Specific heat of pure graphite in the ultra-quantum limit region

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  • Solid StateCommunications,Vol.19,pp. 12411245,1976. PergamonPress. Printedin GreatBritain.



    Groupede recherchesur les semiconducteurset les dilectriqueset Dpartementde physique,UniversitdeSherbrooke,Sherbrooke,Qubec,Canada,J1K 2Rl

    (Received22 December1975byS.Amelinckx)

    Wecalculatetheelectronicspecificheatof puregraphitein theultra-quantumlimit regionfor fields between60 and200kG, at verylow tem-peratures,using theSlonczewskiWeissbandmodelwithvaluesof theenergy-bandparameterswhich are in agreementwith recentmagneto-reflectionexperiments.Theeffectof trigonal warpingof the Fermisur-facesassociatedwith the parameter~ is neglectedin thecalculation.Ourresultsshowthat, for mostof the rangeof fields considered,theelectronicspecific heatC is verynearlyproportionalto boththemagneticfieldstrengthH andthe temperatureT, accordingto the relationC cvHTwitha coefficientaof about0.091pJ/g-at.K2 kG. Theresultsalso indicatethat,at theupperendof themagneticfield range,theC(Ii) curves,at a given T,departprogressively,thoughslightly, fromlinearity with increasingH.

    1. INTRODUCTION the parametery~of the SlonczewskiWeissbandmodel7is neglected,thenear n 0 energylevel for holes

    IN A PRECEDINGpaper, WallaceansiGuptahavecal- (highestvalencebandlevel) is independentof themag-culatedthemagneticsusceptibilityof puregraphiteat netic field,whereasthat for electrons(lowestconduc-liquid-heliumtemperatureandin thepresenceof strong tion bandlevel) is field-dependent.6In addition,puremagnetic fields. They have found that graphite,which graphiteis an intrinsicsemimetal,sothat thehole den-hasa stronglydiamagnetic behaviorfor low magnetic sity is equalto theelectrondensity.fields,23 shouldbecomestronglyparamagneticfor fields In thepresentpaper,we calculatethe electronicgreater than about H = 60kG,4 that is to say, as soon as specific heatof puregraphitein theultra-quantumlimitthe n = 1 energylevelsfor the two higherbranchesof region. The aim of this calculation is twofold: (1) tographite have passed through the Fermi surface, and the determine how the very unusual features of the n = 0so-called ultra-quantum limit region is reached.5 The magnetic energy levels andtheir occupationby thecause of this large change in susceptibility is attributed carriersare reflectedin theelectronicspecific heatandto the very unusual features of theenergy-bandstruc- (2) to examinewhetherthis quantitycouldbeavaluableture of graphite when subjected to strong magnetic tool for getting additional informationaboutthefields. In theultra-quantumlimit region, only the fl = 0 energy-bandstructureof graphite.The methodof calcu-Landaulevels are of specialinterest.As shownby Gupta lation usedis a direct one.The two near n = 0 energyandWallace,6the four n = 0 magneticenergylevelsof levels arefirst found,accordingto basicallythesamegraphitealongtheverticalHKH edgeof the hex- procedureasthat followed in reference1, exceptthatagonal Brillouin zone canbe divided into two low-lying in the presentpaperwe usea newset of valuesfor theor near and two widely separated or far levels. At energy-band parameters which are in agreement withvery low temperatures, thermal excitation to higher recentlow-frequencymagnetoreflectiondata.8Then,levelsis negligible,andonly the two near n = 0 levels, the electronicspecificheatisworkedout,bothanalyti-which intersecttheFermisurface,areoccupiedby the cally andnumerically,from the secondderivativeof thecarriers.If theeffectof trigonalwarpingassociatedwith Helmholtz free energywith respectto temperature,using

    thewell known thermodynamicalformulaC = T(a2F/~ Partof this work wasperformedat the EatonElec. 8T2)H.

    tronicsResearchLaboratory,McGill University,Montreal,Quebec,Canada. 2. ENERGY-BAND STRUCTUREOF GRAPHITEIN

    * On leaveof absencefrom Centrede Recherchessur les THE ULTRA-QUANTUM LiMIT REGION

