Pressure Driven Phase Transition in 1T-TiSe 2 , a MOIPT+DMFT Study
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Pressure Driven Phase Transition in 1T-TiSe , a MOIPT+DMFT Study S. Koley
St. Anthony’s College, Shillong, Meghalaya, 793001, India
The nature of unconventional superconductivity associated with charge density wave order intransition metal dichalcogenides is currently a debated issue. Starting from a normal state elec-tronic structure followed by a charge ordered state how superconductivity in 1T-TiSe arises withapplied pressure is still under research. A preformed excitonic liquid driven ordered state mediatedsuperconductivity is found in broad class of TMD on the border of CDW. Using dynamical meanfield theory with input from noninteracting band structure calculation, I show a superconductingphase appears near about 2 GPa pressure at a temperature of 2 K and this region persists upto 4GPa Pressure. I. INTRODUCTION
Despite its advanced age, superconductivity is still oneof the hottest topic in the strongly correlated materials[1–4]. It has been considered the most extraordinaryand mysterious property of materials for a long time.High-T c superconductors might look like an evergreenresearch theme [5]. The main reason behind this isthe number of opened questions concerning the pairingmechanism which is strongly related to the materials’electronic structure [5]. The basic and till date mostcomprehensive theory of superconductivity was intro-duced by Bardeen, Cooper and Schrieffer [6]. Bardeenet al., discovered that an existing attractive forcebetween two electrons makes energetically favorablebound two electron state and they can act as boson tocondensate without violation of Pauli exclusion principle.The low dimension and other structural symmetry makethe transition metal dichalcogenide (TMD) system likelyto be charge density wave (CDW) ordered. Energy min-imization of these type of electronic system leads to theCDW transition. In the CDW ordered phase the struc-tural periodicity of the system is reorganized to achievestable state, which affects conductivity of the material[7–9]. Ordering in a CDW state and superconductingstate are structurally two different phenomena but theircoexistence and competitive nature is found in many sys-tems [10, 11]. Condensed matter physics has a lots offindings both theoretically and experimentally on thesetwo competing ground states [12, 13]. The findings likeFermi surface instabilities, electron-phonon coupling, or-bital selectivity and antiferromagnetic ground state leadto both CDW and superconducting phase [8, 12, 14, 15].TiSe was one of the first known compounds with CDWground state, and also most studied material till now dueto the puzzling nature of its CDW transition. Here theCDW transition at a temperature 200 K is to a commen-surate state with a 2 × × points that the idea of Fermisurface nesting as a reason for CDW transition is notapplicable here. The normal state of this material is ex-plained as a small indirect band gap semimetal [16, 18–20]. Latest experiments on TiSe like angle resolved pho- toemission spectra [21] explained that the CDW phasein TiSe consists of larger indirect band gap at differ-ent momentum direction of brillouin zone. TheoreticallyCDW here is predicted by electron-phonon coupling andexciton induced orbital selectivity [8], though the CDWground state and low temperature superconductivity stillremains controversial.Recent finding of superconductivity (SC) in doped 1T-TiSe prompted large number of intense research activi-ties [22–26] due to the questionable SC transition follow-ing charge density wave (CDW) [27, 28]. The similarityin the pressure temperature phase diagram of 1T-TiSe with other strongly correlated materials and the semi-metallic behavior of the compound in normal and den-sity wave led to the idea of Overhauser type CDW tran-sition [30], i.e. Bose-Einstein condensation of correlatedexcitons. The superconductivity in Cu x TiSe was foundafter the CDW phase due to doping of Cu confined in aregion around critical doping [22]. Corresponding transi-tion temperatures form a dome-like structure with cop-per doping. Moreover dome formation is mostly foundas the important charatestics of phase diagrams found inheavy fermion compounds, cuprate superconductors andlayered materials [31, 32]. However superconductivity