Nataliya Kalashnyk

Michel Daher Mansour

Joffrey Pijeat

Reḿi Plamont

Xavier Bouju

Teodor Silviu Balaban

Steṕhane Campidelli

Laurence Masson

Sylvain Clair
email: sylvain.clair@univ-amu.fr

Michel Daher

Teodor Silviu

Edge-On Self-Assembly of Tetra-bromoanthracenyl-porphyrin on Silver Surfaces

Molecular self-assembly on surfaces is driven by the range of interactions between the molecules themselves and the substrate. Generally, a face-on structure is favored for aromatic molecules lying !at on the surface. Here, we report on the supramolecular self-assembly of 5,10,15,20-tetrakis(10-bromoanthracen-9-yl)porphyrin on the Ag(111) and Ag(110) surfaces. Well-ordered molecular chains were observed by room-temperature scanning tunneling microscopy on both surfaces. The relatively small size of the unit cell revealed an edge-on con"guration of the porphyrin macrocycles, that is, perpendicular to the surface plane, as confirmed by molecular mechanics calculations. Distinct intermolecular interactions were found on the two surfaces, providing different molecular chain orientations on Ag(111) and Ag(110).

■ INTRODUCTION

The control on the structure of supramolecular architectures on a surface is of prime importance to steer their properties. [1][2][3] For two-dimensional (2D) self-assemblies in a monolayer or submonolayer regime, conformational adaptation is often produced as a response to various stimuli and a transition from 2D planar to three-dimensional (3D) reticular geometry can be observed. [4][5][6][7] Porphyrins are a widely studied class of molecules exhibiting promising electronic, optical, magnetic, or catalytic properties with large interest in materials science for many applications. [START_REF] Smykalla | Metallo)porphyrins for potential materials science applications[END_REF][START_REF] Kuzmin | Structures and Properties of Porphyrin-Based Film Materials Part I. The Films Obtained via Vapor-Assisted Methods[END_REF][START_REF] Chilukuri | Properties, and Reactivity of Porphyrins on Surfaces and Nanostructures with Periodic DFT Calculations[END_REF][START_REF] Diercks | Reticular Electronic Tuning of Porphyrin Active Sites in Covalent Organic Frameworks for Electrocatalytic Carbon Dioxide Reduction[END_REF] Porphyrin-based thin films deposited on a solid surface can be easily processed and investigated at the molecular level by scanning probe microscopy techniques. [START_REF] Gottfried | Surface Chemistry of Porphyrins and Phthalocyanines[END_REF][START_REF] Auwarter | Porphyrins at Interfaces[END_REF][START_REF] Otsuki | STM Studies on Porphyrins[END_REF] They most generally adopt a face-on conformation to optimize their adsorption on a metal or a HOPG surface, but an edge-on geometry can also be observed because of the presence and the size of side groups, the nature of the solvent, or the concentration. [START_REF] El Garah | Molecular Design Driving Tetraporphyrin Self-Assembly on Graphite: a Joint STM, Electrochemical and Computational Study[END_REF][START_REF] Adachi | The Polymorphism of Porphyrin 2D Assemblies at the Liquid-Graphite Interface: the Effect of a Polar Solvent Additive and a Flexible Spacer on the Face-On and[END_REF][START_REF] Sakano | Chronological Change from Face-On to Edge-On Ordering of Zinc-Tetraphenylporphyrin at the Phenyloctane-Highly Oriented Pyrolytic Graphite Interface[END_REF][START_REF] Otsuki | Face-on and Columnar Porphyrin Assemblies at Solid/Liquid Interface on HOPG[END_REF][START_REF] Trelka | Surface Assembly of Porphyrin Nanorods with One-Dimensional Zinc-Oxygen Spinal Cords[END_REF][START_REF] Zhou | Observation of Co-Existence of 'Face-On' and 'Edge-On' Stacking Styles in a Porphyrin Monolayer[END_REF] The edge-on self-assembly is particularly attractive because it allows for charge migration along the columnar structure, [START_REF] Schouten | Charge Migration in Supramolecular Stacks of Peripherally Substituted Porphyrins[END_REF] thus providing particular mesoscopic properties appealing for molecular electronics. More elaborated strategies based on the use of a self-assembled monolayer (SAM) as a support have also been implemented to gain advanced control on the self-assembly of porphyrins. [START_REF] Lombana | In-Solution Patterning of Standing Up Porphyrin Based Nanostructures within Hydrogen Bonded Porous Networks -a Structural Effect of a Host Matrix on Guest Entities[END_REF][START_REF] Otte | Ordered Monolayers of Free-Standing Porphyrins on Gold[END_REF] Tetra-anthracenylporphyrin (TAP) derivatives were first studied for their particular bis-facially blocked geometries because of the perpendicular anthracenyl groups and were tested as catalysts for the reversible coordination of oxygen, [START_REF] Cense | Oxygenation d'une ferroporphyrine encombree sur ses deux faces, la ferromeso-tetra-anthracenyl phorphine[END_REF] the oxidation of PAHs, [START_REF] Harden | Biomimetic iron porphyrincatalysed oxidation of cyclopenta [a] phenanthrenones[END_REF] or as active molecules against HIV infection. [START_REF] Vzorov | Inactivation of human immunodeficiency virus type 1 by porphyrins[END_REF] From a synthetic point of view, the formation of TAP derivatives is a challenge. [START_REF] Sooambar | Synthesis, photophysical, electrochemical, and electrochemiluminescent properties of 5,15-bis(9-anthracenyl)porphyrin derivatives[END_REF][START_REF] Davis | Bis-Anthracene Fused Porphyrins: Synthesis, Crystal Structure, and Near-IR Absorption[END_REF][29] TAP was first synthetized by Cense et al. [START_REF] Cense | Oxygenation d'une ferroporphyrine encombree sur ses deux faces, la ferromeso-tetra-anthracenyl phorphine[END_REF] in 1979 following the method developed by Adler and Longo and the porphyrin was isolated in 0.2% yield. [START_REF] Adler | Mechanistic Investigations of Porphyrin Syntheses. I. Preliminary Studies on ms-Tetraphenylporphin[END_REF] Thereafter, attempts to synthesize TAP using the Lindsey strategy [START_REF] Tohara | An improved synthesis of mesotetraanthryiporphyrin by a kinetically controlled Lindsey reaction[END_REF] or the reaction in inorganic acid [START_REF] Kumar | One-pot general synthesis of metalloporphyrins[END_REF] were reported and barely reached 3% yield. In 1985, Volz and Schaeffer reported a new synthetic method based on the formation of a pyrrole-carbinol intermediate, which cyclotetramerized in propionic acid to form TAP in 6% yield. 33 The strategy of Volz and Schaeffer was also applied by the group of Anderson in 2010 to synthetize a tetra-dimesityloxyanthracenylporphyrin in 10% yield and by Filatov et al. [START_REF] Filatov | Reversible oxygen addition on a triplet sensitizer molecule: protection from excited state depopulation[END_REF] in 2015 to synthesize a tetraphenylanthracenyl porphyrin in 15% yield. The symmetry matching between the anthracene moieties and the macrocycle of porphyrins was exploited by the group of Anderson to form a π-extended porphyrin exhibiting a maximum absorption around 1400 nm after the fusion of the four dimesityloxyanthracene groups on the porphyrin core. [START_REF] Davis | A Porphyrin Fused to Four Anthracenes[END_REF] Recently, we reported the synthesis of 5,10,15,20-tetrakis(10bromoanthracen-9-yl)porphyrin (TBAP, Figure 1a and synthetic details in Supporting Information Figure S1) and we demonstrated its successful reactivity in Suzuki-Miyaura cross-coupling reactions. [START_REF] Pijeat | Synthesis and Suzuki-Miyaura Cross Coupling Reactions for Post-Synthetic Modification of a Tetrabromo-Anthra[END_REF]

