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Calculations for the Linear and

Nonlinear Optical Coefficients of

ZCTC Crystals

Chao Feng1, Guiwu Lu*, 1, Xinqiang Wang2, Peng Zhang1, Guanggang Zhou1, Hongwang Yang1, Yuqiu Jiao1

1 Department of Mathematics and Physics, China University of Petroleum, Beijing 102249, China

2 Institute of Crystal Materials and State Key Laboratory of Crystal Materials, Shandong University, Jinan, 250100, China Received ZZZ, revised ZZZ, accepted ZZZ

Published online ZZZ

Keywords Density functional theory (DFT); second harmonic generation (SHG) coefficients; second-order nonlinear optics; elec-tronic structure

* Corresponding author: e-mail: lugw@https://www.doczj.com/doc/9f10782926.html,, Phone: +86 010 ********, Fax: +86 010 ********,

Web: https://www.doczj.com/doc/9f10782926.html,, https://www.doczj.com/doc/9f10782926.html,, https://www.doczj.com/doc/9f10782926.html,, https://www.doczj.com/doc/9f10782926.html,

The structural, electronic, linear and nonlinear optical properties of ZnCd(SCN)4(ZCTC) single crystal were studied by using quantum-mechanical calculations based on the density functional theory (DFT) and pseudopoten-tial method. The optimized lattice constant can be com-pared with the experimental values when the effects of temperature are considered. Combining with partial den-sity of states of ZCTC crystal, it was found that C2p, N2p and S3p orbitals occur obvious hybrid, which is the main source of the second-order nonlinear optical effect of ZCTC crystal. Then DFT has been used to calculate the hyperpolarizability of ZCTC single crystal, from which the second harmonic generation (SHG) coefficients was calculated using ionic group theory. The calculated SHG coefficients are d14=1.08~1.93 pm/V and d15=4.01~4.79 pm/V. Both of them are in agreement with the tendencies shown by the experimental results (d14=3.2 pm/V and

d15=7.6 pm/V). It was also found that the greatest contri-bution for SHG of ZCTC crystal is from SCN anionic group.

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1 Introduction Currently, second-order nonlinear op-tics continues to be a topical area of research because of its tremendous potentials in optical electronic applications such as electro-optical (EO) switches, ultrafast devices for information processing, storage, computing and et al. [1-5]. Zinc cadmium thiocyanate (ZCTC), ZnCd(SCN)4, has been discovered as a promising UV nonlinear optical (NLO) crystal candidate material for frequency doubling of diode lasers [6-8]. Wang et al. have reported the growth, structural, thermal properties, linear and NLO properties of ZCTC [7-10]. Lu et al. have analysed the Raman spectra of ZCTC crystal by space group theory [11]. However, to our best knowledge, the theoretical study for electronic struc-ture and second harmonic generation (SHG) coefficient of ZCTC crystal has not been reported. In order to explain some experimental phenomena and further understand the NLO properties of this material, it is obvious that the theo-retical calculations may be useful. In recent years, many calculation methods have been used for the calculation of NLO properties [12]. In the present paper, the quantum-mechanical calculations based on the density functional theory (DFT) and pseudopotential method are used to study the structural, electronic, linear optical properties of ZCTC. Then DFT has been used to calculate the hyperpo-larizability of ZCTC single crystal, from which the SHG coefficients have been calculated using anionic group theory. The calculated results are in agreement with expe-rimental data.

2 Computational model and method ZnCd(SCN)4 crystal belongs to tetragonal crystallographic system, I-4 space group, with cell parameters [7]: a = b =1.1135(2)nm, c=0.43760(10)nm , V =0.5426(2)nm3, Z=2, D x=2.510 g/cm3. Its unit cell consists of two kinds of slightly flat-tened tetrahedron: ZnN4 and CdS4 (Fig 1). The most strik-ing features are the －N＝C＝S－ bridges which connect the center atoms of the flattened tetrahedron, Zn and Cd,

2 C. Feng et al.: Calculations for the Linear and Nonlinear Optical Coefficients of ZCTC Crystals

1 2 3 4 5 6 7 8 9

57 and form infinite three dimensional －Zn－N＝C＝S－Cd

－ networks (Fig 1.a).

The geometry optimization for the crystal was per-

formed with the plane wave pseudopotential method [13]

which was successfully used in many fields[14-18]. We

used the generalized gradient approximation (GGA) [19]

for the exchange correlation function by Perdew and Zun-

ger and norm-conserving pseudopotential methods [20] to

describe the valence electron interaction with the atomic

core. In order to ensure convergence in our calculation, the

plane wave cutoff energy of 550eV was selected. The

summation over the Brillouin zone was carried out with a

k-point sampling using a Monkhorstpack grid [21] with pa-

rameters of 2×2×5. The interactions between atoms con-

verge to 0.003eV/nm, and the energy threshold value is

1.0×10-5 eV/atom.

