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PAPER

Ab initio study o

Physics Department, Lorestan University, Kh

lu.ac.ir; omidi_alireza@yahoo.com; Fax: +9

† CCDC 174799. For crystallographic datDOI: 10.1039/c5ra14945b

Cite this: RSC Adv., 2015, 5, 90559

Received 28th July 2015Accepted 16th October 2015

DOI: 10.1039/c5ra14945b

www.rsc.org/advances

This journal is © The Royal Society of C

n optical properties of glycinesodium nitrate: a novel semiorganic nonlinearoptical crystal†

M. Dadsetani* and A. R. Omidi

The electronic structure, and linear and nonlinear optical susceptibilities of crystalline glycine-sodium

nitrate (GSN) has been studied using the full potential linear augmented plane wave method within

density-functional theory. In addition, we have investigated the excitonic effects by means of the

bootstrap exchange–correlation kernel within time dependent density functional theory. The crystal in

question has a band structure with low dispersion which is a characteristic behavior of molecular

crystals. Findings show that the inorganic nitrate group plays a major role in enhancing the optical

response of this semi-organic crystal. Although, GSN shows a smaller nonlinear response, in comparison

with organic crystals, it has a wide range of transparency as well as sufficient anisotropy, which make it

a promising crystal for nonlinear applications. This study show that c(2)yyx is more important in the infrared

region of the spectra, while c(2)yzy possesses the dominant peak in ultraviolet region. In addition to the

high potential of excitonic effects, the investigated crystal shows extremely small wavelengths of

plasmon peaks.

1. Introduction

Since the invention of the laser in 1960, there have beensignicant developments in the eld of nonlinear opticalmaterials. Among the nonlinear optical (NLO)materials, organicmaterials have attracted a great deal of attention, as they havelarge optical susceptibilities, inherent ultrafast response timesand high optical thresholds for laser power compared withinorganic materials.1 During the last decades, extensive studieshave been devoted to achieving a better understanding of factorsthat may lead to acentric crystals with large NLO responses.Recent studies have shown that semi-organic materials, espe-cially from the amino acid family, are promising candidates forNLO applications as they tend to combine the features of organicwith that of inorganic materials.2 They have several attractiveproperties such as high NLO coefficient, high damage threshold,wide transparency range, high mechanical strength and thermalstability. The amino acids have interesting physical and chem-ical properties, since they have chiral symmetry as well as theelectron acceptor carboxyl acid (–COOH) group and the electrondonor amino (–NH2) group. Combinations of amino acidswith other organic and inorganic materials, give rise to prom-inent NLO materials, such as L-arginine phosphate, L-argininehydrochloride, L-histidine tetrauoroborate, L-alanine

orramabad, Iran. E-mail: dadsetani.m@

8-66-33120192; Tel: +98-66-33120192

a in CIF or other electronic format see

hemistry 2015

tetrauoroborate, L-histidine bromide, etc. Glycine is thesimplest amino acid which has a basic role in the structure ofglycine-sodium nitrate (GSN), as a new NLO crystal. The crys-talline GSN (C2H5N2NaO5) has proven to be a suitable candidatefor NLO applications, and the SHG efficiency of this crystal wasfound to be greater than that of potassium dihydrogen phos-phate (KDP) crystal.3 The crystal band structure, the density ofstates and the optical absorption of GSN were studied boththeoretically and experimentally.4 Vijayakumar et al.5 have re-ported the vibrational spectral analysis of this material using FT-Raman and FT-IR spectroscopy. It is found that GSN is thermallystable up to 198 �C.6 Suresh et al.7 have reported the densitymeasurements, the dielectric studies and the dependence ofextinction coefficient (k) and refractive index (n) on the absorp-tion of GSN. In addition, the thermal analyses, the dielectricbehavior, the mechanical strength and the work hardeningco-efficient of GSN were studied by J. Mary Linet and S. JeromeDas, experimentally.8 Also, there are studies about the effect ofPH on crystal growth9 and SHG10 of GSN.

