Size dependent magnetic and optical properties in diamond shaped graphene quantum dots: A DFT study

Journal of Physics and Chemistry of Solids 99 (2016) 34–42

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Size dependent magnetic and optical properties in diamond shaped graphene quantum dots: A DFT study Ritwika Das, Namrata Dhar, Arka Bandyopadhyay, Debnarayan Jana n Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India

art ic l e i nf o

a b s t r a c t

Article history: Received 10 May 2016 Received in revised form 26 July 2016 Accepted 9 August 2016 Available online 10 August 2016

The magnetic and optical properties of diamond shaped graphene quantum dots (DSGQDs) have been investigated by varying their sizes with the help of density functional theory (DFT). The study of density of states (DOS) has revealed that the Fermi energy decreases with increase in sizes (number of carbon atoms). The intermediate structure with 30 carbon atoms shows the highest magnetic moment (8 μB , μB being the Bohr magneton). The shifting of optical transitions to higher energy in smallest DSGQD (16 carbon atoms) bears the signature of stronger quantum confinement. However, for the largest structure (48 carbon atoms) multiple broad peaks appear in case of parallel polarization and in this case electron energy loss spectra (EELS) peak (in the energy range 0–5 eV) is sharp in nature (compared to high energy peak). This may be attributed to π plasmon and the broad peak (in the range 10–16 eV) corresponds to π + σ plasmon. A detail calculation of the Raman spectra has indicated some prominent mode of vibrations which can be used to characterize these structures (with hydrogen terminated dangling bonds). We think that these theoretical observations can be utilized for novel device designs involving DSGQDs. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Density functional theory Graphene Quantum dots Optical properties

1. Introduction The recent resurgence of scientific investigation of two dimensional (2D) materials with strong covalent intralayer bonding along with weak van der Waals interlayer stacking has promoted search for newer materials for device application. 2D materials are important in their own right because of their ultra-thin structures, smooth surfaces, varieties ranges of materials. The emergence of Dirac-like materials such as graphene and topological insulators has been one of the key developments in condensed matter physics in recent times. Graphene, single layer of graphite with excellent conductivity has stimulated the interest of scientists due to the possibilities of several applications in the field of nano-electronics [1,2] and graphene based polymer nanocomposites [3,4]. However, from the electronic device point of view, the major drawback of graphene is its zero band gap. So opening of band gap at Dirac point in graphene is an important task to use it as a logic and high speed switching device. It has been achieved by reducing the dimensionality of graphene, asymmetrical strain engineering [5,6] which breaks the lattice symmetry and also by chemical functionalization of boron (B) and nitrogen (N) atoms [7–9] in the pristine graphene. But in quantum dots (QDs), the confinement takes place in three spatial directions. They are sometimes referred n

Corresponding author. E-mail address: [email protected] (D. Jana).

http://dx.doi.org/10.1016/j.jpcs.2016.08.004 0022-3697/& 2016 Elsevier Ltd. All rights reserved.

as artificial atoms as in both cases the electronic properties can be tailored by changing their size as well as shape. QDs thus provide us a unique opportunity to study the confinement effects of electronic wave function on the property of material. Graphene quantum dots (GQDs) are nanometer-sized (below 20 nm) fragments of graphene having finite Bohr radius [10] and have been successfully fabricated [11–13]. Any GQD possesses graphene lattices inside the dots regardless of the size of the dot. Combining several unique properties associated with graphene with a normal QD, GQDs exhibit extraordinary optoelectronic properties compared to other QDs due to quantum confinement and edge effect [14]. By reducing the sizes of GQDs band gap can be tuned upto 3 eV [15] in contrast to other external approaches and can be used for the applications in the area of LEDs, optical qubits, energy conversion bioanalytic sensors, photoresistor [16–18] etc. Moreover, GQDs from 3D graphene grown by chemical vapor deposition have shown enough for detection of Fe þ 3 ions [19]. Over the past two decades, inorganic semiconductor QDs have attracted a great amount interest due to their optical and electronic properties [20]. But the main disadvantage is their unstability. Rather a GQD has more “molecule like” stable character in which electronic transport is confined in all three spatial dimensions. It is expected that spatial confinement of carriers in GQDs will lead to opening of band gap and excitation under luminescence. The band gap of GQDs is tuned by modifying the size and surface chemistry [15]. The absorption spectra of GQDs indicate a prominent peak at ∼230 nm, which is assigned for π to π *

