From chessboard of bipolarons of size 4a in cubic $\hbox {La}_{7/8}\hbox {Sr}_{1/8}\hbox {MnO}_3$ to stripes of the same bipolarons in layered high $T_c$ cuprates

Abstract

The compound $\hbox {La}_{1-x}\hbox {Sr}_x\hbox {MnO}_3$ exhibits a charge order (CO) state at $x\approx 1/8$ and $T<T_{co}$, which recalls the CO state with a decrease in the temperature of the superconducting transition, $T_c$, observed in all cuprates at this doping value. Local excitations of lattice and magnetic origins measured in the two-dimensional metallic state of $\hbox {La}_{7/8}\hbox {Sr}_{1/8}\hbox {MnO}_3$ reveal the existence of bipolarons of size 4a resulting from structural and antiferromagnetic pairings of hole-rich orbital polarons of size 2a. They are intertwined with hole-poor domains in a disordered state at $T>T_{co}$ which become ordered on a chessboard organized in a 3D-order state of ferromagnetically paired polarons at $T<T_{co}$. Applied to the $\hbox {CuO}_2$ planes of the cuprates of the “214” family, this model produces stripes of bipolarons intertwined with stripes of antiferromagnetically arranged spins, hole-poor, both of size 4a, leading to a spin density wave with a wave vector $\delta =1/8$, a charge density wave with $q=1/4$, the Yamada laws $\delta (x)=x$ and $T_c\propto \delta$ and a decrease of $T_c$ at x=1/8. This work invokes relevance of a bipolaronic origin of high $T_c$ superconductivity, in which bipolarons of size 4a can play a major role.

Introduction

Among manganites, the pseudocubic $\hbox {La}_{1-x}\hbox {Sr}_x\hbox {MnO}_3$ (edge of a cubic lattice unit cell a=3.9Å) appears particularly interesting because its phase diagram shows characteristics typical of cuprates with high superconducting (SC) temperature $T_c$ (see the phase diagrams in the Supplementary Material (SM) of this paper, Ref.¹, Fig. SM-1). In the doping range $0.1 \le x \le 0.15$ around $x=1/8$ and for $T<T_{co}$², this material presents a three-dimensional (3D) charge order state without a consensus reached to date on its origin^{3,4,5,6,7,8,9,10}. A previous study of spin dynamics has shown that the charge order (CO) state arises from a ferromagnetic (F) and metallic state with a two-dimensional (2D) character¹¹. This 2D character is the consequence of the magnetic structure of $\hbox {LaMnO}_3$, which consists of ferromagnetic planes weakly coupled by an antiferromagnetic coupling (AF) in their perpendicular direction^11,12. This 2D metallic state stabilizes at a higher temperature $T_{FM}$, where the charge-induced magnetic coupling is equivalent along the two directions of the MnO bond¹¹. Specifically for this compound with x=1/8, the values $T_{FM}$=181K and $T_{co}$=159K have been determined¹³.

In the same doping range of the phase diagram of all high-$T_c$ cuprates, a depression of superconducting temperature $T_c$(x) is observed^14,15. At the same time, a long range charge density wave (CDW)^16,17 and a spin density wave (SDW) with periods $\approx 4a$ and $\approx 8a$, respectively, occur in $\hbox {La}_{2-x}\hbox {Ba}_x\hbox {CuO}_4$^18,19 and $\hbox {La}_{2-x}\hbox {Sr}_x\hbox {CuO}_4$²⁰ with optimal order at $x=1/8$. Several models have been proposed^{18,21,22,23,24,25,26,27} while the microscopic origin of high $T_c$ superconductivity remains uncovered and the role of CDW which competes with superconductivity is still debated^15,28.

In the present paper, we report new additional phonon acoustic branches in $\hbox {La}_{7/8}\hbox {Sr}_{1/8}\hbox {MnO}_3$, at T$<T_{co}$ and T$>T_{co}$ along the MnO bond directions that lie in restricted ranges of wave vector q in the reciprocal space. Based on a previous study at $x=0.2$ (Refs.^1,29) and considering previous data on magnetic excitations along several symmetry directions (Refs.^11,29), they can be attributed to hole-rich domains of size 4a intertwined with hole-poor domains of the same size organized in a chessboard structure at $T<T_{co}$. They coexist with static charge density waves (CDW) of period 4a ($4a\sqrt{2}$) along [100] ([110]) identified from gaps in the magnetic dispersion law $Dq^2$. The bipolaronic origin of the hole-rich domains appears in the 2D metallic state, where the chessboard is expected to be distorted. We recall that a bipolaron results from the structural and AF pairings of two polarons which are defined in real space by the coupling of one charge with phonons. In the present work, these pairing properties are considered for bipolarons with a hole-rich character that are separated by hole-poor domains. Applying this picture to the layered structure of high $T_c$ cuprates which shares the same orbital structure, one gets a picture of stripes of bipolarons of size 4a intertwined with stripes of spins arranged antiferromagnetically, hole-poor. A superconducting state is expected at $T_c$ from the band created by their fluctuations. At $x=1/8$ as the size of the hole-poor domains between the stripes of bipolarons decreasing with x becomes equal to the size 4a of the bipolarons, the CDW that coexists with the bipolarons acquires a long-range period 4a that produces partial immobilization and ordering of the stripes of bipolarons and, accordingly, a decrease in the value of $T_c$.

The origin of the additional branches of the acoustic phonon was first discussed for $x=0.2$ in $\hbox {La}_{1-x}\hbox {Sr}_x\hbox {MnO}_3$²⁹. There, three acoustic phonon branches with transverse and longitudinal character labeled TA*(q), $\hbox {TA}^{perp}$(q) and LA*(q) have been observed in restricted ranges of q values along [100] + [010] + [001] (see TA* and $\hbox {TA}^{perp}$in Fig. SM-2a of Ref.¹ adapted from Ref.²⁹). They coexist with the longitudinal LA(q) and transverse TA(q) phonon branches of the pseudocubic structure common to the twinned domains. The wave vector q along [100] + [010] + [001], superposed by twining, corresponds to waves propagating along the MnO bond directions and q is expressed in reciprocal lattice units (rlu) of the cubic structure so that q = ($2\pi /a$)[q00] etc (a=3.9Å).

In the problem of charges subjected to multiple competing interactions, the spectrum of acoustic phonon branches may indicate that there exists an effective medium characterized by the LA and TA branches of the pseudocubic structure with the highest intensity, in which hole-rich domains characterized by local excitations appear as additional branches in the restricted range [$q_{min}-0.5]$ along the MnO bond directions. The lower limit $q_{min}$ of the range of q can be used to determine the size of the domains if the spatial distribution of these domains preserves the phase of the excitations. This property is obtained if the domains are in contact along some direction or if they are intertwined with a hole-poor domain of the same size that transmits the phase along some other direction. Both situations are observed in our data at $x=0.2$ and $x=1/8$. These branches are observable if the lifetime of the domain is long compared to the reciprocal frequency of the local phonon excitations, so they are no longer observed in the true metallic compound with $x=0.3$³⁰. We get a weakly inhomogeneous picture with localized (hole rich) and non-localized (hole poor) components for the charge, less accentuated than in the images of charge segregation or phase separation previously used²⁹.

