Abstract
The excitonic insulator phase has long been predicted to form in proximity to a band gap opening in the underlying band structure. The character of the pairing is conjectured to crossover from weak (BCSlike) to strong coupling (BEClike) as the underlying band structure is tuned from the metallic to the insulating side of the gap opening. Here we report the highmagnetic field phase diagram of graphite to exhibit just such a crossover. By way of comprehensive angleresolved magnetoresistance measurements, we demonstrate that the underlying band gap opening occurs inside the magnetic fieldinduced phase, paving the way for a systematic study of the BCSBEClike crossover by means of conventional condensed matter probes.
Introduction
Half a century ago, Mott pointed out that tuning the carrier density of a semimetal towards zero produces an insulating state in which electrons and holes form bound pairs^{1}. It was later argued that such pairing persists even if a semiconducting gap opens in the underlying band structure, giving rise to what has become known as the strong coupling limit of an ‘excitonic insulator’^{2}. These ‘weak’ and ‘strong’ coupling extremes on either side of the band gap opening were subsequently proposed to be manifestations of the same excitonic state of electronic matter^{3,4,5,6,7}. Studies of photoexcited excitons in semiconductors have provided indirect evidence that these two extremes are connected via a crossover^{8,9,10,11}.
The hallmark of an excitonic insulator is the spontaneous formation of a broken symmetry phase in equilibrium that straddles both sides of a band gap opening in the underlying band structure^{4, 5, 7}. On the weak coupling side, electrons and holes pair at the Fermi surface in direct analogy to electronelectron pairing in BardeenSchriefferCooper (BCS) superconductors^{7, 12}. On the strong coupling side, bound electronhole pairs form across a semiconducting gap giving rise to an exciton gas which can subsequently condense. The symmetry broken by the ground state is expected to depend on the specifics of the band structure and can include a BoseEinstein Condensate (BEC) of excitons^{7}, a Wigner crystalline solid^{5} (i.e. a strong coupling variant of a spin or chargedensity wave) or a state with chiral symmetry breaking^{13, 14}. Despite extensive experimental searches for a phase transition into an excitonic insulator phase bridging the weak and strong coupling regimes, only the weak coupling regime has thus far been reported^{15, 16}.
Here we show the quantum limit of graphite^{17,18,19}, by way of temperature and angleresolved magnetoresistance measurements, to host an excitonic insulator phase that evolves continuously between the weak and strong coupling limits in equilibrium. We find that the maximum transition temperature T _{EI} ≈ 9.3 K of the excitonic phase is coincident with a band gap opening in the underlying electronic structure at B _{0} = 46 ± 1 T, which is evidenced above T _{EI} by a thermally broadened inflection point in the magnetoresistance. The overall asymmetry of the observed phase boundary around B _{0} resembles the original theoretical predictions of a magnetic fieldtuned excitonic insulator phase^{4, 5, 7, 20, 21}, suggesting a smooth crossover between the BCS and BEC regimes with increasing magnetic field^{4, 5, 7}.
The sharp phase transitions in quantum limit graphite above 20 T (see Fig. 1) have been the subject of numerous experimental studies^{16, 22,23,24,25}. Our experimental phase boundary (solid black circles in Fig. 1a) is traced from both interplane (see Fig. 1b) and inplane resistance data (see Supplementary Information). While the fieldinduced insulating behavior has been associated with the formation of a fieldinduced densitywave phase^{19, 26,27,28}, the relationship of the densitywave phase to the opening of a band gap in the underlying electronic structure has remained undetermined. In the absence of a direct measurement of the underlying gap, it has been assumed from fixed angle studies performed thus far (i.e. θ = 0°)^{16, 24, 25} that a band gap opening coincides with the upper magnetic field phase boundary of the phase near ≈54 T^{19} (see Fig. 1a). Such an analysis has suggested that the entire magnetic fieldinduced phase lies on the weak coupling BCS side where Landau subbands always overlap (i.e. Fig. 2a)^{26} and where pairs are formed by connecting opposing momentumstates on the Fermi surface.
