- Email: [email protected]
Orbital mixing in solids as a descriptor for materials mapping
Solid State Communications 203 (2015) 31–34
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Orbital mixing in solids as a descriptor for materials mapping Marc Esser a, Volker L. Deringer a, Matthias Wuttig b,c, Richard Dronskowski a,c,n a b c
Institute of Inorganic Chemistry, RWTH Aachen University, Landoltweg 1, 52056 Aachen, Germany Institute of Physics IA, RWTH Aachen University, 52074 Aachen, Germany Jülich–Aachen Research Alliance (JARA-FIT and JARA-HPC), RWTH Aachen University, 52074 Aachen, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 3 November 2014 Accepted 7 November 2014 by S. Das Sarma Available online 20 November 2014
The creation of “maps” for solid-state materials has a long-standing history in condensed matter theory. Here, based on periodic density-functional theory (DFT) output, a heuristic numerical indicator is constructed to assess s–p orbital mixing in materials (or, depending on one's viewpoint, the tendency toward “sp3 hybridization”). Other than before, this now intrinsically includes structural information and the microscopic effects associated with it. The new method provides useful insights to understand physical relationships in composition space and promises to help to identify hitherto unknown material candidates. & 2014 Elsevier Ltd. All rights reserved.
Keywords: C. Material Characterization D. Material design E. First-principles
The rational design of new and improved functional materials is a key challenge for the solid-state sciences. An abundance of candidates and possible compositions exists, however, and the sheer amount of them is too large to oversee, let alone to handle. It has long been advocated to partition the compositional and structural space according to suitably chosen criteria, or, in a more plastic language, to “map out” promising compounds according to their physical or chemical relationship. Indeed, the creation of structure maps to classify materials has a long-standing tradition in condensed-matter theory. Examples are the scheme by Phillips and Van Vechten to predict the crystal structures of the octet ANB8 N compounds [1], the subsequent extension to suboctet compounds by St. John and Bloch [2], and the classification of IV–VI compounds by Littlewood [3]. Building upon this groundwork, one of us (M.W.) has recently proposed a first “treasure map” for phase-change materials (PCMs) [4], which are leading contenders for new information storage and processing technologies [5–8]. Later, it has been suggested that this map can be extended to other classes of functional materials, such as topological insulators and thermoelectrics [7], which would further increase its scope. Similar conclusions were reached very recently when a link between the bonding nature of PCMs and the application in thermoelectrics has been suggested [9]. Finally, PCMs on this map have other emerging applications such as in optical displays [10] or brain-like computing [11]. Hence, further exploration of this map would seem worthwhile, without any doubt.
n
Corresponding author. E-mail address: [email protected] (R. Dronskowski).
http://dx.doi.org/10.1016/j.ssc.2014.11.008 0038-1098/& 2014 Elsevier Ltd. All rights reserved.
Despite its appeal, the above-mentioned map has an important limitation which now needs to be remedied: it uses orbital radii to estimate s–p mixing and ionicity in a heuristic manner, essentially following the scheme of St. John and Bloch [2] and, as a consequence, the structural nature of the compounds under study is missing. This is best seen when looking at an exemplary map of some textbook compounds. Fig. 1 shows a St. John–Bloch plot for two carbon and boron nitride (BN) polymorphs, as well as two rocksalt-type compounds. Diamond, with its dense rigid network, will surely be of different bonding nature than graphite, but they form a single data point in the above-mentioned map because they are all allotropes of elemental carbon and thus possess identical orbital radii. The same is true for the BN polymorphs. In this contribution, we demonstrate that one can include the important structural information intrinsically in such map by replacing the coordinate for the y-axis (s–p orbital difference) by a better suited quantity based on first-principles calculations. To exemplify this concept, we introduce an indicator to quantify orbital mixing in solid materials. The new technique is based on unambiguous density-functional theory (DFT) output and hence applicable for all kinds of valence configurations. This allows one to project out the electron density situated in “sp3” mixed levels in the style of a fat-band plot and thus to assess the degree of orbital mixing as a revised coordinate, as will be demonstrated shortly. In what follows, we rely on periodic DFT simulations in the local density approximation (LDA) [12], using the projector augmented-wave (PAW) method [13] as implemented in VASP [14]. It is known that chemical information can be extracted from plane-wave based functions by fitting a set of atomic orbitals to replace these very plane-wave functions [15] and thus combine the advantages of both widely used computational techniques for
32
M. Esser et al. / Solid State Communications 203 (2015) 31–34
Fig. 1. Orbital-radii based St. John–Bloch plot [2] for textbook solid-state compounds. Different allotropes possess the same orbital radii, and so attain the same data point for s–p mixing and ionicity.