This study tries to give new information about the linear andnonlinear optical properties of GSN within the framework ofDFT theory. To the best of our knowledge, there are no ab initiofull band structure reports on the nonlinear optical response ofGSN. In addition, the mBJ-based calculations for the linearoptical response and the electronic structure have been re-ported for the rst time. It's worth mentioning that, in the mostof theoretical simulations for nonlinear optic, the bulksusceptibilities are calculated from a straightforward tensorsum over the microscopic (molecular) properties,11,12 while

RSC Adv., 2015, 5, 90559–90569 | 90559

RSC Advances Paper

a more comprehensive approach is the full treatment of peri-odic crystals by means of band structure theory, although a fewstudies have used ab initio full band-structure model to calcu-late the linear and nonlinear optical responses of crystals. Also,this study tries to investigate the excitonic effects of GSN crystalusing Time Dependent Density Functional Theory (TDDFT).13,14

Such a study (on GSN crystal) has not been reported earlier. Wehave used bootstrap approximation which is known to giveoptical spectra in excellent agreement with experiments,15 andis computationally less expensive than solving the Bethe Sal-peter equation. The differences between TDDFT and RPAresults show clear signatures of excitonic effects.

Since the nonlinear susceptibilities are very sensitive to theenergy gap, we have used mBJ16 approximation which can effi-ciently improve the band gap and give better band splitting.Studies have shown that the mBJ potential is generally asaccurate in predicting the energy gaps of many semiconductorsas the much more expensive GW method.17

Next section presents the basic theoretical aspects andcomputational details of our study. The calculated electronicstructure and the optical response are presented in Section 3. Lastsection is devoted to the summary and principle conclusions.

2. Computational detailsA. Calculation parameters

Our calculations were performed using the highly accurate all-electron full potential linearized augmented plane wave (FP-LAPW) method based on DFT as implemented in the ELKcode.18 This is an implementation of the DFT with differentpossible approximation for the exchange correlation (xc)potentials. We have used the generalized gradient approxima-tion (GGA) of Perdew–Burke–Ernzerho19 and the modiedBecke–Johnson exchange potential (mBJ),16,17 for the exchange–correlation potentials. The mBJ exchange potential is availablethrough an interface to the Libxc library.20 Basis functions areexpanded in combinations of spherical harmonic functionsinside non-overlapping spheres at the atomic sites (muffin-tinspheres) and in plane waves in the interstitial regions. Themuffin-tin radii for oxygen (O), nitrogen (N), sodium (Na),hydrogen (H) and carbon (C) was taken to be 1.16, 1.16, 2.2, 0.6and 1.18 a.u., respectively. The interstitial plane wave vector cutoff Kmax is chosen such that RmtKmax equals 5 for all the calcu-lations. The convergence parameter RmtKmax, where Kmax is theplane-wave cut-off and Rmt is the smallest of all atomic sphereradii, controls the size of the basis. The valence wave functionsinside the spheres are expanded up to lmax¼ 10 while the chargedensity was Fourier expanded up to Gmax ¼ 14. We found theoptical spectra to be sufficiently well converged with 300k-points in the IBZ, and self-consistency was achieved by aniterative process with energy convergence up to 0.0001 eV.

We have used the FHI-aims code21 for relaxing the atomicpositions and structural parameters. FHI-aims uses numericatom-centered orbitals as the quantum-mechanical basis set:

fiðrÞ ¼uiðrÞr

Ylmðq;fÞ (1)

90560 | RSC Adv., 2015, 5, 90559–90569

Where, Ylm(q,f) are spherical harmonics and the radial ui(r) partsare numerically tabulated. Hence, the bases are very exible andany kind of desired shape can be achieved. This enables accurateall-electron full-potential calculations at a computational costwhich is competitive with plane wave methods. It should benoted that, FHI-aims is an efficient computer program packageto calculate physical and chemical properties of condensedmatter and other cases, such as molecules, clusters, solids,liquids. We have relaxed the structures (including van der Waalscorrections) in a way that every component of the forces actingon the atoms was less than 10�4 eV A�1.

B. Mathematical framework of optical response

We have calculated both linear and non-linear optical spec-troscopy within independent particle picture. In addition,sophisticated calculation of linear optical spectra by solvinglinear response time-dependent density functional theory(TDDFT) equation using bootstrap exchange–correlation kernelis presented. The linear optical properties of matter can bedescribed by means of the transverse dielectric function 3(u).There are two contributions to 3(u), namely, intra-band andinter-band transitions. We ignored the intra-band contribution,since it is important only for metals. The components ofdielectric function were calculated using the traditionalexpression in the random phase approximation (RPA):22–24

3ijðuÞ ¼ dij þ 4pi

u

"� i

U

Xk

Wk

Xcv

1

3vc

pvc

iðkÞpcv jðkÞðuþ 3vc þ ihÞ

þ�pvc

iðkÞpcv jðkÞ�*

ðu� 3vc þ ihÞ

!#(2)

Where the term in the bracket is optical conductivity (atomicunits are used in the above formula). The sum covers allpossible transitions from the occupied to unoccupied states.The term pcv

j denotes the momentummatrix element transitionfrom the energy level c of the conduction band to the level v ofthe valence band at certain k-point in the BZ. ħu is the energy ofthe incident photon and 3vc h 3v � 3c is the differences betweenvalence and conduction eigenvalue. Wk is the weight of k-pointsover the Brillouin zone and U is the unit cell volume.