■ INTRODUCTION

The control on the structure of supramolecular architectures on a surface is of prime importance to steer their properties. [1][2][3] For two-dimensional (2D) self-assemblies in a monolayer or submonolayer regime, conformational adaptation is often produced as a response to various stimuli and a transition from 2D planar to three-dimensional (3D) reticular geometry can be observed. [4][5][6][7] Porphyrins are a widely studied class of molecules exhibiting promising electronic, optical, magnetic, or catalytic properties with large interest in materials science for many applications. [START_REF] Smykalla | Metallo)porphyrins for potential materials science applications[END_REF][START_REF] Kuzmin | Structures and Properties of Porphyrin-Based Film Materials Part I. The Films Obtained via Vapor-Assisted Methods[END_REF][START_REF] Chilukuri | Properties, and Reactivity of Porphyrins on Surfaces and Nanostructures with Periodic DFT Calculations[END_REF][START_REF] Diercks | Reticular Electronic Tuning of Porphyrin Active Sites in Covalent Organic Frameworks for Electrocatalytic Carbon Dioxide Reduction[END_REF] Porphyrin-based thin films deposited on a solid surface can be easily processed and investigated at the molecular level by scanning probe microscopy techniques. [START_REF] Gottfried | Surface Chemistry of Porphyrins and Phthalocyanines[END_REF][START_REF] Auwarter | Porphyrins at Interfaces[END_REF][START_REF] Otsuki | STM Studies on Porphyrins[END_REF] They most generally adopt a face-on conformation to optimize their adsorption on a metal or a HOPG surface, but an edge-on geometry can also be observed because of the presence and the size of side groups, the nature of the solvent, or the concentration. [START_REF] El Garah | Molecular Design Driving Tetraporphyrin Self-Assembly on Graphite: a Joint STM, Electrochemical and Computational Study[END_REF][START_REF] Adachi | The Polymorphism of Porphyrin 2D Assemblies at the Liquid-Graphite Interface: the Effect of a Polar Solvent Additive and a Flexible Spacer on the Face-On and[END_REF][START_REF] Sakano | Chronological Change from Face-On to Edge-On Ordering of Zinc-Tetraphenylporphyrin at the Phenyloctane-Highly Oriented Pyrolytic Graphite Interface[END_REF][START_REF] Otsuki | Face-on and Columnar Porphyrin Assemblies at Solid/Liquid Interface on HOPG[END_REF][START_REF] Trelka | Surface Assembly of Porphyrin Nanorods with One-Dimensional Zinc-Oxygen Spinal Cords[END_REF][START_REF] Zhou | Observation of Co-Existence of 'Face-On' and 'Edge-On' Stacking Styles in a Porphyrin Monolayer[END_REF] The edge-on self-assembly is particularly attractive because it allows for charge migration along the columnar structure, [START_REF] Schouten | Charge Migration in Supramolecular Stacks of Peripherally Substituted Porphyrins[END_REF] thus providing particular mesoscopic properties appealing for molecular electronics. More elaborated strategies based on the use of a self-assembled monolayer (SAM) as a support have also been implemented to gain advanced control on the self-assembly of porphyrins. [START_REF] Lombana | In-Solution Patterning of Standing Up Porphyrin Based Nanostructures within Hydrogen Bonded Porous Networks -a Structural Effect of a Host Matrix on Guest Entities[END_REF][START_REF] Otte | Ordered Monolayers of Free-Standing Porphyrins on Gold[END_REF] Tetra-anthracenylporphyrin (TAP) derivatives were first studied for their particular bis-facially blocked geometries because of the perpendicular anthracenyl groups and were tested as catalysts for the reversible coordination of oxygen, [START_REF] Cense | Oxygenation d'une ferroporphyrine encombree sur ses deux faces, la ferromeso-tetra-anthracenyl phorphine[END_REF] the oxidation of PAHs, [START_REF] Harden | Biomimetic iron porphyrincatalysed oxidation of cyclopenta [a] phenanthrenones[END_REF] or as active molecules against HIV infection. [START_REF] Vzorov | Inactivation of human immunodeficiency virus type 1 by porphyrins[END_REF] From a synthetic point of view, the formation of TAP derivatives is a challen by Cense et al. [START_REF] Cense | Oxygenation d'une ferroporphyrine encombree sur ses deux faces, la ferromeso-tetra-anthracenyl phorphine[END_REF] in 1979 fo Adler and Longo and the yield. [START_REF] Adler | Mechanistic Investigations of Porphyrin Syntheses. I. Preliminary Studies on ms-Tetraphenylporphin[END_REF] Thereafter, attemp Lindsey strategy [START_REF] Tohara | An improved synthesis of mesotetraanthryiporphyrin by a kinetically controlled Lindsey reaction[END_REF] or the r reported and barely reache Schaeffer reported a new formation of a pyrrole-car tetramerized in propionic ac strategy of Volz and Schaeffe Anderson in 2010 to synthe nylporphyrin in 10% yield a synthesize a tetraphenylanth The symmetry matching bet the macrocycle of porphyrin Anderson to form a π-e maximum absorption aroun four dimesityloxyanthracene Recently, we reported the s bromoanthracen-9-yl)porp synthetic details in Suppor we demonstrated its succes cross-coupling reactions. [START_REF] Pijeat | Synthesis and Suzuki-Miyaura Cross Coupling Reactions for Post-Synthetic Modification of a Tetrabromo-Anthra[END_REF]

■ CONCLUSIONS

We studied the self-assembly of TBAP molecules on the Ag(111) and Ag(110) surfaces. The molecules adopt an edgeon configuration and form extended 2D domains of highly ordered chains after annealing. The formation of the chains originates from two distinct processes on theses surfaces. On Ag(111), a columnar structure is formed through π-π interaction between the anthracenyl groups. On Ag(110), the chains are formed through lateral interaction between the molecules, possibly aided by the formation of metal-organic complexes. The TBAP precursor can adopt different 3D configurations because of the many degrees of liberty gained by the frustrated rotation of the four bromoanthracenyl groups. As a result, radically different intermolecular interactions can take place, leading to important modifications in the supramolecular organization. Accordingly, noticeable variations in their overall properties can be expected. [START_REF] Smykalla | Metallo)porphyrins for potential materials science applications[END_REF][START_REF] Diercks | Reticular Electronic Tuning of Porphyrin Active Sites in Covalent Organic Frameworks for Electrocatalytic Carbon Dioxide Reduction[END_REF] Our results should provide inspiration for a guided and modular complex 2D self-assembly on a surface by taking use of a single intrinsically threedimensional functional precursor. TAP derivatives is a challenge. [START_REF] Sooambar | Synthesis, photophysical, electrochemical, and electrochemiluminescent properties of 5,15-bis(9-anthracenyl)porphyrin derivatives[END_REF][START_REF] Davis | Bis-Anthracene Fused Porphyrins: Synthesis, Crystal Structure, and Near-IR Absorption[END_REF][29] TAP was first synthetized by Cense et al. [START_REF] Cense | Oxygenation d'une ferroporphyrine encombree sur ses deux faces, la ferromeso-tetra-anthracenyl phorphine[END_REF] in 1979 following the method developed by Adler and Longo and the porphyrin was isolated in 0.2% yield. [START_REF] Adler | Mechanistic Investigations of Porphyrin Syntheses. I. Preliminary Studies on ms-Tetraphenylporphin[END_REF] Thereafter, attempts to synthesize TAP using the Lindsey strategy [START_REF] Tohara | An improved synthesis of mesotetraanthryiporphyrin by a kinetically controlled Lindsey reaction[END_REF] or the reaction in inorganic acid [START_REF] Kumar | One-pot general synthesis of metalloporphyrins[END_REF] were reported and barely reached 3% yield. In 1985, Volz and Schaeffer reported a new synthetic method based on the formation of a pyrrole-carbinol intermediate, which cyclotetramerized in propionic acid to form TAP in 6% yield. 33 The strategy of Volz and Schaeffer was also applied by the group of Anderson in 2010 to synthetize a tetra-dimesityloxyanthracenylporphyrin in 10% yield and by Filatov et al. [START_REF] Filatov | Reversible oxygen addition on a triplet sensitizer molecule: protection from excited state depopulation[END_REF] in 2015 to synthesize a tetraphenylanthracenyl porphyrin in 15% yield. The symmetry matching between the anthracene moieties and the macrocycle of porphyrins was exploited by the group of Anderson to form a π-extended porphyrin exhibiting a maximum absorption around 1400 nm after the fusion of the four dimesityloxyanthracene groups on the porphyrin core. [START_REF] Davis | A Porphyrin Fused to Four Anthracenes[END_REF] Recently, we reported the synthesis of 5,10,15,20-tetrakis(10bromoanthracen-9-yl)porphyrin (TBAP, Figure 1a and synthetic details in Supporting Information Figure S1) and we demonstrated its successful reactivity in Suzuki-Miyaura cross-coupling reactions. [START_REF] Pijeat | Synthesis and Suzuki-Miyaura Cross Coupling Reactions for Post-Synthetic Modification of a Tetrabromo-Anthra[END_REF] Here, we tackled the adsorption of TBAP on silver surfaces with a longer-term goal to create elaborated supramolecular or covalent 2D superstructures with this original precursor. Indeed, halogenated porphyrins have recently attracted attention for their ability to create a robust nanostructure in an on-surface synthesis approach. 2,[37][START_REF] Ferreira | Supramolecular Ordering and Reactions of a Chlorophenyl Porphyrin on Ag(111)[END_REF][39] We focused on the influence of the large bromoanthracenyl groups on the selfassembly processes directed by a well-defined metal surface. More precisely, we selected the Ag(111) and Ag(110) surfaces (Figure 1b) for their mild reactivity and their two distinct crystallographic structures.

■ METHODS

The TBAP molecules were obtained from PorphyChem company (https://www.porphychem.com) or synthesized following the literature procedure (see Supporting Information, Figure S1). [START_REF] Pijeat | Synthesis and Suzuki-Miyaura Cross Coupling Reactions for Post-Synthetic Modification of a Tetrabromo-Anthra[END_REF] The STM experiments were performed in ultrahigh vacuum (UHV, base pressure in the 10 -10 mbar range) in two different experimental setups. The single-crystal Ag(111) and Ag(110) surfaces were cleaned by several cycles of Ar + bombardment, intermolecular forces (ASE with the extended Huckel the core of the ASED-MO is a powerful numerical to structural properties of vari surfaces or in the gas phas

■ RESULTS AND DISC

The deposition of TBAP leads to the formation of su ing to phase P1 as shown i straight molecular chains w orientations. Most of the c direction of the substrate periodicity inside a chain is The chains are regularly di two adjacent chains, thus th 5 substrate atomic distanc shown in the high-resoluti molecular chains have loca are stacked together and positioned chains, which a Figure 1d, aligned along [11̅ 0]-chains, the periodicit 2.5 substrate atomic distan the STM image in Figu Information, Figure S2a. T chain types suggest a uniq phase P1.

