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Science Drivers—Condensed Matter and Materials Physics

OVERVIEW

Most applications of high magnetic fields to the study of condensed matter systems have been concerned with “hard” condensed matter systems—typically, rigid solids or structures fabricated from such solids. These would include the surfaces of solids and interfaces between solids, small particles of a solid, wires with nanometer-scale diameters, and two-dimensional materials such as graphene, which is regular and practically rigid even though it is just a single layer of carbon atoms.

The properties of hard condensed matter systems are determined by the Coulomb forces between electrons and the constituent ions and by the constant motion of the electrons, dictated by the microscopic laws of quantum mechanics. Scientific research seeks to understand how these fundamental laws lead to huge diversity in the macroscopic behavior of different materials, nanostructures, and devices. With increased understanding comes the ability to design and optimize materials to attain desired technological goals and, on occasion, to conceive of radically new technologies that can have a profound effect on our lives.

Magnetic field studies have been a very important tool for exploring the electronic structure of condensed matter systems. Although applied magnetic fields in many cases have a relatively small effect on the overall electronic structure, they enable experimental techniques that can reveal properties of the underlying electronic structure that would be otherwise inaccessible. In other cases, magnetic field effects can be strong enough to drastically change the nature of the electronic state



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2 Science Drivers—Condensed Matter and Materials Physics Overview Most applications of high magnetic fields to the study of condensed matter sys- tems have been concerned with “hard” condensed matter systems—typically, rigid solids or structures fabricated from such solids. These would include the surfaces of solids and interfaces between solids, small particles of a solid, wires with nano- meter-scale diameters, and two-dimensional materials such as graphene, which is regular and practically rigid even though it is just a single layer of carbon atoms. The properties of hard condensed matter systems are determined by the Coulomb forces between electrons and the constituent ions and by the constant motion of the electrons, dictated by the microscopic laws of quantum mechanics. Scientific research seeks to understand how these fundamental laws lead to huge diversity in the macroscopic behavior of different materials, nanostructures, and devices. With increased understanding comes the ability to design and optimize materials to attain desired technological goals and, on occasion, to conceive of radically new technologies that can have a profound effect on our lives. Magnetic field studies have been a very important tool for exploring the elec- tronic structure of condensed matter systems. Although applied magnetic fields in many cases have a relatively small effect on the overall electronic structure, they enable experimental techniques that can reveal properties of the underlying elec- tronic structure that would be otherwise inaccessible. In other cases, magnetic field effects can be strong enough to drastically change the nature of the electronic state 20

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S c i e n c e D r i v e r s — C o n d e n s e d M at t e r and M at e r i a l s P h ys i c s 21 itself. In either situation, access to higher magnetic fields is important to allow the study of new materials and new phenomena. Magnetic fields couple most strongly to the electrons in a material, either by acting on electrical currents generated by the electrons’ quantum mechanical motion, or by coupling to the magnetic moments arising from the electrons’ intrin- sic spin. These coupling mechanisms are indeed the basis for the majority of experi- ments using high magnetic fields to study the electronic structure of materials. However, the much weaker coupling of magnetic fields to the magnetic moments of nuclear spins is also important, as it is the physical basis for nuclear magnetic resonance (NMR) techniques. Although NMR techniques are most widely used in studies of chemical and biological systems, which will be discussed in Chapters 3 and 4, NMR techniques can also be used to extract information about the elec- trons in condensed matter systems by measuring changes in the nuclear response that arise from magnetic interactions between the nuclei and the electrons. For example, NMR measurements have played a key role in elucidating the fundamen- tal properties of electrons in superconductors, two-dimensional electron systems, and antiferromagnets. The electronic structures of condensed matter systems can vary in many ways. Materials may be either electrical conductors or insulators, whose conductivities may differ by many orders of magnitude, even at room temperature. At low tem- peratures, some materials are superconductors, which can carry currents with no resistance at all. In many systems, both insulators and conductors, electrons on individual atoms can develop magnetic moments, which may order at low tem- peratures in a variety of ways. In other cases, instabilities in the electron system can lead to spatially periodic oscillations in the charge density and/or to displace- ments in atomic positions that change the symmetry of the crystal, and may lead to macroscopic electric dipole moments (ferroelectricity). In some cases, magnetic moments and atomic displacements are coupled, and magnetic fields can be used to influence electric polarizations. Of particular interest are systems that may change from one form of order to another as a function of temperature, pressure, or alloy composition; such materials may have very peculiar properties close to their phase transitions. Magnetic fields may provide a way for studying such phase transitions, and in some cases they may be strong enough to directly influence these transitions. Crystalline materials are typically characterized by an electronic band struc- ture, which specifies a discrete set of allowed electron energies for each possible electron momentum. Metals, which are good conductors of electricity, will typi- cally have a surface in momentum space, known as the Fermi surface, separating occupied electron states, whose energies lie below a cutoff (the Fermi energy), from empty states, whose energies lie above the cutoff. Electrons close to the Fermi sur- face are of particular interest because they play a dominant role in electrical trans- port and many other properties of the material. Strong magnetic fields can lead to

