Article
pubs.acs.org/JPCC
A DFT Study of Linear and Nonlinear Optical Properties
of 2‑Methyl-4-nitroaniline and 2‑Amino-4-nitroaniline Crystals
M. Dadsetani* and A. R. Omidi
Physics Department, Lorestan University, Khorramabad, Iran
ABSTRACT: The electronic structure and linear and nonlinear optical susceptibilities of 2-methyl-4-nitroaniline (MNA) and
2-amino-4-nitroaniline (ANA) crystals have 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. Our calculations show indirect band gaps for MNA and ANA crystals.
Both crystals have band structures with low dispersion which is a characteristic behavior of organic crystals. The crystalline ANA
shows larger band dispersion, compared to MNA, due to the higher intermolecular interactions. Findings show that the
substituent groups play major roles in enhancing the optical response of push−pull organic crystals. On the other hand, the
intermolecular interactions make the band dispersion increase and the optical response, especially the nonlinear one, decrease.
The MNA crystal shows larger values of nonlinear response, since all the constitutive molecules are mainly polarized along the
same axis and there is less overlap between them. Moreover, the considerable below-band-gap anisotropy as well as the high
values of nonlinear susceptibilities make these crystals suitable candidates for nonlinear purposes. In addition to the high
potential of excitonic effects, both crystals have extremely small wavelengths of plasmon peaks. Finally, this study gives reliable
results for the optical spectra in both linear and nonlinear regimes.
1. INTRODUCTION
Since the invention of the laser in 1960, there have been significant developments in the field of nonlinear optical materials.
Nonlinear optical (NLO) materials, defined as materials in
which light waves can interact with each other, are key materials
for the fast processing of information and optical storage
applications. The important development in nonlinear optical
materials occurred in 1970, when Davydov et al. reported a
strong second harmonic generation (SHG) in organic materials
which have electron donor−acceptor groups.1 During the last
decades, extensive studies have been devoted to achieving a
better understanding of factors that may lead to acentric
crystals with large NLO responses. The results of these studies
show that a good electron donor−acceptor group can change
the NLO response, considerably. Therefore, a wide range of
organic materials which have electron donor−acceptor groups
have been synthesized particularly for second-order nonlinear
optics. One of the most common donor−acceptor molecules is
para-nitroaniline (p-NA), since much interest in NLO properties of substituted benzene derivatives stems from the basic
studies on the chemistry and physics of the p-NA molecule.
© 2015 American Chemical Society
The p-NA molecule consists of a benzene ring in which an
electron donor amino (NH2) group is substituted in a paraposition to an electron acceptor nitro (NO2) group. These
opposite ends of the conjugated system lead to maximum
acentricity and large intramolecular charge transfer interactions.
In spite of having an appreciable hyperpolarizability, p-NA crystallizes in the centrosymmetric space group, which restricts the
observation of any macroscopic second-order optical effects.
This drawback has led to the search for other molecular materials. Several closely related derivatives of p-NA, with noncentrosymmetric crystallization, have been found. Replacing
a meta-hydrogen with a methyl group (amino group) gives
2-methyl-4-nitroaniline (2-amino-4-nitroaniline) crystal (Figure 1).
Such chemical substitutions give the maximum eccentricity to
the molecules and provide crystals with noncentrosymmetric
structure which have large values of nonlinear response.
Contrary to 2-amino-4-nitroaniline (for brevity, we name it
ANA), the 2-methyl-4-nitroaniline (MNA) crystal has attracted
Received: June 6, 2015
Published: June 15, 2015
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DOI: 10.1021/acs.jpcc.5b05408
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The Journal of Physical Chemistry C
Currently, the FP-LAPW method provides the most reliable
results within density functional theory.
For the first time, this study tries to investigate the excitonic
effects of MNA and ANA crystals using time dependent density
functional theory (TDDFT).33,34 We have used the Bootstrap
approximation which is known to give optical spectra in excellent agreement with experiments,56 and is computationally less
expensive than solving the Bethe−Salpeter equation. The
differences between TDDFT and RPA results show clear
signatures of excitonic effects.35−37
Since the nonlinear susceptibilities are very sensitive to the
energy gap, we have performed mBJ38 calculations which can
efficiently improve the band gap and give better band splitting.
Studies have shown that the mBJ potential is generally as
accurate in predicting the energy gaps of many semiconductors
as the much more expensive GW method.39
In the present study, we have selected two organic molecules,
with similar chemical formula but different crystal structures, to
observe the effect of molecular arrangement on optical spectra.
Findings show that the linear response is basically moleculedependent but the nonlinear response is crystal-dependent,
since it is very sensitive to the band structure and intermolecular interactions. Contrary to other works, this study
clearly shows that the intermolecular interactions can change
the band dispersion and nonlinear optical response, considerably. According to the results of our study, the higher linear
responses do not necessarily lead to higher nonlinear responses.
In the case of organic crystals, the present work is one of the
few studies which provides an excellent opportunity to compare
theoretical and experimental optical spectra in both linear and
nonlinear regimes. The results of TDDFT calculations show
that the excitonic effects can be very dramatic, particularly near
the band edge. However, this study provides new important
information for the electronic structure, band structure, energy
loss, and linear and nonlinear optical spectra of investigated
compounds. We hope that our work will lead to comprehensive
experimental studies of these compounds which have promising
linear and nonlinear optical susceptibilities.
The next section presents the basic theoretical aspects and
computational details of our study. The calculated electronic
structure and the optical response are presented in section 3.
The last section is devoted to the summary and principle
conclusions.
