Structural Properties of Solid : density functional study

Tatsuki ODA Graduate School of Natural Science and Technology, Kanazawa University Kanazawa, 920-1192, Japan

Abstract the liquid transforms to the solid phases; γ- at 54 K, β-phase at 44 K, and α-phases We have studied structural properties of the at 24 K [5]. These are categorized to the molec- high-pressured solid with using the ular crystal that there are covalent bonds in density functional theory. For the ε-phase, we intramolecule and weak interactions between employed the lattice constants of available ex- intermolecules. The magnetism of molecules is perimental data and obtained the stable struc- preserved in these phases [6, 7] and the α-phase ture of double chain. The structure factor ex- has an anti-ferromagnetic ordering of molecu- tracted from the electron density shows a good lar magnets [8]. In these phases, the rectan- agreement with experimental results. The lo- gular shape of O4 is a typical local structure cal structure of chain was found to have O2 of multi-molecules. The isolated O4 has the pairs, which is consistent with the results of op- ground state of spin singlet in gaseous phase tical measurements. The persistent magnetic [9] and could be found in liquid phase [10, 11]. polarization has never been supported. In the latter phase, the chain of O2 has been implied [12]. This was because the γ-phase had the molecular chains accompanied with the ro- 1 Introduction tational fluctuation of molecules. The of β-phase has the two-dimensional Oxygen molecule has a spin-triplet ground array of molecules, in which the triangular state. The magnetism on molecules is the lattice causes an magnetic frustration among main origin of paramagnetic or antiferromag- anti-ferromagnetic interactions [8]. netic properties of condensed oxygens. It could make complex on the structural property of Imposing the hydrostatic pressure, the α- oxygen. High pressures on oxygen have pro- phase transforms to novel states of structure. moted the collapse of magnetism and the in- The optical measurements implied δ-phase in duction of metallic feature. In the metallic a wide range of pressure at low temperatures phase, the superconducting property was also [13]. Akahama et al. concluded the direct reported [1]. In this report, we present a short transition to ε-phase at 7.6 GPa from the re- review for condensed phases and our recent sults of X-ray diffraction measurements [14]. work on the structural property in ε-phase [2]. From the analysis of spin-polarized neutron This phase appears in a wide pressure range diffraction measurements, Goncharenko et al. of solid oxygen. The atomic positions in the found the phase boundary of different mag- phase has not been determined yet. The mag- netic orders [15]. These recent studies for netic collapse and the metalization has been phase boundary among α-, δ-, and ε- seem to studied at boundaries of the phase[3, 4]. be still controversy. The for low Oxygen forms gas phase at the room tem- pressures is presented in Fig. 1. The boundary peratures and normal pressures. At the low between liquid- and solid-phases could be ex- temperature of 90 K, the gas changes to a para- pected (not shown) in Fig. 1. For ε-phase, magnetic liquid. At further low temperatures, based on the speculation related with mag- netic interactions and triangular local struc- The structural analysis with X-ray diffrac- tures, the noncollinear magnetism has been tion measurement at ε-phase has been per- studied [16]. formed first by Johnson et al. and successively For the high pressures, no by some groups [3, 18, 19]. The magnetic α- has been