electronic-structure calculation. In this paper, the reconstruction of the l-resolved density channels (or orbital nature) is achieved by an analytical projection onto a minimal local basis of Slater-type orbitals [16]. The viability of this plane-wave/PAW based approach has been demonstrated before, especially when it comes to reconstructing bonding information from structurally complex systems such as crystal surfaces [17] or amorphous matter [18]. We briefly recall the utility of this technique by considering fatband plots resolved according to the out-of-plane pz-orbital (Fig. 2b) of graphite, and also to the pz-orbital in diamond. Crosses indicate the course of energy eigenvalues through momentum space; the size of the superimposed circles indicates the pz-projection in arbitrary scaling. On the right, the densities of states (DOS) are displayed as they emerge from the bands. While the pz-contributions in diamond are well distributed over the entire energy range (as apparent from the DOS, which covers the entire Brillouin zone ! and not just one specific path through k -space), in graphite they form the characteristic π-system located around the Fermi edge. Now, instead of looking at single-orbital contributions, it is desirable to evaluate the orbital mixing. For this, we take inspiration by the findings of Pauling [19], who concluded that the valence sand p-orbitals can undergo a unitary basis-set transformation by linear combination of the atomic orbitals ϕ (LCAO) to yield a new directional one-electron function Ψ sp3 ¼ aϕs þbϕpx þ cϕpy þ dϕpz . This new function is commonly referred to as “sp3 hybrid orbital”. The coefficients a–d are bound by the constraint of orthogonality, and their squares have to be normalized to unity, i.e., 2
2
a2 þb þ c2 þ d ¼ 1. Pauling found that the best bonding function results when the squares of all four orbital coefficients assume a value of 14. We now suggest to compare ab initio computed coefficients to these ideal value. To this end, we introduce a heuristic parameter μj. The quantity μj is specific for each atom, band (index j) and ! k -point, and it is derived from the s- and p- orbital projections at this point in reciprocal space. The coefficients are squared and normalized to unity before entering the indicator, satisfying Pauling's constraint. The formula for μj reads 3 2 32 ! ! ! ! 2pmin ð k Þ 56 pmax ð k Þ sð k Þ 7 ð1Þ μj ð k Þ ¼ 4 ! ! 41 ! ! 5: pmax ð k Þ þ sð k Þ pmax ð k Þ þ sð k Þ Here, s is the squared and normalized coefficient of the s-orbital, pmax is the largest squared and normalized p-coefficient and pmin is
Fig. 2. (a) Structural sketch of textbook carbon allotropes, highlighting the out-of-plane pz-orbital in graphite. (b) Computed electronic band structures as indicated by crosses; the size of the superimposed circles shows the pz-contribution (fat-band plots). (c) As before, but weighted with the “sp3” mixing indicator (Eq. (1)). The scaling of the weighting is arbitrary and differs between plots, for illustrative purposes.
the smallest one. The quantity μj is defined such that it equals unity if orbital mixing is complete, and zero if no “sp3” character can be discerned. While we employ Pauling's conclusions about ideal coefficients (see above), we—in sharp contrast—do not adopt the chemical thinking of an unitary basis-set transformation but use the physicists’ notion of orbital mixing by bonding between atoms. Hence, the above definition is deliberately based on canonical orbitals (see below).
M. Esser et al. / Solid State Communications 203 (2015) 31–34
The above-defined quantity can, next, be used to weight the orbital-projected DOS elements, rffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! ! ! ! ð2Þ Mj ðE; k Þ ¼ ∑ μj ð k ÞC njμ ð k ÞC jν ð k Þδ εj ð k Þ E ;
6
The square root in Eq. (2) is introduced to counteract the prior squaring during normalization. The MDOS reveals, in an energy resolved way, the levels which show “sp3” mixing. Like the overall ! DOS, it k -averages over the entire Brillouin zone. The results are given in Fig. 2c. While graphite, as expected, shows virtually no “sp3” mixing, diamond clearly exhibits some mixed levels offside from the special points. The low amount of mixing is to be expected: the canonical orbitals are required to be orthogonal to each other and are thus rather “atom-like” at the high-symmetry points, and this is also reflected in very few occupied states in the MDOS compared to the total DOS. Symmetry can be broken only offside the special points, and thus orthogonality be lifted, allowing the orbitals to mix and form “hybrid” levels. We reiterate that we do not employ hybridization as a unitary basis-set transformation which would lead to a new one-electron function. The results so far suggest that the indicator is able to distinguish between different structures according to the local bonding nature, which is the main goal of this work. To further proceed towards the ab initio creation of structure maps, the obtained information may now be condensed into a single, characteristic value for each polymorph. This is achieved by calculating the percentage of the integrated electron density in mixed levels compared to the total amount of integrated electron density: R ϵF MDOSðEÞ dE sp3 mixing ð%Þ ¼ Rϵ1 100: ð4Þ F 1 DOSðEÞ dE Fig. 3 yields the results of such calculations for the fourth maingroup elements, all in the cubic diamond structure. It also serves to show the influence of the nature of the atom within isostructural materials. As said already, there is small but existent “sp3” mixing in diamond carbon, on the order of 5.8%. Moving down the fourth main group, there is a decline in mixing with atomic number, with densities in mixed levels of the order of E4.6% (Si), 4.3% (Ge) and 4.0% (Sn). This results from the increasingly different spatial extent of valence orbitals with the same quantum number n but different angular momentum l, e.g., 3s and 3p. The valence orbitals in carbon are very similar to each other in terms of radial extent due
diamond structure
sp3 mixing (%)
6
Δsp3 = 1.17
5
Δsp3 = 0.35 Δsp3 = 0.25
4
C 3
Si
Ge
Sn
Fig. 3. Course of “sp ” orbital mixing in the fourth main group computed according to Eq. (4). The connecting lines are guides to the eye only.