Upon annealing at 200 long molecular chains align silver substrate and extends Figure 1e). The interchain periodic unit along the ch two molecules. This can b images acquired with a pa different contrast, as sho Information Figure S3. Thi relationship of 5 substrate direction (see Supporting tentative epitaxial model). (or half of the unit ce intermolecular distance is one is ∼4% shorter. In add shifted along the [11̅ 0] dir two neighboring chains ( direction).

The small periodicity (<1 not compatible with a flat- 22138 Here, we tackled the adsorption of TBAP on silver surfaces with a longer-term goal to create elaborated supramolecular or covalent 2D superstructures with this original precursor. Indeed, halogenated porphyrins have recently attracted attention for their ability to create a robust nanostructure in an on-surface synthesis approach. 2,[37][START_REF] Ferreira | Supramolecular Ordering and Reactions of a Chlorophenyl Porphyrin on Ag(111)[END_REF][39] We focused on the influence of the large bromoanthracenyl groups on the selfassembly processes directed by a well-defined metal surface. More precisely, we selected the Ag(111) and Ag(110) surfaces (Figure 1b) for their mild reactivity and their two distinct crystallographic structures.

■ METHODS

The TBAP molecules were obtained from PorphyChem company (https://www.porphychem.com) or synthesized following the literature procedure (see Supporting Information, Figure S1). [START_REF] Pijeat | Synthesis and Suzuki-Miyaura Cross Coupling Reactions for Post-Synthetic Modification of a Tetrabromo-Anthra[END_REF] The STM experiments were performed in ultrahigh vacuum (UHV, base pressure in the 10 -10 mbar range) in two different experimental setups. The single-crystal Ag(111) and Ag(110) surfaces were cleaned by several cycles of Ar + bombardment, The STM images were acquired i typical tunneling current I T ≈ 0. ≈ ±(1-1.5) V. All images were atomically resolved images of t accuracy of ±5%. Images were WSxM [START_REF] Horcas | A software for scanning probe microscopy and a tool for nanotechnology[END_REF] and Gwyddion 41 softw Molecular mechanics calculati numerical code based on the ato delocalization (ASED-MO) me including dispersive interactions intermolecular forces (ASED+). with the extended Huckel molec the core of the ASED-MO metho is a powerful numerical tool to structural properties of various ty surfaces or in the gas phase. ■ RESULTS AND DISCUSSI Upon annealing at 200 °C, t long molecular chains aligned w silver substrate and extends up to Figure 1e). The interchain distan periodic unit along the chain di two molecules. This can be mo images acquired with a particul different contrast, as shown, Information Figure S3. This con relationship of 5 substrate atom direction (see Supporting Info tentative epitaxial model). The a (or half of the unit cell a) intermolecular distance is abou one is ∼4% shorter. In addition, shifted along the [11̅ 0] direction two neighboring chains (b vec direction).

The small periodicity (<1 nm) not compatible with a flat-lying or synthesized porting Informa-followed by annealing. The precursors were thoroughly degassed prior to deposition onto the atomically clean substrate. STM measurements were performed at room temperature with a commercial Omicron VT-STM system. The STM images were acquired in constant current mode with typical tunneling current I T ≈ 0.1-0.3 nA and sample bias V T ≈ ±(1-1.5) V. All images were subsequently calibrated using atomically resolved images of the pristine surfaces with an accuracy of ±5%. Images were partly treated with the free WSxM [START_REF] Horcas | A software for scanning probe microscopy and a tool for nanotechnology[END_REF] and Gwyddion 41 software programs.

Molecular mechanics calculations were performed using a numerical code based on the atom superposition and electron delocalization (ASED-MO) method [START_REF] Anderson | Electron density distribution functions and the ASED -MO theory[END_REF][START_REF] Bosson | Interactive quantum chemistry: A divide-and-conquer ASED-MO method[END_REF] with an extension including dispersive interactions to account for the long-range intermolecular forces (ASED+). 44 A semiempirical approach with the extended Huckel molecular orbital theory constitutes the core of the ASED-MO method. The so-called ASED+ code is a powerful numerical tool to describe reasonably well the structural properties of various types of molecules adsorbed on surfaces or in the gas phase. [45][START_REF] Villagomez | Self-assembly of enantiopure domains: The case of indigo on Cu(111)[END_REF][47]

■ RESULTS AND DISCUSSION

The deposition of TBAP on Ag(111) at room temperature leads to the formation of supramolecular domains corresponding to phase P1 as shown in Figure 1c. This phase consists in straight molecular chains with two distinct nearly orthogonal orientations. Most of the chains are aligned along the [1̅ 2̅ 3] direction of the substrate and are about 5 nm long. The periodicity inside a chain is 0.8 nm (green dots in Figure 1d). The chains are regularly distributed distanced 1.4 nm between two adjacent chains, thus the interchain spacing corresponds to 5 substrate atomic distances along the [11̅ 0] direction. As shown in the high-resolution STM image in Figure 1d, these molecular chains have locally a well-defined length, and they are stacked together and packed between perpendicularly positioned chains, which are 5 to 10 nm long (blue dots in Figure 1d, aligned along the [11̅ 0] direction). Along these [11̅ 0]-chains, the periodicity is 0.7 nm, thus corresponding to a 2.5 substrate atomic distance. A tentative epitaxial model for the STM image in Figure 1d is reported in Supporting Information, Figure S2a. The similar periodicities along both chain types suggest a unique intermolecular configuration in phase P1.

Upon annealing at 200 °C, the self-assembly evolves into long molecular chains aligned with the [11̅ 0] direction of the silver substrate and extends up to 100 nm or more (phase P2, Figure 1e). The interchain distance is 1.5 nm. In fact, here the periodic unit along the chain direction (a vector) comprises two molecules. This can be more clearly seen in some STM images acquired with a particular tip state and exhibiting a different contrast, as shown, for example, in Supporting Information Figure S3. This consistently produces an epitaxial relationship of 5 substrate atomic distances along the [11̅ 0] direction (see Supporting Information Figure S2b for a tentative epitaxial model). The average intermolecular spacing (or half of the unit cell a) is 0.7 nm and the long intermolecular distance is about 4% larger while the short one is ∼4% shorter. In addition, every second chain is slightly The STM images were acquired in const typical tunneling current I T ≈ 0.1-0.3 n ≈ ±(1-1.5) V. All images were subsequ atomically resolved images of the pris accuracy of ±5%. Images were partly WSxM [START_REF] Horcas | A software for scanning probe microscopy and a tool for nanotechnology[END_REF] and Gwyddion 41 software prog Molecular mechanics calculations we numerical code based on the atom supe delocalization (ASED-MO) method [START_REF] Anderson | Electron density distribution functions and the ASED -MO theory[END_REF]4 including dispersive interactions to acco intermolecular forces (ASED+). 44 A se with the extended Huckel molecular orb the core of the ASED-MO method. The is a powerful numerical tool to describ structural properties of various types of m surfaces or in the gas phase. [45][START_REF] Villagomez | Self-assembly of enantiopure domains: The case of indigo on Cu(111)[END_REF][47]

■ RESULTS AND DISCUSSION

The deposition of TBAP on Ag(111) leads to the formation of supramolecular ing to phase P1 as shown in Figure 1c. straight molecular chains with two dist orientations. Most of the chains are alig direction of the substrate and are ab periodicity inside a chain is 0.8 nm (gre The chains are regularly distributed dist two adjacent chains, thus the interchain s 5 substrate atomic distances along the shown in the high-resolution STM ima molecular chains have locally a well-de are stacked together and packed bet positioned chains, which are 5 to 10 n Figure 1d, aligned along the [11̅ 0] di [11̅ 0]-chains, the periodicity is 0.7 nm, th 2.5 substrate atomic distance. A tentati the STM image in Figure 1d is rep Information, Figure S2a. The similar pe including dispersive interactions to account for the long-range intermolecular forces (ASED+). 44 A semiempirical approach with the extended Huckel molecular orbital theory constitutes the core of the ASED-MO method. The so-called ASED+ code is a powerful numerical tool to describe reasonably well the structural properties of various types of molecules adsorbed on surfaces or in the gas phase. [45][START_REF] Villagomez | Self-assembly of enantiopure domains: The case of indigo on Cu(111)[END_REF][47]

■ RESULTS AND DISCUSSION

The deposition of TBAP on Ag(111) at room temperature leads to the formation of supramolecular domains corresponding to phase P1 as shown in Figure 1c. This phase consists in straight molecular chains with two distinct nearly orthogonal orientations. Most of the chains are aligned along the [1̅ 2̅ 3] direction of the substrate and are about 5 nm long. The periodicity inside a chain is 0.8 nm (green dots in Figure 1d). The chains are regularly distributed distanced 1.4 nm between two adjacent chains, thus the interchain spacing corresponds to 5 substrate atomic distances along the [11̅ 0] direction. As shown in the high-resolution STM image in Figure 1d, these molecular chains have locally a well-defined length, and they are stacked together and packed between perpendicularly positioned chains, which are 5 to 10 nm long (blue dots in Figure 1d, aligned along the [11̅ 0] direction). Along these [11̅ 0]-chains, the periodicity is 0.7 nm, thus corresponding to a 2.5 substrate atomic distance. A tentative epitaxial model for the STM image in Figure 1d is reported in Supporting Information, Figure S2a. The similar periodicities along both chain types suggest a unique intermolecular configuration in phase P1.

