High harmonics from backscattering of delocalized electrons
HHigh harmonics from backscattering of delocalized electrons
Chuan Yu, Ulf Saalmann, and Jan M. Rost
Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany (Dated: February 23, 2021)Electron backscattering is introduced as mechanism to enhance high-harmonic generation in solid-state likesystems with broken translational symmetry. As a paradigmatic example we derive for a finite chain of N atoms the harmonic cut-off through backscattering of electrons in the conduction band from the edges ofthe chain. We also demonstrate a maximum in the yield of the high harmonics from the conduction bandif twice the quiver amplitude of the driven electrons equals the length of the chain. High-harmonic spectraas a function of photon energy are shown to be equivalent if the ratio of chain length to the wavelengthof the light is kept constant. Our quantum results are corroborated by a (semi-)classical trajectory modelwith refined spatial properties required to describe dynamics with trajectories in the presence of brokentranslational symmetry. Since the pioneering experiment by Ghimire et al. [ ] high-harmonic generation (HHG) with strong laser fields appliedto solids has been a focus of experimental and theoreticalresearch with first reviews available [ ] . The so-called“three-step model” [ ] is key to understand the microscopicelectron dynamics of HHG in atoms and molecules semi-classically in terms of classical trajectories [ ] . It has beenadapted successfully for interband HHG in solids [ ] , sug-gesting that fundamental properties of high harmonics areruled by the same basic principles from atoms to the solidstate. On the other hand, a solid-state environment shouldoffer more possibilities to influence these phenomena thanan atom due to the larger structural complexity and vari-ability [ ] . Indeed, under suitable conditions, a solid-state HHG spectrum exhibits several cut-offs [
10, 15 ] dueto the (band-)structured continuum of solid-state electrons,in contrast to the single atomic cut-off.In an atomic context, cut-offs can be extended if thelaser-driven electron acquires a larger momentum throughbackscattering from another atom or ion. This requires alarge distance of the order of the atomic quiver amplitude A /ω between the back-scattering and recombining ion,where A is the peak vector potential and ω the carrierfrequency of the laser. This can theoretically be achieved inlaser-assisted ion-atom collisions with a suitable impact pa-rameter [ ] or for above-threshold ionization in rare-gasclusters with a suitable size, as demonstrated recently in anexperiment [ ] , but not in molecules which are typicallytoo small. Solid-state like systems, on the other hand, caneasily match the spatial requirements set by the quiver am-plitude of conduction-band electrons and any irregularityin their periodicity may give rise to backscattering. Indeed,we will analytically predict and numerically demonstrate inthe following significantly extended HHG cut-offs throughbackscattering of delocalized electrons.To keep the situation as simple as possible we investi-gate HHG from a chain of N atoms with a lattice constant(interatomic distance) of d = [ ] . We will show that ex-tended cut-offs through backscattering from the edge canoccur and that HHG is most efficient if the full excursion of the excited laser-driven electron (twice the quiver ampli-tude x q ) matches the length of the chain, i. e., if N ≈ N q with the latter defined by 2 x q ≡ N q d . Motivated by sim-ple scaling arguments and the (semi-)classical trajectorypicture for interband harmonics, the predicted cut-off andmaximal high-harmonic yield is accurately reflected in theHHG spectra obtained with the laser-driven flux of the 4 N electrons. Apart from small modifications we find the elec-tron dynamics in a chain with N (cid:166)
