Cooper minimum of high-order harmonic spectra from MgO crystal in an ultrashort laser pulse
Yiting Zhao, Xiaoqin Xu, Shicheng Jiang, Xi Zhao, Jigen Chen, Yujun Yang
CCooper minimum of high-order harmonic spectra from MgO crystal in an ultrashort laser pulse
Yiting Zhao , , Xiaoqin Xu , Shicheng Jiang , Xi Zhao , Jigen Chen , ∗ and Yujun Yang † Zhejiang Provincial Key Laboratory for Cutting Tools ,Taizhou University, Taizhou 31800, China Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China State Key Laboratory of Precision Spectroscopy, East China Normal university, Shanghai 200062, China J. R. Macdonald Laboratory, Department of Physics, Kansas State University, Manhattan, Kansas 66506, USA (Dated: November 28, 2019)Cooper minimum structure of high-order harmonic spectra from atoms or molecules has been extensivelystudied. In this paper, we demonstrate that the crystal harmonic spectra from an ultrashort mid-infrared laserpulse also exhibit the Cooper minimum characteristic. Based on the accurate band dispersion and k-dependenttransition dipole moment (TDM) from the first-principle calculations, it can be found that the harmonic spectrafrom MgO crystal along Γ -X direction present a dip structure in the plateau, which is originated from the valleyof TDM by examining the distribution of the harmonic intensity at the k-space. The Cooper minimum featurein crystal HHG will pave a new way to retrieve the band information of solid materials by using HHG from theultrashort mid-infrared laser pulse. PACS numbers:
I. INTRODUCTION
Atoms and molecules irradiated by an intense laser pulsecan produce high-order harmonic generation (HHG) [1–13].Under the influence of the strong laser field, an electron canbe ionized from the bound state, and accelerated in the con-tinuum state, finally it can recombine with the ion and givesrise to the emission of extreme ultraviolet (XUV) radiation[10, 14]. The emitted XUV radiation is closely related to thephotorecombination, thus it could encode the structural infor-mation on the irradiated target and can be used to study struc-tural features of the target and in particular the Cooper mini-mum [15–17], which corresponds to the nodal structure in thebound-free transition matrix element.Because of clear signatures, the photoionization spec-troscopy has been traditionally used to observe the Cooperminimum. In the process of HHG, the photorecombination isessentially the time inverse of photoionization, therefore theCooper minimum should also appear in the harmonic spectrafrom atoms or molecules. The minimum structure in HHGfrom atoms or aligned molecules have been extensively inves-tigated in many works [15–21]. Recently, HHG from solidshas attracted great interest because of significant applicationsin attocecond pulse generation and all-optical reconstructionof the band dispersion of solids [22, 23]. It has demonstratedtheoretically and experimentally that the interband polariza-tion dominates the harmonics above the band gap for MgO,ZnO, and GaAs driven by a mid-infrared laser pulse [24–27].The interband process for solid HHG can be understood by thesemiclasscial recollision model [24, 25, 28–33]: the electronfirstly tunneling excitation from the valence band, then the ac-celeration on the conduction band, finally the electron-holerecombination results in the harmonic photon. Since the har-monic generation from the interband current depends strongly ∗ [email protected] † [email protected] on the transition dipole moment (TDM) [34–36] of the solid,if there exists a zero in the matrix element between the va-lence band and the conduction one, analogous to the harmonicspectra from gaseous media, the Cooper minimum structure isexpected to appear in the harmonic spectra from solids.In this paper, based on the accurate band dispersion andk-dependent TDM from the first-principle calculations, westudy the feature of HHG from MgO crystal in an ultrashortmid-infrared laser pulse. It is found that, the harmonic spectrafrom TDM by the first-principle theory show a clear dip struc-ture, which almost does not depend on the parameters of thedriving laser pulse. Through analyzing the distribution of theharmonic intensity at di ff erent crystal momentums, it is clari-fied that the minimum of TDM leads to the Cooper minimumstructure of HHG spectra. II. THEORY AND MODELSA. Semiconductor Bloch equations
