- Email: [email protected]
Lattice stability and superconductivity of the metallic hydrogen at high pressure
PERGAMON
Solid State Communications 119 (2001) 569±572
www.elsevier.com/locate/ssc
Lattice stability and superconductivity of the metallic hydrogen at high pressure E.G. Maksimov, D.Yu. Savrasov* a
P.N. Lebedev Physical Institute, 117924 Moscow, Russian Federation Received 13 March 2001; accepted 3 July 2001 by L.V. Keldysh
Abstract Ab initio calculations of metallic hydrogen are presented. Investigation of the lattice stability of metallic monoatomic phases con®rms the main part of conclusions obtained previously by Kagan's group in the framework of the perturbation approach to fourth order in the electron±phonon interaction. The results of the ab initio calculation of the critical temperature Tc of the superconducting transition is also presented. It is shown that Tc can reach very high values about 600 K near the lattice instability in respect to shear deformations. q 2001 Published by Elsevier Science Ltd. PACS: 62.20.Kr; 74.25.2q Keywords: A. Metallic hydrogen; D. Phonons; D. Electron±phonon interaction
The problem of the metallic hydrogen attracted considerable interest during many years beginning from the classic study by Wigner and Hantington [1]. In that work it was predicted for the ®rst time that hydrogen which exist under low pressure in a molecular insulating phase should transform to a monoatomic metallic phase under high pressure. The transition pressure predicted in that work turned out to be rather low, Pc 25 GPa: After that there was a lot of theoretical investigations of this point. These calculations lead to a spread in the values Pc over the range 25± 1500 GPa. The experimental investigations do not show the insulator±metal transition at least up to 300 GPa (for details see the recent review Ref. [2]). Nevertheless, the investigations of the metallic hydrogen still attract the attention of the researchers. The reason is related to the predictions [3,4] of very interesting, even exotic, phenomena, which may exist in the metallic hydrogen phase. The ®rst one is the possible existence of a high-temperature superconductivity with critical temperature Tc , 200 K [3]. Another nontrivial idea has been advanced by Kagan [4] about the existence of a metastable metallic phase of the
* Corresponding author. Address: Max-Planck-Institute fur Festkorperforschung, Abteilung Andersen, 70569 Stuttgart, Germany. Tel./fax: 149-711-689-1665. E-mail address: [email protected] (D.Y. Savrasov).
hydrogen at zero pressure. These two ideas were discussed in the recent two works [5,6]. This short communication is also devoted to the investigation of these problems. The comprehensive study of the metastable phase of hydrogen at zero pressure was given by Kagan's group [4] using the perturbation theory to fourth order in the electron± proton interaction. It was shown that there is a minimum on the curve of the total energy E(V) at the volume V corresponding rs < 1:65: The value rs is de®ned as V 4p r a 3 ; N 3 s B
1
where aB is the Bohr radius. It was also demonstrated that for the densities satisfying the inequality rs # 1:65
2
the bulk modulus is positive K0 2V
22 E . 0; 2V 2
3
which ensures stability with respect to long wavelength density disturbances. Moreover, it was shown that the main part of simple crystalline structures like BCC, FCC is unstable with respect to some shear deformation at P 0: It was found that at P 0 monoatomic metallic hydrogen has a tendency towards crystallization in highly anisotropic structures. The minimum of energy is possessed by a family
0038-1098/01/$ - see front matter q 2001 Published by Elsevier Science Ltd. PII: S 0038-109 8(01)00301-5
E.G. Maksimov, D.Yu. Savrasov / Solid State Communications 119 (2001) 569±572
of crystalline structures that is based on the simple hexagonal lattice with ratio c=a , 1 and produces a trigonal `®lamentary' structure with two-dimensional periodicity. The difference between the energies of the family members is very small and comprises the value about 10 K. The other family with a little higher energies is also based on simple hexagonal lattice with c=a , 1 and yields a layerlike structure with ®xed separation of crystal planes but with different proton arrangement in these planes. The Kagan's group has also studied the behavior of the monoatomic hydrogen at the elevated pressure. The anisotropic crystal structures are stable and energetically favored practically over the whole range 1 # rs # 1:65:
4
The simple crystal structures are dynamically unstable up to pressure corresponding the inequality rs $ 1:2:
5
