Theory of strong-field injection and control of photocurrent in dielectrics and wide bandgap semiconductors
Stanislav Yu. Kruchinin, Michael Korbman, Vladislav S. Yakovlev
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a r Theory of strong-field injection and control of photocurrent in dielectrics andwide bandgap semiconductors
S. Yu. Kruchinin, ∗ M. Korbman, and V. S. Yakovlev
1, 2, † Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany Ludwig-Maximilians-Universit¨at, Am Coulombwall 1, 85748 Garching, Germany
We propose a theory of optically-induced currents in dielectrics and wide-gap semiconductorsexposed to a non-resonant ultrashort laser pulse with a stabilized carrier-envelope phase. In or-der to describe strong-field electron dynamics, equations for density matrix have been solved self-consistently with equations for the macroscopic electric field inside the medium, which we modelby a one-dimensional potential. We provide a detailed analysis of physically important quantities(band populations, macroscopic polarization, and transferred charge), which reveals that carrier-envelope phase control of the electric current can be interpreted as a result of quantum-mechanicalinterference of multiphoton excitation channels. Our numerical results are in good agreement withexperimental data.
PACS numbers: 78.20.Bh, 72.80.Sk, 77.22.Ej, 42.50.HzKeywords: dielectric, semiconductor, few-cycle pulse, strong-field physics, ultrafast optics, carrier-envelopephase, coherent control
I. INTRODUCTION
The time it takes electrons in a solid to respond to anexternal electric field is on the order of a fraction of a fem-tosecond. Due to recent progress in ultrafast and attosec-ond science, it is now possible to perform attosecond-scale time-resolved measurements of extremely nonlinearphenomena that occur when intense few-cycle laser pulsesinteract with a solid, and opportunities related to thisprogress have recently begun to be explored. In particu-lar, implicit observations of subcycle temporal structuresassociated with interband tunneling in a dielectric werereported , and effects related to Bloch oscillations in abulk solid were observed .A direct observation of attosecond-scale electron mo-tion was reported by Schiffrin et al. in a very recentpaper demonstrating that electric currents in a fusedsilica sample can be switched and driven by the instan-taneous field of an optical waveform at intensities justbelow the damage threshold. It was argued that, at fieldstrengths where the induced potential difference betweenneighboring unit cells approaches the bandgap energy,the experiment can be interpreted using Wannier–Starkstates. In this paper, we show that these kinds of mea-surements can also be explained within a more conven-tional approach based on interference between differentmultiphoton channels. This relates the results of Ref. 5to the scope of coherent control.Injection and coherent control of electric currents insemiconductors was studied before, both in experimentsand theory . Coherent control was demonstrated in( ω , 2 ω ) schemes based on the interference of one- andtwo-photon excitation pathways induced by a laser pulseand its second harmonic. In such experiments, it is suffi-cient for pulse durations and time delays to be less thanor comparable to the carrier dephasing time ( ∼
100 fs).Now, with the availability of the few-cycle pulses, con-trol over the photocurrent can be achieved within a single laser pulse and on much shorter time scales ( ∼ , and ageneral abstract theory that describes all phase effectsin these terms is available . However, it is not obvi-ous whether concepts developed for relatively weak fieldscan be applied to interpret experiments with intense few-cycle pulses, in which the absorption of many photons isrequired to excite valence-band electrons, and laser fieldamplitude reaches values at which perturbation theory isexpected to break down.To investigate the non-resonant optical injection andcontrol of electronic currents in dielectrics, we developa model based on a self-consistent solution of multibandoptical Bloch equations (OBE) together with equa-tions for the dielectric polarization and field inside thecrystal. The paper is organized as follows. In Sec. II wediscuss our theoretical formalism, the gauge choice, andapproximations that we make calculating the amount ofcharge flowing through a capacitor-like junction. Resultsof our numerical simulations, their interpretation andcomparison to experimental data are given in Sec. III.Sec. IV presents our conclusions. II. THEORY