Time-dependent density-functional theory (TDDFT),25 whichextends density-functional theory into the time domain, isanother method which is able, in principle, to determineneutral excitations of a system. The TDDFT method can handlelarge systems and is, basically, exact. Hence, this study tries tocover both the RPA- and TDDFT-based linear optical responses.The key quantity of TDDFT is the exchange–correlation kernelfxc, which, together with the Kohn–Sham (KS) single-particledensity-response function cs, determines the interacting-particles density-response function c, as follows:

cGG0�1ðq;uÞ ¼ ðcsÞGG0

�1ðq;uÞ � 4p

jG þ qj2 dGG0 � f xcGG0 ðq;uÞ (3)

While cs is constructed using the single-particle states obtainedwith a given approximation to the static exchange–correlationpotential Vxc(r),26 fxc is a true many-body quantity, basically,

This journal is © The Royal Society of Chemistry 2015

Paper RSC Advances

containing all the dynamic exchange–correlation effects ina real interacting system. Although formally exact, the predic-tions of TDDFT are only as good as the approximation of theexchange–correlation kernel. A great amount of effort has beeninvested into the development of approximations to fxc ofsemiconductors and insulators.27–35 In this study, we have usedthe bootstrap approximation for fxc15 which has a wide appli-cability and is computationally less expensive than solving theBethe Salpeter equation. The exact relationship between thedielectric function and the kernel fxc for a periodic solid can bewritten as:

3GG0 �1ðq;uÞ ¼ dGG0 þ vGG0 ðqÞcGG0 ðq;uÞ

¼ dGG0 þ csGG0 ðq;uÞvGG0 ðqÞ�1� �vGG0 ðqÞ

þ f xcGG0 ðq;uÞ�csGG0 ðq;uÞ

��1(4)

Here v(q) is the bare Coulomb potential. All these quantities arematrices on the basis of reciprocal lattice vectors G. The boot-strap exchange–correlation is approximated by

f bootxc ðq;uÞ ¼ � 3�1ðq;u ¼ 0ÞvðqÞ3000 ðq;u ¼ 0Þ � 1

¼ 3�1ðq;u ¼ 0Þc000 ðq;u ¼ 0Þ (5)

Where 30(q,u)¼ 1� v(q)c(q,u) denotes the dielectric function inRPA. The superscript 00 indicates that only the G ¼ G0 ¼0 component is used in the denominator. This coupled set ofequations is solved by rst setting fbootxc ¼ 0 and then solving (4)to obtain 3�1. This is then “bootstrapped” in (5) to nd a newfbootxc , and the procedure is repeated until self-consistencybetween the two equations at u ¼ 0 is achieved.

In addition to the linear response, this study tries to coverthe nonlinear response of investigated crystals. The mathe-matical relations for calculating the second order susceptibili-ties as well as their inter- and intra-band contributions havebeen developed by Sipe and Ghahramani,36 and Aversa andSipe.37 Within the independent particle picture, the complexsecond-order nonlinear optical susceptibility tensors can bewritten as:38–44

cinterijk ð�2u;u;uÞ¼ 1

U

Xnmlk

Wk

8<:

2rnmin~rml

j~rlnko

ðuln � umlÞðumn � 2uÞ �1

ðumn � uÞ

�24~rlmk

n~rmn

i~rnljo

ðunl � umnÞ �~rnl

jn~rlm

k~rmnio

ðulm � umnÞ

359=;

(6)

cintraijk ð�2u;u;uÞ ¼ 1

U

Xk

Wk

(Xnml

umn�2

ðumn � uÞhuln~rnl

jn~rlm

k~rmnio

�uml~rlmkn~rmn

i~rnljoi

�8iXnm

~rnminDmn

j~rnmko

umn2ðumn � 2uÞ

þ 2Xnml

~rnmin~rml

j~rlnkoðuml � ulnÞ

umn2ðumn � 2uÞ

9=; (7)

This journal is © The Royal Society of Chemistry 2015

cmodijk ð�2u;u;uÞ ¼ 1

2U

Xk

Wk

(Xnml

1

umn2ðumn � uÞ

�hunl~rlm

in~rmn

j~rnlko� ulm~rnl

in~rlm

j~rmnkoi

� iXnm

~rnmin~rnm

jDmn

ko

umn2ðumn � uÞ

)(8)

From these formulae (atomic units are used in these rela-tions), we can notice that there are three major contributions toc(2)ijk(�2u,u,u): the inter-band transitions cinterijk (�2u,u,u), theintra-band transitions cintraijk (�2u,u,u), and the modulation ofinter-band terms by intra-band terms cmod

ijk (�2u,u,u), where ns m s l and i, j and k correspond to Cartesian indices.

