High-harmonic spectra of hexagonal nanoribbons from real-space time-dependent Schrödinger calculations
EEPJ manuscript No. (will be inserted by the editor)
High-harmonic spectra of hexagonalnanoribbons from real-space time-dependentSchr¨odinger calculations
Helena Dr¨ueke a and Dieter Bauer b University of Rostock, Germany
Abstract.
High-harmonic spectroscopy is a promising candidate forimaging electronic structures and dynamics in condensed matter byall-optical means and with unprecedented temporal resolution. We in-vestigate harmonic spectra from finite, hexagonal nanoribbons, such asgraphene and hexagonal boron nitride, in armchair and zig-zag configu-ration. The symmetry of the system explains the existence and intensityof the emitted harmonics.
High-harmonic generation (HHG) has been first observed in gases [1,2]. Its non-perturbative nature, featuring a plateau of almost constant high-harmonic yield, wassubsequently explained by the three-step model [3,4]: An electron is removed fromthe atom, propagates under the external field’s influence, and recombines with theatom. The orbital energies of electrons in atoms do not depend on momentum, andthe electron’s dispersion relation in the continuum is shaped parabolically. Therefore,no harmonics are emitted from electrons in the ground state or free electrons, onlyby transitions between bound states or recombination from continuum states back tobound states.To describe HHG in the bulk of solids, the orbital energies and the continuumare replaced by electronic bands [5]. This opens a whole new field of research [6,7,8,9,10,11,12,13]. Analogous to the HHG process in gases, the transition of electronsbetween valence and conduction bands causes high harmonics, called interband har-monics. Intraband harmonics, on the other hand, are produced by the movement ofelectrons in partially filled, non-parabolic bands. Band structures are usually definedfor periodic or infinite solid bulk systems. However, every realistic system has bound-aries, which may cause completely different HHG spectra compared to the bulk [14,15,16,17,18]. Graphene and hexagonal boron nitride (hBN) are two-dimensional ma-terials that possess fascinating features with promising potential applications [19,20]. Their hexagonal structure allows for two different edges: zig-zag and armchair.While graphene consist only of carbon atoms (all identical), hBN is built from boronand nitrogen atoms. Recently, the interaction of intense laser light with graphene got a e-mail: [email protected] b e-mail: [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Will be inserted by the editor into the focus of interest for its prospects to steer electrons at will on ultrafast timescales [21,22,23,24].
The nanoribbons’ atomic nuclei were positioned in a hexagonal lattice, as describedfor the armchair and zig-zag configuration in the following sections 3 and 4. Atomicunits are used throughout this paper unless stated otherwise. The distance betweenneighboring lattice sites was 2 . . (cid:6) A). An effectiveP¨oschl-Teller potential V ( r ) = − (cid:80) i V i cosh ( ε | r − r i | ) with ion potentials V i = 3 . ± V os and screening parameter ε = 2 describe the attractive potentials of the nuclei. Forgraphene ribbons, all atoms are carbon, therefore the additional on-site potential V os = 0. A non-zero on-site potential represents two alternating, different kind ofatoms, such as boron and nitrogen in hBN. At which lattice sites the ion potentialsare increased or decreased by V os is sketched in the following sections.In this work, we did not employ the usual tight-binding approximation com-monly made in condensed-matter theory but have developed a 2D, real-space, time-dependent Schr¨odinger solver for the ab initio simulation of the intense-laser interac-tion with 2D matter. In that way we are able to reveal differences and similarities inHHG spectra as compared to corresponding tight-binding studies, e.g., in Ref. [25].The non-interacting electronic orbitals in our Schr¨odinger solver are defined on a two-dimensional grid of spacing ∆x = ∆y = 0 .
2, which encompasses all lattice sites plusa border of 8 on each side. In contrast to the usual tight-binding description, thisallows us to have electron orbitals that are not only localized at lattice sites but alsobetween them, or free electrons.The electronic eigenstates of the system were found by imaginary-time propagationemploying the Crank-Nicolson method [26]. Starting from a random initialization,imaginary timesteps − .
05i are taken (each step followed by renormalization of thewavefunction) until the ground state is reached and the relative change of the stateis smaller than the threshold of 10 − for two consecutive iterations. To find thehigher-lying states, the workflow is identical, but with an additional (Gram-Schmidt)orthogonalization to all previously found states in each iteration. This gives us allstates of interest of the unperturbed system.Real-time simulations of the interaction of all occupied electronic orbitals with ashort laser pulse were performed with a timestep 0 .
