Topological blocking at the Bi(111) surface due to surface relaxation

The topological characteristics of Bi and its alloys with Sb have fueled intense debate since the prediction of three-dimensional topological insulators. However, a definitive resolution has not been reached to date. Here, we provide theoretical evidence that surface relaxation conceals the underlying bulk topology of pure Bi. Using density-functional-theory calculations for thin Bi(111) films (up to 17 bilayers), we first demonstrate a substantial interbilayer expansion near the surface. Motivated by this finding, we extend our analysis to thick Bi(111) films (up to 250 bilayers) incorporating relaxation layers, within the framework of a empirical tight-binding model with spin-orbit coupling. Our results reveal that these relaxation layers topologically block the emergence of a surface state and significantly suppress the one-particle spectrum of surface states at the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mover accent="true"><a:mi>M</a:mi><a:mo>¯</a:mo></a:mover></a:math> point, thereby obscuring the experimental identification of Bi's topological properties. This phenomenon, which we term “topological blocking,” provides crucial insights into the long-standing difficulty of observing surface states of Bi(111) at the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mover accent="true"><c:mi>M</c:mi><c:mo>¯</c:mo></c:mover></c:math> point. Furthermore, it establishes a framework for understanding and predicting the topological behavior in systems where surface relaxation disrupts the bulk-edge correspondence.