Strain and Ligand Effects
When one metal is combined with another at a surface, its chemical properties can change by two mechanisms. Strain arises because the atoms adopt interatomic distances different from those in their own bulk lattice (tensile when stretched, compressive when squeezed). The ligand effect arises from heterometallic bonding between an atom and neighbors of a different element. The two usually occur together, which makes them hard to separate experimentally. See d-band model and Bimetallic surfaces.
Isolating the ligand effect with subsurface 3d metals (2004)
To separate the ligand effect from strain, Kitchin, Nørskov, Barteau and Chen studied Pt(111) slabs whose second layer was replaced by a 3d metal (Ti to Ni) while the surface Pt layer kept the Pt lattice constant, so it had no lateral strain. Every subsurface metal still broadened and lowered the surface d band, most strongly for the early 3d metals and least for Ni, and the rms band width was proportional to a tight-binding Pt-X matrix element. The ligand effect is thus weak between late transition metals (where strain tends to dominate) and strong between early and late ones, and on its own it weakened H2 and O2 dissociative adsorption, in some cases making it endothermic (Kitchin et al. 2004). Summary: kitchin2004.
A unified matrix-element picture (2004)
Kitchin, Nørskov, Barteau and Chen expressed both effects through a single tight-binding interatomic matrix element for d-d coupling, summed over all nearest neighbors of a surface atom. The identity of the neighbors (ligand effect) enters through element-specific d-orbital length scales, and strain enters through the interatomic distance to the fifth power. Both effects act by changing the d-band width, and their contributions are cumulative: tensile strain narrows the band and raises the d-band center, while strong bonding to the substrate broadens it and lowers the center. For Ni monolayers, strong Ni-W bonding outweighs more than 10% tensile strain on W(110), the two nearly cancel on Ru(0001), and strain dominates on Pt(111), with H2 adsorption energies following the resulting d-band centers (Kitchin et al. 2004). Summary: kitchin2004a.
Tensile strain in metal-terminated carbides (2005)
In early transition metal carbides, incorporating carbon expands the metal sublattice: nearest-neighbor metal distances are 0.17 to 0.33 Å longer than in the pure metals (for example 2.89 Å in Ti versus 3.06 Å in TiC). Kitchin and coworkers attributed the stronger atomic H binding they computed on metal-terminated β-Mo2C, TiC, NbC and TaC surfaces (relative to the parent metals) to this tensile strain, by analogy with strained late transition metals, while noting that subsurface carbon is expected to attenuate the effect (Kitchin et al. 2005). Summary: kitchin2005.
Strain and ligand effects from a solid-state table (2010)
In the tight-binding width model of İnoğlu and Kitchin, ligand effects enter through the characteristic orbital lengths of neighboring atoms and strain through the interatomic distances, so both are captured by one formula parameterized from DFT. For 25% surface impurities in Pd(111), larger 4d and 5d impurities broadened the d band of neighboring Pd atoms, while smaller 3d impurities narrowed it and raised its center toward the Fermi level, which the authors called an electronic strain effect in which Pd d orbitals are effectively under-coordinated (İnoğlu 2010). Summary: inouglu2010.
Strain in bulk 3d perovskites (2011)
Akhade and Kitchin applied uniform strains of -5% to +5% to bulk cubic LaBO3 and SrBO3 (B = Sc to Cu). Tensile strain narrowed and compressive strain broadened the B d band through changes in B-O orbital overlap. Using Harrison’s solid state table they modeled the width as proportional to the d-orbital radius to the 3/2 power over the B-O distance to the 7/2 power, the oxide analogue of the metal matrix-element picture; the width was linear in the inverse 7/2 power of distance for every compound, and the slope scaled with the B d-orbital radius, so Ti-based perovskites were more strain-sensitive than Cu-based ones (Akhade 2011). Summary: akhade2011.
Strain on 3d perovskite surfaces (2012)
For cubic LaBO3 and SrBO3 (001) surfaces strained between -5% and +5%, Akhade and Kitchin found that compressive strain broadens and tensile strain narrows the B-site d band, as in metals and in the bulk perovskites, with relaxation of the top layer weakening the effect. The reactivity response differed from metals: on relaxed LaBO3 surfaces compressive strain strengthened O adsorption and tensile strain weakened it, while on SrBO3 both signs of strain made adsorption slightly more favorable. Compressive strain generally lowered the oxygen vacancy formation energy. Overall the strain effects were small compared with changing the B cation (d-band filling) or the A cation (oxidation state), and were largest for low d filling, so the authors describe strain as a fine-tuning handle (Akhade 2012). Summary: akhade2012.