At $x=0.2$, the value $q_{min}=0.25$ of the additional LA* and TA* branches has determined the linear size 2a of the “hole rich” domains using the relation $\xi ={\pi }/|\textbf{q}_{min}|$ so that the boundary of the Brillouin zone $q_{min}$ = 0.5 rlu corresponds to the smallest distance of coupling $\xi =a$. The maximum intensity observed at $q_{min} = 0.25$ rlu in the TA* branch is a consequence of the stationary character of the local vibrations of size 2a in contact, defining chains along the MnO bond direction³¹. The size 2a corresponds to the ferromagnetic “orbital polaron” introduced to explain the charge-ordered states of manganites ($x=0.25, x=0.5$) in which the charge-phonon coupling is neglected so that polarons or domains of hole-density do not appear^32,33. In such an ionic picture, the $Mn^{3+}$ ion consists of an unoccupied external level showing a doubly degenerate orbital state $e_g$ (${x^2-y^2}$, $3z^2-r^2$) well separated in energy from the inert spin core $S=3/2$³⁴. Whatever its 1D, 2D or 3D character, the orbital polaron has been defined by the orbital states $T^x$, $T^y$, $T^z$ ($T^{\alpha }=3\alpha ^2-r^2$, $\alpha =x,y,z$) of the $Mn^{3+}$ sites with lobes pointing toward the common neighboring site ($Mn^{4+}$) that carries the hole^32,33. At $x=0.2$, the complete superstructure of chains of 1D polarons was determined considering the other transverse acoustic branch $TA^{perp}(q)$ in the perpendicular direction superposed by twining. In contrast to TA* and LA*, this branch is observed in the small range of q values up to the cutoff point $q_c\approx 0.35$ observed with an energy far from TA* by $\Delta =3 meV$. This cutoff not only determines the periodic distribution $2\pi /q_c=3a$ of the chains but can also be interpreted as the zone boundary of a fictitious Brillouin zone defined during the short lifetime of the chain direction. In this way, the excitation at $q_c\approx 0.35$ corresponds to the transverse excitations in phase opposition of the hole-rich and hole-poor domains, intertwined, each with a rigid thickness 1.5a. This thickness value suggests that the hole-rich density extends along the direction perpendicular to the chains or that the p and d orbitals are hybridized. One gets a picture of chains of polarons of thickness 1.5a with periodic distance 3a ($x\approx 0.17$) similar to an electronic crystal liquid³⁵ (see the sketch in Fig. SM-2b of Ref.¹). The same periodic structure of charges appears in the magnetic excitations, as shown below.

We recall that in metallic manganites, along the directions of charge hopping (or tunneling), the magnetic spectrum consists of two distinct parts, depending on the range of q values. A law $Dq^2$ appears at small values of q related to spin waves of the electronic band origin³⁴, and a discrete energy spectrum spreads at larger values of q. This discrete energy spectrum can be explained by considering that the hopping (or tunneling) of the charges has two effects that can be described in an ionic picture as follows: i) it induces a local ferromagnetic coupling between a $Mn^{3+}$ spin and its $Mn^{4+}$ neighbor³⁶, ii) it induces a fluctuation of the anisotropic $e_g$ orbital state of the $Mn^{3+}$ sites, first neighbor of the $Mn^{4+}$ sites as the position of the charge varies. The atomic structure being determined by the orbital states, their fluctuations should be strongly coupled to phonons. At low doping values such as $x=1/8$ and x=0.2 where the charges are strongly correlated, their collective motion induces orbital fluctuations that are also strongly correlated so that they appear in the low energy range as q-dependent phonon branches. Their magnetic character^29,37 is the consequence of the coupling between the orbital and spin degrees of freedom at the $Mn^{3+}$ sites^38,39,40.

At $x=0.2$, the absence of a law $Dq^2$ in the magnetic spectrum along [111] indicates that the hopping (or tunneling) of the charges is restricted to the MnO bond directions (see Fig. SM-4 of Ref.¹). Along these directions, the discrete energy spectrum observed beyond the law $Dq^2$ is divided into two sets of E(q) branches, lying at $E<27$ meV with $q\ge q_{min}\approx 0.35$ rlu and at $E>27$ meV with $q\ge q_{min}=0.25$ rlu (see Fig. SM-3a of Ref.¹). Since these characteristic values q are the same as those of the two additional transverse phonon branches $TA^*$ and $TA^{perp}(q)$, separated in energy by the same value $\Delta =3 meV$ at $q\approx 0.35$, we conclude that the discrete magnetic spectrum arises from the motion or fluctuation of the same superstructure of the hole-rich and hole-poor domains. The two sets of branches E (q) are therefore attributed, for $E>27 meV$ and $q_{min}=0.25$ rlu, to the 1D orbital polarons (hole-rich domains) of size 2a in contact along chains, and, for $E<27 meV$ and $q_{min}\approx 0.35$, to the “hole-poor” domains intertwined with the chains along the perpendicular direction, defining the same thickness 1.5a of the hole-poor domains as the branch $TA^{perp}(q)$. A fully coherent picture appears by considering that, in the upper energy range ($E>27 meV$), the set of three branches agrees with the three expected MnO bond directions of the chains. At $T<T_{FM}$ their magnetic energy that coexists with the phonon energies starts to increase and becomes connected to the law $Dq^2$, highlighting the 1D character of the metallic and ferromagnetic state during the lifetime of the chain direction (see Fig. SM-3 in Ref.¹ adapted from Ref.²⁹).

Fig. 1

(a) Magnetic spectrum along [110] at $T>T_{co}$ (full red circles) and along [111] at T$<T_{co}$ (empty blue triangles). At $T>T_{co}$ we have checked that the [110] direction is equivalent to the [111] direction which was used in the highest energy range to open the available energy window. The two (four) branches observed along [110] ([111]) arise from AF paired (F paired) hole-rich domains fluctuating between the diagonals of squares (cubes). (b) Phonon spectrum determined at $T=300K$ along [110]. The rectangle outlines the additional excitation observed between LA and TA at q=0.125 rlu. The anomalous increase of their intensity is shown in the raw data of panel (c).

