Introduction

In 1968 Neil Ashcroft predicted superconductivity with high critical temperature for hypothetical metallic hydrogen [1]. As an advantage of this candidate for high-\(T_{c}\) superconductors Ashcroft mentioned the high Debye phonon frequency, which enters the preexponent in the BCS formula for \(T_{c}\). But he also estimated, using the spherical Fermi surface and static screening in the Thomas–Fermi model, that in the metallic hydrogen the Coulomb repulsion, represented by the Coulomb constant (pseudopotential) \(\mu ^{*}\), should be very weak in comparison with the Fröhlich phonon attraction described by the phonon coupling constant \(\lambda\). The significance of this observation is obvious if we look at any expression for \(T_{c}\), that takes into account the strong electron-phonon coupling along with the Coulomb repulsion, for example

$$\begin{aligned} T_{c} =\frac{\omega _{0} }{1.2} \exp \left( \frac{-1.04\left( 1+\lambda \right) }{\lambda -\mu ^{*} \left( 1+0.62\lambda \right) } \right) , \end{aligned}$$
(1)

or

$$\begin{aligned} T_{c} =1.14\omega _{0} \exp \left( -\frac{1+\lambda +\lambda _{0} }{\lambda -\mu ^{*} \left( 1+\lambda _{\infty } \right) } \right) . \end{aligned}$$
(2)

These formulas represent results obtained for different models of the phonon spectrum in [2, 3], respectively, both based on the Eliashberg equation accounting for the strong retarded electron–phonon interaction. In the above formulas \(\omega _{0}\) is a characteristic phonon frequency, \(\lambda ,\; \lambda _{0} ,\; \lambda _{\infty }\) are electron-phonon coupling constants defined in respective models, and \(\mu ^{*}\) is the Bogoliubov–Tolmachov–Morel–Anderson pseudopotential.

A new step towards the achievement of the hydrogen-based superconductivity was made again by Ashcroft in 2004, who suggested that the high pressure needed for the formation of metallic hydrogen may be lowered by chemical bond with metal atoms in compounds with a significant predominance of hydrogen atoms [4]. Later this approach was substantiated in [5,6,7,8,9] by the first-principle calculations of the electron band structure and phonon spectra for superhydrides of several lanthanoid elements at megabar pressures, as well as for the vanadium hydrides [10] and hydrogen sulfide \(\hbox {H}_{{3}}\)S [11]. In these calculations very high values of the coupling constant \(\lambda\) and critical temperature \(T_{c}\) were predicted within the Migdal–Eliashberg theory. Finally, this approach was implemented experimentally by two groups led by Hamley (in the USA) [12] and Eremets (in Germany) [13]. Superconductivity at 250 K has been achieved in lanthanum decahydride \(\mathrm {LaH}_{10}\) under high pressure of about 150 GPa.

The strong electron–phonon interaction and high phonon frequencies make the phonon mechanism virtually the sole candidate for the explanation of the high values of the temperature of superconducting transition in superhydrides. At the same time, these high values of \(T_{c}\) were obtained in theoretical calculations [5, 6] with the choice of small values of the effective Coulomb constant \(\sim 0.1\), which in our opinion needs farther justification. We intend to discuss below the effects that the peculiar features of the one-particle electronic spectra of the rare-earth superhydrides may play in the achieving of such Coulomb constant and in the increasing of the values of critical temperature.

It should be noticed that in the same 1968 year one of us (E.A.P.) suggested a possible non-phononic mechanism of high-temperature superconductivity (HTS) [14], where the Cooper pairing of the charge carriers was brought about by interaction with the so-called acoustic plasmons—the collective excitations arising in solids with two subsystems of degenerate charge carriers with significantly different effective masses (light and heavy) [15].

In the Eliashberg theory of superconductivity [16] the electron–phonon interaction constant is determined by the momentum-averaged spectral function \(\alpha ^{2} F\left( \nu \right)\) of this interaction:

$$\begin{aligned} \lambda =2\int _{0}^{\infty }\frac{1}{\nu } \alpha ^{2} F\left( \nu \right) d\nu . \end{aligned}$$
Fig. 1
figure 1

Calculated electronic band structure for \(\mathrm {LaH}_{10}\) at 300 GPa [5]

In the case of the plasmon-mediated interaction between the charge carriers the spectral function of the electron-plasmon interaction (the retarded screened Coulomb interaction) is defined by means of the Coulomb matrix element \(V_{C} \left( q\right)\) and frequency- and momentum-dependent dielectric function \(\varepsilon \left( \nu ,q\right)\):

$$\begin{aligned} {\tilde{S}}_{C} \left( \nu ,q\right) =-\frac{1}{\pi } \mathrm {Im} \, \frac{V_{C} \left( q\right) }{\varepsilon \left( \nu ,q\right) } , \end{aligned}$$

where \(\nu\) and q are transferred frequency and momentum.

