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Lithium halide monolayers: Structural, electronic and optical properties by first principles study
Physica E ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Lithium halide monolayers: Structural, electronic and optical properties by first principles study Mandana Safari a, Pegah Maskaneh a, Atousa Dashti Moghadam a, Jaafar Jalilian b,n a b
Physics Department, Razi University, Kermanshah, Iran Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, P.O. Box 67149-67346, Kermanshah, Iran
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
Structural optimizing represents that unlike graphene-like structures, the cubic face structure is more favorable for alkali halide 2D structures. Nonbonding electron pairs cause a planar buckling for all compounds. Electronic calculations show that all compounds have an indirect energy gap. All compounds are optically transparent in the visible spectrum range.
art ic l e i nf o
a b s t r a c t
Article history: Received 12 October 2015 Received in revised form 27 December 2015 Accepted 19 January 2016
Using first principle study, we investigate the structural, electronic and optical properties of lithium halide monolayers (LiF, LiCl, LiBr). In contrast to graphene and other graphene-like structures that form hexagonal rings in plane, these compounds can form and stabilize in cubic shape interestingly. The type of band structure in these insulators is identified as indirect type and ionic nature of their bonds are illustrated as well. The optical properties demonstrate extremely transparent feature for them as a result of wide band gap in the visible range; also their electron transitions are indicated for achieving a better vision on the absorption mechanism in these kinds of monolayers. & 2016 Elsevier B.V. All rights reserved.
Keywords: Alkali halide Optical transition 2D monolayer Density functional theory
1. Introduction Alkali halides known as ionic compounds are characterized by their highly crystalline nature, high melting points and strong miscibility in polar media [1,2]. These materials can be considered as prototype insulator materials with great technological importance [3–5]. For many years, thermodynamic, elastic, structural and electronic properties of them have been investigated comprehensively. The effect of defects in these compounds has been presented too [6,7]. Also band structure investigations reveal these compounds are recognized as wide-gap insulators that n
Corresponding author. E-mail address: [email protected] (J. Jalilian).
demonstrate their optical transparency in the visible region of the electromagnetic spectrum [8]. This transparency feature, especially in LiF, in high compression makes this material a suitable choice for versatile window material through which to carry out wave profile measurements for shock compression experiments [9]. These compounds generally crystallize in both B1 (NaCl-type) and B2 (CsCl-type) structures. Their phase transitions have been classified as pressure and temperature dependence in first-order kind [7,10]. On the other hand, studying the role of electronic correlation on the boundary between localized and delocalized electronic states become possible through energy level investigation, due to their electronic states kind. Their applications for a range of optical applications are significant besides the interaction mechanisms between the electronic and geometric structure can
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be interestingly demonstrated [11]. LiF, LiCl and LiBr, known as Li halides, have been investigated in bulk phase theoretically [12–14] and experimentally [15,16]. As a result, the energy difference between the top of the valence band and the outermost core levels in LiF, LiCl and LiBr was determined by the electron spectroscopy for chemical analysis (ESCA) technique in the bulk phase. The energy differences obtained between the top of the valence band and the Li 1 s line in LiF, LiCl and LiBr were 49.8 eV, 53.2 eV and 54.1 eV respectively. These results agree rather well with the predictions of the point charge model, including corrections for polarization effects [15]. Since there has been increased attention to the properties of thin films and nanostructures, the great theoretical interest was paid to their nanostructures due to interest in the application of alkali halide systems as sensitive photostimulable storage materials [17]. Low dimensional alkali halide nanocrystals of LiF were investigated structurally and energetically by Bichoutskaia and Pyper in 2006 [18]. The required energy for forming one plane of new contacts in the LiF (2 2 n) chain is essentially constant and independent of the lengths of the two fragments being joined. Also the simulation performed with density functional theory (DFT) and coupled-cluster (CCSD) calculations on the series of (LiF)n ¼ 2,36 neutral clusters showed that nanotube structures with hexagonal and octagonal transverse cross-sections are stabile, and this stability is similar to the typical cubic form of large LiF crystals [19]. These investigations showed that these Li halides in cubic shape have been formed regardless to cross section of nanostructures. Here we focus on Li halide monolayers in structural, electronic and optical aspects to achieve a point of view in these kinds of nanostructures for the first time. Our preferable structure for investigation is cubic structure agreed with previous researches mentioned above.
2. Computational details In order to calculate our results, the first principles study in the framework of the density functional theory [20,21] as implemented in WIEN2k code [22] is used. To achieve reliable results we have performed full potential augmented plane waves plus local orbitals (FPAPW þlo). The generalized gradient approximation energy functional formulated by Perdew–Burke–Ernzerhof (GGA-PBE) [23] has employed as the approximation method for the exchange–correlation energy functional. The cut-off parameter known as RMT Kmax that it has been equaled 7, the value of this parameter demonstrates our accuracy and efficiency and Gmax ¼14 Ry1/2 as another parameter of charge calculations. The optical data of this paper has been performed by using the random phase approximation (RPA) method [24] to gain imaginary part of dielectric function and Kramers–Kronig relations for real part. The number of k-points is 5000 in the first Brillouin zone that leads to 24 22 8 as a good mesh for calculations here.
