Foreward for special issue of philosophical magazine on: topological correlated insulators and SmB₆

Abstract

This Foreward briefly surveys the recent burst of research on the mixed valent insulator SmB₆ in the context of two questions, (1) is SmB₆ in fact the first example of a strongly correlated topological insulator, and (2) if so are there unique features of special interest for both topology and strong correlations? Accordingly, the papers of this special issue are situated within this general rubric.

Keywords:

• Strongly correlated electrons
• mixed valence
• Kondo effect
• insulators
• topological insulator
• samarium hexaboride

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Corrigendum for Foreword for special issue of philosophical magazine on: topological correlated insulators and SmB6

Introduction

Cubic SmB₆ is a strongly correlated material that helped to start the field of homogeneous mixed valence [1–4] when it was found that the Sm ions are present as both Sm²⁺ (4f⁶) and Sm³⁺ (4f⁵), with all the Sm sites being crystallographically equivalent, in spite of a large f-shell Coulomb repulsion that acts to stabilise integer valences in most rare earth solids. There is a large literature of a steady flow of research devoted to this aspect of SmB₆, dating from more than 50 years ago. The material is of high current interest for the testing of an elegant new narrative that has evolved since 2010. This narrative goes as follows.

Early experimental and theoretical work in the late 1970s [5,6] established SmB₆ as the first and perhaps still the foremost example of what came to be known as Kondo insulators [7–9]. The insulating bulk gap arises [6,10] from hybridisation between strongly correlated Sm 4f states and a weakly correlated nominally Sm 5d band (but having substantial B 2p admixture). But the experimental case for its existence as a bulk insulator has always suffered from a major flaw that has until now defied all attempts at explanation. The flaw is that the expected and observed exponential rise of the resistivity and the Hall effect magnitude as the temperature T decreases below the value of its insulating gap is interrupted at around 4 K by a plateau which implies metallic in-gap states but is very difficult to rationalise by invoking an impurity band because its magnitude exceeds fundamental limits for three-dimensional low temperature conduction [5,11].

The proposed resolution [12,13] is that SmB₆ is also a topological insulator (TI), i.e. it has an insulating bulk resulting from an electronic structure having a certain topology that then requires conducting surface states in which the spin direction is locked in a certain way to the momentum. The low T resistivity plateau of SmB₆ would then arise from conduction in these topologically required (protected) surface states, which would evade the tension between its magnitude and three-dimensional transport limits. The elegant TI concept and theory [14–21] was initially set forth for weakly correlated systems, first in two dimensions and then in three dimensions and was experimentally demonstrated for such systems [22–28] very quickly thereafter. Excellent reviews of this exciting period are available, including those referenced here [29–31].

For SmB₆ many questions immediately arise, but two can be singled out. (1) Foremost, is the new narrative correct, i.e. is SmB₆ in fact the first example of a strongly correlated TI? (2) If so, are there unique features that would be generally interesting for either or both of the communities studying topology and strong correlations? The remainder of this Foreward brings out some of the most pressing issues raised by each question and situates the papers of this issue [32–37] within this general rubric. But it is only a minimal introduction to what is already a complex subject with a very large literature. A more detailed overview can be obtained from the online videos of talks, discussions and pedagogical overviews at a recent international conference held at the University of Michigan [38] and in the extensive reference lists of the papers of this issue.

Correctness of TI narrative for SmB₆?

For the first question above, there is certainly enough supportive experimental evidence to justify taking the new narrative seriously. An explosion of research worldwide was triggered in late 2012 by three preprints, two reporting clever transport experiments conclusively showing that the resistance plateau does indeed arise from surface conduction [39,40] and a third [41] reporting tunnelling spectra and arguing for the observation of surface states in the bulk insulating gap. The impact of the three preprints was amplified by an article in Nature News [42]. Surface states crossing the Fermi energy have indeed been observed by angle-resolved photoemission spectroscopy (ARPES) [43–46] for the [1 0 0] surface and two-dimensional quantum oscillations using the de Haas van Alphen (dHvA) effect [47] were found for the [1 0 0] and [1 1 0] surfaces. The dHvA data included so-called Landau level index plots that extrapolate in the infinite magnetic field limit to a non-integer value, indicative of a Berry phase expected for a two-dimensional Dirac electron system, as is expected for a TI surface state.