Figure 1. Molecular structures (a, b) and different views of 2-methyl4-nitroaniline and 2-amino-4-nitroaniline crystals (c, d).
much attention in different studies. The linear and nonlinear
optical properties of MNA have already been the topic of
several experimental2−6 and theoretical7−12 studies. The FT-IR
spectra of crystalline MNA deposited on a poly substrate were
studied by Vallee and co-workers.13,14 Nogueira et al.15 reported
the longitudinal optical (LO) and transversal optical (TO)
wavenumbers of polar vibrations found in the unpolarized FT-IR
reflectivity spectra of MNA crystals. Intra- and intermolecular
electronic transitions were calculated for MNA molecular clusters
by Guillaume et al.16 Okwieka et al. have measured the polarized
FT-IR, Raman, neutron scattering, and UV−visible spectra of
MNA crystal plates, powder, and solutions.17 There are a few
studies on optical spectra of ANA. In one of them, Kolev et al.
studied the properties of 2-amino-4-nitroaniline by means of UV
and linear polarized IR spectroscopy.18
It is worth mentioning that theoretical simulations for nonlinear optics are often used to understand relationships at the
molecular level, and fewer investigations have tackled the prediction and interpretation of the NLO response at the macroscopic level. Early studies only considered the bulk susceptibilities which were calculated from a straightforward tensor sum
over the microscopic (molecular) properties.19,20 Improved
approaches incorporate the local field effects, either by using
Lorentz-like expressions or by evaluating these within electrostatic interaction schemes.21−25 A more comprehensive approach
is the full treatment of periodic crystals by means of band
structure theory, although a few studies have used the ab initio
full band-structure model to calculate the linear and nonlinear
optical responses of crystals. During the past few years, a number
of studies have been published on the linear and nonlinear
optical properties of molecular crystals within band structure
theory.26−30 To the best of our knowledge, this is the first ab
initio full band-structure study that reports the electronic structure and optical properties of MNA and ANA crystals. Thus, we
expect it to be useful for future investigations.
It should be noted that even predicting the linear optical
properties for molecular crystals is quite challenging, due to the
interplay between different strengths of bonding types in molecular crystals (i.e., the strong intramolecular covalent bonds vs
much weaker intermolecular van der Waals and possibly hydrogen bonds). Hence, this study is important and interesting
enough to deserve attention.
In the present study, we adopted density functional theory
(DFT)31,32 to determine the electronic structure and optical
susceptibilities of MNA and ANA crystals using the state-ofthe-art full potential linear augmented plane wave method.
2. COMPUTATIONAL DETAILS
A. 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 ELK code.40 This is an implementation of the DFT with different possible approximations
for the exchange correlation (XC) potentials. We have used the
generalized gradient approximation (GGA) of Perdew−Burke−
Ernzerhoft41 and the modified Becke−Johnson exchange
potential (mBJ)38,39 for the exchange-correlation potentials.
The mBJ exchange potential is available through an interface to
the Libxc library.42 Basis functions are expanded in combinations of spherical harmonic functions inside nonoverlapping
spheres at the atomic sites (muffin-tin spheres) and in plane
waves in the interstitial regions. The muffin-tin radii for oxygen
(O), nitrogen (N), hydrogen (H), and carbon (C) were taken
to be 1.16, 1.16, 0.6, and 1.20 au, respectively. The interstitial
plane wave vector cutoff Kmax is chosen such that RmtKmax equals
3.5 for all of the calculations. The convergence parameter RmtKmax,
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where Kmax is the plane wave cutoff and Rmt is the smallest of all
atomic sphere radii, controls the size of the basis. The valence
wave functions inside the spheres are expanded up to lmax = 10,
while the charge density was Fourier expanded up to Gmax = 14.
Convergence tests have shown that 64 k-points in the irreducible
wedge of the Brillouin zone are sufficient, as more k-points give
no appreciable change in the energy or properties. However, we
have used a dense mesh of 512 k-points for the optical
calculations. Self-consistency was achieved by an iterative process
with energy convergence up to 0.0001 eV in less than 60
iterations. The band structures have been computed on a discrete
k-mesh along high-symmetry directions. We have used 1000
k-points for the band structure and DOS plot.
We have used the FHI-aims code43 for relaxing the atomic
positions and structural parameters. FHI-aims uses numeric
atom-centered orbitals as the quantum-mechanical basis set
φi(r ) =
ui(r )
Ylm(θ , φ)
r
handle large systems and is, basically, exact. Hence, this study
tries to cover both the RPA- and TDDFT-based linear optical
responses. The key quantity of TDDFT is the exchange-correlation
kernel f xc, which, together with the Kohn−Sham (KS) singleparticle density-response function χs, determines the interactingparticles density-response function χ, as follows:51
−1
−1
χGG
(q , ω) = (χ s )GG
′ (q , ω) −
′
(3)
While χ is constructed using the single-particle states
obtained with a given approximation to the static exchangecorrelation potential Vxc(r),32 f xc is a true many-body quantity,
basically, containing all the dynamic exchange-correlation
effects in a real interacting system. Although formally exact,
the predictions of TDDFT are only as good as the approximation of the exchange-correlation kernel. A great amount of
effort has been invested into the development of approximations to f xc of semiconductors and insulators.52−61 In this
study, we have used the Bootstrap approximation for f xc56
which has a wide applicability and is computationally less
expensive than solving the Bethe−Salpeter equation. The exact
relationship between the dielectric function and the kernel f xc
for a periodic solid can be written as
s
(1)
where Ylm(θ, φ) are spherical harmonics and the radial ui(r)
parts are numerically tabulated. Hence, the bases are very
flexible and any kind of desired shape can be achieved. This
enables accurate all-electron full-potential calculations at a computational cost which is competitive with plane wave methods.
