Electromagnetic Response of Two Low Dimensional Systems: Graphene and Optical Fibre

Vyas, Vivek M. (2012) Electromagnetic Response of Two Low Dimensional Systems: Graphene and Optical Fibre. PhD thesis, Indian Institute of Science Education & Research Kolkata.

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Abstract

Mankind has always been curious about nature and naturally available materials. The question what makes different materials different and who decides their properties, has bothered many people from time to time. Many great thinkers also proposed theories and explanations intending to answer this question in a universal manner, but till this date this question is only partially answered. We now understand that the visible matter is made up of atoms, and it is the interplay between them that decides the bulk properties of a material made out them. Atoms interact amongst themselves and arrange in different forms to realise different phases of matter, which although are made up of same ingredients but show different behaviour under external stimulus. Atoms are known to interact with light, but collection of atoms that form a solid, interact with light in a fundamentally different manner. This difference in behaviour allows one to study and differentiate various materials and various phases of the same material. Such a study hopefully brings one closer to aforementioned question. In cases, where one has sufficient knowledge of microscopic working of a given material, it may be possible to give atleast a qualitatively correct theory of electromagnetic response starting from microscopic building blocks. But as one can guess there are many more materials whose microscopic working itself is not clearly known and hence such microscopic approach fails. In some cases, although the microscopic structure is well known but the complexity of the problem of light-matter interaction itself may not allow one to construct such a microscopic theory. In such cases one resorts to phenomenological approach, whereby one tries to construct an effective theory based on broad arguments and experimental evidences. In this thesis, we have tried to understand electromagnetic response of two systems, graphene and optical fibre. For graphene case we follow the former microscopic route whereas for optical fibre the latter one is used. A century ago, Kamerlingh Onnes discovered the phenomenon of superconductivity, and since then it has been a subject of amazement and puzzle for scientists [1]. A microscopic theory explaining this phenomenon came almost half a century later from the work of Bardeen, Cooper and Schrieffer (BCS) [1]. It was showed by Anderson and Nambu, amongst others, that the supercon- ductivity is intimately related to the phenomenon of spontaneous symmetry breaking [2, 3]. In spontaneous symmetry breaking, a system realises a vacuum which is not invariant under certain continuous global symmetries of the Hamiltonian, and leads to creation of gapless Nambu-Goldstone modes. As argued by Anderson and others, in case of BCS theory, this gapless Nambu-Goldstone mode is absorbed by photon to make it massive. It must be pointed out that, although photon becomes massive, the theory always maintains gauge invariance and currents are always conserved. Infact, it was shown by Schwinger that gauge invariance and gauge boson mass can coexist [4]. Above is the so called Anderson-Higgs mechanism. While all experimentally observed supercon- ductors display some or the other type of spontaneous symmetry breaking, there is no fundamental principle indicating that superconductivity can only occur via symmetry breaking. One of the main goals of this thesis is to propose a mechanism of superconductivity without spontaneous symmetry breaking in graphene. The proposed mechanism is demonstrated by constructing two simple models, which are not unphysical and hence can be experimentally realised. Graphene, an atomically thin monolayer of Graphite, consists of carbon atoms connected to one another via a hexagonal network of covalent bonds. It was experimentally isolated in 2004 by Novoselov et. al., and since then it has opened a new way to simulate relativistic physics in condensed matter systems [5]. It was first shown by Wallace that, the valence band touches conduction band at six different points in Brillouin zone, around which electronic dispersion becomes linear [6]. Later Semenoff showed that electronic excitations around these points, which are known as Dirac points, obey Dirac like equation and are relativistic [7]. From symmetry arguments one finds, that only two of these Dirac points are inequivalent, and are often referred to as valleys. So the low energy electronic excitations in Graphene are nothing but two species of massless Dirac fermions defined on a plane moving with Fermi velocity � 106m/s. By suitable choice of substrate or by selective doping, a certain discrete symmetry known as sublattice symmetry can be broken, which results in occurrence of a gap in the electronic spectrum due to which Dirac fermions become massive. Being intrinsically two dimensional, these massive Dirac fermions possess non-trivial spin angular momentum apart from fundamental electronic spin. First two chapters of this thesis are devoted to these discussions. It is observed that the presence of this non-trivial spin is responsible for many interesting phenomena that are discussed in two subsequent chapters. The third chapter deals with two Abelian gauge theory models. In the first model, it is shown that presence of a dynamical Abelian gauge field in Graphene, which couples oppositely to both valley fermions, gives rise to superconductivity. It is observed that this planar Abelian gauge field via virtual fermion loops gets coupled to external electromagnetic field. This coupling is through a topological mixed Chern-Simons term, which is special to three space-time dimensions. It is this unique coupling that paves the way to superconductivity by developing a gauge invariant mass for the external electromagnetic field. Effects like infinite DC conductivity, Meissner effect and persistent currents, all naturally follow from the same. This is in constrast to BCS theory, where Anderson-Higgs mechanism was responsible for developing photon mass. A