Different phases of solid boron under high pressure are studied by first
principles calculations. The $\alpha$-B$_{12}$ structure is found to be stable
up to 270 GPa. Its semiconductor band gap (1.72 eV) decreases continuously to
zero around 160 GPa, where the material transforms to a weak metal. The
metallicity, as measured by the density of states at the Fermi level, enhances
as the pressure is further increased. The pressure-induced metallization can be
attributed to the enhanced boron-boron interactions that cause bands overlap.
These results are consist with the recently observed metallization and the
associated superconductivity of bulk boron under high pressure (M.I.Eremets et
al, Science{\bf 293}, 272(2001)).Comment: 14 pages, 5 figure

A. Jayaraman

C. Mailhiot

D. Li

D. U. Gubser

D. Vanderbilt

D. W. Bullett

E. D. Jemmis

F. H. Horn

F. P. Bundy

F. Siringo

G. J. Ackland

H. J. Monkhorst

H. Werheit

J. C. Jamieson

J. P. Perdew

J. Wittig

Jian Ping Lu

Jijun Zhao

K. J. Dunn

M. C. Payne

M. Fujimori

M. I. Eremets

M. M. Dacorogna

N. Vast

O. Shimomura

R. J. Nelmes

S. Lee

Y. Wang

English

arXiv

Crossref

Pressure-induced metallization in solid boron

Different phases of solid boron under high pressure are studied by first principles calculations. The $\alpha$-B$_{12}$ structure is found to be stable up to 270 GPa. Its semiconductor band gap (1.72 eV) decreases continuously to zero around 160 GPa, where the material transforms to a weak metal. The metallicity, as measured by the density of states at the Fermi level, enhances as the pressure is further increased. The pressure-induced metallization can be attributed to the enhanced boron-boron interactions that cause bands overlap. These results are consist with the recently observed metallization and the associated superconductivity of bulk boron under high pressure (M.I.Eremets et al, Science{\bf 293}, 272(2001))