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22 High Magnetic Field Science and I t s A pp l i c a t i o n in the US oscillations in transport and other electronic properties, whose periods depend on the precise size and shape of the Fermi surface. The ability to extract information about the Fermi surface through measurement of these oscillations is one reason magnetic fields are such an important tool for studying conducting materials. Magnetic fields are particularly vital in the study of superconducting materials. Typically, superconductors will expel magnetic fields up to a certain field strength, denoted the lower critical field, Hc1. Above Hc1, magnetic fields will enter, but the material retains superconducting properties up to an upper critical field, Hc2. Measurement of the critical fields, as a function of temperature and orientation of the sample, gives important information about the underlying parameters of the superconductor. Superconductors with very large values of Hc2 are of special interest, because it is precisely those superconductors that have potential for use in the construction of high-field superconducting magnets. A relatively new direction of high magnetic field research is in the area of soft condensed matter, which encompasses a variety of physical systems that are soft in the sense that they can be easily deformed by mechanical or thermal stress or electric and magnetic fields. Such systems include polymers, gels, colloids, mem- branes, and biological cells or organisms. The binding between molecules in these mostly organic or biological materials (hydrogen bonding, van der Waals, or π-π bonding) is much weaker than in normal solids. High magnetic fields can be used to assemble and align functional, organic or inorganic, nano- and microstructures and to probe their structures, properties, and dynamics, with potential applications in drug delivery, optics, sensors, and nanoelectronics. Applications of high magnetic fields in these experiments make use of the torque exerted by a magnetic field on an object with an anisotropic diamagnetic susceptibility or the force exerted on any object by a strong gradient in field strength. In the remainder of this chapter, a number of examples are discussed where high magnetic fields play a critical role in condensed matter research. Quantum Critical Matter All systems are disordered if the temperature is large enough, with no discern- ible correlations or patterns in time or space for the configurations of the particles that make up the system. In most solid materials, the particles of interest are the electrons, which carry both a charge and a magnetic moment. Interactions among these electrons can lead to instabilities in their overall energy that are resolved in most cases by the establishment of some type of order. Some compounds order magnetically, when the magnetic moments of individual atoms spontaneously take on long-lived and spatially periodic patterns that create internal magnetic fields as the temperature is reduced. In some cases, materials are transformed from metals, where electrons are free to move and carry current, to insulators, where

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S c i e n c e D r i v e r s — C o n d e n s e d M at t e r and M at e r i a l s P h ys i c s 23 they become spatially localized and thus incapable of carrying electrical current. Some metals can become superconducting, where a condensed state is formed that consists of electron pairs with opposite moments and opposite momenta, capable of carrying electrical current without dissipating energy. Order is overwhelmingly favored as the temperature is lowered toward absolute zero, where the system is said to enter its lowest energy, or ground, state. Most systems become ordered at nonzero transition temperatures, but in some cases this order can be suppressed to progressively lower temperatures by varying a parameter such as pressure or the electric or magnetic field, or simply by changing the composition of the material in question. An extremal ordered state occurs when the transition temperature is continuously suppressed to zero temperature, forming a quantum critical point (QCP). Here the system is poised just on the verge of becoming ordered, and in the absence of this order it fluctuates wildly and unpredictably among configurations where order is only present on short length scales and for short times (Hertz, 1976; Millis, 1993; Sachdev, 1999; Sachdev, 2008). Much of the functionality that we demand of modern materials depends on the presence of these collective electronic instabilities, as they lead to different and competing ground states: superconductivity, and full or partial charge, orbital, and magnetic order. Controlling the relative stabilities of these ground states by means of external parameters such as electric or magnetic field, chemical variation, pres- sure, or temperature lends these materials their technological value as novel sen- sors, or as active elements in electrical or mechanical systems. Indeed, the greatest sensitivity to external variables, and the most complex interplay of energy scales, is generically found near QCPs. It is widely believed that materials with the most extremized functionality require the near balance of competing ground states, and examples of families hosting these QCPs have been documented in virtually every class of material where strong electronic correlations are possible. The most celebrated example is the emergence of superconductivity with the extinguishing of magnetic order in f-electron-based heavy fermion compounds (Mathur et al., 1998), as well as in the iron-pnictide and cuprate superconductors. One of the ear- liest QCPs studied involves the interplay of superconductivity and charge density wave instabilities in layered chalcogenides (Morosan et al., 2006), in conductors formed from organic molecules (Jaccard et al., 2001), and even in the A-15 family that hosts the most widely used conventional superconductors Nb3Sn and NbTi (Bilbro and McMillan, 1976). Of particular interest for applications is the symbiosis of ferroelectricity and magnetic order present in multiferroic compounds (Rowley et al., 2010; Kim et al., 2009). Quantum criticality, when ordering is prohibited at any nonzero temperature and occurs only at zero temperature, is increasingly believed to be a central fea- ture of the phase behaviors of virtually every class of correlated electron system from the f-electron-based heavy fermions (Gegenwart, 2008; Von Lohneysen and

