Graphite in 90 T: Evidence for Strong-coupling Excitonic Pairing

The quantizing effect of high magnetic fields upon the motion of electrons perpendicular to the field orientation effectively reduces the electronic dimensionality of metals. Ultimately a sufficiently strong magnetic field confines electrons to their lowest Landau level, dramatically increasing the electronic degeneracy at the Fermi energy. In a 2D system, the fractional quantum Hall effectStormer emerges in this regime, due to electron-electron interactions. In 3D systems, on the other hand, electrons can still move freely parallel to the applied field. The quasi-one-dimensionality of their energy spectrum leaves them vulnerable to various instabilities like charge-density-wave (CDW), spin-density-wave (SDW), excitonic insulator (EI) and valley-density-wave(VDW)Halperin ; MacDonald . As a result, the behavior of the three-dimensional(3D) electron gas in this regime has attracted tremendous attention both experimentally and theoretically Halperin ; MacDonald ; Celli1965 ; Lee1969 ; VDW1987 , albeit with only a limited number of experimental systems with sufficiently low carrier density to reach this limit.

In graphite, beyond its quantum limit (QL) induced by a moderate field applied along the c-axis, several phase transitions have been observedYaguchi ; FauqueBook . In TaPJia2017TaP and TaAsRamshaw2017TaAs , the dominant anomalies in resistivity have been attributed to the field-induced annihilation of Weyl nodes reminiscent of the total evacuation of Dirac valleys in bismuthZhu2017bismuth . However, only in graphite there is compelling thermodynamic evidenceBenoit2017 associated with field-induced activation gapsBenoit2013 , which point to collective electronic phenomena beyond the single-particle picture.

The first experimental observation of an abrupt increase in magnetoresistance around 25 T at low temperature was reported in 1981Tanuma . The dome-like phase diagram in the temperature-field plane was established by the study of in-plane magnetoresistance(I$\perp$c). Around 54 T, the insulating state is destroyedYaguchi . There are several proposals for the identity of the field-induced order, ranging from CDWYoshioka ; ArnoldCDW ; Shindou1 , SDWTakada ; SDWTakahashi , EIAkiba ; ZhuEI and spin nematic excitonic insulatorPan . Recently, a new transition and a second dome in the temperature-field plane ending at 75 T was established by out-of-plane (I$\parallel$c) measurementsBenoit2013 . Signatures of these transitions in ultrasonic attenuation and velocityBenoit2017 (up to 65 T) and in Nernst coefficient Benoit2011 in fields below 45 T have been detected. Exfoliation was also used to study the thickness dependence of the transitionsThickness2018 .

Here, we present a study of magnetoresistance in a non-destructive magnetic fields up to 90.5 T revealing an additional phase and bringing new insight to the identity of the previously-known field-induced states. The contrasting roles of orbital and Zeeman energies can be elucidated by the orientation of the applied field owing to their differing anisotropies.The $\alpha$ and $\beta$ transitions (marking the beginning of the in-plane and out-of-plane insulating response) do not deviate from a cosinusoidal angular dependence over the entire measurement range. By contrast to all other threshold fields, the 54 T transition ($\alpha^{\prime}$) does not follow a cosinusoidal behavior, in contrast to all other threshold fields. This implies that the Zeeman energy plays a central role in the transition from one field-induced insulator to the other one. The combination of temperature dependence and angle dependence of the $\zeta$ transition lead us to identify phase B as a spin-polarized strong-coupling excitonic insulator (likely paring between electron and hole bands with the opposite spin) destroyed by strong magnetic field. The C-phase established afterwards displays a metallic behavior irrespective of the orientation of the charge flow (parallel and perpendicular) respective to the graphene planes. This highlights the peculiarity of the phase B where a magnetoresistanceless in-plane metallicity coexists with out-of-plane activation. Furthermore, the absence of semimetal to full semiconductor gap at the $\zeta$ transition indicates that in phase C the lowest spin-polarized hole and electron levels are still occupied.

