Anharmonic theory of superconductivity in the high-pressure materials
UUnderstanding the Effects of Pressure,Anharmonicity and Phonon Softening on theSuperconducting Critical Temperature
Chandan Setty , Matteo Baggioli ‡ , and Alessio Zaccone (cid:5) Department of Physics, University of Florida, Gainesville, Florida, USA. Instituto de Fisica Teorica UAM/CSIC, c/Nicolas Cabrera 13-15, Universidad Autonoma de Madrid, Cantoblanco,28049 Madrid, Spain. Department of Physics ”A. Pontremoli”, University of Milan, via Celoria 16, 20133 Milan, Italy. Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB30HE Cambridge, U.K. ‡ e-mail: [email protected] * e-mail: [email protected] (cid:5) e-mail: [email protected] ABSTRACT
Electron-phonon superconductors at high pressures have displayed the highest values of critical superconducting temperature T c on record, now rapidly approaching room temperature. A variety of other superconducting materials have been investigatedexperimentally in recent years, unveiling a zoology of different T c trends as a function of applied pressure P . In spite of theimportance of high- P superconductivity in the quest for room-temperature superconductors, a mechanistic understanding ofthe effect of pressure and its complex interplay with phonon anharmonicity and superconductivity is missing, as numericalsimulation studies can only bring system-specific phenomenological insights. Here we develop a theory of electron-phononsuperconductivity under an applied pressure which fully takes into account the anharmonic decoherence of the optical phonons.The results are striking: generic trends of T c with increasing P observed experimentally in both elemental and more complexsuperconductors can be recovered and explained in terms of coherence/incoherence properties of the boson glue providedby anharmonically damped optical phonons within the governing gap equation. In particular, T c first increases, then goesthrough a peak and then decays upon further increasing the ratio Γ / ω , where Γ is the optical phonon damping and ω theoptical phonon energy at zero pressure and momentum. In other words, T c increases with Γ / ω in a regime where phononsbehave like well-defined quasiparticles, and decays with Γ / ω in a regime of strong anharmonic damping where phonons areincoherent (“diffusons"). This framework is able to explain recent experimental observations of T c as a function of pressure incomplex materials (e.g. TlInTe ), where T c first decreases with pressure then goes through a minimum after which it rises withpressure again. In a second scenario, which is experimentally realized in certain strongly correlated metals (e.g. the cuprates),the phonon quasiparticles are not well defined anymore ( Γ (cid:29) ω ), the underlining physics becomes incoherent and the T c dependence on pressure is completely reversed but still predicted by our theoretical model. Introduction
When a crystal lattice is subjected to a (hydrostatic) pressure deformation, its phonon frequencies change in response to thechange of volume, in a way which is controlled by the materials’s Grüneisen parameter, hence by the anharmonicity of thevibration modes. However, the effects of these changes in the phonon frequencies, and of the related anharmonicity, on thesuperconducting properties of a material have largely remained poorly understood. Filling this knowledge gap is an urgentproblem in order to develop an understanding of superconductivity in materials under pressure, which include the highest- T c values recorded so far in the high-pressure hydride materials .On one hand, a large number of experimental works have shown how the superconducting critical temperature T c changes asa function of pressure P for a variety of materials. For elemental superconductors, a commonly observed trend in experimentsis a decrease of T c with increasing P , which has been theoretically predicted upon analyzing the behaviour of the Eliashbergelectron-phonon coupling function α g ( ω ) as a function of P . An increase of P typically shifts the α g ( ω ) distribution tohigher frequencies, thus driving the system into an unfavourable regime for the electron-phonon coupling . This behaviouris at odds with Ashcroft’s famous expectation that an increase of the Debye frequency ω D would necessarily lead to anincrease of T c , which may hold for weak-coupling BCS materials, but not for strong-coupling materials with larger λ values.Also, a notable exception to the above standard rule for elemental superconductors is represented by α -uranium , to be a r X i v : . [ c ond - m a t . s up r- c on ] J u l igure 1. Mechanism of T c enhancement through anharmonicity with two phonon modes. (Left panel) In the absence ofanharmonic