3. Results and discussion In order to explain electronic and optical characteristics of LiF, LiCl and LiBr monolayers, this part is required to achieve the best configuration to find balance situation in structures and reach acceptable results. 3.1. Structural investigation In order to find stable configurations of these monolayers energetically, we have optimized volume as implemented in Birch– Murnaghan equation of state [25] and relax these structures to get closer to minimum force possible exerting to each atom. The
Fig. 1. Lithium halide monolayer: (a) length bond illustrated for x- and y-directions and different angles created by structural optimizing, (b) top-view of monolayer and (c) first Brillouin zone with k-path for energy band structure calculations.
structural optimization results express that the cubic face for these ionic compounds is more preferable than the other configurations, and these results demonstrate good agreement with previous calculations for alkali halide nanotubes [19]. In Fig. 1a, stable structure of these monolayers as a representation of Li halides has been illustrated for better understanding. Fig. 1b is the top-view of these monolayers. The first Brillouin zone for these kinds of structures is shown as well (see Fig. 1c). The optimized parameters, i.e. bond length, buckling factor and angle between each three close atoms in each direction are presented in Table 1 for LiBr, LiCl and LiF. These parameters are identified in Fig. 1 to find the distinction of these variables in these monolayers. It is obvious that these two dimensional structures are not flat and get planar buckling after relaxation set. This phenomenon occurs as a result of the existence of nonbonding electrons in halogen atoms. In the other words, after bond formation, three pairs of nonbonding electrons have broadened in a side and push the other bonding orbital to another side, so this monolayer get buckled naturally. When we investigate a 2D monolayer, these nonbonding electrons located on opposite sides in adjacent atoms to achieve structural stability. Thus buckling formation in these monolayer is a natural reformation. 3.2. Electronic properties In order to show the bond nature of these structures, electron Table 1 The measured angle of every three atoms in the x-direction (θ) and y-direction (ϕ), bond length of structure in the x-direction (ax), bond length of structure in the ydirection (ay) buckling factor for all of under discussion compounds. Compounds
θ (degr)
ϕ (degr)
ax (Å)
ay (Å)
Δ
LiF LiCl LiBr
136.15 132.10 132.16
160.60 151.41 147.01
1.88 2.41 2.57
1.92 2.51 2.70
0.70 0.97 1.04
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Fig. 2. 2D electron charge density distribution for (a) LiF monolayer, (b) LiCl monolayer and (c) LiBr monolayer; magnitude of the charge density shown by colors, blue color at top of the box show the smallest value and white color at bottom of the box is largest value.
density distribution of each compound is indicated in Fig. 2. The highest value of electronegativity in periodic table belongs to fluorine (3.98) and the lowest value of this parameter is for elements of the alkali group, so lithium (0.98) [26] as a first member of this group play donor role in bond. Fluorine as an acceptor almost creates an ionic bond, this kind of bond can be observed obviously in Fig. 3 and localization of electron density distribution around each atom proves this claim. With regard to lower electronegativity of Cl (3.16) and Br (2.96) [26] in comparison with F, it could be inferred that Li–F bond is stronger than the bond of Li–Cl (Li–Br). This provides more strong s bond in LiF than LiCl (LiBr) that cause to more stable s bond in LiF energetically, so the energy level of s energy band goes down more than corresponding bands in LiCl and LiBr, thus we see the energy band gap gets decrease with changing halogen atoms to lower electronegativity. In order to show these different band gaps, total and partial electron density of states (DOS) are investigated in Fig. 3. As it can be seen in Fig. 3a, all of these compounds are insulators. Halogen atoms have a significant role in forming energy levels below the Fermi level (see Fig. 3b) for all of these structures, while both halogen and corresponding lithium in each compound play roles in energy levels that are located above the Fermi level (see Fig. 3c). On the other hand, the density of energy states around the Fermi level in the valence band for Br, Cl and F atoms is more than that of Li atom. It is due to higher electronegativity in halogen atoms and more electrons belonged halogens which are engaged in bond. As it was seen, these compounds are known as members of the insulator group of materials because of their wide band gap. To present a better view, we focus on their band structure here (see Fig. 4). The gap values are identified as 6.693, 5.873 and 5.126 eV for LiBr, LiCl and LiF, respectively. As it is clear from these graphs, these structures have wide indirect band gaps. The conduction band minimum is in the k2 direction while the valence band
3
Fig. 3. Density of states for LiF, LiCl and LiBr monolayers versus energy (eV); (a) total density of states for all compounds, (b) total density of states for halogen atoms and (c) total density of states for lithium atoms in different compounds.
