Vertical temperature-boundary of the pseudogap under the superconducting dome of the Bi2Sr2CaCu2O8+d phase-diagram

TL;DR

This study combines experiments and theory to identify a vertical boundary of the pseudogap in the phase diagram of Bi2Sr2CaCu2O8+d, revealing its disappearance at a specific doping level across all temperatures within the superconducting dome.

Contribution

It provides evidence that the pseudogap boundary is vertical in the phase diagram, linking the pseudogap collapse to a Lifshitz transition in Bi2Sr2CaCu2O8+d.

Findings

Pseudogap disappears at doping pc between 0.222 and 0.226.

The normal-state pseudogap collapses at a Lifshitz transition.

The pseudogap boundary forms a vertical line in the T-p phase diagram.

Abstract

Combining electronic Raman scattering experiments with cellular dynamical mean field theory, we present evidence of the pseudogap in the superconducting state of various hole-doped cuprates. In Bi2Sr2CaCu2O8+d we track the superconducting pseudogap hallmark, a peak-dip feature, as a function of temperature T and doping p, well beyond the optimal one. We show that, at all temperatures under the superconducting dome, the pseudogap disappears at the doping pc, between 0.222 and 0.226, where also the normal-state pseudogap collapses at a Lifshitz transition. This demonstrates that the superconducting pseudogap boundary forms a vertical line in the T-p phase diagram.

Figures

Equations

$${\chi }_{B_{1g}}^{\prime \prime }(\omega )=2$$

$$\times [\mathrm{Im}G(\mathbf{k},{\omega }^{\prime })\mathrm{Im}G(\mathbf{k},\omega +{\omega }^{\prime })$$

$$-\mathrm{Im}F(\mathbf{k},{\omega }^{\prime })\mathrm{Im}F(\mathbf{k},\omega +{\omega }^{\prime })]$$

Full text

Discovered thirty years ago Bednorz and Müller (1986), the copper oxide (cuprate) superconductors have not ceased to arise interest because their critical temperature $T_{\rm c}\,$is incredibly high at ambient pressure in comparison with conventional superconductors. Central to the high-$T_{\rm c}\,$cuprate problem is the challenge to understand the pseudogap (PG) state. In the normal phase, where the PG has been studied extensively, it manifests below a characteristic temperature T^{\ast}\,$$> $T_{\rm c}\,$as a loss of low energy spectral weight in spectroscopic responses Alloul et al. (1989); Warren et al. (1989); Homes et al. (1995); Fedorov et al. (1999); Opel et al. (2000); Fauqué et al. (2006); Bernhard et al. (2008); Chatterjee et al. (2011); Vishik et al. (2012); Sacuto et al. (2013); Sakai et al. (2013); Benhabib et al. (2015); Hashimoto et al. (2015); Mangin-Thro et al. (2014), and indirectly in thermodynamical and transport properties Tallon and Loram (2001); Ando et al. (2004); Daou et al. (2009); Shekhter et al. (2013). Its properties cannot be accounted for by the standard Fermi liquid theory of metals Abrikosov (1988); Nozieres (1997).

An even greater challenge is to establish whether the PG exists in the superconducting phase, and if yes, what its doping dependence is. This is crucial to understand the relation between superconductivity and the pseudogap Anderson (1987); Kotliar and Liu (1988); Varma (1999); Chubukov et al. (2008); Moon and Sachdev (2009), which remains far from being well-understood Norman et al. (2005); Keimer et al. (2015). However, there are only very few probes that can disentangle a pseudogap from a superconducting gap. Note, even when the doping end-point of the normal state PG is known, it is unclear how that extrapolates in the superconducting phase, since it involves crossing a phase boundary. In the absence of an explicit method to identify the PG in the superconducting phase, this can be settled only through normal state extrapolations that require involved data analysis of heat capacity Tallon and Loram (2001) and angle-resolved photo-emission spectra (ARPES) Hashimoto et al. (2015), or of magneto-resistivity and nuclear magnetic resonance measurements Alloul et al. (2010); Daou et al. (2009); Badoux et al. (2016); Zheng et al. (2005); Kawasaki et al. (2010) under application of very high magnetic fields.