We have checked all these statements using nonperturbative approach to work out the total energy of crystals in the framework of the density functional approach [9]. The fullpotential LMTO code [10] was used for numerical calculations. We have con®rmed the main part of the results obtained in the works [4,7,8]. There are only some differences concerning the particular values of the energies of speci®c crystal structures and the values of the parameters rs corresponding, for example, to the range of the dynamical stability of crystal phases. The most prominent difference is connected with the behavior of shear modules of FCC structure. Our value rs 1:05; when this structure becomes stable, is considerably smaller than rs 1:25 obtained in the work [8] and the increasing of the shear modulus C44 under the pressure obtained by us is faster than in the work [8]. The discussion of all these problems will be presented elsewhere later. Here we would like to consider with more details the problem of high-Tc superconductivity in the metallic hydrogen. After the initial work [3] the problem of high-Tc superconductivity was discussed many times [11±17]. The conclusion of the main part of these works was the existence of high value Tc in the metallic hydrogen due to high value of the Debye temperature in this system. This conclusion stems easily from the simplest estimation based on the BCS theory. It this approximation Tc can be written as 1 Tc < 1:14TD exp 2 : 6 l 2 mp Here TD is the Debye temperature, l is the constant of electron±phonon coupling, and m p is the Coulomb pseudopotential. Metallic hydrogen differs from the majority of ordinary metals in the following features which favors high Tc values: p 1. due to the small atomic mass, the value TD 1= M in hydrogen are much high than in other metals;
2. since the hydrogen atom has no internal electron shells, the electron±phonon interaction is merely the Coulomb potential rather than a signi®cantly weaker crystalline pseudopotential; 3. because the electron density in metallic hydrogen is high than in other metals the Coulomb pseudopotential turns out to be rather small. The work of Gupta and Sinha [14] should be mentioned here. They suggested that the proper accounting of the nonlinear screening and nonadiabaticity can reduce the Coulomb electron±proton potential to some type of small pseudopotential with the resulting decrease of the coupling constant l to the small value of about 0.2. The all preceding investigations of the superconductivity in metallic hydrogen were made either in a nonselfconsisting manner or in the framework of the second order perturbation in respect to electron±proton interaction. The question about the actual value of the coupling constant l in metallic hydrogen and its Tc, by this means, still remains unanswered. We use for our investigations ab initio linear-response method developed recently [18,19] which was applied very ef®ciently to study superconducting and kinetic properties in many simple and transition metals [18±20]. This method allows to calculate phonon spectra, matrix elements of electron±phonon interaction, and so-called spectral functions of the electron±phonon interaction a2 F v and a2tr F v: The function a2 F v describes renormalization of the electron mass and relaxation rate in the normal state of metals and superconducting properties in the framework of Eliashberg equations [21,22]. The constant of the electron±phonon coupling l is expressed in terms of the function a2 F v as
l2
Z1 a2 F v dv v 0
The
function
a2tr F v
7 describes
the
dynamical
7000
6000
5000
Frequency, K
570
4000
3000
2000
1000
0 (1,0,0)
(0,0,0)
(1/2,1/2,1/2)
Fig. 1. Calculated phonon spectrum of metallic hydrogen in FCC lattice for rs 1:
E.G. Maksimov, D.Yu. Savrasov / Solid State Communications 119 (2001) 569±572
conductivity s (v ). The details of the calculation of these function can be found in [18,19]. Because the self-consisting calculations of the superconducting properties for complicated crystal structures of metallic hydrogen, existing at considerably low pressure, are very time-consuming, we have made such calculations for the simple FCC structure in the range of its stability r s , 1:05: Here we present the result of our calculations for rs 1 which corresponds the pressure P 2000 GPa. Fig. 1 shows the result of our calculations of the phonon spectrum. We would like to emphasize the existence of a soft phonon mode in this system. The whole branch of transverse phonons is considerably lower than the longitudinal one. It is the consequence of the above mentioned proximity of the dynamical instability of FCC structure in the respect to the shear displacement. Fig. 2 shows the function a2 F v and a2tr F v: Usually, in the ordinary metals these two function are considerably similar [18,19]. Here there is a similar case but the function a2tr F v has larger values at energies corresponding to the transverse phonons. Thus, these phonons give very large contribution to both these functions, that is, to the electron mass renormalization, relaxation rate, and superconducting pairing as well as in the electron scattering accompanying the transport processes. The coupling constant l calculated by the Eq. (7) has the very large value l < 7: It is the result of the existence of the soft phonon mode. We can easily understand this fact using the expression for coupling constant l derived for simple metals [23].
