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Electronic and optical properties of beryllium sulfide monolayer: Under stress and strain conditions
Physics Letters A 380 (2016) 3546–3552
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Electronic and optical properties of beryllium sulfide monolayer: Under stress and strain conditions Jaafar Jalilian ∗ , Mandana Safari Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran
a r t i c l e
i n f o
Article history: Received 1 June 2016 Received in revised form 4 August 2016 Accepted 19 August 2016 Available online 26 August 2016 Communicated by R. Wu Keywords: Beryllium sulfide nanosheet Optical properties Tuning band gap Stress and strain Density functional theory
a b s t r a c t Electronic and optical properties of two-dimensional graphene-like structure of beryllium sulfide (BeS) have been studied in the framework of the density functional theory. Different values of stress and strain are exerted for tuning electronic and optical parameters. The electronic results show that both biaxial stress and strain effects cause band gap reduction with different rates. Also, we have red and blue shifts in the optical absorption spectrum peaks by applying strain and stress, respectively. Our results express that BeS monolayer can be the promising candidate for the future nano-devices. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Vigorous research on graphene and its applications [1,2] has created a fertile ground for the investigation of a broad range of alternative two-dimensional nanomaterials. In the last decade, many groups of two-dimensional semiconductor nanomaterials were studied and applied in the different fields of industry. Some of more famous materials are BN [3,5,4], SiC [6,7], transition metal dichalcogenides [8,9], black phosphorus (BP) [10,11] and zinc oxide [12,13]. Among these categories of nanomaterials, twodimensional II–VI semiconductors have been studied by different groups. Their results presented that these materials in graphenelike structure have a good structural stability [14]. Also, the electronic and optical properties of family of these two-dimensional materials have been studied by computational and experimental approaches [15–17]. Yu and Guo studied two-dimensional beryllium sulfide (BeS) by first principles calculations [18]. They found that this monolayer has a good thermodynamical stability up to 1000 K and is a wide gap semiconductor about of 4.26 eV. Also, they found that armchair edge BeS nanoribbon has semiconducting properties while the zigzag edge is a ferromagnetic metal. Recently, An and colleagues investigated electronic transport properties of graphene-like beryllium sulfide nanoribbons [19]. They obtained that the electrons flow mainly through the two edges of
*
Corresponding author. P.O. Box 67149-67346, fax: +98 9377455595. E-mail address: [email protected] (J. Jalilian).
http://dx.doi.org/10.1016/j.physleta.2016.08.037 0375-9601/© 2016 Elsevier B.V. All rights reserved.
zigzag BeS nanoribbon and they presented that BeS nanoribbon can be the promising candidate for the future nano-devices. On the other hand, exerting stress and strain is one of the effective way to control electronic and optical properties of twodimensional materials. Liu and colleagues studied applying biaxial stress and strain on electronic properties of AlN nanosheet [20]. They claimed that in some values of shear strain along the zigzag direction induces an indirect-to-direct transition in band gap. Different kind of behavior was reported for ZnO graphenelike structure by Kaewmaraya et al. [21]. Their results illustrated that the band gap of the sheet varies almost linearly with uniaxial strain while it shows a parabola-like behavior under homogeneous biaxial strain. Similar studies are performed for different twodimensional materials by first principles calculations [13,22,23]. Hence, we decided to study electronic and optical properties of BeS monolayer by first principles calculations to find more potential applications for this compound. These properties are studied under different biaxial stress and strain conditions. 2. Computational details Electronic and optical properties of beryllium sulfide monolayer have been investigated using first principles study calculations in the framework of the density functional theory as implemented WIEN2k code [24]. Full potential augmented plane waves plus local orbital (FPAPW + lo) has been used to expand basis set functions. The exchange-correlation potential has been treated using generalized gradient approximation presented by Perdew–Burke–
J. Jalilian, M. Safari / Physics Letters A 380 (2016) 3546–3552
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We optimized the lattice constant of BeS monolayer (see Fig. 1b). The obtained equilibrium lattice constant is a = b = 3.45 Å that is in good agreement with previous calculations [18]. After optimizing lattice constants, the force relaxation calculation has been performed to achieve the equilibrium atomic positions. We have found the atomic positions of beryllium and sulfur atoms are completely similar to other graphene-like structures, Be(1/3, 2/3, 0) and S(2/3, 1/3, 0). Also, the relaxation process present that this structure is completely flat and there is not any planar buckling. After these steps, we exerted different values of stress and strain to investigate the variation of electronic and optical properties of BeS monolayer under different structural conditions. Considered values are:
a±δ = 1 ±
δ
aeq ,
100
(2)
where δ = 2, 4, 6 and 8, aeq is the equilibrium lattice constant (free-strain lattice constant) and the positive and negative signs refer to strained and stressed cases, respectively. 3.2. Electronic properties
Fig. 1. (a) The unit call of honeycomb structure of BeS monolayer and its lattice vector, (b) The E–V curve of BeS monolayer.