Fig. 2

Magnetic spectrum along the MnO bond directions. (a) at $T<T_{co}$ E(q) branches of the magnetic spectrum adapted from Ref.¹³ showing a dip at $q=7/16$ rlu in the $E=LA$ and $E=LO$ branches corresponding to the “in-plane” magnetic fluctuations. The upper energy branch occurring at $E=LO'$ at $T<T_{co}$ has been omitted for clarity. (b) Quadratic law $Dq^2$ measured at T=150K (blue circles) adapted from Ref.¹¹ showing an energy gap at $q=1/8$ rlu outlined by an arrow and compared to the $Dq^2$ measured at T=169K (red circles). (c) Maximum of energy observed at $q=0.45$ rlu in the upper energy branches of the magnetic fluctuations. The $q'$ scale is defined by $q'=(0.83)^{-1}q$ (see the text). The E(0) branch, also observed in lattice excitations, but not displayed, can be attributed to a binding energy of the bipolaron to the lattice²⁹. $\Gamma$ is the energy line-width.

Fig. 3

The dispersion branches sketched by continuous lines reveal four distinct experimental situations. (a) an F coupling at the interface between two neighboring F domains, each of size $a\sqrt{3}$ along [111], (b) a F coupling between adjacent F domains of size 2a showing a period 8a. (c) a AF coupling at the interface between two neighboring F domains, each of size $a\sqrt{2}$ along [110]. (d) an AF coupling at the interface between two neighbor F domains of size $2a'$ with $a'=0.25/0.3 a$ and $q'$=q x 0.3/0.25 along the MnO directions of the planes (see the text).

Results: magnetic and lattice excitations at x=1/8

In Sect. "Magnetic excitations in the 3D ferromagnetic state (T < T_co) and in the 2D metallic state (T > T_co) in the hole-rich, hole-poor model" the previously reported magnetic spectra are interpreted in the hole-rich, hole-poor model¹¹, in Sect. "The additional branch of phonon excitations in the 3D ferromagnetic state (T < T_co)", we present new lattice excitations obtained at $T<T_{co}$ and in Sect. "The additional phonon branches observed at T > T_co in the 2D ferromagnetic and metallic state", new lattice excitations obtained at $T>T_{co}$. In the discussion (Sect. "Discussion: A model for high T_c cuprates"), we show how the existence of bipolarons of size 4a can be a key to interpreting some properties of the layered high $T_c$ cuprates, especially those with the simplest structure of the 214 family.

Magnetic excitations in the 3D ferromagnetic state ($T<T_{co}$) and in the 2D metallic state ($T>T_{co}$) in the hole-rich, hole-poor model

In the magnetic spectrum, the new situation at $x=1/8$ with respect to $x=0.2$, is the existence of a law $Dq^2$ along several directions of symmetry. Moreover, the observation of additional acoustic phonon branches in the $[q_{min}-0.5]$ range of the q value at $x=1/8$ as at $x=0.2$ (they are described in the next Section) allows us to interpret the discrete energy magnetic spectrum in terms of domains in direct space.

Figure 1a presents the experimental data along the “diagonal” directions, [111] ($T<T_{co}$ or 3D ferromagnetic state) and [110] ($T>T_{co}$ or 2D metallic state), complemented and improved over the previous report¹¹ whereas Fig. 2c presents those along the MnO bond directions^11,37. Along the diagonal directions, the magnetic spectrum is divided into two sets of branches E(q) $E<27meV$ and $E>27meV$. From our study at $x=0.2$, they are readily attributed to the magnetic excitations of the hole-poor ($E<27$ meV) and hole-rich ($E>27$ meV) domains. This separation in two energy ranges indicates that the excitations of the two types of domain do not interfere or that the hole rich and the hole poor domains are in contact along distinct “diagonal” directions. As expected, the low-energy branches attributed to hole-poor domains lie approximately in coincidence with the acoustic branches reported in Fig. 1b.

In contrast, along the MnO bond directions reported in Fig. 2c, a single set of branches is observed. There, the two types of excitation interfere or, equivalently, the two types of domain are intertwined.

The complete characterization of the hole-rich domains is provided by the value $q_{min}$ and the dispersion with q of the branches. They are described in the 3D charge order ($T<T_{co}$) and the 2D metallic ($T>T_{co}$) states successively.

a) In the 3D charge order state ($T<T_{co}$), i) along the diagonal [111], four branches, numbered from one to four in the figure, exhibit the value $q_{min}$=0.25 rlu (the higher energy branch being poorly defined). Their dispersion with q reveals inflection points at $q=0.375$ rlu, middle of the range [$q_{min}$−0.5]. This dispersion provides evidence of an F coupling at the interface between two adjacent F domains of size $2a\sqrt{3}$. They reveal domains of size $4a\sqrt{3}$, obtained by a pairing F of small domains $2a\sqrt{3}$ in contact along the four diagonals [111]. The dispersion with q expected for a long-range order of such F-paired domains in contact along [111] is obtained by a folding of the Brillouin zone at $q=0.25$ as sketched in Fig. 3a. ii) Along the MnO bond directions where the excitations of the hole-rich and hole-poor domains interfere or are intertwined, the determination of the $q_{min}$ value is masked by the spectacular coincidence of the magnetic spectrum successively with the TA, LA, LO, and LO’ phonon branches for $q\ge 0.25$ (see Fig. 4 of Ref.³⁷). However, the temperature evolution of the branches allows us to determine the value of $q_{min}$= 0.25 rlu along [100] and [110], at the center of the large energy gap observed along these two directions where the intensity of the E(TA) branch jumps on the E(LA) one (see Fig. 4c of Ref.³⁷ and Fig.SM-5 of Ref.¹). The most surprising feature is the departure of the ferromagnetic branches from the acoustic LA(q) and LO(q) branches observed at $q\approx 0.44=7/16$ along the MnO bond directions as seen in Fig. 2a (adapted from Ref.¹³, improved from Fig. 4 of Ref.³⁷). Despite twinning, these two branches can be attributed to the directions [100] and [010] of the (a, b) planes (fluctuations of $T^x$ and $T^y$) since they still exist in the 2D metallic state, and, for $T<T_{co}$ (3D charge order state), they are separated in energy from the new branch with $E=LO'$ which occurs at $T<T_{co}$ readily assigned to the direction [001] (fluctuations of $T^z$). This upper energy branch has been omitted in Fig. 2a for clarity. Using the same analysis as along [111], this dip value in the dispersion is attributed to a folding by eight of the Brillouin zone or a period 8a for the domains lying along the two MnO directions of the planes (see the sketch in Fig. 3b).

From observations along these two symmetry directions, we conclude that the 3D space is filled by large cubes 4ax4ax4a obtained by ferromagnetic pairing of adjacent orbital polarons 2ax2ax2a along their four [111] diagonals (see Fig. 4a,b). All these cubes are identical. There exists a unique arrangement of these large cubes that reconciles the fact that their diagonals are in contact along the four [111] directions, and, at the same time, their sides along the two MnO directions of the (a, b) planes produce a period 8a. As sketched in Fig. 4c, these two properties are realized by considering two families of columns of cubes 4ax4ax4a. One family of cubes differs from the other one by a translation by a or $-a$ along the c axis. In the $MnO$planes, this 3D arrangement determines a chessboard of hole-rich and hole-poor domains of side 4a with an alternation of the two types of domains along the c axis (see Fig. 4d,e). These hole-rich and hole-poor domains are represented by different squares of side 4a in the Fig. 4d,e and correspond to different cross-sections of these cubes by the planes MnO: in the middle of the cube 2ax2ax2a on the Fig. 4a and shifted up or down by the lattice spacing a (a distance between adjacent planes MnO).