Figure 1Unit cell of ZnCd(SCN)4crystal (a, ball and stick

model; b, polyhedral model.)

The coupled-perturbed (CP) method has been adopted

to calculate the NLO coefficients of LiNbO3 and urea crys-

tals, which gives some physical insight into microscopic

mechanis m of NLO effects of crystal materials [22-24].

However, CP method is not suitable for metal organic

compound crystal material (such as ZCTC et al.) [25].

Anionic grouping theory proposed by Chen et al for theo-

retical studying on NLO crystal properties has predicted

the SHG coefficients of many materials such as BNN, Na-

NO2, Na2SbF5 successfully [26-28]. The anionic grouping

theory for NLO coefficient rests on the following two as-

sumptions: 1) the overall SHG coefficient of the crystal is

the geometrical superposition of the microscopic second-

order susceptibility tensors of the anionic groups and it has

nothing to do with the essentially spherical cations; 2) the

microscopic second-order coefficient of the basic anionic

group can be derived from localized molecular orbital cal-

culations of the groups. In an unit cell of ZCTC crystal,

there are two tetrahedrons (ZnN4 and CdS4) and four SCN

groups (Fig 2.), and the macro SHG coefficients can be

calculated from micro hyperpolarizability of ZnN4, CdS4

and SCN ionic groups using anionic group theory.

Figure 2Structure and coordinate of CdS4, SCN and ZnN4 ionic

groups.

In nonlinear optics, the polarization of a molecule in-

duced by an external radiation optical field E

is often ap-

proximated as a creation of an induced dipole moment.

Under the intense polarization condition, one can use a

Taylor series expansion in the electric field components to

demonstrate the dipolar interaction with the external radia-

tion electric field E

. The molecule induced dipole mo-

ment can be written as [29]:

2!3!

i i ij j ijk j k ijkl j k l

E E E E E E

μμαβγ

=+++ (1)

where the subscripts i, j, k, l represent the different

components x, y or z of Cartesian coordinate, respectively,

μ0 is the dipole moment in the absence of the applied elec-

tric field, and α, β, γ are the linear polarizability, first and

second hyperpolarizability tensors.

Calculations were performed at several levels of basis

set to determine the static first hyperpolarizability of the

above three ionic groups. Fig 2 shows the structure and

coordinate which is used for calculations of SCN, ZnN4

and CdS4ionic groups. Taking into account the electron

correlations which play important role in calculation of the

first hyperpolarizabilities [30-32], the first hyperpolariza-

bilities have been calculated by using the Density Func-

tional Theory (DFT) [33-37] with B3LYP exchange and

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Table 1 Lattice constants, bond populations and bond lengths of

ZnCd(SCN)4 crystal

c Present Experiment [8]

a 1.1091 1.1135(2)

c0.5455 0.43760(10)

Bond Bond population Bond length (nm)

S－C 0.86 0.16356 0.1651(5)

C－N 1.75 0.11885 0.1153(6)

Zn－N 0.31 0.19628 0.1967(5)

S－Cd 0.35 0.25945 0.25640(12) correlation functions [38]. B3LYP which including a mix-

ture of Hartree-Fock (HF) exchange with DFT exchange-

correlation was devised by Becke in 1993, it is the most

frequently used functions [39]. Taking into account the in-

fluence of diffuse functions and polarization functions [30-

31], 6-31+G, 6-31+G(d) and 6-311++G(d) basis sets have

been choosen for comparison. Exceptionally, the relativis-

tic effect must be considered for transition metals, so

LanL2DZ basis set has been used for Cd. The convergence

standard of self-consistent field (SCF) is 10-6.

Oudar and Zyss deduced the relationship between ma-

cro and micro second-order nonlinear polarizability of

crystal [40]. According to the characteristics of molecular

crystals, the relationship between second-order nonlinear

polarizability (()2

IJK

χ) and micro second-order nonlinear po-

larizability (which is first hyperpolarizability βijk ) is:

()()()()

123

IJK I J K IJK

Nf f f b

χωωω

= (2)

Where b ijk is the unit cell second-order NLO response

built from the tensor sum of the first hyperpolarizabilities

(βijk) of the constitutive units, which can be wrote as

()()()

cos cos cos

s s s

IJK Ii Jj K k ijk

ijk s

θθθβ

= ?

