A Srinivasulu

] B Shantharjun

] D Vani

] K Chinna Ashalu

] A Mohd

J Wencel-Delord

F Colobert

K Rajender Reddy

Iron Catalysis Iron-Catalyzed Minisci Type Acetylation of N-Heteroarenes Mediated by CH(OEt) 3 /TBHP

Iron-catalyzed acetylation of electron deficient N-heteroarenes has been reported using triethylorthoformate as robust and inexpensive acetyl source. This new method is

Introduction

N-heterocycles, such as quinoline and isoquinoline derivatives, are key structural motifs of natural products and biologically active scaffolds. In particular, quinoline alkaloids have attracted great interest, due to their antitussive, antipsychotic, antispasmodic, anxiolytic, antibacterial, antifugial and anticancer activities. [1] Consequently, design of synthetic approaches allowing rapid and straightforward access to these compounds from simple and non-prefunctionalized precursors is an attractive research field. In this context, Minisci reaction, [START_REF]For Minisci radical alkylation and acylation see: a)[END_REF] implying radical C-H functionalization of electron deficient N-heteroarenes established itself as a method of choice to install alkyl or acyl moieties. Following this seminal work, a variety of protocols allowing radical acylation of electron-deficient N-heteroarenes have been devised. [START_REF]For Minisci radical alkylation and acylation see: a)[END_REF] Accordingly, α-keto acids, [2b] aldehydes, [2c] benzylamines, [2d] aryl methanes [2e] etc., have proven to be appealing coupling partners to rapidly introduce aromatic and aliphatic acyl moieties. In clear contrast, direct installation of the acetyl moiety is much more challenging. Indeed, acetylation is not possible when toluene derivatives or benzylamines substrates are used as aldehyde source and when 2-oxopropanoic acid is selected as acetyl precursor, use of a silver salt in combination with strong oxidants and additives are needed. [START_REF]For Minisci radical alkylation and acylation see: a)[END_REF] Zhu and co-workers discovered that gold is also promising catalyst for such reaction, allowing a radical coupling between a large panel of N-heterocycles and ethanol. [3] Recently, silver-catalyzed acetylation mediated by a mild Selectfluor oxidant was dis-successfully applied for the acetylation of quinolines, isoquinoline, quinoxalines, arylpyridines, bipyridines, and benzothiazole. closed by Baxter. [4] In parallel, a metal-free acetylation of isoquinoline was achieved using α-keto acids as coupling partner, in combination with potassium persulfate oxidant. [5] A rare example of mild protocol employing formaldehyde as coupling partner was disclosed by Antonchick. [6] However, efficiency of this strategy and its generality need for PIFA oxidant and TMSN 3 might be limiting. Finally, an original electrocatalytic approach has been conceived very recently, delivering acetylated heteroarenes in a presence of pyruvic acid. [7] Considering the importance of functionalized N-heterocycles, we surmised that design of new routes towards a large panel of acetylated products, using cheap and abundant metalcatalyst is a challenging research field. Accordingly, herein we report an alternative strategy for acetylation of N-heterocycles, using for the first time triethylorthoformate in combination with TBHP as robust and inexpensive acetyl radical source. Remarkably, the desired reactivity is achieved with a cheap and abundant iron catalyst and the coupling occurs without the need of external solvent, since triethylorthoformate is used as self-solvent: This work follows our previous contribution on the iron catalyzed oxidative cross coupling reactions. [8][9][10][11]

Results and Discussion

We initiated our experimental work by establishing optimal reaction conditions for acetylation of 2-methylquinoline 1a with triethylorthoformate 2. Rapidly it was discovered that when 1a (1 mmol) was treated with an excess amount of 2 in the presence of 3 (3 equiv. of TBHP 70 % aq solution) at 100 °C, 4acetyl-2-methyl quinoline 4a was formed in 74 % yield (Table 1, entry 1). By varying the temperature to 90 °C or 110 °C, the efficiency of the coupling was decreased to 61 % and 65 % respectively (Table 1, entries 2 and3). Subsequently, it was determined that 4 equivalents is an optimal amount of TBHP oxidant (Table 1,entries 4,5). Increasing catalyst loading, 20 mol-%, had slightly harmful impact on the reaction (Table 1 entry 6). By changing the oxidation state of Fe III to Fe II the yield of the product 4a has decreased to 62 % (Table 1,entry 7). Other Fe III catalysts such as ferric chloride hexahydrate or simple ferric chloride delivered 4a in lowered yields of 58 % and 54 % respectively (Table 1, entries 8 and9). Altering the oxidation state of the catalyst from Fe III to Fe II drastically hampered the reaction [4a isolated in 42 % with FeCl 2 and 20 % with Fe(acac) 2 respectively; Table 1, entries 10 and11]. The absence of water in TBHP (TBHP in Decane) did not improve the yield of the product 4a (Table 1,entry 12). No enhancement was achieved by changing the oxidant from TBHP to DTBP (Di tert-butyl peroxide), TBPB (tert-butyl peroxybenzoate), DCP ( Dicumyl peroxide) andH 2 O 2 (Table 1, entries 13-16). The product 4a, was not detected neither in the absence of Fe III perchlorate catalyst nor TBHP (Table 1, entries 17 and18).

Table 1. Screening of reaction conditions. [a] Entry Catalyst (mol-%) Oxidant Temp Yield [equiv.]

[°C]

[%] [b] 1 Fe(ClO 4 ) The generality and substrate scope of the reaction was evaluated using various N-heteroarenes as shown in the Table 2, 4a-4s. Particularly 2-methyl quinolines reacted well with triethylorthoformate and TBHP in the presence of Fe III perchlorate•xH 2 O and afforded the corresponding acetyl quinolines from moderate to good yields (52-75 % yields, Table 2, 4a-4h).

Electron withdrawing halogen substitution at positions 6, 7 and 8 of 2-methyl quinolines furnished the corresponding acetylated products 4b-4f in high yields. Slightly lower efficiency is observed for electron-rich substrates bearing methyl or methoxy motifs at 6 position of 2-methyl quinoline (55 % and 52 % Table 2, 4g and 4h). Acetylation of 2-phenyl quinoline furnished the corresponding product 4i in 72 % yield and the fused aromatic acridine afforded 4j in moderate yield (50 %). Notably, the mixtures of acetylated products were obtained using "not biased" quinoline congeners, with a comparable rate at both, C2 and C4 positions. When 8-methyl quinoline (1k) underwent the acetylation, both regioisomers 4ka (2-acetyl-8-methyl quinoline) and 4kb (4-acetyl-8-methyl quinoline) were delivered in Table 2. Substrate scope of quinolines, isoquinoline, quinoxaline. [a] [a] Reaction Conditions: 1 (1 mmol), 2 (2 mL), 3 TBHP (4 equiv.), Fe(ClO 4 ) 3 •xH 2 O (0.1 mmol), 100 °C, Isolated yields after column chromatography.

44 % and 40 % yields respectively. When 6-methyl quinoline (1l) was subjected to the standard reaction conditions, the mixture of 2-acetylated, 4-acetylated and 2,4-diacetylated products (4la-4lc) were formed. The regioisomers 4la (2-acetyl-6-methyl quinoline) and 4lb (4-acetyl-6-methyl quinoline) were generated in 45 % and 34 % yields respectively. The diacylated product 4lc was isolated only as a minor product in 5 % yield. In the case of simple quinoline, two regioisomers 4ma (2-acetyl quinoline) and 4mb (4-acetyl quinoline) were delivered in 42 % and 44 % yield, while the diacetylated product 4mc was separated as a minor compound in 8 % yield (Table 2, entries 4ma-4mc). Comparable results were obtained starting from quinolines having the electron donating and electron withdrawing groups at C4 position of quinoline giving 4-methyl-2-acetyl quinoline (4n) and 4-chloro-2-acetyl quinoline (4o) (Table 2, 4n, 74 % and 4o, 73 %). When isoquinoline was subjected to the reaction, only 1-acetyl isoquinoline was afforded in good yield (Table 2,4p). In contrast, acetylation of quinoxaline and 2methyl quinoxaline turned out to be less efficient, furnishing the corresponding functionalized products 4q and 4r in 29 % and 26 % yield respectively. Challenging diacetylation of phenanthroline gave C2-symmetric 4s isolated in low, 22 % yield (Table 2).

The substrate scope was extended to pyridines, bipyridines and benzothiazole (Table 3, 6a-6l). When 2-phenyl pyridine 5a was subjected to the standard reaction conditions, 6a was obtained regioselectively but in moderate 37 % yield. [START_REF] Huang | Data of 6a is matching with the reported compound a)[END_REF] Rewardingly, addition of 1 equiv. of pivalic acid enabled drastic im-provement in the reaction efficiency, providing the desired product in 65 % yield. This modified protocol was further employed for acetylation of an array of N-heterocycles (Table 3, entries 6a-6l).