This superstructure of charges can be seen as a structure of “octopolarons” with one charge per 8 Mn sites ($x=1/8$). It provides the origin of the two static peaks previously reported¹³ that indicate the periods 2a (magnetic) and 4a (nuclear) along the c axis^4,13.

Moreover, gaps have been reported at $q=0.125$ rlu which open in the dispersion law $Dq^2$ for ferromagnetic magnons of metallic origin for ferromagnetic excitations, just below $T_{co}$ along the two symmetry directions [100] and [110] of the planes (see Fig. 2b along [100] adapted from Ref.¹¹). We attribute these gaps to new static spin density waves of period 4a (and $4a\sqrt{2}$) along [100] (and [110]) which, in their turn, manifest appearance of charge density waves (CDW) with the same period polarized by the ferromagnetic spin ordered structure of the isolating charge-ordered low temperature phase. These charge density waves are intertwined with the chessboard of hole-rich/hole-poor domains (see the green line sketched for clarity along one MnO direction in Fig. 4e).

b) In the 2D metallic state ($T>T_{co}$), a strong change appears in dispersion with q of the branches of the hole-rich domains. Moreover, the effect of disorder differs depending on whether the domains considered are along the diagonals [110] or along the MnO bond directions of the (a, b) planes. i) Along the diagonal directions [110], two branches numbered 1 and 2 in Fig. 1 have been determined for the hole-rich domains (E>27 meV). A first decrease in intensity is observed at $q=0.3$ rlu (see the arrows) that prevents one from accurately determining $q_{min}$ ($\approx 0.25$ rlu). The modulation of the two branches exhibits a rounded shape with the maximum energy lying at $q\approx 0.375$ rlu in the middle of the range [$q_{min}$−0.5] (see raw data in Fig.SM-6 of Ref.¹). This maximum corresponds to a double-scale AF coupling between the AF spin characteristics of an AF state. They provide evidence of an AF coupling at the interface between two adjacent F domains of size $\approx 2a\sqrt{2}$ and, by the way, domains of size $\approx 4a\sqrt{2}$ distributed with some disorder along the two [110] directions. The dispersion with q expected for a long-range order of AF-paired domains in contact along [110] is sketched in Fig. 3c.

ii) In contrast, along the MnO bond directions, the value $q_{min}$ = 0.3 rlu is clearly defined in Fig. 2c and the modulation of the magnetic branch with the main intensity accurately determines the maximum energy at $q=0.45$ rlu at the middle of the range q=$[0.3-0.6]$. This value agrees with an AF pairing between to adjacent domains of incommensurate size $2a'$ observed in a fictitious Brillouin zone with $a'$ = 0.83a so that the zone boundary appears at $q=0.6$ instead of $q=0.5$ (see Fig. 3d) and note⁴¹. Such a large distortion of the lattice being unphysical, we recall that along this direction the excitations of the hole-rich domains interfere with those of the hole-poor domains. This discommensuration is therefore attributed to the varying thickness values of the intertwined hole-poor domains, lattice-locked, while the size of the hole-rich domains should maintain the size value 4a. The same conclusion has been obtained for the incommensurate CDW observed in high $T_c$ when analyzed in direct space^42,43. This comparison suggests a tight relationship between the size of the domain and the period of a CDW in the 2D metallic state, as observed in the CO state where our data have determined the size 4a for the domains and the period 4a for the CDW. Finally, we emphasize that, in this 2D metallic state, the law of $Dq^2$ is connected to the excitations of the hole-poor domains with a “wave” or unlocalized charge character along the [110] direction and to excitations of bipolarons with a localized character intertwined with the hole-poor domains along the MnO bond directions, reminding of the nodal, anti-nodal dichotomy observed in cuprates at high energy.

The origin of these observations is obtained from our phonon spectrum studies at $T<T_{co}$ and $T>T_{co}$.

Fig. 4

3D superstructure of orbital bipolarons. (a) 3D ferromagnetic orbital polaron 2ax2ax2a defined by the orbitals $T^x$, $T^y$ and $T^z$ lying at the center of each face and pointing to the $Mn^{4+}$site in an ionic picture (cyan circle), adapted from Ref.^33. (b) Cubes of side 4a containing eight charges (octopolarons) defined 1) from magnetic excitations that reveal F domains on the scale $4a\sqrt{3}$ along the four [111] directions resulting from the F pairing of domains of size $2a\sqrt{3}$ along these directions. They are observed thanks to orbital fluctuations that occur during their motion 2) from the additional lattice excitations that reveal bipolaronic domains of size 4a along the 3 MnO directions (12 links) observed during their short lifetime. (c) 3D charge order defined by two families of columns of identical cubes of side 4a sketched with red and blue colors corresponding to a translation by $+a$ or $-a$ along c axis. Each cube is connected to cubes of same family by its four diagonals [111] (see the two dotted-dashed lines along one diagonal [111] of each family). (d) Projection in (a, b) planes of the two families of cubes with bases respectively at the coordinates $z_1$ and $z_2=z_1-a$ or $z_2=z_1+a$. (e) Chessboard of hole-rich and hole-poor domains of size 4a with alternation along the $\textbf{c}$ axis determined by the 3D charge order. The white circles indicate Mn sites. The large blue circles visualize the center of the hole-rich polarons in a given MnO plane. The green line indicates a CDW of period 4a along one MnO bond direction of the planes intertwined with the chessboard of bipolarons also inferred from our experiments. The CDW of period $4a\sqrt{2}$ also observed along the [110] direction is not shown. The large red circle enhances the small red one. The charge density extends up to the oxygen sites sketched by filled brown circles.

Fig. 5

Phonon spectra along [100] determined in longitudinal (red and green color) and transverse (blue color) configurations with: in (a) at 15 K ($T<T_{co}$) one additional branch labeled TA’, and, in (b), at 300 K and 180 K ($T>T_{co}$) four additional branches labeled LO’ and LA’ ($q_{min}\approx 0.3$ rlu), TA’ and TA” ($q_{min}\approx 0.15$ rlu). The $q'$ wave-vector is defined by $q'=(0.3/0.25)^{-1}q$ (see the text). The dashed area indicates that the LA(q) branch splits into three branches at $q\le 0.15$ rlu. In (a) and (b), the continuous and dotted lines are guides to the eye. The dotted vertical lines point to the characteristic values $q_{min}$=0.125 and $q=0.25$ rlu for TA’ ($T<T_{co}$) and $q_{min}$=0.15 for TA’ and TA”, $q_{min}$=0.3 for LO’ and LA’ ($T>T_{co}$). The horizontal arrows represent their estimated error.

