Pseudogap phase of cuprate superconductors confined by Fermi surface topology

The properties of cuprate high-temperature superconductors are largely shaped by competing phases whose nature is often a mystery. Chiefly among them is the pseudogap phase, which sets in at a doping p* that is material-dependent. What determines p* is currently an open question. Here we show that the pseudogap cannot open on an electron-like Fermi surface, and can only exist below the doping p FS at which the large Fermi surface goes from hole-like to electron-like, so that p* ≤ p FS. We derive this result from high-magnetic-field transport measurements in La1.6−xNd0.4SrxCuO4 under pressure, which reveal a large and unexpected shift of p* with pressure, driven by a corresponding shift in p FS. This necessary condition for pseudogap formation, imposed by details of the Fermi surface, is a strong constraint for theories of the pseudogap phase. Our finding that p* can be tuned with a modest pressure opens a new route for experimental studies of the pseudogap.

A central puzzle of cuprate superconductors 1 , the pseudogap is a partial gap that opens in their spectral function, detected most directly by angle-resolved photoemission spectroscopy (ARPES). It opens below a temperature T* that decreases monotonically with increasing hole concentration (doping) p. For example, T* = 130 ± 20 K in La 2-x Sr x CuO 4 (LSCO) at p = 0.15 2 and T* = 75 ± 5 K in La 1.6 −x Nd 0.4 Sr x CuO 4 (Nd-LSCO) at p = 0.20 3 . Transport properties like the electrical resistivity ρ(T) and the Nernst coefficient ν(T) are affected by the opening of the pseudogap and so may be used to detect T*, as previously reported in refs. 4,5 and ref. 6 , respectively.
In Fig. 1a, we show the temperature-doping phase diagram of LSCO, Nd-LSCO and La 1.8-x Eu 0.2 Sr x CuO 4 (Eu-LSCO). We see that all three materials have the same T* up to p~0.17, irrespective of their different crystal structures 6 . Indeed, Nernst data 6 show, for example, that T* = 120 ± 10 K at p = 0.15 in both LSCO and Nd-LSCO, whose structure in that part of the phase diagram is orthorhombic, and T* = 115 ± 10 K at p = 0.16 in Eu-LSCO, whose structure in that part of the phase diagram is tetragonal (LTT) (Supplementary Fig. 1). In Nd-LSCO and Eu-LSCO, T* decreases linearly all the way from p~0.08 to p~0.23 (blue line in Fig. 1a). On its trajectory, the T* line goes unperturbed through the LTT transition at p~0.14 for Eu-LSCO (short green line) and at p~0.19 for Nd-LSCO (short red line). Clearly, the pseudogap mechanism does not care about the crystal structure (Note that it is also robust against disorder 7 ).
The critical doping p* at which the pseudogap phase comes to an end, however, is material-specific. In LSCO, the linear decrease in T* vs p comes to an end at p* = 0.18 ± 0.01 8,9 . In Nd-LSCO, resistivity measurements 4,5 at p = 0.20 and above show that the T* line only comes to an end at p* = 0.23 ± 0.01. Why does T* not continue to track the dashed blue line in Fig. 1 beyond p = 0.17 for LSCO, or beyond p = 0.23 for Nd-LSCO? In Fig. 1e, we plot p* for the three single-layer cuprates LSCO, Nd-LSCO and Bi2201 10 , as a function of p FS , the doping at which the Fermi surface undergoes a change from hole-like to electron-like as determined by ARPES measurements 2,3,11,12 . Within error bars, we observe that p* = p FS , in other words, it appears that what limits p* is the constraint that the pseudogap cannot open on an electron-like Fermi surface.
