Breakdown of the Migdal approximation at Lifshitz transitions with giant zero-point motion in H3S superconductor
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Breakdown of the Migdal approximation at Lifshitz transitions with giant zero-pointmotion in the H S superconductor
Thomas Jarlborg , , Antonio Bianconi , , DPMC, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland RICMASS, Rome International Center for Materials Science Superstripes, Via dei Sabelli 119A, 00185 Rome, Italy Solid State and Nanosystems Physics, National Research Nuclear University MEPhI(Moscow Engineering Physics Institute), Kashirskoye sh. 31, Moscow 115409, Russia, Institute of Crystallography, Consiglio Nazionale delle Ricerche, via Salaria, 00015 Monterotondo, Italy
While 203 K high temperature superconductivity in H S has been interpreted by BCS theoryin the dirty limit here we focus on the effects of hydrogen zero-point-motion and the multibandelectronic structure relevant for multigap superconductivity near Lifshitz transitions. We describehow the topology of the Fermi surfaces evolves with pressure giving different Lifshitz-transitions.A neck-disrupting Lifshitz-transition (type 2) occurs where the van Hove singularity, vHs, crossesthe chemical potential at 210 GPa and new small 2D Fermi surface portions appear with slowFermi velocity where the Migdal-approximation becomes questionable. We show that the neglectedhydrogen zero-point motion ZPM, plays a key role at Lifshitz transitions. It induces an energyshift of about 600 meV of the vHs. The other Lifshitz-transition (of type 1) for the appearing ofa new Fermi surface occurs at 130
GP a where new Fermi surfaces appear at the Γ point of theBrillouin zone here the Migdal-approximation breaks down and the zero-point-motion induces largefluctuations. The maximum T c = 203 K occurs at 160 GPa where E F /ω = 1 in the small Fermisurface pocket at Γ. A Feshbach-like resonance between a possible BEC-BCS condensate at Γ andthe BCS condensate in different k-space spots is proposed. PACS numbers: 74.20.Pq,74.72.-h,74.25.Jb
Introduction
The recent discovery of superconductivity in pressur-ized sulfur hydride metal with critical temperature withT c above 200K [1, 2] has provided experimental evidencethat the coherent quantum macroscopic superconductingphase can occur at a temperature higher than the low-est temperature ever recorded on Earth (-89.2 o C) [3, 4].High temperature superconductivity has been found inother hydrides [5] like pressurized
P H with T c at about100 K [6], and it has been predicted in other hydrideslike yttrium hydride [7]. Superconductivity in pressur-ized hydrides was proposed by Ashchroft and his collab-orators [8, 9]. The Universal Structure Predictor: Evo-lutionary Xtallography, USPEX, code [10] available to-day has allowed to predict the structure and high tem-perature superconductivity in pressurized sulfur hydride[11–13]. The prediction of metallic H S with the Im ¯3 m lattice symmetry [12, 13] has now been confirmed byx-ray diffraction experiments [2] above 120 GPa whileat lower pressure different stoichiometry and structureare found [14, 15]. The low mass of H atom is push-ing the sulfur-hydrogen T u stretching mode and the T u phonons at Γ to high energy, so the energy cut-off forthe pairing interaction ω =150 ±
50 meV . The super-conducting temperature was predicted by employing theAllen-Dynes modified McMillan formula [11–13], and theMigdal Eliashberg formula [16–19]. Most of these workshave used the Eliashberg theory with BCS approxima-tions for isotropic pairing in a single band metal, assum-ing the dirty limit, which reduces a multiband to an ef- fective single band metal, and the Migdal approximation ω /E F <<
1. More advanced theories have used density-functional theory including calculations of the effectiveCoulomb repulsion [20, 21]. Within the Eliashberg theorythe Migdal approximation assumes that the electronicand ionic degrees of freedom can be rigorously separatedin agreement with the Born-Oppenheimer approxima-tion, which is valid for metals where the chemical poten-tial is far away from the band edges. The breakdown ofMigdal approximation was observed in the multi-gap su-perconductor Al x M g − x B [22–25] where the band edgeof the σ band fluctuates across the chemical potential dueto zero point motion [26] The breakdown of the Migdalapproximation in a multigap superconductor is relevantsince it requires the correction to the chemical potentialinduced by pairing below the critical temperature [25] ig-nored in the standard Eliashberg theory. Moreover wherethe Migdal approximation breaks down in a multigapsuperconductor it enetrs in a unconventional supercon-ducting phase where the coexisting multiple condensatescould be either in the Bose Einstein condensate (BEC)regime or in the BEC-BCS crossover regime. In thesecomplex multi gaps superconductors where BEC, BEC-BCS and BCS condensates coexist the exchange inter-action between different condensates [25, 27–29] whichis neglected in the BCS approximations, but could be-come a relevant term both to increase or to stabilize hightemperature superconductivity. This quantum term is acontact interaction, given by the quantum overlap be-tween the condensates, which increases the condensationenergy [25, 27–31] and the critical temperature via theshape resonance, analog to the Fano Feshbach resonancein ultracold gases. It has been found that the critical tem-perature shows the maximum amplification where one ofthe condensates is in the BCS-BEC crossover, which canoccurs on the verge of a Lifshitz transition [3, 4, 30–35]At the Lifshitz transition, a change of the topology of theFermi surface is induced by pressure or doping and it hasbeen shown to control high temperature superconductiv-ity in iron pnictides [30, 31].Since H S is a multiband metal and