is e;$a to/
V H = V = E w = v d B w
[ 10 ]
Bhere V H in e;$ation 10 is the Ha ,ffect otage) v d is the drift veocit% and w is the +idth of the sampe Csing this information) the carrier concentration and the Ha oefficient) R H ) can be cac$ated onsider a cond$cting or semicond$cting sampe) +ith dimensions w ,l ,t ) +here the =∣ B∣ j This meas$red votage across this sampe +i be positive if the magnetic fied is e;$a to B majorit% of charge carriers are positive ! see .ig$re T+o # and negative if the majorit% of charge carriers are negative ! see .ig$re Three #
<
Fig!re Five" :rientation of semicond$cting sampe
The c$rrent densit%)
J ) is e;$a to
J =
I A
=
I wt
=
e n v d A w t
=e n v d ) +here n is the n$mber of
charge carriers per $nit vo$me from ,;$ation 10 the Ha ,ffect otage is given b%/
V H = E w =v d B w =
BI e n t
[ 11 ]
(f the magnetic fied) B ) the Ha ,ffect otage) V H ) the c$rrent) I ) the +idth of the sampe) w ) and the charge of the charge carrier) e ) are *no+n or can be meas$red) then the charge concentration can be cac$ated
n= 4so the Ha oefficient )
R H
IB e t V H
[ 12 ]
) can be cac$ated The Ha oefficient is defined as/
R H ≡
1
ne
=
V H t
[ 13 ]
B I
Therefore) once the Ha oefficient is *no+n) the concentration of charge carriers n can be fo$nd The Ha oefficient can aso be $sed to find the mobiit% of the sampe m and the Ha ange
! The ange of the ,ectric .ied +hich is a combination of the ,ectric .ied prod$ced prod$ced b% the Ha ,ffect and the ,ectric .ied prod$cing the c$rrent thro$gh the sampe#
H = tan
>
−1
m B
$.) E%!i&'ent and et!& The ,;$ipment consists of/ 1 Po+er S$pp% for the eectromagnet ! 0 D 1? ots ) > 4mps# 2 onstant $rrent Po+er S$pp% ! 0 D >0 m4 ) (dea% 0 D 20 m4 # 3 6a$ss meter +ith Ha Probe < Semicond$ctor sampe ! 6e singe cr%sta # mo$nted on P" p-t%pe 6e cr%sta Thic*ness t / 0> mm Bidth w / < mm Length l / ? mm > Eeith% m$timeter +ith m scae for meas$ring Ha ,ffect otage ? Hand-hed m$timeter +ith m4 scae for meas$ring c$rrent thro$gh sampe @ Ha effect apparat$s ! eectromagnet) poe pieces) and piars#
Fig!re i*" Ha ,ffect 4pparat$s D T+o cois of +ire s$rro$nding poe pieces and t+o piars for mo$nting probes
Fig!re even" Ha Probe for 6a$ssmeter
?
Fig!re Eight" Semicond$cting sampe mo$nted on P" board +ith connections for onstant $rrent Po+er S$pp% and otage eas$rement
Fig!re -ine" "oc* diagram of ,xperimenta Set-$p
+.) et!& and ,roced!re 1 o$nt the P" ! +ith mo$nted cr%sta # on one of the piars of the Ha ,ffect 4pparat$s and the Ha probe for the 6a$ss eter on the other piar
Fig!re Ten" Ha ,ffect 4pparat$s +ith probe and P" mo$nted on piars
2 onnect the Ha Probe to the 6a$ss eter 3 onnect the t+o cois of the eectromagnet in series to the Po+er S$pp% @
< onnect the Eeith% $timeter to the iivot Terminas on the P" $timeter !miivot scae#
Fig!re Eleven" Ha ,ffect P"
onnected to m4 eter and in series +ith $rrent ontroed Po+er S$pp%
> onnect the onstant $rrent Po+er S$pp% to the P" +ith the m4 meter in series O/ 0 3E /-4 THE 4REE- TER5-6 FOR ,OT7E 6-8 THE 36C9 TER5-6 FOR -E46T7E /-8ER THE 4REE- ETTER-4 CO-T6-T C/RRE-T ,O0ER /,,. 569E /RE THE 0TCH - THE :.:; 65, ,OTO-.