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24 High Magnetic Field Science and I t s A pp l i c a t i o n in the US Wolfle, 2008), to complex oxides that include cuprates (Broun, 2008), as well as low-dimensional conductors (Jaccard, 2001), and three-dimensional-based metals with magnetism such as the Fe pnictides and chalcogenides (Dai et al., 2009). Very few compounds form with magnetic order possible only at zero temperature, and in most cases it is necessary to use pressures, compositions, or magnetic fields to tune the ordering temperature to zero degrees to form a QCP if magnetic order is continuous, or a quantum end point (QEP) if the magnetic transition becomes discontinuous or first order. Quantum critical (QC) compounds can be exqui- sitely sensitive to disorder, making compositional tuning problematic. Pressure tuning has an appealing simplicity, although the bulky equipment needed for high-pressure measurements may limit experimental access, particularly for ther- modynamic measurements, which are of particular value for understanding how cooperative phases are stabilized at the lowest temperatures. For all these reasons, magnetic field B tuning is increasingly attractive, particularly if it is paired with low temperatures T to span an extended range of B/T. Since ordering temperatures are emergent scales, the fields required to suppress order may be very small, as in YbRh2Si2, where only 0.6 T is required to drive the 0.065 K Neel temperature to zero degrees (Custers et al., 2003), or very large, as in SrCu2(BO3)2, where 20 T is required to induce magnetic order via the Bose-Einstein condensation of dimer triplets (Kageyama et al., 1999). Systems where QCPs dominate have remarkable properties that challenge our understanding of such apparently simple concepts as metallic conduction. When magnetic fields suppress magnetic order in heavy fermions, or superconductivity in YBCO (Sebastian et al., 2010) to expose bare QCPs, the electrical resistivity ρ becomes linear in temperature, although the quadratic temperature dependence of a normal metal is regained when the system is tuned sufficiently far from the QCP (Custers et al., 2003). The violation of the Wiedemann-Franz law in CeCoIn5 near the field-tuned QCP (Tanatar et al., 2007) implies that the entities that carry charge and heat are very different from the familiar conduction electrons found in normal metals. Indeed, the familiar idea that electrical current is carried by individual electrons, or quasiparticles, must itself fail when the strong QC fluctuations limit their lifetime to be less than ħ/E, given by the Uncertainty Principle (Smith et al., 2008). It is not yet known how universal these observations are, and we have yet to even scratch the surface on how we might exploit these unusual metals to manipu- late charge, spin, and heat to provide novel functionalities. These unconventional metals prove to have novel ground states, once the more familiar magnetic order is suppressed (Julian, 1996). Of most interest is the unconventional superconductivity that often is revealed when pressure or composition is used to suppress magnetic or charge order to a temperature of zero degrees. Indeed, this idea has become close to a prescription for finding new families of superconductors (Basov and Chubokov, 2011). Surely the discovery of field-induced superconductivity in ferromagnetic

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S c i e n c e D r i v e r s — C o n d e n s e d M at t e r and M at e r i a l s P h ys i c s 25 UGe2 and URhGe represents the most exotic realization of the expectation that unconventional superconductors are stabilized by the plethora of low-energy exci- tations that proximity to QCPs can afford (Lévy et al., 2005). Superconductivity is not the only instability that is found when magnetic order becomes impossible beyond its QCP. One option, found in systems as diverse as Sr3Ru2O7, Co-doped BaFe2As2, and possibly cuprates, is partial or nematic elec- tronic order where there is a spontaneously broken rotation symmetry that does not involve the breaking of translational symmetry (Fradkin et al., 2010). High magnetic fields have been instrumental for delineating the full phase behaviors for these systems, most notably in URu2Si2 (Figure 2.1). Here, the nature of the “hid- den” order parameter remains uncertain (Kim et al., 2003; Mydosh and Oppeneer, 2011), although it has recently been proposed to be an electronic nematic as well (Okazaki et al., 2011). It seems likely that new types of order will emerge as higher fields and more sophisticated measurement techniques become available. FIGURE 2.1  Sketch of the high magnetic field T-H phase diagrams for URu2Si2 and U(Ru0.96Rh0.04)2Si2. MM, metamagnetic transitions; NFL, non-Fermi liquid; and phase II is colored in orange. SOURCE: Reprinted figure with permission from J.A. Mydosh and P.M. Oppeneer, 2011, Colloquium: Hidden order, superconductivity, and magnetism: The unsolved case of URu 2Si2, Reviews of Modern Physics 83: 1301-1322, Figure 12. Copyright 2011 by the American Physical Society.

