Nonadiabatic dynamics of electron scattering from adsorbates in surface bands

We present a comparative study of nonadiabatic dynamics of electron scattering in quasi-two-dimensional surface band which is induced by the long-range component of the interactions with a random array of adsorbates. Using three complementary model descriptions of intraband spatiotemporal propagation of quasiparticles that go beyond the single-adsorbate scattering approach we are able to identify distinct subsequent regimes of evolution of an electron following its promotion into an unoccupied band state: (i) early quadratic or ballistic decay of the initial-state survival probability within the Heisenberg uncertainty window, (ii) preasymptotic exponential decay governed by the self-consistent Fermi golden rule scattering rate, and (iii) asymptotic decay described by a combined inverse power-law and logarithmic behavior. The developed models are applied to discuss the dynamics of intraband adsorbate-induced scattering of hot electrons excited into the $n=1$ image-potential band on Cu(100) surface during the first stage of a two-photon photoemission process. Estimates of crossovers between the distinct evolution regimes enable assessments of the lifespan of a standard quasiparticle behavior and thereby of the range of applicability of the widely used Fermi golden rule and optical Bloch equations approach for description of adsorbate-induced quasiparticle decay and dephasing in ultrafast experiments.

Figure 1

(Color online) Lowest-order self-energy diagrams which are linear in the concentration of adsorbates. The full lines symbolize electron propagators and dashed lines the adsorbate scattering potential centered at the same site (denoted by small vertex quadrangle).

Figure 2

(Color online) Excited IS-electron scattering or decay rate calculated from Fermi’s golden rule expression (16) for the long-range dipole electron-adsorbate interaction taken from Ref. 17 and $\Theta =0.7\%$. Energy scale in a.u. (left) and meV (right).

Figure 3

(Color online) Plots of the imaginary part ${\Gamma }_{\mathbf{K}}\left(\omega \right)$ (full thick curve) and real part ${\Lambda }_{\mathbf{K}}\left(\omega \right)$ (dashed thick curve) of the self-energy ${\Sigma }_{\mathbf{K}}\left(\omega \right)$ for initial wave vector $K=0$. All energies are measured from the IS-band bottom $\omega =0$. The intersection of ${\Lambda }_{\mathbf{K}}\left(\omega \right)$ with the thin line $\omega -E_{\mathbf{K}}$ below the band bottom determines the energy of bound state in spectrum (19).

Figure 4

(Color online) Same as in Fig. 3 but for initial wave vector $K=0.0316\text{a}\text{.u}\text{.}$ corresponding to electron energy $E_{\mathbf{K}}=13.6\text{meV}$. Note that for this value of initial electron energy there appears three intersections of ${\Lambda }_{\mathbf{K}}\left(\omega \right)$ with the line $\omega -E_{\mathbf{K}}$. The energy of leftmost intersection slightly below the band bottom determines the energy of bound state whereas the ones above the bottom give rise to complex poles (resonances) that give rise to maxima in the band part of spectrum (19).

Figure 5

(Color online) Band or continuous part of spectrum (19) for initial electron momentum $K=0$ or $E_{\mathbf{K}}=0$. The contribution of this part to the total spectral weight is 23.4% and the remainder is located in the bound state located at ${\omega }_{\mathbf{K}}^B=-0.000216\text{a}\text{.u}\text{.}=-5.87\text{meV}$ below the band bottom. The FGR pole is not well defined for $E_{\mathbf{K}}=0$ and the spectrum is dominated by bound state (18) and threshold behavior (20).

Figure 6

(Color online) Illustration of the two-pole structure of the band part of spectrum (19) for initial electron momentum $K=0.0316\text{a}\text{.u}\text{.}$ corresponding to the energy $E_{\mathbf{K}}=13.6\text{meV}$. The contribution of this part to the total spectral weight is 96.3% and the remainder is associated with the pole located at the bound-state energy ${\omega }_{\mathbf{K}}^B=-1.468\times {10}^{-6}\text{a}\text{.u}\text{.}=-0.0399\text{meV}$ below the band bottom. The threshold resonance slightly above the band bottom has a small weight $\left(\simeq 2.3\%\right)$ relative to the FGR-pole-derived peak at $\omega \sim 0.0003\text{a}\text{.u}\text{.}$ $\left(\sim 8\text{meV}\right)$.

Figure 7

(Color online) Same as in Fig. 6 but for initial electron momentum $K=0.0857\text{a}\text{.u}\text{.}$ corresponding to the energy $E_{\mathbf{K}}=100\text{meV}$. The spectrum is dominated by the FGR-pole-derived Lorentzian-shaped maximum whose position is to a good approximation determined by the zero of expression $\left[\omega -E_{\mathbf{K}}-{\Lambda }_{\mathbf{K}}\left(\omega \right)\right]$ in the postlogarithmic region. The bound state slightly below and the threshold resonance slightly above the band bottom (see inset) carry negligible spectral weight.

Figure 8

(Color online) IS-electron survival probability $L_{\mathbf{K}}\left(t\right)$ as a function of time ($1\text{a}\text{.u}\text{.}=2.4189\times {10}^{-2}\text{fs}$ or $1\text{fs}=41.34\text{a}\text{.u}\text{.}$) for initial quasiparticle momentum $K=0$. The dashed curve is the bare FGR decay described by expression (22). Note logarithmic scale on the vertical axis. Notable in the depicted behavior of $L_{\mathbf{K}}\left(t\right)$ is the absence of an intermediate evolution interval during which the survival probability could be well approximated by the FGR decay. Here the early Zeno behavior (23) is directly superseded by the nonexponential dephasing (28) at around $t=8000\text{a}\text{.u}\text{.}$ $\left(\sim 200\text{fs}\right)$.