In the following, we examine whether this connection is accidental or not, by independently probing how p* and p FS evolve under the effect of hydrostatic pressure in Nd-LSCO. With a maximal T c of only 20 K, this cuprate provides a window into the pseudogap phase near its end point at p*, free of superconductivity, down to the T = 0 limit, achieved by applying a magnetic field of 30 T (or greater), whose sole effect is to suppress superconductivity and reveal the underlying normal state (ref. 5 ). Our measurements reveal a large and unexpected downward shift of p* with pressure, which we find to be driven by a corresponding change in p FS , so that the relation p* ≤ p FS is obeyed. This fundamental property has direct and fundamental implications for the mechanism of pseudogap formation.

Results
Determination of p* and p FS . Our study is based on transport signatures of p* and p FS . We begin with p*. When p < p*, the electrical resistivity ρ(T), Nernst coefficient and Hall coefficient R H (T) all exhibit large upturns at low temperature 4,5,6,9 - Phase diagram and correlation between p* and p FS . a Temperaturedoping phase diagram, showing the pseudogap temperature T* measured by ARPES (triangles), electrical resistivity (circles) and Nernst effect (squares), in LSCO (black), Nd-LSCO (red) and Eu-LSCO (green) (adapted from ref. 6 ; see references therein). The diamonds mark the critical doping p* at which the pseudogap phase ends at T = 0 (in the absence of superconductivity), for LSCO (black; p* = 0. 18, refs. 8,9 ) and Nd-LSCO (red; p* = 0. 23, refs. 4,5 ). The short solid lines mark the transition into the lowtemperature tetragonal structure (LTT) for Eu-LSCO (green; ref. 36 and Nd-LSCO (red; ref. 34 ), in the interval where it crosses T*. The grey line is the superconducting critical temperature T c of LSCO. The dashed line is a linear extension of the T* line (blue). The red arrow illustrates the effect of applying hydrostatic pressure to Nd-LSCO: it shifts p* down in doping. b, c, d Sketch of the Fermi surface in the first Brillouin zone (frame), as measured by ARPES in LSCO (ref. 11 ) at p = 0.17 (b) and in Nd-LSCO (ref. 3 ) at p = 0.20 (c) and p = 0.24 (d). The red dashed line is the anti ferromagnetic zone boundary (AFZB), which intersects the hole-like Fermi surface of Nd-LSCO at p = 0.20, but not its electron-like surface at p = 0.24. e Correlation between p* and p FS in single-layer cuprates. p* is measured in the normal state at T = 0, by high-field transport in Nd-LSCO (p* = 0.23 ± 0.01, ref. 5 ) and LSCO (p* = 0.18 ± 0.01, refs. 8,9 ), and by highfield NMR in Bi2201 (p* = 0.38 ± 0.02, ref. 10 ). p FS is measured by ARPES in LSCO (p FS = 0.19 ± 0.01, refs. 2,11 ), Nd-LSCO (p FS = 0.22 ± 0.01; ref. 3 ) and Bi2201 (p FS = 0.41 ± 0.02, ref. 12 ). The red arrow shows the effect of applying pressure to Nd-LSCO (red square). We find that p* and p FS decrease in tandem, preserving the equality p* = p FS and thereby showing that p* is constrained by the condition p* ≤ p FS . The grey shading marks the forbidden region signatures of the pseudogap phase, attributed to a drop in carrier density n from n = 1 + p above p* to n = p below p* 5,9,13 . In Nd-LSCO at p = 0.20, ARPES sees a gap opening at T* = 75 K (ref. 3), precisely the temperature below which the resistivity exhibits an upward deviation from its high-T linear behaviour 4,5 . By contrast, at p = 0.24 the three transport coefficients show no trace of any upturn at low T 4,5,6, , with the resistivity remaining linear down to T → 0 (Fig. 2a), consistent with the absence of a gap in ARPES data 3 .
