Time-resolved photoabsorption in finite systems : A first-principles NEGF approach

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Time-resolved photoabsorption in finite systems : A first-principles NEGF approach Perfetto, E.; Uimonen, Anna-Maija; van Leeuwen, Robert; Stefanucci, G. Perfetto, E., Uimonen, A.-M., van Leeuwen, R., & Stefanucci, G. (2016). Timeresolved photoabsorption in finite systems : A first-principles NEGF approach. In Progress in Non-equilibrium Green’s Functions (PNGF VI) (Article 012004). Institute of Physics Publishing Ltd.. Journal of Physics: Conference Series, 696. https://doi.org/10.1088/1742-6596/696/1/012004 2016 This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: 130.234.74.31 This content was downloaded on 01/08/2016 at 11:34 Please note that terms and conditions apply. Time-resolved photoabsorption in finite systems: A first-principles NEGF approach View the table of contents for this issue, or go to the journal homepage for more 2016 J. Phys.: Conf. Ser. 696 012004 (http://iopscience.iop.org/1742-6596/696/1/012004) Home Search Collections Journals About Contact us My IOPscience Time-resolved photoabsorption in finite systems: A first-principles NEGF approach E. Perfetto1, A.-M. Uimonen2,3, R. van Leeuwen2,4and G. Stefanucci1,4,5 1Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Rome, Italy 2Department of Physics, Nanoscience Center, FIN 40014, University of Jyväskylä, Jyväskylä, Finland 3Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom 4European Theoretical Spectroscopy Facility (ETSF) 5Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, Via E. Fermi 40, 00044 Frascati, Italy E-mail: [email protected] Abstract. We describe a first-principles NonEquilibrium Green’s Function (NEGF) approach to time-resolved photoabsortion spectroscopy in atomic and nanoscale systems. The method is used to highlight a recently discovered dynamical correlation effect in the spectrum of a Krypton gas subject to a strong ionizing pump pulse. We propose a minimal model that captures the effect, and study the performance of time-local approximations versus time-nonlocal ones. In particular we implement the time-local Hartree-Fock and Markovian second Born (2B) approximation as well as the exact adiabatic approximation within the Time-Dependent Density Functional Theory framework. For the time-nonlocal approximation we instead use the 2B one. We provide enough convincing evidence for the fact that a proper description of the spectrum of an evolving admixture of ionizing atoms requires the simultaneous occurrence of correlation and memory effects. 1. Introduction Time-resolved photoabsorption spectroscopy is a cutting edge experimental technique to probe quantum systems in nonstationary states [1–5]. The sample is driven out of equilibrium by a strong pump pulse (whose intensity can exceed 1015 W/cm2) and subsequently probed with a weak ultrashort (down to hundreds of as) pulse. By measuring the energy per unit frequency carried by the transmitted probe as a function of the delay between the pump and probe pulses one obtains the time-dependent photoabsorption spectrum. From this spectrum it is possible to read out information like, e.g., charge/exciton dynamics [6–19], dynamical screening effects [20– 23] and several other nonequilibrium properties [24–33]. For optically thin samples (atoms, molecules and more generally nanostructures) the transmitted probe pulse can be expressed in terms of the time-dependent probe-induced change of the dipole moment [34–41]. The latter is defined as the difference between the time-dependent dipole moment originating from the pump+probe fields and the time-dependent dipole moment originating from the pump only. Both quantities can be calculated from the time-dependent Progress in Non-equilibrium Green’s Functions (PNGF VI) IOP Publishing Journal of Physics: Conference Series 696 (2016) 012004 doi:10.1088/1742-6596/696/1/012004 Content from this work may be used under the terms of theCreativeCommonsAttribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 quantum average of the dipole operator over the many-electron state of the system evolving according to the Schrödinger equation with the appropriate electric field. Due to electron correlations these averages are difficult to compute using the Configuration Interaction (CI) method since the number of configurations scales exponentially with the number of electrons. CI is often implemented in the so called Single Active Electron (SAE) approximation [42, 43] which, by construction, neglects double or higher excitations. An alternative approach, statistical in nature, to tackle the problem is provided by Time-Dependent Density Functional Theory (TDDFT) [44]. In TDDFT the interacting many-body system is mapped onto a fictitious noninteracting system having the same density, and hence the same dipole moment, as the interacting one [45, 46]. Therefore, TDDFT is ideal to study much larger systems than those accessible by CI. However, most TDDFT calculations are performed using the Adiabatic Local Density Approximation (ALDA) for the potential of the fictitious system but, unfortunately, the ALDA potential suffers from the same drawbacks of the SAE approximation in CI [47, 48]. Other critical drawbacks of the ALDA potential in the context of photoabsorption are that (i) the renormalization of the molecular levels due to the nearby presence of a highly polarizable structure or medium is poorly described [49–54] and (ii) single-particle excitation with a longrange charge-transfer character are missed [55–57]. In Ref. [41] we investigated the possibility of using a statistical approach based on NonEquilibrium Green’s Functions (NEGF) [58–62]. The NEGF approach is computationally more expensive than TDDFT but still more convenient than CI. From the theoretical point of view the advantage of NEGF over TDDFT is that the relevant electron-electron scattering processes can be incorporated through a proper selection of Feynman diagrams for the self-energy Σfrom which the Green’s function G=G0+G0ΣGcan be self-consistently calculated, G0being the noninteracting Green’s function. The aformentioned drawbacks of the ALDA potential in TDDFT are absent in NEGF already with rather simple self-energies. It is also worth observing that the self-energy Σ[G]is a functional of the Green’s function and hence the NEGF approach is a nonperturbative approach as it amounts to sum a subset of Feynman diagrams to infinite order in the interaction strength. Actually, the most convenient method to calculate the nonequilibrium Gis not through the Dyson equation G=G0+G0ΣGbut through the so called Kadanoff-Baym Equations (KBE) [58– 62]. The KBE are nonlinear integro-differential equations for the lesser and greater components of G,G<and G>respectively. For a given self-energy the computational cost for the solution of the KBE scales like N3 t[63–71], where Ntis the number of time steps. This poses severe limitations to the maximum propagation time, essentially preventing the use of NEGF for an acceptable frequency resolution of the time-resolved photoabsorption spectra. The scaling of the computational cost can be reduced from N3 tto N2 tusing the Generalized Kadanoff-Baym Ansatz (GKBA) [72]. The GKBA is an ansatz for G≶(t, t′)in terms of the one-particle density matrix ρ(t) = −iG<(t, t), and allows for collapsing the KBE into a single nonlinear integrodifferential equation for the one-time quantity ρ(t). The ansatz is exact for the Hartree-Fock (HF) self-energy and an approximation for correlated self-energies (beyond HF). Nevertheless, this approximation turns out to be extremely accurate in systems with well-defined quasiparticles (like atoms and molecules) and more generally whenever the collision time is much smaller than the quasiparticle lifetime [72]. We wish to emphasize here that within the GKBA the NEGF formalism is converted into a theory for the one-particle density matrix which could provide useful hints for the development of TDDFT functionals beyond the ALDA. In