*Corresponding author. Present address: Department ofPhysics, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto606}8502, Japan. Fax: #81-75-753-3783.E-mail address: yag@scphys.kyoto-u.ac.jp (H. Yaguchi).

Physica B 298 (2001) 546}550

Successive magnetic-"eld-induced electronic phase transitionsin graphite

Hiroshi Yaguchi��*, John Singleton�, Tadao Iwata�

�The Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK�Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai-mura, Ibaraki 319-1195, Japan

Abstract

We have investigated the longitudinal magnetoresistance of graphite in magnetic "elds of up to &55T. We haveobserved that one of the features within the density-wave state is accompanied by a very steep resistance increase. Thelarge resistance increase strongly suggests that all of the occupied Landau levels have become nested. Therefore, wepropose that this transition represents the onset of a spin-density-wave state, where the nesting vectors for the electronand hole subbands are common. We have also found that the re-entrant transition is accompanied by clear hysteresis,suggestive of a "rst order transition. � 2001 Elsevier Science B.V. All rights reserved.

Keywords: Graphite; Field-induce phase transition; Density wave; Magnetoresistance

Since Tanuma et al. [1] "rst observed a dramaticincrease in the magnetoresistance of graphite ina pulsed-"eld experiment, its electron-hole systemhas attracted considerable attention. This resist-ance increase is considered to be due to a phasetransition involving many-body e!ects because it isvery sharp and its onset "eld strongly depends ontemperature [2]. (We shall call this transition the� transition below.) This phenomenon has beenusually discussed in terms of the formation ofa 2k

�-type density wave along the c-axis due to the

one-dimensionality of the energy spectrum causedby Landau quantisation; this theoretical idea wasoriginally proposed by Yoshioka and Fukuyama[3]. As graphite is a semimetal with small carrier

densities of electrons and holes, the carrier system isin its quantum limit in the "eld range of interest(B'20T). In the quantum limit, only the lowestelectron (n"0) and hole (n"!1) Landau sub-bands (each of them spin-split) are occupied and allthe other subbands are far away from the Fermilevel. Under these circumstances, several nestingsacross two Fermi points are possible at su$cientlylow temperatures (Fig. 1). The theoretical inter-pretation also suggests that at higher "elds, other2k

�-type density waves and/or a re-entrant

transition back to the normal state may occur,which stimulated a number of pulsed-high-"eldexperiments [4,5]. In fact, a few additional featuresin the resistance were found at higher magnetic"elds, some of which appear to be attributable tosuccessive transitions such as the predicted 2k

�-

type instabilities. However, it was not until ourrecent measurements up to &55T that de"nitiveevidence for the re-entrant transition was obtained

0921-4526/01/$ - see front matter � 2001 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 3 8 0 - 5

Fig. 1. Landau levels of graphite in a magnetic "eld of 30Tparallel to the c-axis. The nesting vector for the charge densitywave in the n"0, spin-up subband is shown as an example.

Fig. 2. Longitudinal magnetoresistance at various temper-atures. The onset transition (�) and the re-entrant transition (��)at 6.5K are indicated with arrows. The inset shows the trans-verse magnetoresistance at 1.1K for comparison; the �, � and ��transitions are indicated.

[6]. This observation enables the phase boundarybetween the normal state and the proposed "eld-induced density-wave state to be delineated. How-ever, there is still uncertainty about the nature ofthe "eld-induced density-wave state, including thefeatures appearing in the magnetoresistancebetween the � transition (the onset of the density-wave state) and the re-entrant transition i.e. insidethe whole density-wave state. Although the theo-retical interpretation suggests a 2k

�-type density

wave along the c-axis, most of the experimentsdone thus far investigate the transverse (in-plane)resistance rather than the longitudinal (out-of-plane) resistance. An exceptional study is foundin Ref. [7] investigating the longitudinalmagnetoresistance in pulsed "elds of up to &37T.In this paper, we have investigated the mag-

netoresistance of graphite using pulsed magnetic"elds of up to &55T at temperatures down to0.9K. In particular, we focus on results of longitu-dinal magnetoresistance measurements. The sam-ples used were #akes of single crystal graphite (Kishgraphite). Some of them were pristine and the restwere irradiated with fast neutrons of a #ux(E'1MeV) of 5.5�10��/(cm� s) at &503C for 1,2 and 4 h in JAERI JRR-4. (The imbalance of theelectron (n) and hole (p) densities p!n are esti-mated to be 0.7, 1, 2�10��/cm� fromHall measure-ments, respectively.) Electrical contacts were made