Lifshitz transitionscould occur by increasing pressure [3, 4] it has been pro-posed that it is a multigap superconductior near Lifshitztransitions [27–29], where also multi scale phase separa-tion [32] at a the Lifshitz transition could appear, similarto what has been observed in the cuprates [34–37].The coupling between the electronic and the atomiclattice degrees of freedom in sulfur hydrides at zero tem-perature has been neglected in previous calculations ofthe electronic structure assuming a very large Fermi en-ergy. On the contrary in the case of band edges close tothe chemical potential and near Lifshitz electronic topo-logical transitions the zero point motion (ZPM) cannotbe neglected. It is known that the ZPM modifies the bandstructure itself [38–44]. with corrections of the band gapenergy which can be larger than those induced by corre-lation. Moreover, the lattice disorder from ZPM will alsocause band broadenings, as it has been demonstrated inearly investigations [41]. Such effects have been shown tobe important for many properties in several different ma-terials, even if their atomic masses are larger [42–44] andthe lattice disorder can perturb spin waves and phonons[45]. The zero point motion is expected to be large in H Sbecause of the small mass of H atoms, high frequency S-H stretching modes, and the double well potential forhydrogen in H S [3] which is similar to the well studiedcases of ice and biological macromolecules.In this work we discuss the effect of lattice compres-sion and zero point motion on the electronic structureof H S in the pressure range above 120 GPa wherethe metallic Im ¯3 m phase is stabilized by the ZPM ofhydrogen atoms. We first focus on the effect of thelattice compression on the large Fermi surface, identifiedwith No.2 in ref. [3] and on the small hole Fermisurface pockets at Γ identified with No. 4 and No.5 inreference [3]. In particular we discuss the character ofvan Hove singularity (vHs) in the large Fermi surface,giving a peak in the density of states, and how its movestoward the chemical potential by increasing pressure.Above the critical pressure 210 GPa the vHs crossesthe chemical potential and we find that tubular Fermisurface portions with two-dimensional character in thek-space appear, indicating a 3D-2D topological Lifshitztransition of type 2. Second, we study the effect of thezero point motion on the van Hove singularity and wefind a very large energy shift of the order of 600 meVof the vHs due to ZPM. Third in the pressure range120-180 GPa the amplitude of ZPM of hydrogen-sulfurstretching mode is shown to be larger than the S-H bond splitting in the R3m structure therefore it stabilizesthe symmetric Im ¯3 m structure. In this pressure rangewe have studied the Lifshitz transitions of type 1, forthe appearing of new small Fermi surface pocket atΓ at 130 GPa. Finally we compare our results withrecent experiments showing that the isotope coefficientfor the superconducting critical temperature divergesand the critical temperature goes toward zero at 130GPa in agreement with predictions of the BPV theory[27–31] for a Lifshitz transition of type 1. Moreover themaximum critical temperature, 203 K, appears wherethe Fermi energy in the small Fermi hole pocket at Γ isof the order of the pairing interaction ω as predicted inref. [27–31]. The Band Structure
The high pressure phase of metallic H S has the cubicSpace Group: 229 with Im ¯3 m lattice symmetry. The Im ¯3 m lattice structure can be described by the smallBody Centered Cubic (bcc) unit cell, which has been usedby all previous calculations providing the electronic banddispersion in the bcc Brillouin zone (bcc BZ).Here, we use also an alternative simple cubic unit cellmade by 8 atoms per unit cell with a simple cubic Bril-louin zone (sc BZ), which permits an easier comparisonwith a traditional group of superconductors, namely theA15 compounds. In fact A15 compounds have a latticestructure with the cubic Space Group: 223 belonging tothe same ditesseral central class, or galena type, of cu-bic space groups with the same Hermann-Mauguin pointgroup m ¯3 m .We have obtained the band dispersion in the large sim-ple cubic (sc) Brillouin zone which grabs more details ofthe electronic band dispersion in the complex bitruncatedcubic honeycomb lattice. Here S sites form a bcc lattice(exactly as Si in the A15 compound V Si) with linearchains of H on the limits of the sc cell, similar to how thetransition metals form linear chains in A15 superconduc-tors. These results allow a more easy comparison of theelectronic bands between the two type of materials.We show the electronic self-consistent paramagneticbands for the bcc BZ in Fig. 1 and in Fig. 2 for the scBZ. In this last picture the simple cubic unit cell contains8 sites totally. The calculations have been performed bythe linear muffin-tin orbital (LMTO) method [46] and thelocal spin-density approximation (LSDA) in the frame ofstandard methods [47–50]. The basis set goes up through ℓ =2 for S and ℓ = 1 for H. We have found that the S-2p core levels are always far below the valence band re-gion at these pressures (about 12.2 Ry below E F ). TheWigner-Seitz (WS) radii are 0.38 a for S and 0.278 a forH. The k-point mesh corresponds to 11 points between Γand X , or 1331 points totally in 1/8 of the BZ, or withfiner k-point mesh for plots of the bands along symme-try lines. One spin-polarized calculation is made for thelargest volume starting from an imposed ferro-magnetic(FM) configuration. All local FM moments converge to −30−25−20−15−10−5051015 ( E − E F ) ( e V ) H S (bcc BZ)
H NN P Γ FIG. 1: (Color online) The band structure for H S at sym-metry points for the small bcc unit cell with a =5.6 a.u., cor-responding to P = 210 GP a in the bcc Brillouin zone (bccBZ) −30−25−20−15 −10−5051015 ( E − E F ) ( e V ) H S Γ X M Γ R M R X
