Electronic structure and optical properties of Sr₂IrO₄ under epitaxial strain

The crystal structure of SIO with space group (142) I4₁/acd consists of eight formula units in the $\sqrt{2}{a}_{0}\times c$ unit cell [18] as shown in figure 1, with the lattice constants a = b = 5.497 Å and c = 25.798 Å. (An equivalent unit cell with the body-centered tetragonal lattice and four formula units in the basis may also be used.) Note that $a=\sqrt{2}{a}_{0}$. The structure shows a staggered rotation of the IrO₆ octahedra about the c-axis by the angle θ = 11.5°. The unit cell has four IrO₆ planes stacked along the c-axis, each plane consisting of two octahedra, with staggered rotation angles and antiferromagnetic Ir moments. The electronic structure is essentially controlled by the Ir atoms with the 5d⁵ configuration placed in the crystal field of the oxygen octahedra, with the Sr atoms playing a passive role of donating electrons to the system, described by the nominal chemical formula ${\mathrm{Sr}}_{2}^{2-}$ Ir⁴⁺ ${{\rm{O}}}_{4}^{2-}$. The structure can be thought of as IrO₂ layers separated from each other by two intervening SrO planes, making the structure quasi-two-dimensional. This means that the basic electronic structure can be modeled by a single plane of the Ir atoms in a minimal model, which we discuss later.

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Density-functional theory (DFT) with linearized full potential muffin-tin orbital method (LMTO) [19–21] was used to solve the Kohn–Sham equations within the local spin-density approximation for the exchange and correlational functional [22, 23]. The LMTO basis set consisted of spdf orbitals for Ir and Sr and spd orbitals for O, which were augmented outside the muffin-tin spheres by two Hankel functions of two different decay lengths. The semicore states Ir $5p$ and Sr $3p$ were treated as valence electrons. The band calculations were carried out for the optimized structures within the local spin-density approximation including both the SOC and the Hubbard U terms (LSDA + SO + U). We used the Vienna ab initio simulation package (VASP) in projector augmented wave formalism [24] for structural optimization and for computing the magnetic moments.

The magnitude of U used in the DFT calculations depends on the method, since each method uses a different type of basis set. Following our earlier works [25], we have used U = 2.7 eV in the LMTO calculations, while a somewhat smaller U = 2 eV was used in the VASP calculations, which yielded similar band widths and band gaps.

Atomic positions were relaxed in all our calculations within the local spin-density approximation and the force convergence was obtained with a tolerance of 10⁻³ Ryd/Bohr. For the bulk, unstrained structure, we took the experimental unit cell and optimized the positions of the atoms. An important structural parameter is the IrO₆ octahedral rotation angle θ, which we found to be $13.5^\circ$ as compared to the measured angle of 11.5°.

For the epitaxially strained structure, the in-plane lattice constant was changed according to the strain parameter e_xx ≡ (a − a_bulk)/ a_bulk, while the out-of-plane lattice constant was determined from the condition that the cell volume is preserved under strain, so that e_zz ≡ (c − c_bulk)/ c_bulk = −2e_xx. Structural parameters were then optimized for each strain condition, with the fixed lattice constants. We varied the in-plane strain e_xx by ±2%, which is the same order of magnitude as the strains present in the experimental structures, e.g. e_xx ≈ 2% for SIO grown on the GdScO₃ substrate and −2% for the NdGaO₃ substrate [12]. The computed structural parameters are shown in table 1 and figure 2. The trend in the structural parameters shown here is similar to the results obtained from an earlier work using the dynamical mean-field theory [15].

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For test purposes, we also performed a second set of calculations, where we took the in-plane lattice constant to be the same as the substrate as usual, but used the measured c/a ratio from the experiments [12]. No significant changes in the bond angles and bond lengths were found.

The electron bands for the optimized structures under three different strain conditions are shown in figure 3, calculated within the local spin-density approximation with Coulomb interaction and SOC included (LSDA+SO+U). The Fermi surfaces for the doped SIO are shown in figure 4. The results show systematic changes of the band structure features in the gap region, some of which are shown in figure 5. The magnitude of the fundamental gap Δ_g increases with tensile strain, and so does the direct gap Δ_X at the X point, while the direct gap at the Γ point does not change very much. This trend in the gap values may be expected, since tensile strain would reduce the band widths of the LHB and the UHB, while the Coulomb U is relatively unchanged. The opposite happens for the compressive strain. The Γ point beats the trend because the valence top at Γ has a considerable contribution from the oxygen states. The approximately linear variation of the gap with strain is clearly visible in the partial density-of-states shown in figure 6. This trend of the gap variation with strain has already been noted in earlier works [15–17].