Ernzerhof (GGA-PBE) [25]. The value of optimized input parameters in our calculations are R M T K max = 8 and lmax = 10. The Fourier expansion of electron charge density is expanded by G max = 14 Ry1/2 . The convergence criteria in optimization and electronic calculations were set to | E i +1 − E i | ≤ 10−4 Ry on energy of system and |ρi +1 − ρi | ≤ 10−5 on electron charge, respectively. Vacuum spacing is arranged so that the minimum distance between two monolayer in adjacent unit cells is about 15 Å, provided that atoms have negligible interaction at that far distance. Based on Monkhorst–Pack scheme [26], 15 × 15 × 3 and 35 × 35 × 7 k-meshes have been considered in the whole of first Brillouin zone to calculate electronic and optical properties, respectively. The optical data of this paper has been performed by the random phase approximation (RPA) method [27] to gain imaginary part of dielectric function and Kramers–Kronig relations for the real part. 3. Results and discussion 3.1. Structural properties In order to find structural stability of graphene-like structure of beryllium sulfide monolayer (see Fig. 1a), the total energy of compound versus unit cell volume has been calculated using thermodynamical Brich–Murnaghan equation of state [28]:
⎧
E (V ) = E 0 +
2/3 9B 0 V 0 ⎨ V0 ⎩
16
V
3
⎧
+
2/3 9B 0 V 0 ⎨ V0 16
⎩
V
B 0
−1
⎫ ⎬
2 −1
(1)
⎭
6−4
V0 V
⎫ 2/3 ⎬ ⎭
where V 0 is the initial considered volume, V is the deformed volume, B 0 is the bulk modulus, and B 0 is the derivative of the bulk modulus with respect to pressure. This formula presented the variations of total energy of system versus the unit cell volume.
Electronic property is an important part for every compound investigations. In fact, it helps us improve our rough understanding about compounds’ electrical behavior into an acceptable level. “Beryllium” as one of the elements of group II of periodic table can contribute in ionic-covalent bond especially with “sulfur” element. In order to identify the characterization of this compound, we have calculated the fractional ionic character (FIC). This factor can be calculated by formula (1) as it was represented by Xu and Ching for calculation the contribution of ionic and covalent bonds in binary compounds [29].
Q Be − Q S × 100 , F IC = Q +Q Be
(3)
S
where Q Be is affective charge of beryllium that has participated in Be–S bond, and Q S is sulfur’s simultaneously. Obviously, this factor can be represented between 100% and 0%, which denote to ionic and covalent natures, respectively. The fractional ionic character for the Be–S bond is 65% and shows that it keeps covalent nature a bit, yet the ionic nature is more than covalent nature here. Investigating density of states of electrons compared with band structure graph tells us electronic properties of this structure in general. These two graphs are illustrated in Fig. 2. Recognizing its semiconducting property as a consequence of band gap existence is the first point to mention. Band gap value of this nanosheet is about 4.55 eV. On the other hand, its wide band gap is indirect type in K → M direction and tells us how electron transition may occur in electrical conductivity process as well. Clearly, it can be considered as a good agreement with previous data that approves our calculation in general [18]. Moreover, investigation of the partial density of states in Fig. 3 brings about a good view about the main orbitals around the Fermi level. These orbitals simultaneously play a main role in electrical conductivity of compounds. Clearly, s and p orbitals which are situated in almost the similar region of energy, create density of states around the Fermi level. As it is clear from Fig. 3c and 3d, the edge of conduction band minimum (CBM) and of valence band maximum (VBM) are created by p z known as π and π ∗ orbitals of sulfur and beryllium, so it demonstrates that electrical conductivity mostly move through these orbitals after supplying voltage. While the p x + p y stays in plane to create sp 2 hybridized orbitals with s orbital. One of the most important parts of the investigation of every compound is to study the effect of external forces on them such as substrate effect, supplying substrate effect and so on. We
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Fig. 2. (a) The energy band structure of BeS monolayer in free-strained case; (b) Corresponding total density of state for this structure.