l
2
1:51 2 V pl V k l rs ie v2
8
where V pl is the plasma frequency of ions s 4pe2 n V pl : M
9
5
αtr2F(ω)
Spectral functions
4
3
2
α2F(ω)
571
This value can be written for the protons in the form
V pl
1:04 eV: rs3=2
10
Vie2 is the averaged squared screened electron±ion potential Z2kF Vie2
0 Z2kF 0
dqq3 Vie q dqq3 Vie 0
:
11
The value of V pl for our case is about 1 eV. It means that the ratio kV pl2 =v2 l is $ 100: It is one order value large than for alkaline metals [23]. The comparable values of this ratio can exist in some polyvalent metals such as Pb, but the values of Vie2 for these metals are usually smaller than for metallic hydrogen. The numerical solution of the Eliashberg equation with the function a2 F v presented on Fig. 2 gives the very high value of Tc < 600 K. The most important difference between polyvalent metals and metallic hydrogen is the values corresponding to Debye temperatures. The softness of the transverse phonons presented on Fig. 1 is a matter of convention. Indeed, this mode is soft in comparison with the proton plasma frequency and the longitudinal mode, but the averaged transverse phonons frequency is not too small (kv' l , 1000 K) to lead to a high values of Tc 600 K. We also calculated the electroresistivity and thermoconductivity of the metallic hydrogen using the function a2tr F v: Both these properties demonstrate usual metallic type behavior. The corresponding transport coupling constant l tr is even larger than l and equals to 10.3. Certainly, the obtained results concerning high-Tc superconductivity are beyond not only a practical applications but any experimental checking as well. Nevertheless, they can shed additional light on the two important problems widely disputed some time ago. First, this example demonstrates that the potential of the electron±phonon interaction in increasing Tc is very high and has not been fully used at full length up to now. Second, it con®rms the old idea that the high-Tc superconductivity can, under the conditions of a normal electron±phonon mechanism of superconductivity, be obtained in the systems close to structure instability. The values Tc obtained in our work are a few times lager than it was found in many old works [11±17], where the softness of the transverse phonons was absent.
1
Acknowledgements
0 0
1000
2000
3000
4000
5000
6000
7000
Frequency, K
Fig. 2. Calculated spectral and transport spectral function of electron±phonon interaction of metallic hydrogen in FCC lattice for rs 1:
The authors are indebted to V.L. Ginzburg, O.V. Dolgov and Yu.I. Shilov for helpful discussions. The work was partially supported by the grants from ISRC, RFBI and the Russian State Program for Superconductivity.
572
E.G. Maksimov, D.Yu. Savrasov / Solid State Communications 119 (2001) 569±572
References [1] E. Wigner, H.B. Huntington, J. Chem. Phys. 3 (1935) 764. [2] E.G. Maksimov, Yu.I. Shilov, UFN 169 (1999) 1223 (Physics Uspekhi 42 (1999) 1121). [3] N.W. Ashcroft, Phys. Rev. Lett. 21 (1968) 1748. [4] E.G. Brovman, Yu.M. Kagan, A. Kholas, JETP 61 (1971) 2429 (Sov. Phys. JETP, 34 (1971) 1258). [5] C.F. Richardson, N.W. Ashcroft, Phys. Rev. Lett. 78 (1997) 118. [6] S.D. Kaim, N.P. Kovalenko, E.V. Vasiliu, Cond-mat. 0010019 v2. [7] E.G. Brovman, Yu.M. Kagan, A. Kholas, JETP 62 (1972) 1492 (Sov. Phys. JETP 35 (1972) 783). [8] Yu.M. Kagan, V.V. Pushkarev, A. Kholas, JETP 73 (1977) 967 (Sov. Phys. JETP 46 (1977) 511). [9] W. Kohn, L.J. Sham, Phys. Rev. A140 (1965) 1133. [10] S.Yu. Savrasov, D.Yu. Savrasov, Phys. Rev. B46 (1992) 12181. [11] T. Schneider, E. Stoll, Physica 55 (1971) 702. [12] A.C. Switendick, in: D.V. Douglas (Ed.), Superconductivity in d- and f-Band Metals, Plenum Press, New York, 1976, p. 593.
