Determination of the Hole Lifetime from Photoemission: Ti $3d$ States in ${\mathrm{TiTe}}_2$

The electron hole spectral width ${\Gamma }_h$ of Ti $3d$ states in layered ${\mathrm{TiTe}}_2$ and the inverse lifetime $V_i$ of the photoelectron final states are determined from experiment within the one-step theory of photoemission. The condition for the possibility of separating the effects of ${\Gamma }_h$ and $V_i$ is a strongly non-free-electron character of the final states. The resulting drastic changes of the line shape with photon energy are experimentally observed and explained by an ab initio theory. A nonmonotonic dependence of $V_i$ on the final-state energy is observed; it is shown to reveal the real space structure of the complex electron self-energy.

Figure 1

Photon energy dispersion of peak position (a) and sharpness (b) for Ti $3d$ emission, experiment (open and filled circles) versus theory (lines). Theoretical values are obtained with the conservative dependence $V_i(E)$ shown in Fig. 2c. In the initial states graph the dashed line shows the ab initio LDA band structure, and the solid line shows the renormalized band structure used in the photoemission calculation. The final states graph shows real lines of complex band structure contributing to the LEED state for a constant $V_i=2\mathrm{eV}$. The size of the circle shows the squared modulus of the transition amplitude to an individual wave from the initial state defined by $k_{-}$ and $k_{+}$ in the initial states panel. Intervals marked by Greek letters are related to branches of the complex band structure shown in Fig. 3. Right-hand panel compares experimental and theoretical EDCs; spectra are marked by photon energies.

Figure 2

Determination of $V_i$ and ${\Gamma }_h$ from line shape analysis. (a) Calculated peak sharpness $I_m$ for ${\Gamma }_h=1\mathrm{meV}$ (dashed line) and for ${\Gamma }_h=20\mathrm{meV}$ (solid line) with $V_i=2\mathrm{eV}$. (b) Calculated $I_m$ for ${\Gamma }_h=20\mathrm{meV}$ with $V_i=1.5\mathrm{eV}$ ($\hslash \omega <46\mathrm{eV}$) and $V_i=4\mathrm{eV}$ ($\hslash \omega >46\mathrm{eV}$). Circles in (a) and (b) are the experimental curve of Fig. 1b. (c) Conservative dependence $V_i(E)$. (d) Dependence of line shape on ${\Gamma }_h$: solid curves are calculated with ${\Gamma }_h=20\mathrm{meV}$ for $\hslash \omega =51\mathrm{eV}$ and ${\Gamma }_h=40\mathrm{meV}$ for $\hslash \omega =70\mathrm{eV}$ and dashed curves ${\Gamma }_h=1\mathrm{meV}$ for $\hslash \omega =51\mathrm{eV}$ and ${\Gamma }_h=10\mathrm{meV}$ for $\hslash \omega =70\mathrm{eV}$. (e) Dependence of the fitting quality (mean square deviation $R=||I_{\exp }-I_{\mathrm{th}}||$) on ${\Gamma }_h$ for $\hslash \omega =51$, (solid line), 59 (dashed line), and 70 eV (dot-dashed line).

Figure 3

(a) Shows function $V_i(E)$ derived by fitting theoretical line shapes to experimental ones. Circles show values of $V_i$ that lead to a pointwise agreement with the experiment. Thin error bars show relative sensitivity of the line shape to variations of $V_i$. Thick bars show $V_i$ ranges over which the line shape is insensitive to $V_i$, and the calculated spectrum coincides with the experiment to within the measurement error. Dashed curve is the conservative $V_i(E)$. (b),(c) Show penetration depths of some partial waves for $V_i=2\mathrm{eV}$. Vertical extent of the shaded area is proportional to the probability of transition to a wave from the initial state at $ϵ=20$ (b) and 220 meV (c). (d) Compares theory and experiment for $\hslash \omega =65–67\mathrm{eV}$.