Can it be proved that the observed surface states are in fact topologically protected? An almost immediate test [48] using transport was to compare the effects of magnetic and nonmagnetic impurity doping on the resistance plateau, because the existence of the TI state depends on time-reversal symmetry which presumably would be broken by the former but not the latter. This experiment nicely supported the TI narrative. But there are also theory requirements that can be directly tested by ARPES. To be a strong topological insulator (a) an odd number of surface states for the k-space surface unit cell is required, (b) these states should display spin-momentum locking, and (c) these states should be robustly present (topologically protected). Papers in this issue [32,33] and in the literature [49,50] show nicely how the theory works, including the role of time reversal symmetry. For the single-particle bulk gap of SmB₆ the literature shows two general theoretical approaches, one [6,49,51] that constructs the low-lying 4f fermion excitations based on the lowest energy many-body ionic states of Sm²⁺ 4f⁶ (⁷F₀) and of Sm³⁺ 4f⁵ (⁶H_5/2), and another [50,52–54] that proceeds from density functional band theory with further many-body corrections included in some way, e.g. Gutzwiller correlations or dynamic mean field theory (DMFT). Even though the underlying quantum states and wavefunctions are greatly different, nonetheless the resulting six lowest lying bands of fermion excitations from both approaches have the same symmetry properties (apparently coincidentally) and hence lead to the same TI predictions for a gap produced by hybridisation of the 4f fermion exitations to the nominally Sm 5d conduction band. For the surface states, requirement (a) above is accomplished on the [1 0 0] surface by having one state at the surface Brillouin zone point and having states at each of the two points.

Consensus as to the TI origin of the surface states seen in ARPES has not been reached. As predicted, ARPES does indeed find surface states on the [1 0 0] surface at T and , and nowhere else, and those at are quite robust [55], even against extreme oxidation. However, the crucial state (crucial because it renders the total number odd) shows great variability among data-sets and measurement surfaces [55], and some ARPES data [56] have indicated the presence of a trivial Rashba surface state. If such were definitely the only surface state it would immediately rule out the TI narrative. But the variety of surface states found at currently prevents such a strong conclusion, and TI theory does not preclude the additional presence of trivial surface states, a circumstance known to occur for weakly correlated TI materials. Ref. [56] also argues that the metallic surface states observed at actually disperse from a binding energy far below that of the bulk gap and therefore are not of topological origin. Finally, spin-momentum locking (spin texture) for the surface states has been reported from spin-resolved ARPES [57]. This result would be nearly conclusive for the TI scenario, but unfortunately, uncertainty arises because very similar spin-resolved data were also reported [58] for the material YbB₆, which has subsequently been shown [59] to be a non-TI single valent insulator.

The polar nature of the [1 0 0] cleavage plane of SmB₆, which in the bulk has both positive Sm ions and negative B ions, is another source of great uncertainty. Studies of cleaved [1 0 0] surfaces using scanning tunnelling microscopy (STM) [60–62] found a variety of surface morphologies and/or reconstructions, depending on the details of the surface termination. Ref. [34] of this issue provides an excellent overview. The STM results are potentially a problem for the interpretation of ARPES spectra from cleaved [1 0 0] surfaces. Although the spatial resolution of ARPES has been good enough to detect large positional variations in the spectra of cleaved surfaces [55], the resolution is far from being adequate to discriminate on the much smaller length scales of the typical variations found in the STM studies. Ref. [35] of this issue describes detailed work to prepare and characterise sufficiently uniform and unreconstructed [1 0 0] surfaces that they can be reliably studied using photoemission. This paper is also notable for the application of a novel and promising ARPES instrument. Not surprisingly, the tunnelling spectra obtained in STM studies vary with the surface or with surface terminations. Nonetheless these spectra have been interpreted [34,60–62] as providing evidence of the bulk gap and of both surface and bulk states within the gap. Finally, the TI narrative has also been challenged with arguments that relaxation of the [1 0 0] charged surface leads to trivial (non-TI) metallic surface states that are different from those reported in the other ARPES studies and give rise to the surface metallicity [63], or that the relaxation can shift the bands at the surface such that the states crossing the Fermi energy in ARPES spectra actually arise from the bulk nominally Sm 5d states away from the gap [64]. These last two proposals are inconsistent with the preponderance of current ARPES data.