10 well described withthe band structure of the periodic system.For the chain of N atoms we compute the harmonic spec-trum generated per atom S N ( ω ) ∝ N − (cid:12)(cid:12)(cid:12) (cid:82) d t J tot ( t ) W ( t ) e − i ω t (cid:12)(cid:12)(cid:12) , (1)where J tot ( t ) is the total current in the system and W ( t ) is a window function of the laser-pulse-envelope shape forimproving the signal-to-noise ratio. Details of the meth-ods and parameters used as well as the periodic treat-ment for the limit N →∞ can be found in [ ] . Thelaser pulse with frequency ω is linearly polarized alongthe chain and described in dipole approximation by thevector potential A ( t ) = A sin ( ω t / [ n cyc ]) sin ( ω t ) for0 ≤ t ≤ π n cyc /ω and A ( t ) = n cyc = A = λ = N signals that theHHG spectra approach the periodic limit N →∞ . However,in the lower central part of Figs. 1a–c one sees a strongerHHG response which prevails for a certain range of systemssizes. To see this more clearly, Figs. 1d–f show (in red) spec-tra at the system sizes N =
16, 24 and 32, where the HHGresponse for midsize harmonic orders is enhanced. Theseselected system sizes (marked by horizontal dashed lines inFigs. 1a-c) have the widest enhancement region. The en-hancement is particularly evident in comparison to the pe-riodic limit (grey areas). The latter is apparently reached in a r X i v : . [ phy s i c s . a t o m - ph ] F e b
0 10 20 30 40 50 20 40 60 80 S y s t e m s i z e N −10 −7 −4 −1 (a)
0 15 30 45 60 75 30 60 90 120 10 −12 −9 −6 −3 (b)
0 20 40 60 80 100 40 80 120 160 10 −14 −11 −8 −5 (c) −1−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8−1 −0.5 0 0.5 1 k k E n e r g y ( a . u . ) k (units of π /d ) ω ω V V C C (g) −1−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8−1 −0.5 0 0.5 1 k' k (units of π /d ) ω ' V V C C (h) −12 −8 −4
0 10 20 30 40 50 I n t e n s i t y ( a r b . u n i t s ) Harmonic orderPeriodic N = 80 N = 16 (d) −14 −10 −6 −2
0 15 30 45 60 75Harmonic orderPeriodic N = 120 N = 24 (e) −16 −12 −8 −4
0 20 40 60 80 100Harmonic orderPeriodic N = 160 N = 32 (f ) FIG. 1. HHG spectra as a function of harmonic order and system size N for wavelengths of 1600 nm (a), 2400 nm (b) and 3200 nm (c)and for the same wavelengths but at fixed N =
16 (d), N =
24 (e), and N =
32 (f), indicated by horizontal dashed lines in (a-c). Thespectra of the periodic system are shown as shaded area for comparison. The dashed lines indicate the estimates of the 1st and 2ndcut-off for the periodic system at ω and ω , specified in the text. (g) Sketch of the k -space dynamics in the periodic system, for anelectron excited to the bottom of the C -band at A ( t ) ≈ − A . The maximal C - V and C - V band energy differences (i.e., the cut-offs ω and ω ) are achieved at k = A and k =
0. (h) Sketch of the k -space dynamics in the finite system, with an edge backscatteringevent in the C -band at the vacuum level occuring at A ( t s ) = − A . The horizontal dotted line represents the sign change of k ( t ) due tobackscattering. With a subsequent band-gap transition to the C -band, this sketch corresponds to the maximally achievable harmonicfrequency in the backscattering case ω (cid:48) at k (cid:48) (see text). the longest chains considered for each wavelength ( N = [ ] with the valence band V ( k ) : ω = C ( k ) − V ( k ) = k = A = π/ d and ω = C ( k ) − V ( k ) = k =
0. Due to the bandstructure, the lowest valence band V does not actively par-ticipate in the HHG processes. An electron, excited fromthe 2nd valence band V at t , preferentially enters the 1stconduction band C at k ≈ Γ -point ( k =
0) andsubsequently moves with momentum k ( t ) = A ( t ) − A ( t ) + k . (2)This time-dependent k -space motion always holds in the pe-riodic limit, but can be modified by backscattering in finitesystems, as will be discussed below.The enhanced spectra (red in Fig. 1d–f) exhibit a smalldip at the 1st cut-off but the enhancement does not extendto the 2nd cut-off. This observation suggests that the en-hancement is not due to a more efficient mechanism to en-ter C preserving the original k ( t ) . Rather, it must be aprocess which changes k ( t ) . This can be achieved by elas-tic scattering in the presence of a laser field. Indeed, as wewill see, the enhancement is due to electrons in C , beingbackscattered from the edge of the chain.When an electron wavepacket approaches the systemedge, it can either be reflected from it (i.e., being