Controlling ultrafast currents by the non-linear photogalvanic effect
Georg Wachter, Shunsuke A. Sato, Christoph Lemell, Xiao-Min Tong, Kazuhiro Yabana, Joachim Burgdörfer
aa r X i v : . [ phy s i c s . op ti c s ] M a r Controlling ultrafast currents by the non-linear photogalvanic effect
Georg Wachter , ∗ Shunsuke A. Sato , Christoph Lemell ,Xiao-Min Tong , , Kazuhiro Yabana , , and Joachim Burgd¨orfer Institute for Theoretical Physics, Vienna University of Technology, 1040 Vienna, Austria, EU Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan and Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan (Dated: September 24, 2018)We theoretically investigate the effect of broken inversion symmetry on the generation and con-trol of ultrafast currents in a transparent dielectric (SiO ) by strong femto-second optical laserpulses. Ab-initio simulations based on time-dependent density functional theory predict ultrafastDC currents that can be viewed as a non-linear photogalvanic effect. Most surprisingly, the direc-tion of the current undergoes a sudden reversal above a critical threshold value of laser intensity I c ∼ . × W / cm . We trace this switching to the transition from non-linear polarization cur-rents to the tunneling excitation regime. We demonstrate control of the ultrafast currents by thetime delay between two laser pulses. We find the ultrafast current control by the non-linear pho-togalvanic effect to be remarkably robust and insensitive to laser-pulse shape and carrier-envelopephase. In the last decade, ultrafast few-cycle laser pulses withwell-defined carrier-envelope phase (CEP) have becomeavailable providing novel opportunities to explore the ul-trafast and non-linear response matter to strong opticalfields. The study of the induced electronic motion andof the highly non-linear optical response have focussedon rare gas atoms [1], molecules [2] and, more recently,on nanostructures, surfaces and bulk matter [3, 4]. Thedriven electron dynamics can be monitored through op-tical signals [5–9] and through emitted electrons [10–15].Very recently, Schiffrin et al. [16] have demonstrated di-rected electron currents generated inside transparent di-electrics by carefully tailored laser pulses. In turn, the ul-trafast response can characterize the impinging laser field[17]. Currently, avenues are explored to exploit such ul-trafast modulation of electric currents for petahertz-scalesignal processing [18] enabled by the short intrinsic timescale of the electron motion ( ∼ ab-initio simu-lations based on time-dependent density functional the-ory (TD-DFT) predict the generation of strongly non-linear currents in α -quartz that are, in contrast to pre-viously observed currents [16, 17], independent of thedetails of the laser pulse shape. The direction of thecurrents is found sensitive to the instantaneous laser in-tensity. Analysis of the spatio-temporal charge dynamicson the atomic length and time scale allow us to link thisto the transition from non-linear polarization currents todirectional tunneling excitation, the latter being highlysensitive to the alignment between the laser polarizationand the chemical bonds in the crystal. We demonstrate that this transition may be investigated in a pump-probesetup leaving its marks as a change of the direction of thecurrent as a function of the pump-probe delay.Theoretical exploration of ultrafast processes in solidsfaces the challenge to tackle the time-dependent many-body problem. Time-dependent density functional the-ory has emerged as a versatile tool allowing for an ab-initio description of a variety of strong field processesin the solid state [8, 13, 15, 19]. Here, we employ areal-space, real-time formulation of TDDFT [20–25] forthe electronic dynamics induced by strong few-cycle laserpulses in α -SiO ( α -quartz). Briefly, we solve the time-dependent Kohn-Sham equations (atomic units are usedunless stated otherwise) i∂ t ψ i ( r , t ) = H ( r , t ) ψ i ( r , t ) , (1)where i runs over the occupied Kohn-Sham orbitals ψ i .The Hamiltonian H ( r , t ) = 12 ( − i ∇ + A ( t )) + ˆ V ion + Z d r ′ n ( r ′ , t ) | r − r ′ | + ˆ V XC ( r , t )(2)describes the system under the influence of a homogenoustime-dependent electric field F ( t ) of amplitude F alongˆ a with vector potential A ( t ) = − R t −∞ F ( t ′ ) dt ′ in the ve-locity gauge and in the transverse geometry [26] allowingto treat the bulk polarization response of the infinitelyextended system along the polarization direction. Theperiodic lattice potential ˆ V ion is given by norm-conservingpseudopotentials of the Troullier-Martins form [27] repre-senting the ionic cores (O(1s ) and Si(1s )). Thevalence electron density is n ( r , t ) = P i | ψ i ( r , t ) | . Forthe exchange and correlation potential ˆ V XC we employthe adiabatic Tran-Blaha modified Becke-Johnson meta-GGA functional [28]. It accurately reproduces the bandgap ∆ ∼ and yields good agreementwith the experimental dielectric function over the spec-tral range of interest including at optical frequencies [29].The time-dependent Kohn-Sham equations (Eq. 1) aresolved on a Cartesian grid with discretization ∼ . ∼ .