try to figure out the electronic and optical behavior of material in the presence of different structural situations. Supplying stress and strain on crystal lattice parameters can be considered as one of the implication of substrate existence for example. As we discussed in structural part, we exert stress and strain for BeS nanosheet isotropically in plane. The amount of different situations of them are ±2%, ±4%, ±6% and ±8% of strain (setting minus amount of strained cases equal to stressed cases). In Fig. 4, the 2D counter plots of electron density for ±8% (sections 4a and 4c) have been compared with free-strained case on section 4b. As it is clear, strain and stress change the charge distribution of BeS nanosheet. Left panel in every section of Fig. 4 (i.e. 4a, 4b and 4c) illustrates the σ bond situated in the plane, while the right panel of them denotes to p z orbital that can create π bond in structure as well. Obviously, as the stress increases up to 8% the FIC decreases till 60%. This demonstrates that charge contribution between “Be” and “S” goes up a bit more than between them in free-strained case. On the other side, when the strain gets
increase till 8%, the charge contribution goes down and FIC receives to 68%. These cases identified with charge distribution. The value variations of band gap for different percentages of strain and stress are plotted in Fig. 5a. This graph illustrates that BeS nanosheet shows an anisotropic electronic behavior as encounter stress and strain. As it is mentioned, the free-strained case stand in the extremum compared to other cases. The value for ±2%, ±4%, ±6% and ±8% stay smaller (see Table 1 to see the exact value of every case). These data can utilize for that industrial application in which many different cases of band gap are required. Moreover, the band structure shows light signs of changing in σ ∗ and π ∗ energy levels for low percentages of stress and strain while this variation intensifying in 8% and 6% of strained cases. So, we found that while in the stressed cases the direction of band structure stays the same compared with free-strained case, the direction of band gap for the 6% and 8% of strained cases changes from K → M to K → as shown in Fig. 5b (for more details see Table 1). It could be a result of rearranging in compounds orbitals, so these strain variations change σ ∗ stability and make them decrease their energy level along with going up the level of p z orbital. In order to approve this different behavior at the edge of CBM and VBM for strained and stressed cases, we plotted partial density of states in detail for ±8% cases (see Fig. 6). As we discussed, the strained case shows the change in direction of band structure. It accompanied by change in p z and p x + p y partial DOS energetically. Obviously, p z (as π ∗ orbital) goes up in energy levels with respect to p x + p y (as a partial of σ ∗ orbital) for CBM in strained case. While p x + p y (as a partial of σ orbital) comes more down with respect to p z orbital (as π orbital) in VBM. These variations in level of orbital energy occur for both atoms that are involved in bond (Fig. 6). These debates help industry to improve its ability for creating vast range of applications regarding BeS quality. 3.3. Optical properties Optical discussions can create a good opportunity for finding new applications and improving material capability in this field. In order to investigate optical properties of compounds we need to utilize the complex dielectric function. Dielectric function con-
Fig. 3. The partial density of state for BeS monolayer (a) s and p orbitals of beryllium atom, (b) s and p orbitals of sulfur atom, (c) p z and p x + p y orbitals of beryllium atom and (d) p z and p x + p y orbitals of sulfur atom in free-strain BeS nanosheet.
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Fig. 4. Two-dimensional contour plots of the charge densities of Kohn–Sham eigenstates in x– y (left panels) and x–z (right panels) planes corresponding to: (a) 8% of stress, (b) equilibrium lattice parameter and (c) 8% of strain.
εcomplex = ε + i ε .
(4)
The imaginary part is the chief function for indicating interband transition in compounds. Considering occupied |ik and unoccupied | f k states with regard to Fermi–Dirac distribution function as one part could demonstrate how it works in general (equation (3)).
ε αα (ω) =
4π e 2 m2
ω
2 i, f
2dk3
(2π
)3
ik| P α | f k 2 f k 1 − f k i f
× δ E kf − E ki − h¯ ω .
Fig. 5. (a) Graphical illustration of band gap variations of BeS monolayer for different percentages of strain and stress and (b) band gap direction for I-8% of stress and II-8% of strain.
sists of two parts, the real and imaginary parts, which indicates the material response to electromagnetic spectrum as well as definition of how these waves behave in vacuum. The relation between complex dielectric function and its parts is identified by the equation below:
(5)
The imaginary part of dielectric function for BeS nanosheet with respect to energy of incident photon compared with 8% of stress and strain is illustrated in Fig. 7. This function calculated in both x and z directions of polarization. Imaginary part of the dielectric function connected with photon absorption as its equation is. Because of threshold of spectrum, obviously, BeS nanosheet as well as its strain and stress cases (±8%) have semiconducting properties optically. These three cases behave in different trends in x and z directions of photon polarization. This fact indicates the existence of different pattern in photon absorption. It obviously obeys the selection rules and level of energy in band structure. Since absorption spectrum shows the photon absorption in compound, it directly connects with imaginary part of dielectric function. Studying absorption spectrum can confirm the imaginary part of dielectric function and itself brings a good view about how it absorbs electromagnetic spectrum for each photon energy. In order to study the absorption spectrum of these compound, we plotted this spectrum for free-strained case as well as 8% strain and stress in plane in Fig. 8. The first peak of these graphes denotes the first increase in photon absorption, it occurs around 5 eV for every three cases in x direction while it happens in different energy values for z direction. It means, for every three cases, excited electrons which absorb the incident photon with x polarization choose almost the same orbitals for final electron state. It is in contrast with absorption of photons in z polarization. On the other hand, it can be seen that in the second peak, the photon absorption of free-strained case occurs about 8 eV. This peak for the strained case (8%) occurs in higher photon energy (around 9 eV), and it
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Table 1 Calculated electrical band gap values (eV) and band gap directions for ± different percentages of strain of BeS monolayer (minus amount of strained cases consider as stressed cases in general).
