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Ab-initio study of electronic and phononic properties of bulk TiC and its narrow nanowires by density functional theory
Journal Pre-proof Ab-initio study of electronic and phononic properties of bulk TiC and its narrow nanowires by density functional theory
Peiman Amiri, Hamdollah Salehi, Yasamin Loveimi Motlagh PII:
S0921-4526(19)30661-1
DOI:
https://doi.org/10.1016/j.physb.2019.411761
Reference:
PHYSB 411761
To appear in:
Physica B: Physics of Condensed Matter
Received Date:
27 October 2018
Accepted Date:
07 October 2019
Please cite this article as: Peiman Amiri, Hamdollah Salehi, Yasamin Loveimi Motlagh, Ab-initio study of electronic and phononic properties of bulk TiC and its narrow nanowires by density functional theory, Physica B: Physics of Condensed Matter (2019), https://doi.org/10.1016/j.physb. 2019.411761
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Journal Pre-proof
Ab-initio study of electronic and phononic properties of bulk TiC and its narrow nanowires by density functional theory Peiman Amiri, Hamdollah Salehi, 1* and Yasamin Loveimi Motlagh
Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran
Abstract In this article, electronic and phononic properties of TiC bulk and its narrow nanowires are calculated by using Quantum-Espresso/PWSCF computational package in the framework of density functional theory. According to the results, this compound shows a metallic behavior in the bulk structure, but for the smallest diameters of the nanowires, against the bulk, a semiconducting behavior was observed. This deviation becomes negligible at higher diameters. In the following, phonon calculations have been performed on bulk and the first diameter of nanowires with a square cross-section. By calculating the phonon modes of Brillouin zone, the thermodynamic properties of both mentioned structures such as vibrational free energy, the heat capacity
Cv,
Debye temperature , entropy and their variations against temperature T have been obtained
through the quasi-harmonic Debye model. Calculations show that heat capacity behavior of the vibrational lattice is consistent with Debye model. Moreover, the temperature dependence of a Debye temperature indicates that the Debye stiffness in a bulk is greater than the nanowire structure. Keywords: Band structure, Debye temperature, Density of states, Entropy, Heat capacity, Pseudo-potential.
1. Introduction Titanium Carbide (TiC) in a bulk state is a family of metal carbides [1]. The stable structure of this compound is the rock-salt structure with the Fm3m space group. The bulk of TiC structure is highly resistant to abrasion and has a high melting point [2-8]. Among nanoscale materials, much attention has been paid to one-dimensional nanomaterial’s, including nanowires. Which come from a wider set of mineral compounds with different types of 1*Corresponding
author E-mail address: [email protected]
1
Journal Pre-proof crystalline structures, for example, carbides, nitrides and oxides [9]. Carbides are used as nanoscale materials due to their unique compounds with thermal and mechanical properties, high chemical resistance and electromagnetic radiation resistance [10]. The electrical behavior of the bulk structure is different from that of the nanowire. Changing the dimension of a system usually results in new physical properties. Nanostructures and diameter variations have been considerably investigated in recent years [11]. One of the issues which have attracted the attention of many scientists is the study of the thermal and phononic properties of nanostructures, and one of the important features is the phonon frequency spectrum survey. In recent years, many studies have been carried out in this field, and significant results have been obtained on the vibrations of the lattice and the phonon spectra of nanostructures [12]. In 1975, Nickel et al. presented the results of self-consistent calculations (APW) band structuring of some of the TMX-type compounds, including titanium carbide [13]. In 1995, Ahuja et al. obtained the structural, elastic and high-pressure properties of titanium carbide bulk state using FP-LAPW method, and discovered the stability of this compound in salt type [14]. In 2006, Enyashin and his colleagues developed an electronic structure of (100) nanowires of titanium carbide with square cross sections using the DFTB method. In this study, the titanium carbide nanowires are simulated along X-axis with three different diameters. A gap of 0.8 electron volts at the Γ point for the first diameter was reported [9]. In 2016, Jaffari and his colleagues studied the role of diameter effect on the optoelectrical properties of titanium carbide nanowire by using the FP-LAPW method. In this study, the first diameter of this nanowire was announced as a semiconductor compound [11]. In the same year, Xinhui Xia and his colleagues studied the wide-temperature electrochemical energy storage of the Single-Crystalline, Metallic TiC nanowires. In this work, they reported a facile CVD method for one-step synthesis of TiC nanowire arrays directly on flexible carbon cloth and demonstrated their excellent wide-temperature ultrafast EDLC performance [15]. In 2017, by using first-principles calculations based on density functional theory, Victor V. Ilyasov et.al studied the adsorption energy of the oxygen atom, the local atomic structure, and the electronic properties of the defective (001), (111) and (110) surfaces of titanium carbide. They indicated that the adsorption of oxygen at different binding positions leads to a substantial reconstruction of the local atomic structures and their related energy band spectra [16]. In the same year, they studied the local atomic structure, thermodynamics, and electronic properties of the non-stoichiometric W/TixCy(111) systems with different surface reconstructions in comparison to the stoichiometric W/TiC(111) systems using first-principles calculations based on DFT. Their results show that reconstruction mechanisms due to the nucleation of tungsten on the polar TixCy(111) surface are responsible for the physical properties variation of the material [17]. To the best of our knowledge, there is no systematic study on the phononic and thermodynamic properties and temperature dependence of Debye stiffness of TiC(001) nanowires. Comparison of these results with corresponding bulk values is not discussed.