45 a.u. per-pendicular to the polarization direction in a cuboid cellof dimensions 9.28 × × employing a nine-point stencil for the kinetic energy operator and a Bloch-momentum grid of 4 k -points. The time evolution isperformed with a 4 th -order Taylor approximation to theHamiltonian with a time step of 0.02 a.u. including apredictor-corrector step. The solution of Eqs. 1 and 2allows to analyze the time and space dependent micro-scopic vectorial current density j ( r , t ) = | e | X i
12 [ ψ ∗ i ( r , t ) ( − i ∇ + A ( t )) ψ i ( r , t ) + c . c . ](3)as well as the mean current density J ( t ) along the laserpolarization direction F , averaged over the unit cell ofvolume Ω, J ( t ) = 1Ω Z Ω d r j ( r , t ) · F / | F | . (4)The polarization density P ( t ) = R t −∞ J ( t ′ ) dt ′ [30] givesthe charge density D ( t ) transferred by the pulse. Thetotal charge Q will depend also on the details of the ge-ometry of the laser focus and of the collection volume notexplicitly treated in the following.First studies of the short-pulse induced current andcharge transfer in polycrystalline SiO [16, 17, 25] founda sinusoidal dependence on the carrier-envelope phase, φ CE , of the few-cycle electromagnetic field A ( t ) ∼ A ( t ) cos( ω L t + φ CE ) with (cos ) envelope A ( t ) and ω L the carrier frequency of the IR laser. Subcycle controland steering of electrons required exquisite control overthe instantaneous electric field F ( t ). Here, we explore analternate route towards steering, controlling, and switch-ing ultrafast currents that does not rely on φ CE control ofthe instantaneous field but on the instantaneous intensitydependence of a direct (DC) current.Starting point is the observation that the total chargedensity D ( τ p ) transferred at the conclusion of the pulsecan be split into a CEP dependent part with amplitude D CEP ( τ p ) and a residual part D ( τ p ). D CEP ( τ p ) tendsto decrease with increasing pulse length while the mag-nitude of D ( τ p ) increases with the pulse length (Fig. 1).For pulse length exceeding a few optical cycles, D ( τ p )is approximately proportional to the pulse duration anddominates the signal exceeding D CEP by about one orderof magnitude.The charge transferred by the induced DC current, D ( τ p ), features a strongly non-linear scaling with in-tensity | D | ∝ I . or, equivalently, field strength | D | ∝ F . (Fig. 2). The origin of this highly non-linear re-sponse lies in the broken centrosymmetry of the SiO FIG. 1. (Color online) Pulse length dependence of transferredcharge at intensities (a) 1 × W / cm , (b) 5 × W / cm ,and (c) 1 × W / cm , each with cos pulse shape. Carrier-envelope phase (CEP) dependent part D CEP (blue opensquares); CEP independent part D (green full squares);dashed lines: linear slope going through the origin.FIG. 2. (Color online) Carrier-envelope phase independent transferred charge density D (green full squares) as functionof laser intensity (cos pulse with full duration τ p = 20 fs, ~ ω L = 1 . | D | (empty boxes). Powerlaw | D | ∝ I . (dashed line). Electrons move along +ˆ a for I ≤ I c = 3 . × W / cm (upper panel) and along − ˆ a for I & I c (broken line to guide the eye). crystal along the ˆ a direction. In general, generation of adirected flow of charge by a laser field requires a brokeninversion symmetry. For few-cycle laser pulses with well-defined CEP, inversion symmetry is violated by a suit-able choice of φ CE . In the present case, it is no longerthe temporal shape of the laser electric field but the elec-tronic and crystallographic structure of matter the laserinteracts with that causes ultrafast currents. This novelmechanism does not rely on delicate CEP control yet of-fers sub-cycle response and switching.The appearance of a direct current in a homogeneousmedium under illumination, independent of the CEP, andlinearly increasing with pulse duration, can be viewedas a non-linear analogue to the well-known photogal-vanic (PG) effect [31–35] as first discussed qualitativelyby Alon [36]. Conventionally, the lowest order photogal-vanic effect is described as J PG k = β kln F l F ∗ n . (5) J PG is quadratic in the electric field components andlinear in the time-averaged laser intensity I ∝ F l F ∗ l . FIG. 3. (Color online) (a) Time-dependent polarization den-sity P ( t ) (solid green), laser field F ( t ) (black dashed, scaled)for τ p = 20 fs, ω L = 1 . × W / cm .