Band gap Direction
−8%
−6%
−4%
−2%
0
2%
4%
6%
8%
4.38 K →M
4.43 K →M
4.46 K →M
4.55 K →M
4.55 K →M
4.53 K →M
4.51 K →M
4.36 K →
4.07 K →
Fig. 6. The partial density of state for BeS monolayer (a) p z and p x + p y orbitals of beryllium atom and also (b) p z and p x + p y orbitals of sulfur atom in 8% stress, besides (c) p z and p x + p y orbitals of beryllium atom and (d) p z and p x + p y orbitals of sulfur atom in 8% strain.
Fig. 7. The imaginary part of the complex dielectric function for stressed, freestained and strained cases in (a) x direction and (b) z direction of polarizations.
Fig. 8. The absorption spectrum of BeS monolayer in ±8% strained cases (minus amount of strained case consider as stressed case) compared with free-strained case in (a) x and (b) z directions of polarization.
happens in lower photon energy (around 7 eV) in the stressed case energetically as it occurs in the crest of free-strained and strained cases interestingly. This trend helps the industry to optimize the equipments required as it could be available merely by changing the lattice parameter in any way possible. Moreover, Kramers–Kronig relations make the real part of the dielectric function correspond to imaginary part. So the real part is well on the way of characterizing optical properties of material too. It obeys the equation below:
where P r . denotes the principal value of this function. These equations manipulate quantum consideration in optical investigation as a sophisticated way to optical characterization. Fig. 9 illustrates the real part of the dielectric function for energy of incident photon. This graph consists of two panels as each one is for one direction of polarization (x and z directions). This figure is plotted for three cases including free-strained, strained and stressed cases. The starting point of the real part of dielectric function denotes static dielectric constant in the presence of static electric field. As it can be seen, the static dielectric constant stays the same for all of these cases (2.5 for x direction and 1.8 for z direction, regardless of ignorable differences in each case). This trend can be recognized in optical dielectric constant on the other side of graphs (ε (ω → ∞)) that shows the number of 1 for x and z
2 ε α β (ω) = δα β + P r .
∞
π
0
ε α β (ω ) ω dω , ω 2 − ω2
(6)
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graphene-like structure has a good structural stability. Energy band structure presented the BeS nanosheet is an indirect band gap semiconductor. The energy gap value decreases in different trends for stress situations with respect to strain. The threshold of optical absorption spectrum shifts toward higher and lower energy ranges by exerting stress and strain in z direction, but this threshold for x direction almost remains the same. The optical reflectivity for this compound is about 2.5% in visible range of light that denotes to transparency behavior in visible range. The obtained electronic and optical results represent that BeS graphene-like structure can be a good candidate for tunable optoelectronic materials and devices. Acknowledgements Computing resources used in this work were provided by the Nano-Fanavaran of Bistoon, High Performance and Grid Computing Center, Kermanshah, Iran. Fig. 9. The real part of the complex dielectric function for stressed, free-stained and strained cases in (a) x direction and (b) z direction of polarizations.
Fig. 10. The reflectivity spectrum of BeS monolayer in ±8% strained cases compared with free-strained case in (a) x and (b) z directions of polarization.