2. Computational Methods
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Journal Pre-proof All computations for bulk and nanowires titanium carbide were performed in the framework of the density functional theory (DFT) using the Quantum-Espresso computational package [18]. In this computational package, the Kohn-Sham single-particle equations were solved using the pseudo-potential method and the valance electronic wave functions were expanded in a plane-wave basis set. After solving Kohn-Sham equations, the structural and electron properties were determined by calculating physical parameters such as total energy, forces, the density of states and band structure diagram. The pseudo-potentials used in this research for titanium and carbon atoms were made by the Norm Conserving method and its exchangecorrelation function is PBE-GGA. After conducting adequate convergence tests, an energy cutoff of 90 Ry was chosen for plane-wave expansion. The bulk and nanowire Brillouin zone integrations were performed by using Monkhorst-Pack meshes of 10 10 10 and 1110 k points, respectively, along with the Methfessel-Paxton method of smearing with a broadening parameter of 0.001 Ry. All structures were optimized to achieve the minimum energy by accurate relaxation of the atomic position down to forces less than 1 mRy/bohr. In comparison to normal LDA and GGA, in order to improve the exchange and correlation parts of our calculations, HSE functional we applied for related nanostructure computations. In this study, the chosen structural phase of bulk TiC is NaCl, which is the most stable phase of this compound and various nanowire structures with [001] crystallographic direction growth are composed of the bulk by a top-down method in order to preserve the symmetry of the corresponding bulk structure [19]. The free nanowire structures were simulated by using the supercell approach with a vacuum thickness of about 12 Å. The nanowires cross sections are sketched in the X-Y plane of fig. 1. The vacuum is applied along with the two perpendicular directions of this plane.
3
Journal Pre-proof In fig. 1, the structure of titanium carbide in the bulk and nanowire structure is presented by Xcrysden software, respectively.
fig.1. (a) Cross sections of the first (smallest square), second (intermediate square), and third (largest square) sizes of square nanowires studied in this paper. The larger (blue) and smaller (red) circles stand for Ti and C atoms, respectively. The crystalline structure of the titanium carbide compound in (b) bulk and (c) nanowire
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Journal Pre-proof 3. Results of bulk 3.1 Band Structure Fig 2 shows the band structure of the titanium carbide compound with the GGA approximation in the bulk structure. In order to compare the two approximations of GGA and HSE, the band structure of the bulk was also investigated with the HSE approximation as shown in fig. 3. In both diagrams, Fermi energy is set on zero. In both approximations, there is a contribution of the electron density around the Fermi level that cuts off the Fermi surface; therefore no gaps were observed. Thus, it can be concluded that the HSE approximation has no significant effect on the metallic-like behavior of the compound, and the metallic-like property of the compound remains unchanged, which is consistent with other existing results [20]. In both figures, the lowest energy band is related to the orbital 2s of a carbon atom.
(a)
(b)
Fig. 2. The band structure of the titanium carbide compound in the bulk structure with approximation (a) GGA, (b) HSE.
5
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Density of states
The total density of states (DOS) of the electrons in the range of -13 to 8 eV is plotted for the titanium carbide with an approximation of GGA in fig. 3, and the Fermi energy is set on zero. As seen in the figure, there is a contribution of the electron density around the Fermi level, therefore that gap is covered and this leads to a metallic-like behavior for the TiC in a bulk structure. This result is consistent with other existing results as well [20]. .
Fig. 3. Total density of states of titanium carbide within the bulk structure.
3.3 Phononic properties In the following, the phonon dispersion curve will be studied for TiC compound in bulk structure. Fig.4 shows the phonon dispersion curves along with the density of phonon states calculated for the bulk structure of the TiC compound. That the branches of the dispersion diagram of this compound are expected to be equal to 6 in the case of having two atoms in their primitive cell of NaCl structure. As seen in fig. 4, there are 6 branches in the phonon dispersion curve. The three lower branches at Γ point are acoustic branches, among which, the two lower branches are transverse acoustic modes (TA) and the next is the longitudinal
6
Journal Pre-proof acoustic mode (LA), which has the highest frequency in comparison to the two lower branches. The transverse acoustic branches (TA) at the Γ point are degenerate. This compound has a frequency gap in the frequency range of 445 to 520 cm-1. The waves in this region are severely weakened, and light cannot cross the crystal and is fully reflected in this range. The three upper branches at higher frequencies are transverse optical modes (TO) and longitudinal optical modes (LO). According to the phonon dispersion diagram, transverse acoustic and optical modes on the X path are degenerate, and degeneration behavior was removed and reappeared in the path of L . The phonon density of states versus frequency is depicted in fig. 4. According to this figure, the dispersion curve has no negative imaginary frequencies; therefore, dynamically, the titanium carbide compound in the rock salt structure is stable. One of the differences between optical and acoustic modes is that the dispersion is much weaker in the case of optical modes which can be observed from phonon density of states curve as well.
Fig. 4. The phonon dispersion curve (left) along with the density of phonon states (right) for the bulk structure of TiC.
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Journal Pre-proof Fig. 5 represents the projected phonon density of states of Ti and C atoms in the bulk structure of TiC. It is clear that the projected phonon density of states of Ti and C atoms dominate the low and high-frequency domain, respectively. This results from the atomic mass difference that leads to a separation of acoustic and optical phonon branches.
