  In conventional superconductors, magnetic impurities form an impurity band
due to quantum interference of the impurity bound states, leading to
suppression of the superconducting transition temperature. Such quantum
interference effects can also be expected in d-wave superconductors. Here, we
use scanning tunneling microscopy to investigate the effect of multiple
non-magnetic impurities on the local electronic structure of the
high-temperature superconductor
Bi$_{2}$Sr$_{2}$Ca(Cu$_{1-x}$Zn$_{x}$)$_{2}$O$_{8+\delta}$. We find several
fingerprints of quantum interference of the impurity bound states including:
(i) a two-dimensional modulation of local density-of-states with a period of
approximately 5.4 \AA\ along the $a$- and $b$-axes, which is indicative of the
d-wave superconducting nature of the cuprates; (ii) abrupt spatial variations
of the impurity bound state energy; (iii)an appearance of positive energy
states; (iv) a split of the impurity bound state. All of these findings provide
important insight into how the impurity band in d-wave superconductors is
formed.
