Novel magnesium borides and their superconductivity

TL;DR

This study uses ab initio methods to discover new stable magnesium borides and analyze their potential superconducting properties under high pressure conditions.

Contribution

It identifies previously unknown stable Mg-B compounds and predicts their stability ranges and superconducting transition temperatures.

Findings

Discovery of new stable Mg-B compounds MgB3 and Mg3B10.

MgB2 remains stable across 0-200 GPa.

Most magnesium borides have low T_c below 3 K.

Abstract

With the motivation of searching for new superconductors in the Mg-B system, we performed ab initio evolutionary searches for all the stable compounds in this binary system in the pressure range of 0-200 GPa. We found previously unknown, yet thermodynamically stable, compositions MgB$_3$ and Mg$_3$B$_{10}$. Experimentally known MgB$_2$ is stable in the entire pressure range 0-200 GPa, while MgB$_7$ and MgB$_{12}$ are stable at pressures below 90 GPa and 35 GPa, respectively. We predict a reentrant behavior for MgB$_4$, which becomes unstable against decomposition into MgB$_2$ and MgB$_7$ at 4 GPa and then becomes stable above 61 GPa. We find ubiquity of phases with boron sandwich structures analogous to the AlB$_2$-type structure. However, with the exception of MgB$_2$, all other magnesium borides have low electron-phonon coupling constants $\lambda$ of 0.32 to 0.39 and are predicted to have T$_c$ below 3 K.

Figures

Tables

Table 1. Table 1: Computed superconducting T c of different sandwich borides

Structure	MgB2 (P6/mmm)	MgB3 (C2/m)	Mg3B10(Amm2)	MgB4 (C2/m)
P (GPa)	0	31	40	33	0
N(Ef) [states/eV per electron]	0.084	0.064	0.038	0.039	0.044
$\lambda$	0.73*	0.38	0.33	0.32	0.39
$⟨{\omega }_{log}⟩$ (K)	719*	811	843	784	749
Tc (K)	27.6*	2.5	1.0	0.7	2.8

Table 2. Table 2: Optimized structures of MgB 3

Phase	Lattice	Atom	x	y	z
	parameters
C2/m [2 f.u.]	a = 2.998 Å	Mg(4i)	0.7471	0.0000	0.7982
layered	b = 5.109 Å	B1(4g)	0.0000	0.6673	0.0000
at 30 GPa	c = 8.852 Å	B2(8i)	0.5210	0.6717	0.4055
	$\beta$ = 115.30∘
C2/m [4 f.u.]	a=7.959Å	Mg1(4i)	0.4363	0.0000	0.7000
at 50 GPa	b=2.850Å	Mg2(2c)	0.0000	0.0000	0.5000
	c=10.833Å	Mg3(2b)	0.0000	0.5000	0.0000
	$\beta$=116.98 ∘	B1(4i)	0.9041	0.0000	0.1214
		B2(4i)	0.8063	0.0000	0.2382
		B3(4i)	0.7390	0.0000	0.5339
		B4(4i)	0.6943	0.0000	0.6689
		B5(4i)	0.8529	0.0000	0.8334
		B6(4i)	0.7445	0.0000	0.9430
Cmcm [2 f.u.]	a = 2.676 Å	Mg(4c)	0.0000	0.5997	0.2500
at 200 GPa	b = 11.521Å	B1(4c)	0.0000	0.2490	0.2500
	c = 2.668Å	B2(4c)	0.0000	0.8302	0.2500
		B3(4c)	0.0000	0.9667	0.2500

Equations

$$T_c=\frac{⟨{\omega }_{log}⟩}{1.2}exp\left(\frac{-1.04(1+\lambda )}{\lambda -{\mu }^{\ast }(1+0.62\lambda )}\right),$$

$$\omega_{log}=exp\Big{[}\dfrac{2}{\lambda}\int\dfrac{d\omega}{\omega}\alpha^{2}F(\omega)ln(\omega)\Big{]}$$

$$\lambda =2{\int }_0^{\infty }\frac{{\alpha }^{2}F(\omega )}{\omega }d\omega .$$