In Fig. 3, we reproduce published data 5 for ρ(T) (Fig. 3a) and R H (T) (Fig. 3c) in Nd-LSCO at different dopings. The upturns decrease as p approaches p* from below. We determine p* as the doping where the upturns in ρ(T) and R H (T) vanish, giving p* = 0.23 ± 0.01 (Fig. 4). (Note that at that doping ρ(T) displays a slight upturn while R H (T) remains flat. This difference comes possibly from the fact that ρ(T) is sensitive to the total carrier density while R H (T) is balanced by electron-and hole-like contributions, as in the reconstructed Fermi surface just below p* for an anti ferromagnetic scenario 14 . See discussion in ref. 5 ).
We now turn to p FS . In a single-layer cuprate, the Fermi surface changes topology from hole-like at p < p FS (Fig. 1c) to electron-like at p > p FS (Fig. 1d). (p FS is the doping at which the van Hove singularity in the density of states crosses the Fermi level.) ARPES studies show that the Fermi surface is hole-like in LSCO at p = 0.17 11 and Nd-LSCO at p = 0.20 3 , while it is electron-like at p = 0.20 2 and p = 0.24 3 , respectively, so that p FS = 0.19 ± 0.01 in LSCO and p FS = 0.22 ± 0.01 in Nd-LSCO.
Changes under hydrostatic pressure. We now examine the effect of pressure on p FS and p*. Pressure is known to change the crystal structure of Nd-LSCO from LTT at ambient pressure to HTT at P > 4.2 GPa 15 . Our band-structure calculations based on a standard tight-binding model (see 'Methods' section and Supplementary Fig. 2) show that this causes a decrease in the ratio |t'/t|, where t and t' are nearest-and next-nearest-neighbour hopping parameters. It is a property of this model that the Fermi surface goes from hole-like to electron-like with decreasing |t'/t| (at fixed p), consistent with fits to ARPES data on LSCO 2,11 . Pressure applied to Nd-LSCO is therefore expected to reduce p FS (Fig. 5c). Although the maximum pressure in our experiment is 2.0 GPa, it still reduces the CuO 6 octahedron tilt angle of the LTT structure significantly towards the HTT phase 15 , which decreases |t'/t|.
Experimentally, we study the pressure dependence of p FS by looking at R H for p ≥ p*. In LSCO in the regime above p FS~0 .19, R H decreases linearly with doping as the system moves away from p FS ( Supplementary Fig. 3), to eventually become negative above p~0.35 16,17 . Quantitatively, an increase in doping by δp = 0.02 corresponds to a~13% drop in R H (Fig. 5b). Since all there is in this regime is a single large electron-like Fermi surface, Nd-LSCO in the regime above p FS~0 .22 must display a similar linear is seen to deviate upwards from its linear T dependence at high T (black line) below the onset temperature T* of the pseudogap phase (Fig. 1a). b Difference between the data and the linear fits of Supplementary Fig. 6, showing that the onset of the upturn at T* (arrow) is independent of pressure. The sharp kink is due to the structural transition from orthorhombic at high T (LTO) to tetragonal at low T (LTT) (see  (Fig. 3a) and from pressure data at p = 0.21 (Fig. 3b), p = 0.22 (Fig. 2c) and p = 0.23 ( Supplementary Fig. 7). The pseudogap critical point p* is where the upturn goes to zero (arrow). b R H (0) vs doping for ambient pressure (blue), P = 1.0 (purple) and 2.0 GPa (red), obtained from ambient pressure data at p = 0.21, 0.22 and 0.23 (Fig. 3c) for H = 33 T, and p = 0.25 ( Supplementary Fig. 4) for H = 16 T, and from pressure data at p = 0.21 (Fig. 3d) and p = 0.22 (Fig. 2d) for H = 33 T, and p = 0.25 ( Supplementary Fig. 4) at H = 16 T. All lines are a guide to the eye. The effect of pressure is to shift p* down in doping (grey arrows), by roughly dp* = −0.02 for 2.0 GPa. All the error bars reflect the uncertainty on the extrapolation to T = 0 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-02122-x correspond to a similar displacement in doping. In Fig. 5a (see also Supplementary Fig. 4), we see that R H in Nd-LSCO at p = 0.25 decreases by~12% under 2 GPa, which implies that p FS in Nd-LSCO shifts down by δp FS~0 .02 under 2 GPa, based on the LSCO data.