FHI-aims is an efficient computer program package44−50 to calculate physical and chemical properties of condensed matter
and other cases, such as molecules, clusters, solids, and liquids. It
should be noted that the primary production method is density
functional theory and the package is also a flexible framework for
advanced approaches to calculate ground-state and excited-state
properties. However, we have relaxed the structures (including
van der Waals corrections) in a way that every component of the
forces acting on the atoms was less than 10−4 eV/Å.
B. Mathematical Framework of Optical Response. The
linear optical properties of matter can be described by means of
the transverse dielectric function ε(ω). There are two contributions to ε(ω), namely, intraband and interband transitions. We
ignored the intraband contribution, since it is important only
for metals. The components of the dielectric function were
calculated using the traditional expression in the random phase
approximation (RPA)35−37
εij(ω) = δ ij +
+
⎡
4πi ⎢ i
−
ω ⎢⎣ Ω
∑ Wk ∑
k
cv
−1
εGG
′(q , ω) = δGG ′ + vGG ′(q)χGG ′ (q , ω)
s
= δGG ′ + χGG
(q , ω)vGG ′(q){(1
′
xc
s
− [vGG ′(q) + fGG
(q , ω)]χGG
(q , ω)}−1
′
′
(4)
Here v(q) is the bare Coulomb potential. All of these quantities
are matrices on the basis of reciprocal lattice vectors G. The
Bootstrap exchange-correlation is approximated by
f xcboot (q , ω) = −
ε−1(q , ω = 0)v(q)
ε000(q , ω = 0) − 1
=
ε−1(q , ω = 0)
χ000 (q , ω = 0)
(5)
where ε0(q,ω) = 1 − v(q)χ(q,ω) denotes the dielectric function
in the RPA. The superscript 00 indicates that only the G = G′ =
0 component is used in the denominator. This coupled set of
equations is solved by first setting f boot
= 0 and then solving
xc
eq 4 to obtain ε−1. This is then “bootstrapped” in eq 5 to find a
new f boot
xc , and the procedure is repeated until self-consistency
between the two equations at ω = 0 is achieved.
In addition to the linear response, this study tries to cover the
nonlinear response of investigated crystals. The mathematical
relations for calculating the second order susceptibilities as well
as their inter- and intraband contributions have been developed
by Sipe and Ghahramani62 and Aversa and Sipe.63 Generally,
the complex second-order nonlinear optical susceptibility tensors
can be written as64−70
i
j
1 ⎛⎜ pvc (k)pcv (k)
εvc ⎜⎝ (ω + εvc + iη)
(pvci (k)pcvj (k))* ⎞⎤
⎟⎥
(ω − εvc + iη) ⎟⎠⎥⎦
4π
xc
δ − fGG
(q , ω)
2 GG ′
′
|G + q|
(2)
where the term in the brackets is optical conductivity (atomic
units are used in the above formula). The sum covers all
possible transitions from the occupied to unoccupied states.
The term pjcv denotes the momentum matrix element transition
from the energy level c of the conduction band to the level v of
the valence band at certain k-points in the Brillouin Zone (BZ).
ℏω is the energy of the incident photon, and εvc ≡ εv − εc is the
difference between the valence and conduction eigenvalues.
Wk is the weight of k-points over the Brillouin zone, and Ω is
the unit cell volume.
Time-dependent density functional theory (TDDFT),51
which extends density functional theory into the time domain,
is another method which is able, in principle, to determine
neutral excitations of a system. The TDDFT method can
χijkinter ( −2ω; ω , ω)
=
1
Ω
⎧
⎪
j k
i
2rnm
{ rml
⃗ rln⃗ }
1
−
⎪
(ωmn − ω)
⎩ (ωln − ωml)(ωmn − 2ω)
∑ Wk ⎨
nmlk
⎫
⎡ r ⃗k { r ⃗i r ⃗ j }
r ⃗ j { r ⃗ k r ⃗ i } ⎤⎪
⎬
× ⎢ lm mn nl − nl lm mn ⎥⎪
⎢⎣ (ωnl − ωmn)
(ωlm − ωmn) ⎥⎦⎭
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about 4°. On the other hand, 2-amino-4-nitroaniline crystallizes
in the chiral noncentrosymmetric orthorhombic, space group
P212121, again with four molecules in the unit cell. Here a, b,
and c are selected to be along the optical x, y, and z axes,
respectively.18 The X-ray crystallographic data were optimized
by minimizing the forces acting on the atoms. The optimized
lattice parameters for crystalline MNA (a = 7.40 Å, b = 11.70 Å,
c = 8.17 Å, α = γ = 90°, and β = 94.05°) and ANA (a = 3.67 Å,
b = 10.24 Å, c = 17.12 Å, and α = β = γ = 90°) are found to be
in good agreement with the X-ray crystallographic data. As can
be seen in Figure 1, the dipole molecules tend to be along the
z-axis. Hence, in both crystals, εzz exhibits the maximum values
of the dielectric function. On the other hand, the molecular
planes are principally perpendicular to the x-axis which makes
εxx the smallest, since there is very little electronic response out
of the molecular plane. As we go along the y-axis, the polarization of ANA molecules changes alternately which gives rise to
higher intermolecular attractions. In addition, the small intermolecular distance, especially in ANA, provides strong interaction between π-electrons. Altogether, these conditions show
that the crystalline ANA tends to have higher intermolecular
interactions and greater band dispersion.