R. Das et al. / Journal of Physics and Chemistry of Solids 99 (2016) 34–42

transition of π bonds [21]. Several experimental studies have revealed that optical band gap is dependent on size of GQDs [22]. Photoluminescence (PL) behavior is strongly associated with the nanostructures in GQDs and hence emission sites are also affected by edges [22,23]. The calculation employing density functional theory (DFT) have shown that band gap of GQDs increases to ∼2 eV in a GQD consisting of twenty aromatic rings [24]. So far, photophysics of biggest synthesized GQDs with 168 carbon atoms were recently studied theoretically by Schumacher using timedependent DFT (TDDFT)-based approach [25] and they have identified some optical transitions which are dominated by transitions between molecular orbitals. Yazyev [26] has indicated the potential application of GQDs in the field of opto-electronics and spintronics due to their interesting optical and magnetic properties. Theoretical calculation based on tightbinding (TB) model have been employed to study the optical properties and magnetic levels of hexagonal [27] and triangular GQDs [28]. Magnetic levels of hexagonal GQDs display a Hofstadter-butterfly characteristic feature and approach the Landau levels of 2D graphene as the magnetic field increases. Magnetic properties of triangular shaped GQDs embedded in a hexagonal boron nitride (h-BN) sheet have been investigated using DFT technique [29] which suggest about ferrimagnetic spin polarization on each graphene dot with nonzero magnetic moment. Magnetic properties of hexagonal, circular and random shaped GQDs have been explored by Espinosa-ortega et al. [30] using TB approach. They have identified dispersed edge states for the above mentioned GQDs for displaying the diamagnetic nature. Basak et al. [31] have studied linear optical absorption in diamond shaped GQD (DSGQDs) by employing this technique and various excitations from highest occupied molecular orbital (HOMO) to lowest occupied molecular orbital (LUMO) are analyzed. Using DFT calculation Yamijala et al. [32] have examined the structural stability, electronic, magnetic and optical properties of rectangular quantum dots of graphene along with BN and their hybrid by varying several parameters such as sizes and amount of substitution. They have achieved spin polarized HOMO-LUMO gaps of 1.35 eV for a particular system of GQD substituted with B and N without application of any external voltage. Agapito et al. [33] have demonstrated that the antiferromagnetic, ferromagnetic and non-magnetic states of DSGQDs can be controlled selectively by an external electric field using ab initio DFT calculation. So far there is no such literature which elucidates the size dependent magnetic and optical properties of DSGQDs using DFT. As far as experimental studies are concerned, it is quite impossible to synthesize GQD without hydrogen (H) passivation. This is only due to the dangling bonds and so edges undergo reconstruction, hence leading to deformation and unstability of structures. Considering relatively larger graphene quantum dot with 48 carbon atoms, its structural properties have been studied theoretically [34–37] although its experimental synthesis is yet to be done. Karki et al. [36] have shown that on increasing the cluster size, binding energy per atom follows perfect linearity with percentage fraction of H-atom as well as C-atom. A non invasive characterizing technique Raman spectroscopy [38] has been widely used in the study of different properties of the carbon materials [39–41]. Naturally, Raman fingerprint of graphene is described with some specific nature of vibrations associated with structural deformation. Among them, the so-called G-band, a C–C in plane stretching mode are common to all sp2 carbon materials while the D-band is used to characterize disorder in sp2 carbon materials. Motivated from various experimental and theoretical studies in recent years, we concentrate on DSGQD of various sizes. Here three unique structures of DSGQDs are considered with number of carbon atoms 16, 30, 48 respectively which are

35

Fig. 1. Different structures with 16, 30, 48 carbon atoms respectively from top to bottom.

R. Das et al. / Journal of Physics and Chemistry of Solids 99 (2016) 34–42

designated as config1, config2, config3 (shown in Fig. 1) obtained from a 5  5 unit cell of graphene. These structures are interesting as all of them are mixtures of zigzag and armchair edges. So these structures play an important role to study the effect of edge geometry and symmetry. We focus on magnetic and polarization dependent optical properties of DSGQDs of various size to indicate the effect of quantum confinement. In addition, the fundamental relevance to practical applications of the H-terminated DSGQDs inspired us to employ a first principles based Raman spectra study as a non-destructing characterization tool.

