Pressure induced crossing of the core-levels in 5d metals
Alexey A. Tal, Mikhail I. Katsnelson, Marcus Ekholm, Johan Jönsson, Leonid Dubrovinsky, Natalia Dubrovinskaia, Igor A. Abrikosov
PPressure induced crossing of the core-levels in 5d metals
Alexey A. Tal,
1, 2, ∗ Mikhail I. Katsnelson,
3, 4
Marcus Ekholm, Johan Jönsson, Leonid Dubrovinsky, Natalia Dubrovinskaia, and Igor A. Abrikosov
1, 2 Theory and Modeling, IFM-Material Physics, Linköping University, SE-581 83, Linköping, Sweden Materials Modeling and Development Laboratory,National University of Science and Technology ’MISIS’, 119049, Moscow, Russia Radboud University of Nijmegen, Institute for Molecules and Materials,Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands Department of Theoretical Physics and Applied Mathematics,Ural Federal University, Mira str, 19, Ekaterinburg, 620002, Russia Bayerisches Geoinstitut, Universitat Bayreuth, D-95440 Bayreuth, Germany
Pressure induced interaction between core electrons, the core level crossing (CLC) transition hasbeen observed in hcp Os at P 400 GPa [L. Dubrovinsky, et al., Nature 525, 226–229 (2015)]. Inthis work, we carry out a systematic study of the influence of pressure on the electronic structurein all metals of the 5d series (Hf,Ta,W,Re,Os,Ir,Pt,Au) using first-principles electronic structurecalculations. We have found that CLC is a general effect for this series of metals. While in Pt itoccurs at 1500 GPa, at a pressure substantially higher than in Os, in Ir it occurs already at 80 GPa.Moreover, we predict that in Re the CLC transition may appear at ambient pressure. We analyzethe shifts of the CLC transition pressure across the series within the Thomas-Fermi model, andshow that the effect has many common features to the atomic collapse in the rare-earth elements.
The properties of matter are determined by its elec-tronic structure, which is sensitive to external parameterslike pressure, temperature, and chemical composition. Amodification of the electronic structure may lead to aphase transition and a change the material properties,allowing for a synthesis of new materials. The behav-ior of matter at extreme conditions has always drawnattention of a broad research community. For instance,compression may result in qualitative changes of the stateof a solid such as structure, magnetic state and conduc-tivity. The metals of 5d group of periodic table are ofparticular interest for high-pressure study due to theirremarkable properties. Hafnium (Hf) has attracted greatscientific and technological interest due to the position ofits d-band in the middle of a broad sp band, which hasan impact on its electronic and superconducting proper-ties [1–4]. Instability of hcp phase of Hf at high pres-sure was proven both theoretically and experimentally[5, 6]. Tantalum is stable even at the pressure of hun-dreds of GPa [7, 8]. Even though the stability at higherpressure is still under debate [9, 10], only hcp phasewas observed experimentally. Ta shows high chemicaland thermodynamical stability, having a melting pointat ambient pressure of about 3950K [11] and it is widelyused in the microelectronics industry for producing in-tegrated circuits. The strength of tungsten are of con-siderable importance for optimizing the design and op-eration of high-pressure apparatus. Pressure calibrationin diamond anvil cells is largely based on equations ofstate derived from shock data for standard materials suchW, Mo, Cu [12, 13]. Neither theoretical calculations norexperimental observations suggest that W may suffer astructural transition under pressure. Rhenium (Re) wasstudied under extremely high pressures up to 600 GPa and showed no structural transformations in the consid-ered pressure range. That makes Re a good candidate forultra-high pressure calibration[14–17]. In the recent pa-per Dubrovinsky et al. [18] have presented experimentson osmium compression up to over 770 GPa in diamondanvil-cell. Even though hcp phase turned out to be stablein the