    TrsBassesTemperatures,C.N.R.S.,andServiceBassesTemperatures,CentredEtudesNuclaires,Grenoble, The energylevelsin a magneticfield H alongtheFrance. verticaledgeof thehexagonalBrilloum zonewere



    calculatedby McClure2andInoue9from theSlonczewskiWeissbandmodel.7If trigonal warpingis ENERGY (eV)neglected (y~ = 0),1.6 theMcClureInouesecularequa-tion is, for a givenmagneticquantumnumbern:

    (n+~)Q= ~EE3) [EE1 + EE21 00I0~ _~)2 (1 + p)2 -0020 [~-~oi~.oI7

    E2 = 7i17 + ~7~[2 E

    3 = ~72F2, V = 74F/70, -QO5O~~f[EE3/E_E, E_E2\12+ Q2~~2, (1) Eewhere iL 2 (~(1 v)2 (1 + ~)2)j -~--j -0.040 EhQ = 3y~a~/2r~,E~= ~+ ~v~i+ ~75F2, 212 ~kzCo_

    0 7r/8 1714 3 ,r/8 lr/zF = 2 cosx, x = ~k~c

    0. K REDUCED WAVE VECTOR HHere,a0 is the in-planelatticeparameter(= 2.456A), C0 Fig. 1. Plot of thetwo near n = 0 Landauenergyis the lattice parameterin thez direction,perpendicular levels of graphite against position x = ~k~coalong theto thebasalplane(= 6.708A), r~= (hC/IeI!~I)bs

    2iS the vertical Brillouin zone edgeHKH for a magneticfield ofcyclotronradius,and ~ and the7, arethe energy-band H = 200 kG. ThecurvesofEe(X) (lowestconductionparameterswhosebestvaluesare sofar:8 bandlevel)andof Eh(x)(highest valencebandlevel) are

    calculatedfor the SlonczewskiWeissbandmodel withL\ = 0.002eV, theparametervalues:Yo = 3.124eV, 7i = 0.377eV,

    72 = 0.021eV, ~ = 0.120eV, -y~ 0.003eV, and70 = 3.124eV, ~ . = 0.377eV, 72 = 0.021eV, ~ = 0002eV. The effect of trigonal warpingof the

    Fermi surfaces associated with the parameter 7~iS= 0.276 eV, ~ = 0.120eV, 7~= 0.003eV. innored(y~= 0).


    In theultra-quantumlimit region,only theknowledgeof magnitude,asshown in equation(2).b0 As is evidentthe two near magneticenergylevelsasa function of from equations(3) and(5), Eh is independentof themagneticfield andpositionk~alongtheverticalHKH magneticfield, whereasEe is field-dependent.Specifi-zoneedgeis of interest.Puttingn = 0 in equation(1) cally,Ee increaseswith H, but its dependenceon fieldthe near Landaulevel for holesEh (highestvalence is foundto beweakerthan that obtainedin Wallaceandbandlevel) is found immediatelyto be Guptaspaper,1mainlybecauseof the different choice

    of theset of valuesusedfor theenergy-bandparameters.Eh = 472F. (3) Formostof the rangeof k~alongtheHKH edgeof

    The near n = 0 Landaulevel for electronsEe(lowest theBrillouin zone,the field-dependentpart of Ee (in eV)conductionbandlevel) is thengivenby one of the three is approximatelygivenbysolutionsof theremainingequation: 1 Q[74


    2 27k ~(72 7s)I j 3.6x l05H,7o J7i

    + [E~E2+ E2E3+ E3E~ Q(l + v

    2)]E (4) whereHis in kG. The two near n = 0 Landauenergylevels,calculatedfrom equations(3) and (5) for the set

    ~E~E2E3 q [(1 + v)

    2E1+ (I v)

    2E2] } = 0. of bandparametersof equation(2), not including7~,

    andfor a magneticfield ofH = 200kG,areillustrated

    However,sincethis level isclosetoEh, it canbe shown in Fig. 1.by a simpleperturbationcalculationthat


    2 HEATQ1L~+ 6 ~Ee ~7

    2F2+ ~ ~ ~,(5) Forgraphite,the Helmholtzfree energyperunit

    Q + 711 volume in theultra.quantumlimit region canbewritten

    wherep2, which is small as compared to unity, has been in the formneglected.Theexpressionfor Ee in equation(5) is essen- F = N~t kBT ~ J D(E