Upon annealing at 200 °C, the self-assembly evolves into long molecular chains aligned with the [11̅ 0] direction of the silver substrate and extends up to 100 nm or more (phase P2, Figure 1e). The interchain distance is 1.5 nm. In fact, here the periodic unit along the chain direction (a vector) comprises two molecules. This can be more clearly seen in some STM images acquired with a particular tip state and exhibiting a different contrast, as shown, for example, in Supporting Information Figure S3. This consistently produces an epitaxial relationship of 5 substrate atomic distances along the [11̅ 0] direction (see Supporting Information Figure S2b for a tentative epitaxial model). The average intermolecular spacing (or half of the unit cell a) is 0.7 nm and the long intermolecular distance is about 4% larger while the short one is ∼4% shorter. In addition, every second chain is slightly shifted along the [11̅ 0] direction so that the unit cell includes two neighboring chains (b vector, aligned with the [1̅ 1̅ 2] direction).

The small periodicity (<1 nm) along the molecular chains is not compatible with a flat-lying adsorption of the porphyrins. 2a) with a size compatible with the interchain spacing (1.5 nm) but not with the average intermolecular spacing (0.7 nm). To elucidate the supramolecular configuration of the chain structure, we performed gas-phase molecular mechanics calculations of a TBAP columnar structure for which the porphyrin macrocycle plane was set parallel for all molecules and perpendicular to the chain direction associated with the e 3 molecular axis (see Figure 2ac). In this way, the anthracenyl groups could adequately rotate to create π-stacking interaction between the molecular units. We found an average optimum distance between two neighboring porphyrin cycles of 0.68 nm, in good agreement with the experimental results. For comparison, the ide stacking distance in a columnar porphyrin crystal is clos nm, [START_REF] Castella | The first asymmetrically beta-polysubstituted porphyrin-based hexagonal columnar liquid crystal[END_REF] while a stacking distance of 0.4-0.7 nm was obse the edge-on self-assembly of other metal-free [START_REF] Adachi | The Polymorphism of Porphyrin 2D Assemblies at the Liquid-Graphite Interface: the Effect of a Polar Solvent Additive and a Flexible Spacer on the Face-On and[END_REF][START_REF] Otsuki | Face-on and Columnar Porphyrin Assemblies at Solid/Liquid Interface on HOPG[END_REF] or m porphyrin [START_REF] Sakano | Chronological Change from Face-On to Edge-On Ordering of Zinc-Tetraphenylporphyrin at the Phenyloctane-Highly Oriented Pyrolytic Graphite Interface[END_REF][START_REF] Zhou | Observation of Co-Existence of 'Face-On' and 'Edge-On' Stacking Styles in a Porphyrin Monolayer[END_REF][START_REF] Schouten | Charge Migration in Supramolecular Stacks of Peripherally Substituted Porphyrins[END_REF][START_REF] Auwarter | Assembly and Manipulation of Rotatable Cerium Porphyrinato Sandwich Complexes on a Surface[END_REF] derivatives.

The calculations revealed a tendency for the molec dimerize by locally adjusting the distance betwe porphyrin planes. As a result, within a dimer, the neigh porphyrin cores are located 0.56 nm apart. Remarkably structure, the distance between the anthracenyl planes r rather constant (0.35 nm). This is obtained thank distortion of the C-C bonds joining the anthracenyl gr the porphyrin macrocycle, inducing a slight in-plane r alternatively to the one or to the other end of the co structure (Figure 2d). The structure and the size modeled structure are thus in good agreement w experimental results on the Ag(111) surface, with the e 3 the molecule aligned with the [11̅ 0] direction, that is, th direction. Furthermore, the tendency to dimerize obse the gas-phase simulation is compatible with the fact tha are two molecules per unit cell in the supramolecular p measured by STM.

To modulate the molecule-substrate interactio performed additional deposition experiments on the Ag(110) surface. Similarly, an ordered structure was after room-temperature deposition, consisting mainly i chains, 3 or 4 nm in length, aligned along the [11̅ 0] su direction (phase P3, Figure 3a,b). The terminations chains are systematically aligned with the perpendicula direction, indicating an important interchain interaction chain endings. The Journal of Physical Chemistry C pubs.acs.org/JPCC form a square pattern compatible with the t with the average elucidate the supraucture, we performed lations of a TBAP yrin macrocycle plane endicular to the chain r axis (see Figure 2aould adequately rotate the molecular units. tance between two , in good agreement with the experimental results. For comparison, the ideal π-π stacking distance in a columnar porphyrin crystal is close to 0.4 nm, [START_REF] Castella | The first asymmetrically beta-polysubstituted porphyrin-based hexagonal columnar liquid crystal[END_REF] while a stacking distance of 0.4-0.7 nm was observed in the edge-on self-assembly of other metal-free [START_REF] Adachi | The Polymorphism of Porphyrin 2D Assemblies at the Liquid-Graphite Interface: the Effect of a Polar Solvent Additive and a Flexible Spacer on the Face-On and[END_REF][START_REF] Otsuki | Face-on and Columnar Porphyrin Assemblies at Solid/Liquid Interface on HOPG[END_REF] or metalloporphyrin [START_REF] Sakano | Chronological Change from Face-On to Edge-On Ordering of Zinc-Tetraphenylporphyrin at the Phenyloctane-Highly Oriented Pyrolytic Graphite Interface[END_REF][START_REF] Zhou | Observation of Co-Existence of 'Face-On' and 'Edge-On' Stacking Styles in a Porphyrin Monolayer[END_REF][START_REF] Schouten | Charge Migration in Supramolecular Stacks of Peripherally Substituted Porphyrins[END_REF][START_REF] Auwarter | Assembly and Manipulation of Rotatable Cerium Porphyrinato Sandwich Complexes on a Surface[END_REF] derivatives. The calculations revealed a tendency for the molecules to dimerize by locally adjusting the distance between the porphyrin planes. As a result, within a dimer, the neighboring porphyrin cores are located 0.56 nm apart. Remarkably, in this structure, the distance between the anthracenyl planes remains rather constant (0.35 nm). This is obtained thanks to a distortion of the C-C bonds joining the anthracenyl group to the porphyrin macrocycle, inducing a slight in-plane rotation alternatively to the one or to the other end of the columnar structure (Figure 2d). The structure and the size of the modeled structure are thus in good agreement with the experimental results on the Ag(111) surface, with the e 3 axis of the molecule aligned with the [11̅ 0] direction, that is, the chain direction. Furthermore, the tendency to dimerize observed in the gas-phase simulation is compatible with the fact that there are two molecules per unit cell in the supramolecular phase as measured by STM.

To modulate the molecule-substrate interaction, we performed additional deposition experiments on the open Ag(110) surface. Similarly, an ordered structure was found after room-temperature deposition, consisting mainly in short chains, 3 or 4 nm in length, aligned along the [11̅ 0] substrate direction (phase P3, Figure 3a,b). The terminations of the chains are systematically aligned with the perpendicular [001] direction, indicating an important interchain interaction at the chain endings. Upon deposition at 300 °C, the chains markedly elongate and can reach 40 nm in length (Figure 3c,d). The periodicity along the chains is 1.4 nm and corresponds to 5 substrate atomic distances along the [11̅ 0] direction. The parallel chains are located 1.2 nm apart, corresponding to 3 substrate atomic distances along the [001] direction (see Supporting Information Figure S2c for a tentative epitaxial model).

Here again, we can exclude a flat-lying configuration because of the small size of the unit cell. However, the interchain distance is significantly shorter than in P1 and P2 phases on Ag(111) (1.2 nm vs 1.4 nm), so that the columnar structure is also not compatible. Rather, the side extension of TBAP (1.44 nm, Figure 2a) is consistent with the periodicity along the chain direction (a in Figure 3d, 1.4 nm). In fact, this distance is very close to the interchain distance in phase P2 on Ag(111). We thus suggest a molecular model where the porphyrin macrocycle plane is aligned with the chain direction (with the e 3 axis of TBAP perpendicular to [11̅ 0], i.e., the chain direction). Consequently, the STM contrasts along the chains have completely different origins on the two surfaces. Figure 4 shows that the apparent contrast of the chain formation is aligned with (or perpendicular to) the porphyrin planes on Ag(110) (or Ag(111)), respectively. We excluded the possibility of a tilted configuration (with the e 3 axis neither aligned nor perpendicular to the chain direction) because of the high symmetry of the STM contrast with respect to the chain direction. The particularity of the P3/P4 phases on Ag(110) consists in the longer distance between the molecules in the columnar direction (e 3 axis) for which no favorable π-π interaction between the anthracenyl groups is allowed. If we consider the epitaxial interaction as the main driving force for the self-assembly, a denser (5 × 2) superstructure (instead of the observed (5 × 3) structure) would produce an intermolecular distance between the porphyrin planes of 0.8 nm, slightly different from what is observed in P1 and P2 (0.7 nm). It is suggested that a different adsorption configuration and a different conformation of the rotating anthracenyl groups is responsible for this behavior. Not the intermolecular interaction alo sufficiently strong so that extended formed. This is particularly demon which all chain endings are perfect short chains are found after room-te both Ag(111) and Ag(110) surfaces interaction of the molecules with elongation of the chains after annea the increase of diffusivity at higher t possibly activated by a conformatio cenyl) or chemical (loss of Br) chan After room-temperature depositio found another phase (phase P5, Fi organized domains and coexisting metastable phase disappeared up 100 °C. The structure here is still chains aligned along the [11̅ 0] direc pattern. The periodicity along the substrate atomic distances, that is 20% but the periodicity of the molecula direction remains the same, that is, 1 distances) (see Supporting Informat domains, the TBAP molecules have t P3 and P4, but they are simply posit their e 1 axis and not any more a direction. Two kinds of rhombic (labeled A and B in Figure 5), dep core shift between two adjacent chai distances along [11̅ 0]. The formation confirms the presence of substant actions along [11̅ 0] and also in symmetry directions.