Fig!re T#elve" onstant $rrent Po+er S$pp%
? The 6a$ss eter is ver% sensitive (t m$st be caibrated to 5ero before it is $sed To do this the probe m$st be moved a+a% from an% magnets or so$rces of magnetic fieds) s$ch as transformers and c$rrent carr%ing +ires S+itch :n the 6a$ss eter and pace the ha probe a+a% from the eectromagnet and from a po+er s$ppies Seect the F1 range of the 6a$ss eter and $sing the 5ero adj$stment *nob of the 6a$ss eter) adj$st the reading of the 6a$ss eter as 5ero Do not switch on the electromagnet at this stage. @ Go+ chec* to see that the votage across the sampe is 5ero +hen the magnetic fied is 5ero S+itch :G the constant c$rrent and set the c$rrent to > m4 in the constant c$rrent so$rce A
Eeep the magnetic fied at 5ero as recorded b% the 6a$ss eter Do not switch on the electromagnet at this stage. A Set the votage range of the m$timeter at 0-200m (f needed set the votage as recorded b% the m$timeter to be 5ero b% adj$sting the 5ero set pot provided on the P" b% $sing a scre+driver Bhen the c$rrent of > m4 is passed the cr%sta +itho$t appication of the magnetic fied the Ha otage as recorded b% the m$timeter sho$d be 5ero 4dj$st the 5ero set pot on% +ith the permission of the ab instr$ctor Do not switch on the electromagnet at this stage. = "ring the c$rrent reading of the constant c$rrent so$rce to ero b% adj$sting the *nob of the constant c$rrent so$rce 10 S+itch :G the eectromagnet ! 12 ) 3 4 ax # 11 Seect the range of the 6a$ss eter to F10 and meas$re and record the magnet f$x densit% at the center bet+een the poe pieces The tip of the Ha Probe and the cr%sta sho$d be paced bet+een the center of the poe pieces The poe pieces sho$d be as cose to the cr%sta and to the tip of the Ha Probe ,OE ,ECE HO/8 -E7ER TO/CH THE CRT6 OR THE T, OF THE H6 ,RO3E. The 'agnetic fl!* densit sho!ld be 'ore than 1;:: 4a!ss for cond!cting this e*&eri'ent. 12 o not change the c$rrent in the eectromagnet Eeep the magnetic fied constant thro$gho$t the entire experiment 13 ar% the c$rrent thro$gh the constant c$rrent so$rce in ver% sma intervas bet+een 0 m4 and 20 m4 (t is s$ggested that %o$ start +ith a reading near 20 m4 and then decrease the c$rrent b% intervas of abo$t 10 m4 and record the data o not exceed 20 m4 or %o$ +i destro% the cr%sta ompete 1> trias at ro$gh% 10 m4 intervas 8ecord the Ha ,ffect otage and $rrent for each interva in the tabe provided 1< Pot V H vers$s I 1> omp$te the the ratio of Ha ,ffect otage divided b% the $rrent in the co$mn provided V H mV I mA
1? omp$te the mean of the ratio of Ha ,ffect otage divided b% the $rrent 1@ Cse this mean to comp$te the Ha oefficient R H =
V H
t
I
B
1A (f the Ha oefficient is positive) the materia is a p-t%pe semicond$ctor materia (f the Ha oefficient is negative) the materia is a n-t%pe semicond$ctor materia 1= omp$te the oncentration of harge arriers n=
1
e R H
−19
e =1.602 X 10
C
20 eas$re the distance bet+een contact points $sed to meas$re the otage !L# omp$te the 8esistivit% of the sampe r =
V L w t L I
21 omp$te the mobiit%
m=
R H r
22 omp$te the Ha 4nge −1 H =tan m B =
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;.) Observations
w
Bidth of Specimen
/ IIIIIIIIIIIIIIIIIIIIIIII m
l / IIIIIIIIIIIIIIIIIIIIIII m
Length of Specimen
Thic*ness of Specimen
t / IIIIIIIIIIIIIIIIIIIIIII m
istance bet+een ontact Points for otage 8eadings L / IIIIIIIIIIIIIIIIIIIIIII m agnetic .$x ensit% Trial Number
B / IIIIIIIIIIIIIIIIIIIIIII 6a$ss J IIIIIIIIIIIIII F 10-< Tesa Hall Voltage
Current
I mA
V H mV
Ratio of Hall Volatge to Current,
V H I
1 2 3 < > ? @ A = 10 11 12 13 1< 1> ean a$e of 8atio/IIIIIIIIIIIIIIIIIIIIIII W Ha oefficient
R H =
V H
t
I
B
/IIIIIIIIIIIIIIIIIIIIIIIm-1
T%pe of r%sta ! n-t%pe or p-t%pe # / IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII arrier oncentration
10
n=
1
e R H
/ IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII carriers per m3
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w
Bidth of Specimen
/ IIIIIIIIIIIIIIIIIIIIIIII m
