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

In recent years, there have been increasing experimental evidence suggesting the existence of an electronic nematic state in a number of unconventional $d$-wave superconductors. Interestingly, it is expected that the long-range nematic order can coexist with $d$-wave superconductivity. We analyze the in-plane upper critical field $H_{c2}$ after taking the influence of nematic state 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 novel two-fold oscillation pattern by a weak nematic order. Moreover, such effect is much more significant at higher temperatures. These behaviors are measurable and may be used to probe the predicted coexisting nematic order in $d$-wave superconductors. In addition, we show that the concrete angular dependence of $H_{c2}$ and the positions of its maximum can be strongly affected by several parameters, including temperature and $T_c$.