arX

iv1

912

0964

9v2

[ph

ysic

sop

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7 M

ar 2

020

DRAFT

How to obtain complex transition dipole moments satisfying crystal symmetry and

periodicity from ab-initio calculations

Shicheng Jiang12 Chao Yu1 Jigen Chen3 Yanwei Huang2 Ruifeng Lu1dagger and C D Lin4lowast1 Department of Applied Physics Nanjing University of Science and Technology Nanjing 210094 P R China

2 State Key Laboratory of Precision Spectroscopy

East China Normal University Shanghai 200062 China3 Zhejiang Provincial Key Laboratory for Cutting Tools Taizhou University Taizhou 31800 China and

4 J R Macdonald Laboratory Department of Physics

Kansas State University Manhattan Kansas 66506 USAlowast

(Dated March 10 2020)

Transition dipole moments (TDM) between energy bands of solids deserve special attention nowa-days as intense lasers can easily drive non-adiabatic transitions of excited electron wave packetsacross the Brillouin zones The TDM is required to be continuous satisfying crystal symmetryand periodic at zone boundaries While present day ab-initio algorithms are powerful in calculatingband structures of solids they all introduced random phases into the eigenfunctions at each crystalmomentum k In this paper we show how to choose a ldquosmooth-periodicrdquo gauge where TDMs can besmooth versus k preserving crystal symmetry as well as maintaining periodic at zone boundariesBased on band structure and TDMs in the ldquosmooth-periodicrdquo gauge calculated from ab-initio al-gorithms we revisit high-order harmonic generation from MgO which exhibits inversion symmetryand ZnO which has broken symmetry The symmetry properties of TDMs with respect to k ensurethe absence of even-order harmonics in system with inversion symmetry while the TDM in thelsquosmooth-periodicrdquo gauge for ZnO is shown to enhance even harmonics that were underestimated inprevious simulations These results reveal the importance of correctly treating the complex TDMsin nonlinear laser-solid interactions which has been elusive so far

I INTRODUCTION

In quantum mechanics band theory is the foundationfor understanding the structure of solids and their in-teractions with lights To interpret a plethora of experi-ments on all kinds of condensed materials ab-initio com-puter codes have been developed and the values of suchpackages are well recognized In recent years with theadvent of intense lasers and their nonlinear interactionswith solids such ultrafast phenomena like high-orderharmonic generation (HHG)[1ndash5] laser-induced chargetransfer[6ndash9] Bloch oscillation[10ndash12] laser-controlleddielectrics [13 14] and ultrafast renormalization [15ndash17] have been widely investigated The equations ofmotion based on band theory and crystal-momentumrepresentation for example semiconductor Bloch equa-tions (SBEs)[18ndash20] and other extended forms[21 23]have already been used to interpret these nonlinear ultra-fast phenomena However true quantitative comparisonbetween theoretical results with experiments remains achallenge despite that the band structure and transitiondipoles were calculated from advanced ab initio codesIn the interaction of strong lasers with solids non-

adiabatic transitions of electrons between bands are im-portant The excited electron wave packet can even goacross the first Brillouin zone When the carrier is mov-ing along a path in the k-space the wavepacket willacquire a dynamical phase and a geometry phase[2526] Previously geometric phase was mostly considered

lowast Emaillowast cdlinphysksuedu Emaildagger rflunjusteducn

only for closed paths for ensuring the theory is gauge-independent However as shown recently[28 29] inthe SBEs method for high-order harmonic generationin solids gauge invariance is achieved when the correctphase of the dipole transition moment (TDM) is includedin the SBEs The combination of geometry phase anddipole phase is well-defined whether the system is openor closed[29] Both of these two phases will be encodedin the macroscopic polarization and thus the photonicsignal[27]In order to describe the interaction of strong laser fields

with solids in a finite k space including the phase accu-mulated by the moving wavepacket it requires that (1)the TDMs should be calculated accurately for the wholefirst Brilloin zone (2) the TDMs with k-dependent phaseshould be continuous and periodic with respect to k Us-ing ab-initio software to calculate accurate absolute val-ues of TDMs Yu et al[24] were able to obtain improvedHHG spectra However since all ab-initio algorithmscalculated the eigenfunctions at each k separately ran-dom phases are introduced at each k point and the phaseof the TDM is not smooth and continuous By not-ing the importance of the TDM phase Jiang et al[30]obtained the smooth phase analytically using the tight-binding model They were able to reproduce the orienta-tion dependence of the HHG spectra from ZnO first re-ported by S Ghimire et al [1] and S Gholam-Mirzaei etal[2] However the tight-binding model is too primitiveand there remains quantitative discrepancies with exper-iments It is desirable to obtain ambiguity-free phase forthe TDM calculated from ab-initio algorithmsThe problem of random phase in TDM has been well

recognized and studied for a long time[31ndash34] and so-

DRAFT

2

lutions have been suggested However as pointed outby Yakovlev[35] they either do not ensure the period-icity with respect to k or require the evaluation of theso-called covariant derivatives Recently a ldquosmooth pro-cedurerdquo suggested in Ref [34] has been used in har-monic generation in solids[28 36 37] Here we will showthat this widely used ldquosmooth procedurerdquo is not robustThe method will introduce Zakrsquos phase[38] into the eigen-functions which would then break the periodicity of theTDMs

In this paper we will obtain smooth TDM phase thatalso satisfies periodic conditions of the crystal by intro-ducing what we will called the lsquosmooth-periodicrdquo gaugeto distinguish it from the ldquoperiodic gaugerdquo We first sum-marize in Section II the rdquosmooth procedurerdquo We thenaddress systems with (Section III) and without (SectionIV) inversion symmetry In this lsquosmooth-periodicrdquo gaugethe symmetry properties of k-dependent eigenfunctionsand TDMs for system with inversion symmetry are re-tained Such symmetry properties would ensure the ab-sence of even-order harmonics driven by long pulses evenif multi-band excitations are included Using such newTDMs we calculated the harmonic spectra of MgO andfound improved agreement with earlier experiment Sim-ilarly we also revisited the HHG spectra of ZnO in thedirection with broken symmetry While even harmon-ics of ZnO were predicted in Jiang et al[30] the signalswere much too weak relative to the odd harmonics WithTDMs calculated using the rdquosmooth-periodicrdquo gauge theeven harmonics were greatly enhanced and become com-parable to odd harmonics We comment that previouslyeven harmonics were modeled in terms of Berry Curva-ture With the rdquosmooth-periodicrdquo gauge the latter isa consequence of treating correctly the TDM that satis-fies smooth and periodic conditions of crystals A shortconclusion is given in Section V to end this paper

II THE COMMONLY USED ldquoSMOOTH

PROCEDURErdquo AND ITS DEFICIENCY

The commonly used ldquosmooth procedurerdquo was de-scribed in details [34] by Hjelm and coworkers It is neces-sary to show this method briefly here The Bloch wave-function is expressed as Φm(k r) = eikrum(k r) whereum(k r) is periodic um(k r) = um(k r +R) with R be-ing the lattice spacing In most of the ab-initio softwaresthe periodic part is expanded by plane waves um(k r) =sum

h a(k+Gh)eiGhr In principle the Bloch functions are

periodic in k space Φm(k r) = Φm(k +G r) Here G isthe reciprocal lattice vector Since the eigenfunctions areobtained separately for different k points which leads torandom phases ϕm(k) uprime

m(k r) = um(k r)eiϕm(k) Notethat ϕm(k) is randomly generated and is discrete withrespect to k

In the ldquosmooth procedurerdquo a complex number zm(k)

is defined by

zm(k) = |zm(k)| eiαm(k) = 〈uprimem(k r)|uprime

m(k +∆k r)〉 (1)

A new wavefunction is constructed by

uprimeprimem(k +∆k r) = uprime

m(k +∆k r)eminusiαm(k) (2)

By renaming the function uprimeprimem(k +∆k r) according to

uprimeprimem(k +∆k r) rarr uprime

m(k +∆k r) (3)

the same procedure can be repeated to the next pointuprimem(k+2∆k r) When this procedure goes over the first

Brillouin zone the phase-modified wavefunction becomecontinuous with respect to k We rename the final wave-function after the ldquosmooth procedurerdquo as us

m(k r) to dis-tinguish it from the original one generated by ab-initio

softwareSince ∆k is small

〈uprimem(k r)|uprime

m(k +∆k r)〉

asymp e∆k〈um(kr)|nablakum(kr)〉+i(ϕm(k+∆k)minusϕm(k)) (4)

which leads to

uprimeprimem(k+∆k r) = um(k+∆k r)eminus∆k〈um(kr)|nablakum(kr)〉+iϕm(k)

(5)As the procedure of Eq (1-3) goes through the pathk0 rarr k0 + k the phase-modified wavefunction becomes

uprimem(k r) = um(k r)e

iint

k

k0dκDmm(κ)

eiϕm(k0) (6)

where Dmm(k) = i 〈um(k r)|nablakum(k r)〉 is the BerryconnectionTo conclude this method forces the wavefunction to

be continuous with respect to k and the new Berry con-nection Ds

mm(k) = i 〈usm(k r)|nablaku

sm(k r)〉 = 0 Mean-

while at the same time this method introduces a phase

Θm(k) =int k

k0

dκDmm(κ) + ϕm (k0) to the eigenfunction

The additional phase Θm(k) will break the periodic-ity of the eigenfunction In the followed two sectionswe will provide different methods to deal with the non-periodicity for systems with and without inversion sym-metry

III SYSTEMS WITH INVERSION SYMMETRY

The eigenfunction um(k r) =sum

h a(k + Gh)eiGhr is

defined in the ldquoperiodic gaugerdquo[39] As shown by Zak[38]

the Zakrsquos phase γ =int k+G

kdκDmm(κ) is equal to zero or π

in theldquoperiodic gaugerdquo Thus for system with inversionsymmetry the simplest way for the ldquosmooth procedurerdquois to extend it to the second Brillouin zone In this wayphase difference between the starting point k0 and k0 +2G is zero or 2π which means that the periodicity ofeigenfunctions in k space is 2GIn this section rock-salt MgO with inversion symme-

try is taken as example to explain our method Fig

DRAFT

3

1(b) shows the band structure of MgO along (-10-1)rarr Γ(000)rarr(101) Figs 1(c)-(h) are the correspond-ing TDMs calculated by the ldquosmooth procedurerdquo betweendifferent pairs of bands The eigenfunctions are calcu-lated by DFT package in VASP[40] using the Perdew-Burke-Ernzeroff GGA functional The cutoff energy ofplane wave is 500 eV The k-point grids of 20times20times20with none-zero weight in the first Brillouin zone and 400points with zero weight along the one dimensional pathare used Since the DFT simulation underestimates theband gap the conduction bands are shifted to get bet-ter agreement with the experimental gap 78 eV As ex-pected both the energy bands and TDMs are periodicwith 2G