decoherence ( D = (Middle panel) Weak anharmonic decoherence ( D ∼ D ∗ )sensitizes the phase of the S- and aS- processes and enables them to act coherently and enhance the effective coupling ofelectrons and phonons leading to strong Cooper pairs. (Right panel) For very strong anharmonicity ( D (cid:29) D ∗ ), the S- and aS-processes are only weakly sensitive to their phases making them effectively indistinguishable while acting to reduce theeffective coupling of electrons and phonons leading to weak pairing.discussed in detail below, while another puzzling material such as bismuth is known to have a very low T c at ambient pressure(on the order of the mK) and a decent T c (7 − . A similar surprising trend is observed uponintroducing structural disorder into crystalline solids which may lead to the enhancement of T c (as is the case for Bi and Be) inthe amorphous state, an effect which has been recently explained in Ref. as due to the favourable effect of disorder-inducedsoft transverse phonon modes on the pairing.Exploring the effect of P on more complex non-elemental materials has led to a zoology of trends of T c as a function of P .On the other hand, numerical simulations have provided invaluable quantitative insights into the phonon dispersion relations,and into the structural stability of superconducting compounds, including many new materials. Numerical calculations alsoallow one to estimate the anharmonicity of the various phonon modes involved, by comparing fully anharmonic calculationswith harmonic calculations . Despite these efforts, a mechanistic picture of high pressure effects on the superconductingstate which takes into account anharmonic decoherence of the bosonic glue is missing.As a matter of fact, early theoretical approaches ignored phonon anharmonicity while more recent works on the high- T c hydrides only take into account the phonon energy renormalizations neglecting anharmonic decoherence. The latter is keyto properly describe the effect of pressure on phonon-mediated Cooper pairing, as we will show in the following.In this paper, we develop such theory by working with a gap equation derived from the Migdal-Eliashberg equations inthe weak coupling BCS limit. Crucially, to mediate Cooper pairing, we implement optical phonon propagators which containthe effect of an external applied pressure and the resulting anharmonic decoherence via the optical phonon damping . Theanalytical theory is able to provide predictions that allow one to disentangle the complex interplay between pressure-inducedchanges of optical phonon energy and anharmonic decoherence, and their effects on the T c .Different physical regimes are predicted, which include (i) monotonic decrease of T c with P ; (ii) non-monotonic trend witha minimum, in conjunction with optical phonon softening, which qualitatively explains recent experiments in TlInTe ; (iii)non-monotonic trend with a maximum in a regime of incoherent phonons where the quasi-particle picture breaks down. heoretical framework Optical phonon energy under pressure
We start by analyzing the effect of external pressure on the optical phonons of a crystal lattice. The main effect of pressure is toinduce a negative volume change of the material. The change of volume, in turn, is related to a change of phonon frequency,through the Grüneisen parameter, γ = − d ln ω (cid:48) / d ln V , via : ω (cid:48) ( V ) ω (cid:48) P = = (cid:18) VV (cid:19) − γ , (1)where ω (cid:48) P = refers to optical phonon energy at zero ambient pressure. The above relations apply to individual phonon modeswith frequency ω (cid:48) .The volume change is related to the change of pressure as described by the Birch-Murnaghan equation of state , which isderived based on nonlinear elasticity theory, and provides an expression for P ( V ) . Upon replacing V with ω (cid:48) in (1), one obtainsthe following relation between the optical phonon frequency ω (cid:48) and the applied pressure : P ( X ) = b (cid:16) X − X (cid:17) (cid:2) + η ( − X ) (cid:3) , (2)with X ≡ ( ω (cid:48) / ω (cid:48) P = ) γ / . Upon inverting the above Eq.