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The fragility of the chiral surface current of ($d_{zx}+id_{yz}$)-wavesuperconductor, a potential candidate for Sr$_2$RuO$_4$, against surfaceroughness is demonstrated utilizing the quasiclassical Eilenberger theory.Comparing the chiral surface currents of ($d_{zx}+id_{yz}$)-wave and($p_{x}+ip_{y}$)-wave pairings, we conclude the chiral current for($d_{zx}+id_{yz}$)-wave SC is much more fragile than that for the($p_x+ip_y$)-wave one. The difference can be understood in terms of the orbitalsymmetry of the odd-frequency Cooper pairs arising at the surface. Our resultsshow the ($d_{zx}+id_{yz}$)-wave scenario can explain the null spontaneousmagnetization in Sr$_2$RuO$_4$ experiments.

Destruction of surface states of ($d_{zx}+id_{yz}$)-wave superconductor by surface roughness: application to Sr$_2$RuO$_4$