this work we briefly describe the NEGF+GKBA approach to time-resolved photoabsorption spectroscopy and apply it to model Hamiltonians with few bound states and a continuum of states. Our purpose is to shed light on a recently discovered dynamical correlation effect in a gas of Krypton atoms [28, 41]. In the experiment the gas is perturbed by a strong near infrared pump pulse responsible for the ionization of some of the atoms. During the action of the pump the Progress in Non-equilibrium Green’s Functions (PNGF VI) IOP Publishing Journal of Physics: Conference Series 696 (2016) 012004 doi:10.1088/1742-6596/696/1/012004 2 neutral gas evolves into an admixture of Kr atoms and Krn+ions with n= 1,2,3, . . .. By probing the admixture with a weak, extreme ultraviolet attosecond pulse having a delay τwith respect to the pump pulse one can monitor the evolution of the photoabsorption spectrum as a function of τ. The development of absorption peaks in different spectral regions are fingerprints of the presence of Kr ions with different charge. Our calculations have shown that the HF self-energy as well as other time-local self-energies, e.g., the Markovian second-Born (M2B), capture at most the absorption of Kr1+ [41]. Thus, time-local approximations to the self-energy do not describe the absorption of multiply ionized Kr atoms. Instead the full (nonlocal in time) second-Born (2B) approximation correctly captures the absorption of Kr1+ and Kr2+ as well as the retardation in the onset of the absorption peak of Kr2+ with respect to the onset of the absorption peak of Kr1+. In this work we introduce a simple model Hamiltonian to clarify the role of timenonlocal correlations in the formation of an admixture of multiply ionized atoms. The simplicity of the model also allows for obtaining an analytic expression of the exact exchange-correlation potential of static DFT, and hence to perform exact adiabatic TDDFT simulations. We show that similarly to the HF and the M2B approximations also the exact adiabatic approximation of TDDFT fails in reproducing the absorption peaks of multiply ionized atoms. This provides a conclusive evidence for the fact that a proper description of the time-resolved spectrum of an evolving admixture of ions requires the simultaneous inclusion of correlations and memory effects. 2. Time-resolved photoabsorption spectrum We consider a finite system like an atom or a molecule driven out of equilibrium by a pump pulse. Choosing the pump frequency appropriately electrons can be expelled from the system, thus changing its absorption properties. Our goal is to study the photoabsorption spectrum of the system during the ionization process. For this purpose we assign a single-particle basis consisting of a certain number of orthonormal bound wavefunctions {ϕi(r)}and of a continuum of extended wavefunctions {ϕk(r)}. Let ˆcασ (ˆc† ασ) be the annihilation (creation) operator for an electron with spin σ=↑,↓in ϕα(r), where αcan be either a bound index or a continuum index. For ionization processes from the valence shells Auger recombination are unlikely to occur. Furthermore, as we are not interested in the energy of the photoelectrons, we ignore the Coulomb repulsion between bound and extended electrons. The equilibrium Hamiltonian then reads ˆ Heq =X k,σ ǫkˆc† kσ ˆckσ +X ij,σ hij ˆc† iσ ˆcjσ +1 2X ijmn σσ′ vijmnˆc† iσ ˆc† jσ′ˆcmσ′ˆcnσ.(1) Without any loss of generality we have chosen the set {ϕk(r)}to be the set of continuum eigenstates with energy ǫkof the single particle Hamiltonian h(r) = −∇2 2+Vn(r),Vn(r)being the nuclear potential. The one- and two-electron integrals appearing in Eq. (1) are defined according to hij ≡Rdrϕ∗ i(r)h(r)ϕj(r)and vijmn ≡Rdrdr′ϕ∗ i(r)ϕ∗ j(r′)ϕm(r′)ϕn(r)/|r−r′|. The interaction between light and matter is treated in the dipole approximation. Let E(t)be the electric field and dαβ ≡Rdrϕ∗ α(r)rϕβ(r)the matrix elements of the dipole vector. The Hamiltonian of the light-matter interaction reads ˆ HLM(t) = X αβ,σ [E(t)·dαβ] ˆc† ασ ˆcβσ (2) where the sum runs