To study the pressure dependence of p*, we measured the highfield normal-state resistivity under pressure up to 2 GPa. Our data on LSCO below p* show that the pseudogap is unaffected by pressure, in the sense that the amplitude of the low-T upturn in the resistivity at p = 0.143 (ref. 9 ) is unchanged ( Supplementary  Fig. 5). In Nd-LSCO at p = 0.15, our measurements ( Fig. 2b and Supplementary Fig. 6) also show that T* is independent of pressure, as previously observed in YBa 2 Cu 3 O y (ref. 18 ). So a pressure of 2 GPa does not tune T* directly. Nevertheless, in Nd-LSCO just below p* we observe a dramatic effect of pressure on the pseudogap: at p = 0.22, 2.0 GPa completely eliminates the low-T upturn in the resistivity (Fig. 2c), resulting in a linear T behaviour characteristic of the regime at p* and above (refs. 4,8 ). A suppression of the low-T upturn is also seen at p = 0.21 (Fig. 3b). This large and unexpected effect of pressure on the pseudogap is our main experimental finding.
In Fig. 3, we compare the effect of pressure on the resistivity and Hall coefficient of Nd-LSCO with the effect of doping. (Note that unlike for YBa 2 Cu 3 O y , pressure does not change the doping in Nd-LSCO, which is set by the Sr content.) We see that pressure has the same effect as doping, consistent with a lowering of p* induced by pressure. Quantitatively, p* moves down from 0.23 at P = 0 to 0.21 at P = 2.0 GPa (Fig. 4). A downward shift δp* = −0.02 in 2.0 GPa is the same shift (δp FS~− 0.02) that is observed for p FS . We infer that it is the downward movement of p FS that constrains p* to move down with pressure, thereby showing that the condition p* ≤ p FS must hold.

Discussion
This elucidates why the pseudogap phase of LSCO ends at a lower doping than in Nd-LSCO, for indeed p* = p FS in LSCO, within error bars (Fig. 1e). In most cuprates, p FS is much higher, as in the single-layer material Bi2201, for example, where p FS = 0.41 ± 0.02 (ref. 12 ). Remarkably, p* is nearly as high, with p* = 0.38 ± 0.02 in Bi2201 (ref. 10 ), as illustrated in Fig. 1e.
Our finding that p* ≤ p FS is consistent with the known properties of all cuprates. In particular, it holds true for all known single-layer cuprates, including not only those in Fig. 1e, but also Tl 2 Ba 2 CuO 6+δ , for example, where at p~0.3 the Fermi surface is hole-like and there is no pseudogap 19 . It also holds for bi-layer cuprates, such as Bi 2 Sr 2 CaCu 2 O 8+δ , in the sense that the pseudogap opens only once 20 or after 21 both Fermi surfaces (bonding and anti-bonding) have become hole-like 22 . The close proximity of p* to a van Hove singularity (at p FS ) may have some impact on the physics of the pseudogap. However, since the velocity vanishes at the van Hove point, we do not expect any singularity in the transport properties. Moreover, in the presence of substantial c-axis dispersion, of the order of t z~2 0 meV in Nd-LSCO, the divergence in the density of states gets cut off at low temperature 23 .