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3.1.1. Strain induced ${\rm{\Gamma }}\mbox{--}X$ crossover

The calculated spin and orbital magnetic moments with and without strain are listed in table 2. We note that the total energy obtained with spin moments constrained along the planar direction ($\hat{x}$) has a lower energy than the spin moments constrained along $\hat{z}$, consistent with the $\hat{x}$ orientation of the magnetic moments. Table 2 shows the calculated magnetic moments using VASP with the LSDA+U+SO functional.

There has been much interest in the magnetic moments in SIO, in particular on the ratio μ_l/μ_s [15–17, 27–29], since a ratio of two indicates the spin-orbital entanglement of the wave function and a deviation from this value is indicative of a mixture between J_eff = 1/2 and 3/2 sectors, as suggested from table B1. For the unstrained material (e_xx = 0), the calculated total moment μ_l + μ_s = 0.38μ_B is in good agreement with values obtained from the magnetic susceptibility measurements (0.5μ_B) [30] as well as from previous calculations (0.36μ_B) [1]. As already mentioned, the d⁵ configuration of Ir can be thought of as a single hole in the t_2g manifold, leading to the cubic-field values, μ_l = 2/3μ_B and μ_s = 1/3μ_B, as seen from table B1. When a tetragonal field is present as in the case of SIO, the $| {J}_{\mathrm{eff}},m\rangle$ states get mixed among themselves, and the magnetic moments can be substantially altered from the cubic-field value.

From our calculations, we find μ_l/μ_s ≈ 2.6 (table 2) for the unstrained structure, in general agreement with earlier calculations [15, 16] as well as with a recent measurement using non-resonant magnetic x-ray diffraction which obtained the value μ_l/μ_s ≈ 2.5. An earlier x-ray absorption measurement [27] yielded the ratio μ_l/μ_s ≈ 1.1; the reason for the discrepancy between the two measurements is unclear.

The magnetic moment ratio as well as its variation with strain can be approximately described by invoking a tetragonal crystal field for the single ion in the presence of the spin–orbit coupling. For more accurate description, a renormalized spin–orbit coupling has been invoked [15]. The results for the single ion in the tetragonal field are given in appendix B. For the spin moment aligned along $\hat{x}$, which is the case for SIO, we have μ_l/μ_s = 2 + 4ξ/3, where ξ = ε/λ is the ratio of the tetragonal field to the spin–orbit coupling strength λ ≈ 0.4 eV. For the unstrained case, ε ≈ 0.14 eV [15, 31], so that the ratio μ_l/μ_s = 2.47, in reasonable agreement with the DFT result stated in table 2. If we use the results of [29] which suggest that the magnitude of ε increases (decreases) by about 0.1 eV for tensile (compressive) epitaxial strain of 2%, then μ_l/μ_s = 2.8 for the tensile case and 2.1 for the compressive case, which more or less explains the DFT calculated trend shown in table 2.

The calculated absorption spectra for bulk SIO without strain are shown in figure 7 for light polarization along the plane (E ∥plane), which is also compared to the measured data [12]. The spectra show three distinct peaks which can be understood from the transitions between the J_eff = 1/2 and J_eff = 3/2 states as shown in figure 8. The J_eff = 3/2 states are split into two as seen from table 1 due to the tetragonal crystal field V_t = 2ε/3. In the experiment, only two distinct peaks are seen (these are labeled α and β), while the weaker γ peak is missing, possibly due to instrumental broadening.

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We did not find any significant asymmetry in α(ω) for polarization directions within the plane. However, there is a large difference whether E∥ plane or E∥z, as discussed later.