FIG. 2: (Color online) The band structure for H S at symme-try points for the simple cubic large unit cell, in the simplecubic BZ at a pressure of 210
GP a zero, which shows that FM and FM spin fluctuations areunlikely. The low-lying s-band on S is very similar to theSi-s band in the A15-compound V Si [47], but the highpressure in H S makes it to overlap with the S-p band.In contrast, the Si-p band in V Si is separated from theSi-s band. The separation is never complete in H S , butit is visible as a dip in the DOS at 13 eV below E F for −25 −20 −15 −10 −5 0 500.511.522.5 H S (E−E F ) (eV) D O S ( c e ll ⋅ e V ) − a=5.6 a.u.a=5.8 a.u.a=6.0 a.u.a=6.2 a.u. FIG. 3: (Color online) The total DOS for H S with variablelattice constants between 5.6 and 6.2 a.u. the largest lattice constants, see Fig. 3.An approximate value for the pressure, P , calculatedas a surface integral using the virial theorem [51], is usefulto get insight to the relative contribution from differentatoms. These total pressures ( P ) amount to 0.98 and -0.3Mbar at the two extreme lattice constants. The partialS- and H-pressures increases much more on H than on Swhen the lattice constant is reduced. This together withthe charge transfers, indicate that the S sublattice is morecompressible than that of H. A charge transfer from S toH at high pressure will enforce the hardening of H. Moreprecise values of P , shown in Table I, are obtained fromthe volume derivative of the calculated total energies, andwe obtain 180 GPa for a =5.5 a.u. and 73 GPa for a =6.1a.u., which agree well with ref. [16]. TABLE I: Lattice constants a , DOS at E F (in units ofstates/eV/Cell), pressure P , the Hopfield parameter NI in eV / ˚ A for S and H, λ , and T C estimated from the McMillanformula. The phonon moments are 600 and 1800 K for S andH, respectively, and ω log is 1500 K. a (a.u.) N ( E F ) P (GPa) NI ( S ) NI ( H ) λ T C (K)5.4 1.00 300 5.6 4.3 1.0 1455.6 0.99 215 5.5 4.3 1.0 1435.8 0.90 150 4.5 3.4 0.81 1096.0 0.88 95 4.1 3.1 0.73 936.2 0.90 50 3.9 2.8 0.70 73 The electronic numerator
N I of the electron-phononcoupling constant λ = N I /K (where the force constant K = M ω is taken from experiment) is calculated inthe Rigid Muffin-Tin Approximation (RMTA) [52, 53].The matrix elements for d − f -scattering in S and p − d scattering in H are missing, which leads to some under-estimation of the total λ . Note also that LMTO usesoverlapping Wigner-Seitz (WS) spheres with no contri-bution to I from the interstitial region. The values of N I on H and S are reasonably large in comparison tothe values in transition metals. The low local DOS iscompensated by larger I and the scattering to p-states,while in the A15’s large scattering arises mainly from d-states. Our N ( E F ), shown in Table I, are in agreementwith the work of Papaconstantopoulos et al. [16], whichquite unusually are largest at low volume and large P .This is because of the band edges that cross E F and be-come occupied at high P . But for even larger volume N ( E F ) behaves normally again and it increases when P goes down, as shown by Papaconstantopoulos el al. [16].The matrix elements I show strong P -dependences, and N I (the so-called Hopfield parameter) increases steadilywith P in agreement with ref. [16]. The absolute valueof λ is smaller than in ref. [16]. This can partly be dueto the smaller basis in our case, and also because of theuse of WS-spheres in LMTO instead of non-overlappingMT-sphere geometry in LAPW [16].In the estimation of T c from the McMillan equationanother uncertainty concerns the Coulomb repulsion µ ∗ .Many T C -calculations use µ ∗ =0.11-0.13, but theories forcalculation of µ ∗ are approximate or unreliable [54]. Re-tardation can make µ ∗ large and screening makes it smallor even negative depending on the band width and thephonon frequency [54]. Here we use an empirical formulafor µ ∗ proposed by Bennemann-Garland [55], which leadsto a small value of the order 0.03. Thus, we caution thateven if our calculated T c from the McMillan formula willbe large it is very approximate, similar to what has beenconcluded in the other works. Our T c ’s are of the order145 and 75 K between the two extreme lattice constantswhen the total coupling strength goes from 1.0 and 0.7.These T c ’s are reduced to 95 and 40 K between the ex-treme values of a if µ ∗ is 0.13. The different contributionsto the total λ from each S and H atoms are comparable.Moreover, the total λ itself is not unusually large. Us-ing the McMillan formula, valid only for a single effectiveband BCS system in the dirty limit, the high T c is mainlybecause of the large phonon frequency pre-factor ( ω log )in the equation.In order to investigate the pressure effects on the elec-tronic structure we show in Fig. 3 the different total DOSfor different lattice constant a of the perovskite structurechanging from 6.2 to 5.6 a.u. The total DOS at E F perS-atom is about 3.5 ( Ry ) − and 1 ( Ry ) − per H-site,compared to the order 20 per V in V Si or in elementaryV and Nb. This is not surprising because of the largewidth of the wide band in H S having its bottom at 2Ry below the chemical potential, see Fig. 3, while forincreasing number of (d-) electrons in transition metalA15 compounds the total band width is more like 2/3 ofa Ry [47].The charge within the H WS-sphere increases from 1.3 to 1.4 el./H when the lattice constant decreases from6.2 to 5.6 a.u., and the H-s charge goes from 0.95 to1.0. This fact justifies somewhat the use of LSDA forH even though atomic H with exactly 1 electron is bestdescribed by the Hartree potential only. The resultsshow a very large effect of the pressure on the lowestdispersive bands with H s character, and the energy shiftof the narrow peak of DOS at the chemical potentialdue to the van Hove singularity. The shift is small inthe figure because of the large energy scale.
