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In recent years, there have been increasing experimental evidence suggestingthe existence of an electronic nematic state in a number of unconventional$d$-wave superconductors. Interestingly, it is expected that the long-rangenematic order can coexist with $d$-wave superconductivity. We analyze thein-plane upper critical field $H_{c2}$ after taking the influence of nematicstate into account, and find that the four-fold oscillation of angle-dependent$H_{c2}$ in a pure $d$-wave superconducting state is turned into a noveltwo-fold oscillation pattern by a weak nematic order. Moreover, such effect ismuch more significant at higher temperatures. These behaviors are measurableand may be used to probe the predicted coexisting nematic order in $d$-wavesuperconductors. In addition, we show that the concrete angular dependence of$H_{c2}$ and the positions of its maximum can be strongly affected by severalparameters, including temperature and $T_c$.

Novel behavior of upper critical field due to nematic order in $d$-wave superconductors