Volume44A,number3 PHYSICSLETTERS 4 June1973

LATTICE PARAMETERSOF n- AND p-TYPE Bi2Te3

J.O. BARNES andJ.A. RAYNECarnegie-MellonUniversity,Pittsburgh,Pennsylvania,USA

and

R.W. URE,Jr.Universityof Utah, SaltLakeCity, Utah, USA

Received24 April 1973

Precisionlatticeparametermeasurementshavebeenmadeon Bi2 Te3 overawide rangeof carrierdensities.Forn-typematerialc/a is lessthanthecritical value.Jii~74A simplemodelfor thevariationof c anda is presented.

Attempts[1,2] havebeenmadeto correlatetheelectronicpropertiesof Bi2Te3 with thegeometryofits unit’cell. In particular,it hasbeendemonstratedthat thereexistsa critical valueof (c/a)cth= ~J195/4= 6.98212.Thiscritical ratio correspondsto thecondi-

~ 30.49tion that the(122) rhombohedralplane,perpendicular-ly bisectingtheassociatedreciprocallattice point,passesthrough(333)point. Previousdatasuggestthe I I ~empiricalresultthatBi2Te3 is eithern or p-type, de- 3048

pendingon whetherc/a ~ (c/a)~.Thepresentwork I I I I I0.1 0.3 0.5 I 3 5wasundertakento testthis conclusionover a wide

rangeof carrier densities,usingwell-characterised (ol

singlecrystals./Datawere obtainedonzone-levelledsinglecrystals /

4,388 - /of n- andp-typeBi2Te3 containinganexcessor deficit /

of tellurium. Thecarrier densities,~ determinedfrom the saturationvalueof theHall coefficientat -~

4,3864.2K, areshownin table 1. X-ray measurementsat296°Kwere madewith a PickerdiffractometerusingmolybdenumK~,radiation,reflectionsup to(00.81) —

and(66.0)beingemployedto determinetheapparentlatticeparametersc anda respectively.Thecorrected I I I

0,1 0.3 0.5 I 3 5valuesof c anda for thesesamples,obtainedfrom (iO’°cm

3)

computerfits of the datato a Nelson-Rileyextrapola-tion function,aregiven in table I. (51

From thetableit canbe seenthat boththe c anda Fig. 1. Variation of (a) thec-parameterand(b) thea-param-parametersfor p-typeBI

2Te3 areindependentof car- eterfor n-typeBI2Te3 asafunctIonof totalcarrierconcentra-rier concentrationwithin experimentalerror.Our val- tion °H~Thedashedandfull curvesarethecalculatedvaria-

tions for the one-andtwo-bandmodelsdiscussedin thetext.uesare in reasonableagreementwith thebestdatapre- Thearrowscorrespondto thevalueof thec anda-parametersviously availablefor p-typematerial[3]. The resulting for thep-typesamplewith n = 1.8 x 1018 cm

3.

c/a ratio is 6.9505,which is indeedlessthan (c/a)~~.In thecaseof n-typeBi

2Te3,both fig. 1 and table 1showthat the latticeparametersvary systematically

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Volume 44A, number3 PHYSICSLETTERS 4 June1973

Table I overlapelectrons.SinceS13 is negative,it is clear thatLatticeparametersof Bi2Te3 at 296 K asafunction of carrier thesestrainscorrespondto an expansionalongthea-density*, axis anda contractionalongthe c-axis.

Lattice parameter* The dashedlines in fig. I showthe expectedvaria-tions of thelattice parameters,computedfrom (I)

Carrier °H c (A) a ** (A) usingthe measuredelasticmoduli of Bi2Te3 andtype (1018cm’

3) choosingE= 11.7 eV to give a fit to the c-paraneter

p 1.8 30.4959±0.0009 4~f1~35 datafor n less than 4 X 1018 cni3. This value ofE20 30.4961±0.0008 4.3876±0.0006 seemsto be physicallyreasonableandalso givesa

plausiblerepresentationof the resultsfor thea-param-n 0.7 30.4931±0.0016 -~ eterover the samerangeof n. However, theseare

1.6 30.4923±0.0011 4.3859±0.0005 clearly a discrepancybetweentheory andexperiment3.7 30.4856±0.0011 4.3855±0.00085.0 30.4850±0.0007 for higher carrierconcentrations. -

6.8 30.4900±0.0010 4.3866±0.0009 The reasonfor this discrepancyis probablydueto16 30.4847±0.0011 4.3871±0.0005 the existenceof a secondsetof conductionbandmm-25 30.4816±0.0015 4.3860±0.0007 ima [51,which lie 32 meV abovethemain ellipsoidal52 30.4831±0.0007 - minima andwhichbeginto fill for n greaterthan

4 X 1018 cm’3. If we ignoretheoverlapstraindueto* Quotederrorsarerms valuesobtainedfrom fitting proce- .

dure.Estimatedtotal errorsare shownin figure. the secondband,thepnncipalchangein theanalysis** Missing entriescorrespondto thin samplesin which this is to alter the dependenceof n on the total carrier

parameterunmeasurable. density ~ The full lines in fig. I showthe resultsof

a revisedcalculation,usingthevariationof n with n11

with carrier concentration.The maximumc/a ratio, obtainedfrom dHvA andHall effect data[5]. Al-which occursat low carrier densities,is 6.9 522andis though the fit to experimentis quite good, theagree-alsoless than(c/a)~~.It must be concluded,therefore, rnentmay not be significantdueto our neglectofthat theempiricalcorrelationdiscussedaboveis not overlapeffectsfrom thesecondband.However,sinceapplicableto our samplesof n-typematerialandis the locationsof theassociatedminima within thezoneprobablygenerallyinvalid. arenot known,considerationof theseeffetcsis not

Thevariationwith carrier density of the lattice pa- possibleat present.It neverthelessseemsreasonablyrametersfor n-typeBi2Te3 canbe understoodin certain that thebehaviorof the latticeparametersistermsof a simple zoneoverlapmodel for theband determinedby electroniceffectsof the type discussed.structure[4]. In this modeltheprincipalconductionbandminima aretreatedasoverlapregionsinvolving asphericalenergysurfaceandJones-typezonefaces Referenceswith normalsperpendicularto thec-axis.The strains

resultingfrom this overlaparegiven by [11 H.H. Soonpaa,J.Appl. Phys.34 (1963)3600.— — 121 H.H.Soonpaa,J.Chem.Phys.Sol.25(1964)1107.

= (S11 + S12)En; sc/c = ~I3 En, (1) [3] M.H. Francombe,Brit. J. Appi. Phys.9 (1958)415.[41 H. Jones,Proc. Roy. Soc.(London)A147(1934)396.

where~ ~12’ S13 arecomponentsof the appropri- [5] R.B. Mallinson,J.A. Rayneand R.W. Ure,Jr.,Phys.Rev.ate compliancematrix,E is themeanelectronenergy 175 (1968) 1049.referredto thezonecenterandn is the density of

216