In this article, we present evidence that the PG develops in the SC state of different under-doped compounds, showing that it is a universal property of cuprates. In the case of Bi2Sr2CaCu2O8+δ (Bi-2212), we are able to follow the PG evolution with doping under the superconducting dome. We show that the pseudogap end is a vertical line in the $T-p$ phase diagram within a narrow range of doping $0.222<p_{c}<0.226$ Not (a), the doping level where a Lifshitz transition from a hole-like to an electron-like Fermi surface takes place in the underlying electronic structure Kaminski et al. (2006); Benhabib et al. (2015). Our experimental findings are analyzed within the cellular dynamical mean-field theory (CDMFT) applied to the two-dimensional Hubbard model.

Before presenting our results, we shall explain how we identify the PG in the superconducting state. In this case, the PG manifests itself as a dip in the electronic background of the anti-nodal Raman response at frequencies higher than the pair breaking peak (PP). This PP-dip structure results from the interplay between the PG and the SC gap, and can be smoothly connected to the PG appearing in the electronic spectrum above $T_{\rm c}\,$Loret et al. (2016).

In order to illustrate this point, we discuss the cartographies of the difference between the anti-nodal ($B_{1g}\,$) Raman response $\chi(\omega,T)-\chi(\omega,T=250\,K$) of an over-doped and under-doped Bi-2212 single crystals over a wide range of temperature and energy (Fig. 1). Details of the Raman experiments, the crystal characterization and the doping setting are given in Supplementary Material (SM).

We focus first on the overdoped compound (Fig.1 (a)) for which there is no PG. At low temperature below $T_{\rm c}\,$, two distinct zones can be clearly seen: a blue (dark gray) zone at low energy representative of a loss of spectral weight and a red (gray) one around $2\Delta$ associated with the spectral weight transfer into the PP as expected when a SC gap is opening Klein and Dierker (1984); Devereaux and Hackl (2007). The redistribution is partial because there is no sum rule in Raman scattering in contrast to that in optics Devereaux and Hackl (2007). Above $T_{\rm c}\,$, this spectral weight redistribution disappears and there is no sign of the PG phase at this doping level.

On the other hand, the cartography of the under-doped compound (Fig.1 (b)) is sharply different. Below $T_{\rm c}\,$the PP is surrounded by two regions of spectral weight depressions (blue/dark gray), and spectral weight is transferred to the PP also from energies higher than $2\Delta$ Loret et al. (2016) leaving a spectral depression (blue/dark gray): This is the dip associated to PG in SC phase. Above $T_{\rm c}\,$, the dip (centered around $\omega/2\Delta\simeq$ 1.7) disappears while the loss of spectral weight below $\omega/2\Delta\simeq$ 1.5 persists and it merges with the normal-state PG Opel et al. (2000); Venturini et al. (2002); Guyard et al. (2008); Sacuto et al. (2011, 2013). The PP-dip feature is therefore the hallmark of the PG in the SC phase, which has been also confirmed theoretically by a CDMFT analysis on the Hubbard model Loret et al. (2016). We note that in the normal phase the PG spectral depression completely vanishes nearby $210$ K (which corresponds to $T^{\ast}\,$). A thorough analysis Not (b) not shown here, reveals this depletion is connected to a hump increasing in high energy electronic background (green/yellow zone around 3 of $2\Delta$), suggesting a transfer of spectral weight over a large frequency range as expected from CDMFT studies Sakai et al. (2013); Gull et al. (2013) and also detected by other studies Li et al. (2012); Bernhard et al. (2008).

Our first key result is that the PP-dip feature is present in several families of hole-doped cuprates, as shown in Fig. 2. Here the anti-nodal ($B_{1g}\,$) Raman responses $\chi^{\prime\prime}_{B_{1g}\,}(\omega,T)$ are displayed over a wide frequency range in the SC state and in the normal state (just above $T_{\rm c}\,$) for four distinct slightly under-doped cuprates: HgBa2CuO4+δ (Hg-1201), YBa2Cu3O7-δ (Y-123), Bi2Sr2CaCu2O8+δ and HgBa2Ca2Cu3O8+δ (Hg-1223). The electronic background, which is superposed on various phononic peak contributions, strongly depends on the material. In order to disentangle the material-dependent features and put into evidence the universal character of PP-dip structure we have subtracted the normal-state Raman spectra (just above $T_{\rm c}\,$) from the superconducting ones as displayed in the insets of Fig.2(a)-(d).