    1) Log[l + ~ dL1,tially similar to thatpreviouslyobtainedby WallaceandGupta, but takesproperaccountof the fact that the where (6)bandparameters72 and7~do not havethesame


    N=E fD(E,)f(EjdE, (7) C(J/Km3)16 - ELECTRONIC

    Fermienergy,kB is Boltzmannsconstant,T is theabsol- 14 -is the totalnumberof electronsp unit volume,pis the NEAT axis) T 4.2K 7ute temperature,i = e or h andrepresentsthebranch (H / indexwhichspecifiesoneof thetwo near n = 0 mag- 12 -

    ~TTICEneticenergylevels(Ee andEh) workedout in thepre-vious section,f(~is the FenniDiracdistributionfunc- 10 ESTIMATEDSPECIFICtion, andD(E,) is thedensityof statesperunit volume I SPECIFIC HEATandenergyof the ith branchgivenby 8 - 1 421 1.792 339.0 I2 dxx 1:2 I 0.0562 I 10.63

    D(E1) = ir

    2c0r~ (8) 6 - 1041 0.0021 I 0.39351

    (K) I(m~~at.K)I(JA(,i~)Iwherean extrafactorof 4 hasbeenincludedtoaccount ~for thespin degeneracyandthefact that thereare two 1: 1.2KnonequivalentHKH axesin theBrillouin zone. 2 ~

    The electronicspecificheatis thenobtainedfrom

    2j 0 I I~C = T ~ (9) 60 80 100 120 140 160 180 200

    H(kG)After carryingout thesecondderivativeofFwith res-pect to T, accordingto equation(9), andmakinguseof Fig. 2. ElectronicspecincheatC of graphitein theultra-equations(7) and(8), we find the following result: quantumlimit region asa function of magneticfield H,

    at thetemperatures4.2, 1.2, and0.4K. Thecurvesare

    = ~ (~J~ [v1(x)]2~ calculatedfrom equation(10), using theenergy-bandir2c0r~ [1 + e3~i~]2 parameterslisted in thecaptionof Fig. I. Estimated

    valuesof the lattice specificheatof graphite,at each/ ap ~ J y1(x)~ ) temperature,are indicatedfor reference.

    + I\akBT) [1 +eYi~]2j (10)

    respectively.It is easyto seethat sucha conditioniswherey1(x) = [E1(x) p]/kBT, the limits onx are from metwhen the lengthofx for the lowestoccupiedcon-0 to ir/2, andthe quantity(aP/akB7), determinedfrom ductionbandlevel isequalto that for thehighestun-equation(7) by theconditionN/~T= 0, is givenby occupiedvalencebandlevel.



    In order to obtaintheelectronicspecific heatCofe~~ \ y.(x) ~ ~ / graphitein theultra-quantumlimit regionas a function ( 1 [1 + eYi~]2)/(~ J& [1 + eYi~]2) - of magneticfield andtemperature,the integrationsonx

    (11) in equations(10), (11), and(13) havebeendonenumeri-

    The only thing that remainsto be calculated is the Fermi cally. In Fig. 2, we have plotted the variation of Cwithenergyp. Thiscanbeeasily doneby noting that intrinsic H for magneticfields rangingfrom 60 to 200 kG andatgraphiteis compensatedso that theholedensityhasto thetemperatures0.4, 1.2, and4.2K. As we cansee,thebe equal to the electrondensity.12In theultra- resultsclearly indicatethat, for mostof therangeofquantumlimit region,the conditionfor p is then fields considered,C isvery nearlyproportionalto both

    themagneticfield andthe temperature,andcanbe wellJD(Ee)f(Ee) ~3~Ee= JD(Eh)[l ~f(Eh)} dEh, (12) representedby therelation

    C~7(ll)T~atfT, (14)which canbe written in the form,with the help of equa- with a coefficientaof about0.091pJ/g-at.K2kG. Ontion (8), the otherhand,it is worthmentioningthat, at theupper

    J dXf.f[Ee(X)] +f[Eh(x)]} = ~, (13) endof the magneticfield range,thecurvesofCvsH,ata givenT, departprogressively,thoughslightly, from

    whereEh(x) andEe(x) aregivenby equations(3) and(5), linearity with increasingmagneticfield.