over all bound and continuum states. The time-evolution of any many-body state |Ψ(t)iis therefore governed by the Schrödinger equation id dt |Ψ(t)i=ˆ H(t)|Ψ(t)iwhere ˆ H(t) = ˆ Heq +ˆ HLM(t)(3) Progress in Non-equilibrium Green’s Functions (PNGF VI) IOP Publishing Journal of Physics: Conference Series 696 (2016) 012004 doi:10.1088/1742-6596/696/1/012004 3 is the total Hamiltonian. Let E(t)be the pump field and e(t)the probe field. The pump field is typically a fewcycle pulse with frequency centered around the transition from a valence state to a low-energy continuum state. Thus for E=Ewe can ignore the terms in the sum of Eq. (2) where both indices α, β are continuum indices. On the contrary the main effect of the probe field is to promote transitions in the bound sector. Therefore we can restrict the sum in Eq. (2) to bound indices α, β =i, j for E=e. It follows that for E=Ewe have ˆ HLM(t) = ˆ HP(t)≡X ij,σ [E(t)·dij] ˆc† iσ ˆcjσ +X ik,σ [E(t)·dik] ˆc† iσˆckσ + H.c.,(4) whereas for E=E+ewe have ˆ HLM(t) = ˆ HP&p(t)≡ˆ HP(t) + X ij,σ [e(t)·dij] ˆc† iσˆcjσ.(5) These approximations are based on physical considerations and simplify the presentation. However, they are not necessary for the theory developed in the next section. We are interested in calculating the time-resolved photoabsorption spectrum S(ω)for frequencies corresponding to transitions between bound many-particle states. Using the convention that quantities with a tilde denote the Fourier transform of the corresponding timedependent quantities, the spectrum for the frequencies of interest is given by S(ω) = −2Im hω˜ e∗(ω)·˜ dp(ω)i(6) where dpis the probe-induced change of the time-dependent average of the dipole operator ˆ d=Pij,σ dij ˆc† iσ ˆcjσ. In order to calculate dpone has to calculate the time-dependent average dP&p(t)of the dipole operator with ˆ HLM =ˆ HP&p, the time-dependent average dP(t)of the dipole operator with ˆ HLM =ˆ HPand then subtract the two dp(t)≡dP&p(t)−dP(t).(7) We observe that ˜ dp(ω)is not proportional to ˜ e(ω)since the system is out of equilibrium. The spectrum S(ω)depends on the entire function e(t)and not only on its frequency component ˜ e(ω). In particular S(ω)depends on the delay between the pump and probe pulses. 3. NEGF+GKBA approach In this section we describe the NEGF+GKBA approach to calculate dP&p. The same approach can be used for dPby setting e= 0. The average dP&p(t) = −2iPij dijG< ji(t, t)where G<(t, t)is the equal-time lesser component of the Green’s function G(z, z′)with arguments on the Keldysh contour. The latter is a matrix function with indices in both the bound and extended sector and it satisfies the equation of motion id dz −hHF(z)G(z, z′) = δ(z, z′) + Zd¯zΣ(z, ¯z)G(¯z, z′).(8) In accordance with the considerations of the previous section the HF hamiltonian in Eq. (8) reads hHF,αβ(z) =        δkk′ǫk(α, β) = (k, k′) E(z)·dik (α, β) = (i, k) E(z)·dki (α, β) = (k, i) hij + [E(z) + e(z)] ·dij +Pmn wimnjρnm(z) (α, β) = (i, j) (9) Progress in Non-equilibrium Green’s Functions (PNGF VI) IOP Publishing Journal of Physics: Conference Series 696 (2016) 012004 doi:10.1088/1742-6596/696/1/012004 4 with wimnj ≡2vimnj −vimjn and ρnm(z) = −iGnm(z, z+) = −iG< nm(t, t)independently of the branch of the contour where zlies. The correlation self-energy Σis a functional of Gand has nonvanishing matrix elements only in the bound sector. As for the calculation of dP&pwe only need Gwith bound indices we project Eq. (8) onto the bound sector and find X mid dz δim −hHF,im(z)Gmj(z, z′)−X kE(z)·dikGkj (z, z′) =δ(z, z′)δij +X mZd¯zΣim(z, ¯z)Gmj(¯z, z′).(10) Similarly we can obtain an equation for Gkj. Taking into account Eq. (9) we find id dz −ǫkGkj(z, z′)−X mE(z)·dkmGmj (z, z′) = δ(z, z′).(11) Next we define the noninteracting Green’s function gkas the solution of id dz −ǫkgk(z, z′) = δ(z, z′)(12) with Kubo-Martin-Schwinger boundary conditions, and rewrite Eq. (11) in integral form Gkj(z, z′) = Zd¯z gk(z, ¯z)X mE(¯z)·dkmGmj(¯z, z′).