backscat-tered) or leak out of the system (i.e., being ionized).Backscattering (ionization) will be dominant if its mean en-ergy is lower (higher) than the vacuum level. In the clas-sical three-step description, backscattering at a time t s is assumed to be elastic, resulting in a sign change of the in-stantaneous momentum k ( t > t s ) = A ( t ) − A ( t s ) + A ( t ) − k . (3)In a solid-state system the electron (and the accompa-nying hole) suffer the momentum kick while moving ontheir respective band with dispersion E ( k ) . This is illus-trated in Fig. 1h for the electron. In general, the bandenergy at backscattering must be below the vacuum level E = k s = C ( k s ) = A through (unperturbed) interaction with the laser field, themaximal final momentum is k (cid:48) = k s + A − π/ d in the firstBrillouin zone (BZ) leading to the recombination energy ω (cid:48) = C ( k (cid:48) ) − V ( k (cid:48) ) = extended through backscattering . Indeed, this corre-sponds to harmonic order 35, 53, 70 for the wavelengths λ = C -band via a subsequent band-gaptransition. Hence, edge backscattering suggests itself as apathway to high-energy states in analogy to backscatteredelectrons from an ion in the atomic context. There, how-ever, backscattering only leads to higher photo-electron en-ergies [
17, 20 ] , but not to larger cut-offs in HHG. This ismainly due to the fact that the electron’s wavefunction inthe atomic context is usually spatially localized on the ion(playing the role of the hole) and the continuum electronwavepacket. The lacking overlap prevents recombinationnecessary for HHG between the energetic electron far away −20 −15 −10 −5
0 0.3 0.6 0.9 1.2 (a) N / N q = 1.65 I n t e n s i t y ( a r b . u n i t s ) Harmonic energy (a.u.) N = 16, λ = 1600 nm N = 24, λ = 2400 nm N = 32, λ = 3200 nm 10 −15 −12 −9 −6 −3
0 1 2 3 4 5 6 7 8 (b) Y N ( a r b . u n i t s ) N / N q λ = 1600 nm λ = 2400 nm λ = 3200 nm FIG. 2. The HHG spectra from Fig. 1d–f as function of photon en-ergy (a) and their integrated yield Y N beyond the 1st cut-off ω asa function of scaled system size N / N q (b). The vertical dashed linein panel (a) represents the edge-backscattering cut-off at ω (cid:48) = from the ion available for recombination. In solid-state likesystems, on the other hand, we deal with spatially delo-calized Bloch electrons, for which overlap of electron-holewavefunctions can be more easily achieved [ ] . More-over, electrons reflected by the edges of an extended sys-tem continue to move inside the system, allowing them torecombine with significant wavefunction overlap. There-fore, backscattering represents a promising mechanism forincreasing the energy of solid-state harmonics.As a next step we work out which role the spatial exten-sion of the chain plays for backscattering. To this end wevary in Fig. 2 the wavelength of the light while keeping thevector potential fixed. The latter ensures that the dynam-ics in momentum space, and in particular the energy gainthrough backscattering depending on A as discussed so far,remains the same while through the variation of the wave-length the quiver amplitude x q ∝ A λ changes linearly, re-sulting in different scales for the spatial dynamics. Hence,locking the ratio of chain length versus wavelength N /λ in addition to an identical A should provide similar condi-tions for the high-harmonics-generating electron dynamicsand we expect similar spectra, provided the harmonic yieldis not shown as a function of the harmonic order but of theharmonic energy instead, as done in Fig. 2a. The similarityof the three spectra with the extended cut-off at ω (cid:48) = N q /λ where the largest enhancement of thehigh harmonic yield for edge backscattering occurs. For thispurpose we integrate the harmonic yield in the spectral re-gion of enhancement, Y N = (cid:82) ω d ω S N ( ω ) . That the curveslevel off for large N simply reflects convergence to the peri-odic limit without edge backscattering. That all three inte-grated yields have a similar shape over the entire scaledrange of N illustrates the universality of the underlyingstrong-field dynamics of delocalized electrons provided thatmomentum