Other issues with ARPES and dHvA are the following. It has not yet been proved that any of the observed surface states are in fact the ones responsible for the plateau conduction. So far there is no paper in the literature making a quantitative connection between the surface conductivity and surface state masses, lifetimes and Fermi surface sizes observed in ARPES or dHvA. For the [1 1 0] plane, magnetothermoelectric measurements [65] indicate large mass surface states while the states observed in dHvA are much lighter. In this connection, recent theory [66] proposes a mechanism (mentioned further below) for the presence of both light and heavy mass surface states on the same surface. A related problem is that the surface states reported from dHvA have much smaller Fermi surface sizes than the prominent and robust states found in ARPES. The correspondence between dHvA pockets and the smaller pockets seen by ARPES is definitely better but still not as close as desirable. There is, however, general agreement between ARPES and dHvA on rather light surface state masses for the [1 0 0] plane. Of course, achieving the desired consistency of transport, dHvA and ARPES is also complicated by the fact that transport and dHvA are performed with oxidised sample surfaces and ARPES with vacuum-cleaved samples, with only the ARPES state being robust against oxidation.

There remain other important issues with transport. There is variability in the results of transport measurements for the plateau conductivity [38] and the realisation of a need to make such measurements in a way that does not inadvertently measure multiple surfaces. The consequent use [67,68] of the Corbino geometry has corrected previous inferences from the Hall effect of carrier densities that are unphysically large for two-dimensional transport. Two different bulk sample preparation methods are common, flux growth and floating zone [69], and there is some sample dependence of measurement results, for example some studies finding that a true plateau resistivity in the latter occurs only below 0.5 K [70,71]. For the flux-grown samples, it is important to be sure that measured results are not affected by Al inclusions [47]. So far there is no documented sample dependence for ARPES, but it is also true that systematic studies have not been made. The topic of sample preparation should also include a mention of work to prepare and characterise thin films of SmB₆ [72]. A final transport issue concerns the position of the chemical potential in the bulk gap. ARPES and tunnelling spectroscopy show the gap magnitude to be roughly 20 meV, whereas the much smaller bulk resistivity activation energy of 3–5 meV, documented in great detail in Ref. [36] of this issue, suggests that the chemical potential lies much closer to a band edge than would be possible to understand for a bulk gap with no bulk impurity states in the gap, based on usual solid-state theory. T-dependent ARPES, which can also directly observe the conduction band (see below) shows that the difference cannot be explained as the result of direct and indirect gaps. No clear understanding of what determines the chemical potential position has yet emerged.

Special aspects of TI narrative for SmB₆

Various aspects of the TI narrative for SmB₆ make it novel from both the topological and the strongly correlated viewpoints. To begin, another part of the new narrative is that SmB₆ is also unusual and potentially important because its bulk is a true insulator. The experiments for weakly correlated three-dimensional TI materials have often required great ingenuity and/or sophisticated theoretical arguments to overcome the effects of bulk impurity conduction rivaling that of the surface. In contrast, SmB₆ offers the possibility of studying the surface conduction rather easily at temperatures that are easily accessible, and indeed a great variety of studies can be cited. Some studies have been made from the viewpoint of simply trying to ascertain the origin of the plateau and some from the viewpoint that likely the plateau does indeed have a topological origin. A pre-TI period study [73] of the pressure dependence of the resistivity and Hall effect led to the prescient conclusion that the in-gap states giving rise to the plateau do not arise from impurities but are intrinsic (albeit thought at the time to be of bulk origin but acknowledging the consequent tension with fundamental three dimensional transport limits). In the recent post-TI period, there are studies of the effects on the resistivity plateau of bulk doping, as reported in great detail in Ref. [36] of this issue, and by other workers [74], studies of the effects of surface preparation [75], including subsurface cracks that can create hidden surface area [76], detailed magnetotransport studies [67,77] and a gating study [68]. The conclusions of these studies do not all neatly agree, but the studies indicate a great richness that is nicely revealed by the ease with which the surface conduction can be studied.