observed up to 96 GPa [3], above phase is transformed to ε-phase around 8 GPa which the system of ε-phase transforms to [14]. The α- and ε-phases are analyzed to metallic ζ-phase. The early work on electronic monoclinic phases classified by the same space structure calculation for solid oxygens was de- group (C2/m), as shown in Fig. 1 [20]. The voted to the study on insulator-metal transi- molecular axes point in the direction normal to tion [17]. At the transition pressure, where ab-plane. This is consistent with both of avail- the discontinuity in the X-ray diffraction pro- able experimental and theoretical results. The files was found, the vibron frequency has been unit cell of ε-phase is dimerized along both a- still observed [18]. This is the signature of and b-axes at the transition from that of α- O2 molecule. The structure of ζ-phase is not phase, resulting to eight molecules in the cell. definitively determined, however the theoret- The most impressive feature is an appearance ical approach for the high-pressured crystal of the new (31¯1¯) line in the diffraction profile proposed the prototype of crystal structure [14]. This is accompanied by the (31¯0) line, which has a good agreement in X-ray diffrac- whose d-value (distance between neighboring tion profiles [4]. diffraction planes) crosses around 50 GPa with The feature of ε-phase is an occupation over that of the (22¯0) diffraction line [18]. a wide range of pressures in the phase diagram. The infrared and Raman spectra of op- This means a stability of atomic and electronic tical measurements have provided the local structures. So far, however, the atomic posi- structure information. Angnew et al. pro- tion of ε-phase has never been determined yet. posed a simple chain model of O4 unit to ex- It could be interesting to know a form of molec- plain their data and Gorelli et al. found the ular arrangement, because solid oxygen have new peak in the far infrared region, confirm- the peculiar properties which the other solid ing the existence of dimerized unit of oxygen of diatomic molecules never has. The determi- molecules [21, 22, 23]. This O4 unit seems to nation of structure in ε-phase will prompt the be non-magnetic unlike the unit in the liquid study on solid oxygen considerably. or gaseous phases [9, 11]. The recent study based on the density functional theory (DFT)[24] predicted the herringbone-type chain structure(space group Cmcm) in dense oxygen [25]. In this study, stability of the phase was investigated by estimating enthalpies accurately from first- principles. The assumption for size of unit cell, however, may not be commensurate with the results of X-ray diffraction. We proposed a new crystal structure which models the ε-phase. This study was based also on the density functional approach [24], but performed with referring the result of X-ray diffraction closely. Taking into account this Figure 1: Phase diagram of oxygen. As priority of study, we used only total energies a boundary between α- and δ- phases, the and atomic forces for structural optimization. boundaries specified by the curve 1 and curve 2 Our proposed structure, which has a double were proposed on the base of the X-ray diffrac- chain consisting O2 pairs, shows lower ener- tion analysis [14] and the spin-polarized neu- gies than the herringbone-type chain structure tron diffraction [18], respectively. at high pressures and consistencies with exper- Table 1: Lattice constants at the pressures for 3 calculations. 2 P(2,2) 1 (GPa) Lattice Constants P(1,1) a (A)˚ b (A)˚ c (A)˚ β (deg.) 0 P(1,3) 9.6a 8.248 5.768 3.814 117.66 -1 b -2 19.7 7.705 5.491 3.642 116.2 -3