The optical absorption in the low energy range occurs due to transitions within the Ir ${{t}}_{2g}$ manifold, transitions that are however not dipole allowed due to the optical selection rules (δ L = ±1 for plane-polarized light). In this subsection, we discuss the optical transition matrix elements and conclude that they become dipole allowed due to the admixture of the Ir p states into the Ir d bands. To get the dipole matrix elements, we need to estimate two things, as indicated from equation (4) below: (i) The amount of the Ir p–d admixture η, which we estimate from perturbation theory as well as from density-functional results and (ii) The optical matrix element for dipole-allowed transition between the Ir p and d orbitals $\langle p| \hat{e}\cdot {\bf{p}}| d\rangle$, which we estimate from an effective hydrogenic model.

The d–d transitions are non-zero in the crystal only because there is deviation from the spherical symmetry, the so called crystal field, which also produces splitting between the ${{t}}_{2g}$ and the e_g states [32]. The crystal field not only splits the Ir (5d) orbitals, but it also mixes the Ir (5p) and (4f) orbitals into the 5d states, so that the d–d optical transitions have non-zero matrix element. Alternatively, such transitions may be equivalently described to be due to the transfer between Ir d atomic orbitals to the adjacent atoms because of the wave function overlap, e.g. within the linear combination of atomic orbitals model.

We now estimate the dipole matrix elements, which are non-zero due to the p−d and f–d mixing for the Ir atom. We first consider the p−d mixing, which as we shall see makes the larger contribution. Symbolically, this is given by

$$\begin{eqnarray}\begin{array}{rcl}{M}_{{dd}} & = & \langle {\psi }_{c{\bf{k}}}| \hat{e}\cdot {\bf{p}}| {\psi }_{v{\bf{k}}}\rangle \\ & = & \langle d+\eta p| \hat{e}\cdot {\bf{p}}| d+\eta p\rangle \approx 2\eta \times \langle p| \hat{e}\cdot {\bf{p}}| d\rangle .\end{array}\end{eqnarray} \tag{ 4 }$$

The admixture η of the Ir p states into the Ir d bands, although small due to the large p−d energy difference, is nevertheless significant enough and is largely responsible for the optical transition in SIO.

The magnitude of η can be estimated from the partial density of states (PDOS) in the band calculations in the energy range of the Ir d bands. With the definition for the PDOS, ${\rho }_{d}(\varepsilon )={\sum }_{i}{\left|\langle {\psi }_{i}| d\rangle \right|}^{2}\delta (\varepsilon -{\varepsilon }_{i})$ and ${\rho }_{p}(\varepsilon )={\sum }_{i}{\left|\langle {\psi }_{i}| p\rangle \right|}^{2}\delta (\varepsilon -{\varepsilon }_{i})$, and with the wave functions ψ ≈ d + ηp, one can estimate the average value of the admixture from the two PDOS in the energy range of interest. The PDOS, computed using the linearized augmented plane wave (LAPW) method¹ as implemented in the Wien2k code [33], are shown in figure 9. We have obtained the value of η from the ratio of the integral of the two PDOS in the energy window of the Ir ${{t}}_{2g}$ bands, viz. from −2 to 1 eV. Denoting these two integrals over this energy range by ${\rho }_{p}\equiv \int {\rho }_{p}(\varepsilon ){\rm{d}}\varepsilon$ and ${\rho }_{{d}}\equiv \int {\rho }_{{d}}(\varepsilon ){\rm{d}}\varepsilon$, respectively, we find the result: η² ≈ ρ_p/ρ_d ≈ 0.19 × 10⁻² or η ≈ 4%. We find a similar admixture of the $4f$ states, ρ_f/ρ_d ≈ 0.2 × 10⁻², indicating a nearly equal $4f$ admixture ${\eta }_{4f}\approx 4 \%$ into the Ir (5d) bands.

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Alternatively, the strength of the admixture η can be estimated from perturbation theory. From the second order perturbation theory, we have

$$\begin{eqnarray}&&\eta =3\times \langle p| {V}_{{cf}}| d\rangle /({\varepsilon }_{d}-{\varepsilon }_{p}),\end{eqnarray} \tag{ 5 }$$

where the factor of three is from the degeneracy of the p states. If we further approximate $\langle p| {V}_{{cf}}| d\rangle \sim \langle d| {V}_{{cf}}| d\rangle \approx {{\rm{\Delta }}}_{{cf}}$, admittedly a crude approximation, and take the ${{t}}_{2g}$ –e_g splitting Δ_cf ∼ 2 eV, and use the result ε_d − ε_p ∼ 43 eV for Ir from the standard Atomic tables [34], we get the perturbation theory result of η ≈ 14 %, which is a factor of three too high as compared to the DFT result. However, given the crudeness of the approximation we made for the matrix elements in the perturbation theory, the order of magnitude agreement is reasonable. In the following, we shall use the DFT value for the p–d admixture η ≈ 4 %.