with silver paste. The resistance as a function ofmagnetic "eld was measured by a conventionalfour-contact method, employing a lock-in tech-nique with an alternating current of typically&200�A at 200kHz. High magnetic "elds of up to&55T, generated using a pulsed magnet witha pulse duration of &7ms, were applied along thec-axis. Low temperatures were reached by means of�He and �He inserts. Care was taken to make surethat the electric current carried by the sample waslow such that the transport was Ohmic.Fig. 2 shows the longitudinal (out-of-plane)

magnetoresistance in "elds of up to &55T at tem-peratures between 0.9 and 6.5K. The 6.5K traceexhibits a resistance increase with an onset of&40T and a sharp bend at &50T. The formerrepresents the � transition, the transition to thedensity-wave state. The latter is the re-entranttransition (labelled �� in Fig. 2 after Ref. [6]) backto the normal phase mentioned above. The� transition moves towards lower "elds with de-creasing temperature whilst the re-entranttransition moves towards higher "elds. The "eldpositions of these transitions are in good agreement

H. Yaguchi et al. / Physica B 298 (2001) 546}550 547

Fig. 3. Transverse magnetoresistance traces of neutron-irra-diated specimens. The � transition and the re-entrant (��)transition at 4.2K are indicated with arrows. Whilst the� transition and the re-entrant (��) transitions are rather robust,the � transition disappears.

with results of the transverse magnetoresistancemeasurements in Ref. [6]. These two transitions areobserved at all the temperatures in Fig. 2 althoughthe � transition becomes less clear at lower temper-atures.Another striking feature is seen at relatively low

temperatures: the very steep resistance increaseseen between the � transition and the re-entranttransition. Although such a large resistance in-crease is not observed in the transverse mag-netoresistance, this increase probably correspondsto the � transition seen in the transverse mag-netoresistance (see inset to Fig. 2) for the followingreasons: (1) Both occur in close regions of the B}Tplane. (2) Both become more pronounced with de-creasing temperature whilst the � transition is at-tenuated. In Ref. [7], the longitudinal magneto-resistance was investigated up to &37T and thesteep resistance was "rst observed at low temper-atures. Besides this, it was found that this steepresistance is accompanied by non-Ohmic transport,which may be interpreted as the sliding of a depin-ned density wave along the c-axis. On the otherhand, in Ref. [8] non-Ohmic transport in the trans-verse magnetoresistance was found in the phaseimmediately after the � transition, which leads tothe � transition being interpreted as a density-wavetransition parallel to the ab-plane rather than to thec-axis. As the directions of the two density wavesare very di!erent, these observations of di!erentnon-Ohmic transport suggest two distinct phases.In view of the directions of the density waves, the� transition "ts the theoretical interpretation sug-gesting the occurrence of a 2k

�-type instability

along the c-axis. Also, the resistance increase is toolarge, if any of the Landau subbands remain unnes-ted in the phase whose onset is the � transition. Inorder for all the Landau subbands to be nestedsimultaneously, the only possible nesting vectorcommon to all the occupied subbands is that for thespin density wave (SDW) i.e. connecting the twoFermi points via k

�"0 for the electron subband

and via k�"�/c in the extended zone representa-

tion for the hole subbands, or vice versa.We can also show another piece of evidence that

the � transition is likely to correspond to the SDW.As graphite is a semimetal, the concentration ofelectrons and that of holes are ideally the same. The

charge neutrality condition n"p holds true. Sincethe energy spectrum is one dimensional owing toLandau quantisation, this charge neutrality condi-tion may be expressed as

k���t

#k���s

#k�����t

#k�����s

"2�/c. (1)

Hence, the charge neutrality condition ensures thatthe nesting vectors for the SDWs in the electron(n"0) and hole (n"!1) Landau subbands aretranslationally equivalent. In fact, it is reported thatin samples in which this charge neutrality conditionis not valid, the transition � does not appear at leastin "elds of up to &40T and that, in contrast, the� transition is rather robust against the violation ofthe charge neutrality [9]. (Such samples were pre-pared by fast-neutron irradiation; the neutron ir-radiation creates defects in the samples, some ofwhich work as acceptors.) As the formation of theSDW relies on the charge neutrality condition, thisexperimental fact may support the attribution ofthe � transition to the SDW transition. In thiscontext, we have extended to &53T measurementson neutron-irradiated specimens. Fig. 3 shows thetransverse magnetoresistance at several temper-atures for two neutron-irradiated samples withdi!erent neutron dosages, which were chosen from

