Dani Z.
Metin
,
Lukas
Hammerschmidt
and
Nicola
Gaston
*
The MacDiarmid Institute for Advanced Materials and Nanotechnology, Department of Physics, University of Auckland, New Zealand. E-mail: n.gaston@auckland.ac.nz; Tel: +64 9 923 4427
First published on 24th October 2018
Atomically thin gallium layers have recently been experimentally produced via solid–melt exfoliation, and show promise as robustly metallic 2D materials for electronic applications. However, the extent to which the experimental technique can be extended to other metals relies on understanding how the 2D structures relate to the bulk form of gallium, which is itself unique as an elemental ‘molecular metal’. We relate the experimentally formed 2D materials to the theoretically predicted ‘bilayer gallium’ which has previously been shown to be stable in vacuum at the nanoscale, via density functional theory calculations. We also study the variation of electronic structure with lattice strain to confirm the extent to which the metallicity will be robust on a wide range of substrate materials.
However, the use of gallium raises questions about the generality of this approach for synthesising 2D metals, as the bulk metal is itself highly unique: long known to be the only elemental example of a ‘molecular metal’, it exhibits a rich phase diagram,8 with respect to size,9–12 temperature13 and pressure dependence.14,15 Under standard conditions, it adopts the α-phase,16,17 an 8-atom unit cell in which the atoms are arranged in pairs, within a rhombohedral lattice. This is sometimes referred to as a dimeric structure18–20 giving rise to the description of the metal as a molecular solid: this is a conceptually satisfying description, as it gives some explanation of the low melting temperature of the metal. Estimating the metallic contribution to the cohesive energy of the bulk crystal by subtracting the dimer bond energy from the total energy of the bulk gives an intermolecular energy that correlates with the low melting temperature; in contrast, the low melting temperatures of mercury and caesium correlate with their total cohesive energies. In summary, the relationship between the structure of gallium and its low melting temperature has been well established.
The electronic properties of the metal, however, are equally influenced by – if not responsible for – its unusual structure. Composed of covalently bonded-dimers, the metallic bonding is highly anisotropic, existing in the plane perpendicular to the average alignment of the dimers. In consequence, the conductivity of gallium metal has long been known to have a 2D character.21,22
At the nanoscale, the relative stability of different polymorphs can change dramatically. In the case of gallium, this has been demonstrated both experimentally, and computationally, in studies of the finite temperature behaviour. The experimental observations of the higher-than-bulk melting temperatures of gallium clusters23–26 contain numerous features in the observed specific heat signatures of solid–solid transitions. These have since been elucidated through extensive first-principles molecular dynamics simulations27–31 which have also provided the first theoretical explanation of why the melting temperatures in these clusters are so high: relatively stable two-dimensional structures form in the liquid phase and lower its entropy, stabilising the nanoscale liquid relative to the solid and elevating the melting temperature compared to the bulk.32
Following the discovery and extensive investigation of graphene33–35 other atomically-thick two-dimensional materials have attracted much attention, due to their auspicious mechanical, thermal, optical, and electronic properties.2–7 In this work, we look to place the experimental discovery of 2D gallium into the context of its previously known 2D character to answer the question: to what extent could other elemental metals be used to construct similar structures of reduced dimensionality?
The two-dimensional structure first found in first principles molecular dynamics studies of clusters,36 is a relaxed (010) bilayer surface of bulk α-gallium. The structure of bulk α-gallium is displayed in Fig. 1a with the covalent pairs of gallium atoms oriented vertically and the metallic conduction along horizontal planes.37 It is along this conducting plane, and by slicing through the covalent pairs of atoms within bulk α-gallium, that the most stable surface structure of gallium is formed: the (010)-gallium surface where, despite the significance of the dimeric bonds for many of the properties of bulk gallium, these bonds are severed to allow significant relaxation, at the surface, resulting in a strengthening of the metallic character.
