Investigation of valence band reconstruction methods for attosecond streaking data from surfaces

P. Drmota; D. Greening; J. P. Marangos; J. W. G. Tisch

doi:10.1364/OE.27.009394

1. Introduction

The attosecond streaking technique [1–3] evolved from an attosecond laser pulse characterisation method [4] into a powerful tool for the study of the photoionization of matter providing direct insight into ultrafast electron dynamics [5–9]. Cavalieri et al. [7] extended the scope of attosecond streaking from gases to solid surfaces. Subsequent experiments [8–11] specifically addressed photoemission delays and microscopic screening of the streaking field inside the solid. Current algorithms that are widely employed for the analysis of streaking data from both gases and solids are based on the theory of atomic gas-phase streaking [12, 13]. An important open question is whether these reconstruction methods based on atomic systems are valid when applied to attosecond streaking at surfaces. The transition from atoms to solids entails the inclusion of the electronic band structure and the spatial inhomogeneity of the near-infrared (NIR) field [14]. Moreover, inelastic scattering, plasmon excitations and space charge build-up can influence the electron signal [7]. Analyses of experimental data generally avoided a full theoretical description and reverted to modifications of established methods known from atomic streaking. These methods include the centre-of-energy (COE) analysis [15–17], generalized projection algorithms such as complete reconstruction of attosecond bursts (CRAB) [17, 18], and a fit of Gaussian wavepackets within the wavepacket approximation of atomic streaking theory [7–9], which we denote GWP in the following. We identify systematic errors that arise in the treatment of simulated surface streaking data with atomic streaking theory-based algorithms and relate them to the spatial NIR inhomogeneity that necessitates for surface streaking a more general mathematical structure than atomic streaking. In particular, we investigate the effect of reconstructing valence band streaking simulations in the jellium model with CRAB, GWP and the COE analysis on the reconstruction of photoelectron wavepackets and the NIR field.

2. Theoretical approach

In the single-active-electron approximation, the probability amplitude for a direct transition (see Fig. 1(a)) from initial state Ψ_i to final state Ψ_f due to an extreme ultraviolet (XUV) pulse with vector potential A_X is expressed in the Coulomb gauge and in atomic units (a.u.) by

T_{f \leftarrow i} = \int_{- \infty}^{\infty} 〈 Ψ_{f} (t) | A_{X} (t) \nabla | Ψ_{i} (t) 〉 d t,

where ∇ is the nabla operator. As a function of final-state momentum k_f, T_f←i(k_f) is equivalent to the spectrum b_k of a streaked photoelectron wavepacket ∭b_k exp[ik · r − i|k|t] d³k at position r. An initial valence band state with energy ∊_i is modelled as a Bloch wave with wavevector k_i‖ parallel to the surface and stationary wavefunction ψ_i(z) with wavevector

k_{i ⊥} = - \hat{z} i \frac{\partial}{\partial z} ψ_{i}

along the surface normal in the z direction [19–21]:

Ψ_{i} (r, t) \propto ψ_{i} (z) exp [i k_{i ‖} \cdot r_{‖} - i ∊_{i} t] .

The final state with energy ∊_f is heuristically described by a wavefunction decaying inside the solid with the inelastic mean-free path (IMFP) Λ and accumulating a Volkov phase Φ_f over time:

Ψ_{f} (r, t) \propto ψ_{f} (z) h_{Λ} (z) exp [i k_{f ‖} \cdot r_{‖} - i ∊_{f} t - i Φ_{f} (r, t)] .

For the damped envelope we use h_Λ(z) = Θ(z) + Θ(−z) exp [z/(2 cos(θ)Λ)] [19–23], with angle θ from the surface normal and unit-step function Θ(z). The generalization of the Volkov phase to inhomogeneous streaking laser fields with vector potential A_L [19]

Φ_{f} (r, t) = \int_{τ}^{t} (A_{L} (r, t^{'}) \cdot k_{f} (r) + A_{L} {(r, t^{'})}^{2} / 2) d t^{'},

with the ionization time τ, is approximate and only valid for step-wise constant potentials if the Jacobian

(\frac{\partial A_{L}}{\partial x}, \frac{\partial A_{L}}{\partial y}, \frac{\partial A_{L}}{\partial z})

is negligible. For photoelectrons ejected by XUV radiation, the influence of higher orders in the semi-classical Wentze-Kramers-Brillouin approximation [24] is small, justifying Eq. (4) despite variations of the NIR field over the extent of a photoelectron wavepacket.

