Hydrogenation, dehydrogenation of $\alpha$-tetragonal boron and its transition to $\delta$-orthorhombic boron

Boron bulk crystals are marked by exceptional structural complexity and unusual related physical phenomena. Recent reports of hydrogenated $\alpha$-tetragonal and a new $\delta$-orthorhombic boron B$_{52}$ phase have raised many fundamental questions. Using density functional theory calculations it is shown that hydrogenated $\alpha$-tetragonal boron has at least two stable stoichiometric compositions, B$_{51}$H$_{7}$ and B$_{51}$H$_{3}$. Thermodynamic modeling was used to qualitatively reproduce the two-step phase transition reported by Ekimov et al. [J. Mater. Res. 31, 2773 (2016)] upon annealing, which corresponds to successive transitions from B$_{51}$H$_{7}$ to B$_{51}$H$_{3}$ to pure B$_{52}$. The so obtained $\delta$-orthorhombic boron is an ordered, low-temperature phase and $\alpha$-tetragonal boron is a disordered, high-temperature phase of B$_{52}$. The two phases are connected by an order-disorder transition, that is associated with the migration of interstitial boron atoms. Atom migration is usually suppressed in strongly bound, covalent crystals. It is shown that the migration of boron atoms is likely to be assisted by the migration of hydrogen atoms upon annealing. These results are in excellent agreement with the above mentioned experiment and they represent an important step forward for the understanding of boron and hydrogenated boron crystals. They further open a new avenue to control or remove the intrinsic defects of covalently bound crystals by utilizing volatile, foreign atoms.


I. INTRODUCTION
Crystalline boron is the last monatomic system for which the phase diagram is not fully determined yet and even the number of phases is not certain. Currently only three, αrhombohedral (R), β-R, and γ-orthorhombic (O) boron, are generally accepted, but more than ten phases were reported before. However, the field underwent great progress, recently (for a review see Ref. 1).
Among the various reported allotropes is α-tetragonal boron (α-T boron), which is a quite complex material. The α-T phase was first synthesized in 1943. 2 Its structure was thought to be B 50 , where each unit cell is composed of four icosahedra and two interstitial atoms. 3 But theorists questioned the existence of B 50 because the structure is short of 10 electrons to completely fill up its valence states. 4 Later, experiments showed that the apparent α-T B 50 crystals were actually containing carbon and nitrogen impurities [5][6][7] and the stability of these impurity-containing B 50 C 2 and B 50 N 2 crystals was also demonstrated by DFT calculations. 8,9 However, Hayami and Otani theoretically showed that α-T boron can be stabilized if the boron content is changed to B 52 . 10 At the same time several experimental works, mostly employing high-pressure high-temperature (HPHT) methods, reported the synthesis of α-T B 52 . [11][12][13][14][15][16][17] The HPHT synthesis was not used in the early days, and it is therefore likely that α-T boron is thermodynamically stable at high pressure and high temperature. 18 Unfortunately, owing to the lack of accurate information about the crystal structures, it is not clear if the various reported samples represent the same phase or if they contain impurities. A recent theoretical study by Uemura et al. has identified the characteristic features of pure α-T boron B 52 . 19 It was shown that the occupation of interstitial sites as well as non-stoichiometric compositions, i.e. non-integer number of atoms per primitive unit cell, are crucial for the stability of the system. Their results could then be used to define a family of α-T boron systems. 11 Here we are concerned with hydrogenated α-T boron and its relation to a new orthorhombic boron phase. Hydrogenated α-T boron crystals, with compositions B 51.5 H m , were prepared by Ekimov el al. 11,20 Their samples were synthesized by thermal decomposition of decaborane at high temperature (T ∼ 1100 − 1300 • C) and high pressure (p = 8 − 9 GPa) and the initial H content was m = 7.7. This is exceptionally high for a crystalline semiconductor. Afterwards they annealed the crystals at ambient pressure and observed two phase transitions as T was raised. At T = 450 • C, m reduced to m = 4.7 and at T = 700 • C, hydrogen was fully released. Interestingly, during the second step the tetragonal lattice underwent a transition to an orthorhombic lattice. In the following we will call this structure δ-O boron in order to distinguish it from the more established γ-O boron. The formal composition B n of δ-O boron is n = 52, although X-ray analysis showed n to be in the range from 51.6 to 52. This structure was theoretically predicted by Hayami and Otani 10 , and Zhu. 21 However, their calculations were limited to the primitive unit cell and therefore miss two important aspects of boron crystals which were found to be crucial for the stability of α-T boron: the effect of disorder and non-stoichiometric compositions. 19 Similar investigations on δ-O are necessary before it can be considered as a new boron allotrope.
In this paper, the structure of hydrogenated α-T boron, the process of dehydrogenation, and the phase transition to δ-O boron are studied theoretically. The study of hydrogenation/dehydrogenation is of general importance for material research, and our results are potentially useful for hydrogen-related technologies, such as hydrogen storage. 22,23 Our insights on the phase transition are of central importance for ongoing efforts to create the phase diagram of boron. The paper is organized as follows. Section II B describes the crystal structure and calculation methods, along with several definitions used throughout this paper. In Sec. III, a structural study on hydrogenated α-T boron is given. In Sec. IV, the process of dehydrogenation is studied. In Sec. V, the phase transition and its implications are discussed. Finally a summary is given in Sec. VI.