Natural graphite samples were commercially obtained. In-plane magnetoresistance was measured with a standard four-contacts set-up. Out-of-plane magnetoresistance was measured using two pairs of electrodes attached to the top and bottom of a sample, as illustrated in the inset of the Fig.1. In both cases, the magnetic field was predominantly oriented along the c-axis of the sample, with $\theta$ is defined to be the angle between the field orientation and c-axis. High-field magnetoresistance measurements were performed both in WHMFC, Wuhan and NHMFL, los Alamos. The results were found to be consistent among several samples. In WHMFC, the current was applied by a NI-5402 signal generator worked set to 100 kHz and voltage was recorded by a NI-5105 high-speed digitizer worked at 4 MHz. A digital phase lock-in method was used to extract the magnetoresistance. High magnetic field measurements up to 90.5 T were performed at NHMFL, los Alamos. The 100 T magnet consists an inner and outer magnet. The outer magnet is driven by a generator to produce field between 0 and 37 T at 3 s total width , followed by a faster (15 ms) capacitor bank driven pulse to 90.5 T. The Nernst effect was measured by the standard one-heater-two-thermometer method in a dilution refrigerator equipped with a piezo-rotator in a 17 T superconducting magnet on highly orientated pyrolytic graphite (HOPG) and kish graphite samplesZhuNernstGraphite ; SM .

Fig.1(a) shows field-dependence of in-plane $\rho_{xx}$ and out-of-plane $\rho_{zz}$ magnetoresistance, up to 90.5 T at 1.4 K. $\rho_{zz}$ rises by one order of magnitude at the $\beta$ transition. It drops before rising again in the vicinity of 54 T, signaling the existence of a second insulating phase named B Benoit2013 . It drops again at 75 T and becomes relatively flat afterward upon the destruction of the phase B. The in-plane resistivity, $\rho_{xx}$, presents a kink at the $\alpha$ transition and a small enhancement at the $\beta$ transition, becomes flat and then shows a dramatic rise above 75 T. Fig. 1(b) displays the field-dependence of anisotropy ratio $\rho_{xx}$/$\rho_{zz}$. The three field-induced cascading phases can be clearly delineated: the phase A refers to the one between the $\alpha$ and $\alpha^{\prime}$ anomalies, the phase B exists between $\alpha^{\prime}$ and $\zeta$ anomalies and the phase C starts at $\zeta$ and continues up to the highest explored magnetic field. The zero-field anisotropy ratio of the natural graphite sample studied here was found to be around 0.002, comparable to previous reportsNautreAnisotropy ; SM Upon the application of the magnetic field, $\rho_{xx}$/$\rho_{zz}$ steadily increases and attains 0.3 at 7.5 T above which the quantum limit is reached. Then it decreases steeply following the $\beta$ and $\alpha^{\prime}$ transition, recovering a more quasi-2D behavior. Finally, following the $\zeta$ transition, the system tends to become quasi-3D again. We note that different types of graphite present different anisotropy ratiosBenoit2013 , and even samples of the same type can be different in their zero-field anisotropiesSM . In spite of these differences, however, the evolution of anisotropy with magnetic field was found to be reproducible SM .

To map out the complete phase diagram, in addition to measuring $\rho_{xx}$ up to 65 T with a regular pulsed magnet( Fig. 2 (a)), we measured $\rho_{xx}$ up to 90.5 T at different temperatures (Fig.2 (b)). In Fig.2 (a), curves are shifted for clarity and transitions are identified. Arrows indicate the two new transitions $\zeta_{2}$ and $\zeta^{\prime}$ by our measurements, which encircle a new phase named phase C. Symbols representing $\zeta_{1}$ are taken from the previous $\rho_{zz}$ dataBenoit2013 . This leads to the field-temperature phase diagram of Fig.2 (c). The phase A and B were known previouslyBenoit2013 ; Benoit2017 . In all three phases, the temperature dependence of $\rho_{xx}$ remains metallic. In phase A, $\rho_{zz}$ is insulating-like while $\rho_{xx}$ is metallicSM ; in phase B, $\rho_{zz}$ is also insulating-like while $\rho_{xx}$ is metallic, but the activation gap is smallerBenoit2013 . In the new phase C, $\rho_{zz}$ is almost constantBenoit2013 while $\rho_{xx}$ is metallic. Interestingly, the activation gap in B-phase is smaller than that in A-phase, in contrast to the upper temperature bound of Phase C being greater than B-phase.