(b) Time-dependent non-linear current ∆ J NL (Eq. 6, redsolid). (c) α -quartz lattice with broken inversion symmetry(ˆ a → − ˆ a ). “Larger” atoms are closer to the reader thansmall atoms; labeled atoms form a helix along ˆ a direction.(d-e) Snapshot of the current density in an ˆ a -ˆ c cut planegoing through the central oxygen at a time 1 (cid:13) (d), 2 (cid:13) (e),dash-dotted rectangle in (c). Angles between electric fieldand O-Si bond are γ = 25 . γ = 51 . n ( τ p ) inplane cut through the oxygen marked by a star in (c) forintensity 1 × W / cm . For linearly polarized light, the photogalvanic tensor β kln associated with the two-wave mixing in the second-order susceptibility χ (2) kln (0; ω, − ω ) is nonzero only in non-centrosymmetric crystals [37]. Microscopically, a varietyof mechanisms may contribute to the PG effect such asasymmetric excitation, scattering, or recombination ofelectrons and electronic defects [32]. One important re-alization is the so-called “shift current” [37–39] due tothe shift between the center of charge of the valence elec-trons and the excited electrons in the conduction band.This shift current has been predicted to be important inseveral semi-conductors [37, 40] and has been first exper-imentally verified for ferroelectrics [39].The present non-linear generalization of photogalvanic effects is obviously a signature of strong-field interactionwith matter. This is underscored by the surprising ob-servation of current reversal as a function of laser inten-sity (Fig. 2). We find a critical value of current reversalat I c = 3 . × W / cm . Electrons move along the+ˆ a direction for low intensities I < I c while they prop-agate along − ˆ a direction for higher intensities I & I c .We elucidate the microscopic mechanism for this rever-sal by analysis of the spatio-temporal charge dynamics.At lower intensities I < I c , the multi-photon driven non-linear polarization currents lead to a localized accumu-lation of charge in between the Si-O bonds as displayedin the time-averaged density fluctuations at the conclu-sion of the laser pulse (Fig. 3f). This implies the forma-tion of an induced atomic-scale dipole around the oxygenatoms, i.e. a displacement of the center of charge by ver-tical excitation, resembling the shift current mechanismof the standard photogalvanic effect but generalized tohigher order reflected in the non-linear intensity scalingof | D | ∼ I . . For higher laser intensities I > I c thedominant charge transfer mechanism is excitation of thetilted conduction band by tunneling. Tunneling signifi-cantly depends on the local potential landscape in tun-neling direction. We find tunneling is enhanced when thebond direction is aligned with the laser field as evidencedby a strongly asymmetric current density at times nearthe maxima of the electric field (Fig. 3d,e). Tunneling ex-citation is more efficient along − ˆ a where the O-Si bond ismore closely aligned with the laser field ( γ = 25 . a direction tunneling is suppressed because ofthe larger angle ( γ = 51 . ∝ exp( − / √ I ), the transition is quite abrupt suggestingits potential for femtosecond current switching.For high intensities I > I c , the charge transfer showssub-cycle time structure. The time-dependence of thetunneling current can be conveniently analyzed by thenon-linear response contribution ∆ J NL ( t ) after subtract-ing the linear-response current scaled to the instanta-neous field,∆ J NL ( t ) = J ( t ) − π Z ∞−∞ e − iωt σ ( ω ) F ( ω ) (6)with the conductivity σ ( ω ) determined for low intensity I < I c [26]. During the rise time of the pulse (Fig. 4)∆ J NL is still ≈ − P ( t ) = R t −∞ dt ′ J ( t ′ ) isin phase with F ( t ), the current spikes are in phase with F ( t ) as expected for tunneling excitation. At later times(from -1 fs), ∆ J NL remains in phase with, but becomes FIG. 4. (Color online) (a) Laser-induced transferred chargedensity as a function of the peak-peak delay ( I = 2 . × W / cm , I = 0 . × W / cm , ω L = 1 . τ p = 20 fs).Red squares, dashed line: TD-DFT simulations; solid line:model Eq. 7. Inset: magnified data around ∆ t ∼ I ( t ) = F ( t ) ( c/ π ) for a given delay. Insets: pulseshape after superposition of pump and probe pulse F ( t ) at∆ t = −