With the discovery of HTS in cuprates in 1986 several authors [17,18,19] suggested plasmonic mechanism as a reason for increased \(T_{c}\). On the other hand, Abrikosov in [20] emphasized the crucial role of the so called “extended saddle points”, observed near the Fermi level in the cuprate metal-oxides [20, 21], stating that they should be responsible for high values of \(\textit{T}_{c}\) even in the case of standard phonon mechanism. We have shown in a series of publications [22,23,24,25] that significant anisotropy of the Fermi velocity along quasi-2D Fermi surface (FS) of cuprates, caused by these features of the one-particle spectra, should bring about the emergence of acoustic plasmon branch even in the case of a single-connected FS, when the roles of “heavy” and “light” carriers are played by the quasiparticles, belonging to the FS in the vicinity of the “extended saddle points” and away from them, respectively. As the result, due to the symmetry of the Fermi-surface, the acoustic plasmonic excitations contribute in cuprates to the Cooper pairing in the d-wave channel.

Fig. 2
figure 2

Open Fermi-surface of one of the conduction bands with eight “necks” at L-points of the fcc crystal’s BZ, shown within the cubic cell of the reciprocal bcc lattice, with BZ shaded in gray. Also shaded in gray one of the planes, traversing four L- and two X-points of the BZ, used to define direction (through angle \(\varphi\)) of quasimomentum q lying within the hexagonal edge of BZ around L-point

Though the rare-earth superhydrides are completely different from HTS cuprate metal-oxides, there is some similarity in single-electron spectrum feature near Fermi level in both cases. The electronic bands of \(\mathrm {LaH}_{10}\) at pressure of 300 GPa, calculated in [5], are shown in Fig. 1 (Fig. 6b in [5]). Of the bands, crossing the Fermi level, of special interest for us is the one showing almost “flat” dispersion in the vicinity of the Fermi level. This band is shown schematically in Fig. 2 and has open FS with eight “necks” at \(\mathrm {L}\)-points of the fcc Brillouin zone (BZ). A slight “doping” with hydrogen vacancies may lead to a downward shift of the Fermi level (FL) without affecting the band structure. In this case additional hole pockets in the form of oblate ellipsoids may appear inside the “necks” of the FS. All these band features are due to the hybridization of La 4f and H 1s states near the FL in the vicinity of the L-point of the BZ [5]. The existence of such a band in the electronic spectrum of \(\mathrm {LaH}_{10}\) allowed us to put forward in [26] a supposition about the appearance of the acoustic plasmon excitations in the collective electronic spectra of this system and their influence on the superconducting critical temperature.

In the present paper we are going to expand upon the points made in [26] and in the next section will give the detailed analytical expression for the polarization operator for the system with uniaxially anisotropic effective mass. In Sect. 2 we introduce a simple analytical model for the conduction band describing the mentioned peculiarities and present the results of the numerical calculations of polarization operator, dielectric function and acoustic plasmon dispersion for such a band. We also study the effects of the small changes of the Fermi level and emphasize their potential role in the increasing of the critical temperature. In the final section we summarize the main results of this paper.

Polarization operator

The branches of the collective electronic spectrum are determined by zeroes of the frequency- and momentum-dependent dielectric permittivity (dielectric function)

$$\begin{aligned} \varepsilon \left( \nu ,q\right) =1+V_{C} \left( q\right) \Pi \left( \nu ,q\right) , \end{aligned}$$

which implies that we need to know the retarded polarization operator (PO),

$$\begin{aligned} \Pi \left( \nu ,q\right) {=}\frac{2}{\hbar ^{2} } \int \frac{d^{3} p}{\left( 2\pi \right) ^{3} } \frac{f\left( p+q\right) -f\left( p\right) }{\nu {-}\xi \left( p{+}q\right) {+}\xi \left( p\right) {+}i\delta }, \end{aligned}$$
(3)

in our case—in the presence of an open Fermi surface, described above. In (3) \(\xi\) denotes the quasiparticle energy, f stands for Fermi distribution function and \(\delta\) is a positive infinitesimal value.