(2) to obtain ω (cid:48) as a function of P , it is clear that ω (cid:48) is a monotonicallyincreasing function of P , with the increase being modulated by anharmonicity through γ . Also, b = B / γ , with B the bulkmodulus, while η = ( / )( − B (cid:48) ) with B (cid:48) = dB / dP .In the above relations, the frequency ω (cid:48) refers to the real part of the phonon dispersion relation, whereas the imaginary partof the dispersion relation is related to the phonon damping coefficient Γ (the inverse of the phonon lifetime), as follows ω = ω − i ω Γ + O ( q ) , ω (cid:48) ≡ Re ( ω ) = (cid:113) ω − Γ + O ( q ) , Γ ≡ Im ( ω ) + O ( q ) . (3)Hence, ω (cid:48) denotes the renormalized phonon energy measured e.g. in Raman scattering (i.e. the Raman shift), while Γ representsthe linewidth of the Raman peak. Let us emphasize that these expressions are at leading order in the momentum q and higherorder corrections O ( q ) are neglected at this stage.We now introduce a key dimensionless parameter for the subsequent analysis D ≡ Γ / ω , (4)which quantifies the degree of coherence of the phonon. Low values of D signify high coherence of the phonons, which can thusbe treated as approximately independent quasiparticles, whereas large D values correspond to incoherent vibrational excitationsin the diffusive regime (“diffusons” in the language introduced by Phil Allen, Feldman and co-workers ). The schematicpicture that will emerge from the subsequent theoretical analysis is anticipated in Fig.1.In the following section, we introduce the theoretical framework for the Cooper pairing and we will start by consideringhow the superconducting critical temperature T c varies as a function of D . Gap equation with anharmonic phonon damping
For a generic Fermionic Matsubara frequency ω n and momentum k , we denote the gap function as ∆ ( i ω m , k ) . With a constantcoupling g , the gap equation can be derived from the Eliashberg equations in the one-loop ( weak coupling) approximation, andtakes the form ∆ ( i ω n , k ) = g β V ∑ q , ω m ∆ ( i ω m , k + q ) Π ( q , i ω n − i ω m ) ω m + ξ k + q + ∆ ( i ω m , k + q ) , (5)where β is the inverse temperature and V is the volume. In Matsubara frequency space, we choose the pairing mediator to be adamped optical phonon given by the bosonic propagator Π ( q , i Ω n ) = Ω ( q ) + Ω n + Γ ( q ) Ω n , (6)where Ω n is the bosonic Matsubara frequency, Ω ( q ) = ω + α q is the phonon dispersion, and the damping factor, Γ ( q ) ≡ D ω ,is a constant independent of momentum for high-frequency optical phonons . In accordance with the Klemens formula , uasiparticle ( coherent ) incoherent ~ / D ~ a + a D e a D D * D T c Figure 2.
An illustration of the two regimes present in our model: the “coherent” regime where the critical temperature growswith D ≡ Γ / ω and the “incoherent” regime, where the functional dependence is inverted. In the “coherent” regime, the opticalphonons behave like independent quasiparticles with frequencies renormalized by anharmonicity, whereas in the incoherentregime the quasiparticle coherence breaks down due to the large anharmonic damping.one can also include an additional prefactor, 1 + e ω / T − , in the damping term Γ ( q ) to account for a temperature depen-dent phonon linewidth. We find that this has a negligible effect on the results discussed below. The factor D controls thestrength of the damping term and may change with pressure. The leading order contribution to the square of the dispersionis Ω ( q ) (cid:39) ω + vq where v = ω α . This is the first momentum correction which was neglected in Eq.(3). Assuming anisotropic, frequency-independent gap ∆ ( i ω n , k ) ≡ ∆ , we can set the external frequency and momentum to zero without any lossof generality. Converting the resulting summation into an energy integral, the gap equation becomes1 = ∑ ω m (cid:90) ∞ − µ λ T d ξ [ v ξ + M + ω m − D ω m ω ] [ ω m + ξ + ∆ ] , (7)where M = µ v + ω . Here we have defined the effective coupling constant λ = N ( ) g , N ( ) is the density of states at theFermi level, and µ is the chemical potential. We can now utilize the energy integral identity (cid:82) ∞ − ∞ d ξ ( z ξ + s )( ξ + r ) = π sr ( s + z r ) inthe limit of large chemical potential to yield the gap equation1 = ∑ ω m λ π T (cid:0) M + ω m − D ω m ω (cid:1)(cid:112) ω m + ∆ [( M + ω m − D ω m ω ) + ( ω m + ∆ ) v ] . (8)We can now perform the final Matsubara sum after seeking a condition for T c by setting ∆ =
0. Defining p = ω D + iv and Q ± = (cid:16) p ± (cid:112) p − M (cid:17) leads to an equation for T c that can be numerically solved given by − M (cid:48) = ψ (cid:18) (cid:19) + (cid:34) (cid:26) p (cid:48) − Q (cid:48) + Q (cid:48) + − Q (cid:48)− ψ (cid:18) − Q (cid:48) + π T (cid:48) c (cid:19) + − p (cid:48) + Q (cid:48)− Q (cid:48) + − Q (cid:48)− ψ (cid:18) − Q (cid:48)− π T (cid:48) c (cid:19) + c . c (cid:27) + { D → − D } (cid:35) , (9)where ψ ( x ) is the digamma function, the primed quantities are dimensionless and are defined as Q (cid:48)± ≡ Q ± √ λ and so on. Results
Schematic T c dependence on optical phonon energy and anharmonicity Upon numerically solving Eq.(9) for a constant damping coefficient Γ , we can study the evolution of T c as a function of thedimensionless parameter D ≡ Γ / ω . The trend is shown in Fig.2. At low D values, T c increases with D , then goes through a P π T c / λ / Figure 3.