II. CRYSTAL STRUCTURE, DEFINITIONS, AND METHOD
A. Crystal structure The basic unit cell of α-T-type crystals contains four icosahedra and two interstitial atoms (B 50 ). To ease comparison with the idealized α-T B 50 in previous works, we describe our systems within the high-symmetry space group P 4 2 /nnm, although the real structures have lower symmetry. There are several interstitial sites in α-T boron and they are illustrated in Fig. 1. The site names are given by their Wyckoff notations in parenthesis, such as (2b) or (4c). Among them, the (2b) site is a fully occupied site (FOS) and the occupying atoms can be considered as a part of the B 50 host crystal. Others are partially occupied sites (POS), which could also be considered as defects. The lowest-energy structure of pure α-T boron is B 52 , i.e., the basic B 50 structure with two additional (4c)-site atoms. 10,19 There are four symmetry-equivalent sites for (4c) in α-T boron. When two atoms occupy two (4c) sites in the same ab plane, we call it the in-plane configuration. When they occupy two (4c) sites in different ab planes, we call it the out-ofplane configuration (see Fig. 2). The lowest-energy structure of pure α-T boron is B 52 in the out-of-plane configuration 10,19 and in this paper pure α-T boron is always related to this structure, unless otherwise stated.
The formula unit of hydrogenated α-T boron is B n H m ; henceforth, n and m are used to indicate B and H contents, respectively. Unfortunately, the chemical composition, given by n and m, is not quite certain. When Ekimov et al. reported the synthesis of pure α-T boron, n = 51.5 was specified. 11 However, hydrogen inclusion was later discovered by mass spectroscopy, and two kinds of compositions BH 0.15 and BH 0.09 were found by annealing.
Furthermore, the B content varied from 51.5 to 52.0 during the annealing. 20 Whether or not the difference in the B content is due to experimental error is unclear. If we assume n = 51.5 for the parent crystal, then BH 0.15 and BH 0.09 correspond to m = 7.7 and 4.6, respectively.
These are the compositions mentioned in the introduction.

B. Computational methods
The electronic structures of the considered systems were studied by density functional theory (DFT) using the pseudopotential method and the Osaka2k code. 24 It uses the parameterization by Perdew and Zunger for the local density approximation (LDA) 25 where B 52 is assumed to be in the lowest-energy structure of pure it is common to consider the normalized formation energy Now, let j refer to a specific configuration of H atoms. The experimental observed hydrogen content is the ensemble (thermal) average m , where different configurations j contribute to m through their formation energies E f,mj . The ensemble average is obtained through calculating the partition function Z, where g mj is the multiplicity of the j-th configuration of the H content m and β = 1/k B T is inverse to temperature with the Boltzmann constant k B . In Eq. (4), the configurations j for a given value of m are grouped and the summation over a group is denoted by z m , where F m is the partial free energy for the H content m. The entropic contributions to F m are then entirely from the configurational degrees of freedom. Finally, the mean value m is obtained by In order to study the annealing of B n H m at ambient pressure, Eq. (1) is modified such that solid hydrogen H (s) is replaced by molecular hydrogen in the gas phase 1/2 H (g) 2 and its partial pressure p is a parameter. As the density of hydrogen gas is much smaller than that of the solid, we have to replace the total energy E[H (s) ] in Eq. (2) by the chemical potential of hydrogen gas µ H 2 . For µ H 2 , we use the ideal gases expression where c = 5/2, R is the gas constant, and v is the molecular volume. Real gas corrections to Eq. (7) (van der Waals or the inclusion of the latent heat of phase transitions) are estimated to be negligible for the considered energy scales. The mj-th component of the free energy F mj (T, p) of the j-th H configuration and m H atoms is then obtained by For the determination of the partition function Z in Eq. (4), this free energy F mj (T, p) is used instead of E f,mj and f = F/(n + m) indicates the normalized free energy.