In Fig 2 we present the HHG spectra calculated bysemiconductor Bloch equations (SBEs) with one valenceband (band 2) and two conduction bands (band 3 andband 4) included The spectra show two plateaus withthe right side of the dashed line being from the recombi-nation of electron-hole pair from the second conductionband (band 4) to the valence band (band 2)[27] To theleft which is due to recombination from band 3 the greenarrow marks a minimum which is similar to the Cooperminimum in atoms Such minimum originates from theminimum of the absolute value of dipole moment betweenband 2 and band 3 This minimum has also been foundin the TD-DFT simulation in Ref [41] Thus it wouldbe of interest to see if this minimum can be observedin experiments if the detected photon energy range canbe extended[27 42 43] This kind of minimum will bediscussed in details in another paper[44]

FIG 1 (a) Structure of rock-salt MgO Red and green ballsare O and Mg respectively (b) Band structures along theΓminusX axis (c)-(e) Real (black line) and imaginary (red line)part of TDMs generated by the ldquosmooth procedurerdquo in theextended Brillouin zone

FIG 2 HHG spectra from MgO calculated by 1D three-bandSBEs The equations are solved in the extended Brillouinzone and all the elements used in the SBEs model are fromFig 1 Laser parameter 30 fs 1300nm 1times1013 Wcm2

FIG 3 Comparison between experimental and calculatedCEP-dependent HHG from MgO (a) is reprinted from Fig2(c) of [27] (b) is the calculated spectrum (c) and (d) arereprinted from Fig 1(a) of [27] (e) and (f) are the calculatedspectra along Γ-X (cut from (b)) Laser parameters 1700nm 10fs 4times1013 Wcm2

DRAFT

4

TABLE I Properties of TDMs for different ldquoparitiesrdquo of eigen-functions

us

m(k r) us

n(k r) Ds

m(k)

us

m(minusk r) =

us

m(kminusr)

us

n(minusk r) =

us

n(kminusr)

Ds

mn(minusk) =

minusDs

mn(k)

us

m(minusk r) =

us

m(kminusr)

us

n(minusk r) =

minusus

n(kminusr)

Ds

mn(minusk) =

Ds

mn(k)

us

m(minusk r) =

minusus

m(kminusr)

us

n(minusk r) =

us

n(kminusr)

Ds

mn(minusk) =

Ds

mn(k)

us

m(minusk r) =

minusus

m(kminusr)

us

n(minusk r) =

minusus

n(kminusr)

Ds

mn(minusk) =

minusDs

mn(k)

From the analysis of Section II after the ldquosmooth pro-cedurerdquo the newly derived eigenfunction us

m(k r) sat-isfies the strict periodic boundary condition us

m(k r) =ei2Grus

m(k + 2G r) and usm(k r) = us

m(k r + R) Onestill would like to know what are the symmetry proper-ties of the TDMs for systems with inversion symmetryThe results are summarized in Table I while the deriva-tion is given in the Appendix As shown in details in theAppendix When the periodic functions us

m and usn have

the same parity the TDM between band m and n is anodd function with respect to k When they have oppo-site parity the TDMs between them is an even functionwith respect to k In a three-band model there are manypathways to generate excitations Take the MgO exam-ple one possibility is to choose a path band2 rarr 3 rarr 4The corresponding macroscopic polarization is given by

P (t) sim P (k0 t) + P (minusk0 t)

= D24(k0 +A(t))D43(k0 +A(t))D32(k0 +A(t))E(t)2

+D24(minusk0 +A(t))D43(minusk0 +A(t))D32(minusk0 +A(t))E(t)2

+ cc(7)

Here E(t) and A(t) are electric field and vector potentialrespectively If the driven laser is a long pulse E(t +T2) = minusE(t) and A(t + T2) = minusA(t) where T is theoptical cycle of the laser Thus we can get

P (t+ T2) sim P (k0 t+ T2) + P (minusk0 t+ T2)

= D24(k0 minusA(t))D43(k0 minusA(t))D32(k0 minusA(t))E(t)2

+D24(minusk0 minusA(t))D43(minusk0 minusA(t))D32(minusk0 minusA(t))E(t)2

+ cc

(8)

By comparing Eq (7) and (8) and using the propertieslisted in TABLE I [ the parities of the wavefunctions forthese three bands are shown in Fig 5(b)] we can find thatP (t) = minusP (t+ T2) The odd parity of macroscopic po-larization similar to the case for an atomic target guar-antees that no even-order harmonics in the spectra Ifthe parities of TDMs have not accounted for odd sym-metric macroscopic polarization is not present In otherwords because of the parities of TDMs coupling of mul-tiple bands cannot generate even harmonics if the systemhas inversion symmetry

FIG 4 Black line is the same as fig 3(c) harmonic spec-trum calculated using correct dipole moments (Dc1c2

(k) isodd Dvc2

(k) and Dvc1(k) are even functions of k) Red line

The same as the black line except that Dc1c2(k) is changed

to an even function of k artificially The subscripts v c1 c2represent the valence band first conduction band and secondconduction band respectively The intensities of the spectrain the right side of the vertical solid line were multiplied by afactor of 4

Using the more accurate band structure and dipole mo-ments in particular the newly dipole phases constructedin the present rdquosmooth-periodicrdquo gauge we can improvethe simulation reported in [27] where the dipole momentshave been set to be a constant Comparison of CEP-dependent spectra between experimental data and oursimulation is presented in Fig 3(a) and (b) The laserparameters used in the simulation can be found in thecaption Many features in the experimental data are re-produced in the present simulation (1) In Fig3(a) theslope of photon energy versus the CEP has been repro-duced in Fig 3(b) (2) Both experiment and simulationshow two plateaus in the same photon energy region (3)In Fig 3(b) our simulation indicates a minimum around13 eV which is consistent with experiment even thoughit is near the low energy end of the experimental datathus it could also be due to detector efficiency in the en-ergy region Note that this minimum also appears in Fig2 at the 15th order harmonic Comparing with the sim-ulations reported in [27] we have witnessed significantimprovement in the present simulation

Figs 3(c-f) compare the HHG spectra at two CEPs05 π and 00 between experiment and the present sim-ulation It is clear that the sine-like pulse (CEP=05π) would generate sharper discrete harmonics while acosine-like pulse would produce relatively flatter onesSuch results are in agreement with the measurements

In this article we are concerned with how harmonicspectra are affected if the parity and periodicity of tran-sition dipole moments are not correctly accounted forIn many prior calculations approximations were made

DRAFT

5

in which the k-dependent dipole moments were taken tobe its absolute values This means that it is an even func-tion with respect to k In the three-band model for MgOusing the method presented here Dc1c2(k) which is thecoupling between the two conduction bands is an oddfunction If we arbitrarily change it to an even functionhow the HHG spectra would be altered Fig 4 showsthe original spectra copied from Fig 3(c) (in black lines)and compare it to HHG spectra (in red lines) if Dc1c2(k)is changed to an even function For the first plateauharmonics no significant changes occur since the firstplateau is due to recombination of electrons from the firstconduction band to the valence band For the harmonicsin the 2nd plateau we can see the change as the harmonicpeaks are shifted Based on Eqs (7) (8) and the analy-sis in [27] the peaks in the secondary plateau are givenby ω = (2n minus 1 minus θπ)ω0 where ω0 is the frequency ofthe driving pulse and the phase θ is the phase differencebetween the two pathways to reach the conduction band2 By changing the parity of Dc1c2(k) artificially fromodd to even the peaks in the 2nd plateau will be givenby ω = (2nminus θπ)ω0 This can explain the shift of thepeaks in the secondary plateau shown in Fig 4 Thusif the parity of the dipole moment has wrong parity thegenerated harmonic spectra will be quite different

We conclude this section by illustrating the differencebetween gas phase and solids as shown in Fig 5 Inthe gaseous medium with inversion symmetry a triangu-lar system will never be formed because of the parity ofthe wavefunction For crystal with inversion symmetrywhere energy level is expanded into a band a trianglesystem can be formed While the parity of the TDMswill prevent the generation of even-order harmonics

FIG 5 Illustration of transition paths of electrons for gasesand solids with inversion symmetry For the gas phase a tri-angular system will never be formed because the transitionbetween states with same parity is forbidden While for solidcase where the energy levels are extended into bands transi-tion is between bands at these k points away from Γ are notforbidden The symmetry properties of the TDMs will ensurethe absence of even order optical signal

IV SYSTEM WITH BROKEN SYMMETRY

For system with broken symmetry the Zakrsquos phasecan be any value Thus the method above for systemwith inversion symmetry is not valid anymore Notethat in the periodic gauge the Berry connection is pe-riodic Dmm(k) = Dmm(k + G) which means it can beexpanded as

Dmm(k) = gm(k) + σm (9)

where gm(k) =sum+infin

n=1 f1(n) cos(nLk) + f2(n) sin(nLk) isthe rdquoACrdquo component and σm is a constant which can beregarded as the rdquoDCrdquo component The DC componentwill lead to divergence of the introduced additional phase

Θm(k) =int k

k0

dκDmm(κ) + ϕm (k0) We do not need to

care about the AC part because this component wouldnot influence the periodicity the continuity and the fi-nal observable physical quantities [28] If the ldquosmoothprocedurerdquo is carried out for the first Brilloin zone theDC-induced non-periodic phase of the TDMs betweenband m and n is (σn minus σm)k

FIG 6 Red line is the k-dependent dipole phase generatedby ldquosmooth procedurerdquo Such phase is not periodic because ofthe DC term in the additionally introduced phase The blackline is the k-dependent dipole phase after the DC componentis taken away as introduced in this article

Here we take the direction Γ minus A of wurtzite ZnO asan example In Fig 6 the red line is the k-dependentdipole phase for the first Brillouin zone obtained fromVASP using a ldquosmooth procedurerdquo As stated abovethe dipole phase cannot be ensured to be periodic be-cause of the DC component However it is easy to getthe slope by (σn minus σm) = (αmn(πL)minus αmn(minusπL))GHere αmn(plusmnπL) are the phase of TDMs which are readfrom the data generated by ldquosmooth procedurerdquo shownby the red line in Fig 6 We can then get the periodicdipole phase by subtracting the DC part off Dp

mn(k) =Ds

mn(k)ei(σmminusσn)k which is shown by the black line in

Fig 6 for ZnO At the same time the Berry connection

DRAFT

6

is changed from zero to Dpmm(k) minus Dp

nn(k) = σm minus σnWith that all the elements in the SBEs model are pe-riodic and continuous with respect to k This meansthat the equation of motion for carriers can be solved ina finite k space In Fig 7 the calculated HHG spec-trum (blue line) by solving two-band SBEs model usingTDMs obtained by our ldquosmooth-periodicrdquo procedure iscompared to the experimental data (green line) We alsopresent the spectrum (red line) by solving two-band SBEsincluding only dipole phase obtained from tight-bindingmodel Even though tight-binding model can approxi-matively reproduce the orientation-dependent feature ofHHG spectra usually it is too primitive to produce therelative strength between odd and even order harmonicsUsing accurate TDMs obtained from ab-initio softwarewith the help of our ldquosmooth-periodicrdquo procedure theexperimental spectra of ZnO first reported in S Ghimireet al [1] has finally been satisfactorily reproduced theo-retically