Here, n denotes the valence states,m denotes the conductionstates, and l denotes all states (l s m, n). Two kinds of transi-tions take place: one of them is vcc0 which involves one valenceband (v) and two conduction bands (c and c0) and the secondtransition is vv0c which involves two valence bands (v and v0) andone conduction band (c). The symbolsDnm

i(~k) and {rnmi(~k)rml

j(~k)}are dened as follows:

Dnmi(~k) ¼ vnn

i(~k) � vmmi(~k) (9)�

rnmi

~k

rml

j

~k

�¼ 1

2

rnm

i

~k

rml

j

~k

þ rnm

j

~k

rml

i

~k

(10)

Where~vnmi is the i component of the electron velocity (given as

~vnmi(k) ¼ iunm(k)rnm

i(~k)). The position matrix elements betweenband states n andm, rnm

i(~k), are calculated from themomentum

matrix element ~pnmi using the relation:45 rnmið~kÞ ¼ pnmið~kÞ

imunmð~kÞ;

where the energy difference between the states n and m aregiven by ħunm ¼ ħ(un � um) and Wk is the weight of k-points.

As mentioned before, GSN is a non-centrosymmetric crystalwhich has eight second order nonlinear susceptibilities: yxy,yyx, yzy, zxx, zxz, zyy, zzx and zzz. According to the results of thisstudy, c(2)yzy possesses the dominant peaks of nonlinear response.

3. Results and discussionA. Crystal structure, electronic and linear optical properties

Fig. 1 shows the unit cell as well as different views of GSN crystalstructure. The asymmetric unit of GSN consists of one glycinemolecule, one sodium cation and one nitrate anion. As can beseen in Fig. 1A, the glycine molecules are located between layersof NaNO3 chains and linked to sodium nitrate by strong intra-molecular hydrogen bonds of N–H–O type. The zwitterion formsof glycine provide an antiparallel arrangement and non-centrosymmetric crystalline growth. Although the polarity ofglycine molecules changes alternately, they cannot nullify eachother completely. In addition, the crystal seems as innitechains in the ac plane. The NLO activity of crystal can beattributed to the structural organization of innite chains ofhighly polarity entities connected in a head to tail arrangement

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Fig. 1 The crystal structure along y-axis (A), the crystal structure along x-axis (B), the unit-cell of crystal (C).

RSC Advances Paper

in GSN. Experimental studies have shown that GSN hasa monoclinic non-centrosymmetric crystalline structure withspace group Cc. The optimized lattice parameters for crystallineGSN (a ¼ 14.329 A, b ¼ 5.2662 A, c ¼ 9.1129 A, a ¼ g ¼ 90� andb ¼ 119.10�) are found to be in good agreement with the X-raycrystallographic data.46,47 The crystallographic a and b axes areparallel to the optical x and y axes, respectively, but the c-axisdeviates from the z-axis by about 29�. The polarization axis ofglycine molecule is very close to the y-axis; in addition, themolecular plane of nitrate group is mainly located in the b–cplane, hence 3yy exhibits the maximum values of dielectricfunction. On the other hand, the planes containing glycine andnitrate groups are mainly perpendicular to the x-axis giving thesmallest values of 3xx, since there is very little electronicresponse out of the molecular plane.

In Table 1, we have presented the respective geometricalparameters such as bond lengths and bond angles of optimizedstructure. In spite of small deviations, the optimized values ofbond angles and bond lengths are very close to the experimentalvalues. For example, the calculated C–H and N–H bonds areslightly longer than their experimental counterparts. Due to thestrong N–H–O hydrogen bonding of NH3 group, with the oxygenof the carbonyl group, the optimization procedure (includingvan der Waals correction) gives smaller values for the bondangle C9–N12–H14. On the other hand, the attraction betweennitrate group (NO3) and sodium (Na) atom makes the bondangle Na5–O6–C7 smaller than experimental value. It should benoted that, the nitrate group has planar form in both experi-mental and optimized regimes.