Remarkably, in the phase B, the resistivity is almost constant at T = 0.6 K. According to our field-rotated data, in the phase B, plotted as a function of Bcos$\theta$ all $\rho_{xx}$ curves collapse on top of each other (See Fig. 3). The magnitude of the in-plane resistance (7.5 $\Omega$) combined to the thickness of the samples (24$\mu m$) yields a resistance of $500,000\Omega$ or 0.06$e^{2}/h$ per layer, similar to what was previously reported in the case of kish graphiteBenoit2013 . The presence of an activation gap along the c-axis suggests in-plane edge transport Benoit2013 ; Bernevig2007 in this bulk material. In this picture, the rising of $\rho_{xx}$ with increasing field in the phase C would be due to the destruction of the c-axis activation , which ends layer-decoupling and edge transport.

In graphite spin-orbit coupling is weak and band masses are light, consequently the Zeeman energy $\frac{1}{2}g\mu_{B}B$ is small compared to the cyclotron energy $\hbar\omega_{c}$FuseyaPRL2016 . This shows up in the angle-dependent Nernst data measured at 0.6 K and up to 17 T (See Fig. S1 in the supplement). In contrast to the case of bismuth ZhuPRB2011 , the angle dependence of the Nernst peaks corresponding to the evacuation of Landau levels is smooth and close to a cosine. The Fermi surface of graphite, composed of two very anisotropic ellipsoids has a large mass anisotropy (1:7 for the electron ellipsoid, and 1:9 for the hole ellipsoidSchneider ). Because of this large anisotropy, and the smallness of the Zeeman splitting, the angular dependence of the frequency of quantum oscillations up to 60 degrees is close to a cosine fit Schneider .

In this context, the combination of angle-dependent $\rho_{zz}$ data at 1.4 K up to 90.5 TZhuEI and the angle-dependent Nernst data up to 17 T reveals a number of important features regarding the role of Zeeman and cyclotron energies in the field-induced cascade. Fig. 4 shows how the onset fields for different transitions evolve as the field is tilted off the c-axis. As seen in the Fig. 4, except the $\alpha^{\prime}$ transition, all the others follow a perfect cosine behavior (See the Fig.3 and SMSM for raw data). Interestingly, in Fig. 4(b), the ratio of Zeeman to orbital energies represented from the low field Landau level spectrum by (B_up-B_down)/(B_up+B_down) presents an angle-dependence much weaker than cosinusiodal and strikingly similar to the $\alpha^{\prime}$ transition at 54 T for $\theta=0$. This indicates that the driving force behind the $\alpha^{\prime}$ transition, which separates two field-induced insulating states, is the Zeeman energy.