Hydrogen Zero Point Motion Effects
Usually the atomic velocities ( v i ) from the vibrationsare much slower than the electronic velocities, and theelectronic structure can relax adiabatically at all times.Therefore, the electronic structure calculations usuallycan neglect v i , as in ’frozen-phonon’ calculations. How-ever it is well-known that the lattice of real materialsbecomes distorted at large T (’thermal disorder’) andsome distortion remains at T =0 due to the ’zero pointmotion’, ZPM. Thermal disorder and ZPM will modifyand broaden the bands compared to the case of a perfectlattice because the potentials at different sites are not ex-actly the same. Each atomic (i) lattice position deviatesfrom its average position by u ( t, T ), because of the exci-tation of phonons The time average of u ( T ) for harmonicuncorrelated vibrations are well represented by a Gaus-sian distribution function with width (FWHM) < u > ,which tends to ~ ω/K at low T (ZPM) and to 3 k B T /K at high T , see ref. [56].The Debye temperature for H phonons ( ∼ K ) insulfur hydride is much higher than T c and the disorderamplitude from ZPM is almost the same as at T = T c .The maximum superconducting gap (2∆) would be ∼ T =0 in the large Fermi surface, i.e., the bandNo.2 using the notations to identify the 5 bands crossingthe Fermi level introduced in ref. [3]. With the parame-ters as in the calculation of λ , and ∆ = V , the amplitudefor the potential modulation for phonons [57], one canestimate that u ∆ , the displacement of phonons that leadto superconductivity should be ∼ H S (having a unitcell of 64 atoms) compared with alternative perturbativeapproaches for evaluation of the band energy changes asfunction of displacement (u) based on the Allen-Heine-Cardona (AHC) method, which is suitable for simple sys-tems like carbon nanotubes [38, 39]. While AHC typeapproach may seem more sophisticated, when it is ap-plied to complex 3-D materials it is more important toconsider good statistics and rely on experimental infor-mation on force constants [42]-[45]. The important pointof our approach is that a large supercell is used in order −25 −20 −15 −10 −5 0 5051015 H S a=5.6 a.u.(E−E F ) (eV) D O S ( c e ll ⋅ e V ) − ordereddisordered FIG. 4: (Color online) The DOS for H S for cubic 64-sitesupercells. The (blue) thin line is the DOS for the perfectlyordered supercell. The (red) heavy line shows the DOS forthe disordered supercell with u S =0.01a and u H =0.033a ZeroPoint Motion to have a good statistics for the individual u of each atomso that the bands energy changes as function of increas-ing disorder can be determined properly. The averageatomic displacements and their 3-D distribution are de-termined from the phonon spectrum and atomic massesand it does not make sense to attempt an ab-initio cal-culation of u using frozen zone-boundary phonons, andtight-binding method.Calculations for disorder within supercells with 64atoms for FeSi [42], or even less (48 atoms) for purplebronze [43], have shown that different generations of in-ternal disorder in the cells produce similar results (as longas the disorder amplitude < u > are the same). Sym-metry makes the bands identical in different irreducibleBrillouin zones (IBZ) for ordered supercells, and thesebands have their exact correspondence in the bands ofthe small cell. But this symmetry is lost if the atoms aredisordered, and the bands have to be determined in halfof the BZ.With a force constant K = M ω of 7 eV/˚A we obtainan average amplitude < u > of the order 0.15 ˚Afor ZPM.This is close to 10 percent of the H-H distance, whichaccording to the Lindemann criterion suggests that theH sublattice is near melting [56]. The < u > -amplitudefor the S sublattice is normal, because of its large mass,and it is probable that the rigid S-lattice is important forthe stability of the structure in which the H-atoms arerather loosely attached to their ideal positions.The ZPM generates different shifts of the Madelungpotential at different sites. This modifies the band en-ergies (∆ ǫ ) and leads to ’fluctuations’ (there is a spread of the eigenvalues in different IBZ) of ∆ ǫ , in particularat the band edges. The low-T energy band fluctuationsin materials with narrower band widths are known to beabout 20 meV for u in the range 0.03-0.04 ˚A[42]-[45].From an extrapolation of these values to the conditionsin H S (larger < u > ) we estimate that the band energyfluctuation can be larger than 150 meV for H-bands.The band No.2 of H S in Fig. 3 is very wide, about2 Ry for the high lying valence bands, and the S-s bandoverlaps with the S-p band. This makes the band dis-persion and Fermi velocities high in large portions of thek space, where the effects of energy band broadening inthese points of the k-space is less important. However inportions of the K-space around the Γ-X-M path in the scBZ, where the van-Hove singularity crosses the chemicalpotential the Fermi velocity is getting small and strongdynamical fluctuations controlled by the zero point lat-tice fluctuations are expected to be relevant.The electronic calculations for the ZPM in H S atP=210 GPa, a =5.6 a.u., have been carried out usinga large supercell in which the lattice is disordered. Eachatom is assigned randomized displacements, u x , u y , u z insuch a way that the distribution of all displacement am-plitudes ( | u | ) has a bell shaped (Gaussian) distributionwith the FWHM width equal to the averaged displace-ment amplitude ( < u > ) at was described before and inref. [42]. In Fig. 4 we show the energy renormaliza-tion of the DOS due to the calculated zero point motion(ZPM) . Here we consider a 2x2x2 extension of the cubicunit cell with 64 atoms totally which permits to calculate u from 192 displacement vectors, which is a reasonablygood statistics for calculating < u > . As was discussedabove, it is sufficient to do the calculation for one disor-dered configuration when the supercells are large with atleast 48 atoms. Because of the large mass difference be-tween S and H we here allow larger < u > for H ( u H ) thanfor S ( u S ). For small < u > it can be shown that cor-relation of vibrational movements and anharmonic termsare small [56]. But u H is large, and in the generationof a disordered configuration