In order to calculate the current and polarization in-duced by an ultrashort pulse, we solve the density-matrix equations in the independent particle approxi-mation. Unlike the time-dependent Schr¨odinger equa-tion , this approach takes into account Pauli block-ing of interband transitions and it can be extendedto account for electron-electron scattering and interac-tion with the bath. The approximation of independentparticles is reasonable for ultrafast strong-field phenom-ena since the carrier-field interaction is much strongerthan the carrier-carrier interaction. Another argumentin favor of this approximation is that the net currentonly depends on the total electronic momentum, whichis not changed by electron-electron interaction . Ontimescales much shorter than a period of lattice oscilla-tions, we can also neglect the electron-phonon interac-tion. For longitudinal optical (LO) phonons in typicalsemiconductors and dielectrics, the oscillation period isabout tens of femtoseconds.In the basis of Bloch states, the Hamiltonian of theelectronic subsystem of a dielectric interacting with alaser field can be written as H = H + H int ( t ) , (1) H = X ℓ ∈ CB ǫ e ℓ c † ℓ c ℓ + X j ∈ VB ǫ h j d † j d j , (2) H int ( t ) = X ℓ ,ℓ ∈ CB M ℓ ℓ ( t ) c † ℓ c ℓ + X j ,j ∈ VB M j j ( t ) d † j d j + X ℓ,j h M ℓj ( t ) c † ℓ d † j + M ∗ ℓj ( t ) d j c ℓ i , (3)where ℓ = { f, k ′ , s ′ } and j = { i, − k , s } are compositeindices that include a band index ( i and f ), crystal mo-mentum ( k and k ′ ) and spin ( s and s ′ ).In the velocity gauge (VG), the amplitude of interac-tion with the optical field can be written as M VG fi ( k ′ , k , t ) = − σ fi em A ( t ) δ ( k ′ − k )[ δ fi ~ k + p fi ( k )]+ δ ( k ′ − k ) δ fi e A ( t )2 m , (4)where A ( t ) is the vector potential in a medium, indices i and f enumerate all valence and conduction bands, σ fi is a band-specific sign function defined as σ fi = (cid:26) + , i, f ∈ VB , − otherwise , (5)and p fi ( k ) = − i ~ Ω Z Ω d r u ∗ f, k ( r ) ∇ r u i, k ( r ) (6)is the momentum matrix element between Bloch ampli-tudes u i, k , Ω being the volume of an elementary cell.The term proportional to A ( t ), can be eliminated inthe dipole approximation by the following gauge trans-formation :Ψ ′ ( t ) = Ψ( t ) exp i ~ e m t Z −∞ A ( t ′ ) dt ′ . (7) The expression for the interaction amplitude in thelength gauge can be written as M LG fi ( k ′ , k , t ) = − σ fi e E ( t )[ δ fi i ∇ k + ξ fi ( k )] δ ( k ′ − k ) , (8)where E ( t ) is the electric field in the medium, and ξ fi ( k ) = iΩ Z Ω d r u ∗ f, k ( r ) ∇ k u i, k ( r ) . (9)In the present paper, we use the velocity gauge be-cause it results in a system of dynamic equations in whichdifferent k -states are uncoupled. In contrast, the cor-responding length-gauge equations have a term propor-tional to ∇ k δ ( k ′ − k ), which couples different k -pointsand introduces singularities, and matrix elements ξ fi ( k ),which are not uniquely defined. These problems withthe length gauge in the description of optical phenomenain solids are well-known, and have been solved in recenttheoretical treatments .Nevertheless, the treatment of the crystal polarizationin the velocity gauge has a few important drawbacks,discussed, for example, in Ref. 23. First, the solutionof dynamic equations requires a large number of bands,only a few of which contain a significant carrier popula-tion at the end of a simulation. Second, a certain sumrule for momentum matrix elements must be satisfied .Violation of these conditions leads to serious numericalartifacts and unphysical results. In other words, a de-scription of strong-field phenomena in the velocity gaugerequires an accurate solution of the stationary problem.Equations in the length gauge have an advantage of lessstrict requirements to the quality of the eigenproblem so-lution, so that the energy spectrum and optical matrixelements could be calculated with more simple and roughmethods, or even used as adjustable parameters .Despite these problems, a fully ab initio approachbased on time-dependent density functional theory inthe velocity gauge was developed and successfullyapplied to describe the non-perturbative polarization re-sponse of solids to a strong field. Such calculations arevery resource-demanding and require a high-performancesupercomputer.Our present approach is based on the numerical solu-tion of density-matrix equations for a one-dimensionallattice and has very moderate computational require-ments. Yet it allows us to reproduce important featuresof laser-matter interaction on the ultrashort time scalefor intensities near the damage threshold, and it givesgood agreement with experimental data.Starting from the definitions of two-point density-matrix elements, ρ ( k ) jℓ ≡ h d j, − k c ℓ, k i , ρ ( k ) ℓℓ ′ ≡ h c † ℓ, k c ℓ ′ , k i ,ρ ( k ) jj ′ ≡ h d † j, − k d j ′ , − k i , Eqs. (1)–(4), and the Liouville–von Neumann equation dρdt = 1 i ~ [ H, ρ ] , one obtains the following system of equations for each k in the Brillouin zone: ddt ρ jℓ = 1 i ~ " X j ′ (cid:16) E jj ′ ρ j ′ ℓ − M ℓj ′ ρ j ′ j (cid:17) + X ℓ ′ (cid:16) E ℓℓ ′ ρ jℓ ′ − M ℓ ′ j ρ ℓ ′ ℓ (cid:17) + M ℓj ,ddt ρ ℓℓ ′ = 1 i ~ " X ℓ ′′ (cid:16) E