FIG 7 HHG spectra from ZnO Green line is the experi-mental data blue line is calculated by two-band SBEs withelements obtained from ab-initio software with the help of thepresent ldquosmooth-periodicrdquo procedure The red line is fromcalculations where the dipole phases are calculated from thetight-binding model The red line and experimental data arecopied from our previous work[30] To present clear compar-ison the spectra are shifted vertically

V CONCLUSION REMARKS

Although ab-initio software has been widely used toinvestigate electronic properties nowadays the randomphase generated in the algorithms prevents its applica-tion to calculate non-adiabatic dynamics especially whenthe external field is a strong laser In this article we firstshow that the commonly used ldquosmooth procedurerdquo can-not ensure the periodicity of the wavefunction Secondwe provide two different methods to overcome this de-fect for systems with and without inversion symmetry

Because our approaches ensure continuity and periodic-ity of all the elements used in the equations of motionthe gauge resulting by the transformation of our methodscan be referred to as a ldquosmooth-periodicrdquo gauge to distin-guish it from rdquoperiodicrdquo gauge used by Resta [39] Basedon this gauge HHG spectra from solids with and withoutinversion symmetry are revisited It is emphasized thatsymmetry properties are the key factors to ensure the ab-sence of even order harmonics for systems with inversionsymmetry With the accurate TMDs with dipole phaseand Berry connection the HHG spectrum from ZnO withbroken symmetry is also improved greatly The TMDsintroduced in this work is fundamental to all applicationsrelating to optical signals from solids such as laser wave-form control banddipole reconstruction and detectingdynamic information

ACKNOWLEDGMENT

This work was supported by the NSF of JiangsuProvince (Grant No BK20170032) and NSF of China( 11704187 11974112 11975012) CDL was supportedin part by the Chemical Sciences Geosciences and Bio-sciences Division Office of Basic Energy Sciences Officeof Science US Department of Energy under Grant NoDE-FG02-86ER13491 SJ also thanks for the support bythe Project funded by China Postdoctoral Science Foun-dation No 2019TQ0098 SJ thanks for the fruitful dis-cussion with Prof Chengcheng Liu from Beijing Instituteof Technology and Dr Prasoon Saurabh from East ChinaNormal University

APPENDIX DERIVATION OF THE PARITIES

OF TRANSITION DIPOLE MOMENTS

Both usm(k r) and ukp

m (k r) satisfy the kmiddotp equation

(

minus1

2nabla2

r + V (r)minus ik middot nablar

)

us(kp)mk (r) =

(

Em(k)minusk2

2

)

us(kp)mk (r)

(A1)

where ukpm (k r) is assumed to satisfy

ukpm (minusk r) = ukplowast

m (k r) (A2)

When the system has inversion symmetry

ukpm (minusk r) = plusmnukp

m (kminusr)Dkpmm(k) = 0 (A3)

usm(k r) must be related to ukp

m (k r) through a gaugetransformation eg

usm(k r) = ukp

m (k r)eiβ(k) (A4)

DRAFT

7

As stated in the main text after the ldquosmooth procedurerdquothe Berry connection

Dsmm(k) = i 〈us

m(k r)|nablakusm(k r)〉 = 0 (A5)

By inserting Eq (A4) into Eq (A5) we have

Dsmm(k) = i

lang

ukpm (k r)eiβ(k)|nablak

(

ukpm (k r)eiβ(k))rang

= Dkpmm(k)minusnablakβ(k) = 0 (A6)

It is easy to prove that Dkpmm(k) is a real number and an

even function with respect to k

Dkpmm(minusk) = Dkp

mm(k) (A7)

In order to ensure Dsmm(k) = 0 β(k) = odd function +

const Further if the system has inversion symmetryβ(k) = const Thus us

m(k r) also has

usm(minusk r) = plusmnus

m(kminusr) (A8)

Using Eq (A8) we can get all the properties listed inTable I

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[2] S Gholam-Mirzaei J Beetar and M Chini Appl PhysLett 110 061101 (2017)

[3] G Vampa et al Nature (London) 522 462 (2015)[4] T T Luu et al Nature 521 498 (2015)[5] M Hohenleutneret al Nature (London) 523 572

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Rev B 87 115201 (2013)[8] T Paasch-Colberg et al Nat Photon 8 214 (2014)[9] G Wachter et al Phys Rev Lett 113 087401 (2014)

[10] O Schubert et al Nat Photonics 8 119 (2014)[11] L Li P Lan X Liu L He X Zhu O D Mucke P

Lu Optics Express 26 23844 (2018)[12] L Liu J Zhao J M Yuan Z X Zhao Chin Phys B

28 114205 (2019)[13] M Schultze et al Nature 493 75 (2013)[14] M Schultze et al Sicence 346 1348 (2014)[15] A Chernikov C Ruppert H M Hill A F Rigosi T

F Heinz Nat Photon 9 466 (2015)[16] L Meckbach T Stroucken and S W Koch Appl Phys

Lett 112 061104 (2018)[17] L Meckbachet al arXiv preprint arXiv190308553

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and Electronic Properties of Semiconductors Fivth Edi-tion World Scientic Publishing Company 2009

[21] G Vampa et al Phys Rev B 91 064302 (2015)

[22] Z Wang et al Nat Commun 8 1686 (2017)[23] L Meckbach T Stroucken and S W Koch Phys Rev

B 97 0354425 (2018)[24] C Yu et al Phys Rev A 94 013846 (2016)[25] M V Berry Proc R Soc London Ser A 392 45 (1984)[26] Y G Yao et al Phys Rev Lett 92 037204 (2004)[27] Y S You et al Opt Lett 42 1816 (2017)[28] J Li et al Phys Rev A 100 043404 (2019)[29] L Yue M Gaarde Imperfect Recollisions in High-

Harmonic Generation in Solids arXiv preprintarXiv200104626 (2020)

[30] S C Jiang et al Phys Rev Lett 120 253201 (2018)[31] K S Virk J E Sipe Phys Rev B 76 035213 (2007)[32] R W Nunes X Gonze Phys Rev B 63 155107 (2001)[33] I Souza et al Phys Rev B 69 085106 (2004)[34] U Lindefelt HE Nilsson and M Hjelm Semicond Sci

Technol 19 1061 (2004)[35] V S Yakovlev M S Wismer Comput Phys Commun

217 82-88 (2017)[36] M X Wu et al Phys Rev A 91 043839 (2015)[37] M Du C Liu Y Zheng Z Zeng R li Phys Rev A

100 043840 (2019)[38] J Zak Phys Rev Lett 62 2747 (1989)[39] R Resta J Phys Condens Mat 12 R107 (2000)[40] G Kresse and J Hafner Phys Rev B 47 558 (1993)[41] N Tancogne-Dejean et al Nat Commun 8 745 (2017)[42] Y S You D A Reis and S Ghimire Nat Phys 13

345 (2017)[43] Y S You et al Opt Lett 44 530 (2019)[44] Y Zhao et al arXiv191112092

DRAFT

3

1(b) shows the band structure of MgO along (-10-1)rarr Γ(000)rarr(101) Figs 1(c)-(h) are the correspond-ing TDMs calculated by the ldquosmooth procedurerdquo betweendifferent pairs of bands The eigenfunctions are calcu-lated by DFT package in VASP[40] using the Perdew-Burke-Ernzeroff GGA functional The cutoff energy ofplane wave is 500 eV The k-point grids of 20times20times20with none-zero weight in the first Brillouin zone and 400points with zero weight along the one dimensional pathare used Since the DFT simulation underestimates theband gap the conduction bands are shifted to get bet-ter agreement with the experimental gap 78 eV As ex-pected both the energy bands and TDMs are periodicwith 2G

In Fig 2 we present the HHG spectra calculated bysemiconductor Bloch equations (SBEs) with one valenceband (band 2) and two conduction bands (band 3 andband 4) included The spectra show two plateaus withthe right side of the dashed line being from the recombi-nation of electron-hole pair from the second conductionband (band 4) to the valence band (band 2)[27] To theleft which is due to recombination from band 3 the greenarrow marks a minimum which is similar to the Cooperminimum in atoms Such minimum originates from theminimum of the absolute value of dipole moment betweenband 2 and band 3 This minimum has also been foundin the TD-DFT simulation in Ref [41] Thus it wouldbe of interest to see if this minimum can be observedin experiments if the detected photon energy range canbe extended[27 42 43] This kind of minimum will bediscussed in details in another paper[44]

FIG 1 (a) Structure of rock-salt MgO Red and green ballsare O and Mg respectively (b) Band structures along theΓminusX axis (c)-(e) Real (black line) and imaginary (red line)part of TDMs generated by the ldquosmooth procedurerdquo in theextended Brillouin zone

FIG 2 HHG spectra from MgO calculated by 1D three-bandSBEs The equations are solved in the extended Brillouinzone and all the elements used in the SBEs model are fromFig 1 Laser parameter 30 fs 1300nm 1times1013 Wcm2

FIG 3 Comparison between experimental and calculatedCEP-dependent HHG from MgO (a) is reprinted from Fig2(c) of [27] (b) is the calculated spectrum (c) and (d) arereprinted from Fig 1(a) of [27] (e) and (f) are the calculatedspectra along Γ-X (cut from (b)) Laser parameters 1700nm 10fs 4times1013 Wcm2

DRAFT

4

TABLE I Properties of TDMs for different ldquoparitiesrdquo of eigen-functions

us

m(k r) us

n(k r) Ds

m(k)

us

m(minusk r) =

us

m(kminusr)

us

n(minusk r) =

us

n(kminusr)

Ds

mn(minusk) =

minusDs

mn(k)

us

m(minusk r) =

us

m(kminusr)

us

n(minusk r) =

minusus

n(kminusr)

Ds

mn(minusk) =

Ds

mn(k)

us

m(minusk r) =

minusus

m(kminusr)

us

n(minusk r) =

us

n(kminusr)

Ds

mn(minusk) =

Ds

mn(k)

us

m(minusk r) =

minusus

m(kminusr)

us

n(minusk r) =

minusus

n(kminusr)

Ds

mn(minusk) =

minusDs

mn(k)

