β‑Rhombohedral Boron: At the Crossroads of the Chemistry of Boronand the Physics of FrustrationTadashi Ogitsu,*,† Eric Schwegler,† and Giulia Galli‡

†Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, California 94550, United States‡University of California, Davis, California 95616, United States

*S Supporting Information

CONTENTS

1. Introduction A2. β-Rhombohedral Boron: From Its Identification to

the Establishment of Partial Occupancy C3. The Crystal Structure of β-Rhombohedral Boron E4. Thermodynamic Stability of β-Rhombohedral

Boron G5. Geometrical Frustration in β-Rhombohedral

Boron J5.1. Specific Heat and Entropy of β-Boron L

6. Electronic Structure of β-Rhombohedral Boron M6.1. The Electronic Requirement of β-Boron

without POS Rationalized with B12 and B28Clusters M

6.2. DFT Simulations Begin to Explain the Natureof POS O

6.3. Atomic Density and Favored POS Config-urations: What Are the Implications? Q

7. Anomalous Properties of β-Rhombohedral Boron R8. Issues Related to the Third Law of Thermody-

namics S9. Prospect of Boron Research TAssociated Content T

Supporting Information TAuthor Information T

Corresponding Author TNotes TBiographies U

Acknowledgments UReferences U

1. INTRODUCTION

At the beginning of the 21st century, the properties of most ofthe elementsat least the ones with stable forms that can beisolated at ambient conditionsare well understood. Thefundamental properties of elements, including the thermody-

namic stability of their allotropes and polymorphs, have beencompiled and published in many books.1 Only the fifth elementof the periodic table, boron, has eluded a complete character-ization, and its thermodynamic stability at ambient pressure hasnot yet been established by experiments (see Figure 1).2

According to ref 1a, “boron displays a large family of allotropes,16 in total, which is second in size only to sulfur”, although sofar only four of them, α-rhombohedral boron, β-rhombohedralboron, γ-boron, and T-192, have been confirmed to bethermodynamically stable.2,3 In spite of the important progressof the past few years in understanding the phase diagram ofboron, there are still conflicting views in the literature, e.g.,concerning the stability of two of its allotropes at low pressure,β- and α-rhombohedral,4 and on the existence of the allotropeT-50.5

At ambient pressure, liquid boron solidifies into the β-rhombohedral phase,3a,6 indicating that this phase is thethermodynamically stable one at high temperature (T). β-Boron has an extremely complex structure, with more than 300atoms per hexagonal unit cell, that consists of a combination oficosahedra and fused icosahedra; a phase transition from β-

Received: August 24, 2012

Figure 1. Phase diagram of elemental boron published in recentreports. These reports yield a consistent energy ordering of thedifferent phases (α-boron, β-boron, γ-B28, T-192); however, largeuncertainties remain on the location of phase boundaries. Red lines arefrom ref 2, the blue line is from ref 3b, and the green lines are from ref3c.

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boron to other phasesby cooling or annealing at ambientpressurehas never been reported. In fact, early reportssuggested that β-boron is likely to be the thermodynamicallystable phase of boron for all temperatures below melting atambient pressure,7 down to at least ∼1400 K.4b,8 Howeverthese reports lack a quantitative comparison of the thermody-namic stability of different phases, due to the lack of robustmeasurements of formation enthalpies. The high meltingtemperature and chemical inertness of β-boron make itchallenging to accurately estimate its enthalpy of formationfrom calorimetric measurements.Nevertheless, β-boron would have been accepted as the de

facto stable phase at all temperatures at ambient pressure if theso-called electronic requirementa closed shell−electronicstructureand the thermodynamic requirementa perfectcrystal structurewere clearly satisfied, but these requirementsare apparently violated9 in β-boron. For example, experimentshave revealed that β-rhombohedral boron possesses a macro-scopic amount of intrinsic defects9c for which no ordering hasbeen observed. In addition, although an electron count doesnot reveal a closed shell structure, the transport properties ofthe system appear to be similar to those of a nearly closedshell−electronic structure solid. In particular β-boron “behaves”like a p-type semiconductor,10 with a very low electronicconductivity9c,11 and an optical gap of about 1.5 electron volts(eV).9c,12

Theoretical and computational approaches have played acrucial role in understanding boron and its compounds. Earlymolecular orbital (MO) calculations elucidated two keyconcepts in the chemistry of boron. First, the peculiarmolecular structure of diborane has been shown to arise fromthe formation of three-center, two-electron (3c2e) bonds.13

Second, MO theory predicted the charge state of B12H12−2,5b

which was later confirmed by the successful synthesis ofdipotasium salt, K+

2B12H12−2.14 As we shall show, those two

concepts3c2e bonds and the presence of slightly electrondeficient (2e) boron icosahedra in elemental boronare key tounderstanding the nature of boron allotropy.The second wave of theoretical developments came about

with density functional theory (DFT) and high-performancecomputers, which made it possible to perform most of thecalculations on the bulk structures of boron available to date.The first theoretical study of the electronic structure of β-rhombohedral boron9b did not include intrinsic defects andresulted in an incomplete occupation of the valence band, andits author concluded: “Hence, a deformed structure occurs inwhich some atoms are randomly displaced into interstitialsites.”9b This suggested that intrinsic defects play a role indetermining the electronic structure of boron and in satisfyingelectronic requirements.The mechanism of self-doping caused by the intrinsic defects

was not fully understood until 2007−2009,4f,i,k although it hadbeen discussed in earlier studies.4d,15 It is now understood thatthe interplay between the imbalance of electron requirementsbetween the B12

5b,14 and B284d,k,15c,16 units present in β-

rhomohedral boron, the local intrinsic instability of B28,4k,16 and

the conversion of two-center, two-electron (2c2e) bonds to3c2e bonds due to the presence of self-interstitials, all lead to anearly perfect closed shell−electronic structure.4k The defectsare necessary to stabilize β-boron; this means that in β-borondefects have a negative formation energy.4e,f,i,k,15b,16a,17 Withinfirst principles−DFT calculations, β-boron is found to be themost energetically favorable allotrope of boron at ambient

pressure and at all temperatures below melting, when amacroscopic amount of defects is present.4f,i,k

Many defect configurationsincluding various combinationsof vacancies and self-interstitialsexhibit exactly the sameenergy, leading to a disordered and degenerate ground state forβ-boron, with a macroscopic amount of residual entropy. Themechanism that leads to the multiplicity of geometricalconfigurations is called “geometrical frustration”,4l a propertyexhibited also by ice and magnetic−spin ice materials.18

Proton disorder in ice and spin disorder in magneticpyrochlore materials may be described with the same type ofHamiltonian used to describe defects in boron, that of aferromagnetic Ising model on corner−sharing tetrahedra.18g

Within a nearest neighbor−interaction approximation, thisHamiltonian has an exactly degenerate and disordered groundstate with a macroscopic amount of residual entropy.18g

The calculated specific heat of spin ice as a function oftemperature, obtained by using a ferromagnetic Ising modelwith nearest neighbor (NN) interactions, shows excellentagreement with experiments, thus supporting the existence of adegenerate ground state with macroscopic entropy.18h How-ever, the use of a more accurate spin−spin interactiondipole−dipole interactionand of an advanced cluster−updatealgorithm, shows that the dipole Ising model exhibits a phasetransition to an ordered phase at very low temperature (below0.2 K).18h The spin−spin correlation measured by neutronscattering is in better agreement with a dipole Ising model thanwith an NN Ising model, suggesting that the dipole Ising modelis a better approximation of spin ices; however, a phasetransition to an ordered phase has never been observed in realmaterials.18h Therefore, it was tentatively concluded that theobserved residual entropy of spin ices is a nonequilibriumproperty due to the slow dynamics at low temperature, and thatsuch systems are likely to have a nearly degenerate groundstate, not an exactly degenerate one.18h

In this review, the description of boron ground-stateproperties parallels the interpretation of the residual entropyadopted in the case of spin ices: an exact degeneracy is unlikelyto exist in a real system. In other words, elemental boron isdescribed as nearly degenerate, although the model Hamil-tonian for defects in β-boron appears to have an exactlydegenerate ground state. The identification of a model IsingHamiltonian that describes the defects’ configurations wasinstrumental to understand how the macroscopic (near)degeneracy can be realized.18c,g,19

As the case of spin ice shows, the comparison of the specific-heat profile obtained from theoretical models and experimentalmeasurements plays an important role in understanding howgeometrical frustration leads to a nearly degenerate groundstate with macroscopic residual entropy.18g In the case ofboron, unfortunately, no attempt has been made to measure theimpact on the specific heat of different defect configurations.Therefore, the near degeneracy in the configuration of defectshas not yet been confirmed experimentally. Many experimentalstudies reported evidence of defect diffusion in β-boron, whichwas considered as one of the leading causes of the anomalies inthe transport properties10b,12d,20 and in internal frictiondata.20a,j,k It was also shown that the optical absorption of β-boron exhibits a dramatic change as a function of temperatureat about 150−200 K.20l,m Below this temperature range, only afew absorption peaks appear in the optical gap, but theirnumber greatly increases above ∼200 K. A structural phasetransition was suspected to occur at this T, but it was not

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evident from X-ray diffraction measurements.20a We note thatthe type of phase transition suspected in this work probably wasa disorder-to-order phase transition as X-ray diffractionmeasurements will effectively detect such a type of transition.Evidence for another type of phase transition based oncalorimetric measurements at T = 550 K was reported in2011, although the nature of this transition is yet to bedetermined.21

Given that many of the anomalies in the transport propertiesare observed at 150−200 K, it was first suggested that they areprobably due to the diffusion of defects.20a,l,n Later observa-tions, however, seem inconsistent with the notion of defectdiffusion in boron. For example, the occupation rates of thepartially occupied sites (POS) show clearly measurabledifferences depending on the synthesis conditions, and thedifference in the occupation rates persisted for more thanseveral years after annealing at ambient conditions.9c If boroninterstitials diffused below a temperature of 150 K,20n theyshould, it seems, reach an equilibrium occupation configurationwithin a reasonable time scale: if so, the occupation rates wouldbe the same in different samples, after a sufficiently longannealing time. Apparently, this is not the case.To resolve this issue, the activation barriers between many

different occupation configurationswhose total energies arenearly degeneratewere determined by first principles−DFTcalculations.4l This revealed a few boron-hopping paths withlow enough activation energies to allow for defect diffusion atlow temperature. However, some of the activation energieswere found to be on the order of several eV, thus unlikely toallow for atomic diffusion at low temperature. Their presencemight explain the persistent difference in the occupation ratesof POS in samples obtained by different annealing procedures.The lowest-energy activation mechanism involves only defectlocations belonging to the same crystallographic site; therefore,in this case defect diffusion does not change the occupationrate.4l Among the configurations visited during diffusion, thereare geometrical arrangements responsible for introducingsignificant gap levels in the electronic band structure of β-boron.4l

Several recent articles have reviewed the properties of boron,including the chemistry of borane and boride,22 the phasediagram of elemental boron under pressure,23 the chemistry ofboron clusters,24 experimental measurements on β-boron,25

and the electronic and mechanical properties of boron andboron-rich compounds.26 This review focuses on the recenttheoretical advances in describing the properties of β-rhombohedral boron. The other allotropes, α-rhombohedral,T-50 (or T-192), and γ-B28, were reviewed by Albert andHillebrecht in 2009;22b therefore, in this article, only newdevelopments since 2009 will be briefly discussed (α-boron,26,27 γ-B28,

3d,28 T-505l−n). A comprehensive collectionof experimental data can be found in ref 29.We will first describe the crystal structure of β-rhombohedral

boron and the partial occupancy of defect sites (that we call,following the literature, partially occupied sites or POS). Wewill then discuss β-boron’s thermodynamic stability, which inrecent years has been extensively investigated by theorists, andgeometrical frustration. We will then present an analysis of theelectronic properties. We conclude with a few comments on thethird law of thermodynamics and on prospects of future boronresearch.