2. Computational methodology Our present investigation on DSGQDs employs DFT as implemented in SIESTA package [42–44]. Here we have made use of spin polarized Generalized Gradient Approximation (GGA) according to Perdew, Burke, Ernzerhof (PBE) [45] parametrization for the exchange-correlation functional, double ζ plus polarized basis set and energy cut off 400 Ry. The sampling of the Brillouin zone was done for the supercell with the equivalent of a 24  24  1 Monkhorst-Pack (MP) [46] of k point grid. All the simulations are performed by using the diagonalization method. To avoid the interaction emerging from periodic boundary conditions in all three dimensions a supercell was used with a height of 20 Å to consider enough vacuum. The relaxation of the atomic positions of all three structures was performed with forces on individual atoms smaller than 0.001 eV/Å (not shown in figure). The convergence criteria for energy of the self-consistent field (SCF) cycle is taken to be 10  5 eV. C–C bond length after relaxation becomes 1.424 Å which is very close to experimentally achieved data [15]. In general, optical properties of any system can be calculated with the help of complex dielectric function: ϵ(ω) = ϵ1(ω) + i ϵ2(ω) where ϵ1(ω) and ϵ2(ω) are real and imaginary part of the dielectric function respectively. For long wavelength limit (q → 0) the imaginary part of the dielectric function is given by [47–49].

ϵ2(ω) =

2e 2 π Ωϵ 0

∑

2 → → 〈ψKVB| u . r |ψKCB〉 δ(EKCB − EKVB − ω)

K , CB, VB

(1)

(2)

n and k are the real and imaginary part of refractive index. They are related to ϵ1 and ϵ2 as 1

⎛ 2 ⎞2 ϵ1 + ϵ22 + ϵ1 ⎟ n(ω) = ⎜ ⎜ ⎟ 2 ⎝ ⎠

(4)

Now, by knowing expression of (3) and (4) we can further compute

R(ω) =

(n − 1)2 + k 2 2kω 1 dR(ω) , α(ω) = , RM (ω) = 2 2 c ℏ R ( ω ) dω (n + 1) + k

(5)

where c represents the speed of light in vacuum. Reflectivity modulation is a very convenient way to analyze the characteristic feature of the reflectivity spectrum of a material as it has both positive and negative values. The loss function (defined below) measures the collective excitation of any structure. It is an effective approach to investigate the structure of carbon materials with high spatial and energy resolution.

⎛ ϵ2 1 ⎞ ⎟⎟ = L(ω) = Im⎜⎜ − ⎝ ϵ(ω) ⎠ ϵ12 + ϵ22

(6) 2

Here we neglect the coupling between sp states and pz state as we are only interested in low energy regime (q → 0) of the spectra. Geometry optimization and corresponding frequency calculations of the H-terminated DSGQDs have been carried out using Becke's three parameter with Lee–Yang–Parr non local electron correlation (B3LYP) of the DFT and 6–31G(d) basis set as implemented in Gaussian 09W software package. Absence of the imaginary frequencies for all the structure support the stability [35] and confirm the global energy minima. With chosen functional/basis set (B3LYP/6–31G(d)) a scaling factor 0.9614 is introduced [52,53] to get the actual wave number (WN) from the computational data. For further methodical details related with Raman study the readers are referred to the literature of graphene work [41].

3. Results and discussion 3.1. Electronic properties

In the above expression ω is the frequency of electro-magnetic (EM) wave in unit of energy, Ω represents the volume of the cell, ϵ0 is the free space permittivity. CB and VB represent the con→ → duction band and valence band respectively. u and r denote the polarization vector and position vector of EM field. Here the selfconsistent ground state DFT energies and eigen functions have been inserted into the dipolar transition matrix elements via Kohn–Sham (KS) formalism [42–44]. For the purpose of optical calculation 0.2 eV optical broadening were applied. The data for ϵ2(ω) are generated by SIESTA code and with the help of Kramers– Kronig (KK) transformation [50,51] data of ϵ1(ω) are produced. Other relevant optical properties like absorption coefficient α(ω), reflectivity R(ω), reflectivity modulation RM (ω) and loss function L (ω) are investigated in this work. The complex refractive index (N͠ ) of a material is related by the expression N͠ = ϵ(ω) where

