Thermodynamic Critical Field Research Articles

Changing the type of superconductivity by magnetic and potential scattering

By evaluating the upper and thermodynamic critical fields, $H_{c2}$ and $H_c$, and their ratio $H_{c2}/H_c $ at arbitrary temperatures, we argue that situations are possible when a type-II material is transformed to type-I by adding magnetic impurities.

Properties of the pressure-induced superconducting state in trihydrides ScH3 and LaH3

The present paper discusses the thermodynamic properties of the superconducting state which gets induced in the following compounds due to the influence of the high pressure: ScH3 ( p = 18 GPa) and LaH3 (p = 11 GPa). Using the Eliashberg formalism, for the Coulomb pseudopotential equal to 0.1, the following quantities have been determined: the maximum critical temperature ( K and K), the energy gap at the Fermi surface (G), the free energy (F), the thermodynamic critical field (HC), and the specific heat (both for the superconducting (CS) and the normal (CN) state). Then, the dimensionless parameters, namely, , , and , have been obtained. Those ratios deviate from the predictions of the BCS theory in the case of the studied compounds: , , , , , and . In the last step, it has been proven that the ratio of the electron effective mass () to the electron band mass (me) takes relatively high values, which are weakly dependent on the temperature. For example, it has been shown that and for .

The Eliashberg study of the electron–phonon superconductivity in [formula omitted] compound

The superconductivity in YSn3 is induced by the electron–phonon interaction. The most relevant thermodynamic parameters of the superconducting state for the considered compound have been determined in the framework of the Eliashberg formalism. On the basis of the numerical calculations, it has been shown that the critical temperature (TC) is equal to 6.52K. The obtained result accurately reproduces the experimental value of TC, which is equal to ∼7K. Next, the low-temperature value of the order parameter (Δ(0)) has been determined, as well as the thermodynamic critical field (HC(0)), and the specific heat jump at the critical temperature (ΔC(TC)). It has been found that the dimensionless ratios: RΔ≡2Δ(0)/kBTC,RH≡TCCN(TC)/HC2(0), and RC≡ΔC(TC)/CN(TC), where CN denotes the specific heat for the normal state, take the values fairly exceeding the predictions of the BCS theory. In particular, it has been obtained: 3.97, 0.149 and 2.19, respectively. Let us note that the theoretical result for RC is in agreement with the existing experimental data.

Study of the superconducting phase in silicene under biaxial tensile strain

The electron-doped silicene under the influence of the biaxial tensile strain is predicted to be the phonon-mediated superconductor. By using the Eliashberg formalism, we investigate the thermodynamic properties of the superconducting silicene in the case when the tension is 5% and the electron doping equals 3.5×1014cm−2. Under such conditions, silicene monolayer is expected to exhibit the highest superconducting transition temperature (TC). In particular, based on the electron–phonon spectral function and assuming a wide range of the Coulomb pseudopotential values (μ⋆∈〈0.1,0.3〉) it is stated that the superconducting transition temperature decreases from 18.7K to 11.6K. Similar behavior is observed in the case of the zeroth temperature superconducting energy gap at the Fermi level: 2Δ(0)∈〈6.68,3.88〉 meV. Other thermodynamic parameters differ from the predictions of the Bardeen–Cooper–Schrieffer theory. In particular, the ratio of the energy gap to the critical temperature changes in the range from 4.14 to 3.87. The ratio of the specific heat jump to the specific heat in the normal state takes the values from 2.19 to 2.05, and the ratio of the critical temperature and specific heat in the normal state to the thermodynamic critical field increases from 0.143 to 0.155. It is also determined that the maximum value of the electron effective mass equals 2.11 of the electron band mass.

Phonon-mediated superconductivity in compressed NbH4 compound

There has been much interest in the superconducting properties of metal hydrides for the last few years. The hydrogen-rich compounds are regarded as a way of hydrogen metallization at lower pressures than that required for pure hydrogen. This paper deals with the thermodynamic properties of the superconducting state in niobium hydride (NbH4) at 300 GPa. All theoretical calculations have been conducted within the framework of the strong-coupling Eliashberg theory. It has been stated that the critical temperature (T C ) is equal to 49.57 K when the Coulomb pseudopotential takes the commonly accepted value of 0.1. In the considered case, the ratio of the energy gap to the critical temperature (RΔ ≡ 2Δ(0)/k B T C), the ratio of the specific heat jump to the heat of the normal state (RC ≡ ΔC(T C)/C N(T C)) and the ratio connected with the thermodynamic critical field (R H ≡ T C C N(T C )/H 2 C(0)) are equal to 3.91, 2.12 and 0.154, respectively. Above results differ significantly from the values predicted by the weak-coupling Bardeen-Cooper-Schrieffer theory, which omits a very important role of the strong-coupling and phonon retardation effects.

