establishing hagall potential

8/13/2019 Establishing Hagall Potential

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Establishing a Hall Potential in Silver

via an Orthogonal Magnetic Field to

Deduce the Density of Accumulated

Negatively Charged Particles

Created by Brian Hallee

Partnered by Joseph Oxenham

Performed October 29, 2010


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the theory section' it is #orth noting that Hall7s discovery has s$a#ned

several related discoveries on the /uantum scale%8 "hese ne# /uantum0

based Hall effects have not only found their #ay into many modern

a$$lications you li!ely use every day' they have also hel$ed to narro# do#n

$hysical constants -i%e% the fine structure constant. to one $art in a billion%9

Only three decades ago #as the Hall Effect generali4ed to the /uantum

scale% "he :integer; /uantum hall effect #as first $redicted by three

$hysicists in ),+< #ho' in turn' doubted their o#n calculations% Nonetheless'

five years later' a team led by Dr% =laus von =lit4ing discovered that the Hall

conductivityincreased by' #hat $hysicists no# grant as' exact integers%


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so$histicated voltmeters for accurate measurements at this scale% "y$ically'

they find a home in all ty$es of sensing e/ui$ment such as s$eed' $ressure'

current' or fluid0flo# sensing% "hey are also li!ely to be $aired u$ #ith a

$otentiometer for a$$lications that re/uire a s#itch to be robust -eg% electric

guns.%+5hile one #ould be inclined to believe that $hysicists #ould have'

more or less' :closed the boo!; on the Hall Effect by stretching and

generali4ing it as far as is $hysically feasible over the $ast )@@ years' this is

not the case% Currently' heavy research is involved in understanding the

$hysics behind the fractional/uantum Hall Effect discovered in ),*1% As

common sense #ould suggest' the discovery of this $henomenon stemmed

from the observations of Hall ste$s #ith fractional /uantum numbers% "he

brunt of the understanding of this occurrence comes from the utili4ation of

:#ave0functions;% Ho#ever' as this article undergoes construction' heavy

research remains in the realm of :/uasi0$articles; and their aty$ical

fractional charge%

"heoretical asis

Fortunately' as scientists of

the t#enty0first century' #e

are able to a$$ly the electron

and $roton to our conce$tual

e(am$les to fully gras$ the

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Figure 1: A semiconductor e($eriencing the HallEffect


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$henomenon and a$$ly mathematics to it% "he derivation of the hall voltage

is relatively sim$le an only re/uires a firm understanding of current'

magnetic force' and Ne#ton7s second la#% 5e #ill refer to figure ) /uite

fre/uently' and it is only fitting that #e inform you of the meaning behind the

variables it contains at the start%

I B "he current $assing through the semiconductor -Natural convention callsfor this to designate $ositive charge movement-$rotons. De$icted in the diagram 5e #ill observe shortly that #e are only

concerned #ith electron movement in this e($eriment%FmB "he magnetic force e(erted on the moving charge carriers

B B "he magnetic field

dTB "he thic!ness of the $late -&n our derivation' #e #ill sim$ly designate

this as T.

DB "he height of the $late

FEB "he electric force

VH B "he Hall oltage% -"o differentiate this from regular $otential and

velocity' #e #ill use Hv.

&f #e further besto# another variable to the figure' d -An infinitesimally

small length s$anning the :height; of the bloc!.' #e can state that a charge

dG moves through the length d over a time dt% 3nfortunately for Hall' this

is #here his toils #ould have ended' as #e are ready to a$$ly the notion of a

:charged $article;' or electron' to the derivation% At this $oint' the volume

d contains an amount of charge dG' or the number of individual charged

$articles times the volume they occu$y% "hus'

Charge in d BdG B nd -eqn. 1)

5e can infer from figure one that the volume d is e/ual to "Dd%

"herefore'

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dGBn"Dd -eqn. 2)

&f #e divide e/uation 1 by the differential time #e s$o!e of earlier' #e

obtainI

dGdt B n"Dddt J & B n"Dvd -eqn. 3)