The normalized critical temperature with D = αω and ω given by the formula Eq.(2). α increases from red topurple.maximum after which it then decays sharply upon further increasing D . The maximum appears around D ∗ ∼ O ( ) with its exactvalue determined by the microscopic parameter M . This corresponds exactly to the scale at which the real and the imaginarypart of the phonon dispersion relation become comparable ( Γ ∼ ω ) and the phonons turn into quasi-localized “ diffusons ”. Inthis sense, this is analogous to the Ioffe-Regel crossover scale .The mechanistic picture shown in Fig.1 can be used to understand the non-monotonic dependence of T c upon the anharmonicdecoherence parameter D . To begin, we note that in the absence of D , the gap equation in Eq. 7 has even terms only in theMatsubara frequency transfer ω m . Hence both constructive Stokes (S-) and destructive anti-Stokes (aS-) processes, which emitand absorb energy respectively, contribute to the gap equation equivalently. However, when D is non-zero, Eq. 7 is sensitiveto the sign of the energy transfer, thereby distinguishing the two processes. From this property, it is clear that the energyintegral and Matsubara summations in Eqs. 7 and 8 lead to terms that are proportional to D in the numerator of the gap equation.Provided D (cid:46) D ∗ , this effectively increases the electron-phonon coupling λ and hence the Cooper pair binding energy. Forvalues of D much larger than D ∗ , the phonons are extremely damped and S- and aS- processes again contribute approximatelyequally to the gap equation, thus reducing the effective electron-phonon coupling.At low values of damping Γ (low- D regime) and Γ / ω (cid:28)
1, the real part of the dispersion relation dominates over theimaginary part, and the phonons behave like coherent quasiparticles with well-defined momentum k . In the opposite regime oflarge anharmonic damping Γ / ω (cid:29) D ), we have that Im ω > Re ω , hence the phonons lose their coherence andthe quasiparticle approximation breaks down. These two regimes correspond to two different Cooper pairing regimes. Oneregime we call the “coherent” regime (because here phonons behave like coherent quasiparticles), where T c correlates positivelywith anharmonic damping (hence where damping enhances T c ). The second regime we call “inchoerent” and here, instead, T c decreases with further increasing the anharmonic damping. Notice that, in the coherent regime, T c increases (decreases) asthe optical phonon energy ω decreases (increases), whereas the opposite trends apply in the incoherent regime. This impliesthat the effect of pressure can either promote or depress superconductivity depending on the underlying physics of the opticalphonons in a given lattice.The theoretical prediction in Fig.2 can be fitted with the following simple functions T c ( D ) ∼ a + a D e a D for D < D ∗ ( coherent ) , T c ( D ) ∼ D − for D > D ∗ ( incoherent ) , (10)with a n > T c on theexternal pressure P . This conceptual schematization will be shown in the next sections to hold a number of consequences for adeeper mechanistic understanding of the effect of pressure on superconductivity in complex materials. ◆◆◆◆◆◆◆◆ ◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆ ◆ ◆◆◆ ◆◆◆ ◆◆◆◆ P [ GPa ] N o r m a li z e d R a m a n S h i ft
128 130 132 134 136 138 14002468101214 ω ' [ cm - ] P [ G P a ] Figure 4.