directions. On the other hand, the abnormal region of propagation identifies with sheer fall about 4 eV for the first fall of x direction, while it occurs in 8–10 eV energy region for z direction. These abnormal regions of propagation correspond to plasmonic frequency in compounds, so we expect to increase the change in reflectivity percentage as it is found out from Fig. 10. This figure is plotted for both directions of polarization in two panels as usual pattern for all optical graphs. The initial percentage of reflectivity of x direction starts about 5%. It decreases even more for z direction till 2.5%. The reflectivity spectrum stays low in the visible range of electromagnetic spectrum that shows the importance of them as a transparent structure too. Besides, in high range of energy, reflectivity increases dramatically till 30% in x direction and 40% in z direction. The stress and strain make it occurs in different range of energy than of free-strained case. It may create an interesting situation about optical properties application that just strain and stress can bring different range of applications for BeS monolayer. 4. Conclusion In summary, electronic and optical properties of BeS graphenelike structure have been studied by full potential augmented plane waves plus local orbital in the framework of the density functional theory. The structural results show that beryllium sulfide in
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Electronic and optical properties of beryllium sulfide monolayer: Under stress and strain conditions Jaafar Jalilian ∗ , Mandana Safari Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran
a r t i c l e
i n f o
Article history: Received 1 June 2016 Received in revised form 4 August 2016 Accepted 19 August 2016 Available online 26 August 2016 Communicated by R. Wu Keywords: Beryllium sulfide nanosheet Optical properties Tuning band gap Stress and strain Density functional theory
a b s t r a c t Electronic and optical properties of two-dimensional graphene-like structure of beryllium sulfide (BeS) have been studied in the framework of the density functional theory. Different values of stress and strain are exerted for tuning electronic and optical parameters. The electronic results show that both biaxial stress and strain effects cause band gap reduction with different rates. Also, we have red and blue shifts in the optical absorption spectrum peaks by applying strain and stress, respectively. Our results express that BeS monolayer can be the promising candidate for the future nano-devices. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Vigorous research on graphene and its applications [1,2] has created a fertile ground for the investigation of a broad range of alternative two-dimensional nanomaterials. In the last decade, many groups of two-dimensional semiconductor nanomaterials were studied and applied in the different fields of industry. Some of more famous materials are BN [3,5,4], SiC [6,7], transition metal dichalcogenides [8,9], black phosphorus (BP) [10,11] and zinc oxide [12,13]. Among these categories of nanomaterials, twodimensional II–VI semiconductors have been studied by different groups. Their results presented that these materials in graphenelike structure have a good structural stability [14]. Also, the electronic and optical properties of family of these two-dimensional materials have been studied by computational and experimental approaches [15–17]. Yu and Guo studied two-dimensional beryllium sulfide (BeS) by first principles calculations [18]. They found that this monolayer has a good thermodynamical stability up to 1000 K and is a wide gap semiconductor about of 4.26 eV. Also, they found that armchair edge BeS nanoribbon has semiconducting properties while the zigzag edge is a ferromagnetic metal. Recently, An and colleagues investigated electronic transport properties of graphene-like beryllium sulfide nanoribbons [19]. They obtained that the electrons flow mainly through the two edges of
*
Corresponding author. P.O. Box 67149-67346, fax: +98 9377455595. E-mail address: [email protected] (J. Jalilian).
http://dx.doi.org/10.1016/j.physleta.2016.08.037 0375-9601/© 2016 Elsevier B.V. All rights reserved.
zigzag BeS nanoribbon and they presented that BeS nanoribbon can be the promising candidate for the future nano-devices. On the other hand, exerting stress and strain is one of the effective way to control electronic and optical properties of twodimensional materials. Liu and colleagues studied applying biaxial stress and strain on electronic properties of AlN nanosheet [20]. They claimed that in some values of shear strain along the zigzag direction induces an indirect-to-direct transition in band gap. Different kind of behavior was reported for ZnO graphenelike structure by Kaewmaraya et al. [21]. Their results illustrated that the band gap of the sheet varies almost linearly with uniaxial strain while it shows a parabola-like behavior under homogeneous biaxial strain. Similar studies are performed for different twodimensional materials by first principles calculations [13,22,23]. Hence, we decided to study electronic and optical properties of BeS monolayer by first principles calculations to find more potential applications for this compound. These properties are studied under different biaxial stress and strain conditions. 2. Computational details Electronic and optical properties of beryllium sulfide monolayer have been investigated using first principles study calculations in the framework of the density functional theory as implemented WIEN2k code [24]. Full potential augmented plane waves plus local orbital (FPAPW + lo) has been used to expand basis set functions. The exchange-correlation potential has been treated using generalized gradient approximation presented by Perdew–Burke–
J. Jalilian, M. Safari / Physics Letters A 380 (2016) 3546–3552
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We optimized the lattice constant of BeS monolayer (see Fig. 1b). The obtained equilibrium lattice constant is a = b = 3.45 Å that is in good agreement with previous calculations [18]. After optimizing lattice constants, the force relaxation calculation has been performed to achieve the equilibrium atomic positions. We have found the atomic positions of beryllium and sulfur atoms are completely similar to other graphene-like structures, Be(1/3, 2/3, 0) and S(2/3, 1/3, 0). Also, the relaxation process present that this structure is completely flat and there is not any planar buckling. After these steps, we exerted different values of stress and strain to investigate the variation of electronic and optical properties of BeS monolayer under different structural conditions. Considered values are:
a±δ = 1 ±
δ
aeq ,
100
(2)
where δ = 2, 4, 6 and 8, aeq is the equilibrium lattice constant (free-strain lattice constant) and the positive and negative signs refer to strained and stressed cases, respectively. 3.2. Electronic properties
Fig. 1. (a) The unit call of honeycomb structure of BeS monolayer and its lattice vector, (b) The E–V curve of BeS monolayer.