ℓ ′ ℓ ′′ ρ ℓℓ ′′ − E ℓ ′′ ℓ ρ ℓ ′′ ℓ ′ (cid:17) + X j (cid:16) M ℓ ′ j ρ ∗ jℓ − M ∗ ℓj ρ jℓ ′ (cid:17) ,ddt ρ jj ′ = 1 i ~ " X j ′′ (cid:16) E j ′ j ′′ ρ jj ′′ − E j ′′ j ρ j ′′ j ′ (cid:17) + X ℓ (cid:16) M ℓj ′ ρ ∗ jℓ − M ∗ ℓj ρ j ′ ℓ (cid:17) , (10)where E ℓℓ ′ = δ ℓℓ ′ ǫ ℓ + M ℓℓ ′ , and indices ℓ and j enumerate conduction and valencebands, respectively. Quasimomentum indices are omittedhere, for simplicity.The charge carriers generated by an intense laser pulseinduce a strongly nonlinear response. If the characteristicsize of the sample in the direction of polarization is smallenough, then the field inside the crystal is affected by thesurface charge distribution, which should be taken intoaccount by solution of semiconductor Maxwell–Blochequations . However, a full numerical solution ofthese equations in a three-dimensional case presents arather hard computational problem, so we resort to asimplified model of the dielectric polarization.In the experiment reported in Ref. 5, the fused silicasample was exposed to a few-cycle pulse with a centralphoton energy ~ ω L ≈ . ∼ µ m) but much thicker than the electron’sde Broglie wavelength. In this case, the effects of quan-tum confinement are negligible, and thus the stationarystates are described by Bloch wavefunctions. An externalfield polarizes the sample, and the induced surface chargecreates a screening field. From the condition of continu-ity of the normal component of the electric displacementfield at the surface between two media D , ⊥ = D , ⊥ itfollows that the electric field E ( t ) inside the sample isconnected with the laser field E L ( t ) and macroscopic po-larization P ( t ) by the following relation (SI units): E ( t ) = E L ( t ) − P ( t ) /ε . (11)From the definition ˙ A = − E , we obtain the equation for A u A u SiO I SD E L zyx Figure 1. (Color online) A schematic representation of themetal-dielectric junction exposed to the electric field E L ofthe laser pulse. Courtesy of A. Schiffrin. the vector potential inside the dielectric: ddt A ( t ) = − E L ( t ) + P ( t ) /ε . (12)The macroscopic polarization satisfies the equation ddt P ( t ) = J ( t ) , (13)where the charge current density is given by the followingexpression : J ( t ) = − em V Z V d r X s (cid:20) ~ i ( ∇ r − ∇ r ′ ) + e A ( t ) (cid:21) × ρ ( r ′ , s ′ , r , s ; t ) (cid:12)(cid:12)(cid:12)(cid:12) r ′ = r ,s ′ = s (14)where averaging is done over a volume V , s and s ′ arespin indices of initial and final states, respectively, and m is the free-electron mass.From this equation we obtain J ( t ) = 2 em Z BZ d k (2 π ) (X f,i σ fi ρ fi ( k , t ) × [ δ fi ~ k + Re p fi ( k )] − N VB e A ( t ) ) , (15)where indices f and i enumerate all valence and conduc-tion bands, the sign function σ fi is defined by Eq. (5),the multiplier 2 accounts for the spin degeneracy, and N VB is the number of valence bands.We solve Eqs. (10), (12), and (13) self-consistently fora given laser field E L ( t ), energy spectrum ǫ i ( k ), and mo-mentum matrix elements p fi ( k ).As long as the external field is not strong enough togenerate a significant number of electron-hole pairs, theinduced screening field can be evaluated from the linearpolarization response: D = ε E . In the approximationof an instantaneous and linear, we evaluate the relativepermittivity ε at a central laser frequency ω L and, fromEq. (11), we obtain A ( t ) = A L ( t ) /ε. (16)In this case only Eqs. (10) for the density matrix mustbe solved.In the next sections, we discuss the comparison of thesetwo approaches, and we refer to the model that includesdielectric screening self-consistently [Eqs. (10), (12),and (13)] as OBE/SCDS, and to the linear screeningmodel [Eqs. (10) and (16)] just as OBE. III. RESULTS AND DISCUSSION
It is well known that nonparabolicity of electronicbands plays an important role in strong-field phenom-ena , and their description requires an accurate solutionof the stationary problem for the entire Brillouin zone.We calculate the stationary electronic levels ǫ i, k and mo-mentum matrix elements p fi ( k ) using the plane-wavepseudopotential method .The eigenvalue problem for the crystal is written as (cid:20) − ~ m ∇ + U ( r ) (cid:21) φ n, k ( r ) = ǫ n, k φ n, k ( r ) , where φ n, k ( r ) = u n, k ( r ) exp(i kr ) is the wave functionof an electron, u n, k ( r ) is a Bloch amplitude, and U ( r )is a periodical lattice potential. Since u n, k ( r ) has theperiodicity of the lattice, it can be expanded in a Fouriersum: u n, k ( r ) = X m C ( k ) nm exp(i K m r ) , (17)where K m are the reciprocal-lattice vectors.Assuming that the laser pulse is linearly polarized, wemay reduce the problem to one spatial dimension andsolve the density-matrix equations for the k -states thatbelong to a certain direction in the Brillouin zone. Inthis paper, we consider a one-dimensional lattice withthe pseudopotential U ( z ) = c (1 − tanh c z ) . (18)We assume that the laser field polarization is parallelto the [001] direction of α -quartz, for which the latticeconstant is a k ≡ c = 5 . c = − . c = 0 . E g ≈ m c ≈ . m [see Fig. 2(a)]. Thesevalues are in good agreement with the results for the Γ–A direction in the Brillouin zone of α -quartz obtainedwith more complex and rigorous treatments .It is worth noting that diagonal momentum matrix el-ements between the full Bloch functions P ii ≡ ~ k + p ii ( k )quickly approach zero at the boundaries of the Brillouin (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6)(cid:7) (cid:1) (cid:8) (cid:4) (cid:9) (cid:10) (cid:11)(cid:12)(cid:13)(cid:13)(cid:12)(cid:13)(cid:14)(cid:13) (cid:1) (cid:2)(cid:2) (cid:3)(cid:4) (cid:15) (cid:5) (cid:11)(cid:12) (cid:11)(cid:13)(cid:16)(cid:17) (cid:13) (cid:13)(cid:16)(cid:17) (cid:12) (cid:6) (cid:7) (cid:18)(cid:8)(cid:19)(cid:10) (cid:20)(cid:12)(cid:20)(cid:14)(cid:20) (cid:1) (cid:2)(cid:2) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5)(cid:1) (cid:6) (cid:5) (cid:7) (cid:8)(cid:9)(cid:5)(cid:10)(cid:8)(cid:9)(cid:5)(cid:11)(cid:9)(cid:9)(cid:5)(cid:11)(cid:9)(cid:5)(cid:10) (cid:3) (cid:4)(cid:4) (cid:5)(cid:6) (cid:12) (cid:7) (cid:8)(cid:13) (cid:8)(cid:9)(cid:5)(cid:14) (cid:9) (cid:9)(cid:5)(cid:14) (cid:13) (cid:2) (cid:1)(cid:15)(cid:1)(cid:16) (cid:2) (cid:1)(cid:15)(cid:1)(cid:17)(cid:2)(cid:18)(cid:7) Figure 2. (Color online) (a) Band structure obtained vianumerical solution of the eigenvalue problem for the one-dimensional periodic potential given by Eq. (18). (b) Di-agonal matrix elements of the momentum operator P ii ≡ ~ k + p ii ( k ) versus k for the topmost valence and lowest con-duction bands. zone, i.e. electrons are slowing down in this region [seeFig. 2(b)].To produce results that could be compared with ex-perimental data, we must specify a proper connectionbetween our one-dimensional calculations and physicalvalues defined for a three-dimensional crystal. We as-sume that the electromagnetic wave is polarized alongthe z -axis and the current density slowly changes withinthe xy -plane. Then the current density J ( t ) in a three-dimensional (3D) crystal is connected with its 1D coun-terpart via the effective cross-section S of the three-dimensional Brillouin zone, J ( t ) = Z BZ d k (2 π ) j ( k , t ) = n S (2 π ) π/a k Z − π/a k dk z π j ( k z , t ) , where j ( k z , t ) is the one-dimensional analog of the ex-pression in curly brackets from Eq. (15), and n is the unitvector in the direction of the current. In a first approxi-mation, the parameter S can be estimated directly fromthe properties of the lattice for a given material. For α -quartz with a hexagonal lattice and current directionalong the [001] direction, we get S SiO = 3 √ (cid:18) πa ⊥ (cid:19) ≈ . , where a ⊥ ≡ a = 4 . S can be found moreprecisely from the requirement that the OBE solutionshould give a correct linear response in a low-intensityregion. With a given linear susceptibility χ (1) and po-larization calculated from Eq. (10), we have S SiO ≈ .
43 at. u., which is in good agreement with the aboveestimation.For numerical integration of Eqs. (10), (12), and (13),we discretized the k -space and applied the fourth-orderRunge–Kutta method with a constant stepsize. At thehighest field intensities considered in this paper, numer-ical convergence was achieved with 18 bands and 201 k -points. The requirement of a large number of bands isa consequence of the velocity gauge drawbacks discussedearlier in Sec. II.In subsequent calculations, we use the following ex-pression for the vector potential of the laser field: A L ( t ) = − A θ ( τ L − | t | ) cos (cid:18) πt τ L (cid:19) sin( ω L t + ϕ CE ) , (19)where ω L is the central frequency of the laser pulse, θ ( x )is the Heaviside step function, A = E /ω L and E arethe amplitudes of vector potential and electric field in thevacuum, respectively, ϕ CE is the carrier-envelope phase.Total pulse duration τ L is related to FWHM of | A ( t ) | as τ L = πτ FWHM / [4 arccos(2 − / )] ≈ . τ FWHM .Let us start the discussion of our numerical results withthe distribution of populations in the first conductionband at the moment of time when the electromagneticfield is over. Fig. 3 shows that the few-cycle pulse gener-ates an asymmetric band population distribution in thecrystal momentum space, which causes the appearanceof macroscopic current.We argue that these symmetry-breaking effects of thewaveform with a well-defined CE phase emerge due to theinterference of different multiphoton excitation pathways.This phenomenon is schematically depicted on Fig. 4,where arrows of different colors describe different mul-tiphoton channels that interfere with each other. Note,that excitation pathways might have the same numberof photons with different frequencies, as well as differentnumbers of photons. In this picture, physical observablesbecome