N͠ = n(ω) + ik(ω)

1

⎛ 2 ⎞2 ϵ1 + ϵ22 − ϵ1 ⎟ k(ω) = ⎜ ⎜ ⎟ 2 ⎝ ⎠

3.1.1. Density of states (DOS) We have investigated density of states (DOS) as well as projected density of states (PDOS) of DSGQDs. Occurrence of zero DOS always play an important role to explain various properties of 2D systems [54,55]. There are several number of zero DOS regions in case of config1 but for config2 only one zero DOS region is observed (shown in Fig. 2). Similarly number of zero PDOS regions also decreasing as size of DSGQDs become larger which is reflected

30

20 15 10 5 0 -10

(3)

config1 config2 config3

25 DOS (eV-1)

36

-5

0 E - EF (eV)

5

Fig. 2. Density of states for three configurations.

10

R. Das et al. / Journal of Physics and Chemistry of Solids 99 (2016) 34–42

8

config1 config2 config3

10

ε1||(ω) (arb. unit)

-1 PDOS (eV )

20

0 -10 -20 -10

-5

0

5

10

4 2 0 -2

Fig. 3. Projected density of states (2P orbital) for three configurations. Positive Y axis indicates up spin states and negative Y axis indicates down spin states.

config1 config2 config3

2.206–9.124 2.200–8.090 2.645–7.353

5

10 15 20 Energy (eV)

25

30

config1 config2 config3

3 2 1

⊥

6.918 5.890 4.708

ε1 (ω) (arb. unit)

Range of zero PDOS Difference (eV) (eV)

0

4

Table 1 Fermi energy and zero PDOS regions for 2S orbital. Configuration name Fermi energy EF (eV)

config1 config2 config3

6

E - EF (eV)

4.493 4.086 3.611

37

from Fig. 3. No zero DOS or PDOS regions are observed for largest structure config3, which is consistent with absence of zero energy state for larger structures [31]. For all three configurations finite DOS have been achieved at the Fermi energy. Non-zero DOS at Fermi energy for all configurations in GQD reflects their participation in novel transport and thermodynamic properties like specific heat at the nanoscale. Importance of zero PDOS in 2P orbital for config1 and config2 also will be reflected in its optical properties. Study of PDOS for 2S orbital gives more interesting results. For all three configurations broad region of zero PDOS have been found (see Table 1 for comparison). Regions of zero PDOS is decreasing with increase of sizes. Flat regions of zero PDOS of nearly 7 eV, 6 eV, 5 eV have been noticed for config1, config2, config3 respectively. It is seen that with increase in size of DSGQDs, the electron levels become closer [35]. In config1 quantum effect becomes significant. The quantum confinement effect becomes less prominent with increment of

||

ε2 (ω) (arb. unit)

-1

10

15

20

25

30

Energy (eV) Fig. 5. Real part of the dielectric function for different polarizations.

number of atoms in DSGQDs and hence gap in PDOS of 2S orbital decreases. Moreover, lesser flat regions are due to decrement in the spacing of energy level and increment in the delocalization of electrons in larger DSGQDs. From this it can be conjectured that for infinitely large DSGQDs flat PDOS region will vanish. A plot of EF vs 1/L2 (L being the length of the perimeter of the respective configuration) of three different configurations (not shown) gives a reasonable straight line fit to the data shown in Table 1. 3.2. Optical properties

4

10x10x1 30x30x1 60x60x1

3

Before discussing the features associated with dielectric functions, we schematically show the convergence of the imaginary part of dielectric functions ϵ2 with respect to energy for different values of k points (10  10  1, 30  30  1, 60  60  1) in Fig. 4. It is observed that the values obtained exactly coincide with each other. Henceforth, for any optical calculation related with the structures, we restrict to 10  10  1 meshes.

2 1 0

0

0

5

10 15 Energy (eV)

20

25

Fig. 4. Effect of k-point mesh size on the variation of the imaginary part of dielectric function (ϵ2) for parallel polarization.