sp3 mixing (%)
Ω j
C (diamond)
5
μ; ν
which, after reciprocal-space integration, yields what we call the mixing-projected DOS (MDOS): Z ! ! ∑Mj ð k Þ d k : ð3Þ MDOSðEÞ ¼
33
4
BN (cub.)
3
AgCl
2 1 0
NaCl BN (hex.)
C (graphite) 0
0.5
1
1.5
2
Ionicity Fig. 4. (Color online) Exemplary map using the new approach: the St. John–Bloch plot of Fig. 1 has been re-created based on Eqs. (4) and (5). Comparing the two maps, it is clear to see that the distinction of allotropic or polymorphic structures is achieved by the newly introduced ab initio quantity spanning the vertical axis.
to “missing” 1p-orbitals, which would partially shield the 2porbitals from the nuclear charge, while the 2s-orbital gets shielded by the 1s-orbital. This leads to a relative contraction of the 2s-, and simultaneous expansion of the 2p-orbitals; a unique effect among carbon's homologues. In fact, the difference in spatial extent of the s- and p-orbitals for the quantum numbers n ¼2 and 3 and its consequences for the 〈sjs〉, 〈pjp〉 and 〈sjp〉 overlap integrals upon bonding nicely explains the strongly different interatomic distances in simple diatomics, say, N2 and P2 [20]. Before re-calculating the map given in Fig. 1, the abscissa needs to be specified. Inspired by the previous map of PCMs [4], we will use the ionicity Δχ which we quantify by
Δχ ¼
∑ j mj χ j ∑i mi χ i : ∑i mi ∑ j mj |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} anions
ð5Þ
cations
where χi denotes the electronegativity according to the Allred– Rochow scale [21], and m is the number of atoms of type i. The new map is finally depicted in Fig. 4. Clearly, it distinguishes between allotropes: diamond as the “sp3” archetype is now situated in the upper left corner, marking the highest degree of orbital mixing of all phases mapped, while graphite resides in the lower left with virtually no “sp3” mixing at all. Naturally, both allotropes are not ionic. Likewise, the BN polymorphs are separated according to their local bonding nature. As ionic counterparts to graphite and diamond, they are situated more to the right side of the map. Further to the right side of the map (Fig. 4), NaCl serves as an example of strongly ionic bonding, and its 901 bonding angles are at variance with the tetrahedral network in diamond. Nonetheless, there is a miniscule amount of orbital mixing of 0.8% to be found occurring somewhere in the Brillouin zone. AgCl is isostructural to NaCl but less ionic, thus positioned roughly between the C allotropes on the left and NaCl on the right side of the map. The charge transfer is smaller in AgCl than in NaCl, and thus there is a larger covalent bonding character, which facilitates a higher amount of orbital mixing. Due to the heuristic nature of the present implementation, the absolute values for each material are arbitrary, such that focus should be on the relative positions of the materials on the map. Many other quantum-chemical methods could potentially serve as indicators for the local bonding nature, e.g., McWeeny's optimum hybrids [22] or Ruedenberg's localized atomic and
34
M. Esser et al. / Solid State Communications 203 (2015) 31–34
molecular orbitals [23]. Indeed, the approach of natural bond orbital (NBO) analysis [24] has recently been generalized to periodic systems, using a sophisticated plane-wave/PAW based projection technique [25]. Naturally, calculations employing a basis-set of sp3-hybrids yield for bulk silicon a fully sp3 hybridized electron structure, while the approach based on canonical orbitals reveals 4.6% orbital mixing. This emphasizes that the focus should be on trends, not absolute numbers. The use of other ab initio indicators for generating materials maps may be subject of future studies. In conclusion, we have presented a way to incorporate structural information in materials maps, using quantum-chemical indicators (as exemplified here) based on first-principles calculations. A numerical indicator has been developed to directly access the amount of “sp3” orbital mixing in solid-state materials. The band structure and electronic DOS can be weighted with this indicator, thus producing a DOS that is projected onto these mixed levels, dubbed MDOS. The integrated MDOS can be compared to the total electron count such that one obtains the percentage of electron density situated in mixed levels. By plotting this percentage against ionicity, new materials maps can be created which separate polymorphs according to their bonding nature for the first time. This opens up new possibilities in the search for functional materials, as demonstrated before [4].