For preliminary analytical estimates, the polarization operator for such a Fermi surface was represented as a superposition, with a certain weight, of the polarization operator for a sphere and the contributions from eight “necks” of the Fermi surface, which were approximated by one-sheet hyperboloids of revolution in the vicinity of the \(\mathrm {L}\)-points of the Brillouin zone.

For this purpose, we have extended the Lindhard formula for the 3D PO [27] to the case of anisotropic uniaxial effective mass and obtained a general analytical expression for a band, characterized by dispersion

$$\begin{aligned} \xi \left( \vec {p}\right) =\frac{p_{\bot }^{2} }{2m_{\bot } } \pm \frac{p_{\parallel }^{2} }{2m_{\parallel } }, \end{aligned}$$
(4)

which corresponds to a Fermi surface in the form of a spheroid or hyperboloid of revolution, depending on the sign in (4). Here \(p_{\bot }\) is the transvers 2D component of the momentum in the isotropic plane, \(p_{\parallel }\) is the momentum component along the third (rotational) axis, while \(m_{\bot }\) and \(m_{\parallel }\) are the respective masses. The result of integration in eq. (3) with spectrum (4) over the transverse momentum \(d^{2} p_{\bot }\) may be written in the form

$$\begin{aligned} \Pi \left( \nu ,\vec {q}\right)&=\frac{m_{\bot } p_{F\bot } \alpha _{\pm } \gamma _{\pm } }{\pi ^{3} \cos ^{2} \vartheta } \nonumber \\&\quad \cdot \left\{ 1+ \frac{1}{4z_{\pm } \alpha _{\pm } \gamma _{\pm } } \int _{-\gamma _{\pm } }^{\gamma _{\pm } }\left[ \mathrm {sgn}\left( A_{\pm ,1} \right) \sqrt{A_{\pm ,1}^{2} -B^{2} } \right. \right. \nonumber \\&\quad -\left. \left. {\mathop {\mathrm{sgn}}\nolimits }\left( A_{\pm ,2} \right) \sqrt{A_{\pm ,2}^{2} -B^{2} } \right] dx \right\} ; \end{aligned}$$
(5)

here we have introduced several functions dependent on the parameter of mass anisotropy \(\eta =\, \sqrt{{m_{\bot } \big / m_{\parallel } } }\) and polar angle \(\vartheta\) (\({-\pi \big / 2} \le \vartheta \le {\pi \big / 2}\)), measured from the plane perpendicular to the rotation axis (\(q_{\bot } \equiv q\cos \vartheta\), \(q_{\parallel } \equiv q\sin \vartheta\)):

$$\begin{aligned} \begin{array}{rcl} {A_{\pm ,1} } &{} {=} &{} {\pm \eta ^{2} x\cdot \sin \vartheta -z_{\pm } \alpha _{\pm } -u_{\pm } ;} \\ {A_{\pm ,2} } &{} {=} &{} {\pm \eta ^{2} x\cdot \sin \vartheta +z_{\pm } \alpha _{\pm } -u_{\pm } ;} \\ {B^{2} } &{} {=} &{} {\left( 1+\eta ^{2} x^{2} \right) \cos ^{2} \vartheta ,} \end{array} \end{aligned}$$

where \({\alpha _ \pm } = {\cos ^2}\vartheta \pm {\eta ^2}{\sin ^2}\vartheta\), \({z_ \pm } = \frac{\displaystyle {q{{\mid {{\alpha _ \pm }} \mid }^{{1 \big / 2}}}}}{\displaystyle {2p_{F \bot }^{\left( \pm \right) }}}\), \({u_ \pm } = \frac{\displaystyle \nu }{\displaystyle {q{{\mid {{\alpha _ \pm }} \mid }^{{1 \big / 2}}}V_{F \bot }^{\left( \pm \right) }}}\), \(p_{F\bot }^{\left( \pm \right) }\) and \(V_{F\bot }^{\left( \pm \right) }\)—Fermi momentum and velocity, lying in transversal plane, and ± signs correspond to those in (4). For the spheroid FS the integration limit \(\gamma _{+}\) in (5) is ultimately determined by the maximal value of the Fermi momentum in the direction of the rotation axis, \(p_{F\parallel }^{\left( +\right) } =\eta ^{-1} \cdot p_{F\bot }^{\left( +\right) }\), and reads as \(\gamma _{+} =\eta ^{-1}\). In the case of the open FS in the form of one-sheet hyperboloid the choice of \(\gamma _{-}\) is somewhat arbitrary, as formally there is no such a limiting value as \(p_{F\parallel }^{\left( -\right) }\). In reality this maximal value may not exceed the size of the BZ and should be determined by the “length” of the “necks” of the FS within the BZ.