Pyridines bearing para-halogen substituted phenyl moieties underwent the reaction smoothly; 4-F-C 6 H 4 , 4-Cl-C 6 H 4 , 4-Br-C 6 H 4 pyridines furnished the corresponding products 6b-6d in better yields than electron rich substrates containing motifs such as 4-OMe-C 6 H 4 and3-Me-4-OMe-C 6 H 3 groups (Table 3, 6e and6f ). When 2-OMe-5-F-C 6 H 3 was reacted under standard acetylation conditions, the expected product 6g was accessed in 42 % yield. 2-naphthyl pyridine was converted into the functionalized product 4-acetyl-2-naphthyl pyridine (6h) in 48 % yield (Table 3,6h). Acetylation of the disubstituted substrate 2,6-diphenyl pyridine delivered 4-acetyl-2,6-diphenyl pyridine 6i in 41 % yield and 2,2-bipyridine and 4,4-bipyridines were converted into the expected products 6j and 6k sluggishly. Under the standard reaction conditions, benzothiazole underwent the acetylation and afforded the 2-acetyl benzothiazole in moderate yield 42 % (Table 3,6l). However, acetylation of 4-nitrophenyl pyridine, 2-methyl pyridine and simple pyridine failed. Up on successful acetylation on benzothiazole, then benzoxazole was also subjected to the standard reaction conditions but deceivingly no desired product was formed. In order to find out acetyl source as well as mechanism, several control experiments were carried out (Scheme 1). Reactions with triethylorthoacetate (Scheme 1a) and triethylorthopropionate (Scheme 1b) under optimized reaction conditions with 4-methyl quinoline, provided the same 2-acetyl-4-methyl quinoline (4n). Similalry tripropylorthoformate (Scheme 1c) and tributylorthoformate (Scheme 1d) under similar conditions resulted in 2-propionylated and 2-butanoylated products respectively. The above reactions suggest that acyl functionality is generated from the respective alkoxy groups of the orthoformates. For example we assume that HC(OEt) 3 generates the acetaldehyde in situ, and further converts into acetyl radical in presence of TBHP (Scheme 2). This was further established with the reaction with acetaldehyde where the desired product (4n) in 30 % yield was observed (Scheme 1e). The formation of desired product was not observed with Ethanol (Scheme 1f ). Radical mechanism was established by radical scavenger (TEMPO) where the desired product (4n) was isolated only 10 % yield (Scheme 1g).

Based on the control experiments and literature precedents, [13] the following mechanism is proposed. Triethylorthoformate generates the acetyl radical in presence of TBHP (Scheme 2). In parallel, iron perchlorate co-ordinates N-atom of heteroarene A and forms more electron deficient intermediate B. The acetyl radical present in the reaction medium attacks at the more electrophilic C1 position of the heteroarene intermediate B to provide the corresponding radical cation C. The hydrogen abstraction from the radical cation C by the tBuO radical which releases from TBHP followed by elimination of tBuOH leads to the rearomatized intermediate D. [14] Furthermore, the final product E is furnished together with the regeneration of catalyst D (Scheme 2).

In conclusion, we have developed an iron-catalyzed acetylation of quinolines, isoquinolines and quinoxalines using triethylorthoformate/TBHP system as acetyl source. This novel acetylation method is extended to pyridines, bipyridines and benzothiazole. We have demonstrated an operationally simple method using commercially available and cheaper catalyst Fe(ClO 4 ) 3 as well as robust triethylorthoformate and TBHP.

Experimental Section

General Procedure for Iron Catalyzed Acetylation of 4a-4s with Triethylorthoformate and tert-Butyl Hydroperoxide: In a reaction vessel, Fe(ClO 4 ) 3 •xH 2 O (0.1 mmol) and N-heteroarenes (1a-s) (1 mmol) were dissolved in 2 mL of triethylorthoformate (2). To the above mixture tert-butyl hydroperoxide 70 % aq solution (4 equiv.) was added slowly over a period of 2 minutes. The reaction mixture was stirred at 100 °C for 12 hours. The progress of the reaction was monitored by TLC. After completion of the reaction, the reaction mixture was allowed to come to room temperature and extracted with ethyl acetate. The organic layer was washed with brine, water and dried with anhydrous Na 2 SO 4 . The organic layer was concentrated under reduced pressure and afforded the crude product. The crude product was purified by silica gel column chromatography using petroleum ether/ethyl acetate (4:1 ratio) mixture as eluent and the pure product was analyzed by 1 H NMR, 13 C NMR and ESI-HRMS.

General Procedure for Iron Catalyzed Acetylation of 6a-6l with

Triethyl Orthoformate and tert-Butyl Hydroperoxide: In a reaction vessel, Fe(ClO 4 ) 3 . xH 2 O (0.1 mmol), Pivalic acid (1.1 mmol) and 2-aryl pyridines (5a-i) or bipyridines 5j and 5k or benzothiazole 5l (1 mmol) were dissolved in 2 mL of triethyl orthoforrmate (2). To the above mixture tert-butyl hydroperoxide 70 % aq solution (4 equiv.) was added slowly over a period of 2 minutes. The mixture was stirred at 100 °C for 12 hours. The progress of the reaction was monitored by TLC. After completion of the reaction, the reaction mixture was allowed to come to room temperature and extracted with ethyl acetate. The organic layer was washed with brine, water and dried with anhydrous Na 2 SO 4 . The organic layer was concentrated under reduced pressure and afforded the crude product. The crude product was purified by silica gel column chromatography using petroleum ether/ethyl acetate (4:1 ratio) mixture as eluent. The pure products were analyzed by 1 H NMR, 13 C NMR and ESI-HRMS.

Scheme 2 .

 2 Scheme 2. Proposed mechanism.

Eur. J. Org. Chem. 2019, 1815-1819 www.eurjoc.org © 2019 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

Acknowledgements

A. S. thanks to CSIR-India, B. S. thanks UGC-India and D. V. thanks to DST-India for the financial support in the form of SRFs and JRF. Dr. KCA, Dr. KRR and Dr. FC thanks to IFCPAR -India (CEFIPRA) for financial support (5305-1). KRR also thanks the reviewers for their critical comments and suggestions. CSIR-IICT Commun. No: IICT/Pubs./2018/187. References


Aabid Mohd

Thippani Anitha

] Kallu

Rajender Reddy
email: rajender@iict.res.inrajenderkallu@gmail.com

Joanna Wencel-Delord

Françoise Colobert
email: francoise.colobert@unistra.frwenceldelord@unistra.fr

P-Stereogenic Phosphonates via Dynamic Kinetic Resolution: A Route towards Enantiopure Tertiary Phosphine Oxides

Keywords: P-stereogenic phosphonate, O-P coupling, Sulfoxide, Dynamic kinetic resolution, Tertiary phosphine oxide

 Due to the versatility of this diastereoselective approach, this method greatly contributed to the design of P-stereogenic ligands such as DiPAMP or Mini-Phos, and their applications in asymmetric catalysis (Scheme 1a). [6] Later Jugé showed that chiral amines such as ephedrine were also very powerful chiral auxiliaries delivering the corresponding P-stereogenic oxazaphospholidine in high yield and diastereoselectivity (Scheme 1b). [7] Similar strategies developed by Han in 2013 and Framery in 2015 involved

7836

induction is achieved, several diastereopure phosphonates can be afforded either by crystallization or flash chromatography. Thus accessed optically pure P-stereogenic precursors may be used as appealing building blocks to rapidly assembly original privileged scaffolds as illustrated via the synthesis of a chiral ligand such as PAMPO. the use of respectively a benzoxazaphosphinine-2 oxide [2g] and an oxaphospholidine obtained from D-glucosamine [8] as chiral phosphinyl transfer agents. Importantly an alternative protocol using enantioselective deprotonation of aryldimethyl phosphine boranes when using alkyllithium reagents in combination with (-)-sparteine chiral ligand was disclosed by Evans (Scheme 1c). [9] Regarding the great importance of P-stereogenic compounds in pharmaceutical and agrochemistry, but also their key importance in asymmetric synthesis, the development of new synthetic routes for the synthesis of P-stereogenic compounds is still a tempting challenge in organic chemistry. In particular, designing a strategy towards a P-stereogenic "platform", featuringastrongmodularityandfacileconvertibilityintoanarrayofenantiopure molecules would be a significant advance. Following this challenge, enantioenriched P-stereogenic phosphonates bearing two different alkoxy substituents appear as excellent candidates that could undergo the chemoselective and sequential functionalization of both alkoxy substituents, paving the way towards a broad array of enantiopure tertiary phosphine oxides. Besides, the absolute configuration of the newly accessed scaffolds could also be controlled by a simple modification of the order of the sequential functionalization. However despite their great synthetic value, enantioenriched P(O)R 3 (OR 1 )(OR 2 ) scaffolds remain elusive (Scheme 1d).