2. β-RHOMBOHEDRAL BORON: FROM ITSIDENTIFICATION TO THE ESTABLISHMENT OFPARTIAL OCCUPANCY

In this section, we review how the crystal structure of β-rhombohedral boron and its partial occupancy were established.We also briefly discuss the discoveries of other majorallotropes: α-rhombohedral,30 α-tetragonal (or T-50),5c−g,j,k

β-tetragonal (T-192),5g,31 and γ-B28.2,3b,32

Elemental boron was successfully isolated in 1900−1910,33and diffraction patterns were first taken in the 1930s.34

Methods for synthesizing boron were established from 1940through the 1950s,3a,4a,5a,c,30,35 which revealed the presence ofvarious polymorphs and allotropes. Most of these allotropeshave complicated crystal structures based on icosahedra, whichwere only identified in the 1950s.4a,5a,30a Except for the simplestformsα-rhombohedral boron30a and, perhaps, γ-B28, whosestructure was determined in 20092,3buncertainties remainabout the structure and physical properties of boron’sallotropes.The first crystal of α-tetragonal boron was identified in

1951,5a and it was called T-50, from the 50 atoms in itstetragonal unit cell. In 1955, based on the electron require-ments of the B12H12 cluster estimated from MO theory, it wasreported that the T-50 structure is 10 electrons short of fillingall bonding orbitals (or valence bands).13b This suggested that apure form of T-50 would not be stable.5b In 1958, additionalexperiments5c seemed to confirm the existence of T-50, butfrom the late 1960s to the mid-1970s it was demonstrated thatonly B48B2C2 or B48B2N2 compounds were in fact synthesi-zed.5d−f,36 It was later reported that the pure form of tetragonalboron has a structure similar to α-AlB12 with partial occupancy,and it was described as B212B12B2.5.

5g,31b This structure is acombination of entwined icosahedra, regular icosahedra, andinterstitials; this phase, first reported in 1960,5g,31a,b was namedT-192. The controversy around the T-50 phase continued formany years, and theoretical studies eventually supported thenotion that T-50 can only be stabilized in the presence ofimpurities.5j,k Very recently, three independent groups5l−n havereported on the identification of this phase as pure elementalboron and have suggested that T-50 appears as a metastablephase due to the Ostwald step rule.37

In 1957, Sands and Hoard announced the identification ofthe β-rhombohedral phase, which belongs to the space groupR3 m with a lattice parameter (a) of 10.12 Å, a rhombohedralangle (α) of 65°28′, and an approximate atomic density of 108atoms per rhombohedral cell.4a In 1960, Hoard and Newkirkreported that many of the synthesized pure-boron solids fromstudies completed in 1930−1950 were likely to contain the β-rhombohedral phase.4b In the early 1960s, the thermodynamicstability of β-rhombohedral boron at high temperature wasestablished,3a,6 and some reports suggested that it wasthermodynamically stable at all temperatures, but theseconclusions were based on indirect evidence.4a,b To date, noexperiment has unequivocally determined the thermodynamicstability of this phase.In 1962, X-ray analysis provided a partially correct report of

the atomic positions of β-rhombohedral boron.38 In 1963, themain crystallographic features of the system were reported: thestructure was described as being composed of 12 halficosahedra surrounding a larger icosahedron (84 atoms) and20 more atoms connecting these B84 units in adjacent unit cells(see Figure 2).7 In 1970, an additional crystallographic position,

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B16, was identified by Hoard et al.9a This work also reportedpartial occupancy at B13 and B16 sites, but with a slightinconsistency in the atomic densities estimated from fitting X-ray derived positions, and from flotation-measurements.9a Twoother independent studies from about the same period, byGeist et al.39 and Callmer,40 refined the atomic positions in β-boron; the former did not report the presence of partialoccupancy but the latter did. However the lattice parametersand the atomic positions from these three papers from the1970s are in excellent agreement with each other. Donohue1a

suggested that the partial occupancy reported by Hoard et al.“may possibly [be] due to the somewhat lower purity of thesample used....” He adds: “It is also difficult to understand whythe standard errors of Hoard et al. are only one-third, on theaverage, those of Geist et al. The fewer number of reflectionsused by the latter by no means can account for thisdiscrepancy.”In 1988, Slack et al.9c,41 reported the presence of four more

POS based on a study that carefully investigated the correlationbetween the impurities and the synthesis conditions and theirinfluence on the occupation rates of POS, and on the atomicdensities estimated by different approaches; such investigationturned out to be crucial for later theoretical studies. Thisresearch team also suggested a few possible short-rangecorrelations between the occupation configurations of POS.In addition Slack et al. reported the electric conductivity of β-boron; the impurity-free limit extrapolated from severalmeasurements confirmed the semiconducting properties ofthe system (resistivity on the order of 10−8 Ω cm). It is fair tosay that the work of Slack established the presence of partialoccupancy in elemental boron.The α-rhombohedral phase of boron was identified shortly

after the discovery of the β-rhombohedral one.30 This structureconsists of a single icosahedron in a slightly distorted cubiccell,30b with the C3 axis of the icosahedron aligned to the c-axisof the unit-cell. This geometrical arrangement indicates that theformation of three 3c2e external bonds (called delta bonds atthe time) and of 2c2e external bonds may yield a closed shell−electronic structure, which was later confirmed.42

It was reported very recently that the composition of the α-boron unit cell may not be restricted to a single B12 but couldalso have two additional boron interstitial atoms.43 This newstructural model reconciles the known discrepancy between themeasured high frequency IR-active phonon and theoretical

predictions. However, these new sites need to be partiallyoccupied in order to be consistent with the reported density ofα-boron.In the past decade, several theoretical studies4c−e,h,15c,17 have

addressed the relative thermodynamic stability of the α- and β-rhombohedral phases. Contrary to the earlier suggestions byexperimentalists, some theoretical calculations reported that α-boron should be the stable phase at low temperature andambient pressure, but these results were obtained with anapproximation that only takes into account the partialoccupancy of B13 and B164c−e,15c,17 sites. Later, threeindependent theoretical studies showed that incorporating thePOS originally omitted makes β-rhombohedral more stablethan α-rhombohedral boron.4f,i,k

The crystal structure of a new high-pressure phase of boron,γ-boron, first reported in 2008,32a,44 consists of icosahedra andinterstitials,2,3b and is characterized by an ionic character.2,28k

(However, we note that this report is likely to be a rediscoveryof the phase originally reported in 1965 by Wentorf.3b,d,45).This property originates from the electron deficiency of theicosahedra,5b,14 which seems to prove the universal ioniccharacter of icosahedra-based phases of elemental bor-on.2,5b,14,15b,16a,28k,46 An isostructural phase transition of γ-boron was proposed by Zarechnaya et al. in 2010;28b however,later theoretical and experimental studies consistently rejectedthis view.28m,o

At the time of writing this review, the crystal structure of thesuperconducting high-pressure phase discovered in 200147 hasnot yet been identified; theory suggests it may be the α-Ga(Cmca) structure.48 Although some studies49 suggest differentscenarios, enthalpy comparisons indicate that the α-Gastructure is indeed the most likely candidate for the structureof the superconducting phase.2,48a Interestingly, differentapproaches based on compression of α-boron27b and Li dopingof α-boron,27d,e,50 originally proposed by Gunji et al. in 1993,50a

have also successfully observed a phase transition to asuperconducting phase.27a,b,d,50h We note that the super-conducting phase obtained by compression of α-boron retainsits original crystal symmetry most likely as a metastable state,with metallization achieved by band overlap.27b,50h

Before turning to the discussion of the detailed structure ofβ-rhombohedral boron, it is interesting to draw a comparisonbetween the boron crystal structures and those of the othergroup III elements. In 2003, Haussermann et al.48a calculatedthe phase diagram of group III elementsboron, aluminum,and galliumand compared the α-rhombohedral (bct and fcc)and α-Ga crystal structures. Interestingly, in the case of boron,they found that the α-Ga structure, rather than bct or fcc, is themost stable phase above 74 GPa. They also identified the stablepressure ranges for the α-rhombohedral phase of aluminum andgallium. In 2004, Ma et al.48b studied the superconductivity ofthe α-Ga phase of boron, and they reported that the pressuredependence of the electron−phonon coupling constant isconsistent with experimental observations. Consequently, theysuggested that the α-Ga phase might be a good candidate forthe high-pressure metallic phase of boron.Extensive work on the Zintl phase p-block intermetallic

compounds showed that group III elements form icosahedralclusters. Therefore, one might think that the ability to formsuch clusters is a general property of trivalent elements with 2sand 1p valence electrons, which raises the question: Why doesonly elemental boron form icosahedra-based crystal structures?The α-rhombohedral phases for aluminum and gallium appear

Figure 2. The crystal structure of β-rhombohedral boron in terms ofB84 and B10 units. (a) B84 consists of a central B12 icosahedron (blue)surrounded by 12 half icosahedra (pink). (b) B84 units in adjacentrhombohedral unit cells connected via B10 cluster units (gold). Thisview is from the c-axis (perpendicular to the page). These layeredstructures stick to each other along this axis, and interstitial boronatoms (not drawn) lie between B10 clusters connecting the layers.

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in the negative pressure range.48a It is perhaps the localizationlength scale of the valence orbital that singles out boron fromthe rest of group III elements.

3. THE CRYSTAL STRUCTURE OF β-RHOMBOHEDRALBORON

In 1974, Donohue1a described the crystal structure of β-rhombohedral boron by writing: “The structure consists of anenormously complicated framework of boron atoms whichcontains, however, some simple and beautiful features whichwould have made Kepler leap with joy.” This refers to Kepler’s16th-century study of packing two-dimensional space withpentagons.51

The crystal structure of β-rhombohedral boron was firstdescribed based on one B84 unit, 2B10 cluster units, and oneinterstitial.7,9a The B84 unit comprises a central icosahedronsurrounded by 12 half icosahedra (see Figure 2a). The outershell of B84 consists of 60 boron atoms creating the samestructure as that of a C60 fullerene molecule.52 B84 units inadjacent unit cells are connected via B10 cluster units (seeFigure 2b). The interstitial atom (in Figure 2) is locatedbetween two B10 clusters. In this review, we use an alternativeexplanation of the crystal structure based on B12 icosahedra andB28 triply fused icosahedra,53 which makes it easier torationalize the relationship between the atomic and electronicstructure.The rhombohedral cell of idealized β-boron with 105 boron

atoms per rhombohedral unit-cell consists of an outer layer of20 icosahedra8 at the corners and 12 along the edges of thecelland two triply fused icosahedra in the middle, which areconnected by an interstitial boron atom (see Figure 3a,b). Thecluster units may be connected by conventional 2c2e bonds(see Figure 3c). Many reports have summarized the latticeparameters of β-boron,4k and they are all in good agreement.

For example, one well-accepted report7 provides the followingparameters: a = 10.145 ± 0.015 Å and α = 65°17′ ± 8′. Thisrhombohedral angleslightly larger than the one for a cubiclattice, which is 60°allows this system to accommodate theB10 clusters, which are part of the B28 triply fused icosahedra(see Figures 2b and 3b). As we will discuss later, B28 plays animportant role in counter-balancing the electron deficiency ofB12.Five crystallographic POS are compatible with rhombohedral

symmetry (space group R3 m) and are described below. (Thisreview uses the standard naming scheme for the crystallo-graphic sites.9c)B13 is a vacancy site in the B28 triply fused icosahedra unit

(see Figure 4). There is a simple reason for the presence of B13vacancies: the total number of electrons in B28 (3 × 28) is toolarge for a closed shell−electronic structure when all of theexternal bonds are formed. The vacancy formation in triply

Figure 3. The crystal structure of β-rhombohedral boron in terms of B12 icosahedra (green) and B28 triply fused icosahedra (gold). The gray linesdenote the rhombohedral unit cell. (a) Eight icosahedra (green) are located at the corners of the unit cell, and 12 icosahedra are located along themiddle of each edge of the unit cell. (b) Two B28 triply fused icosahedra (gold) and the interstitial atom (gold) are located in the middle of the unitcell. (c) The icosahedra and the fused icosahedra are directly bonded to each other, with large holes between the different building units.

Figure 4. The B13 vacancy sites (red spheres) in a B28 cluster unit.Side view (left) and bottom view (right).