Left:
The normalized Raman shift (proportional to ω (cid:48) ) as a function of pressure and its fit with an empiricalfunction. The value at zero pressure is ≈
128 cm − . Data taken from Right:
Comparison between the best empirical fit of (shown in the left panel) and the Eq. 2 in terms of ω ≈ ω (cid:48) , ( Γ (cid:28) ω ). The parameters are set to the values shown in Eq.(11).In Fig.2 we assumed that the pairing is mediated by high-frequency optical phonons near the Debye frequency ω D forwhich the Klemens model gives a simplified (constant) anharmonic damping coefficient Γ = D ω . In the more general case, theKlemens damping is given by Γ = αω , where α is a prefactor which depends on the microscopic physics which governs thedecay of the optical phonon into two acoustic phonons. Notably, α ∼ γ , where γ is the lattice Grüneisen parameter introducedabove. The latter is a function of the interatomic potential , hence of the electronic orbital/bonding physics, and can be easilycomputed, for a given phonon mode in a give material, from first principles .Using this more general Klemens formula for a generic optical phonon, we obtain the trends shown in Fig.3. A lineardecreasing trend of T c as a function of P is predicted by our theory for the coherent-phonon regime, in agreement with manyexperimental data sets in the literature for simple (e.g. elemental) superconductors . Theoretical analysis of superconductivity in TlInTe at high pressure In this section, we explore the potential of the above framework to rationalize recent experimental data where highly non-trivial(e.g. non-monotonic) dependencies of T c upon P have been observed, and for which a theoretical explanation is lacking. Westudy the paradigmatic case of TlInTe , for which accurate experimental data are available for the optical phonon energy andthe anharmonic damping, as measured by Raman scattering, and also for T c , as a function of pressure.We start by fitting the experimental data for the optical phonon energy (renormalized by anharmonicity) ω (cid:48) as a function ofpressure, displayed in Fig.4 left panel, by means of Eq.2, and we get: b = , γ = . , η = . , ω (cid:48) P = ≈ ω , P = =
127 cm − . (11)The fitting is shown in Fig.4 right panel, where the frequency values refer to ω (cid:48) . The latter has been obtained by using Eq.3 incombination with Eq.2. The optical mode energy increases upon increasing P in a conventional way up to P = Γ is observed upon further increasing P , as shown inFig. 4 left panel.The increase of anharmonicity with pressure is clearly evidenced by the behaviour of the Raman peak linewidth Γ , as shownin Fig.5 left panel. Notice that the percentile growth of the linewidth under pressure is much larger than that of the normalizedRaman shift. In this sense, the material is characterized by giant anharmonicities and the damping effects are fundamental.Here, in the same panel, different empirical trends are shown, alongside the experimental data which manifest a significantscatter. In general, Γ (cid:28) ω for this system, such that this case belongs to the “coherent” regime discussed in the previoussection and in Fig.2. Indeed, we checked that ω (cid:48) and ω differ by only about 0 .
01% at all P values. These different trends for Γ have been implemented, alongside the fitted optical phonon energy ω (cid:48) from Fig.4, into our theoretical gap-equation frameworkfor the prediction of T c presented in the previous section of this paper. The resulting theoretical T c trends are shown in Fig.5 incomparison with the experimental T c data from Ref. , as a function of the applied pressure.All the Γ trends in Fig. 5 left panel clearly lead to the same qualitative dependence of T c on P , with a minimum. Thephysics behind this trend is explained by our theoretical framework: at low P the T c decreases because of the increase in P , igure 5. Left:
The normalized linewidth and three different sets of fits. The zero pressure value is taken as 3 . − . Datataken from . Right:
The corresponding theoretical calculations for the critical temperature (solid lines) are compared with theexperimental data (symbols). The colors of the theoretical curves for T c match the respective models for the linewidth in the leftpanel.which induces an increase of the optical phonon frequency ω (cid:48) or ω . The subsequent phonon softening leads to the minimumand to an inversion of the trend: upon further increasing the pressure the T c starts to rise. This is due to the fact that lower ω values lead to a Stokes/anti-Stokes constructive interference (in the presence of anharmonic damping), which enhances theCooper pairing, . This behaviour, with a minimum in T c is independent of the particular Γ trend with P , and in fact occurseven for Γ constant with P .The role of the Γ trend with P is to control the position of the minimum as a function of pressure. Also, importantly, thepresence of a rise in Γ leads to a stronger rise after the minimum, which confirms that in the “coherent” regime the T c can bestrongly enhanced by the anharmonic damping, as discussed in the context of Fig.2. This finding has deep implications forhigh- T c hydrogen-based materials, where the anharmonic damping of the optical phonons can be significant and may be tunedby the material design. Positive pressure effect on T c in the cuprates and other anomalous materials Observation of T c rising with the applied pressure P , i.e. dT c / dP >
