Coulomb Pseudopotential Research Articles

The superconducting phase of calcium under the pressure at 200 GPa: The strong-coupling description

The thermodynamic parameters of the superconducting state in calcium under the pressure at 200GPa have been determined. The numerical analysis by using the Eliashberg equations in the mixed representation has been conducted. It has been stated that the critical temperature (TC) decreases from 36.15K to 20.79K dependently on the assumed value of the Coulomb pseudopotential (μ⁎∈〈0.1,0.3〉). Next, the order parameter near the temperature of zero Kelvin (Δ(0)) has been obtained. It has been proven that the dimensionless ratio 2Δ(0)/kBTC decreases from 4.25 to 3.90 together with the growth of μ⁎. Finally, the ratio of the electron effective mass to the electron bare mass (me⁎/me) has been calculated. It has been shown that me⁎/me takes the high value in the whole range of the superconducting phase's existence, and its maximum is equal to 2.23 for T=TC.

Quasiparticle relaxation across the multiple superconducting gaps in the electron-doped BaFe1.85Co0.15As2

We study the quasiparticle relaxation dynamics in the electron-doped superconductor, BaFe1.85Co0.15As2 (BFCA) by using optical spectroscopy in both time and frequency domains. The estimate of the electron−phonon (el–ph) coupling constant from quasiparticle relaxation rate in time domain gives a value of ∼0.16 that is much too small to give a TC of 25 K. We obtain a negative value of the Coulomb pseudopotential that implies that the el−ph interaction alone is not strong enough to induce a superconducting transition in BFCA. More importantly, the electron-boson spectral function determined from optical scattering rate in frequency domain does not resemble the measured phonon density of states and has a dominant peak (at 53 meV) beyond the phonon frequencies. Our results support that the BFCA is a multi-gap s-wave superconductor and that its electron pairing is based on a spin-mediated pairing mechanism.

Thermodynamic investigations of high-pressure superconducting state in CaLi2 at 45 GPa

The thermodynamic properties of the high-pressure superconducting phase in CaLi2 have been studied. A pressure value of 45 GPa has been chosen, at which the critical temperature reaches the maximum (TC=12.9K). The following results have been obtained. (i) The critical temperature cannot be calculated using the classical analytical formulas (McMillan or Allen–Dynes). (ii) The critical value of the Coulomb pseudopotential is high: μC⋆=0.23. Generalizing the results, the function μC⋆(p) has a high value (μC⋆(p)>0.15) in practically the entire pressure range from 28 to 60 GPa. (iii) The remaining fundamental thermodynamic parameters differ from the Bardeen, Cooper, and Schrieffer (BCS) values. In particular, the zero-temperature energy gap to the critical temperature: R1≡2Δ(0)/kBTC=3.95; the ratio of the specific heat for superconducting (CS) and the normal (CN) state: R2≡(CS(TC)−CN(TC))/CN(TC)=2.01; and the parameter connected with the zero-temperature thermodynamic critical field: R3=TCCN(TC)/HC2(0)=0.158. Finally, it has been shown that the value of the electronic effective mass is high: me⋆/me>2.

A first-principles prediction of two-dimensional superconductivity in pristine B2C single layers

Based on first-principles lattice dynamics and electron-phonon coupling calculations, B2C sheet is predicted to be a two-dimensional (2D) phonon-mediated superconductor with a relatively high transition temperature (Tc). The electron-phonon coupling parameter calculated is 0.92, and it is mainly contributed by low frequency out-of-plane phonon modes and electronic states with a {\pi} character. When the Coulomb pseudopotential is set to 0.10, the estimated temperature Tc is 19.2 K. To be best of our knowledge, B2C is the first pristine 2D superconductor with a Tc higher than the boiling point of liquid helium.

Thermodynamics of the Superconducting State in Calcium at 200 GPa

The thermodynamic parameters of the superconducting state in Calcium under the pressure at 200 GPa were calculated. The Coulomb pseudopotential values (μ ⋆) from 0.1 to 0.3 were taken into consideration. It has been shown that the specific heat’s jump at the critical temperature and the thermodynamic critical field near zero Kelvin strongly decrease with μ ⋆. The dimensionless ratios r 1≡ΔC(T C)/C N(T C) and \(r_{2}\equiv T_{\mathrm{C}}C^{N}(T_{\mathrm{C}})/H^{2}_{\mathrm{C}}(0)\) significantly differ from the predictions based on the BCS model. In particular, r 1 decreases from 2.64 to 1.97 with the Coulomb pseudopotential; whereas r 2 increases from 0.140 to 0.157. The numerical results have been supplemented by the analytical approach.