3.2.1. Dielectric functions To study optical properties of these DSGQDs analysis of ϵ1 and ϵ2 are very important. For parallel polarization, two plasma frequencies (ωp) have appeared for config1 and config2 whereas there is no plasma frequency for config3. In config1 two ωps emerge at positions 2.482 eV and 2.709 eV as shown in Fig. 5. There is a slight shifting of ωps for config2 and the positions are 2.862 eV and 3.007 eV. However, for perpendicular polarization the reverse situation is observed. Here plasma frequency occurs at 19.737 eV and 20.322 eV for config3 only.

38

R. Das et al. / Journal of Physics and Chemistry of Solids 99 (2016) 34–42

3

||

2 1

0

5

10 15 20 Energy (eV)

25

0

⊥

2 1 15

20 Energy (eV)

25

0.5

0

5

10 15 20 Energy (eV)

4

3

0 10

1

30

config1 config2 config3

4

config1 config2 config3

1.5

L (ω) (arb. unit)

0

ε2⊥(ω) (arb. unit)

2

config1 config2 config3

L (ω) (arb. unit)

ε2||(ω) (arb. unit)

4

30

25

30

config1 config2 config3

3 2 1 0

10

15

20 Energy (eV)

25

30

Fig. 6. Imaginary part of the dielectric function for different polarizations.

Fig. 7. EELS function for different polarizations.

The static real part of dielectric function ( ϵ1(0)) has maximum value for config2 in both type of polarization as depicted from Fig. 5 whereas minimum value have occurred for config1. The oscillatory behavior is restricted in the range 0–5 eV for parallel direction. In perpendicular direction a sharp peak with high intensity has been noticed. Here oscillations are mainly restricted in the energy range 10–30 eV. Analyzing ϵ2 (shown in Fig. 6) spectra, most intense peak have appeared at positions 2.367 eV, 2.012 eV and 1.187 eV for config1, config2, config3 respectively indicating the tendency of red shifting with increasing size for parallel polarization. The zero value of ϵ2 occurs only for config1 within the range 8.807 eV to 9.401 eV. This suggests the strong confinement effect in config1. We can split the whole excitation spectrum in two different regions, one belongs to smaller energy window within 0–9 eV and another one corresponds to relatively higher energy scale within 12–18 eV. Transitions from ground state to first excited state is attributed from longer wavelength excitations while the shorter wavelength excitations illustrate the transitions from ground to second excited state which are consistent with the reported data of Kwon et al. [22]. Considering perpendicular polarization most significant peaks have emerged at positions 20.034 eV, 19.837 eV, 19.660 eV showing red shifting nature with increase in size. These excitations may be due to π + σ plasmon. A hump like characteristics has also been noticed for all three configurations in the range 12–16 eV showing a blue shifted nature with increase in size of DSGQDs which may be due to π plasmon excitation.

quantum dots for parallel as well as perpendicular polarization. In parallel polarization an overall blue shift of the peaks is observed with increase of the size of quantum dots. Sharp peaks are observed in the energy range 0–10 eV for config1 and config2, whereas a hump is noticed for config3 within same range. Zero EELS region of config1 reflecting from ϵ2 data is consistent with the 2P PDOS graph shown in Fig. 3. It can be observed from Fig. 7 that there is a tendency of broadening of peaks from config 1 to config 3. It elucidates the fact that for larger size larger is the effective energy loss which again ensures about confinement of smallest DSGQDs. For config3, EELS peak in the low energy range (0–5 eV) is due to π plasmon excitation and the broad peak (in the range 10–16 eV) corresponds to π + σ plasmon excitation. Both the peaks confirm the fact that as the size of DSGQDs increases, it tends to behave like single sheets of graphene [56,57] illustrating the bulk feature. High energy (12–25 eV) excitations are observed for perpendicular polarization. Most intense peak for all three structures have appeared at almost same position around 20 eV. There is a tendency of blue shift within 12–16 eV whereas the red shifting nature of peaks occurs within 22–25 eV. It is to be noted that the intensity of most significant peak is proportional to the size of DSGQDs.