References [1] [2] [3] [4] [5] [6] [7] [8]
[9] [10] [11] [12] [13] [14]
[15] [16] [17] [18]
Acknowledgments
[19] [20]
We thank Stefan Maintz for technical help and Professor Roald Hoffmann (Cornell) for insightful remarks. This work was supported by the Deutsche Forschungsgemeinschaft (SFB 917 “Nano-switches”) and the Studienstiftung des deutschen Volkes (scholarship to V.L.D.).
[21] [22] [23] [24] [25]
J.C. Phillips, J.A. Van Vechten, Phys. Rev. Lett. 22 (1969) 705–708. J.St. John, A.N. Bloch, Phys. Rev. Lett. 33 (1974) 1095–1098. P.B. Littlewood, J. Phys. C: Solid State Phys. 13 (1980) 4855–4873. D. Lencer, M. Salinga, B. Grabowski, T. Hickel, J. Neugebauer, M. Wuttig, Nat. Mater. 7 (2008) 972–977. M. Wuttig, N. Yamada, Nat. Mater. 6 (2007) 824–832. D. Lencer, M. Salinga, M. Wuttig, Adv. Mater. 23 (2011) 2030–2058. M. Wuttig, S. Raoux, Z. Anorg. Allg. Chem. 638 (2012) 2455–2465. Earlier theoretical work on crystalline PCMs from our laboratories includes: M. Wuttig, D. Lüsebrink, D. Wamwangi, W. Wełnic, M. Gilleßen, R. Dronskowski, Nat. Mater. 6 (2007) 122–128; W. Zhang, A. Thiess, P. Zalden, R. Zeller, P.H. Dederichs, J.-Y. Raty, M. Wuttig, S. Blügel, R. Mazzarello, Nat. Mater. 11 (2012) 952–956. S. Lee, K. Esfarjani, T. Luo, J. Zhou, Z. Tian, G. Chen, Nat. Commun. 5 (2014) 3525. P. Hosseini, C.D. Wright, H. Bhaskaran, Nature 511 (2014) 206–2011. C.D. Wright, Y. Liu, K.I. Kohary, M.M. Aziz, R.J. Hicken, Adv. Mater. 23 (2011) 3408–3413. D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566–569. P.E. Blöchl, Phys. Rev. B 50 (1994) 17953–17979. G. Kresse, J. Furthmüller, Comput. Mater. Sci. 6 (1996) 15–50; G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169–11186; G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758–1775. D. Sanchez-Portal, E. Artacho, J.M. Soler, Solid State Commun. 95 (1995) 685–690. S. Maintz, V.L. Deringer, A.L. Tchougréeff, R. Dronskowski, J. Comput. Chem. 34 (2013) 2557–2567. V.L. Deringer, R. Dronskowski, Chem. Sci. 5 (2014) 894–903. V.L. Deringer, W. Zhang, M. Lumeij, S. Maintz, M. Wuttig, R. Mazzarello, R. Dronskowski, Angew. Chem. Int. Ed. 53 (2014) 10817–10820. L. Pauling, J. Am. Chem. Soc. 53 (1931) 1367–1400. T.A. Albright, J.K. Burdett, M.-H. Whangbo, Orbital Interactions in Chemistry, second ed., John Wiley & Sons, Hoboken, New Jersey, 2013. A. Allred, E. Rochow, J. Inorg. Nucl. Chem. 5 (1958) 264–268. R. McWeeny, G. Del Re, Theor. Chim. Acta 10 (1968) 13–22. C. Edmiston, K. Ruedenberg, Rev. Mod. Phys. 35 (1963) 457–465. J.P. Foster, F. Weinhold, J. Am. Chem. Soc. 102 (1980) 7211–7218. B.D. Dunnington, J.R. Schmidt, J. Chem. Theory Comput. 8 (2012) 1902–1911.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Orbital mixing in solids as a descriptor for materials mapping Marc Esser a, Volker L. Deringer a, Matthias Wuttig b,c, Richard Dronskowski a,c,n a b c
Institute of Inorganic Chemistry, RWTH Aachen University, Landoltweg 1, 52056 Aachen, Germany Institute of Physics IA, RWTH Aachen University, 52074 Aachen, Germany Jülich–Aachen Research Alliance (JARA-FIT and JARA-HPC), RWTH Aachen University, 52074 Aachen, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 3 November 2014 Accepted 7 November 2014 by S. Das Sarma Available online 20 November 2014
The creation of “maps” for solid-state materials has a long-standing history in condensed matter theory. Here, based on periodic density-functional theory (DFT) output, a heuristic numerical indicator is constructed to assess s–p orbital mixing in materials (or, depending on one's viewpoint, the tendency toward “sp3 hybridization”). Other than before, this now intrinsically includes structural information and the microscopic effects associated with it. The new method provides useful insights to understand physical relationships in composition space and promises to help to identify hitherto unknown material candidates. & 2014 Elsevier Ltd. All rights reserved.