Full text

Superconductivity, high pressure, evolutionary algorithm, density functional theory

Introduction

Tremendous efforts have been put to design conventional superconductors with higher and higher critical temperatures Choi et al. (2009); Dewhurst et al. (2003); Andrzejewski (2008); Mehl et al. (2001); Cava et al. (1994). It is also the main focus of theoretical and experimental studies to determine that how high the superconducting transition temperature Tc can be pushed in binary and ternary boron-compounds. For instance, theoretical works predicted thermodynamically unstable CaB2 to be superconducting at $\sim$50 K Choi et al. (2009) and hole-doped LiBC to have Tc of 65 K Dewhurst et al. (2003). Ternary Mo2Re3B with Tc= 8.5 K Andrzejewski (2008), CuB2-xCx (Tc $\sim$ 50 K) Mehl et al. (2001) and multiple-phase bulk sample of yttrium-palladium-boron-carbon (Tc= 23 K Cava et al. (1994)) are important boron-based superconductors.

The unexpected discovery of superconductivity in MgB2 with high Tc = 39 K Nagamatsu et al. (2001) has triggered a flurry of publications. In previous studies, superconductivity in MgB2 has been thoroughly investigated Welp et al. (2003); Goncharov and Struzhkin (2003); Choi et al. (2002); Brinkman et al. (2002); Mazin et al. (2002). The isotope effect demonstrated the phonon-mediated nature of superconductivity in this compound Hinks et al. (2001). Although doping is usually expressed as a hope to enhance the desired properties, carbon-doped MgB2 (Mg(B0.8C0.2)2) has a lower Tc = 21.9 K Ribeiro et al. (2003). Aluminum, with one more electron than magnesium, was reported to be an unfit candidate for partial substitution for magnesium (Mg1-xAlxB2)Slusky et al. (2001). This shows that increasing electron concentration suppresses superconductivity of magnesium diboride.

Elemental magnesium Li et al. (2010) and boron Oganov et al. (2009) have been shown to exhibit unexpected chemistry under high pressure, raising the motivation of studying their compounds. Moreover, materials composed of light atoms could make good conventional superconductors. The Mg-B system was subject to some explorations of superconductivity Tomita et al. (2001); Ma et al. (2009); Yakıncı et al. (2002). Stability of boron-rich magnesium borides, e.g., MgB7, MgB12 and Mg*∼5B44* has been extensively studied by experiment at ambient pressure Pediaditakis et al. (2010). Borides of similar metals, e.g., Ca-B  Shah and Kolmogorov (2013) and Li-B  Hermann et al. (2012) and stability of 41 metal borides Geest and Kolmogorov (2014) were studied and new compounds were shown to appear at high pressure. High-pressure phase of MgB2 (KHg2-type structure) was reported to be a poor metal with no superconductivity, highlighting the main role of delocalized bonding of the boron honeycomb layers in the superconducting properties of MgB2 with AlB2-type structure Ma et al. (2009).

To date, there is no comprehensive and systematic theoretical research into the stability and properties of magnesium borides at high pressure. Here, with the knowledge of the important role of magnesium Ribeiro et al. (2003), crucial existence of honeycomb boron layers Ma et al. (2009) and substantial effect of electron concentration Slusky et al. (2001), we present results of extensive computational searches for stable magnesium borides MgxBy and their superconductivity.

Methods

Ab initio variable-composition evolutionary method USPEXOganov and Glass (2006); Oganov et al. (2010, 2011); Zhu et al. (2012) was applied to the Mg-B system at 0, 30, 50, 75, 100, 150 and 200 GPa. This method has the capability of finding possible compositions and the corresponding stable and metastable structures at given pressures, and successfully predicted new phases of MgB2 at high pressure Ma et al. (2009) and new stable phases of different systems like Na-Cl, boron and *Na-He * Zhang et al. (2013); Oganov et al. (2009); Dong et al. (2017). High-temperature superconductivity in hydrogen-rich compounds, e.g., Sn-H Esfahani et al. (2016) and Ge-H Esfahani et al. (2017) were also studied. In this method, we created initial generation of structures and compositions randomly with up to 28 atoms in the primitive cell. Subsequent generations were obtained using heredity, transmutation, softmutation, and random symmetric generator Lyakhov et al. (2013).

