PHYSICAL REVIEW B 101, 115421 (2020) Graphene-mediated interaction between hydrogen adsorbates Keian Noori , 1 Hillol Biswas, 2 Su Ying Quek , 1, 2 and Aleksandr Rodin 3, 1 1 Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, 117546, Singapore 2 Department of Physics, National University of Singapore, 2 Science Drive 3, 117542, Singapore 3 Yale-NUS College, 16 College Avenue West, 138527, Singapore (Received 25 November 2019; accepted 27 February 2020; published 18 March 2020) Interaction between adsorbed hydrogen atoms in graphene is studied using a combination of DFT and the path integral formalism. Our results reveal a complex nonmonotonic interaction profile. We show that the strength and sign of the interaction are dictated by the adsorbate arrangement, as well as the system doping. The path integral approach given here allows one to compute energies and densities in an efficient manner without relying on exact diagonalization. DOI: 10.1103/PhysRevB.101.115421 I. INTRODUCTION Atomic-scale manipulation of graphene and graphenelike materials has been the subject of a number of theoretical and experimental works in recent years. On one hand, there is the possibility of replacing individual atoms in the lattice by a different species to modify the electronic properties of the host [1–6]. On the other hand, it is possible to deposit adatoms on top of the bulk system in a highly controlled nondestructive manner to create magnetic states [7,8]. When multiple adatoms are present, they interact via their host’s electronic states. This interaction modifies the energy of the system and causes the adatoms to attract or repel. Studies focusing on pairs of impurities in graphene [9–11] have revealed that the sign of the interaction depends on the sublattice arrangement of the adatoms, as well as their separa- tion and energies. Understanding this interaction is useful in determining whether the impurities will tend to aggregate or disperse for the given system parameters. Most of the analytic work done on this subject employs the Green’s function formalism with the Dirac-cone continuum model. While this does give the correct general picture, it is useful to have a method that preserves the underlying lattice symmetry since the cone approximation is less applicable at small impurity separations. One approach is to use the hopping Hamiltonian and perform an exact diagonalization to extract the interaction energy and the impurity-induced charge density. The downside of this method is its numerical cost: To avoid finite-size effects and, at the same time, be able to explore a variety of impurity configurations, the system needs to be quite large. The purpose of the present work is to provide a computationally-efficient method for calculating interaction energies and induced charge densities in graphene. While we focus on hydrogen adsorbates, similar to the experi- mental setup of Ref. [8], our approach works equally well for substitutional impurities. The computational complex- ity of our method depends on the number of adatoms and not their separation—an advantage over the exact diagonal- ization approach. A similar formalism for computing the impurity-induced charge density and impurity interaction was first employed in Ref. [12]. In our work, however, we rely on fewer approximations and generalize the result to an arbitrary number of impurities. We use two complementary approaches in this study. For small impurity separations, we employ density functional theory (DFT) as it provides accurate ground-state charge den- sities and total energy differences for our model system. With increasing distance between the impurities, DFT becomes prohibitively expensive due to large unit cells. Therefore, we turn to the path integral formalism to describe the system. By including the entire Brillouin zone in our path integral calculations, we ensure that the symmetries of the system are respected, allowing for a direct comparison between the two approaches to demonstrate both qualitative and quantitative agreement. Using path integrals, we recover the expected Friedel oscillations and, for systems of two impurities, show that the sign of the interaction depends not only on the impurity arrangement but also on the system doping. The key novelty of the path integral approach described here is its ability to treat arbitrary impurity configurations, allowing for the use of numerical energy minimization algorithms to study dispersion or aggregation of multiple adatoms. II. DFT RESULTS We begin by exploring the effect of the adsorption of a single hydrogen (H) atom impurity on a 12 × 12 supercell of pristine, planar (unrelaxed) graphene. It is generally accepted [8,13–16] that introducing a hydrogen adsorbate to graphene leads to sp 3 hybridization of the host atom, effectively decou- pling its p z orbital from the rest of the lattice. This produces a localized state close to the Dirac point. We observe that even in the absence of the lattice deformation, which typically accompanies this hybridization, hydrogen adsorption gives rise to a state with an energy above the Dirac point, as shown in Fig. 1. The adsorbed H adatom modifies the charge density of pristine graphene such that the induced density oscillates in a concentric manner around the impurity, as shown in Fig. 2. 2469-9950/2020/101(11)/115421(9) 115421-1 ©2020 American Physical Society