Fig. 6

(a) and (b): Comparison of the spectra measured in the transverse configuration at values $q=0.2$, $q=0.25$ and $q=0.3$ at $T<T_{co}$ (14K, upper panel) and $T>T_{co}$ (180K, lower panel, with spectra shifted by 25 counts for clarity). This comparison shows that, at $T<T_{co}$, the TA’ branch exhibits a maximum intensity at $q=0.25$ and that the LA branch acquires a large intensity in the transverse configuration for $q\ge 0.25$. (c) and (d): Comparison of raw data obtained in longitudinal configuration at $q=0.35$, for $T<T_{co}$ (14K left panel) and $T>T_{co}$ (180K, right panel), showing the occurrence of the LA’ mode at $T>T_{co}$.

The additional branch of phonon excitations in the 3D ferromagnetic state ($T<T_{co}$)

At $T<T_{co}$ the values of the lattice parameter in the pseudocubic structure are very close to each other and the Jahn-Teller effect is nearly suppressed^3,12,13.

Figure 5a shows the existence of an additional phonon branch TA’(q) lying between the TA(q) and LA(q) branches observed in transverse configuration along the [100]+[010]+[001] directions superimposed by twining. This branch TA’ gives evidence for the two characteristic wave vectors $q=0.125$ and $q=0.25$. First, this branch disappears at $q_{min}$=0.125 +/−0.025 rlu (see raw data in Ref.¹, Fig. SM-7 and Fig. SM-8). Near this wave vector the longitudinal branch LA seems to be repulsed from TA’ suggesting interaction of these modes with an anti-crossing behavior as shown in Fig. 5a. This value $q_{min}=0.125$ reveals that the cubes 4ax4ax4a observed by magnetic excitations have a charge-phonon coupling origin. The wave vector $q=0.25$ rlu is outlined by the highest intensity $I_{max}$ of the branch TA’. This maximum intensity occurs at $T\le T_{co}$ (see the comparison of the raw spectra of the TA’ branch in Fig. 6a and b). In our previous study at $x=0.2$ (see Introduction), the maximum intensity of the TA* branch observed at $q_{min}=0.25$ has been related to stationary excitations of 1D orbital polarons of size 2a in contact along chains. In the same way, in the present 3D case where the TA’ branch is nearly q-independent for $q\ge 0.25$ this maximum intensity is attributed to nearly stationary excitations of 3D orbital polarons of size 2a in contact along the three MnO directions.

In conclusion, the lattice excitations determine large cubes of side 4a consisting of eight cubes corresponding to orbital polarons of size 2a. This is the same picture as that obtained by magnetic excitations and observed during their motion. We present on the Fig. 4b the two identical cubes outlining ferromagnetic (Fig. 4b1) and nuclear (Fig. 4b2) coupling between polarons. This ordered state of ferromagnetic orbital polarons is responsible for the coincidence of the magnetic and acoustic lattice excitation, which also manifests itself by a new intensity of LA in the transverse configuration for $q\ge 0.25$ (compare the raw data of Fig. 6a and b, and of Fig.SM-9 in Ref.¹). However, because of twinning, the fundamental information on the period 8a along the two MnO directions of the planes is missing, which in contrast outlines the powerful probe of the charge correlations provided by the orbital fluctuations (magnetic excitations).

The proposed 3D superstructure unlocks the ordering vectors (1/8 1/8 1/2) and (1/8 1/8 1/4), corresponding to the periodicity of, respectively, magnetic (F) and lattice (nuclear) coupling. To our knowledge these ordering vectors have not been reported in the literature, probably because of their low intensity as compared to the other observed ordering vectors. This may be a challenge for future experiments to confirm the proposed vectors with the (1/8 1/8) in-plane component.

The additional phonon branches observed at $T>T_{co}$ in the 2D ferromagnetic and metallic state.

At $T>T_{co}$ the lattice parameters of the pseudocubic structure exhibit well-distinct values and a large Jahn-Teller effect is observed^3,12,13. Figure 5b presents evolution of the lattice dynamics above the CO transition. There, three additional branches are observed in the restricted range of values q that give evidence for the two characteristic wave vectors $q=0.15$ and $q=0.3$. Along the [100]+[010]+[001] directions of q, LA (q) behaves differently depending on the range of the wave vector. For $q<q_{min}$=0.15 rlu, the three branches indicate that LA(q) is distinct for the three directions of the MnO bond, superposed by twinning (see the raw data shown for $T>T_{co}$ and $T<T_{co}$ at $q = 0.15$ in Fig. SM-10 of Ref.¹). For $q>q_{min}$=0.15 rlu, a single branch LA(q) remains that becomes broad at $q=0.25$ rlu. The same wave vector $q_{min}\approx 0.15 +/-0.05$ rlu manifests itself as the point where the two other transverse branches TA’ and TA” come out. In fact, due to LA contamination, only the value $q_{min}$ of the branch TA” can be experimentally determined (see the raw data in Fig. SM-11 of Ref.¹).

Another characteristic wave vector appears at twice this value. The value $q_{min}\approx 0.3$ points to the emergence of the additional LA’(q) branch just below the LA(q) main phonon branch (compare the raw data of Fig. 6-d and Fig. 6-c for $q=0.35$. See raw data at several q values in Fig. SM-12, Ref.¹ and compare with the [110] direction in Fig. SM-13, Ref.¹). Also, the additional TA” branch apparently splits into two branches at $q\ge 0.3$ (see the raw data of Fig. 6b at $Q=(2,0,0.3$)).

The data collected above $T_{co}$ point to an “apparent” incommensurate size $a'$ in the 2D metallic regime scaled with respect to the lattice parameter a as $a' = (0.25/0.3)a \approx 0.83a$, with the corresponding zone boundary $q'_{zb}=(0.25/0.3)^{-1}q_{zb}$ of a fictitious Brillouin zone. With this new zone boundary $q'_{zb}=0.5$ equivalent to $q_{zb}=0.6$, the observed concave dispersion curve of LA’(q) makes sense if extended up to this zone boundary, as shown by the horizontal dotted line in Fig. 5b. It reveals an inflection point in the middle of the [0.25–0.5] range of $q'$ (or the [0.3–0.6] range of q) and therefore a new elastic force at the interface between two adjacent domains of size $2a'$. As for the magnetic excitations where the same value $q_{min}=0.3$ is observed⁴² we attribute this discommensuration to the varying sizes of the hole-poor domains, lattice-locked, intertwined with the hole-rich domains which maintain their size 4a. These observations provide a full characterization of hole-rich bipolarons of size 4a, resulting from the structural and antiferromagnetic pairings of hole-rich orbital polarons of size 2a in a metallic state.