From the analysis of Section II after the ldquosmooth pro-cedurerdquo the newly derived eigenfunction us

m(k r) sat-isfies the strict periodic boundary condition us

m(k r) =ei2Grus

m(k + 2G r) and usm(k r) = us

m(k r + R) Onestill would like to know what are the symmetry proper-ties of the TDMs for systems with inversion symmetryThe results are summarized in Table I while the deriva-tion is given in the Appendix As shown in details in theAppendix When the periodic functions us

m and usn have

the same parity the TDM between band m and n is anodd function with respect to k When they have oppo-site parity the TDMs between them is an even functionwith respect to k In a three-band model there are manypathways to generate excitations Take the MgO exam-ple one possibility is to choose a path band2 rarr 3 rarr 4The corresponding macroscopic polarization is given by

P (t) sim P (k0 t) + P (minusk0 t)

= D24(k0 +A(t))D43(k0 +A(t))D32(k0 +A(t))E(t)2

+D24(minusk0 +A(t))D43(minusk0 +A(t))D32(minusk0 +A(t))E(t)2

+ cc(7)

Here E(t) and A(t) are electric field and vector potentialrespectively If the driven laser is a long pulse E(t +T2) = minusE(t) and A(t + T2) = minusA(t) where T is theoptical cycle of the laser Thus we can get

P (t+ T2) sim P (k0 t+ T2) + P (minusk0 t+ T2)

= D24(k0 minusA(t))D43(k0 minusA(t))D32(k0 minusA(t))E(t)2

+D24(minusk0 minusA(t))D43(minusk0 minusA(t))D32(minusk0 minusA(t))E(t)2

+ cc

(8)

By comparing Eq (7) and (8) and using the propertieslisted in TABLE I [ the parities of the wavefunctions forthese three bands are shown in Fig 5(b)] we can find thatP (t) = minusP (t+ T2) The odd parity of macroscopic po-larization similar to the case for an atomic target guar-antees that no even-order harmonics in the spectra Ifthe parities of TDMs have not accounted for odd sym-metric macroscopic polarization is not present In otherwords because of the parities of TDMs coupling of mul-tiple bands cannot generate even harmonics if the systemhas inversion symmetry

FIG 4 Black line is the same as fig 3(c) harmonic spec-trum calculated using correct dipole moments (Dc1c2

(k) isodd Dvc2

(k) and Dvc1(k) are even functions of k) Red line

The same as the black line except that Dc1c2(k) is changed

to an even function of k artificially The subscripts v c1 c2represent the valence band first conduction band and secondconduction band respectively The intensities of the spectrain the right side of the vertical solid line were multiplied by afactor of 4

Using the more accurate band structure and dipole mo-ments in particular the newly dipole phases constructedin the present rdquosmooth-periodicrdquo gauge we can improvethe simulation reported in [27] where the dipole momentshave been set to be a constant Comparison of CEP-dependent spectra between experimental data and oursimulation is presented in Fig 3(a) and (b) The laserparameters used in the simulation can be found in thecaption Many features in the experimental data are re-produced in the present simulation (1) In Fig3(a) theslope of photon energy versus the CEP has been repro-duced in Fig 3(b) (2) Both experiment and simulationshow two plateaus in the same photon energy region (3)In Fig 3(b) our simulation indicates a minimum around13 eV which is consistent with experiment even thoughit is near the low energy end of the experimental datathus it could also be due to detector efficiency in the en-ergy region Note that this minimum also appears in Fig2 at the 15th order harmonic Comparing with the sim-ulations reported in [27] we have witnessed significantimprovement in the present simulation

Figs 3(c-f) compare the HHG spectra at two CEPs05 π and 00 between experiment and the present sim-ulation It is clear that the sine-like pulse (CEP=05π) would generate sharper discrete harmonics while acosine-like pulse would produce relatively flatter onesSuch results are in agreement with the measurements

In this article we are concerned with how harmonicspectra are affected if the parity and periodicity of tran-sition dipole moments are not correctly accounted forIn many prior calculations approximations were made

DRAFT

5

in which the k-dependent dipole moments were taken tobe its absolute values This means that it is an even func-tion with respect to k In the three-band model for MgOusing the method presented here Dc1c2(k) which is thecoupling between the two conduction bands is an oddfunction If we arbitrarily change it to an even functionhow the HHG spectra would be altered Fig 4 showsthe original spectra copied from Fig 3(c) (in black lines)and compare it to HHG spectra (in red lines) if Dc1c2(k)is changed to an even function For the first plateauharmonics no significant changes occur since the firstplateau is due to recombination of electrons from the firstconduction band to the valence band For the harmonicsin the 2nd plateau we can see the change as the harmonicpeaks are shifted Based on Eqs (7) (8) and the analy-sis in [27] the peaks in the secondary plateau are givenby ω = (2n minus 1 minus θπ)ω0 where ω0 is the frequency ofthe driving pulse and the phase θ is the phase differencebetween the two pathways to reach the conduction band2 By changing the parity of Dc1c2(k) artificially fromodd to even the peaks in the 2nd plateau will be givenby ω = (2nminus θπ)ω0 This can explain the shift of thepeaks in the secondary plateau shown in Fig 4 Thusif the parity of the dipole moment has wrong parity thegenerated harmonic spectra will be quite different

We conclude this section by illustrating the differencebetween gas phase and solids as shown in Fig 5 Inthe gaseous medium with inversion symmetry a triangu-lar system will never be formed because of the parity ofthe wavefunction For crystal with inversion symmetrywhere energy level is expanded into a band a trianglesystem can be formed While the parity of the TDMswill prevent the generation of even-order harmonics

FIG 5 Illustration of transition paths of electrons for gasesand solids with inversion symmetry For the gas phase a tri-angular system will never be formed because the transitionbetween states with same parity is forbidden While for solidcase where the energy levels are extended into bands transi-tion is between bands at these k points away from Γ are notforbidden The symmetry properties of the TDMs will ensurethe absence of even order optical signal

IV SYSTEM WITH BROKEN SYMMETRY

For system with broken symmetry the Zakrsquos phasecan be any value Thus the method above for systemwith inversion symmetry is not valid anymore Notethat in the periodic gauge the Berry connection is pe-riodic Dmm(k) = Dmm(k + G) which means it can beexpanded as

Dmm(k) = gm(k) + σm (9)

where gm(k) =sum+infin

n=1 f1(n) cos(nLk) + f2(n) sin(nLk) isthe rdquoACrdquo component and σm is a constant which can beregarded as the rdquoDCrdquo component The DC componentwill lead to divergence of the introduced additional phase

Θm(k) =int k

k0

dκDmm(κ) + ϕm (k0) We do not need to

care about the AC part because this component wouldnot influence the periodicity the continuity and the fi-nal observable physical quantities [28] If the ldquosmoothprocedurerdquo is carried out for the first Brilloin zone theDC-induced non-periodic phase of the TDMs betweenband m and n is (σn minus σm)k

FIG 6 Red line is the k-dependent dipole phase generatedby ldquosmooth procedurerdquo Such phase is not periodic because ofthe DC term in the additionally introduced phase The blackline is the k-dependent dipole phase after the DC componentis taken away as introduced in this article

Here we take the direction Γ minus A of wurtzite ZnO asan example In Fig 6 the red line is the k-dependentdipole phase for the first Brillouin zone obtained fromVASP using a ldquosmooth procedurerdquo As stated abovethe dipole phase cannot be ensured to be periodic be-cause of the DC component However it is easy to getthe slope by (σn minus σm) = (αmn(πL)minus αmn(minusπL))GHere αmn(plusmnπL) are the phase of TDMs which are readfrom the data generated by ldquosmooth procedurerdquo shownby the red line in Fig 6 We can then get the periodicdipole phase by subtracting the DC part off Dp

mn(k) =Ds

mn(k)ei(σmminusσn)k which is shown by the black line in

Fig 6 for ZnO At the same time the Berry connection

DRAFT

6

is changed from zero to Dpmm(k) minus Dp

nn(k) = σm minus σnWith that all the elements in the SBEs model are pe-riodic and continuous with respect to k This meansthat the equation of motion for carriers can be solved ina finite k space In Fig 7 the calculated HHG spec-trum (blue line) by solving two-band SBEs model usingTDMs obtained by our ldquosmooth-periodicrdquo procedure iscompared to the experimental data (green line) We alsopresent the spectrum (red line) by solving two-band SBEsincluding only dipole phase obtained from tight-bindingmodel Even though tight-binding model can approxi-matively reproduce the orientation-dependent feature ofHHG spectra usually it is too primitive to produce therelative strength between odd and even order harmonicsUsing accurate TDMs obtained from ab-initio softwarewith the help of our ldquosmooth-periodicrdquo procedure theexperimental spectra of ZnO first reported in S Ghimireet al [1] has finally been satisfactorily reproduced theo-retically

FIG 7 HHG spectra from ZnO Green line is the experi-mental data blue line is calculated by two-band SBEs withelements obtained from ab-initio software with the help of thepresent ldquosmooth-periodicrdquo procedure The red line is fromcalculations where the dipole phases are calculated from thetight-binding model The red line and experimental data arecopied from our previous work[30] To present clear compar-ison the spectra are shifted vertically

V CONCLUSION REMARKS

Although ab-initio software has been widely used toinvestigate electronic properties nowadays the randomphase generated in the algorithms prevents its applica-tion to calculate non-adiabatic dynamics especially whenthe external field is a strong laser In this article we firstshow that the commonly used ldquosmooth procedurerdquo can-not ensure the periodicity of the wavefunction Secondwe provide two different methods to overcome this de-fect for systems with and without inversion symmetry

Because our approaches ensure continuity and periodic-ity of all the elements used in the equations of motionthe gauge resulting by the transformation of our methodscan be referred to as a ldquosmooth-periodicrdquo gauge to distin-guish it from rdquoperiodicrdquo gauge used by Resta [39] Basedon this gauge HHG spectra from solids with and withoutinversion symmetry are revisited It is emphasized thatsymmetry properties are the key factors to ensure the ab-sence of even order harmonics for systems with inversionsymmetry With the accurate TMDs with dipole phaseand Berry connection the HHG spectrum from ZnO withbroken symmetry is also improved greatly The TMDsintroduced in this work is fundamental to all applicationsrelating to optical signals from solids such as laser wave-form control banddipole reconstruction and detectingdynamic information

ACKNOWLEDGMENT

This work was supported by the NSF of JiangsuProvince (Grant No BK20170032) and NSF of China( 11704187 11974112 11975012) CDL was supportedin part by the Chemical Sciences Geosciences and Bio-sciences Division Office of Basic Energy Sciences Officeof Science US Department of Energy under Grant NoDE-FG02-86ER13491 SJ also thanks for the support bythe Project funded by China Postdoctoral Science Foun-dation No 2019TQ0098 SJ thanks for the fruitful dis-cussion with Prof Chengcheng Liu from Beijing Instituteof Technology and Dr Prasoon Saurabh from East ChinaNormal University

APPENDIX DERIVATION OF THE PARITIES

OF TRANSITION DIPOLE MOMENTS

Both usm(k r) and ukp

m (k r) satisfy the kmiddotp equation

(

minus1

2nabla2

r + V (r)minus ik middot nablar

)

us(kp)mk (r) =

(

Em(k)minusk2

2

)

us(kp)mk (r)