Ernzerhof (GGA-PBE) [25]. The value of optimized input parameters in our calculations are R M T K max = 8 and lmax = 10. The Fourier expansion of electron charge density is expanded by G max = 14 Ry1/2 . The convergence criteria in optimization and electronic calculations were set to | E i +1 − E i | ≤ 10−4 Ry on energy of system and |ρi +1 − ρi | ≤ 10−5 on electron charge, respectively. Vacuum spacing is arranged so that the minimum distance between two monolayer in adjacent unit cells is about 15 Å, provided that atoms have negligible interaction at that far distance. Based on Monkhorst–Pack scheme [26], 15 × 15 × 3 and 35 × 35 × 7 k-meshes have been considered in the whole of first Brillouin zone to calculate electronic and optical properties, respectively. The optical data of this paper has been performed by the random phase approximation (RPA) method [27] to gain imaginary part of dielectric function and Kramers–Kronig relations for the real part. 3. Results and discussion 3.1. Structural properties In order to find structural stability of graphene-like structure of beryllium sulfide monolayer (see Fig. 1a), the total energy of compound versus unit cell volume has been calculated using thermodynamical Brich–Murnaghan equation of state [28]:
⎧
E (V ) = E 0 +
2/3 9B 0 V 0 ⎨ V0 ⎩
16
V
3
⎧
+
2/3 9B 0 V 0 ⎨ V0 16
⎩
V
B 0
−1
⎫ ⎬
2 −1
(1)
⎭
6−4
V0 V
⎫ 2/3 ⎬ ⎭
where V 0 is the initial considered volume, V is the deformed volume, B 0 is the bulk modulus, and B 0 is the derivative of the bulk modulus with respect to pressure. This formula presented the variations of total energy of system versus the unit cell volume.
Electronic property is an important part for every compound investigations. In fact, it helps us improve our rough understanding about compounds’ electrical behavior into an acceptable level. “Beryllium” as one of the elements of group II of periodic table can contribute in ionic-covalent bond especially with “sulfur” element. In order to identify the characterization of this compound, we have calculated the fractional ionic character (FIC). This factor can be calculated by formula (1) as it was represented by Xu and Ching for calculation the contribution of ionic and covalent bonds in binary compounds [29].
Q Be − Q S × 100 , F IC = Q +Q Be
(3)
S
where Q Be is affective charge of beryllium that has participated in Be–S bond, and Q S is sulfur’s simultaneously. Obviously, this factor can be represented between 100% and 0%, which denote to ionic and covalent natures, respectively. The fractional ionic character for the Be–S bond is 65% and shows that it keeps covalent nature a bit, yet the ionic nature is more than covalent nature here. Investigating density of states of electrons compared with band structure graph tells us electronic properties of this structure in general. These two graphs are illustrated in Fig. 2. Recognizing its semiconducting property as a consequence of band gap existence is the first point to mention. Band gap value of this nanosheet is about 4.55 eV. On the other hand, its wide band gap is indirect type in K → M direction and tells us how electron transition may occur in electrical conductivity process as well. Clearly, it can be considered as a good agreement with previous data that approves our calculation in general [18]. Moreover, investigation of the partial density of states in Fig. 3 brings about a good view about the main orbitals around the Fermi level. These orbitals simultaneously play a main role in electrical conductivity of compounds. Clearly, s and p orbitals which are situated in almost the similar region of energy, create density of states around the Fermi level. As it is clear from Fig. 3c and 3d, the edge of conduction band minimum (CBM) and of valence band maximum (VBM) are created by p z known as π and π ∗ orbitals of sulfur and beryllium, so it demonstrates that electrical conductivity mostly move through these orbitals after supplying voltage. While the p x + p y stays in plane to create sp 2 hybridized orbitals with s orbital. One of the most important parts of the investigation of every compound is to study the effect of external forces on them such as substrate effect, supplying substrate effect and so on. We
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Fig. 2. (a) The energy band structure of BeS monolayer in free-strained case; (b) Corresponding total density of state for this structure.
