Understanding the $L_{2,3}$ x-ray absorption spectra of early $3d$ transition elements

In this work we calculate the x-ray absorption spectra at the $L_{2,3}$ edges of the early $3d$ elements by solving the Bethe-Salpeter equation. We focus our discussion on the origin of the observed deviation of the branching ratio between $L_2$ and $L_3$ edges from its statistical value. Using the absorption edge of Ca in ${\text{CaF}}_2$ we show that the deviation is related to the mixing between the excitations from $2p_{1/2}$ and $2p_{3/2}$ core levels. Furthermore we find that the mixing is triggered by the exchange term of the electron-hole Hamiltonian. This term does not depend on the dielectric function, therefore such coupling is also present in metals explaining the high values of the branching ratios observed in the metals Ca and Ti. The calculated $\text{Ti}{L}_{2,3}$ spectra of ${\text{SrTiO}}_3$ and the anatase and rutile phases of ${\text{TiO}}_2$ reproduce even all fine details of the experimental spectra.

Figure 1

(Color online) (a) Experimental ${\text{CaF}}_2$ $L_2$ (right) and $L_3$ (left) edges (Ref. 16) and calculations within ground-state DFT, using a supercell DFT core-hole approach and with the current BSE method. (b) BSE calculations when certain parts of the Hamiltonian are set to zero. For the line labeled as BSE $\left(L_2\right)$ and BSE $\left(L_3\right)$ the excitations from $2p_{3/2}$ and $2p_{1/2}$ core levels are not allowed to interact, for BSE $\left(L_{2,3},H^{dir}=0\right)$ the direct Coulomb term of the Hamiltonian is set to zero, and for BSE $\left(L_{2,3},H^x=0\right)$ the exchange part is neglected. (c) Decomposition of the spectrum into contributions involving squared matrix elements from $2p_{1/2}$ and $2p_{3/2}$ states as well as the coherent cross terms between these excitations.

Figure 2

(Color online) The calculated (solid line) and experimental $L_{2,3}$ edges of fcc Ca (experiment taken from Ref. 25).

Figure 3

(Color online) Calculated (full line) and experimental (dotted curve, taken from Ref. 26) $\text{Ti}{L}_{2,3}$ edges in (a) ${\text{SrTiO}}_3$, (b) ${\text{TiO}}_2$ anatase, and (c) ${\text{TiO}}_2$ rutile within the BSE formalism. In the upper panels we show also bars indicating the oscillator strength of the excitons while in the lower panels the corresponding binding energies are displayed.

Figure 4

(Color online) Decomposition of the $L_{2,3}$ edge of ${\text{SrTiO}}_3$ calculated within BSE into contributions involving (a) excitations from $2p_{1/2}$ and $2p_{3/2}$ states (b) excitations into $d\text{−}{e}_g$ and $d\text{−}{t}_{2g}$ states as well as their coherent cross terms.

Figure 5

(Color online) (a) Decomposition of the conduction-band components of the BSE eigenstates contributing to the calculated $L_3$ edge of ${\text{SrTiO}}_3$. The quantity $D\left(\epsilon \right)$ is defined as: $D\left(\epsilon \right)={\sum }_i{\sum }_{k,h,e}{A}_{k,h,e}^2\times f_i\times \delta \left(\epsilon -{\epsilon }_{e\mathbf{k}}\right)$, where $f_i$ is the oscillators strength and the sum runs over the states contributing to the A and B peaks of the $L_3$ edge, respectively. $D\left(\epsilon \right)$ is essentially the density of states of the conduction bands extracted from these peaks. (b) The ground-state $\text{Ti}3d\text{−}{e}_g$ and $\text{Ti}3d\text{−}{t}_{2g}$ DOS of ${\text{SrTiO}}_3$.

Figure 6

(Color online) $L_{2,3}$ edges calculated for (a) MnO and (b) CoO. The single-particle states has been calculated using on-site hybrid exchange energy functionals (Ref. 27).

Figure 7

(Color online) The dependence of the $L_{2,3}$ branching ratio in rutile ${\text{TiO}}_2$ as a function of the spin-orbit splitting of the $2p_{1/2}$ and $2p_{3/2}$ states compared to experimental and calculated ratios of the various $3d$ metals.