Condensed matter theory is a massive field to which no book or books can do full justice. Every chapter in this book is possible material for a book or books. So it is clearly neither my intention nor within my capabilities to give an overview of the entire subject. Instead I will focus on certain techniques that have served me well over the years and whose strengths and limitations I am familiar with.

My presentation is at a level of rigor I am accustomed to and at ease with. In any topic, say the renormalization group (RG) or bosonization, there are treatments that are more rigorous. How I deal with this depends on the topic. For example, in the RG I usually stop at one loop, which suffices to make the point, with exceptions like wave function renormalization where you need a minimum of two loops. For non-relativistic fermions I am not aware of anything new one gets by going to higher loops. I do not see much point in a scheme that is exact to all orders (just like the original problem) if in practice no real gain is made after one loop. In the case of bosonization I work in infinite volume from the beginning and pay scant attention to the behavior at infinity. I show many examples where this is adequate, but point to cases where it is not and suggest references. In any event I think the student should get acquainted with these more rigorous treatments after getting the hang of it from the treatment in this book. I make one exception in the case of the two-dimensional Ising model where I pay considerable attention to boundary conditions, without which one cannot properly understand how symmetry breaking occurs only in the thermodynamic limit.

This book has been a few years in the writing and as a result some of the topics may seem old-fashioned; on the other hand, they have stood the test of time.

Ideally the chapters should be read in sequence, but if that is not possible, the reader may have to go back to earlier chapters when encountering an unfamiliar notion.

I am grateful to the Aspen Center for Physics (funded by NSF Grant 1066293) and the Indian Institute of Technology, Madras for providing the facilities to write parts of this book.

Research on magnetic carbon nanostructures is a young and challenging field of materials science. It is in the process of widening our understanding of both items combined in the title of this book, of magnetism, and of carbon nanostructures. Magnetism in Carbon Nanostructures is taken here in the most general sense, which includes intrinsic magnetism as well as magnetism imported into a carbon nanostructure by a magnetic guest species. While effects of the latter type have been extensively studied, the former is the subject of lively current debate. It comprises a broad spectrum of magnetic phenomena not arising from conventional sources of magnetism in materials, i.e. the transition metals of the periodic table, but arising instead from structural irregularities, from dimensional reduction, defects and disorder. Equally persuasive as the promise of novel insight into the nature of magnetism and the basics of carbon nanostructures is the prospect of technological innovation offered by carbon-based magnetic materials. These materials are lighter than their metallic analogs. Also, they are expected to be available at lower fabrication cost and to be more efficient in terms of power consumption. In particular, during the past two decades, a wide range of actual and potential applications of carbon nanostructure magnetism have been identified, extending from fullerene-based magnetic resonance imaging (MRI) contrast agents to zigzag graphene nanoribbon (zGNR) transmission elements in nano-spintronics.

This text aims at a synoptic presentation of these advances, highlighting theory, experiment, and nanotechnological perspectives. Its main objective is to unite what is currently scattered over numerous articles in journals and conference proceedings, or found in individual chapters of multi-author monographs. Beyond these items, a great number of very instructive survey articles, illuminating various facets of carbon nanostructure magnetism, have been published. Likewise, excellent books on magnetism in condensed-matter systems or on carbon nanostructures (most notably on graphene, the topic of several extensive monographs that have appeared during the past few years) are available to readers, many of them covering some of the topics presented in this work. Yet, as far as I see, a comprehensive treatment of the subject, addressing the principles and the practice of carbon nanostructure magnetism in a single volume, is still lacking in this multitude of materials.

This chapter combines brief summaries of computational as well as experimental procedures that have been instrumental for the study of nanostructure magnetism. All of the methods sketched here have been successfully applied to magnetic carbon nanostructures. None of them, however, is specific to this type of materials. The models and techniques included in this chapter are pertinent to condensed matter systems or have, in some cases, an even wider scope of usage, including matter at the molecular scale. They thus have the status of standard methods. Procedures more specifically adjusted to exploring magnetism in carbon-based materials, such non-local measurement of spin signals in carbon transmission elements, will be introduced later in their distinct theoretical or experimental contexts.

Computational Methods

In the first part of this chapter we survey a variety of methods that have proved to be efficient in modeling carbon nanostructure magnetism. Procedures based on the tight-binding and the Hubbard model, as sketched in Section 3.1, have played important roles in clarifying the electronic architecture of graphene and its derivatives. Tight-binding schemes have successfully addressed salient phenomena pertaining to these materials, among them the nature of electronic states at the Fermi level of graphene as massless Dirac Fermions. Lifting the methodological constraints typical for these methods, such as the restriction of interatomic coupling to next-neighbor sites, or the representation of a given lattice site by a single electron, leads to more general theories of electronic structure. These are the topic of Section 3.2, which deals with ab initio and density functional theory (DFT) methods. In all cases covered in this survey, emphasis is placed on the capacity of a given computational approach to describe magnetic states and to represent magnetic properties.

The Tight-Binding and the Hubbard Model

The theory of one- or two-dimensional carbon nanostructures, represented respectively by the nanotube and the graphene paradigms, has been largely developed by use of the tight-binding or the Hubbard model. In this section we will introduce some general traits of these approaches, insofar as they are of relevance to this text. Both models rest on the tight-binding premise. According to this condition, the system in its ground state may be described as a lattice populated with atoms in a hydrogen-like configuration.

After the review of spin interactions in carbon nanostructures in Chapters 7 and 8, and the summary of mechanisms and material properties essential for spintronics in the previous chapter, the present chapter will combine these two domains. In the first place, we will attempt to clarify why carbon nanostructures are of major interest as potential carrier materials in spintronics circuits. Further, we will characterize the physical effects that make these structures relevant as elements of spin networks. Since devices based on these effects have been conceived but are still quite far from the manufacturing stage, our account will be about currently discussed models of these devices, and their experimental examination. We focus on two classes of materials: graphene and carbon nanotubes.