In Tables 1 and 2, we have presented the respective geometrical parameters such as bond length and bond angle of
optimized structure. These tables show that the calculated
C−H bonds are slightly longer than their experimental counterparts. Due to the electron withdrawing effects of the nitrogroup, those C−C bonds which are along the long molecular
axis (i.e., C2−C3 and C5−C6) are shorter than other C−C
bonds. In addition, the C−C bonds which are near the aminogroup (C1−C2 and C1−C6) are longer than those which are
near the nitro-group (C3−C4 and C4−C5). Such differences
show that the benzene ring (as a molecular bridge) plays a
major role in electron delocalization. Due to the near dipolar
structure, those bond lengths which are located in the metaposition (C2−N7 in ANA and C2−C7 in MNA) are longer
than those located in the para-position (C1−N11). The optimization procedure (including van der Waals corrections)
makes the ANA molecule be nonplanar, although the paraamino group tends to be planar, due to the high intermolecular
interactions. Except for substitute positions, all the optimized
bond angles agree well with the experimental values. For
example, the bond angles of the meta-amino group (C2−N7−
H17, C2−N7−H17, and H18−N7−H17 in ANA) and the
bond angle of the methyl-group (C2−C7−H17 in MNA) have
more deviation compared to other angles. In addition, the
electron donating and electron withdrawing characteristics of
substituent groups distort the symmetry of the benzene ring,
yielding ring angles smaller and larger than 120° at the substitute positions.
In what follows, the electronic band structure and total and
partial densities of electron states are presented. Figure 2 shows
the calculated electronic band structure and the corresponding
total DOS of investigated crystals. As can be seen, the mBJ
approximation pushes the valence bands to lower energies and
the conduction bands to higher energies, yielding improved
results for the band gap. As a result of low intermolecular
interactions, both the GGA and mBJ approximations give band
structures with low dispersions. Moreover, when we go from
GGA to mBJ, an empty region with no energy bands appears in
the range 4−6 eV. Despite close similarity, ANA shows more
band dispersion (due to the higher intermolecular interactions).
It should be noted that mBJ gives a better band gap and better
χijkintra ( −2ω; ω , ω)
=
1
Ω
⎧
⎪
∑ Wk ⎨∑
⎪
⎩ nml
k
−2
ωmn
k i
[ωln rnl⃗ j { rlm
⃗ rmn
⃗ }
(ωmn − ω)
j
k
⃗ }
nm{Δmn rnm
2
nm ωmn (ωmn − 2ω)
j
k
i
− ωml rlm
⃗ { rmn
⃗ rnl⃗ }] − 8i ∑
+ 2∑
nml
j k
⃗ rln⃗ }(ωml
nm{ rml
ωmn 2(ωmn −
⇀
ri
⇀
ri
− ωln) ⎫
⎬
⎪
2ω) ⎭
⎪
(7)
χijkmod ( −2ω; ω , ω)
=
1
2Ω
⎧
⎪
∑ Wk ⎨∑
⎪
k
⎩ nml
1
i
j k
[ωnl rlm
⃗ { rmn
⃗ rnl⃗ }
ωmn 2(ωmn − ω)
j k
− ωlm rnl⃗ i { rlm
⃗ rmn
⃗ }] − i ∑
nm
i
j
k
rnm
⃗ { rnm
⃗ Δmn} ⎫
⎬
2
ωmn (ωmn − ω) ⎪
⎭
⎪
(8)
From these formulas (atomic units are used in these relations),
we can notice that there are three major contributions to
inter
χ(2)
ijk (−2ω, ω, ω): the interband transitions χijk (−2ω, ω, ω), the
intra
intraband transitions χijk (−2ω, ω, ω), and the modulation of
interband terms by intraband terms χmod
ijk (−2ω, ω, ω), where n ≠
m ≠ l and i, j, and k correspond to Cartesian indices.
Here, n denotes the valence states, m denotes the conduction
states, and l denotes all states (l ≠ m, n). Two kinds of transitions take place: one of them is vcc′ which involves one valence
band (v) and two conduction bands (c and c′), and the second
transition is vv′c which involves two valence bands (v and v′)
and one conduction band (c). The symbols Δinm(k)⃗ and {rinm(k)⃗
⃗ are defined as follows
rjml(k)}
i
i
(k ⃗) − vmm
(k ⃗ )
Δinm(k ⃗) = vnn
1 i ⃗ j ⃗
j
i
i
{rnm
(k ⃗)rmlj(k ⃗)} = (rnm
(k )rml(k ) + rnm
(k ⃗)rml
(k ⃗))
2
(9)
(10)
where vin⃗ m is the i component of the electron velocity (given as
⃗ The position matrix elements between
vin⃗ m(k) = iωnm(k)rinm(k)).
⃗ are calculated from the momentum
band states n and m, rinm(k),
⃗
matrix element p⃗inm using the relation71 rinm(k)⃗ = (p⃗inm(k)/
⃗
imωnm(k)), where the energy differences between the states n
and m are given by ℏωnm = ℏ(ωn − ωm) and Wk is the weight of
k-points.
As mentioned before, both ANA and MNA are noncentrosymmetric crystals. ANA has six second order nonlinear
susceptibilities, xyz, xzy, yxz, yzx, zxy, and zyx, while MNA has
14 components, xxx, xxz, xyy, xzx, xzz, yxy, yyx, yyz, yzy, zxx,
(2)
zxz, zyy, zzx, and zzz. Our calculations show that χ(2)
zyx and χxyz
(2)
are the dominant components of ANA but χzzz is the dominant
one in MNA.