Fig. 1 (a) Schematic diagram of photoionization from the Au(111) surface [25]. (b) The dashdotted and dashed lines in the upper panel show the amplitude and phase of the physical vector potential as a function of the distance from the Au(111) image plane at z_im. In the lower panel, the jellium model potential is shown alongside with circles indicating atomic surface layers of Au(111) and the dashed dampening envelope h_Λ of the final state Ψ_f.

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To better understand the impact of using atomic streaking-related reconstruction methods for surface streaking, we find the relevant approximations that transform the surface description of streaking into its atomic counterpart [12,13]. In the length gauge and using the dipole approximation for the XUV electric field E_X = −∂A_X/∂t only, the transition probability amplitude Eq. (1) reads $- i \int_{- \infty}^{\infty} 〈 Ψ_{f}^{LG} (t) | r \cdot E_{X} (t) | Ψ_{i}^{LG} (t) 〉 d t$ , where $Ψ_{f}^{LG} (r, t) = Ψ_{f} (r, t) exp (- i \int_{0}^{r} A_{L} (r^{'}, t) \cdot d r^{'})$ and $Ψ_{i}^{LG} (r, t) = Ψ_{i} (r, t)$ (since A_L = 0 for the initial state), yielding

T_{f \leftarrow i} \propto \int_{- \infty}^{\infty} E_{X} (t) \cdot 〈 Ψ_{f} (t) | r exp (i \int_{0}^{r} A_{L} (r^{'}, t) \cdot d r^{'}) | Ψ_{i} (t) 〉 d t .

As it is impossible to separate the Volkov phase inside the final state Eq. (3) from the matrix element in Eq. (5) because A_L and the wavevector k_f⊥ depend on the position r, the model has to be restricted to spatially homogeneous streaking fields and free-electron-like final states. Assuming grazing incident light fields with polarization perpendicular to the surface for simplicity, we identify the dipole moment

d_{f \leftarrow i} (t) \propto \int_{- \infty}^{\infty} ψ_{f}^{*} (z) h_{Λ} (z) z exp [i z A_{L ⊥} (t)] ψ_{i} (z) d z

and note that Eq. (5) obtains the mathematical structure of atomic gas-phase streaking [12,13], which enables an accurate frequency resolved optical gating [26]-type reconstruction of the spectrogram. The approximate correspondence outlined above has been exploited to analyse data from experimental surface streaking studies [7–9,11,17].

3. Numerical methods

We model the surface by a potential step of height V₀ (see lower panel in Fig. 1(b)), which is popularly known as the jellium model from the field of ion-surface collisions [27–29]. More recently, it was successfully applied to streaking on Al(111) [30] and Mg(0001) [22] surfaces, as well as to RABBITT (reconstruction of attosecond beating by interference of two-photon transitions) on the Ag(111) and Au(111) surfaces [21]. Siek et al. [16] verified numerically that continuum states of cleaved WSe₂ can be adequately represented within the jellium approximation. We model the initial and final states in accordance with Baggesen and Madsen [31]. The incident NIR field is parametrized by an 800 nm, 10¹² W/cm², 5 fs full width at half maximum (FWHM) Gaussian pulse with π/4 carrier-envelope phase and is described at the surface by gradually connecting the sum of incident and Fresnel-reflected wave to the Fresnel-transmitted wave with the matching function μ(z) = (1 + tanh[1 − z/(2δ)]) /2 (see upper panel in Fig. 1(b)) in the style of [14], where δ is the skin depth. We use the relative permittivity −20.278 + 2.071i [32] of Au at 800 nm for 87° incidence relative to [111]. Grazing incidence, polarization of the light fields perpendicular to the surface and electron detection normal to the surface are assumed. For weak and chirped Gaussian XUV pulses much shorter than the NIR period, Φ_f can be expanded around τ to quadratic order [33] and terms of order $A_{L}^{2}$ can be neglected. Then, the transition probability amplitude Eq. (1) evaluates to

T_{f \leftarrow i} \propto A_{0 X} \int_{- \infty}^{\infty} ψ_{f}^{*} (z) h_{Λ} (z) g (z) \frac{d ψ_{i} (z)}{d z} d z,

g (z) = exp (- \frac{Ω {(z)}^{2}}{4 a (z)}) / \sqrt{a (z)},

where

a (z) = T_{X}^{- 2} 2 ln 2 + i [β_{X} + k_{f} (z) E_{L} (z, τ) / 2]

and Ω(z) = ∊_f − ∊_i − ∊_X + k_f(z)A_L(z, τ) with streaking electric field E_L = −∂A_L/∂t, XUV amplitude A_0X, energy ∊_X, FWHM T_X and chirp β_X. Away from the surface, a streaked electron wavepacket (SEWP) is observed as a superposition of plane waves