5e utili4ed the fact that charge $er0unit time is current' and distance $er0

unit time is velocity to achieve e/uation 8% "he subscri$t d for velocity

denotes the :drift; nature of this movement% arring unnecessary

electrodynamics' the drift velocity arises from the mostly random motion of

electrons% Although #e tend to believe that all electrical $henomena o$erate

at relativistic s$eeds' this is sim$ly not the case for mean electron

movement in a #ire% 5hen an electron field is $resent' they #ill tend to drift'

on the order of )@09

mKs' to#ards their destination due to the large amount

:roadbloc!s; -nuclei' inter0molecular forces' etc%. they must overcome%* "his

drift velocity #ill come in handy as our derivation $rogresses% "hus #e re0

#rite e/uation 8 as follo#sI

vd B &n"D -eqn. 4)

Ne(t' #e consider the magnetic field and its effects on our stream of charged

$articles% "he general e/uation governing magnetic force on a charged

system is #ritten as follo#sI

F B /v ( -eqn. 5)

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efore #e move any further' #e must denote #hat charges are actually of

focus in our e($eriment% "o solve this' #e #ill treat as an a(iom the fact that

semi0conductors -such as Silver' our metal of choice. allo# for negative

charge movement #ithin their structure%, &n layman7s terms' #hen a current

develo$s in our sam$le' #e can safely assume that electrons are the cause

of this' and they move in the o$$osite direction of standard convention%

"hus' #hen a$$lying e/uation < both /uantitatively and /ualitatively' #e

must ta!e into account the negative charge attributed to electrons% 5e #ill

begin #ith the /ualitative argument to su$$ort figure ) and its' thus far'

mysterious charge accumulations%

F B 0 0i ( 0! B L -eqn. )

E/uation ? is strictly concerned #ith the vector analysis of e/uation


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continue our /uantitative derivation to arrive at a clean formula to $redict

#hat this $otential might be% "a!ing the absolute value of e/uation


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Finally' substituting in e/uation 9 and multi$lying by D' #e arrive at the Hall

oltage formulaI

Hv B D&n"D B &n" -eqn. 12)

5hile e/uation )1 loo!s about as clean as clean gets' therein lies a $roblem

#ith the term in $arenthesis% N can' by no means available today' be

measured directly or even guessed% "y$ically' there are some#here on the

order of )@)@

electrons that accumulate on the surface of a semiconductor at

lo# currents% Ho#ever' #hile #e mentioned earlier that direct measurement

of Hall Potentials re/uire rather $recise' high0/uality e/ui$ment' #e #ere

able to utili4e such gear in our attem$t at this lab% Naturally' magnetic field

is easily measureable #ith a standard gauss meter% "hus' our notation used

in e/uation )1 has become clear' as &n" re$resents the slo$e of t#o

measurable /uantities% "he most efficient #ay to solve for n' our ultimate

goal' is to measure the Hall Potential at different current and field strengths

and obtain the slo$e of the subse/uent gra$h% From this' the charge carrier

density in the accumulation can be found using sim$le algebraI

n B &slo$e " #eqn. 13)

"o $lace n into $ers$ective' electrons #ield a mass on the order of )@08)!g

#hile the mass of silver roughly @%< !g% "hus' #e #ill e($ect the density of

$articles in the accumulation to be verylarge%

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A$$aratus

5hile the

theory behind

the Hall

Potential may

have come

across as

relatively

sim$le' the

e/ui$ment used

to

e($erimentally achieve it #as not% "his lab utili4ed a sle# of meters' a

standalone' dedicated' #ater0cooled' DC $o#er source' and a 80"esla 5al!er

Scientific electromagnet as seen in figure 1% Starting #ith the meters' #e

#ere re/uired to monitor the Hall oltage' current' and field strength

simultaneously% "hus' #e #ere given a voltmeter that read voltage to the

order to )@0


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ammeter' and a gaussmeter% All #ere $re0assembled and $re0calibrated as

#e arrived to carry out the lab $rocedure% A $hoto of the meters can be

vie#ed in figure 8 belo#% "he $o#er su$$ly $layed an im$ortant role in our

ability to follo# through #ith this lab% As noted in the lab handout,' at

ma(imum current the electromagnet consumes *


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in the Hall device% Conse/uently' #e should e($ect the negative charge