(13) Substitution of this result into Eq. (10) leads to X mid dz δim −hHF,im(z)Gmj(z, z′) = δ(z, z′)δij +X mZd¯zΣion(z, ¯z) + Σ(z, ¯z)imGmj(¯z, z′), (14) where we defined the ionization self-energy according to Σion,ij(z, z′) = X kE(z)·dikgk(z, z′)E(z′)·dkj.(15) Although the ionization self-energy resembles the embedding self-energy in quantum transport [73–75] its physical origin is different, see below. Equation (14) is a closed equation for the Green’s function in the bound sector. Subtracting from it the adjoint of Eq. (14) and setting z=z+we obtain an equation for the one-particle density matrix ρij (t)with indices in the bound sector −id dtρ(t) + [hHF(t), ρ(t)] = i[I(t) + Iion(t)] −H.c.. (16) where the collision integral I(t)reads I(t) =Zt −∞ dt′Σ<(t, t′)G>(t′, t) + Σ>(t, t′)G<(t′, t),(17) and the ionization integral Iion(t)is defined as I(t)with Σion in place of Σ. Due to the presence of the off-diagonal in time G≶in the collision and ionization integrals Eq. (16) is not a closed equation for ρ. The GKBA is an approximation that relates G≶(t, t′)to ρ(t)and ρ(t′)[72], Progress in Non-equilibrium Green’s Functions (PNGF VI) IOP Publishing Journal of Physics: Conference Series 696 (2016) 012004 doi:10.1088/1742-6596/696/1/012004 5 thereby transforming Eq. (16) into a closed integro-differential equation for ρ. According to the GKBA G<(t, t′) = −GR(t, t′)ρ(t′) + ρ(t)GA(t, t′),(18a) G>(t, t′) = + GR(t, t′)¯ρ(t′)−¯ρ(t)GA(t, t′),(18b) with ¯ρij =δij −ρij. It is worth noting that the GKBA results are extremely sensitive to the choice of the quasiparticle propagator GR(t, t′) = [GA(t′, t)]†. For Σ=0(no correlations) the Green’s functions G≶that solve the equation of motion (10) can be written as in Eqs. (18) provided that GR(t, t′) = −iθ(t−t′)The−iRt t′d¯ t hHF(¯ t)i(19) is the HF propagator. The numerical simulations presented in Ref. [41] show that for atomic systems the HF propagator yields accurate results in the correlated case too. Similar accuracies were recently reported for Hubbard clusters [76] and for the two-dimensional Anderson–Hubbard model [77]. A nice property of the HF propagator is that it guarantees the satisfaction of all basic conservation laws provided that the self-energies is conserving [78]. The results presented in this work have been obtained using Eq. (19). Nevertheless, it should be emphasized that the performance of the HF propagator is not always this good for all systems. In open systems, e.g., in molecular junctions attached to metallic electrodes, it is pivotal to include relaxation effects due to electron collisions and to the presence of the electrodes (electronic bath) [79]. Before closing this section let us discuss and simplify the ionization self-energy. From Eq. (12) we have g≶ k(t, t′) = −if≶(ǫk)e−iǫk(t−t′)(20) with f<(ǫk)the Fermi function and f>= 1 −f<. As the energy of the extended states is larger than the chemical potential we have f<(ǫk) = 0 and f>(ǫk) = 1. This implies that the lesser ionization self-energy vanishes. On the other hand the greater component reads Σ> ion,ij(t, t′) = −iX kE(t)·dike−iǫk(t−t′)E(t′)·dkj =E(t)←→ σij(t−t′)E(t′),(21) where in the last equality we defined the dyad ←→ σij(t) = −iPkdike−iǫktdkj. In Fourier space ←→ σij(ω) = −2πi X k dik δ(ω−ǫk)dkj ≈2iX k dik Im 1 ω−ǫk+iηdkj,(22) where ηis a positive constant of the order of the level spacing of the continuum states. We already mentioned that the pump pulse is a few cycles pulse centered around some ionization frequency ωP. Therefore, Σ> ion is dominated by those terms in ←→ σ(t−t′)that oscillate at frequency ǫk≃ωP. Then, for simplicity, we implement a time-local approximation ←→ σij(ω)≈←→ σij(ωP) which implies ←→ σij(t−t′) = ←→ σij(ωP)δ(t−t′). Substituting this result into Eq. (21) yields Σ> ion(t, t′) = −iδ(t−t′)Γ(t),(23) where Γij(t) = iE(t)←→ σij(ωP)E(t)(24) is a self-adjoint positive-definite matrix for all times t. Inserting this result into the ionization integral the equation of motion (16) for ρbecomes −id dtρ(t) + [hHF(t), ρ(t)] −i{Γ(t), ρ(t)}=iI(t)−H.c.