In what follows, the electronic band structure, and total andpartial densities of electron states are presented. Fig. 2 showsthe calculated electronic band structure and the correspondingtotal DOS, of investigated crystals. As can be seen, the mBJapproximation push the valence bands to lower energies andthe conduction bands to higher energies, yielding improvedresults for the band gap. As a result of low intermolecularinteractions, both GGA and mBJ approximation give bandstructures with low dispersions. Moreover, when we go fromGGA to mBJ, an empty region with no energy bands appears inthe range of 4–6 eV. In the rest of this study, we will report ourresults using mBJ only, since it gives better band gap and better

90562 | RSC Adv., 2015, 5, 90559–90569

band splitting. The results of our calculations using mBJ (GGA)approximation gives a fundamental direct band gap value of5.32 eV (3.38 eV) at G, while J. Hemandez-Paredes andcolleagues4 have reported the band gap value of 2.98 (2.99) eVwith LDA (GGA) approximation, which is smaller than ourresult. Suresh et al. have evaluated the band gap by theextrapolation of the linear part of optical absorption (a) spec-trum.7 They found the band gap to be 4.0 eV, which is smallerthan the calculated value with mBJ approximation. In otherexperimental study,8 one can see a sharp decrease (at around l

¼ 240 nm) in absorption spectra of GSN crystal which showsgood agreement with calculated band gap within mBJapproximation.

In sum, the GSN crystal has a wide range of transparencywhich is desirable for NLO applications, since the absorptionsnear the fundamental or second harmonic signals will lead tothe loss of the conversion of SHG.

In order to show the role that different groups play, we havereported the partial density of states for functional groups,separately. As can be seen in Fig. 3, the p-states of NO3 playmajor roles in the top of valence bands and the bottom ofconduction bands. According to this gure, the inorganicnitrate group has important role in the optical activity of GSNcrystal, since the main peaks of optical response come fromelectron transitions from the highest valence bands to thelowest group of conduction bands. As expected for molecularcrystals, glycine and NO3 groups create quasi-separated states.It is worth mentioning that, compared to other groups, thesodium atom has negligible contribution to the top valencebands and the bottom of conduction bands. In addition, thisgure clearly shows that the glycine and nitrate groups exist asdipolar structures, since they have a sharp electron-donatingpeak just below the band gap and a sharp electron-withdrawing peak above the band gap. In what follows, theimaginary and real parts of dielectric function and the differentcomponents of refractive indices are represented. Assumingthat they give a small contribution to the dielectric functions,we ignored the indirect inter-band transitions involving scat-tering of phonons.48–51 The calculated imaginary (32

ii) and realparts (31

ii) of dielectric functions are presented in Fig. 4. As canbe seen, the imaginary components have two main structures:

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Table 1 Optimized values of GSN crystal using FHI-aims code

Bond angle Optimized values (�) Experimental dataa (�) Bond length Optimized values (A) Experimental dataa (A)

O2–N1–O4 119.592 119.066 N1–O2 1.2700 1.2410O2–N1–O3 119.742 120.464 N1–O3 1.2632 1.2350O3–N1–O4 120.664 120.467 N1–O4 1.2647 1.2465Na5–O6–C7 126.746 130.951 Na5–O6 2.4457 2.4099O6–C7–O8 126.106 126.025 O6–C7 1.2692 1.2419O8–C7–C9 115.954 116.186 C7–O8 1.2670 1.2472O6–C7–C9 117.928 117.759 C7–C9 1.5241 1.5199C7–C9–H11 108.863 109.209 C9–H10 1.0951 0.9696C7–C9–N12 112.730 111.924 C9–H11 1.098 0.9701C7–C9–H10 110.005 109.250 C9–N12 1.4845 1.4802H10–C9–H11 107.150 107.967 N12–H13 1.0599 0.8899H10–C9–N12 109.340 109.224 N12–H14 1.0509 0.8896H11–C9–N12 108.586 109.183 N12–H15 1.0313 0.8898C9–N12–H13 109.702 109.493 Na5–O4 2.6324 2.6146C9–N12–H14 111.476 109.482 Na5–O2 2.5710 2.6473C9–N12–H15 109.364 109.547H13–N12–H14 110.633 109.461H13–N12–H15 107.167 109.414H14–N12–H15 108.386 109.430

a Ref. 46 and 47.