A. Bonding and vibrational properties of interstitial hydrogen
For pure α-T boron the characteristics of interstitial sites were analyzed previously. 19 It was found that up to a composition n = 52 the (4c) site is the most preferable POS, followed by (8h) and (8i). The occupation of (4g) is negligible and (2a) is the least preferable one (however, this site is important for the inclusion of N or C atoms).
The site occupancies of hydrogenated α-T boron were measured by Ekimov et al. 11 Since H atoms cannot be detected by X-ray diffraction, only the occupancies of B atoms were measured to be: 100% (2b), 31% (4c), 6% (4g). An apparent feature of hydrogenated α-T boron is the occupation of the (4g) site, which has never been reported before for pure α-T boron.
In order to determine the preferable hydrogen sites, we calculated the formation energies as listed in Table I. For all H sites, e f decreases when the B content increases from 50 to 51. We will explain this results in the discussion below. Let us first focus on the B 51 H case, where the interstitial B atom is located at a (4c) site. e f of the (8j) site is by far smallest, which is interesting because (8j) is not very relevant for pure α-T boron. The next most However, the value is significantly reduced from that of pure α-T boron (76 meV/atom) 19 so that we cannot fully ignore this cite. The site at the center of the icosahedron (ico) can be excluded from further consideration because e f is far too high.
These differences in the formation energy arise from the different bonding environments of interstitial sites in B 50 H and B 51 H. Table II   Hydrogenation could thus be a practical method to control the in-gap states of α-T boron, which cannot be achieved by varying the B content only, as explained above. Furthermore, it could be possible to identify these states by optical spectroscopy. Since the number of in-gap states in pure and hydrogenated α-T boron differs, optical spectroscopy could help to distinguish corresponding crystals (another method is to measure the lattice parameters, as discussed in Sec. III D).

C. Stability of hydrogenated α-tetragonal boron
In order to learn about the energetic stability and the thermodynamic properties of hy- As expected from the analysis in Sec. III A, the (8j) sites are the most preferential ones for H occupation, and hence this result is reasonable. Within this range the lowest-energy configuration is m = 3 (three (8j)-sites are occupied). As seen in Fig. 3, this corresponds to completely filled valence bands, with leaving the in-gap states unoccupied.
For m > 4, other sites come to participate, although (8j)-sites are still major sites. This series is indicated by a blue line. The lowest-energy state is found at m = 7 and the H configuration is: five (8j) and two (4g) sites (also see Fig. 3). Interestingly, the next preferential sites to occupy after (8j) are (4g) and not (8h) or (8i) sites, even though the low formation energies in Table I suggest that (8h) and (8i) sites should follow. The reason for this is found by using the empirical rule from above, namely, placing a H atom close to a site where covalent bonds are already formed causes e f to increase. When (8j) sites are already occupied, additional (8h) or (8i)-site occupation increases e f because the (8h) and (8i) sites are close to (8j) site. Therefore (4g)-sites are more favorable to be occupied in that situation.
A notable observation in Fig. 4 is that there are two local minima at m = 5 and m = 7 that nicely correspond to the experimentally observed H content of the crystals for which a phase transition occurred in the annealing experiment, namely 4.6 and 7.7. 20 This correspondence is discussed in more detailed in the next section.
With these results we can calculate the thermodynamic average m at finite tempera-