l / IIIIIIIIIIIIIIIIIIIIIII m
Length of Specimen
Thic*ness of Specimen
t / IIIIIIIIIIIIIIIIIIIIIII m
istance bet+een ontact Points for otage 8eadings L / IIIIIIIIIIIIIIIIIIIIIII m agnetic .$x ensit% Trial Number
B / IIIIIIIIIIIIIIIIIIIIIII 6a$ss J IIIIIIIIIIIIII F 10-< Tesa
Current
I mA
Distance between
Two Points between which
Hall Voltage
V L mV
V
is easure!. l m
Resisti"it# V L w t r = I L m
1 2 3 < > ? @ A = 10 11 12 13 1< 1> ean a$e of 8atio/IIIIIIIIIIIIIIIIIIIIIII W obiit% Ha 4nge 11
m=
R H r
/IIIIIIIIIIIIIIIIIIIIIII m 2 V −1 s −1
H =tan
−1
B m
/ IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
<.) 6nalsis 1Bhat are the experimenta errors that %o$ enco$ntered +hie cond$cting this experimentK
2Bhat are %o$r $ncertainties in the meas$rement of the Ha ,ffect otage) the c$rrent thro$gh the sampe) the magnetic fied and the distance bet+een the contact pointsK
3,stimate the maxim$m $ncertaint% in comp$ting the Ha oefficient
12
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=.) -ot >!st for F!n ( 1: &oints E*tra )" Prior to the deveopment of Ha effect transd$cers) there +ere vario$s other +a%s to meas$re a magnetic fied onsider a coi of +ire carr%ing and aternating c$rrent 4 magnetic fied +ith a reative% arge magnit$de +i deveop in the center of the coi This magnetic fied +i aternate at the same fre;$enc% as the aternating c$rrent r$nning thro$gh the coi (n order to meas$re this fied +e can $se a sma 7search9 coi and +atch the votage ind$ced in this sma coi The magnetic fied +i be proportiona to the votage ind$ced 8eca that .arada% eft $s a mode of ho+ eectricit% can be ind$ced from magnetic f$x that changes +ith time
=− N
d B dt
=− N
d B⋅ A =− N d ∣ B∣∣ A∣ cos dt dt
Bhere G is the n$mber of t$rns) 4 is the cross sectiona area of the coi) " is the magnit$de of the magnetic fied) and Q is the ange bet+een the fied and the area vector (n o$r case) +e +i *eep the ever%thing constant except for the aternating magnetic fied (n this j$st for f$n) %o$ are given an oscioscope) a f$nction generator) a coi to provide a magnetic fied) vario$s cabes) and a search coi Yo$r job is to get a ro$gh idea of ho+ the magnit$de of the magnetic fied aro$nd a c$rrent carr%ing coi varies as the distance from the coi varies Cse the e;$ipment provided 6ood $c*
Ta*e ten readings abo$t a centimeter apart and meas$re the magnetic fied aong the center of the coi) on either side of the coi a*e a ;$ic* pot of %o$r data $sing a spreadsheet id %o$ get +hat %o$ expectedK
,xpain) in detai) ho+ %o$ meas$red the magnetic fied
13
1<
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,reab Hall Effect Honor?s ab at a constant speed v The magnetic 14 cond$cting rod is p$ed thro$gh a magnetic fied B fied is into the page and the rod is p$ed in the positive x-direction ra+ the forces fet b% the eectron and the proton) d$e to eectric fieds and magnetic fieds) as the cond$cting rod is p$ed thro$gh the magnetic fied 4t first) there is on% a magnetic fied
at a constant speed v The magnetic 24 cond$cting rod is p$ed thro$gh a magnetic fied B fied is into the page and the rod is p$ed in the positive x-direction 4s the positive and negative charges move to the top and bottom of the rod) an eectric fied evoves ra+ the forces fet b% the eectron and the proton) d$e to eectric fieds and magnetic fieds) as the cond$cting rod is p$ed thro$gh the magnetic fied
@@@@
,
1>
at a constant speed v The magnetic 34 cond$cting rod is p$ed thro$gh a magnetic fied B fied is into the page and the rod is p$ed in the positive x-direction 4s the positive and negative charges move to the top and bottom of the rod) an eectric fied evoves ,rove that the votage ind$ced across the rod) +hen a stead% state condition is reached) is e;$a to ∣∣=∣ B∣∣v∣ l ) +here is the ength of the rod
<(f the magnit$de of the magnetic fied is ∣ B∣= 2T ) and the speed of the rod is ∣v∣=10 m / s ) and the ength of the rod is 10 meters) +hat is the potentia difference ind$c ed across the rodK
1?