c Pressure (GPa) 33 7.39 5.23 3.53 115.6 -4 P(3,3) c 54.5 7.13 4.99 3.43 115.3 -5 c 71 7.00 4.83 3.36 115.1 -6 c -7 90 6.89 4.70 3.31 114.8 40 60 80 100 120 Energy Cut-off (Ry) a Ref. [14], b Ref. [19], c read from Fig 2 in Ref. [18] Figure 2: Energy cut-off dependences of pres- imental results. sure tensor. Four non-vanishing elements are presented only. The experimental lattice con- stants of 0.96 GPa were used [14]. 2 Method THz) and the latter is larger than the exper- For the density functional approach, we used imental value (5.12 eV) by 17 % [30]. These planewave basis to represent wavefunctions properties are similar to the previous results in and electron densities and ultrasoft-type pseu- density functional approaches [4, 25, 28]. The dopotentails [26, 27] to include core-valence in- lattice constants used in this work were ex- teractions. The 1s states were included in the tracted in the X-ray diffraction measurements core, while the other states were described ex- at five points of pressure (9.6, 19.7, 33, 54.5, plicitly. In the construction of pseudopoten- 71, 90 GPa) [14, 18, 19]. and listed in Table 1. tials, we took the cut-off radius of 1.15 a.u. with including a d-symmetry local orbital [28]. The energy cut-offs of 100 and 400 Ry were taken for wavefunction and electron density, -31.981 respectively. This level of cut-off shows the -31.982 convergence of internal pressures as well as the total energy of systems. Figure 2 represents -31.983 the energy cut-off dependence of pressure ten- -31.984 sor which was calculated with the lattice con- Energy (Ha) -31.985 stants at 0.96 GPa [14]. The k-point sampling of 4×4×4 meshes for the eight-molecular unit -31.986 cell was used. It is essentially enough to com- -31.987 1.16 1.18 1.2 1.22 1.24 1.26 1.28 pare the total energies for two types of atomic Bond Length (Angstrom) configuration at fixed volumes for semiconduc- tors. The exchange and correlation energy was treated in the generalized gradient approxima- Figure 3: Total energies with respect to bond tion (GGA) proposed by Perdew and Wang length. (PW91) [29]. The ground state of molecule corresponds to the spin-polarized state with a magnetiza- 3 Result and Discussion tion of 2 µB and shows a bond length of 1.22 A(see˚ Fig. 3), a vibrational frequency of 46.7 We consider the molecular arrangement of THz, and a binding energy of 5.99 eV. The two monoclinic α-phase [8, 20], as an ideal struc- former are in good agreement with the corre- ture, in which molecules form two rectangular sponding experimental values (1.21 A,˚ 47.39 sublattices in the ab-plane and the molecular u1 u2 ∆v ∆x (a) type 1 (b) type 2 1' 7 Figure 4: Distortion of molecules in ab-plane 8' 6 for (a) type 1 and (b) type 2. 3 8 b 2 5 4 1 axis is perpendicular to the plane. The c-axis forms the angle of about 116 degree with the a- a axis [3]. This ideal structure shows the extinc- (a) 9.6 GPa tion of (31¯0) diffraction for the structure fac- tor. To examine this diffraction line in struc- ture factor, we performed the simple simula- tion in which we assumed the charge density of Gaussian type on atomic positions. As a result, it was found that to induce the experi- mental features we could make the distortion, alternate chain slidings along [110] direction, (b) 54.5 GPa (c) 90 GPa represented in Fig. 4(a) (calls as type 1), in Figure 5: Molecular arrangements obtained at which the molecules are taken off from the (a) 9.6 GPa, (b) 55 GPa, and (c) 90 GPa. The (31¯0) planes alternatively. In contrast to this, thin and thick lines are drawn for guide of eyes. the atomic configuration of previous work pre- The couple of parameters, u1 and u2, specifies sented as type 2 in Fig. 4(b), shows the extinc- the distances of neighboring chains along the tion of (31¯0) and forms two strong diffraction [110] direction. lines, (200) and (201¯), around the (001) line. As shown Figs. 4(a) and (b), the internal coor- dinates, ∆v and ∆x, for these types of distor- GPa. We call the optimized structures type tion determine the individual atomic positions 3 configuration. In the dimerized chain, the and we could use initial atomic configurations individual molecule has two nearest neighbors for optimization. for the bond angle with being nearly normal in At the pressure of 9.6 GPa, the internal coor- ab-plane and there exist the rectangular O4’s dinates of ∆v and ∆x are separately optimized (pairs of O2). These structural features com- to be 0.037 and 0.067. The successive opti- pletely coincide with results of the optical ex- mizations revealed that the distortion of type periments [21, 22]. 1 induced an instability to dimerized chains of The electronic structure is common to the the [110] direction. This dimerization of chain crystals of type 2 and 3 in forming the en- essentially provides a new picture of inter- ergy gap at the Fermi level. The dimeriza- nal molecular arrangement for ε-phase, which tion of molecules in rectangular shape forms would correspond to a simple chain model pre- the bonding and anti-bonding orbitals which viously proposed [21, 22], but has never been consist of the couple of π∗ molecular orbitals. associated with the crystal structure. Resulting to a non-bonding orbital on each To search a lower energy geometry, we con- molecule, in crystal, this orbital connects to structed the initial configuration with mixed the non-bonding orbital in the other neighbor- distortions, for example, ∆v = 0.04 and ∆x = ing molecule. On the connection there is a 0.03. Starting optimizations from these kinds difference between type 2 and 3, however al- of configuration, we found that the low energy most the same size of energy gap (bonding- configuration presented in Fig. 5 (a) at 9.6 antibonding splitting) is formed. The total en- ergies of type 3 are as low as those of type 2 Table 2: Distances of neighboring molecules. (the herringbone-type chain structure). Figure The value of dmn specifies the distance of m’th 6 shows the total energy differences between and n’th molecules numbering in Fig. 5 (a). type 3 and type 2 at five pressures calculated. ˚ The new structure of type 3 is slightly lower in pressure distance (A) 0 energy than the type 2 at the high pressures. (GPa) d12 d23 d34 d36 d28 This shows a contrast with the previous result 9.6 2.10 2.14 2.13 2.89 2.76 in which the distortion of type 2 is stable only 54.5 1.99 2.04 2.01 2.31 2.28 at lower pressures [25]. 90 1.99 2.08 1.99 2.10 2.20