Paper RSC Advances

(i) below 11 eV and (ii) above 11 eV. The dominant peaks arelocated in part (i), and there is large anisotropy here. Allcomponents have a sharp peak (a) around 6.2 eV which (incomparison with Fig. 3) comes from the electron transitionsbetween a-valence bands to the e-conduction bands (a / e). Asmentioned before, the high intensity of 32

yy(u), in part (i), can beattributed to the fact that the polarization vector of glycinemolecules are very close to the y-axis; in addition to themolecular planes of nitrate groups which are located in the b–cplane. On the other hand, the molecular planes of glycine and

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nitrate groups are mainly perpendicular to the x-axis whichmakes 32

xx(u) smallest. The peak b (in 32xx(u) and32

zz(u)) islocated around 7.7 eV and comes from the electron transitionsbetween b-valence bands to the f-conduction bands (b / f).Both 32

xx(u) and 32yy(u) have another sharp peak (g) at part (i),

which is located around 10.15 eV and come from d / f tran-sitions. There is a wide hump around 17.5 eV as well as severalsmall peaks at part (ii). The weakness of structures at this part,compared to part (i), can be attributed to the fact that 32(u)scales as 1/u2. Furthermore, the investigated crystal shows large

RSC Adv., 2015, 5, 90559–90569 | 90563

Fig. 2 Our results for the band structure and total DOS of GSN crystal using m-BJ and GGA approximations.

RSC Advances Paper

values of 3(0)/3(N), whose deviation from one is a sign of thepolarity of materials.

The variations of refractive indices of investigatedcompound are represented in Fig. 5. According to this gures,

Fig. 3 Calculated density of states for the different functional groupsof GSN crystal.

Fig. 4 Calculated imaginary (32ii) and real parts (32

i) of dielectricfunction for GSN crystal using mBJ approximation.

90564 | RSC Adv., 2015, 5, 90559–90569

we can see considerable anisotropy, particularly in non-absorbing region. It should be noted that this anisotropy isnecessary for phase-matching conditions. The GSN crystalmainly shows the uniaxial, rather than biaxial, behavior. Forexample, our calculations give the static values of 1.35, 1.49 and1.48 for nxx(0), nyy(0) and nzz(0), respectively. We also have rep-resented the variations of refractive indices with wavelength. Ascan be seen, nxx < nyy z nzz and there is sufficient anisotropy forwavelengths above 200 nm, which make crystal promisingcandidate for SHG in IR-VIS and UV-VIS regions. Suresh et al.7

have measured the refractive index of GSN crystal by Brewster'sangle method using He–Ne laser of wavelength 632.8 nm. Theirexperimental result (n ¼ 1.510) is very close to our theoreticalresults (nyy ¼ 1.5078 and nzz ¼ 1.5019).

B. Nonlinear optical response

In this part of our study, we present our results for the nonlinearoptical properties of GSN. It should be mentioned that, thephysical interpretation of nonlinear results is very difficult,since in addition to valence–valence and conduction–conduc-tion transitions, both inter- and intra-band transitions canparticipate in nonlinear procedure. In addition, the nonlinearsusceptibilities are very sensitive to slight changes of banddispersion, due to the 2u and u resonances which appear in theimaginary and real parts of c(2).

The absolute values of nonlinear susceptibilities are rep-resented in Fig. 6. As can be seen, the maximum values ofnonlinear response are located around 3 eV and 6.5 eV. Therst peak is more important, since it is located below the bandgap. Furthermore, c(2)

yyx and c(2)zyy have larger values of nonlin-

earity at energy values under 1 eV and in the range of 1–2.5 eV,respectively. While, c(2)

yzy possesses the dominant peak around3 eV. The imaginary and real parts of c(2)

yzy as well as u/2u intra-and inter-band contributions to this element is presented inFig. 7. According to this gure, the main peak which is locatedbelow the band gap, mainly comes from 2u intra-bandcontribution, while both u intra-band and u inter-bandfactors play important role in that sharp peak which islocated above the band gap (around 6.5 eV). As can be seen, the2u-resonances start to contribute at energies below the bandgap (E # Eg/2), while the u-contributions come at frequenciesabove the band gap. This gure also shows oppositesymmetrical patterns for the u- and 2u-resonances at higher

This journal is © The Royal Society of Chemistry 2015

Fig. 5 Theoretical results for the refractive indices of GSN crystal.

Paper RSC Advances

energies which make the high-energy nonlinear response,decrease. Although GSN shows lower values of nonlinearresponse, in comparison to organic crystals,52 it has sufficientanisotropy and wide range of transparency which make itpromising crystal for SHG in IR-VIS and VIS-UV regions. Forexample, the experimental measurements of GSN10 haveshown that this crystal has a high grade of SHG efficiency atl ¼ 1064 nm.