Based on the solution of two-band semiconductor Blochequations (SBEs) [37–40], we investigate the interaction of anultrashort mid-infrared laser pulse with MgO crystal. Atomicunits are used throughout this article, unless stated otherwise.A linearly polarized laser field is propagated along the Γ -X di-rection of MgO, and the corresponding SBEs [26, 41–43] canbe read ∂ p cv ( k , t ) ∂ t = − i ( E c ( k ) − E v ( k ) − i / T ) p cv ( k , t ) + i (cid:2) ρ c ( k , t ) − ρ v ( k , t ) (cid:3) F ( t ) · D cv ( k ) + F ( t ) · (cid:53) k p cv ( k , t ) (1) ∂ρ v ( k , t ) ∂ t = − (cid:2) F ( t ) · D cv ( k ) p cv ( k , t ) (cid:3) + F ( t ) · (cid:53) k ρ v ( k , t ) (2) a r X i v : . [ phy s i c s . a t m - c l u s ] N ov ∂ρ c ( k , t ) ∂ t = (cid:2) F ( t ) · D cv ( k ) p cv ( k , t ) (cid:3) + F ( t ) · (cid:53) k ρ c ( k , t ) (3)Here, E v ( k ) ( E c ( k )) is the dispersion of the highest va-lence (lowest conduction) band contributing to HHG , ρ v ( k , t ) ( ρ c ( k , t )) is the population in the corresponding band, p cv ( k , t ) and D cv ( k )are the microscopic interband polarizationand the transition dipole moment between the conduction andvalence bands, respectively. F ( t ) = ˆ ε F ( t ) is the laser electricfield with a Gaussian envelope and ˆ ε being the polarizationdirection. T is the interband dephasing time. In this paper,T is set to a quarter-cycle of the driving laser field.The intraband current J intra ( t ) because of the motions of thecarriers in the bands under a laser pulse is given by J intra ( t ) = (cid:88) λ = c , v (cid:90) BZ v λ ( k ) ρ λ ( k , t ) d k (4)where v λ ( k ) = (cid:79) k E λ ( k ) is the group velocity and λ is theband index. The interband polarization J inter ( t ) from the re-combination of the electron with the hole can be expressedby J inter ( t ) = ∂∂ t (cid:90) BZ D cv ( k ) p cv ( k , t ) d k + c . c . (5)In this work we are interested in the harmonic spectrum,which is proportional to the absolute square of the projectionof the Fourier-transformed total current onto the laser polar-ization direction, S HHG ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞−∞ [ J intra + J inter ] e i ω t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (6) B. Band structure and transition dipole moment
The interband and intraband currents depend significantlyon the band structure and TDM [24, 35]. By using theVienna Abinitio Simulation Package (VASP) code [44, 45],accurate k-dependent energy bands and TDM are achievedhere. Geometric optimizations of MgO crystal with symmetrygroup Fm m were performed within generalized gradient ap-proximation (GGA) in the parametrization of Perdew-Burke-Ernzerhof (PBE). The energy cuto ff was set to be 400 eV, anda k-point Monkhorst pack mesh of 10 × ×
10 was used inthe Brillouin zone for electronic structure calculations. Theband dispersion and TDM of MgO along the Γ -X directionwere calculated by the HSE06 hybrid function with parame-ter AEXX = D cv ( k ) = i (cid:104) Φ c ( k ) | p | Φ v ( k ) (cid:105) [ E c ( k ) − E v ( k )] (7) Because MgO crystal has inversion symmetry, the TDM be-tween the lowest conduction band and the highest valenceband is a real and even function [41], as presented by the redsolid line in Fig. 1(b).In most of previous works about solid HHG, the TDM wascalculated by the first-order k · p theory [35], D cv ( k ) = id [ ε c (0) − ε v (0)][ E c ( k ) − E v ( k )] (8)which is valid when the carriers in conduction or valencebands are mostly populated at Γ . However, under the inter-action of a strong laser pulse, electrons (holes) may travelthrough the entire Brillouin zone, thus the TDM from the first-order k · p approximation is not applicable to this situation. Forcomparison, the TDM from the first-order k · p theory is alsoshown by the blue dot dashed line in Fig. 1(b). Obviously,TDM in the first-principle case exists minima at k = ± π/ a ( a = et al . proved that, theshape of the k-dependent TDM plays an important role in har-monic generation [35]. Therefore, we will discuss how thedi ff erence in shapes of TDMs e ff ects the HHG spectra. E n e r g y ( e V ) CB1 VB (a) -1.0 -0.5 0.0 0.5 1.0024 D cv ( a . u . ) k ( /a ) k p theory first-principles (b) Fig. 1: (a) The lowest conduction band and highest valence band ofMgO along the Γ -X direction; (b) k-dependent dipole moments fromthe first-principle calculations (red solid curve) and first-order k · p theory (blue dot dashed curve). III. COOPER MINIMUM STRUCTURE OF CRYSTALHHG