Zhao, Jijun

Lu, Jian Ping

Carolina Digital Repository

arXiv:cond-mat/0109550v1  [cond-mat.mtrl-sci]  28 Sep 2001Pressure-induced metallization in solid boronJijun Zhao ∗ and Jian Ping Lu †Department of Physics and Astronomy, University of North Carolina at Chapel Hill, NC27599-3255(October 31, 2018)AbstractDifferent phases of solid boron under high pressure are studied by firstprinciples calculations. The α-B12 structure is found to be stable up to 270GPa. Its semiconductor band gap (1.72 eV) decreases continuously to zeroaround 160 GPa, where the material transforms to a weak metal. The metal-licity, as measured by the density of states at the Fermi level, enhances asthe pressure is further increased. The pressure-induced metallization can beattributed to the enhanced boron-boron interactions that cause bands over-lap. These results are consist with the recently observed metallization and theassociated superconductivity of bulk boron under high pressure (M.I.Eremetset al, Science293, 272(2001)).71.20.Nr, 61.50.Ah, 62.50.+pTypeset using REVTEX1It is well known that many semiconductors and insulators undergo nonmetal-metal transi-tions upon applying high pressure.1–5 Most of these metallization are associated with a struc-tural transition from low coordination insulator to a high coordination metallic phase.1,3–5 Insome case, the pressure-induced metallization can be attributed to the increased interatomicinteractions which lead to the band overlap.2 Recently, pressure-induced metallization andsuperconductivity were found in solid boron above 160 GPa.6 There is no direct experimen-tal evidence whether the nonmetal-metal transition is caused by the band gap closure or astructural transition into the metallic bct phase.7 Moreover, the superconducting transitiontemperature increases from 6 K at 175 GPa to 11.2 K at 250 GPa. This is in contrastto typical sp metals such as Al, where the Tc decreases with pressure.8,9 In this letter wepresent results of first principles calculations on solid boron under high pressure for differentphases (α-B12, bct, and fcc). Our results suggest that α-B12 phase is stable and maintainsits rhombohedral structure up to 270 GPa. Around 160 GPa, a nonmetal-metal transitionis found in α-B12 phase. Further increases in pressure leads to an enhanced density of statesat the Fermi level. Charge density analysis shows that the pressure-induced metallizationis mainly due to the increased interatomic interactions that broaden the band width. Wepropose that the recently observed metallization6 can be understood in terms of the bandgap closure.Under ambient pressure, solid boron exists in various complex crystal structures withicosahedral B12 cluster as a common structural component. The B12 icosahedrons can be in-terlinked by strong covalent bonds in different ways to form different polymorphs such as theα-rhombohedral B12 (α-B12), α-tetragonal B50, or the β-rhombohedral B105 (β-B105).10,11,15Among these crystal structures α-B12 has the simplest form with one B12 cluster per rhom-bohedral unit cell. Previous experimental studies on solid boron include the structuralanalyses10, bulk modulus11, Raman spectrum12, and electron density distribution13. On thetheoretical side, the structural, electronic and vibrational properties of solid boron (mostly inα-B12 phase) have been calculated by first principle methods.7,12,14,15 In particular, a struc-tural transition from the insulating α-rhombohedral to metallic body-centered-tetragonal2(bct) structures was predicted to occur at the pressure of 210 GPa.7 But no detailed analy-sis on nonmetal-metal transition were done.3FIGURESFIG. 1. Optimized α-B12 rhombohedral crystalline structure shows strong bonding betweenicosahedral B12 clusters. Present GGA calculations give the cell constant a=4.98 Å and cell angleα=58.2◦, which compare well with experimental values of 5.06 Å , 58.2◦10. (See Table I for thedetailed structural parameters.)Our density functional calculations is based on the plane-wave pseudopotentialmethod16,17. The density functional is treated by the generalized gradient approximation(GGA)18 with the exchange-correlation potential parameterized by Wang and Perdew19. Weuse a ultrasoft nonlocal pseudopotential to model the ion-electron interactions20. The kineticenergy cutoff for the plane-wave basis is 240 eV. Large set of Monkhorst-Pack k meshes21 (8×8×8, 14×14×14, and 14×14×14 for α-B12, bct, and fcc lattice respectively) are used tosample the Brillouin zone. For a given external pressure, both the cell parameters and theatomic positions are fully optimized.The optimized crystalline structure of α-B12 is shown in Fig.1. It is a network of B12 clus-ters with two kinds of intercluster covalent bonds (d1 and d2 in Table I) between neighboringicosahedrons. The calculated cohesive energy 6.95 eV/atom is comparable to the experi-mental value 5.81 eV.22 The indirect gap 1.72 eV from our GGA calculation agrees with theestimated experimental value (2.0 eV)23 better than the previous LDA result (1.43 eV).144Table I summarizes the structural parameters of α-B12 under different pressures. Experi-mental and previous theoretical results are included for comparison. Satisfactory agreementis found between present calculations and previous works. As shown in the table, uponrelaxation the α-B12 maintains its rhombohedral lattice structure up to high pressure. Thecell angle α between the lattice vectors does not change with pressure, while the latticeconstant smoothly decreases. All the interatomic bonds are compressed with most notablechanges in the long intercluster bonds (d2). This leads to a dramatic modification on theelectronic states (to be discussed later).5TABLESTable I. Summary of structural parameters of α-B12 under different pressures (0 ∼ 400GPa, given in bracket). a denotes the lattice parameter; α describes the angle between therhombohedral lattice vectors; r is the mean radius of the B12 icosahedron; d1 and d2 are theshort and long intercluster bond lengths. Available experimental values10,11 and previoustheoretical results12,15 are included for comparison.