Finally, it should be noted that in t of phase P3, the apparent contrast a randomly distributed with either a co (as highlighted in Figure 3b). Thi dehalogenation of the molecules. halogen bonding, [START_REF] Teyssandier | Halogen Bonding in Two-Dimensional Crystal Engineering[END_REF] the formation along the apparent chain direction adatoms cannot be excluded on Ag usually observed for halogenated pre surfaces. 2,[51][START_REF] Barton | Formation of Organometallic Intermediate States in On-Surface Ullmann Couplings[END_REF][53] This issue could also the rhombic metastable phase P5, co at room temperature but with larger The Journal of Physical Chemistry C pubs.acs.org/JPCC markedly elongate d). The periodicity nds to 5 substrate The parallel chains 3 substrate atomic (see Supporting axial model). nfiguration because ver, the interchain and P2 phases on lumnar structure is ion of TBAP (1.44 riodicity along the fact, this distance is se P2 on Ag(111). ere the porphyrin direction (with the ], i.e., the chain ts along the chains surfaces. Figure 4 chain formation is rphyrin planes on e excluded the the e 3 axis neither ection) because of ith respect to the P3/P4 phases on een the molecules no favorable π-π s is allowed. If we in driving force for ructure (instead of uld produce an yrin planes of 0.8 in P1 and P2 (0.7 tion configuration anthracenyl groups is responsible for this behavior. Noteworthily, the strength of the intermolecular interaction along [001] (e 3 axis) is sufficiently strong so that extended well-ordered chains are formed. This is particularly demonstrated in phase P3 for which all chain endings are perfectly aligned. The fact that short chains are found after room-temperature deposition on both Ag(111) and Ag(110) surfaces suggests a rather strong interaction of the molecules with the silver surfaces. The elongation of the chains after annealing may be facilitated by the increase of diffusivity at higher temperature, but it is also possibly activated by a conformational (rotation of anthracenyl) or chemical (loss of Br) change.

After room-temperature deposition on Ag(110), we also found another phase (phase P5, Figure 5), with large wellorganized domains and coexisting with phase P3. This metastable phase disappeared upon slight annealing at 100 °C. The structure here is still composed of molecular chains aligned along the [11̅ 0] direction but with a rhombic pattern. The periodicity along the [11̅ 0] direction is of 6 substrate atomic distances, that is 20% larger than in phase P3, but the periodicity of the molecular chains along the [001] direction remains the same, that is, 1.2 nm (3 substrate atomic distances) (see Supporting Information Figure S2d). In these domains, the TBAP molecules have the same orientation as in P3 and P4, but they are simply positioned further apart along their e 1 axis and not any more aligned along the [001] direction. Two kinds of rhombic domains are observed (labeled A and B in Figure 5), depending on the porphyrin core shift between two adjacent chains of ±2 substrate atomic distances along [11̅ 0]. The formation of ordered 2D domains confirms the presence of substantial intermolecular interactions along [11̅ 0] and also in the other domain highsymmetry directions.

Finally, it should be noted that in the high-resolution images of phase P3, the apparent contrast at the chain termination is randomly distributed with either a concave or a convex shape (as highlighted in Figure 3b). This may indicate a partial dehalogenation of the molecules. Therefore, rather than halogen bonding, [START_REF] Teyssandier | Halogen Bonding in Two-Dimensional Crystal Engineering[END_REF] the formation of metal-organic links along the apparent chain direction through insertion of Ag adatoms cannot be excluded on Ag(110), similar to what is usually observed for halogenated precursors on various metal surfaces. 2,[51][START_REF] Barton | Formation of Organometallic Intermediate States in On-Surface Ullmann Couplings[END_REF][53] This issue could also explain the observation of the rhombic metastable phase P5, composed of longer chains at room temperature but with larger intermolecular distances r configurations on r models are super-

ion, for which the metal-organic hieved yet. embly of TBAP molecules on the rfaces. The molecules adopt an edgerm extended 2D domains of highly ealing. The formation of the chains nct processes on theses surfaces. On structure is formed through π-π nthracenyl groups. On Ag(110), the ugh lateral interaction between the by the formation of metal-organic precursor can adopt different 3D the many degrees of liberty gained by the four bromoanthracenyl groups. As t intermolecular interactions can take t modifications in the supramolecular , noticeable variations in their overall ted. [START_REF] Smykalla | Metallo)porphyrins for potential materials science applications[END_REF][START_REF] Diercks | Reticular Electronic Tuning of Porphyrin Active Sites in Covalent Organic Frameworks for Electrocatalytic Carbon Dioxide Reduction[END_REF] Our results should provide d modular complex 2D self-assembly use of a single intrinsically threerecursor.

TENT

  The Journal of Physical Chemistry C Methods intermong different molecular chain orientations on Ag(111) and Ag(110).

Figure 1 .

 1 Figure 1. Schematic drawing of the (a) TBAP molecule and (b) Ag(111) and Ag(110) surfaces. (c) STM image of the self-assembly of TBAP on Ag(111) after room-temperature deposition (P1 phase) and (d) corresponding zoomed-in image. (e) P2 phase obtained after annealing at 200 °C and (f) corresponding zoomed-in image. The unit cell is indicated in (f) with a = 1.4 nm and b = 3.0 nm. Scale bars (c,e) 12 nm and (d,f) 2 nm.

  The deposition of TBAP on A leads to the formation of supram ing to phase P1 as shown in Fig straight molecular chains with tw orientations. Most of the chains direction of the substrate and periodicity inside a chain is 0.8 n The chains are regularly distribut two adjacent chains, thus the inte 5 substrate atomic distances al shown in the high-resolution ST molecular chains have locally a are stacked together and pack positioned chains, which are 5 Figure 1d, aligned along the [1 [11̅ 0]-chains, the periodicity is 0. 2.5 substrate atomic distance. A the STM image in Figure 1d Information, Figure S2a. The sim chain types suggest a unique in phase P1.

Figure 1 .

 1 Figure 1. Schematic drawing of the (a) TBAP molecule and (b) Ag(111) and Ag(110) surfaces. (c) STM image of the self-assembly of TBAP on Ag(111) after room-temperature deposition (P1 phase) and (d) corresponding zoomed-in image. (e) P2 phase obtained after annealing at 200 °C and (f) corresponding zoomed-in image. The unit cell is indicated in (f) with a = 1.4 nm and b = 3.0 nm. Scale bars (c,e) 12 nm and (d,f) 2 nm.

  molecule and (b) the self-assembly of sition (P1 phase) hase obtained after ed-in image. The 3.0 nm. Scale bars pubs.acs.org/JPCC Article followed by annealing. The precurs degassed prior to deposition onto substrate. STM measurements were temperature with a commercial Omic

Figure 1 .

 1 Figure 1. Schematic drawing of the (a) TBAP molecule and (b) Ag(111) and Ag(110) surfaces. (c) STM image of the self-assembly of TBAP on Ag(111) after room-temperature deposition (P1 phase) and (d) corresponding zoomed-in image. (e) P2 phase obtained after annealing at 200 °C and (f) corresponding zoomed-in image. The unit cell is indicated in (f) with a = 1.4 nm and b = 3.0 nm. Scale bars (c,e) 12 nm and (d,f) 2 nm.

AP

  molecule and (b) e of the self-assembly of deposition (P1 phase) P2 phase obtained after zoomed-in image. The d b = 3.0 nm. Scale bars Indeed, the 4 Br atoms in the molecule form a square pattern 1.4 nm wide (Figure

Figure 2 .

 2 Figure 2. Molecular mechanics calculations. (a,b) Structure of the relaxed isolated TBAP molecule. (c,d) Gas-phase columnar structure. A dimerization of the porphyrin cores is obtained, while the distance between the anthracenyl planes is kept constant. The C-Br bond of the bromoanthracenyl groups is alternatively tilted by an angle α 1 = 8.2°and α 2 = -9.5°.

Figure 3 .

 3 Figure 3. STM images of the self-assembly of TBAP on Ag(1 After room-temperature deposition (P3 phase). (b) Corres zoomed-in image. The chain endings are imaged with either a or a convex shape, as highlighted by the purple cartoon in the of the image. (c) P4 phase obtained after annealing to 300 Corresponding zoomed-in image. The unit cell is a = 1.4 nm 1.2 nm. Scale bars (a,c) 7 nm, (b,d) 2 nm.

.Figure 3 .

 3 Figure 3. STM images of the self-assembly of TBAP on Ag(110). (a) After room-temperature deposition (P3 phase). (b) Corresponding zoomed-in image. The chain endings are imaged with either a concave or a convex shape, as highlighted by the purple cartoon in the bottom of the image. (c) P4 phase obtained after annealing to 300 °C. (d) Corresponding zoomed-in image. The unit cell is a = 1.4 nm and b = 1.2 nm. Scale bars (a,c) 7 nm, (b,d) 2 nm.

Figure 4 .

 4 Figure 4. Comparison of the intermolecular configurations on Ag(111) and Ag(110). The tentative molecular models are superimposed on STM images. Scale bars: 1 nm.

Figure 5 .

 5 Figure 5. STM image of the metastable d on Ag(110) after room-temperature dep symmetric. Scale bar: 5 nm.

Figure 5 .

 5 Figure 5. STM image of the metastable domains (phase P5) obtained on Ag(110) after room-temperature deposition. A and B are mirror symmetric. Scale bar: 5 nm.