In contrast to LA, the branch TA of the effective medium is still observed at any value of q. There, the two additional branches TA’(q) and TA”(q) observed above and below the branch TA with $q_{min}$=0.15 can be attributed to the two expected transverse excitations, “out-of-plane” and “in-plane”, of the bipolarons. A splitting of this latter branch occurs on the scale $q \ge 0.3$ of the polaron, showing that the transverse coupling does not play the same role as the longitudinal coupling in the pairing of the polarons. A similar anomaly is observed in the first optical branch reported in Fig. 5b which corresponds to the vibrations of the cubes of the La/Sr atoms with respect to those of the Mn sites, obtained by translation of the vector (1/2, 1/2, 1/2)a. By the way, this latter anomaly provides the thickness value a of the bipolarons along the c axis. These incommensurate values of $q_{lim}$ being the same as those of the LA’ branch, they characterize the same bipolarons of size 4a along the two MnO bond directions of the planes.

Actually the best proof of the in-plane $q=(1/8,1/8)$ structure of the bipolarons observed at $T<T_{co}$ appears along the diagonal directions [110] + [101] + [011] of the pseudocubic structure reported in Fig. 1b. There a strong anomaly is observed between the branches TA and LA, common to the twin domains, at the value $q=0.125$ outlined by a rectangle in this figure (see raw data in Fig. 1c). At this value of q, the LA and TA excitations exhibit enhanced intensities and shifted energies, corresponding to a tendency to a crossing behavior. This anomaly reveals a periodic modulation of the charge-phonon coupling at $q=(1/8,1/8,0)$ and therefore appears as a precursor state of the in-plane structure of the ordered bipolarons.

We notice that the spatial characteristics (size, distance) of the bipolarons obtained at $x=1/8$ in the basal planes are the same as those of the ferromagnetic “platelets” observed at $x=0.06$ in a disordered state, static or quenched^44,45. The main difference between the two doping values $x=0.06$ and $x=1/8$ appears along the axis c where the thickness value obtained at $x=0.06$ (7Å $\approx 2a$) is twice as great as its value, a, at $x=1/8$⁴⁵ leading to the common origin of hole-rich bipolarons at the two doping concentrations.

In conclusion of this section, the additional branches of acoustic phonons observed in the range of large values of q ($q_{lim}\ge 0.125$) at $T<T_{co}$ along the directions of the MnO bonds agree with the existence of large cubes 4ax4ax4a resulting from the ordering of orbital polarons 2ax2ax2a. Corresponding magnetic excitations that explicitly visualize hole-poor domains reveal the hole-rich character of the orbital polarons (2a), their F coupling in forming the domains of size 4a, and their intertwining with the hole-poor domains along the two MnO bond directions of the planes (period 8a). They determine a chessboard of hole-rich domains 4a x 4a in each (a, b) plane with alternation of hole-rich and hole-poor domains along the c axis. The full 3D charge ordered state is characterized by two families of same cubes of octopolarons of side 4a, obtained by considering the magnetic excitations also along the diagonals [111]. This superstructure of charges is intertwined with CDW of collective charge origin indicated by gaps in the $Dq^2$ law of the spin waves. The bipolaronic origin of the domains 4a is provided by the dispersion with q of the additional lattice and magnetic excitations observed in the 2D metallic state. As discussed in layered cuprates concerning the CDW, the apparent incommensurate size of the bipolarons (~0.83x4a what corresponds to $q_{lim}=0.15$), can be attributed to their intertwining with hole-poor domains of various sizes, lattice locked. These observations go beyond the pictures obtained from previous work that describe the charge inhomogeneous state either as a superstructure of polarons⁴, a transition between two orbital structures⁵, or a Peierls transition¹⁰.

Discussion: a model for high $T_c$ cuprates

The advantage of the magnetic excitations in $\hbox {La}_{7/8}\hbox {Sr}_{1/8}\hbox {MnO}_3$ for getting the local magnetic correlations, charge-induced, along several symmetry directions is lost in cuprates in which the spins exhibit an antiferromagnetic structure. Moreover, the collective 3D character of these local excitations required for their observations cannot be expected in the layered structure that is necessary to obtain superconductivity. The present observations in $\hbox {La}_{7/8}\hbox {Sr}_{1/8}\hbox {MnO}_3$ therefore provide a unique opportunity to obtain a realistic model of bipolarons in the $\hbox {CuO}$ planes of cuprates that share the same structure of hybridized orbitals $p-d$ as in the $\hbox {MnO}$ planes.

Several studies have outlined similarities in the anomaly of the oxygen bond stretching mode observed at $E\approx 60$ meV^{30,46,47,48,49,50} and on Fermi surfaces^{51,52,53,54,55} in which a strong charge-phonon coupling has been observed and possible local lattice distortions⁵⁶. The direct correspondence between the nodal, anti-nodal dichotomy observed at high energy in cuprates and the present low excitations of bipolarons has been outlined in Section II-I. In $\hbox {La}_{7/8}\hbox {Sr}_{1/8}\hbox {MnO}_3$, the minimization of the Coulomb energy is obtained by the alternation along the c axis between the “hole rich” and “hole poor” domains of the chessboards. The same should be true for the biaxial structure of stripes in which the bipolarons should exhibit a linear size 4a. Therefore, testing the present model of bipolarons at $x=1/8$ in the $\hbox {CuO}$ planes implies consideration of a 2D superstructure of stripes of “hole-rich” bipolarons of linear size 4a in the plane, distant by 2a along the stripe (one orbital polaron) intertwined with stripes of AF-arranged spins, “hole-poor”, of same width 4a as sketched in Fig. 7a. The charge density, which is known to lie mainly on oxygen sites, is sketched by large “on-site” circles for simplicity. The size 4a of the hole-rich bipolaron that determines the superconducting coherence length $\xi =1.5$ nm is twice that of the Jahn-Teller bipolarons (2a) previously considered^57,58. At $x=1/8$, the stripes of the bipolarons should induce a modulation of the AF spin structure that leads to four superstructure peaks of scattering intensity in the wave vector $\delta$(x=1/8)=1/8, along [100] and [010] away from the position of the AF peak $(\pi ,\pi )$ or (1/2, 1/2) in our notation. As discussed for Jahn-Teller bipolarons⁵⁸, the coherence of the AF spin structure across the stripes of bipolarons could be preserved thanks to the AF pairing (singlet state) of the bipolarons.

Fig. 7

Model of ordered state of bipolarons in some high $T_c$cuprates. (a) At $x=1/8$: the stripes of hole-rich bipolarons of size $\approx 4a$ are intertwined with stripes of AF spins (hole-poor) of same size $\approx 4a$ along the CuO bond directions. For simplicity, the charge density is sketched by a large circle on the Cu center of the polarons instead of on the four surrounding oxygen atoms. The CDW of period 4a is sketched by a green line. (b) Alternation of the hole-rich (bipolarons) and hole-poor (AF spins) stripes every two planes that can be related to a short range period of four layers or 2c parameters for the CDW along c as observed in $\hbox {La}_{2-x}\hbox {Ba}_x\hbox {CuO}_4$, from Ref.²⁴. (c) Left hand size: co-existence between the two types of stripes structure of bipolarons, namely along x or y, with an alternation along $\textbf{c}$ (blue and red lines as at $x\ge 0.055$), and along the diagonal $x+y$ (cyan lines as at $x\le 0.05$). In the latter case ($x<0.055$), zig-zag chains of bipolarons are suggested by us.