3. RESULTS AND DISCUSSION
A. Crystal Structure and Electronic and Linear Optical
Properties. The molecular and crystalline structures of
2-methyl-4-nitroaniline (MNA) and 2-amino-4-nitroaniline
(ANA) are represented in Figure 1. Studies have shown that
2-methyl-4-nitroaniline crystallizes in the monoclinic system,
space group Ia, with four molecules in the unit cell.72 The
crystallographic a and b axes are parallel to the optical x and y
axes, respectively, but the c-axis deviates from the z-axis by
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Table 1. Optimized Values of MNA Crystal by FHI-aims Code
a
bond angle
optimized values (deg)
experimentala data (deg)
bond length
optimized values (Å)
experimentala data (Å)
C2−C1−C6
C1−C6−C5
C6−C5−C4
C5−C4−C3
C4−C3−C2
C3−C2−C1
H12−N11−H13
H13−N11−C1
N11−C1−C6
C1−C6−H10
H10−C6−C5
C6−C5−H9
H9−C5−C4
C5−C4−N14
C4−N14−O15
O15−N14−O16
O16−N14−C4
N14−C4−C3
C4−C3−H8
H8−C3−C2
C3−C2−C7
C7−C2−C1
C2−C1−N11
C1−N11−H12
H19−C7−H17
H19−C7−H18
H18−C7−H17
H19−C7−C2
H18−C7−C2
H17−C7−C2
118.395
122.422
118.954
119.398
122.606
118.203
115.835
118.060
120.963
117.722
119.856
122.798
118.247
118.996
120.269
120.304
119.423
121.606
118.214
119.179
122.503
119.284
120.626
123.821
108.932
107.985
106.362
108.909
112.318
112.195
119.440
121.138
118.841
121.267
120.162
119.145
119.995
119.996
119.498
119.437
119.424
120.568
120.591
119.387
118.524
121.800
119.672
119.344
119.873
119.966
121.419
119.438
121.052
120.010
109.493
109.454
109.477
109.487
109.470
119.477
C1−C2
C2−C3
C3−C4
C4−C5
C5−C6
C6−C1
H10−C6
H9−C5
H8−C3
C7−C2
C1−N11
N11−H13
N11−H12
C4−N14
N14−O15
N14−O16
C7−H19
C7−H18
C7−H17
1.4148
1.3782
1.3910
1.3660
1.3699
1.4074
1.0584
1.0749
1.0564
1.4822
1.3462
0.9910
1.0149
1.4160
1.2426
1.2216
1.0660
1.0050
1.0992
1.3913
1.3811
1.3986
1.3873
1.3776
1.4043
0.9302
0.9303
0.9299
1.5046
1.3583
0.8596
0.8599
1.4332
1.2307
1.2326
0.9596
0.9604
0.9604
Reference 72.
separately. As can be seen in Figure 3, the total DOS mainly
comes from the p-states of the benzene ring and nitro group.
The nitro and amino groups have important roles in the
bottom of conduction bands and top of valence bands, respectively. It is worth mentioning that, compared to its counterpart
(the meta-amino group of ANA), the methyl group has a
negligible contribution to the top valence bands and the
bottom of conduction bands. On the other hand, in comparison
to the para-amino of ANA, the amino group of MNA has more
contributions to the top valence bands. In addition, this figure
indirectly gives a clear illustration for the electron withdrawing
and electron donating effects. As can be seen, the acceptor
group (such as the nitro group) has a higher density of states at
conduction bands, since it can absorb the excited electrons,
while the methyl and amino groups, which have a negligible
density of states at conduction bands, tend to be donors. The
electron delocalization in the benzene ring is well pictured in
this figure, since the benzene ring itself injects the electrons
into the conduction bands and absorbs the exited electrons
simultaneously. As a result of this figure, the benzene ring and
the substituent groups play major roles in producing the main
peaks of optical spectra, since these peaks mainly come from
the low-energy transitions between the higher valence bands
and the lower conduction bands. In what follows, the imaginary
and real parts of the dielectric function and the different components of refractive indices are represented. Assuming that
they give a small contribution to the dielectric functions, we
ignored the indirect interband transitions involving scattering of
band splitting. Thus, we will report our results using mBJ only.
The mBJ-based calculations show that MNA (ANA) possesses
a fundamental indirect band gap by about 2.63 eV (1.83 eV) in
the ΓC−ZV (XC−ΓV) direction, while the smallest direct bandto-band transition is in the ΓC−ΓV (ΓC−ΓV) direction by about
2.67 eV (2.14 eV). The transmittance spectra of MNA crystal
were measured by Levine et al.2 According to the results of that
study, the onset of transmittance appears at around λ ∼ 450 nm
= 2.76 eV which is very close to the band gap calculated in the
present study. As can be seen, due to the higher overlap between
π-orbitals along the x-axis, the band dispersion along Γ−Z and
Γ−Y is much smaller than the band dispersion along Γ−X. The
ANA crystal shows more band dispersion, since there is smaller
distance and hence higher intermolecular overlapping. In order
to understand the role that van der Waals interaction plays in
crystal structures, two optimization calculations have been performed: (i) with van der Waals correction and (ii) without van
der Waals correction. Findings show that this factor is very
important for optimization, since, when we ignore the van der
Waals correction, the relaxed structure shows larger intermolecular distance, higher band gap, and lower band dispersion,. For
example, the GGA approximation gives the band gap value of
1.23 (1.72) eV for ANA with (without) van der Waals correction. In sum, van der Waals interactions can change the band
dispersion and hence the optical response (particularly, the
nonlinear one) considerably.