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fused icosahedra was first rationalized in the case of Ga28clusters, which appear in Zintl-phase intermetallic compound-s.16a,54 All recent first-principles studies on bulk β-rhombohe-dral boron confirm that full occupancy of the B13 site is notenergetically favored.4c−k,15c,17

The structure of β-boron includes several additional POSwhose occupation rates have been reported, e.g., in ref 9c (seeTable 1). A B16 self-interstitial exists in the middle of the

hexagonal ring connecting the B12 icosahedra (see Figure 5).B19 and B20 interstitials lie in the hexagonal rings connectingB12 and B28 (see Figures 6 and 7). B17 and B18 interstitials arenext to each other and surround the B13 vacancy sites (seeFigure 8). We will see that these locationsB16, B19, and B20being at the center of hexagonal rings, and B17 and B18

surrounding B13 vacancy sitesare key to understanding theeffect of POS on the stabilization and electronic structure of β-boron. Ultimately, the knowledge of POS locations provides acoherent picture of the observed number of partially occupiedsites and of the atomic density of 320 atoms per hexagonal cell;it also helps explain the observed electronic propertiesthat ofa p-type semiconductor with a very low electronic con-ductivitywithout requiring any long-range ordering of thearrangement of partially occupied sites. Finally, it explains whythe observed fluctuation in the occupation rates does not affectthe electronic properties, as we will discuss in detail in sectionVI. As a side note, we point out that there are additional

Table 1. The Occupation Ratesa Determined by Theory4k

and Experiments9c

B13 B16 B17 B18 B19 B20 energy meV

1_A 79.1 30.5 4.2 4.2 8.3 0.7 0.771_B 69.4 30.6 13.9 13.9 0 0 6.701_C 66.7 27.8 16.7 16.7 0 0 6.051_D 80.6 29.2 2.8 2.8 4.2 4.2 3.611_E 70.8 30.6 12.5 12.5 0 0.7 3.322_A 70.8 31.9 12.5 12.5 0 0 3.272_B 79.2 33.3 4.2 4.2 6.9 0 2.992_C 76.4 33.3 6.9 6.9 4.2 0 2.882_D 70.8 30.6 12.5 12.5 0 0.7 2.732_E 72.2 33.3 11.1 11.1 0 0 2.512_F 73.6 31.9 9.7 9.7 2.8 0 2.432_G 73.6 33.3 9.7 9.7 1.4 0 2.33MG57 77.7 25.8 3.2 5.8 7.2 0 N/AEP 74.5 27.2 8.5 6.6 6.8 3.2 N/AMG197 73.0 28.4 9.7 7.4 7.0 2.5 N/A

aThe theory used a self-consistent fitting cycle between the Ising typemodel Hamiltonian and the DFT−total energy calculations. The labels1_X and 2_X (X = A-G) correspond to the first fitting cycle and thesecond fitting cycle, respectively. MG57, EP, and MG197 correspondto samples synthesized in different conditions.9c Reproduced withpermission from ref 4k. Copyright 2009 American Chemical Society.

Figure 5. B16 interstitial POS (blue spheres) exist at the center ofhexagonal rings connecting B12 icosahedra, as seen from differentangles (left and right).

Figure 6. The B19 POS (purple spheres) are at the center ofhexagonal rings connecting B12 icosahedra (green bonds) and B28triply fused icosahedra (gold bonds), as seen from the top (left) andthe side (right). Only the icosahedra at z = c/2 are drawn.

Figure 7. The locations of B20 POS (purple spheres) are at the centerof hexagonal rings connecting the B12 icosahedra and B28 triply fusedicosahedra, as seen from the top (left) and side (right). For the sake ofclarity, only the upper half of B20s, and the associated B12 icosahedraare drawn.

Figure 8. The B17 (magenta spheres) and B18 (purple spheres) POSsurround the B13 vacancy (red spheres) sites, as seen from the side(left) and the bottom (right), with only the upper half drawn in thelatter view.

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interstitial sites that are known to accommodate foreignimpurities.55

Another important aspect of the POS layout in β-boron is itsconnection to a two-dimensional kagome lattice. In fact, thePOS form a double layer−expanded kagome lattice (seeFigures 9−11). This lattice plays an important role in

determining the thermodynamic properties of boron and inexplaining the observed anomalies in the temperature depend-

ence of the transport properties. In 1958, Frank and Kaspernoticed the close relationship between rhombohedral symmetryand a kagome network.56 In 1977, Hughes et al. reported thatthe icosahedra in β-boron at z = c/2 (where c is the length ofthe rhombohedral unit-cell along the z-axis) form a kagomenetwork, and added: “Finally, and dramatically, the B28-(C3v)unit ... occupies the same tetrahedral hole in the β-B105-(R3 m)structure, preserves one of the threefold axes of Fd3m, andproduces the observed rhombohedral symmetry”57 (see Figure12). However, it was only in 2010 that the impact of a kagomenetwork in the crystal structure on boron thermodynamicproperties was pointed out.4l

We end this section by introducing the most recent additionto the peculiar “structural” properties of β-boron. On the basisof comparisons of IR- and Raman-active phonons of 10B-enriched, 11B-enriched and natural β-boron samples, it wasdemonstrated that shifts of certain phonon modes can beexplained only if the distribution of 10B and 11B are notcompletely random.58

4. THERMODYNAMIC STABILITY OFβ-RHOMBOHEDRAL BORON

In this section, we review the studiesfirst experimental andthen theoreticalof the thermodynamic stability of β-rhombohedral boron. As already noted, liquid boron has beenobserved to solidify only into the β-rhombohedral boronphase.3a,6 Moreover, when heated, α-rhombohedral borontransforms to the β-rhombohedral phase30b via well-definedphases at T = 1643, 1863, and 1913 K.8a Several experimentalstudies7,8b indicated that the α-to-β phase transition isirreversible, and Carlsson8b pointed out that “a large increasein the time required for the transformation with decreasingtemperature was observed indicating a large activation energy.This means that a practical limit for observing the trans-formation may be about 1000 C.” This indicates that, within thetemperature range studied in ref 8b, the free energy of β-rhombohedral boron is lower than that of α-rhombohedralboron, but slow kinetics precluded the determination of therange of stability of α-rhombohedral boron at lower temper-ature. Machaldze observed the α-to-β phase transition over therange 1723−1985 K, at ambient pressure.59 The measured heatof transformation, ΔH = −1.05 kcal/mol, indicates that thetransformation is exothermic. In 2011, Parakhonskiy et al.performed high-pressure and high temperature experimentsand identified a α−β coexistence points at high pressure.3c Onthe basis of a linear extrapolation, they proposed that the α−βphase boundary at P = 0 GPa is at T = 933(50) K.In 1992, Mailhiot et al.60 reported the first extensive study of

boron phase boundaries based on first principles−DFTcalculations. They computed enthalpy curves of variousconceivable polymorphs of boron and of the α-rhombohedralphase up to 2-fold volume compression. They consideredseveral polymorphs: body-centered-tetragonal (bct), diamondstructure, simple cubic (sc), face-centered cubic (fcc), andhexagonal closed pack (hcp). At low pressure, the α-rhombohedral phase was found to be the most stable. Ittransforms to bct at 210 gigapascals (GPa) and to fcc at 360GPa. Inspired by theoretical predictions on the high Tcsuperconductivity of the lightest elements61 and by the high-pressure phase diagram predicted by Mailhiot et al.,60 Eremetset al.47 compressed β-rhombohedral boron close to 260 GPaand found that it exhibits superconducting properties above160 GPa. The crystal structure of the superconducting phase,

Figure 9. The kagome network of icosahedra (green spheres) and thetwo-dimensional POS network create a double layer−expandedkagome lattice. For clarity, only B13 (red), B16 (blue), and B19(purple) POS are shown. The gold spheres are placed at the locationof interstitial atoms connecting two triply fused icosahedra. Half of aPOS network is depicted on the far side (thin bonds and smallspheres), and the rest is on the near side (thick bonds and largerspheres) relative to the kagome network of icosahedra.

Figure 10. The location of POS in the double layer-expanded kagomelattice. B13 (red), B16 (blue), B17 (green), B18 (orange), B19(purple), and B20 (gray). The thinner bonds and the smaller spheresare on the far side, and the thicker bonds and the large spheres are onthe near side.

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however, has not yet been experimentally identified. On thebasis of the predicted crystal structures, bct or fcc,Papaconstantpoulos and Mehl49a and Bose et al.49b calculatedthe superconducting transition temperature as a function ofpressure, which showed the opposite trend as compared to theexperiments.47

The stability of boron at low P and T is still controversial;theoretical studies to date have all treated POS differently,which makes the comparison of results quite challenging. Whendiscussing theoretical findings, it is important to keep in mindthat the total energy of boron decreases with occupation of thePOS. Therefore, the POS should be incorporated in thecalculations as completely and as realistically as possible inorder to provide good approximations to the real structure.In 2002, Zhao and Lu62 reported the first theoretical study of

α- and β-rhombohedral boron based on DFT calculations.They focused on identifying the high-pressure superconductingphase, and they did not give any detail on the phase boundarybetween α- and β-boron at zero pressure. Haussermann et al.48a

briefly mentioned that, in the absence of POS, α-boron is 20meV/atom lower in energy than β-boron, but they noted thatan accurate comparison must include the POS.

In 2004, Masago et al.17 first reported on the relative stabilityof α- and β-boron, where partial occupancy was considered,although only in an approximate manner: the B17−B20interstitials were ignored. These authors showed that inclusionof the B13 vacancy and B16 interstitial, in fact, reduces theinternal energy of β-boron. They also calculated the Helmholtzfree energies of α- and β-rhombohedral boron as the sum of theDFT total energies and the POS-configurational free energy atfinite temperature. Their results indicate a transition temper-ature of Tα−β = 200 K under equilibrium conditions, muchlower than the T at which the α−β transition was observedexperimentally (1400 K). Consequently, they stated: “Takingphonon effects into account, we may obtain the calculatedtransformation temperature in good agreement with experi-ment.” In 2006, the same group4e published a revised transitiontemperature, Tα−β = 970 K, with the phonon contributionadded to the free energy, which was estimated from Γ-pointphonons calculated with a DFT based frozen phonon method(as a reminder, the experimental transition temperatures4b,8,30b

referred to in refs 4e and 17 do not correspond to the onesunder equilibrium conditions). In 2005, Prasad et al.4d

performed DFT−total energy calculations on α- and β-rhombohedral boron, with no POS taken into account. Theirresults were consistent with those of previous studies: α-boronwas found to be more stable than β-boron at T = 0 K.In 2007, Shang et al.4h studied the thermodynamic and

mechanical stability of α- and β-rhombohedral boron usingfirst-principles DFT total-energy and phonon calculations.They took into account the approximate occupations of thePOS, and they also considered four different atomic densities:104, 105, 106, and 111 atoms per rhombohedral cell. TheirDFT−total energy results were consistent with previous work:α-boron was found to be the stable phase at low temperature,and β-boron the stable one at high temperature. Shang et al.also pointed out that β-boron without any defects, or POS, ismechanically unstable at high temperature, although it isthermodynamically stable. They argued that introducingdefects, as detected in real samples, improves on the mechanicalstability. The total energy of β-boron as a function of atomicdensity indicates that the minimum atomic density is obtainedfor 105−110 atoms per rhombohedral cell. In 2009,

Figure 11. The quasi two-dimensional network of POS, (2B28)B, an inverted pair of triply fused icosahedra (gold bonds), and the rhombohedralunit-cell, as seen from the top (left) and side (right).

Figure 12. Kagome network of icosahedra at z = c/2 in β-boron(green bonds). The holes in the kagome network are occupied by aninverted pair of triply fused icosahedra (gold bonds).