05 using, again, Crank-Nicolsonpropagation. The pulse was a 4-cycle sin -shaped laser pulse of frequency ω = 0 . λ (cid:39) . µ m) and polarized along the ribbon. The electronic dipoles were recordedat each time step during the laser pulse. Harmonic spectra were calculated as theabsolute square of the Fourier transform of the recorded dipoles, multiplied by asymmetric Hann window [27,28]. First, we investigate a hexagonal nanoribbon in the armchair configuration. A totalof 24 lattice sites are arranged in the shape of four hexagons as shown in Figure 1. For V os = 0, the armchair ribbon is symmetric about the horizontal as well as the verticalaxis through the center. The introduction of an on-site potential deepens the blue(square) sites’ potentials while making the orange (circle) ones shallower. This causesa left-right asymmetry, while the top-bottom symmetry is conserved. Note that thelines drawn in Figure 1 connect nearest neighbors. In tight-binding calculations (such ill be inserted by the editor 3 as in Ref. [25]), hopping takes place along these lines. However, in our simulationbased on the time-dependent Schr¨odinger equation, electronic wavefunctions are notrestricted to move along these lines but may propagate in the entire plane. Fig. 1.
Armchair ribbon: At the blue (square) lattice sites, the potential depth is increasedby the on-site potential, V blue = V avg + V os . The orange (circle) sites correspond to theshallower potentials of depth V orange = V avg − V os . The asymmetry in the potential leads to an asymmetry in the orbitals. Figure 2 (a-d) show the highest occupied (a and c) and lowest unoccupied (b and d) orbitalswithout (a and b) and with (c and d) on-site potential. The orbitals without on-sitepotential are horizontally and vertically symmetric (as is the potential), and there isonly a small bandgap between the occupied and unoccupied states. With an on-sitepotential, the occupied orbitals are localized on the sites with deeper potentials andtherefore have a decreased energy. The unoccupied orbitals are localized on the siteswith shallower potentials and therefore have increased energy. This leads to a bandgapbetween the occupied and unoccupied orbitals, which, for V os (cid:38) .
2, grows linearlywith the on-site potential (Figure 2 (e)). In contrast to tight-binding methods, ourapproach allows us to calculate an arbitrary number of orbitals of increasing energy.The next state above the conduction band is a ”free” electron, i.e., not localized onthe ribbon but still inside the simulation box with reflecting boundary conditions.The incoming laser field is linearly polarized along the armchair ribbon. All emit-ted harmonics are linearly polarized in the same direction. The emission of harmonicspolarized in the perpendicular direction requires a top-bottom asymmetry in the sys-tem, which the armchair ribbon does not possess, regardless of on-site potential. Thebandwidths and bandgaps ( ∆E intra , ∆E min , and ∆E max ) from Figure 2 (e) explainthe most important features of the harmonic spectra shown in Figure 3. Intrabandharmonics are only present at harmonic energies below the width of the valence band ∆E intra . Interband harmonics can be observed between the minimum ∆E min andmaximum ∆E max bandgap between the valence and (first) conduction band. Abovethe two bands are the box states (marked as gray lines in Figure 2 (e)), which arenot localized on the ribbon, and whose energies are determined by the size of thesimulation box. Only the energies of the four lowest box states are shown, but manymore lie above them. Transitions to these box states cause harmonics above ∆E max .In an experiment, there are no box states (unless it is performed in a cavity), buttransitions to higher bands or the continuum would also cause harmonics beyond themaximum bandgap. These can not be described in tight-binding approximation withone atomic orbital per site because then the energy difference between states is boundfrom above by ∆E max (see Ref. [25]). In the zig-zag configuration, a total of 26 lattice sites are arranged in six hexagons, asshown in Figure 4. On the orange sites, the on-site potential decreases the potential
Will be inserted by the editor(a)(b)(c)(d) 00.020.040.060.080.1 | ψ i | V os -0.6-0.4-0.20 E n e r g y E (e) valence bandconduction bandbox states∆ E intra ∆ E intra ∆ E min ∆ E max (a)(b) (c)(d) Fig. 2.
Orbitals and orbital energies of the armchair ribbon. (a) - (d) Orbitals of thearmchair ribbon. (a) Highest occupied orbital without on-site potential ( V os = 0). (b) Lowestunoccupied orbital without on-site potential ( V os = 0). (c) Highest occupied orbital with on-site potential V os = 0 .