(A1)

where ukpm (k r) is assumed to satisfy

ukpm (minusk r) = ukplowast

m (k r) (A2)

When the system has inversion symmetry

ukpm (minusk r) = plusmnukp

m (kminusr)Dkpmm(k) = 0 (A3)

usm(k r) must be related to ukp

m (k r) through a gaugetransformation eg

usm(k r) = ukp

m (k r)eiβ(k) (A4)

DRAFT

7

As stated in the main text after the ldquosmooth procedurerdquothe Berry connection

Dsmm(k) = i 〈us

m(k r)|nablakusm(k r)〉 = 0 (A5)

By inserting Eq (A4) into Eq (A5) we have

Dsmm(k) = i

lang

ukpm (k r)eiβ(k)|nablak

(

ukpm (k r)eiβ(k))rang

= Dkpmm(k)minusnablakβ(k) = 0 (A6)

It is easy to prove that Dkpmm(k) is a real number and an

even function with respect to k

Dkpmm(minusk) = Dkp

mm(k) (A7)

In order to ensure Dsmm(k) = 0 β(k) = odd function +

const Further if the system has inversion symmetryβ(k) = const Thus us

m(k r) also has

usm(minusk r) = plusmnus

m(kminusr) (A8)

Using Eq (A8) we can get all the properties listed inTable I

[1] S Ghimire A D DiChiara E Sistrunk P Agostini LF DiMauro and D A Reis Nat Phys 7 138 (2011)

[2] S Gholam-Mirzaei J Beetar and M Chini Appl PhysLett 110 061101 (2017)

[3] G Vampa et al Nature (London) 522 462 (2015)[4] T T Luu et al Nature 521 498 (2015)[5] M Hohenleutneret al Nature (London) 523 572

(2015)[6] A Schiffrin et al Nature 493 70 (2013)[7] S Yu Kruchinin M Korbman V S Yakovlev Phys

Rev B 87 115201 (2013)[8] T Paasch-Colberg et al Nat Photon 8 214 (2014)[9] G Wachter et al Phys Rev Lett 113 087401 (2014)

[10] O Schubert et al Nat Photonics 8 119 (2014)[11] L Li P Lan X Liu L He X Zhu O D Mucke P

Lu Optics Express 26 23844 (2018)[12] L Liu J Zhao J M Yuan Z X Zhao Chin Phys B

28 114205 (2019)[13] M Schultze et al Nature 493 75 (2013)[14] M Schultze et al Sicence 346 1348 (2014)[15] A Chernikov C Ruppert H M Hill A F Rigosi T

F Heinz Nat Photon 9 466 (2015)[16] L Meckbach T Stroucken and S W Koch Appl Phys

Lett 112 061104 (2018)[17] L Meckbachet al arXiv preprint arXiv190308553

(2019)[18] D Golde et al Phys Rev B 77 075330 (2008)[19] T T Luu H J Worner Phys Rev B 94 115164 (2016)[20] H Haug S W Koch Quantum Theory of the Optical

and Electronic Properties of Semiconductors Fivth Edi-tion World Scientic Publishing Company 2009

[21] G Vampa et al Phys Rev B 91 064302 (2015)

[22] Z Wang et al Nat Commun 8 1686 (2017)[23] L Meckbach T Stroucken and S W Koch Phys Rev

B 97 0354425 (2018)[24] C Yu et al Phys Rev A 94 013846 (2016)[25] M V Berry Proc R Soc London Ser A 392 45 (1984)[26] Y G Yao et al Phys Rev Lett 92 037204 (2004)[27] Y S You et al Opt Lett 42 1816 (2017)[28] J Li et al Phys Rev A 100 043404 (2019)[29] L Yue M Gaarde Imperfect Recollisions in High-

Harmonic Generation in Solids arXiv preprintarXiv200104626 (2020)

[30] S C Jiang et al Phys Rev Lett 120 253201 (2018)[31] K S Virk J E Sipe Phys Rev B 76 035213 (2007)[32] R W Nunes X Gonze Phys Rev B 63 155107 (2001)[33] I Souza et al Phys Rev B 69 085106 (2004)[34] U Lindefelt HE Nilsson and M Hjelm Semicond Sci

Technol 19 1061 (2004)[35] V S Yakovlev M S Wismer Comput Phys Commun

217 82-88 (2017)[36] M X Wu et al Phys Rev A 91 043839 (2015)[37] M Du C Liu Y Zheng Z Zeng R li Phys Rev A

100 043840 (2019)[38] J Zak Phys Rev Lett 62 2747 (1989)[39] R Resta J Phys Condens Mat 12 R107 (2000)[40] G Kresse and J Hafner Phys Rev B 47 558 (1993)[41] N Tancogne-Dejean et al Nat Commun 8 745 (2017)[42] Y S You D A Reis and S Ghimire Nat Phys 13

345 (2017)[43] Y S You et al Opt Lett 44 530 (2019)[44] Y Zhao et al arXiv191112092

DRAFT

4

TABLE I Properties of TDMs for different ldquoparitiesrdquo of eigen-functions

us

m(k r) us

n(k r) Ds

m(k)

us

m(minusk r) =

us

m(kminusr)

us

n(minusk r) =

us

n(kminusr)

Ds

mn(minusk) =

minusDs

mn(k)

us

m(minusk r) =

us

m(kminusr)

us

n(minusk r) =

minusus

n(kminusr)

Ds

mn(minusk) =

Ds

mn(k)

us

m(minusk r) =

minusus

m(kminusr)

us

n(minusk r) =

us

n(kminusr)

Ds

mn(minusk) =

Ds

mn(k)

us

m(minusk r) =

minusus

m(kminusr)

us

n(minusk r) =

minusus

n(kminusr)

Ds

mn(minusk) =

minusDs

mn(k)

From the analysis of Section II after the ldquosmooth pro-cedurerdquo the newly derived eigenfunction us

m(k r) sat-isfies the strict periodic boundary condition us

m(k r) =ei2Grus

m(k + 2G r) and usm(k r) = us

m(k r + R) Onestill would like to know what are the symmetry proper-ties of the TDMs for systems with inversion symmetryThe results are summarized in Table I while the deriva-tion is given in the Appendix As shown in details in theAppendix When the periodic functions us

m and usn have

the same parity the TDM between band m and n is anodd function with respect to k When they have oppo-site parity the TDMs between them is an even functionwith respect to k In a three-band model there are manypathways to generate excitations Take the MgO exam-ple one possibility is to choose a path band2 rarr 3 rarr 4The corresponding macroscopic polarization is given by

P (t) sim P (k0 t) + P (minusk0 t)

= D24(k0 +A(t))D43(k0 +A(t))D32(k0 +A(t))E(t)2

+D24(minusk0 +A(t))D43(minusk0 +A(t))D32(minusk0 +A(t))E(t)2

+ cc(7)

Here E(t) and A(t) are electric field and vector potentialrespectively If the driven laser is a long pulse E(t +T2) = minusE(t) and A(t + T2) = minusA(t) where T is theoptical cycle of the laser Thus we can get

P (t+ T2) sim P (k0 t+ T2) + P (minusk0 t+ T2)

= D24(k0 minusA(t))D43(k0 minusA(t))D32(k0 minusA(t))E(t)2

+D24(minusk0 minusA(t))D43(minusk0 minusA(t))D32(minusk0 minusA(t))E(t)2

+ cc

(8)

By comparing Eq (7) and (8) and using the propertieslisted in TABLE I [ the parities of the wavefunctions forthese three bands are shown in Fig 5(b)] we can find thatP (t) = minusP (t+ T2) The odd parity of macroscopic po-larization similar to the case for an atomic target guar-antees that no even-order harmonics in the spectra Ifthe parities of TDMs have not accounted for odd sym-metric macroscopic polarization is not present In otherwords because of the parities of TDMs coupling of mul-tiple bands cannot generate even harmonics if the systemhas inversion symmetry

FIG 4 Black line is the same as fig 3(c) harmonic spec-trum calculated using correct dipole moments (Dc1c2

(k) isodd Dvc2

(k) and Dvc1(k) are even functions of k) Red line

The same as the black line except that Dc1c2(k) is changed

to an even function of k artificially The subscripts v c1 c2represent the valence band first conduction band and secondconduction band respectively The intensities of the spectrain the right side of the vertical solid line were multiplied by afactor of 4

Using the more accurate band structure and dipole mo-ments in particular the newly dipole phases constructedin the present rdquosmooth-periodicrdquo gauge we can improvethe simulation reported in [27] where the dipole momentshave been set to be a constant Comparison of CEP-dependent spectra between experimental data and oursimulation is presented in Fig 3(a) and (b) The laserparameters used in the simulation can be found in thecaption Many features in the experimental data are re-produced in the present simulation (1) In Fig3(a) theslope of photon energy versus the CEP has been repro-duced in Fig 3(b) (2) Both experiment and simulationshow two plateaus in the same photon energy region (3)In Fig 3(b) our simulation indicates a minimum around13 eV which is consistent with experiment even thoughit is near the low energy end of the experimental datathus it could also be due to detector efficiency in the en-ergy region Note that this minimum also appears in Fig2 at the 15th order harmonic Comparing with the sim-ulations reported in [27] we have witnessed significantimprovement in the present simulation

Figs 3(c-f) compare the HHG spectra at two CEPs05 π and 00 between experiment and the present sim-ulation It is clear that the sine-like pulse (CEP=05π) would generate sharper discrete harmonics while acosine-like pulse would produce relatively flatter onesSuch results are in agreement with the measurements

In this article we are concerned with how harmonicspectra are affected if the parity and periodicity of tran-sition dipole moments are not correctly accounted forIn many prior calculations approximations were made

DRAFT

5

in which the k-dependent dipole moments were taken tobe its absolute values This means that it is an even func-tion with respect to k In the three-band model for MgOusing the method presented here Dc1c2(k) which is thecoupling between the two conduction bands is an oddfunction If we arbitrarily change it to an even functionhow the HHG spectra would be altered Fig 4 showsthe original spectra copied from Fig 3(c) (in black lines)and compare it to HHG spectra (in red lines) if Dc1c2(k)is changed to an even function For the first plateauharmonics no significant changes occur since the firstplateau is due to recombination of electrons from the firstconduction band to the valence band For the harmonicsin the 2nd plateau we can see the change as the harmonicpeaks are shifted Based on Eqs (7) (8) and the analy-sis in [27] the peaks in the secondary plateau are givenby ω = (2n minus 1 minus θπ)ω0 where ω0 is the frequency ofthe driving pulse and the phase θ is the phase differencebetween the two pathways to reach the conduction band2 By changing the parity of Dc1c2(k) artificially fromodd to even the peaks in the 2nd plateau will be givenby ω = (2nminus θπ)ω0 This can explain the shift of thepeaks in the secondary plateau shown in Fig 4 Thusif the parity of the dipole moment has wrong parity thegenerated harmonic spectra will be quite different