maximum happens in a direction between k2 and k3. This band type variation from direct type in the bulk phase to indirect type in monolayer structures maybe lie in the interaction of pz orbital from every atom in the bonds. However, this orbital locates perpendicular to monolayer plate, the buckling phenomenon has been engaged it with other orbital trans-shapes and it has formed an indirect energy band gap subsequently. 3.3. Optical properties Optical properties of solids are determined by the response of their electrons to the time-dependent electromagnetic interference caused by the incident beam. In order to investigate the optical properties of the structure, optical spectra such as the real and imaginary parts of the dielectric tensor, the absorption and reflectivity functions are calculated. The complex dielectric function is dependent on the frequency and is directly associated with the electronic band structure of the crystals. This function consists of two parts that called imaginary and real parts of dielectric function. According to the appropriate elements in the transition matrix, the imaginary part of the dielectric function is defined as below: αβ εimaginary (ω) =
4π e 2 m2ω2
∑ ∫ dk〈ck |pα |vk 〉〈vk |p β |ck 〉δ (Eck − Evk − ω). c, v
(1)
These two parts are related to each other through the Kramers– Kronig relations. The real part of the dielectric function is obtained correspondingly:
εreαβ (ω) = δαβ +
2 Pr . π
∫0
∞
αβ ω′εim (ω′)
ω′ 2 − ω2
dω′,
(2)
where Pr. denotes the principal value [27]. Because of the importance of these functions in the optical properties study, the real part of the dielectric function versus the incident photon energy is calculated for three directions of electric field polarization (the electric field is polarized parallel to the alkali halides monolayer
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Fig. 4. Energy band structure for (a) LiF, (b) LiCl and (c) LiBr compounds. The selected k-vectors were presented in Fig. 1.
Fig. 5. Real part of complex dielectric function versus incident photon energy for all compounds in each direction.
(x-axis and y-axis) and perpendicular to the alkali halides monolayer (z-axis)) for LiBr, LiCl and LiF and it is shown in Fig. 5. As it can be seen in Fig. 5, the real part of the dielectric function is identified for LiF, LiCl and LiBr. The values of dielectric function for zero energy called static dielectric constant have been obtained for LiBr, LiCl and LiF and have been summarized in Table 2. As it is clear from this table, static dielectric constant [28] values have not significant variations and also in high energy converge to the same value. This value famed as optical dielectric constant comes close to 1 in amount as one can see in figures (see Fig. 5). In all of three directions of E field ( E∥x, E∥x and E∥z ) the summits get decrease in intensity and happen at lower energies
Table 2 The value of static dielectric constant for all of 3 compounds, for three directions of E field polarization. Electric field direction
LiF
LiCl
LiBr
Parallel direction ( E∥x ) Parallel direction ( E∥y ) Perpendicular direction ( E∥z )
1.04 1.04 1.04
1.07 1.07 1.06
1.89 1.92 1.74
when atomic size of halogens increases simultaneously. For LiCl and LiBr structures, we can see sharp falls that happen in every direction of polarization. This despite the fact that sharp fall does
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not occur for LiF practically and the spectrum gets decrease over a range of energy and finally achieves a minimal value at 18 eV energy. It is obvious that photons of the high energy region involve with this structure in a wide range of energy. Moreover, some plasmonic frequencies (minimal value of the function is corresponding to plasmonic frequency in compounds) can be recognized as a consequence. However, this trend keeps going in LiCl monolayer, the range of reaction region is smaller and the number of plasmonic frequencies is lower too. These quantities for LiBr go down even more. These sharp falls in LiCl, LiBr and the decreasing trend in LiF are consequences of subjecting on this sort of photons to create π and π + σ plasmons. The distinguishable behavior of parts of the dielectric function, here real part (see Fig. 5), in each direction of polarization denotes to anisotropic behavior of these structures in these three directions. These different behavior will be understood for all spectra that comes as follow too. The percentages of reflectivity spectra in terms of energy (eV) in all three directions of polarization of E field (parallel, x and y axis, and perpendicular to the monolayer plane) are illustrated in Fig. 6 for these three compounds. Reflectivity spectrum behaves corresponding to the real part of the dielectric function. Its maximum value as a peak happens where the sharp fall does. It is obvious that all of these compounds have a neglectable per cent of the reflection spectrum in the lower range of energy as well as the visible spectrum. LiF reflectivity starts with less than 0.02% per cent and keep remaining around till after near ultra violet spectrum for all three directions of light polarization. LiCl monolayer behaves transparently very close to LiF. This can be known as a result of the small size of Li, F and Cl atoms and also low thickness of nanostructures here. The reflection value hard reaches to 0.18% for other ranges and shows its transparency feature as mentioned above. The reflectivity percentages of LiBr monolayer in all three directions start around 2.5% and achieve to nearby 25% approximately. Hence, with increasing photon energy and decreasing wavelength after the visible range of energy, less photons pass
5
through the monolayer, and reflectivity percentages for structures with larger atoms get increasing trend. The imaginary part of the dielectric function is illustrated for LiF, LiCl and LiBr for three directions of polarization in Fig. 7. The imaginary part is related to the photon absorption in compounds, so investigating its threshold and summits helps us to know every compound's absorption in detail. In order to achieve this purpose, we need to identify every energy level in band structure for each monolayer. In Fig. 8 band structures of LiF and LiBr labelled in number and distinguished with different colors In order of better recognition also the atomic orbital of each atom that contribute in each energy level summarized in table below (see Table 3). This table not only demonstrates the importance of halogen orbitals in band structures, but also proves the ionic nature of these compounds with less contribution of Li atom. In Fig. 9 the absorption spectra of these monolayers are shown for three directions of polarization for these three monolayers. As it can be seen, these structures' thresholds are located in those energies that are equal to energy gap values. Also the main peaks in the spectra of LiF, LiCl and LiBr labelled with P1, P2 and P3 respectively. In other to achieve a better vision about the absorption mechanism and identify the specific energy level that involves in electron transitions in each specific range of energy for incident photon, we have provided three tables (see Tables 4–6). The most probable transitions for each region (P1, P2 and P3) with some sub-peaks have been demonstrated in these tables (Tables 4–6). The numbers that utilized here are based on those numbers are illustrated in Fig. 8 as we mentioned above. Also the contribution of each orbital in each energy level brought about here, is specified in Table 3. As a result of these data, we can conclude that halogen atoms allocate the most contribution of the valence band while lithium energy levels go to the conduction band because of empty level in p orbital. These compounds form ionic band gap so we see localized orbitals and less contribution of lithium orbitals in valence band. As it has been mentioned above,
Fig. 6. Reflectivity spectrum versus incident photon energy for all compounds in each direction.