Structure relaxations were carried out using VASP package Kresse and Furthmüller (1996) in the framework of density functional theory (DFT) adopting PBE-GGA (Perdew-Burke-Ernzerhof generalized gradient approximation) Perdew et al. (1996). The projector augmented-wave approach (PAW) Blöchl (1994) with [He] core for both Mg and B atoms was used to describe the core electrons and their effects on valence orbitals. A plane-wave kinetic energy cutoff of 600 eV and dense Monkhorst-Pack k-points grids with reciprocal space resolution 2$\pi$ $\times$ 0.03 Å*-1* was used Monkhorst and Pack (1976). Phonon frequencies and electron-phonon coupling (EPC) were calculated using Quantum ESPRESSO Giannozzi et al. (2009). PBE-GGA functional is used for this part. A plane-wave basis set with a cutoff of 60 Ry gave a convergence in energy with a precision of 1 meV/atom. For electron-phonon coupling, a 6$\times$6$\times$2, 6$\times$6$\times$4 and a 4$\times$4$\times$4 q-point meshes were used for C2/m-MgB3, Amm2-Mg3B10 and C2/m-MgB4, respectively. Denser k-point meshes, 12$\times$12$\times$4, 12$\times$12$\times$8 and 8$\times$8$\times$8 were used for the convergence checks of the EPC parameter $\lambda$.

Results

Search for stable compounds

Pressure can stabilize new or destabilize the known compounds, and a proper sampling of all promising compositions is needed. In Fig.1(a)., the enthalpies of formation $\Delta$Hf per atom (with respect to the stable structures of elemental magnesium and boron) are shown in the convex hull form as obtained from all possible compounds. Convex hull gives all thermodynamically stable compositions of a multicomponent system, and their enthalpies of formation (per atom). The convex hull (see Fig.1(a)) includes all thermodynamically stable states, while unstable ones will always appear above it. The distance of an arbitrary compound above the tieline of the convex hull is a measure of its instability, as it shows the decomposition energy of that compound into the nearest stable compounds. The convex hull construction shows that boron-rich compounds are stabilized at high pressure.

Taking our predicted structures/compounds and experimentally known large-cell structures of MgB7, Mg*∼5B44*, MgB12 (all three compounds feature B12-icosahedra, and for the latter two, we constructed ordered approximants of disordered experimental structures

• for MgB12 containing 388 atoms in the unit cell), we computed the phase diagram of the Mg-B system. At pressures studied here, MgB2, MgB3, Mg3B10, MgB4, MgB7 and MgB12 have stability fields, making the phase diagram (Fig. 1(b)) very rich. A recent list of 41 metal borides presented in Ref Geest and Kolmogorov (2014) at 0 and 30 GPa, clearly demonstrates metal borides often have a variety of stable phases at high pressure.

On increasing pressures metastable compounds, MgB3 and MgB6, get closer to the tieline. Our calculations indicate that at 54 GPa, MgB3 reaches stability and forms the C2/m structure. Unlike MgB3, MgB6 cannot compete with other compounds and remains metastable throughout the entire pressure range (0 to 200 GPa).

Although MgB6 emerges as a metastable compound from our calculations, we still studied it, keeping in mind recent observation of superconductivity in YB6 Souma et al. (2005). Moreover, there is experimental evidence for MgB6 as a non-equilibrium phase Li et al. (2002).

MgB2

Some of us studied high-pressure phases of MgB2 using the evolutionary algorithm USPEX Ma et al. (2009). Our results accord well with that study, as the phase transition happens at 190 GPa. The transition from AlB2-type structure (Fig.2) with space group P6/mmm to KHg2-type structure with space group Imma, completely destroys superconductivity. The role of B-B $\pi$-bonded network and charge transfer from Mg to B atoms are explained as having major role in superconducting properties Ma et al. (2009); Mazin et al. (2002).