The second part is the estimation of the dipole matrix element ${M}_{{pd}}\equiv \langle p| \hat{e}\cdot {\bf{p}}| d\rangle$ appearing in equation (4), which we do from an effective hydrogenic model [35, 36]. In this model, the atomic wave functions are described as hydrogenic wave functions, but with an orbital dependent effective nuclear charge Z. Thus

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Psi }}}_{{nlm}} & = & {R}_{{nl}}(r){Y}_{{lm}}({\rm{\Omega }}),\\ {R}_{{nl}}(\rho ) & = & {N}_{{nl}}(Z){\rho }^{l}{{\rm{e}}}^{-\rho /2}{L}_{n+l}^{2l+1}(\rho ),\end{array}\end{eqnarray} \tag{ 6 }$$

where $\rho =(2Z/{{na}}_{0})r$. Note that Z here is an effective atomic number, which takes into account the screening of the core electrons and depends on the principal quantum number n also. We take the value [35] Z = 18.7 appropriate for Ir n = 5 orbitals (5p and 5d). The integration can be performed analytically to yield the result M_pd for the plane polarized light:

$$\begin{eqnarray}&&{M}_{{pd}}=\langle {{\rm{\Psi }}}_{510}| {p}_{z}| {{\rm{\Psi }}}_{520}\rangle =-0.25i\end{eqnarray} \tag{ 7 }$$

in atomic units (viz. ℏ = 1, Bohr radius a₀ = 1, m_e = 1/2, Energy unit = 1 Ryd). Note that for atoms, the direction of polarization does not matter due to spherical symmetry. Plugging in the estimated magnitudes of M_pd and η in equation (4), we find the d–d matrix element due to the admixture of Ir (5p) into the Ir (5d) states

$$\begin{eqnarray}&&| {M}_{{dd}}^{p}| =2\eta \times | {M}_{{pd}}| =2\times 4 \% \times 0.25=2\times {10}^{-2}.\end{eqnarray} \tag{ 8 }$$

A similar calculation for the matrix element due to the admixture with the Ir (4f) orbitals yields a number, which is an order of magnitude smaller than $| {M}_{{dd}}^{p}|$, viz.

$$\begin{eqnarray}&&| {M}_{{dd}}^{f}| =2{\eta }_{f}\times | {M}_{{fd}}| =2\times 4 \% \times 0.064=5\times {10}^{-3},\end{eqnarray} \tag{ 9 }$$

where we estimated the 4f admixture η_f from figure 9 and used the effective atomic number Z = 38.3 for the Ir 4f orbitals [35]. Thus the Ir (4f) orbitals contribute a much smaller amount as compared to the Ir (5p) orbitals. The estimate given by equation (8), M_dd ∼ 10⁻², is very comparable to the average matrix element M_cv ∼ 10⁻² obtained from the band calculations (figure 10), indicating that the admixture of the Ir (5p) orbitals is indeed responsible for the optical absorption.

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4.3.1. Uniform strain

The calculated optical absorption for uniform compression (uniform pressure) is compared to the same for bulk SIO without any compression in figure 11. Apart from a shift in the peak positions to lower energies, caused by a reduction in band gap due to the increase of the band width due to compression, we note that the overall absorption coefficient is increased for both light polarizations. This can be explained from our above argument of Ir 5p–5d admixture, because a larger crystal field upon compression mixes the Ir 5p more into the Ir 5d bands, making the optical matrix element larger. Indeed, following the same logic, as the lattice constant is increased, there is less and less admixture of the Ir (5p) orbitals, and in the limit of infinite lattice constant, the admixture vanishes (perfect spherical symmetry), so that optical absorption would be zero as would be expected for the $d\to d$ transition due to the dipole selection rules.