0.04 tion results in two distances between neighbor- 0.03 ing chains. The relaxation of these distances

0.02 against the increasing pressure could be char-

0.01 acterized by an equal rate of decrease for the

0 lattice constants, a and b. In Fig. 8, we present pressure dependence of the two distances, u1 -0.01 and u2 (see Fig. 5(a)). The shorter distance Total Energy (eV/.) -0.02 shows the gradual decrease as increasing pres- -0.03 sure, while the longer one decreases rapidly at -0.04 0 20 40 60 80 100 low pressures. This rapid decrease, which is an Pressure (GPa) indirect evidence of chain structure along [110] direction, is in good agreement with the previ- Figure 6: Total energies of the type 3 configu- ous analysis of X-ray diffraction measurement ration with respect to that of type 2 at calcu- at low pressures, which resulted in the similar lated pressures. decrease of a and b [18].

At higher pressures, the double chain struc- ture is preserved. The structures of 54.5 and 1.215

90 GPa are presented in Fig. 5 (b) and (c), 1.210 respectively. In the double chain, however, a novel change in arrangement of molecules oc- 1.205 curs, namely, some angles between neighbor- 1.200 ing bonds are decreased to an angle less than 90 degree. At high pressures near the transi- 1.195 Bond lenght (Angstrom) tion point to ζ-phase, the molecular arrange- 1.190 ment within ab-plane becomes similar to that 1.185 of α-phase. The distances between neighboring 0 20 40 60 80 100 molecules are listed in Table 2. The distances Pressure (GPa) connecting nearest neighbors (d12, d23, and d34 in the table) are shorter than the others. The Figure 7: Pressure dependence of intramolec- largeness of d36 indicates a novel arrangement ular bond length. of molecules within the double chain. The pressure dependence of d23 is not monotonic, The profile of X-ray diffraction measurement showing a complex feature of molecular relax- is related with the structure factor contributed ation. The bond length in each molecule lin- from the electron density. To see the compari- early decreases between 1.212 and 1.187 A˚ with son with experimental data, the structure fac- respect to pressure, as presented in Fig. 7. tors, estimated from the calculations, are pre- Our proposed crystal structure has the pair sented in Fig. 9. The extinction law in the of chains, which are stacked in the direction profiles is in good agreement with the avail- normal to the [110] direction. This dimeriza- able experimental data [3, 14, 18]. The profiles 2.7 (2-20) 2.6 1.2 (001) (3-10)(2-2-1)(40-1) (002) (3-1-1) (400) 90 GPa 2.5 1

2.4 71 GPa 0.8

2.3 54.5 GPa 0.6 33 GPa 2.2 (5-1-1)

Intensity (arb. unit) 0.4 distance (Angstrom) 2.1 19.7 GPa 0.2 2 (1-30) (200) (020) (400) (040) 9.6 GPa 0 1.9 5 10 15 20 25 0 20 40 60 80 100 2 θ (degree) Pressure (GPa)