0, has been historically relatively rare, until the discoveryof the high- T c cuprates. Before the cuprates, not many materials displayed an enhancement of T c with P , a notable case beingthat of α -uranium . Interestingly, α -uranium presents a vibrational density of states (VDOS) which is very rich of softvibrational modes , traditionally attributed to anharmonicity in crystals , although their origin in α -uranium is still underdebate . In general, the well known existence of phonon anomalies in α -uranium , suggests that it is highly possible thatthe observed rise in T c with P (followed by a maximum) is due to incoherent vibrational excitations, which would support thescenario of the “incoherent” regime predicted by our theory in Fig.2.Right after the discovery of the cuprates, a positive T c vs P correlation has been observed in various high- T c cupratematerials , including the first cuprate, the La-Ba-Cu-O compound . Although the actual mechanism of superconductivityin the cuprates is still under debate (and possibly not related to phonons dynamics), some researchers believe that the pairingcould be mediated by (yet to be identified) bosonic excitations that live in the CuO layer .Regardless of the ultimate nature of the bosonic “glue” at work in the cuprates, the fundamental point, for our purposes, isthat due to strong interactions/correlations both the bosonic and fermionic quasiparticles are destroyed and no sharp peaksare observed in scattering experiments. Definitive proofs of this mechanism are the violation of the Mott-Ioffe-Regel (MIR)criterion which is experimentally observed at large temperatures and the anomalous thermal diffusion supported by recentstudies .The positive pressure effect on T c observed in high- T c cuprates could thus be explained within our proposed framework, asdue to highly energetic but not quasiparticle-like bosonic excitations in the “incoherent” regime of our Fig.2. . As a consistencycheck, the positive correlation of T c with pressure has been indeed observed in the underdoped cuprates Bi Sr CuO + δ (Bi2201), which supports the theoretical framework presented above. Instead, in the overdoped cuprates Bi Sr CaCu O + δ (Bi2212) the dependence of T c on P displays a minimum, just like in Fig.5, which suggests that the overdoped materials belongto the “coherent” regime. his analysis thus provides a rationale for tuning the pressure effect on T c in the cuprates via the oxygen doping, by explainingthe positive correlation in the underdoped materials with the breadown of the quasiparticle picture for the bosonic “glue”. Conclusions
We presented a comprehensive theory of the pressure effect on Cooper pairing in superconductors where the pairing is mediatedby generic bosonic excitations. Our theory is based on solving the gap equation with a bosonic propagator. A specific calculationis presented for optical phonons which takes into account: (i) the anharmonicity of the phonon via the Klemens’ damping, (ii)the effect of pressure on the phonon frequency.The predictions of this theory are rich and far-reaching. First of all, the theory identifies two fundamental regimes as afunction of the dimensionless ratio D between anharmonic phonon damping and phonon frequency. At low values of this ratio, T c is strongly enhanced by anharmonicity and at the same decreases with increasing pressure. At large values of the D ratio(after a maximum), where the phonons are no longer well-defined quasiparticles, T c instead correlates positively with pressureand is lowered by anharmonicity (see Fig.2).In the more conventional case of systems in the “coherent phonon” regime, where phonons are coherent quasiparti-cles, the standard linearly decreasing correlation between T c and P is recovered using the Klemens theory of anharmonicdamping for optical phonons. The theory, however, is able to describe also more complex materials. In particular, it pro-vides a qualitative description of recent experimental data on TlInTe for which phonon frequencies, anharmonic phonondamping and T c were all measured experimentally. The theory predicts that T c initially decreases with P as a consequenceof the optical phonon energy increasing with P , but then goes through a minimum, as the optical phonon starts to softenand to become more anharmonic, after which it rises with P . The predicted behaviour is well supported by the experimental data.This theoretical picture provides a comprehensive mechanistic rationale for the pressure effect on superconductivity, byphysically describing different regimes of negative/positive pressure effect on T c across a variety of materials, from simpleelemental superconductors to complex unconventional materials. By clarifying the deep interplay between anharmonicity of thebosonic glue and pressure effects on the pairing mechanism, the theory provides new guidelines for material design, which mayprove useful for discovering and/or engineering new materials with enhanced T c . Acknowledgements
M.B. acknowledges the support of the Spanish MINECO’s “Centro de Excelencia Severo Ochoa” Programme under grantSEV-2012-0249. C.S. is supported by the U.S. DOE grant number DE-FG02-05ER46236. A.Z. acknowledges financial supportfrom US Army Research Laboratory and US Army Research Office through contract nr. W911NF-19-2-0055.
Author contributions
The first two authors, C.S. and M.B., contributed equally to this work. C.S., M.B. and A.Z. designed research, C.S. developedthe analytical calculations and code with inputs from M.B. and A.Z., M.B. developed the graphics, A.Z. wrote the paper withinputs from C.S. and M.B.
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