Ultrafast dynamics and decoherence of quasiparticles in surface bands: Preasymptotic decay and dephasing of quasiparticle states

We develop a many-body description of ultrafast electron dynamics in surface bands appropriate for studying relaxation of hot electrons and holes excited in the processes of one- and two-photon photoemission and inverse photoemission from surfaces. The description is based on the formalism for calculation of quasiparticle survival probabilities combined with self-consistent treatment of the electronic response of the system. We show that the calculation of survival amplitudes which carry information on the quasiparticle decay and decoherence can be conveniently mapped onto the problem of renormalization of quasiparticles by the interactions with bosonized excitations constituting the system heatbath. Applying this approach to the benchmark $\mathrm{Cu}\left(111\right)$ surface we are able to assess the regimes of preasymptotic non-Markovian quasiparticle dynamics in surface bands and locate transitions to the regime of exponential decay governed by the modified Fermi golden rule-type of transition rates. The general validity of these findings enables us to establish borderlines between different regimes of ultrafast electronic relaxation and on that basis to introduce a simple interpolation scheme for modeling of quasiparticle decay in the course of spectroscopic measurements.

Figure 1

(Color online) Plot of electron densities residing in the Shockley surface state, $\mid {\varphi }_{SS}^{\left(L\right)}\left(z\right){\mid }^2$, on the left (111) surface, and in the first image potential state, $\mid {\varphi }_{IS}^{\left(R\right)}\left(z\right){\mid }^2$, on the right (111) surface of a 31 atomic layer thick Cu slab (shaded region) with interlayer spacing $d_0=3.943\text{a.u}$.

Figure 2

(Color online) Intensity plot of the image state intraband component of the slab electronic density excitation spectrum ${\mathcal{S}}_{\mathrm{IS},\mathrm{IS}}(\mathbf{Q},\omega )$, shown as a function of energy $\omega$ and momentum $Q$ of the excitations.

Figure 3

(Color online) Cuts of the image state intraband component of the excitation spectrum ${\mathcal{S}}_{\mathrm{IS},\mathrm{IS}}(\mathbf{Q},\omega )$ from Fig. 2 taken along the planes $Q=\mathrm{const}$ as indicated in the inset. SP and MP denote the peaks associated with monopole surface plasmon and higher order multipole plasmon mode, respectively.

Figure 4

(Color online) Intensity plot of the surface state intraband component of the slab electronic density excitation spectrum ${\mathcal{S}}_{\mathrm{SS},\mathrm{SS}}(\mathbf{Q},\omega )$, shown as a function of energy $\omega$ and momentum $Q$ of the excitations. Note that the maxima of ${\mathcal{S}}_{\mathrm{SS},\mathrm{SS}}(\mathbf{Q},\omega )$ are not plotted beyond the upper limit of vertical scale.

Figure 5

(Color online) Intensity plot of the IS-SS interband component of the slab electronic density excitation spectrum ${\mathcal{S}}_{\mathrm{IS},\mathrm{SS}}(\mathbf{Q},\omega )$, shown as a function of energy $\omega$ and momentum $Q$ of the excitations.

Figure 6

(Color online) Schematic of contributions to the amplitude of 2PPE from a surface state band via the intermediate IS-band channel. Thick full and dashed lines denote the IS and SS propagators, respectively, the full thin line denotes the electron propagator in a final photoelectron state above the vacuum level, and wiggly lines illustrate the boson propagator $\mathcal{D}$. Electron-photon field interaction matrix elements are denoted by filled circles and pump and probe photon pulses are delayed by the time interval $\tau$. The photoelectron terminates in the final state $\mid \mathbf{f}⟩$ at the observation time $t_{obs}>t+\tau$.

Figure 7

(Color online) Intensity plot of ${\mathcal{W}}_{\mathrm{IS},\mathrm{IS}}(\mathbf{K},\mathbf{Q},\omega )$ defined in Eq. (37) and representing the unbounded component of the integrand in Eq. (30) over the regions in $(Q,\omega )$-phase space of the metal excitations which give major contributions to intraband-induced decoherence of an image potential state electron with initial $K=0$.

Figure 8

(Color online) Intensity plot of ${\mathcal{W}}_{\mathrm{IS},\mathrm{SS}}(\mathbf{K},\mathbf{Q},\omega )$ defined in Eq. (37) and representing the unbounded component of the integrand in Eq. (30) over the regions in $(Q,\omega )$-phase space of the metal excitations which give major contributions to electron decoherence in interband transitions from the image potential band to the surface state band with initial $K=0$. Due to the Pauli exclusion principle only the electron interband transitions into the unoccupied SS-band states for which $Q>K_F^{SS}$ are allowed.

Figure 9

(Color online) Intensity plot of ${\mathcal{W}}_{\mathrm{SS},\mathrm{SS}}(\mathbf{K},\mathbf{Q},\omega )$ defined in Eq. (37) and representing the unbounded component of the integrand in Eq. (30) over the regions in $(Q,\omega )$-phase space of the metal excitations which give major contributions to intraband-induced decoherence of a hole in the surface state band with initial $K=0$. Here only the hole intraband transitions into the states below the Fermi energy in the SS band for which $Q<K_F^{SS}\simeq 0.1\text{a.u.}$ are allowed.

Figure 10

(Color online) Intra- and inter-band components and total survival probability $L_{\mathbf{K},\mathrm{IS}}\left(t\right)$ for an electron promoted into the first image state band on $\mathrm{Cu}\left(111\right)$ with initial state wave vector $K=0.01\text{a.u.}$ (see legend for assignments). Asymptotic decay given by the interpolation fit of Eq. (42) and “bare” FGR exponential decay law are shown by thin dash-double dotted and dash-dotted curves, respectively.

Figure 11

(Color online) Same as in Fig. 10 for $K=0.05\text{a.u}$.

Figure 12

(Color online) Intra- and inter-band components and total survival probability $L_{\mathbf{K},\mathrm{SS}}\left(t\right)$ for a hole created in the surface state band on $\mathrm{Cu}\left(111\right)$ with initial state wave vector $K=0.01\text{a.u.}$ (see legend for assignments). Asymptotic decay given by the interpolation fit of Eq. (42) and “bare” FGR exponential decay law are shown by thin dash-double dotted and dash-dotted curves, respectively.

Figure 13

(Color online) Same as in Fig. 12 for $K=0.05\text{a.u}$.

Figure 14

(Color online) (a) Feynman diagram for the second order self-energy of the $n$th band electron in GWA. Dotted lines denote Coulomb interactions (A4) and the bubble represents the response function (3). (b) Equivalent diagram in the electron-boson interaction formulation. Filled dots denote matrix elements $V_{\mathbf{Q}}$ and the wiggly line the boson propagator $D_{n,n^{\prime };n^{\prime },n}(\mathbf{Q},{\omega }^{\prime })$ given by Eqs. (A5, A9), respectively.