In order to show the role that different groups play, we have
reported the partial density of states for functional groups,
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Table 2. Optimized Values of ANA Crystal by FHI-aims Code
a
bond angle
optimized values (deg)
experimentala data (deg)
bond length
optimized values (Å)
experimentala data (Å)
C2−C1−C6
C1−C6−C5
C6−C5−C4
C5−C4−C3
C4−C3−C2
C3−C2−C1
H12−N11−H13
H13−N11−C1
N11−C1−C6
C1−C6−H10
H10−C6−C5
C6−C5−H9
H9−C5−C4
C5−C4−N14
C4−N14−O15
O15−N14−O16
O16−N14−C4
N14−C4−C3
C4−C3−H8
H8−C3−C2
C3−C2−N7
N7−C2−C1
C2−C1−N11
C1−N11−H12
H18−N7−H17
H18−N7−C2
H17−N7−C2
119.068
121.324
118.955
121.129
120.545
118.973
116.046
120.552
119.990
118.638
120.073
121.315
119.726
119.551
119.543
121.405
119.042
119.290
119.627
119.779
120.751
120.227
120.938
122.898
109.024
110.383
113.970
119.461
120.831
118.789
121.821
119.979
119.107
119.945
120.050
120.169
119.562
119.607
120.599
120.611
119.165
118.837
121.890
119.267
118.994
120.004
120.017
121.120
119.736
120.369
120.006
120.048
119.982
119.709
C1−C2
C2−C3
C3−C4
C4−C5
C5−C6
C6−C1
H10−C6
H9−C5
H8−C3
N7−C2
C1−N11
N11−H13
N11−H12
C4−N14
N14−O15
N14−O16
N7−H18
N7−H17
1.4373
1.3835
1.4059
1.4044
1.3764
1.4202
1.0914
1.0863
1.0869
1.4110
1.3508
1.0263
1.0218
1.4179
1.2654
1.2576
1.0249
1.0231
1.4277
1.3785
1.3965
1.3870
1.3829
1.4052
0.9301
0.9307
0.9301
1.4219
1.3670
0.8603
0.8601
1.4411
1.2460
1.2377
0.8600
0.8595
Reference 18.
Figure 2. Calculated band structure and total DOS of MNA and ANA crystals using m-BJ and GGA approximations.
phonons.26−30 Recent studies have shown that, although without
phonon contributions, DFT calculations can generate experimental results very well.73 The calculated imaginary and real
parts of principle components of dielectric functions are
presented in Figure 4. There is a high degree of similarity
between the linear optical spectra of MNA and ANA crystals.
Three spectral structures (A, B, and C) in the imaginary parts
of the dielectric function can be seen. The dominant peaks are
located in part A, and there is large anisotropy here. As mentioned before, the high intensity of εzz
2 (ω) can be attributed to
the fact that the molecular polarizations of MNA and ANA
molecules are mainly along the z-axis, which results in the
strong light−matter interaction. It should be noted that the
ANA crystal shows higher values of the dielectric function,
since its long molecular axis is closer to the z-axis. On the other
hand, the molecular planes are mainly perpendicular to the
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Figure 3. Calculated density of states for the functional groups of MNA and ANA molecules.
surrounding molecules, which seems as opposite rows of
dipoles in the y−z plane, enhances the molecular polarizability
and hence the linear optical response of ANA crystal. On the
other hand, the molecular packing along the x-axis increases the
band dispersion and gives smaller values of nonlinear response.
Furthermore, both crystals show large values of ε(0)/ε(∞),
whose deviation from one is a sign of the polarity of materials,
but it is a little higher in ANA.
The variations of refractive indices of investigated compounds are represented in Figure 5. According to this figure, we
can see considerable anisotropy again, particularly in the
nonabsorbing region. It should be noted that this anisotropy is
necessary for phase-matching conditions. The MNA crystal has
been shown to be phase-matchable for SHG at λ ∼ 1064 nm2.
Figure 5 shows a biaxial, rather than uniaxial, behavior in both
crystals. For example, our calculations for crystalline MNA
(ANA) give the static values of 1.55, 1.83, and 2.18 (1.52, 1.85,
and 2.55) for nxx(0), nyy(0), and nzz(0), respectively. We also
have represented the experimental spectra of refractive indices.
In the case of MNA,6 the numerical data agree well with the
experimental results. As can be seen, both theoretical and experimental results show similar patterns for the dispersion of
refractive indices. In addition to the noticeable differences
(nxx < nyy < nzz), both works agree that nzz (nxx) has a higher
(lower) rate of growth compared to other components. It
should be noted that the scissor correction can give a much
better agreement between calculated results and experimental
data. In the case of crystalline ANA, we could not find valid
experimental data for a conclusive comparison. However, according
to this figure, both MNA and ANA crystals have sufficient anisotropy in the IR-VIS region, which makes them promising crystals
for SHG.
Figure 4. Calculated imaginary and real parts of the principle
components of dielectric function for MNA and ANA crystals within
the mBJ approximation.
x-axis, which makes εxx
2 (ω) smallest. The values of peaks in
parts B and C are much smaller than the main peak of part A,
although there are more small peaks in part B. The weakness of
structures at parts B and C, compared to A, can be attributed to
the fact that ε2(ω) scales as 1/ω2. The crystalline ANA has two
peaks at part A; the major peak comes from the transitions
between the flat part (T−Y−Γ−Z) of the top valence band
to the low conduction band. Due to the small dispersion of
band structure as well as the small volume of the unit cell, the
corresponding peak is very sharp. On the other hand, the minor
peak can be attributed to the transitions from the other flat part
(S−X−U−R) of the top valence band to the lowest conduction
bands. As shown in Figure 1, the electric field produced by the
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Figure 5. Theoretical and experimental results for the refractive indices of MNA and ANA crystals.