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26 High Magnetic Field Science and I t s A pp l i c a t i o n in the US The electronic states of the system itself may also be in transition at the QCP, where the system may fluctuate between very different states such as a local- ized moment antiferromagnet and a nonmagnetic metal, where order persists only in the form of strong electronic correlations with limited range and lifetime (Figure 2.2). This dual role for the QCP as an electronic delocalization transition implies that there is an associated Fermi surface volume change. Indeed, Hall effect measurements on YbRh2Si2 (Paschen et al., 2004) and quantum oscillation studies of CeIn3 (Harrison et al., 2007) and CeRhIn5 (Shishido et al., 2005) have provided evidence for this quantum critical breakdown of the Fermi surface. It is clear that quantum oscillation measurements like the ones carried out on f-electron-based heavy fermions have been instrumental for establishing the link between QC and electronic delocalization, and the application of these ideas to other types of QC systems is an important area of future effort. High fields and low temperatures are particularly needed to resolve the heavy mass parts of the Fermi surface, which seem to be most strongly impacted by proximity to the QCP (McCollam et al., 2005). Our understanding of quantum critical matter is driven forward by the con- tinuous discovery of new materials with new and remarkable properties. The explo- ration of the phase behaviors of these new compounds is crucial, and there is an underlying expectation that there is an overarching phase diagram, with individual compounds representing various regimes of this master phase diagram. Magnetic fields, especially if they can be combined with other variables like pressure, are important not only for tuning the strength of order but also as a thermodynami- cally relevant scaling variable. There is every reason to believe that the availability of high fields will lead to the discovery of new types of QCPs and ordered phases, both in existing materials and in those that are yet to be discovered. Innovations in high-field measurement techniques over the past decade have greatly accelerated progress. It is not enough to use high fields to access a novel ordered phase. Also needed is an experimental description of the thermodynamic and transport properties of this new phase. New techniques for carrying out mea- surements of the specific heat (Jaime et al., 2000), magnetocaloric effect (Kohama et al., 2010), and magnetostriction (Jaime et al., 2012) in pulsed fields have greatly increased the types of basic information available about these high-field phases, enabling a full thermodynamic analysis. Many of these innovations have been made possible by the availability of long-pulse magnets. In this vein, the availability of higher field magnets for neutron scattering experiments can similarly be expected to be transformative. One of the most significant technical advances in pulsed field measurement techniques has been the application of focused ion beams (FIB) for shaping single crystals (Moll et al., 2010). Electrical resistivity measurements have been limited in the past by eddy current heating, while low signal-to-noise ratios made measure- ments on good conductors problematic. Both of these problems can be largely

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S c i e n c e D r i v e r s — C o n d e n s e d M at t e r and M at e r i a l s P h ys i c s 27 FIGURE 2.2  (Below) Schematic phase diagram near a QCP. QCPs distort the fabric of the phase dia- grams creating a V-shaped phase of quantum critical matter fanning out to finite temperatures from the quantum critical point. As matter is tuned to quantum criticality, ever-larger droplets of nascent order develop. On length-scales greater than these droplets, electrons propagate as waves. Inside the droplet, the intense fluctuations radically modify the motion of the electron and may lead to its break- ing up into its constituent spin and charge components. (Above) Physics inside the V-shaped region of the phase diagram probes the interior of the QCPs (D), whereas the physics in the normal metal (N) or antiferromagnet (A) reflects their exterior. If, as we suspect, quantum critical matter is universal, then no information about the microscopic nature of the material penetrates into the droplets. Making an analogy with a black hole, the passage from noncritical to critical quantum matter involves crossing a “material event horizon.” Experiments that tune a material from the normal metal past a QCP force electrons through the “horizon” in the phase diagram into the interior of the quantum critical matter, from which they ultimately reemerge through a second horizon on the other side into a new universe of magnetically ordered matter. SOURCE: Reprinted by permission from Macmillan Publishers Ltd.: Nature. P. Coleman and A.J. .Schofield, 2005, Quantum criticality, Nature 433: 226-229. Copyright 2005.