MgB3

MgB3, one of the new high-pressure compounds, lies 5 meV/atom above the MgB2-MgB7 tieline at 50 GPa. It becomes stable at 54 GPa and remains stable until 130 GPa in the C2/m phase. Finally Cmcm structure becomes more favorable than all other possible structures up to 200 GPa. AB3 is interestingly a common stoichiometry for metal borides as reported for WB3 Liang et al. (2013), MnB3 Niu et al. (2014) and NaB3Geest and Kolmogorov (2014), however, MgB3 has not been studied yet, neither computationally nor experimentally.

MgB3 stabilizes at high pressure, while CaB3 Shah and Kolmogorov (2013) and LiB3 Hermann et al. (2012) are not stable even at high pressure. Structural information for the predicted stable MgB3 phases is provided in Table 2 and in Fig. 3. The metastable layered C2/m phase at pressures below 43 GPa is important, since it has graphene-like hexagonal boron pattern, which may be a hint of a potentially superconducting phase.

MgB4

MgB4 has a remarkable reentrant behavior: this compound is thermodynamically stable in the pressure range 0-4 GPa, then becomes unstable to decomposition into other borides, and then is again thermodynamically stable at pressures >61 GPa (Fig. 1(b)). Below we consider lowest-enthalpy phases corresponding to this composition (see Fig. 4).

The Pnma phase of MgB4 is stable at ambient pressure in accord with theoretical Geest and Kolmogorov (2014) and experimental Yan et al. (2007) results and remains the most favorable phase up to 31 GPa. Unlike all the other MgB4 phases and most of magnesium borides at different pressure conditions, which are metallic, Pnma-MgB4 is a semiconductor. The predicted phase diagram shows that at 31 GPa the semiconducting state breaks down, and MgB4 transforms into a metallic C2/m (similar to AlB2-type) structure. C2/m has the lowest enthalpy in a narrow pressure range from 31 to 36 GPa. From 36 to 60 GPa, the P$\bar{1}$ phase becomes more favorable, and at very high pressures (60 - 200 GPa), high symmetry structure, I4/mmm, becomes stable.

The main feature of I4/mmm and P$\bar{1}$ phases is prisms of boron that hold one or two magnesium atoms. Having boron double-layers (in comparison with P6/mmm-MgB2), the C2/m structure can be described as boron sandwich of this composition. Boron sandwiches have graphene-like layer(s) of boron, intercalated by magnesium atoms. Phonon calculations were performed to check the dynamical stability throughout the Brillouin zone. We did not find any dynamical instability (see Fig. 10 and Fig. S9, ESI). Due to high density of states (DOS) at the Fermi level (N(Ef)), high-pressure phases of MgB4 can be potential candidates for superconductivity. Electron-phonon coupling (EPC) calculations revealed that among MgB4 phases, only layered C2/m-MgB4 is a superconductor.

MgB4 has analogous stoichiometry to many AB4 systems, e.g. MnB4 Meng et al. (2012); Niu et al. (2014), CrB4 Niu et al. (2012), CaB4 Yahia et al. (2008); Liu et al. (2010) and so forth. AB4 structures are mostly orthorhombic or tetragonal with 20 atoms per cell. Some of these structures are in BaAl4-type structure with space group I4/mmmBruzzone (1969). By removing Mg from the prisms, one observes a pattern similar to the $\alpha$-Ga structure of boron Oganov et al. (2009).

Increasing pressure, we see emergence of a graphene-like boron double-layered phase (MgB2 has a simple hexagonal AlB2-type structure, which is a single-layered phase of this type). The extra layer is located 1.7 Å from the first layer and displaced by 0.8 Å (A$\alpha\beta$ A$\alpha\beta$ …, A represents Mg and $\alpha$,$\beta$ are B layers). At 36 GPa, some boron blocks were formed with a pattern of 1 and 2 magnesium atoms per block. Finally, at a higher pressure 60 GPa, the body-centered tetragonal BaAl4-type structure (space group I4/mmm), which is widely adopted among AB4 intermetallic compounds, forms. In I4/mmm structure, magnesium is located in the center of the truncated rectangular prisms made of boron atoms. This structure is similar to Cmcm-MgB3, in which, there are two magnesium atoms located in each of the truncated rectangular prisms (see Fig. 3(b). and Fig. 5(a).).