To address this synthetic challenge, we hypothesized that a direct and stereoselective O-P coupling could be envisioned between a phenol bearing a chiral auxiliary in ortho-position and a commercially available or easily accessible racemic Hphosphinate as an organophosphorus precursor. Interestingly, as H-phosphinates are prone to epimerisation, [4b] such a coupling could potentially occur via Dynamic Kinetic Resolution Mechanism (DKR) thus allowing conversion of a racemic product into a high value-added diastereoenriched P-compound. As far as the choice of the chiral auxiliary is concerned, we surmised that a sulfoxide moiety could be an efficient source of Scheme 1. Synthesis of P-stereogenic products. chirality. Indeed, the sulfoxide auxiliary precursors 1) are straightforwardly accessible on large scale from cheap precursors, 2) may be easily installed on the phenol coupling partner and 3) their high important optical stability under various conditions should warrant recycling of the phenol-sulfoxide motif while performing final functionalization of the P-stereogenic product (Scheme 1e). Thus accessed optically pure phosphonates could potentially be transformed into a variety of original P-stereogenic scaffolds by using organolithium or Grignard's reagents with a concomitant release of the enantiopure phenol bearing chiral sulfoxide. Besides, if chiral phenol is used as the auxiliary, the diastereomeric P(O)R(OR 1 )(OAr) obtained present two sites for further functionalization and if chemoselective substitution of (OAr) and (OR 1 ) can be conducted, a particularly large panel of P-stereogenic compounds could be accessed (Scheme 1e).

Targeting the foreseen O-P couplings, we drew the inspiration from the Atherton Todd (AT) reaction, a classical synthetic method to prepare phosphoramidates, phosphonates and other organophosphorus derivatives via X-P coupling between an organophosphorus precursor including H-phosphinate/Hphosphonate, and a nucleophile such as amine/alcohol. [10] This reaction typically occurs in carbon tetrachloride and requires addition of a base, usually a trialkylamine. The (AT) reaction proceeds through inversion of configuration on P atom (Scheme 2). [11] Indeed the first step corresponds to the formation of a chlorophosphate intermediate and proceeds through Scheme 2. Atherton Todd reaction.

retention of configuration on P atom, while the second step i.e. nucleophilic substitution takes place with inversion of configuration on P atom.

Following these considerations, we have designed phenols 2A-C, bearing p-tolyl sulfinyl and the more hindered tert-butylsulfinyl and adamantly moieties respectively as chiral inductors for a diastereoselective O-P coupling reaction. Accordingly the enantiopure phenols 2 bearing the sulfinyl auxiliary were synthesized in high overall yields following a general strategy involving 1) the protection of o-bromophenol as a MOM ether, 2) a bromine/ lithium exchange followed by condensation with (S)-p-tolylmenthyl sulfinate (A), (S)-(tert-butyl)-tertbutane thiosulfinate (B) or adamantan-1-yl-adamantane-1-sulfinothioate (C) and finally 3) deprotection with HCl and crystallization (Scheme 3). [12] With these chiral phenols in hand, we first performed the coupling between the commercially available racemic ethylphenyl H-phosphinate 1a and (S)-2-(p-tolylsulfinyl) phenol 2A. Under the standard reaction conditions, [1 equivalent of (S)-2-(p-tolylsulfinyl) phenol 2A, 2 equivalents of triethylamine in carbon tetrachloride at room temperature under air] and using 2 equivalents of ethylphenyl H-phosphinate 1a, the desired O-P coupling reaction proceeded well, providing the corresponding product 3aA in 83% isolated yield, but with a low 55:45 diastereomeric ratio measured by 31 P NMR of the crude reaction mixture (Table 1, entry 1). Although the reaction is equally efficient at lower temperature, ie. -20 °C, no improvement in diastereoselectivity was observed (Table 1, entry 2). This modest chiral induction prompted us to use the more hindered (S)-2-(tert-butylsulfinyl)phenol 2B and its coupling with ethylphenyl H-phosphinate 1a proceeded with full conversion furnishing 3aB within 3 h and with an increased diastereomeric ratio of 71:29 (entry 3). Detailed optimization of the reaction conditions revealed that: 1) the coupling was highly water-sensitive and thus, the diastereoeselectivity was improved up to 83:17 while adding molecular sieves (entry 4); 2) the replacement of toxic CCl 4 with THF, toluene, dioxane, Et 2 O or DME in presence of 10 equivalents of CCl 4 did not hamper the reactivity and full conversion of 1a to 3aB was observed despite a slightly decreased stereoselectivity (dr 77:23, entries 5-9). Of note is that THF was selected as the solvent of choice for further screening of the other reaction parameters; 3) decreasing reaction's concentration positively impacted the chiral induction and the optimal results in terms of efficiency and diastereoselectivity, ie. dr of 83:17 and full conversion within 2 h, were reached for a concentration of 0.05 M (entries 10-11); 4) the use of 4 equivalents of diisopropylethylamine instead of Et 3 N allowed to get an excellent dr of 90:10 although prolonged reaction time up to 24 h was required (entry 12). Of note is that other bases, such as DBU, DMAP and 2,6-lutidine performed poorly (entries 13-15). Finally, to further increase the stereoinduction during Scheme 3. Synthesis of enantiopure phenols 2. [c,d] tBu Et 3 N 10 THF 2 > 95 83:17 11 [c,e] tBu Et 3 N 10 THF 6 90 83:17 12 [c,d,f ] tBu iPr 2 EtN 10 THF 24 > 95 90:10 13 [c,d] tBu DBU 10 THF 1 > 95 50:50 14 [c,d] tBu DMAP 10 THF 2 85 50:50 15 [c,e] tBu lutidine 10 THF 2 0 -16 [c,d,f ] Adam iPr 2 EtN 10 THF 24 67 90:10 17 [c,d,f,g] tBu iPr 2 EtN 10 THF 24 80 83:17

[a] yield determined by 1 H NMR. [b] -20 °C. [c] With molecular sieves (75mg/mL). [d] 0.05 M. [e] 0.025 M. [f] iPr 2 EtN (4 equiv.). [g] this AT reaction we evaluated the potential of even more sterically demanding phenol-sulfoxide 2C, bearing an adamantyl substituent at S-atom. Disappointingly, no improvement in the chiral induction was observed, while the overall efficiency of the coupling decreased (entry 16). Accordingly, under the optimized reaction conditions, that is, using 2 equivalents of racemic ethylphenyl H-phosphinate, 1 equivalent of enantiopure (S)-2-(tert-butylsulfinyl)phenol, 10 equivalents of carbon tetrachloride, 4 equivalents of diisopropylethylamine and 75 mg/mL of molecular sieves (MS) in THF under argon at room temperature for about 24 h, the maximum diastereomeric ratio of 90:10 was achieved with 85% yield. Subsequently, we hypothesized that the low configurational stability of the H-phosphinates should facilitate their in situ racemisation in the reaction mixture. Accordingly, the overall transformation might occur via Dynamic Kinetic Resolution scenario. In order to validate this assumption, this O-P coupling was performed using equimolar amounts of 1a and 2B. As ex-pected, we were delighted to obtain the desired product in 80% yield and 83:17 dr unambiguously suggesting the dynamic kinetic resolution outcome (Table 1, entry 17). Changing several reaction parameters did not allow improving the diastereoselectivity. This DKR scenario could be explained based on recently reported DFT studies of the racemization of alkylphenyl H-phosphinates. [13] They proposed that in presence of an alcohol (i.e. our enantiopure ortho sulfinylphenol 2), pentavalent intermediates may be generated allowing the racemization through pseudorotation (Scheme 4). With the optimized conditions in hand, the scope of this was explored using different racemic H-phosphinates, [4b,14] secondary phosphine oxides [14f ] as well as differently substituted (S)-2-(tert-butylsulfinyl) phenols. [12b] The AT reaction proceeds smoothly for a wide variety of H-phosphinates as well as for the secondary phosphine oxides. In general, as the steric hindrance of the alkoxy group attached to P atom in H-phosphinates increases (3aB-3gB, Table 2), yields and diastereoselectivities of the reaction decrease. More sterically hindered H-phosphinates such as isopropyl-, cyclohexyl-, tert-butyl-, and adamantylphenyl H-phosphinates were converted into the corresponding AT products 3dB (64%, 77:23 dr), 3eB (67%, 60:40 dr), 3fB (60% NMR yield, 60:40 dr) and 3gB (68%, 50:50 dr) with lower yields and diastereoselectivities. Of note is that this surprisingly low chiral induction observed for the coupling of the adamantyl-substituted 1g does not arise from the slow racemization of the H-phosphinate (increased configurational stability of 1g at room temperature may be expected) as no improvement was reached when performing the coupling in presence of 2 equivalents of 1g and decomposition occured while conducting the reaction at 80 °C.