The PP-dip feature is clearly observable at and above $2\Delta$ in all the compounds considered. The PP-dip disappears just above $T_{\rm c}\,$(see black curves) as reported in our previous work Loret et al. (2016). The key observation here is that the presence of the PP-dip is independent of the number of copper-oxide layers: one layer (Hg-1201), two layers (Y-123 and Bi-2212) and three layers (Hg-1223), hence cannot be ascribed to a coupling between copper-oxygen planes Damascelli et al. (2003).

We can use then the PP-dip as a hallmark to track the PG inside the SC dome with doping and find the doping level for which the pseudogap disappears. In Fig. 3 we display the difference between the SC $B_{1g}\,$Raman responses of Bi-2212 at $12$ K and the one just above $T_{c}$ for several doping levels. The following are our main results concerning the $T-$doping evolution of the PP-dip feature. (i) The PP-dip survives in the over-doped region. (ii) The dip, which is the PG feature in the superconducting phase, reduces with increasing doping. The continuation of the PG in the superconducting overdoped regime is in agreement with earlier worksZheng et al. (2005); Kawasaki et al. (2010); Vishik et al. (2012); Hashimoto et al. (2015). (iii) Most interestingly, the PP-dip feature disappears in a narrow doping range between $p=0.222$ and $p=0.226$.

We now analyze how the PP-dip feature evolves as a function of doping comparing with the CDMFT results and examine if this gives a consistent picture. We calculate the Raman spectra of the two-dimensional Hubbard model with parameters appropriate for hole-doped cuprates: The (next-)nearest-neighbor transfer integral $t\sim 0.3{\rm~{}eV}$ ($t^{\prime}=-0.2t$) and the onsite Coulomb repulsion $U=8t$. The CDMFT is implemented on a 2$\times$2 cluster and is solved with a finite-temperature extension of the exact diagonalization method Caffarel and Krauth (1994); Capone et al. (2007); Liebsch and Ishida (2012); Sakai et al. (2016a). Within the bubble approximation, the $B_{1g}\,$Raman response is calculated through

[TABLE]

with \gamma_{B_{1g}\,}$$=$$\frac{1}{2}[\cos(k_{x})-\cos(k_{y})] and $f(\omega)$ being the Fermi distribution function. Here, the normal ($G$) and anomalous ($F$) Green’s functions calculated with the CDMFT are interpolated in the momentum space Kyung et al. (2006). This approximation is quite robust in the anti-nodal region, which includes the cluster momenta $\mathbf{K}=(0,\pm\pi)$, $(\pm\pi,0)$, and will not affect our conclusions on the $B_{1g}\,$Raman response. The 2$\times$2 CDMFT well portrays the richness of phases appearing in hole-doped cuprates, including the Mott insulator, the anti-ferromagnetism, the $d$-wave SC and the PG stateMaier et al. (2005); Kotliar et al. (2006); Tremblay et al. (2006); Kancharla et al. (2008); Ferrero et al. (2009); Sordi et al. (2010); Gull et al. (2010, 2013); Gull and Millis (2015). In particular within CDMFT the PG originates from a singularity in the self-energy close to the Mott transition. This singularity evolves in the superconducting state, determining a prominent peak structure in the superconducting pairing function Sakai et al. (2016a, b) and, as outcome, the PP-dip structure in the Raman $B_{1g}\,$spectra Loret et al. (2016). However, the optimal doping for which $T_{\rm c}\,$is maximal is $p^{th}_{opt}\approx 0.08-0.10$, which is smaller than the one ($p_{opt}\approx 0.16$) in experiments. A quantitative comparison with experiments is therefore not possible and we restrict ourselves to a qualitative one.