MgB6

Although MgB6 is predicted to be stable with respect to decomposition to the elements (Mg and B) Wang et al. (2014), it is not stable against decomposition into elemental boron and MgB4 (see Fig. 1(a) and (b)). Furthermore, in an experimental study at ambient pressure, MgB6 was not found as an individual phase Moiseev and Ivanovskii (2005).

Since intercalated graphite AC6 (A = Mg, Ca, Sr, Ba) Calandra and Mauri (2006); Pan et al. (2011); Kim et al. (2006) is superconducting, we searched for the lowest enthalpy MgB6 phases. We observed a hexagonal distorted triple-layered phase, which is the lowest in enthalpy, in the pressure range 15-28 GPa that intrigued us. MgB6 forms a recently predicted phase at ambient pressure and remains in this Cmcm structure until 15 GPa Wang et al. (2014), above which a triple-layered structure has lowest enthalpy until 28 GPa (Fig. 6(c) and (d)). Between 28 GPa and 88 GPa, the R-3m structure becomes more favorable, and eventually, very high pressure imposes a pattern similar to I4/mmm-MgB4 into P21/m-MgB6 (see Fig. 6(a).); This pattern emerges in the pressures greater than 90 GPa in both MgB6 and MgB4. We found the Cmcm structure to be a semiconductor in agreement with the previous report Wang et al. (2014), whereas the rest of the phases are metallic. The semiconductor-metal transition Cmcm $\rightarrow$ P21/c happens at 15 GPa.

Mg3B10

Mg3B10, a boron-rich compound, stable above 55 GPa, has a monoclinic (space group C2/m) phase. Above 83 GPa this phase transforms into the P2/m phase (Fig. 1(b)). Metastable Amm2-Mg3B10, which we predict to have the lowest enthalpy among Mg3B10 phases in the pressure range 30-42 GPa, has a layered sandwich structure and is superconducting (Fig. 7(c) and (d)).

Superconductivity

Kolmogorov et al., proposed metal sandwiches consisting of one or more layers of metal and a graphene-like layer of boron i.e., MS-2 and MS-4 with single hexagonal layer of B Kolmogorov and Curtarolo (2006). In our study we found, boron sandwiches, new structures with one layer of metal atoms alternating with multiple boron layers. Interestingly, boron sandwich structures are ubiquitous here. For example, in MgB3, there is a layered structure with space group C2/m below 43 GPa (see Fig. 9.) featuring $\alpha A\beta\gamma B$… stacking of B-Mg layers ($A$ and $B$ denotes Mg and $\alpha\beta\gamma$ are B layers). C2/m structure of MgB4 in 31-36 GPa, Amm2 structure of Mg3B10 in 30-42 GPa and P21/c structure of MgB6 in 15-28 GPa (see colored areas in Fig. 1(b).) also feature boron sandwiches.

Boron sandwiches are layered structures with stackings of [MgB2] and/or [MgB4] blocks (see Fig. 9.). For example, Mg3B10 can be represented as a [MgB2][MgB4][MgB4]… sequence of layers, and MgB3 can be represented as a [MgB2][MgB4]… . Superconductivity in MgB2 is mostly related to the boron layers, i.e. B-B $\sigma$ and $\pi$-bonded network, therefore, sandwich borides with hexagonal boron layers might have superconducting properties. We checked this by electron-phonon coupling calculations. Eliashberg spectral function ($\alpha^{2}F$) calculations lead to the results depicted in Fig. 10. and listed in Table 1. The electron-phonon coupling constants ($\lambda$) for different structures at given pressures, logarithmic averaged phonon frequencies ($\omega_{log}$) and superconducting transition temperatures (Tc) are also provided (for more information, see the ESI). Density of states at Ef listed in Table 1 shows Tc is higher for boron sandwiches with higher N(Ef) per electron. One can see from the projected density of states that 2p states of B atoms, located in planar nets, dominate the DOS at the Fermi level (see Fig. 8. Bands structures and total DOS of other sandwich borides are also provided in the ESI, Fig. S3, S6 and S10)