Notes

  ation is available free of charge at 10.1021/acs.jpcc.0c05908.

	TBAP, tentative epitaxial models for
	and P5, and additional STM images
	)
	ATION
	arseille Univ, Universitéde Toulon,
	eille, France; orcid.org/0000-0002-
	lvain.clair@univ-amu.fr
	Aix Marseille Univ, Universitéde Toulon,
	eille, France r -Aix Marseille Univ, CNRS, CINAM,
	sitéParis-Saclay, CEA, CNRS, NIMBE,
	ur-Yvette, France
	arseille Univ, CNRS, Centrale Marseille,
	ce
	S-CNRS, Universitéde Toulouse, 31055
	rcid.org/0000-0001-7827-3496 -Aix Marseille Univ, CNRS, Centrale
	eille, France
	UniversitéParis-Saclay, CEA, CNRS,
	191 Gif-sur-Yvette, France; orcid.org/
	1
	ix Marseille Univ, CNRS, CINAM,
	ation is available at:
	021/acs.jpcc.0c05908
	ibuted equally.

tion

https://dx.doi.org/10.1021/acs.jpcc.0c05908 J. Phys. Chem. C 2020, 124, 22137-22142

https://dx.doi.org/10.1021/acs.jp J. Phys. Chem. C 2020, 124, 221

Acknowledgments

along the chain direction, for which the metal-organic bonding would not be achieved yet.

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

L. Nony and F. Bocquet are gratefully acknowledged for their support in image treatment. This work was partly funded by the ANR program project MAGMA (ANR-16-CE29-0027-01) and by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (Labex NanoSaclay, reference: ANR-10-LABX-0035).

■ CONCLUSIONS

We studied the self-assembly of TBAP molecules on the Ag(111) and Ag(110) surfaces. The molecules adopt an edgeon configuration and form extended 2D domains of highly ordered chains after annealing. The formation of the chains originates from two distinct processes on theses surfaces. On Ag(111), a columnar structure is formed through π-π interaction between the anthracenyl groups. On Ag(110), the chains are formed through lateral interaction between the molecules, possibly aided by the formation of metal-organic complexes. The TBAP precursor can adopt different 3D configurations because of the many degrees of liberty gained by the frustrated rotation of the four bromoanthracenyl groups. As a result, radically different intermolecular interactions can take place, leading to important modifications in the supramolecular organization. Accordingly, noticeable variations in their overall properties can be expected. [START_REF] Smykalla | Metallo)porphyrins for potential materials science applications[END_REF][START_REF] Diercks | Reticular Electronic Tuning of Porphyrin Active Sites in Covalent Organic Frameworks for Electrocatalytic Carbon Dioxide Reduction[END_REF] Our results should provide inspiration for a guided and modular complex 2D self-assembly on a surface by taking use of a single intrinsically threedimensional functional precursor.

■ ASSOCIATED CONTENT

* sı Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcc.0c05908. Synthetic route of TBAP, tentative epitaxial models for phases P1, P2, P4, and P5, and additional STM images for phase P2 (PDF) TBAP molecules on the molecules adopt an edgeed 2D domains of highly e formation of the chains ses on theses surfaces. On is formed through π-π l groups. On Ag(110), the interaction between the rmation of metal-organic can adopt different 3D egrees of liberty gained by moanthracenyl groups. As cular interactions can take tions in the supramolecular e variations in their overall ur results should provide complex 2D self-assembly single intrinsically threeailable free of charge at cs.jpcc.0c05908.

tative epitaxial models for d additional STM images

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

L. Nony and F. Bocquet are gratefully acknowledged for their support in image treatment. This work was partly funded by the ANR program project MAGMA (ANR-16-CE29-0027-01) and by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (Labex NanoSaclay, reference: ANR-10-LABX-0035).


Mathias Marconi

Fabrice Raineri

Ariel Levenson

Alejandro M Yacomotti
email: alejandro.giacomotti@c2n.upsaclay.fr

Julien Javaloyes

Si H Pan

Abdelkrim El Amili

Yeshaiahu Fainman

Mesoscopic limit cycles in coupled nanolasers

Two coupled semiconductor nanolasers exhibit a mode switching transition, theoretically characterized by limit cycle -or mode-beating-oscillations. Their decay rate is vanishingly small in the thermodynamic limit, i.e. when the spontaneous emission noise β-factor tends to zero. We provide experimental evidence of mesoscopic limit cycles -with ∼ 10 3 intracavity photons-through photon statistics measurements. We first show that the order parameter quantifying the limit cycle amplitude can be reconstructed from the mode intensity statistics. As a main result we observe a maximum of the averaged amplitude at the mode switching, accounting for limit cycle oscillations. We finally relate this maximum to a dip of mode cross-correlations, reaching a minimum of g

(2) ij = 2/3, which we show to be a mesoscopic limit. Coupled nanolasers are thus an appealing testbed for the investigation of spontaneous breaking of time-translation symmetry in presence of strong quantum fluctuations.

How quantum fluctuations affect nonequilibrium periodic orbits? This question, intimately related to the spontaneous breaking of time translation symmetry, has strongly motivated a large community of physicists in the last few years. Although the spontaneous time symmetry breaking is well known in classical nonlinear dynamical science [START_REF] Guckenheimer | Nonlinear oscillations, dynamical systems, and bifurcations of vector fields[END_REF], its realization in the quantum world has been a subject of debate. In a seminal paper [START_REF] Wilczek | [END_REF], F. Wilczek pointed out the existence of quantum periodic motion in a time-invariant Hamiltonian, launching a new field of research known as time crystals: the time counterparts of spatial crystals, for which the continuous spatial translation symmetry is spontaneously broken. Since then many efforts have been devoted to understand and implement time crystals in different domains such as condensed matter and QED systems (see, e.g., Ref. [3] for a review). [4]. In contrast to Floquet time crystals -or π-spin glasses- [5][6][7] subjected to a periodic forcing, in BTC's the Hamiltonian is time-independent, and it is the continuous time symmetry which is spontaneously broken in a small though macroscopic fraction of a many body quantum system. The prediction is that, in the thermodynamic limit, a periodic solution emerges whose decay rate tends to zero, i.e. the amplitude of the oscillations becomes constant in time. Such a divergent time scale is related to a closure of a Liouvillian gap in the thermodynamical limit, hence to a dissipative phase transition [8]. In a BTC this decay rate is associated to the lowest eigenvalue with nonzero imaginary part [4]. Hence, the persistent oscillations are associated to the spontaneous symmetry breaking since they only take place in the thermodynamic limit. The model developed in Ref. [4] accounts for cooperative emission in cavities, which can be realized by cold atoms in a cavity subjected to Raman driving. In addition, a number of many body limit cycles could be classified as BTCs [9][10][11]. An important conclusion is that the existence of BTCs is to be experimentally tested, since limit cycles might not survive in the presence of fluctuations.

Recently, F. Iemini et al. have proposed a new class of dissipative time crystals called boundary time crystals (BTC's)

In this work we propose coupled nanolasers (examples of recent realizations can be found in Refs. [12][13][14]) as testbeds for limit cycles subjected to strong quantum noise -in this case due to spontaneous emission-and provide experimental evidence on the existence of limit cycles with a thousand photons inside the cavities.

Lasers are fascinating workbenches to study nonequilibrium statistical mechanics. A paradagmatic example, realized since the early days of laser theory, is the second order phase transition at the oscillation threshold in the thermodynamic limit, i.e. for vanishingly small spontaneous emission β factor [15][16][17]; intracavity photon number scales as β -1 , which can be identified as the thermodynamic parameter [17]. Here we explore limit cycle oscillations that emerge as mode beating when the two eigenmodes of the two coupled nanolasers operate simultaneously. Specifically, this occurs at a mode switching transition between the bonding and anti-bonding modes of a photonic crystal molecule [18].

We consider the photon statistics around a switching transition from the bonding (B) to the anti-bonding (A) (b) Bloch sphere, showing noiseless limit cycles close to a switching point (Fa 1,2 (t) = 0 in Eqs. 1) ; blue trajectories correspond to orbits increasing pumping power from P/P0 = 6.008 (fixed point B: x = 1), 6.010, 6.012, 6.016 (limit cycles, x > 0), 6.020 (perfect mode beating limit cycle: x ≈ 0) , 6.024, 6.028, 6.032 (limit cycles, x < 0), and 6.036 (fixed point A: modes of a nanolaser dimer formed by two evanescently coupled semiconductor nanocavities (coupling constant K, see Fig. 1a) [12,18] as the pump is increased. At the switching point, the high energy (blue-shifted, here B) mode switches off, and simultaneously the fundamental (red-shifted, here A) mode switches on [18].