Next we explored the modification of the aromatic or aliphatic substituent attached to P atom in H-phosphinate. With a naphthyl group the AT product 3hB was obtained in similar yield and diastereomeric ratio (72%, 82:18 dr). However with sterically hindered cyclohexyl group the yield decreased drastically and a slightly lower diastereoselectivity was obtained as well (3iB, 34%, 78:22 dr). Moreover, in case of ethyl methylphosphinate,1j, the corresponding product 3jB was furnished in good yield and moderate diastereoselectivity (3jB, 80%, 70:30 dr). Finally, substitution pattern of the phenol-sulfoxides 2 was investigated. In case of substrates 2D-F, the reaction proceeded equally well, providing the corresponding products 3aD (64%, 77:23 dr), 3aE (78%, 83:17 dr), 3bE (58%, 80:20 dr), 3dE (52%, 79:21 dr) and 3aF (68%, 56:44 dr) bearing methoxy or bromine substituent in ortho-or para-positions. Nevertheless, phenol bearing a methyl group in ortho position was unreactive in this O-P coupling, even at higher temperature, showing that the more hindered phenols were difficult to react with a H-phosphinate substrate. Secondary phosphine oxides also underwent the O-P coupling reaction smoothly delivering the corresponding products, 3kB (67%, 55:45 dr) and 3lB (75%, 51:49 dr) with drastically lower diastereoselectivities. Subsequently, recrystallization of 3aB was targeted and a monocrystal structure was obtained (Figure 1). Structural X-ray diffraction analysis allowed determination of the absolute configuration of the major diastereomer as (S S ,S P ). The absolute configurations of other major diastereomers were attributed accordingly. Successively, our efforts have focused on post-modifications of these P-stereogenic compounds. Towards this goal both diastereomers of 3gB were separated by silica gel chromatography delivering the optically pure (S S ,S P )-3gB and (S S ,R P )-3gB in synthetically useful yields (33 and 35% yield of each diastereomer). With the diastereomerically pure P-precursors in hand, the postfunctionalization was attempted, using MeMgBr as nucleophile. Rewardingly, fully chemoselective transformation took place as selective cleavage of the sulfoxyde-phenyl 2B occured, delivering 4 in 64% yield and 90:10 er. Importantly, the chiral auxiliary 2B could be recovered in 93% yield and excellent optical purity (ee 99:1) (Scheme 5). Furthermore, in order to showcase the synthetic utility of this methodology, we have embarked on designing a new protocol for the synthesis of PAMPO, key precursor for the synthesis of the seminal DIPAMP ligand. Towards this end diastereomerically pure sample of (S S ,R P )-3lB was prepared via silica gel chromatography. This starting material was subsequently reacted with o-AnMgBr in THF and at 0 °C, thus furnishing (S P )-PAMPO in excellent yield of 84%. The optical purity of PAMPO, ee > 98 % was confirmed by chiral HPLC analysis. Besides, the specific optical rotation measurement {[α] D 20 = -9.74 (c = 0.5, CHCl 3 )}, by comparison with literature values, [2b,15] indicates the S-configu-

Scheme 4 .

 4 Scheme 4. Speculative mechanism of the racemization of alkylphenyl H-phosphinates.

Figure 1 .

 1 Figure 1. X-ray crystal structures of 3aB (S S , S P ).

Scheme 5 .

 5 Scheme 5. Post-functionalization of 3gB and 3lB.

Table 1 .

 1 Atherton Todd O-P coupling reaction. Scheme of Table 1 in the middle of the page

	Entry	R	Base	CCl 4 (x equiv.)	solvent	Y hours	Yield [a] [%]	dr
	1	pTol	Et 3 N	-	C C l 4	24	90	55:45
	2 [b]	pTol	Et 3 N	-	C C l 4	24	90	58:42
	3	tBu	Et 3 N	-	C C l 4	3	> 95	71:29
	4 [c]	tBu	Et 3 N	-	C C l 4	2	> 95	83:17
	5 c	tBu	Et 3 N	10	THF	1	> 95	77:23
	6 c	tBu	Et 3 N	10	Toluene	1	> 95	77:23
	7 c	tBu	Et 3 N	1 0	E t 2 O	1	> 95	77:23
	8 c	tBu	Et 3 N	10	dioxane	1	> 95	77:23
	9 c	tBu	Et 3 N	10	DME	1	> 95	77:23
	10

Table 2 .

 2 Scope for O-P coupling reaction.

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Eur. J. Org. Chem. 2019, 7836-7841 www.eurjoc.org © 2019 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

Eur. J. Org. Chem. 2019, 7836-7841 www.eurjoc.org © 2019 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 7840ration; this observation is consistent with the expected mechanism of the post-functionalization, occurring with inversion of the configuration at P-stereocenter. Besides, 2B was recovered as a single enantiomer in extremely high yield (94%). In summary we have developed a new diastereoselective O-P coupling. The use of the sulfoxide as chiral auxiliary allows performing this P-H functionalization under modified Atherton-Todd conditions, at room temperature and in absence of a catalyst. The diastereoselectivity observed is moderate to high but the diastereomeric compounds may be separated in some cases, thus providing optically pure P-stereogenic scaffolds. Importantly, these new P-stereogenic scaffolds are interesting precursors for many other P-stereogenic molecules as post-modifications may be conducted. Accordingly, this new methodology allows synthesis of PAMPO, the key precursor of P-stereogenic ligands in very efficient and straightforward method.

Acknowledgments

We thank the Indo-French Centre for the Promotion of Advanced Research (IFCPAR/CEFIPRA) for financial support. A. M. is very grateful to CEFIPRA for a doctoral grant. We are also very grateful to Dr. Lydia Karmazin et Dr. Corinne Bailly for singlecrystal X-ray diffraction analysis.


C Carbillet

V Cherkez

M A Skvortsov

M V Feigel'man

F Debontridder

L B Ioffe

V S Stolyarov

K Ilin

M Siegel

D Roditchev

T Cren

C Brun

Spectroscopic evidence for strong correlations between local superconducting gap and local Altshuler-Aronov density-of-states suppression in ultrathin NbN films

HAL is

I. INTRODUCTION

Non-magnetic disorder was initially thought to have little effect on the superconducting properties of conventional thin films, as far as time-reversal symetry is preserved [START_REF] Anderson | Theory of dirty superconductors[END_REF][START_REF] Abrikosov | On the theory of superconducting alloys: I. The electrodynamics of alloys at absolute zero[END_REF] . Decades of intense experimental and theoretical works have ended to the opposite conclusion showing that beyond a critical disorder all systems transit either to a metallic or to an insulating state 3,[START_REF] Feigel'man | Fractal superconductivity near localization threshold[END_REF] . For thin films that are not single-crystals 5 but consist of coupled nanocrystals two different classes of systems were historically considered. A qualitatively different superconducting behavior was observed between so-called granular and homogeneous disordered thin films. In granular ones a poor electrical coupling between the nanocrystals makes the film a disordered array of Josephson junctions 3,[START_REF] Efetov | Phase transition in granulated superconductors[END_REF][START_REF] Orr | Local superconductivity in ultrathin Sn films[END_REF][START_REF] White | Destruction of superconductivity in quench-condensed two-dimensional films[END_REF][START_REF] Jaeger | Threshold for superconductivity in ultrathin amorphous gallium films[END_REF] . Their superconducting properties are controlled by weak Josephson couplings between phases of local superconducting order parameters attributed to individual nanocrystals. In homogeneous ones the nanocrystals are much better electrically coupled to each other. Their superconducting properties are controlled by the subtle interplay between the non-magnetic disorder distribution and electron-electron interactions [START_REF] Feigel'man | Fractal superconductivity near localization threshold[END_REF][10][11][START_REF] Haviland | Onset of superconductivity in the two-dimensional limit[END_REF] .

Nevertheless experimental and theoretical studies performed in the last ten years enabled to show that this historical categorization between granular and homogeneous thin films is lacking of an important link between them. At low disorder in homogeneously disordered films macroscopic measurements show that T c and ∆ monotonously decrease with rising disorder corresponding to k F l decreasing from a value much larger than one toward unity [START_REF] Feigel'man | Fractal superconductivity near localization threshold[END_REF][10][11][START_REF] Haviland | Onset of superconductivity in the two-dimensional limit[END_REF] (k F being the Fermi wavevector and l the elastic electron mean-free path). This effect has been satisfactorily explained by a reduction of the electronic screening upon increasing disorder which reduces the diffusion constant D and is called the "fermionic" mechanism [START_REF] Finkel | stein[END_REF]14 . Many films like, for example, Bi, Pb, MoGe, NbSi behave this way and follow the so-called Finkelstein scenario down to the almost complete destruction of superconductivity 10,11,[15][16][17] . However a different behavior can also take place for sufficiently large disorder in nominally homogeneous films, i.e. typically for 1 ≤ k F l ≤ 3: in amorphous InO x , TiN or NbN the single-particle gap probed by tunneling remains finite and decouples from the decreasing T c , while a pseudogap emerges in a significant temperature range above T c [18][19][20][21] . These latter features are believed to correspond to the so-called "bosonic" scenario, where disorder localizes single-electron wavefunctions and provoke an "emergent granularity" in the superconductor [START_REF] Feigel'man | Fractal superconductivity near localization threshold[END_REF]22,23 . These emergent superconducting puddles were revealed by scanning tunneling microscopy/spectroscopy (STM/STS) in TiN, InO x and NbN 18,20,21,24,25 . From moderate to strong disorder, the long-range phase coherence between the emergent neighboring superconducting islands gradually weakens and eventually destroys, leaving the material with residual incoherent pairing properties 26,27 .

From the above considerations one sees that in fact most films are expected to present an interplay between "fermionic" and "bosonic" effects for strong enough disorder before superconductivity disappears. Nevertheless there is a lack of experimental data combining macroscopic and local measurements on the same system to carefully explore this interplay. Such a detailed study was conducted in NbN 21 and enabled to draw a phase diagram, where at low disorder the fermionic mechanism dominates, while bosonic effects develop at stronger disorder, as exposed above.