4. (d) Lowest unoccupied orbital with on-site potential V os = 0 . V os . The orbitalenergies of the orbitals shown in (a) - (d) are marked. The arrows mark the bandwidth ofthe valence band ( ∆E intra ) and the minimum and maximum bandgap between the valenceand (first) conduction band ( ∆E min and ∆E max ).0 25 50 75 100Harmonic order10 − − D i p o l e s t r e n g t h ( a r b . u . ) (a) V os = 0 V os = 0 . O n - s i t e p o t e n t i a l V o s (b)∆ E intra ∆ E min ∆ E max − − D i p o l e s t r e n g t h ( a r b . u . ) Fig. 3.
Harmonic spectra of the armchair ribbon in parallel direction. (a) Harmonic spec-tra without (bold orange line, V os = 0) and with (blue line, V os = 0 .
4) on-site potential.(b) Harmonic spectra as a function of harmonic order and on-site potential. depth, while on the blue sites, it deepens the potential. The on-site potential causesa top-bottom asymmetry but no left-right asymmetry.As for the armchair ribbon, the asymmetry of the potential leads to decreasedenergies of states in the valence band, localized at the deeper sites, and increasedenergies of states in the conduction bands, localized at the shallower sites (see Fig-ure 5). The minimum bandgap increases almost linearly with the on-site potential,the bandwidths of both bands decrease.The parallelly polarized harmonics (Figure 6 (a)) are present with and without on-site potential. The bandwidth and bandgaps can explain the cutoffs of both intra- andinterband harmonics. Perpendicular harmonics (Figure 6 (b)) with on-site potential ill be inserted by the editor 5
Fig. 4.
Zig-zag ribbon: At the blue (square) lattice sites, the potential depth is increased bythe on-site potential, V blue = V avg + V os . The orange (circle) sites correspond to the shallowerpotentials of depth V orange = V avg − V os .(a)(b)(c)(d) 00.0250.050.0750.10.125 | ψ i | V os -0.6-0.4-0.20 E n e r g y E (e) valence bandconduction bandbox states∆ E intra ∆ E intra ∆ E min ∆ E max (a)(b) (c)(d) Fig. 5.
Orbitals and orbital energies of the zig-zag ribbon. (a) - (d) Orbitals of the zig-zag ribbon. (a) Highest occupied orbital without on-site potential ( V os = 0). (b) Lowestunoccupied orbital without on-site potential ( V os = 0). (c) Highest occupied orbital withon-site potential V os = 0 .
4. (d) Lowest unoccupied orbital with on-site potential V os = 0 . V os . The orbitalenergies of the orbitals shown in (a) - (d) are marked. The arrows mark the bandwidth ofthe valence band ( ∆E intra ) and the minimum and maximum bandgap between the valenceand (first) conduction band ( ∆E min and ∆E max ). agree with these cutoffs, as well. Without an on-site potential, there is no top-bottomasymmetry, and therefore almost no harmonics perpendicular to the laser are ob-served. Transitions to the box states lead to weak harmonic emission above ∆E max . The introduction of an on-site potential in hexagonal nanoribbons causes lower ener-gies for occupied states and higher energies for unoccupied states. The valence band’sbandwidth decreases, and the minimum and maximum bandgaps between the valenceand conduction bands increase. These three energies explain the overall features inharmonic spectra for different on-site potentials. Intraband harmonics are only presentat energies below the valence bandwidth. Interband harmonics are present at ener-gies between the minimum and maximum bandgap. For a laser polarized along theribbon, the resulting harmonics are polarized in the same direction unless a non-zeroon-site potential causes a top-bottom asymmetry, which is only possible in the zig-zagribbon.
Will be inserted by the editor0 25 50 75 100Harmonic order00.20.40.60.81 O n - s i t e p o t e n t i a l V o s (a)∆ E intra ∆ E min ∆ E max E intra ∆ E min ∆ E max − − D i p o l e s t r e n g t h ( a r b . u . ) Fig. 6.
Harmonic spectra of the zig-zag ribbon as a function of on-site potential in (a) par-allel and (b) perpendicular polarization.
The results of this paper provide valuable verification of simpler tight-bindingmodels [25]. Our approach is not limited to a fixed number of states (grouped inbands), and our results account for transitions to even higher bands or the continuum.However, these transitions are expected to play an important role only in the gener-ation of higher harmonics beyond the cutoff ∆E max , leading to higher-order plateauswith decreasing yield (see, e.g., [29]). On the other hand, tight-binding approachescapture the essential mechanisms underlying high-harmonic generation up to ∆E max ,are computationally much less demanding and thus can be used to investigate muchlarger systems.The datasets generated and analyzed during this study are available atdoi:10.17605/OSF.IO/8RTFU [30]. References
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