In the following, we have presented the linear opticalspectra in comparison with nonlinear one, to show the inter-esting similarities between them. Unlike the linear spectra, thefeatures in the nonlinear spectra are very difficult to identifyfrom the band structure, because of the presence of 2u and u

resonances. Generally, whenever the inter-band peaks appear,the intra-band peaks appear simultaneously. Since themagnitude of inter-band transitions are realizable from 32(u),one could expect the nonlinear structures to be realized fromthe features of 32(u). Hence, we nd it useful to compareabsolute values of nonlinear susceptibilities with the imagi-nary parts of dielectric function, as a function of both u and 2u(Fig. 8). This gure shows signicant similarities betweenlinear and nonlinear spectra. The colored arrows indicate theagreement between nonlinear and linear peak positions (asa function of both u and 2u). For example, in the below-bandgap region, both linear and nonlinear structures have twopeaks (the red and green arrows). The blue arrow shows thatboth linear and nonlinear regimes have a sharp peak around

Fig. 6 Absolute values of cijk(2), in units of (pm V�1), for GSN crystal.

Fig. 7 The imaginary and real parts of cyzy(2) as well as u/2u intra- and

inter-band contributions to it in units of (pm V�1).

This journal is © The Royal Society of Chemistry 2015

6.2 eV. Furthermore, this gure also shows that the below-band gap nonlinear structures originate from the 2u-reso-nances, while that nonlinear sharp peak which is located justabove the band gap (the blue arrow) mainly comes from theu-resonances. It can be seen that, the nonlinear structureslocated in the range of 7–10 eV mainly come from the 2u-resonances, but the sharp peak around 10 eV, mainly originate

RSC Adv., 2015, 5, 90559–90569 | 90565

Fig. 8 Linear optical response in comparison with nonlinear one forGSN crystal.

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from the u-resonances. Finally, this gure shows that when wemove from the linear regime to the nonlinear one, the low-energy peaks are enhanced and shied to lower energies, butthe high-energy peaks tend to be small. As mentioned before,the molecular crystals possess band structure with smalldispersions, particularly around the band gap, whichenhances the two photon absorption at lower energies. On theother hand, increase in band dispersion at higher energies,makes the two photon absorption diminish. Another reasonfor this reduction could be explained by the fact that c2(u)

Fig. 9 The RPA and TDDFT results for the imaginary (32ii) and real (31

ii) p

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scales as 1/u2. Due to valence–valence and conduction–conduction transitions, it is almost impossible to predict theexact behavior of nonlinear response, although the generalbehavior can be recognized from the combination of 32(u) and32(2u).

Finally, we can estimate the values of rst order hyper-polarizabilities (tensor bijk) of GSN molecule by using, theexpression (bijk ¼ cijk/Nf

3) given in ref. 53 and 54. Here, N is thenumber of molecules per cm3 and f is the local eld factorwhich its value is varying between 1.3 to 2.0. The calculatedvalue for byyx of GSN (at l z 1064 nm) is found to be 2.23 �10�30 (esu).

C. TDDFT calculations and excitonic effects

We have used the bootstrap kernel15 (within the framework oftime dependent DFT) to investigated the excitonic effects inGSN crystal. Recent studies show that TDDFT kernels (such asbootstrap kernel) which have a long range 1/q2 contribution inthe long-wavelength limit are able to capture the excitonformation in solids29–34 with low computational cost.

The results of RPA and TDDFT calculations, for the imagi-nary and real parts of dielectric function of GSN crystal, arerepresented in Fig. 9. One can see a strong signature of excitoniceffects in bulk GSN, by comparing red-shied TDDFT resultswith those of RPA. It is clear that, despite the extremely goodoverall agreement between RPA and TDDFT results, the boot-strap procedure tends to enhance the low-energy structures. Inaddition, there is a slight red-shi in going from RPA to TDDFTcalculations. Although the excitonic effects have minor roles athigher energies, 3(xx) shows considerable deviations at energyvalues between 10–18 eV.

At the end part of this section, we have shown the electronenergy loss spectra of GSN crystal in Fig. 10. The energy lossfunction, L(u) ¼ Im[�1/3] is an important factor which illus-trates the energy loss of a fast electron traversing in a material.

arts of dielectric function of GSN crystal.

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Fig. 10 The RPA and TDDFT results for the energy loss spectra of GSN crystal.