The critical temperature of superconductivity is estimated from the Allen-Dynes modified McMillan equationAllen and Dynes (1975):

[TABLE]

where $\omega_{log}$ is the logarithmic average phonon frequency and $\mu^{*}$ is the Coulomb pseudopotential,

[TABLE]

The EPC parameter $\lambda$ is defined as integral involving the spectral function $\alpha^{2}F$:

[TABLE]

Here we used $\mu^{*}$ = 0.10 for Coulomb’s pseudopotential, as a reasonable value for most materials Richardson and Ashcroft (1997); Xie et al. (2010); Zhou et al. (2011). After MgB2, C2/m-MgB4 has the highest Tc of 2.8 K at zero pressure (it is metastable at 0 GPa). In C2/m-MgB4, high-frequency phonons, mostly by boron atoms, contributes 80.5% to the total EPC parameter, and low-frequency modes are mainly from magnesium vibrations with 19.5% contribution. Sandwich borides, in general, have high phonon density between 200 to 400 cm*-1*, however, Eliashberg spectral function indicates a poor electron-phonon coupling in this range. Logarithmic average phonon frequencies $\langle\omega_{log}\rangle$ is comparable to that of P6/mmm-MgB2, however, much weaker electron-phonon coupling and lower densities of states at the Fermi level result in very low transition temperatures 0.7-2.8 K. (For more information about phonon band structures, phonon density of states, Eliashberg spectral function and electronic band structures of these phases, see the ESI)

Directly relevant to superconductivity of sandwich borides is the value of DOS at the Fermi level. For example, the DOS is 0.044 states/eV per electron for C2/m-MgB4. This is about half the value of the MgB2 which is 0.084 states/eV per electron. We can see a trend of increasing Tc when we have higher DOS (values are listed in Table 1). However, other parameters are essential as well. Since logarithmic average phonon frequencies are almost equal, outstanding MgB2 superconductivity can be related to the higher density of states at the Ef mainly from boron p-states and stronger electron-phonon coupling parameter $\lambda$ = 0.73 mainly affected by lower frequency modes. $\lambda$ of other boron sandwiches is about half of the value of MgB2 (see values listed in Table 1), which due to exponential dependence of Tc on $\lambda$, the Tc value of MgB2 is about 10 times higher than other magnesium borides.

Conclusions

Using ab initio evolutionary structure search, we have extended our previous study of MgB2 to other possible Mg-B compounds up to megabar pressures. A remarkable variety of candidate high-pressure ground states has been identified. In this systematic study, under pressures from 0 to 200 GPa, we have found 6 stable compounds, i.e., MgB2, MgB3, MgB4, Mg3B10, MgB7 and MgB12. Interestingly, MgB7 and MgB12, which are reported to be stable at ambient pressure, are not competitive at very high (above 90 GPa) pressure. In all compounds, at sufficiently high pressures sandwich borides give way to structures with three-dimensional topology.

Most of the predicted stable phases are metallic. No magnesium-rich phases are stable. By decreasing pressure to 0 GPa, the Tc value of C2/m-MgB4 is enhanced and reaches 2.8 K. The importance of layered structures at the boron-rich end of the Mg-B phase diagram is noteworthy. The valence bands close to and below the Ef are dominated by boron p-states in layered structures. Therefore, EPC calculations are performed and revealed Mg-B sandwich borides are superconducting with Tc of 2.5, 1.0 and 0.7 K for C2/m-MgB3, Amm2-Mg3B10 and C2/m-MgB4 at 31, 40 and 33 GPa, respectively.

Acknowledgements

We thank DARPA (grant W31P4Q1210008), the Government of Russian Federation (14.A12.31.0003) and the Foreign Talents Introduction and Academic Exchange Program (B08040). X.F.Z thanks the National Science Foundation of China (grant no. 11174152), the National 973 Program of China (grant no. 2012CB921900), and the Program for New Century Excellent Talents in University (grant no. NCET-12-0278).