In common theories, 2D bipolarons are expected to form a narrow band and, being bosons, to condense into a superconducting state in a small band filling, following the $T_c\propto n_b(T)/m^*$ relation where $m^*$ is the effective mass in the plane^{59,60,61,62,63,64,65}. The density of bipolarons $n_b(T)$ is believed to occur in the pseudo-gap phase and to reach the form $n_b\propto x$ at the $T_c$ value⁶³ in agreement with the universal law for $T_c$ determined by muon relaxation experiments⁶⁶.

The consequences of this model of bipolarons in the dependencies on doping of i) CDW(x), SDW(x), $T_c$(x), ii) on the downward dispersion of the hourglass spectrum and iii) in the extension of this model to low doping are considered successively, specifically for the “214” family, which exhibits the simplest structure.

i) In the inhomogeneous spin-charge model sketched in Fig. 7a, the spin density wave (SDW) $\delta$=1/8 results from the intertwining of the AF-arranged spin stripes and the stripes of hole-rich bipolarons with the same width 4a where the involved AF spin correlations keep in phase. In fact, the appearance of a CDW accompanied by lattice distortion is another possible consequence of hole doping with charge-lattice coupling, with a collective charge origin⁵⁹. Therefore, the universal “1/8” anomaly could correspond to the peculiar situation that occurs when the size of the hole-poor domain intertwined with the bipolarons becomes equal to the size of the hole-rich bipolarons (4a). As in Sect. "The additional branch of phonon excitations in the 3D ferromagnetic state (T < T_co)" for $\hbox {La}_{7/8}\hbox {Sr}_{1/8}\hbox {MnO}_3$, this situation allows the stabilization of a long-range CDW with $q_{CDW}$ = 1/4 rlu. This modulated charge density expected to be coupled to lattice distortion, intertwined with the localized effect of the charges or bipolarons, is another consequence predicted from charge phonon coupling⁵⁹. It should favor long-time and long-range spatial correlations, leading to a decrease of the bulk value $T_c$^{14,15,17,67,68}.

The most spectacular effect is observed in the LTT $\hbox {La}_{7/8}\hbox {Ba}_{1/8}\hbox {CuO}_4$¹⁸ where the value of $T_c$ measured by the Meissner effect drops to $\approx 4K$ and increases when a disorder is introduced^{24,67,69,70,71}. The observation of a period of four layers (two lattice parameters) along the $\textbf{c}$ axis for the CDW agrees with the existence of hole-rich stripes intertwined with hole-poor stripes in the planes²⁴. As sketched in Fig. 7b, the alternation required to minimize the Coulomb energy along the c axis includes every two planes (1+3, 2+4 etc. as shown in the Fig. 7b), with parallel and shifted stripes. In this compound, the variation with temperature of $q_{CDW}$ from $q_{CDW}$ = 0.235 rlu, long range, at low temperature to $q_{CDW}$=0.3 rlu, short range, at higher temperature⁴⁹, recalls the temperature variation of the $q_{min}$ value of the lattice and magnetic excitations of bipolarons reported in the present study of $\hbox {La}_{7/8}\hbox {Sr}_{1/8}\hbox {MnO}_3$ (see Sect. "Magnetic excitations in the 3D ferromagnetic state (T < T_co) and in the 2D metallic state (T < T_co) in the hole-rich, hole-poor model") outlining the relationship between the wave vector $q_{CDW}$ interpreted in direct space⁴³ and the scale 4a of the bipolarons.

Due to their intertwining, the ordering of hole-rich bipolarons at $x=1/8$ also optimizes the correlations of the SDW. This is observed in $\hbox {La}_{2-x}\hbox {Sr}_x\hbox {CuO}_4$ where, at $x=1/8$, a long-range static component of the SDW is observed which coexists with a dynamic short-range component¹⁴. A similar effect is obtained by applying a magnetic field⁷². At $x=1/8$, this intertwining yields the relation $\delta =q_{CDW}/2$. The present model of ordered stripes of bipolarons (period 8a) stabilized by a CDW of period 4a may provide a physical origin of the $q=1/8$ rlu charge modulation observed in the vortex halo of Bi2212 cuprate compound by applying a magnetic field⁷³ what was interpreted as a pair density wave²⁷. The intertwining between the CDW of period 4a, the SDW of period 8a and the presumed order of the PDW with a specific scale 8a has also been established in $\hbox {La}_{2-x}\hbox {Sr}_x\hbox {CuO}_4$ doped with Fe where an SDW coexists with the CDW⁷⁴.

ii) Turning to the hourglass magnetic spectrum, we recall that the magnetic fluctuations of the low energy downward dispersion curve which departs from the (1/2$\pm$ $\delta$,1/2$\pm$ $\delta$) magnetic peaks are strongly anisotropic^75,76,77. The downward dispersion reveals therefore a new coupling between the AF spin stripes arising from a narrow band of bipolarons in the metallic and superconducting state with a long range character. It competes with the local magnetic coupling of super-exchange origin inside the hole-poor stripes of AF spins with corresponding dispersion curve of energy shifted up at higher energy²⁶ above $E_{cross}$. This long range coupling between spins should be responsible for the disappearance, at $x\approx 0.03$⁷⁸, of the AF peak at ($\pi ,\pi$) characteristic of the undoped parent compound. The magnetic energy gap that opens at the bottom of the downward dispersion curve for $T<T_c$^14,26 is consistent with a coupling between the spin and charge order parameters in the superconducting state.

iii) The present model of stripes of bipolarons can be extended to low doping as follows. The parameter $\delta(x)$ corresponds to the periodic distance $d=a/\delta$ between the stripes of bipolarons. The linear variation of $\delta(x)$ approximately observed in $\hbox {YBa}_2\hbox {Cu}_3\hbox {O}_{6+x}$ and well obeyed in $\hbox {La}_{2-x}\hbox {Sr}_x\hbox {CuO}_4$ and $\hbox {Bi}_{2+x}\hbox {Sr}_{2-x}\hbox {CuO}_6$^20,79,80 therefore corresponds to the decrease of this periodic distance from infinity at $x=0$ to 8a at $x=1/8$, as the planes are progressively filled by stripes of bipolarons with doping. Consequently, the band of bipolarons intertwined with hole-poor domains is filled linearly with bipolarons by doping, leading to the two linear relations $\delta(x)=x$ and $T_c \propto \delta$²⁰.