    ~ #~(eV) in equation(10) givesquite a negligible contributiontoFER MI ENERGY the electronicspecific heat,andcancompletelybe

    ignoredto a verygoodapproximation.As a result,the-o.or7~ Tfield and temperaturedependencesof C, aswell asitsmagnitude,cansimply beunderstoodfrom the con--0.0180siderationof only thefirst term of equation(10). An

    -0.0185approximateexpressionfor this termcanbe obtainedanalyticallyby usingalow-temperatureasymptoticex-

    -0.0190pansionaroundp (typically, at T = 4.2K, H = 100kG,kBT/IpI 2 x l02 ~ 1), andfurther assumingthat Ee(x)

    -0.0195may bereasonablywell approximatedby Eh(x) [seeequations(3) and(5), andFig. 1]. Thefinal resultis

    -0.020(I I I I I I I 1I4k~\F1r2_____ kBT______~iizl, (15)60 90 120 150 80 210 C 2~lT2cr2) 6 2( 72) (/.1 ~ p \ I

    Fig. 3. Plot of the Fermi energyp of graphitein the I \272/ ~ jultra-quantumlimit regionasa function of magnetic Lfield H, at T= 4.2K. The curve is calculatedusingthe wherethe factor2 takesthe i summationin equationparameterslistedin thecaptionof Fig. I (10) into account.Forthe magneticfields andtempera-

    Fromthe literature,1315we note thatmuchex- tureswith which we areconcernedin thepresentpaper,theexpressionfor C in equation(15) is foundto agreeperimentalwork hasbeendoneregardingthe tempera- with thenumericalevaluationof equation(10) tolessture dependenceof thespecific heatof graphitebelow than2%. If we neglecttheslight variationsof theFermiliquid-helium temperaturein the absenceof a magneticfield. The resultsof the measurementsclearly showthat energywithH andT, equation(15)clearly indicates

    thatC is directlyproportionalto boththe magneticfielda linearterm,7T, representingtheelectroniccontri-bution to thespecific heat,canbe identifiedin all cases. strengthandthetemperature.Suchresultsare readily

    understood.In fact, sincetheconditionsof high degener-More specifically,by measurementsdown to 0.4K, VanderHoevenetal. 14,15haveobtaineda valueof 7 = 13.8 acy are satisfied,theusual formula for theelectronic.tJ/g-at.K2for apuresinglecrystalof graphite.As we specificheatin termsof density of statesat the Fermienergyapplies,andwe have7canseefrom Fig. 2, sucha value of 7 is quiteconsistentwith our theoreticalcalculations.In fact, we find that C = ~ir2k~D(p)T. (16)7(11), asdefinedin equation(14), variesasafunction ofmagneticfield from about5.45pJ/g.at.K2forH = 60 kG The electronicdensityof statesat the Fermienergyinto about 18.6pJ/g-at.K2for H= 200 kG. Thiscanalso this caseis givenbybe consideredas a clear indicationthat, in theultra-

    ~) 1 1/ i/2/ \1/21 (17)quantumlimit region, the purelyelectroniccontribution D(p) = (~~r~)L 72~~_) (1 -~ Ito the specificheatof graphiteshouldbe easilyob-servableexperimentally.In this respect,we havemdi- L 272 272/ jcatedin Fig. 2, for reference,an estimatedvalueof the which increaseslinearly with increasingH. If we nowlattice specificheatat eachtemperature;1315 it is taketheeffect of thevariationof the Fermienergywithassumedthat it will notvarysignificantly withmagnetic magneticfield (seeFig. 3) into accountin equation(17),field.16 it is easyto seethat slight departuresfrom this linearity