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Fig. 7. Imaginary part of complex dielectric function versus incident photon energy for all compounds in each direction.
Fig. 8. Illustrates: different energy levels of valence and conduction bands for (a) LiF and (b) LiBr. For LiCl and LiBr, the electronic band structure was similar.
these compounds indicate anisotropic feature in every direction. So it is obvious that these electron transitions occurred in a different manner for every direction. 4. Conclusion In this paper, we have studied the structural, electronic and optical properties of lithium halide monolayers (LiF, LiCl, LiBr) in
the framework of first principles study as implemented in WIEN2k code. These compounds show high ionic nature in bond as a result of having one of the most different in elements’ electronegativity specially in LiF. These compounds can form and stabilize in cubic shape unlike other famous graphene-like monolayers with the well-known member, graphene. The bond length in the x-direction and y-direction are different, also it leads to the electronic symmetry breaking of orbitals in plane. So we can call these
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M. Safari et al. / Physica E ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Table 3 The atomic orbitals that contributed in every level of energy in monolayer band structure. Lithium halides
1
2
px of halogen py of halogen pz of halogen s of halogen s of Lithium p of Lithium
n n
n
3
4
n
n n
5
6
n n
n
7
n
8
9
10
11
12
13
14
n
n n
n
7
Table 4 The atomic orbitals that contributed in every level of energy in monolayer band structure. Direction
LiF
P1
P2
P3
E∥x
3, 5, 6 → 7, 8
3, 4, 5, 6 → 9
3, 4, 5, 6 → 10, 11 4, 5, 6 → 12
E∥y
4, 5, 6 → 7 3, 5, 6 → 8
3, 5, 6 → 9 1→8
1, 4, 6 → 10 5, 6 → 11, 12 6 → 13
E∥z
1, 2, 3, 4, 5, 6 → 7 2, 3, 4 → 8
2, 3 → 9 3, 4 → 10
1 → 10, 11 5 → 12
n n n
n
n n
n
compounds as anisotropic structures. The type of band structure changes from direct type in the bulk samples to indirect one in under-discussion monolayers. These compounds are extremely transparent as a result of wide band gap and optical properties demonstrate their interaction with photons. As it was mentioned, the symmetry breaking as a result of forming these monolayers gives rise to different behaviors in x-, y- and z-directions of electric field polarization. Also, their electron transitions are indicated for achieving a better vision about the absorption mechanism in these kinds of monolayers. The most probable transitions for each region (P1, P2 and P3) with some sub-peaks is demonstrated as well as the contribution of each orbital in each energy level. As a result of these data, we can conclude that halogen atoms allocate the most contribution of the valence band while lithium energy levels go to the conduction band because of existing unoccupied p orbital for lithium. These compounds form ionic bonds, so we see localized orbitals and less contribution of lithium orbitals in the valence band. The indirect band gap is a result of growing up in the energy level of the pz orbital for halogen atoms with changing in dimension from the bulk phase to the 2D monolayer.
Table 5 The atomic orbitals that contributed in every level of energy in monolayer band structure. Direction
LiCl
P1
P2
P3
E∥x
5→7 5, 6 → 8
1, 2 → 8 3, 4, 5, 6 → 9
2, 3, 5, 6 → 11 4, 5, 6 → 12, 13 2, 3, 4, 5 → 10
E∥y
4, 5, 6 → 7 5, 6 → 8
1, 3, 4 → 8 3, 4, 5, 6 → 9 3→7
1, 2, 4 → 10 6 → 11, 12, 13, 14 4, 5 → 13
E∥z
2, 3, 4, 6 → 7 3, 4 → 8
2→8 4→9
1, 2 → 9 1, 4 → 10 3, 4 → 11 6 → 12, 13
Fig. 9. Optical absorption spectrum versus incident photon energy for all compounds in each direction.
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Table 6 The atomic orbitals that contributed in every level of energy in monolayer band structure. Direction
LiBr
P1
P2
P3
E∥x
4, 5 → 7
5→7 6→8 4→9
2 → 10 4 → 12 6 → 13
E∥y
4, 6 → 7
3, 5 → 7 3, 4, 5, 6 → 8 4→9
5 → 11, 12 6 → 13 1 → 10
E∥z
4, 5, 6 → 7
3 → 7, 8 4 → 8, 13, 14
3, 4 → 13 4, 6 → 14
Acknowledgment Computing resources used in this work have been provided by the Nano Fanavaran Bistoon, High Performance and Grid Computing Center, Kermanshah, Iran.
