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Excitation of bond-alternating spin-1/2 Heisenberg chains by tunnelling electrons

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 394005 (http://iopscience.iop.org/0953-8984/26/39/394005) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 147.8.31.43 This content was downloaded on 15/06/2017 at 16:49 Please note that terms and conditions apply.

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 394005 (7pp)

doi:10.1088/0953-8984/26/39/394005

Excitation of bond-alternating spin-1/2 Heisenberg chains by tunnelling electrons J-P Gauyacq1 and N Lorente2,3 1

  Institut des Sciences Moléculaires d'Orsay, ISMO, Unité mixte CNRS-Universite Paris-Sud, UMR 8214, Bâtiment 351, Universite Paris-Sud, 91405 Orsay Cedex, France 2   ICN2—Institut Catala de Nanociencia i Nanotecnologia, Campus UAB, 08193 Bellaterra, Barcelona, Spain 3   CSIC—Consejo Superior de Investigaciones Cientificas, ICN2 Building, 08193 Bellaterra, Barcelona, Spain E-mail: [email protected] Received 14 February 2014, revised 8 April 2014 Accepted for publication 8 April 2014 Published 12 September 2014 Abstract

Inelastic electron tunneling spectra (IETS) are evaluated for spin-1/2 Heisenberg chains showing different phases of their spin ordering. The spin ordering is controlled by the value of the two different Heisenberg couplings on the two sides of each of the chain's atoms (bond-alternating chains). The perfect anti-ferromagnetic phase, i.e. a unique exchange coupling, marks a topological quantum phase transition (TQPT) of the bond-alternating chain. Our calculations show that the TQPT is recognizable in the excited states of the chain and hence that IETS is in principle capable of discriminating the phases. We show that perfectly symmetric chains, such as closed rings mimicking infinite chains, yield the same spectra on both sides of the TQPT and IETS cannot reveal the nature of the spin phase. However, for finite size open chains, both sides of the TQPT are associated with different IETS spectra, especially on the edge atoms, thus outlining the transition. Keywords: IETS-STM, magnetic excitations, Heisenberg chain (Some figures may appear in colour only in the online journal)

1. Introduction

Among such systems, the interplay between the various interactions at play leads to a rich range of different magnetic phases in low-dimensional magnetic structures. Topological quantal phase transitions (TQPT) appear in quantal systems at vanishing temperature when a parameter is changed. Recently, these transitions received a lot of attention. Indeed, they do not seem to be associated with breaking a symmetry and thus cannot be describable within Landau's theory [6], but they rather involve a non-local order, a so-called topological order (see e.g. a discussion in [7]). They have been invoked in various systems, in particular in the context of the Quantum Spin Hall effect and of spin lattices (see recent examples in [8–14] and references therein). Among these, the bond-alternating spin-1/2 Heisenberg chain [15–20] has recently been studied in detail and shown to exhibit a TQPT, associated to non-local string order [20] This system is appealing because of its simplicity and it yields a toy system on which to further analyse TQPT properties.

Inelastic electron tunneling (IET) presents extraordinary properties thanks to the extreme spatial localization of the tunneling current and the high accuracy in determining the injected electron energy [1–5]. When an inelastic channel opens, the conduction properties of the tunneling junction change, and a sharp transition of the current is observed that typically translates into an abrupt jump of the tunneling conductance. In this way, vibrations were able to be detected in molecules in a tunneling junction [1]. More recently, the advent of very low-temperature (VLT) scanning tunneling microscopes (STM) together with inbuilt magnetic fields, have upgraded the study of IET to the realm of the very low-energy excitations of magnetic systems [2]. Hence, the STM has become a tool that can examine the most intimate structure of matter through the study of local excitation spectra. It is then interesting to explore the possibilites of this tool on new systems, of increasing complexity. 0953-8984/14/394005+7$33.00

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© 2014 IOP Publishing Ltd  Printed in the UK

J-P Gauyacq and N Lorente

J. Phys.: Condens. Matter 26 (2014) 394005

information on the spectra of systems close to a TQPT. In the present work, we study the excitation of a finite-size bondalternating spin-1/2 Heisenberg chain by tunnelling electrons injected into one of the atoms of the chain. The emphasis is put on the way this excitation is modified when the chain parameters cross the TQPT boundary. Both finite-size open chains and rings of atoms are considered, the latter in an attempt to model infinite periodic chains.