Actually, the relation $\delta (x)= x$ holds even at $x<0.055$ where the SDW rotates along one diagonal direction (with $\delta$ expressed in rlu units) and allows a regime of co-existence of the two types of direction^79,80. In the present model, the rotation corresponds to a decrease in $\delta$ by the factor 1/$\sqrt{2}$ in Å units. It corresponds to an increase of the periodic distance of the stripes of bipolarons by the factor $\sqrt{2}$ so that bulk metallic and superconducting properties disappear. The coexistence of two regimes for the stripes, straight (along the CuO bond) and diagonal, corresponds to the situation sketched in Fig. 7c, left-hand side, where the stripes of bipolarons that run along $(x+y)$ are nucleated at the crossing points between the stripes that run along x and along y. The continuity of the observations shown by the ARPES experiments along the nodal and antinodal directions⁵¹ has led to the proposal of a “staircase” for the CDW⁸¹. In the present picture, the continuity suggests rather a zigzag chain of bipolarons (see the sketch in Fig. 7c) where one charge is shared by two bipolarons as in the chessboard of $\hbox {La}_{7/8}\hbox {Sr}_{1/8}\hbox {MnO}_3$. In the limit of the large concentrations, the change in behavior observed at $x\approx 0.16-0.17$ in cuprates⁸² as in $\hbox {La}_{1-x}\hbox {Sr}_x\hbox {MnO}_3$ means that the present hole-rich, hole-poor model is no longer valid. This conclusion agrees with the existence of a minimal distance of approach between bipolarons as previously observed in a Ca-doped manganite with $x=0.08$⁴⁴.

In summary, the local excitations of the lattice and magnetic origins observed in pseudocubic $\hbox {La}_{7/8}\hbox {Sr}_{1/8}\hbox {MnO}_3$ reveal that the structural, magnetic and electric transition observed at $T=T_{co}$ can be interpreted as a transition from hole-rich bipolarons, AF paired, with a size 4a intertwined with hole-poor domains of varying size at $T>T_{co}$ towards a 3D charge-ordered state. In the planes, this later consists of a chessboard of F-coupled hole-rich bipolarons 4ax4a intertwined with hole-poor domains of the same size, leading to a period 8a along the two MnO bond directions of the planes. This chessboard of bipolarons is intertwined with CDW arising from a collective charge effect detected by gaps in the $Dq^2$ laws. We believe that the picture of the 2D metallic state is valid for the metallic $\hbox {CuO}$ planes of cuprates that share the same hybridized $p-d$ orbital structure. The layered structure implies replacing the chessboard of bipolaron by stripes of bipolarons, so that one gets stripes of hole-rich bipolarons intertwinned with AF spin stripes, hole-poor, rotated by 90 degrees from plane to plane along the c axis. This hole-rich, hole-poor model which results from the competition between attractive and repulsive forces on the charges makes possible a large charge-phonon coupling, necessary to obtain a high critical temperature. As in manganites, the universal “1/8” anomaly should correspond to the peculiar situation where the hole-poor domains intertwined with the hole-rich bipolarons reach the size $\approx 4a$ of the bipolarons. In this way, their fluctuations are reduced by the stabilization of a long-range CDW with wave vector $q_{CDW}=1/4$ rlu that coexists with bipolarons of size 4a, leading to their ordering with a period 8a revealed by a spin density wave of the same period. This model, consistent with the universal law for $T_c$⁶⁶, can be extended to low doping, leading to Yamada’s law $\delta (x)=x$ and $T_c\propto \delta$ observed in $\hbox {La}_{2-x}\hbox {Sr}_x\hbox {CuO}_4$. Our model of bipolarons advanced in the present paper for the 214 family with ordered stripes should be also valid for highly disordered cuprates such as Bi-based compound Bi2212⁴³. The existence of a 4a scale in the whole pseudo-gap phase of this Bi-based cuprate with a filling process observed by direct imaging⁴³ reinforces our confidence in the reliability and general significance of our model.

Although the origin of the pairing is still debated in cuprates, there exists an approach that the pairing occurs in direct space and that the bosons fill the space by doping. The magnetic fluctuations pairing mechanism of superconductivity in cuprates has been long considered as the most probable contender. Nevertheless up to now this mechanism has not been unanimously and undoubtedly approved as the only valid candidate. Our approach for lattice-mediated pairing of pre-formed charge objects (bipolarons) does not exclude other mechanisms of superconducting pairing such as spin-fluctuation-mediated interactions but may play its role in one row with the others.

After finishing this manuscript, we have learned of a direct imaging experiment at very low doping in a Bismuth-based cuprate family⁸³. The authors show the existence of 4ax4a plaquettes the number of which increases with doping. However, they also reveal the existence of an internal anisotropic texture of charge density on oxygen sites inside the large plaquette 4ax4a and its role for setting the direction of the stripe. Their observation means that the stripes consist of 4ax4a domains aligned in a row (see dashed lines in Fig. 7a) in place of anisotropic domains 4ax2a. This outlines the role of the size 4a characteristic to the present bipolaronic model and the periodicity 8a inherent to doping x=1/8. As commented by Zaanen⁸⁴, we are faced with complex effective pairs, that certainly cannot be fully resolved by a unique scale of superconducting pairing but likely by two distinct scales. It cannot be excluded that the 4a scale given by copper sites (plaquettes) and a 2a scale represented by oxygen sites compose a tentative platform for realisation of this approach.

Methods

Sample preparation

A single crystal of $\hbox {La}_{7/8}\hbox {Sr}_{1/8}\hbox {MnO}_3$ was grown by the ICMMO “Institut de Chimie moleculaire et des materiaux d’Orsay” at Orsay University (France). The single crystal, twinned, used for inelastic neutron experiments is a cylinder with 40 mm of height and 4 mm of diameter. Complementary measurements of ac susceptibility, resistivity, lattice parameters and oxygen positions as a function of temperature have been previously reported in Ref.¹³. At $T=T_{co}$, the structural phase transition evolves from a monoclinic structure ($T>T_{co}$) to a triclinic structure ($T<T_{co}$), both structures being close to the orthorhombic one.

Inelastic neutron scattering experiments

The excitations in $\hbox {La}_{1-x}\hbox {Sr}_x\hbox {MnO}_3$ at $x=1/8$ were measured by inelastic neutron scattering at the three-axis neutron spectrometers (TAS) 4 F installed at the cold neutron source and 2 T at the thermal neutron source of the Laboratoire Léon Brillouin (Orphée reactor, Centre d’Etudes de Saclay, France) and the thermal neutron TAS-IN8 (Thermes) at the Institut Laue-Langevin (Grenoble, France)⁸⁵. The spectrometers largely used open geometry with focussing pyrolytic graphite analysers and monochromators (reflection PG002 at 4F, 2T, IN8) and a silicon monochromator (reflection Si111 at IN8). Depending on the studied energy range, filters of higher monochromatic harmonics were installed in the scattered beam (polycrystalline Be-filter at 4F, oriented PG filters at 2T, IN8).