Superconducting properties of BaBi3

We report the superconducting properties of single crystals of the intermetallic compound BaBi3, whose crystal structure is perovskite related. The superconducting transition temperature ( K) was obtained from heat capacity measurements. Using the measured values for the critical fields , and the specific heat C, we estimate the thermodynamic critical field Hc(0), coherence length ξ(0), Debye temperature and coupling constant λep. and λep suggest that BaBi3 is a weakly coupled superconductor. Electronic band structure calculations show a complex Fermi surface and a moderately high DOS at the Fermi level. Further analysis of the electronic specific heat shows that the superconducting properties are dominated by s-wave gap.

Anisotropic criteria for the type of superconductivity

The classical criterion for classification of superconductors as type-I or type-II based on the isotropic Ginzburg-Landau theory is generalized to arbitrary temperatures for materials with anisotropic Fermi surfaces and order parameters. We argue that the relevant quantity for this classification is the ratio of the upper and thermodynamic critical fields, $H_{c2}/H_c$, rather than the traditional ratio of the penetration depth and the coherence length, $\lambda/\xi$. Even in the isotropic case, $H_{c2}/H_c$ coincides with $\sqrt{2}\lambda/\xi$ only at the critical temperature $T_c$ and they differ as $T$ decreases, the long known fact. Anisotropies of Fermi surfaces and order parameters may amplify this difference and render false the criterion based on the value of $\kappa=\lambda/\xi$.

Thermodynamics of the CaLi2 Superconductor in the Vicinity of Structural Phase Transition

Eliashberg formalism is used to investigate the thermodynamic properties of the high-pressure superconducting phase of the CaLi2 compound. In particular, our calculations are conducted in the vicinity of the C2/c → P21/c pressure-induced structural phase transition. We show that, in the considered case, the value of the Coulomb pseudopotential is high and equals 0.26. Moreover, we give the analysis of the thermodynamic parameters such as the superconducting transition temperature (TC), the energy gap at the Fermi level (2Δ(0)), the thermodynamic critical field (HC), and the specific heat of superconducting (CS) and normal (CN) states. We emphasize that the characteristic dimensionless ratios RΔ ≡ 2Δ(0)/kBTC, RH ≡ TCCN(TC)/HC2(0), and RC ≡ ΔC(TC)/CN (TC), have values that are beyond the predictions of the BCS theory in the case of the considered material. In particular, RΔ = 3.85, RH = 0.161, and RC = 1.86. Furthermore, it is proved that the effective electron mass is high and equals 2.02me, where me denotes the bare electron mass.

High temperature superconducting properties of atomic hydrogen at 802 GPa

The Eliashberg equations have been used to calculate the thermodynamic parameters of the superconducting state in the metallic atomic hydrogen at 802GPa. It has been shown that the superconducting transition temperature (TC) is in the range from 332.7K to 259.4K, depending on the assumed Coulomb pseudopotential: μ⋆∈〈0.1,0.2〉. Due to the strong-coupling and retardation effects, the values of the order parameter (Δ), the thermodynamic critical field (HC), and the specific heat for the normal (CN) and superconducting (CS) state differ significantly from the predictions of the BCS theory. As a result, the dimensionless ratios RΔ≡2Δ(0)/kBTC, RH≡TCCN(TC)/HC2(0), and RC≡(CS(TC)−CN(TC))/CN(TC) take the values: RΔ(μ⋆)∈〈4.70,4.51〉, RH(μ⋆)∈〈0.130,0.132〉, and RC(μ⋆)∈〈2.57,2.50〉.

Thermodynamic Critical Magnetic Field for Chlorine Halide Superconductor at High Pressure

Thermodynamic Critical Magnetic Field for Chlorine Halide Superconductor at High Pressure

Influence of lithium doping on the thermodynamic properties of graphene based superconductors

The superconducting phase in graphene can be induced by doping its surface with lithium atoms. In this paper, it is shown that the critical temperature (TC) for the LiC6 and Li2C6 compounds changes from 8.55 K to 21.83 K. The other thermodynamic parameters—the order parameter (Δ), the specific heat for the superconducting (CS) and the normal (CN) state and the thermodynamic critical field (HC)—differ from the predictions of the Bardeen–Cooper–Schrieffer theory. In particular, the ratio RΔ ≡ 2Δ(0)/kBTC is equal to and . Additionally, the quantities RC ≡ ΔC(TC)/CN(TC) and take the values , and , . Finally, it is shown that the electron effective mass at TC is high and equals: and .