Fig. 5. Projected phonon density of states of C (brown solid line) and Ti (blue dashed line) atoms for the bulk TiC structure.
4. Results of Nanowires 4.1 Band Structure Normal GGA approximation leads to a gap of about 0.1 eV for the first radius of the defined nanowires. In comparison to the other theoretical work [9], this calculated value seems to be low. In order to improve the semiconducting behavior of the nanostructures, the HSE approximation was applied to the calculation of electronic structures of the nanowires. Fig. 6 shows the band structure diagrams for all three nanowires diameter with the HSE approximation along the growth direction of nanowires. In this figure, the Fermi energy is set on zero. According to the band structure diagram, the lowest energy band under Fermi level which is between -12 and -10 eV, is related to orbital 2s of a carbon atom. In the lowest 8
Journal Pre-proof energies of a valence band, there is a gap of about 4 to 4.5 eV. Other energy bands under Fermi level are due to the hybrid bond between 3d orbital of the titanium atom and the 2p orbital of a carbon atom that is responsible for the main bond between the titanium and carbon atoms. According to the results obtained for the first and second diameters of the nanowire, a direct gap of about 0.8 eV and an indirect gap of about 0.1 eV, respectively, are observed. The gaps decrease by increasing the diameter due to the reduction of surface effects and quantum limitations so that for the third diameter of the nanowire, the gap is covered and the behavior of the nanostructure is similar to that of related bulk structure.
(a)
(b)
(c)
Fig. 6. Calculated band structure of the first (a), second (b) and third (c) diameter of the nanowires.
4.2 Density of States As mentioned in the calculations part, nanowires of titanium carbide compound are simulated in three different diameters that are elongated in a z crystallographic direction. Fig. 7 shows the total density of states for all three diameters of titanium carbide nanowires. As seen in the figure, the gap of the first diameter is about 0.8 eV. By increasing the diameter of the
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Journal Pre-proof nanowires, the calculated gaps will reduce towards corresponding bulk value. Therefore, for higher diameters, the gap suddenly vanishes. In contrast to a bulk behavior, the first and second diameters have a semiconductor behavior which contributes to quantum limitations and surface effects which are most important for lower radii. In the third diameter, the gap is covered and the nanowire shows a metallic behavior as in the corresponding bulk compound.
(a)
(b)
(C) Fig. 7. Total density of states of (a) the first; (b) the second and (c) the third diameter of TiC nanowires.
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Journal Pre-proof 4.3 Phononic properties Due to computational limitations, phonon calculations were performed only for the first diameter of the nanowire. Fig 8 shows a dispersion curve and corresponding phonon density of states of the above-mentioned nanowire. In the phonon dispersion curve, there are 54 branches which are related to the presence of 18 atoms in the cell. The top 51 branches are optical modes, and the other three lower branches are acoustic modes. However, due to the degeneracy phenomena, in fig. 8 only two acoustic branches are visible. Moreover, when the maximum phonon frequencies are limited to about 150 cm-1, the frequency oscillations of the acoustic branches decrease and the number of optical branches grows, therefore the group velocity and the slope of the acoustic branches decrease and the thermal conductivity decreases further.
Fig. 8. Phonon dispersion curve (left) along with the phonon density of states (right) for the first diameter of the nanowire.
5. The study of thermodynamic properties in a bulk and a nanowire structure 11
Journal Pre-proof The quasi-harmonic approximation (QHA) is applied to obtain the thermodynamic functions including vibrational free energy, heat capacity, Debye temperature and vibrational entropy [21]. In the following, the results will be presented for vibrational free energy, thermal capacity, Debye temperature and vibrational entropy in the studied bulk and nanowire structures.
5.1 Vibrational free energy The free energy is made of two parts: F T , V E V FPh T , V , where E V denotes the ground state energy and FPh T , V is the phonon free energy at a given unit cell volume
V . The phonon free energy can be explained by the following equation [22]: ∞
[
𝐹𝑃ℎ(𝑇,𝑉) = 𝑘𝐵𝑇∫0 g(𝜔)ln 2sinh
( )]𝑑𝜔 . ℏ𝜔 2𝑘𝐵𝑇
(1)
Where V shows the volume dependent phonon frequencies and g stands for the phonon DOS. The vibrational free energy calculated at zero temperature and zero pressure for the titanium carbide compound in both bulk and nanowire structures are listed in Table (1), and their variation diagrams as a temperature parameter are shown in fig. 9a.
Table. 1. Vibrational free energy values for bulk and first diameter of the nanowire structures.
Type of structure
Free energy (Ry / cell)
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0.011
The first diameter of the
0.010
nanowire
According to fig. 9a, free energies of both bulk and nanowire structure decrease at 280 K and 230 K, respectively; and at 1000 K, they reach -0.02 Ry/cell and -0.24 Ry/cell for the bulk and nanowire structure, respectively.
Fig. 9. The Comparison of vibrational free energy per atom (a), the heat capacity per atom (b), Debye temperature per atom (c) and diagram of entropy changes (d) for the bulk (blue dash-point lines) and the first diameter of the nanowire (solid red lines) at various temperatures.
5.2 Heat capacity at constant volume At the constant volume, the heat capacity, by using calculated phonon DOS, is given by [21]: 13
Journal Pre-proof ћ𝜔
Cv =
∞ rkB∫0 dω
exp (k 𝑇) ћ𝜔 2 B 2 k B𝑇 ћ𝜔 [exp k 𝑇 ― 1]
( )
g(ω)
( )
.