3.2.2. Electron energy loss spectra (EELS) In Fig. 7, we depict the variation of EELS spectra with size of

3.2.3. Reflectivity and reflectivity modulation We have schematically shown the variation of reflectivity ( R(ω)) as a function of frequency for parallel as well as perpendicular polarization in Fig. 8. Maximum R(ω) at zero frequency is obtained for config2. It is interesting to note that here zero R(ω) at non zero frequency is observed for all three structures. Occurrence of zero reflectivity is very rare. For nitrogen doped carbon

60

config1 config2 config3

0.1

0

5

10 15 20 Energy (eV)

0.3

⊥

0.2

R

0.1 0 10

15

20 Energy (eV)

25

0 -20 -40 0

5

10 15 20 Energy (eV)

30

25

30

config1 config2 config3

40

config1 config2 config3

0.4

20

-60

30

⊥

R (ω) (arb. unit)

0.5

25

M(ω) (arb. unit)

0

39

config1 config2 config3

40 R || M(ω) (arb. unit)

0.2

||

R (ω) (arb. unit)

R. Das et al. / Journal of Physics and Chemistry of Solids 99 (2016) 34–42

30 20 10 0 -10 -20 10

15

20 Energy (eV)

25

30

Fig. 9. Reflectivity modulation for different polarizations.

Fig. 8. Reflectivity spectra for different polarizations.

nanotubes zero reflectivity has been observed [49]. This zero reflectivity region obeys the formula 2n = n2 + k 2 + 1. But region of zero R(ω) decreases with the increase of size. This result correlates with decrease of the range of PDOS region. All structures possess finite R(ω) within 0–6 eV but config1 has the highest R(ω) rather than config2 and config3. From these it can be concluded that config1 is optically transparent. Within the range 6–12 eV, R(ω) is very low for all configurations. In case of perpendicular polarization R(ω) is zero at zero frequency. These results are consistent with the graph of EELS depicted in Fig. 7. We have also studied reflectivity modulation RM (ω) (graph is depicted in Fig. 9). Config1 has large RM (ω) in the range 4–6 eV, 8– 10 eV, 13–15 eV as R(ω) is very poor in these regions. This indicates that config1 can act as a good tuning device in those particular ranges. Config2 and config3 can not response properly in reflectivity modulation with respect to config1. In perpendicular polarization the appearance of similar situation in config1 indicates its best modulating capacity. It can be used as a good signal modulator in the high energy regime. For this structure highest modulation arises at position 21.547 eV where reflectivity is exactly zero and after that RM (ω) is attenuating gradually. All these discussions clearly suggest that config1 can be used as a good reflector as well as good modulator. 3.2.4. Optical absorption In case of parallel polarization multiple peaks have appeared for all structures (shown in Fig. 10) which may be due to edge states. Most prominent peaks and a flat region of absorption arise for config1 only which again ensure quantum confinement in

smallest DSGQD [31]. DSGQDs have continuous absorption spectra in the UV–visible region [19] due to overlap of electronic absorption caused by closely spaced electronic energy levels. Major peaks in energy range 0–7 eV and corresponding wavelength in nm for all the structures are given below in Table 2. It is evident that config2 shows a peak around 5.35 eV (232.33 nm) which is consistent with Ref. [21]. As size increases, absorption spectrum becomes broader which matches with polycyclic aromatic hydrocarbon (PAH) ions absorption spectra [58]. From the above analysis it is clear that the absorption peak positions and strength can be tuned by the size of DSGQDs. Graphs of perpendicular polarization are more prominent than parallel ones. Only one significant peak having slightly red shifting nature has been observed for all three structures [31]. Intensity of this peak increases enormously as size of DSGQDs increases [31]. For config1 absorption starts with a very weak feature near 12 eV while the most intense absorption occurs at energy around 20 eV. Peak around 20 eV in the optical absorption can be used to diagnose the morphology (size) of DSGQDs. 3.2.5. Raman study Based on the above optical properties, all structures demand some features to be associated with Raman spectra. Hence, in addition to our above mentioned studies, we have also explored the structural anisotropy of these diamond shaped H-graphene clusters by the first principles based 1st order Raman spectroscopy study. The additional H atoms passivate the dangling bonds and help us producing stable structures required in the vibrational studies [41]. Interestingly, the measured C–C bond lengths and corresponding angles of the optimized structures indicate that the

40

R. Das et al. / Journal of Physics and Chemistry of Solids 99 (2016) 34–42

α||(ω) (arb. unit)

20

Table 3 Detailed assignment of fundamental normal mode vibrations of C16H10 , C30H14 and C 48H18 clusters. Here [H], [Hx], α, β represent vibration of hydrogen atoms, expansion and contraction of hexagons, in plane stretching, in plane twisting respectively.