for u H /a =0.05, as for theexpected ZPM, several pairs of H come too close to eachother. Therefore, in order to avoid large anharmonic ef-fects at this stage, we calculate the electronic structurefor a supercell with u S /a =0.01 (which is close to the ex-pected ZPM for S) and u H /a =0.033 (which is 2/3 of theexpected ZPM for H).The resulting DOS for the disordered lattice is shownin Fig. 4. The results show a strong effect of ZPM onthe energy of the narrow peak of the DOS just below thechemical potential. This peak has a large fraction of H s-and p-states, while the peak further down (3-4 eV below E F ) has less H-character and is less affected by disor-der. The width of the DOS peak with large character ofH-states gets wider in the disordered case, which is ex-pected, but we see also a large energy renormalization ofthis narrow peak. This peak which was mostly below thechemical potential in the ordered lattice is now pushedup above the chemical potential because of the lattice −1.5 −1 −0.5 0 0.5 1 1.500.511.522.533.5 (E−E F ) (eV) D O S ( c e ll ⋅ e V ) − a=6.2 a.u.a=6.0 a.u.a=5.8 a.u.a=5.6 a.u.a=5.4 a.u.H S FIG. 5: (Color online) The total DOS functions in Fig. 3 ona finer energy scale near the chemical potential as a functionof the lattice constant. Each DOS curve is separated by 0.5( cell eV ) − units for the sake of visibility. ZPM. The shift at the top of the peak is of the order of600 meV (see Fig. 4) and the peak becomes broader inthe disordered case.Our simple approach to treat the ZPM has beenbased on the assumption that the frozen disorder inour supercell calculation is representative for ZPM. Ifso we may ask how much ZPM would change in theRMTA value of λ and the McMillan estimate of T c .The total DOS at E F is almost the same (7.3 and7.4 states/eV /cell for ordered and disordered case,respectively) as can be seen in Fig. 4. The λ ′ s and T ′ c s are calculated to be 0.88 and 0.86, and 122 K and 118 K ,for ordered and disordered cases, respectively. Thusthere is a small reduction of λ and T c from disordereven if the two N ( E F ) are comparable. However, onecan note that the peak in the DOS has moved frombeing below the chemical potential in the ordered caseto be above when ZPM is taken into account indicatinga 600 meV energy shift of the van Hove singularity withzero point lattice fluctuations involving H atoms. In theremaining sections we discuss some details about theLifshitz transitions as a function of pressure. Lifshitz Transitions as a function of pressure
The energy shift of the van Hove singularity (vHz) canbe followed by looking at the shift of the narrow peakin the total DOS near the chemical potential at differentlattice parameters a which is shown in Fig. 5.This peak is mostly due sulfur orbital contributionsas shown by the partial DOS functions of the vHs forthe case of lattice parameter a = 5 . a.u. shown in Fig. FIG. 6: (Color online) The partial DOS with different atomicand orbital symmetry for H S at the lattice constant 5.4 a.u..,the (red) thin lines are for S and the (blue) heavy lines for H. −2 −1.5 −1 −0.5 0 0.5 1 1.5 200.511.5 (E−E F ) (eV) D O S ( c e ll ⋅ e V ) − H S a=5.6 a.u. band 8band 9band 10band 11Total
FIG. 7: (Color online) The partial DOS for H S at the fixedlattice constant 5.4 a.u. from different bands crossing thechemical potential in the simple cubic Brillouin Zone.
6. The S-p orbital and S-d orbital contribute to about30 and 16 percent of the total DOS respectively, there-fore the vHs is mainly due sulfur atoms. Both the to-tal charge and the ℓ -character at E F on H are mostly s (60-65 percent), and the hybridization between the s-electron and states on the S atoms away from the chainsis large. This large on-site hybridization is favorable forlarge dipole matrix-element contributions to the electron-phonon coupling. Fig. 7 shows the band decompositionof the total DOS in the cubic 8-site unit cell in the scBZ into 4 bands classified as No.8, No 9, No. 10, No.11. The vHs and the peak in the DOS is due only toband 10 which gives the largest Fermi surface and it isdue to the flat dispersing bands with low Fermi veloc-ity near X and M points giving a van Hove singularity.Fig. 8 shows the pressure dependence of the vHs in band10 which goes through the chemical potential at about210 GPa where a is about 5.6 a.u.. This result shows −1.5 −1 −0.5 0 0.5 1 1.500.511.522.533.544.55 (E−E F ) (eV) D O S ( c e ll ⋅ e V ) − a=6.2 a.u.a=6.0 a.u.a=5.8 a.u.a=5.6 a.u.a=5.4 a.u. FIG. 8: The evolution of the partial DOS from band 8-11in the simple cubic Brillouin Zone as function of the latticeconstant. Only bands 9 and 10, shown by thin and heavycontinuos lines respectively, contribute much to the total DOSat the chemical potential E F . that the van Hove singularity approaches the chemicalpotential at 210 GPa but it remains near the chemicalpotential in the energy range of the energy cut off of thepairing interaction in the pressure range showing hightemperature superconductivity. Where the vHs crossesthe chemical potential a Neck-Disrupting Lifshitz Tran-sition, of type 2, occurs, Here the topology of the largeFermi surface changes because of the appearing of smallpieces of tubular 2D Fermi surfaces connecting the largepetals as discussed in the reference [3]. In these tubularportions the Fermi velocity is small therefore the Migdalapproximation breaks down. On the contrary in the largepetals the Fermi energy is much larger and the Migdalapproximation is valid.In order to associate the crossing of the chemical po-tential of the narrow peak in the DOS with the Lifshitztransitions on the topology of the Fermi surfaces we haveplotted the electronic bands in a narrow energy rangenear the chemical potential for both the sc BZ and bccBZ in Fig. 9 and Fig. 10. These band plots show thatseveral Lifshitz transitions appear for increasing pressure.From Fig. 9 it is seen that a local band maximum crosses E F in the sc BZ at about 2/3 of the Γ − M distance when a is ∼ N and H in Fig. 10 in the bands for the bcc BZ. This Lifshitztransition occurs at P = 210 GP a . The energy differ-ence E between the chemical potential and this local −2.5−2−1.5−1−0.500.511.522.5 ( E − E F ) ( e V ) Γ X M R a=5.4 a.u.a=5.6 a.u.a=5.8 a.u.a=6.0 a.u.a=6.2 a.u.