try to figure out the electronic and optical behavior of material in the presence of different structural situations. Supplying stress and strain on crystal lattice parameters can be considered as one of the implication of substrate existence for example. As we discussed in structural part, we exert stress and strain for BeS nanosheet isotropically in plane. The amount of different situations of them are ±2%, ±4%, ±6% and ±8% of strain (setting minus amount of strained cases equal to stressed cases). In Fig. 4, the 2D counter plots of electron density for ±8% (sections 4a and 4c) have been compared with free-strained case on section 4b. As it is clear, strain and stress change the charge distribution of BeS nanosheet. Left panel in every section of Fig. 4 (i.e. 4a, 4b and 4c) illustrates the σ bond situated in the plane, while the right panel of them denotes to p z orbital that can create π bond in structure as well. Obviously, as the stress increases up to 8% the FIC decreases till 60%. This demonstrates that charge contribution between “Be” and “S” goes up a bit more than between them in free-strained case. On the other side, when the strain gets
increase till 8%, the charge contribution goes down and FIC receives to 68%. These cases identified with charge distribution. The value variations of band gap for different percentages of strain and stress are plotted in Fig. 5a. This graph illustrates that BeS nanosheet shows an anisotropic electronic behavior as encounter stress and strain. As it is mentioned, the free-strained case stand in the extremum compared to other cases. The value for ±2%, ±4%, ±6% and ±8% stay smaller (see Table 1 to see the exact value of every case). These data can utilize for that industrial application in which many different cases of band gap are required. Moreover, the band structure shows light signs of changing in σ ∗ and π ∗ energy levels for low percentages of stress and strain while this variation intensifying in 8% and 6% of strained cases. So, we found that while in the stressed cases the direction of band structure stays the same compared with free-strained case, the direction of band gap for the 6% and 8% of strained cases changes from K → M to K → as shown in Fig. 5b (for more details see Table 1). It could be a result of rearranging in compounds orbitals, so these strain variations change σ ∗ stability and make them decrease their energy level along with going up the level of p z orbital. In order to approve this different behavior at the edge of CBM and VBM for strained and stressed cases, we plotted partial density of states in detail for ±8% cases (see Fig. 6). As we discussed, the strained case shows the change in direction of band structure. It accompanied by change in p z and p x + p y partial DOS energetically. Obviously, p z (as π ∗ orbital) goes up in energy levels with respect to p x + p y (as a partial of σ ∗ orbital) for CBM in strained case. While p x + p y (as a partial of σ orbital) comes more down with respect to p z orbital (as π orbital) in VBM. These variations in level of orbital energy occur for both atoms that are involved in bond (Fig. 6). These debates help industry to improve its ability for creating vast range of applications regarding BeS quality. 3.3. Optical properties Optical discussions can create a good opportunity for finding new applications and improving material capability in this field. In order to investigate optical properties of compounds we need to utilize the complex dielectric function. Dielectric function con-
Fig. 3. The partial density of state for BeS monolayer (a) s and p orbitals of beryllium atom, (b) s and p orbitals of sulfur atom, (c) p z and p x + p y orbitals of beryllium atom and (d) p z and p x + p y orbitals of sulfur atom in free-strain BeS nanosheet.
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Fig. 4. Two-dimensional contour plots of the charge densities of Kohn–Sham eigenstates in x– y (left panels) and x–z (right panels) planes corresponding to: (a) 8% of stress, (b) equilibrium lattice parameter and (c) 8% of strain.
εcomplex = ε + i ε .
(4)
The imaginary part is the chief function for indicating interband transition in compounds. Considering occupied |ik and unoccupied | f k states with regard to Fermi–Dirac distribution function as one part could demonstrate how it works in general (equation (3)).
ε αα (ω) =
4π e 2 m2
ω
2 i, f
2dk3
(2π
)3
ik| P α | f k 2 f k 1 − f k i f
× δ E kf − E ki − h¯ ω .
Fig. 5. (a) Graphical illustration of band gap variations of BeS monolayer for different percentages of strain and stress and (b) band gap direction for I-8% of stress and II-8% of strain.
sists of two parts, the real and imaginary parts, which indicates the material response to electromagnetic spectrum as well as definition of how these waves behave in vacuum. The relation between complex dielectric function and its parts is identified by the equation below:
(5)
The imaginary part of dielectric function for BeS nanosheet with respect to energy of incident photon compared with 8% of stress and strain is illustrated in Fig. 7. This function calculated in both x and z directions of polarization. Imaginary part of the dielectric function connected with photon absorption as its equation is. Because of threshold of spectrum, obviously, BeS nanosheet as well as its strain and stress cases (±8%) have semiconducting properties optically. These three cases behave in different trends in x and z directions of photon polarization. This fact indicates the existence of different pattern in photon absorption. It obviously obeys the selection rules and level of energy in band structure. Since absorption spectrum shows the photon absorption in compound, it directly connects with imaginary part of dielectric function. Studying absorption spectrum can confirm the imaginary part of dielectric function and itself brings a good view about how it absorbs electromagnetic spectrum for each photon energy. In order to study the absorption spectrum of these compound, we plotted this spectrum for free-strained case as well as 8% strain and stress in plane in Fig. 8. The first peak of these graphes denotes the first increase in photon absorption, it occurs around 5 eV for every three cases in x direction while it happens in different energy values for z direction. It means, for every three cases, excited electrons which absorb the incident photon with x polarization choose almost the same orbitals for final electron state. It is in contrast with absorption of photons in z polarization. On the other hand, it can be seen that in the second peak, the photon absorption of free-strained case occurs about 8 eV. This peak for the strained case (8%) occurs in higher photon energy (around 9 eV), and it
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Table 1 Calculated electrical band gap values (eV) and band gap directions for ± different percentages of strain of BeS monolayer (minus amount of strained cases consider as stressed cases in general).