config1 config2 config3

16 12

Structures Unscaled WN ( v¯k ) (cm  1)

8

0

5

10

15

20

25

30

412.22

39.92

Breathing like

C30H14

597.97 1100.49 1275.91 1441.22 1640.05 1686.36 301.29

50.75 35.24 377.52 234.16 252.39 251.00 211.06

Breathing like [H] [Hx] α β α Breathing like

C 48H18

915.57 1225.66 1322.84 1412.29 1544.58 1624.74 1646.00 1680.49 238.01

247.27 1631.01 1472.00 2072.47 614.64 1626.13 4342.92 2152.89 514.38

[Hx] [H] [Hx] α α β α α Breathing like

1008.32 1210.29 1257.51 1317.11 1354.94 1379.16 1391.12 1495.95 1625.03

953.84 5018.13 921.40 2968.03 3091.49 2143.16 4415.30 3057.64 22985.30

[Hx] [H] [H] α α [Hx] α α G-like

Energy (eV) config1 config2 config3

40 30 20

⊥

α (ω) (arb. unit)

50

10 0 10

15

20 Energy (eV)

25

Assignment

C16H10

4 0

Scattering activity (RK) (Å4 amu  1)

30

Fig. 10. Absorption coefficient for different polarizations.

Table 2 Major absorption peaks within the energy range 0–7 eV for three configurations. Configuration name Major peaks in energy (eV)

Major peaks in wavelength (nm)

config1

501.21 284.44 253.67 547.57 437.67 306.15 255.23 232.33 204.77 293.16 207.16

config2

config3

2.48 4.37 4.90 2.27 2.84 4.06 4.87 5.35 6.07 4.24 6.00

ratio of sp2 carbon atoms to H atoms increases with increasing cluster size. However, it is noteworthy to mention that for all these 2D structures, out of 3N-6 internal modes (where N is the total number of atoms), some of them are Raman active modes corresponding to different types of vibration of the covalent bonds. Details of the Raman fingerprints of the clusters are shown in Table 3 and depicted in Fig. 11 (some modes far above 2000 cm  1 are intensionally neglected as they are due to hydrogen only). It is clear from the figure that change in cluster size leads to a significant qualitative change in the Raman spectra. Not only intensity of the Raman spectra gradually increases with the cluster size but also the position (in cm  1) as well as the nature of vibration. For the C16H10 structure several Raman active modes with

Fig. 11. Raman spectra (investigated with fitted Lorentzian envelope with FWHM equals to 5 cm  1) of diamond shaped H-graphene clusters (a) C16H10 (b) C30H14 (c) C 48H18. Atomic displacement vectors for G-like vibrational modes are shown in the corresponding inset boxes.

R. Das et al. / Journal of Physics and Chemistry of Solids 99 (2016) 34–42

low but comparable intensity are observed. Whereas, Raman spectrum of the C48H18 structure is predominated by a high intense peak at 1562.30 cm  1. In this context, it is interesting to point out that Panchakarla et al. [59] have experimentally demonstrated the existence of a high intense Raman peak (the G band) around 1569 cm  1 for the graphene accomplished by the arc-discharge method in a hydrogen atmosphere. So, it is noteworthy that C48H18 structure is likely to explore the bulk properties of graphene. Hence displacement vectors corresponding to this mode are observed carefully and depicted in the inset of Fig. 11(c). Counter directional in-plane vibration of the non-equivalent sp2 carbon atoms (responsible for C–C stretching mode) helps us to predict this mode as the G like band for any sp2 network as well [59]. Likewise, search of similar G-like vibration for C16H10 and C30H14 structures leads towards two particular types of vibrations at 1621.27 cm  1 and 1582.46 cm  1, which are shown in the inset of Fig. 11 (a) and (b). These two types of vibrations along with the above mentioned mode of C48H18 lead us to the following conclusions. Position (in cm  1) of the above mentioned G-band like stretching mode is found to be red shifted with gradual increase in cluster size. Intensity of these modes increases with increase in cluster size. Finally, unlike the other structures, the predominated Raman mode of C48H18 cluster is found to be G-band like. This is well expected as the number of sp2 carbon atoms increases with increase in cluster size reflecting bulk feature of infinite graphene sheet. Our observation can predict the lower limit of the number of atoms for which DFT results can be fitted well with that of the bulk properties of graphene. A detailed assignment of the fundamental normal modes are also listed in Table 3. 3.3. Magnetic properties Magnetic moments (μ) of these three structures give finite values like 4μB and 8μB for config1 and config2 respectively where μB is Bohr magneton. But in case of config3 a low value magnetic moment ( 0.6μB ) appears. Occurrence of such a high value of magnetic moment can be explained by Hund's rule. In Ref. [54], it has been shown that according to Hund's rule, Coulomb exchange interaction plays an important role to align spins in case of trigonal graphene nanodisk. For config1 number of atoms at edge (completely exterior position) is 10 and for config2 it is 14. So it is ex1 pected to give a magnetic moment of 2 × 10μB = 5μB and

