Critical Point for Bose–Einstein Condensation of Excitons in Graphite

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PHYSICS subbands. A rich phase diagram consisting of distinct field- induced phases emerges above ∼20 T, which was documented A D by previous studies (25–28). Our focus, here, is the identification 0.5 of the peak transition temperature as the cradle of the excitonic 15.6 K instability. Theoretical calculations by Takada and Goto (21, 29), based 12.7 K on the Slonczewski–Weiss–McClure (SWM) model (30, 31) of B 0.0 9.8 K band structure and including self-energy corrections, predicted (a.u.) 0.4 9.5 K that the electron spin-up and the hole spin-down subbands simul- yx 9.2 K taneously cross the Fermi level at ∼53 T. The first result of the S 9K 8.2 K

present study is to conﬁrm such a simultaneous crossing of the (a.u.) 0.3 9.8 K -0.5 7.3 K

two subbands occurring at a slightly different field, namely 47 yx 9.5 K T, and its coincidence with the peak critical temperature of the S 0.2 9.2 K 30 40 50 field-induced phase. 9.0 K B (T) Fig. 1 presents a sketch of the proposed scenario. As the 40 45 50 55 magnetic field increases, the gap between the spin-up subband B (T) E of electrons and the spin-down subband of holes changes sign. 16 C S At a critical field of 47 T not only the gap vanishes but also 2.0 yx S the two subbands empty. For fields exceeding 47 T, the forma- yx0 14 35.2 K tion of electron–hole pairs costs a finite energy. When the field 1.5 is sufficiently larger than 47 T, electron–hole pairs break up. 30.2 K (K) 12 T

Tuning down the temperature, on the other hand, will increase (a.u.) 25.3 K 1.0 the thermal de Broglie wavelength. The BEC condensation will yx 19.6 K 10 S occur when the exciton wavelength becomes comparable to the 15.6 K interexciton distance, set by carrier concentration (4). The criti- 0.5 12.7 K 8 cal temperature of 9.2 K would be the degeneracy temperature 9.8 K of bosonic excitons in our picture. 0.0 40 42 44 46 48 50 52 20 30 40 50 60 We carried out a study of the Nernst effect (32) in pulsed fields B (T) on Kish graphite samples. The Nernst effect has proved to be B (T) an extremely sensitive probe of Landau spectrum in compensated semimetals like graphite. Quantum oscillations are most Fig. 2. Experimental signature of a critical point. (A) The sketch of the prominent in the Nernst response and dominate the nonoscil- setup. (B) The Nernst signal presents a structure near 47 T. The peak above lating background (33). However, measuring a Nernst signal in 9.2 K is substituted by two distinct anomalies below. There is a jump in the Nernst signal followed by a fall as a function of increasing magnetic field. pulsed fields is challenging and a study of URu2Si2 (34) is the (C) The broad peak centered at 47 T gradually fades away upon warming. unique known case prior to the one presented here. Details of Dashed lines are guides for the eyes. Curves are shifted for clarity. (D) The the experimental setup are given in SI Appendix. evolution of the Nernst signal with warming over a broader temperature Fig. 2 shows our Nernst data near the critical point (see SI range. (E) The Nernst anomalies in the (B, T) plane bifurcate at B = 47 T and Appendix for data extended up to 60 T). The signal smoothly T = 9.2 K. evolves upon cooling. Below 4 K, the low-field anomaly becomes similar to what was reported in a previous study of the Nernst effect below 45 T (35). Our extended data reveal additional information. First of all, the peak near 47 T is the only one detected up A B < 47 T B = 47 T B > 47 T to 60 T. This indicates that the evacuation of the two Landau subbands occurs simultaneously. In other words, the separation Δ in magnetic field is too small to be detected by experiment. This E interpretation is consistent with the vanishing Hall conductivity F observed near 47 T (36). The second observation is that this peak suddenly splits to two distinct anomalies when T < 9.2 K (Fig. 2 B, D, and E). Finally, Δ < 0 Δ = 0 Δ > 0 it is remarkable that the Nernst peak disappears for T > 35 K Increasing magnec field (Fig. 2C). This temperature dependence allows us to quantify the T T high-field effective mass. B T > 9.2 K = 9.2 K < 9.2 K Fig. 3 compares the temperature dependence of the Nernst d peaks near 8 T and near 47 T. What sets the thermal evo- Λ lution of the amplitude of a quantum oscillation is the effective mass and the B/T ratio. The heavier the electrons, the faster the decay of the oscillating signal with warming. The d > Λ d ~ Λ d < Λ larger the B/T ratio, the slower the decay. In this context, Decreasing temperature it is striking to see that the high-field peak vanishes faster with warming than the low-field one. Quantitatively, using the Lifshitz–Kosevich formula for thermoelectric quantum oscilla- Fig. 1. A critical point in (field, temperature) plane. (A) As magnetic field is tions, Ω(T)osc ∝ [αX coth(αX ) − 1]/ sinh(αX ) (37, 38), where increased, the gap ∆ between the electron spin-up and the hole spin-down 2 ∗ α = 2π kB /e~ and X = m T /B, we find that the effective mass subbands evolves from negative to positive. Both subbands are evacuated ∗ ∗ at 47 T and ∆ = 0. Two other subbands with the same level index and oppo- m47T = 0.48 m0 and m8T = 0.05 m0 (Fig. 3), where m0 is the site spin polarities remain occupied. (B) As the temperature is lowered, the bare electron mass. The low-field value is consistent with the thermal de Broglie wavelength Λ becomes longer. At 9.2 K, it becomes mass extracted from Shubnikov–de Haas measurements (18). comparable to the interexciton distance d and BEC of exciton occurs. Thus, there is a 10-fold field-induced enhancement in cyclotron