Keywords: C. Material Characterization D. Material design E. First-principles
The rational design of new and improved functional materials is a key challenge for the solid-state sciences. An abundance of candidates and possible compositions exists, however, and the sheer amount of them is too large to oversee, let alone to handle. It has long been advocated to partition the compositional and structural space according to suitably chosen criteria, or, in a more plastic language, to “map out” promising compounds according to their physical or chemical relationship. Indeed, the creation of structure maps to classify materials has a long-standing tradition in condensed-matter theory. Examples are the scheme by Phillips and Van Vechten to predict the crystal structures of the octet ANB8 N compounds [1], the subsequent extension to suboctet compounds by St. John and Bloch [2], and the classification of IV–VI compounds by Littlewood [3]. Building upon this groundwork, one of us (M.W.) has recently proposed a first “treasure map” for phase-change materials (PCMs) [4], which are leading contenders for new information storage and processing technologies [5–8]. Later, it has been suggested that this map can be extended to other classes of functional materials, such as topological insulators and thermoelectrics [7], which would further increase its scope. Similar conclusions were reached very recently when a link between the bonding nature of PCMs and the application in thermoelectrics has been suggested [9]. Finally, PCMs on this map have other emerging applications such as in optical displays [10] or brain-like computing [11]. Hence, further exploration of this map would seem worthwhile, without any doubt.
n
Corresponding author. E-mail address: [email protected] (R. Dronskowski).
http://dx.doi.org/10.1016/j.ssc.2014.11.008 0038-1098/& 2014 Elsevier Ltd. All rights reserved.
Despite its appeal, the above-mentioned map has an important limitation which now needs to be remedied: it uses orbital radii to estimate s–p mixing and ionicity in a heuristic manner, essentially following the scheme of St. John and Bloch [2] and, as a consequence, the structural nature of the compounds under study is missing. This is best seen when looking at an exemplary map of some textbook compounds. Fig. 1 shows a St. John–Bloch plot for two carbon and boron nitride (BN) polymorphs, as well as two rocksalt-type compounds. Diamond, with its dense rigid network, will surely be of different bonding nature than graphite, but they form a single data point in the above-mentioned map because they are all allotropes of elemental carbon and thus possess identical orbital radii. The same is true for the BN polymorphs. In this contribution, we demonstrate that one can include the important structural information intrinsically in such map by replacing the coordinate for the y-axis (s–p orbital difference) by a better suited quantity based on first-principles calculations. To exemplify this concept, we introduce an indicator to quantify orbital mixing in solid materials. The new technique is based on unambiguous density-functional theory (DFT) output and hence applicable for all kinds of valence configurations. This allows one to project out the electron density situated in “sp3” mixed levels in the style of a fat-band plot and thus to assess the degree of orbital mixing as a revised coordinate, as will be demonstrated shortly. In what follows, we rely on periodic DFT simulations in the local density approximation (LDA) [12], using the projector augmented-wave (PAW) method [13] as implemented in VASP [14]. It is known that chemical information can be extracted from plane-wave based functions by fitting a set of atomic orbitals to replace these very plane-wave functions [15] and thus combine the advantages of both widely used computational techniques for
32
M. Esser et al. / Solid State Communications 203 (2015) 31–34
Fig. 1. Orbital-radii based St. John–Bloch plot [2] for textbook solid-state compounds. Different allotropes possess the same orbital radii, and so attain the same data point for s–p mixing and ionicity.