Superconductivity of calcium in phase VI

The properties of the superconducting state of calcium in phase VI were analyzed. By using the imaginary axis Eliashberg equations it has been shown, that the Coulomb pseudopotential reaches the high value equal to 0.215. In the considered case, the critical temperature is not properly described by the Allen–Dynes formula and it should be calculated with an use of the modified expression. In the paper the exact solutions of the Eliashberg equations on the real axis were also obtained. On this basis it was stated, that the effective potential of the electron–electron interaction is attractive for the frequencies lower or equal to the maximum phonon frequency. Then, the dimensionless parameter 2 Δ(0)/ k B T C = 4.10 was calculated. In the last step it has been proven, that the ratio of the electron effective mass to the bare electron mass is high and reaches its maximum equal to 2.36 for the critical temperature.

Properties of the superconducting state in molecular metallic hydrogen under pressure at 347 GPa

The thermodynamic properties of the superconducting state induced in metallic molecular hydrogen under the influence of pressure 347 GPa were determined. In particular, it has been shown that the critical temperature ( T C ) changes in the range from 120 to 90 K for μ ⁎ ∈ 〈 0.08 , 0.15 〉 , where μ ⁎ is the value of the Coulomb pseudopotential. Next, the energy gap near the temperature of zero Kelvin ( 2 Δ ( 0 ) ) was calculated. It has been stated, that the dimensionless ratio 2 Δ ( 0 ) / k B T C slightly decreases with the increase of μ ⁎ from 3.98 to 3.84. In the last step, the ratio of effective electron mass ( m ⁎ e ) to the bare electron mass ( m e )) was determined. It has been proved that m ⁎ e / m e takes its highest value equal to 1.96 for T= T C .

Energy scale of the electron-boson spectral function and superconductivity inNpPd5Al2

The energy scale ${\ensuremath{\Omega}}_{0}$ of the electron-boson spectral function in the heavy-fermion, $d$-wave superconductor ${\text{NpPd}}_{5}{\text{Al}}_{2}$ is predicted on the basis of Eliashberg theory calculations. Assuming a spectral function shape typical for antiferromagnetic spin fluctuations, and imposing constraints provided by the experimental values for the critical temperature and the low-temperature energy gap, one obtains values of ${\ensuremath{\Omega}}_{0}$ of about 2--2.5 meV, slightly dependent from the strength of the Coulomb pseudopotential. These values are in excellent agreement with the characteristic magnetic fluctuations energy estimated from NMR measurements of the nuclear-spin-lattice relaxation time at the Al site. The calculated temperature dependence of the upper critical field, the local spin susceptibility, and the nuclear-spin-lattice relaxation rate is also in good agreement with available experimental data, showing that a coherent description of the superconducting state can be obtained assuming that the electron pairing in ${\text{NpPd}}_{5}{\text{Al}}_{2}$ is mediated by antiferromagnetic fluctuations. We finally report predictions for the London penetration depth, the energy dependence of the tunneling differential conductance at different temperatures, and the temperature dependence of the energy gap.

The superconducting state in metallic hydrogen under pressure at 2000 GPa

In this study, we have analysed the properties of the superconducting phase in metallic hydrogen under pressure at 2000 GPa. In particular, we have calculated the dependence of the basic thermodynamic quantities on the value of Coulomb pseudopotential ( μ ∗ ). The main results are the following: (i) The critical temperature ( T C ) is large even for high values of μ ∗ . (ii) The values of the ratios R 1 ≡ 2 Δ ( 0 ) / k B T C , R 2 ≡ Δ C ( k B T C ) / C N ( k B T C ) and R 3 ≡ T C C N ( k B T C ) / H C 2 ( 0 ) differ from the BCS values. (iii) The electron effective mass is large and reaches its maximum value for T = T C . Additionally, we have given the analytical formulas for T C , R 1 and R 2 which reproduce results originating from exact numerical calculations very well.

Interband interactions and three-square-well potentials on the superconductivity of MgB 2

Superconductivity in magnesium diboride, MgB 2, is described within the framework of Bogoliubov–Valatin two-band and two-gap formalism, employing a three-square-well potential model. The expressions for the superconducting transition temperature, T c , and the isotope exponent, β , are derived. Numerical calculations with the expressions newly reproduce near-experimental T c ( = 38.50 K ) and β ( = 0.38 ) . The influences of the interband attractive electron–acoustic ( λ π σ ) and electron–optical ( k π σ ) phonons as well as repulsive Coulomb pseudopotential ( μ π σ ∗ ) on T c and β are sketched. It is concluded that λ π σ and k π σ have near-identical elevating effects on T c and β but μ π σ ∗ has reverse effects, similar to the effect of interband impurity scattering.