whole pressure range, authors have discovered twopeculiarities in c/a behavior under pressure. The firstone appeared at ~150 GPa and was attributed to the so-called electronic topological transition (ETT) [19], whilethe second one at ~440 GPa was explained by the cross-ing of core-levels (CLC). The CLC is a novel type of theelectronic transition, and requires systematic investiga-tion in order to obtain full understanding of its nature.A stability of the fcc iridium under pressure has been de-bated for years. A formation of the complex superlatticein iridium under pressure of 59 GPa has been reported[20], however, in other experimental and theoretical stud-ies such a structure has not been observed [21]. Platinumis also widely used as a pressure standard and is knownto be stable in the fcc structure up to 600 GPa [22, 23].The uniqueness of gold and its important role in modernscience is closely related to its exceptional stability tochemical reactions, extreme pressures and temperatures[24, 25]. Gold in the fcc structures become unstable in fa-vor of the hcp structure only under pressure of 240 GPaand at elevated temperatures.[26]. Thus these metalsshow structural stability under high pressure, they arenon-magnetic and may be good candidates for investiga-tion of the transitions of another kind.It is well-known that chemical bonding in solids ismainly due to valence electrons while core-electrons areoften considered "frozen" and do not influence macro-scopic properties of the materials. However, strong re- a r X i v : . [ c ond - m a t . m t r l - s c i ] S e p arrangements of 5p and 4f states at the CLC transitionmay affect the valence electrons due to non-local natureof the electron interactions and therefore could indirectlyinfluence the structural properties as shown in [18]. Inthis Letter we analyze the behavior of the core-levels in5d-metals under pressure by means of ab initio calcu-lations and study the influence of the structure on thecore-levels interplay.For our calculations we used WIEN2k code [27] withLDA functional and k-mesh of 32x32x32 k-points in fccand bcc structures and 39x39x21 in hcp structure. Theradius of real-space muffin-tin sphere varied under differ-ent pressures, while the product K max · R MT was keptequal to 10. The spin-orbit interaction is included varia-tionally.Calculations of the density of states (DOS) have beencarried out in 5d-metals from Hf to Au. DOS calcula-tions at zero pressure were performed for the structures,experimentally proven to be stable at ambient pressure.The results are shown in Fig.1 (left). DOS under pres-sure was calculated for the structures observed at highestpressures reported for the respective elements experimen-tally. In fact, only two 5d-metals Hf and Au are unam-biguously known to undergo a structural transformationunder pressure. Starting with Hf, we see that 4f and5p levels are far from each other. However, with the in-crease of the atomic number (Z), 4f and 5p levels bothshift towards higher binding energies. Importantly, 4flevels shift faster than 5p states. Thus 4f levels "outrun"5p levels. In particular, 4 f / levels cross 5 p / alreadyin Re and finally in Pt both 4 f / and 4 f / electrons laybelow 5p electrons. Table I: Data used for pressure estimations. Values with thesuperscripts are taken from the corresponding experimentalstudies. The rest is calculated in this study by fitting theequation of state with Vinet equation.Hf Ta W Re Os Ir Pt Aubcc* hcp bcc bcc hcp hcp fcc fcc fcc hcp*V P
114 113 a b c d e f g h a b c d e f g h ; bulk modulus is in GPa; a Reference [5]; b Reference [8]; c Reference [28]; d Reference[14]; e Reference [18]; f Reference [20]; g Reference [29]; h Reference [30];