In terms of two-band and three-band SBEs, we firstly ex-amine the dependence of crystal HHG spectra from the first-principle calculations on the driving laser intensity, as shownin Figs. 2(a) and 2(b), respectively. Here, the 3 fs / . × W / cm to 4 . × W / cm .One can see that, as the increase of the laser intensity, HHGspectra in both cases are almost same and the harmonic e ffi -ciency is gradually enhanced; when the peak intensity of thelaser pulse is stronger than 1 . × W / cm , the harmonicspectra from the three-band model appear a clear two-plateaustructure. In particular, harmonic spectra from the two casesexhibit an apparent minimum near 16 eV in the first plateau.In the following, for better explaining the origin of the dip P ho t on e n e r g y ( e V ) I ( W cm ) -3.0-1.7-0.331.0 (a) Two band P ho t on e n e r g y ( e V ) I ( W cm ) -3.0-1.7-0.331.0 (b) Three band Fig. 2: Dependence of crystal HHG spectra from the two-band (a)and three-band (b) models on the driving laser intensity. The dura-tion, wavelength and CEP of the driving laser pulse are 3 fs, 1600nm, and 0, respectively. structure, we focus on harmonic spectra from the two-bandSBEs.Based on the real TDM from the first-principle calculations,the red solid, black short-dash-dotted and green dash-dottedlines in Fig. 3 present harmonic spectra of MgO crystal inthe ultrashort laser pulse, which are generated by the totalcurrent, intraband current and the interband polarization, re-spectively. The laser peak intensity of the incident laser pulseis 3 . × W / cm (about 1 . V / Å), which is lower thanthe damage threshold of MgO. It is clear that harmonics be-low / above the bandgap are dominated by intraband / interbandcurrents, which agree with the recent results for ZnO, MgO,and GaAs in mid-infrared laser pulses [28, 46–49]. Figure3 also shows the harmonic spectrum (blue dashed line) fromthe TDM based on the first-order k · p theory. In this case,intensities of harmonics in the plateau are almost same. How-ever, for the spectrum based on the first-principle calculations,there exists an obvious minimum structure when the photonenergy is 16 eV. This distinction in both cases indicates thatthe TDM (cid:48) s shape has a significant e ff ect on the crystal har-monic spectra. -6 -3 I n t en s i t y ( a r b . un i t s ) Photon energy (eV) total_kp theory total_first-principles intra_first-principles inter_first-principles
Fig. 3: Harmonic spectra of MgO from TDMs calculated by first-principle (red solid) and first-order k · p theories (blue dashed); theblack short dash-dotted and green dash-dotted curves are harmonicspectra produced by intraband and interband currents from TDMwith the first-principle calculations, respectively. The laser param-eters are the same as in Fig. 2. Next, we check the influence of laser parameters to the dipstructure in harmonic spectra from the real TDM based onthe first-principle calculations. Figs. 4(a)-4(d) show the HHGspectra of MgO crystal in laser pulses with di ff erent inten-sity, wavelength, duration, and CEP, respectively. It is foundthat, the minimum structure at harmonic spectra is almost in-dependent of parameters of the driving laser pulses. Becauseharmonics in the plateau are mainly originated from the inter-band polarization, it is natural to deduce that the dip structureis related to the characteristic of the real TDM of MgO crys-tal. In the following, for intuitively clarifying the minimumfeature, we focus on the HHG spectrum from the ultrashortlaser pulse, as shown by the red solid line in Fig. 3. -6 -3 -6 -3 A A A (a) (b) =800 nm =1600 nm =2400 nm (c) FWHM=10 fs FWHM=20 fs FWHM=30 fs I n t e n s i t y ( a r b . un i t s ) Photon energy (eV) (d)