(2)
B
Where r is the number of degrees of freedom in the unit cell, kB shows the Boltzmann constant, g denotes the phonon DOS of the unit cell, demonstrates the phonon frequency and refers to the Planck constant. The limits of Cv at 0 K and high temperature are 0 and rkB, respectively. The heat capacity per atom of the bulk and the nanowire structure of the titanium carbide are plotted against various temperatures in fig. 9b. Due to harmonic approximation effects at temperatures lower than 800 K , the heat capacity, Cv, is strongly dependent on the temperature. However, at higher temperatures, the harmonic approximation effect of the Debye model is extinguished, and Cv is converged to the classical Dulong-Petit limit. For the bulk structure with two atoms in a primitive cell, this classical limit is 6R; and in the nanowire structures with 18 atoms, this limit converges to a constant value of 54R. In order to compare the heat capacity of bulk to heat capacity of the nanowire, the calculated values of nanowire are divided by 9. The heat capacities at 300 K for the bulk and nanowire structures are 4.20R (J/mol.K) and 41.46R (J/mol.K), respectively. According to this figure, the heat capacity of the nanowire structure is greater than the bulk structure, which originates from the more vibrational states of the nanowire in comparison to the bulk structure.
5.3 Debye temperature The Debye temperature is an important physical parameter of solids, which defines a division line between quantum-mechanical and classical behavior of the vibrational lattice
14
Journal Pre-proof energies [23]. In Debye approximation, the temperature dependence of the heat capacity is properly introduced by using a parameter named Debye temperature which is related to a maximum phonon frequency (𝜔𝐷) of the dispersion curve diagram. The Debye temperature is defined as: 𝛩𝐷 =
ℏ𝜔𝐷 𝑘B
.
(3)
Within the harmonic theory, the vibrational frequency is proportional to the square root of the stiffness, therefore, 𝛩𝐷 can be assumed as a criterion of stiffness, and called Debye stiffness [24]. This parameter can define the total resistance of the material against phonon excitations including optical modes as well. In the Debye model heat capacity at constant volume is expressed as follows [24]: 𝛩
Cv =
𝑇 9𝑁𝐴kB(𝛩𝐷)∫0𝑇 dω
( ).
g(ω) 𝜔
ћ𝜔 k B𝑇
(4)
For a defined temperature, Debye temperature is determined by fitting the theoretical Cv and using this formula. Fig. 9c shows the Debye temperature at various temperatures for the bulk and nanowire structure. At high temperatures, this parameter is assumed to be constant; however, in the low-temperature domain, a valley can be seen in the related bulk diagram, where the Debye temperature reaches its lowest value. According to the figure, the Debye temperature of the bulk structure at a high-temperature range converges to a constant value, and at 70 K, it reaches its lowest value (727.21 K). On the other hand, nanowire indicates a completely different behavior from the bulk structure. The Debye temperature of the nanowire structure suddenly increases by increasing the temperature up to 250 K, and after this temperature, it decreases. This different behavior can be attributed to quantum confinement effects. The minimum value of the Debye temperature at zero degrees of K is 120.16 K for a nanowire structure.
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Journal Pre-proof In the nanowire structure of TiC, the temperature dependence of Debye temperature is found to be much higher than that of the bulk structure; this can be easily related to the shape of phonon DOS as shown in figs 4 and 8. Since Debye temperature is correlated to the Debye stiffness and has an inverse relation with heat capacity, it can be concluded that TiC bulk structure has a larger Debye stiffness and smaller heat capacity than that of nanowire structure.
5.4 Entropy For finite temperatures, the vibrational entropy is given by [25]: ∞
S(T) = rkB∫0 dω g(ω)
{( )[coth ( ) ― 1] ― 𝐿𝑛 [1 ― exp ( ― )]}. ћ𝜔 2kB𝑇
ћ𝜔 2kB𝑇
ћ𝜔 k B𝑇
(5)
Fig. 9d shows the diagram of vibrational entropy versus temperature for the structures of the bulk and the nanowire of the titanium carbide. As expected, entropy increases by increasing the temperature in both structures. According to the diagram, at room temperature, the amount of entropy for the bulk structure and the first diameter of the nanowire reach the value of 3.17 kB and 36.68 kB, respectively. The higher amount of entropy of nanowire than bulk TiC can be attributed to more vibrational states of nanowire than the bulk structure.
6. Conclusions In this paper, the electron and phonon properties of titanium carbide compound in bulk and nanowire structures were studied. Calculations were performed within density functional 16
Journal Pre-proof framework by using the pseudo-potential method. Based on the calculations of the density of states, the bulk structure shows a metallic behavior and the band structure calculations confirm this result. For the nanowire structure, due to lowering the size, which is accompanied by increasing surface effects and quantum limitations, the compound loses its metallic property and becomes a semiconductor. In the first diameter of the nanowire, the band gap was estimated to be 0.8 eV; however, by increasing the diameter of the nanowire, the band gap decreased towards corresponding bulk behavior. The investigation of the phonon modes shows the existence of a frequency gap for the bulk structure, but for the nanowire structure, this gap is not observed. Moreover, the study of thermal properties indicates that the vibrational free energies at zero temperature and zero pressure for the titanium carbide compound in both bulk and first diameter of nanowire structures are 0.011 and 0.094 Ry/cell, respectively. The free energies for the bulk and nanowire structures, decreases at 280 K and 230 K, respectively, and at 1000 K, they reach -0.02 Ry/cell and -0.24 Ry/cell for the bulk and nanowire structures, respectively. The phonon heat capacity behavior for the calculated structures is according to a Debye model. Since the temperature dependence of Debye temperature in nanowire structure of TiC is much more than that of the bulk structure, the Debye stiffness is greater in bulk structure than in nanowire structure. The Debye temperature in the structure of the nanowire shows completely different behavior from the bulk structure. The reason for this different behavior can be attributed to surface effects and quantum limitation. The amount of entropy increases by increasing temperature in both structures. The entropy values at room temperature for bulk and nanowire structures are 3.17 kB and 36.68 kB,
17
Journal Pre-proof respectively. Since nanowire has more vibrational states than bulk does, nanowire exhibits larger entropy than bulk structure.