Figure 6. Absolute values of χ(2)
ijk in units of (pm/V) for MNA and ANA crystals (left). Calculated imaginary and real parts of the dominant nonlinear
susceptibilities (right).
B. Nonlinear Optical Response. The mathematical
relations of nonlinear susceptibilities are more complicated
than their linear counterparts, and they are much more sensitive
to slight changes of band dispersion. Regarding numerical methods,
the k-space integration must be performed more carefully and
more conduction bands must be taken into account to reach a
reasonable accuracy. In addition, the nonlinear susceptibilities
are very sensitive to the band gap value, due to the 2ω and ω
resonances which appear in the imaginary and real parts of χ(2).
Physical interpretation of nonlinear results is very difficult, since,
in addition to valence−valence and conduction−conduction
transitions, both inter- and intraband transitions can participate
in the nonlinear procedure.
As mentioned before, the MNA and ANA crystals have
14 and 6 elements of nonlinear susceptibilities, respectively.
The absolute values of nonlinear susceptibilities, as well as the
imaginary and real parts of the dominant components, are
represented in Figure 6. The crystalline MNA shows much
higher values of nonlinear response compared to ANA. For
(2)
example, the dominant components of ANA (χ(2)
zyx and χzxy)
have smaller values compared to the dominant component of
MNA (χ(2)
zzz), since all the MNA molecules are mainly polarized
along the z-axis and there is less overlap between them. In
addition, the crystalline ANA shows larger band dispersion
which makes the nonlinear response smaller. In the case of
MNA, the upper valence bands and the lower conduction bands
behave as parallel lines giving enhanced two-photon absorption.
The fact that the molecular packing along the x-axis gives rise
to the smaller values of first hyperpolarizability and the nonlinear susceptibility has been approved by other studies.12,16
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In the same way, our study shows that the molecular packing
along the x-axis increases the band dispersion and hence
reduces the nonlinear response, especially in ANA crystal.
According to the results of our study, the crystals in question
have sufficient anisotropy and high values of nonlinear response
which make them promising candidates for SHG in the IR-VIS
region. In addition to our results, the experimental results2 have
shown that the MNA crystal is transparent at both of the
fundamental wavelengths in the infrared region (λ = 1064 nm)
and the second harmonic wavelengths in the visible region (λ =
532 nm).
It should be noted that different studies2,3,5,9,10 have used
different directions for MNA molecule, but all of them agree
that d33 = (1/2)χzzz (when the long molecular axis tends to be
along the z-axis) is the dominant component. As pointed out
by Levin et al.,2 d33 is phase matchable using the usual index
birefringence and has the enormous figure of merit of 2000
times larger than LiNbO3. In Figure 7, we have plotted the
Figure 8. Linear optical response in comparison with the nonlinear
one for MNA and ANA crystals (the vertical dashed line indicates the
border of the band gap).
In addition, this figure shows that the below-band-gap
nonlinear structures originate from the 2ω resonances, while
the nonlinear small peak which is located just above the band
gap (the brown arrow) mainly comes from the ω resonances. It
can be seen that the nonlinear structures at energy values above
8 eV come from the 2ω resonances but those structures located
in the range 5−6.5 eV mainly originate from the ω resonances.
In the case of ANA, this figure shows that both ω and 2ω
resonances contribute to the nonlinear optical small peaks
around 3 eV. Finally, this figure shows that, when we move
from the linear regime to the nonlinear one, the low-energy
peaks are enhanced and shifted to lower energies but the highenergy peaks tend to be small. As mentioned before, the molecular crystals possess a band structure with small dispersions,
particularly around the band gap, which enhances the twophoton absorption at lower energies. On the other hand,
increase in band dispersion at higher energies makes the twophoton absorption diminish. Another reason for this reduction
could be explained by the fact that χ2(ω) scales as 1/ω2. As a
result of valence−valence and conduction−conduction transitions, it is almost impossible to predict the exact behavior of
nonlinear response, although the general behavior can be
recognized from the combination of ε2(ω) and ε2(2ω).
Finally, we can estimate the values of first order hyperpolarizabilities (tensor βijk) of MNA and ANA molecules by
using the expression (βijk = χijk/Nf 3) given in refs 74 and 75.
Here, N is the number of molecules/cm3 and f is the local field
factor with a value varying between 1.3 and 2.0. The calculated
values for βzzz of MNA and βzxy of ANA (at λ ≈ 1064 nm) are
found to be 42 × 10−30 (esu) and 35 × 10−30 (esu), respectively. This is very interesting to notice that the calculated value
for βzzz (of MNA) is the same as the experimental result
measured by Levine et al.2
C. TDDFT Calculations and Excitonic Effects. Recent
studies show that TDDFT kernels (such as Bootstrap kernel)
which have a long-range 1/q2 contribution in the long-wavelength
limit are able to capture the exciton formation in solids,54,60 with
low computational cost.