CEP-dependent if the pulse is short enough, i. e.when its spectral width allows for existence of multipho-ton channels with odd and even numbers of photons forthe same k -point in the Brillouin zone. The interferenceof optical excitation pathways might be constructive for k and destructive for − k . When the laser pulse is over,electron-phonon collisions become the dominant interac-tion, which quickly restores the symmetry of electronicpopulation distribution and make the current disappear.Figs. 3(a) and (b) show that the maximum populationdistribution is shifted from the center, and the electronexcitations spread over the whole Brillouin zone whenthe field intensity is increased. This situation is stronglycontrary to the case of a weak field, where most of thecharge carriers are situated around the extremal pointsin the Brillouin zone. At a relatively low field ampli-tude of E ∼ E ∼ . In addition to that, the interaction of electronswith the screening field induced by quantum beats re-sults in a phenomenon analogous to stimulated emission.The induced screening field E ( t ) oscillates out of phasewith respect to the polarization P ( t ). It drives transi-tions from the conduction band to the valence band, andthus, decreases the amplitude of quantum beats.The time dependence of polarization [Fig. 5(b)] showsthat the self-consistent screening model predicts low-frequency oscillations that persist after the laser pulse.These oscillations occur as a result of collective electronmotion driven by the surface charge field. For the pa-rameters of our simulations, the frequency of these os-cillations is in the terahertz region (e. g. 50 THz for E = 2 . . Also, there is an important analogy with theoptical rectification effect .Since we do not consider relaxation phenomena (radia-tive or non-radiative), these plasma oscillations do notdecay with time in our simulations. Their amplitudemight even grow, if the screening field is large enoughto induce interband transitions. Also, the dielectric po-larization model does not take into account the energyloss due to the emission of electromagnetic radiation, sothat the dielectric acts like a resonator for the inducedcurrents. Consequently, the results of our present simu-lations are valid only within a small interval of a few fem-toseconds after the laser pulse. A more accurate descrip-tion of the strong-field phenomena requires the completesolution of semiconductor Maxwell–Bloch equations thattake into account electron-electron and electron-phononinteractions, as well as electromagnetic radiation fromaccelerated charge carriers.In the experiment reported by Schiffrin et al. in Ref. 5, (cid:1)(cid:2) (cid:3) (cid:4) (cid:1) (cid:5) (cid:2)(cid:6) (cid:2) (cid:2) (cid:4)(cid:7)(cid:4)(cid:8)(cid:4)(cid:9)(cid:10) (cid:1) (cid:1)(cid:2) (cid:11) (cid:4) (cid:1) (cid:5) (cid:2)(cid:1)(cid:8)(cid:12) (cid:2) (cid:2) (cid:4)(cid:7)(cid:4)(cid:1)(cid:4)(cid:9)(cid:10) (cid:1) (cid:3) (cid:4)(cid:13)(cid:4)(cid:2) (cid:3) (cid:4)(cid:14)(cid:4)(cid:2)(cid:15)(cid:16)(cid:17) (cid:1)(cid:2) (cid:12) (cid:4) (cid:1) (cid:5) (cid:2)(cid:1)(cid:8)(cid:12) (cid:4) (cid:18)(cid:18) (cid:4)(cid:18)(cid:4) (cid:3) (cid:4)(cid:18)(cid:10) (cid:5) (cid:2) (cid:2)(cid:19)(cid:8) (cid:2)(cid:19)(cid:3) (cid:2)(cid:19)(cid:11) (cid:2)(cid:19)(cid:20) (cid:1) (cid:2) (cid:2) (cid:4)(cid:7)(cid:4)(cid:8)(cid:19)(cid:6)(cid:4)(cid:9)(cid:10) (cid:1) (cid:1)(cid:2) (cid:3) (cid:4) (cid:1) (cid:5) (cid:2)(cid:1)(cid:6)(cid:7) (cid:2) (cid:2) (cid:4)(cid:8)(cid:4)(cid:1)(cid:4)(cid:9)(cid:10) (cid:1) (cid:3) (cid:4)(cid:11)(cid:4)(cid:2) (cid:3) (cid:4)(cid:12)(cid:4)(cid:2)(cid:13)(cid:14)(cid:15) (cid:1)(cid:2) (cid:16) (cid:4) (cid:1) (cid:5) (cid:2)(cid:6)(cid:16)(cid:3)(cid:17) (cid:2) (cid:2) (cid:4)(cid:8)(cid:4)(cid:6)(cid:4)(cid:9)(cid:10) (cid:1) (cid:1)(cid:2) (cid:6) (cid:4) (cid:1) (cid:5) (cid:2)(cid:6)(cid:16)(cid:3)(cid:17)(cid:1)(cid:2) (cid:4) (cid:18)(cid:18) (cid:4)(cid:18)(cid:4) (cid:3) (cid:4)(cid:18)(cid:10) (cid:5) (cid:2) (cid:2)(cid:19)(cid:6) (cid:2)(cid:19)(cid:16) (cid:2)(cid:19)(cid:3) (cid:2)(cid:19)(cid:17) (cid:1) (cid:2) (cid:2) (cid:4)(cid:8)(cid:4)(cid:6)(cid:19)(cid:20)(cid:4)(cid:9)(cid:10) (cid:1) Figure 3. (Color online) Distributions of the lowest conduction band population n c after the laser pulse ( λ L = 800 nm,FWHM = 4 fs, φ CE = 0) for different field amplitudes, calculated (a) with OBE and (b) with OBE/SCDS. Populations forpositive and negative crystal momenta are shown with the dashed and solid lines, respectively. E n e r gy Crystal momentum cv Figure 4. (Color online) A schematic representation of multi-photon channel interference at different points in the Brillouinzone. the measurements yielded the total transferred charge Q P in a fused silica junction defined as Q P = A eff q P , (20) where q P = P ( t → ∞ ) ≡ ∞ Z −∞ J ( t ) dt (21)is the