In this work we show that, in contrast to the common belief, the emergent gap inhomogeneities can be solely explained by the Finkelstein mechanism, where electron interactions play a crucial role. Our claim is supported by the fact that using scanning tunneling microscopy/spectroscopy in NbN thin films, we found a direct spatial cross-correlation between local superconducting gap maps and the mapping of the Altshuler-Aronov suppression of the local densityof-states (LDOS). Through appropriate modeling we demonstrate that both effects can be linked to local variations of the film's resistance. Due to an interplay between disorder and electrons interaction, these resistivity variations drive corresponding spatial variations of the normal-state STS spectra and the local energy gap values.

II. EXPERIMENTAL RESULTS: LOCAL STS MEASUREMENTS OF THE GAP AND V-SHAPE LDOS

Our samples consist of ultrathin NbN films grown exsitu on sapphire, structured in nano-crystals of lateral size 2-5 nm 28 . Special care was taken in order to minimize the thickness of the surface oxyde layer: the freshly grown samples were introduced into our STM set-up in less than 24 hours after their growth. We selected films having a T c ≈ 0.25 T bulk c = 3.8 K, so that seeming "bosonic" properties like gap inhomogeneities and pseudogap features have already developed. The nominal film thickness is about 2.1 nm. We studied their local superconducting properties by scanning tunneling microscopy/spectroscopy (STM/STS) at 300 mK, using PtIr tips, in an ultrahigh vacuum homemade setup. The presented dI/dV measurements were obtained by numerical derivation of single I(V ) curves. T c was extracted from in situ 4-points electrical resistivity mesurements performed in the STM stage during the same run as the STM/STS measurements. The temperature dependence of the resitivity before the superconducting transition is typical of 2D disordered superconducting films and is shown in Fig. 1a.

STM topography measurements show that the film sur- face is very flat (see Fig. 2a). The size of the observed nanoscale structures correlates well with the size of the nanocrystals characterized independently by transmission electron microscopy (TEM) experiments. A characteristic dI/dV spectrum measured locally is shown in Fig. 1b. It presents a fully gapped LDOS with well-defined superconducting coherence peaks. We also note that ingap states are also present, most probably induced by paramagnetic defects at the NbN/sapphire interface. For energies larger than the ones of the coherence peaks, a strong background is seen instead of recovering a standard normal DOS, as in the Bardeen-Cooper-Schrieffer (BCS) case. Fig. 1c enables characterizing this background: it presents a dI/dV spectrum measured on the same location but over a wider energy range about 20 times larger than ∆. We see that the tunneling LDOS is strongly reduced over an energy range much larger than ∆. This LDOS suppression is typical of systems where electronic correlations combined to disorder reduce the tunneling DOS at the Fermi level (E F ) 29,[START_REF] Altshuler | Electron-Electron Interactions in Disordered Systems[END_REF] , an effect first theoretically explained by Altshuler and Aronov. In the following we refer to this effect as AA. Fig. 1c shows that for e|V | > ∆ the measured LDOS follows a power-law as a function of the applied bias voltage, dI/dV ∝ V α . Please note that here the fit is truly performed using the functional dependence V α without any offset: this curve goes to zero at zero voltage. Nevertheless since the fit is performed in the energy region outside the gap, data points exactly around zero voltages are not plotted in this figure. Such a behavior has also being evidenced for instance in highly disordered metallic thin films close to the insulating transition [START_REF] Richardella | [END_REF] .

Theoretically such a dependence is expected in the framework of nonperturbative extension [START_REF] Ingold | Single Charge Tunneling[END_REF][33][34] of the Altshuler-Aronov theory 29,[START_REF] Altshuler | Electron-Electron Interactions in Disordered Systems[END_REF] . Let us note that this kind of effect is also referred to as dynamical Coulomb block- ade effect in mesoscopic physics [START_REF] Ingold | Single Charge Tunneling[END_REF][33][34] . Interestingly, it was shown that the effect of probing the tunneling DOS of an ultrasmall tunnel junction in a 2D system with electronelectron interaction is equivalent to dynamical Coulomb blockade formalism with an ohmic environment 34 . We have already used such an equivalence when modeling the superconducting proximity effect between a Pb single nano-crystal and a 2D metallic homogeneously disordered Pb film [35][36][37] .

The main finding of our work is a systematic spatial correlation between the local energy gap value ∆(r) and the strength of the AA LDOS suppression, quantified by the exponent α(r) in the power law dI/dV (r) ∝ V α(r) . At each square nanometer of the 300 × 250 nm 2 area shown in Fig. 2a, the local gap value was extracted from the peak-to-peak distance in each dI/dV (r) curve. The extracted map of the gap values is shown in Fig. 2b. The energy gap ranges between 1 and about 1.5 meV. The histogram of the gap values is shown in panel 2e and is close to a Gaussian distribution. The spatial distribution of ∆(r) shows emerging local inhomogeneities seen as lighter and darker puddles. Using the scale bar shown in the topography Fig. 2a, one sees that local puddles having a constant gap identified by a certain local color are smaller than 50 nm. Such a behavior was reported previously in similar systems for comparable disorder strength, but with comparatively less clean and less dense STS spectra 18,24,25,38 . ∆ = 1.2 meV is used below as the average gap value of our film.

We call supergrain a local puddle with a nearly constant gap. We define l ∆ as the characteristic size of such supergrain. We found l ∆ ≈ 27 nm which is much larger than the low-T coherence length ξ(0) ≈ 5 nm. l ∆ corresponds to a correlation length characterizing the spatial extent over which local gap values are correlated. A pre-cise way to extract it is to use the spatial dependence of the radial averaging ρ ∆ (r) of the 2D autocorrelation function of ∆(r) (details about the numerical computations of the correlation and cross-correlation functions can be found in the Appendices section). The spatial decay of ρ ∆ (r) and l ∆ are shown in Fig. 3b.

We provide now for the first time a local mapping of the disorder distribution combined to electron interactions which together are responsible for the spatial distribution of the gap inhomogeneities. At each location the gap value ∆(r) was measured, the local AA exponent α(r) was also extracted in the energy range [5;30] mV, separately for positive and negative energies. We then took for α(r) the average between the values fitted for positive and negative energies. The slight asymmetry in our spectra between positive and negative energies probably comes from a non-constant tip DOS.

The 2D map of the exponent α(r) is presented in panel 2c and its histogram in panel 2f. Most of the values are situated between 0.26 and 0.43. It is seen that the exponent map also presents emerging granularity, with some resemblance with the gap map. Nevertheless the grainy structure of the exponent map looks sharper than the one of the gap map. This suggests that the correlation length l α of α(r) should be smaller than l ∆ . The correlation length l α is extracted following a similar procedure as for ∆(r) above. The spatial dependence of the radial averaging ρ α (r) of the 2D autocorrelation function of α(r) is shown in Fig. 3a. l α also is indicated and yields l α ≈ 18 nm. This value is indeed about two thirds of l ∆ . In order to simplify the visual comparison between the gap map and the exponent map, we used the same color scale for both maps but with a reversed order, which makes it easier to see anti-correlations. This means that if statistically anti-correlations dominate between ∆(r) 2c. This linecut starts at a bright puddle with larger local gap and smaller exponent. 33 nm away it ends at a dark puddle with smaller gap and larger exponent. Each spectrum is spaced by 3 nm from the previous one. Vertically each spectrum is offset by one for clarity including the first spectrum which is the bottom one. Each dI/dV (r) curve is superposed with its fitted curve (shown by red lines) using a power law b × V α(r) . Set-point for spectroscopy V = -10 mV, I = 150 pA. and α(r) one expects to see at the same place r bright puddles in both maps: where a large gap exists (bright puddle) a low α value should also be found (bright puddle). The situation is similar for darker puddles. Looking precisely at both maps Fig. 2b and Fig. 2c and trying to compare the shape and color of local puddles, one sees that anti-correlations are indeed present.

In order to mathematically quantify the crosscorrelations existing between ∆(r) and α(r) we computed their 2D normalized cross-correlation function, which is shown in panel 2g. A very large anti-correlation of about -0.55 at coincident points is found. This confirms that the larger is the local energy gap value ∆(r), the smaller is the local AA exponent α(r) and vice-versa, as illustrated on panels 2h and i. The dI/dV (r) spectrum with a large ∆(r) has a flatter AA background (red curve), and vice versa (blue curve). To extract the spatial extent over which local cross-correlations exist, we calculated the radial averaging ρ cross (r) of the 2D cross-correlation function presented in Fig. 2g. Its spatial dependence is shown on panel Fig. 3c. One sees that the typical correlation length of the cross-correlations is l cross ≈ 25 nm. This is larger than l α and close to l ∆ .