[13] R.P. Gupta, S.K. Sinha, in: D.V. Douglas (Ed.), Superconductivity in d- and f-Band Metals, Plenum Press, New York, 1976, p. 583. [14] D.A. Papaconstantopoulos, B.M. Klein, Ferroelectrics 17 (1977) 307. [15] T.W. Barbee, A. Garcia, M.L. Cohen, Nature (1984) 340, see also page 369. [16] L.G. Caron, Phys. Rev. B9 (1974) 5025. [17] Yu.I. Shilov, N.R. Ivanov, Fiz. Tver. Tela 37 (1994) 1473 (Phys. Solid State, 37 (1994) 1473). [18] S.Yu. Savrasov, D.Yu. Savrasov, Phys. Rev. B54 (1996) 16470. [19] E.G. Maksimov, D.Yu. Savrasov, S.Yu. Savrasov, UFN 167 (1997) 337 (Phys. Usp., 40 (1997) 317). [20] E.G. Maksimov, D.Yu. Savrasov, O.V. Dolgov, S.Yu. Savrasov, Solid State Commun. 106 (1998) 409. [21] G.M. Eliashberg, ZETP 38 (1960) 966 (Sov. Phys. JETP 11 (1960) 696). [22] G.M. Eliashberg, ZETP 39 (1960) 1437 (Sov. Phys. JETP, 12 (1961) 1000). [23] D. Scalapino, in: R.D. Parks (Ed.), Superconductivity, vol. 1, Marcel Dekker, New York, 1969, p. 449.
Solid State Communications 119 (2001) 569±572
www.elsevier.com/locate/ssc
Lattice stability and superconductivity of the metallic hydrogen at high pressure E.G. Maksimov, D.Yu. Savrasov* a
P.N. Lebedev Physical Institute, 117924 Moscow, Russian Federation Received 13 March 2001; accepted 3 July 2001 by L.V. Keldysh
Abstract Ab initio calculations of metallic hydrogen are presented. Investigation of the lattice stability of metallic monoatomic phases con®rms the main part of conclusions obtained previously by Kagan's group in the framework of the perturbation approach to fourth order in the electron±phonon interaction. The results of the ab initio calculation of the critical temperature Tc of the superconducting transition is also presented. It is shown that Tc can reach very high values about 600 K near the lattice instability in respect to shear deformations. q 2001 Published by Elsevier Science Ltd. PACS: 62.20.Kr; 74.25.2q Keywords: A. Metallic hydrogen; D. Phonons; D. Electron±phonon interaction
The problem of the metallic hydrogen attracted considerable interest during many years beginning from the classic study by Wigner and Hantington [1]. In that work it was predicted for the ®rst time that hydrogen which exist under low pressure in a molecular insulating phase should transform to a monoatomic metallic phase under high pressure. The transition pressure predicted in that work turned out to be rather low, Pc 25 GPa: After that there was a lot of theoretical investigations of this point. These calculations lead to a spread in the values Pc over the range 25± 1500 GPa. The experimental investigations do not show the insulator±metal transition at least up to 300 GPa (for details see the recent review Ref. [2]). Nevertheless, the investigations of the metallic hydrogen still attract the attention of the researchers. The reason is related to the predictions [3,4] of very interesting, even exotic, phenomena, which may exist in the metallic hydrogen phase. The ®rst one is the possible existence of a high-temperature superconductivity with critical temperature Tc , 200 K [3]. Another nontrivial idea has been advanced by Kagan [4] about the existence of a metastable metallic phase of the
* Corresponding author. Address: Max-Planck-Institute fur Festkorperforschung, Abteilung Andersen, 70569 Stuttgart, Germany. Tel./fax: 149-711-689-1665. E-mail address: [email protected] (D.Y. Savrasov).