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4.3.2. Epitaxial strain

The change of the optical absorption with epitaxial strain is shown in figure 12. With compressive epitaxial strain (e_xx < 0 and e_zz > 0), the distances in the plane are reduced, leading to a larger optical absorption for E ∥ plane as argued for the uniform compression case. The same effect leads to an overall reduction of the optical absorption for E $\parallel \hat{z}$. For the absorption with E ∥ plane, all three peaks α, β, and γ are still there, but the peaks are red (blue) shifted with compressive (tensile) epitaxial strain, consistent with the band structure changes under strain shown in figure 3.

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4.3.3. Optical anisotropy

The polarization dependence of the optical absorption has been reported in the literature [14], where a significant anisotropy is found in the optical spectra. The absorption is significantly reduced for the polarization E $\parallel \hat{z}$ as compared to E ∥ plane in the low energy region corresponding to the transition within the Ir ${{t}}_{2g}$ manifold, while for the higher energy region, the difference in the relative strength is not as drastic. This is reproduced quite well from our calculations as seen from figure 13, where we also compare with the existing experiments. The calculated spectra is for the unstrained structure, while the experimental spectra is for a system with non-uniform strain, the only case for which experimental results are available for both polarizations. The anisotropy is directly attributable to the differences in the matrix element $| \hat{e}\cdot {{\bf{M}}}_{{cv}}({\bf{k}})|$ (equation (2)), since the variation of the refractive index n(ω) with energy or polarization is relatively weak, as indicated from figure 14.

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As seen from equation (A.5), the hopping integrals are in general complex, and of course there is no problem working with the complex hoppings. However, it is convenient to make the NN integral real by a gauge transformation [39]. In the transformation, one simply adds a multiplicative phase factor to the orbital definitions, viz.

$$\begin{eqnarray}&&\begin{array}{l}| {\widetilde{e}}_{1j}\rangle \to | {\widetilde{e}}_{1j}\rangle {{\rm{e}}}^{{\rm{i}}{\varepsilon }_{j}\phi /2}\\ | {\widetilde{e}}_{2j}\rangle \to | {\widetilde{e}}_{2j}\rangle {{\rm{e}}}^{-{\rm{i}}{\varepsilon }_{j}\phi /2},\end{array}\end{eqnarray} \tag{ A.7 }$$

where ε_j = ±1 for j = A/B sublattices. With the choice $\phi ={\tan }^{-1}({t}^{{\prime} }/t)$, where $t,{t}^{{\prime} }$ are given in equation (A.5) for the 1NN hopping, the new hopping integral becomes real, viz. $t\,+\,{{\rm{i}}{t}}^{{\prime} }\to \tilde{t}$, where $\tilde{t}={\left({t}^{2}+{t}^{{\prime} 2}\right)}^{1/2}$. This transformation leaves all same-sublattice hoppings real, while the opposite-sublattice hoppings including t₄ and beyond continue to remain complex. One can choose to work with such complex hoppings, or else, simply ignore the imaginary parts, since hoppings between far-away neighbors are small anyway. We denote the gauge-transformed basis (A.7) as $| {e}_{1}\rangle$ and $| {e}_{2}\rangle$ in equation (A.1) and the hopping amplitudes (real) as t_i as indicated in figure A1.

The antiferromagnetic lattice structure together with the fact that $| {\widetilde{e}}_{1}\rangle$ and $| {\widetilde{e}}_{2}\rangle$ subspaces do not mix leads to the 2 × 2 TB Hamiltonian in the momentum space

$$\begin{eqnarray}H({\bf{k}})=\left(\begin{array}{cc}-{\rm{\Delta }}+{h}_{11} & {h}_{12}\\ {h}_{12}^{* } & {\rm{\Delta }}+{h}_{11}\end{array}\right),\end{eqnarray} \tag{ A.8 }$$

where ${h}_{11}=4{t}_{2}\cos {k}_{x}\cos {k}_{y}+2{t}_{3}(\cos 2{k}_{x}+\cos 2{k}_{y})$, ${h}_{12}=2{t}_{1}(\cos {k}_{x}+\cos {k}_{y})+4{t}_{4}(\cos 2{k}_{x}\cos {k}_{y}\,+\cos 2{k}_{y}\cos {k}_{x})$, a₀ = 1 here, and Δ = U/2 is the staggered field. Diagonalization readily yields the energies of the upper and lower J_eff = 1/2 Hubbard bands to be

$$\begin{eqnarray}&&{\varepsilon }_{\pm }({\bf{k}})={h}_{11}({\bf{k}})\pm \sqrt{{{\rm{\Delta }}}^{2}+{h}_{12}^{2}({\bf{k}})}.\end{eqnarray} \tag{ A.9 }$$