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Author’s name
Affiliation
-----------------------------------------------------------------------------------------------------------Peiman Amiri , Assiatant professor of shahid chamran university of ahvaz,Ahvaz, Iran Hamdollah Salehi, Associated professor of shahid chamran university of ahvaz,Ahvaz, Iran. Yasamin Loveimi Motlagh, Graduated student of shahid chamran university of ahvaz,Ahvaz, Iran It should be mentioned that the authors declare no conflict of interest.
Peiman Amiri, Hamdollah Salehi, Yasamin Loveimi Motlagh PII:
S0921-4526(19)30661-1
DOI:
https://doi.org/10.1016/j.physb.2019.411761
Reference:
PHYSB 411761
To appear in:
Physica B: Physics of Condensed Matter
Received Date:
27 October 2018
Accepted Date:
07 October 2019
Please cite this article as: Peiman Amiri, Hamdollah Salehi, Yasamin Loveimi Motlagh, Ab-initio study of electronic and phononic properties of bulk TiC and its narrow nanowires by density functional theory, Physica B: Physics of Condensed Matter (2019), https://doi.org/10.1016/j.physb. 2019.411761
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
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Ab-initio study of electronic and phononic properties of bulk TiC and its narrow nanowires by density functional theory Peiman Amiri, Hamdollah Salehi, 1* and Yasamin Loveimi Motlagh
Department of Physics, Faculty of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran
Abstract In this article, electronic and phononic properties of TiC bulk and its narrow nanowires are calculated by using Quantum-Espresso/PWSCF computational package in the framework of density functional theory. According to the results, this compound shows a metallic behavior in the bulk structure, but for the smallest diameters of the nanowires, against the bulk, a semiconducting behavior was observed. This deviation becomes negligible at higher diameters. In the following, phonon calculations have been performed on bulk and the first diameter of nanowires with a square cross-section. By calculating the phonon modes of Brillouin zone, the thermodynamic properties of both mentioned structures such as vibrational free energy, the heat capacity
Cv,
Debye temperature , entropy and their variations against temperature T have been obtained
through the quasi-harmonic Debye model. Calculations show that heat capacity behavior of the vibrational lattice is consistent with Debye model. Moreover, the temperature dependence of a Debye temperature indicates that the Debye stiffness in a bulk is greater than the nanowire structure. Keywords: Band structure, Debye temperature, Density of states, Entropy, Heat capacity, Pseudo-potential.
1. Introduction Titanium Carbide (TiC) in a bulk state is a family of metal carbides [1]. The stable structure of this compound is the rock-salt structure with the Fm3m space group. The bulk of TiC structure is highly resistant to abrasion and has a high melting point [2-8]. Among nanoscale materials, much attention has been paid to one-dimensional nanomaterial’s, including nanowires. Which come from a wider set of mineral compounds with different types of 1*Corresponding
author E-mail address: [email protected]
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Journal Pre-proof crystalline structures, for example, carbides, nitrides and oxides [9]. Carbides are used as nanoscale materials due to their unique compounds with thermal and mechanical properties, high chemical resistance and electromagnetic radiation resistance [10]. The electrical behavior of the bulk structure is different from that of the nanowire. Changing the dimension of a system usually results in new physical properties. Nanostructures and diameter variations have been considerably investigated in recent years [11]. One of the issues which have attracted the attention of many scientists is the study of the thermal and phononic properties of nanostructures, and one of the important features is the phonon frequency spectrum survey. In recent years, many studies have been carried out in this field, and significant results have been obtained on the vibrations of the lattice and the phonon spectra of nanostructures [12]. In 1975, Nickel et al. presented the results of self-consistent calculations (APW) band structuring of some of the TMX-type compounds, including titanium carbide [13]. In 1995, Ahuja et al. obtained the structural, elastic and high-pressure properties of titanium carbide bulk state using FP-LAPW method, and discovered the stability of this compound in salt type [14]. In 2006, Enyashin and his colleagues developed an electronic structure of (100) nanowires of titanium carbide with square cross sections using the DFTB method. In this study, the titanium carbide nanowires are simulated along X-axis with three different diameters. A gap of 0.8 electron volts at the Γ point for the first diameter was reported [9]. In 2016, Jaffari and his colleagues studied the role of diameter effect on the optoelectrical properties of titanium carbide nanowire by using the FP-LAPW method. In this study, the first diameter of this nanowire was announced as a semiconductor compound [11]. In the same year, Xinhui Xia and his colleagues studied the wide-temperature electrochemical energy storage of the Single-Crystalline, Metallic TiC nanowires. In this work, they reported a facile CVD method for one-step synthesis of TiC nanowire arrays directly on flexible carbon cloth and demonstrated their excellent wide-temperature ultrafast EDLC performance [15]. In 2017, by using first-principles calculations based on density functional theory, Victor V. Ilyasov et.al studied the adsorption energy of the oxygen atom, the local atomic structure, and the electronic properties of the defective (001), (111) and (110) surfaces of titanium carbide. They indicated that the adsorption of oxygen at different binding positions leads to a substantial reconstruction of the local atomic structures and their related energy band spectra [16]. In the same year, they studied the local atomic structure, thermodynamics, and electronic properties of the non-stoichiometric W/TixCy(111) systems with different surface reconstructions in comparison to the stoichiometric W/TiC(111) systems using first-principles calculations based on DFT. Their results show that reconstruction mechanisms due to the nucleation of tungsten on the polar TixCy(111) surface are responsible for the physical properties variation of the material [17]. To the best of our knowledge, there is no systematic study on the phononic and thermodynamic properties and temperature dependence of Debye stiffness of TiC(001) nanowires. Comparison of these results with corresponding bulk values is not discussed.