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28 High Magnetic Field Science and I t s A pp l i c a t i o n in the US overcome using FIB to shape single crystal samples to have small cross sections, and by tailoring the sample dimensions and current to target the needed voltage levels for a successful measurement (Figure 2.3). FIB processing also allows the deposition of high-conductivity contacts, avoiding surface oxidation issues. It is possible that FIB processing of samples will also make high-pressure resistance measurements less challenging, possibly opening the door to the more routine use of high pressures and high dc fields for the exploration of QC matter. A range of new experimental applications can be contemplated, including quantum oscilla- tion measurements (Sebastian et al., 2012), the implementation of devices where the sample charge and spin transport can be modified independently of the mag- netic field, and even the simultaneous measurement of electrical transport with other quantities such as thermal conductivity or specific heat. One of the main constraints on pulsed magnets has been the need to provide slow rise times to avoid eddy current heating and similar experimental considerations. FIB has the potential to provide better control over these parameters, making experiments in magnets with faster rise times and thus higher peak fields feasible. It is generally believed that the next generation of higher field pulsed magnets for user science would require new innovations in the development of high strength/high conduc- tivity materials for the magnet conductors. Implementation of FIB processing may mean that much higher pulsed fields could be available for user science soon, with the promise of expansion to even higher fields as improved magnet conductors become available. FIGURE 2.3  Four-probe resistance bars for simultaneous c-axis and ab-plane resistivity measure- ments carved out of a SmFeAsO0.7F0.25 single crystal using the FIB. (Top) A crystal is positioned on a substrate with the c axis pointing perpendicular to the plane. The dashed volume indicates the original crystal that is removed during FIB cutting, leaving only the lamellar standing. (Bottom) The lamella is transferred to another substrate and flipped, so that its c axis is now aligned in the plane (short edge). Most of this lamella (dashed line) is again removed, leaving only the small current path standing (violet). Eight platinum leads are deposited onto the crystal edges (all other colors) that are connected to the resistance bars by narrow (~800 nm for c axis) crystal bridges. The common current is injected through the yellow contacts and traverses two c-axis resistance bars (blue and red voltage contacts) and one along the ab plane (green and violet contacts). Dimensions of resistance bars: length ~35 µm (ab plane), ~5 µm (c axis), cross section ~1.5 µm2. SOURCE: Reprinted by permission from Macmillan Publishers Ltd.: Nature Materials. P.J.W. Moll, R. Puzniak, F. Balakirev, K. Rogacki, J. Karpinski, N.D. Zhigadlo, and B. Batlogg, 2010, High magnetic-field scales and critical currents in SmFeAs(O,F) crystals, Nature Materials 9:628-633. Copyright 2010.

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S c i e n c e D r i v e r s — C o n d e n s e d M at t e r and M at e r i a l s P h ys i c s 29

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30 High Magnetic Field Science and I t s A pp l i c a t i o n in the US High Magnetic Field Studies of Low-Dimensional, Frustrated, and Quantum Magnets In conventional magnetic materials, such as ferromagnets and antiferromag- nets, a quantity of immediate interest is the temperature at which there is an onset of long-range order among the atomic moments, or spins. Above this “critical” temperature, Tc, the spins are disordered in space and fluctuate randomly in time as a result of thermal fluctuations. Below Tc, the spins freeze into a static pattern with long-range order, wherein the identification of the orientation of a spin any- where in the sample defines the orientation of all others. The simplest theory for predicting Tc is the so-called “mean field theory.” In mean field theory, the mag- nitude of Tc is proportional to the interaction energy between neighboring spins and to the number of neighbors. An applied magnetic field will also change Tc in different ways depending on the type of long-range order and depending on the magnitude of the applied field. In order to assess whether a field is high enough to affect a change in Tc, one converts field strength to temperature by considering the energy difference between a spin that is aligned and one that is antialigned with the field. This energy is given by gμBH, where g ~ 2, μB is the Bohr magneton (atomic unit of magnetism), and H the applied field. For H equal to 1 tesla, this energy is equivalent to a thermal energy of 1.3 K. Thus, in order to substantially affect a long-range ordered state, a field with strength equivalent to Tc must be applied. For most single investigators using helium-cooled superconducting solenoids, a field strength of 10 tesla is readily obtainable, thus allowing the modification of materials with Tc up to only ~13 K. In this section the committee considers classes of magnets where Tc is suppressed compared to predictions based on mean field theory. In some cases, this is accompanied by a smaller field scale, required to alter the ordered state. The effects of low-dimensionality, frustration, and quantum zero point motion all conspire to suppress the mean field Tc in magnets by promoting fluctuations that destabilize the order. Such factors not only reduce Tc but can also give rise to new, previously unexpected states of matter as well as exotic excitations. Low dimensionality, frustration, and quantum zero point motion are all fixed by the material’s composition and crystal structure so that their effects in suppressing long-range order can only be inferred from intermaterials comparisons. Applied magnetic fields offer a way to tune these effects, either by stabilizing them (as for ferromagnetic fluctuations) or destabilizing them (as for antiferromagnetic fluctuations). Thus, magnetic fields are essential tools for studying new phases of magnetic matter, and the committee discusses the role of high magnetic fields in understanding exotic states.