FIG. 9: (Color online) The band structure for H S along X − M − Γ − R for a for the simple cubic double cell (sc BZ) withthe lattice parameter changing between 5.4 and 6.2 a.u. −2.5−2−1.5−1−0.500.511.522.5 ( E − E F ) ( e V ) Γ PN NH (bcc BZ)H S FIG. 10: (Color online) The variation of the band structurenear the chemical potential plotted for the small bcc unit cellof one formula unit of H S (bcc BZ) as a function of the latticeconstants between 5.4 and 6.2 a.u.. The colors of the bandscorrespond with different lattice parameters as in Fig. 9. Thelabels of symmetry points are as for bcc BZ. band maximum which is associated with the vHs goesfrom -200 to +100 meV when a decreases from 6.2 to 5.6a.u.. This gives a neck disrupting Lifshitz transition inthe Fermi surface at 210 GP a where the neck disappearsat low pressure in the N-H direction [20]. The Fermi sur-face neck appears at the point where the narrow peaks inthe DOS crosses the chemical potential. Another band isseen to be approaching the chemical potential in Fig. 9between X and M when the pressure goes up. However,this potential band crossing (which is not seen on a sym- FIG. 11: (Color online) The upper panel a shows the S-H bond length in the pressure range between 120 and 180GPa where without the hydrogen ZPM, the R3m structurewas expected to be stable with the S-H bond splitting into along (blue squares) and short (violet squares) sulfur-hydrogenbond. The amplitude of the calculated ZPM of the S-H bondis indicated by the red error bars [3, 4]. The S-H amplitudeof the zero point motion, ZPM, is larger than the S-H split-ting in the range 130-180 GPa therefore in this pressure rangethe ZPM stabilizes the Im ¯3 m structure [4] in agreement withexperiments [2]. The panel b shows the Fermi energy E F inthe small hole Fermi pocket at Γ as a function pressure. Thetop of this band crosses the chemical potential at 130 GPagiving the Lifshitz transition of type 1 for the appearing of anew Fermi surface. The position of the vHs E remains belowthe chemical potential but it remains in the energy range ofpairing interaction. Panel c shows the variation of the ex-perimental isotope coefficient calculated from data in ref. [2]which shows a divergence from 0.3 at 180 GPa to 1.5 at 135GPa, which is not predicted by the BCS theory. The criticaltemperature decreases toward zero with a decrease of about60 K in a range of 30 GPa, beteen 160 GPa and 130 GPawhich is not predicted by the BCS theory. Both phenomenaare predicted by the general theory of multigap superconduc-tivity near a Lifshitz transition [3, 4]. metry line in the band plots for the small bcc cell) willnot reach E F to make a FS pocket unless P is increasedeven more.In the pressure range 120 < P <
160 GPa, where theonset of superconductivity occurs followed by the rapidincrease of the critical temperature there are Lifshitztransitions, of type 1, for the appearing of new Fermisurface spots at Γ. There are 3 bands pushed up by in-creasing pressure which cross the chemical potential andthree small closed Fermi surfaces appear. In fact Fig. 9.Fig. 10, show that the tops of these bands at Γ are allabove the chemical potential at the highest P while thetops of these 3 bands bands are below E F at low P .In Fig. 11 we have plotted in panel a the S-H bondlength as a function of pressure and the amplitude of thecalculated spread of this bond length due to the hydro-gen zero point motion indicated by the red error bars.The USPEX theory [12, 13] predicts the second orderphase transition from Im ¯3 m to R3m structure at 180GPa where the sulfur atoms remain in the same sites ofthe bcc unit cell while hydrogen ions are frozen in one ofthe two minima of their double well potential due to thehydrogen bond like in ice structural transitions. On thecontrary this transition is forbidden in the pressure range120-180 GPa since the quantum zero point motion ampli-tude which is larger than the difference between the shortand long hydrogen bonds expected in the R3m structure,Therefore the ZPM stabilizes the Im ¯3 m structure in thepressure range 130-180 GPa [3] in agreement with recentexperiments [2].The panel b in Fig. 11 shows the Fermi energy in thesmall hole Fermi pocket at Γ as a function pressure. Thetop of this band crosses the chemical potential for a pres-sure larger than 130 GPa. At this pressure the Lifshitztransition of type 1 for the appearing of a new Fermi sur-face occurs. In fact above 130 GPa a new small Fermipocket appears at the Γ point of the Brillouin zone. TheFermi energy E F remains smaller than the energy cut offof the pairing interaction below 160 GPa. Therefore Inthe pressure range between 130 and 160 GPa the Migdalapproximation breaks down for electron pairing in thesmall Fermi pocket at the Γ point of the Brillouin Zone.Finally panel c of Fig, 11 shows the variation of theexperimental isotope coefficient in this pressure rangecalculated by recent data [2] which shows a regularincrease with decreasing pressure from 0.3 at 180 GPa to1.5 at 135 GPa. The divergence of the isotope coefficientapproaching the Lifshitz transition at 130 GPa is notpredicted by the BCS theory using standard Midgalapproximation and an effective single band but it ispredicted in the frame of general theory of multigapsuperconductiivity near a Lifshitz transition [4] This issupported by the decrease of the critical temperature ofabout 40 K in H S and of about 60 K in D S while inthe BCS calculations the variation of the critical tem-perature over this 30 GPa range is predicted to be of theorder of 10 K. In fact in the multigaps superconductorsat a Lifshitz transition the critical temperature goestoward zero (like at the Fano-Feshbach anti-resonance)at the appearing of a new nth Fermi surface whereelectrons in the nth band have zero energy E F n =0 atthe band edge forming a BEC condensate, while thecritical temperature has a maximum (Fano-Feshbachresonance) where the electrons have a Fermi energy ofthe order of the pairing energy forming a condensate inthe BEC-BCS crossover [4]. Conclusions In H S the onset and the maximum superconductingcritical temperature, 203 K, are controlled by pressure,like in cuprates where the onset and the maximum valueof the critical temperature 160 K is reached by tuningthe lattice misfit strain at fixed doping [58].The calculated band structure for an ordered H S lat-tice as a function of pressure clearly shows multiple Lif-shitz transitions for appearing of new Fermi surface spotsin the pressure range showing high T c superconductivity,which together with quantum hydrogen zero point mo-tion puts the system beyond the Migdal approximation.New Fermi surface spots appear at the Γ point at 130GPa pressure where the onset of the high critical tem-perature appears. It is possible that the appearing newFermi surface spots drive the negative interference effectin the exchange interaction between multiple gaps [27]contributing to the suppression of the critical temper-ature [3]. This is supported by the isotope coefficientwhich diverges at 130 GPa reaching a value of 1.5 [3],see Fig. 11, well beyond the predictions of single bandEliashberg theory. The divergence of the isotope coeffi-cient observed here has been already observed in cuprates[59, 60] providing a clear experimental indication for aunconventional superconductivity near a Lifshitz transi-tion [4].Increasing the pressure to 210 GPa a van Hove sin-gularity crosses the chemical potential giving a Lifshitztransition for opening a neck. Moreover the vhs remains near the chemical potential within the energy range ofthe energy cutoff for the pairing interaction over thefull pressure range between 210 and 260 GPa. We showthat the quantum zero point hydrogen fluctuations in adouble well [3] typical of hydrogen bond and involvingthe T u phonon stretching mode, has strong effect on theelectronic states near the Fermi level. The quantum hy-drogen zero point motion, induces fluctuations of the 600meV of the energy position of the vHs. The zero pointamplitude of the S-H stretching mode, involving the T u phonon, stabilizes the Im ¯3 m structure in the pressurerange 130-180 GPa and induces large fluctuations of thesmall Fermi surface pockets at Γ . In conclusion we haveshown the presence of large displacement amplitudes ofZPM. Single phonon waves can be disturbed by latticequantum zero point disorder [45] but superconductivityseems to resist to perturbations from ZPM in H S. Onthe other hand, it is also seen that the DOS peak at E F seems to pass through E F with the large zero pointmotion. Finally more work is needed to investigatethe variation of the Fermi level E F in different Fermisurfaces with different H isotopes which change the zeropoint motion amplitude. Author contribution statement ”T. J. and A.B. wrote the main manuscript text andprepared figures. All authors reviewed the manuscriptand contributed equally to the work”
Additional Information
The authors declare thatthey have no competing financial interests.