and spatial dynamics is equivalent. Most inter-esting in the context of backscattering is the sharp rise andmaximum of Y N which occurs close to N / N q = N q = x q / d ,where the length N q d of the chain equals the full quiver excursion 2 x q of the excited electron.We note that while the momentum scale A , the vectorpotential, is a property of the light only, this is not the casefor the spatial scale x q , the quiver amplitude, which de-pends also on the band structure. With the instantaneousmomentum given by Eq. (2), the position-space motion of aBloch electron in band B reads ∆ x B ( t ) ≡ x B ( t ) − x B ( t ) = (cid:82) tt d t (cid:48) dd k B ( k ) (cid:12)(cid:12) k = k ( t (cid:48) ) . (4)Within the Kane band approximation [
22, 23 ] , an ex-plicit expression for x q can be given which is even ana-lytically solvable if the electron moves with initial condi-tion A ( t ) = x q =[ A / ( m ∗ ω )] arctan ( a ) / a , with a = A / k ∗ , where m ∗ is theeffective mass of the electron and k ∗ the band’s momentumscale [ ] .Finally, we discuss how the HHG time-frequency profile(shown in Fig. 3) obtained by Gabor transforming the quan-tum current in Eq. (1), can be mapped onto classical tra-jectories from Eq. (4) for the electron-hole pair. This isfor at least two reasons not straightforward. (i) Even inthe periodic-system case, the semi-classical three-step tra-jectory picture with more than one conduction band hasonly been demonstrated successfully for situations wherethe maximal electron momentum exceeds the BZ boundary(2 A > π/ d ) and therefore takes the excited electron acrossthe gap between C and C and eventually gaps betweenhigher bands [ ] . (ii) It is unclear if the backscatteringmechanism can be described adequately with trajectories.Hence, the more intricate circumstances of backscatteringprovide also a chance to refine the semi-classical trajectorymodel for HHG, which is, however, beyond the scope of thepresent work. Here, we only sketch the feasibility of such
4 4.5 5 5.5 6 0.8 1.2 H a r m o n i c e n e r g y ( a . u . ) − − − (a) Periodic
4 4.5 5 5.5 6Time (units of cycle) 0.8 1.2 H a r m o n i c e n e r g y ( a . u . ) − − − (b) N = 24 4.4 4.6 4.8 5 5.2 − − − − x ( u n i t s o f d ) (c)
4 4.2 4.4 4.6 4.8 − − x ( u n i t s o f d ) Time (units of cycle) (d)
FIG. 3. HHG time-frequency profile at λ = N =
24 (b), respectively. Theblack dots are traces obtained from the semi-classical electron-hole recollision model, without (a) and with (b) backscattering.(c) and (d) provide representative trajectories forming the tracesin (a) and (b) in real space. In the backscattering case, an electron-hole separation of δ L = d is chosen as initial condition. Thedashed lines in (d) indicate the system edges taken as ± N d / an enterprise.Regarding the periodic case (i) we relax the (standard)condition of tunneling at the Γ -point ( k =
0) and allow asmall interval | k | ≤ π/ d about k = C . Do-ing so, the black HHG traces in the time-frequency planeresult for the 2nd HHG plateau (Fig. 3a) with the corre-sponding electron-hole trajectories shown in Fig. 3c. Asthe electron is much lighter than the hole (manifest inthe smaller energy dispersion of V compared to C inFig. 1g), its excursion on the way to recombination is largerthan the one of the hole. Moreover, electron and holemove initially into different directions as their respectiveband dispersions have opposite sign. As one can see inFig. 3c for different electron and hole trajectories starting atthe position zero, respectively, the standard semi-classicalcondition ∆ x C ( t r ) − ∆ x V ( t r ) = t r with high harmonic emission en-ergy C ( k ) − V ( k ) linked to time through the dependence k = k ( t ) .Note, that starting the trajectories at x = x =
0. In thesolid-state context with delocalized electrons, however, theelectron and associated hole trajectories can be born andrecombine with an arbitrary separation δ L = ∆ CV ( t ) = ∆ CV ( t r ) with ∆ CV ( t ) ≡ x C ( t ) − x V ( t ) and centered aboutanother location but x =
0, which is just another way tofulfill the standard semi-classical recombination condition [