∑∑ (3)

In the equation (2) and (3), N is the number of the mo-

lecules per unit volume of crystal, N=N g/V, N g is the num-

ber of molecular in a crystal cell, V is the volume of a crys-

tal cell. ()s

θ are the angles between the coordinate system

of molecular and the coordinate system of crystal, and f I (ω)

is the local field factor. ZCTC crystal belongs to tetragonal

crystallographic system, so the local field factor is Lorentz

local field factor [41].

()()

()

22/3

I I

f n

ωω

=+ (4)

Where n I (ω) is the refractive index of the crystal in I

direction.

In practice, experimental researchers have been accus-

tomed to use SHG coefficient (d IJK) instead of the second-

order nonlinear polarizability (()2

IJK

χ). The relationship be-

tween d IJK and ()2

IJK

χ is:

()2

IJK IJK

dχ

= (5)

The SHG coefficients consist of 18 tensor element.

According to the symmetry of I4space group, its non-

zero element listed in equation (6) [42].

1415

2425

313236

000d d0

d000d d0

d d000d

(6)

According to Kleinman symmetry [42] (which was de-

vised by Kleinman in 1960), d24=d31=d32=d15, d36=d25=d14,

so there are only d14 and d15 is non-zero value in the matrix

of SHG coefficients of ZCTC crystal.

The CASTEP program [43] has been employed to per-

form the optimized geometries, the electronic structure de-

terminations, and, consequently, the optical calculations.

The nonlinear optical properties are calculated by a pro-

gram which is compiled by ourselves.

3 Re sults and discussion

3.1 Geometry structure The lattice constants and

bond lengths of optimized structure of ZnCd(SCN)4 crystal

are listed in Table 1, in which experimental values are also

included [8]. Obviously, the calculated lattice constants of

ZnCd(SCN)4 deviate from the experimental values slightly.

Wang et al. [11] reported the lattice constant a decreasing

and c increasing with temperature. Our calculated values

are obtained at T= 0, but experimental results are meas-

ured at room temperature, so we think that the deviation of

lattice constants comes from the different temperatures be-

tween calculations and experiments. In addition, the

ZnCd(SCN)4 cell is constructed using the experimental lat-

tice constants and atomic coordinates, and the total energy

(-11617.33eV) is higher than the optimized structures’ (-

11618.41 eV), that is to say, the optimized structure is

more stable. It should be noted that if the calculated bond

lengths are comparable with experimental values, then the

deviation of the lattice constants would result in changes of

bond angles of flattened tetrahedral ZnN4 and CdS4, which

can assure the formation of infinite three dimensional －Zn

－N＝C＝S－Cd－ networks.

3.2 Electronic structure Fig 3. shows the band

structure and density of states (DOS) spectra of

ZnCd(SCN)4crystals. From Fig 3(a) we can see that the

energy band structure of ZCTC crystal mainly consists of

four parts: the K region near -16.76eV; the J region near -

13.21eV; the valence band I between -7.49eV and -

0.076eV, and the conduction band H located above 3.74

eV. It’s obvious that both of the valence band maximums

and the conduction band minimums are all in G point (the

center of Brillouin), that is to say, ZCTC crystal has ob-

vious direct bandgap structural feature. The bandgap value

is 3.55eV. This value is smaller than the experimental

bandgap (4.28eV [7]). It is well known that the bandgap

calculated by the DFT is usually smaller than the experi-

mental data. A semiempirical correction for this is neces-

4 C. Feng et al.: Calculations for the Linear and Nonlinear Optical Coefficients of ZCTC Crystals 1

Table 2The atomic population and the Mulliken charges for

ZnCd(SCN)4 crystal

Population Charge (e)

C 2s1.062p2.95-0.02

N 2s1.592p3.90-0.49

S 3s1.813p4.26-0.07

Zn 3d9.944s0.204p0.51 1.34

Cd 4d9.985s0.545p0.500.98

sary to obtain meaningful results. A scissors operator [44-

45] (△=0.73eV) is hence used to shift upward all the con-

duction bands in order to agree with measured values of

the bandgap.

Figure 3(a) Band structure of ZnCd(SCN)4. (b) total DOS of

ZnCd(SCN)4.

Figure 4Partial density of states (PDOS) of ZnCd(SCN)4 crystal.