If we choose for the hyperboloid case \(\gamma _{-} =\eta ^{-1}\), just as for the spheroid one, then the universal “anisotropic Lindhard” expressions for the real and imaginary parts of polarization operator read as:

$$\begin{aligned} \mathop {\mathrm{Re}}\nolimits \Pi _{3D}^{\left( \pm \right) }\left( {\nu ,\vec q} \right)&= \frac{{\sqrt{{m_ \bot }{m_\parallel }}\, p_{F \bot }^{\left( \pm \right) }}}{2{\pi ^3}} \nonumber \\&\quad \cdot \left[ {1 + \frac{{ {1 - {{\left( {{z_ \pm } + {u_ \pm }} \right) }^2}} }}{{4{z_ \pm }}}\ln \mid {\frac{\displaystyle {1 - {z_ \pm } - {u_ \pm }}}{{1 + {z_ \pm } + {u_ \pm }}}} \mid } \right. \nonumber \\&\quad -\left. {\frac{{ {1 - {{\left( {{z_ \pm } - {u_ \pm }} \right) }^2}} }}{{4{z_ \pm }}}\ln \mid {\frac{{1 + {z_ \pm } - {u_ \pm }}}{\displaystyle {{z_ \pm } - {u_ \pm } - 1}}} \mid } \right] ; \end{aligned}$$
(6)
$$\begin{aligned} \mathop {\mathrm{Im}}\nolimits \Pi _{3D}^{\left( \pm \right) }\left( {\nu ,\vec q} \right)&= \frac{{\sqrt{{m_ \bot }{m_\parallel }} \, p_{F \bot }^{\left( \pm \right) }}}{{4{\pi ^2}}} \nonumber \\&\quad \cdot \left[ {\frac{{1 - {{\left( {{u_ \pm } - {z_ \pm }} \right) }^2}}}{{{z_ \pm }}}\theta \left( {1 - {{\left( {{u_ \pm } - {z_ \pm }} \right) }^2}} \right) } \right. \nonumber \\&\quad -\left. {\frac{{1 - {{\left( {{u_ \pm } + {z_ \pm }} \right) }^2}}}{{{z_ \pm }}}\theta \left( {1 - {{\left( {{u_ \pm } + {z_ \pm }} \right) }^2}} \right) } \right] , \end{aligned}$$
(7)

where \(\theta \left( x\right)\) is the Heaviside step function.

In the case of one-sheet hyperboloid, if \(\vartheta \rightarrow \pm \mathop {\mathrm{arccot}}\nolimits \left( \eta \right)\) (the momentum belongs to the asymptotic conical surface), we have \(\alpha _{-} \rightarrow 0\), \(\mathop {\mathrm{Im}}\nolimits \Pi _{3D}^{\left( - \right) } \rightarrow 0\) and \(\mathop {\mathrm{Re}}\nolimits \Pi _{3D}^{\left( - \right) } \rightarrow {\sqrt{m_{\bot } m_{\parallel }}\, p_{F \bot }^{\left( - \right) } \big / 2\pi ^{3} }\).

Finally, if for \({\Pi _{3D}^{\left( -\right) } \big / \gamma _{-} }\) we formally set \(\gamma _{-} \rightarrow 0\), then for \(\vartheta =0\) the classical Stern result [28] for 2D degenerate electron gas with \(m_{2D} =m_{\bot }\) and \(p_{F}^{(2D)} =p_{F \bot }^{\left( - \right) }\) is obtained from (5):