Total energy

−2 −3 −4 −5

Ring, N=12 Chain, N=12 Chain, N=11 Ring, N=10

−6 −7

0

50

100

150

200

250

2.  System and structure We consider finite-size bond-alternating spin-1/2 Heisenberg chains described by the following Hamiltonian (only couplings between first neighbours along the chain are considered):

300

350

N /2

H = ∑ ( JS2k · S2k − 1 + J ′ S2k · S2k + 1) , (1)

Angle (degrees) Figure 1. Total energy of the ground state of rings and open chains

⎯→ ⎯

⎯→ ⎯

⎯→ ⎯

⎯→ ⎯

⎯→ ⎯

k=1

where Si is the local spin 1/2 at site i of the chain. J and J′ are the bond-alternating Heisenberg couplings. Both chains with even and odd numbers of atoms, N, are considered (below they are labelled even chains and odd chains, resp.). In the case of even N chains, we considered open chains (the site N + 1 does not exist) and rings (the site N + 1 is identical to site 1). Rings with a periodic boundary position allow mimicking the case of infinite chains; however, the convergence of the chain properties with the number of atoms depends on the chain structure [27, 29, 30]. We also considered the case of open chains with an odd number of atoms. A Zeeman term associated with an extremely small external magnetic field, B, along the quantization axis, is added in order to control the character of the various degenerate states in the system. We used the same modelling as in [20] i.e. we defined the couplings by:

of N spin-1/2 atoms, as functions of the parameter θ (in degrees) that defines the asymmetry between Heisenberg couplings on the two sides of every chain atom, see equation (2). The energy unit is equal to J0, the Heisenberg coupling at the TQPT. The vertical axis lines indicate the positions of the phase transitions; the TQPT is at θ = 45° (identical couplings). The ferromagnetic (FM) phase extends from θ = 180° to θ = 270°.

A chain of magnetic atoms adsorbed on a surface can be observed by STM. Developments of VLT high-resolution STM further allows to study inelastic processes associated with tunnelling: it is thus possible to determine how a tunnelling electron can excite the magnetic degrees of freedom of a chain of atoms [21–25]; which states are excited and how efficiently? Simultaneously, the spectroscopic analysis of the excitation processes reveals the magnetic structure of the studied nano-objects, in particular the exchange couplings and anisotropies of the chain components. [21, 23–26] In this context, it is interesting to see how an electron can excite a bond-alternating Heisenberg chain, and how the excitation is modified when going through the TQPT, thus extending the transition analysis to excitation processes. Discussion of the excitation of Heisenberg chains by tunnelling electrons [27] showed local effects and also non-local aspects. Indeed, the local spin-flip of the atom into which the tunnelling electron is injected leads to non-local magnon excitation in ferromagnetic chains. In contrast, the excitation of anti-ferromagnetic chains presents a strong non-local aspect [27, 28]: anti-ferromagnetic chains are strongly-coupled, highly-correlated systems [29, 30], the ground state is a mixture of a large number of configurations of local spins along the chain, so that injection of a single electron in a given atom leads to a rich excitation spectrum via correlation-induced processes, i.e. via a non-local excitation process. In this way, excitation by tunnelling electrons probes the correlated and non-local character of a Heisenberg chain and could yield information on the structure of bond-alternating chains. Calculations with finite-size bond-alternating spin-1/2 Heisenberg chains show that excited states can be good approximations of the corresponding states of infinite systems [15]. Hence, a study using finite chains can give valuable

(2) J = J0 2 cos(θ ); J ′ = J0 2 sin(θ ). This choice sets the energy unit equal to J0, the value of the Heisenberg coupling in the AFM case (θ  =  45°), i.e. at the TQPT. With this parameterization, Wang et al [20] discussed the chain structure in relation with string orders. Three phases appear in the infinite system [20]: a ferromagnetic (FM) phase for 180° 

2 Heisenberg chains by tunnelling electrons.

Inelastic electron tunneling spectra (IETS) are evaluated for spin-1/2 Heisenberg chains showing different phases of their spin ordering. The spin ord...
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