electronic-structure calculation. In this paper, the reconstruction of the l-resolved density channels (or orbital nature) is achieved by an analytical projection onto a minimal local basis of Slater-type orbitals [16]. The viability of this plane-wave/PAW based approach has been demonstrated before, especially when it comes to reconstructing bonding information from structurally complex systems such as crystal surfaces [17] or amorphous matter [18]. We briefly recall the utility of this technique by considering fatband plots resolved according to the out-of-plane pz-orbital (Fig. 2b) of graphite, and also to the pz-orbital in diamond. Crosses indicate the course of energy eigenvalues through momentum space; the size of the superimposed circles indicates the pz-projection in arbitrary scaling. On the right, the densities of states (DOS) are displayed as they emerge from the bands. While the pz-contributions in diamond are well distributed over the entire energy range (as apparent from the DOS, which covers the entire Brillouin zone ! and not just one specific path through k -space), in graphite they form the characteristic π-system located around the Fermi edge. Now, instead of looking at single-orbital contributions, it is desirable to evaluate the orbital mixing. For this, we take inspiration by the findings of Pauling [19], who concluded that the valence sand p-orbitals can undergo a unitary basis-set transformation by linear combination of the atomic orbitals ϕ (LCAO) to yield a new directional one-electron function Ψ sp3 ¼ aϕs þbϕpx þ cϕpy þ dϕpz . This new function is commonly referred to as “sp3 hybrid orbital”. The coefficients a–d are bound by the constraint of orthogonality, and their squares have to be normalized to unity, i.e., 2
2
a2 þb þ c2 þ d ¼ 1. Pauling found that the best bonding function results when the squares of all four orbital coefficients assume a value of 14. We now suggest to compare ab initio computed coefficients to these ideal value. To this end, we introduce a heuristic parameter μj. The quantity μj is specific for each atom, band (index j) and ! k -point, and it is derived from the s- and p- orbital projections at this point in reciprocal space. The coefficients are squared and normalized to unity before entering the indicator, satisfying Pauling's constraint. The formula for μj reads 3 2 32 ! ! ! ! 2pmin ð k Þ 56 pmax ð k Þ sð k Þ 7 ð1Þ μj ð k Þ ¼ 4 ! ! 41 ! ! 5: pmax ð k Þ þ sð k Þ pmax ð k Þ þ sð k Þ Here, s is the squared and normalized coefficient of the s-orbital, pmax is the largest squared and normalized p-coefficient and pmin is
Fig. 2. (a) Structural sketch of textbook carbon allotropes, highlighting the out-of-plane pz-orbital in graphite. (b) Computed electronic band structures as indicated by crosses; the size of the superimposed circles shows the pz-contribution (fat-band plots). (c) As before, but weighted with the “sp3” mixing indicator (Eq. (1)). The scaling of the weighting is arbitrary and differs between plots, for illustrative purposes.
the smallest one. The quantity μj is defined such that it equals unity if orbital mixing is complete, and zero if no “sp3” character can be discerned. While we employ Pauling's conclusions about ideal coefficients (see above), we—in sharp contrast—do not adopt the chemical thinking of an unitary basis-set transformation but use the physicists’ notion of orbital mixing by bonding between atoms. Hence, the above definition is deliberately based on canonical orbitals (see below).
M. Esser et al. / Solid State Communications 203 (2015) 31–34
The above-defined quantity can, next, be used to weight the orbital-projected DOS elements, rffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! ! ! ! ð2Þ Mj ðE; k Þ ¼ ∑ μj ð k ÞC njμ ð k ÞC jν ð k Þδ εj ð k Þ E ;
6
The square root in Eq. (2) is introduced to counteract the prior squaring during normalization. The MDOS reveals, in an energy resolved way, the levels which show “sp3” mixing. Like the overall ! DOS, it k -averages over the entire Brillouin zone. The results are given in Fig. 2c. While graphite, as expected, shows virtually no “sp3” mixing, diamond clearly exhibits some mixed levels offside from the special points. The low amount of mixing is to be expected: the canonical orbitals are required to be orthogonal to each other and are thus rather “atom-like” at the high-symmetry points, and this is also reflected in very few occupied states in the MDOS compared to the total DOS. Symmetry can be broken only offside the special points, and thus orthogonality be lifted, allowing the orbitals to mix and form “hybrid” levels. We reiterate that we do not employ hybridization as a unitary basis-set transformation which would lead to a new one-electron function. The results so far suggest that the indicator is able to distinguish between different structures according to the local bonding nature, which is the main goal of this work. To further proceed towards the ab initio creation of structure maps, the obtained information may now be condensed into a single, characteristic value for each polymorph. This is achieved by calculating the percentage of the integrated electron density in mixed levels compared to the total amount of integrated electron density: R ϵF MDOSðEÞ dE sp3 mixing ð%Þ ¼ Rϵ1 100: ð4Þ F 1 DOSðEÞ dE Fig. 3 yields the results of such calculations for the fourth maingroup elements, all in the cubic diamond structure. It also serves to show the influence of the nature of the atom within isostructural materials. As said already, there is small but existent “sp3” mixing in diamond carbon, on the order of 5.8%. Moving down the fourth main group, there is a decline in mixing with atomic number, with densities in mixed levels of the order of E4.6% (Si), 4.3% (Ge) and 4.0% (Sn). This results from the increasingly different spatial extent of valence orbitals with the same quantum number n but different angular momentum l, e.g., 3s and 3p. The valence orbitals in carbon are very similar to each other in terms of radial extent due
diamond structure
sp3 mixing (%)
6
Δsp3 = 1.17
5
Δsp3 = 0.35 Δsp3 = 0.25
4
C 3
Si
Ge
Sn
Fig. 3. Course of “sp ” orbital mixing in the fourth main group computed according to Eq. (4). The connecting lines are guides to the eye only.