Very large tensile strain on 3d metals en route to monoxides (2014)
Expanding fcc 3d metals to the volume of their rock-salt monoxides imposes very large tensile strains (26% for Mn, 19% for Ni, 17% for Cu). For Ti this narrowed the d band, lowered its center and weakened O binding; for Mn the d band split into bonding and antibonding parts with little hybridization to O; for Ni binding strengthened as conventional strain arguments predict, but for Cu it weakened, which the authors attribute to strains up to four times larger than in earlier Cu studies (Xu 2014).
Summary: xu2014.
Strain and ligand effects in doped rutile oxides (2015)
Xu and Kitchin applied the same decomposition to oxides, computing O adsorption at the (110) 5cus site for all 30 dopant-host pairs of RuO2, RhO2, PdO2, OsO2, IrO2 and PtO2. The sum of the ligand effect (changing the neighboring cations) and the strain effect (changing the lattice constant) reproduced the direct doping effect, and the ligand effect was up to an order of magnitude larger. In contrast to metals, the oxide ligand effect works through charge transfer between dopant and host cations, mediated by lattice oxygen, and its sign and size track the difference in effective d-orbital radius of the pair. Strain changed the t2g band without charge transfer, but the adsorption response was element specific, with only Pd and Pt binding O more weakly under compression (Xu 2015). Summary: xu2015b; see also Doped oxides.
Strain versus crystal structure in rutile dioxides (2015)
Xu and Kitchin compared isotropically strained rutile RuO2, RhO2, IrO2 and PtO2 with metastable polymorphs (anatase, brookite, columbite, pyrite) of the same oxides. Strained rutile gave smooth trends of O adsorption and vacancy formation energy with volume, but the polymorphs did not follow them, so strain alone does not explain polymorph reactivity. In rutile IrO2, strain of up to 15% changed the Ir-O bond lengths uniformly without changing bond angles and mostly altered the eg band, leaving the t2g states near the Fermi level and the O adsorption energy almost unchanged, while columbite IrO2 at a similar volume distorted the octahedron, redistributed the t2g states and shifted O adsorption by about 0.15 eV. Strain also made vacancy formation considerably more exothermic, which the authors read as a sign of structural instability (Xu 2015). Summary: xu2015a; see also Oxide polymorphs.
Bounding ligand effects in CuPd alloy sites (2015)
To estimate how an fcc hollow site behaves inside a CuPd alloy, Boes and co-workers embedded each Cu/Pd site configuration in a pure Cu slab and in a pure Pd slab and varied the slab lattice constant to represent the alloy (strain set by Vegard’s law). The two hosts bracket the ligand effect: Pd subsurface layers made H adsorption more favorable, but by less than about 0.05 eV, leaving trends with lattice constant nearly unchanged. Comparison with experiment favored the Pd-embedded estimates, suggesting Pd ligand effects are significant (Boes et al. 2015). Summary: boes2015a.
Strain on isolated dopant atoms (2018)
In single atom alloys, a dopant smaller than its Au or Ag host experiences effective tensile strain, which together with weak electronic mixing gives a narrow d-band peak near the Fermi level. Pd in the smaller Cu lattice is under compressive strain, its d states lose the sharp feature, and CuPd was less reactive than pure Cu for water and NO dissociation, which the authors attribute partly to this strain (Thirumalai 2018). Summary: thirumalai2018.
Quantifying strain, ligand, charge transfer and resonance contributions (2024)
Huang, Wang, Kamanuru, Achenie, Kitchin and Xin kept the same tight-binding picture (hopping integrals set by d-orbital radii and interatomic distances) inside a theory-infused neural network and used Shapley values to split the d-band center shift of 12,165 fcc surface sites into separate contributions. Strain and ligand effects each average about 0.15 eV, charge transfer about 0.01 eV, and a band-shape resonance effect about 0.20 eV. For Pt monolayers on fcc(111) substrates strain dominates, and the strain and ligand contributions correlate linearly with the substrate’s bulk lattice constant and d-orbital radius. A learned bond-length factor shows that relaxation partly counteracts imposed strain, with tensile systems contracting and compressive ones expanding (Huang et al. 2024). Summary: huang2024.