Fig 3(b) is DOS spectrum of ZCTC crystal, and it is al-

so divided into H, I, J, K four groups which corresponds to

band structure. Comparing with the partial density of states

(PDOS) spectrum of ZnCd(SCN)4crystal (Fig 4), we can

conclude the peak K located between -17.81 and -15.87 eV

mainly comes from C2s and N2s orbitals; the peak J lo-

cated between -14.06 and -12.10 eV, mainly comes from

S3s while C2s, C2p orbitals are also included. The peak I

has a broader range between -8.33 and 0.91 eV, in which

Cd4d, Zn3d, C2p, N2p and S3p orbitals are included. The

electronic states of C2p and N2p located between 2.63 and

6.22 eV, which make up the peak H.

In orbitals have obvious hybrid, the SHG effect comes

mainly from electron transition between valence band

maximums and conduction band minimums [28]. Combin-

ing with PDOS of ZnCd(SCN)4 crystal in Fig 4, we can see

that valence band maximums and conduction band mini-

mums mainly consist of C2p, N2p and S3p hybrid orbitals,

and both of Zn3d and Cd4d are in the valence band mini-

mums. So C2p, N2p and S3p orbital hybrid is the main

source of the SHG effect of ZCTC crystal, and Cd and Zn

positive ions provide the complex structure of ZCTC crys-

tal.

Since the ionicity of bonds is a great factor affecting

the physical properties of materials, we calculated the

charge transfer, atomic populations and bond populations

of ZCTC. Bond populations calculated by means of the

Mulliken analysis showed in Table 1, and the atomic popu-

lation and Mulliken charges are listed in Table 2. Cd atoms

bonded with S atoms possess 0.98 positive charge, so the

electron have obvious transfer from Cd to S. However, the

S atoms only possess 0.07 negative charge, so the electrons

from S atoms must transfer to adjacent C atoms which pos-

sess 0.02 positive charge, and the electrons further transfer

to N atoms whose Mulliken charge is 0.49 negative charge.

Zn atoms possess 1.34 positive charges in tetrahedron of

ZnN4, and the electrons from Zn atoms also transfer to N

atoms. Since the N2s orbitals also lose electrons (see Table

2), the electrons from other atoms are distributed in N2p

orbitals. Besides the transfer among different atoms, elec-

trons al so transfer among the different orbitals of the same

kind of atoms. The atomic population in Table 2 displays

these informations. Especially for C atoms, the electrons

from 2s transfer to 2p, and the configuration is2s12p3

which is the same with diamond.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

Figure 5 Electron density projection of ZnCd(SCN)4 crystal.( a, projection plane which is 0.2184 nm from (100) surface; b, pro-jection plane which is 0.473 nm from (010) surface.)

The bond population indicates the overlap degree of the electron cloud of two bonding atoms and can be used to access covalent or ionic nature of a chemical bond. For the bond population, the lowest and highest values imply that the chemical bond exhibits strong ionicity and covalency, respectively. The bond population of the C –N bond are the highest (Table 1), which also can be concluded from the PDOS, the C2s2p hybrid with N2s2p semicores strongly, which indicates stronger covalency. In the flattened tetra-hedral CdS 4, only a few S3p electrons hybrid with the tran-sition metal 4d semicores around the Fermi level, which results in the Cd –S bond population is very small and e x-hibits strong ionicity. The Zn –N bond population is also very small and shows strong ionicity. Electron density pro-jection can show overlap degree of electron cloud more clearly. Fig 5(a) is the electron density projection of a plane apart 0.2184 nm from (100) surface, and Fig 5(b) is the electron density projection of a plane apart 0.473 nm from (010) surface. From Fig 5(a) we can see that the over-lap degree of electron cloud of Zn-N and Cd-S are very low, which imply strong ionicity. From Fig 5(b) we can see that the overlap degree of electron cloud of C-N and C-

S are very high, which imply strong covalency.

Figure 6 Absorption spectrum of ZnCd(SCN)4 crystal.