Fig.1 (right) demonstrates DOS of 5d metals uponcompression. We provide DOS for structures which areknown from experiment to be stable at high pressure.In fact, the crystal structure has very little effect on thecore level crossing transitions, as will be discussed below.Applied pressure causes broadening of p and s levels, so that they form rather broad bands due to overlap of thewave-functions. As it was shown in [18] bulk modulusand its derivative calculated from Birch-Murnaghan fit-ting theoretically may give very inaccurate pressure es-timate for the highly reduced volumes. Thus, for pres-sure estimation we used experimental bulk moduli andtheir pressure derivatives for cases where there were ex-perimental measurements as shown (see Table I). In Hf,Ta, W broadening of 5p levels due to pressure does notlead to overlapping of 4f and 5p levels even at highestpressures considered in this study. On the contrary, inRe 4 f / and 5 p / are overlapping at zero pressure andcompression just causes a broadening of 5p levels. Sincehcp Hf is unstable under high pressure and transformsfirst to ω phase, and then to bcc phase, we provide DOSfor hcp and bcc structures for approximately the samevolume reduction. One can clearly see that the structuredoes not have strong effect on the behavior of core-levelsenergies under pressure. We observed CLC of 4 f / and5 p / in fcc Os at the same pressure as in [18]. In Ir 4 f / and 5 p / states are very close, thus relatively small com-pression up to 80 GPa results in a CLC. These pressuresare substantially lower then in Os, which makes it Ir ex-tremely interesting in terms of a detailed experimentalstudy of the CLC transition. In Pt 4f level lay below 5pand the distance between them is rather big. ThereforeCLC transition in Pt occurs at pressure above 1.5 TPa.Our calculations suggest that one should not expect tosee a CLC in Au at any realistic pressures for either fccor hcp structures. Analysis of the whole series of DOS for5d-metals shows that in addition to broadening pressurecauses shifts of all the core states towards higher bindingenergies and the shift is more pronounced for higher val-ues of Z. It is important to stress that the phenomenonof core-level crossing is strongly linked to the fact thatf-level shift faster towards higher binding energies thenp-level with increasing atomic number. Let us explainthis effect.In the early work M. Goeppert Mayer [31] demon-strated how atomic collapse appears in the rare-earthelements. She showed that the effective f-electrons po-tential suffers drastic changes with the increase of atomicnumber Z. In order to understand the behavior of f andp-levels in 5d-metals we analyzed the atomic potentialbehavior with the change of Z in the Thomas-Fermi ap-proximation for the effective potential: V = − e r [1 + ( Z − φ ( r/µ )] + h π m l ( l + 1) r (1)The approximation suggested by Tietz [32] was usedfor the Thomas-Fermi function φ ( r/µ ). φ ( x ) = 1(1 + αx ) (2)where x = r/a , a = 0 . Z − / and α = 0 . H a f n i u m T an t a l u m BCC 0 10 20 30 T ung s t en BCC 0 10 20 30 R hen i u m HCP 0 10 20 30 O s m i u m HCP 0 10 20 30 I r i d i u m FCC 0 10 20 30 P l a t i nu m FCC 0 10 20 30 -6 -5 -4 -3 -2 -1 0 G o l d E-Ef, RyFCC H a f n i u m HCP 255 GPa 0 10 BCC 331 GPa 0 10 20 30 T an t a l u m BCC 373 GPa 0 10 20 30 T ung s t en BCC 1204 GPa 0 10 20 30 R hen i u m HCP 1153 GPa 0 10 20 30 O s m i u m HCP 476 GPa 0 10 20 30 I r i d i u m FCC 80 GPa 0 10 20 30 P l a t i nu m FCC 1543 GPa 0 10 20 30 -6 -5 -4 -3 -2 -1 0 G o l d E-Ef, RyFCC 796 GPa 0 10 -6 -5 -4 -3 -2 -1 0E-Ef, RyHCP 381 GPa
Figure 1: (left) Electronic density of states (DOS) for 5d-metals from Hf to Au at zero pressure; (right) DOS for 5d metalsfrom Hf to Au in structures experimentally known to be stable at high pressure. For each element the pressure shown in thefigure correspond either to the pressure of the core level crossing transition or to the highest pressure considered in this study. -2000-1500-1000-500 0 0.5 1 ℓ =3 ℓ =1 V , H a r t ee r, Bohr radius 7080 A t o m i c nu m be r Figure 2: Thomas-Fermi potentials for orbital numbers l=1(p-electrons) and l=3 (f-electrons). Potentials are plotted forseries of Z form 70 to 80. The color of the line corresponds toZ, as shown on the right palette.