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56 High Magnetic Field Science and I t s A pp l i c a t i o n in the US Z numbers.) These topological phases are strictly defined only in systems which preserve time-reversal symmetry—that is, systems in the absence of an external magnetic field, which do not have frozen local magnetic moments. Nevertheless, application of strong magnetic fields can be a strong tool for manipulating or probing these systems. Originally it was thought that topological insulators could exist only in two- dimensional systems or in systems that were essentially a stack of weakly connected layers. However, we now know that there are different kinds of topological insula- tor that can exist in genuinely 3D systems. In fact, topological insulators can exist, mathematically, in any dimension, with appropriate classes of symmetry restric- tions, and a complete classification of these phases has been developed, at least for models of weakly interacting electrons in a periodic potential. Concrete examples of Z2 topological insulators have been realized in both 2D and 3D. All these materials share a common trait—that is, a strong spin-orbit coupling, which causes band inversion and flips the parity of the valence band. Coincidentally, the nature of the edge/surface state of a topological insulator is dictated by the spin-orbit coupling, so that for a given momentum direction, only one direction is allowed for the electron spin. This type of spin polarization is often called “helical,” although the spin direction is typically perpendicular to the direction of motion, in contrast to the case of a helical massless particle in high- energy physics, such as a neutrino. An interesting consequence of the helical spin polarization is the existence of dissipationless spin current at the edge/surface of a topological insulator. This property could have important implications for applica- tions in spintronics. Also interesting is the fact that the edge/surface state consists of massless Dirac fermions, whose existence is guaranteed by the time-reversal symmetry. This aspect gives the two-dimensional surface states of a 3D topological insulator a close relation to graphene, but in topological insulators there is neither valley nor spin degeneracy, reducing the number degrees of freedom of the Dirac fermions to one quarter of those in graphene. Transport studies of helically spin-polarized surface Dirac fermions inhabit- ing the surface of 3D topological insulators are a promising research frontier. One would expect various novel physics, including dissipationless spin current and topological protection from backscattering, to show up in transport properties of topological surface states. However, transport studies of the surface states have proved a challenge because of the coexistence of bulk transport channels due to doping by defects in available topological insulator samples. Nevertheless, experi- ments on thin samples in high magnetic fields, carried out at NHFML, were able to distinguish the surface from the bulk contributions. Surface quantum oscilla- tions have been successfully observed, and the Dirac nature of the surface states has already been elucidated. Recent improvements in materials preparation have also contributed to a separation of bulk and surface contributions. More recently,

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S c i e n c e D r i v e r s — C o n d e n s e d M at t e r and M at e r i a l s P h ys i c s 57 the discovery of an intrinsically bulk-insulating material, Bi2Te2Se, gave a big boost to the surface transport studies. Furthermore, by improving the bulk-insulating properties of Bi2Te2Se in a Bi2-xSbxTe2-ySey solid-solution system, researchers have already succeeded in preparing bulk single crystals showing surface-dominated transport (see Figure 2.15). Also, MBE growth of strained HgTe thick films has successfully provided 3D topological insulator samples showing surface-dominated transport. Using those samples, we may anticipate that novel quantum transport phenomena, including the fractional quantum Hall effect in topological surface states, will become an exciting realm of high magnetic field science. The Dirac nature of the surface states of topological insulators has also been observed in STM and optical experiments involving high magnetic fields. In the next 10 years, those new types of high magnetic field experiments will become increasingly more important for the exploration of novel physics associated with nondegenerate Dirac fermions. Intriguingly, it has been shown that the quantum field theory of topological insulators resembles that of a hypothetical particle in high-energy physics, called the “axion,” and this theory leads to various interesting predictions such as the quantization of the magnetoelectric effect, the appearance of an image magnetic monopole, and the half-integer quantum Hall effect. All those phenomena are yet FIGURE 2.15  (Left) Basic crystal structure of a most promising topological-insulator material, the tetradymite Bi2-xSbxTe2-ySey system. This material exhibits a highly bulk-insulating character and a tunable surface Dirac fermions. (Center) Since the surface transport is significant in this material, the apparent resistivity at cryogenic temperatures decreases when the sample thickness is reduced; the thickness dependence shown here suggests that the transport is 70 percent due to surface in the 8-µm thick crystal. (Right) Shubnikov-de Haas oscillations in high magnetic fields up to 45 tesla observed in this material reveal the Dirac nature of the surface state. SOURCE: (Left and center) reprinted figures with permission from A.A. Taskin, Z. Ren, S. Sasaki, K. Segawa, and Y. Ando, 2011, Observation of Dirac holes and electrons in a topological insulator, Physical Review Letters 107:016801, Figure 1(a) and Figure 4. Copyright 2011 by the American Physical Society. (Right) Courtesy of N. Phaun, Prince- ton University, adapted from J. Xiong, Y. Luo, Y. Khoo, S. Jia, R.J. Cava, and N.P. Ong, 2012, High-field Shubnikov–de Haas oscillations in the topological insulator Bi 2Te2Se, Physical Review B 86:045314.