To show the robustness of our results we present in Fig. 4 a linecut of twelve dI/dV (r) spectra measured along the line indicated by a white rectangle in Fig. 2c. This linecut starts at a bright puddle with larger local gap and smaller exponent. It corresponds to the bottom dI/dV spectrum. 33 nm away this linecut ends at a dark puddle with smaller gap and larger exponent. The end corresponds to the top dI/dV spectrum. Each spectrum is spaced by 3 nm from the previous one. It is seen that each individual dI/dV (r) curve is well-fitted by a power law dI/dV (r) = b × V α(r) , where b is a constant.

One could naively think that a direct one-to-one correspondence could exist between the spatial distribution of the NbN nano-crystals encoded in the STM topography and the gap map or AA exponent map. In fact we found no cross-correlation between the local gap or exponent values and the topography of the probed area (see panel 2d). This is in agreement with our previous work 38 . It could be partly due to the inhomogeneous oxidation of the NbN surface if this process would change the one-to-one correspondence between the NbN crystallites distribution and the STM topography. This nevertheless seems unlikely regarding the size of the nanostructures seen in STM which agree well with the size of the nanocrystals measured from TEM experiments.

In contrast, the present results furnish a new way of characterizing locally and quantitatively the underlying disorder distribution combined to electron interaction effects which lead to the appearance of supergrains of size l ∆ . Indeed our results provide a new type of information that was not reported before, to our knowledge. For instance in Refs. 20 or 25, the focus of the STS studies is set on the comparison between local gap values and local peak height values.

We finish this experimental section by giving the numerical values of relevant microscopic parameters that will be used hereafter in our theoretical analysis. The low-temperature coherence length can be estimated as ξ(0) = D/∆ ≈ 5.2 nm. It corresponds to the diffusion coefficient D ≈ 0.5 cm 2 /s obtained from dB c2 (T )/dT data close to T c , see Ref. 28. It is also consistent with our own electrical resistivity measurements in perpendicular high-magnetic field 38 on similar films. Taking for the elastic mean free path l ≈ 0.5 nm 28 , we find for the elastic scattering time: τ = l 2 /3D ≈ 1.7 × 10 -15 s.

III. THEORETICAL MODELING

Our theoretical explanation of the observed results is based on the standard understanding of:

(i) the Altshuler-Aronov anomaly 29,[START_REF] Altshuler | Electron-Electron Interactions in Disordered Systems[END_REF] , properly generalized beyond perturbation theory for both weak and strong coupling 33,39 ;

(ii) the disorder-induced suppression of T c in superconducting films [START_REF] Finkel | stein[END_REF]14 .

Both effects have been described for a spatially homogeneous case. Both have thus to be properly generalized to an inhomogeneous situation, since this corresponds to the case of our experimental results.

The main physical mechanism leading to the suppression of T c in dirty superconductors is through the enhancement of Coulomb repulsion, which in turn reduces the effective attraction in the Cooper channel. An important distinction should here be made between the case of 2D and 3D superconductors. In the 3D case, the whole effect can be simply described in terms of the Coulomb pseudopotential 40 . However, our NbN ultrathin film is a 2D superconductor. Here, in contrast to the 3D situation, one cannot reduce the effect of Coulomb interaction to the modification of just a single Cooper coupling constant. In 2D one should instead track the whole energy dependence of the effective Cooper coupling, which changes considerably within the relevant energy scale of T c . This requires a more sophisticated analysis, which can be done via a renormalization group technique developed by Finkelstein [START_REF] Finkel | stein[END_REF]14 .

In 2D, the effects (i) and (ii) listed above are essentially determined by the film resistivity. Since we want to describe an inhomogeneous situation for these two effects, we assume a model, where the local resistivity ρ(r) fluctuates with the correlation length l α . With particular details relegated to Appendices, we summarize below the main findings.

A. Link between the tunneling LDOS suppression and the local resistance

According to Refs. 29,30,32,33,39 the tunneling DOS suppression in an interacting diffusive normal metal, ν(E), is determined by the spreading resistance between the diffusive scale and the field propagation scale (see Appendix B 1). Relevant eV values (dozens of mV) are much smaller than /τ ≈ 0.33 eV, justifying well the diffusive regime. In the 2D geometry, the log-normal dependence of ν(E) 33,39 can be approximated by a power law with a weakly energy-dependent exponent

α(V ) = R 2πR Q ln ω 0 eV , (1)
where R Q = h/e 2 = 25.8 kΩ is the resistance quantum and ω 0 is the plasma frequency. Taking ω 0 ≈ 10 eV from Ref. 28, V = 10 mV (typical voltage scale for experimental determination of α, see Fig. 1c) and R = 6.8 kΩ (maximal sheet resistance R max before the transition, see Fig. 1a), we obtain α = 0.29. This value is quite close to the mean value 0.35 measured experimentally. A small discrepancy could be related to the use of an underestimated R max that is significantly affected by thermal amplitude superconducting fluctuations, whereas the Altshuler-Aronov effect involves larger frequencies, where this suppression is less efficient. We conclude that the mean value of the AA exponent α(r) is well described by the theory developed in Refs. 33,39.

The key point in our explanation of inhomogeneity seen in α(r) and ∆(r) is an assumption that they both originate from fluctuations of the local 2D resistivity ρ(r) = R + δρ(r). Using the concept of the spreading resistance to calculate δα(r, V ), we obtain (see Appendix B 2):

δα(r, V ) = δρ(r) 2πR Q ln eV D/l 2 α . (2)
Using the experimentally measured dispersion of the AA exponent σ α = 0.028 extracted from panel 2f, its correlation length l α ≈ 18 nm and taking V = 10 mV, we find eV /( D/l 2 α ) ≈ 70 and hence obtain the sheet resistance dispersion σ ρ = 1.1 kΩ. This value corresponds to 16 % of R max . This dispersion value among local resistivities seems quite reasonable regarding the moderate but not too strong disorder existing in our NbN thin films.

We would like here to comment on the justification and meaning associated with the definition of a local resistivity. The correlation length l α ≈ 18 nm of the fluctuations of α exceeds well the electronic mean free path l ≈ 0.5 nm. This implies that our sample is deep in the diffusive regime for the considered energy scale corresponding to α and its variations, thus the local resistivity can be defined. Since the spatial fluctuations of the AA exponent are mainly controlled by the diffusive length scale (the diffusive length L is related to the energy E through the relation L = D/E), these fluctuations enable us to probe the local film resistivity.

B. Link between the local gap inhomogeneities and the local resistances

The additional suppression of the superconducting order parameter in regions with larger local resistivity is explained by the Finkelstein mechanism. In general, one should distinguish between spatial fluctuations coming from small (q < q D ) and large (q > q D ) momenta, where q D = ω D /D with ω D the Debye frequency. For our NbN film we estimate ω D ≈ 300 K, q D ≈ 1 nm -1 . The low-momentum contribution is given by the usual logcube 2D expression

δ∆(r) ∆ ≈ - δρ(r) 6πR Q ln 3 ω D ∆(0) , (3)
cut off at the Debye frequency (see Appendix C 1). The high-momentum contribution can be expressed in terms of the renormalization of the BCS coupling constant λ(r). Substituting the resistivity dispersion σ ρ = 1.1 kΩ into Eq. ( 3), we obtain for the relative gap dispersion σ ∆ / ∆ = 0.074, whereas the experimental value is 0.057 (see panel 2e). Note that Eq. ( 3) contains a large degree of uncertainty: the precise value of ω D being unknown, σ ∆ can only be determined with roughly 50 % accuracy. Nevertheless, the fact that Eq. ( 3) nearly describes the measured gap dispersion indicates that the highmomentum contribution to spatial fluctuations of ∆(r) is absent. This case could physically correspond to a situation, where the 2D resistivity would spatially fluctuate due to the variations of interface resistance between neighboring crystallites (that would lead to small q < q D momenta contribution), while the 3D resistivity within each crystallite would remain identical (no change in the contribution of the q > q D momenta).

IV. DISCUSSION

We have shown that local probe spectroscopic studies are able to demonstrate a direct relation between the properties of the electronic excitation spectrum pertinent to the lowest-energy scale: the superconducting gap (∼ 1 meV) on one hand, and to much higher energy scales (few dozens of meV) relevant to the Coulomb anomaly induced by the Altshuler-Aronov effect on the other hand. This supports the relevance of the "fermionic" mechanism and Finkelstein theory for superconductivity suppression by moderate disorder in NbN thin films not only at the global scale but also locally.

We characterized the nanoscale low-temperature granularity that develops in the superconducting NbN film and have shown that electron-electron interactions play a crucial role in determining this precise granularity. A natural generalization then would be to study and understand the temperature evolution of the inhomogeneity of ∆(r) and α(r), in particular around and above T c , to connect to the pseudogap phenomenon and its current understanding in various materials. Indeed, we point out that for the present ultrathin film the ratio ∆(0)/k B T c ≈ 3.7 is twice larger than the BCS theory prediction 1.76, and significantly larger than the value 2.1 known for thicker and less disordered NbN films 28 , fabricated using the same set-up, procedures and recipees than in the present work.