Other unique aspects of SmB₆ revolve around the bulk gap. In spite of the apparent simplicity of the hybridisation gap theory, the gap nonetheless has an intrinsically many-body nature, in contrast to the case of the weakly correlated TI materials. This many-body character is made clear experimentally by the fact that the gap closes with increasing temperature, as directly observed by two T-dependent ARPES studies [78,79]. The details of the spectra inspired some discussion in these two papers of the relative roles of spin fluctuations, i.e. the Kondo aspects, and of charge fluctuation, i.e. the mixed valence aspects, for the bulk gap. Because the bulk valence mixing is far from integer, the implied Kondo temperature is likely so large that there is no clear separation of the energy scales of spin and charge fluctuations, as would normally be understood for Kondo physics. This distinction may not be very important at the level of making an effective Fermi liquid model for treating the topological effects [12,13], but it has been argued that the distinction could be much more profound [80]. In any case, it is very interesting and currently unique to SmB₆ to observe [78] the evolution in the surface states with increasing temperature as the hybridisation breaks down and the bulk conduction band penetrates the gap region to disperse all the way to the Fermi level. DMFT for models [66,81] and also combined with band theory [78] shows some of the observed T-dependencies of the surface states and/or the bulk gap.

Another manifestation of the many-body nature of SmB₆ is that the bulk gap harbours a collective excitation that is observed in inelastic neutron scattering [82] and STM tunnelling spectra [61]. This excitation is understood to be a spin exciton and detailed theoretical analysis [83] shows that the k-dependence of the excitation is compatible with a single-particle band structure of the form needed for the TI scenario. This excitation may be of great importance for the transport of electrons in the surface states, by providing a scattering mechanism that would not be found in a weakly correlated system and that is proposed to interfere with the topological protection [84]. Planar tunnelling spectroscopy [85] has been interpreted as providing evidence for this scattering. Ref. [33] of this issue points out that such exciton states may also have a role in the nuclear magnetic resonance relaxation rate observed at low temperatures.

A final phenomenon which mixes the strongly correlated and topological aspects of SmB₆ is that the bulk and surface valences of rare earth mixed valent and heavy Fermion materials are typically different. This is important because, although the existence of TI surface states depends on the bulk gap, the details of the actual surface states depend on the surface itself. The surface valence change has two origins, (a) that the hybridisation on the surface tends to be reduced because the number of near neighbours of a surface rare earth atom is reduced relative to the bulk, and (b) that the 4f state binding and affinity energies, relative to the Fermi energy, are often different for the bulk and the surface. These effects are well known and documented in the literature for rare earth materials as cited below (and also for strongly correlated transition metal compounds like V₂O₃ [86]). Theoretical thinking [66,87–89] for SmB₆ to date has focused on the first of these. From a Kondo perspective, reduced hybridisation reduces the surface Kondo temperature relative to that of the bulk, and it has been proposed [87,89] that this leads to a surface breakdown of the Kondo effect and a consequent change in the surface states to have smaller masses than would otherwise be expected, as found in dHvA and ARPES. Another treatment [88] emphasising reduced surface coordination also finds the surface state Dirac point pushed into the valence band and a consequent decrease in the surface state mass. Most recently, a model DMFT calculation [66] has substantiated the Kondo breakdown idea for a Sm-terminated surface, has provided a mechanism for both light and heavy surface states on the same surface, and has treated T-dependent effects for the bulk gap and surface states.