accumulation to reside at the to$ of the device% 5e shall return to this to$ic

in the discussion section #hen #e hy$othesi4e $ositive charge carrier

movement and its effects%

Procedure

Due to the very sensitive' -and not to mention ex$en%ive.' nature of

the e/ui$ment $resent for this e($eriment' the $rocedure #e follo#ed #as

rather rigorous and static% &n other #ords' the only acce$table methodology

to use in $erforming this e($eriment #as to follo# that stated in the lab

re$ort% &n order to ensure $ro$er startu$ of the electromagnet' Dr% reg

atta -&nstructor' Professor of Physics. made a brief a$$earance to oversee

our $rocedure% "o begin' #e s#itched on the $o#er stri$ that served the

voltmeter' ammeter' and gauss meter and follo#ed this by s#itching on the

three res$ective meters% Ne(t' #e s#itched on the DC $o#er su$$ly and the

#ater coolant that served it% efore doing this' ho#ever' it #as of utmost

im$ortance that the coarse' medium' and fine !nobs #ere all set to 4ero on

the $o#er su$$ly% "his is due to the fact that a sudden increase or decrease

in $o#er to the electromagnet could cause severe damage to the internal

com$onents% As #e activated the #ater coolant' the $o#er su$$ly re/uired

monitoring until a light designating sufficient #ater flo# #as galvani4ed%

Once this occurred' #e #ere able to s#itch on the current su$$ly and set it

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Figure 3: -From left to right. oltmeter' Ammeter' Current source' aussmeter


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to


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#e ensured that both the current and DC $o#er #as set to 4ero% At this

moment' all devices #ere turned off and the #ater coolant #as sto$$ered%

Sam$le Calculations

Figure 9 de$icts the gra$h #e achieved from the data of our first run%

oo!ing at our coefficient of determination' #e see that our data almost

$erfectly fits the linear relationshi$ as $redicted by e/uation )1% & feel as

though it is im$ortant' at this $oint' to e($ress the ex$ected value for the

charge carrier density in units of electrons0$er0meter cubed in order to

com$are sam$le values to it% "hat value isI ?%,()@1*em8% ou may vie# our

e($erimental values for charge carrier density in the a$$endi( of this re$ort%

"he formula #e utili4ed to achieve those values is sho#n belo#I

neB?SOPE*I1+'A*IA1+QQ9)@?)%? ( )@0), -).

5e #ish to chec! this algorithm by $erforming a sam$le calculation of our

o#n% &n this e(am$le & #ill use the data $oints B )%@" and B )%)" from

our first run using & B


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@%*RBm@%)" J mB @%*R@%)" B *RAm1

oo!ing at figure )' #e see that our slo$e #as *%1 RAm1' so #e can be sure

our methodology is correct so far% Ne(t' #e use the data $oint B )%@" and

our ne#found slo$ to solve for I

,%1RB*RAm1)%@"L J B ,%1R*RA"m1 B )%)


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astly' #e a$$ly the fact that one electron e(hibits a charge of )%?@()@0),C%

"hus' #e divide -9. by this value to achieve our sought after value in units of

em8I

ne B )%


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bet#een t#o conductors t#o im$ortant $henomena occurI Heat flo#' and

#hat is coined as the &eebec' effect.)) 5e are only concerned #ith the

latter% "he Seebec! effect concerns the fact that energetic electrons #ill

move from the hot unction to the cooler one #hile $ushing some of the

lesser energy electrons #ith them% &n turn' this causes a $otential difference

bet#een the hot and cold ends' and' alas' a thermocou$le $otential is born%

Considering the fact that not only did #e $erform this lab early in the

morning' but #e #ere the first to $erform it that day' #e li!ely had to

com$ensate for more of this $otential than one normally #ould% i!e#ise'

this may have been a source of some of the error #e did garner as the

unctions and conductors heated u$ over the course of the e($eriment% ou

might notice that our first run returned a value a bit lo#er than the other

t#o% & $ostulate that this is due to the heating of the #ires over the first run'

in turn' thro#ing off our initial

com$ensation%

Ne(t #e consider the situation in

#hich $ositive charge carriers #ould

have moved through the Hall Device%

"echnically' this effect trans$ires

from the movement of :holes; in the

metal not $rotons% Ho#ever' for

sim$licity' #e #ill use the term

Page 6 )*

Figure : "he Hall Effect Device


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subse/uently allo#ed for a greater amount of electrons to be $ushed to the