hydrogen at zero pressure. These two ideas were discussed in the recent two works [5,6]. This short communication is also devoted to the investigation of these problems. The comprehensive study of the metastable phase of hydrogen at zero pressure was given by Kagan's group [4] using the perturbation theory to fourth order in the electron± proton interaction. It was shown that there is a minimum on the curve of the total energy E(V) at the volume V corresponding rs < 1:65: The value rs is de®ned as V 4p r a 3 ; N 3 s B
1
where aB is the Bohr radius. It was also demonstrated that for the densities satisfying the inequality rs # 1:65
2
the bulk modulus is positive K0 2V
22 E . 0; 2V 2
3
which ensures stability with respect to long wavelength density disturbances. Moreover, it was shown that the main part of simple crystalline structures like BCC, FCC is unstable with respect to some shear deformation at P 0: It was found that at P 0 monoatomic metallic hydrogen has a tendency towards crystallization in highly anisotropic structures. The minimum of energy is possessed by a family
0038-1098/01/$ - see front matter q 2001 Published by Elsevier Science Ltd. PII: S 0038-109 8(01)00301-5
E.G. Maksimov, D.Yu. Savrasov / Solid State Communications 119 (2001) 569±572
of crystalline structures that is based on the simple hexagonal lattice with ratio c=a , 1 and produces a trigonal `®lamentary' structure with two-dimensional periodicity. The difference between the energies of the family members is very small and comprises the value about 10 K. The other family with a little higher energies is also based on simple hexagonal lattice with c=a , 1 and yields a layerlike structure with ®xed separation of crystal planes but with different proton arrangement in these planes. The Kagan's group has also studied the behavior of the monoatomic hydrogen at the elevated pressure. The anisotropic crystal structures are stable and energetically favored practically over the whole range 1 # rs # 1:65:
4
The simple crystal structures are dynamically unstable up to pressure corresponding the inequality rs $ 1:2:
5
We have checked all these statements using nonperturbative approach to work out the total energy of crystals in the framework of the density functional approach [9]. The fullpotential LMTO code [10] was used for numerical calculations. We have con®rmed the main part of the results obtained in the works [4,7,8]. There are only some differences concerning the particular values of the energies of speci®c crystal structures and the values of the parameters rs corresponding, for example, to the range of the dynamical stability of crystal phases. The most prominent difference is connected with the behavior of shear modules of FCC structure. Our value rs 1:05; when this structure becomes stable, is considerably smaller than rs 1:25 obtained in the work [8] and the increasing of the shear modulus C44 under the pressure obtained by us is faster than in the work [8]. The discussion of all these problems will be presented elsewhere later. Here we would like to consider with more details the problem of high-Tc superconductivity in the metallic hydrogen. After the initial work [3] the problem of high-Tc superconductivity was discussed many times [11±17]. The conclusion of the main part of these works was the existence of high value Tc in the metallic hydrogen due to high value of the Debye temperature in this system. This conclusion stems easily from the simplest estimation based on the BCS theory. It this approximation Tc can be written as 1 Tc < 1:14TD exp 2 : 6 l 2 mp Here TD is the Debye temperature, l is the constant of electron±phonon coupling, and m p is the Coulomb pseudopotential. Metallic hydrogen differs from the majority of ordinary metals in the following features which favors high Tc values: p 1. due to the small atomic mass, the value TD 1= M in hydrogen are much high than in other metals;
2. since the hydrogen atom has no internal electron shells, the electron±phonon interaction is merely the Coulomb potential rather than a signi®cantly weaker crystalline pseudopotential; 3. because the electron density in metallic hydrogen is high than in other metals the Coulomb pseudopotential turns out to be rather small. The work of Gupta and Sinha [14] should be mentioned here. They suggested that the proper accounting of the nonlinear screening and nonadiabaticity can reduce the Coulomb electron±proton potential to some type of small pseudopotential with the resulting decrease of the coupling constant l to the small value of about 0.2. The all preceding investigations of the superconductivity in metallic hydrogen were made either in a nonselfconsisting manner or in the framework of the second order perturbation in respect to electron±proton interaction. The question about the actual value of the coupling constant l in metallic hydrogen and its Tc, by this means, still remains unanswered. We use for our investigations ab initio linear-response method developed recently [18,19] which was applied very ef®ciently to study superconducting and kinetic properties in many simple and transition metals [18±20]. This method allows to calculate phonon spectra, matrix elements of electron±phonon interaction, and so-called spectral functions of the electron±phonon interaction a2 F v and a2tr F v: The function a2 F v describes renormalization of the electron mass and relaxation rate in the normal state of metals and superconducting properties in the framework of Eliashberg equations [21,22]. The constant of the electron±phonon coupling l is expressed in terms of the function a2 F v as