2 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.2012811117 Wang et al. Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 h eito asdb hi lnae lisi emtyi small. is geometry ellipsoid elongated their by caused deviation The d wavelengths, 6.45 electronic Electrons Holes Carrier the along field magnetic (24) with c-axis frequencies, graphite Effect in Alphen electrons Haas–van and de holes The 1. along Table surface Fermi of de of radius set the extensive (24), most data the Alphen to Haas–van According anisotropic. both are mass effective an (41). to mass bare lead the by which than explained larger interactions, quantitatively tempera- account be critical to can superfluid difference taking the The with K). compared condi- (2.17 be BEC ture to The (40) K. K 5.9 3.1 to atom. at Broglie corresponds He equal de each which become of tion scales thermal mass length the the two taking and The estimated nm be can 0.358 wavelength is liquid distance In interatomic wavelength. Broglie de thermal measurements DOS. heat the specific of enhancement recent field-induced a that find Note (39) carriers. of mass extract enhancement. mass to 10-fold allows a anomalies reveals two and the mass of tempera- effective The the magnitude (B) the the warming. of to upon dependence contrast quickly in ture away warming, upon fades and K which limit 35 peak, to quantum high-field up near survives signal peak low-field Nernst The the T. 47 of component oscillatory The (A) 3. Fig. age al. et Wang B A = hsalw h unicto fteaeso xrmlorbit, extremal of areas the of quantification the allows This length Broglie de the and distance interexciton the case, our In the below falls distance inter-Boson the when occurs BEC Syx (a. u.) S (a. u.)