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58 High Magnetic Field Science and I t s A pp l i c a t i o n in the US to be experimentally discovered, and an important prerequisite to the realization of those phenomena is to open a gap in the surface state. It has been proposed that high magnetic fields will be useful for this purpose, and the higher the field, the larger the gap, the easier the observation. This provides excellent opportunities for high magnetic field science to play a major role in the discovery of fundamentally new physics. In a topological superconductor, the energy gap that protects the topological phase is actually the superconducting gap. The low-energy excitations at the bound- ary of a topological superconductor are exotic quasi-particles called Majorana fermions, which are their own antiparticles, and which should have a number of unusual properties. Notably, it has been proposed that, in appropriate situations, where Majorana excitations can be localized and kept well separated from each other, they could be useful for fault-tolerant quantum computing. A promising way to materialize a topological superconductor is to induce electron pairing in the edge/surface state of a topological insulator via proximity effect of a conventional superconductor. One might also use a strongly spin-orbit- coupled semiconductor like InSb or InAs instead of topological insulators, but in this case a magnetic field is required to quench the spin degrees of freedom without suppressing the superconducting state. Recently there have been encour- aging developments in realizing analogous topological superconductivity in 1D hybrid structures, where a nanowire of InSb or InAs is proximity coupled to a bulk superconductor. Topological superconductivity may also be found in natural bulk supercon- ductors when the superconducting order parameter is parity odd. For example, Sr2RuO4 is likely to have a time-reversal breaking, quasi-2D topological supercon- ducting state, which would host Majorana fermions in the half-quantized vortices in magnetic fields. Also, CuxBi2Se3 was recently shown to be a time-reversal- invariant 3D topological superconductor, which hosts intriguing helical Majorana fermions on the surface. It is expected that more topological superconductors will be discovered in natural compounds or in artificially constructed hybrid systems. Considering the important roles that high magnetic fields have played in supercon- ductivity research, we may expect that topological superconductors will naturally provide new exciting opportunities for high magnetic field science. High Magnetic Fields in Soft Matter Research As was mentioned in the overview to this chapter, applications of high mag- netic fields to soft condensed matter research take advantage of the torques and forces on materials that can be exerted by a magnetic field. These may be used to align molecules or other small objects while they are being studied by one or

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S c i e n c e D r i v e r s — C o n d e n s e d M at t e r and M at e r i a l s P h ys i c s 59 another experimental probe, or they can be used to control materials preparation, such as crystal growth, in a desired way. Magnetic Alignment The main interaction with magnetic fields is caused by diamagnetism, the appearance of a weak magnetic moment in a material in opposition to an external magnetic field. This phenomenon is present in all materials and arises from the deformation of the electron orbits in atoms and molecules by the Lorentz force. To magnetically manipulate nanostructures, one takes advantage of the anisotropy of a material’s diamagnetic response, which grows rapidly with size. For example, the increase in the magnetic energy by an object with N atoms or molecules is D E = N c B2 / 2m0 where μ0 is the free space permeability and c the susceptibility tensor per molecule. This energy may vary considerably depending on the relative orientation between the molecule and the magnetic field. Typically, for a single molecule, |χ| is minis- cule (<10−7) for all orientations, hence even at B = 100 T, the difference in energy between different orientations is negligible. However, for a benzene aggregate with 105 molecules (but which is still only ~ 5 nm in size), it becomes energetically favorable at room temperature to align to the direction of the magnetic field at B = 20 T. Similarly, nanoscale chemical aggregates or biological cells can be oriented using high magnetic fields and studied in situ or ex situ. Since the orientational torque is quadratic in B, high magnetic fields are required to study nanometer-sized aggregates (Maret and Dransfield, 1985). In supramolecular chemistry and liquid crystals, relatively weak van der Waals forces or π-π bonding are involved in the process of self-assembly of targeted molecular building blocks into larger structures, like vesicles, dendrites, fibers, wires, etc. Magnetic fields may be used to probe these intermolecular interac- tions by determining the internal structure that is important for their properties. For instance, dye molecules can aggregate in solution in different geometries like stacked or herringbone structures, and their optical response strongly depends on the stacking. One type of aggregate shifts optical absorption or emission to the red while the other shifts it to the blue. Therefore it is possible to tune the absorption in a desired way. The stacking occurs usually in a liquid environment, and it is important to determine the structure under these conditions, which makes stand- ard microscopy or light-scattering techniques hard to use. In a magnetic field the aggregates orient in a specific way with respect to the field while at the same time, optically, the light polarization with respect to this direction can be varied. In this way, the optical absorption along different axes can be determined in an in situ

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60 High Magnetic Field Science and I t s A pp l i c a t i o n in the US experiment. Molecular materials are considered as a possible element in photo- voltaic devices, and it is important to match their optical properties to the optical spectrum of the sun, which makes a detailed knowledge of their structure necessary. Another example is the study of optical properties of nanotubes in solution. A magnetic field readily aligns these tubes along the field axis and allows the study of their properties. In this way it has been possible to observe B-periodic Aharonov Bohm oscillations in the optical absorption of large-diameter carbon nanotubes as a function of magnetic field whenever an integer number of flux quanta fit in the cross section of the tube (Zaric et al., 2004). It is also possible to use the magnetic forces to deform aggregate shapes like spheres or dendrites, and the deformation can be determined through optical experiments. In this way molecular forces on a nanometer scale can be quantita- tively determined in a noninvasive manner. It is important to be aware of these diamagnetic forces. For example, molecular alignment in high fields produces observable effects in NMR spectra, which in principle can be used as constraints on the structures of macromolecules such as proteins. It is of particular interest that the magnetic field experienced by a nano-object is practically identical to the applied field. Although it is easily possible to deform many objects with an electric field, the large values of typical dielectric constants mean that the local electric field may be very different from the externally applied field, which renders any quantitative analysis very difficult in that case. Apart from aligning aggregates in solution with magnetic fields and studying them in situ, one may also use the fields to order matter in solution and then fix it in some way and study the ordered material ex situ afterwards. In general all col- lective states of soft matter like liquid crystals are very easily affected by magnetic orientation since domains in liquid crystalline material contain many molecules. Cooling liquid crystals in a field through the nematic phase transition leads to perfectly oriented materials since the first nuclei at the transition temperature can easily rotate in the magnetic field and aggregate after alignment. The cooled liquid crystalline film is fully transparent, free from domain boundaries, and shows a very high degree of birefringence. Strongly polarizing optically transparent films are useful optical components. Another example is that superior conducting polymers can be made from cylindrical stacks of coronene molecules that aggregate in solu- tion and are then deposited on a substrate by evaporating the solution in a field. The resulting perfectly aligned stacks show a conductivity two orders of magnitude higher than the nonaligned stack, not aligned in a field. Such an increased mobility is very important for organic transistors (Shklyarevsky et al., 2005).