548 H. Yaguchi et al. / Physica B 298 (2001) 546}550

Fig. 4. Temperature dependence of hysteresis around there-entrant transition. The inset demonstrates the peak-"elddependence of the hysteretic behaviour at 0.9K. The arrowsindicate the direction of pulsed-"eld sweep.

the same batches as those used in Ref. [9]. Al-though, as reported in Ref. [9], the transition "eldof � shifts towards higher "elds with increasingneutron dosage, the � transition appears even in thesample with the highest dose. On the other hand,the � transition does not appear in any of the threeneutron-irradiated samples (see inset to Fig. 2 forcomparison). We have con"rmed that the� transition does not occur even in "elds of up to&53T.Finally, we report hysteretic behaviour around

the re-entrant transition, suggestive of a "rst ordertransition. Fig. 4 shows the behaviour of the longi-tudinal magnetoresistance around the re-entranttransition. At 0.9K, on the rising branch of thepulsed "eld, the re-entrant transition occurs at&53T and on the falling branch at &52.3T; thereis nearly triangular-shaped hysteresis loop between&51.5 and 53T. This hysteresis clearly persists atleast to 3.1K. The inset to Fig. 4 shows the depend-ence of 0.9K resistance behaviour on the peak-"eld

strength, which illustrates that the hysteresis is notdue to a poor signal-to-noise ratio or an artifact ofpulsed-"eld measurements. Before discussing thishysteresis further, we describe how the re-entranttransition has been understood. In Ref. [6], wehave observed the re-entrant transition back to thenormal phase and attributed this re-entranttransition to that a Landau level responsible for thedensity-wave state crosses the Fermi level. Morestrictly, the re-entrant transition "eld exactly co-incides with the crossing "eld only at T"0. At"nite temperatures, the re-entrant "eld is slightlylower than the crossing "eld; the re-entranttransition is due to the narrowing of the occupiedwidth of the relevant Landau level.Although the observed hysteresis in Fig. 4 is

suggestive of a "rst order transition, the "eld-induced density wave transition is described withinthe framework of BCS-type mean-"eld theory. Infact, the phase boundary with the normal phase iswell reproduced by a BCS-type formula [6]. Alsothe � transition seems not to show hysteresis [2].These facts support that the re-entrant transitionitself is of second order despite of the observedhysteresis around the re-entrant transition. Onepossible interpretation of the hysteresis we wouldraise is that the band structure is hysteretic uponsweeping magnetic "elds upwards and downwardsacross the crossing "eld at su$ciently low temper-atures. This apparently surprising interpretationcould be realistic if one considers self-energy e!ects.In this case, discontinuous occupation of the rel-evant Landau level will take place at the crossing"eld owing to the self-energy e!ects, leading to there-entrant transition being of "rst order. Recently,Takada and Goto [10] have calculated the renor-malised band structure based on the Slonczewski-Weiss-McClure (SWM) model [11,12] by takinginto account the self-energy due to many-bodye!ects. Takada and Goto's calculations predict thatthe crossing "eld is about 53T and shows goodagreement with the experiments in Ref. [6], whilstconventional band calculations based on the SWMmodel fail to explain this crossing "eld. To ourknowledge, without considering these self-energycorrections, the position of the crossing "eld andthe rather weak temperature dependence of there-entrant "eld cannot be accounted for [6]. This

H. Yaguchi et al. / Physica B 298 (2001) 546}550 549

suggests the importance of the self-energy correc-tions in the vicinity of the crossing "eld or there-entrant "eld.In summary, we have observed a very steep and

large increase in the longitudinal magnetoresis-tance within the "eld-induced density-wave state.We propose that this increase is due to the forma-tion of a 2k

�-type SDW, where all the occupied

Landau subbands are nested. We have obtainedevidence in support of the above proposal frommagnetoresistance measurements on neutron-irradiated graphite. We have also found that there-entrant transition is accompanied by clearhysteresis, suggestive of a "rst order transition. Wepoint out that this could be related to discontinu-ous occupation of the relevant Landau level due toself-energy e!ects.

Acknowledgements

We wish to thank Y. Iye for supplying the graph-ite samples used in the present study. We are very

grateful to H. Jones, T. Hickman and W. J. Sier-tsema for producing and maintaining the pulsed"eld facility at the University of Oxford. H. Y.appreciates support from the Japan Society for thePromotion of Science (Postdoctoral Fellowshipsfor Research Abroad). Other aspects of the presentwork are supported by EPSRC (UK).

References

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