Band gap Direction
−8%
−6%
−4%
−2%
0
2%
4%
6%
8%
4.38 K →M
4.43 K →M
4.46 K →M
4.55 K →M
4.55 K →M
4.53 K →M
4.51 K →M
4.36 K →
4.07 K →
Fig. 6. The partial density of state for BeS monolayer (a) p z and p x + p y orbitals of beryllium atom and also (b) p z and p x + p y orbitals of sulfur atom in 8% stress, besides (c) p z and p x + p y orbitals of beryllium atom and (d) p z and p x + p y orbitals of sulfur atom in 8% strain.
Fig. 7. The imaginary part of the complex dielectric function for stressed, freestained and strained cases in (a) x direction and (b) z direction of polarizations.
Fig. 8. The absorption spectrum of BeS monolayer in ±8% strained cases (minus amount of strained case consider as stressed case) compared with free-strained case in (a) x and (b) z directions of polarization.
happens in lower photon energy (around 7 eV) in the stressed case energetically as it occurs in the crest of free-strained and strained cases interestingly. This trend helps the industry to optimize the equipments required as it could be available merely by changing the lattice parameter in any way possible. Moreover, Kramers–Kronig relations make the real part of the dielectric function correspond to imaginary part. So the real part is well on the way of characterizing optical properties of material too. It obeys the equation below:
where P r . denotes the principal value of this function. These equations manipulate quantum consideration in optical investigation as a sophisticated way to optical characterization. Fig. 9 illustrates the real part of the dielectric function for energy of incident photon. This graph consists of two panels as each one is for one direction of polarization (x and z directions). This figure is plotted for three cases including free-strained, strained and stressed cases. The starting point of the real part of dielectric function denotes static dielectric constant in the presence of static electric field. As it can be seen, the static dielectric constant stays the same for all of these cases (2.5 for x direction and 1.8 for z direction, regardless of ignorable differences in each case). This trend can be recognized in optical dielectric constant on the other side of graphs (ε (ω → ∞)) that shows the number of 1 for x and z
2 ε α β (ω) = δα β + P r .
∞
π
0
ε α β (ω ) ω dω , ω 2 − ω2
(6)
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graphene-like structure has a good structural stability. Energy band structure presented the BeS nanosheet is an indirect band gap semiconductor. The energy gap value decreases in different trends for stress situations with respect to strain. The threshold of optical absorption spectrum shifts toward higher and lower energy ranges by exerting stress and strain in z direction, but this threshold for x direction almost remains the same. The optical reflectivity for this compound is about 2.5% in visible range of light that denotes to transparency behavior in visible range. The obtained electronic and optical results represent that BeS graphene-like structure can be a good candidate for tunable optoelectronic materials and devices. Acknowledgements Computing resources used in this work were provided by the Nano-Fanavaran of Bistoon, High Performance and Grid Computing Center, Kermanshah, Iran. Fig. 9. The real part of the complex dielectric function for stressed, free-stained and strained cases in (a) x direction and (b) z direction of polarizations.
Fig. 10. The reflectivity spectrum of BeS monolayer in ±8% strained cases compared with free-strained case in (a) x and (b) z directions of polarization.
directions. On the other hand, the abnormal region of propagation identifies with sheer fall about 4 eV for the first fall of x direction, while it occurs in 8–10 eV energy region for z direction. These abnormal regions of propagation correspond to plasmonic frequency in compounds, so we expect to increase the change in reflectivity percentage as it is found out from Fig. 10. This figure is plotted for both directions of polarization in two panels as usual pattern for all optical graphs. The initial percentage of reflectivity of x direction starts about 5%. It decreases even more for z direction till 2.5%. The reflectivity spectrum stays low in the visible range of electromagnetic spectrum that shows the importance of them as a transparent structure too. Besides, in high range of energy, reflectivity increases dramatically till 30% in x direction and 40% in z direction. The stress and strain make it occurs in different range of energy than of free-strained case. It may create an interesting situation about optical properties application that just strain and stress can bring different range of applications for BeS monolayer. 4. Conclusion In summary, electronic and optical properties of BeS graphenelike structure have been studied by full potential augmented plane waves plus local orbital in the framework of the density functional theory. The structural results show that beryllium sulfide in
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