Rather than being coincident with the upper magnetic field boundary of the phase, we show here that the band gap opening in the underlying electronic structure lies in close proximity to the magnetic field at which the transition temperature is maximum, therefore exhibiting the signature characteristics of an excitonic insulator phase^{4, 5, 7, 20}. Experimental evidence for the band gap opening at a magnetic field ≈46 T, substantially below the upper boundary of the fieldinduced insulating phase near 54 T, is provided by a point of inflection in the interplane electrical resistance R _{ zz } at temperatures above T _{EI} (solid blue circles in Fig. 1). In the absence of ordering, the sudden emptying of electron and hole states upon opening of the band gap (E _{g}) is expected to result in a discontinuity (i.e. a step) in the electrical resistivity in the limit of zero temperature^{5}. When broadened by the FermiDirac distribution at finite temperatures and by other factors such as fluctuations and a finite relaxation time, this becomes a point of inflection. The thermal evolution of the width of the peak in the derivative ∂R _{ zz }/∂B in Fig. 1c shows that the point of inflection becomes increasingly sharp and steplike on lowering the temperature towards T _{EI}, making it consistent with a discontinuity at B _{0} in the underlying band structure at low temperatures. Importantly, no high temperature feature is seen to occur in R _{ zz } at 54 T, where the band gap was previously assumed to open^{16, 19, 24, 25}, suggesting its shifting to the lower magnetic field value of B _{0} ≈ 46 T by the effects of electronic correlations^{19}.
It should be noted that once densitywave ordering sets in at temperatures below T _{EI} (onset indicated by filled circles in Fig. 1), the development of insulating behavior in R _{ zz } (onset indicated by open circles) implies that the electronic structure must become almost entirely gapped. We find the maximum in R _{ zz } within the insulating phase to be located at a very similar field \({B}_{0}^{^{\prime} }\approx 47\,{\rm{T}}\) to B _{0} (see Figs 1 and 3a and Supplementary Information). One possible explanation for the insulating behavior below T _{EI} is provided by the densitywave excitonic insulator scenario depicted in Fig. 2c,d, whereby the densitywave is primarily hosted by the minorityspin states but also induces a secondary gap to open on the majorityspin Fermi surface (the electronic densityofstates of the majorityspin Fermi surface at 46 T being significantly smaller than that of the minorityspin Fermi surface).
The key experimental evidence for the band gap opening between minorityspin electron and hole bands of graphite (shown in Fig. 2a,b) is provided by our angleresolved measurements shown in Fig. 3. Because the spin and orbital contributions to E _{g} have differing dependences on the orientation of the magnetic field in layered materials, angleresolved measurements (see Fig. 3) enable the spin and orbital contributions to be selectively tuned. The inflection in R _{ zz } at B _{0} ≈ 46 T and the maximum in R _{ zz } within the excitonic insulator phase at \({B}_{0}^{^{\prime} }\approx 47\,{\rm{T}}\) in Figs 1b and 3a are both observed to shift in field on increasing the polar angle θ between the magnetic field and the crystalline caxis. Their angledependences match the behavior expected for the opening of this band gap
between the lowest Landau levels of minorityspin electrons and holes due to the competition between quasitwodimensional Landau quantization and isotropic Zeeman splitting. Here, E _{g} is positive for B > B _{0} and negative (corresponding to a band overlap) for B < B _{0} (see schematic in Fig. 2a,b). The first term on the righthandside (in which \({m}^{\ast }\) is an effective mass that characterizes the splitting between the lowest electron and hole Landau levels^{19}) results from orbital quantization within the twodimensional honeycomb layers, the second term (in which \({g}^{\ast }\) is the effective gfactor, which is approximately isotropic in graphite, and μ _{B} is the Bohr magneton) results from the Zeeman coupling of the magnetic field to the electron spin while the third (E _{0}) is a constant relating to the interplane electronic band structure of graphite. (Equation (1) is strictly valid only at B ≈ B _{0} where a singularity in the minority spin electronic densityofstates causes it to dominate the total densityofstates^{19}, and at \(\theta \mathop{ < }\limits_{ \tilde {}}60^\circ \) where the effect of the interlayer dispersion on the orbital quantization is negligible^{29}. When the magnetic field is reduced to B cos θ ≈ 25 T (where the onset of the fieldinduced phase occurs), the electronic densityofstates of the minority and majority spin components are similar^{19} causing the effect of the Zeeman term to be negligible. In this limit, the field induced spindensity wave (or charge densitywave state) is BCSlike^{26} and its onset depends only on the total electronic densityofstates, which depends on B cos θ to leading order^{23}).