Graphene Spintronics

Graphene is of high interest as nanoelectronic material. Among the features of graphene that motivate this interest are experimentally confirmed electron mobilities as high as 2.00 105 cm2V−1s−1 [347], which is about a hundred times higher than typical mobilities of electrons in silicon. This suggests that graphene-based transistors will function at substantially higher speeds than elements of conventional electronics, while being more efficient in terms of power consumption. Test devices have been shown to be capable of operating at hundreds of gigahertz [348]. Another graphene property of direct relevance for nanotechnological applications is its gate-tunable charge carrier density, resulting from the double-cone structure of the energy surfaces close to the Dirac points. Further, in terms of heat resistance, graphene transistors are distinctly superior to the tools of presently existing electronics, as they have been shown to perform at temperatures higher by 20– 30 ◦C than their silicon counterparts [349]. The challenge of the semi-metallic character of two-dimensional periodic graphene crystals can be met in various ways. Thus, one may tailor a finite band gap in controlled ways by modifying the graphene sheet with adsorbates. In addition, manufacturing electronic devices from the pristine graphene crystal requires its dimensional reduction, associated with the formaton of a finite band gap (see Section 4.5).

The previous three chapters presented a survey of typical carbon nanosystems in two, one and zero dimensions, namely graphene, carbon nanotubes and fullerenes, respectively. None of these systems is intrinsically magnetic.Magnetism, however, can arise through dimensional reduction of structures with one or more dimensions. Thus, as one truncates the infinite graphene sheet, one may generate graphene nanoribbons of the zigzag type (zGNRs), as introduced in Section 4.5.1. In the first section of this chapter, we will show that localized states at the zGNR edges give rise to antiferromagnetic ground states in these systems, and review experimental confirmation for this assertion. The analysis of zigzag edges will be extended to vacancies in graphene-based structures that lack magnetic edge states, such as aGNRs. In this context, we will emphasize the importance of Lieb's theorem as a principle underlying many magnetic phenomena in carbon nanostructures, albeit not all. Further, we will describe specific one-dimensional substructures in graphene-based systems, namely zGNR edge excitations (7.2.3), and topological line defects (7.3).

Intrinsic magnetism in truncated carbon nanotubes of the zigzag type (zSWCNTs) will be discussed in analogy to zGNR magnetism. Section 7.4.1 deals with aggregates of several zSWCNTs. The magnetic effects in these complexes turn out to be rooted in the electronic features of carbon surfaces with negative Gaussian curvature. We conclude with some remarks on the controversial topic of intrinsic magnetism in pure fullerene systems.

Systems Derived from Graphene

In what follows, we review magnetic effects associated with the rupture of the pristine graphene network, as it arises from the formation of edges, or vacancies, or voids as aggregations of vacancies. The focus is here on zigzag edge structures as the preferred loci of unpaired electrons, and thus the substructures of the graphene lattice from which spin interactions originate.

Magnetism in Zigzag Graphene Nanoribbons

Zigzag graphene nanoribbons (see Section 4.5.1) exhibit magnetism in their electronic ground states. In the following we will consider the physical origin of this phenomenon. Once more, we associate the x (y) coordinate with the periodic (finite) direction. Inspecting the zGNR ground state solutions (see Eqs. (4.71)– (4.77)), we found that the graphene wave function must be modified to allow for edge-localized states.

This chapter highlights magnetic phenomena in complex extended carbon nanosystems beyond the prototypes fullerenes, carbon nanotubes, and graphene. We begin this survey with carbon frameworks based on structural motives with hyperbolic geometry, as introduced in Section 7.4.1, where elementary units with negative Gaussian curvature were discussed. These nanostructures, termed carbon nanofoams [569], combine extreme lightness with a high degree of stability. Among their numerous astonishing materials properties is paramagnetic, and, at sufficiently low temperature, ferromagnetic behavior. The relation between the distinct geometric features of carbon nanofoam and its magnetic characteristics is the subject of Section 13.1.

Carbon nanospheres, as considered in Section 13.2, may be understood, in some sense, as the geometric complements of carbon nanofoams, as they consist of closed carbon surfaces and thus derive from fullerenes. We focus here on nanosphere aggregates which are composed of nanospheres nested into each other, carbon nano-onions. The magnetism of these architectures derives from magnetic units embedded in their interior. These impurities range from clusters of transition metal atoms to magnetic nanocrystals.

Sections 13.3 and 13.4 deal with various forms of nanodiamond and nanographite. The latter has been demonstrated to display magnetic features that differ dramatically from bulk graphite. In this context, we introduce activated carbon fibers (ACFs), extended microporous structures composed of graphite nanoparticles. Their magnetism is governed by the simultaneous presence of different exchange mechanisms whose strengths deviate substantially from each other. ACFs are also capable of accommodating metal clusters with finite magnetic moments. As host systems, they determine both the spatial distribution and the mutual magnetic coupling of these species.

Comparison is made with another porous carbon allotrope, zeolite-templated carbon (ZTC). ZTCs are obtained as negative replicas of zeolite compounds. The different design of the ACF and ZTC matrices gives rise to markedly different magnetic lattices in the two materials.

In the final section of this chapter we comment on magnetism in amorphous carbon.

Nanoporous Carbon Magnets

Carbon nanofoam is a cluster-assembled allotrope of carbon. It was first fabricated by laser ablation of glassy carbon in an argon atmosphere at a pressure of about 1–100 Torr [569, 570, 574].