χ (t) = \int_{- \infty}^{\infty} T_{f \leftarrow i} (∊) exp (i ∊ t) / N d ∊

with normalisation

N = {[\int_{- \infty}^{\infty} {| T_{f \leftarrow i} (∊) |}^{2} d ∊]}^{1 / 2}

. The group delay is defined as

\frac{d}{d ∊} arg [T_{f \leftarrow i} (∊)]

[5,34]. From Eq. (6) and by assuming a spatially homogeneous streaking field, we arrive at T_f←i ∝ exp[−Ω²/(4a)], which enables us to convert the deviation Δ∊(τ) of the COE from its unstreaked value ∊₀ to

A_{L} (τ) = - {[2 (∊_{0} + V_{0})]}^{- 1 / 2} Δ ∊ (τ) .

Equation (8) is equivalent to the classical description of streaking [13] up to a small correction due to the different background potential inside and outside the solid. For use with the following three analysis methods (COE, GWP and CRAB), a set of spectrograms from a single valence band state was computed for different δ and β_X by numerically integrating Eq. (6). We choose ∊_i = −5 eV, ∊_X = 115 eV, T_X = 400 as, V₀ = 9.88 eV [21] and Λ = 0.391 nm [35].

3.1. COE

The COE analysis serves to retrieve the NIR waveform and to resolve phase differences and temporal shifts between traces from different initial states within a streaking spectrogram. We adjust the unstreaked central energy ∊₀, amplitude A_0L, durationT_L, phase φ_L, temporal shift Δt and frequency ω_L of the NIR field parametrization A_L(t+Δt) = A_0L exp [−2 ln 2(t/T_L)²] cos(φ_L+ω_Lt) to fit Eq. (8) to the trace of first moments of a spectrogram.

3.2. GWP

The strong-field approximation for atomic streaking spectrograms [12,13] resembles the absolute square of the Fourier transform of the wavepacket (WP) η(t) with phase −∊_it + Φ_f(t) [36,37]. A single-level spectrogram then takes the form of the wavepacket approximation (WPA)

S_{i} (∊_{f}, τ) \approx {| \int_{- \infty}^{\infty} η (t - τ) exp [i (∊_{f} - ∊_{i}) t + i Φ_{f} (t)] d t |}^{2} .

For multi-level emission, a spectrogram is given by the incoherent sum over N initial states,

\sum_{ℓ = 1}^{N} S_{ℓ} (∊_{f}, τ)

. We follow the fitting procedure of Cavalieri et al. [7] and Neppl et al. [8,9] and fit the amplitude f_WP, FWHM T_WP, chirp β_WP and temporal shift Δt of the WP η(t + Δt) = f_WP exp [−4 ln 2(t/T_WP)² − iβ_WPt²] and the amplitude and phase of the NIR field to the input spectrogram.

3.3. CRAB

CRAB [38] maps the phase retrieval problem encountered in atomic streaking spectroscopy onto the well-known frequency resolved optical gating (FROG) [26] technique. FROG-CRAB retrieves the gate G(t) = exp[iΦ_f(t)] and the pulse P(t) = η(t) [39], which is the unstreaked WP in Eq. (9). We use the application Attogram [40] and distinguish unconstrained (blind) CRAB and CRAB with fixed NIR field, which we denote CRAB^*. To extract temporal WP shifts from the CRAB reconstruction, we numerically evaluate the spectrally averaged group delay [5].

4. Results

4.1. Wavepacket analysis

The fitting procedure GWP restricts η(t) in Eq. (9) to have a Gaussian envelope. Moreover, the way spectrograms are generated in GWP using the WPA Eq. (9), the magnitude of a SEWP χ(t) ≈ η(t) exp[iΦ_f(t)] remains undistorted (see filled datapoints in Fig. 2) and the SEWP phase arg[χ(t)] is approximated by the Volkov phase Φ_f(t + τ) (see empty datapoints in Fig. 2). Here we construct explicitly SEWPs in the jellium model (see lines in Fig. 2) to validate these two GWP properties. If the extent of the SEWP is comparable to the scale of spatial NIR variations or if T_X ≪ T_L is violated, the SEWP structure can be affected. We observe this effect only for unrealistically small skin depths (δ < 0.05 nm) for ionization at extrema of the NIR amplitude, where the SEWP magnitude becomes asymmetric (compare Figs. 2(a) and 2(b)).