l2
Z1 a2 F v dv v 0
The
function
a2tr F v
7 describes
the
dynamical
7000
6000
5000
Frequency, K
570
4000
3000
2000
1000
0 (1,0,0)
(0,0,0)
(1/2,1/2,1/2)
Fig. 1. Calculated phonon spectrum of metallic hydrogen in FCC lattice for rs 1:
E.G. Maksimov, D.Yu. Savrasov / Solid State Communications 119 (2001) 569±572
conductivity s (v ). The details of the calculation of these function can be found in [18,19]. Because the self-consisting calculations of the superconducting properties for complicated crystal structures of metallic hydrogen, existing at considerably low pressure, are very time-consuming, we have made such calculations for the simple FCC structure in the range of its stability r s , 1:05: Here we present the result of our calculations for rs 1 which corresponds the pressure P 2000 GPa. Fig. 1 shows the result of our calculations of the phonon spectrum. We would like to emphasize the existence of a soft phonon mode in this system. The whole branch of transverse phonons is considerably lower than the longitudinal one. It is the consequence of the above mentioned proximity of the dynamical instability of FCC structure in the respect to the shear displacement. Fig. 2 shows the function a2 F v and a2tr F v: Usually, in the ordinary metals these two function are considerably similar [18,19]. Here there is a similar case but the function a2tr F v has larger values at energies corresponding to the transverse phonons. Thus, these phonons give very large contribution to both these functions, that is, to the electron mass renormalization, relaxation rate, and superconducting pairing as well as in the electron scattering accompanying the transport processes. The coupling constant l calculated by the Eq. (7) has the very large value l < 7: It is the result of the existence of the soft phonon mode. We can easily understand this fact using the expression for coupling constant l derived for simple metals [23].
l
2
1:51 2 V pl V k l rs ie v2
8
where V pl is the plasma frequency of ions s 4pe2 n V pl : M
9
5
αtr2F(ω)
Spectral functions
4
3
2
α2F(ω)
571
This value can be written for the protons in the form
V pl
1:04 eV: rs3=2
10
Vie2 is the averaged squared screened electron±ion potential Z2kF Vie2
0 Z2kF 0
dqq3 Vie q dqq3 Vie 0
:
11
The value of V pl for our case is about 1 eV. It means that the ratio kV pl2 =v2 l is $ 100: It is one order value large than for alkaline metals [23]. The comparable values of this ratio can exist in some polyvalent metals such as Pb, but the values of Vie2 for these metals are usually smaller than for metallic hydrogen. The numerical solution of the Eliashberg equation with the function a2 F v presented on Fig. 2 gives the very high value of Tc < 600 K. The most important difference between polyvalent metals and metallic hydrogen is the values corresponding to Debye temperatures. The softness of the transverse phonons presented on Fig. 1 is a matter of convention. Indeed, this mode is soft in comparison with the proton plasma frequency and the longitudinal mode, but the averaged transverse phonons frequency is not too small (kv' l , 1000 K) to lead to a high values of Tc 600 K. We also calculated the electroresistivity and thermoconductivity of the metallic hydrogen using the function a2tr F v: Both these properties demonstrate usual metallic type behavior. The corresponding transport coupling constant l tr is even larger than l and equals to 10.3. Certainly, the obtained results concerning high-Tc superconductivity are beyond not only a practical applications but any experimental checking as well. Nevertheless, they can shed additional light on the two important problems widely disputed some time ago. First, this example demonstrates that the potential of the electron±phonon interaction in increasing Tc is very high and has not been fully used at full length up to now. Second, it con®rms the old idea that the high-Tc superconductivity can, under the conditions of a normal electron±phonon mechanism of superconductivity, be obtained in the systems close to structure instability. The values Tc obtained in our work are a few times lager than it was found in many old works [11±17], where the softness of the transverse phonons was absent.