inthese high temperature superconductors is explained asa competing order with antiferromagnetic ground state[32, 33] whereas in 1T-TiSe there is a novel type of su-perconductivity, where emergence of SC ordering is notrelated to any magnetic degrees of freedom [24, 34]. Al-ternatively the SC state close to the CDW state of theparent compound has no effect on each other and thesuperconductivity here is predicted as a conventional SCphase driven by phonon [23]. Around the transition tem-perature the dome formation with doping is then ex-plained as a consequence of the shifting of chemical po-tential above Fermi level caused by the doping inducedelectrons [23]. Though these issues have been explainedpreviously with different theoretical views but there islack of any consistent picture how CDW state evolve intosuperconducting state with pressure. Main goal of thispaper is to find out a consistent theory describing CDWstate to superconducting state transition. What is be-hind the normal to density wave ordering in 1T-TiSe has been already explained [8] from a preformed exci-tonic liquid (PEL) view which causes orbital selectivityand CDW phase. This PEL view [9] is now proved tobe a novel alternative to conventional theories for TMD.I will start from this PEL view to find out high pres-sure phase at high temperature and then I have added asymmetry breaking term to the Hamiltonian to get lowtemperature ordered states. II. METHOD
For 1T-TiSe , normal state followed by incoherent CDWstate with two particle instability is explained fromLCAO + dynamical-mean field theory (DMFT) calcu-lation of a two band extended Hubbard model [8]. Thetwo band model I used to find normal state is H = X k ,l,m,σ ( t lm k + µ l δ lm ) c † k lσ c k mσ + U X i,l = a,b n il ↑ n il ↓ + U ab X i n ia n ib + g X i ( A i + A † i )( c † ia c ib + h.c )+ ω X i A † i A i It has been observed that in the normal state of 1T-TiSe one-particle inter-band hopping is ineffective sothe ordered states must now arise directly as two-particleinstabilities. The two particle interaction, obtained tosecond order is proportional to t ab , which is morerelevant in ordered low T region. The interaction is H res ≃ − t ab χ ab (0 , P ,σσ ′ c † iaσ c jbσ c † jbσ ′ c iaσ ′ , with χ ab (0 ,
0) the inter-orbital susceptibility calculated fromthe normal state DMFT results. Now the new effectiveHamiltonian is H = H n + H HFres , where H n = P k,ν ( ǫ k,ν +Σ ν ( ω ) − E ν ) c † k,ν c k,ν + P a = b, ( k ) t ab ( c † k,a c k,b + h.c. ), with ν = a, b . The residual Hamiltonian H HFres is found bydecoupling the intersite interaction in a generalised HFsense. Now this will produce two competing instabilities,one in particle-hole channel representing a CDW andanother in particle-particle (SC) channel. The HFHamiltonian is H HFres = − p P h i,j i ,a,b,σ,σ ′ ( h n i,a i n j,b + h n j,b i n i,a − h c † i,a,σ c † j,b,σ ′ i c j,b,σ ′ c i,a,σ + h.c. ) (where p isproportionality constant). The CDW phase can befound directly from the particle hole instability with theorder parameter ∆ CDW ∝ h n a − n b i . I studied here thesuperconducting phase with the two particle instabilityin particle-particle channel (with parametrized p=0.1)and the superconducting order parameter can be calcu-lated from ∆ sc ∝ h c † i,a,σ c † j,b,σ ′ i which yields multibandspin-singlet SC.The input for DMFT is taken from the pressure de-pendent LDA DOS which can be calculated from band-width pressure relation D = D exp (∆ p/K ) (where D is bandwidth, D is set 1 for reference value, K is bulk modulus and ∆ p is change in pressure) [44, 45]. Thesuperconducting order parameter ∆ sc is self-consistentlycomputed inside DMFT from the off-diagonal element ofthe matrix Green’s function and it is found that for super-conductivity the inter-orbital pairing amplitude is negli-gible due to the difference in their band energies, so I willbe considering only the intra-orbital pairing amplitude.The total Hamiltonian now ( H = H n + H ( HF ) res ) will besolved within multi orbital iterated perturbation theory(MOIPT)+ DMFT following earlier approaches [7, 8, 38–40]. III. DYNAMICAL MEAN FIELD THEORY