$$\begin{aligned} \mathop {\mathrm{Re}}\nolimits \Pi _{2D} \left( \nu ,\vec {q}\right)= & {} \frac{m_{\bot } p_{F\bot } }{\pi ^{3} } \left\{ 1+ \frac{1}{2z} \right. \cdot \nonumber \\&\left. \left[ \mathrm {sgn}\left( u{-}z\right) \theta \left( \left( u{-}z\right) ^{2} {-}1\right) \sqrt{\left( u{-}z\right) ^{2} {-}1} \right. \right. - \nonumber \\&{\left. \left. \mathrm {sgn}\left( u{+}z\right) \theta \left( \left( u{+}z\right) ^{2} {-}1\right) \sqrt{\left( u{+}z\right) ^{2} {-}1} \right] \right\} }; \end{aligned}$$
(8)
$$\begin{aligned} \mathop {\mathrm{Im}}\nolimits \Pi _{2D} \left( \nu ,\vec {q}\right)= & {} \frac{m_{\bot } p_{F\bot } }{2\pi ^{2} } \cdot \nonumber \\&\left\{ \theta \left( 1-\left( u-z\right) ^{2} \right) \sqrt{1-\left( u-z\right) ^{2} } \right. \nonumber \\&-\left. \theta \left[ 1-\left( u+z\right) ^{2} \right] \sqrt{1-\left( u+z\right) ^{2} } \right\} . \end{aligned}$$
(9)

Note that the expressions, derived here, may be applied to the description of dynamic screening in degenerate multi-valley semiconductors (ellipsoid of revolution) and on the surfaces (1,1,1) of noble metals (one-sheet hyperboloid).

For the parameters extracted from the calculated single-particle spectra, we constructed polarization operator as a superposition of the polarization operator for a sphere and the contributions from eight “necks” of the Fermi surface. It appears, that even such a crude model approximates the polarization operator quite satisfactory (dielectric and spectral functions obtained in this way are shown with thin lines on Fig. 4 below) and confirms the possibility of the existence of acoustic plasmons in a certain range of solid angles.

Numerical calculation of the polarization operator and plasmon dispersion

For a more detailed numerical calculations, we propose the following approximation of the band dispersion in the vicinity of the \(\mathrm {L}\)-points in the Brillouin zone, which models the result of hybridization of La 4f and H 1s states:

$$\begin{aligned} {E_L}\left( {{p_ \bot },{p_\parallel }} \right) {=} {\left\{ \begin{array}{ll} {\xi _r}\left[ {\cos \left( {\pi {\frac{{p_ \bot }}{{{p_0}}}}} \right) {-} 1}\right] {-} {\frac{{p_\parallel ^2}}{{2{m_\parallel }}}}{,}&{}{p_ \bot }{\le }{p_0},\\ {\xi _r}\left[ {{\frac{{{\pi ^2}}}{2}{{\left( {\frac{{{p_ \bot }}}{{{p_0}}}{-} 1} \right) }^2}}{-} 2}\right] {-} {\frac{{p_\parallel ^2}}{{2{m_\parallel }}}}{,}&{}{p_ \bot }{>}{p_0}; \end{array}\right. } \end{aligned}$$

here the momentum component \(p_\parallel\) is along the \(\Gamma\) – \(\mathrm {L}\) direction, while \(p_{\bot }\) lies in the plain of the hexagonal edge of BZ. This expression is merged with a usual tight-binding-approximation formula, based on s orbitals, away from the \(\mathrm {L}\)-points:

$$\begin{aligned} {E_{TB}}= & {} - \frac{{{\varepsilon _{TB}}}}{4}\left[ \cos \left( {\frac{1}{2}{k_x}a} \right) \cos \left( {\frac{1}{2}{k_y}a} \right) + \cos \left( {\frac{1}{2}{k_y}a} \right) \cos \left( {\frac{1}{2}{k_z}a} \right) \right. \nonumber \\&+\left. \cos \left( {\frac{1}{2}{k_z}a} \right) \cos \left( {\frac{1}{2}{k_x}a} \right) \right] , \end{aligned}$$
(10)

where \(\varepsilon _{TB}\approx 8\) eV is the width of the band and a is the lattice constant of the cubic fcc lattice.

The obtained model band dispersion and density of states (DOS) are shown in Fig. 3, alongside with these characteristics for the “row” tight-binding band (10). We should mention in passing that aside from providing conditions for the appearance of the collective acoustic charge-density excitations, which we are going to discuss now, the one-particle spectrum feature that we are describing here also provides for the increased DOS near the FL. The higher DOS mean higher values of phonon coupling constant, thus increasing superconducting critical temperature.

