Dilute antiferromagnetism in magnetically doped phosphorene

 

A. Allerdt,¹ A. E. Feiguin¹

* a.feiguin@northeastern.edu

Department of Physics, Northeastern University, Boston, Mas-sachusetts MA 02115, USA.

Received: 19 September 2017, Accepted: 12 October 2017
Edited by: K. Hallberg
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.090008

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Abstract

We study the competition between Kondo physics and indirect exchange on monolayer black phos-phorous using a realistic description of the band structure in combination with the density matrixrenormalization group (DMRG) method. The Hamiltonian is reduced to a one-dimensional problemvia an exact canonical transformation that makes it amenable to DMRG calculations, yielding exactresults that fully incorporate the many-body physics. We find that a perturbative description of theproblem is not appropriate and cannot account for the slow decay of the correlations and the completelack of ferromagnetism. In addition, at some particular distances, the impurities decouple formingtheir own independent Kondo states. This can be predicted from the nodes of the Lindhard function.Our results indicate a possible route toward realizing dilute anti-ferromagnetism in phosphorene.

Keywords

Density Matrix Renormalization Group; Kondo; RKKY; Anti Ferromagnetism; 2D Materials; Phosphorene

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I. Introduction

Phosphorene, a single layer of black phosphorus, is one of the many allotropes of the element. Others include red, white, and velvet phosphorus. Of these, black is the most thermodynamically stable and least reactive [1]. Its crystalline structure resembles that of graphite, whose 2D form is famously known as graphene, and can similarly be fabricated into 2D layers. The main qual-itative distinction lies in the fact that phosphorene has a "puckered" hexagonal structure which is responsible for opening a gap in the band structure. In addition, the bonding structure is vastly different, since phosphorene does not have sp² bonds, but is composed of 3p orbitals [2, 3].

Successful production of monolayer black phosphorus has been achieved only in the last few years, with the first publications concerning its single layer properties appearing in 2014 [4-7]. While being a semiconductor with a small direct band gap in bulk form (0.3 eV) [8], the gap increases as the number of layers is decreased.

Since its appearance, there have been many proposed promising applications and exotic properties. Besides being a semiconductor with a band gap in the optical range, it has ample flexibility and high carrier mobility [4, 9]. Further, it also possesses interesting optical properties such as absorbing light only polarized in the armchair direction, indicating a possible future as a linear polarizer [10]. Its stable excitons present possi-ble applications in optically driven quantum computing [11]. Interest on its applications continues to grow. For a comprehensive review, we refer to Ref. [11].

Our understanding of magnetic doping in phosphorene is still in its infancy. A thorough study of metal adatoms adsorbed on phosphorene was performed in Ref. [12] where a variety of structural, electronic and magnetic properties emerge. The authors show the binding energies are twice the amount in graphene. A DFT study has proposed possible chemical doping by means of adsorption of different atoms, ranging from n-type to p-type, as well as transition metals with finite magnetic moment [13]. Another conceivable method of moving the Fermi level is to induce strain on the lattice, causing the level to move into the conduction band [14]. The case of magnetic impurities is considerably non-trivial, since one has to account for the Kondo effect [15] with the impurity being screened by the conduction spins in the metallic substrate, and the RKKY interaction [16–18], an effective indirect exchange between impurities mediated by the conduction electrons.

These two phenomena are expected to be present and compete in phosphorene when the Fermi level is not sitting in the gap. The RKKY interaction in phosphorene with the inclusion of mechanical strain is investigated in Ref. [14]. However, all their calculations are based on second order perturbation theory and ignore the effects that Kondo physics can induce. In this work, we present numerical results for two Kondo impurities in phosphorene that capture the full many-body physics and we discuss the competition between Kondo and indirect exchange.

II. Model and numerical method

The Hamiltonian studied in this work is the twoimpurity Kondo model, generically written as:

 

 

IV. Conclusions

By means of a canonical transformation and exact numerical calculations using the DMRG method, we have studied the competition between Kondo and RKKY physics on phosphorene. The method is numerically exact, and even though it is limited to finite systems, these can be very large, of the order of a hundred lattice sites and more. Our results highlight the non-perturbative nature of the RKKY interaction and the non-trivial absence of ferromagnetism. This remains an outstanding question and should stimulate more research in this direction. It is possible that by adding a repulsive interaction between conduction electrons, the system may acquire a net magnetic moment, a behavior of this type has been observed in graphene doped with hydrogen defects [28–30]. According to Lieb’s theorem [31], this is expected to occur for bipartite lattices when two impurities are on the same sublattice. Even though phosphorene has four sublattices (and hence, the theorem does not rigorously apply), the system may still realize similar physics. Unfortunately, our approach can only describe non-interacting/quadratic Hamiltonians and we cannot prove this conjecture. On the other hand, the dominant anti-ferromagnetism at all dopings and distances (more robust than in graphene [21]) indicates a route toward realizing dilute 2D anti-ferromagnetism with phosphorene.

Acknowledgements - The authors are grateful to the U.S. Department of Energy, Office of Basic Energy Sciences, for support under grant DE-SC0014407.

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