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Siberchcot63 studied the equation of state of α- and β-rhombohedral boron using DFT−total energy calculations,where β-boron was modeled with 105 and 106 atoms perrhombohedral cell. The model with 106 atoms produced resultscloser to experiments. In addition, at 0 GPa, the total energy ofthe 106-atom model of β-boron was 4 meV/atom higher inenergy than that of the α-rhombohedral phase; the zero point−motion energy (ZPE) was not taken into account.In 2007, van Setten et al.4f reported the first theoretical DFT

study on β-rhombohedral boron at the experimental atomicdensity, where all of the POS discovered by Slack et al.9c wereconsidered. Starting with 106 atoms per unit cell, they studiedseveral stable-occupation configurations. They found that a B19occupation next to a B13 vacancy gives rise to one of the moststable structures at this atomic density; based on thisinformation, they constructed the occupations of the POSwith 320 atoms per hexagonal cell. With 106 atoms per unitcell, the most stable configuration included a B16 occupied siteand no B13 vacancy; at 320 atoms per hexagonal cell theoccupation configuration containing a B19−B13 interstitialvacancy−pair configuration is the most stable. Moreover, thosetwo configurations have essentially the same total energy peratom. Van Setten et al. further optimized the cell parameters ofthe stable 106-atoms system, which brought the total energy ofβ-boron 1 meV/atom above that of α-boron. They alsocalculated the phonon density of states of α- and β-boron usingthe frozen phonon technique and estimated the quantum ZPE.When added to the total energy, one finds that the free energyof β-boron is lower than that of α-boron. Van Setten et al. alsoestimated the configurational free energy of POS in β-boronand compared the Helmholtz free energies of α- and β-boron,concluding that at 0 GPa, the β-rhombohedral phase is thethermodynamically stable phase of elemental boron at alltemperatures below the melting point.In 2008, Widom and Mihalkovic4i reported on yet another

ground state−POS structure of β-rhombohedral boron basedon DFT−total energy calculations. Their results agree withthose of van Setten et al. on the thermodynamic stability of β-rhombohedral boron, but they disagree on the most stablePOS-occupation configuration. Widom and Mihalkovic pro-posed that the ground state of β-boron should have a uniqueand ordered occupation of POS that does not violate the thirdlaw of thermodynamics, and they predicted a superlatticestructure of POS. In their search for the optimal configuration,they started from systems with 107 atoms in the rhombohedralcell and built superlattice structures with systems containing320 atoms in the hexagonal cell. Contrary to van Setten et al.,the stable POS-occupation configurations found by Widom andMihalkovic consisted of the occupation of a B17−B18 pair nextto a B13 vacancy pair. They also calculated the POSconfiguration’s specific heat and the entropy of the solid as afunction of temperature, which revealed a peculiar result: therewas no clear sign of a first-order transition from the high-temperature, disordered configuration to a low-temperature,ordered one. Such a transition must exist to attain an ordered,ground-state POS configuration. It should be noted that such aphase transition has never been detected in measured specificheat64 data.In 2009, Ogitsu et al.4k reported the first results of a global−

phase space search of POS-configurations in β-rhombohedralboron. We remind the reader that the astronomical number ofpossible ways to occupy the POS poses one of the greatestchallenges for theoretical studies of the properties of β-

rhombohedral boron. For example, within the hexagonal cell,there are the following possible POS-occupation configura-tions:4k

⎡⎣⎢

⎤⎦⎥

12623

Therefore, it is impossible to conduct a thorough study basedon the “hand picking” strategy used in refs 4f and i.Using cluster expansion (CE) techniques and DFT−total

energy calculations, Ogitsu, et al. self-consistently derived amodel Hamiltonian65 and performed a global−phase spacesearch of the POS-occupation configurations. They found manyPOS configurations with nearly the same total energy (nearlydegenerate) that are more stable than α-boron (see Figure 13).

Each of those configurations contained POS types found inprevious work, namely, B13−B19 vacancy-interstitial pairs andB13−B13−B17−B18 vacancy−vacancy−interstitial−interstitialpairs. In addition, a B13−B20 vacancy-interstitial pair wasfound in certain configurations. Most interestingly, no long-range ordering was observed in any of the stable POSconfigurations found in the optimization.The most important result we want to emphasize in this

section is that β-rhombohedral boron has the lowest DFTenergy among all the other allotropes and polymorphsconsidered so far, if all of the POS found by Slack et al.9c aretaken into account and the occupation configurations areoptimized appropriately, at the experimental atomic density,320 atoms per hexagonal cell.4f,i,k Therefore, β-rhombohedralboron is likely to be the ground state of elemental boron, and itis likely to be the stable phase at ambient pressure for alltemperatures below melting. We note that all of the quasi-degenerate stable structures found in aforementioned calcu-lations4k exhibit short-range vacancy-interstitial correlations.As mentioned earlier, based on a linear extrapolation of the

high pressure and temperature experimental results, Para-khonskiy et al. have recently proposed3c that α-boron becomesmore stable than β-boron below T = 933(50) K. Ogitsu andSchwegler pointed out that66 the α−β phase boundary obtained

Figure 13. Relative ab initio DFT total energies, including the zero-point energy contribution, of α-rhombohedral boron (black triangle),perfect β-rhombohedral boron (red triangle, 105 atoms in therhombohedral unit cell), and β-rhombohedral boron with only B13and B16 POS17 (green diamond). Blue circles and yellow squares arethe ab initio DFT total energies of the hR1280 systems.4k Reproducedwith permission from ref 4k. Copyright 2009 American ChemicalSociety.

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from DFT simulation results4k,l is in reasonable agreement withthe Parakhonskiy et al.’s coexistence points, within theexperimental uncertainty, if one takes into account that phononfrequencies computed in the absence of POS are likely anunderestimate of those of the real material.66 In addition wenote that a recent study of the specific heat of α- and β-boroncombined with the heat of transformation from α- to β-boronpublished in 200559 led to a tentative conclusion that β-boronis thermodynamically stable at all temperature belowmelting.64e

In the next section, we describe the cluster expansion (CE)model Hamiltonian developed in ref 4k, which led to thefinding of a nearly degenerate ground state in β-boron.

5. GEOMETRICAL FRUSTRATION INβ-RHOMBOHEDRAL BORON

The general form of a CE Hamiltonian is

∑ ∑ ∑= + + + ···H C US J S S T S S Si

i ii j

i j i ji j k

i j k i j k,

,, ,

, ,(1)

where C, Ui, Ji,j, Ti,j,k are interaction parameters; whenexpressing the internal energy of a binary alloy, Si takes twovalues, −1 or 1, corresponding to the two elements in the alloyat the crystal lattice site i. When applying eq 1 to thedescription of POS occupation of β-rhombohedral boron, thetwo values of Si, −1 or 1, correspond to unoccupied (vacancy)or occupied (interstitial) sites, respectively. The highest order−interaction term considered in ref 4k was a three-body termthat was limited to a B13−B13−B13 interaction within anindividual B28 unit. For the pair-interaction term, a cutoff radiusof 3.5 Å was selected, which is slightly larger than the secondnearest neighbor distance.Ogitsu et al.4k used systems consisting of 105, 106, and 107

atoms in the rhombohedral cell and generated 2,591independent configurations to fit the interaction parameters,C, Ui, Ji,j, and Ti,j,k to DFT total energies. Using theseparameters, a Hamiltonian for a supercell composed of 1280atoms was constructed, and Monte Carlo−simulated annealingcycles were performed; then DFT structural optimizations ofthe atomic positions and the cell parameters of the resultingPOS structures were carried out, and the optimized totalenergies were used to improve the CE Hamiltonian. Two fittingcycles provided a series of POS−occupation configurations, asshown in Figure 13. The discrepancy between the DFT totalenergies and CE energies was less than a few meV/atom.Interestingly, if the Hamiltonian is constructed with only a

pair interaction, J, it reduces to a spin Ising model. For thisreason, a CE Hamiltonian is sometimes called a generalizedIsing model. The CE boron Hamiltonian provides an analogybetween the partial occupancy in β-boron and spin magnetism,and the interstitial-vacancy correlation in β-boron correspondsto the antiferromagnetic (AF) correlation in spin magnetism.Remarkably, the POS problem of β-boron is found to be welldescribed by an AF spin Ising model!In 2010, Ogitsu et al.4l identified an underlying model of the

defects (POS) configurations of β-boron, which showed thatthe POS form a quasi two-dimensional network consisting oftwo layers of an expanded kagome lattice.67 This discovery iscrucial in understanding the observed near degeneracy in theoccupation of POS in β-boron. The expanded kagome latticesometimes called a star latticeis based on a hexagonal latticewith its corners decorated with triangles (see Figure 1d of ref

4l). When these triangles are expanded and their corners touch,the lattice geometry becomes that of a kagome lattice (seeFigure 2 of ref 4l).For some Ising models,68 exact solutions have been found,

and some of them are known to have unique properties: Anexactly degenerate and disordered ground state with amacroscopic amount of residual entropy due to geometricalfrustration, which is, in essence, a conflict between interactionand lattice geometry. We remind readers that the AF Isingmodel on a square lattice has a checkerboard-type orderedground state. Examples of geometrical frustration include AFIsing models on a triangle lattice69 and on a kagome lattice.70

The essence of ground-state degeneracy is often explainedusing three Ising spins at the corners of a triangle interacting viaan AF−nearest neighbor interaction.In the following, unless otherwise noted, we limit the range

of the interaction to nearest-neighbor sites for simplicity.We denote the Ising Hamiltonian by

∑=H JS Si j

i j, (2)

Si denotes the spin state at the site i, and it takes values of 1or −1. J is the interaction parameter. When J > 0, an AF pairwill provide an energy J × (1) × (−1) = −J. On a triangle, oncean AF pair is formed, the total energy (E) does not depend onthe state of the third spin. For example, with a configurationup−down−down, E = −J − J + J = −J, and with up−down−up,E = −J + J − J = −J. This system has a degenerate ground state.A remarkable property is that, as the lattice size grows, the

number of energetically equivalent configurations grows

Figure 14. The α−β phase lines (THEORY) reported in ref 66 andrecent high-pressure experimental measurements (EXP).3c Theexperiments were conducted at the P,T points represented by thesymbols:3c α-boron was found to be stable at the crossings. β-Boronwas found to be stable at the open circles. The filled diamondsrepresent the inferred experimental α−β coexistence line. The linescorrespond to theoretical α−β phase boundaries.66 The dashed linewas obtained without a scaling correction to the PDOS of β-boron.The dashed-dot line was obtained with the maximum scalingcorrection so that the α−β phase line intersects P = 0 GPa at T = 0K. The solid line was obtained by applying a scaling correction thatcorresponds to an increase in the ZPE of β-boron of approximately +5meV/atom. The γ-phase and T-phase are omitted from this plot forclarity. This figure was published in Tadashi Ogitsu and EricSchwegler. The α−β phase boundary of elemental boron. Solid StateSciences, 14 (11-12), 1598−1600. Copyright 2012 Elsevier MassonSAS. All rights reserved.

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exponentially. Therefore, the entropy of the system describedby eq 2 has a finite value at 0 K. Note that if the number ofdegenerate configurations does not grow exponentially, theentropy of a macroscopic system will vanish even in thepresence of degeneracy. For an AF Ising model on a trianglelattice or a kagome lattice, the entropy S (T = 0 K) is0.323R69b,d and 0.5R,70 respectively, where R is the gasconstant. Triangular and kagome lattices do not exhibit aphase transition, and their spin configuration is alwaysdisordered, regardless of the temperature.In ref 4l, it was shown that an arbitrary ground-state

configuration of an AF Ising model on a kagome lattice can bemapped onto that of an AF Ising model on an expandedkagome lattice. It was also shown that the POS-occupationHamiltonian of β-borona nearest neighbor AF IsingHamiltonian on an expanded kagome latticepossessesgeometrical frustration. This model does not exhibit a phasetransition, and the solution is completely disordered at alltemperatures.The CE Hamiltonian that describes the POS-occupation

configuration of β-boron is substantially more complicated thanthe simplest form used in eq 24k,l (hereafter called the β-boronIsing model Hamiltonian). The Hamiltonian includes 40interaction parameters, a complex lattice geometry, and twolayers of expanded kagome lattices connected via an invertedpair of B28’s. Consequently, the properties of the simple Isingmodel that are inherited in the β-boron Ising modelHamiltonian are not immediately apparent, but appropriategroupings of the POS elucidate the nature of geometricalfrustration in this system.4l

The relative occupation rates of POS and their total energiesobtained from self-consistent optimization cycles using CEMCand DFT−total energy calculations were somewhat surpris-ing.4k First, the calculations suggested many nearly degeneratePOS-occupation configurations, with no favored long-rangeorder when using a supercell containing 1280 boron atoms (2× 2 × 3 of the rhombohedral unit-cell). Second, the relativeoccupation rates vary across a wide range but with a smallenergy change. Third, occupations of either a B17−B18 pairingor B19 alone are not particularly favored. Instead, theseoccupations almost always appeared simultaneously in a singlesupercell calculation. Therefore, on average, the number ofB17−B18 pairs is smaller than one per rhombohedral cell,which is reflected in the concentration of holes being less thanone in the electronic structure of the rhombohedral cell. Thelow-energy structures had well-defined band gaps, with an Eg =0.8 eV in the lowest one, which is in good agreement withexperiments when the underestimate of the band gap due to theLDA is taken into account. Moreover, the number of observedhole states in the low-energy structures was always the same asthe number of B17−B18 pairs, when the favored short-rangecorrelations were fully developed, which supports thehypothesis that the local rehybridization around the interstitialPOS dictates β-boron electronic requirements.Clearly boron’s preference for 3c2e bonds4k,13a,c plays a

crucial role in stabilizing β-boron. One interesting question thatarises in trying to understand POS geometrical arrangements isthe following: Why do POS form a 2D and not a 3D network?β-Boron has many ring-type interconnections between itsbuilding units, B12 and B28. As we discussed earlier, thehexagonal ring-type interconnections occur between interstitialsites, leading to perfect self-doping via the conversion of 2c2ebonds to 3c2e bonds; therefore, one might speculate that

interstitials are the preferred POS. Very interestingly, not all ofthe hexagonal rings are occupied, and in the hexagonal ring thePOS occur approximately within an X−Y plane at z = 0 and z =c/2, thereby making a quasi two-dimensional network (seeFigures 11 and 15). If these locations within two-dimensional

planes were not favored over the rest of hexagonal rings, thePOS network would not form a double layer-expanded kagomelattice. Instead, it would form a three-dimensional network (seeFigure 15).The two-dimensional POS network probably arises because

of the role played by the self-interstitials, B17−B20, thatsaturate the dangling bonds created by B13 vacancies. Theother hexagonal ringslocated far from the B13 sites (redhexagons in Figure 15 marked A, which connect two B12s and aB28)would not efficiently saturate the dangling bonds (seeFigures 21 and 22). Therefore the B13-vacancy formation,whose mechanism Tillard-Charbonnel et al.16a rationalized, is acrucial component of geometrical frustration in elementalboron.A crucial role is also played by B16 sites. As a few reports

discussed, occupation at B16 compensates for the electrondeficiency of the B12 icosahedra and counters the imbalance ofelectron deficiency between B12 and B28.