This approximation is believed to reproduce the shiftsproperly enough to distinguish the trends [31].Fig.2 shows how potential V changes with atomic num-ber. We considered series from Z=70 to 80. It is clearthat the effective potential for f-electrons is more sen-sitive to Z. Indeed, and with the increase of Z both pand f-potentials become deeper but the f-potential shiftsfaster and f-levels will at some point outrun the p-levels.This behavior is clearly seen in our first-principles resultsin Fig. 1. Two other effects seen in Fig. 1 are the pressureinduced broadening, strongly pronounced for the p-levels,and a slight pressure induced up-shift of the states in en-ergy, which is an effect of the higher spatial localizationof the corresponding potential walls upon the decreaseof the interatomic distances. These two effects lead toa crossing of the core levels for elements, in which theyturned out to be sufficiently close to each other at ambi-ent pressure.Let us finally consider two elements, Ir and Re, whereour calculations predict the CLC transition at relativelylow pressure, and analyze the experimental informationavailable in literature. In the fcc metals, like Ir, the CLCtransition cannot be searched from the changes of c/a lat-tice parameters ratio, in contrast to hcp Os [18]]. How-ever, electronic transitions may lead to the changes of compressibility (such possibility has been discussed forOs [18]). X-ray diffraction experiments performed byCerenius and Dubrovinsky [20] on fcc iridium showed dis-tortion of the structure by the appearance of the addi-tional diffraction peaks at the pressure exceeding 59 GPa,which was attempted to be explained by the formationof a superlattice. However, the superstructures were notobserved in other experiments or in theoretical calcula-tions. Moreover, a possible change of compressibility as afunction of pressure was observed in recent experimentson Ir-rich Ir-Os alloy [33]. All this points to a possibleelectronic transitions in these systems. In Fig.3 we showthe band structure of Ir at P=0 GPa and at P=80 GPa,covering the pressure range of interest for this element.One can see that the presented bandstructure does notpoint to any Fermi-surface topology change and there-fore the behavior can not be explained by an ETT. Onthe other hand, in our calculations the CLC has beenclearly shown at the pressure of 80 GPa, and it may givean explanation for the observed peculiarity. In fact, thissituation is remarkably similar to the one, seen in Osat P ≈
400 GPa [18]. Of course, new experimental stud-ies should be carried out for Ir. On the other hand, apossibility of the structural transition in this metal cancomplicate the task. For that reason, CLC could be veri-fied experimentally in alloy systems, for example in Ir-Osalloy. Considering Re, the CLC transition is predicted tooccur at already at ambient pressure. In this respect, it isinteresting to point out that the results of investigationsof the equation of state of Re are not totally consistent:while reported bulk moduli [14, 18, 34] are basically thesame within the uncertainty of the measurements, thereported pressure derivatives of the bulk moduli are sub-stantially different. Therefore, Re is another interestingcandidate for the studies of the CLC transition. More-over, because we predict it to occur at ambient pressure,spectroscopic methods can be employed as well.In summary, we have investigated the pressure depen-dence of the electronic structure of 5d metals from Hfto Au. We predict that the newly discovered core-levelcrossing transition in Os also appears in Re, Ir, and Pt.We identify mechanisms that lead to the CLC transitionsin the 5d-metals, which have many common features tothe atomic collapse in the rare-earth elements. The un-derstanding of the CLC obtained in the present work al-lows us to expect to finding this phenomenon in variety ofdifferent systems. The fact that the CLC is predicted formetals used as pressure standards indicates that furtherinvestigations of this phenomenon are important.The work was financially supported by the Knutand Alice Wallenberg Foundation through Grant No.2012.0083, the Swedish Government Strategic ResearchArea Grant Swedish e-Science Research Centre (SeRC)and “Strong Field Physics and New States of Matter”from Knut and Alice Wallenbergs Foundation. I.A.A. isgrateful for the support provided by the Swedish Foun- ambient atom 0 size 0.20
