Large differential attosecond delays in solid state photoemission

Abstract

Time-resolved photoelectron spectroscopy provides access to the electronic structure and non-equilibrium electron dynamics in matter. At solid surfaces photoemission dynamics can be investigated on its natural time scale by measuring attosecond time delays of emitted electrons. Photoelectrons with final state energies of several tens of eV need tens to hundreds of attoseconds to be released into the vacuum. Competing effects determine the emission dynamics and, hence, the full picture of the process is still under debate. The rather large energy differences between the final states probed in commonly reported relative photoemission delays obscure their complex fine structure and hinders the interpretation of the measurements. Here we report differential attosecond delays $τ_{\mathrm{DAD}}$, i.e., relative photoemission delays for energetically close-lying spin-orbit split states. Differential attosecond delays on the order of 30 to 100 as for Bi 5d, Te 4d, and Se 3d core level photoemission from Bi$_2$Te$_3$ and Bi$_2$Se$_3$ can neither be attributed to intra-atomic delays, nor to ballistic transport and subsequent emission. Instead, calculations based on the one-step photoemission theory reveal that photoemission delays vary strongly on the energy scale of the spin-orbit splitting and quantitatively match experimental observations. This strong variation arises from multiple scattering at the surface leading to final states that involve both evanescent and propagating Bloch waves. Their relative amplitudes vary strongly affecting thereby the timing of the photoemission event since evanescent and propagating components exhibit inherently different dynamics.

Figures (7)

• Figure 1: Determination of differential attosecond delays $\boldsymbol{\tau_{\mathrm{DAD}}}$ using spin-orbit split initial states. An XUV pulse excites photoelectrons from an atom-like initial state subject to spin-orbit splitting with the energy difference $E_{\mathrm{SO}}$ here indicated for d-type initial core levels. The two photoemission final states (indicated by green and orange arrows) differentially probe $t_{\mathrm{esc}}(E_{\mathrm{kin}})$ and yield $\tau_{\mathrm{DAD}}$. For one-step photoemission theory the TR LEED states are indicated as thin black lines in the continuum. They are derived for a one-dimensional finite Kronig-Penney model in the vicinity of a bulk band gap. The Kronig-Penney model parameters are chosen to mimic the 1D single-electron potential shown in the lower part. The $\tau_{\mathrm{OSTEWS}}(E_{\mathrm{kin}})$ derived within one-step photoemission theory using the Eisenbud-Wigner-Smith delay formalism Kuzian2020 is shown as red solid line in the left inset together with the inelastic electron lifetime $\tau_{\mathrm{inel}}(E_{\mathrm{kin}})$ (blue dotted line), which varies only weakly with energy. The emission from spin-orbit split initial states leads to two spectral contributions in the photoelectron spectrum (right inset panel). Their relative spectral weight reflects the initial state degeneracy.
• Figure 2: RABBITT measurement for Bi$\boldsymbol{_2}$Te$\boldsymbol{_3}$.a, XUV spectrum (magenta) after reflection off a broadband multilayer XUV-mirror with a reflectivity indicated by the gray shaded area. b, High statistics Bi$_2$Te$_3$ XUV-only photoelectron spectrum (data points) generated by the XUV-spectrum plotted in a. The error bars correspond to the standard deviation of the underlying Poissonian counting statistics. The fit model function (gray solid line) is the sum of four Voigt profiles for Bi 5d$_{5/2}$ (green), Bi 5d$_{3/2}$ (orange), Te 4d$_{5/2}$ (red) and Te 4d$_{3/2}$ (blue), each convoluted with the XUV spectrum (magenta) shown in a and the inelastic background calculated as iterative Shirley-Proctor-Sherwood background (pink). c, RABBITT spectrogram on Bi$_2$Te$_3$. Solid connection lines between b and c indicate the principal bands of the respective photoemission channels in the RABBITT spectrogram. The dashed lines indicate the corresponding 58th sidebands with arrows that symbolize the interfering NIR-pulse induced pathways involved.
• Figure 3: Comparison of theoretical and observed differential attosecond delays.a and b, $\tau_{\mathrm{DAD}}$ as function of the XUV photon energy for Bi$_2$Te$_3$ and Bi$_2$Se$_3$, respectively. The XUV photon energy here corresponds to the even harmonic between the two odd harmonics employed for RABBITT spectroscopy. The black dashed and solid lines correspond to $\tau_{\mathrm{DAD}}$ calculated in the OSTEWS theory assuming $V_{\mathrm{opt}}$ of 1 and 2, respectively. The gray shaded areas along each line reflect a $\pm \qty{1}{\eV}$ uncertainty of the absolute energies of the involved Kohn-Sham quasi-particle initial and final state energies derived in LDA based DFT. The data points (in red) are the $\tau_{\mathrm{DAD}}$ observed in the RABBITT experiments. They are determined by the weighted mean values of all experiments and error bars correspond to the weighted standard deviations.
• Figure 4: Extracted delays of 213 individual RABBITT measurements on Bi$\boldsymbol{_2}$Te$\boldsymbol{_3}$. Each row shows a pair of RABBITT delay differences $\tau_n - \tau_m$ between two photoemission channels $n$ and $m$, with $n,m \in \left\{ \mathrm{Bi 5d}_{5/2}, \mathrm{Bi 5d}_{3/2}, \mathrm{Te 4d}_{5/2}, \mathrm{Te 4d}_{3/2} \right\}$. If the difference is positive, photoemission channel $n$ is emitted delayed with respect to channel $m$. a-f, All extracted delay values as a function of time after cleaving a fresh sample. The color indicates the sample, and the symbol indicates a position on the sample surface where multiple measurements were performed. Horizontal error bars correspond to the duration of the measurement and vertical error bars correspond to Eq. \ref{['eq:Eq5']}. The black line indicates the weighted mean value and the gray shaded area the weighted standard deviation. g-l, Weighted mean of the delays of equivalent measurements, i.e. measurements performed in the same position on the sample. m-r, Occurrence of measured delay values with histogram bins of 5 width. The black curve indicates a normal distribution corresponding to the weighted mean value $\langle \tau \rangle$ (Eq. \ref{['eq:Eq6']} in Methods) and weighted standard deviation $\sigma$ (Eq. \ref{['eq:Eq7']} in Methods).
• Figure 5: Extracted delays of 187 individual RABBITT measurements on Bi$\boldsymbol{_2}$Se$\boldsymbol{_3}$. Each row shows a pair of RABBITT delay differences $\tau_n - \tau_m$ between two photoemission channels $n$ and $m$, with $n,m \in \left\{ \mathrm{Bi 5d}_{5/2}, \mathrm{Bi 5d}_{3/2}, \mathrm{Se 3d}_{5/2}, \mathrm{Se 3d}_{3/2} \right\}$. If the difference is positive, photoemission channel $n$ is emitted delayed with respect to channel $m$. a-f, All extracted delay values as a function of time after cleaving a fresh sample. The color indicates the sample, and the symbol indicates a position on the sample surface where multiple measurements were performed. Horizontal error bars correspond to the duration of the measurement and vertical error bars correspond to Eq. \ref{['eq:Eq5']}. The black line indicates the weighted mean value and the gray shaded area the weighted standard deviation. g-l, Weighted mean of the delays of equivalent measurements, i.e. measurements performed in the same position on the sample. m-r, Occurrence of measured delay values with histogram bins of 5 width. The black curve indicates a normal distribution corresponding to the weighted mean value $\langle \tau \rangle$ (Eq. \ref{['eq:Eq6']} in Methods) and weighted standard deviation $\sigma$ (Eq. \ref{['eq:Eq7']} in Methods).