Fig. 4: Harmonic spectra from the real TDM by using first-principlecalculations in di ff erent laser pulses. (a) the duration, wavelength andCEP are 10 fs, 1600 nm, and 0, respectively; (b) the duration, peakintensity and CEP are 10 fs, 2 . × W / cm , and 0, respectively;(c) the intensity, wavelength and CEP are 2 . × W / cm , 1600nm, and 0, respectively; (d) the intensity, wavelength and durationare 2 . × W / cm , 1600 nm, and 10 fs, respectively. In order to further understand the emission process in crys-tal HHG, time-frequency analyses of the harmonic spectra forfirst-principle and first-order k · p theory cases are presentedin Figs. 5(a) and 5(b). In both cases, time-frequency dia-grams of HHG are similar, harmonics beyond the bandgap areprimarily caused by one quantum path. This result is furtherconfirmed by the harmonic photon energy vs the emission in-stant calculated from the semiclassical recollision model, asshown in the purple circle curve from Fig. 5. It means that theharmonic photon above the band gap and the crystal momen-tum at the emission instant has a one-to-one correspondencein the ultrashort laser pulse. Furthermore, the time-frequencydistribution in Fig. 5(a) shows one hole at the photon energywith 16 eV, which directly correspond to the dip in the har-monic spectrum (the red solid curve) in Fig. 3. In contrast tothe first-principle case, there is no hole at 16 eV in the HHGtime-frequency distribution from the first-order k · p theory,as shown in Fig. 5(b). The above results further testify thatthe minimum structure of the harmonic spectrum is closelyconcerned with the TDM of the crystal.The e ffi ciency of HHG from the interband current is propor-tional to the population of the electron (hole) and the TDM P ho t on ene r g y ( e V ) Time (a.u.) -4.0-3.3-2.7-2.0 (a) First-principle P ho t on ene r g y ( e V ) Time (a.u.) -4.0-3.3-2.7-2.0 (b) kp theory
Fig. 5: Time-frequency distributions of the HHG corresponding tothe red solid (a) and blue dashed (b) curves in Fig. 3. The pink circlecurve is the photon energy vs the emission time from the semiclas-sical recollision model. The laser parameters are the same as in Fig.2. between conduction and valence bands at the recombinationtime t r . It can be observed that, the emission instant is 225a.u. from Fig. 5 when the energy of the harmonic photon isequal to 16 eV. To clearly address the physics of the dip struc-ture in harmonic spectra, we examine populations of elec-trons in the conduction band at this emission instant. Figure6 shows electronic populations of the conduction band for thefirst-principle and first-order k · p cases, respectively. FromFigs. 1(a) and 1(b), one can find that the bandgap betweentwo bands is exactly equal to 16 eV for the crystal momen-tum at k = ± π/ a . In Fig. 6, populations at these crystalmomentums for the emission (cid:48) s instant 225 a.u. are markedby the cross of dashed lines. For both cases, populations atk = ± π/ a with t r =
225 a.u. are no essential di ff erence. Thismeans that the dip structure of the HHG spectrum is almostindependent of the electronic population at the recombinationinstant. -0.60.00.6 K ( a ) (a) k p theory
150 200 250 300 350 400-0.60.00.6
Time(a.u.) (b) First principle
Fig. 6: Electronic populations of the conduction bands for the dipolemoments from first first-order k · p (a) and first-principle (b) calcula-tions, respectively. Now that we know the dip structure of the harmonic spec-trum is related to the k-dependent TDM, the contribution fromdi ff erent crystal momentums to harmonics above the bandgapshould be analyzed. Figs. 7(a) and 7(b) provide a compari-son between distributions of harmonic intensities at di ff erentk from the first first-order k · p theory and first-principle calcu- lations, respectively. Here, in order to distinctly reveal distri-butions of harmonics intensities, we focus on harmonics pro-duced at the main emission times from 200 a.u. to 250 a.u.. Inthe case of the real TDM from the first-principle calculations,the amplitude value of TDM near k = ± π/ a is close to zeroas shown by the orange solid curve in Fig. 7(b), which resultsin no distribution of the harmonic intensity , as presented inFig. 