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Lithium halide monolayers: Structural, electronic and optical properties by first principles study Mandana Safari a, Pegah Maskaneh a, Atousa Dashti Moghadam a, Jaafar Jalilian b,n a b
Physics Department, Razi University, Kermanshah, Iran Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, P.O. Box 67149-67346, Kermanshah, Iran
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
Structural optimizing represents that unlike graphene-like structures, the cubic face structure is more favorable for alkali halide 2D structures. Nonbonding electron pairs cause a planar buckling for all compounds. Electronic calculations show that all compounds have an indirect energy gap. All compounds are optically transparent in the visible spectrum range.
art ic l e i nf o
a b s t r a c t
Article history: Received 12 October 2015 Received in revised form 27 December 2015 Accepted 19 January 2016
Using first principle study, we investigate the structural, electronic and optical properties of lithium halide monolayers (LiF, LiCl, LiBr). In contrast to graphene and other graphene-like structures that form hexagonal rings in plane, these compounds can form and stabilize in cubic shape interestingly. The type of band structure in these insulators is identified as indirect type and ionic nature of their bonds are illustrated as well. The optical properties demonstrate extremely transparent feature for them as a result of wide band gap in the visible range; also their electron transitions are indicated for achieving a better vision on the absorption mechanism in these kinds of monolayers. & 2016 Elsevier B.V. All rights reserved.
Keywords: Alkali halide Optical transition 2D monolayer Density functional theory
1. Introduction Alkali halides known as ionic compounds are characterized by their highly crystalline nature, high melting points and strong miscibility in polar media [1,2]. These materials can be considered as prototype insulator materials with great technological importance [3–5]. For many years, thermodynamic, elastic, structural and electronic properties of them have been investigated comprehensively. The effect of defects in these compounds has been presented too [6,7]. Also band structure investigations reveal these compounds are recognized as wide-gap insulators that n
Corresponding author. E-mail address: [email protected] (J. Jalilian).
demonstrate their optical transparency in the visible region of the electromagnetic spectrum [8]. This transparency feature, especially in LiF, in high compression makes this material a suitable choice for versatile window material through which to carry out wave profile measurements for shock compression experiments [9]. These compounds generally crystallize in both B1 (NaCl-type) and B2 (CsCl-type) structures. Their phase transitions have been classified as pressure and temperature dependence in first-order kind [7,10]. On the other hand, studying the role of electronic correlation on the boundary between localized and delocalized electronic states become possible through energy level investigation, due to their electronic states kind. Their applications for a range of optical applications are significant besides the interaction mechanisms between the electronic and geometric structure can
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be interestingly demonstrated [11]. LiF, LiCl and LiBr, known as Li halides, have been investigated in bulk phase theoretically [12–14] and experimentally [15,16]. As a result, the energy difference between the top of the valence band and the outermost core levels in LiF, LiCl and LiBr was determined by the electron spectroscopy for chemical analysis (ESCA) technique in the bulk phase. The energy differences obtained between the top of the valence band and the Li 1 s line in LiF, LiCl and LiBr were 49.8 eV, 53.2 eV and 54.1 eV respectively. These results agree rather well with the predictions of the point charge model, including corrections for polarization effects [15]. Since there has been increased attention to the properties of thin films and nanostructures, the great theoretical interest was paid to their nanostructures due to interest in the application of alkali halide systems as sensitive photostimulable storage materials [17]. Low dimensional alkali halide nanocrystals of LiF were investigated structurally and energetically by Bichoutskaia and Pyper in 2006 [18]. The required energy for forming one plane of new contacts in the LiF (2 2 n) chain is essentially constant and independent of the lengths of the two fragments being joined. Also the simulation performed with density functional theory (DFT) and coupled-cluster (CCSD) calculations on the series of (LiF)n ¼ 2,36 neutral clusters showed that nanotube structures with hexagonal and octagonal transverse cross-sections are stabile, and this stability is similar to the typical cubic form of large LiF crystals [19]. These investigations showed that these Li halides in cubic shape have been formed regardless to cross section of nanostructures. Here we focus on Li halide monolayers in structural, electronic and optical aspects to achieve a point of view in these kinds of nanostructures for the first time. Our preferable structure for investigation is cubic structure agreed with previous researches mentioned above.
2. Computational details In order to calculate our results, the first principles study in the framework of the density functional theory [20,21] as implemented in WIEN2k code [22] is used. To achieve reliable results we have performed full potential augmented plane waves plus local orbitals (FPAPW þlo). The generalized gradient approximation energy functional formulated by Perdew–Burke–Ernzerhof (GGA-PBE) [23] has employed as the approximation method for the exchange–correlation energy functional. The cut-off parameter known as RMT Kmax that it has been equaled 7, the value of this parameter demonstrates our accuracy and efficiency and Gmax ¼14 Ry1/2 as another parameter of charge calculations. The optical data of this paper has been performed by using the random phase approximation (RPA) method [24] to gain imaginary part of dielectric function and Kramers–Kronig relations for real part. The number of k-points is 5000 in the first Brillouin zone that leads to 24 22 8 as a good mesh for calculations here.