Theoretically, lasers can be described by a quantum master equation using the density matrix approach [START_REF] Scully | Quantum theory of the laser -density operator approach[END_REF][START_REF] Takemura | [END_REF]. Much more simplified models have been used in the past: among them, the semiclassical laser theory -which neglects quantum fluctuations-has the status of a mean field model in statistical mechanics [17], which is a rough approximation for semiconductor nanolasers. A more realistic description needs to incorporate spontaneous emission fluctuations produced by the semiconductor emitters (such as quantum wells, QWs), which can be added to the semiclassical model in the form of Langevin noise terms. Two coupled nanolasers containing QWs can be thus modeled by the following nonlinear coupled stochastic differential equations [12,14,18]:

ȧ1,2 = 1 + iα 2 G 1,2 -κ a 1,2 + (γ + iK) a 2,1 + F a1,2 (t) (1) ṅ1,2 = P -γ tot n 1,2 -G 1,2 |a 1,2 | 2 (2)
where |a i,j | 2 = I i,j and n are normalized as the photon and carrier numbers in the cavities, respectively, κ is the cavity loss rate, α the Henry factor, P the pump rate and γ tot is the total carrier recombination rate. The complex inter-cavity coupling constant quantifies frequency (K) and loss (γ) splitting as a result of the evanescent coupling. G 1,2 = γ β(n 1,2 -n 0 ) is the gain, γ is the two-level radiative recombination rate and n 0 the carrier number at transparency. F ai (t) are Langevin noise terms accounting for spontaneous emission with rate R sp = βF P Bn 2 1,2 /V a where B is the bimolecular radiative recombination rate, F P the Purcell factor and V a the volume of the active medium. We make the common assumption of uncorrelated (white) noise, i.e. F µ (t)F ν (t ) = 2D µν δ(t -t ), where D µν are the following diffusion coefficients:

2D aiai = 2D a * i a * i = 0, 2D aia * i = 2D a * i ai = R sp
, and zero otherwise. Importantly, the spontaneous emission factor β is related to the intracavity saturation photon number as I sat = γ tot /γ β; β -1 can thus be identified as the thermodynamic parameter, since the characteristic photon number scales as β -1 . AA and green: g BA . Black: mean value, and grey: variance of x. Solid lines show the results using the full statistics; dashed lines compute the moments from g The two linear eigenmodes of Eqs. 1 are a B = (a 1 + a 2 )/ √ 2 and a A = (a 1 -a 2 )/ √ 2, corresponding to bonding and anti-bonding modes of the coupled cavities system, respectively (Fig. 1a). As it has been shown elsewhere [14], the dynamics of this system can be separated in two subset of variables: the total intensity and carrier number on one side, and the relative intensities and phases of the cavities on the other side, which can be recast on the Bloch sphere as θ = 2 arctan

I 2 /I 1 ∈ [0, π] and Φ = ψ 1 -ψ 2
, where a j = I j exp (iψ j ). Remarkably, the x-coordinate of the Bloch sphere is nothing but the mode population imbalance, x = (I B -I A )/(I B + I A ), where I B and I A are the intensities of the two eigenmodes. Above laser threshold, the laser molecule operates in the mode with higher net gain, which in our case has been designed to be the B-mode. Experimentally, a switching transition is observed, where the B-mode switches off, and the A-mode switches on as the pump power is ramped up [18]. Indeed, Eqs. 1-2 show mode switching dynamics. Interestingly, the mode switching transition is mediated by the emergence of a limit cycle in the thermodynamic limit, β -1 → ∞ [18]: the B-mode loses stability expelling a limit cycle at a first Hopf bifurcation (x = 1, Fig. 1b); these oscillations account for mode beating. The limit cycle amplitude rapidly increases up to a perfect mode beating situation in which both modes have the same intensity (dual-frequency laser), and each cavity intensity experiences 100%-contrast oscillation (x = 0, Fig. 1b). Further increasing the pump parameter the limit cycle shrinks and coalesces at a second Hopf bifurcation, leading to a stable fixed point on the Bloch sphere corresponding to the A-mode (x = -1, Fig. 1b).

The limit cycle oscillations are long-lasting solutions of the mean field limit: the amplitude decay rate tends to zero. In the presence of noise, fluctuations increase the decay rate. We have quantified such an effect through simulations of the Langevin equations with different β-factors accounting for different spontaneous emission rates. In Fig. 1c-d we show the ensemble average of the cavity 1-intensity, I 1 for β = 1.7 × 10 -5 (Fig. 1c) and β = 1.7 × 10 -2 (Fig. 1d). In these simulations the initial conditions correspond to a maximum of photon number in cavity 1, i.e. Φ = 0 or π, and θ ≤ π/2. Figure 1c corresponds to a macroscopic laser cavity: the effect of noise on the limit cycle is small, and the amplitude does not decay in the whole time window used for the calculations. However, in Fig. 1d we observe a drastic reduction in the decay time of the amplitude; still, the limit cycle undergos thousands of oscillations during the damping time. It is important to point out that the period of oscillations is T = π/K = 0.26 in units of the cavity photon lifetime, corresponding to a frequency of f = 545 GHz in our example of strongly coupled cavities. Such a high frequency combined with a low output photon number, rule out the possibility of the direct observation of the limit cycle. However, we will show below that the order parameter accounting for the limit cycle formation can be quantified through the mode intensity statistics.

By construction, the limit cycle amplitude on the Bloch sphere of Fig. 1b is

A = 1 -x 2 (3)
which is the natural order parameter for the limit cycle. Equation 3 simply states that the limit cycle vanishes for single mode operation, x = ±1, and reaches a maximum order of A = 1 for the "meridian" limit cycle (Ψ = π/2) in the thermodynamic limit. In Fig. 1e-f we show the mean value and variance of the order parameter as a function of the pump. Clearly, A reaches a maximum at P/P 0 ≈ 6.02 for β = 1.7 × 10 -5 (Fig. 1e) with two fluctuation maxima at the bifurcation points. In the "nanolaser" case, β = 1.7 × 10 -2 , A still presents a maximum at the mode switching point, but its value is smaller, A ≈ 0.8, and the pumping range for nonzero

A is broadened with respect to the "macroscopic" case. This last point is important since the nanolaser regime enhances the pumping interval of existence of the limit cycle. Note that the statistics of A can be obtained through the mode-intensity statistics. In the following we will present our experimental results showing the increase of the limit cycle amplitude at the mode switching point, together with an increase of the amplitude fluctuations at both sides, which is the signature of a limit cycle bifurcation. Figure 2a shows the experimental time traces of the two eigenmodes. Modes are detected in the far field, in such a way that their emission can be spatially separated. Mode-intensities are then simultaneously measured using two fast (600 MHz-bandwidth) APD photodetectors as the pump power is ramped up (ramp duration= 6 ns). The time series have been used to reconstruct the statistics of the mode population imbalance (Fig. 2b, right axis). It can be observed that x has a step-like varia-tion with a zero-crossing -that we refer to as the switching point, P s -as the pump power is increased. The full statistics of x can be used to compute the statistics of A. In Fig. 2c we show the mean value A together with the variance (∆A) 2 . We observe a maximum of A ≈ 0.83 at the switching point, in good agreement with the predictions of the Langevin-semiclassical model (Fig. 1f). In addition, there is a peak of (∆A) 2 at each side of the switching point, also in agreement with the model. Therefore, our measurements reveal the emergence of a limit cycle, even though the direct measurement of the time oscillations of one cavity-intensity cannot be done due to both the extremely high oscillation frequency (of the order of hundreds of GHz) together with the weak output signals (in the sub-µW range).

We point out that the intensity fluctuation dynamics becomes extremely slow at the switching point. As a result the fluctuations could be accurately measured with our APD detectors. Indeed, the time-width of autocorrelation functions is typically 2 -3 ns at the switching point, while the timescales of the system are ∼ 10 ps for the photons, and 200 ps for the charge carriers in the QWs. Such a timescale stretching is seemingly due to the critically slowing down of the dynamics close to bifurcation points, which is also predicted by the model. It turns out that the proximity to the bifurcation point is translated into the real part of an eigenvalue approaching zero, which corresponds to a dramatic increase of the timescale. This can be theoretically confirmed in the limit of very strong coupling, where the dynamics can be reduced to a 1D Fokker-Planck equation. We have computed the spectrum of the Fokker-Planck operator, which shows a first exited eigenvalue that vanishes at the switching point (Sec. V, Supplementary Material). In this K 1 limit the double bifurcation structure leading to the limit cycle (Fig. 1b) is degenerated to a single parameter P = P s , and only the stochastic dynamics of x is considered. We expect that the Fokker-Planck operator of the full Langevin-semiclassical model should possess, at each Hopf bifurcation, two eigenvalues with nonzero imaginary parts and their real part tending to zero as β → 0. We relate these features to a gapless Liouvillian spectrum having a nonzero imaginary part of a quantum master equation description, as it has been predicted for a large class of limit cycles such as BTCs [4].