The increase of the ∆(0)/k B T c ratio with the sheet resistance can be understood within the usual mechanism of strong thermal phase fluctuations 41 , which are known to be important in 2D superconducting systems with small superfluid density (suppressed due to strong disorder). Indeed, the width of the Gaussian thermal fluctuational region in 2D, as seen from conductivity measurements, can be estimated by comparing the Aslamazov-Larkin paraconductivity σ 2D AL = (e 2 /16 ) ln(T /T c0 ) and normal-state conductivity as Gi = (e 2 /16 )R , where T c0 is the mean-field transition temperature. Taking into account that phase fluctuations are about three times stronger than Gi 42 , we obtain an estimate for the renormalized T c , which coincides with the Berezinskii-Kosterlitz-Thouless transition: (T c0 -T c )/T c0 ≈ 3 Gi. With the normal-state resistance R ≈ 7 KOhm (see Fig. 1a), this gives T c0 ≈ 1.4T c . Thus we conclude that the major part of the increase of the ∆(0)/k B T c ratio can be understood as being due to fluctuation suppression of the measured T c with respect to the mean-field value T c0 , which should be compared to the gap width ∆(0). The same conclusion was reached in Ref. 21, where much thicker (actually 3D) NbN films were studied. The above explanation of the enhanced ∆(0)/k B T c and pseudogap phenomenon (discussed in more details in the previous paper 38 ) refers to the "bosonic" mechanism of phase fluctuations, while local correlation between Altshuler-Aronov exponent α(r) and the lowtemperature gap value ∆(r) are nicely explained within the "fermionic" mechanism. In that sense NbN films seem to demonstrate simultaneous action of different superconductivity suppression mechanisms. It would be interesting to see if strongly disordered NbN films exhibit a double-gap behavior found recently in InO x films [START_REF] Dubouchet | [END_REF] . Other methods such as the ones addressing local coherence through scanning Josephson microscopy 44 by mapping the local critical current might also shed important new light 45 , when compared to other local spectroscopic properties such as the local gap map or peak height map. Finally, addressing the LDOS measurements and its mesoscopic fluctuations near T c in disordered thin films similar to ours would also help testing and comparing with recent theoretical predictions [46][47][48] .

V. CONCLUSION

In conclusion, we have demonstrated sharp spatial anti-correlations in highly resistive NbN thin films between the local magnitude of the superconducting energy gap ∆(r) and the value of the exponent α(r) characterizing the local density-of-states suppression at the Fermi level due to the Altshuler-Aronov effect induced by Coulomb repulsion. The correlation lengths characteristic of the emergent inhomogeneities at T T c are l ∆ ≈ 27 nm for ∆(r) and l α ≈ 18 nm for α(r), while the cross-correlation length l cross between ∆(r) and α(r) is l cross ≈ 25 nm. Our results are in agreement with the predictions of a local version of the Finkelstein theory for the "fermionic" mechanism of superconductivity suppression by disorder. This behavior is generally expected for any 2D superconducting materials satisfaying: (a) a strong Coulomb interaction between conduction electrons, (b) a spatial scale l α characteristic of the inhomogeneities of the Altshuler-Aronov density-of-states suppression much larger than the superconducting coherence length ξ(0). Our results suggest that in NbN films the emerging granularity can be fully described by a "fermionic" mechanism. Additional theoretical and experimental work is needed to see if the same claim is valid for other recently discovered 2D disordered materials, including monolayer or bilayer dichalcogenides 49,50 , bilayer graphene 51 or interface oxydes 52 .

No. 20-12-00361 and by the Russian Academy of Sciences Program "Low temperature physics".

APPENDIX A: NUMERICAL COMPUTATIONS OF THE AUTO-CORRELATION AND CROSS-CORRELATION FUNCTIONS

The auto-correlation functions calculated from the gap map presented in Fig. 2b and the exponent map in Fig. 2c are computed using the 2D expression (A1) with periodic boundary conditions to treat the finite size spectroscopic data. We chose such periodic boundary conditions for simplicity. Nevertheless, these latter conditions do not affect the global behavior of the auto-correlation function since the correlation lengths that are extracted from the dependence of the auto-correlation functions are about 1/10 -1/20 of the image size. Thus the spatial dependence of the auto-correlation function is not affected by boundary conditions in our case.

In order to ensure proper normalization, the autocorrelation functions are calculated by subtracting the mean value of the variable of interest and by dividing by the standard deviation σ of this variable. For instance for the local gap values ∆(r) the 2D auto-correlation function ρ ∆ (r) is defined by

ρ ∆ (r) = r (∆(r ) -∆ )(∆(r -r) -∆ ) N σ 2 ∆ , (A1)
where N is the total number of pixels in the ∆(r) gap map. The 2D auto-correlation functions were calculated in this way for both ∆(r) and α(r) data presented in Fig. 2. Since ρ ∆ (r) and ρ α (r) were rather isotropic we decided to circularly average them to extract the correlation lengths l ∆ and l α . The result of the circular averaging of ρ ∆ (r) and ρ α (r) is presented in Fig. 3, where the spatial dependence of ρ ∆ (r) and ρ α (r) is plotted as a function of the distance r. The correlation lengths are determined in such a way that the correlation functions remain above the noise level close to zero. The 2D cross-correlation function ρ cross (r) between ∆(r) and α(r) is calculated in the same way as above using also periodic boundary conditions with the following formula:

ρ cross (r) = r (∆(r ) -∆ )(α(r -r) -α ) N σ ∆ σ α . (A2)
ρ cross (r) is plotted in panel g of Fig. 2. The 2D crosscorrelation function between ∆(r) and the topography Z(r) presented in panel a of Fig. 2 is calculated in the same way, α(r) in (A2) being replaced by Z(r). Finally since ρ cross (r) has a rather isotropic behavior, as seen on Fig. 2g, it was circularly averaged in order to extract the cross-correlation length l cross from the spatial dependence of ρ cross (r).

APPENDIX B: ALTSHULER-ARONOV ANOMALY IN INHOMOGENEOUS FILMS

We consider the tunneling density of states ν(E) in a diffusive normal film, where the local resistivity fluctuates with the correlation length a. The mean free path l, the film thickness d and other parameters satisfy the experimentally relevant set of inequalities: l d D/E a. Here and below we use the system of units with k B = = 1.

Homogeneous film

The theory of the Altshuler-Aronov anomaly, also called zero-bias anomaly, 29,[START_REF] Altshuler | Electron-Electron Interactions in Disordered Systems[END_REF][START_REF] Ingold | Single Charge Tunneling[END_REF]33,39,[START_REF] Finkel'stein | Influence of Coulomb interaction on the properties of disordered metals[END_REF] predicts that the Coulomb-induced suppression of the tunneling density of states can be written in the form ν(E) = ν 0 e -S(E) , where ν 0 is the bare density of states at the Fermi level in the absence of interaction. The function S(E) can be considered as the gauge phase correlation function 39,[START_REF] Finkel'stein | Influence of Coulomb interaction on the properties of disordered metals[END_REF] or the instanton action 33 . In the case of diffusive electron motion it can be written in the form

S(E) = 2 R Q 1/τ E dω ω R(ω), (B1)
where R Q = h/e 2 is the resistance quantum and R(ω) is the spreading resistance between the diffusive length r in (ω) = D/ω and electromagnetic field propagation length r out (ω) = Dω 0 /ω 2 , where ω 0 is the plasma frequency. In the 2D geometry with a constant sheet resistance R ,

R(ω) = R 2π ln r out (ω) r in (ω) = R 4π ln ω 0 ω . ( B2
)
Taking the integral in Eq. (B1) we arrive at the standard result for homogeneous films:

S homo (E) = R 4πR Q ln 2 ω 0 E -ln 2 ω 0 τ . (B3)
For practical purposes, the slow log-normal energy dependence of ν(E) predicted by Eq. (B3) can be replaced by the power law ν(E) ∝ E α , with an almost energyindependent index α(E) = -∂S homo (E)/∂ ln E, leading to Eq. (1).

Inhomogeneous film

Consider now the Altshuler-Aronov anomaly in an inhomogeneous film. We will treat inhomogeneity in a simple model, where the local sheet resistance can be written as ρ(r) = R + δρ(r), with δρ(r) being a Gaussian noise with zero average and the correlation length a. The spreading resistance R(r 0 , ω) now depends on the tunneling position r 0 and can be estimated using logarithmic dependence of resistance in 2D. Therefore, with logarithmic accuracy one may calculate R(r 0 , ω) replacing ρ(r) by ρ(r 0 ) for |r -r 0 | < a and by R for |r -r 0 | > a.

In the experimentally relevant case, E D/a 2 , and therefore r in (ω) a. Then the spreading resistance R(r, ω) can be estimated as

R(r, ω) = ρ(r) 2π ln a r in (ω) + R 2π ln r out (ω) a (B4)
Substituting that into Eq. (B1), we get

S(r, E) = S homo (E) + δρ(r) 4πR Q ln 2 1/τ D/a 2 -ln 2 E D/a 2 , ( B5
) where S homo (E) is given by Eq. (B3). Taking the derivative, we see that the AA exponent fluctuates according to Eq. ( 2), where we identify a with l α .