3. Results and discussion In order to explain electronic and optical characteristics of LiF, LiCl and LiBr monolayers, this part is required to achieve the best configuration to find balance situation in structures and reach acceptable results. 3.1. Structural investigation In order to find stable configurations of these monolayers energetically, we have optimized volume as implemented in Birch– Murnaghan equation of state [25] and relax these structures to get closer to minimum force possible exerting to each atom. The
Fig. 1. Lithium halide monolayer: (a) length bond illustrated for x- and y-directions and different angles created by structural optimizing, (b) top-view of monolayer and (c) first Brillouin zone with k-path for energy band structure calculations.
structural optimization results express that the cubic face for these ionic compounds is more preferable than the other configurations, and these results demonstrate good agreement with previous calculations for alkali halide nanotubes [19]. In Fig. 1a, stable structure of these monolayers as a representation of Li halides has been illustrated for better understanding. Fig. 1b is the top-view of these monolayers. The first Brillouin zone for these kinds of structures is shown as well (see Fig. 1c). The optimized parameters, i.e. bond length, buckling factor and angle between each three close atoms in each direction are presented in Table 1 for LiBr, LiCl and LiF. These parameters are identified in Fig. 1 to find the distinction of these variables in these monolayers. It is obvious that these two dimensional structures are not flat and get planar buckling after relaxation set. This phenomenon occurs as a result of the existence of nonbonding electrons in halogen atoms. In the other words, after bond formation, three pairs of nonbonding electrons have broadened in a side and push the other bonding orbital to another side, so this monolayer get buckled naturally. When we investigate a 2D monolayer, these nonbonding electrons located on opposite sides in adjacent atoms to achieve structural stability. Thus buckling formation in these monolayer is a natural reformation. 3.2. Electronic properties In order to show the bond nature of these structures, electron Table 1 The measured angle of every three atoms in the x-direction (θ) and y-direction (ϕ), bond length of structure in the x-direction (ax), bond length of structure in the ydirection (ay) buckling factor for all of under discussion compounds. Compounds
θ (degr)
ϕ (degr)
ax (Å)
ay (Å)
Δ
LiF LiCl LiBr
136.15 132.10 132.16
160.60 151.41 147.01
1.88 2.41 2.57
1.92 2.51 2.70
0.70 0.97 1.04
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Fig. 2. 2D electron charge density distribution for (a) LiF monolayer, (b) LiCl monolayer and (c) LiBr monolayer; magnitude of the charge density shown by colors, blue color at top of the box show the smallest value and white color at bottom of the box is largest value.
density distribution of each compound is indicated in Fig. 2. The highest value of electronegativity in periodic table belongs to fluorine (3.98) and the lowest value of this parameter is for elements of the alkali group, so lithium (0.98) [26] as a first member of this group play donor role in bond. Fluorine as an acceptor almost creates an ionic bond, this kind of bond can be observed obviously in Fig. 3 and localization of electron density distribution around each atom proves this claim. With regard to lower electronegativity of Cl (3.16) and Br (2.96) [26] in comparison with F, it could be inferred that Li–F bond is stronger than the bond of Li–Cl (Li–Br). This provides more strong s bond in LiF than LiCl (LiBr) that cause to more stable s bond in LiF energetically, so the energy level of s energy band goes down more than corresponding bands in LiCl and LiBr, thus we see the energy band gap gets decrease with changing halogen atoms to lower electronegativity. In order to show these different band gaps, total and partial electron density of states (DOS) are investigated in Fig. 3. As it can be seen in Fig. 3a, all of these compounds are insulators. Halogen atoms have a significant role in forming energy levels below the Fermi level (see Fig. 3b) for all of these structures, while both halogen and corresponding lithium in each compound play roles in energy levels that are located above the Fermi level (see Fig. 3c). On the other hand, the density of energy states around the Fermi level in the valence band for Br, Cl and F atoms is more than that of Li atom. It is due to higher electronegativity in halogen atoms and more electrons belonged halogens which are engaged in bond. As it was seen, these compounds are known as members of the insulator group of materials because of their wide band gap. To present a better view, we focus on their band structure here (see Fig. 4). The gap values are identified as 6.693, 5.873 and 5.126 eV for LiBr, LiCl and LiF, respectively. As it is clear from these graphs, these structures have wide indirect band gaps. The conduction band minimum is in the k2 direction while the valence band
3
Fig. 3. Density of states for LiF, LiCl and LiBr monolayers versus energy (eV); (a) total density of states for all compounds, (b) total density of states for halogen atoms and (c) total density of states for lithium atoms in different compounds.