√ yx λ 0.0 0.4 0.8 1.2 2π 0.00 0.05 0.10 0.15 F hra vlto fteNrs nmle n h fetv mass. effective the and anomalies Nernst the of evolution Thermal suigta h oeadeeto em ufcsaecylinders. are surfaces Fermi electron and hole the that assuming , 010203040 7891011 F 4.7 ⊥ (T ) λ ⊥ n h nepril distances, interparticle the and , 96K 19.6 K 35.2 A ⊥ d (10 exciton 6.15 4.49 B m 12 T cm *=0.48 (T) (K) Λ = −2 04 50 40 30 ) 56K 15.6 K 30.2 exciton 4 m e o xml,the example, for He, m 7Tpeak T 47 peak T 8 λ /1.38 ⊥ *=0.05 0 52 45 (nm) 27K 12.7 K 25.3 4 cusat occurs (4) F d ⊥ ⊥ m yusing by , for , A d 0 ⊥ ⊥ 21 18 (nm) the , 0.0 0.2 0.4 nepril itnei h aa ln,wihi 8n o both for nm 18 is which plane, basal the estimated In in be the distance can (31). yields interparticle oscillations scales 90 quantum length of exceed frequency relevant The to out- two unambiguously. of the estimated plane. plane, ratio is basal the basal masses the and the in-plane anisotropic in very than to also longer of-plane is times mass nine effective to The seven is c-axis the ekndb eraigDSadeovstwr C-yeweak-coupling BCS-type is a condensate toward behavior. the evolves and boundary, by DOS lower decreasing the set by weakened Along value falls gap. energy a field-induced binding the the to boundary, below higher saturates the Along and boundaries. field behavior upper BCS the Eq. follow the corre- (−1 to it ceases the temperature, ature of high evacuation (Inset At simultaneous levels. point. the critical to the (black) sponds green (below) by shown tempera- above anomalies Nernst critical symbols the of the in bifurcation the near and triggers equal BEC helium become liquid scales ( in length ture. wavelength two Broglie Bril- The de (C the graphite. the field. and of magnetic factor) thickness finite numerical the in over so extend graphite. remains and the in along other zone zone each louin Brillouin the touch the along bands graphite and hole in pockets dispersion Fermi Band hole (B) and Electron (A) 4. Fig. T D C A (K) K H H 10 15 100 0 5 length (nm) h hs iga ftefis edidcdpaei graphite. in phase field-induced first the of diagram phase The D) 0.1 10 h Ipaei etoe ntodfeetwy tislwrand lower its at ways different two in destroyed is phase EI The 1. addseso,lnt cls n h onaiso h Iphase. EI the of boundaries the and scales, length dispersion, Band 1 H' 01 20 15 10 5 0 h omna h rtclpit hr h rtcltemper- critical the where point, critical the near zoom The ) 02 04 50 40 30 20 10 H' H'