The TB parameters obtained from fitting equation (A.9) to the density-functional bands for the optimized structure with no strain are, in units of eV: U = 0.65, t₁ = −0.095, t₂ = 0.015, t₃ = 0.035, and t₄ = 0.01. The TB fit to the DFT bands are shown in figure 3 as dotted lines. An important feature of the band structure is the occurrence of the conduction minimum at the M point of the Brillouin zone, which the electrons would occupy in the electron doped system. It can be easily shown that for this to happen, the TB parameters must satisfy the condition [7]

$$\begin{eqnarray}&&{t}_{3}\gt {t}_{2}/2\gt -{t}_{1}^{2}/U,\end{eqnarray} \tag{ A.10 }$$

which is clearly satisfied by our parameters given above. Furthermore, the eccentricity of the elliptical energy contours around the M point in the conduction band, as seen in figure 4, is given by [25] $e\equiv {\left(1-{r}^{2}\right)}^{1/2}$, where the axis ratio $r={\left[(2{t}_{3}-{t}_{2})/\left({t}_{2}+2{t}_{3}+4{\left({t}_{1}-2{t}_{4}\right)}^{2}/U\right)\right]}^{1/2}\approx 0.6$ for SIO.

We determine the TB parameters under strain condition with the following ansatz. Using the calculated angle $\theta \approx 13^\circ$ and the hopping expressions equations (A.5) and (A.6), there is a one to one correspondence between t_i and ${V}_{\pi }^{i}$ for the ith neighbor. We obtain the values of ${V}_{\pi }^{i}$ for different neighbors under no strain condition. In order to compute the hopping integrals under strain conditions, we back substitute ${V}_{\pi }^{i}$ and the DFT optimized angles under strain into equations (A.5) and (A.6). In addition to the change of the angles, distances between atoms also change. Taking the variation of V_π to follow Harrison's R⁻⁵ scaling with distance R and including the effect of θ from equations (A.5) and (A.6), we can compute the TB hopping parameters t_i under the epitaxial strain conditions. We assume that strain enters the hopping parameters only via change of the hopping distances and the rotation angle θ.

With this simple TB model, we find that the important trends of the band structure under strain are described qualitatively correctly, though not quantitatively with a factor of two to three discrepancy as compared to the DFT results.

In particular, the model can qualitatively describe the ${\rm{\Gamma }}\mbox{--}X$ crossover of the valence band top under strain (hole pocket in the doped structure), predicted from the DFT calculations. From the TB energy expression equation (A.9), we readily find the Γ–X energy difference for the valence band

$$\begin{eqnarray}&&{E}_{{\rm{\Gamma }}X}={\varepsilon }_{-}({\rm{\Gamma }})-{\varepsilon }_{-}(X)=8{t}_{2}+{\rm{\Delta }}-{\left[{{\rm{\Delta }}}^{2}+16{\left({t}_{1}+2{t}_{4}\right)}^{2}\right]}^{1/2}.\end{eqnarray} \tag{ A.11 }$$

For our TB parameters, this is almost zero consistent with the DFT results. The strain dependence of E_ΓX is obtained from the TB expression (A.11), viz.

$$\begin{eqnarray}&&{\rm{\Delta }}{E}_{{\rm{\Gamma }}X}=\displaystyle \frac{\partial {E}_{{\rm{\Gamma }}X}}{\partial R}{\rm{\Delta }}R+\displaystyle \frac{\partial {E}_{{\rm{\Gamma }}X}}{\partial \theta }{\rm{\Delta }}\theta \approx 1.65\ \mathrm{eV}\times \ {e}_{{xx}},\end{eqnarray} \tag{ A.12 }$$

where we have used the fact that the distance change between the atoms in the plane is simply Δ R/R = e_xx, and also the DFT result for the change in angle with strain, viz. Δ θ ≈ (−1.5 rad ) e_xx, obtained from table 1 assuming a linear strain dependence, since the strain is small. It turns out that the two terms in equation (A.12) contribute nearly half each to the final result, so that both the angle and distance changes are important for the description of the Γ–X crossover.