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S c i e n c e D r i v e r s — C o n d e n s e d M at t e r and M at e r i a l s P h ys i c s 61 Magnetic Levitation A very different application of diamagnetism is magnetic levitation. Since diamagnetic materials gain energy in a magnetic field, they experience a force F (per unit volume) toward the lower field region, given by F = (c/2m0) grad B2 which is proportional to the gradient of the square of the field strength. In a magnet with a vertical bore, this force acts against gravity and enables complete levita- tion of most diamagnetic organic materials for B ~ 16-20 T and gradients ~ 100 T/m (Beaugnon and Tournier, 1991). Such levitation, apart from being a striking demonstration of diamagnetism of living organisms, also allows study of materi- als under microgravity or artificially variable gravity, such as crystal growth under weightless conditions or fluid dynamical problems as a function of gravitational acceleration (Berry and Geim, 1997). An example of fluid dynamics is the study of cryogenic liquids like the ones used as rocket fuel. Understanding the behavior of these liquids in weightless conditions is important in rocket design, and such studies can easily be done with magnetic levitation. Finally, many experiments have been done in microgravity in an attempt to improve the crystal quality of, in particular, protein crystals. High-quality protein crystals are essential for protein structure determination, which is an essential tool for the pharmaceutical application, for example. The idea is that in weight- less conditions the convection plume occurring around the growing crystals is suppressed. This plume arises because the solution near the growing crystal has a different density since it is depleted because of the molecules that are deposited on the surface. The reduction of the growth velocity is known to lead to higher crystal quality. It is possible to suppress the growth plume also in high magnetic fields under the condition that the grad B2 value suppresses the buoyancy of the depleted solution. This condition often requires much higher fields than the usual levitation condition (see Figure 2.16). In summary, there are many as yet unexplored areas in research on soft matter in high magnetic fields since it is not widely recognized that at high magnetic fields diamagnetic energies may become important. Since these forces increase with B2, higher fields will rapidly make them even more important and opportunities for unexpected results will be even greater. This area of application is interdisciplinary par excellence since it requires physicists, chemists, and possibly even biologists to work together.

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62 High Magnetic Field Science and I t s A pp l i c a t i o n in the US FIGURE 2.16  An image of the growth plume of growing lysozyme protein crystals at effective gravity G = 1 (normal gravity) and in a very high magnetic field gradient, showing that with field-induced effectively zero gravity (G = 0) convection can be suppressed and that gravity can even be inverted (G = 0.08). SOURCE: Reprinted with permission from M.C.R. Heijna, P.W.G. Poodt, K. Tsukamoto, W.J. de Grip, P.C.M. Christianen, J.C. Maan, J.L.A. Hendrix, W.J.P. Van Enckevort, and E. Vlieg, 2007, Magnetically controlled gravity for protein crystal growth, Applied Physics Letters 90:264105. Copy- right 2007, American Institute of Physics. Concluding Comments High magnetic fields are a critical research tool in many areas of condensed matter and materials physics. Of the many examples of research in high magnetic fields cited in this chapter, a large fraction were carried out at facilities run by NHFML. The experiments were carried out by users from a large variety of uni- versities and research institutions based in the United States and in other coun- tries around the world. Those users may have brought with them samples, and in some cases measuring devices, prepared at their home institutions. In addition to magnet time, NHFML would have supplied technical support, as well as measure- ment instruments and cryogenic facilities for much of this work. The availability of higher magnetic fields, in both dc and pulsed modes, will be very important for continued progress in this area. (Specific magnet recommendations relevant to this research will be discussed in Chapter 7.) The use of high magnetic fields in soft matter research (biological cells, molecular aggregates, vesicles, polymers, some of them discussed in Chapter 3) has great promise but is largely underexploited. A close collaboration between the large facility and strongly interested chemists and biologists is necessary to fulfil these promises.

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