3.3 Linear optical properties Fig 6 displays the ab-sorption spectrum of ZCTC crystals. It can be seen that the ab-sorption peaks of ZCTC crystal have higher absorption coeffi-cients in the visible range and near-UV range. We can use the model of the interaction between photons and electrons to inter-pret the different optical properties. The transitions between oc-cupied and unoccupied states are caused by the electric field of the photons. The spectrum caused by these excitations can be ex-plained as a joint DOS between the valence and conduction bands and the different electronic distributions would result in different optical spectra. For the absorption spectrum of ZCTC crystal (Fig 6), the peaks A –E correspond to the electrons transition from the valence band (region I in Fig 3.) to the conduction band (region H in Fig 3.). The peak A located at 221nm (the energy is 5.619eV) is the highest, and it is correspond to the electron transition from the valence band maximums (peak 1 in Fig 3(b).) to the conduc-tion band minimums (peak 2 in Fig 3(b).). The energy difference between them is about 5.26eV. The peak B located at 151 nm (the energy is 8.19eV) is correspond to the electron transition from the valence band (peak 3 in Fig 3(b).) to the conduction band min i-mums (peak 2 in Fig 3(b).). The energy difference between them is about 8.4eV. The peak C located at 123 nm (the energy is 10.05eV) is correspond to the electron transition from the valence band (peak 4 in Fig 3(b).) to the conduction band minimums (peak 2 in Fig 3(b).). The energy difference between them is about 9.98eV. The peak D located at 107 nm (the energy is 11.6eV) is correspond to the electron transition from the valence band (peak 5 in Fig 3(b).) to the conduction band minimums (peak 2 in Fig 3(b).). The energy difference between them is about 12.3eV. The peak E located at 73.3 nm (the energy is 16.9eV) is correspond to the electron transition from the valence band (peak 6 in Fig 3(b).) to the conduction band minimums (peak 2 in Fig 3(b).). The energy difference between them is about 18eV.

3.4 Nonlinear optical properties The results of SHG coefficients which were calculated by different basis sets and the experimental data are listed in Table 3. As shown in Table 3, the calculated d 14=1.08~1.93 pm/V and d 15=

4.01~4.79 pm/V, both of them are in agreement with the experimental values 3.2 pm/V and 7.6 pm/V [46].

The molecular NLO properties originate from the light-induced polarization of a molecule, namely, the light-induced electron cloud change. Thus, the different basis sets may influence on the B3lyp calculated results. Diffuse functions were taken into account in all of the three basis sets because of its importance in the calculation of first hyperpolarizability [30-31]. The SHG coefficient values calculated with the 6-31+G(d) basis set are better agree-ment with the experimental data than the 6-31+G basis set. It is demonstrated that the polarization functions are also important in the B3lyp calculation of the first hyperpolari-zability. However, B3lyp calculation with the triple-zeta basis set 6-311++G(d) are not more better than using the 6-31+G(d) basis set. Hence, it is reasonable and reliable to calculate the first hyperpolarizability using the 6-31+G(d) basis set for the sake of lower computational cost.

6 C. Feng et al.: Calculations for the Linear and Nonlinear Optical Coefficients of ZCTC Crystals

1 2 3 4 5 6 7 8 9

Table 3 the first hyperpolarizability and SHG coefficients value of ZnCd(SCN)4 crystal

Table 4 listed the final SHG coefficient values of dif-

ferent ionic group of ZCTC crystal. The greatest contribu-

tion for the SHG coefficients of ZCTC crystal is SCN ionic

group (d14=-0.98~-1.96 and d15=-2.89~-3.81). Relative to

SCN ionic group, the contribution of CdS4 (d14=-0.0028~-

0.17 pm/V and d15=-0.031~-0.22 pm/V) and ZnN4

(d14=0.0024~0.011 pm/V and d15=-0.71~-1.62 pm/V) ionic

group is very small. Therefore, it can be concluded that,

the SHG coefficients of ZCTC crystal mainly from the part

of anionic group which is agreement with Chen’s anionic

grouping theory. This is also consistent with conclusion by

PDOS analysis (see Fig 4).

4 Conclusions The structural, electronic, linear and

NLO properties of ZCTC single crystal were studied by us-

ing the quantum-mechanical calculations based on the DFT

and pseudopotential method. The optimized lattice con-

stant can be compared with the experimental values when

the effects of temperature are considered.

According to the optimized structure, the band struc-

ture, DOS and PDOS of ZCTC crystal were calculated.

ZCTC crystal has obvious direct bandgap structural feature

and the bandgap value is 3.55eV. Combin ing with partial

density of states of ZCTC crystal, it was found that C2p,

N2p and S3p orbitals have obvious hybrid, which is the

main source of the second-order NLO effect of ZCTC

crystal.

The bond populations and electron density projection

of ZCTC crystal were calculated too. It was concluded that

the S-C and C–N bonds show stronger covalency, and Zn–

N and Cd–S bonds exhibit strong ionicity.

At last the hyperpolarizability of ZCTC single crystal

was calculated, from which the SHG coefficients was cal-

culated using ionic group theory. The calculate SHG coef-

ficients are d14=1.08~1.93 pm/V and d15=4.01~4.79 pm/V,

both of them is in agreement with the experimental results

(d14=3.2 pm/V and d15=7.6 pm/V), and the greatest contri-

bution for SHG of ZCTC crystal is from SCN anionic

group.

Acknowledgements This work was supported by key pro-

ject from education ministry of China (108023).

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