2. Computational Methods
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Journal Pre-proof All computations for bulk and nanowires titanium carbide were performed in the framework of the density functional theory (DFT) using the Quantum-Espresso computational package [18]. In this computational package, the Kohn-Sham single-particle equations were solved using the pseudo-potential method and the valance electronic wave functions were expanded in a plane-wave basis set. After solving Kohn-Sham equations, the structural and electron properties were determined by calculating physical parameters such as total energy, forces, the density of states and band structure diagram. The pseudo-potentials used in this research for titanium and carbon atoms were made by the Norm Conserving method and its exchangecorrelation function is PBE-GGA. After conducting adequate convergence tests, an energy cutoff of 90 Ry was chosen for plane-wave expansion. The bulk and nanowire Brillouin zone integrations were performed by using Monkhorst-Pack meshes of 10 10 10 and 1110 k points, respectively, along with the Methfessel-Paxton method of smearing with a broadening parameter of 0.001 Ry. All structures were optimized to achieve the minimum energy by accurate relaxation of the atomic position down to forces less than 1 mRy/bohr. In comparison to normal LDA and GGA, in order to improve the exchange and correlation parts of our calculations, HSE functional we applied for related nanostructure computations. In this study, the chosen structural phase of bulk TiC is NaCl, which is the most stable phase of this compound and various nanowire structures with [001] crystallographic direction growth are composed of the bulk by a top-down method in order to preserve the symmetry of the corresponding bulk structure [19]. The free nanowire structures were simulated by using the supercell approach with a vacuum thickness of about 12 Å. The nanowires cross sections are sketched in the X-Y plane of fig. 1. The vacuum is applied along with the two perpendicular directions of this plane.
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Journal Pre-proof In fig. 1, the structure of titanium carbide in the bulk and nanowire structure is presented by Xcrysden software, respectively.
fig.1. (a) Cross sections of the first (smallest square), second (intermediate square), and third (largest square) sizes of square nanowires studied in this paper. The larger (blue) and smaller (red) circles stand for Ti and C atoms, respectively. The crystalline structure of the titanium carbide compound in (b) bulk and (c) nanowire
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Journal Pre-proof 3. Results of bulk 3.1 Band Structure Fig 2 shows the band structure of the titanium carbide compound with the GGA approximation in the bulk structure. In order to compare the two approximations of GGA and HSE, the band structure of the bulk was also investigated with the HSE approximation as shown in fig. 3. In both diagrams, Fermi energy is set on zero. In both approximations, there is a contribution of the electron density around the Fermi level that cuts off the Fermi surface; therefore no gaps were observed. Thus, it can be concluded that the HSE approximation has no significant effect on the metallic-like behavior of the compound, and the metallic-like property of the compound remains unchanged, which is consistent with other existing results [20]. In both figures, the lowest energy band is related to the orbital 2s of a carbon atom.
(a)
(b)
Fig. 2. The band structure of the titanium carbide compound in the bulk structure with approximation (a) GGA, (b) HSE.
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Density of states
The total density of states (DOS) of the electrons in the range of -13 to 8 eV is plotted for the titanium carbide with an approximation of GGA in fig. 3, and the Fermi energy is set on zero. As seen in the figure, there is a contribution of the electron density around the Fermi level, therefore that gap is covered and this leads to a metallic-like behavior for the TiC in a bulk structure. This result is consistent with other existing results as well [20]. .
Fig. 3. Total density of states of titanium carbide within the bulk structure.
3.3 Phononic properties In the following, the phonon dispersion curve will be studied for TiC compound in bulk structure. Fig.4 shows the phonon dispersion curves along with the density of phonon states calculated for the bulk structure of the TiC compound. That the branches of the dispersion diagram of this compound are expected to be equal to 6 in the case of having two atoms in their primitive cell of NaCl structure. As seen in fig. 4, there are 6 branches in the phonon dispersion curve. The three lower branches at Γ point are acoustic branches, among which, the two lower branches are transverse acoustic modes (TA) and the next is the longitudinal
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Journal Pre-proof acoustic mode (LA), which has the highest frequency in comparison to the two lower branches. The transverse acoustic branches (TA) at the Γ point are degenerate. This compound has a frequency gap in the frequency range of 445 to 520 cm-1. The waves in this region are severely weakened, and light cannot cross the crystal and is fully reflected in this range. The three upper branches at higher frequencies are transverse optical modes (TO) and longitudinal optical modes (LO). According to the phonon dispersion diagram, transverse acoustic and optical modes on the X path are degenerate, and degeneration behavior was removed and reappeared in the path of L . The phonon density of states versus frequency is depicted in fig. 4. According to this figure, the dispersion curve has no negative imaginary frequencies; therefore, dynamically, the titanium carbide compound in the rock salt structure is stable. One of the differences between optical and acoustic modes is that the dispersion is much weaker in the case of optical modes which can be observed from phonon density of states curve as well.