1
Acknowledgements
0 0
1000
2000
3000
4000
5000
6000
7000
Frequency, K
Fig. 2. Calculated spectral and transport spectral function of electron±phonon interaction of metallic hydrogen in FCC lattice for rs 1:
The authors are indebted to V.L. Ginzburg, O.V. Dolgov and Yu.I. Shilov for helpful discussions. The work was partially supported by the grants from ISRC, RFBI and the Russian State Program for Superconductivity.
572
E.G. Maksimov, D.Yu. Savrasov / Solid State Communications 119 (2001) 569±572
References [1] E. Wigner, H.B. Huntington, J. Chem. Phys. 3 (1935) 764. [2] E.G. Maksimov, Yu.I. Shilov, UFN 169 (1999) 1223 (Physics Uspekhi 42 (1999) 1121). [3] N.W. Ashcroft, Phys. Rev. Lett. 21 (1968) 1748. [4] E.G. Brovman, Yu.M. Kagan, A. Kholas, JETP 61 (1971) 2429 (Sov. Phys. JETP, 34 (1971) 1258). [5] C.F. Richardson, N.W. Ashcroft, Phys. Rev. Lett. 78 (1997) 118. [6] S.D. Kaim, N.P. Kovalenko, E.V. Vasiliu, Cond-mat. 0010019 v2. [7] E.G. Brovman, Yu.M. Kagan, A. Kholas, JETP 62 (1972) 1492 (Sov. Phys. JETP 35 (1972) 783). [8] Yu.M. Kagan, V.V. Pushkarev, A. Kholas, JETP 73 (1977) 967 (Sov. Phys. JETP 46 (1977) 511). [9] W. Kohn, L.J. Sham, Phys. Rev. A140 (1965) 1133. [10] S.Yu. Savrasov, D.Yu. Savrasov, Phys. Rev. B46 (1992) 12181. [11] T. Schneider, E. Stoll, Physica 55 (1971) 702. [12] A.C. Switendick, in: D.V. Douglas (Ed.), Superconductivity in d- and f-Band Metals, Plenum Press, New York, 1976, p. 593.
[13] R.P. Gupta, S.K. Sinha, in: D.V. Douglas (Ed.), Superconductivity in d- and f-Band Metals, Plenum Press, New York, 1976, p. 583. [14] D.A. Papaconstantopoulos, B.M. Klein, Ferroelectrics 17 (1977) 307. [15] T.W. Barbee, A. Garcia, M.L. Cohen, Nature (1984) 340, see also page 369. [16] L.G. Caron, Phys. Rev. B9 (1974) 5025. [17] Yu.I. Shilov, N.R. Ivanov, Fiz. Tver. Tela 37 (1994) 1473 (Phys. Solid State, 37 (1994) 1473). [18] S.Yu. Savrasov, D.Yu. Savrasov, Phys. Rev. B54 (1996) 16470. [19] E.G. Maksimov, D.Yu. Savrasov, S.Yu. Savrasov, UFN 167 (1997) 337 (Phys. Usp., 40 (1997) 317). [20] E.G. Maksimov, D.Yu. Savrasov, O.V. Dolgov, S.Yu. Savrasov, Solid State Commun. 106 (1998) 409. [21] G.M. Eliashberg, ZETP 38 (1960) 966 (Sov. Phys. JETP 11 (1960) 696). [22] G.M. Eliashberg, ZETP 39 (1960) 1437 (Sov. Phys. JETP, 12 (1961) 1000). [23] D. Scalapino, in: R.D. Parks (Ed.), Superconductivity, vol. 1, Marcel Dekker, New York, 1969, p. 449.