transferred charge density, and A eff is the effectivecross section of the active volume, estimated as ∼ × − m .This simple definition is directly applicable if theoryincludes all necessary relaxation mechanisms, both ra-diative and non-radiative, and the integral of charge cur-rent density over time takes a finite value. In order tocompare our numerical results with the experiment, weneed to estimate the value of integral (21), taking intoaccount the applicability limitations due to the absenceof relaxation mechanisms in our OBE/SCDS model. As-suming that the current quickly decays after the laserpulse, we estimate the integral (21) by the value of po-larization right after the pulse. Thus, we can define thetransferred charge density as an average value of the po-larization taken in a time interval that is larger than theperiod of quantum beats T b and smaller than the pe-riod of the long-wave polarization oscillations T P after (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6)(cid:1) (cid:7) (cid:6) (cid:8) (cid:9)(cid:10)(cid:6)(cid:11)(cid:10)(cid:10)(cid:6)(cid:11) (cid:2) (cid:1)(cid:2)(cid:12)(cid:13)(cid:8)(cid:9)(cid:14)(cid:10) (cid:9)(cid:15) (cid:10) (cid:15) (cid:14)(cid:10) (cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:20) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6)(cid:1) (cid:7) (cid:6) (cid:8) (cid:9)(cid:10)(cid:6)(cid:11)(cid:10)(cid:10)(cid:6)(cid:11)(cid:2)(cid:3)(cid:8) (cid:16)(cid:17)(cid:18) (cid:1) (cid:2)(cid:3)(cid:4)(cid:3)(cid:3)(cid:5)(cid:2)(cid:3)(cid:4)(cid:3)(cid:3)(cid:6)(cid:3)(cid:3)(cid:4)(cid:3)(cid:3)(cid:6)(cid:3)(cid:4)(cid:3)(cid:3)(cid:5) (cid:1) (cid:3) (cid:1)(cid:7)(cid:1)(cid:8)(cid:1)(cid:9)(cid:10) (cid:1) (cid:2) (cid:1) (cid:11) (cid:12) (cid:13) (cid:14) (cid:4)(cid:1) (cid:15) (cid:4) (cid:16) (cid:2)(cid:3)(cid:4)(cid:3)(cid:5)(cid:2)(cid:3)(cid:4)(cid:3)(cid:6)(cid:3)(cid:3)(cid:4)(cid:3)(cid:6)(cid:3)(cid:4)(cid:3)(cid:5) (cid:1) (cid:3) (cid:1)(cid:7)(cid:1)(cid:6)(cid:1)(cid:9)(cid:10) (cid:1) (cid:1) (cid:2)(cid:3)(cid:4)(cid:3)(cid:17)(cid:3)(cid:3)(cid:4)(cid:3)(cid:17) (cid:3) (cid:1)(cid:11)(cid:18)(cid:19)(cid:16)(cid:17) (cid:8)(cid:3) (cid:8)(cid:17) (cid:6)(cid:3) (cid:6)(cid:17) (cid:1) (cid:3) (cid:1)(cid:7)(cid:1)(cid:6)(cid:4)(cid:17)(cid:1)(cid:9)(cid:10) (cid:1) (cid:11)(cid:14)(cid:16) Figure 5. (Color online) (a) Polarization versus time calcu-lated with OBE (top panel) and with OBE/SCDS (bottompanel), E = 2 V/˚A, λ L = 800 nm, ϕ CE = 0. (b) Polarizationversus time for OBE (dotted line) and OBE/SCDS (solid line)at the end of the laser pulse. For the field intensities close tothe damage threshold, the low-frequency oscillations in polar-ization response are observed. the laser pulse at the highest considered field intensity, q P = 1∆ t τ L +∆ t Z τ L P ( t ) dt, T b ≪ ∆ t ≪ T P . In the subsequent calculations, we assume that ∆ t = 1 fs.To convert the calculated values from atomic to SI units, we use the following relation: q P (C / m ) = ea S (2 π ) × q P (at . u . ) ≈ . × q P (at . u . ) , where a B is the Bohr radius.Fig. 6 shows typical dependencies of the transferredcharge density on the absolute value of the carrier-envelope phase. The most striking difference between theresults obtained with OBE (left panel) and those withOBE/SCDS (right panel) is the large shift (about 0 . π )of the CE phase that maximizes the transferred charge.This phase shift has a simple explanation: in the modelwith constant screening the phase of the field inside thecrystal is exactly the same as that of the laser field. Inthe self-consistent screening model, the field inside themedium receives an additional phase shift from the po-larization generated by the macroscopic charge currentand quantum beats. (cid:1) (cid:1) (cid:2) (cid:3) (cid:4)(cid:5) (cid:6)(cid:7) (cid:2) (cid:8) (cid:9) (cid:10) (cid:11) (cid:12) (cid:13)(cid:5)(cid:14)(cid:11)(cid:13)(cid:5)(cid:14)(cid:4)(cid:5)(cid:5)(cid:14)(cid:4) (cid:2) (cid:8)(cid:15) (cid:2)(cid:3)(cid:16)(cid:17)(cid:18)(cid:9)(cid:19)(cid:12)(cid:5) (cid:5)(cid:14)(cid:11) (cid:5)(cid:14)(cid:20) (cid:5)(cid:14)(cid:21) (cid:5)(cid:14)(cid:22) (cid:4)(cid:3)(cid:17)(cid:12) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4)(cid:5) (cid:6)(cid:7) (cid:2) (cid:8) (cid:9) (cid:10) (cid:11) (cid:12) (cid:13)(cid:5)(cid:14) (cid:15)(cid:13)(cid:5)(cid:14)(cid:11)(cid:15)(cid:5)(cid:5)(cid:14)(cid:11)(cid:15)(cid:5)(cid:14)(cid:15) (cid:2) (cid:8)(cid:16) (cid:2)(cid:3)(cid:17)(cid:18)(cid:19)(cid:9)(cid:20)(cid:12)(cid:5) (cid:5)(cid:14)(cid:11) (cid:5)(cid:14)(cid:21) (cid:5)(cid:14)(cid:22) (cid:5)(cid:14)(cid:23) (cid:4)(cid:3)(cid:24)(cid:12)(cid:3)(cid:24)(cid:12) Figure 6. (Color online) Transferred charge density q P ver-sus carrier-envelope phase obtained from (a) OBE and (b)OBE/SCDS calculations with the following values of the fieldamplitude: 1 .
15, 1 .
2, 1 .
25, 1 .
3, and 1 .