41

for config2 also. But α becomes unity for config3 and hence this formula does not work. Therefore, any configuration having α value much greater than unity is predicted to possess vanishingly small value of magnetic moment. Another significant explanation can be given to make a realization of value regarding magnetic moment. The corresponding value of β (in increasing size) are 0.5, 1, 1.5 respectively. From our data, it can be interpreted that μ is proportional to β for config1 and config2 and maximum value of μ is achieved when β is equal to 1. As β is doubled from config1 to config2 the value of moment is also doubled. In case of config3 and for further increment of size of DSGQDs number of atoms in completely interior is greater than number of atoms in exterior which are situated at edge positions and hence they give rise to lower value of moment. It is expected that formation of local moments are due to edge atoms.

4. Conclusions In summary, we have examined electronic, optical, magnetic characterization of DSGQDs using ab initio DFT calculation. This study has revealed that there is a considerable range of zero PDOS for every structure which implies existence of some forbidden region and restriction of number of states per unit energy level. Decrease of Fermi energy with increasing size proves that the effective density of electron is reduced. It is also noticeable that config2 is the most magnetic among all three structures due to its highest magnetic moment. All structures possess plasma frequency either parallel or perpendicular direction leading to future application of DSGQDs in photonics. Vanishing of ϵ2 value is reported only for config1. From our results it is clear that quantum confinement effect is very prominent for smallest structure. Red shifting nature of ϵ2 is observed for perpendicular polarization and peaks appear at 20.034 eV, 19.837 eV, 19.660 with increase in size. Peaks of perpendicular polarization are more prominent and distinguishable than parallel ones. In the Raman spectra appearance of G like vibration for the largest cluster confirms its bulk nature. As DSGQDs are highly anisotropic system, spectra depending on polarization of electromagnetic wave interact with system. Therefore, DSGQDs may be considered as building block for novel optoelectronic device to trigger further research in nanoelectronics industry.

1

× 14μB = 7μB which nearly match with our numerical data. Si2 milar arguments have been put forwarded by Chowdhury et al. [55,60] for different types of silicene nanodisk system. With increase of atoms in DSGQD (in config3) one might expect to have mere magnetic moments in compared to config1 and config2. But due to saturation effect and emergence of bulk features, the magnetic moment for config3 decreases significantly less than 1 μB approaching the non-magnetic feature of graphene nanosheet (bulk counterpart). Now we define N1, N2, N3 where N1 is number of atoms in completely exterior position not shared by edge, N2 is number of atoms in completely interior positions, N3 is number of atoms in exterior positions but shared by edge. Therefore, to characterize N the shape and size of DSGQD, we define two parameters α = N2 1

and β =

N2 . N3

Now concentrating on config1 and config2, a simple

formula can be framed like

μ = (N − (N1 + N2))μB

when α < 1.

where N is the total number of atoms for a configuration and value of α becomes 0.20 and 0.57 for config1 and config2 respectively. For config1 after substituting value of N ¼16, N1 = 10, N2 = 2 becomes 4μB by using above formula. Similarly this formula is valid

Acknowledgments This work is funded by the DST-FIST, DST-PURSE, Government of India. Two of authors (RD-IF140283 and ND-IF150670) gratefully acknowledge DST, Govt. of India for providing financial assistance through INSPIRE Fellowship scheme. We want to thank anonymous reviewers for their critical comments and suggestions for improvement of the quality of the paper.

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