Spin Polarization and Attosecond Time Delay in Photoemission from Spin Degenerate States of Solids

After photon absorption, electrons from a dispersive band of a solid require a finite time in the photoemission process before being photoemitted as free particles, in line with recent attosecond-resolved photoemission experiments. According to the Eisenbud-Wigner-Smith model, the time delay is due to a phase shift of different transitions that occur in the process. Such a phase shift is also at the origin of the angular dependent spin polarization of the photoelectron beam, observable in spin degenerate systems without angular momentum transfer by the incident photon. We propose a semiquantitative model which permits us to relate spin and time scales in photoemission from condensed matter targets and to better understand spin- and angle-resolved photoemission spectroscopy (SARPES) experiments on spin degenerate systems. We also present the first experimental determination by SARPES of this time delay in a dispersive band, which is found to be greater than 26 as for electrons emitted from the sp-bulk band of the model system Cu(111).

Figure 1

(a) Two phase-shifted transitions interfere in the photoemission process to build up the photoelectron wave function. (b) Experimental setup with relevant geometry. The given coordinate system is in the sample frame.

Figure 2

(a) Cu(111) $h\nu$ dispersion for $E_b$ close to the Fermi level. (b) Band map at $h\nu =130\mathrm{eV}$ with the sp bulk CB. Solid lines indicate the $E_b$ where the spin-resolved MDCs have been measured. The actual set was performed in a random sequence in order to prevent any time-related artifact. (c) orientation in space of the measured $\mathit{P}$ at $k^{-}$ (the one at $k^{+}$ is opposite). The reaction plane is tilted by $\psi =51°$ [24] from the atomic reaction plane $xz$. (d) 3D spin resolved MDC along $x$ measured 0.2 eV below the Fermi level with $h\nu =130\mathrm{eV}$ $\pi$ polarized. The total intensity and the three polarization spatial components are shown. (e) Plot of $P(E_b)$ for the two spin signals $k^{-}$ and $k^{+}$; (f) measurement repeated for different $h\nu$; (g) 3D spin-resolved EDC of the $3p$ core levels with same $E_k$ and $\theta$ as the measurement in (d).

Figure 3

(a) Summary band map of spin polarization $P$ from MDCs measured along $k_x$ with $h\nu =130\mathrm{eV}$ $\pi$ polarized; (b) $P$ from KKR calculations performed for similar experimental parameters; (c) the three calculated spatial components of $P$. Their complexity does not permit us to unambiguously fix a direction along which (b) could be projected in order to plot it in a red-blue color scale.