35 V/˚A.
As a justification of our model, we provide a compar-ison of our results with the measurements published inRef. 5. Since a self-consistent evaluation of the screeningfield is essential for very strong fields, we discuss only theOBE/SCDS model. Fig. 7(a) shows very good agreementbetween theory and experiment up to E = 2 . (cid:1) (cid:1) (cid:2) (cid:3)(cid:4) (cid:5) (cid:6) (cid:7)(cid:7)(cid:8)(cid:9)(cid:10)(cid:10)(cid:8)(cid:9)(cid:11) (cid:2) (cid:7) (cid:2)(cid:3)(cid:12)(cid:13)(cid:14)(cid:6)(cid:10) (cid:10)(cid:8)(cid:9) (cid:11) (cid:11)(cid:8)(cid:9)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:18)(cid:22)(cid:23)(cid:24)(cid:25)(cid:15)(cid:13)(cid:26)(cid:5)(cid:27)(cid:26)(cid:3)(cid:28)(cid:6) (cid:1) (cid:1) (cid:2) (cid:3)(cid:4) (cid:5) (cid:6) (cid:7)(cid:8)(cid:7)(cid:7) (cid:9)(cid:10)(cid:7)(cid:8)(cid:7)(cid:11)(cid:12)(cid:7)(cid:8)(cid:11)(cid:13)(cid:11) (cid:2) (cid:7) (cid:2)(cid:3)(cid:14)(cid:15)(cid:16)(cid:6)(cid:11) (cid:11)(cid:8)(cid:10) (cid:9) (cid:9)(cid:8)(cid:10)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:20)(cid:24)(cid:25)(cid:26)(cid:27)(cid:17)(cid:15)(cid:28)(cid:5)(cid:29)(cid:28)(cid:11)(cid:8)(cid:30)(cid:2)(cid:31)(cid:2)(cid:11)(cid:7) (cid:32)(cid:30) (cid:2) (cid:2) (cid:7)(cid:11)(cid:11) (cid:3)(cid:33)(cid:6) Figure 7. (Color online) (a) Intensity scan of the transferredcharge maximized over CEP: comparison of OBE/SDCS cal-culations with experimental data. (b) Results of fitting thecalculated transferred charge dependence (solid line) and ex-perimental data on a double-logarithmic plot with the powerfunction of field amplitude. tween 5- and 6-photon channels. In this case, Q P is ex-pected to scale as E , and we indeed find that thefunction ∝ E gives a good fit for E . . n -photon absorption p n ( ω ) ∼ F [ I n ( t )]exceeds 90% of its peak value. Here I ( t ) is the cycle-averaged intensity.Even though a longer pulse creates more charge carri-ers, the transferred charge decreases with the pulse du-ration for both low and high field amplitudes (Figs. 8(a)and (b), respectively). This can be explained by the de-creasing overlap of multiphoton channels (see the bottompanel in Fig. 8).Second, we consider the field amplitude scans of thetransferred charge density for the pulses with differentcentral frequencies. Fig. 9 shows that the numericallycalculated transferred charge for the scans in the region Figure 8. (Color online) Top panel: CEP-optimized trans-ferred charge density versus FWHM of the laser pulse for high-and low-field amplitudes. Bottom panel: spectral overlap for5-, 6-, and 7-photon excitation probabilities as a function ofFWHM. (cid:1) (cid:1) (cid:2) (cid:3) (cid:4)(cid:5) (cid:6)(cid:7) (cid:2) (cid:8) (cid:9) (cid:10) (cid:11) (cid:12) (cid:5)(cid:13)(cid:5)(cid:5)(cid:14)(cid:4)(cid:5)(cid:13)(cid:5)(cid:15)(cid:16)(cid:5)(cid:13)(cid:7)(cid:15)(cid:11)(cid:13)(cid:16) (cid:2) (cid:5) (cid:2)(cid:3)(cid:17)(cid:9)(cid:18)(cid:12)(cid:4) (cid:4)(cid:13)(cid:16) (cid:11) (cid:11)(cid:13)(cid:16) (cid:2) (cid:5)(cid:4)(cid:4) (cid:1) (cid:1)(cid:2) (cid:3)(cid:19) (cid:20) (cid:2)(cid:21)(cid:2)(cid:19) (cid:22) (cid:9)(cid:16)(cid:13)(cid:7)(cid:12) (cid:2) (cid:5)(cid:23) (cid:1) (cid:1)(cid:2) (cid:3)(cid:19) (cid:20) (cid:2)(cid:21)(cid:2)(cid:19) (cid:22) (cid:9)(cid:15)(cid:12)
Figure 9. (Color online) Double-logarithmic plot of CEP-optimized transferred charge density and its fit with powersof E versus field amplitude for different central frequenciesof the laser pulse (FWHM = 4 fs). E n +1 thatcorrespond to the interference between n and ( n +1) pho-ton channels. Despite the success of this interpretation,it is clear that this simple power law cannot be validunder all circumstances. For example, interference be-tween more than two multiphoton channels, transitionsto higher conduction bands, and non-perturbative phe-nomena affect the scaling law. IV. CONCLUSIONS
We have presented a quantum-kinetic approach for thedescription of strong-field non-resonant injection of pho-tocurrents in dielectrics and wide bandgap semiconduc-tors. In this model, the appearance of a non-zero trans-ferred electric charge and its dependence on the carrier-envelope phase are intrinsically related to an asymme-try of electronic population distribution in k -space. Thisasymmetry, which also remains after the laser pulse, canbe explained by the interference of different multiphotonexcitation channels. We have presented two argumentssupporting this interpretation. First, we varied the band-width of the laser pulse and found that the amount oftransferred charge is largely determined by the overlapof the dominant excitations channels. Second, we haveshown