By combining the angle-dependence and the temperature-dependence of the threshold fields , we attempt to address three questions regarding the two phases: i) How do electrons and holes, which pair up relate to each other? Are they the occupied and unoccupied states of same Landau sub-band (the DW scenario) or occupied states of electron and hole sub-bands (as in the EI case)? ; ii) Is the electron-hole condensate in the weak-coupling or in the strong-coupling limit?; iii) Do pairing electrons and holes have parallel or opposite spins? Above the QL at 7.5 T, only four sub-bands (electrons and holes, each with up and down spins) remain occupied. Phase A, relatively easy to access by regular pulsed magnetic field, was intensely studied before. Its threshold magnetic field ($\alpha$) is strongly temperature-dependent and its angular dependence follows a cosine dependence. The temperature dependence can be described by a weak-coupling BCS formulaYoshioka ; Yaguchi ; FauqueBook , where the density of state (DOS) is set by the Landau level degeneracy. Its perfect cosine field-dependence (Fig. 4(b)) is to be contrasted by the angle-dependence of a Landau level, like the one evacuated at 7.5 T (Fig. 4(b)). The comparison suggests that the Zeeman energy does not play any role in driving the $\alpha$ transition. This would be true no matter the mutual orientation of spins of electrons and holes. Even with opposite spins, the Zeeman energies of pairing quasi-particle would cancel out leaving the orbital term with an exactly cosine angular dependence. The same can be said about the $\beta$ phase transition. Phase B was discovered using a 80 T magnet Benoit2013 . The $\alpha^{\prime}$ transition, separating A- and B-phases has almost no temperature dependence. It is accompanied by the depopulation of at least one and most probably two Landau levels ZhuEI ; Yaguchi ; Akiba ; Benoit2017 ; ArnoldCDW . According to the present study, the angle dependence of this $\alpha^{\prime}$ transition, which separates two insulators with different gapsBenoit2013 points to Zeeman energy as its driving source. One possibility is that the nature of the condensate is different in the two phases (see SMSM for a more detailed discussion). The $\zeta$ transition field separating phase B and C has almost no temperature dependence and its angle dependence follows a cosine behaviour. Up to 90.5, both $\rho_{xx}$ and $\rho_{zz}$ remain metallic and the Hall coefficient is nullZhuEI . As a consequence, one can rule out the evacuation of the two remaining Landau sub-levels and the subsequent semi-metal to semi-conductor transition previously postulatedArnoldCDW . Therefore; the field-induced destruction of the phase C cannot be attributed to any change in the Density-of-States in the vicinity of a Landau level evacuation. If the electrons and holes which pair up belong to different sub-bands, their different Fermi velocities would provide an opportunity for a strong magnetic field to unbind them through a purely orbital effect. An alternative scenario invoking spin quantum fluctuations destabilizing the EI was recently proposed by Pan and et al. Pan . Neither of these scenarios for the field-induced destruction of phase C would work in the case of a density wave.

In summary, using a very strong magnetic field, we found an additional phase transition above 75 T and dubbed it phase C. By monitoring the angle-dependence of the multiple fields inducing phase transitions, we found that the threshold fields for all transitions, save one ($\alpha^{\prime}$), show exact cosine-dependence in angle. The in-plane magnetoresistance in phase B indicates edge transport. There is no insulating state even at 90 T and the field-induced destruction of phase B is not accompanied by the evacuation of any Landau levels. The combination of these facts leads us to conclude that the phase B is an excitonic insulator, a spin-polarized condensate of electron-hole pairs.

Z. Z. acknowledges useful discussions with Ryuichi Shindou. This work in China was supported by the 1000 Youth Talents Plan, the National Science Foundation of China (Grant No. 11574097 and No. 51861135104), The National Key Research and Development Program of China (Grant No.2016YFA0401704). Z. Z. was also supported by directors funding grant number 20120772 at LANL. N. H. and R. D. M. acknowledges support from the USDOE BES ‘Science of 100T’ program. The National High Magnetic Field Laboratory-PFF facility is funded by the National Science Foundation Cooperative Agreement Number DMR-1157490 and DMR-1644779 , the State of Florida and the U.S. Department of Energy. In France, this work was part of QUANTUM LIMIT projects funded by Agence Nationale de la Recherche. B. F. acknowledges support from Jeunes Equipes de l’Institut de Physique du Collège de France (JEIP). K. B. was supported by China High-end foreign expert program, 111 Program and Fonds-ESPCI-Paris.

Supplemental Material for “Graphite in 90 T: Evidence for Strong-coupling Excitonic Pairing”