Fig. 3
figure 3

Dispersion along the symmetry directions of BZ (a) and DOS (b) of the model conduction band, used in numerical calculations (thick lines). The dashed lines in the figure show the dispersion and DOS, respectively, of the simple band in the tight-binding approximation, used in construction of the model band

On Fig. 4 the results of numerical calculation of dielectric permittivity and spectral function of the screened Coulomb interaction for the model band described above, are shown with thick lines. In this computations we first calculated the imaginary part of the polarization function (3) and then obtained the real part of the dielectric function by means of the Kramers–Kronig relation. The computation of the energy-surface integrals for the imaginary part of (3) was done by covering the BZ with a mesh of tetrahedral cells (see e.g. [29, 30]), with the density of mesh point been adaptively increased when approaching the integration surface. The number of tetrahedra crossed by the integration surface was in the range of \(10^{7}\) in the highly parallelized GPU computation. The quasi-momentum \(q=0.05{\pi \big / a}\) for this calculation lies in the plain of the hexagonal edge of the Brillouin zone and its direction is given by angle \(\varphi =30{}^\circ\) (see Fig. 2).

Fig. 4
figure 4

The frequency dependences of the real (solid curve) and imaginary (dashed curve) parts of the dielectric permittivity (a) and the spectral function of the electron–electron interaction (b). Thick lines represent the numerical calculation in the framework of our model band dispersion, while the functions resulting from the analytical calculation of polarization operator as superposition of contributions for spherical and hyperboloid parts, described in Sect. 2, are shown with thin lines

There is a low-frequency zero (at about 0.12 eV in this case), corresponding to the peak in spectral function, and describing the additional branch of collective charge excitations (the high-frequency zero of the dielectric function, corresponding to the usual plasma frequency, lies far beyond the frequency range of this Figure). This additional branch appears due to the significant Fermi-velocity anisotropy of the conduction band, caused by “flat” portions of the band spectrum, as was discussed in the Introduction, which can be seen from Fig. 5, where we compare the dielectric permittivity and spectral function, calculated for our model band (solid curve on Fig. 3), with those of a conduction band of an fcc metal, obtained within the tight-binding approximation (dashed curve on Fig. 3). It should be noted that the acoustic plasmonic branch appears only in a portion of the phase space, and for some directions of quasimomentum it is much more pronounced then in the others. One of this favorite directions is defined by the diagonal of the cubic cell of the reciprocal bcc lattice, so for the Fig. 5 we made calculation with the same model band as for the Fig. 4, but with quasimomentum lying in this direction—to emphasize the well-defined additional zeroes of dielectric function for our model band (thick curves) and their absence in the case of a basic tight-binding band.

Fig. 5
figure 5

Comparison of the dielectric permittivity (a) and spectral function of the screened Coulomb interaction (b) for the model conduction band of \(\mathrm {LaH}_{10}\) (thick curves) and conduction band of an fcc metal in the tight-binding approximation (thin curves) (quasi-momentum here is directed along the diagonal of the cubic cell of the reciprocal bcc lattice, see Fig. 2). Solid and dashed curves on (a) correspond to the real and imaginary parts of dielectric permittivity, correspondingly

Finally, dispersion and attenuation of acoustic plasmons with momenta lying in the plane of the hexagonal edge of the Brillouin zone are shown on Fig. 6. The frequency values were determined by the positions of the maximum of the spectral function for corresponding momenta and attenuation was estimated as the half-width of the plasmonic peak.

Fig. 6
figure 6

Dispersion (dashed curve) and attenuation (shadowing) of the acoustic plasmon for transferred quasi-momenta in the plain of the hexagonal edge of the Brillouin zone. Dots represent calculated values, while dashed line is just a guide for the eye

Fig. 7
figure 7

Open Fermi-surface of the same conduction band as in Fig. 2, but with FL lowered by 0.08 eV. We can see the appearance of additional sheets of the FS in the form of small spheroidal “lenses” inside the “necks” in the vicinity of the L-points

In Introduction we have mentioned a possibility for the composition of a superhydride to deviate slightly from stoichiometry. For example, a small number of hydrogen vacancies in lanthanum decahydride, \(\mathrm {LaH}_{10-x}\) with \(x \ll 1\), should lead to a slight lowering of the Fermi level with no significant effect on the band structure. Up until now we have dealt with the FL position corresponding to the ideal \(\mathrm {LaH}_{10}\), but here, toward the end of the paper, we’ll touch upon the effects connected to the change in the position of the FL.