that, for fields up to E ∼ . E , where the concentration of excited charge carriers reaches a level where our mean-field descriptionbecomes inappropriate and plasma oscillations start todetermine the polarization response.We have also compared two models for the evaluationof the screening field due to charges appearing on sur-faces of a mesoscopic structure exposed to a laser pulse:a model assuming an instantaneous linear dielectric re-sponse, and a more rigorous one where the polarizationresponse is evaluated self-consistently with quantum dy-namics. We have found that the self-consistent evalua-tion of the screening field is essential for an accurate de-scription of CEP effects. For instance, these two modelspredict different dependencies of the transferred chargeon the carrier-envelope phase and, thus, the different val-ues of φ CE which maximize the current. V. ACKNOWLEDGMENTS
We gratefully acknowledge A. Schiffrin, N. Karpowicz,T. Paasch-Colberg, and Prof. F. Krausz for fruitful dis-cussions. This work is supported by the DFG Cluster ofExcellence: Munich-Centre for Advanced Photonics. ∗ [email protected] † [email protected] M. Gertsvolf, M. Spanner, D. M.Rayner, and P. B. Corkum,J. Phys. B: At. Mol. Opt. Phys. , 131002 (2010). A. V. Mitrofanov, A. J. Verhoef, E. E. Serebryannikov,J. Lumeau, L. Glebov, A. M. Zheltikov, and A. Baltuˇska,Phys. Rev. Lett. , 147401 (2011). S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F.DiMauro, and D. A. Reis, Nature Physics , 138 (2011). S. Ghimire, A. D. DiChiara, E. Sistrunk, U. B.Szafruga, P. Agostini, L. F. DiMauro, and D. A. Reis,Phys. Rev. Lett. , 167407 (2011). A. Schiffrin, T. Paasch-Colberg, N. Karpowicz,V. Apalkov, D. Gerster, S. M¨uhlbrandt, M. Korb-man, J. Reichert, M. Schultze, S. Holzner, J. V. Barth,R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I.Stockman, and F. Krausz, Nature , 70 (2013). G. Kurizki, M. Shapiro, and P. Brumer,Phys. Rev. B , 3435 (1989). R. Atanasov, A. Hach´e, J. L. P. Hughes, H. M. van Driel,and J. E. Sipe, Phys. Rev. Lett. , 1703 (1996). A. Hach´e, Y. Kostoulas, R. Atanasov, J. L. P.Hughes, J. E. Sipe, and H. M. van Driel,Phys. Rev. Lett. , 306 (1997). J. Rioux and J. Sipe, Physica E , 1 (2012). T. T. Nguyen-Dang, C. Lefebvre, H. Abou-Rachid, andO. Atabek, Phys. Rev. A , 023403 (2005). T. Nakajima and S. Watanabe,Phys. Rev. Lett. , 213001 (2006). V. Roudnev and B. D. Esry,Phys. Rev. Lett. , 220406 (2007). H. Haug and S. W. Koch,
Quantum theory of the opticaland electronic properties of semiconductors (World Scien-tific, 2004). F. Rossi and T. Kuhn, Rev. Mod. Phys. , 895 (2002). H. Bachau, A. N. Belsky, P. Martin, A. N. Vasil’ev, andB. N. Yatsenko, Phys. Rev. B , 235215 (2006). M. Korbman, S. Yu.. Kruchinin, and V. S. Yakovlev, NewJ. Phys. , 013006 (2013). V. M. Axt and T. Kuhn, Rep. Prog. Phys. , 433 (2004). S. Datta,
Electronic Transport in Mesoscopic Systems (Cambridge University Press, 1995). C. Jacobini,
Theory of Electronic Transport in Semicon-ductors: A pathway from Elementary Physics to Nonequi-librium Green Functions (Springer, 2010). B. H. Bransden and C. J. Joachain,
Quantum Mechanics (Pearson Education Limited, 2000). E. I. Blount (Academic Press, 1962) pp. 305–373. C. Aversa and J. E. Sipe, Phys. Rev. B , 14636 (1995). K. S. Virk and J. E. Sipe, Phys. Rev. B , 035213 (2007). T. Otobe, M. Yamagiwa, J.-I. Iwata, K. Ya-bana, T. Nakatsukasa, and G. F. Bertsch,Phys. Rev. B , 165104 (2008). K. Yabana, T. Sugiyama, Y. Shinohara, T. Otobe, andG. F. Bertsch, Phys. Rev. B , 045134 (2012). O. Hess and T. Kuhn, Phys. Rev. A , 3347 (1996). H. Giessen, A. Knorr, S. Haas, S. W. Koch, S. Linden,J. Kuhl, M. Hetterich, M. Gr¨un, and C. Klingshirn,Phys. Rev. Lett. , 4260 (1998). J. Rammer,
Quantum Transport Theory (Perseus Books,1998). M. L. Cohen and J. R. Chelikowsky,
Electronic Structureand Optical Properties of Semiconductors (Springer, 1988). P. M. Schneider and W. B. Fowler,Phys. Rev. Lett. , 425 (1976). J. R. Chelikowsky and M. Schl¨uter,Phys. Rev. B , 4020 (1977). E. Gnani, S. Reggiani, R. Colle, and M. Rudan,IEEE Transactions on Electron Devices , 1795 (2000). T. M. Fortier, P. A. Roos, D. J. Jones, S. T.Cundiff, R. D. R. Bhat, and J. E. Sipe,Phys. Rev. Lett. , 147403 (2004). R. Kersting, K. Unterrainer, G. Strasser, H. F. Kauffmann,and E. Gornik, Phys. Rev. Lett. , 3038 (1997). M. Bonitz, J. F. Lampin, F. X. Camescasse, andA. Alexandrou, Phys. Rev. B , 15724 (2000). G. Meinert, L. B´anyai, P. Gartner, and H. Haug,Phys. Rev. B , 5003 (2000). M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich,Phys. Rev. Lett. , 446 (1962). A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang,D. Bliss, J. Larkin, and M. Alexander,Applied Physics Letters64