The phonon-mediated superconductivity in [formula omitted] sheet

The values of the thermodynamic parameters of the superconducting state induced by the electron–phonon interaction in the B2C sheet have been determined. It has been found that the critical temperature (TC) can change in the range from 20.4K to 13K for the Coulomb pseudopotential (μ⋆) ranging from 0.1 to 0.2. Generalization of the results for the atomic layer demonstrated that the maximum value of TC is equal to 25K. Next for the B2C sheet, the order parameter (Δ), the thermodynamic critical field (HC), and the specific heat for the superconducting (CS) state and the normal (CN) state have been calculated. The obtained results show that the dimensionless thermodynamic ratios, RΔ≡2Δ(0)/kBTC, RH≡TCCN(TC)/HC2(0), and RC≡(CS(TC)−CN(TC))/CN(TC), deviate from the predictions of the classical BCS theory. In particular RΔ(μ⋆)∈〈3.87,3.79〉, RH(μ⋆)∈〈0.155,0.157〉, and RC(μ⋆)∈〈1.67,1.62〉. The numerical calculations have been conducted in the framework of the Eliashberg formalism.

Estimation of the superconducting parameters for silane at high pressure

The properties of the superconducting state in the Cmca phase of silane ( SiH4 ) at the pressure of 250 GPa have been considered. In particular, the critical temperature (TC), the free energy difference between the superconducting and normal state (ΔF ≡ FS - FN), the thermodynamic critical field (HC) and the specific heat jump (ΔC ≡ CS - CN) have been determined. It has been shown that the dimensionless ratios: [Formula: see text] and RC ≡ ΔC(TC)/CN(TC) assume the values: RH(μ⋆) ∈ 〈0.147, 0.154〉 and RC(μ⋆) ∈ 〈2.20, 1.79〉, where the symbol μ⋆ denotes the Coulomb pseudopotential and μ⋆∈ 〈0.1, 0.3〉.

Investigations of the superconducting states of noncentrosymmetric LaPdSi3and LaPtSi3

The noncentrosymmetric superconductors LaPdSi3 and LaPtSi3 have been studied with magnetization, specific heat, resistivity and muon spin rotation/relaxation (muSR) measurements. These crystallize in the tetragonal BaNiSn3 structure and superconductivity is observed at Tc = 2.65 K for LaPdSi3 and Tc = 1.52(6) K for LaPtSi3. The results are consistent with both compounds being weakly coupled, fully gapped superconductors but muSR measurements reveal that LaPdSi3 is a bulk type-I superconductor while LaPtSi3 is a type-II material with a Ginzburg-Landau parameter of kappa = 2.49(4). This is further supported by specific heat measurements, where the transition in an applied field is first-order in LaPdSi3 but second-order in LaPtSi3. The electronic specific heat in the superconducting state was analyzed using an isotropic s-wave model that gave Delta/kTc = 1.757(4) for LaPdSi3 and 1.735(5) for LaPtSi3. The temperature dependence of the effective penetration depth of LaPtSi3 was extracted from muSR measurements and was fitted giving Delta/kTc = 1.60(8) and lambda(0) = 239(3) nm. A critical field of Bc(0) = 182.7 G was obtained for LaPdSi3 from muSR measurements, which is in good agreement with the calculated thermodynamic critical field.

Magnetic flux density and the critical field in the intermediate state of type-I superconductors

To address unsolved fundamental problems of the intermediate state (IS), the equilibrium magnetic flux structure and the critical field in a high-purity type-I superconductor (indium film) are investigated using magneto-optical imaging with a three-dimensional vector magnet and electrical transport measurements. The least expected observation is that the critical field in the IS can be as small as nearly 40% of the thermodynamic critical field ${H}_{c}$. This indicates that the flux density in the bulk of normal domains can be considerably less than ${H}_{c}$, in apparent contradiction with the long-established paradigm, stating that the normal phase is unstable in fields below ${H}_{c}$. Here we present a theoretical model consistently describing this and all other properties of the IS. Moreover, our model, based on a rigorous thermodynamic treatment of the observed equilibrium flux structure in a tilted field, allows for a quantitative determination of the domain-wall parameter and the coherence length, and provides new insight into the properties of superconductors.

Superconductivity in oxygen deficient LaNiAsO1−δ (δ = 0–0.3)

LaNiAsO itself is a superconductor with a Tc of ∼2.7K. Oxygen deficiency could be introduced up to a level of δ=0.3 with retaining the host structure and increase the Tc to ∼4.1K at δ=0.3. The Tc of LaNiAsO1−δ monotonically increases with increasing δ and changes negligibly when δ>0.3. Using the measured values of the lower critical field Hc1, the upper critical field Hc2, and the specific heat Cp, we estimated the thermodynamic critical field Hc(0), coherence length ξ(0), penetration depth λ(0), the Ginzburg–Landau parameter κ(0), and the electron–phonon coupling constant λep for LaNiAsO0.7 with an optimal Tc. The specific heat data follow a BCS-like exponential decrease below Tc, indicating a fully gapped superconductivity. In addition, the replacement of La by Sm failed to enhance the Tc. The many differences distinguish LaNiAsO from the 1111-type Fe-based superconductors, thus highlighting the importance of Fermi surface topology for superconductivity.