The strong fluctuations in the limit cycle amplitude come from the strong, non-gaussian fluctuations of the mode population imbalance x. In order to further investigate the nature of such fluctuations, we first point out that the semiclassical model predicts exponential equilibrium distributions for x in the K 1 limit [14] (see Supplementary Material, Sec. V), namely

ρ eq (x; Λ) = N e -Λx , (4)
where N = Λ/ (2 sinh Λ); Λ can be approximated as a linear function of the pump close enough to the switching point, Λ ∼ (P/P s -1)I/β 2 (see Sec. V, Supp. Material). The experimental statistical distributions of x are shown in Fig. 3. Flat distributions can be observed at the switching point, in agreement with the theoretical prediction in the mesoscopic regime (Eq. 4), with Λ > 0 for P < P s , Λ = 0 at P = P s , and Λ > 0 for P > P s . Usually, the experimentally accessible quantity is the photon correlations rather than the order parameter. Nevertheless, both of them are related. First we note that the zero time-delay mode cross-correlations reach a limit of g BA (τ = 0) = 2/3 in the mesoscopic limit cycle regime. This 2/3 limit can be easily deduced from the relation between the second order coherence and the two lowest order moments of x. Under the hypothesis of decorrelated total intensity I = I B + I A and xfluctuations, Ix = I x , it can be shown that

g (2) BA = g (2) II 1 -x 2 1 -x 2 , ( 5
)
where we have removed τ = 0 to simplify the notation. We will further assume that the total intensity fluctuations are poissonian, hence g

II = 1, in agreement with our measurements (Fig 3). The ideal case of a flat statistical distribution of x leads to x = 0 and x 2 = 1/3; hence g

(2) BA = 2/3. This theoretical prediction is also in good agreement with our experimental results: in Fig. 2 (middle panel, left axis) we show g

(2) AA , g (2)
BB and g

BA . Importantly, g

BA ≈ 0.7 is less that unity at the switching point, revealing mode anti-correlations. The mode cross-correlation minimum, min[g

(2)
BA ], is strongly influenced by noise. In Fig. 4 we display numerical simulations decreasing the system size from β -1 = 5.9 × 10 4 to 5.9, showing a clear crossover between the macroscopic and the mesoscopic regimes. Interestingly, the cross-correlation functions have a double dip structure in the macroscopic regime, whereas there is a single dip in the mesoscopic regime, in particular for β = 0.017 corresponding to our experimental situation. Now, our measurements using classical APD detectors can be compared to photon correlation measurements using single photon detectors, for which the time resolution is as short as 50 ps for single nanowire single photon detectors (SNSPDs) in the telecommunication band. Cross correlations measurements using SNSPDs under pulse pumping also show a minimum of g BA ≈ 0.7 close to the switching point (Sec. II, Supplementary Material).

The cross-correlation is related to the limit cycle amplitude as

A 2 ≈ g (2)
BA at the switching point (Eq. 5). In addition, neglecting limit cycle amplitude fluctuations we easily get A ≈ g (2) BA (6) close to the switching point; both are shown in Fig. 4 for comparison. We point out that g The experimental result is marked with a red symbol.

to the uncorrelated limit, hence the non-trivial statistical information is contained in the depth of the crosscorrelation dip (green area in Fig. 4). Since min[g

BA ] approaches unity in the thermodynamic limit (see Fig. 4, bottom-right panel), no significant statistical information can be extracted from these measurements in the macroscopic regime. In contrast, the cross-correlation function is a good statistical indicator in the mesoscopic regime, where g

AB < 1. However, it is important to point out that small modal cross-correlations (g

(2) ij < 1 with i = j)
is not a sufficient condition for the presence of limit cycles. Indeed, mode anticorrelated fluctuations have already been reported in other photonic systems such as VCSELs or micropillar lasers, but no limit cycle dynamics has been reported in those examples. For instance, polarization switching dynamics in VCSELs have been performed in the past, showing strong mode anticorrelation [START_REF] Willemsen | [END_REF][22][23], with reported cross correlation functions below g

(2) ij = 1/2 [24]. More recently, g

ij < 1 has been shown at the polarization switching of bimodal micropillar lasers [25], which has been explained as a statistical mixture of a thermal and a coherent state [26].

In conclusion we have shown that mesoscopic limit cycles emerge at the mode switching of a nanolaser dimer. Such limit cycles are mode beating oscillations when two eigenmodes operate simultaneously. This has been possible thanks to photon statistics measurements, which allowed us to compute the order parameter A which is the amplitude of the limit cycle oscillation. We have shown that a maximum of A is observed at the mode switching point, together with two maxima of the order parameter fluctuations at each side of the mode transition, which are signatures of limit cylce bifurcations in the presence of noise, as predicted by a Langevin-semiclassical model. We conjecture that this scenario may support vanishing eigenvalues of the Liouvillian within a quantum master equation description, with a nonzero imaginary part, which has been recently shown to characterize a large family of many body limit cycles [4]. In addition, we have related the order parameter to photon correlation measurements, and show that the mesoscopic limit cycle regime is associated with a 2/3 limit of the mode cross-correlations. Therefore, a coupled nanolaser system proves useful as a testbed for the investigation of limit cycles subjected to strong quantum noise, and the spontaneous breaking of time translation symmetry.

We acknowledge enlightening discussions with A. Biella, Z. Denis and C. Ciuti. This work has been partially funded by the "Investissements d'Avenir" program (Labex NanoSaclay, Grant No. ANR-10-LABX-0035) and the ANR UNIQ DS078.

FIG. 1 .

 1 FIG.1. Mode beating limit cycles in two strongly coupled nanolasers subjected to spontaneous emission noise. (a) Schematics of two evanescently coupled laser nanocavities, with coupling strength K. Bottom frame: SEM image of the utilized InP photonic coupled lasers; right: the two modes of the photonic molecule (B: bonding, A: antibonding) are split in energy. (b) Bloch sphere, showing noiseless limit cycles close to a switching point (Fa 1,2 (t) = 0 in Eqs. 1) ; blue trajectories correspond to orbits increasing pumping power from P/P0 = 6.008 (fixed point B: x = 1), 6.010, 6.012, 6.016 (limit cycles, x > 0), 6.020 (perfect mode beating limit cycle: x ≈ 0) , 6.024, 6.028, 6.032 (limit cycles, x < 0), and 6.036 (fixed point A: x = -1); other parameters (Eqs. 1-2) are K = 12κ, γ = 0.05κ, α = 7, γ = 2.2 GHz, γtot = 5 GHz, κ = 140.84 GHz, n0 = 10 18 cm -3 × Va with Va = 0.016 × 10 -12 cm 3 . (c)-(d) Ensemble average of the intensity in the cavity 1 for P/P0 = 6.015; the average is taken over 100 different noise realizations and the same initial conditions (see text). Insets: zoom over few oscillation periods. (e)-(f) Mean value and fluctuations of the order parameter as a function of the pump power P normalized to the transparency pump P0. (c) and (e) β = 1.7 × 10 -5 and Va = 16 × 10 -12 cm -3 . (d) and (f) β = 0.017 and Va = 0.016 × 10 -12 cm 3 . Other parameters of the Langevin terms are FP = 1.03 and B = 3 × 10 10 cm 3 s -1 .

  FIG.1. Mode beating limit cycles in two strongly coupled nanolasers subjected to spontaneous emission noise. (a) Schematics of two evanescently coupled laser nanocavities, with coupling strength K. Bottom frame: SEM image of the utilized InP photonic coupled lasers; right: the two modes of the photonic molecule (B: bonding, A: antibonding) are split in energy. (b) Bloch sphere, showing noiseless limit cycles close to a switching point (Fa 1,2 (t) = 0 in Eqs. 1) ; blue trajectories correspond to orbits increasing pumping power from P/P0 = 6.008 (fixed point B: x = 1), 6.010, 6.012, 6.016 (limit cycles, x > 0), 6.020 (perfect mode beating limit cycle: x ≈ 0) , 6.024, 6.028, 6.032 (limit cycles, x < 0), and 6.036 (fixed point A: x = -1); other parameters (Eqs. 1-2) are K = 12κ, γ = 0.05κ, α = 7, γ = 2.2 GHz, γtot = 5 GHz, κ = 140.84 GHz, n0 = 10 18 cm -3 × Va with Va = 0.016 × 10 -12 cm 3 . (c)-(d) Ensemble average of the intensity in the cavity 1 for P/P0 = 6.015; the average is taken over 100 different noise realizations and the same initial conditions (see text). Insets: zoom over few oscillation periods. (e)-(f) Mean value and fluctuations of the order parameter as a function of the pump power P normalized to the transparency pump P0. (c) and (e) β = 1.7 × 10 -5 and Va = 16 × 10 -12 cm -3 . (d) and (f) β = 0.017 and Va = 0.016 × 10 -12 cm 3 . Other parameters of the Langevin terms are FP = 1.03 and B = 3 × 10 10 cm 3 s -1 .

FIG. 2 .

 2 FIG. 2. Experimental time traces as the pump power is quasistatically ramped up in time. Top panel: Intensity traces for bonding (blue) and antibonding (red) modes measured with 600 MHz-APD detectors. Pump ramp duration=6 ns. Thick lines: average corresponding to 10 4 time traces. Middle panel: second order correlations (left axis) and the two lowest moments of the mode population imbalance x (right axis). Blue: g (2) BB , red: g (2)

( 2 )

 2 ij (Eqs. S3-S5 of the Supplementary Material), thus neglecting correlations between I and x. Bottom panel: two first moments of the order parameter A. Blue: mean value, A ; yellow: variance, (∆A) 2 .

FIG. 3 .

 3 FIG.3. Joint histograms P (x, Itot; t) corresponding to the rumping-up intensity traces of Fig.2. Frames correspond to increasing pump powers at different times: from bottom-right to top-left corners t=6712.7, 6713.1, 6713.5, 6713.9, 6714.3, 6714.7. It can be observed that nearly flat x-statistics occur at the switching point, i.e. t=6713.5-6713.9. x-distributions are exponentially decaying from x = 1 to x = -1 for small pump powers (t=6712.7, top-left), and from x = -1 to x = 1 for large pump powers (t=6714.7, bottom-right).

FIG. 4 .

 4 FIG.4. Minimum of the second order cross correlations (green) and maximum of the squared mean value of the limit cycle amplitude (blue squares) between bonding and antibonding modes of a laser dimer for increasing system size β -1 . The thermodynamic and mesoscopic limits are indicated with dashed horizontal lines. Green line: guide to the eye. Inset: zero time delay intensity correlations g (2) ij [(i, j) = (B, B): blue, (A, A): red, and (B, A): green lines] as a function of the pump parameter around the mode switching point, showing: a double dip structure in the macroscopic regime (bottom right), and a single dip in the mesoscopic regime (top left). The experimental result is marked with a red symbol.