It is striking that a minute change in the Fermi surface, smaller than that illustrated between Fig. 1c (p slightly below p FS ) and Fig. 1d (p slightly above p FS ), can switch off a gap of magnitudẽ 20 meV (ref. 3 ). This extreme sensitivity of the pseudogap on the details of the Fermi surface suggests that for the pseudogap to form it is necessary that the Fermi surface intersects the anti ferromagnetic zone boundary (AFZB), the dashed line in Fig. 1d, as proposed in refs. 20,24 . This is precisely what happens when the doping drops below p FS . There is empirical evidence that this intersection may indeed be a crucial element of the pseudogap mechanism. First, this AFZB is where in k-space the separation of ungapped and (pseudo)gapped states occurs, as detected by quasiparticle interference in scanning tunnelling microscopy 21 . In other words, the AFZB is the pseudogap phase's organising principle in k-space: it defines the so-called 'Fermi arcs'. Note that when the pseudogap turns on (with decreasing T or p), the k-space area contained by the ungapped states (between the Fermi arcs and the AFZB) goes from A~1 + p to A~p (ref. 21 ). Second, upon crossing p* from above, the carrier density measured by the Hall number goes from n = 1 + p to n = p (refs. 5,13 ), consistent with A~n. The simplest way to obtain a loss of 1.0 hole per planar Cu atom is to reconstruct the large hole-like Fermi surface (with 1 + p holes) by folding it about the AFZB, which produces four small nodal hole pockets (with p holes) 14 . This can be achieved either by AF order with a wavevector Q = (π, π) or by an Umklapp surface coincident with the AFZB, as in the YRZ model 25 (Supplementary Fig. 4). The dashed line is a linear fit through the data at low pressure, whose slope is −6 % GPa −1 .
b Sketch of the doping dependence of R H in LSCO ( Supplementary Fig. 3), normalised by its value at p FS (blue line). The red line illustrates the effect of shifting p FS down in doping by an amount δp FS = −0.02 (grey arrow): R H decreases by~13 % (red arrow). This is the amount by which R H decreases in Nd-LSCO under 2.0 GPa, for p > p* (a). c Sketch of the theoretical dependence of p FS on the band-structure parameter t'/t. For a given value of t'/t, p FS is given by the blue line, which separates the regions where the cuprate Fermi surface is electron-like (above) and hole-like (below). Our band-structure calculations for Nd-LSCO yield t'/t = −0.171 in the LTT structure (black) and t'/t = −0.152 in the HTT structure (red) ( Supplementary Fig. 2), so that p FS is expected to decrease (grey arrow) under pressure, as observed experimentally phase forms 26 . The issue of why there is a pseudogap in holedoped cuprates and not in electron-doped cuprates, however, is an open question. If the pseudogap phase needs states near (π, 0) to form, the question is at what energy do those states have to be relative to ε F ? We presume that they should be within the pseudogap energy Δ PG of ε F . This implies that for p just above p FS , the pseudogap phase can still form. This introduces some 'width' to the criterion p* ≤ p FS . In a similar vein, a 3D dispersion of the Fermi surface in the k z (or c) direction will also give some 'width' to the criterion, since at a given doping the Fermi surface can be electron-like at some k z value but still hole-like at some other k z value.
The requirement that a pseudogap cannot form in a cuprate with an electron-like Fermi surface imposes a stringent constraint on theories of the pseudogap phase 25,[27][28][29][30][31] . In the YRZ model 25 , a pseudogap forms because carriers undergo Umklapp scattering, inherited from the Mott insulator, at points where the Fermi surface intersects the AFZB; this model therefore agrees with our proposed constraint. Our findings are also broadly consistent with spin-fermion models that hinge on the hot spots that lie at the intersection of Fermi surface and AFZB 27,28 . More specifically, two recent theoretical studies 29,32 find that a pseudogap only opens on hole-like Fermi surfaces and that p* ≤ p FS for a wide range of band-structure parameters, even in the strong-coupling regime where the anti ferromagnetic correlations responsible for the pseudogap are short ranged. On the other hand, in scenarios characterised by a wavevector Q = (0, 0), the AFZB plays no special role and there is then no obvious reason for the constraint to be effective. This would seem to rule out nematic order 31 and intraunit cell magnetic order 30,33 as possible drivers of the pseudogap phase. Instead, these orders would be secondary instabilities 6 .