The change of the FS as the result of a shift of the Fermi level by \(-0.08\) eV is illustrated in Fig. 7. The main result of this change is the appearance of the additional hole-like “pockets” of the Fermi surface, which implies farther increase in the system’s anisotropy, not only through the variation of the Fermi velocity over the singly-connected FS, but also through the variation of the masses on the various sheets of the FS. As we mentioned earlier, such anisotropy facilitates the appearance of acoustic plasmons.

Fig. 8
figure 8

The frequency dependencies of the real (solid curves) and imaginary (dashed curves) parts of the dielectric permittivity (a) and the spectral function of the electron–electron interaction (b). Here thick lines represent the calculation with the FS shifted by − 0.08 eV, while the thin lines correspond to the “stoichiometric” \(\mathrm {LaH}_{10}\) (they are the same as the thick lines of Fig. 4)

The dielectric permittivity and Coulomb spectral function calculated with this FL position are shown on Fig. 8. We see that the maximum of the resulting plasmonic spectral function drifts to the left with redistribution of its weight towards the lower frequencies, simultaneously contributing to the increase of the electron–electron attraction constant and decrease of the Coulomb repulsion constant.

Conclusion

The experimental measurement of the exponent of the isotope effect was performed in [13] with substitution of deuterium for hydrogen in \({{{{{\mathop {\mathrm{LaD}}\nolimits } }_{10}}} \big / {{{{\mathop {\mathrm{LaH}}\nolimits } }_{10}}}}\). The result was rather close to the classical value \(\beta ={1\big / 2}\) obtained in the model of the pure Fröhlich electron-phonon attraction. On the one hand, this is a strong experimental suggestion that the electron-phonon interaction plays the predominant role in superconductivity of the rare-earth superhydrides. On the other hand, using the formulas for isotope-effect exponent, which are derived from (1) and (2) (see [2, 31], respectively):

$$\begin{aligned} \beta= & {} \frac{1}{2} \left[ 1-\frac{\left( 1+\lambda \right) \left( 1+0.62\lambda \right) \mu ^{*2} }{\left[ \lambda -\mu ^{*} \left( 1+0.62\lambda \right) \right] ^{2} } \right] ; \\ \beta= & {} \frac{1}{2} \left\{ 1-\frac{\left( 1+\lambda _{\infty } \right) \left( 1+\lambda +\lambda _{0} \right) \mu ^{*2} }{\left[ \lambda -\mu ^{*} \left( 1+\lambda _{\infty } \right) \right] ^{2} } \right\} , \end{aligned}$$

we may conclude that the effective Coulomb constant \(\mu ^{*}\) must be negligibly small. Our analytical and numerical results suggest that low-frequency acoustic collective electronic excitations (acoustic plasmons) may appear in superhydrides due to the peculiarities of the one-particle band spectra of these compounds. We believe that the existence of these excitations bring about the additional suppression of the Coulomb repulsion, thus constituting the “plasmonic” mechanism of the increasing of \(T_{c}\). As was mentioned in [26], in \(\mathrm {LaH}_{10}\) the preconditions for such mechanism arise due to the hybridization of La 4f and H 1s states near the Fermi level in the vicinity of the \(\mathrm {L}\)-point of the BZ [5]. It should be noted that these conditions are absent in the cases of another lanthanide superhydrides’ electron spectra, such as \(\mathrm {YbH}_{10}\) and \(\mathrm {LuH}_{8}\) under the same megabar pressure calculated in [8], with essentially lower predicted \(T_c\) of 102.1 K and 86.2 K, respectively. On the other hand, the conditions for the “plasmonic” mechanism in \(\mathrm {LaH}_{10}\) appear to be even more favorable in the case of a slight downward sift of the FL (induced by the presence of a small amount of hydrogen vacancies in the decahydride structure) due to the enhancement of the low-frequency part of acoustic plasmon spectral function.

Of course, the overall Coulomb interaction remains repulsive and the plasmonic attraction is effective in a limited portion of the phase volume. Nevertheless, the interaction is isotropized in all directions due to the s-wave symmetry of pairing, which obviously occurs in the case of superhydrides under pressure (in contrast with the HTS cuprates demonstrating d-wave pairing), and the additional plasmon attraction leads to the suppression of the Coulomb pseudopotential and to the increase in the superconducting critical temperature of superhydrides such as \(\mathrm {LaH}_{10}\).