Constraints onTcfor superconductivity in heavily boron-doped diamond

Calculations of electron-phonon coupling are performed for boron-doped diamond structures without electronically compensating defects over a wide range of boron concentration. The effects of boron substitutional disorder are incorporated through the use of randomly generated supercells, leading to a disorder-broadened distribution of results. After averaging over disorder, this study predicts a maximum bulk ${T}_{c}$ near $55\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ for boron concentrations between 20% and 30%, assuming the validity of the simple structural model used and a Coulomb pseudopotential of ${\ensuremath{\mu}}^{*}=0.12$. Considering only the largest electron-phonon coupling values of the distribution, superconductivity may still percolate through the material at higher temperatures, up to $80\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, through the regions of large coupling. A synthesis path is proposed to experimentally access this class of materials.

Possible mechanism of superconductivity inPuCoGa5probed by self-irradiation damage

Measurements of the electrical resistivity of a polycrystalline PuCoGa{sub 5} sample reveal significant modifications of the superconducting properties as a function of time, due to the increase of defects and impurities resulting from self-irradiation damage. More than four years of aging were necessary to detect a deviation from linearity in the time dependence of the critical temperature. The observed behavior is understood in the framework of the Eliashberg theory, confirming the 'dirty' d-wave character which was already suggested by nuclear magnetic resonance. We show that experimental data accumulated so far can be well reproduced by assuming a phononic mechanism for superconductivity, with reasonable values of the electron-phonon coupling and Coulomb pseudopotential. Further experiments are then required to assess the role of spin fluctuations in stabilizing the superconducting state in this compound.

Absence of superconductivity down to 80 mK in graphite intercalated BaC 6

Following recent theoretical and experimental investigations of superconductivity in the graphite intercalated compounds CaC 6 and SrC 6, we report on a very low temperature magnetisation study on BaC 6 down to 80 mK. The data show no trace of superconductivity even at fields as low as 0.7 Oe. Using a McMillan parametrisation of the BCS parameters, we conclude that the Coulomb pseudopotential is expected to be as large as 0.19 in both BaC 6 and SrC 6, i.e. 40% larger than in CaC 6. As an alternative scenario, we argue that extrinsic effects such as intercalant disorder may depress superconductivity in both BaC 6 and SrC 6, as in the case of CaC 6.

Asymptotic behavior of Eliashberg gap function in the complex plane and its implications for the Coulomb pseudopotential in superconductors

The energy gap function in the Eliashberg formalism draws a spiral in the complex plane. A simple, efficient numerical scheme is devised to extend that spiral towards the bottom of the Fermi sea for lead, mercury, tin, and a model superconductor. The spiral is confirmed to converge to a constant value with a finite negative real part. The asymptotic behavior is explored to reveal a V-shaped dependence of the Coulomb pseudopotential ${\ensuremath{\mu}}^{*}$ on the electron cutoff frequency in Pb and Hg, with a strong growth tendency when the cutoff frequency is large. A reference formula due to Allen and Dynes correctly predicts the later growth trend of ${\ensuremath{\mu}}^{*}$ at large cutoff frequencies but overestimates its rate. Previous values of ${\ensuremath{\mu}}^{*}$ in the literature have to be viewed critically.

The thermodynamic properties of the [formula omitted] superconductor: The two-band Eliashberg equations

In the present paper it was shown that the critical value of the Coulomb pseudopotential ( μ C ∗ ) for MgB 2 is 0.03578; the calculated value of μ C ∗ essentially differs from the value given by G.A. Ummarino et al. [G.A. Ummarino, R.S. Gonnelli, S. Massidda, A. Bianconi, Physica C 407 (2004) 121]. Assuming μ C ∗ = 0.03578 , for the selected values of temperature, the order parameters for σ and π band ( Δ σ and Δ π ) as well as the wave function renormalization factors ( Z σ and Z π ) were calculated. The ratios R a ≡ 2 Δ a ( 0 ) / k B T C were estimated, where a is the band index and T C is a value of the critical temperature. Values of R σ = 4.24 and R π = 1.54 were obtained. In the present work a total normalized density of states function was also calculated. In the last step the values of the wave function renormalization factors that define effective mass of electrons from the σ and π band were estimated for T C ; [ Z σ ] ω = 0 = 2.32 and [ Z π ] ω = 0 = 1.63 . Obtained results were compared with experimental data and with the other theoretical predictions.

Theoretical study of even denominator fractions in graphene: Fermi sea versus paired states of composite fermions

The physics of the state at even denominator fractional fillings of Landau levels depends on the Coulomb pseudopotentials, and produces, in different GaAs Landau levels, a composite fermion Fermi sea, a stripe phase, or, possibly, a paired composite fermion state. We consider here even denominator fractions in graphene, which has different pseudopotentials as well as a possible fourfold degeneracy of each Landau level. We test various composite fermion (CF) Fermi sea wave functions [fully polarized, SU(2) singlet, SU(4) singlet] as well as the paired composite fermion states in the $n=0$ and 1 Landau levels and predict that (i) paired states are not favorable, (ii) CF Fermi seas occur in both Landau levels, and (iii) an SU(4) singlet composite fermion Fermi sea is stabilized in the appropriate limit. The results from detailed microscopic calculations are generally consistent with the predictions of the mean field model of composite fermions.