α a (Å) r (Å) d1 (Å) d2 (Å)Exp.10 (0 GPa) 58.2◦ 5.06 1.69Exp.11 (0.5 GPa) 58.0◦ 5.07 1.70 1.67 2.00Theor.12 (0 GPa) 58.2◦ 4.98 1.68 1.65 1.98Theor.15 (0 GPa) 58.1◦ 5.03 1.69 1.67 2.00This work (0 GPa) 58.2◦ 4.98 1.70 1.67 1.99This work (100 GPa) 58.0◦ 4.56 1.56 1.51 1.74This work (200 GPa) 58.1◦ 4.34 1.49 1.43 1.63This work (300 GPa) 58.2◦ 4.19 1.44 1.38 1.56This work (400 GPa) 58.3◦ 4.06 1.41 1.34 1.5160 100 200 300 4000.00.51.01.5ofccbctα-B12 Relative enthapy (eV/atom)Pressure (GPa)4 5 6 7-7-6-5fccbctα-B12Atomic volume (A3/atom)Energy (eV/atom)FIG. 2. Relative enthalpy of bct and fcc phase with respect to α-B12. The crossing of bct andα-B12 phase occurs at about 270 GPa. No cross between fcc and bct phases is found up to 400GPa. Insert: cohesive energy per atom of α-B12, bct, fcc phases as a function of atomic volume.To investigate the pressure-induced structural transition we optimized both the cell pa-rameters and atomic positions for α-B12, bct, fcc phases under each external hydrostaticpressure. Both the total energy and the enthalpy for each phase were calculated up to 400GPa. The results are shown in Fig.2. At the ambient pressure, we find 0.82 eV energydifference between the equilibrium α-B12 and the bct phase, and 1.45 eV difference betweenthe α-B12 and the fcc phase. Upon applying pressure, α-B12 remains the low energy struc-ture but the enthalpy of bct phase becomes lower than that of α-B12 at 270 GPa. Such acrossing indicates a thermodynamic instability of α-B12 and a possible structural transitionto the bct phase. The enthalpy of the fcc phase crosses that of the α-B12 at the pressure of315 GPa, but it is never lower than that of the bct phase up to the highest pressure (400GPa) studied.Previous LDA calculations predicted a structural transition from α-B12 to bct at 2107GPa.7 We have also preformed independent LDA calculations and obtained a similar tran-sition pressure of 200 GPa. Our GGA transition pressure is about 60 GPa higher than theLDA results. As GGA calculations provide a better agreement with experiments in the bandgap than LDA calculations, we believe that the GGA transition pressure (270 GPa) is morereliable. In the previous LDA calculations, a further transition from bct to fcc was predictedto occur at 360 GPa.7 We have not observed such a transition up to 400 GPa in both LDAand GGA calculations. The discrepancy may due in part to the fact that our calculationsare fully optimized in both the lattice parameters and the atomic positions, while the c/aratio of the bct structure was fixed in the previous calculations.7-3 -2 -1 0 1 20 GPa240 GPa200 GPa160 GPa120 GPa Density of State (arb. unit)Energy (eV)FIG. 3. The electronic density of states of α-B12 under different pressures. Gaussian broadeningof 0.05 eV is used. The broadening of both the valence and the conduction bands with pressureleads to the band gap closure and the formation of metallic states around 160 GPa.From above discussions we conclude that it is unlikely that the experimentally observed8nonmetal to metal transition at 160 GPa6 is due to the structural transition into the metallicbct phase. The alternative explanation for the metallization is the band gap closure uponexternal pressure, similar to that happened in iodine.2 As shown in Table I, upon applyingpressure, the α-B12 maintains its rhombohedral lattice structure with compressed interclusterand intracluster bonds. The strengthened boron-boron interactions broaden both the valenceand conduction bands, eventually leads to the band gap closure and the development ofmetallic states near the Fermi level. This effect is clearly shown in Fig.3, where we present thedensity of states for several different pressures. We perform further analysis of the electrondensity. Upon applying pressure, both the two-center intercluster bonds and the three-centerintracluster bonds13–15 are greatly strengthened while the two-center intracluster bonds areless sensitive. This can be seen in the contour plot of the total charge distributions fordifferent pressures. Fig.4 shows an example of the contour plot, which includes the two-center intercluster and intracluster bonds. A much higher charge density in the area of theintercluster bonds is found at higher pressures. Similar effect was also observed in the three-center intracluster bonds. Thus, we conclude that the enhancement of both intercluster andthree-center intracluster bonding leads to pressure-induced metallization.9FIG. 4. The total charge density contour plot for α-B12 phase under different pressures: 0 GPa(upper), 160 GPa (middle), 240 GPa (down). The current slice includes two-center intercluster andintracluster bonds. The representative sites for boron atoms are labeled as B in upper plot. Red,yellow, green, blue colors indicate decreasing electron density. A significantly enhanced chargedensity (red color) in the area of the intercluster bond can be seen in the case of higher pressures.A clearer picture of nonmetal-metal transition can be seen in the pressure dependentof the band gap shown in Fig.5. From extrapolation, the band gap closure occurs around160 GPa. Experimentally, it was found that β-B105 undergo a nonmetal to superconductortransition at about 160 GPa6. Although our calculations were based on α-B12 phase, webelieve that our results are qualitatively comparable to the experiments because both phaseshave the same icosahedral structural component.100 50 100 150 200 2500.00.40.81.21.6  Band Gap (eV)Pressure (GPa)N(EF ) (arb. unit)FIG. 5. The band gap (filled dots) and the electronic density of states N(EF ) (open squares) atthe Fermi level for α-B12 as functions of pressure. The extrapolated pressure for band gap closureis 160 GPa. In the metallic phase, N(EF ) increases rapidly with pressure at first, then tampersoff around 240 GPa.In the metallic phase of α-B12 our calculations show that density of states N(EF ) at theFermi level increases rapidly with pressure from 160 GPa to 240 GPa, beyond which it sat-urates (see Fig.5). The enhancement of metallicity with pressure can be directly associatedwith the increase of superconducting transition temperature Tc from 6 K at 175 GPa to 11.2K at 250 GPa.6 In the phonon mediated superconductivity, the Tc can be obtained as24,25Tc =ωln1.2exp(−1.041 + λλ− µ∗ − 0.62λµ∗),where ωln is the characteristic phonon frequency, λ is the electron-phonon coupling strength,µ∗ is the screened coulomb repulsive interaction. The electron-phonon coupling λ is relatedto the density of states at the Fermi level N(EF ) byλ = N(EF ) < I2 > /(M < ω2 >),where < I2 > is the average square of the electron-phonon matrix element and < ω2 >is averaged phonon frequency.25 Typically for sp metal such as aluminum, < ω2 > in-crease with pressure and N(EF ) decreases, while µ∗ is insensitive.9 Thus, Tc decreases withpressure.8,9 On the contrary, in solid boron, Tc was found to increases substantially with11pressure (175∼250 GPa).6 Our calculations show that such increase in Tc can be understoodby the dramatic increase in the density of states N(EF ) with pressure after the metallizationof α-B12. In addition, we also have calculated the N(EF ) and its pressure dependence for themetallic bct phase. The N(EF ) for bct phase is much higher than that of α-B12 phase and itdoes not show a clear pressure dependence. This result further supports our interpretationthat the observed metallization in boron may not originate from a structural transition fromα-B12 to bct phase.In summary, we have performed first principles calculations to investigate the structuraland electronic properties of solid boron under high pressure. Structural transition from α-B12 to bct phase is found to occur at about 270 GPa. The semiconductor gap (1.72 eV atambient pressure) decreases upon external pressure. Band gap closure and a nonmetal-metaltransition is found around 160 GPa. As the external pressure further increase (160 GPa to240 GPa), the density of states at Fermi level increase dramatically. We conclude that theexperimentally observed metallization in boron might be due to the band gap closure insteadof a structural transition into the metallic bct phase.This work is supported by the U.S. Army Research Office (Grant DAAG55-98-1-0298).We acknowledge computational supports from the North Carolina Supercomputer Center.∗: zhaoj@physics.unc.edu†: jpl@physics.unc.edu12REFERENCES1 J.C.Jamieson, Science139, 129(1963); J.Wittig and B.T.Matthias, Science160, 92(1968).2O.Shimomura, K.Takemura, Y.Fujii, S.Minomura, M.Mori, Y.Noda, Y.Yamada,Phys.Rev.B18, 715(1978).3K.J.Dunn, F.P.Bundy, J.Appl.Phys.51, 3246(1980)” F.P.Bundy, K.J.Dunn,Phys.Rev.B22, 3157(1980).4A.Jayaraman, Red.Mod.Phys.55, 65(1983).5G.J.Ackland, Rep.Prog.Phys.64, 483(2001).6M.I.Eremets, V.V.Struzhkin, H.K.Mao, R.J.Hemley, Science293, 272(2001).7G.Mailhiot, J.B.Grant, A.K.McMahan, Phys.Rev.B42, 9033(1990).8D.U.Gubser, A.W.Webb, Phys.Rev.Lett.35, 104(1975).9M.M.Dacorogna, M.L.Cohen, P.K.Lam, Phys.Rev.B34, 4865(1986).10G.Will, B.Keifer, B.Morosin, G.A.Slack, Mater.Res.Soc.Symp.Proc.97, 151(1987).11R.J.Nelmes, J.S.Loveday, D.R.Allan, J.M.Besson, G.Hamel, P.Grima, S.Hull,Phys.Rev.B47, 7668(1993).12N.Vast, S.Baroni, G.Zerah, J.M.Besson, A.Polian, M.Grimsditch, J.C.Chervin,Phys.Rev.Lett.78, 693(1997).13M.Fujimori, T.Nakata, T.Nakayama, E.Nishibori, K.Kimura, M.Sakata,Phys.Rev.Lett.82, 4452(1999).14 S.Lee, D.M.Bylander, L.Kleinman, Phys.Rev.B42, 1316(1990).15D.Li, Y.N.Xu, W.Y.Ching, Phys.Rev.B45, 5895(1992).16The density functional plane-wave pseudopotential calculations are performed by using13CASTEP. CASTEP is an ab initio program with plane-wave pseudopotential packagedistributed by Accelrys Inc: http://www.accelrys.com/.17M.C.Payne, M.T.Teter, D.C.Allen, T.A.Arias, J.D.Joannopoulos, Rev.Mod.Phys.64,1045(1992).18 J.P.Perdew and Y.Wang, Phys.Rev.B45, 13244(1992).19Y.Wang and J.P.Perdew, Phys.Rev.B43, 8911(1991).20D.Vanderbilt, Phys.Rev.B 41, 7892(1990).21H.J.Monkhorst, J.D.Pack, Phys.Rev.B13, 5188(1976).22C.Kittle, Introduction to Solid State Physics, 7th ed., (Wiley, New York, 1996).23 F.H.Horn, J.Appl.Phys.30, 1611(1959).24 P.B.Allen, R.C.Dynes, Phys.Rev.B12, 905(1975).25W.L.McMillan, Phys.Rev.167, 331(1968).14

Pressure-induced metallization in solid boron

Abstract

Similar works

Full text

Available Versions

Crossref

Carolina Digital Repository