Fig. 4. The phonon dispersion curve (left) along with the density of phonon states (right) for the bulk structure of TiC.
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Journal Pre-proof Fig. 5 represents the projected phonon density of states of Ti and C atoms in the bulk structure of TiC. It is clear that the projected phonon density of states of Ti and C atoms dominate the low and high-frequency domain, respectively. This results from the atomic mass difference that leads to a separation of acoustic and optical phonon branches.
Fig. 5. Projected phonon density of states of C (brown solid line) and Ti (blue dashed line) atoms for the bulk TiC structure.
4. Results of Nanowires 4.1 Band Structure Normal GGA approximation leads to a gap of about 0.1 eV for the first radius of the defined nanowires. In comparison to the other theoretical work [9], this calculated value seems to be low. In order to improve the semiconducting behavior of the nanostructures, the HSE approximation was applied to the calculation of electronic structures of the nanowires. Fig. 6 shows the band structure diagrams for all three nanowires diameter with the HSE approximation along the growth direction of nanowires. In this figure, the Fermi energy is set on zero. According to the band structure diagram, the lowest energy band under Fermi level which is between -12 and -10 eV, is related to orbital 2s of a carbon atom. In the lowest 8
Journal Pre-proof energies of a valence band, there is a gap of about 4 to 4.5 eV. Other energy bands under Fermi level are due to the hybrid bond between 3d orbital of the titanium atom and the 2p orbital of a carbon atom that is responsible for the main bond between the titanium and carbon atoms. According to the results obtained for the first and second diameters of the nanowire, a direct gap of about 0.8 eV and an indirect gap of about 0.1 eV, respectively, are observed. The gaps decrease by increasing the diameter due to the reduction of surface effects and quantum limitations so that for the third diameter of the nanowire, the gap is covered and the behavior of the nanostructure is similar to that of related bulk structure.
(a)
(b)
(c)
Fig. 6. Calculated band structure of the first (a), second (b) and third (c) diameter of the nanowires.
4.2 Density of States As mentioned in the calculations part, nanowires of titanium carbide compound are simulated in three different diameters that are elongated in a z crystallographic direction. Fig. 7 shows the total density of states for all three diameters of titanium carbide nanowires. As seen in the figure, the gap of the first diameter is about 0.8 eV. By increasing the diameter of the
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Journal Pre-proof nanowires, the calculated gaps will reduce towards corresponding bulk value. Therefore, for higher diameters, the gap suddenly vanishes. In contrast to a bulk behavior, the first and second diameters have a semiconductor behavior which contributes to quantum limitations and surface effects which are most important for lower radii. In the third diameter, the gap is covered and the nanowire shows a metallic behavior as in the corresponding bulk compound.
(a)
(b)
(C) Fig. 7. Total density of states of (a) the first; (b) the second and (c) the third diameter of TiC nanowires.
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Journal Pre-proof 4.3 Phononic properties Due to computational limitations, phonon calculations were performed only for the first diameter of the nanowire. Fig 8 shows a dispersion curve and corresponding phonon density of states of the above-mentioned nanowire. In the phonon dispersion curve, there are 54 branches which are related to the presence of 18 atoms in the cell. The top 51 branches are optical modes, and the other three lower branches are acoustic modes. However, due to the degeneracy phenomena, in fig. 8 only two acoustic branches are visible. Moreover, when the maximum phonon frequencies are limited to about 150 cm-1, the frequency oscillations of the acoustic branches decrease and the number of optical branches grows, therefore the group velocity and the slope of the acoustic branches decrease and the thermal conductivity decreases further.
Fig. 8. Phonon dispersion curve (left) along with the phonon density of states (right) for the first diameter of the nanowire.
5. The study of thermodynamic properties in a bulk and a nanowire structure 11
Journal Pre-proof The quasi-harmonic approximation (QHA) is applied to obtain the thermodynamic functions including vibrational free energy, heat capacity, Debye temperature and vibrational entropy [21]. In the following, the results will be presented for vibrational free energy, thermal capacity, Debye temperature and vibrational entropy in the studied bulk and nanowire structures.
5.1 Vibrational free energy The free energy is made of two parts: F T , V E V FPh T , V , where E V denotes the ground state energy and FPh T , V is the phonon free energy at a given unit cell volume
V . The phonon free energy can be explained by the following equation [22]: ∞
[
𝐹𝑃ℎ(𝑇,𝑉) = 𝑘𝐵𝑇∫0 g(𝜔)ln 2sinh
( )]𝑑𝜔 . ℏ𝜔 2𝑘𝐵𝑇
(1)
Where V shows the volume dependent phonon frequencies and g stands for the phonon DOS. The vibrational free energy calculated at zero temperature and zero pressure for the titanium carbide compound in both bulk and nanowire structures are listed in Table (1), and their variation diagrams as a temperature parameter are shown in fig. 9a.
Table. 1. Vibrational free energy values for bulk and first diameter of the nanowire structures.