Defining B _{0} as the field at which the band gap opens (i.e. E _{g} = 0), Equation 1 produces a linear dependence of 1/B _{0} on cos θ, with an offset of −\(({m}^{\ast }/{m}_{{\rm{e}}}){g}^{\ast }/2\). On plotting the 1/B _{0} data versus cos θ in Fig. 3b, the intercept of the fitted solid green line yields \(({m}^{\ast }/{m}_{{\rm{e}}}){g}^{\ast }/2\approx 0.284\) (where m _{e} is the free electron mass). The near coincidence of \({B}_{0}^{^{\prime} }\) below T _{EI} with B _{0} above T _{EI} suggests that it can be used to provide an independent estimate of \(({m}^{\ast }/{m}_{{\rm{e}}}){g}^{\ast }/2\) (see Supplementary Information). On plotting the \(1/{B}_{0}^{^{\prime} }\) data versus cos θ in Fig. 3b, the intercept of the fitted dotted green line yields \(({m}^{\ast }/{m}_{{\rm{e}}}){g}^{\ast }/2\approx 0.352\). The average 0.32 ± 0.03 of the two intercepts (indicated by an X symbol in Fig. 3b) is similar to the value ≈0.37 expected from the known parameters of graphite (\({g}^{\ast }=2.5\) ^{30} and \(({m}^{\ast }/{m}_{{\rm{e}}})=0.3\) ^{18, 19}). (This effective mass parameter corresponds to the magnetic fielddependence of the energy difference between n = 0 (electron) and n = −1 (hole) Landau levels^{19}, and is larger than the effective mass (≈0.05 m _{e}) of the electron and hole pockets).
Our measurements identify the band gap opening in the underlying electronic structure to coincide with the maximum T _{EI} of the asymmetric excitonic phase boundary (black circles in Fig. 1a), resembling theoretical predictions made in the high magnetic field limit^{20} (red line). The physical situation can therefore be described as follows: electronhole pairing for B < B _{0} occurs at the Fermi surface in momentumspace in accordance with a BCSlike transition into a weakly coupled spin or chargedensity wave phase^{26,27,28} (schematic in Fig. 2c). Such behavior has been confirmed experimentally by the observation of an exponential increase in the transition temperature with increasing magnetic field^{31}. At B ≈ B _{0}, however, singularities in the electronic densityofstates at the top of the minorityspin hole band and bottom of the minorityspin electron band coincide with the chemical potential, causing strongly bound minorityspin pairs to greatly outnumber weakly bound majorityspin pairs and therefore dominate the thermodynamics. When B > B _{0}, the minorityspin pairing takes place across a band gap, thereby becoming local excitonic in nature^{2} (schematic in Fig. 2d). Pairing across a band gap is predicted to give rise to an increasingly dilute density of excitons as the magnetic field is increased^{4, 5, 7, 20}. The exciton gap function, Δ, is expected to approach zero at the upper extremity of the phase (near ≈ 54 T in Fig. 1a). The total minorityspin energy gap, which will determine the thermally activated transport properties of such a correlated electron state, is given by the band gap and correlation gap added in quadrature \({E}_{{\rm{a}}}=\sqrt{{E}_{{\rm{g}}}^{2}+{{\rm{\Delta }}}^{2}}\). This gap becomes comparable to the band gap E _{g} when the exciton density vanishes^{4, 5, 7, 20}. Such behavior is demonstrated in Fig. 3c by the evolution of an activation gap within the excitonic insulator phase, obtained from Arrhenius plots of R _{ zz } ^{25} (see Supplementary Information), that continues to grow in the region B > B _{0}, and then intersects with E _{g} on approaching the upper phase boundary. The point of intersection provides a lower bound estimate of ≈3 meV for the exciton binding potential energy, which is expected to be similar to the value of Δ at the peak transition temperature^{4, 5, 7}. On estimating Δ from E _{g} given by Equation (1) and experimental E _{a} data in Fig. 3c, we find Δ to peak near B _{0} and then collapse rapidly to zero at 52 T, which is consistent with a scenario in which an Excitonic phase forms around a band gap opening^{4, 5, 7, 20}.