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X Z W K E F E n e r gy ( e V ) Figure 3: (left) Band structure of iridium at ambient pressure; (right) band structure of iridium under pressure of 80 GPa; dation for Strategic Research (SSF) program SRL GrantNo. 10-0026. The support by the grant from the Min-istry of Education and Science of the Russian Federation(Grant No. 14.Y26.31.0005) is gratefully acknowledged.The calculations were performed on resources providedby the Swedish National Infrastructure for Computing(SNIC) at the National Supercomputer Center (NSC). ∗ Electronic address: [email protected][1] J. C. Duthie and D. G. Pettifor, Phys. Rev. Lett. , 564(1977).[2] H. L. Skriver, Phys. Rev. B , 1909 (1985).[3] J. S. Gyanchandani, S. C. Gupta, S. K. Sikka, andR. Chidambaram, J. Phys. Condens. Matter , 6457(1999).[4] Y. K. Vohra and P. T. Spencer, Phys. Rev. Lett. , 3068(2001).[5] R. Hrubiak, V. Drozd, A. Karbasi, and S. K. Saxena, J.Appl. Phys. , 4 (2012).[6] Y. Hao, J. Zhu, L. Zhang, H. Ren, and J. Qu, Philos.Mag. Lett. , 61 (2011).[7] J. a. Moriarty, J. F. Belak, R. E. Rudd, P. Söderlind,F. H. Streitz, and L. H. Yang, J. Phys. Condens. Matter , 2825 (2002), ISSN 0953-8984.[8] H. Cynn and C.-S. Yoo, Phys. Rev. B , 8526 (1999).[9] Y. Yao and D. D. Klug, Phys. Rev. B - Condens. MatterMater. Phys. , 1 (2013).[10] L. Burakovsky, S. P. Chen, D. L. Preston, a. B. Be-lonoshko, a. Rosengren, a. S. Mikhaylushkin, S. I. Simak,and J. a. Moriarty, Phys. Rev. Lett. , 1 (2010). [11] A. Dewaele, M. Mezouar, N. Guignot, and P. Loubeyre,Phys. Rev. Lett. , 29 (2010).[12] A. Dewaele, P. Loubeyre, and M. Mezouar, Phys. Rev.B - Condens. Matter Mater. Phys. , 1 (2004), ISSN01631829.[13] A. D. Chijioke, W. J. Nellis, and I. F. Silvera, J. Appl.Phys. , 0 (2005), ISSN 00218979.[14] S. Anzellini, A. Dewaele, F. Occelli, P. Loubeyre, andM. Mezouar, J. Appl. Phys. , 043511 (2014).[15] C. S. Zha, W. a. Bassett, and S. H. Shim, Rev. Sci. In-strum. , 2409 (2004).[16] L. Dubrovinsky, N. Dubrovinskaia, V. B. Prakapenka,and A. M. Abakumov, Nat. Commun. , 1163 (2012),ISSN 2041-1723.[17] a. K. Verma, P. Ravindran, R. S. Rao, B. K. Godwal,and R. Jeanloz, Bull. Mater. Sci. , 183 (2003).[18] L. Dubrovinsky, N. Dubrovinskaia, E. Bykova, M. Bykov,V. Prakapenka, C. Prescher, K. Glazyrin, H.-P. Lier-mann, M. Hanfland, M. Ekholm, et al., Nature (2015),ISSN 0028-0836.[19] I. M. Lifshitz, Soviet Physics JEPT , 1130 (1960).[20] Y. Cerenius and L. Dubrovinsky, J. Alloys Compd. ,26 (2000).[21] S. Grussendorff, N. Chetty, and H. Dreysse, J. Phys. Con-dens. Matter , 4127 (2003).[22] S. Ono, J. P. Brodholt, and G. David Price, J. Phys.Chem. Solids , 169 (2011).[23] a. B. Belonoshko and a. Rosengren, Phys. Rev. B - Con-dens. Matter Mater. Phys. , 1 (2012).[24] B. Hammer and J. K. Norskov, Why gold is the noblestof all the metals (1995).[25] D. Batani, A. Balducci, D. Beretta, A. Bernardinello,T. Löwer, M. Koenig, A. Benuzzi, B. Faral, and T. Hall,Phys. Rev. B , 9287 (2000), ISSN 0163-1829. [26] L. Dubrovinsky, N. Dubrovinskaia, W. a. Crichton, a. S.Mikhaylushkin, S. I. Simak, I. a. Abrikosov, J. S. DeAlmeida, R. Ahuja, W. Luo, and B. Johansson, Phys.Rev. Lett. , 1 (2007).[27] P. Blaha et al., WIEN2K, An Augmented Plane WavePlus Local Orbitals Program for Calculating CrystalProperties (Technical Univ, Vienna, 2001).[28] D. He and T. Duffy, Phys. Rev. B , 4 (2006).[29] M. Matsui, E. Ito, T. Katsura, D. Yamazaki, T. Yoshino,A. Yokoyama, and K. I. Funakoshi, J. Appl. Phys. ,013505 (2009). [30] C. Bercegeay and S. Bernard, Phys. Rev. B , 214101(2005).[31] M. G. Mayer, Phys. Rev. , 184 (1941), ISSN 0031899X.[32] T. Tietz, J. Chem. Phys. , 789 (1956).[33] K. V. Yusenko, E. Bykova, M. Bykov, S. a. Gromilov,A. V. Kurnosov, C. Prescher, V. B. Prakapenka, M. Han-fland, S. van Smaalen, S. Margadonna, et al., Journal ofAlloys and Compounds , 155 (2015), ISSN 09258388.[34] R. Jeanloz, B. K. Godwal, and C. Meade, Nature349