We conclude this section by illustrating the differencebetween gas phase and solids as shown in Fig 5 Inthe gaseous medium with inversion symmetry a triangu-lar system will never be formed because of the parity ofthe wavefunction For crystal with inversion symmetrywhere energy level is expanded into a band a trianglesystem can be formed While the parity of the TDMswill prevent the generation of even-order harmonics

FIG 5 Illustration of transition paths of electrons for gasesand solids with inversion symmetry For the gas phase a tri-angular system will never be formed because the transitionbetween states with same parity is forbidden While for solidcase where the energy levels are extended into bands transi-tion is between bands at these k points away from Γ are notforbidden The symmetry properties of the TDMs will ensurethe absence of even order optical signal

IV SYSTEM WITH BROKEN SYMMETRY

For system with broken symmetry the Zakrsquos phasecan be any value Thus the method above for systemwith inversion symmetry is not valid anymore Notethat in the periodic gauge the Berry connection is pe-riodic Dmm(k) = Dmm(k + G) which means it can beexpanded as

Dmm(k) = gm(k) + σm (9)

where gm(k) =sum+infin

n=1 f1(n) cos(nLk) + f2(n) sin(nLk) isthe rdquoACrdquo component and σm is a constant which can beregarded as the rdquoDCrdquo component The DC componentwill lead to divergence of the introduced additional phase

Θm(k) =int k

k0

dκDmm(κ) + ϕm (k0) We do not need to

care about the AC part because this component wouldnot influence the periodicity the continuity and the fi-nal observable physical quantities [28] If the ldquosmoothprocedurerdquo is carried out for the first Brilloin zone theDC-induced non-periodic phase of the TDMs betweenband m and n is (σn minus σm)k

FIG 6 Red line is the k-dependent dipole phase generatedby ldquosmooth procedurerdquo Such phase is not periodic because ofthe DC term in the additionally introduced phase The blackline is the k-dependent dipole phase after the DC componentis taken away as introduced in this article

Here we take the direction Γ minus A of wurtzite ZnO asan example In Fig 6 the red line is the k-dependentdipole phase for the first Brillouin zone obtained fromVASP using a ldquosmooth procedurerdquo As stated abovethe dipole phase cannot be ensured to be periodic be-cause of the DC component However it is easy to getthe slope by (σn minus σm) = (αmn(πL)minus αmn(minusπL))GHere αmn(plusmnπL) are the phase of TDMs which are readfrom the data generated by ldquosmooth procedurerdquo shownby the red line in Fig 6 We can then get the periodicdipole phase by subtracting the DC part off Dp

mn(k) =Ds

mn(k)ei(σmminusσn)k which is shown by the black line in

Fig 6 for ZnO At the same time the Berry connection

DRAFT

6

is changed from zero to Dpmm(k) minus Dp

nn(k) = σm minus σnWith that all the elements in the SBEs model are pe-riodic and continuous with respect to k This meansthat the equation of motion for carriers can be solved ina finite k space In Fig 7 the calculated HHG spec-trum (blue line) by solving two-band SBEs model usingTDMs obtained by our ldquosmooth-periodicrdquo procedure iscompared to the experimental data (green line) We alsopresent the spectrum (red line) by solving two-band SBEsincluding only dipole phase obtained from tight-bindingmodel Even though tight-binding model can approxi-matively reproduce the orientation-dependent feature ofHHG spectra usually it is too primitive to produce therelative strength between odd and even order harmonicsUsing accurate TDMs obtained from ab-initio softwarewith the help of our ldquosmooth-periodicrdquo procedure theexperimental spectra of ZnO first reported in S Ghimireet al [1] has finally been satisfactorily reproduced theo-retically

FIG 7 HHG spectra from ZnO Green line is the experi-mental data blue line is calculated by two-band SBEs withelements obtained from ab-initio software with the help of thepresent ldquosmooth-periodicrdquo procedure The red line is fromcalculations where the dipole phases are calculated from thetight-binding model The red line and experimental data arecopied from our previous work[30] To present clear compar-ison the spectra are shifted vertically

V CONCLUSION REMARKS

Although ab-initio software has been widely used toinvestigate electronic properties nowadays the randomphase generated in the algorithms prevents its applica-tion to calculate non-adiabatic dynamics especially whenthe external field is a strong laser In this article we firstshow that the commonly used ldquosmooth procedurerdquo can-not ensure the periodicity of the wavefunction Secondwe provide two different methods to overcome this de-fect for systems with and without inversion symmetry

Because our approaches ensure continuity and periodic-ity of all the elements used in the equations of motionthe gauge resulting by the transformation of our methodscan be referred to as a ldquosmooth-periodicrdquo gauge to distin-guish it from rdquoperiodicrdquo gauge used by Resta [39] Basedon this gauge HHG spectra from solids with and withoutinversion symmetry are revisited It is emphasized thatsymmetry properties are the key factors to ensure the ab-sence of even order harmonics for systems with inversionsymmetry With the accurate TMDs with dipole phaseand Berry connection the HHG spectrum from ZnO withbroken symmetry is also improved greatly The TMDsintroduced in this work is fundamental to all applicationsrelating to optical signals from solids such as laser wave-form control banddipole reconstruction and detectingdynamic information

ACKNOWLEDGMENT

This work was supported by the NSF of JiangsuProvince (Grant No BK20170032) and NSF of China( 11704187 11974112 11975012) CDL was supportedin part by the Chemical Sciences Geosciences and Bio-sciences Division Office of Basic Energy Sciences Officeof Science US Department of Energy under Grant NoDE-FG02-86ER13491 SJ also thanks for the support bythe Project funded by China Postdoctoral Science Foun-dation No 2019TQ0098 SJ thanks for the fruitful dis-cussion with Prof Chengcheng Liu from Beijing Instituteof Technology and Dr Prasoon Saurabh from East ChinaNormal University

APPENDIX DERIVATION OF THE PARITIES

OF TRANSITION DIPOLE MOMENTS

Both usm(k r) and ukp

m (k r) satisfy the kmiddotp equation

(

minus1

2nabla2

r + V (r)minus ik middot nablar

)

us(kp)mk (r) =

(

Em(k)minusk2

2

)

us(kp)mk (r)

(A1)

where ukpm (k r) is assumed to satisfy

ukpm (minusk r) = ukplowast

m (k r) (A2)

When the system has inversion symmetry

ukpm (minusk r) = plusmnukp

m (kminusr)Dkpmm(k) = 0 (A3)

usm(k r) must be related to ukp

m (k r) through a gaugetransformation eg

usm(k r) = ukp

m (k r)eiβ(k) (A4)

DRAFT

7

As stated in the main text after the ldquosmooth procedurerdquothe Berry connection

Dsmm(k) = i 〈us

m(k r)|nablakusm(k r)〉 = 0 (A5)

By inserting Eq (A4) into Eq (A5) we have

Dsmm(k) = i

lang

ukpm (k r)eiβ(k)|nablak

(

ukpm (k r)eiβ(k))rang

= Dkpmm(k)minusnablakβ(k) = 0 (A6)

It is easy to prove that Dkpmm(k) is a real number and an

even function with respect to k

Dkpmm(minusk) = Dkp

mm(k) (A7)

In order to ensure Dsmm(k) = 0 β(k) = odd function +

const Further if the system has inversion symmetryβ(k) = const Thus us

m(k r) also has

usm(minusk r) = plusmnus

m(kminusr) (A8)

Using Eq (A8) we can get all the properties listed inTable I

[1] S Ghimire A D DiChiara E Sistrunk P Agostini LF DiMauro and D A Reis Nat Phys 7 138 (2011)

[2] S Gholam-Mirzaei J Beetar and M Chini Appl PhysLett 110 061101 (2017)

[3] G Vampa et al Nature (London) 522 462 (2015)[4] T T Luu et al Nature 521 498 (2015)[5] M Hohenleutneret al Nature (London) 523 572

(2015)[6] A Schiffrin et al Nature 493 70 (2013)[7] S Yu Kruchinin M Korbman V S Yakovlev Phys

Rev B 87 115201 (2013)[8] T Paasch-Colberg et al Nat Photon 8 214 (2014)[9] G Wachter et al Phys Rev Lett 113 087401 (2014)

[10] O Schubert et al Nat Photonics 8 119 (2014)[11] L Li P Lan X Liu L He X Zhu O D Mucke P

Lu Optics Express 26 23844 (2018)[12] L Liu J Zhao J M Yuan Z X Zhao Chin Phys B

28 114205 (2019)[13] M Schultze et al Nature 493 75 (2013)[14] M Schultze et al Sicence 346 1348 (2014)[15] A Chernikov C Ruppert H M Hill A F Rigosi T

F Heinz Nat Photon 9 466 (2015)[16] L Meckbach T Stroucken and S W Koch Appl Phys

Lett 112 061104 (2018)[17] L Meckbachet al arXiv preprint arXiv190308553

(2019)[18] D Golde et al Phys Rev B 77 075330 (2008)[19] T T Luu H J Worner Phys Rev B 94 115164 (2016)[20] H Haug S W Koch Quantum Theory of the Optical

and Electronic Properties of Semiconductors Fivth Edi-tion World Scientic Publishing Company 2009

[21] G Vampa et al Phys Rev B 91 064302 (2015)

[22] Z Wang et al Nat Commun 8 1686 (2017)[23] L Meckbach T Stroucken and S W Koch Phys Rev

B 97 0354425 (2018)[24] C Yu et al Phys Rev A 94 013846 (2016)[25] M V Berry Proc R Soc London Ser A 392 45 (1984)[26] Y G Yao et al Phys Rev Lett 92 037204 (2004)[27] Y S You et al Opt Lett 42 1816 (2017)[28] J Li et al Phys Rev A 100 043404 (2019)[29] L Yue M Gaarde Imperfect Recollisions in High-

Harmonic Generation in Solids arXiv preprintarXiv200104626 (2020)

[30] S C Jiang et al Phys Rev Lett 120 253201 (2018)[31] K S Virk J E Sipe Phys Rev B 76 035213 (2007)[32] R W Nunes X Gonze Phys Rev B 63 155107 (2001)[33] I Souza et al Phys Rev B 69 085106 (2004)[34] U Lindefelt HE Nilsson and M Hjelm Semicond Sci

Technol 19 1061 (2004)[35] V S Yakovlev M S Wismer Comput Phys Commun

217 82-88 (2017)[36] M X Wu et al Phys Rev A 91 043839 (2015)[37] M Du C Liu Y Zheng Z Zeng R li Phys Rev A

100 043840 (2019)[38] J Zak Phys Rev Lett 62 2747 (1989)[39] R Resta J Phys Condens Mat 12 R107 (2000)[40] G Kresse and J Hafner Phys Rev B 47 558 (1993)[41] N Tancogne-Dejean et al Nat Commun 8 745 (2017)[42] Y S You D A Reis and S Ghimire Nat Phys 13