sp3 mixing (%)
Ω j
C (diamond)
5
μ; ν
which, after reciprocal-space integration, yields what we call the mixing-projected DOS (MDOS): Z ! ! ∑Mj ð k Þ d k : ð3Þ MDOSðEÞ ¼
33
4
BN (cub.)
3
AgCl
2 1 0
NaCl BN (hex.)
C (graphite) 0
0.5
1
1.5
2
Ionicity Fig. 4. (Color online) Exemplary map using the new approach: the St. John–Bloch plot of Fig. 1 has been re-created based on Eqs. (4) and (5). Comparing the two maps, it is clear to see that the distinction of allotropic or polymorphic structures is achieved by the newly introduced ab initio quantity spanning the vertical axis.
to “missing” 1p-orbitals, which would partially shield the 2porbitals from the nuclear charge, while the 2s-orbital gets shielded by the 1s-orbital. This leads to a relative contraction of the 2s-, and simultaneous expansion of the 2p-orbitals; a unique effect among carbon's homologues. In fact, the difference in spatial extent of the s- and p-orbitals for the quantum numbers n ¼2 and 3 and its consequences for the 〈sjs〉, 〈pjp〉 and 〈sjp〉 overlap integrals upon bonding nicely explains the strongly different interatomic distances in simple diatomics, say, N2 and P2 [20]. Before re-calculating the map given in Fig. 1, the abscissa needs to be specified. Inspired by the previous map of PCMs [4], we will use the ionicity Δχ which we quantify by
Δχ ¼
∑ j mj χ j ∑i mi χ i : ∑i mi ∑ j mj |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} anions
ð5Þ
cations
where χi denotes the electronegativity according to the Allred– Rochow scale [21], and m is the number of atoms of type i. The new map is finally depicted in Fig. 4. Clearly, it distinguishes between allotropes: diamond as the “sp3” archetype is now situated in the upper left corner, marking the highest degree of orbital mixing of all phases mapped, while graphite resides in the lower left with virtually no “sp3” mixing at all. Naturally, both allotropes are not ionic. Likewise, the BN polymorphs are separated according to their local bonding nature. As ionic counterparts to graphite and diamond, they are situated more to the right side of the map. Further to the right side of the map (Fig. 4), NaCl serves as an example of strongly ionic bonding, and its 901 bonding angles are at variance with the tetrahedral network in diamond. Nonetheless, there is a miniscule amount of orbital mixing of 0.8% to be found occurring somewhere in the Brillouin zone. AgCl is isostructural to NaCl but less ionic, thus positioned roughly between the C allotropes on the left and NaCl on the right side of the map. The charge transfer is smaller in AgCl than in NaCl, and thus there is a larger covalent bonding character, which facilitates a higher amount of orbital mixing. Due to the heuristic nature of the present implementation, the absolute values for each material are arbitrary, such that focus should be on the relative positions of the materials on the map. Many other quantum-chemical methods could potentially serve as indicators for the local bonding nature, e.g., McWeeny's optimum hybrids [22] or Ruedenberg's localized atomic and
34
M. Esser et al. / Solid State Communications 203 (2015) 31–34
molecular orbitals [23]. Indeed, the approach of natural bond orbital (NBO) analysis [24] has recently been generalized to periodic systems, using a sophisticated plane-wave/PAW based projection technique [25]. Naturally, calculations employing a basis-set of sp3-hybrids yield for bulk silicon a fully sp3 hybridized electron structure, while the approach based on canonical orbitals reveals 4.6% orbital mixing. This emphasizes that the focus should be on trends, not absolute numbers. The use of other ab initio indicators for generating materials maps may be subject of future studies. In conclusion, we have presented a way to incorporate structural information in materials maps, using quantum-chemical indicators (as exemplified here) based on first-principles calculations. A numerical indicator has been developed to directly access the amount of “sp3” orbital mixing in solid-state materials. The band structure and electronic DOS can be weighted with this indicator, thus producing a DOS that is projected onto these mixed levels, dubbed MDOS. The integrated MDOS can be compared to the total electron count such that one obtains the percentage of electron density situated in mixed levels. By plotting this percentage against ionicity, new materials maps can be created which separate polymorphs according to their bonding nature for the first time. This opens up new possibilities in the search for functional materials, as demonstrated before [4].