2, 7, 8 ] ∆ x C ( t r ) − ∆ x V ( t r ) = ∆ CV ( t r ) − ∆ CV ( t ) = k ( t ) onthe vector potential A ( t ) akin to continuum electron trajec-tories in an atomic situation, but also due to sign changesin the dispersion d B / d k of the energy bands, a feature notpresent in atoms.We turn now to the situation of backscattering (ii), illus-trated with a chain of N =
24 in Figs. 3b, d. As one sees rightaway, the time-frequency traces differ drastically from theperiodic situation. We can describe the traces with classicaltrajectories modified in two aspects compared to the stan-dard scenario in Fig. 3c. Firstly, if the energy of the elec-tron is below the vacuum level, C ( t s ) <
0, backscatteringtakes place by elastic reflection of the trajectories at thechain edges (shown dashed in Fig. 3d). This means thatfor t > t s Eq. (3) holds instead of (2). Secondly, the elec-tron and the associated hole trajectory are born and re-combine with a spatial separation δ L centered about a spe-cific location in the chain which is no longer arbitrary butnecessary to give the traces in Fig. 3b in agreement withthe quantum time-frequency profile. The spatial separa-tion δ L implies that not the entire length of the chain is available for the high-harmonics-generating quiver dynam-ics of electron and hole trajectories. This is consistent withthe fact that the integrated high-harmonic yield Y N peakssystematically for values of N / N q slightly larger than onein Fig. 2b. At the present level of a simple classical trajec-tory description, the spatial separation δ L is a parameterwhose value emerges through comparison with the (quan-tum) time-frequency pattern of HHG. In the classical limit λ → ∞ , we expect this parameter to become irrelevant asits scaled value δ L /λ tends to zero. Indeed, careful inspec-tion of Fig. 2b reveals that the offset from N / N q = λ .To summarize, we have established electron backscatter-ing as a mechanism to extend the cut-off for solid-state likeharmonics in systems with an inherent length scale due tobroken translational symmetry. For simplicity and consis-tency, we have chosen to demonstrate and analyze backscat-tering with finite chains of atoms solving the many-electrondynamics based on density functional theory. This has al-lowed us to link the quiver amplitude of the driven electronto the extension of the system, revealing that one achievesthe highest integrated harmonic yield beyond the 1st cut-off of the periodic system, if twice the quiver amplitude isapproximately equal to the length of the atomic chain. Theband energy at the momentum where backscattering takesplace must be below the vacuum level of the system, oth-erwise ionization dominates reflection. This is a universalcondition for the extended cut-off, which takes, howeverdifferent values depending on the band structure.Akin to the time-frequency profile of standard interbandhigh harmonics, also harmonics due to backscattering canbe described in terms of a simple trajectory picture withelastic reflection from the edges of the atomic chain anda finite distance of electron and hole trajectories at birthand recombination. Since spatial properties become rele-vant for systems with broken translational symmetry, theywill be interesting and helpful to refine semi-classical tra-jectory descriptions.Backscattering as introduced here has close analogies inextended atomic systems. However, in the latter it leadsonly to higher energies in laser-driven photo-ionization (of-ten termed above-threshold ionization), but not to largerhigh-harmonic cut-offs, since the localized electrons inatomic systems lack the ability for overlap of electron am-plitudes at large distances which is possible for the delocal-ized electrons in solid-state like systems. Other sources ofbreaking the periodicity of the solid-state system, such asimpurities, domain walls or grain boundaries, may also in-duce backscattering and ensuing effects on HHG. Work inthis direction is underway.CY acknowledges discussion with Lars Bojer Madsen inthe early stage of this work. [ ] S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. Di-Mauro, and D. A. Reis,
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