345 (2017)[43] Y S You et al Opt Lett 44 530 (2019)[44] Y Zhao et al arXiv191112092

DRAFT

5

in which the k-dependent dipole moments were taken tobe its absolute values This means that it is an even func-tion with respect to k In the three-band model for MgOusing the method presented here Dc1c2(k) which is thecoupling between the two conduction bands is an oddfunction If we arbitrarily change it to an even functionhow the HHG spectra would be altered Fig 4 showsthe original spectra copied from Fig 3(c) (in black lines)and compare it to HHG spectra (in red lines) if Dc1c2(k)is changed to an even function For the first plateauharmonics no significant changes occur since the firstplateau is due to recombination of electrons from the firstconduction band to the valence band For the harmonicsin the 2nd plateau we can see the change as the harmonicpeaks are shifted Based on Eqs (7) (8) and the analy-sis in [27] the peaks in the secondary plateau are givenby ω = (2n minus 1 minus θπ)ω0 where ω0 is the frequency ofthe driving pulse and the phase θ is the phase differencebetween the two pathways to reach the conduction band2 By changing the parity of Dc1c2(k) artificially fromodd to even the peaks in the 2nd plateau will be givenby ω = (2nminus θπ)ω0 This can explain the shift of thepeaks in the secondary plateau shown in Fig 4 Thusif the parity of the dipole moment has wrong parity thegenerated harmonic spectra will be quite different

We conclude this section by illustrating the differencebetween gas phase and solids as shown in Fig 5 Inthe gaseous medium with inversion symmetry a triangu-lar system will never be formed because of the parity ofthe wavefunction For crystal with inversion symmetrywhere energy level is expanded into a band a trianglesystem can be formed While the parity of the TDMswill prevent the generation of even-order harmonics

FIG 5 Illustration of transition paths of electrons for gasesand solids with inversion symmetry For the gas phase a tri-angular system will never be formed because the transitionbetween states with same parity is forbidden While for solidcase where the energy levels are extended into bands transi-tion is between bands at these k points away from Γ are notforbidden The symmetry properties of the TDMs will ensurethe absence of even order optical signal

IV SYSTEM WITH BROKEN SYMMETRY

For system with broken symmetry the Zakrsquos phasecan be any value Thus the method above for systemwith inversion symmetry is not valid anymore Notethat in the periodic gauge the Berry connection is pe-riodic Dmm(k) = Dmm(k + G) which means it can beexpanded as

Dmm(k) = gm(k) + σm (9)

where gm(k) =sum+infin

n=1 f1(n) cos(nLk) + f2(n) sin(nLk) isthe rdquoACrdquo component and σm is a constant which can beregarded as the rdquoDCrdquo component The DC componentwill lead to divergence of the introduced additional phase

Θm(k) =int k

k0

dκDmm(κ) + ϕm (k0) We do not need to

care about the AC part because this component wouldnot influence the periodicity the continuity and the fi-nal observable physical quantities [28] If the ldquosmoothprocedurerdquo is carried out for the first Brilloin zone theDC-induced non-periodic phase of the TDMs betweenband m and n is (σn minus σm)k

FIG 6 Red line is the k-dependent dipole phase generatedby ldquosmooth procedurerdquo Such phase is not periodic because ofthe DC term in the additionally introduced phase The blackline is the k-dependent dipole phase after the DC componentis taken away as introduced in this article

Here we take the direction Γ minus A of wurtzite ZnO asan example In Fig 6 the red line is the k-dependentdipole phase for the first Brillouin zone obtained fromVASP using a ldquosmooth procedurerdquo As stated abovethe dipole phase cannot be ensured to be periodic be-cause of the DC component However it is easy to getthe slope by (σn minus σm) = (αmn(πL)minus αmn(minusπL))GHere αmn(plusmnπL) are the phase of TDMs which are readfrom the data generated by ldquosmooth procedurerdquo shownby the red line in Fig 6 We can then get the periodicdipole phase by subtracting the DC part off Dp

mn(k) =Ds

mn(k)ei(σmminusσn)k which is shown by the black line in

Fig 6 for ZnO At the same time the Berry connection

DRAFT

6

is changed from zero to Dpmm(k) minus Dp

nn(k) = σm minus σnWith that all the elements in the SBEs model are pe-riodic and continuous with respect to k This meansthat the equation of motion for carriers can be solved ina finite k space In Fig 7 the calculated HHG spec-trum (blue line) by solving two-band SBEs model usingTDMs obtained by our ldquosmooth-periodicrdquo procedure iscompared to the experimental data (green line) We alsopresent the spectrum (red line) by solving two-band SBEsincluding only dipole phase obtained from tight-bindingmodel Even though tight-binding model can approxi-matively reproduce the orientation-dependent feature ofHHG spectra usually it is too primitive to produce therelative strength between odd and even order harmonicsUsing accurate TDMs obtained from ab-initio softwarewith the help of our ldquosmooth-periodicrdquo procedure theexperimental spectra of ZnO first reported in S Ghimireet al [1] has finally been satisfactorily reproduced theo-retically

FIG 7 HHG spectra from ZnO Green line is the experi-mental data blue line is calculated by two-band SBEs withelements obtained from ab-initio software with the help of thepresent ldquosmooth-periodicrdquo procedure The red line is fromcalculations where the dipole phases are calculated from thetight-binding model The red line and experimental data arecopied from our previous work[30] To present clear compar-ison the spectra are shifted vertically

V CONCLUSION REMARKS

Although ab-initio software has been widely used toinvestigate electronic properties nowadays the randomphase generated in the algorithms prevents its applica-tion to calculate non-adiabatic dynamics especially whenthe external field is a strong laser In this article we firstshow that the commonly used ldquosmooth procedurerdquo can-not ensure the periodicity of the wavefunction Secondwe provide two different methods to overcome this de-fect for systems with and without inversion symmetry

Because our approaches ensure continuity and periodic-ity of all the elements used in the equations of motionthe gauge resulting by the transformation of our methodscan be referred to as a ldquosmooth-periodicrdquo gauge to distin-guish it from rdquoperiodicrdquo gauge used by Resta [39] Basedon this gauge HHG spectra from solids with and withoutinversion symmetry are revisited It is emphasized thatsymmetry properties are the key factors to ensure the ab-sence of even order harmonics for systems with inversionsymmetry With the accurate TMDs with dipole phaseand Berry connection the HHG spectrum from ZnO withbroken symmetry is also improved greatly The TMDsintroduced in this work is fundamental to all applicationsrelating to optical signals from solids such as laser wave-form control banddipole reconstruction and detectingdynamic information

ACKNOWLEDGMENT

This work was supported by the NSF of JiangsuProvince (Grant No BK20170032) and NSF of China( 11704187 11974112 11975012) CDL was supportedin part by the Chemical Sciences Geosciences and Bio-sciences Division Office of Basic Energy Sciences Officeof Science US Department of Energy under Grant NoDE-FG02-86ER13491 SJ also thanks for the support bythe Project funded by China Postdoctoral Science Foun-dation No 2019TQ0098 SJ thanks for the fruitful dis-cussion with Prof Chengcheng Liu from Beijing Instituteof Technology and Dr Prasoon Saurabh from East ChinaNormal University

APPENDIX DERIVATION OF THE PARITIES

OF TRANSITION DIPOLE MOMENTS

Both usm(k r) and ukp

m (k r) satisfy the kmiddotp equation

(

minus1

2nabla2

r + V (r)minus ik middot nablar

)

us(kp)mk (r) =

(

Em(k)minusk2

2

)

us(kp)mk (r)

(A1)

where ukpm (k r) is assumed to satisfy

ukpm (minusk r) = ukplowast

m (k r) (A2)

When the system has inversion symmetry

ukpm (minusk r) = plusmnukp

m (kminusr)Dkpmm(k) = 0 (A3)

usm(k r) must be related to ukp

m (k r) through a gaugetransformation eg

usm(k r) = ukp

m (k r)eiβ(k) (A4)

DRAFT

7

As stated in the main text after the ldquosmooth procedurerdquothe Berry connection

Dsmm(k) = i 〈us

m(k r)|nablakusm(k r)〉 = 0 (A5)

By inserting Eq (A4) into Eq (A5) we have

Dsmm(k) = i

lang

ukpm (k r)eiβ(k)|nablak

(

ukpm (k r)eiβ(k))rang

= Dkpmm(k)minusnablakβ(k) = 0 (A6)

It is easy to prove that Dkpmm(k) is a real number and an

even function with respect to k

Dkpmm(minusk) = Dkp

mm(k) (A7)

In order to ensure Dsmm(k) = 0 β(k) = odd function +

const Further if the system has inversion symmetryβ(k) = const Thus us

m(k r) also has

usm(minusk r) = plusmnus

m(kminusr) (A8)

Using Eq (A8) we can get all the properties listed inTable I

[1] S Ghimire A D DiChiara E Sistrunk P Agostini LF DiMauro and D A Reis Nat Phys 7 138 (2011)

[2] S Gholam-Mirzaei J Beetar and M Chini Appl PhysLett 110 061101 (2017)

[3] G Vampa et al Nature (London) 522 462 (2015)[4] T T Luu et al Nature 521 498 (2015)[5] M Hohenleutneret al Nature (London) 523 572

(2015)[6] A Schiffrin et al Nature 493 70 (2013)[7] S Yu Kruchinin M Korbman V S Yakovlev Phys

Rev B 87 115201 (2013)[8] T Paasch-Colberg et al Nat Photon 8 214 (2014)[9] G Wachter et al Phys Rev Lett 113 087401 (2014)

[10] O Schubert et al Nat Photonics 8 119 (2014)[11] L Li P Lan X Liu L He X Zhu O D Mucke P

Lu Optics Express 26 23844 (2018)[12] L Liu J Zhao J M Yuan Z X Zhao Chin Phys B

28 114205 (2019)[13] M Schultze et al Nature 493 75 (2013)[14] M Schultze et al Sicence 346 1348 (2014)[15] A Chernikov C Ruppert H M Hill A F Rigosi T

F Heinz Nat Photon 9 466 (2015)[16] L Meckbach T Stroucken and S W Koch Appl Phys

Lett 112 061104 (2018)[17] L Meckbachet al arXiv preprint arXiv190308553

(2019)[18] D Golde et al Phys Rev B 77 075330 (2008)[19] T T Luu H J Worner Phys Rev B 94 115164 (2016)[20] H Haug S W Koch Quantum Theory of the Optical

and Electronic Properties of Semiconductors Fivth Edi-tion World Scientic Publishing Company 2009

[21] G Vampa et al Phys Rev B 91 064302 (2015)

[22] Z Wang et al Nat Commun 8 1686 (2017)[23] L Meckbach T Stroucken and S W Koch Phys Rev