On the experimental side, the ability to continuously suppress and restore the pseudogap with pressure in a given sample without changing the doping or disorder level provides a promising avenue to study the pseudogap state, compatible with a range of probes such as Raman, optics, X-ray and neutron scattering.

Methods
LSCO samples. Single crystals of La 2−x Sr x CuO 4 (LSCO) were grown by the fluxzone technique, with nominal Sr concentrations of x = 0.145 at Hokkaido University and x = 0.18 at the University of Tokyo. Samples for resistivity measurements were cut in the shape of small rectangular platelets, of typical dimensions 1 mm × 2 mm × 0.5 mm, with the smallest dimension along the c axis. Contacts were made using H20E silver epoxy, diffused by annealing. The hole concentration (doping) p for our nominal x = 0.145 sample was determined using the doping dependence of the (tetragonal to orthorhombic) structural transition temperature, T LTO , which is detected in the resistivity as a small but sharp kink. This yields p = 0.143 (ref. 9 ). For our sample with x = 0.18, we take p = x.
Nd-LSCO samples. Single crystals of La 2−y−x Nd y Sr x CuO 4 (Nd-LSCO) were grown at the University of Texas at Austin with a Nd content y = 0.4, using a travellingfloat-zone technique, and cut from boules with nominal Sr concentrations x = 0.15, 0.21, 0.22, 0.23 and 0.25. The samples were prepared for transport measurements as described above for the LSCO samples. For all five samples, the hole concentration p is given by p = x, with an error bar ±0.003. The samples labelled here p = 0.21, 0.22, 0.23 and 0.24 are the same samples as those studied in ref. 5 ; the sample labelled here p = 0.25 is a new sample, with p = 0.25 ± 0.003.
Resistivity and Hall measurements. The electrical transport measurements were performed via a standard four-point low AC technique using an SR830 lock-in amplifier and a Keithley 6221 current source. A current of typically 2 mA was applied within the CuO 2 planes and the magnetic field along the c-axis. The longitudinal and Hall resistances R xx and R xy were measured at Sherbrooke in steady fields up to 16 T and at the NHMFL in steady fields up to 45 T. The Hall resistance R xy is obtained by reversing the field and anti-symmetrizing the data, as R xy (H) = (R xy ( + H) − R xy (−H))/2.
Application of pressure. Pressure was applied on our samples using a miniature non-magnetic piston-cylinder cell. The pressure medium is Daphne oil 7474, which remains liquid at all pressures measured here at 300 K, ensuring a high degree of hydrostaticity. The internal pressure is measured both at room temperature and at 4.2 K, using either the fluorescence of a small ruby chip or a Sn manometer. The values quoted throughout are the low temperature pressures. The error bar on all the pressure values is ± 0.05 GPa, which comes from the uncertainty in measuring the position of the fluorescence peaks. For each measurement, the cell was cooled slowly (<1 K min −1 ) to ensure a homogeneous freezing of the pressure medium.
Band-structure calculations. Band-structure calculations were performed by using the full potential augmented plane wave band method, implemented in the WIEN2k package. We used the local density approximation for the exchangecorrelation potential. We used 1000 k-points inside the first Brillouin zone. The convergence of total energy with respect to the number of k-points was checked to have a precision better than 0.007 eV per formula unit. We calculated the electronic structure of La 2 CuO 4 using the structural parameters of Nd-LSCO measured experimentally by X-ray diffraction for both ambient pressure (LTT structure; ref. 34 and P = 4.2 GPa (HTT structure; ref. 15 ). In the case of P = 4.2 GPa (HTT), in order to determine the internal position of atoms inside the CuO 6 octahedron, we assumed an isotropic contraction of the CuO 6 octahedra from hydrostatic pressure. Experiments on La 2 CuO 4 have shown this assumption to be valid 35 . Tight-binding hopping parameters were obtained by fitting Cu(d x2-y2 )-driven bands for highsymmetry points of the anti ferromagnetic zone boundary.