7(b). In the first-order k · p theory case, TDM has biggervalues near k = ± π/ a , which induces a clear distribution ofthe harmonic intensity, as observed from Fig. 7(a). Aboveall, it can be demonstrated that the valley shape of the TDMfrom the first-principle calculations results in the dip structurein the harmonic spectra. -0.9 -0.6 -0.3 0.0 0.3 0.6 0.981012141618 P ho t on ene r g y ( e V ) k ( a ) -0.3-0.030.20.5 (a) k p theory T D M ( a . u . ) P ho t on ene r g y ( e V ) k ( a ) -0.7-0.30.030.4 (b) First-principles T D M ( a . u . ) Fig. 7: The contribution of di ff erent crystal momentums to harmon-ics based on the first first-order k · p (a) and first-principle calculations(b), respectively. The orange solid curves are the TDMs from the twocases, and the pink dashed curve is the bandgap between two bands.The laser parameters are the same as in Fig. 2. Finally, we explore dependences of amplitudes of TDMsfrom the first-principle calculations and the first-order k · p theory with the bandgap, as shown by the green short-dottedand black dotted curves in Fig. 8. The corresponding har-monic spectra in both cases are also presented in Fig. 8. Inthe case of the first-order k · p theory, TDM and the harmonicspectrum near 16 eV have no minimum structure. However,for the case of the first-principle calculations, the TDM (cid:48) s val-ley at 16 eV directly coincides with the minimum of the har-monic spectrum. Thereby, we can draw a conclusion that, be-cause the amplitude of TDM between valence and conductionbands exists zero values, harmonic spectra from MgO crystalalso exhibit the Cooper minimum structure, which is similarto harmonic spectra from gaseous media. IV. SUMMARY
In conclusion, we have demonstrated that harmonic spectraof MgO crystal in the ultrashort laser pulse have the Cooperminimum structure. By comparing harmonic spectra fromTDMs of the first-order k · p theory and the first-principle cal-culations, it is confirmed that, the shape of TDM plays an im-portant role in the generation of the HHG spectrum, and thevalley of the real TDM from the first-principle calculationslead to the dip structure near 16 eV in the harmonic spectrafrom MgO crystal. More importantly, by taking the valley-dipcorrespondence as the benchmark, the emitted photon energyand the crystal momentum have a one-to-one match, and the -9 -6 -3 -8 -5 -2 Log D cv ( a . u . ) I n t en s i t y ( a r b . un i t s ) HHG_kp theory HHG_first-principles
Photon energy (eV)
TDM_first-principles
TDM_kp theory
Fig. 8: Values of TDMs from the first-principle calculations (greenshort-dotted line) and the first-order k · p theory (black dotted line)versus the bandgap, and the corresponding harmonic spectra shownby red solid and blue dashed curves in Fig. 3. intensity of the harmonic from an ultrashort laser pulse is ap- proximately proportional to the square of the TDM’s value.Thereby, the k-dependent bandgap and TDM between valenceand conduction bands are hoped to be mapped by harmonicswith energies above the minimum bandgap, which will pavea new way to the all-optical reconstruction of the electronicband structure by taking advantage of the crystal HHG. ACKNOWLEDGEMENT
The authors sincerely thank Prof. Ruifeng Lu for providingthe code. J. G. Chen is supported by the National Natural Sci-ence Foundation of China under Grant No. 11975012. Projectsupported by the National Key R&D Program of China (GrantNo. 2017YFA0403300), the National Natural Science Foun-dation of China (Grant Nos. 11774129, 11627807), the JilinProvincial Research Foundation for Basic Research, China(Grant No. 20170101153JC) and the Science and Technologyproject of the Jilin Provincial Education Department (GrantNo. JJKH20190183KJ). [1] P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte,and G. D. Tsakiris, Nature , 267-271 (2003).[2] G. Sansone et al ., Science , 443-446 (2006).[3] E. Goulielmakis et al ., Science , 1614-1617 (2008).[4] J. G. Chen, Y. J. Yang, J. Chen, and B. B. Wang, Phys. Rev. A , 043403 (2015).[5] J. G. Chen, R. Q. Wang, Z. Zhai, J. Chen, P. M. Fu, B. Wang,and W. M. Liu, Phys. Rev. A , 033417 (2012).[6] B. Zhang, J. M. Yuan, and Z. X. Zhao, Phys. Rev. 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