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Fig. 1 Structures of (a) bulk α-gallium, illustrating the (010) plane (b) bilayer gallium from bulk α-gallium and (c) optimized bilayer gallium. |
Simply cutting through the bulk structure (see Fig. 1b) of α-gallium, as described above, leads to a two-layered surface that undergoes a strong relaxation (see Fig. 1c). The obtained structure is pseudo-hexagonal, whereby three of the six atoms that make up a hexagon are shifted vertically. This leads to a surface with an out-of-plane honeycomb structure parallel to the vacuum plane. Notably, and in contrast to the case of the relationship of graphene to graphite, this structure is formed by slicing through covalent bonds, rather than weak van der Waals bonds. This structure is the same as the most stable structure described by Kochat et al., as one of their two gallenene structures.1
In the following, we examine the metallic character of this surface using Density Functional Theory (DFT) which we validate against state of the art hybrid calculations, to ensure that the metallic character is robust with respect to the inclusion of exact exchange, which has not been done in previous work. We also examine the electronic response to lattice deformation of a range of symmetries.
Bilayer gallium was optimized with starting bond lengths of 2.69 Å between nearest Ga–Ga neighbours and an interplanar distance of 1.31 Å, taken from the bulk lattice. The cell volume was allowed to relax during optimization. A criterion of 10−4 eV was used for convergence within the electronic SC loop. An energy difference of 10−3 eV between steps was required for breaking of the ionic relaxation loop. After optimization, bond lengths between Ga–Ga nearest neighbours was 2.71 Å and the interplanar distance remained at 1.31 Å. As illustrated by Fig. 1, bilayer gallium is derived from the (010) surface of the bulk phase of α-gallium. The optimized structure has an interplanar distance, d = 1.31 Å. Along the y axis in bilayer gallium, there is 20.3 Å of vacuum space in order to prevent interaction between adjacent unit cells in the y axis within the periodic boundary conditions.
For optimizations and single point calculations, a 14 × 1 × 14 gamma-centered k-grid was employed. For band structure calculations, sampling was done within the cubic Brillouin zone.
The effect of variation of the lattice parameters on the density of states (DOS) is studied to assess the response of the electronic structure, and metallicity, of 2D gallium to induced lattice strain. Changes in the bond lengths are investigated by varying the a, and c lattice parameters (in the x- and z-directions of the lattice), as a percentage of their equilibrium lengths, and the interplanar (buckling) distance, d (in the y-direction), of the bilayer gallium structure.
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Fig. 3 Projected band structure and density of states for gallium bilayer with equilibrium interplanar distance. |
The material is clearly metallic. In the region within 2 eV of the Fermi energy, the bands are almost linear between the Γ and X point, and this is also seen around the M point. At the point of crossing, these bands are offset from the Fermi energy by only 0.04 eV.
The effect of changing r1, the bond distance of gallium atoms within the plane, is shown in Fig. 4. This is changing the first coordination sphere for gallium in the optimized structure. The interplanar distance remains unchanged.
The DOS of the equilibrium structure itself is highly metallic, with few features of note other than a strong dip at 2 eV below the Fermi level, below which the states are primarily of s-character, and above which they are of p-character. Upon compression of r1, the DOS changes only weakly at EF, by values of −0.258, −0.0495 and 0.5145 for changes in bond length by −2%, −5% and −10%, respectively. The most significant change is a closing of the gap between s- and p-character states at 2 eV below the Fermi level, due to the compression.
On the other hand, when r1 is extended, the gap between the s- and p-character states opens, resulting in a overall increase of the density of states between 1 eV above and below the Fermi level. This results in changes in the DOS value at EF of 0.3485, 1.0285 and −0.007 for extension by 2%, 5% and 10%, respectively.
As r2 is compressed, the DOS at EF changes by 0.308, −0.2145 and −0.0505 for changes in bond length by −2%, −5% and −10%, respectively. When r2 is expanded, the DOS at EF changes by 0.387, 0.758 and 1.675 for extension by 2%, 5% and 10%, respectively. As with compression and extension of r1, the effect of compression and extension of r2 results in most obvious changes at the pseudogap between the s- and p-character states, at 2 eV below the Fermi level. However in this case the changes are more significant, in particular seen for the extreme case of 10% compression, at which point this gap has been completely suppressed. As a consequence, we can conclude that the electronic structure of 2D gallium is more strongly dependent on this out-of-plane component than on the in-plane features of the structure, however, since this effect occurs only on a feature of the electronic structure well below the Fermi level it has relatively few implications for the metallicity of the material.