The CDMFT $B_{1g}\,$Raman responses in the SC state for increasing doping levels at a low temperature $T=0.005t$ are plotted on Fig.4. Comparing with the experimental Raman spectra of Fig. 3, we observe that the PP and dip trends are well reproduced by the CDMFT until $p^{th}=0.178$, which in the CDMFT phase diagram corresponds to a highly over-doped point. Both the theoretical and experimental dip-depth (pointed out by red arrows) reduce, while the distance in energy between the PP (black arrow) and dip-bottom (red arrows) stays roughly constant as a function of doping. At $p^{th}=0.178$, the theoretical dip, if still present, is very weak (see Fig. 4). A Lifshitz transition is presumably located just above $p^{th}=0.178$ because the spectral function at the anti-nodal point is almost symmetric (see SM). Above $p^{th}=0.178$ the convergence of the CDMFT noticeably slows down and we could not obtain a well-converged solution. The closeness to the van Hove singularity might be one of the reasons why it becomes technically harder and harder to obtain converged CDMFT solutions. A more detailed comparison between experiment and theory will be published elsewhere.

We finally show that at a doping $p_{c}$ which is between $p=$0.222 and 0.226, the PP-dip, and hence the PG, disappears from low temperatures ($T\approx 10$ K) to $T_{\rm c}\,$. Fig. 5(a) and (b) display respectively the $T-$dependent subtracted $B_{1g}\,$Raman response ($\chi"(T)-\chi"(T_{0})$) of the OD62 ($p=0.222$) and OD58 ($p=0.226$) compounds, where $T_{0}$ is a temperature just above $T_{c}$, and is 70 K and 60 K, respectively. While the OD62 compound still displays a dip between 600 and $1200\,{\rm cm}^{-1}$ for $T<$ $T_{\rm c}\,$, shown as a negative contribution in the close-up of Fig. 5(c), the OD58 compound displays no dip over an equivalent temperature-range, as shown by the positive contribution in the closeup of Fig. 5(d). This proves that the PG in Bi-2212 ends on a vertical line inside the SC dome of the $T-p$ phase diagram, which can be drawn between $p=0.222$ and $0.226$ (see Fig. 6).

Our result does not show any reentrant behaviour of the pseudogap inside the superconducting dome, in contrast to that conjectured in Vishik et al. (2012); Hashimoto et al. (2015), but rather a straight line, at least down to 12 K (well below that of Ref.Hashimoto et al. (2015)). Note, our conclusions are based entirely on anti-nodal studies, unlike that of Ref. Vishik et al. (2012). Concerning the PG end-point in the normal state, our earlier results Benhabib et al. (2015) are in good agreement with antinodal ARPES analysisVishik et al. (2012); Hashimoto et al. (2015). In this last case ARPES and Raman probe both antinodal quasiparticles.

Our results strongly suggest that the superconducting PG, as in the normal state, is sensitive to the topology of the underlying Fermi surface since close to $p_{c}$ a Lifshitz transition takes place from a hole-like to an electron-like Fermi surface Benhabib et al. (2015). Furthermore, if the PG disappearance were a phase transition, it would be a first order one. This is expected for a Lifshitz transition of electrons coupled to a lattice Hackl and Vojta (2008); Kee and Kim (2005).

On the theory side, the relation between the pseudogap and the Lifshitz transition is not a well settled issue. The slowing down of the CDMFT solution approaching the van Hove doping level in concomitant with a strong decreasing of the dip depth is compatible with the experimental scenario, though future CDMFT improvements are needed to settle this issue.

In conclusion, we have shown that the peak-dip structure in the Raman $B_{1g}\,$spectra, which is the hallmark of the PG in the SC phase, is a universal feature of the hole-doped cuprates. Following the PP-dip evolution with doping and temperature in the case of Bi-2212, we show that the pseudogap persists on the over-doped side before disappearing abruptly and its end draws a vertical line in the $T-p$ phase diagram just in between $p=0.222$ and $p=0.226$. This corresponds to the same doping range where the normal-state pseudogap collapses, following up a Lifshitz transition of the Bi-2212 anti-bonding band, where the Fermi surface changes from holelike to electronlike.