B 97 0354425 (2018)[24] C Yu et al Phys Rev A 94 013846 (2016)[25] M V Berry Proc R Soc London Ser A 392 45 (1984)[26] Y G Yao et al Phys Rev Lett 92 037204 (2004)[27] Y S You et al Opt Lett 42 1816 (2017)[28] J Li et al Phys Rev A 100 043404 (2019)[29] L Yue M Gaarde Imperfect Recollisions in High-

Harmonic Generation in Solids arXiv preprintarXiv200104626 (2020)

[30] S C Jiang et al Phys Rev Lett 120 253201 (2018)[31] K S Virk J E Sipe Phys Rev B 76 035213 (2007)[32] R W Nunes X Gonze Phys Rev B 63 155107 (2001)[33] I Souza et al Phys Rev B 69 085106 (2004)[34] U Lindefelt HE Nilsson and M Hjelm Semicond Sci

Technol 19 1061 (2004)[35] V S Yakovlev M S Wismer Comput Phys Commun

217 82-88 (2017)[36] M X Wu et al Phys Rev A 91 043839 (2015)[37] M Du C Liu Y Zheng Z Zeng R li Phys Rev A

100 043840 (2019)[38] J Zak Phys Rev Lett 62 2747 (1989)[39] R Resta J Phys Condens Mat 12 R107 (2000)[40] G Kresse and J Hafner Phys Rev B 47 558 (1993)[41] N Tancogne-Dejean et al Nat Commun 8 745 (2017)[42] Y S You D A Reis and S Ghimire Nat Phys 13

345 (2017)[43] Y S You et al Opt Lett 44 530 (2019)[44] Y Zhao et al arXiv191112092

DRAFT

6

is changed from zero to Dpmm(k) minus Dp

nn(k) = σm minus σnWith that all the elements in the SBEs model are pe-riodic and continuous with respect to k This meansthat the equation of motion for carriers can be solved ina finite k space In Fig 7 the calculated HHG spec-trum (blue line) by solving two-band SBEs model usingTDMs obtained by our ldquosmooth-periodicrdquo procedure iscompared to the experimental data (green line) We alsopresent the spectrum (red line) by solving two-band SBEsincluding only dipole phase obtained from tight-bindingmodel Even though tight-binding model can approxi-matively reproduce the orientation-dependent feature ofHHG spectra usually it is too primitive to produce therelative strength between odd and even order harmonicsUsing accurate TDMs obtained from ab-initio softwarewith the help of our ldquosmooth-periodicrdquo procedure theexperimental spectra of ZnO first reported in S Ghimireet al [1] has finally been satisfactorily reproduced theo-retically

FIG 7 HHG spectra from ZnO Green line is the experi-mental data blue line is calculated by two-band SBEs withelements obtained from ab-initio software with the help of thepresent ldquosmooth-periodicrdquo procedure The red line is fromcalculations where the dipole phases are calculated from thetight-binding model The red line and experimental data arecopied from our previous work[30] To present clear compar-ison the spectra are shifted vertically

V CONCLUSION REMARKS

Although ab-initio software has been widely used toinvestigate electronic properties nowadays the randomphase generated in the algorithms prevents its applica-tion to calculate non-adiabatic dynamics especially whenthe external field is a strong laser In this article we firstshow that the commonly used ldquosmooth procedurerdquo can-not ensure the periodicity of the wavefunction Secondwe provide two different methods to overcome this de-fect for systems with and without inversion symmetry

Because our approaches ensure continuity and periodic-ity of all the elements used in the equations of motionthe gauge resulting by the transformation of our methodscan be referred to as a ldquosmooth-periodicrdquo gauge to distin-guish it from rdquoperiodicrdquo gauge used by Resta [39] Basedon this gauge HHG spectra from solids with and withoutinversion symmetry are revisited It is emphasized thatsymmetry properties are the key factors to ensure the ab-sence of even order harmonics for systems with inversionsymmetry With the accurate TMDs with dipole phaseand Berry connection the HHG spectrum from ZnO withbroken symmetry is also improved greatly The TMDsintroduced in this work is fundamental to all applicationsrelating to optical signals from solids such as laser wave-form control banddipole reconstruction and detectingdynamic information

ACKNOWLEDGMENT

This work was supported by the NSF of JiangsuProvince (Grant No BK20170032) and NSF of China( 11704187 11974112 11975012) CDL was supportedin part by the Chemical Sciences Geosciences and Bio-sciences Division Office of Basic Energy Sciences Officeof Science US Department of Energy under Grant NoDE-FG02-86ER13491 SJ also thanks for the support bythe Project funded by China Postdoctoral Science Foun-dation No 2019TQ0098 SJ thanks for the fruitful dis-cussion with Prof Chengcheng Liu from Beijing Instituteof Technology and Dr Prasoon Saurabh from East ChinaNormal University

APPENDIX DERIVATION OF THE PARITIES

OF TRANSITION DIPOLE MOMENTS

Both usm(k r) and ukp

m (k r) satisfy the kmiddotp equation

(

minus1

2nabla2

r + V (r)minus ik middot nablar

)

us(kp)mk (r) =

(

Em(k)minusk2

2

)

us(kp)mk (r)

(A1)

where ukpm (k r) is assumed to satisfy

ukpm (minusk r) = ukplowast

m (k r) (A2)

When the system has inversion symmetry

ukpm (minusk r) = plusmnukp

m (kminusr)Dkpmm(k) = 0 (A3)

usm(k r) must be related to ukp

m (k r) through a gaugetransformation eg

usm(k r) = ukp

m (k r)eiβ(k) (A4)

DRAFT

7

As stated in the main text after the ldquosmooth procedurerdquothe Berry connection

Dsmm(k) = i 〈us

m(k r)|nablakusm(k r)〉 = 0 (A5)

By inserting Eq (A4) into Eq (A5) we have

Dsmm(k) = i

lang

ukpm (k r)eiβ(k)|nablak

(

ukpm (k r)eiβ(k))rang

= Dkpmm(k)minusnablakβ(k) = 0 (A6)

It is easy to prove that Dkpmm(k) is a real number and an

even function with respect to k

Dkpmm(minusk) = Dkp

mm(k) (A7)

In order to ensure Dsmm(k) = 0 β(k) = odd function +

const Further if the system has inversion symmetryβ(k) = const Thus us

m(k r) also has

usm(minusk r) = plusmnus

m(kminusr) (A8)

Using Eq (A8) we can get all the properties listed inTable I

[1] S Ghimire A D DiChiara E Sistrunk P Agostini LF DiMauro and D A Reis Nat Phys 7 138 (2011)

[2] S Gholam-Mirzaei J Beetar and M Chini Appl PhysLett 110 061101 (2017)

[3] G Vampa et al Nature (London) 522 462 (2015)[4] T T Luu et al Nature 521 498 (2015)[5] M Hohenleutneret al Nature (London) 523 572

(2015)[6] A Schiffrin et al Nature 493 70 (2013)[7] S Yu Kruchinin M Korbman V S Yakovlev Phys

Rev B 87 115201 (2013)[8] T Paasch-Colberg et al Nat Photon 8 214 (2014)[9] G Wachter et al Phys Rev Lett 113 087401 (2014)

[10] O Schubert et al Nat Photonics 8 119 (2014)[11] L Li P Lan X Liu L He X Zhu O D Mucke P

Lu Optics Express 26 23844 (2018)[12] L Liu J Zhao J M Yuan Z X Zhao Chin Phys B

28 114205 (2019)[13] M Schultze et al Nature 493 75 (2013)[14] M Schultze et al Sicence 346 1348 (2014)[15] A Chernikov C Ruppert H M Hill A F Rigosi T

F Heinz Nat Photon 9 466 (2015)[16] L Meckbach T Stroucken and S W Koch Appl Phys

Lett 112 061104 (2018)[17] L Meckbachet al arXiv preprint arXiv190308553

(2019)[18] D Golde et al Phys Rev B 77 075330 (2008)[19] T T Luu H J Worner Phys Rev B 94 115164 (2016)[20] H Haug S W Koch Quantum Theory of the Optical

and Electronic Properties of Semiconductors Fivth Edi-tion World Scientic Publishing Company 2009

[21] G Vampa et al Phys Rev B 91 064302 (2015)

[22] Z Wang et al Nat Commun 8 1686 (2017)[23] L Meckbach T Stroucken and S W Koch Phys Rev

B 97 0354425 (2018)[24] C Yu et al Phys Rev A 94 013846 (2016)[25] M V Berry Proc R Soc London Ser A 392 45 (1984)[26] Y G Yao et al Phys Rev Lett 92 037204 (2004)[27] Y S You et al Opt Lett 42 1816 (2017)[28] J Li et al Phys Rev A 100 043404 (2019)[29] L Yue M Gaarde Imperfect Recollisions in High-

Harmonic Generation in Solids arXiv preprintarXiv200104626 (2020)

[30] S C Jiang et al Phys Rev Lett 120 253201 (2018)[31] K S Virk J E Sipe Phys Rev B 76 035213 (2007)[32] R W Nunes X Gonze Phys Rev B 63 155107 (2001)[33] I Souza et al Phys Rev B 69 085106 (2004)[34] U Lindefelt HE Nilsson and M Hjelm Semicond Sci

Technol 19 1061 (2004)[35] V S Yakovlev M S Wismer Comput Phys Commun

217 82-88 (2017)[36] M X Wu et al Phys Rev A 91 043839 (2015)[37] M Du C Liu Y Zheng Z Zeng R li Phys Rev A

100 043840 (2019)[38] J Zak Phys Rev Lett 62 2747 (1989)[39] R Resta J Phys Condens Mat 12 R107 (2000)[40] G Kresse and J Hafner Phys Rev B 47 558 (1993)[41] N Tancogne-Dejean et al Nat Commun 8 745 (2017)[42] Y S You D A Reis and S Ghimire Nat Phys 13

345 (2017)[43] Y S You et al Opt Lett 44 530 (2019)[44] Y Zhao et al arXiv191112092

DRAFT

7

As stated in the main text after the ldquosmooth procedurerdquothe Berry connection

Dsmm(k) = i 〈us

m(k r)|nablakusm(k r)〉 = 0 (A5)

By inserting Eq (A4) into Eq (A5) we have

Dsmm(k) = i

lang

ukpm (k r)eiβ(k)|nablak

(

ukpm (k r)eiβ(k))rang

= Dkpmm(k)minusnablakβ(k) = 0 (A6)

It is easy to prove that Dkpmm(k) is a real number and an

even function with respect to k

Dkpmm(minusk) = Dkp

mm(k) (A7)

In order to ensure Dsmm(k) = 0 β(k) = odd function +

const Further if the system has inversion symmetryβ(k) = const Thus us

m(k r) also has

usm(minusk r) = plusmnus

m(kminusr) (A8)

Using Eq (A8) we can get all the properties listed inTable I

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