Figure 8: Pressure dependence of two dis- Figure 9: Structure factors at the pressures tances between neighboring chains. calculated as a function of the diffraction angle of measurement. The wave length of X-ray, λ = 0.4817 A˚ was used. closely reproduce the features of experimental data at the relatively high pressures. As pres- appears at the range of 33∼55 GPa. sure increases, the intensity of (31¯1)¯ and (31¯0) reduces and the intensity of (401¯) grows. The Due to the constraint of periodic boundary crossings of (31¯0) with (22¯0), and (13¯0) with condition, the infinite chain exits in the crystal (400) are clearly shown in the calculation, as structure of our model. This recalls a possibil- observed in the work of Weck et al.. In the ity of sliding along the chain. The alternate × present work, some weak lines are expected in distortion of sliding, resulting in a 4 4 super- the range of low angles at the low pressures, lattice of ideal unit cell (structure of α-phase), where no indication exits in the correspond- did not improve the profile of structure fac- ing experimental diffraction profiles. This fact tors at low pressure. Another candidate of im- and some differences in relative intensity be- provement is a variation of long period. The tween observed and calculated profiles at the existence of modulation is not denied by the re- low pressures (strong intensity of (31¯1)¯ and stricted information deduced from our present weak of (401¯) at 9.6 GPa in Fig. 9, compared work. with experimental results [14]) imply that our proposed molecular arrangement presented in The ε-phase is bound on the magnetic Fig. 5 (a) are not completely enough to de- phases at the low pressures. The recent neu- scribe details of the structural property. tron diffraction measurement revealed that the At low pressures, due to the property of magnetic alignment changed at the boundary molecular crystal, the intermolecular interac- between α- and δ-phases [8, 15]. In both of tion is important for determining detailed ge- phases, the antiferromagnetic alignment along ometry of molecules. The discrepancy between a [110] direction exists. In general, there is calculation and experiment will be attributed no relation between two phases on first or- to the shortcoming that the DFT do not con- der transition except for their energetics. It tain the many-electron effect on weak molec- is interesting, however, to speculate the role ular pair interaction like a Van Der Waals in- of magnetic interaction. The chain of herring- teraction [31]. The DFT is also incomplete in bone type (type 2) is nonmagnetic, while the electron correlation effects on electronic struc- single chain of [110] before the transition can ture. The band gap is considerably underes- preserve their magnetic state. This magnetic timated; direct gap(Γ point) of 0.88 eV and interaction might stabilize the linear structure indirect gap (Γv − Ac) of 0.75 eV at 9.6 GPa along [110] direction. Successively, the dimer- (experiment: 2.4 eV around 10 GPa) [32]. The ized chain becomes nonmagnetic due to the overlap between valence and conduction bands hybridization of non-bonding orbitals between chains. The noncollinear magnetic scheme [9] V. Aquilanti, D. Ascenzi, M. Bartolomei, [33, 34], which efficiently searches the low- D. Cappelletti, S. Cavalli, M. D. C. Vi- est energy configuration of magnetic system, tores, and F. Pirani, Phys. Rev. Lett. 82, found the ground state to be the same non- 69 (1999). magnetic configuration described here. The proposal of noncollinear magnet for ε-phase [10] G. N. Lewis, J. Am. Chem. Soc. 46, 2027 has never been supported in our calculations (1924). [4, 16]. [11] T. Oda, and A. Pasquarello, Phys. Rev. Lett. 89, 197204 (2002); ibid., J. Phys.: 4 Summary Condens. Matter 15 S89 (2003); ibid., Phys. Rev. B 70, 134402 (2004). We gave a short review on the structural prop- erties for condensed phases. We presented [12] A. P. Brodyanskii, Yu. A. Freiman, and calculated results for crystal structures of ε- A. Jezo˙ wski, J. Phys.: Condens. Matter phase, with use of the DFT and assumption 1, 999 (1989). of the experimental lattice constants. We pro- posed the new structure which has the dimer- [13] M. Santoro, F. A. Gorelli, L. Ulivi, R. ized chain of O2 molecules. The local structure Bini, and H. J. Jodl, Phys. Rev. B 64, (rectangular O2 pair) is consistent with results 064428 (2001). of the optical measurements and structure fac- tors of crystal almost agree with the profiles [14] Y. Akahama and H. Kawamura, O. Shi- B 64 of X-ray diffractions. The double chain along momura, Phys. Rev. , 054105 (2001). [110] direction would be used to refine the anal- [15] I. N. Goncharenko, O. L. Makarova, and yses in high-pressured oxygens. N. Ulivi, Phys. Rev. Lett. 93, 05502 [C,D class; 29000K (A), 3000K (B)] (2004).