How to cite this article : Jarlborg, T. and Bianconi,A. Breakdown of the Migdal approximation at Lifshitztransitions with giant zero-point motion in H S super-conductor. Sci. Rep. 6, 24816; doi: 10.1038/srep24816(2016) [1] Drozdov, A.P. et al. Conventional superconductivity at203 K at high pressures. Nature , 73 (2015)[2] Einaga, M. et al. Crystal structure of 200 K-Superconducting phase of sulfur hydride system PreprintarXiv:1509.03156 (2015)[3] Bianconi, A., Jarlborg, T. Superconductivity above thelowest Earth temperature in pressurized sulfur hydride.EPL (Europhysics Letters) , 37001 (2015)[4] Bianconi, A., Jarlborg, T. Lifshitz transitions and zeropoint lattice fluctuations in sulfur hydride showing nearroom temperature superconductivity. Novel Supercon-ducting Materials , 37; DOI:10.1515/nsm-2015-0006(2015),[5] Eremets, M. I., Trojan, I. A. , Medvedev, S. A., Tse,J. S., Yao, Y. Superconductivity in Hydrogen DominantMaterials: Silane Science , 1506 (2008)[6] Drozdov, A. P., Eremets, M. I., Troyan, I. A. Supercon- ductivity above 100 K in P H at high pressures PreprintarXiv:1508.06224 (2015)[7] Y. Li, et al. Pressure-stabilized superconduc-tive yttrium hydrides, Scientific Reports , 9948;DOI:10.1038/srep09948 (2015)[8] Ashcroft, N.W. Symmetry and higher superconductiv-ity in the lower elements. In Bianconi, A. (ed.) Symme-try and Heterogeneity in High Temperature Supercon-ductors. vol. 214 of NATO Science Series II: Mathemat-ics, Physics and Chemistry, (Springer Netherlands,2006).[9] Babaev, E., Sudbo, A., Ashcroft, N.W. A superconductorto superfluid phase transition in liquid metallic hydrogen.Nature , 666-668 (2004)[10] Zurek, E., Hoffmann, R., Ashcroft, N. W., Oganov, A.R. and Lyakhov, A. O. A little bit of lithium does a lotfor hydrogen. Proceedings of the National Academy of Sciences , 17640-17643 (2009).[11] Li, Y., Hao J., Liu H., Li Y., Ma Y. The metallization andsuperconductivity of dense hydrogen sulfide The Journalof Chemical Physics , 174712 (2014)[12] Duan, D.,et al. Pressure-induced metallization of dense(H S)2H with high-T c superconductivity. Sci. Rep. ,6968; DOI:10.1038/srep06968 (2014).[13] Duan, D., et al. Pressure-induced decomposition of solidhydrogen sulfide. Phys. Rev. B , 180502 (2015).[14] Li, Y. et al. Dissociation products and structures ofsolid H S at strong compression. Physical Review B ,020103 (2016).[15] Ishikawa, T. et al. Superconducting H S phase in sulfur-hydrogen system under high-pressure. Scientific Reports , 23160 (2016).[16] Papaconstantopoulos, D., Klein, B. M., Mehl, M. J.,Pickett, W. E. Cubic H S around 200 GPa: an atomichydrogen superconductor stabilized by sulfur Phys. Rev.B , 184511 (2015).[17] Errea, I., et al. Hydrogen sulphide at high pressure:a strongly-anharmonic phonon-mediated superconductorPhys. Rev. Lett. , 157004 (2015).[18] Durajski, A. P., Szczesniak, R., Li Y. Non-BCS thermo-dynamic properties of H S superconductor, Physica C:Superconductivity and its Applications , 1 - 6 (2015)[19] Quan, Y., Pickett, W. E. van Hove singularities and spec-tral smearing in high temperature superconducting H3SPhysical Review B 93, 104526 (2016).[20] Flores-Livas, J. A., Sanna, A., Gross, E. K. U. High tem-perature superconductivity in sulfur and selenium hy-drides at high pressure. The European Physical JournalB , 1 (2016).[21] Akashi, R., Kawamura, M., Tsuneyuki, S., Nomura,Y., Arita, R. First-principles study of the pressure andcrystal-structure dependences of the superconductingtransition temperature in compressed sulfur hydrides.Phys. Rev. B , 224513 (2015)[22] Ummarino, G.A., Gonnelli, R.S., Massida, S., Bianconi,A. Two-band Eliashberg equations and the experimental T c of the diboride Mg − x Al x B . Physica C: Supercon-ductivity , 121 (2004).[23] Campi, G., et al. Study of temperature dependent atomiccorrelations in MgB . The European Physical JournalB - Condensed Matter and Complex Systems, , 15(2006).[24] Boeri, L., Cappelluti, E., Pietronero, L. Small Fermienergy, zero-point fluctuations, and nonadiabaticity in MgB Physical Review B , 012501 (2005).[25] Innocenti D., et al., Resonant and crossover phenomenain a multiband superconductor: Tuning the chemical po-tential near a band edge. Phys. Rev. B , 184528 (2010).[26] Yildirim T., et al., Giant Anharmonicity and NonlinearElectron-Phonon Coupling in MgB : A Combined First-Principles Calculation and Neutron Scattering StudyPhysical Review Letters T c superconductivity in a superlattice of quantum stripesSolid State Communications , 369 (1997).