IV. TWO-STEP TRANSITION DURING DEHYDROGENATION
In this section, we investigate the dehydrogenation of B 51 H m crystals by annealing, as performed by Ekimov et al. 20 In their experiments the as-grown B 51 H 7.7 crystals were subject to annealing at ambient pressure and a two-step phase transition was observed. In the first step the hydrogen content was reduced to m = 4.6 at T a1 = 600 − 720 K (the mean value is 660 K). Further raising the temperature led to the complete release of H at T a2 = 820 − 970 K (the mean value 900 K).
The partial free energy f m calculated by using Eqs. (7) and (8)  is an open channel for the interstitial atoms to move. This removal may be called "thermal degassing". 36 For silicon clathrates, thermal degassing of sodium atoms was achieved by heating in vacuum. This process occurred at a certain temperature: T = 635 K for Na 24 Si 136 clathrate 37,38 and 400 K for Na 4 Si 24 clathrates 36 . In these clathrates, sodium atoms occupy definite sites (implying long-range order) and therefore a sharp phase transition is observed.
Sung et al. proposed the idea that new silicon allotropes might be obtained by degassing the foreign species from a clathrate compound. 39 The present study extents the idea of thermal degassing to a more universal tool for material syntheses. revealed that they are identical, due to nearly vanishing structural and energetic differences.
Because of the excellent consistency between the experimental and the theoretical structures, we conclude that the δ-O structure indeed exits. Then an important question is: why and  To address this question, let us explain why the lattice distortion is sensitive to the atom configuration of (4c) boron sites. As seen in Fig. 2 This increase is consistent with our finding that the orthorhombic distortion can only occur for n = 52. However, where the extra B atoms come from is unclear. One possibility may be the agglomeration of B atoms during the initial crystal growth, that could provide mobile B atoms.
Another puzzling point is the apparent ordering of POS during the annealing process. The fact that the parent crystal B 51 H 7 has tetragonal symmetry is consistent for multiple reasons.
First, since the energy difference of different H configurations is small, a random occupation is to be expected for high-T synthesis. Second, the non-stoichiometrc composition of n = 51.5 also supports the tetragonal symmetry. 19 Third, the gain in the formation the energy for the T → O transformation (as shown in Table III) is also very small (less than 2 meV/atom).
This is on the edge of the accuracy of the GGA functional. Thus the T and O structures may be considered as quasi-degenerate. In this case the the free energy at high temperatures is dominated by entropic contribution and therefore a random occupation of the (4c) interstitial B atoms is very likely to occur. On the other hand, it is certain from the discussion above that the (4c) interstitial B atoms are ordered in the out-of-plane configuration after the T→O transition. Because the structural difference between δ-O and α-T boron is only the order of the two (4c)-site atoms, this transformation is an order-disorder transition. Such a transition is suggested theoretically by Widom and Huhn for boron-carbide, although so far there is no experimental evidence. 40,41 For metallic alloys, order-disorder transitions are wellknown: for example, CuZn alloys exhibit a transition at T od = 465 • C, which are associated with the structural transformation from BCC to a CsCl-type structure. 42 But, it is rare to observe an order-disorder transition in strongly bound, covalent crystals.
We modeled the order-disorder transition with a free-energy model. The details of the calculation are given in Appendix A. As shown in Fig. 7, the order-disorder transition temperature T od is 1260 K. This phase transition identifies the ordered δ-O boron to be the low-temperature modification and α-T boron to be the high-temperature modification of B 52 ; the latter is stabilized by the configurational entropy of disordered POS. Therefore, it is no surprise that only α-T crystals were obtained in all the HPHT syntheses reported, conditions. Figure 7 finally reveals the puzzles related to the transition. One of them is that two different types of transitions, hydrogen release and the ordering of interstitial atoms, apparently occur simultaneously. At first glance, the two seem to be unrelated and therefore should occur at different temperatures T dh (or T a2 in the annealing experiment) for complete dehydrogenation and T od for the order-disorder transition. And indeed Fig. 7 shows that T dh is lower than T od . Theoretically, a crystal should transform between O and T at T dh . But, migration of atoms in solids does not occur at low temperatures, due to their low mobility. Therefore this order-disorder transition was not observed before. Here we propose that the role of dehydrogenation is to stimulate the migration of B atoms through the motion of H atoms upon release. Then the temperature difference T od − T dh is the driving force of the transition. Moving B atoms may not be at all ruled out for boron-rich crystals; in β-rhombohedral boron, it is currently speculated that B atoms migrate at a moderate temperatures. 43,44 . Hence, it is possible that the motion of H atoms enhances the migration of B atoms.
Another puzzle is the process of the ordering. To obtain the ordered out-of-plane configu-ration from B 51 H 3 during annealing, not only new B atoms but also atoms at (4c) sites must migrate. This could be difficult because these atoms already form strong covalent bonds with their neighbors. However, (4c) sites are interstitial sites. In silicon the diffusion of impurities is enhanced by accompanying migration of intrinsic defects, vacancies or self interstitials. 45 The H sites in B 51 H 3 are close to (4c) B sites, so that removing a H atom at a (8j) site is likely to affect the occupation of a nearby (4c) B atom. Therefore, we hypothesize that the T→O transition occurs because the migration of B atoms is assisted by the migration H atoms upon hydrogen release. So the three types of changes, the evacuation of H atoms, the migration of interstitial B atoms, and the lattice distortion T→O work cooperatively.
Lastly, we surmise about another cooperative role that hydrogen might have during the growth process. As stated above, the temperature of 1100 − 1300

VI. CONCLUSION
We have studied the structure and thermodynamic stability of hydrogenated α-tetragonal boron, its dehydrogenation process and the transition to δ-orthorhombic boron B 52 with den- connected by an order-disorder transition, that is associated with the ordering of interstitial boron atoms at (4c)-sites. This ordering implies boron atoms to migrate, which is usually hindered in a strongly bound, covalent crystal. Our analysis reveals that the migration of boron atoms is likely to be assisted by the migration of hydrogen atoms upon annealing and we refer to this mechanism as "cooperative atom migration".
Our results represent an important step forward for the understanding of boron and hydrogenated boron crystals. We determined the atomic structure of stable hydrogenated boron compounds and revealed structural and thermodynamic relations between α-tetragonal boron and δ-orthorhombic boron. These findings are also in excellent agreement with the experiments of Ekimov et al. 20 The present study opens an new avenue to control or remove the intrinsic defects of covalently bound crystals by utilizing volatile, foreign atoms. This was previously considered to be difficult.