To explore superconducting regime of 1T-TiSe , I usedthe MOIPT+DMFT with pressure dependent DOS. Theanomalous Green’s function: F ( k, τ ) ≡ −h T τ c k ↑ ( τ ) c − k ↓ (0) i satisfying F ( − k, − τ ) = F ( k, τ ) for s wave pairingis introduced in the low temperature and high pressurephase of 1T-TiSe . Now in the presence of superconduct-ing pairing the one-particle Green’s function isˆ G ( k, τ ) ≡ −h T τ Ψ( k, τ )Ψ † ( k, i = (cid:18) G ( k, τ ) F ( k, τ ) F † ( k, τ ) − G ( − k, − τ ) (cid:19) (1)and all the interactions are described through self en-ergy matrix,ˆΣ( k, iω n ) = (cid:18) Σ( k, iω n ) S ( k, iω n ) S ⋆ ( k, − iω n ) − Σ ⋆ ( k, iω n ) (cid:19) (2)where ω n = (2 n + 1) π/β are fermionic Matsubara fre-quencies and S ( iω n ) contains pairing information.Within the infinite dimensional symmetry formalism thek-dependence of anomalous Green’s function will bethrough ǫ k and the self energy is purely k independent[41], so Σ = Σ( iω n ) . Furthermore, since the SC or-der parameter is considered real in this system, then S ( iω n ) = S ⋆ ( − iω n ). Now the full lattice Green’s func-tion can be calculated using Dyson’s equation asˆ G − ( k, iω n ) = (cid:18) iω n + µ − ǫ k iω n − µ + ǫ k (cid:19) − ˆΣ( iω n )(3)= (cid:18) iω n + µ − ǫ k − Σ( iω n ) − S ( iω n ) − S ( iω n ) iω n − µ + ǫ k + Σ( − iω n ) (cid:19) The self-consistency is obtained by putting the conditionthat the impurity Green’s function coincides with theonsite Green’s function of the lattice.Since in TiSe the normal state and the CDW orderedstate is having major dependence on Ti-d and Se-pstate and preformed excitons drive the incoherent nor-mal state to coherent CDW state I will now extend theGreen’s function here to take the inter-orbital hoppingand Coulomb interaction into account and will put or-bital indices on the Green’s function and self energy.Following the earlier formalism [38, 41] to take inter-orbital hopping in Green’s function I will now rewritethe full orbital dependent Green’s function asˆ G ν − ( k, iω n ) = (cid:18) G − νν − S ( iω n ) − S ( iω n ) − G − ⋆νν (cid:19) where G − νν = iω n + µ − ǫ kν − Σ ν ( iω n ) − ( t νν ′ − Σ νν ′ ) ( iω n + µ − ǫ kν ′ − Σ ν ′ ( iω n ))Where ν ( ν ′ stands for a(b),b(a) bands. In the aboveequation Σ ν , Σ ν ′ , and Σ νν ′ are calculated following ear-lier procedure [8]. The electron lattice interaction andinter-orbital coulomb interaction contributes in orbitaldependent self energy. Now to solve the above mentionedmodel within DMFT I have redesigned the IPT which isalready an established technique for the repulsive Hub-bard model in paramagnetic phase [41, 42]. Though itis an approximate method it gives good qualitative ar-guement with the more exact methods to solve repul-sive Hubbard model [8, 38, 43]. Also IPT has been suc-cessfully applied [40] in single band attractive Hubbardmodel. Reasonably this can be used in present contextalso. The IPT technique for superconductivity is con-structed in a way that it should successfully reproducethe self energy leading order terms in both the weak cou-pling limit and large frequency limit and it must be exactin the atomic limit. Considering all this the IPT self en-ergy is computed for superconductivity.In the atomic limit, the second order self energy van-ishes as discussed earlier [40]. Within this modifiedIPT approximation I have solved DMFT equation tak-ing U aa = U bb = U =0.5 eV and U ab =0.1 (This valueswere calculated reproducing the normal state and CDWstate) and the order parameter is computed from the offdiagonal element of the matrix Green’s function in pres-ence of an infinitesimal symmetry breaking field. IV. RESULT AND DISCUSSION
I will show how the new DMFT approach described aboveexplains a range of experimental data. DMFT renormal-izes the LDA band position depending on its occupationby the intra and inter-orbital Hartree terms in self energyand dynamical correlations generically transfer spectralweight in larger energy scale. DMFT many body den-sity of states (DOS) for Ti-d and Se-p band is shown attwo different pressure in Fig.1 at three different tempera-tures. These three temperatures correspond to predictednormal state (250 K), CDW state (100 K), [8] and su-perconducting state (2 K). A clear ‘orbital selective’ gapis found in the CDW state spectral function at both 1.1GPa and 2.3 GPa but in the superconducting state thenumber density in the Fermi level increases significantly.Also at 250 K temperature the orbital selectivity van-ishes. Interestingly with increasing pressure orbital se-lective pseudogap of the CDW phase increases as well as (a) D O S ( a . u . ) Energy (eV) 2 K100 K250 K (b) D O S ( a . u . ) Energy (eV) 2 K100 K250 K