Fig. 2 SEWPs emitted at the minimum (min) and the maximum (max) of the NIR field for β_X = 0 fs⁻², (a) δ = 0.03 nm and (b) δ = 0.2 nm. The Gaussian magnitude and Volkov phase assumed by GWP (datapoints) are compared to jellium model calculations (lines).

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The results of the CRAB^* analysis using the first-moment COE of a particular spectrogram and using the sum of exact incident input and reflected NIR waves are equivalent. Both GWP and CRAB slightly overestimate the chirp (see Fig. 3(a)), which can be attributed to the effect of the streaking electric field, which appears in conjunction with β_X in Eq. (7). Regarding WP width (see Fig. 3(b)), we observe broadening of the WPs of approximately 165 as (see “Jellium WP” lines in Fig. 3(b)) compared to the incident XUV pulse duration. The “Jellium WP” reference lines in Fig. 3 are determined in post-processing from unstreaked jellium WPs, as this corresponds to the quantity η(t) retrieved by both CRAB [39] and GWP. GWP is highly accurate and insensitive to δ for spectrograms with β_X = 0 (see filled circles in Fig. 3(b)). CRAB and CRAB^* are skin depth independent as well, but reconstruct incorrect WP widths (see filled diamonds and squares in Fig. 3(b)). For β_X ≠ 0, all three methods are sensitive to δ (see empty datapoints in Fig. 3(b)) and both CRAB analyses converge towards the correct WP width for near-homogeneous streaking fields (δ > 0.5 nm). Due to the rudimentary microscopic description of the surface, the limit of small δ does not trivially correspond to a perfectly screened metal. CRAB^* is less susceptible to β_X than CRAB and GWP (see distances between empty and filled datapoints of same shape in Fig. 3(b)), suggesting that the degrees of freedom allowing CRAB to construct arbitrary WPs and/or NIR fields, which are known to cause non-uniqueness problems [41], are a source of reconstruction error. Results of temporal shifts shown in Fig. 3(c) are within reasonable agreement with the spectrally averaged group delay of the unstreaked jellium WP reference.

Fig. 3 Atomic streaking-reconstruction results from single valence level jellium spectrograms with varying δ. Panel (a) shows retrieved WP chirp, (b) the corresponding WP FWHMs. The reference lines “Jellium WP” are determined directly from jellium simulations. Panel (c) shows temporal shifts of reconstructed electron WPs compared to the jellium reference. Panels (d) and (e): Retrieved values of the NIR field from COE, GWP and CRAB analyses are close to the sum of incident and Fresnel-reflected NIR waves (see dashdotted lines) and significantly different from the Fresnel-transmitted wave (see dashdotdotted lines). The inset in (e) contains a magnified view on the area marked by the dashed rectangle.

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4.2. NIR analysis

The NIR field reconstruction results of CRAB, GWP and COE shown in Figs. 3(d) and 3(e) suggest that the NIR field amplitude and phase are dominated by the combined incident and Fresnel-reflected fields, which holds true for significantly different IMFPs and δ. In contrast to the situation of WP parameter retrievals, retrieved NIR field amplitudes are dependent on δ in the absence of XUV chirp (see filled datapoints in Fig. 3(e)). This can be understood from the behaviour of the matching function μ(z), which dominates Im[a(z)] in Eq. (7) if β_X is small.

4.3. Band structure

Gagnon and Yakovlev [42] showed generally that a central energy uncertainty causes an error in retrieved WP widths. We construct multi-level spectrograms $\sum_{ℓ = 1}^{N} S_{ℓ} (∊_{f}, τ)$ with N ∈ {2, 3, 4} energetically overlapping levels using the WPA Eq. (9) to exclude surface effects in subsequent reconstructions. We extract the FWHM of WPs from their CRAB reconstruction and additionally fit them with multi-level GWP. Using T_WP = 0.6 fs, ∊_0,ℓ distributed around 75 eV with 2 eV separation between levels and the amplitude f_WP,ℓ linearly decreasing from f_WP,N = 1 (highest energy) to 0 < f_WP,1 ≤ 1 (lowest energy), bandwidths of simulated spectra range from 4.3 eV to 8.2 eV (see Fig. 4(a) for examples at f_WP,1 = 0.5).