4k,15b In addition to theB16 sites, the bottom of a pyramid made of four B12s (see thered hexagon labeled BP in the lower right part of Figure 15)could also connect the B12s. The B16s are located at the side ofa pyramid, and they form a triangle in the X−Y plane. If thebottom of the pyramid (BP) were occupied, the POS networkwould not be an expanded kagome lattice. It remains unclearwhy B16 can be occupied but BP is not. Nonetheless, B16occupation is consistent with the fact that the B19−B16interaction is repulsive (AF), because the distance between theBP site and the closest B19 is smaller than that of the favoredB19−B16 occupation configuration, where a B16 is occupiedon one of the far sides of an occupied B19 triangle (see Figure16). From these observations, it seems reasonable to assumethat occupying the far side of B16 (as compared to the BP site)provides a more homogeneous charge distribution, whichmakes the system more stable.

Figure 15. B12s (green) and B28s (gold) icosahedra are connected byhexagonal rings (red and blue), which can exhibit perfect self-dopingby converting 2c2e bonds into 3c2e bonds. Only the blue hexagons arePOS; the red ones are not occupied at all.

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5.1. Specific Heat and Entropy of β-Boron

In order to understand the consequences of geometricalfrustration on the entropy of boron, Ogitsu et al.4l performedreplica exchange Monte Carlo simulations on the β-boron Isingmodel for a wide range of temperatures, from below 1 K to 108

K, and calculated the constant volume−specific heat as afunction of temperature. The entropy of this model as afunction of temperature was calculated by standard thermody-namic integration, where infinite temperature was chosen as thereference:

∫= = ∞ + ′ ′′

∞S T S T T

C TT

( ) ( ) d( )

T (3)

These numerical simulations confirmed that the β-boronIsing model Hamiltonian does not show a phase transition atany temperature, and the POS-occupation configurations arecompletely disordered at all temperatures under equilibriumconditions.In real β-boron samples, X-ray diffraction measurements

show no long-range ordering or any sign of a structural phasetransition in the occupations of the POS. Therefore, one mightbe tempted to conclude that the thermodynamic properties ofthe frustrated β-boron Ising model would explain the nature ofthe occupation of the POS in β-boron, but the followingobservation adds a twist. When Slack et al.9c performed X-raystructural analysis, they used three samples that had been left atambient conditions for long periods of time: roughly 2, 5, and17 years. In spite of such a long “annealing” time, those samplesshowed a small yet measurable difference in the relative POS-occupation rates, indicating that the relaxation time scale ofself-interstitials is extremely long.We note that diffusion of interstitial atoms over a wide range

of temperature does not necessarily contradict the observationslisted above. A recent study on hopping barriers of POSrevealed the following:4l The barrier within the samecrystallographic site, B16, is low enough for interstitials todiffuse even below 200 K, while ones between differentcrystallographic sites are high enough to prevent them fromhopping at around ambient temperature. Hopping betweendifferent crystallographic sites would lead to a change of the

occupation rate of POS, while hopping between the same sitesdoes not. This may explain why there is seemingly contra-dictory experimental evidence regarding the diffusion ofinterstitials at a relatively low temperature (ambient T andbelow). At higher temperature, T = 550 K, the recent specificheat measurements revealed that hopping of interstitials withbarriers as high as 4.75 eV can take place.21

Specific-heat measurements are the most direct experimentalmethods to study the nature of geometrical frustration, becausethey reflect the number of states accessible at a giventemperature. Even at low temperature, a macroscopic numberof states exist in a system with geometrical frustration. For asolid, the measured specific heat consists of a phonon,configurational and an electronic contribution. β-Boron isknown to be a semiconductor, and, at least at a low temperature(for example, below 200 K where the optical absorption showsa minimum amount of gap levels20l) one may ignore theelectronic contribution. In this case, by subtracting thevibrational contribution from the total specific heat, one canobtain the configurational specific heat. Figure 17 depicts the

configurational specific heat and the entropy of β-boron as afunction of temperature. The figure also shows the total specificheatthe sum of the lattice-phonon contribution and the POS-configurational contributionalong with experimental values.The overall agreement between the experiments and theory isexcellent, but the contribution from the POS configuration isvery small compared to the lattice-phonon one, and aconfirmation of POS diffusion at low temperature by specificheat measurements has yet to be accomplished.The Arrhenius plot of the conductivity of β-boron is known

to show two kinks, T ∼ 200 and 550 K,10a,12c,d,21,71 (related toanomalous behaviors discussed later), and the slope of ln σ(1/T) significantly increases at each kink. It is also known that thenumber of electronic states within the band gap increasessignificantly above T ∼ 200 K.20l,m Therefore, it is natural to

Figure 16. One of the ground state−POS configurations obtained inab initio calculations, where the AF correlation between the sites B13,B19, and B16 can be seen. Site BP, which is not a POS, is located atthe center of a B16 triangle (slightly off the plane). The distancebetween the B19 interstitial and the B16 (or BP) sites is the longest forfavored sites and the shortest for unoccupied sites, consistent with theAF (repulsive) interaction between B19 and B16 (BP) sites.

Figure 17. The configurational specific heat (a) and the entropy (b) ofthe POS. The theoretical total specific heatthe sum of phonon andconfigurational specific heatand experimental values (c).4l Note: theβ-boron’s electric conductivity is known to rise significantly at hightemperature (see text for more detail), therefore, most likely theelectronic contribution needs to be considered in the high temperaturerange. Omitting the electronic contribution is an approximation validonly at relatively low temperatures. Reproduced with permission fromref 4l. Copyright 2010 American Physical Society.

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expect that the electronic contribution to the specific heat willalso increase at higher temperatures, much more so than in anormal semiconductor. In fact, it was reported that the hightemperature specific heat of β-boron64b exceeds the Dulong-Petit value of 3R.64e

The precise determination of the specific heat of β-boron forall temperatures below melting, together with its electroniccontribution, is highly desirable for the establishment of thethermodynamic properties of this material.

6. ELECTRONIC STRUCTURE OF β-RHOMBOHEDRALBORON

In this section, we discuss the electronic structure of β-boron,particularly in relation to the stable POS occupationconfiguration presented in the previous section. We first offera simplified explanation and then give a detailed analysis.As mentioned earlier, boron displays a p-type semiconduct-

ing electronic property10,72 with an optical-excitation gap of1.4−1.6 eV.9c,12c,d,73 A semiconductor-like band structurepersists in β-rhombohedral boron when POS are not included;however, the valence bands are not fully occupied. The numberof hole states is 15 per hexagonal cell (three times therhombohedral unit cell); they could be filled if one added fiveboron atoms to the hexagonal cell. In fact, the experimentallydetermined atomic density of β-boron, 320 atoms-per-hexagonal-cell, contains 5 atoms more than those of a cellwithout POS, 315 per hexagonal cell. However, it is known thatthe number of interstitial atoms per cell in β-boron is morethan five since B28 units show a significant amount of vacanciesat B13 sites. Surprisingly, neither an introduction of B13vacancies, nor an introduction of interstitials alter the numberof valence states except for one case, that is, if sites B17−B18are occupied, which introduces an additional valence state perpair. The tendency of boron to form 3c-2e bonds plays a crucialrole in preserving the number of valence states uponintroduction of interstitials.4k

The interstitial sites in β-boron act as almost perfect self-dopants. Consequently, at the atomic density of 320 atoms perhexagonal cell, β-boron becomes a p-type semiconductor. The

hole states responsible for the p-type character originate fromthe B17−B18 pair occupation.We note that, most likely, the following observations74 are

indirect evidence of the stabilization mechanisms of β-boron viaPOS occupation. It was reported in 2012 that n-type doping ofβ-boron is compensated by a change of POS occupation suchthat a semiconductor-to-metal transition is prevented,74d andsimilar phenomena have been known for some time.74a−c

However, doping by certain types of transition metals does notlead to a simple charge transfer mechanism,75 and in these casesthe above argument used for n-doped samples is unlikely toapply.A detailed electronic band diagram based on an extensive

amount of experimental data is presented in ref 25 (andreferences therein): there is a band gap of about 1.5 eV withtwo clear gap levels near the valence band top and six traplevels. It was proposed that the gap levels originate fromPOS;15a however, the detail has yet to be fully understood.These authors of ref 24 characterized the electronic structure ofβ-boron as that of a crystal with defects, while an “amorphousconcept” was proposed by other groups.76

In the following, we give more details about boron’selectronic structure, including a historical perspective. Notethat the electron-counting rules commonly used in borane andboride chemistry are not discussed here, but review articles thatcover the topic are available.24,77 Instead, the concept ofmaximally localized Wannier functions (MLWF)78 will be usedto analyze the chemical bonding.

6.1. The Electronic Requirement of β-Boron without POSRationalized with B12 and B28 Clusters

We first consider the preference of boron to form icosahedra,clusters made of 12 boron atoms.5b,14 Note that a B12icosahedra cluster is not stable as an isolated form, because aclosed shell−electronic structure may be obtained only whenexternal bonds are formed. Most interestingly, if all of theexternal bonds are 2c2e bonds, this unit lacks two electrons tobe a perfect closed shell−electronic structure, which suggeststhat a B12 boron icosahedron might display an anionic characterin the condensed phase.5b The counterpart, a cation-like boron

Figure 18. Schematic diagrams of the electron requirements of building units B12 (left), 2(B28)B (center), and the solid phase β-hR105 (right)without POS; the diagrams illustrate how the total electron requirement of 320 electrons per unit cell can be described by units of B12 and 2(B28)B.