Thermodynamic properties of the lead superconductor: The new value of the Coulomb pseudopotential

In the framework of the Eliashberg approach, the selected thermodynamic properties of the lead superconductor are analyzed. The new value of the Coulomb pseudopotential ( μ ∗ ) is assumed, μ ∗ = 0.1295 . In particular, the wave function renormalization factor ( Z ) and the order parameter ( Δ ) for different values of the temperature are calculated. It is shown that the electron–phonon interaction strongly enhances the effective electron mass; [ | Z | ] max = 2.84 . Next are calculated: the free energy difference between the superconducting and the normal state ( Δ F ), the thermodynamical critical field ( H C ), the specific heat in the normal state ( C N ) and the specific heat in the superconducting state ( C S ). Finally, the ratios R 2 ≡ ( C S − C N ) / C N and R 3 ≡ ( T C C N ) / H C 2 are obtained. The numerical results have shown that R 2 = 2.6344 and R 3 = 0.1293 . The presented theoretical values of the ratios agree well with the experimental data.

D-wave superconductivity from electron-phonon interactions

I examine electron-phonon mediated superconductivity in the intermediate coupling and phonon frequency regime of the quasi-two-dimensional Holstein model. I use an extended Migdal-Eliashberg theory that includes vertex corrections and spatial fluctuations. I find a d-wave superconducting state that is unique close to half filling. The order parameter undergoes a transition to s-wave superconductivity on increasing filling. I explain how the inclusion of both vertex corrections and spatial fluctuations is essential for the prediction of a d-wave order parameter. I then discuss the effects of a large Coulomb pseudopotential on the superconductivity (such as is found in contemporary superconducting materials like the cuprates), which results in the destruction of the s-wave states, while leaving the d-wave states unmodified.

A pseudopotential approach to the superconducting state parameters of (Ni33Zr67)1−x M x (M=Ti, V, Co, Cu) ternary metallic glasses

Superconductivity in ternary metallic glasses has been investigated using the model pseudopotential approach, which has been found quite successful in explaining superconductivity in metals, binary alloys and binary glasses. It is observed that this simple methodology successfully explains superconducting behaviour of ternary glasses without requiring the solution of Dirac equation for a many body problem or estimation of various interactions as required in ab-initio pseudopotential theory. In the present work superconducting state parameters of fourteen metallic glasses of (Ni-Zr)-M system (M=Ti, V, Co, Cu) have been determined in the BCS-Eliashberg-McMillan framework. It is observed that addition of V, Co, and Cu as the third element (M) to a binary metallic glass (Ni33 Zr67) causes the parameters λ,T c, α, andN 0 V to decrease, and Coulomb pseudopotential (μ*) to increase with concentration of M, showing that the presence of third element (M) causes suppression of superconducting behaviour of the alloy. The decrease inT c with increasing concentration of third element (M) may be attributed to the modifications in density of states at the Fermi levelN(E F), and probable changes in the band structure of the alloy due to addition of the third element (M). Slight difference is noticed when Ti is added to the Ni33 Zr67 alloy. In this caseT c rises initially and then decreases with concentration of M, showing a peak at aboutx=0.05. This indicates that on addition of Ti, 3d states grow near the Fermi level and hence contribute substantially toN(E F), favouring superconducting behaviour in this case. The present results forT c show an excellent agreement with the experimental data. QuadraticT c equations have been proposed, which provide successfully theT c values of ternary metallic glasses under consideration.

Effect of local field correction in screening dependence of superconducting state parameters for Be90Al10 and Be70Al30 metallic glasses

AbstractThe superconducting state parameters (SSP) viz. electron–phonon coupling strength (λ), Coulomb pseudopotential (μ*), transition temperature (Tc), isotope effect exponent (α) and interaction strength (N0V) for two metallic glasses, i.e. Be90Al10 and Be70Al30, have been investigated by extending the BCS–Eliasberg–McMillan formalism for the metallic glass superconductors incorporating the effect of local field correction in dielectric screening functions which in turn become prominent in determining superconducting state parameters of the metallic glasses under consideration. We have employed eleven forms of dielectric screening functions – viz. Hartree, RPA, Modified Hubbard, Overhauser, Geldart–Vosko, Animalu, King–Cutler, SCS, Vashistha–Singwi, Ichimaru–Utsumi, and Farid et al. – in conjunction with Ashcroft's potential to predict superconducting state parameters. It is observed that the values of SSP are quite sensitive to the form of dielectric screening functions. In conclusion the present computed values of SSP for the RPA form of dielectric screening are in good agreement with the experimental values and other theoretical results for both glassy materials, which confirms the validity of the proposed formalism. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)