Fig. 4 (a) We create multi-level spectrograms by incoherently averaging over an ensemble of electrons with varying central energies (see dotted lines adding up to a solid curve) and common single-electron FWHM T_WP. (b) CRAB retrievals of band spectrograms underestimate T_WP = 0.6 fs (see dashed line) in accordance with the Fourier transform limit T_F of the band spectrum. Multi-level GWP with known level structure converges exactly.

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From the properties of the Fourier transform ℱ in Eq. (9), we expect retrieved WP widths to be greater than the Fourier transform limit, corresponding to the FWHM T_F of b(t) = |ℱ⁻¹B(∊)|², with the unstreaked and unchirped band spectrum $B (∊) = \sum_{ℓ = 1}^{N} {S_{ℓ} (∊, - \infty) |}_{β_{X} = 0}$ . Figure 4(b) demonstrates that CRAB retrieves widths that are well below the single-WP duration T_WP (see dashed line in Fig. 4(b)), but close to and within the Fourier transform limit T_F (see white area in Fig. 4(b)) of the total unstreaked and unchirped band spectrum. Multi-level GWP trivially recovers the WP duration exactly in all cases (see crosses in Fig. 4(b)).

5. Discussion and conclusion

In this article, the applicability of selected atomic gas-phase streaking retrievals to surface streaking data is investigated. It is shown theoretically that two severe approximations – spatially homogeneous streaking field and free-electron-like final state – have to be made in order to transform the quantum description of surface streaking into its atomic counterpart, on which popular retrieval methods are based. Firstly, we find that it is necessary to account for the band structure of a solid by incoherently averaging over an ensemble of electrons [42] using a multi-level retrieval. Otherwise, retrieved WP durations underestimate the true single-electron duration, in accordance with the Fourier transform limit of the band spectrum. Secondly, the streaking field amplitude and phase reconstructed from COE, GWP and CRAB analyses of simulated surface streaking spectrograms are dominated by the combined incident and Fresnel-reflected waves, which suggests that the main streaking effect takes place within a distance from the surface much smaller than the IMFP. Thirdly, streaked electron WPs extracted from jellium model calculations are insignificantly deformed by the streaking field for realistic δ, which supports the assumption of Gaussian WPs made in GWP. Moreover, temporal shifts of WPs are retrieved with attosecond precision using the COE, GWP and CRAB approaches. Lastly, WP chirp is consistently overestimated by retrievals as a consequence of the streaking electric field at the surface mimicking the effect of positive XUV chirp and retrieved WP widths exhibit errors depending on β_X and δ. Based on our error analysis, performing CRAB^* with the NIR field fixed to the COE of experimental data could improve the reconstruction result. The surface description employed in our simulations, the jellium model, is potentially not accurate for complex structures, such as the composite materials WSe₂ [16] and WO₃ [17]. However, we point out that more complicated wavefunction models cannot change the appearance of retrieval errors due to the mathematical structure imposed by the spatial inhomogeneity of the streaking field, and are likely to compound those errors. Therefore, the above results do not yet provide a justification for applying atomic streaking theory-based methods to surface streaking data. The present analysis does not exclude that relative quantities, such as temporal shifts derived from phase differences or relative WP broadening, can be accurately retrieved from surface streaking measurements with atomic streaking theory-based algorithms modified to account for the band structure. Rather, by the fact that wavepacket retrievals from GWP and both CRAB and CRAB^* become sensitive to the NIR skin depth with the same scaling, it emphasises the significance of distinguishing between surface streaking and atomic streaking, whose differences go beyond the band structure and the algorithmic implementation of the reconstruction method. Specialized surface streaking reconstruction algorithms supporting the spatial properties of the NIR field at the surface have to be developed and would aid to resolve a long-lasting controversy over the importance of NIR screening in the strong-field regime [8,9,14,22,31], in addition to eliminating systematic error sources, such as those reported in this article.

Funding

EPSRC/Defence Science and Technology Laboratory (DSTL) Multidisciplinary University Research Initiative (MURI) (EP/N018680/1).

Acknowledgments

We gratefully acknowledge helpful discussions with Professor Uwe Thumm and Dr. Marcelo J. Ambrosio.

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Investigation of valence band reconstruction methods for attosecond streaking data from surfaces

Abstract

1. Introduction

2. Theoretical approach

3. Numerical methods

3.1. COE

3.2. GWP

3.3. CRAB

4. Results

4.1. Wavepacket analysis

4.2. NIR analysis

4.3. Band structure

5. Discussion and conclusion

Funding

Acknowledgments

References

Cited By

Figures (4)

Equations (9)

Optics Express