APPENDIX C: GAP FLUCTUATIONS IN INHOMOGENEOUS FILMS

If the correlation length of the local resistivity fluctuations exceeds the superconducting coherence length, a ξ(0), then the spectral gap at a given point r is determined by the order parameter ∆ in a homogeneous system with the resistivity ρ = ρ(r). The dependence of ∆ on ρ results from the Coulomb repulsion enhanced in a diffusive media [START_REF] Altshuler | Electron-Electron Interactions in Disordered Systems[END_REF] . Assuming the effect can be treated perturbatively, we will rederive the known expression [START_REF] Yu | Fluctuation shift of the transition temperature of thin superconducting films[END_REF][START_REF] Maekawa | Localization effects in twodimensional superconductors[END_REF][START_REF] Takagi | Anderson localization and superconducting transition temperature in two-dimensional systems[END_REF][START_REF] Fukuyama | Superconducting transition temperature of dirty thin films in weakly localized regime[END_REF] focusing on proper cutoffs of energy and momentum integrals.

Order parameter suppression in homogeneous films

Coulomb suppression of superconductivity 53 can be described in terms of the energy-dependent correction to the Cooper channel interaction constant 47 :

δλ ε,ε = - 1 2πν (dq) 1 Dq 2 + ε + ε , ( C1
)
where ν is the 3D density of states at the Fermi energy per one spin projection, both Matsubara energies, ε and ε , are assumed to be larger than T c , and

(dq)f (q 2 ) ≡ 1 d dq x dq y (2π) 2 ∞ n=0 f [q 2 x + q 2 y + (πn/d) 2 ].
(C2) Perturbative account of δλ ε,ε in the normal state reproduces the log-cube shift of T c 54-57 . In the superconducting state, it leads to the shift of the order parameter compared to the clean case. In the low-temperature limit (T T c ), this shift is given by 47

δ∆ ∆ 0 = ω D ∆ dε ε dε ε δλ ε,ε , (C3)
where ∆ 0 is the gap value in the clean system. This formula generalizes the BCS relation δ∆/∆ = δλ/λ 2 to the case of an energy-dependent coupling constant. Since the energies ε and ε in Eq. (C3) are limited by ω D , one can represent the general expression (C1) as δλ ε,ε ≈ δλ low ε,ε + δλ high , (C4)

where δλ low ε,ε is the energy-dependent contribution from low (q < q D ) momenta and δλ high is the energyindependent contribution from high (q > q D ) momenta:

δλ low ε,ε = - 1 2πν
(dq) θ(q D -q) Dq 2 + ε + ε , (C5)

δλ high = - 1 2πν
(dq) θ(q -q D ) Dq 2 , (C6)

and q D = ω D /D is determined by the Debye frequency.

The high-momentum contribution δλ high is just a correction to the Cooper channel constant. According to Eq. (C3) it leads to the perturbative suppression of the order parameter: δ∆ high /∆ = δλ high /λ 2 . This expression can be extended beyond the perturbation theory using the BCS relation that results in the renormalization of the bare gap: ∆ 0 → ∆ 1 , where

∆ 1 = ∆ 0 exp[1/λ -1/(λ + λ high )]. (C7)
Once the high-momentum contribution is incorporated into ∆ 1 , the perturbative shift of the gap due to the lowmomentum part δλ low ε,ε should be determined by Eq. (C3) with ∆ 0 replaced by ∆ 1 .

In order to estimate the low-momentum contribution δλ low ε,ε for our NbN films, we have to compare ω D ≈ 300 K and the transverse Thouless energy D(π/d) 2 ≈ 950 K. Since the latter is larger, one can leave only the n = 0 mode in Eq. (C2), leading to the 2D contribution (the 2D resistivity is ρ = R Q /4πνDd) δλ low ε,ε = -

ρ 2πR Q ln ω D max(ε, ε ) (C8)
and hence to the following shift of ∆:

δ∆ low ∆ 1 = - ρ 6πR Q ln 3 ω D ∆ . ( C9
)
For our NbN films with ρ = R max = 6.8 kΩ, ω D ≈ 300 K and ∆ = 12 K, the low-momentum contribution to the gap suppression according to Eq. (C9) is given by δ∆ low /∆ 1 ≈ 0.55, which is at the border of applicability of the perturbation theory. At the same time, λ high in Eq. (C5) is determined by the 3D diffusion which should be cut off at q ∼ 1/l: δλ high ∼ -(ρ/π 2 R Q )(d/l) ∼ 0.1, yet with an unknown prefactor. Depending on the latter, ∆ 1 /∆ 0 is in the range of several tens of percent, that is a bit too small to describe experimental data.

Inhomogeneous films

In Sec. C 1, we calculated the gap suppression δ∆ with respect to the clean case (ρ → 0). To get gap fluctuations δ∆(r) in an inhomogeneous system with long-range fluctuations of ρ(r), we simply replace ρ by δρ(r):

δ∆(r) ∆ = - δρ(r) 6πR Q ln 3 ω D ∆ + δλ high (r) ln 2 ω D ∆ , (C10)
where δλ high (r) contains the large-momenta contribution:

δλ high (r) = - 2d δρ(r) R Q (dq) θ(q -q D ) q 2 . (C11)
Equation (C10) is derived in the model, where the resistivity [both the 3D ρ 3 (x, y) and the 2D ρ(x, y) related via ρ 3 = ρ/d] exhibits small spatial fluctuations. In a more realistic model presumably applicable to our NbN films, these are the crystalline interfaces that are the sources of resistance fluctuations at scales larger that a few nm. For such a model of structural disorder, highmomentum fluctuations of δ high (r) are absent, and one arrives at Eq. ( 3).

FIG. 1 :

 1 FIG. 1: (Color online) (a) Square resistance of a 2.1 nm thick NbN film, measured in the same stage where STM is performed. Tc is defined when R reaches zero, leading to Tc = 3.8 K. (b) Typical local dI/dV (V ) spectrum measured at T = 300 mK. Set-point for spectroscopy V = -10 mV, I = 150 pA. (c) Local dI/dV (V ) spectrum measured at the same location and temperature as in (b) but on a larger voltage scale. The large depletion of the LDOS seen around EF has a characteristic V-shape due to electron-electron interactions enhanced by disorder. The red (lighter) curve shows a power-law fit of this V-shape dependence, according to the relation dI/dV (V ) = b × V α where b is a constant.

FIG. 2 :

 2 FIG. 2: (Color online) Spatial cross-correlations between variations of local the superconducting gap ∆(r) and the Altshuler-Aronov exponent α(r). Measurements done at 300 mK in zero magnetic field. (a) STM topography of a 300 × 250 nm 2 area of the NbN sample (height variation Z in nm). Scanning parameters: V = -50 mV and I = 50 pA. (b) Color-coded map showing the local energy gap ∆(r) (in meV) measured in area (a) by STS. (c) Color-coded map showing the local exponent α(r) in area (a). (d) Color-coded map showing the cross-correlation between the topography Z(x, y) shown in (a) and the gap map ∆(x, y) shown in (b). No cross-correlations are found. (e) Histogram of ∆(r) values occuring in (b). (f) Histogram of α(r) values occuring in (c). (g) Color-coded map showing the cross-correlation between the gap map ∆(x, y) shown in (b) and the exponent map α(x, y) shown in (c). A strong spatial anti-correlation is found: where local electron-electron interactions are stronger (larger α(x, y)) ∆(x, y) is smaller and vice-versa. (h): Representative dI/dV spectra of the regions seen in (b). Brighter (respectively darker) spectra are measured in brighter (darker) regions of panel (b). (i) Same spectra as in (h) but over an energy scale 20 to 30 times larger than ∆. Brighter (respectively darker) spectra are measured in brighter (darker) regions of panel (c).

FIG. 3 :

 3 FIG. 3: (Color online) Spatial decay of the correlation functions calculated from the radial averaging of the 2D auto-correlation or cross-correlation functions from the data shown in Fig. 2. Each profile starts at the center of each map (corresponding to the origin 0) and extends up to 100 nm. (a) Spatial dependence of ρα(r) defined as the correlation function of the Altshuler-Aronov exponent α(r). (b) Spatial dependence of ρ∆(r) defined as the correlation function of the superconducting energy gap ∆(r). (c) Spatial dependence of ρcross(r) defined as the correlation function of the 2D cross-correlation between the local Altshuler-aronov exponent α(r) and superconducting energy gap ∆(r).

FIG. 4 :

 4 FIG. 4: (Color online) Linecut of twelve dI/dV (r) spectra (shown by black curves) measured along the line indicated by a white rectangle in Fig.2c. This linecut starts at a bright puddle with larger local gap and smaller exponent. 33 nm away it ends at a dark puddle with smaller gap and larger exponent. Each spectrum is spaced by 3 nm from the previous one. Vertically each spectrum is offset by one for clarity including the first spectrum which is the bottom one. Each dI/dV (r) curve is superposed with its fitted curve (shown by red lines) using a power law b × V α(r) . Set-point for spectroscopy V = -10 mV, I = 150 pA.

Acknowledgments

We thank D. S. Antonenko for useful discussions. This work was supported by the French ANR project SUPER-STRIPES under the reference number ANR-15-CE30-0026, by the Russian Scientific Foundation under Grant