Quantitative determination of spin-dependent quasiparticle lifetimes and electronic correlations in hcp cobalt

We report on a quantitative investigation of the spin-dependent quasiparticle lifetimes and electron correlation effects in ferromagnetic hcp Co(0001) by means of spin- and angle-resolved photoemission spectroscopies. The experimental spectra are compared in detail to state-of-the-art many-body calculations within the dynamical mean-field theory and the three-body scattering approximation, including a full calculation of the one-step photoemission process. From this comparison we conclude that although strong local many-body Coulomb interactions are of major importance for the qualitative description of correlation effects in Co, more sophisticated many-body calculations are needed in order to improve the quantitative agreement between theory and experiment, in particular, concerning the linewidths. The quality of the overall agreement obtained for Co indicates that the effect of nonlocal correlations becomes weaker with increasing atomic number.

Figure 1

(Color online) (a) Geometry of the experiment. (b) Sixfold symmetry observed in the LEED pattern of the hcp Co(0001) surface. (c) Sketch of the SBZ of the hcp lattice. In (a) and (c), the magnetization orientation is also indicated.

Figure 2

(Color online) (a) Experimental spin-integrated photoemission spectra of hcp Co(0001) measured with $p$ polarization in normal emission along the $\Gamma \text{A}$ direction of the bulk Brillouin zone at different excitation energies (steps of 2 eV). The curves are labeled by selected photon energies and by the wave vectors in units of $\Gamma \text{A}=0.77{\text{Å}}^{-1}$ ($\Gamma$ point at $x=0$ and A point at $x=1$). (b) Corresponding calculation obtained by the $\text{LSDA}+\text{DMFT}+1\text{SM}$ method for hcp Co with an in-plane magnetization along the $\left[1\overline{1}00\right]$ direction. [(c) and (d)] Spin-integrated band-structure calculations for two configurations of the magnetization, out-of-plane [along [0001] direction, green (light) lines] and in-plane [along $\left[1\overline{1}00\right]$ direction, black (dark) lines]. In (c), LSDA calculations are shown, and in (d) fully relativistic valence band states obtained self-consistently in the SPRKKR formalism by suppressing the imaginary part of the self-energy $\text{Im}{\Sigma }_{\text{DMFT}}$.

Figure 3

(Color online) Spin-resolved spectra of hcp Co(0001) for different polarizations. [(a) and (b)] Experiment [upward (black (dark)] triangles: majority spin states; downward [red (light)] triangles: minority spin states. [(c) and (d)] $\text{LSDA}+\text{DMFT}+1\text{SM}$ theory [black (dark) and red (light) lines for majority and minority spin electrons, respectively]. [(a) and (c)] For $p$ and [(b) and (d)] $s$ polarization. Additional upward (black) [downward (red)] triangles on top of the spectra indicate majority [minority] bulk states while vertical black (red) bars majority (minority) spin surface-related features. The curves are labeled by the photon energies (right) and the wave vectors (left) for both the hcp $\left(\Gamma \text{A}=0.77{\text{Å}}^{-1}\right)$ and fcc $\left(\Gamma \text{L}=1.54{\text{Å}}^{-1}\right)$ lattices for comparison.

Figure 4

(Color online) Quasiparticle spectral functions of hcp Co and the photoemission peak positions obtained from the spin-resolved measurements of Fig. 3 for different polarizations (◇ for $p$ and ○ for $s$ polarization). The results are obtained by the $\text{LSDA}+\text{DMFT}$ [(a) and (b)], the $\text{LSDA}+3\text{BS}$ [(c) and (d)] and the LSDA [(e) and (f)] methods for majority [(a), (c), and (e)], and minority spin [(b), (d), and (f)] electronic states, respectively. The symmetries of the different states are indicated.

Figure 5

(Color online) Comparison between experimental and theoretical spectra at the $\Gamma$ [(a) and (b), $\text{h}\nu =72\text{eV}$] and A [(c) and (d), $\text{h}\nu =48\text{eV}$] points for $p$ polarization. (a) Spin-integrated experimental spectra [thick (black) symbols], single-particle LSDA-based calculation including surface effects [thick dotted (green) line], and $\text{LSDA}+\text{DMFT}+1\text{SM}$ spectra [thick solid (green) line]. (b) Spin-dependent $\text{LSDA}+\text{DMFT}+1\text{SM}$ calculations for different $U$ values. Black (dark) [red (light)] color denote majority (minority) spin and green (lighter) color the corresponding spin-integrated spectrum. Thin dotted, dashed, thick, and thin solid lines for $\text{U}=1.5\text{eV}$, 2 eV, 2.5 eV, and 3 eV, respectively. In (c) and (d), analogous data as in (a) and (b). Symmetry labels in the two consecutive Brillouin zones of the hcp lattice are also indicated.

Figure 6

(Color online) Fits to spin-resolved valence band spectra at $x=0.62\Gamma \text{A}$ $\left(h\nu =56\text{eV}\right)$ of the hcp Co(0001) bulk Brillouin zone. [(a) and (b)] For $p$ and [(c) and (d)] for $s$ polarization. Black (dark) [red (light)] lines are the results of the fits to majority (minority) spin spectra, which are indicated by upward (downward) triangles and arrows, respectively. Thin lines correspond to the different Lorentzian functions and backgrounds used in the fits. Thick lines indicate the final fit of each spectrum obtained by using Eq. (2). Symmetry labels, surface resonance (SR) and surface state (SS) features are indicated.

Figure 7

(Color online) Comparison between the experimental (symbols) and theoretical (lines) imaginary part of the self-energy of hcp Co(0001) for (a) majority [black (dark) color] and (b) minority [red (light) color] spin electrons. The experimental data points only contain the electronic and impurity scattering contributions to the linewidth. On the right hand side, labels to identify the different $k$ points are also given for both hcp and fcc lattices. Full and open symbols are for $p$ and $s$ polarization, respectively. The theoretical calculations correspond to $\text{Im}{\Sigma }_{\text{DMFT}}$ for $U=2.5\text{eV}$ (thick dotted lines) and $U=3\text{eV}$ (thin dotted lines) and to $\text{Im}{\Sigma }_{3\text{BS}}$ for $U=2.5\text{eV}$ (solid lines).