EN Van der Pauw method 范德堡法WikiWord文件下载.docx
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∙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