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Generally, the energy loss spectra show two main structures.The low-energy peaks (under 10 eV) can be attributed to theinter-band transitions between valence and conduction bands.Hence, there is a clear correspondence between the loss peakpositions with those of Im-3ii, but the loss peaks have slightblue-shis.55 The second main structure of the loss spectra isa wide peak around 27 eV, which corresponds to the collectiveplasmon excitations. The plasmon peaks correspond to theabrupt reduction of 32(u) and to the zero crossing of 31(u). Ascan be seen in this gure, the RPA structures are very close tothose of TDDFT, but the excitonic effects make the low-energypeaks enhance. In both RPA and TDDFT regimes, Lxx hasa lower intensity of plasmon peak, while Lzz and Lyy have closevalues of intensity.

Like to linear regime, the excitonic effects also can changethe optical spectra in nonlinear regime. For example, as shownfor the linear optical spectra, the excitonic effects not only canenhance the non-linear optical spectra, especially at lowerenergies, they can shi the onset of nonlinear response to lowerfrequencies.56–58

To sum up, we have used a full ab initio treatment forhandling nonlinear response of periodic crystals within theframework of band structure theory. This study gives reliabledispersions for the linear and nonlinear optical spectra inboth absorbing and non-absorbing regions. In addition,since the experimental measurement of nonlinear suscepti-bilities is expensive and somewhat cumbersome, suchstudies provide an extremely useful guide for research onnonlinear organic crystals. This study provides new valuableinformation about the optical properties of investigated solidwhich are a major topic, both in basic research and forindustrial applications. According to our study, the investi-gated crystal has wide range of transparency as well assufficient anisotropy, in the non-absorbing region, which isimportant for phase matching. So, GSN crystal can beconsidered as proper candidate for SHG in the IR-VIS andVIS-UV regions. It should be noted that, our calculations havebeen conducted based on van der Waals interactions, whichplay major roles in the band structure and optical response ofmolecular crystals.

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4. Conclusions

We have studied the electronic structure as well as the linearand nonlinear optical properties of GSN crystal using the FP-LAPW method within the framework of DFT and TDDFT. Ourcalculations, based on mBJ and GGA approximations, give riseto the direct band gap for this crystal. Regarding the crystalstructure, the planes containing nitrate groups and glycinemolecules mainly tend to be perpendicular to the x-axis, hence3xx shows the smallest values of dielectric function. This studyshows that the inorganic nitrate group plays major role inenhancing the optical response of GSN, since the top of valencebands and the bottom of conduction bands comes from the p-states of nitrate group. In spite of having band structure withlow dispersion, due to the high value of band gap, GSN showssmaller nonlinear response, compared to organic crystals.Despite this, it has high thermal stability, sufficient anisotropyand wide range of transparency, which make it promisingcrystal for SHG. Among the nonlinear optical susceptibilities,c(2)yzy possesses the dominant peak in non-absorbing region ofspectra, while c(2)yyx is more important in infrared region. Find-ings show that the 2u-resonance intra-band factor plays animportant role in the below-band-gap structures of nonlinearspectra. This study shows that there is a signicant correspon-dence between the peak positions of linear spectra (as a func-tion of both u and 2u) with those of nonlinear one.

According to the results of our study, the excitonic effects canproduce red-shied enhanced structures, in both linear andnonlinear regimes. Finally, both DFT and TDDFT calculationsfor the energy loss spectra yield plasmon peaks around 27 eV.

Appendix

GSN

Glycine sodium nitrate DFT Density functional theory TDDFT Time dependent density functional theory NLO Nonlinear optic GGA Generalized gradient approximation mBJ Modied Becke–Johnson exchange potential RPA Random phase approximation

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KS

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Kohn–Sham

32

ii

Imaginary parts of dielectric function 31

ii

Real parts of dielectric function nii The i-component of refractive index Lii The i-component of energy loss function pnm

j

Momentum matrix elements Wk Weight of k-points over the Brillouin zone U The unit cell volume fxc Exchange–correlation kernel Vxc(r) Exchange–correlation potential fbootxc (q,u) Bootstrap exchange–correlation kernel c(2)ijk(�2u,u,u) Second-order nonlinear optical susceptibilities cinterijk (�2u,u,u) Inter-band contribution to nonlinear

susceptibilities

cintraijk (�2u,u,u) Intra-band contribution to nonlinear

susceptibilities

cmodijk (�2u,u,u) Modulation of inter-band terms by intra-band

terms in nonlinear procedure

bijk First order hyperpolarizabilities rnm

i(~k)

Position matrix elements between band statesn and m

~vnmi

The i component of the electron velocity

Dnmi(~k)

The difference between electron-velocities of

states n and m

ħunm The energy difference between the states n and

m

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