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Fig. 6 Densities of states for bilayer gallium with compression and extension along the Z vector. The equilibrium values of r1 and r2 are given in the inset. |
As r1 and r2 are simultaneously compressed or extended, the DOS at EF shows the same small shifts previously described; in this sense, the directionality of the compression or extension matters less than the effect on the interatomic distances. As previously noted, it is the pseudogap region that occurs around 2 eV below the Fermi energy that is most sensitive, in particular to shortening of the interatomic distances. However these effects have no real implications for the metallicity of the structure, which remains robust.
Straining the lattice in the x-direction results in clear changes of the band structure. At the M point, the two bands nearest to EF in the equilibrium structure shift down with compression of r1, and up with extension of r1, thereby demonstrating the sensitivity of the electronic properties of bilayer gallium to bond strain. However, when the lattice strain is induced in the y-direction, the opposite trend is seen for r2 than r1; the two bands which nearly touch at the M point in the equilibrium structure are shifted upwards in the case of compression of r2, and are shifted downwards in the case of extension of r2. Lattice strain in the z-direction causes least shift of these bands, likely due to a cancelling effect of changing both r1 and r2. Compression of r1 causes a downwards shift of the M-point band crossing, and compression of r2 causes a shift upwards.
In summary, it is the variation of r2 – the out of plane bond length – to which the band structure is most sensitive. This is also the dimension in which structural change is most directly induced by the interaction with a substrate. In particular, the pseudogap at ca. −2 eV is lost completely at 10% compression, while it is strongly opened at 10% extension in this direction. However the nature of the bands around the Fermi level is relatively unchanged.
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Fig. 8 Band structure for bilayer gallium derived from HSE06 functional with inclusion of exchange (a) default, 20% (b) 50% (c) 70% exchange and (d) HF calculation (100% exchange). |
The band structure remains relatively unperturbed around the Fermi energy, for increasing percentages of exact exchange. However the Hartree–Fock calculation shows significant differences in the conduction bands: most notably, a gap opens at the M-point and between the Γ- and X-points. Even in this limiting case, however, the metallicity of two-dimensional gallium is retained, due to the bands crossing the Fermi energy at the Z-point.
The question remains, whether this robust metallicity could be expected to be found for other 2D metals in similar structures, in particular for those with similar electronic structure but much higher thermal stability, such as aluminium. The answer lies in the particular ability of gallium to combine covalent and metallic bonding – and the recognition that the extent of covalent bonding in the metallic lattice is itself anisotropic.
Analysis of changes in the Electron Localisation Function (ELF) (see ESI†) with the induced strain examined above provides the following picture. In the optimised bilayer structure, for an isosurface value of 0.6, the areas of localised electron density are connected along the x-direction, demonstrating that the covalent character is a contribution to the r1 bonds, but not to the out of plane r2 bonds, which remain more metallic in character. This is a direct inversion of the relative orientation of the covalent and metallic bonds in the α-gallium bulk structure, but results from reconstruction of the surface as it is obtained by cutting through the covalent bonds that exist in the bulk.
As r1 is lengthened, the covalent nature of these bonds is weakened, as shown by the disconnection of the isosurface between individual atoms. However the sensitivity of the electronic structure to the out of plane distance, d, is explained by the induced overlap in localised electron density that occurs once the bilayer is compressed sufficiently in this direction, which may be interpreted as corresponding to the introduction of covalency into the r2 bond. The recognition that the degree of covalent character in the interatomic bonds within the two-dimensional character varies with structural change leaves open the possibility that similar structures might be prepared from other elemental metals, given the right precursor structure.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8cp05280h |
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