[16] R. Gebauer, S. Serra, G. L. Chiarotti, References S. Scandolo, S. Barroni, and E. Tosatti, Phys. Rev. B 61, 6145 (2000). [1] K. Shimizu, K. Suhara, M. Ikumo, M. I. Eremets, and K. Amaya, Nature (Lon- [17] M. Otani, K. Yamaguchi, H. Miyagi, and 393 don) , 767 (1998). N. Suzuki, J. Phys.: Condens. Matter 10, [2] T. Oda et al., in preparation. 11603 (1998). [3] Y. Akahama, H. Kawamura, D. H¨auser- [18] G. Weck, P. Loubeyre, and R. LeToullec, mann, M. Hanfland, and O. Shimomura, Phys. Rev. Lett. 88 035504 (2002) Phys. Rev. Lett. 74, 4690 (1995). [19] S. W. Johnson, M. Nicol, and D. Schiferl, [4] S. Serra, G. Chiarotti, S. Scandolo, and E. J. Appl. Cryst. 26, 320 (1993). Tosatti, Phys. Rev. Lett. 80, 5160 (1998). [20] C. S. Barrett, L. Meyer, and J. Wasser- 23 [5] G. C. DeFotis, Phys. Rev. B , 4714 man, J. Chem. Phys. 47, 592 (1967). (1981). [21] S. F. Angnew, B. I. Swanson, and L. H. [6] A. Perrier and H. Kamerlingh Onnes, Jones, J. Chem. Phys. 86 5239 (1987). Phys. Comm. Leiden, 139c-d, 25 (1914). [7] E. Kanda, T. Haseda, and A. Otsubo, [22] G. A. Gorelli, L. Ulivi, M. Santoro, and 83 Physica 20, 131 (1954). R. Bini, Phys. Rev. Lett. 4093 (1999). [8] R. J. Meier and R. B. Helmholdt, Phys. [23] G. A. Gorelli, L. Ulivi, M. Santoro, and Rev. B 29, 1387 (1984). R. Bini, Phys. Rev. B 63 104110 (2001). [24] P. Hohenberg and W. Kohn: Phys. Rev. 136 (1964) B864 ; W. Kohn and L. J. Sham: Phys. Rev. 140 (1965) A1133.

[25] J. B. Neaton and N. W. Ashcroft, Phys. Rev. Lett. 88, 205503 (2002)

[26] D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).

[27] A. Pasquarello, K. Laasonen, R. Car, C. Lee, and D. Vanderbilt, Phys. Rev. Lett. 69, 1982 (1992); K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Van- derbilt, Phys. Rev. B 47, 10142 (1993).

[28] B. Militzer, F. Gygi, and G. Galli, Phys. Rev. Lett. 91, 265503 (2003).

[29] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992).

[30] K. P. Huber and G. Herzberg, Molecu- lar Spectra and Molecular Structure: IV. Constants of Diatomic Molecules, (Van Nostrand Reinhold Company, New York, 1979), p. 490.

[31] K. Nozawa, N. Shima, and K. Makoshi, J. Phys. Soc. Jpn. 71, 377 (2002).

[32] S. Desgreniers, Y. K. Vohra, and A. L. Ruoff, J. Phys. Chem. 94, 1117 (1990).

[33] R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985).

[34] T. Oda, A. Pasquarello and R. Car, Phys. Rev. Lett. 80, 3622 (1998).