FIG. 1. (Color Online) Density of states at three differenttemperature and two pressure, (a) 1.1 GPa and (b)2.3 GPa.This two different pressure at lower temperature representsCDW state and superconducting state. Red color stands forDOS of ‘a’ band and green color stands for ‘b’ band. DOS fordifferent temperatures are shifted in the y-axis. in the SC phase the number density at the Fermi levelincreases. The increasing number density at the Fermilevel also supports that not only increased pressure butalso doping the system will help in superconductivity.Surprisingly the spectral functions are not showing anyconventional energy gap in the Fermi level. In 1T-TiSe SC phase is following an unconventional CDW state andan excitonic normal state. So superconductivity herecan also be associated with unusual gap. In this sce-nario momentum-resolved one particle spectral functioncan prove if there is presence of any energy gap in theFermi surface.In Fig.2 I show DMFT one-particle spectral functions, A a,b ( k , ω ) = − Im G a,b ( k , ω ) /π at M point at differentpressure. Absence of infrared Landau Fermi liquid quasi-particle poles indicates existence of a incoherent excitonicfeatures in E k,a,b which is associated with a pseudogapin angle resolved photoemission spectra (ARPES) line-shapes in CDW state. Here I have chosen ‘M’ point onlybecause in the CDW state a ‘quasi-particle peak’ (orig-inated due to electron phonon coupling) is found exper-imentally and theoretically [8, 46] at this point. Belowthe superconducting transition temperature a clear ‘gap’can be discernible in ARPES lineshapes at M point atboth 2.3 GPa and 4.2 GPa pressure. This superconduct- I n t en s i t y ( a . u . ) Energy (eV)1.1 Gpa2.5 Gpa4.2 Gpa5.0 Gpa
FIG. 2. (Color Online) Theoretical ARPES spectra at ‘M’point in different pressure at 2 K temperature. As pressureincreases from CDW state to normal state crossing a super-conducting region the spectral density at Fermi level changes.A gap can be detected at the Fermi level at 2.5 and 4.2 GPai.e. in the superconducting state. R e s i s t i v i t y ( no r m a li z ed ) Temperature (K)1.1 GPa1.5 GPa2.3 Gpa3.8 GPa4.2 GPa1.1 GPa2.3 GPa3.8 GPa 0 0.1 0.2 0 2 4 6
FIG. 3. (Color Online) Temperature dependent DMFT re-sistivity at different pressure. DMFT results agree well withprevious experimental results (represented by three types ofcolored points for three different pressure after A.F. Kusmart-seva, et al.[48]). The inset shows the resistivity at low tem-perature at 3.8 GPa pressure in the superconducting phase.Resistivity numerical data is normalized with respect to max-imum value ing ‘gap’ closes with increasing pressure as 1T-TiSe goesthrough a transition into a metallic state. Existence ofsuch type of superconducting gap at M point was alsofound earlier in doped 1T-Tise [26].Next I have computed transport properties of thedichalcogenide in high pressure. DMFT resistivity here iscalculated from dc conductivity. Remarkably the DMFTresistivity (Fig.3) at different pressure shows a very goodaccord with the earlier experimental resistivity data [48].At low pressure region upto 1.1 GPa the resistivity curveresembles with the curve of ambient pressure. The CDWpeak is prominent and resistivity increases from nor-mal state to CDW state and decreases afterwards. Thestrong peak in the resistivity curve signaling the CDWtransition becomes gradually weak with the increase inpressure, and the maxima in - dρ ( T ) /dT also moves tolower temperature. In the low pressure phase resistiv- ity above transition temperature shows a semimetallicbehavior and the same is suspected well below the tran-sition temperature[47]. Further with increase in pres-sure in the high temperature region 1T-TiSe undergoesmetallic behavior where resistivity manifests almost lin-ear dependence with temperature in full qualitative ac-cord with experiment[48]. Also with increasing pressurethe CDW peak in resistivity decreases and is destroyedcompletely above 2 GPa pressure and superconductivityis observed at about 2 K temperature. This excellentlydescribes experimental resistivity data [48] at differentpressure (Fig.3). In the inset of (Fig.3) the low temper-ature part of resistivity at 3.8 Gpa pressure is shown inan