Figure 9

(Color online) Same as in Fig. 8 but for initial quasiparticle momentum $K=0.0857\text{a}\text{.u}\text{.}$ corresponding to the energy $E_{\mathbf{K}}=100\text{meV}$. Here a strong interference between exponential and postexponential dephasings that cause survival collapse extend over the interval $\sim 70000–130000\text{a}\text{.u}\text{.}$ $\left(\sim 1750–3250\text{fs}\right)$. The inset shows the early Zeno behavior (23).

Figure 10

(Color online) Same as in Fig. 9 but for initial quasiparticle momentum $K=0.172\text{a}\text{.u}\text{.}$ corresponding to the energy $E_{\mathbf{K}}=400\text{meV}$. The survival collapse is not reached on the time scale of the plot $\sim 3400\text{fs}$. Inset: Blow up of the early Zeno decay (23) shown on the time scale of 400 a.u. $\left(\sim 10\text{fs}\right)$.

Figure 11

(Color online) Evolution of the phase $\varphi \left(t\right)$ (modulo $2\pi$) of the survival amplitude [see Eqs. (6, 3)] of IS electron for initial momentum $K=0.0857\text{a}\text{.u}\text{.}$ corresponding to the energy $E_{\mathbf{K}}=100\text{meV}$ shown in the range $90000<t<130000\text{a}\text{.u}\text{.}$ $\left(\sim 2200<t<3000\text{fs}\right)$. In this interval the corresponding survival probability shown in Fig. 9 exhibits a survival collapse and crossover from the FGR to the asymptotic decay. The initial quasiparticle phase grows linearly with time as $-i\varphi \left(t\right)=-i{E}_{\mathbf{K}}t$ but in the crossover interval starts oscillating around a finite asymptotic value. In this interval the initial quasiparticle identity is lost.

Figure 12

(Color online) Single realization of a potential $V\left(x,y\right)$ describing randomly distributed scattering centers depicted in an area of $300\times 300\text{Bohr}\text{radius}$ squared. The concentration is $c=0.0003\text{a}\text{.u}\text{.}$ The white solid lines depict the functions $\text{Re}\text{exp}\left(iKx\right)$, where $K=0.0836$ and $K=0.1672\text{a}\text{.u}\text{.}$ (see text).

Figure 13

(Color online) Survival probability $L_{\mathbf{K}}\left(t\right)$ for $K=0$ at low density of the scattering centers, $c=0.0003\text{a}\text{.u}\text{.}$ The three curves correspond to three different configurations of the scattering centers. The dot-dashed line is the fitted FGR decay $\text{exp}\left(-2\Gamma t\right)$ with $2\Gamma =0.52\times {10}^{-4}\text{a}\text{.u}\text{.}$ (see text and Fig. 2). The inset shows the computed early decay (symbols) and fits (full line) to quadratic Zeno behavior (23) which yields ${\tau }_Z=988\text{a}\text{.u}\text{.}=23.9\text{fs}$. The dot-dashed line is the same as in the main plot.

Figure 14

(Color online) Same as in Fig. 13 but for $K=0.0836\text{a}\text{.u}\text{.}$ The dot-dashed line is the fitted FGR decay $\text{exp}\left(-2\Gamma t\right)$ with $2\Gamma =1.31\times {10}^{-4}\text{a}\text{.u}\text{.}$ (see text and Fig. 2).

Figure 15

(Color online) Same as in Fig. 14 but for $K=0.1672$. The dot-dashed line is the fitted FGR decay $\text{exp}\left(-2\Gamma t\right)$ with $2\Gamma =0.86\times {10}^{-4}\text{a}\text{.u}\text{.}$ (see text and Fig. 2).

Figure 16

(Color online) Numerical values of the Zeno time as a function of concentration of scatterers. Data are extracted from simulations of the quasiparticle survival probability (31) using expression (23). The full curve is a power-law fit to numerical values which yields ${\tau }_Z=0.339c^{-0.527}\text{fs}$, in very good agreement with the analytical result ${\tau }_Z\propto c^{-1/2}$ derived in Sec. 2. The inverted triangle is the value of ${\tau }_Z$ obtained in Sec. 2 for the concentration $c=0.0003$ $\left(\Theta =0.7\%\right)$.

Figure 17

(Color online) (a) The momentum spectrum of the wave function at time $t=0$ (small open circle), and at time $t=80\text{fs}$. The initial wave function corresponds to the wave vector magnitude $K=0$. (b) Same as in (a) but for the wave function with initial wave vector magnitude $K=0.1672\text{a}\text{.u}\text{.}$ (small open circle) pointing in the $x$ direction. The radius of the big $K^{\prime }$ circle is equal to initial $K$ (see text for details).

Figure 18

(Color online) Survival probabilities of three eigenstates diagonalizing the adsorbate-free lattice Hamiltonian for initial energies $E$ of 0, 100, and 400 meV (shaded lines). The number of lattice sites is $4000000$, and the number of adsorbates is $28000$ $\left(\Theta =0.7\%\right)$. The dashed lines are fits to form (22). The inset shows the early Zeno decays for initial $E_K=0\text{meV}$ (triangles), $E_K=100\text{meV}$ (circles), and $E_K=400\text{meV}$ (squares), which all follow the universal curve (23) independent of $E_K$.

Figure 19

(Color online) Survival probabilities of states diagonalizing the adsorbate-free lattice Hamiltonian for $E=400\text{meV}$ and three different adsorbate coverages ($\Theta =0.7\%$, $\Theta =3.5\%$, and $\Theta =7\%$) as denoted in the figure. The number of lattice sites is $4000000$. The inset shows the decay rates in the exponential (FGR) regime (dashed lines in the body of the figure) as functions of the coverage of defects for initial energies of 100 and 400 meV.