Connection between in-plane upper critical field $H_{c2}$ and gap symmetry in layered $d$-wave superconductors revisited

Abstract

Angle-resolved upper critical field $H_{c2}$ provides an efficient tool to probe the gap symmetry of unconventional superconductors. We revisit the behavior of in-plane $H_{c2}$ in $d$-wave superconductors by considering both the orbital effect and Pauli paramagnetic effect. After carrying out systematic analysis, we show that the maxima of $H_{c2}$ could be along either nodal or antinodal directions of a $d$-wave superconducting gap, depending on the specific values of a number of tuning parameters. This behavior is in contrast to the common belief that the maxima of in-plane $H_{c2}$ are along the direction where the superconducting gap takes its maximal value. Therefore, identifying the precise $d$-wave gap symmetry through fitting experiments results of angle-resolved $H_{c2}$ with model calculations at a fixed temperature, as widely used in previous studies, is difficult and practically unreliable. However, our extensive analysis of angle-resolved $H_{c2}$ show that there is a critical temperature $T^{\ast }$: in-plane $H_{c2}$ exhibits its maxima along nodal directions at $T<T^{\ast }$ and along antinodal directions at $T^{\ast }<T<T_c$. The concrete value of $T^{\ast }$ may change as other parameters vary, but the existence of $\pi /4$ shift of $H_{c2}$ at $T^{\ast }$ appears to be a general feature. Thus a better method to identify the precise $d$-wave gap symmetry is to measure $H_{c2}$ at a number of different temperatures, and examine whether there is a $\pi /4$ shift in its angular dependence at certain $T^{\ast }$. We further show that Landau level mixing does not change this general feature. However, in the presence of Fulde-Ferrell-Larkin-Ovchinnikov state, the angular dependence of $H_{c2}$ becomes quite complicated, which makes it more difficult to determine the gap symmetry by measuring $H_{c2}$.

12 pages, 11 figures

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