We have used the bootstrap kernel56 (within the framework
of time dependent DFT) to investigate the excitonic effects in
MNA and ANA crystals. Both RPA and TDDFT calculations
for the imaginary and real parts of the dielectric function of
investigated crystals are represented in Figure 9. We see a
strong indication of excitonic effects in bulk MNA and ANA by
comparing our TDDFT results with those of RPA. It is clear
that, despite the extremely good overall agreement between
Figure 7. Theoretical and experimental dispersion of d33 = (1/2)χ(2)
zzz in
units of (pm/V) for MNA crystal.
theoretical and experimental5 dispersion of |χ(2)
zzz| for crystalline
MNA. As can be seen, the results herein corroborate well with
the experimental results. For example, both of them show large
(small) values of nonlinear response at lower (higher) wavelengths. Although our results give rise to a steeper curve, scissor
shift correction can reduce the rate of growth and give better
agreement. While the scissor correction gives only a blue shift
to the linear response, it can change the nonlinear spectra, considerably. As shown in other studies,28,29 the nonlinear response
is highly sensitive to the scissor correction, since valence−
valence−conduction and valence−conduction−conduction
transitions are both active in the nonlinear regime. Unlike the
linear spectra, the features in the nonlinear spectra are very
difficult to identify from the band structure, because of the
presence of 2ω and ω resonances. Generally, whenever the
interband peaks appear, the intraband peaks appear simultaneously. Since the magnitude of interband transitions is
realizable from ε2(ω), one could expect the nonlinear structures
to be realized from the features of ε2(ω). Hence, we find it
useful to compare absolute values of nonlinear susceptibilities
with the imaginary parts of the dielectric function, as a function
of both ω and 2ω (Figure 8). This figure shows significant
similarities between linear and nonlinear spectra. The colored
arrows indicate the agreement between nonlinear and linear
peak positions (as a function of both ω and 2ω). For example,
at below-band-gap spectra, both linear and nonlinear structures
have a main peak (the black arrows), although ANA has
another small peak just below the band gap (green arrow).
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Figure 9. Calculated imaginary and real parts of εxx, εyy, and εzz within the framework of RPA and TDDFT for MNA and ANA crystals.
RPA and TDDFT results, the bootstrap procedure tends to
enhance the low-energy structures. In addition, there is a slight
red-shift in going from RPA to TDDFT calculations. Although
the excitonic effects have minor roles at higher energies, they
give considerable high-energy deviations (energy values
between 10 and 18 eV) only along the x-axis.
At the end part of this section, we have shown the electron
energy loss spectra for MNA and ANA crystals in Figure 10.
The energy loss function, L(ω) = Im[−1/ε], is an important
factor which illustrates the energy loss of a fast electron
traversing in a material. Generally, the energy loss spectra show
two main structures. The low-energy peaks (under 10 eV) can
be attributed to the interband transitions between valence and
conduction bands. Hence, there is a clear correspondence
between the loss peak positions with those of Im εii, but the
loss peaks have slight blue-shifts. It is important to note that,
when an interband transition gives rise to a feature in Im εii at
the frequency ωk, it gives rise to a peak in the loss function,
occurring at a larger frequency, where ω̃ k2 = ωk2 + ( f k2ωp2/γ).76
The second main structure of the loss spectra is a wide peak
around 27 eV (28 eV) which corresponds to the collective
plasmon excitations in MNA (ANA) crystal. The plasmon
peaks correspond to the abrupt reduction of ε2(ω) and to the
zero crossing of ε1(ω). As can be seen in this figure, the RPA
Figure 10. RPA and TDDFT results for the energy loss spectra of
MNA and ANA crystals.
structures are very close to those of TDDFT, but the excitonic
effects make the low-energy peaks enhance. In both crystals, Lxx
has a lower intensity of the plasmon peak, since the x-axis is
mainly perpendicular to the molecular planes. We also note that
different components of MNA tend to have different values of
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plasmon intensity (Lxx < Lzz < Lyy), while ANA shows close
values for Lzz and Lyy.
To sum up, we have used a full ab initio treatment for
handling the nonlinear response of periodic crystals within the
framework of band structure theory. This study gives reliable
dispersions for linear and nonlinear optical spectra in both
absorbing and nonabsorbing regions. In addition, since the
experimental measurement of nonlinear susceptibilities is
expensive and somewhat cumbersome, such studies provide
an extremely useful guide for research on nonlinear organic
crystals. In spite of the structural similarity between ANA and
MNA molecules, the intermolecular interactions can modify the
crystalline band structure and optical response, considerably.
Hence, the intermolecular interactions should be considered as
important factors in the nonlinear response of organic crystals.
According to the results of this study, the higher values of linear
response do not necessarily lead to higher values of nonlinear
response, since the crystalline ANA shows higher dielectric
response, larger band dispersion, and lower nonlinear response.
This study shows that the linear response is mainly part-dependent,
but the nonlinear response is basically bulk-dependent, since it is
very sensitive to the molecular arrangement and intermolecular
interactions. This study provides new valuable information about
the optical properties of investigated solids which are a major topic,
both in basic research and for industrial applications. This work
shows that both MNA and ANA crystals have sufficient anisotropy
in the nonabsorbing region which is important for phase matching.
According to our study, the investigated crystals are transparent
at both the fundamental wavelengths in the infrared region and
second harmonic wavelengths in the visible region. Thus, both
MNA and ANA crystals can be considered as proper candidates for
SHG in the IR-VIS region. Moreover, our calculations show that
van der Waals interactions play major roles in the band structure
and optical response of organic crystals. Finally, a satisfactory
coincidence of both the theoretical prediction and the experimental
results was achieved. This confirms the capability of our computer
simulation.
which make them suitable candidates for SHG in the IR-VIS
region.
The TDDFT calculations show that the excitonic effects have
a very dramatic influence on the optical properties of investigated semiconductors, particularly near the band edge. Furthermore, both DFT and TDDFT calculations for the energy loss
spectra yield a plasmon peak around 27 (28) eV for MNA (ANA)
crystal. Due to the extremely small wavelengths of plasmon peaks,
the crystals in question can be useful in plasmon-based electronic,
computer chips and high-resolution lithography and microscopy.
This study reproduces the experimental results very well; in
addition to the molecular hyperpolarizability, our calculations give
reliable results for both the linear and nonlinear spectra.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Phone/fax: +98-66-33120192.
Notes
The authors declare no competing financial interest.
■
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