The stability of the excitonic insulator phase centered around B _{0} depends on the effective strength of the interactions determining the binding energy. The combination of anisotropic orbital and isotropic Zeeman contributions to E _{g} (as defined by Equation 1) shifts the opening of the band gap and hence the optimal transition temperature of the exitonic insulator to lower values of the component of magnetic field perpendicular to the planes, B _{0} cos θ, as θ is increased (Figs 3 and 4). The reduced optimal transition temperature of the excitonic insulator phase and its reduced extent in B cos θ at higher angles suggest that the maximum pairing strength at E _{g} = 0 is weakened at higher angles by the reduction in Landau level degeneracy, caused by the singularity in the densityofstates being shifted to lower values of B cos θ. The angledependent measurements hence provide an experimental means of tuning the pairing strength in a condensed matter system, independent of the electron gas density, analogous to that achieved in cold atomic gases^{32}.
While the nature of the broken symmetry in quantumlimit graphite has remained an open question^{16, 24,25,26,27,28}, our observation of the maximum transition temperature at the field B _{0} implies that the broken symmetry accompanying its formation bridges opposing limits of the phase diagram in which excitons are strongly and weakly bound. Beyond the proposed formation of a densitywave in the lowfield, weakcoupling limit, the possibilities for the broken symmetry in the excitonic phase include a BoseEinstein condensate of excitons^{7}, a Wigner crystalline^{5} or supersolid^{33} state of excitons, or a state with chiral symmetry breaking^{13, 14}. One way of forming a reconstructed electronic dispersion^{19} typical of an excitonic phase^{4, 5, 20} is a spinordered phase with an interplane component to the ordering vector of Q _{ z } = π/c (shown schematically in Fig. 2c,d to couple electrons and holes of opposite spin). This has the attractive property of producing broken translational symmetry along the caxis, as expected for a crystalline exciton phase^{5}, while leaving the inplane mobility of the electrons and holes intact and open to the possibility of superfluid^{7} or supersolid behavior^{33}. This same Q _{ z } vector also nests the majorityspin bands, which must ultimately be important for increasing the resistivity of the high field state.
Our observation of an ordered excitonic phase nucleating around the opening of a band gap, suggests that graphite is an attractive material for investigating exotic ordered states in ultralow density electronic systems^{17, 19, 34} with poorly screened coulomb interactions^{1}. The nature of the broken symmetry in the excitonic insulator phase and whether the onset of the insulating phase precedes or is coincident with it remains an open question. In particular, there exists a second fieldinduced phase at higher magnetic fields centered on ≈70 T, as reported by Fauqué et al.^{25}, raising the possibility that this is a second excitonic insulator phase involving only the majorityspin carriers (the upper phase between ≈55 and 75 T also being evident in Fig. 3c). The similarity in shape of the second magnetic fieldinduced phase to that at low fields suggest that it may be centered around a band gap opening between the majorityspin Landau subbands at ≈70 T. Further measurements of R _{ zz } at higher temperatures around 70 T and angleresolved measurements made at higher magnetic fields ought to reveal whether or not a second majorityspin band gap opening occurs at this field.
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Acknowledgements
Z.Z. and R.D.M. thank K. Behnia and B. Fauqué for fruitful discussions while N.H. and R.D.M. thank A.V. Balatsky for useful discussions. This research performed under the DOE BES ‘Science at 100 tesla’ at the magnet lab. which is supported by NSF Cooperative Agreement No. DMR1157490.
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Z.Z., R.D.M., F.F.B. and N.H. performed the measurements, Z.Z., A.S. and N.H. wrote the manuscript, and B.J.R. and K.A.M. performed additional measurements contained in the Supplementary Information.
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Correspondence to Z. Zhu or N. Harrison.
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