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cluster, can be realized in the triply fused icosahedron, B28,which has an excess number of electrons for the number ofchemical bonds involved15b,16a (see Figure 18).In 1955, Longuet-Higgins and Roberts rationalized the

electronic requirements of boron icosahedra.5b On the basis ofMO theory, they concluded that B12H12 has an open shell−electronic structure; therefore, it is not stable as a neutralmolecule. They suggested that an ion, B12H12

−2, might exist.Evarhardt, Crawford, and Lipscomb reached a similarconclusion.13b Five years later, Pitochelli and Hawthornesynthesized K+2(B12H12)

−2, which confirmed that B12H12−2 is

indeed stable.14 The electronic structure of the B28 cluster unitwas not fully understood until 2000 (see Figure 19).16a

The electronic structure of the triply fused icosahedra, theB28 unit, was explained through extensive research onintermetallic compounds, such as K4Na13Ga49.57,

16a,54a,d−g,80 inwhich an isostructural cluster unitthe triply fused icosahedramade of gallium atoms (2Ga28)Gaforms. A description of Gaclusters found in intermetallic compounds helps to understandB28 in boron. In a subgroup of these compounds, namely, p-block intermetallic−Zintl phase compounds, the charge transferbetween low-Z metal and p-block element such as gallium issupposed to be complete. Therefore, one may deduce thecharge state of the cluster in the compound form, and a verysimilar stoichiometry is found in some salts, such asK+

2B12H12−2.14

Extended Huckel calculations on an isolated Ga28H18 clustershowed that, when the external bonds of Ga28 are completedwith 2c2e bonds, they have more than enough electrons to fillout the bonding orbitals, which means that the triply fusedicosahedra, Ga28, are cationic. Therefore, they are unlikely to bepresent in pure form in a Zintl phase−intermetallic compoundin a reducing environment. Interestingly, the triply fusedicosahedra in those compounds have vacancies at the atomicsites corresponding to the B13 vacancy site in a B28 unit of β-rhombohedral boron.16a The local/global bonding/antibondingcharacter of an electronic state associated with this vacancy sitein Ga28 was analyzed using a crystal orbital overlap population

(COOP) approach or its molecular version, MOOP, whichshowed the following.16a First, without the vacanciescorresponding to B13, the valence electrons will occupyglobally antibonding electronic states. Second, the vacancysites are in a triangular configuration, and the HOMO band haslocally antibonding character within the triangle. These siteshave globally bonding character, such as that between B13 andits surrounding atoms. Removal of a vacancy atom does notchange the number of bonding states in Ga28; therefore, anintroduction of a vacancy at this site lowers the Fermi level ofthe system and reduces frustration by removing the localantibonding orbital. Unfortunately, studies of Zintl phases didnot confirm a precise electron count for (2Ga28)Ga becausethose intermetallic compounds tend to have additionalvacancies, which makes it difficult to compare the theoreticalelectron count and the experimental value estimated from thematerial’s stoichiometry. Nevertheless, a qualitative explanationexists: Perfect Ga28 will have an excess of electrons occupyingglobal/local antibonding orbitals, which appear around thebottom of the conduction band, and the introduction ofvacancies at the B13 sites will stabilize the system by notoccupying the global antibonding orbitals and by removing thelocal antibonding orbitals close to the top of the valence band.In 2001, Jemmis and Balakrishnarajan15b confirmed an accurateelectron count for (2B28)B based on a combination of anempirical electron-counting rule and MO cluster calculations.When all of the external 2c2e bonds are formed, (2B28)B hasthree more electrons than needed for a closed-shell structure.There are effectively four B12 units in a unit-cell of β-boron,each of which lack two electrons to be a closed shell, and thatmakes a total of eight electrons short per cell for the icosahedra.With one (2B28)B unit per rhombohedral cell, the total electroncount for β-rhombohedral boron without partial occupancy is−8 + 3 = −5 relative to a closed shell, which is in perfectagreement with an earlier estimate based on bulk-phase,electronic band−structure calculations (see Figure 18). Jemmisand Balakrishnarajan also showed that one vacancy per B28 unitat B13 lowers the total energy of an isolated (2B28)BH36 cluster

Figure 19. Density of states (DOS), projected DOS and crystal orbital overlap population (COOP) of the Ga28 cluster that appear in the Zintl phase− intermetallic compound K4Na13Ga49.57.

16a The Ga28 DOS (left) dominates the contribution to the total DOS near the Fermi level. The COOPanalysis (middle) shows that the observed vacancies at A, B, and C (or A′, B′, and C′) (right) are due to local antibonding orbitals and to the excessnumber of bonding electrons of Ga28.

16a Reproduced with permission from ref 16a. Copyright 2000 American Chemical Society.

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(to be precise it should be (B27)B(B28)H36 or (2B27)BH36),where the electron requirements, or the number of bondingorbitals, do not change by introducing a vacancy at the B13 site.These researchers also proposed that since B12 and B28 haveopposite ionic characters, moving a B13 atom to a B16interstitial site may reconcile the imbalance in the electronrequirements (or ionicity), since B16 sites are surrounded byB12 icosahedra. They further suggested that five additionalatoms per hexagonal cell occupying the POS identified by Slacket al.9c will make β-rhombohedral boron a closed shell−electronic structure. The suggestion that interstitials areresponsible for filling the valence bands of β-boron, wasmade earlier by Werheit and Schmechel in 1999.15a Note thatfive atoms per hexagonal cell correspond to 5/3 atoms perrhombohedral cell, which add exactly five electrons perrhombohedral cell. The discrepancy between the experimentalobservation of β-boron being semiconducting and earliertheoretical work9b seemed to have been reconciled, despitethe lack of an explanation for why the additional self-interstitialsdo not change the number of valence states per unit-cell. Thework of ref 15b did not discuss whether partial occupancy ispurely an entropic effect at high temperatures or if itsoccurrence lowers the internal energy of the solid.The first theoretical electronic-band structure calculation on

β-boron, without taking partial occupancy into account, waspublished in 1982,9b and it suggested an incomplete valencebandfive electrons per rhombohedral cell short of a closed-shell structure. Such an insufficient number of valence electronsshould generate a metallic character rather than a semi-

conducting one. Most importantly, such a crystal structurewould be unstable because of the incomplete formation ofchemical bonds, which seemed to have been the case with theT-50 allotrope. Bullett9b suggested that the disorder reportedexperimentally, e.g., the presence of partial occupancy, might beat the origin of the discrepancy between the experimentalobservations and the early theoretical calculations of theelectronic requirement (i.e., a stable allotrope should have aclosed shell−electronic structure).9b Note that B17−B20interstitials were not identified until 1988;9c therefore, therewas not enough evidence available to Bullett to preciselyidentify the mechanism leading to the formation of a closedshell electronic structure.In 1991, King attempted to explain how β-boron could form

a closed shell system, and he proposed a scenario involving achange in the number of chemical bonds.54c

6.2. DFT Simulations Begin to Explain the Nature of POS

In 2002, Imai et al.81 reported the first electronic band structureof β-boron with POS, based on DFT; the precise locations ofthe atoms with POS for their three β-boron models were notavailable.As mentioned before, in 2007, van Setten et al.4f showed that

a B19 occupied site near a B13 vacancy significantly reduces theinternal energy of β-boron, because it yields a nearly closedshell−electronic structure, but the band gap, Eg = 0.35 eV, issignificantly smaller than the experimental value, about Eg = 1.5eV, even considering the well-known underestimate due to theapproximation used for the exchange-correlation energy.

Figure 20. A schematic illustration describing how the occupation of B16 interstitials (gray spheres, upper right) self-dope β-rhombohedral boron, asdescribed by maximally localized Wannier functions (MLWFs).4k The hole states in the valence band of β-boron (red area in the plot of electronicdensity of states, middle) correspond to the inter-B12 2c2e bonding (red isosurfaces, upper left). Upon occupation of a B16 site, three 2c2e bonds(red isosurfaces) are converted to three 3c2e bonds (blue isosurfaces). Each B16 interstitial supplies three valence electrons without changing thenumber of valence bands (or bonding states).

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In 2008, Widom and Mihalkovic4i found a different stablestructure based on B17 and B18 occupations next to a B13vacancy pair. Their band structure has one hole state perrhombohedral cell with Eg ∼ 0.8 eV, which is in betteragreement with experiments. In refs 4f and i, the mechanismthat keeps the number of valence states almost constant wasnot described.In 2009, Ogitsu et al.4k studied the local bonding properties

of the configurations identified by their CEMC search, by usingmaximally localized Wannier functions (MLWF) and showedtwo types of rehybridization mechanisms: One keeps thenumber of valence states unchanged and the other introducesan additional valence state (see Figures 20−22).We note that to rationalize the chemical bonding in borane

and boride, researchers have developed many types of electron-counting rules.15b,82 Those approaches were very powerfultools for rationalizing or predicting the stable forms of boronclusters. However in β-boron, POS are located at the linksbetween the cluster units, so the separation into fragments withuniquely defined electron counts faces great challenges. This iswhy an alternative approach based on MLWFs, which is bettersuited for describing condensed phases,4k was adopted.The MLWFs are related to the eigenfunctions of Kohn−

Sham orbitalsthe solution of the DFT Hamiltonianby aunitary transformation, where the unitary matrix is defined so asto minimize the sum of the spreads of the transformedfunctions (Wannier Functions).83 Note that conventionalWannier Functions are expressed as the Fourier transforms ofthe Bloch states, where each function is localized within theprimitive unit-cell. Upon successful “maximal localization,” onemight often relate a MLWF to a chemical bond based on thelocation of the center of the MLWF and its spread. ThereforeMLWFs generalize the concept of Boys orbitals used inquantum chemistry.84 A MLWF is “occupied” by two electrons

when it is constructed from the occupied Bloch states, andwhen spin degeneracy is assumed. When a MLWF appears tobe well localized between two atoms, it is natural to assume thata 2c2e-type covalent bond is formed. This is the case indiamond and silicon where each MLWF is localized at one offour sp3 covalent bonds. Therefore, given the locations of thecenters of MLWFs and the spreads in diamond and silicon, onecan unambiguously identify the types of bonding configurationsin these systems.4k A 3c2e bond of boron can be described as abonding orbital consisting of a linear combination of threeatomic orbitals, where the bonding charge is concentrated atthe center of three atoms.13c

When the bonding orbital becomes delocalized and thenumber of bonds in a system becomes incommensurate with itsgeometrical feature (such as the number of edges and/or thenumber of faces of the cluster) the bonding-propertyinformation from the MLWFs is limited, perhaps reflectingthe full quantum mechanical nature of chemical bonding. Inthis case, the number of bonding electrons can still beestimated from the MLWFs, but an individual MLWF cannotbe interpreted as a unique chemical bond. For example, aB12H12 cluster has 13 intracluster bonding orbitals, a numberthat is incommensurate with the 30 edges and/or 20 faces.5b,13b

Those bonding orbitals are delocalized over the cluster, andsometimes interpreted as metallic bonds.13c

In bulk β-rhombohedral boron, the number of MLWFsbelonging to a B12 is 13, which is in agreement with empiricalelectron-counting rules.4k Here the criterion for “belonging to aB12” are so defined: The centers of MLWFs are located insidethe polyhedron defined by the planes that are normal to thevertices of the B12 and passing through the corner of B12. Wenote that the electron count for (2B28)B in bulk β-boronobtained from MLWF was confirmed4k to be in agreement withthe results of Jemmis et al.,15b who used an empirical electron-

Figure 21. The stabilization mechanism due to the formation of the B13 vacancy and the B19 interstitial pair is described using MLWFs.4k A B13vacancy is favored due to the local antibonding character of the orbitals (see Figure 19); dangling bonds (red isosurfaces) surround the B13 vacancy(left). Occupation of the B19 interstitial site that is closest to the B13 vacancy stabilizes the system by saturating the dangling bonds and byconverting three 2c2e bonds to three 3c2e bonds (blue isosurfaces surrounding the B19 interstitial, right). In this case, the B19 occupation adds threevalence electrons without changing the number of valence states (or bonding states). The energy gain due to such a vacancy-interstitial pair, in thelanguage of spin magnetism, corresponds to an antiferromagnetic interaction between B13 and B19 sites.

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counting rule and MO calculations on the (2B28)B clusterfragment.The main results from a MLWF analysis are the following.

When the centers of hexagonal rings (B16, B19, and B20) areoccupied, three 2c2e bonds are converted to three 3c2e bonds,but the number of bonding states does not change (see Figures20 and 21). Boron’s preference for 3c2e bonds4k,13a,c plays acrucial role in stabilizing β-boron. On the other hand, whenB17 and B18 are occupied in a paired configuration, thenumber of bonding states increases by one (see Figure 22). TheB17−B20 occupations saturate the dangling bonds in the B28unit created by the B13 vacancy, thus favoring the locationsclose to B13 (see Figures 21 and 22). The occupation of thehexagonal-ring site leads to perfect self-doping. The threeelectrons per boron atom increase the energy of the Fermi levelwithout changing the number of valence bands, and at theexperimental atomic density320 atoms per hexagonal cellβ-boron should become a closed shell−electronic structuresystem. On the other hand, a B17−B18 pair occupationintroduces an additional valence band, promoting a hole state.Therefore, it is important to determine the optimal relativeoccupation rates between B17, B18, and B19 (or B20) to makecomparisons with the experimental observations of a p-typesemiconducting character10,72 with a very low conductivity andan optical gap of about 1.5 eV.9c,12c,d,73

In summary, the electron counts for B125b,14 and (2B28)-

B15b,16a estimated from the cluster-fragment approach are notaltered in the bulk phase.4k,9b The introduction of a B13vacancy does not change the electron count for B28.

