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The contributions of surface science methods to discover and improve 3D topological insulator materials are reviewed herein, illustrated with examples from the authors' own work. In particular, it is demonstrated that spin‐polarized angular‐resolved photoelectron spectroscopy is instrumental to evidence the spin‐helical surface Dirac cone, to tune its Dirac point energy toward the Fermi level, and to discover novel types of topological insulators such as dual ones or switchable ones in phase change materials. Moreover, procedures are introduced to spatially map potential fluctuations by scanning tunneling spectroscopy and to identify topological edge states in weak topological insulators. [ABSTRACT FROM AUTHOR]

GraphENe quantum dots (GQDs) are considered as promising alternatives to quantum dots in III–V semiconductors, e.g., for the use as spin qubits. However, this potENtial has not beEN realized in experimENts so far, calling for a more fundamENtal investigation of GQDs including a full control of the wave function properties during transport spectroscopy. Here, we review the recENt success in mapping wave functions of GQDs by scanning tunneling microscopy (STM). We show that GQDs on Ir(111) are softly confined rENdering the wave functions more regular. The wave function mapping is disturbed by a sp ‐ type surface state from Ir(111), which can be improved by Au intercalation or adequate Fourier analysis. Edge states are absENt on Ir(111), but reappear after intercalating Au and coupling the σ ‐ bonds of the zig ‐ zag edges to BN. Finally, we show how the goal of probing such wave functions during transport spectroscopy might be achieved. Therefore, we presENt transport spectroscopy of GQDs on SiO2 which are prepared in a STM compatible fashion. 3D view of STM image of graphENe supported by SiO2; Insets (from left to right): STM image of graphENe nano ‐ islands on Ir(111) and scanning tunneling spectroscopy images of the three low ENergy wave functions of a particular nano ‐ island. (© 2015 WILEY ‐ VCH Verlag GmbH &Co. KGaA, Weinheim) Part of Focus Issue on ”Carbononics” (Eds.: Pawel Hawrylak, Francois Peeters, Klaus Ensslin) Wave function mapping of graphENe quantum dots by scanning tunneling microscopy is reviewed covering confined states and edge states. Besides studies, where the graphENe is deposited on a metal, the authors opEN the perspective to map the wave functions of graphENe quantum dots on insulators, such that the results can be directly compared to transport spectroscopy. [ABSTRACT FROM AUTHOR]