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PHYSICS electrons and 21 nm for holes (Table 1). Thus, the interexciton Table 2. A description of the samples used in this study d = 19.5 ± 2 nm distance in the basal plane is exciton . The exci- Sample Type Dimension (mm3) ton mass would be twice the cyclotron mass resolved at 47 T; therefore, mexciton = 0.96 m0. Using these numbers, one finds K1 Kish 1×0.95×0.04 that the BEC condition dexciton = Λexciton/1.38 (4) is satisfied K2 Kish 0.95×0.9×0.02 when T = 8 K. This is remarkably close to the critical tempera- K3 Kish 1.1×1.2×0.06 ture of 9.2 K detected by our experiment. As seen in Fig. 4C , in graphite the two length scales are two orders of magnitude longer 4 one-dimensional spectrum is a generic feature of the three- than in He and in both the experimentally observed critical dimensional electron gas confined to its lower Landau level, temperature is close to where the BEC is expected. exciton formation is not. In most semimetals with heavy atoms, In a compensated semimetal, charge neutrality does not the electric permittivity is large. Therefore, Coulomb attraction impede a concomitant evolution of the density of electrons and between holes and electrons is attenuated, hindering the forma- holes with increasing magnetic field across the quantum limit. tion of excitons. The electric permittivity in bismuth, for example, This is indeed what happens in semimetallic bismuth at high is 20 times larger than in graphite (see SI Appendix for more magnetic fields: at 30 T, the carrier density increases to more discussion). Another difference between graphite and bismuth than five times its zero field value (42). However, this is unlikely is the evolution of mass and carrier density across the quan- to happen in graphite because of its band structure (30, 31) tum limit. The unavoidable enhancement in DOS due to Landau (Fig. 4 A and B). When carriers are confined to the lowest Lan- level degeneracy leads to an increase in carrier density in bismuth dau subbands, the DOS steadily increases due to the degeneracy and an increased mass in graphite. As a consequence, the latter of Landau levels. In a compensated metal, this can occur either becomes a strongly correlated electron system at high magnetic by an enhancement in the concentration of electrons and holes, fields. by an enhancement in mass, or a combination of both. Now, in One open question is the origin of the larger Nernst signal in graphite (in contrast to bismuth), electron and hole ellipsoids the EI state in the vicinity of the critical point. Any quantita- are aligned parallel to each other and their dispersion is simi- tive analysis, however, requires a more complete set of data in lar. Moreover, and crucially, both electron and hole bands are order to quantify the magnitude of the transverse thermoelectric half-filled along the kz . As a result, the room for any significant conductivity αxy , whose amplitude reflects the ratio of entropy to modification of the Fermi wave-vector along the orientation of magnetic flux (32, 46). Availability of DC fields above the present magnetic field and a change in carrier density is small. Thus, ceiling of 45 T would lead to multiprobe studies of the critical our analysis safely assumed that carrier density does not change point unveiled by the present study. between 7 T and 47 T. In summary, we carried out pulsed-field Nernst measurements The boundaries of the EI phase shown in Fig. 4D are strik- in graphite up to 60 T. We found a 47 T anomaly in the Nernst ingly similar to the theoretical expectations (8, 43). The left response and identified it as the result of the simultaneous evac- (low-field) boundary evolves to a mean-field expression for crit- uation of two Landau subbands, an electron-like and a hole-like ∗ B∗ ical temperature Tc(B) = T exp(− B ), which is a BCS-type one. The Nernst anomaly suddenly bifurcates to two distinct 1 formula kBTc(B) = 1.14EFexp(− ) (18, 44) and the evo- anomalies marking the boundaries of the field-induced state N (EF)V lution of the DOS with magnetic field governs the evolution of below 9.2 K. We showed that the BEC condensation temperature the phase transition. This expression fails as the critical point of excitons is expected to occur close to this temperature. is approached, leading to the saturation of the critical temperature. In contrast, on the right (high-field) side, the destruction Materials and Methods of the field-induced state is abrupt and the critical temperature is Samples. The Kish graphite samples we used in the experiment were pinned to a magnetic field of 53 T. This field does not correspond obtained commercially. The summary of sample information is listed in to the evacuation of any Landau level, as shown by the absence Table 2. of any anomaly in our data. In the BEC scenario, it corresponds to the unbinding of the electron–hole pair by magnetic field (see The Nernst Measurement under Pulsed Field. The measurement of Nernst the sketches in the Fig. 4D). effect under pulsed field was performed in WHMFC in Wuhan. The signal Note that only at B = 47 T the critical temperature cor- was recorded by high-speed digitizer PXI-5922 made by National Instrument responds to the degeneracy temperature of excitons and the running at 2-MHz rate. More detailed information to obtain the authentic transition is, strictly speaking, a BEC condensation. When the Nernst signal is provided in SI Appendix. magnetic field exceeds 47 T, the exciton binding energy becomes lower than the band gap and the order is destroyed by unbind- Data Availability. All data are available within the paper or SI Appendix. ing. On the other hand, decreasing the magnetic field diminishes ACKNOWLEDGMENTS. This work was supported by the National Key the DOS, the screened Coulomb attraction between electrons Research and Development Program of China (Grant 2016YFA0401704), the and holes, and the transition occurs well below the degeneracy National Science Foundation of China (Grants 51861135104 and 11574097), temperature. and Fundamental Research Funds for the Central Universities (Grant 2019kfyXMBZ071). In France, it was supported by the Agence Nationale A BEC picture of field-induced phase transition would explain de la Recherche (ANR-18-CE92-0020-01 and ANR-19-CE30-0014-04) and its presence in graphite in contrast to its absence in other by Jeunes Equipes de l0Institut de Physique du College` de France. Z.Z. semimetals pushed beyond the quantum limit (45). While the acknowledges useful discussions with Ryuichi Shindou and Yuanchang Li.

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