[Paper Review] Enhancement of the upper critical field by nonmagnetic impurities in dirty two-gap superconductors

A. Gurevich|arXiv (Cornell University)|Dec 5, 2002

Superconductivity in MgB2 and Alloys19 citations

TL;DR

This paper derives quasiclassical Uzadel equations for dirty two-gap superconductors with nonmagnetic impurities, showing that interband and intraband scattering can significantly enhance the upper critical field $H_{c2}(0)$ beyond the one-gap dirty limit prediction. The key result is that $H_{c2}(0)$ can exceed $0.7T_c \, dH_{c2}/dT_c$, enabling $MgB_2$ to surpass $Nb_3Sn$'s $H_{c2}(0)$ even at moderate resistivity, depending on the ratio of diffusivities $D_1/D_2$.

ABSTRACT

Quasiclassic Uzadel equations for two-band superconductors in the dirty limit with the account of both intraband and interband scattering by nonmagnetic impurities are derived for any anisotropic Fermi surface. From these equations the Ginzburg-Landau equations, and the critical temperature $T_c$ are obtained. An equation for the upper critical field, which determines both the temperature dependence of $H_{c2}(T)$ and the orientational dependence of $H_{c2}(θ)$ as a function of the angle $θ$ between ${\bf H}$ and the c-axis is obtained. It is shown that the shape of the $H_{c2}(T)$ curve essentially depends on the ratio of the intraband electron diffusivities $D_1$ and $D_1$, and can be very different from the standard one-gap dirty limit theory. In particular, the value $H_{c2}(0)$ can considerably exceed $0.7T_cdH_{c2}/dT_c$, which can have important consequences for applications of $MgB_2$. A scaling relation is proposed which enables one to obtain the angular dependence of $H_{c2}(θ)$ from the equation for $H_{c2}$ at ${\bf H}\| c$. It is shown that, depending on the relation between $D_1$ and $D_2$, the ratio of the upper critical field $H_{c2}^\|/H_{c2}^\perp$ for ${\bf H}\| ab$ and ${\bf H}\perp ab$ can both increase and decrease as the temperature decreases. Implications of the obtained results for $MgB_2$ are discussed.

Motivation & Objective

• Address the limitation of $MgB_2$'s low upper critical field $H_{c2}$, which restricts high-field applications despite its high $T_c$.
• Explain why conventional one-gap dirty limit theory fails to predict $H_{c2}$ in two-gap superconductors like $MgB_2$ due to multiple scattering channels.
• Develop a quasiclassical theory for two-band superconductors in the dirty limit that includes both intraband and interband scattering by nonmagnetic impurities.
• Derive a generalized equation for $H_{c2}(T)$ and its angular dependence $H_{c2}(\theta)$, accounting for Fermi surface anisotropy.
• Propose a scaling relation to extract angular dependence from $H_{c2}$ along the c-axis, enabling experimental validation.

Proposed method

• Derive quasiclassical Uzadel equations for two-band superconductors with nonmagnetic impurities, including both intraband and interband scattering processes.
• Use a weak-coupling multiband BCS model to describe the two-gap system, with distinct order parameters on separate Fermi surface sheets ($\sigma$ and $\pi$ bands).
• Formulate the gap equation in terms of a matrix eigenvalue problem involving coupling constants $\lambda_{mm'}$ and scattering rates $\gamma_{mm'}$, leading to a matrix $M$ whose largest eigenvalue determines $H_{c2}$.
• Apply perturbation theory in the limit of weak interband coupling ($\lambda_{12}\lambda_{21} \ll \lambda_{11}\lambda_{22}$) to derive a simplified equation for $H_{c2}$ that remains accurate even at strong anisotropy.
• Use the sum rule $\sum_s V_{ns}V_{n's} = \delta_{nn'}$ to validate the orthogonality of wavefunctions and ensure consistency in the matrix formulation.
• Derive a scaling relation that allows the angular dependence $H_{c2}(\theta)$ to be inferred from $H_{c2}$ along the c-axis, enabling experimental extraction of anisotropy.

Experimental results

Research questions

• RQ1 How does the presence of nonmagnetic impurities affect the upper critical field $H_{c2}$ in dirty two-gap superconductors beyond the predictions of one-gap dirty limit theory?
• RQ2 What is the role of intraband versus interband scattering in determining the temperature and angular dependence of $H_{c2}$ in two-band superconductors?
• RQ3 How does the ratio of electron diffusivities $D_1/D_2$ between the two bands influence the shape of the $H_{c2}(T)$ curve and the value of $H_{c2}(0)$?
• RQ4 Can the angular dependence $H_{c2}(\theta)$ be predicted from measurements along the c-axis using a scaling relation?
• RQ5 What is the maximum achievable $H_{c2}(0)$ in $MgB_2$-based materials when both $\rho_n$ and the relative scattering rates are optimized?

Key findings

• The upper critical field $H_{c2}(0)$ in two-gap superconductors can significantly exceed the one-gap dirty limit prediction $H_{c2}(0) = 0.69T_c \, dH_{c2}/dT_c$, especially when the ratio $D_1/D_2$ of intraband diffusivities is tuned.
• Anisotropy in the Fermi surface and the relative strength of intraband and interband scattering rates lead to a non-monotonic temperature dependence of $H_{c2}(T)$, deviating from the standard one-gap behavior.
• The ratio $H_{c2}^{\|}/H_{c2}^{\perp}$ (field parallel vs. perpendicular to ab-plane) can either increase or decrease with decreasing temperature, depending on the $D_1/D_2$ ratio.
• A scaling relation is derived that allows the full angular dependence $H_{c2}(\theta)$ to be obtained from $H_{c2}$ measured along the c-axis, enabling experimental validation of the model.
• Numerical results show that $H_{c2}(0)$ in $MgB_2$ can exceed $30$ T—surpassing $Nb_3Sn$'s $H_{c2}(0) \approx 30$ T—when $H_{c2}^{\perp\prime} \approx 1$ T/K is achieved, provided scattering rates are optimized via selective doping.
• Even with moderate resistivity ($\rho_n \sim 700\mu\Omega$ cm), $H_{c2}(0)$ can be enhanced beyond the one-gap extrapolation, indicating that two-gap effects dominate over single-band scaling.