EN Van der Pauw method 范德堡法Wiki.docx

EN Van der Pauw method 范德堡法Wiki.docx

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ENVanderPauwmethod范德堡法Wiki

VanderPauwmethod范德堡法

FromWikipedia,thefreeencyclopedia

ThevanderPauwMethodisatechniquecommonlyusedtomeasuretheresistivityandtheHallcoefficientofasample.Itspowerliesinitsabilitytoaccuratelymeasurethepropertiesofasampleofanyarbitraryshape,solongasthesampleisapproximatelytwo-dimensional（i.e.itismuchthinnerthanitiswide）,solid（noholes）,andtheelectrodesareplacedonitsperimeter.ThevanderPauwMethodemploysafour-pointprobeplacedaroundtheperimeterofthesample,incontrasttothelinearfourpointprobe:

thisallowsthevanderPauwmethodtoprovideanaverageresistivityofthesample,whereasalineararrayprovidestheresistivityinthesensingdirection.[1]Thisdifferencebecomesimportantforanisotropicmaterials,whichcanbeproperlymeasuredusingtheMontgomeryMethod,anextensionofthevanderPauwMethod.

Fromthemeasurementsmade,thefollowingpropertiesofthematerialcanbecalculated:

∙Theresistivityofthematerial

∙Thedopingtype（i.e.whetheritisaP-typeorN-typematerial）

∙Thesheetcarrierdensityofthemajoritycarrier（thenumberofmajoritycarriersperunitarea）.Fromthisthechargedensityanddopinglevelcanbefound

∙Themobilityofthemajoritycarrier

ThemethodwasfirstpropoundedbyLeoJ.vanderPauwin1958.[2]

Contents

∙1Conditions

∙2Samplepreparation

∙3Measurementdefinitions

∙4Resistivitymeasurements

o4.1Basicmeasurements

o4.2Reciprocalmeasurements

o4.3Reversedpolaritymeasurements

o4.4Measurementaccuracy

o4.5Calculatingsheetresistance

∙5Hallmeasurements

o5.1Background

o5.2Makingthemeasurements

o5.3Calculations

∙6Othercalculations

o6.1Mobility

∙7Footnotes

∙8References

Conditions

Therearefiveconditionsthatmustbesatisfiedtousethistechnique:

[3]

1.Thesamplemusthaveaflatshapeofuniformthickness

2.Thesamplemustnothaveanyisolatedholes

3.Thesamplemustbehomogeneousandisotropic

4.Allfourcontactsmustbelocatedattheedgesofthesample

5.Theareaofcontactofanyindividualcontactshouldbeatleastanorderofmagnitudesmallerthantheareaoftheentiresample.

Samplepreparation

InordertousethevanderPauwmethod,thesamplethicknessmustbemuchlessthanthewidthandlengthofthesample.Inordertoreduceerrorsinthecalculations,itispreferablethatthesampleissymmetrical.Theremustalsobenoisolatedholeswithinthesample.

Somepossiblecontactplacements

Themeasurementsrequirethatfourohmiccontactsbeplacedonthesample.Certainconditionsfortheirplacementneedtobemet:

∙Theymustbeontheboundaryofthesample（orasclosetoitaspossible）.

∙Theymustbeinfinitelysmall.Practically,theymustbeassmallaspossible;anyerrorsgivenbytheirnon-zerosizewillbeoftheorderD/L,whereDistheaveragediameterofthecontactandListhedistancebetweenthecontacts.

Inadditiontothis,anyleadsfromthecontactsshouldbeconstructedfromthesamebatchofwiretominimisethermoelectriceffects.Forthesamereason,allfourcontactsshouldbeofthesamematerial.

Measurementdefinitions

∙Thecontactsarenumberedfrom1to4inacounter-clockwiseorder,beginningatthetop-leftcontact.

∙ThecurrentI12isapositiveDCcurrentinjectedintocontact1andtakenoutofcontact2,andismeasuredinamperes（A）.

∙ThevoltageV34isaDCvoltagemeasuredbetweencontacts3and4（i.e.V4-V3）withnoexternallyappliedmagneticfield,measuredinvolts（V）.

∙Theresistivityρismeasuredinohms⋅metres（Ω⋅m）.

∙Thethicknessofthesampletismeasuredinmetres（m）.

∙ThesheetresistanceRSismeasuredinohmspersquare（Ω/sqor

）.

Resistivitymeasurements

Theaverageresistivityofasampleisgivenbyρ=RS⋅t,wherethesheetresistanceRSisdeterminedasfollows.Forananisotropicmaterial,theindividualresistivitycomponents,e.g.ρxorρy,canbecalculatedusingtheMontgomerymethod.

Basicmeasurements

Tomakeameasurement,acurrentiscausedtoflowalongoneedgeofthesample（forinstance,I12）andthevoltageacrosstheoppositeedge（inthiscase,V34）ismeasured.Fromthesetwovalues,aresistance（forthisexample,R12,34）canbefoundusingOhm'slaw:

Inhispaper,vanderPauwshowedthatthesheetresistanceofsampleswitharbitraryshapescanbedeterminedfromtwooftheseresistances-onemeasuredalongaverticaledge,suchasR12,34,andacorrespondingonemeasuredalongahorizontaledge,suchasR23,41.TheactualsheetresistanceisrelatedtotheseresistancesbythevanderPauwformula

Reciprocalmeasurements

Thereciprocitytheorem[1]tellsusthat

Therefore,itispossibletoobtainamoreprecisevaluefortheresistances

and

bymakingtwoadditionalmeasurementsoftheirreciprocal