A decrease in the Kondo temperature on the surface implies also a shift of the surface Sm valence toward Sm³⁺ (4f⁵), and X-ray absorption spectroscopy (XAS) [74] studies have reported such a shift. Ref. [37] of this issue reports an extensive and confirming study of this surface valence shift using Sm core level photoemission spectroscopy on cleaved [1 0 0] surfaces, finding such a shift for both the surface and a subsurface region, and a dependence on the time after cleave which is discussed in terms of the relaxation of the charged cleaved surface, an issue mentioned already above.

As pointed out in Ref. [37], the surface valence shift toward 4f⁵ is in fact rather surprising relative to past work. In mechanism (b) above, the magnitudes of the surface 4f binding and affinity energies relative to the Fermi level are usually increased and decreased, respectively, typically by several tenths of an eV, which favours an increased surface 4f occupation relative to that of the bulk, i.e. opposite to the reported XAS result for SmB₆. The usual direction of the energy shift is understood in a very general way [90] as having the same origin as for mechanism (a) above, that the reduced surface coordination also causes a reduced cohesive energy for the surface. In this usual understanding, if increased occupation favours the magnetic state, e.g. increased 4f¹ on the surface of Ce materials at the expense of 4f⁰, both mechanism (a) and (b) can work together for a reduced surface Kondo temperature. But for the opposite occupation change, e.g. increased 4f¹⁴ on the surface of Yb materials at the expense of 4f¹³, the two mechanisms compete for increasing or decreasing the surface Kondo temperature. The usual direction of energy shift (favouring increased 4f occupation) has easily been seen in photoemission spectroscopy of core levels generally and of 4f states in Ce [91], Yb [92] and Tm [93] materials, and also in early studies of Sm metal [94,95] where the bulk valence is 4f⁵ and the surface has both 4f⁵ and 4f⁶ Sm atoms, i.e. a surface valence shift toward 4f⁶. Similarly, an early photoemission study of SmB₆ [95] seemed to show a surface shift of its 4f⁶ → 4f⁵ transition away from the Fermi energy toward deeper binding energy, but with the surface valence the same as the bulk, interpreted as a sign of surface inhomogeneous mixed valence. But the use in that study of surfaces produced by argon ion sputtering of a polycrystalline sample certainly introduces great uncertainty relative to the current cleaved single-crystal studies. Nonetheless, a recent photoemission study [55] of single crystal SmB₆ also reports 4f⁶ → 4f⁵ transitions surface shifted to deeper binding energy for certain regions of a cleaved [1 0 0] surface. Countering these results, there is one other Sm case of the opposite surface valence shift, i.e. toward 4f⁵, reported for heavy Fermion SmOs₄Sb₁₂ [96]. Thus, it seems that there may be more to understand about the origins and directions of surface valence shifts for SmB₆ (and apparently for Sm materials in general) and their implications for the surface states.

One final topic nominally lies outside the scope of this special issue, but may turn out to be relevant and in any case is very exciting and deserving of mention here. There is a report [97] of dHvA quantum oscillations that are interpreted as showing a bulk three-dimensional Fermi surface, in spite of an acknowledgement that the bulk is an electrical insulator. The implication is that the bulk of SmB₆ harbours a very novel low temperature many-body state leading to a charge neutral bulk Fermi surface. This report has of course aroused great interest and also raises a question as to how it relates to the earlier dHvA study [47] that reported two-dimensional surface states. Each of two recently published papers [98,99] offer both theoretical and experimental perspectives on this new and very different possibility for exciting physics in SmB₆.

In summary, the current situation for the TI narrative in SmB₆ can be described as a question of whether the many detailed uncertainties signal that the narrative is fundamentally flawed, or whether the understanding of the origin of the bulk gap is sound enough that the generality of the topological theory can protect the narrative against these details such that they will ultimately be resolved and fitted into the TI scenario, possibly in ways that expand the understanding of both the topology and the strong correlations. In any case, the new burst of work on SmB₆ promises to lead in one way or another to a much better understanding of this fascinating paradigm material.

Disclosure statement

No potential conflict of interest was reported by the author.