extended frame.In Fig.4 I show the normalized optical conductivity asa function of temperature and pressure. An extendedshoulder like (right inset of Fig.4) feature is observed atlow temperature in the optical conductivity which re-duces with increasing temperature to 6 K in the CDWstate. This shoulder like feature is due to the supercon-ducting gap found in DMFT ARPES spectra. Thoughthere is still some sign of gap in optical conductivity at1 GPa pressure but the off diagonal Green’s functionsshown in the inset display a comparatively larger integralthan the superconducting ones which concludes that thegap is due to the CDW fluctuations. Optical conductiv-ity is shown here till 0.7 eV without losing any importantfeature because I have considered only two band closer tothe FL. Presence of superconducting gap in the spectralfunction and observation of sharpness of the low energyoptical conductivity is highly contrary to each other: itis not a Drude peak because at lowest temperature alsothe compound shows orbital selective non Fermi liquidbehavior in DMFT. Moreover it is due to the coherencewhich sets in due to the superconducting order. If I checkthe off-diagonal Green’s function at three different pres-sure at 2 K which also gives superconducting order pa-rameter, the decrease in both the peaks is perceptiblewith increasing pressure. This detectable change in or-der parameter confirms the presence of phase transitionat particular pressure and temperature.DMFT Fermi surface (FS) map at different temperatureis given in Fig.5. FS in 1T-TiSe does not rebuild acrosstransition from normal state to CDW state in 1T-TiSe .This was one of the pillar to suggest existence of a dy-namically fluctuating excitonic liquid at high tempera-ture which is being coherent at low temperature i.e. thesystem undergoes transition into CDW state. But stillenough information can be collected from FS in transi-tion from CDW to superconducting phase. The bandpockets spread out at normal state and, more impor-tantly, a brighter ring structure appears at the M pointin the CDW state. Now below superconducting tran-sition temperature a much brighter elliptic structure inthe middle of the hexagonal arms can be observed. Sincethese are found in the superconducting state so it may bea direct consequence of phase transition. Further the FSalso shows isotropic s-wave superconducting gap. More- O p t i c a l c ondu c t i v i t y ( no r m a li z ed ) Energy (eV) 2 K, 1 Gpa2 K, 2 Gpa2 K, 4 Gpa10 K, 1 Gpa10 K, 2 Gpa10 K, 4 Gpa 0.1 0.2 0 0.3 O p t c ondu c t i v i t y Energy (eV) 0 7e-05-1.5 0 1.5 I m F ( ω ) Energy (eV)
FIG. 4. (Color Online) Optical conductivity (normalized withrespect to the highest value) at three different pressure and attwo temperature for 1T-TiSe . The same (lower inset) at lowenergy presenting extended shoulder like feature appearingdue to superconducting gap. Upper inset shows correspondingoff diagonal Green’s function at 2 K temperature. over in the superconducting state there are electronlikepockets which develop from the sinking of band belowFL. In case of doped samples of the same similar struc-ture has been found in FS [26]. Now if this theory ofsuperconductivity in this TMD is to be right then thesuperconducting order parameter which is determinedself consistently within DMFT has to follow its natureof (1 − T /T c ) / . In Fig.6 main panel I have showntemperature dependent superconducting order param-eter which is calculated self consistently from DMFT:∆ sc ∝ h c a ↓ c a ↑ i = − | p | π R −∞ Im F ( ω ) dω . The order pa-rameter is following its nature and becomes almost zeronear 2K, i.e at the transition temperature. Thus suchquantitative argument built upon a novel PEL view be-tween DMFT results and previous theoretical and experi-mental results support strongly the idea of a dynamicallyfluctuating excitonic liquid at high T giving way to a low- T superconducting ordered state.However, to be a dependable theory the same formu-lation must also comprehensively describe transport aswell. In DMFT, this task is simplified: it is an excel-lent approximation to compute transport co-efficients di-rectly from the DMFT propagators G a,b ( k, ω ) [49], since(irreducible) vertex corrections rigorously vanish for one-band models, and turn out to be surprisingly small evenfor the multi-band case. In Fig. 3 resistivity