maximum happens in a direction between k2 and k3. This band type variation from direct type in the bulk phase to indirect type in monolayer structures maybe lie in the interaction of pz orbital from every atom in the bonds. However, this orbital locates perpendicular to monolayer plate, the buckling phenomenon has been engaged it with other orbital trans-shapes and it has formed an indirect energy band gap subsequently. 3.3. Optical properties Optical properties of solids are determined by the response of their electrons to the time-dependent electromagnetic interference caused by the incident beam. In order to investigate the optical properties of the structure, optical spectra such as the real and imaginary parts of the dielectric tensor, the absorption and reflectivity functions are calculated. The complex dielectric function is dependent on the frequency and is directly associated with the electronic band structure of the crystals. This function consists of two parts that called imaginary and real parts of dielectric function. According to the appropriate elements in the transition matrix, the imaginary part of the dielectric function is defined as below: αβ εimaginary (ω) =
4π e 2 m2ω2
∑ ∫ dk〈ck |pα |vk 〉〈vk |p β |ck 〉δ (Eck − Evk − ω). c, v
(1)
These two parts are related to each other through the Kramers– Kronig relations. The real part of the dielectric function is obtained correspondingly:
εreαβ (ω) = δαβ +
2 Pr . π
∫0
∞
αβ ω′εim (ω′)
ω′ 2 − ω2
dω′,
(2)
where Pr. denotes the principal value [27]. Because of the importance of these functions in the optical properties study, the real part of the dielectric function versus the incident photon energy is calculated for three directions of electric field polarization (the electric field is polarized parallel to the alkali halides monolayer
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Fig. 4. Energy band structure for (a) LiF, (b) LiCl and (c) LiBr compounds. The selected k-vectors were presented in Fig. 1.
Fig. 5. Real part of complex dielectric function versus incident photon energy for all compounds in each direction.
(x-axis and y-axis) and perpendicular to the alkali halides monolayer (z-axis)) for LiBr, LiCl and LiF and it is shown in Fig. 5. As it can be seen in Fig. 5, the real part of the dielectric function is identified for LiF, LiCl and LiBr. The values of dielectric function for zero energy called static dielectric constant have been obtained for LiBr, LiCl and LiF and have been summarized in Table 2. As it is clear from this table, static dielectric constant [28] values have not significant variations and also in high energy converge to the same value. This value famed as optical dielectric constant comes close to 1 in amount as one can see in figures (see Fig. 5). In all of three directions of E field ( E∥x, E∥x and E∥z ) the summits get decrease in intensity and happen at lower energies
Table 2 The value of static dielectric constant for all of 3 compounds, for three directions of E field polarization. Electric field direction
LiF
LiCl
LiBr
Parallel direction ( E∥x ) Parallel direction ( E∥y ) Perpendicular direction ( E∥z )
1.04 1.04 1.04
1.07 1.07 1.06
1.89 1.92 1.74
when atomic size of halogens increases simultaneously. For LiCl and LiBr structures, we can see sharp falls that happen in every direction of polarization. This despite the fact that sharp fall does
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not occur for LiF practically and the spectrum gets decrease over a range of energy and finally achieves a minimal value at 18 eV energy. It is obvious that photons of the high energy region involve with this structure in a wide range of energy. Moreover, some plasmonic frequencies (minimal value of the function is corresponding to plasmonic frequency in compounds) can be recognized as a consequence. However, this trend keeps going in LiCl monolayer, the range of reaction region is smaller and the number of plasmonic frequencies is lower too. These quantities for LiBr go down even more. These sharp falls in LiCl, LiBr and the decreasing trend in LiF are consequences of subjecting on this sort of photons to create π and π + σ plasmons. The distinguishable behavior of parts of the dielectric function, here real part (see Fig. 5), in each direction of polarization denotes to anisotropic behavior of these structures in these three directions. These different behavior will be understood for all spectra that comes as follow too. The percentages of reflectivity spectra in terms of energy (eV) in all three directions of polarization of E field (parallel, x and y axis, and perpendicular to the monolayer plane) are illustrated in Fig. 6 for these three compounds. Reflectivity spectrum behaves corresponding to the real part of the dielectric function. Its maximum value as a peak happens where the sharp fall does. It is obvious that all of these compounds have a neglectable per cent of the reflection spectrum in the lower range of energy as well as the visible spectrum. LiF reflectivity starts with less than 0.02% per cent and keep remaining around till after near ultra violet spectrum for all three directions of light polarization. LiCl monolayer behaves transparently very close to LiF. This can be known as a result of the small size of Li, F and Cl atoms and also low thickness of nanostructures here. The reflection value hard reaches to 0.18% for other ranges and shows its transparency feature as mentioned above. The reflectivity percentages of LiBr monolayer in all three directions start around 2.5% and achieve to nearby 25% approximately. Hence, with increasing photon energy and decreasing wavelength after the visible range of energy, less photons pass
5
through the monolayer, and reflectivity percentages for structures with larger atoms get increasing trend. The imaginary part of the dielectric function is illustrated for LiF, LiCl and LiBr for three directions of polarization in Fig. 7. The imaginary part is related to the photon absorption in compounds, so investigating its threshold and summits helps us to know every compound's absorption in detail. In order to achieve this purpose, we need to identify every energy level in band structure for each monolayer. In Fig. 8 band structures of LiF and LiBr labelled in number and distinguished with different colors In order of better recognition also the atomic orbital of each atom that contribute in each energy level summarized in table below (see Table 3). This table not only demonstrates the importance of halogen orbitals in band structures, but also proves the ionic nature of these compounds with less contribution of Li atom. In Fig. 9 the absorption spectra of these monolayers are shown for three directions of polarization for these three monolayers. As it can be seen, these structures' thresholds are located in those energies that are equal to energy gap values. Also the main peaks in the spectra of LiF, LiCl and LiBr labelled with P1, P2 and P3 respectively. In other to achieve a better vision about the absorption mechanism and identify the specific energy level that involves in electron transitions in each specific range of energy for incident photon, we have provided three tables (see Tables 4–6). The most probable transitions for each region (P1, P2 and P3) with some sub-peaks have been demonstrated in these tables (Tables 4–6). The numbers that utilized here are based on those numbers are illustrated in Fig. 8 as we mentioned above. Also the contribution of each orbital in each energy level brought about here, is specified in Table 3. As a result of these data, we can conclude that halogen atoms allocate the most contribution of the valence band while lithium energy levels go to the conduction band because of empty level in p orbital. These compounds form ionic band gap so we see localized orbitals and less contribution of lithium orbitals in valence band. As it has been mentioned above,
Fig. 6. Reflectivity spectrum versus incident photon energy for all compounds in each direction.