[28] Bianconi, A. Feshbach shape resonance in multiband su-perconductivity in heterostructures. Journal of Supercon-ductivity , 625 (2005).[29] Guidini, G., Perali, A. Band-edge BCS-BEC crossover ina two-band superconductor: physical properties and de-tection parameters. Supercond. Sci. Technol. , 124002 (2014).[30] Caivano, R., et al. Feshbach resonance and mesoscopicphase separation near a quantum critical point in multi-band FeAs-based superconductors, Superconductor Sci-ence and Technology , 014004 (2009).[31] Bianconi, A., Quantum Materials: Shape Resonances inSuperstripes Nature Physics , 536 (2013).[32] Bianconi, A., Poccia, N., Sboychakov, A. O.,Rakhmanov, A. L., Kugel, K. I. Intrinsic arrestednanoscale phase separation near a topological Lifshitztransition in strongly correlated two-band metals. Super-conductor Science and Technology , 024005 (2015).[33] Kugel, K., Rakhmanov, A., Sboychakov, A., Poccia, N.,Bianconi, A. Model for phase separation controlled bydoping and the internal chemical pressure in differentcuprate superconductors. Physical Review B , 165124(2008).[34] Bianconi, A., et al. Coexistence of stripes and supercon-ductivity: T c amplification in a superlattice of supercon-ducting stripes Physica C: Superconductivity , 1719(2000).[35] Bianconi, A. Superstripes Int. J. Mod. Phys. B , 3289(2000).[36] Poccia, N., et al., Optimum inhomogeneity of local latticedistortions in La Cu O y Proceedings of the NationalAcademy of Sciences , 15685 (2012).[37] Campi, G., et al., Inhomogeneity of charge-density-waveorder and quenched disorder in a high-Tc superconductorNature , 359 (2015).[38] Capaz, R. B., Spataru, C. D., Tangney, P., Cohen, M. L.,Louie, S. G. Temperature dependence of the band gap ofsemiconducting carbon nanotubes. Physical Review Let-ters , 036801 (2005).[39] Gonze, X., Boulanger, P., Ct, M. Theoretical approachesto the temperature and zero-point motion effects on theelectronic band structure. Annalen der Physik ,168-178 (2011).[40] Cannuccia, E., Marini, A., Zero point motion effect onthe electronic properties of diamond, trans-polyacetyleneand polyethylene Eur. Phys. J. B , 320 (2012).[41] McKenzie, R.H., Wilkins, J.W., Lattice fluctuation dis-order in quasi-one dimensional materials Phys. Rev. Lett. , 1085 (1992).[42] Jarlborg, T. Electronic structure and properties of pureand doped FeSi from ab initio local-density theory, Phys.Rev. B , 15002 (1999).[43] Jarlborg, T., Chudzinski, P., Giamarchi, T. Effects ofthermal and spin fluctuations on the band structure ofpurple bronze Li Mo O Phys. Rev. B , 235108(2012).[44] Jarlborg, T. Role of thermal disorder for magnetism andthe transition in cerium: Results from density-functionaltheory Phys. Rev. B , 184426 (2014).[45] Jarlborg, T. Electronic structure and properties of su-perconducting materials with simple Fermi surfaces J. ofSupercond. and Novel Magn., , 1231 (2014)[46] Barbiellini, B., Dugdale, S.B., Jarlborg, T. The EPMD-LMTO program for electron positron momentum den-sity calculations in solids, Comput. Mater. Sci. , 287(2003).[47] Jarlborg, T., Manuel A.A., Peter, M., Experimental andtheoretical determination of the Fermi surface of V Si,Phys. Rev. B , 4210 (1983).[48] Jarlborg, T., Moroni, E. G., Grimvall, G. Transition in Ce from temperature-dependent band-structure calcula-tions Phys. Rev. B , 1288 (1997).[49] Moroni, E. G., Grimvall, G., Jarlborg, T. Free EnergyContributions to the hcp-bcc Transformation in Transi-tion Metals, Phys. Rev. Lett. , 2758 (1996).[50] Jarlborg, T., Bianconi, A. Fermi surface reconstruc-tion of superoxygenated La Cu O superconductors withordered oxygen interstitials, Phys. Rev. B , 054514(2013).[51] Pettifor, D. Theory of energy bands and related prop-erties of 4d-transition metals. II. The electron-phononmatrix element, superconductivity and ion core enhance-ment, J. Phys. F: Metal Phys. , 1009 (1977).[52] Gaspari, G.D., Gyorffy, B.L. Electron-Phonon Interac-tions, d Resonances, and Superconductivity in TransitionMetals, Phys. Rev. Lett. , 801 (1972).[53] Dacorogna, M., Jarlborg, T., Junod, A., Pelizzone,M., Peter, M. Electronic structure and low-temperatureproperties of V(x)Nb(1-x)N alloys J. Low Temp. Phys. , 629 (1984).[54] Bauer, J., Han, J. E., Gunnarsson, O. Retardation ef-fects and the Coulomb pseudopotential in the theory ofsuperconductivity Phys. Rev. B , 054507 (2013).[55] Bennemann, K. H., Garland, J. W., in ’Superconductiv-ity in d- and f-band Metals’, Rochester 1971, Ed. A.H. Douglas (AIP New York, 1972), p. 103.[56] Grimvall, G. Thermophysical properties of materials