curve atdifferent pressure is shown which matches well with pre-vious experiment both qualitatively and quantitatively.Furthermore I have also fitted the resistivity curves from3K, above superconducting transition, to 30 K well be-low T CDW to find out the actual procedure from CDWto superconducting transition. Until 4 GPa pressure theresidual resistivity (see inset of Fig.6) manifests a largechange. Whereas only some marginal change in residualresistivity is found after superconducting phase. Detailstudy of the resistivity temperature exponent n, calcu-lated from fitting the resistivity with R ( T ) = R + AT n ,shows also huge change across the pressure range where (a) −π π k x −π π k y -0.2 0 0.2 0.4 0.6 0.8 1 (b) −π π k x −π π k y -0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 (c) −π π k x −π π k y -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 FIG. 5. (Color Online) DMFT Fermi surface map in the(a)superconducting, (b)CDW and (c)normal state. superconductivity appears and at the pressure wheretransition temperature is maximum in the dome this ex-ponent ‘n’ shows local minimum. Outside the supercon-ducting dome the value of ‘n’ remains about 3.0, whichis in stark conflict with the idea of electron-electron orelectron-phonon scattering which gives value of n as 2or 5 respectively. This fact is rather unusual but thesame is found in experiment also [48]. Wilson [50] pro-posed this higher power law temperature dependence ofthe resistivity as scattering from low density band to ahigher one. Though at the superconducting pressure re-gion this exponent dovers around a value of 2.6 whichrepresents a quantum critical scenario and quantum fluc-tuation around that region.In the LDA bands of TiSe there is negative indirectband gap. Increased pressure or doping closes the gap ∆ sc Temperature (K) 0.5 1 1 2 3 4 5 R ( no r m a li z ed ) Pressure (Gpa) 3 1 2 3 4 5 E x ponen t Pressure (Gpa)
FIG. 6. (Color Online) Temperature dependency of super-conducting order parameter in the main panel. The left in-set shows the residual resistivity (normalized with respect tothe highest value) at different pressure. The residual resis-tivity shows good agreement with the previous experimentalresult (represented by black points after A.F. Kusmartseva,et al.[48]). The right inset shows the resistivity exponent atdifferent pressure. and some of the low energy levels cross the Fermi en-ergy. Thus with increasing pressure CDW order is alsodestroyed. Increasing the number density at the FL en-hances superconducting pairing correlations in a system.It has been observed that with increasing pressure thenumber density at the FL (D(E f )) increases. Such an in-crease in D(E f ) in turn increases superconducting tran-sition temperature T c in many systems as expected alsofrom the formula K B T c = ~ ωexp − / [ D ( E f ) V ]. Whereasincrease in pressure beyond 4 GPa closes the supercon- ducting gap found at M point and the dichalcogenidebehaves like a metal. V. CONCLUSION
In the light of these DMFT results, it can be concludedthat with pressure the preformed excitons in the normalstate drives the compound to undergo a CDW supercon-ducting phase transition. Though conventional s-wavegap is absent in the density of states but ARPES spectrashows the presence of superconducting gap at M point,which is confirmed in FS plots also. An electronlikepocket is found to grow with applied pressure in the su-perconducting phase of this dichalcogenide. My approachhighlights the role of pressure associated with excitoniccorrelation in small carrier density systems. Based upona combination of high pressure LDA bandstructure and aDMFT treatment of dynamical interband excitons cou-pled to phonons, I have shown that this calulation de-scribes a wide variety of features in this system. I havepresented both qualitative and quantitative scenario forthe emergence of unconventional superconductivity as adirect outcome of exciton phonon coupling at differentpressure. Also in other parent or impurity doped TMDssuperconductivity arises from normal states following aCDW order the present scenario should be relevant to allthese cases.SK acknowledges useful discussion and collaboration onsimilar systems with Arghya Taraphder. [1] H. Hosono and K. Kuroki, Physica C , 399 (2015).[2] A. L. Ivanovskii,
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