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Fig. 7. Imaginary part of complex dielectric function versus incident photon energy for all compounds in each direction.
Fig. 8. Illustrates: different energy levels of valence and conduction bands for (a) LiF and (b) LiBr. For LiCl and LiBr, the electronic band structure was similar.
these compounds indicate anisotropic feature in every direction. So it is obvious that these electron transitions occurred in a different manner for every direction. 4. Conclusion In this paper, we have studied the structural, electronic and optical properties of lithium halide monolayers (LiF, LiCl, LiBr) in
the framework of first principles study as implemented in WIEN2k code. These compounds show high ionic nature in bond as a result of having one of the most different in elements’ electronegativity specially in LiF. These compounds can form and stabilize in cubic shape unlike other famous graphene-like monolayers with the well-known member, graphene. The bond length in the x-direction and y-direction are different, also it leads to the electronic symmetry breaking of orbitals in plane. So we can call these
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M. Safari et al. / Physica E ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Table 3 The atomic orbitals that contributed in every level of energy in monolayer band structure. Lithium halides
1
2
px of halogen py of halogen pz of halogen s of halogen s of Lithium p of Lithium
n n
n
3
4
n
n n
5
6
n n
n
7
n
8
9
10
11
12
13
14
n
n n
n
7
Table 4 The atomic orbitals that contributed in every level of energy in monolayer band structure. Direction
LiF
P1
P2
P3
E∥x
3, 5, 6 → 7, 8
3, 4, 5, 6 → 9
3, 4, 5, 6 → 10, 11 4, 5, 6 → 12
E∥y
4, 5, 6 → 7 3, 5, 6 → 8
3, 5, 6 → 9 1→8
1, 4, 6 → 10 5, 6 → 11, 12 6 → 13
E∥z
1, 2, 3, 4, 5, 6 → 7 2, 3, 4 → 8
2, 3 → 9 3, 4 → 10
1 → 10, 11 5 → 12
n n n
n
n n
n
compounds as anisotropic structures. The type of band structure changes from direct type in the bulk samples to indirect one in under-discussion monolayers. These compounds are extremely transparent as a result of wide band gap and optical properties demonstrate their interaction with photons. As it was mentioned, the symmetry breaking as a result of forming these monolayers gives rise to different behaviors in x-, y- and z-directions of electric field polarization. Also, their electron transitions are indicated for achieving a better vision about the absorption mechanism in these kinds of monolayers. The most probable transitions for each region (P1, P2 and P3) with some sub-peaks is demonstrated as well as the contribution of each orbital in each energy level. As a result of these data, we can conclude that halogen atoms allocate the most contribution of the valence band while lithium energy levels go to the conduction band because of existing unoccupied p orbital for lithium. These compounds form ionic bonds, so we see localized orbitals and less contribution of lithium orbitals in the valence band. The indirect band gap is a result of growing up in the energy level of the pz orbital for halogen atoms with changing in dimension from the bulk phase to the 2D monolayer.
Table 5 The atomic orbitals that contributed in every level of energy in monolayer band structure. Direction
LiCl
P1
P2
P3
E∥x
5→7 5, 6 → 8
1, 2 → 8 3, 4, 5, 6 → 9
2, 3, 5, 6 → 11 4, 5, 6 → 12, 13 2, 3, 4, 5 → 10
E∥y
4, 5, 6 → 7 5, 6 → 8
1, 3, 4 → 8 3, 4, 5, 6 → 9 3→7
1, 2, 4 → 10 6 → 11, 12, 13, 14 4, 5 → 13
E∥z
2, 3, 4, 6 → 7 3, 4 → 8
2→8 4→9
1, 2 → 9 1, 4 → 10 3, 4 → 11 6 → 12, 13
Fig. 9. Optical absorption spectrum versus incident photon energy for all compounds in each direction.
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Table 6 The atomic orbitals that contributed in every level of energy in monolayer band structure. Direction
LiBr
P1
P2
P3
E∥x
4, 5 → 7
5→7 6→8 4→9
2 → 10 4 → 12 6 → 13
E∥y
4, 6 → 7
3, 5 → 7 3, 4, 5, 6 → 8 4→9
5 → 11, 12 6 → 13 1 → 10
E∥z
4, 5, 6 → 7
3 → 7, 8 4 → 8, 13, 14
3, 4 → 13 4, 6 → 14
Acknowledgment Computing resources used in this work have been provided by the Nano Fanavaran Bistoon, High Performance and Grid Computing Center, Kermanshah, Iran.
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