(25) Progress in Non-equilibrium Green’s Functions (PNGF VI) IOP Publishing Journal of Physics: Conference Series 696 (2016) 012004 doi:10.1088/1742-6596/696/1/012004 6 where the curly brackets signify an anticommutator. From Eq. (25) we clearly see that the ionization self-energy is responsible for taking away particles from the system. We also see that the draining of particles is effective only during the pump since Γ(t)vanishes for E(t) = 0. This corresponds to the intuitive picture that no ionization occurs after the pump pulse has passed through the system. 4. Numerical Results In this Section we present numerical results on a simple model system in order to illustrate the performance of various approximations in describing the strong-field multiple ionization and the subsequent formation of a mixture of different ions. The model consists of two levels, one describing a 3dorbital (level 1) and the other describing a 4porbital (level 2) of a Kr atom. The pump couples level 1 and 2 via the dipole matrix dij =ˆ xd0(1−δij), and at the same time couples level 2 to the continuum states, thus generating a ionization dyad ←→ σij(ωP) = −iσ0δi2δj2. The remaining quantities of the model are hij =εiδij and vimnj =Uimδij δmn, with i, j = 1,2. The numerical values of the model parameters have been chosen to closely reproduce energy- and time-scales of the first-principle simulations of Ref. [41]. These values are all reported in the caption of Fig. 1. Equation (25) has been solved within two different approximations, namely HF and 2B. In the first case the collision integral I(t)is identically zero, while within 2B we have [61] I(2B) ik (t) = X nm pq sr X j virpnwmqsj Zt −∞ dt′hG< nm(t, t′)G< pq(t, t′)G> sr(t′, t)G> jk(t′, t) +G> nm(t, t′)G> pq(t, t′)G< sr(t′, t)G< jk(t′, t)i.(26) The real-time propagation has been performed with a time step ∆ = 0.1a.u. and the spectra have been obtained by a discrete Fourier transform of the probe-induced dipole with Nt= 104 time-steps. The atom, initially in its ground state with both levels filled, is ionized by a few-cycle pump E(t) = ηPE(t)where E(t) = E0sin2(πt/∆P) sin(ωPt). The expelled charge as a function of time is diplayed in Fig. 1. During the ionization process the occupation of level 1 remains roughly constant, meaning that the atom is loosing charge mainly from level 2. We also notice that the time-evolution of the expelled charge calculated within HF and 2B is essentially the same. This fact could suggest that the system is weakly correlated and that HF well captures the physics of the problem. However, the expelled charge is just one possible observable. Below we show that the HF and 2B photoabsorption spectra exhibit qualitative different features (although the local density in these two approximations agrees with high accuracy). After a time τthe system is probed with an ultrashort pulse e(t) = ηpe(t), with e(t) = e0sin2(π(t−˜τ)/∆p) sin(ωp(t−˜τ)) for ˜τ < t < ˜τ+ ∆p. Here τ= ˜τ−(∆P−∆p)/2is the time distance between the maxima of the pump and probe pulses. The frequency ωpof the probe is chosen to promote the intra-atomic excitation between the levels 1 and 2. The transient absorption spectrum within the HF approximation is displayed in the top-left panel of Fig. 2. Only one absorption peak appears, corresponding to the absorption of a singly ionized atom. We verified that the only effect of increasing the intensity of the pump is a shift in the position of the peak. We also varied the frequency and duration of the pump but we found no sign of absorption peaks due multiply ionized atoms. A single-peak absorption spectrum is also observed within the M2B approximation [41] (not shown), thus suggesting that the solution of Eq. (25) without the inclusion of memory effects in the self-energy cannot describe multiple ionization. However, the occurrence of multiple ionization (and subsequent absorption) for a strong enough pump is physically expected [28]. Progress in Non-equilibrium Green’s Functions (PNGF VI) IOP Publishing Journal of Physics: Conference Series 696 (2016) 012004 doi:10.1088/1742-6596/696/1/012004 7