    Figure 3 showsthevariationof theenetgyof the mustoccuras a function of H, especiallyat thehighestFermilevel p of graphiteasa function of magneticfield fields considered.This, combinedwith the factthatin theultra-quantumlimit regionat T = 4.2K. As we Ee(X) becomesmoreandmoredistinct fromEh(x) withcansee,p is a monotonicandslowly varyingfunctionof increasingH, immediatelyexplainswhy at agivenT theHin this region,12goingfrom about 0.0200eV for curvesof CvsH departprogressively,thoughslightly,H= 60kG to about0.0178eV for H= 200kG. On from linearity at theupperendof themagneticfieldthe otherhand,p is foundnot to dependsignificantly range.upon temperature.The latterpoint is clearly reflectedinthe quantity(ap/akBT) [equation(1 1)], which is Very Acknowledg,nents The authorwishesto thanksmall. Actually, at T= 4.2K, the numericalevaluation ProfessorP.R. Wallacefor fruitful discussions,andof equation(11)yields(aP/akBT) 2.5 x iO~for ProfessorM.S. Dresselhausfor providing reference8H= 60kG,and i0~for H= 200kG, respectively, prior to publication.He alsowishesto thankhiscol-Becauseof thesmallnessof (a~/ak~T),thesecondterm leaguesin the PhysicsDepartmentat the Universityof


    Sherbrookefor their stimulatinghospitality.Particularly,heis very gratefulto ProfessorL.G. Caronwho providedthefinancialsupportduring thefinal stagesof thepresentwork, andProfessorM.J. Aubin for a criticalreadingofthemanuscript andfor fruitful suggestions.Finally, theCommissionfranco-qubcoisea la RechercheScien-tifique et Technologiqueis acknowledgedfor financialsupportduring the authorsstayat McGill University.


    1. WALLACE P.R. & GUPTA O.P.,SolidStateCommun. 15, 1577 (1974).

    2. McCLURE J.W.,Phys.Rev. 119,606(1960).

    3. SHARMA M.P., JOHNSONL.G. & McCLURE J.W.,Phys.Rev. B9, 2467 (1974).

    4. To the authorsknowledge,sucha largechangein themagneticsusceptibilityof graphiteunderstrong-mag-netic-fieldconditionsat very low temperaturehasnot yet beenobservedexperimentally.

    5. WOOLLAM J.A.,Phys.Rev.B3, 1148(1971).

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    10. In reference1, WallaceandGuptahaveusedthefollowing valuesfor theenergy-bandparametersof graphite[SCHROEDERP.R.,DRESSELHAUSM.S. & JAVAN A., in ThePhysicsofSemimetalsandNarrow-GapSemiconductors, (Editedby CARTERD.L. & BATE R.T.), p. 139.PergamonPress,Oxford (1971);WOOLLAM J.A.,Phys.Rev.Lett. 25, 810 (1970);McCLURE J.W.,inProc. 10thBiennialCarbon Coni.p. 294,Bethlehem,Pennsylvania(1971)]: ~ = 0.025eV, Yo = 2.85eV, 7~= 0.31eV 72 7~= 0.0185eV,

    = 0.29eV, and y~= 0.18eV. It is easyto seethat theexpressionwe obtain forEe in equation(5) of thepresentpaperexactlyreducesto that given in equation(4) of WallaceandGuptaspaperif the bandpara-meters72 andls are assumedto be of thesamemagnitude.

    11. The field-dependentpart of thenear n = 0 energylevel for electronsEe, asgiven in referenceI, is inexact.In fact, thecorrectexpressionshould be

    (2 74)Q70 7i

    which is approximatelyequalto (for the set of bandparametersof reference10) 4.5 x l05H (kG) eV.

    12. McCLURE J.W. & SPRYW.J.,Phys.Rev. 165, 809 (1968).

    13. KEESOMP.H.&PEARLMAN N.,Phys. Rev. 99, 1119(1955).

    14. VAN DER HOEVEN B.J.C., Jr.& KEESOM P.H.,Phys.Rev. 130, 1318 (1963).

    15. VAN DER HOEVEN B.J.C., Jr., KEESOM PH., McCLURE J.W. & WAGONERG., Phys. Rev. 152, 796(1966).

    16. DELHAES P., LEMERLE M.Y. & BLONDET-GONT~G., Cr. hebd. sanc., Paris 272, 1285 (1971).

    17. HARRISON W.A., Solid State Theory,p. 239.McGraw-Hill, NewYork (1970).


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