References [1] [2] [3] [4] [5] [6] [7] [8]
[9] [10] [11] [12] [13] [14]
[15] [16] [17] [18]
Acknowledgments
[19] [20]
We thank Stefan Maintz for technical help and Professor Roald Hoffmann (Cornell) for insightful remarks. This work was supported by the Deutsche Forschungsgemeinschaft (SFB 917 “Nano-switches”) and the Studienstiftung des deutschen Volkes (scholarship to V.L.D.).
[21] [22] [23] [24] [25]
J.C. Phillips, J.A. Van Vechten, Phys. Rev. Lett. 22 (1969) 705–708. J.St. John, A.N. Bloch, Phys. Rev. Lett. 33 (1974) 1095–1098. P.B. Littlewood, J. Phys. C: Solid State Phys. 13 (1980) 4855–4873. D. Lencer, M. Salinga, B. Grabowski, T. Hickel, J. Neugebauer, M. Wuttig, Nat. Mater. 7 (2008) 972–977. M. Wuttig, N. Yamada, Nat. Mater. 6 (2007) 824–832. D. Lencer, M. Salinga, M. Wuttig, Adv. Mater. 23 (2011) 2030–2058. M. Wuttig, S. Raoux, Z. Anorg. Allg. Chem. 638 (2012) 2455–2465. Earlier theoretical work on crystalline PCMs from our laboratories includes: M. Wuttig, D. Lüsebrink, D. Wamwangi, W. Wełnic, M. Gilleßen, R. Dronskowski, Nat. Mater. 6 (2007) 122–128; W. Zhang, A. Thiess, P. Zalden, R. Zeller, P.H. Dederichs, J.-Y. Raty, M. Wuttig, S. Blügel, R. Mazzarello, Nat. Mater. 11 (2012) 952–956. S. Lee, K. Esfarjani, T. Luo, J. Zhou, Z. Tian, G. Chen, Nat. Commun. 5 (2014) 3525. P. Hosseini, C.D. Wright, H. Bhaskaran, Nature 511 (2014) 206–2011. C.D. Wright, Y. Liu, K.I. Kohary, M.M. Aziz, R.J. Hicken, Adv. Mater. 23 (2011) 3408–3413. D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566–569. P.E. Blöchl, Phys. Rev. B 50 (1994) 17953–17979. G. Kresse, J. Furthmüller, Comput. Mater. Sci. 6 (1996) 15–50; G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169–11186; G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758–1775. D. Sanchez-Portal, E. Artacho, J.M. Soler, Solid State Commun. 95 (1995) 685–690. S. Maintz, V.L. Deringer, A.L. Tchougréeff, R. Dronskowski, J. Comput. Chem. 34 (2013) 2557–2567. V.L. Deringer, R. Dronskowski, Chem. Sci. 5 (2014) 894–903. V.L. Deringer, W. Zhang, M. Lumeij, S. Maintz, M. Wuttig, R. Mazzarello, R. Dronskowski, Angew. Chem. Int. Ed. 53 (2014) 10817–10820. L. Pauling, J. Am. Chem. Soc. 53 (1931) 1367–1400. T.A. Albright, J.K. Burdett, M.-H. Whangbo, Orbital Interactions in Chemistry, second ed., John Wiley & Sons, Hoboken, New Jersey, 2013. A. Allred, E. Rochow, J. Inorg. Nucl. Chem. 5 (1958) 264–268. R. McWeeny, G. Del Re, Theor. Chim. Acta 10 (1968) 13–22. C. Edmiston, K. Ruedenberg, Rev. Mod. Phys. 35 (1963) 457–465. J.P. Foster, F. Weinhold, J. Am. Chem. Soc. 102 (1980) 7211–7218. B.D. Dunnington, J.R. Schmidt, J. Chem. Theory Comput. 8 (2012) 1902–1911.