13 fs (weak pulse before strong pulse) and at +10 fs. proportional to the laser field, indicative of a conductor-like linear response J ( t ) = σ D F ( t ) with a Drude (freecarrier-like) conductivity σ D for the tunneling-inducedelectron population in the conduction band.The present analysis of the non-linear photogalvanicDC current suggests that the key control parameter isthe instantaneous intensity I ( t ) rather than the cycle av-eraged intensity in the conventional photogalvanic effector the instantaneous value of the field F ( t ) in the CE-phase controlled AC current. This sensitivity to I ( t ) canbe explored in a pump-probe setting, in which the in-stantaneous intensity can by manipulated by the delaybetween pump and probe pulses. The pump-probe de-lay may therefore serve as knob for fast charge trans-fer by the non-linear photogalvanic effect. To demon-strate this control we choose the intensity of both pumpand probe pulse to be subcritical ( I , < I c ) with pumpintensity I = 2 . × W / cm and probe intensity I = 0 . × W / cm . However, the superimposed fieldsgive rise to a maximum intensity of I max = 2 . I =5 . × W / cm above I c . The sign and amplitude of theinduced current is controlled by the time delay betweenthe laser pulses (Fig. 4). For large positive and nega-tive delays, the transferred charge saturates at the samepositive value. In contrast, for near-zero delay ∆ t = 0where the maximum intensity is attained, the DC currentswitches direction and the transferred charge becomesnegative. Remarkably, during the period of strong over-lap the modulation of the DC current occurs for delays onthe sub-fs time scale resulting from the strongly varyingmaximum instantaneous laser intensity as a function of pump-probe delay (Fig. 4b). Assuming that the chargetransfer is governed by the central peak of the combinedlaser pulse, a simple estimate in analogy to Eq. 5 predicts D (∆ t ) = sgn( I c − I max (∆ t )) β NL I max (∆ t ) . , (7)where sgn denotes the sign function and I max (∆ t ) isthe maximum instantaneous laser intensity for pump-probe delay ∆ t (Fig. 4b). In Eq. 7, we denote the non-linear generalization of the photogalvanic tensor by β NL .This simple model reproduces the temporal variation of D (∆ t ) in the full TD-DFT calculations remarkably well,underlining that the maximum instantaneous laser fielddrives the non-linear photogalvanic effect through tun-neling near the field maximum.In conclusion, we predict a non-linear extension of thephotogalvanic effect into the strong-field regime givingrise to ultrafast DC currents in insulators illuminated bymulti-femtosecond laser pulses. We observe a stronglynon-linear intensity dependence and even a reversal ofthe induced currents above a critical intensity I c asso-ciated with the transition from non-linear polarizationcurrents to tunneling excitation. The charge transferis rather insensitive to details of the laser pulse shapeand carrier-envelope phase but strongly dependent onthe maximum instantaneous field strength. The lattermay be controlled by the pump-probe delay in a two-pulse setup giving rise to a distinct sign change in thetransferred charge as function of the pump-probe delay.The non-linear photogalvanic effect opens up opportuni-ties for light-field controlled femtosecond charge separa-tion with relatively modest requirements on the drivinglaser. Even many-cycle pulses without CEP stabiliza-tion can be used as the lattice structure instead of theCEP is employed to break the inversion symmetry alongthe laser polarization axis. The non-linear photogalvaniceffect is conceptually simpler than the CEP dependentcharge transfer since no elaborate steering of the conduc-tion band electrons is necessary. Therefore, the effectis robust against changes in the laser pulse parameters.This may be advantageous in particular for optical in-terconnects based on surface plasmon propagation [18]where the pulse shape and duration of a surface plasmonwave packet is difficult to control. The sharp thresh-old intensity I c for reversal of the current may providea simple route towards femtosecond current switchingand, moreover, a sensitive intensity calibration for laserpulses that directly measures the maximum electric fieldstrength in the material. Finally, the photogalvanic effectmay also be investigated by associated terahertz emission[41].This work was supported by the FWF (Austria), SFB-041 ViCoM, SFB-049 Next Lite, doctoral college W1243,and P21141-N16. G.W. thanks the IMPRS-APS for fi-nancial support. 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