topological infinite order Berezinskii-Kosterlitz-Thouless phase transition is seen to occur at a certain finite temperature, due to which the Abelian gauge field develops singularities. It is found that these singularities contribute to electromagnetic response of the system, and precisely cancel the superconducting contribution. So, the Berezinskii-Kosterlitz-Thouless phase transition results in a superconductor- insulator transition, which marks loss of superconductivity. Graphene samples obtained in reality are of finite size, with a well defined boundary. It is shown, in case of arm-chair edged boundary, that the boundary supports dissipationless and gapless chiral modes. The presence of mixed Chern- Simons coupling in bulk is seen to play the pivotal role in establishing this bulk-boundary interplay. In fact, it is shown that the theory living on the boundary is identical to Schwinger model on a circle. It is worth emphasizing, that since in the above model superconductivity does not occur because of spontaneous symmetry breaking, there exists no local order parameter characterising ordered/disordered phase. The second model deals with gapped Graphene under a constraint that, the currents generated in response to some interaction from both the valley fermions are always equal. Interestingly, the theory in presence of this local constraint is actually an Abelian gauge theory. Ward-Takahashi iden- tities following from this local gauge invariance are seen to yield surprising consequences. Firstly, it implies confinement of Dirac fermions, the fundamental elementary excitation of the system. Dirac fermions cease to propagate and do not show up as poles in scattering matrix. Secondly, fermion- antifermion bound pair (exciton) is seen to be allowed by the theory as a propagating mode. Contrary to belief that the theory would describe an insulator, it is found that indeed the theory shows superconductivity at low temperatures. By explicit calculations, it is seen that it possesses infinite DC conductivity, shows Meissner effect and flux quantisation. It is seen that, the Lagrange multiplier field introduced to implement the above constraint behaves like a Nambu-Goldstone mode of BCS theory, and plays the central role in realising superconductivity. However, unlike BCS theory, here Meissner effect and flux quantisation occur not due to Anderson-Higgs mechanism, but rather due to topological Chern-Simons coupling. After a certain finite temperature, it is observed that spontaneous proliferation of singularities in Lagrange multiplier field takes place via Berezinskii- Kosterlitz-Thouless phase transition, which marks the superconductor-insulator transition. In case of finite size Graphene with armchair edges, the existence of dissipationless and gapless chiral edge modes is seen as a consequence of superconductivity in bulk. It is worth mentioning that, above proposed mechanism reminds one of Schwinger model: mass- less QED in two space time dimensions, where photon becomes dynamically massive and is con- fined [8]. A natural question arises, whether the above discussed models can actually be realised in Graphene or not. Longitundinal optical phonons couple to both valley fermions in Graphene like a vector field but with a relative minus sign, and one wonders, whether they can be used to realise the first model [9]. It is shown that although the dispersion of these modes is different from that of a massless mode, still by careful manipulation they can be effectively modelled by an Abelian gauge theory. However, the Abelian theory hence obtained is massive and by explicit calculation of response functions, it is shown that it fails to serve the purpose. Unlike above microscopic calculation, the last chapter of this thesis deals with a phenomenolog- ical description of electromagnetic response of an optical fibre. As is well known, electromagnetic response of any given material is encoded in Maxwell’s theory by presence of permittivity and permeability of the given material. These response functions are functions of space-time and if the external electromagnetic field is sufficiently weak, then one can show, using model scattering calculations, that they are independent of electromagnetic field as well. However, if the fields are sufficiently intense then multiphoton processes take place and even a model scattering calcula- tions become involved. One can still study effective dynamics by allowing response functions to be functions of fields and by choosing an appropriate ansatz for these response functions. Now that one has knowledge of response functions, the dynamics can be understood easily by solving Maxwell’s equations. Since response functions now are functions of fields themselves, one finds that Maxwell’s equations are no longer simple linear equations. These nonlinear equations hold all the information regarding the dynamics and hence any exact solution to these is of significant value. In most situations in real life, one does not face these nonlinearities in light-matter interactions. However, with the advent of high power lasers and optical fibres, one can easily work in domain where these nonlinear responses are important [10]. Further in an optical fibre, the dynamics of guided modes occurs only along axial direction, and hence the problem gets considerably simplified and effectively boils down to solving a nonlinear equation of motion for electromagnetic field in 1+1 dimensions. In case when one is interested in pulse propagation, with a typical pulse width being of the order of nanoseconds, the resultant equation and its solutions called solitons have been well studied [11]. However, if one goes to even shorter time scales of around hundred femtoseconds, then the nonlinearities are too difficult to allow an exact solution to equation of motion in general. We have shown that in certain situations one can indeed solve the equation, to get a new class of exact soliton solutions of bright, dark and kink type. Further, it is shown that these solitons can be unidirectional or chiral, which means that they only propagate along one direction. It is also found that these solutions possess non-trivial phase dynamics which may provide them extra stability.

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