We are grateful to A. Georges, A.J. Millis, S. Borisenko, and Louis Taillefer for fruitful discussions. Correspondences and requests for materials should be addressed to A.S. (alain.sacuto@univ-paris-diderot.fr), M.C. (marcello.civelli@u-psud.fr) and S.S (shiro.sakai@riken.jp). S.S. is supported by JSPS KAKENHI Grant No. JP26800179 and JP16H06345; B.L. is supported by the DIM OxyMORE, Ile de France.

I SUPPLEMENTAL MATERIAL

Details of the Bi-2212 single crystal characterization

The Bi-2212 single crystals were grown by using a floating zone method. Wen et al. (2008); Mihaly et al. (1993) The critical temperature $T_{c}$ for each crystal has been determined from magnetization susceptibility measurements at a $10$ Gauss field parallel to the c-axis of the crystal. A complementary estimate of $T_{c}$ was achieved from electronic Raman scattering measurements by defining the temperature from which the $B_{1g}\,$superconducting pair breaking peak disappears. The level of doping $p$ was defined from $T_{c}$ using Presland and Tallon’s equationPresland et al. (1991): $1-T_{c}/T_{c}^{max}=82.6(p-0.16)^{2}$. In the over-doped regime, we have established a relationship between $T_{c}$ and the $2\Delta$ pair breaking peak. This give us another reliable way for directly estimate $T_{\rm c}\,$from $2\Delta$, and then evluate $p$ via the Presland and Tallon’s equation, see details of section C in the SM of Ref. Benhabib et al. (2015).

Details of the Raman experiments

Raman experiments have been carried out using a JY-T64000 spectrometer in single grating configuration using a 600 grooves/mm grating and a Thorlabs NF533-17 notch filter to block the stray light. The spectrometer is equipped with a nitrogen cooled back illuminated 2048x512 CCD detector. Two laser excitation lines were used: 532 nm and 647.1 nm from respectively a diode pump solid state laser and a Ar+/Kr+ mixed laser gas. Measurements between 10 and 290 K have been performed using an ARS closed-cycle He cryostat. This configuration allows us to cover a wide spectral range (90 cm*-1* to 2200 cm*-1*) in one shot. The $B_{1g}\,$-symmetry Raman response is obtained from crossed light polarizations along the Cu-O bond directions, giving us access to the anti-nodal region of the momentum space where the $d$-wave superconducting gap is maximal and the pseudogap sets in. All the spectra have been corrected for the Bose factor and the instrumental spectral response. They are thus proportional to the imaginary part of the Raman response function $\chi^{\prime\prime}(\omega,T)$ Sacuto et al. (2011, 2013). In order to have reliable comparisons, all the Bi-2212 crystals have been measured in exactly the same experimental conditions. Special care has been devoted to cover a wide spectral range in one shot and maintain the laser spot at the same location on the crystal surface during the run in temperature. This makes reliable the intensities of the electronic Raman background from low to high energy without any adjusting the spectra from one temperature to another. We have checked that each spectrum is reproducible.

Raman responses of Bi-2212 in the superconducting and normal states

In Figure 7 are reported the Raman responses of Bi-2212 single crystals for several doping levels in the superconducting ($T=12\,K$) and the normal state just above $T_{\rm c}\,$. The doping range extends from the under-doped to the over-doped regime.

CDMFT spectral function

In Figure 8 is plotted the spectral function $A(\textbf{k},\omega)$ at the antinodal point $\textbf{k}=(\pi,0)$, calculated with the CDMFT for various doping levels $p^{th}$. The calculation was done at $T=0.005t$ in the superconducting state. At small $p^{th}$, the spectra show a strong electron-hole asymmetry: The Bogoliubov peaks are much stronger for $\omega<0$ than $\omega>0$. This asymmetry results from the underlying normal-state bare electronic structure, where the van Hove singularity is located below the Fermi energy. As $p^{th}$ increases, the asymmetry decreases as expected. Because the spectrum at $p^{th}=0.18$ is close to a symmetric one, we estimate that the Lifshitz transition point, where the underlying van Hove singularity crosses the Fermi energy, is located just above $p^{th}=0.18$.