15b,16a

Consequently, β-boron, without POS, has an insufficientnumber of valence electronsfive electrons per rhombohedralcellfor a closed shell−electronic structure. These results4k arein agreement with the proposed electron counts of β-boronusing the cluster fragments15b,16a and older electronic−bandstructure calculations.9b

6.3. Atomic Density and Favored POS Configurations:What Are the Implications?

As we discussed in the previous section, the stable B19occupation does not change the electron count, and a B17−B18 occupation increases the number of bonding orbitals (orvalence bands) by one.4k As a consequence, the relative stabilityof these two occupation configurations depends on thechemical potential of the system, since they have a differentnumber of bonding orbitals. At a lower atomic density, thesystem has fewer valence electrons, thus favoring a B19occupation over a B17−B18 occupation;4f the oppositepreference arises at the higher atomic density.4i

The occupancy of the B13 site also impacts the structure ofother units in β-boron. For example, if the Fermi level is lowenough so that the HOMO−LUMO orbitals of the B28 clusterunit (which have local antibonding character) are not occupied(see Figure 19), then the B13 site is occupied. In that case, thenearby B17−B20 interstitials, which serve to terminate thedangling bonds created by a B13 vacancy, are not presenteither. Accordingly, B17−B20 occupations are not the mostfavorable structure in the 106-atoms systems, while the B19occupation becomes the most stable since it forms the closedshell−electronic structure at the experimental atomic density,106 2/3 atoms per rhombohedral cell.4f

The atomic density affects the density of the B13 vacanciesvia the location of the Fermi level with respect to the HOMO−LUMO levels of B28, which in turn dictate the B17−B20occupations.4k

These considerations lead to an interesting hypotheticalsituation: If B28 did not exhibit a local intrinsic instability at theB13 site, β-rhombohedral boron could have been a stable andelectron-precise semiconductor with only one type of POS,B16, at the atomic density of 320 atoms per hexagonal site;indeed the presence of a B16 occupation leads to perfect self-doping by converting the bonding from 2c2e to 3c2e.

Figure 22. The stabilization mechanism due to the formation of a complex B13 vacancy pair and a B17−B18 interstitial pair is described usingMLWFs.4k Two B13 vacancy pairs (two red spheres, left) in the opposing B28 units may be introduced to stabilize the (2B28)B unit (see Figure 19),but at the expense of introducing dangling bonds (red isosurfaces, left). These dangling-bond states can be completely saturated by the introductionof a B17−B18 interstitial pair (purple spheres, right), leading to the introduction of an additional bonding state. In this case, no clear formation of3c2e bonds is observed. This stabilization mechanism does not provide perfect self-doping and leaves hole states in the valence band at theexperimental atomic density of 320 atoms per hexagonal cell.

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Therefore, the local frustration within the B28 unit apparentlycontributes to the complexity of β-rhombohedral boron.Another interesting aspect of the stable occupation of the

POS is that the ab initio/CEMC optimization procedureresulted in the persistent presence of B17−B18 pairoccupations. This increases the number of valence bands,leading to a small amount of hole states, which prevents thesebands from being perfectly electron precise.4k If the trueground-state POS configuration of β-boron is required to beelectron precisea perfect semiconductorthe atomic densitymust be slightly higher than 320 atoms per hexagonal cell.

7. ANOMALOUS PROPERTIES OF β-RHOMBOHEDRALBORON

Many experimental studies have reported that β-boron exhibitsanomalous transport and optical properties.Tsagareishvili et al.20a performed internal-friction experi-

ments for a wide range of temperatures and reported the mostdirect evidence of diffusion of boron defects. They measuredthe inelastic response of the system during periodicaldistortions. The origin of an inelastic response is usuallyinterpreted as originating from defect diffusion, which dissipatesenergy. In ref 20a, clear peaks are reported at temperatures of150 K and 530 K. The activation energies, Ea, for those peakswere determined from their temperature dependencies, Ea =0.13 eV and Ea = 1.3 eV, respectively. The authors state that nofirst order−phase transition could be detected in the samples ofβ-rhombohedral boron by thermal or X-ray analysis. Later, theyreported more peaks with some evidence that impurities mightcontribute to the inelastic response.20k,85 These experimentalobservations seem to support the notion of boron-defectdiffusion over a wide range of temperatures.Werheit and Franz20l,m reported a dramatic change in the

optical absorption of β-boron within the band gap around 150−180 K. Below this temperature range, one clear absorption peakappears at E = 0.4 eV above the valence-band top; above thistemperature range, many trap levels appear over the entirerange of the band gap. Exposure to light changes the absorptionprofile, and the corresponding relaxation time scale was foundto be very long, on the order of several hours or more at aboutT = 100 K. Consequently, the sample had to be left in the darkfor a long time before making measurements. Werheit andFranz suggested that the peculiar behavior of β-boron’s opticalproperties may be explained by the diffusion of boroninterstitials.Lonc and Jacobsmeyer86 measured magnetoresistance as a

function of temperature and magnetic field. The results areinteresting and puzzling. Single-crystal measurements shownegative (positive) magnetoresistance in low (high) magneticfield. For a fixed magnetic field, the temperature dependence ofthe magnetization shows a peak located at ∼100 K. On theother hand, polycrystalline measurements showed only positivemagnetoresistance, with the same peak position of themagnetization as in the single-crystal measurements. Usingoptical excitation at T = 77 K to set up the initial conditions ofthe sample, Zareba and Lubomirska20b measured the thermallystimulated current and observed peaks at T = 117 and 214 K.The peak at 117 K appears after a long duration of opticalexcitation. Geist20g reported on the temperature dependence ofthe photoconductivity and on the photoelectron paramagneticresonance (photo-EPR); the former showed different temper-ature profiles at high and at low temperatures. Moreover, Geistnoted that the concentration of the gap level at E = 0.23 eV

above the valence band top was about one state perrhombohedral cell. He also suggested that the peculiarity ofhis observation might be related to the incommensuratenumber of electrons and bonds in β-boron. Tavadze et al.20i

measured the temperature dependence of the dielectric loss at50 Hz and found a peak at T = 213 K with Ea = 0.18 eV. Theyobserved another peak at T = 233 K with Ea = 0.23 eV, whichcompletely disappeared by annealing at T = 978 K with H2.Young, Oliver, and Slack20c measured the temperature andfrequency dependence of microwave elastic losses, whichshowed a steep rise at a temperature of about 100 K. Kubleret al.20e measured the magnetic susceptibility over a wide rangeof temperatures, and their results indicate that impurities arenot likely to be responsible for the observed gap levels.In addition, the following experimental reports might be

related to boron-defect diffusion. Golikova’s team measured thetemperature-dependent resistivity (ρ) and found that the slopeof log ρ/(1/T) changes at a temperature of about 200 K. Theysuggested that the conduction mechanism of β-boron is that ofa Mott type-variable range hopping.71b Moreover, the photo-current as a function of illuminance measured at varioustemperatures clearly showed distinct behaviors below andabove T = 200 K.87

In 1993, Werheit, Laux, and Kuhlman12d reported on thedetail of the optical absorption and its relaxation behavior, andfrom this they interpreted the anomalies observed in variousphysical quantities at around T = 500−600 K, including theinternal-friction peaks, elastoresistance, thermal expansion,electrical conductivity, thermal conductivity, and EPR.20a,e,f,88

Although it is not clear if all of those observations share thesame origin, the POS-hopping mechanism reported in 20104l

might be responsible for some of the anomalies, but furtherinvestigations are needed to come to robust conclusions.In 2004, Werheit and Wehmoller20n reported on the

stochastic jumping behavior of the time dependence of thedc conductivity and the photoconductivity, and they attributedthose observations to boron-defect diffusion. In 2009, Werheitand Kummer89 described fractional photoconductivity glowexperiments and argued that their results support the notion ofoptically induced structural changes in the POS. In 2012, thespecific heat of β-boron measured by differential scanningcalorimetry was reported.21 On the basis of these results, it wassuggested that B13 atoms may start to diffuse at about T = 550K. The estimate of ref 21 on the activation barrier, Ea = 4.75 eV,is consistent with theoretical values reported on B13diffusion.4l,21

The other types of reported anomalies are large capaci-tance,90 the pressure effect on dc conductivity,91 the isotopeeffect on the mechanical properties,85c,92 the kink in the verylow temperature specific heat,64d and an isotope effect onphonons that indicates nonstatistical isotope distribution.58

Note that the kink in the specific heat might originate fromlocalized Einstein phonons due to disorder in the POS; thekink in the specific heat was observed at very low T (∼1 K) andis unlikely related to defect diffusion.The most relevant explanation of the observed anomalous

properties of β-boron comes from recent theoretical studies,4k,l

which showed that the lattice of POS exhibit geometricalfrustration that leads to a macroscopic amount of states nearthe ground state (or near degeneracy). More specifically, thereare a large number of boron defect configurations, which maybe dynamically explored via their diffusion even at lowtemperatures if the barriers separating those configurations

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are low enough. Ogitsu et al. reported that there are at least twolow-activation paths: one between adjacent B16 sites andanother between adjacent B19 and B20 sites, with barriers of0.25 and 0.5 eV, respectively.4l These findings suggested thatthe low-temperature excitation at ∼150−180 K is related toB16−B16 interstitial hopping, and the higher temperatureexcitations at ∼500−600 K are related to B19−B20 interstitialhopping (as mentioned above, Hoffmann and Werheitsuggested that this excitation is due to B13 diffusion with anactivation barrier of Ea = 4.75 eV21). The most stable POSconfigurations4k showed a very clear band gap with a smallamount of hole states in the valence bands. It was also found4l

that introducing unfavorable B16 occupation, close to B19,adds a significant amount of gap levels, although it only raisesthe total energy by several meV/atom. Therefore, theinterstitial atoms at B16 sites could hop within B16 sites overa wide range of temperatures, and the hopping betweenenergetically degenerate B16 sites could take place at T < 150K. At higher temperatures, the B16 interstitial might be able tohop into a B16 next to a B19 occupation or next to a B17−B18pair, which would introduce gap levels.4l

Finally, we note that nearly identical mechanisms underliethe near-degeneracy of β-boron defects and frustrated spinmagnets, where a variety of experimental and theoreticalmethods have been employed to understand the nature of theproblem. We suggest that approaches similar to those used forspin magnets might play a crucial role in understanding thepartial occupation’s effect on boron’s mechanical and electronicproperties under various physical conditions. For example,applying a magnetic field to a frustrated spin magnet is apowerful tool to investigate the nature of frustration. In the caseof boron, changing the chemical potential of electronscorresponds to the application of a magnetic field to a spinmagnet. Given that pressure changes boron’s chemicalpotential, pressure-dependent experiments might expose thenature of boron-defect diffusion. In fact, Zhang et al.91

demonstrated that applying pressure to boron alters its electricconductivity and the temperature dependence of theconductivity.