to$% Ho#ever' loo!ing more carefully at e/uations )1 and )8' #hile the slo$e

varies linearly #ith current' the value of nehas an inverse relationshi$ #ith

this increased current% "hus' in theory' the increase in current should be

cancelled out by the increase in slo$e' and neshould remain constant%

"herefore' returning to our original $rediction' #e feel as though the

variance in our first run #as due to a cold a$$aratus e(hibiting thermocou$le

$otentials%

One interesting $henomena #e #ere unable to e($eriment #ith in our lab

session is the moving of the conductor itself% Naturally' e/uation ? is only

concerned #ith the movement of the electrons relative to the electro magnet

-or' s$ecifically' the electric field.% "hus' #e can hy$othesi4e that if #e #ere

to move the entire Hall Device in the o$$osite direction of the current -in our

case' to#ard the bac! of the electromagnet. at the exact s$eed as the

electron drift s$eed' then the motion of the electrons #ould be 4ero relative

to the field% &f this is the case' then e/uation ? 4eros out and #e have a

magnetic force of e(actly 4ero% Succinctly' if #e move the Hall device at an

e/ual and o$$osite velocity relative to the electrons inside' #e #ill not

observe a Hall $otentialV Achieving this even near0$erfectly in the lab setting

#ould be rather difficult% Ho#ever' the theory further solidifies the relation

bet#een magnetic fields and charge carriers%

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Although ),thcentury scientific advancement hinders us from directly

com$aring our value of electron density #ith Hall7s' #e can /ualitatively

a$$reciate the rigor involved in his finding of such a remar!ably small

$otential difference% Nonetheless' the effect has found a$$lications far and

#ide due to its ability to give a clean on0off signal% 5hile the classical Hall

Effect' at this $oint' may be fully understood' its e(tensions into the /uantum

#orld #ill li!ely be a to$ic of heated debate and research for years to come%

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!or"s CitedCain' F% -1@@9' March ).% (ecord for urt*e%t +alaxy i% ro'en -ain%

Tetrieved October 8@' 1@)@' from 3niverse "odayI

htt$IKK###%universetoday%comK,89+Krecord0for0furthest0gala(y0is0

bro!en0againK

eorgia State 3niversity% -n%d%.% /*e all ffect% Tetrieved October 8@' 1@)@'

from Hy$erPhysicsI htt$IKKhy$er$hysics%$hy0

astr%gsu%eduKhbaseKmagneticKhall%html

Honey#ell% -n%d%.% all ffect &en%in and -$$lication% Tetrieved October 8@'

1@)@' from Honey#ellI

htt$IKKcontent%honey#ell%comKsensingK$rodinfoKsolidstateKtechnicalKhall

boo!%$df

Hugh D% oung' T% A% -1@@+.% niver%ity P*y%ic%.Pearson Addison05esley%

atta' D% % -n%d%.% /*e all ffect in &ilver ote%.FrostburgI Frostburg State

3niversity%

Microstar aboratories% -1@@,.% /*ermocou$le old unction%% TetrievedOctober 8@' 1@)@' from Microstar aboratoriesI

htt$IKK###%mstarlabs%comKsensorsKthermocou$le0cold0unctions%html

Giu' % -),,+' A$ril 1+.% a%ic% of all ffect6 i%tory% Tetrieved October 8@'

1@)@' from 2ohns Ho$!ins 3niversityI

htt$IKK###%$ha%hu%eduKW/iuymK/heKnode)%html

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ENDNO"ES

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APPEND& 0 D&SC CON"EN"S

TOO"D&TEC"OT

A9 X HAEFFEC"%DOC

"HEOFF&C&AM&CTOSOF"5OTDATEPOT"CONCETN&N"HEHAEFFEC"&N

S&ET

HAEFFEC"SPTEADSHEE"%S

"HEM&CTOSOF"ECESPTEADSHEE"CON"A&N&N"HETA5DA"AEN"ETED

D3T&N"HECO3TSEOF"HEA

HA+%PN

F&3TE) 3SED&N"H&SATEPOT"

F&3TE1%PN

F&3TE1 3SED&N"H&SATEPOT"

&MY)1+,%2P

F&3TE8 3SED&N"H&SATEPOT"

&MY)1++%2P

F&3TE< 3SED&N"H&SATEPOT"

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) Ed#in Hall -)*+,.% ZOn a Ne# Action of the Magnet on Electric CurrentsZ%-merican ournal

of 7at*ematic%-American 2ournal of Mathematics' ol% 1' No% 8. 2-8.I 1*+X

,1% doiI)@%18@+K18?,19


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)) Microstar aboratories% -1@@,.% /*ermocou$le old unction%% Tetrieved October 8@' 1@)@'from Microstar aboratoriesI htt$IKK###%mstarlabs%comKsensorsKthermocou$le0cold0

unctions%html

establishing hagall potential

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