On the thermodynamic properties of the Rb3C60 superconductor

The thermodynamic properties of the superconducting state in Rb3C60 fulleride have been studied in the framework of the Migdal–Eliashberg approach. In the first step, the high physical value of the Coulomb pseudopotential has been predicted (μC★=0.33), which corresponds to the screened Coulomb potential UC=0.61eV. Next, the temperature dependence of the order parameter and the wave function renormalization factor has been calculated. On the basis of the obtained results, the ratio of the energy gap to the critical temperature (RΔ≡2Δ(0)/kBTC), the ratio of the specific heat jump to the normal state specific heat (RC≡ΔCTC/CNTC), and the parameter connected with the thermodynamic critical field (RH≡TCCNTC/HC2(0)) have been estimated. It has been shown that the above parameters significantly differ from the BCS predictions: RΔ=4.06,RC=2.15, and RH=0.145. Finally, the temperature dependence of the electron effective mass (me★) has been presented; me★ assumes maximum at the critical temperature: 2.27me, where me denotes the electron band mass. In the paper, the obtained theoretical results have been also compared with the experimental data.

Superconducting state above the boiling point of liquid nitrogen in the GaH3 compound

The thermodynamic parameters of the superconducting state in the GaH3 compound under a pressure of 120 GPa have been determined. A wide range of Coulomb pseudopotential values has been taken into account: μ⋆∈〈0.1,0.2〉. It has been shown that, for the considered values of μ⋆, the superconducting state is characterized by a critical temperature which is higher than the boiling temperature of liquid nitrogen: TC∈(122.99,89.78) K. Other parameters significantly deviate from the predictions of BCS theory. In particular, the ratio of the energy gap to the critical temperature (RΔ ≡ 2Δ(0)/kBTC) changes in the range from 4.15 to 4.03. The ratio of the specific heat jump to the specific heat in the normal state (RC ≡ ΔC(TC)/CN(TC)) takes values from 1.87 to 1.58. The parameter , where HC is the thermodynamic critical field, is characterized in the following manner: RH∈〈0.147,0.150〉. Finally, it has been proven that the maximum value of the electron effective mass () equals 2.66me, where the symbol me denotes the electron band mass.

Dirichlet boundary condition for the Ginzburg-Landau equations

As is well known, the Ginzburg Landau phenomenological theory described with a good accuracy the thermodynamic properties of a superconducting material. The system of two coupled nonlinear differential equations is completed with the usual Neumann boundary condition as long as is considered a superconductor insulator interface. In this paper, we solve the Ginzburg Landau equations for a circular geometry containing a half-circular pillar defect and considering the unusual superconducting Dirichlet boundary condition. This choice, leading to take the extrapolation de Gennes length equal to zero. Our results point that, the thermodynamic critical fields, magnetization, free energy and vorticity, depend on the chosen boundary condition.

Elaboration of the α-model derived from the BCS theory of superconductivity

The single-band α-model of superconductivity (Padamsee et al 1973 J. Low Temp. Phys. 12 387) is a popular model that was adapted from the single-band Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity mainly to allow fits to electronic heat capacity versus temperature T data that deviate from the BCS prediction. The model assumes that the normalized superconducting order parameter Δ(T)/Δ(0) and therefore the normalized London penetration depth λL(T)/λL(0) are the same as in BCS theory, calculated using the BCS value αBCS ≈ 1.764 of α ≡ Δ(0)/kBTc, where kB is Boltzmann’s constant and Tc is the superconducting transition temperature. On the other hand, to calculate the electronic free energy, entropy, heat capacity and thermodynamic critical field versus T, the α-model takes α to be an adjustable parameter. Here we write the BCS equations and limiting behaviors for the superconducting state thermodynamic properties explicitly in terms of α, as needed for calculations within the α-model, and present plots of the results versus T and α that are compared with the respective BCS predictions. Mechanisms such as gap anisotropy and strong coupling that can cause deviations of the thermodynamics from the BCS predictions, especially the heat capacity jump at Tc, are considered. Extensions of the α-model that have appeared in the literature, such as the two-band model, are also discussed. Tables of values of Δ(T)/Δ(0), the normalized London parameter Λ(T)/Λ(0) and λL(T)/λL(0) calculated from the BCS theory using α = αBCS are provided, which are the same in the α-model by assumption. Tables of values of the entropy, heat capacity and thermodynamic critical field versus T for seven values of α, including αBCS, are also presented.