8. ISSUES RELATED TO THE THIRD LAW OFTHERMODYNAMICS

In closing we remind the readers of the various versions of thethird law of thermodynamics. According to Linder,93 four formsof the third law of thermodynamics can be found in theliterature:(1) The original Nernst’s Heat Theorem: limT→0KΔS(T) = 0(2) Planck’s form: limT→0K(S(T)/N) = 0; N is number of

particles(3) Lewis’ form: “Every substance has a finite positive

entropy, but at the absolute zero of temperature the entropymay become zero, and does so become in the case of perfectcrystalline substances.”94

(4) Unattainability of absolute zero temperature.Despite this variety of options, Linder states that none is

“completely satisfactory from a phenomenological point ofview.”Schwabl95 formulates the third law as statements 1 and 4

with the condition that the constant value of entropy isindependent of parameters, such as pressure and volume.Within this formulation, the presence of a macroscopic amountof residual entropy does not contradict the third law.Landsberg96 provides a similar view. Gokcen and Reddy97

state that Nerst’s heat theorem is the third law, and theypropose the convention of taking zero entropy as the constantvalue, because many experimental observations, such as theentropies of allotropes, consistently show a convergence to aunique value. Dugdale98 presents a similar view. Kaufman99

states that statement 4 is the third law, and he discusses how itleads to statement 3. Although none of these textbooks describePlanck’s form as the definitive third law of thermodynamics,some older references100 do.We note here that the discussion of boron given in this

review is based on Planck’s form partly because, contrary towhat is described in the various textbooks, Planck’s form seemsmore commonly accepted as discussed in the article by Kox:101

“Nowadays it is the view of many that quantum theory providesa solid basis for the statement that at absolute zero all systemshave zero entropy and that one can formulate the Third Law inthis way. But others still confine themselves to the statementthat one cannot reach absolute zero. In spite of thesedifferences, however, there does seem to be a general consensusthat the Third Law is of a different, less fundamental and moreexperimental character than the other laws of thermodynamics.”This view arose, perhaps, partly from extensive studies on ice

and spin ice, where somewhat controversial studies on thesesystems resulted in a tentative consensus: The presence of zero-point entropy is likely to be a nonequilibrium property.18h Forconvenience of the readers, we briefly summarize the studies ofclassical frustrated systems.In 1935, Pauling18c successfully explained the reported

residual entropy18b of ice based on the proton-disordermodel.18a An H2O molecule has six independent protonconfigurations in the tetrahedral coordination of ice, and one-quarter of those have protons in the adjacent water molecules,thus satisfying the “ice rule” of Bernal and Fowler.18a Theresulting number of configurations (W) for ice consisting of Nwater molecules is W = (6/4)N, and the corresponding residualentropy is S/N = R ln 3/2 ∼ 0.405R, which seemed inagreement with the early experimental value of 0.44R.18b

However, in 1982, Tajima et al.102 showed that KOH-doped iceexhibits a structural phase transition at T = 72 K, where up to70% of the residual entropy is removed, suggesting thepresence of an ordered phase at low temperatures. In 2005,Singer et al.103 demonstratedbased on first-principles DFTtotal-energy calculationsthat the ordered phase of ice, namedice XI, below T = 90 K has a lower free energy than that ofproton-disordered ice, Ih.The studies on the residual entropy of ice, continuing over

seven decades, support the notion that a macroscopic amountof residual entropy at T = 0 K is a nonequilibrium property dueto slow dynamics at low temperatures, and direct experimentalevidence of a low-temperature ordered phase has beenprovided.102,104

Information about the macroscopic residual entropy in spinmagnetic−pyrochlore materials, or spin ice, evolved much likethe studies of ice over the years, but the current consensus isnot as definitive because of the lack of a direct experimentaldiscovery of an ordered phase. For spin ices, a very simplemodel Hamiltoniana spin Ising modelplayed a crucial rolein understanding the nature of the problem. The mostimportant aspect, perhaps, is the existence of exact analyticalsolutions for certain types of Ising models,68−70 where in somecases, a degenerate ground state with a macroscopic residualentropy can be rigorously proven. Such models include the AFIsing model on a triangle lattice,69 and the AF Ising model on a

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kagome lattice.70 A possible relation between an Ising modeland the proton-disorder problem in ice was pointed out byAnderson in 1956.19

In 1997, Harris et al.18g showed that the spin magnetism inpyrochlore, Ho2Ti2O7, and the proton-disorder problem of icecan be mapped on a ferromagnetic Ising model on corner-sharing tetrahedra; hence, those magnetic pyrochlore materialsare called spin ices. This Ising model with nearest neighborinteraction is exactly degenerate and disordered with amacroscopic amount of residual entropy, and the temperatureprofile of the specific heat of this model is in excellentagreement with that of the real material.18h However, byintroducing more realistic, long-range dipole−dipole interac-tions to this model and by using a more advanced clusterupdate−Monte Carlo algorithm the presence of an orderedground state was revealed.18h Although the predicted phasetransition has never been observed by experiments, themeasured short-range, spin−spin correlation from neutron-diffraction experiments agrees with a dipole Ising model, ratherthan an NN Ising model.18h This led to the consensus that theobserved residual entropy in the magnetic pyrochlore materialsis likely to be a nonequilibrium property caused by the slowdynamics at low temperatures.18h In this case, Planck’s form ofthe third law is likely to be consistent with the observation.These are the two well-known cases, where the low

temperature thermodynamic properties of real systems havebeen studied extensively, and the implications to the third lawof thermodynamics have been being discussed vigorously,which probably contributed to the popular view that Planck’sform is the third law. This view, although not rigorously proven,necessitated the description of an allotrope of boron: theground state of this system is nearly degenerate (although thesimulation results indicate otherwise), and there should be anordered ground state (although there is no existing directexperimental evidence).Among of the various versions of third law described in

Linder’s textbook,93 the case of elemental boron is consistentwith Nernst’s Heat Theorem (1), Lewis’ form (3) and theunattainability principle (4) but not with Planck’s form (2). Inthis context, a statement such as “an allotrope stabilized bydefects should have an ordered ground state due to the thirdlaw” is the least favorable one. Nevertheless, the trueconnection between elemental boron and the third law ofthermodynamics is yet to be fully understood.

9. PROSPECT OF BORON RESEARCHDetermination of the phase diagram of elemental boron hasbeen a very active area of research, yet the location of the phaseboundaries is still quite uncertain (see Figure 1). The majorchallenges on this subject stem from the intrinsic properties ofelemental boron: complex crystal structures of boron allotropesthat are separated by large kinetic barriers4b,5b,7 and thestructural and electronic properties that have strong tendencyto accommodate impurities.5e−g,23,105 The large kinetic barrieris a serious obstacle for the experimental determination ofphase boundaries. Direct mapping of the phase boundaries inhigh-pressure and temperature experiments has not beensuccessful at low pressure and temperature due to slowkinetics.2,3b,c,5n Use of a catalyst such as Pt might help to solvethe problem;3c however, a careful assessment of the impactsuch an impurity has on the thermodynamic stability ofdifferent phases will be necessary.5e−g,23,105 A determination ofthe Gibbs free energy profile from calorimetric measurements

could be an alternative approach, which requires large enoughhigh quality samples27c,d,64e,106 and an accurate estimate of theenthalpy difference between phases (heat of transformation).59

There are a few recent developments in solving the formerproblem,27c,d,64e while the latter is still a major obstacle. Withintheory, an assessment of the accuracy of approximations used inprevious works, e.g., local or semilocal descriptions of theelectronic exchange-correlation energy,107 will be desired. Amore precise yet computationally expensive method such asquantum Monte Carlo (QMC) may be applicable even to thelarge boron cells thanks to rapid improvements in computingpower.108 Although QMC will be able to provide an accurateestimate of the enthalpy differences between allotropes at T = 0K, the computed phonon contribution to the free energy109 willinevitably have some uncertainty due to the presence of POS. Acombination of theory (e.g., QMC) and experiment (e.g.,specific heat measurements) might be the most realisticapproach applicable in the near future to enable an accuratedetermination of the phase diagram of boron.Using our simple model4k,l that accurately describes the

POS-occupation problem in β-boron, it would be interesting topursue future investigations of the impact of POS-occupationon the various transport properties of boron. In particular, it islikely that the lack of a precise microscopic model forrepresenting the POSs in boron has prevented researchersfrom fully exploiting the potential uses for boron in electronicdevices.As the analogy with spin-magnetic systems suggests,

modifying the chemical potential of boronfor example, byapplying pressuremight provide a way to tune the relativepreference of B19 and B17−B18 occupations, which mightreveal a new ordered phase. Combined with precisecalorimetric measurements, this might enable fine-tuning ofthe electric/magnetic properties of elemental boron throughpartial occupations. Moreover, boron’s unique photoconduc-tance, large capacitance, and magnetoresistance might becontrolled with doping, which could lead to boron-basedspintronic devices. In addition, recent and rapid developmentsin the theory of boron-based nanomaterialsspecifically in lowdimensional structuresare already showcasing diverse andunique properties that might lead to novel devices.110

The peculiar properties that hindered the development ofboron-based opto-magneto-electronic devices could become anadvantage as we learn to control the properties of borondefects. Now, the door is wide open, and only a few moreobstaclesall identifiedmust be negotiated to reach theopportunities in sight.

ASSOCIATED CONTENT

*S Supporting Information

3D images of the figures in wrl format. This material is availablefree of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION

Corresponding Author

*E-mail: ogitsu@llnl.gov.

Notes

The authors declare no competing financial interest.

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Biographies

Tadashi Ogitsu received his Ph.D. in Materials Science in 1994 fromthe University of Tsukuba in Japan. His thesis research with Prof.Makoto Okazaki was focused on theoretical investigations of CVDdiamond growth mechanisms. In 1994, he joined the Institute for SolidState Physics (ISSP) and the Materials Design and CharacterizationLaboratory at the University of Tokyo as a research associate, where hedeveloped a parallel first-principles DFT code, which he used to studyfullerenes, high pressure phases of materials, and quantum effects oflight nuclei with Prof. Shinji Tsuneyuki, and contributed to thecommon use supercomputing facility at ISSP with Prof. HajimeTakayama. In 1998, he joined the National Center for Super-computing Applications at the University of Illinois at Urbana-Champaign as a postdoc where he studied quantum effects and particlestatistics in the solid phases of hydrogen with Prof. David M. Ceperleyand Prof. Richard M. Martin. In 2001, he joined Lawrence LivermoreNational Laboratory as a staff scientist. Since then, he has worked onvarious research subjects such as high pressure physics, ultrafast non-equilibrium physics, and materials and interfaces for energyapplications such as carbon aerogels for supercapacitor applicationsand photoelectrochemical hydrogen production.

Eric Schwegler received his Ph.D. in physical chemistry in 1998 fromthe University of Minnesota, following undergraduate degrees incomputer science and chemistry from Southwestern University. Histhesis research was focused on the development of linear scalingelectronic structure theory. In 1998, he joined Lawrence LivermoreNational Laboratory as a postdoctoral researcher, and in 2005 he wasappointed as the Quantum Simulations Group Leader in theCondensed Matter and Materials Division. The Quantum SimulationGroup is focused on the application of state-of-the-art quantumsimulation tools to understand and predict the properties of a widerange of materials, including semiconductor-water interfaces, metallic

alloys, liquids and solids under high pressures and temperatures, nano-confined fluids, and the aqueous solvation of ions and molecules.

Giulia Galli is Professor of Chemistry and Physics at the University ofCalifornia, Davis. She holds a Ph.D. in Physics from the InternationalSchool of Advanced Studies in Trieste, Italy. Prior to joining the UCDfaculty, she was the head of the Quantum Simulations group at theLawrence Livermore National Laboratory. She is a Fellow of theAmerican Physical Society (APS) and was elected chair of the Divisionof Computational Physics of the APS in 2006−2007. She is therecipient of a Department of Energy award of excellence in technicalcomputing (2000) and the Science and Technology Award from theLawrence Livermore National Laboratory (2004). She is currentlychair of the Extreme Physics and Chemistry of Carbon Directorate ofthe Deep Carbon Observatory (https://dco.gl.ciw.edu/). Her researchactivity is focused on quantum simulations of systems and processesrelevant to condensed matter physics, physical chemistry, materialsand nano-science (http://angstrom.ucdavis.edu/).

ACKNOWLEDGMENTSDuring the course of our research on boron, we discussed thetopic with many researchers, who greatly helped understandingthe extraordinary nature of elemental boron: Dr. Seiji Mizuno(University of Hokkaido, Japan), Prof. Yukitoshi Motome, Dr.Masafumi Udagawa, Prof. Kaoru Kimura, Dr. Shinji Todo, Prof.Koji Fukushima, Prof. Shinji Tsuneyuki, Prof. Kazuo Ueda(University of Tokyo, Japan), Prof. Syoji Yamanaka (HiroshimaUniversity, Japan), Dr. Takao Mori (NIMS, Japan), Prof. MaryAnne White, Mr. Anthony Cerqueira (Dalhousie University,Canada), Prof. Nicola Marzari (Ecole Polytechnique Federalede Lausanne, Switzerland), Prof. Manfred Sigrist, Prof. MatthiasTroyer (Eidgenossiche Technische Hochschule Zurich, Swit-zerland), Dr. Jonathan Yates (Oxford University), Prof. IvoSouza (Universidad Del Pais Vasco, Spain), Dr. RoderichMoessner (Max Planck Institute Dresden, Germany), Prof.Francois Gygi, Dr. Leonardo Spanu (University of CaliforniaDavis, USA), and Dr. John Reed. T.O. acknowledges assistanceof Liam Krauss (Lawrence Livermore National Laboratory) ingenerating some of figures.

Part of this work was performed under the auspices of theU.S. Department of Energy by Lawrence Livermore NationalLaboratory under Contract No. DE-AC52-07NA27344 andpartially supported by DOE/SciDAC under Grant No. DE-FG0206ER46262.

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Chemical Reviews Review

dx.doi.org/10.1021/cr300356t | Chem. Rev. XXXX, XXX, XXX−XXXY