Type of structure
Free energy (Ry / cell)
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0.011
The first diameter of the
0.010
nanowire
According to fig. 9a, free energies of both bulk and nanowire structure decrease at 280 K and 230 K, respectively; and at 1000 K, they reach -0.02 Ry/cell and -0.24 Ry/cell for the bulk and nanowire structure, respectively.
Fig. 9. The Comparison of vibrational free energy per atom (a), the heat capacity per atom (b), Debye temperature per atom (c) and diagram of entropy changes (d) for the bulk (blue dash-point lines) and the first diameter of the nanowire (solid red lines) at various temperatures.
5.2 Heat capacity at constant volume At the constant volume, the heat capacity, by using calculated phonon DOS, is given by [21]: 13
Journal Pre-proof ћ𝜔
Cv =
∞ rkB∫0 dω
exp (k 𝑇) ћ𝜔 2 B 2 k B𝑇 ћ𝜔 [exp k 𝑇 ― 1]
( )
g(ω)
( )
.
(2)
B
Where r is the number of degrees of freedom in the unit cell, kB shows the Boltzmann constant, g denotes the phonon DOS of the unit cell, demonstrates the phonon frequency and refers to the Planck constant. The limits of Cv at 0 K and high temperature are 0 and rkB, respectively. The heat capacity per atom of the bulk and the nanowire structure of the titanium carbide are plotted against various temperatures in fig. 9b. Due to harmonic approximation effects at temperatures lower than 800 K , the heat capacity, Cv, is strongly dependent on the temperature. However, at higher temperatures, the harmonic approximation effect of the Debye model is extinguished, and Cv is converged to the classical Dulong-Petit limit. For the bulk structure with two atoms in a primitive cell, this classical limit is 6R; and in the nanowire structures with 18 atoms, this limit converges to a constant value of 54R. In order to compare the heat capacity of bulk to heat capacity of the nanowire, the calculated values of nanowire are divided by 9. The heat capacities at 300 K for the bulk and nanowire structures are 4.20R (J/mol.K) and 41.46R (J/mol.K), respectively. According to this figure, the heat capacity of the nanowire structure is greater than the bulk structure, which originates from the more vibrational states of the nanowire in comparison to the bulk structure.
5.3 Debye temperature The Debye temperature is an important physical parameter of solids, which defines a division line between quantum-mechanical and classical behavior of the vibrational lattice
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Journal Pre-proof energies [23]. In Debye approximation, the temperature dependence of the heat capacity is properly introduced by using a parameter named Debye temperature which is related to a maximum phonon frequency (𝜔𝐷) of the dispersion curve diagram. The Debye temperature is defined as: 𝛩𝐷 =
ℏ𝜔𝐷 𝑘B
.
(3)
Within the harmonic theory, the vibrational frequency is proportional to the square root of the stiffness, therefore, 𝛩𝐷 can be assumed as a criterion of stiffness, and called Debye stiffness [24]. This parameter can define the total resistance of the material against phonon excitations including optical modes as well. In the Debye model heat capacity at constant volume is expressed as follows [24]: 𝛩
Cv =
𝑇 9𝑁𝐴kB(𝛩𝐷)∫0𝑇 dω
( ).
g(ω) 𝜔
ћ𝜔 k B𝑇
(4)
For a defined temperature, Debye temperature is determined by fitting the theoretical Cv and using this formula. Fig. 9c shows the Debye temperature at various temperatures for the bulk and nanowire structure. At high temperatures, this parameter is assumed to be constant; however, in the low-temperature domain, a valley can be seen in the related bulk diagram, where the Debye temperature reaches its lowest value. According to the figure, the Debye temperature of the bulk structure at a high-temperature range converges to a constant value, and at 70 K, it reaches its lowest value (727.21 K). On the other hand, nanowire indicates a completely different behavior from the bulk structure. The Debye temperature of the nanowire structure suddenly increases by increasing the temperature up to 250 K, and after this temperature, it decreases. This different behavior can be attributed to quantum confinement effects. The minimum value of the Debye temperature at zero degrees of K is 120.16 K for a nanowire structure.
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Journal Pre-proof In the nanowire structure of TiC, the temperature dependence of Debye temperature is found to be much higher than that of the bulk structure; this can be easily related to the shape of phonon DOS as shown in figs 4 and 8. Since Debye temperature is correlated to the Debye stiffness and has an inverse relation with heat capacity, it can be concluded that TiC bulk structure has a larger Debye stiffness and smaller heat capacity than that of nanowire structure.
5.4 Entropy For finite temperatures, the vibrational entropy is given by [25]: ∞
S(T) = rkB∫0 dω g(ω)
{( )[coth ( ) ― 1] ― 𝐿𝑛 [1 ― exp ( ― )]}. ћ𝜔 2kB𝑇
ћ𝜔 2kB𝑇
ћ𝜔 k B𝑇
(5)
Fig. 9d shows the diagram of vibrational entropy versus temperature for the structures of the bulk and the nanowire of the titanium carbide. As expected, entropy increases by increasing the temperature in both structures. According to the diagram, at room temperature, the amount of entropy for the bulk structure and the first diameter of the nanowire reach the value of 3.17 kB and 36.68 kB, respectively. The higher amount of entropy of nanowire than bulk TiC can be attributed to more vibrational states of nanowire than the bulk structure.
6. Conclusions In this paper, the electron and phonon properties of titanium carbide compound in bulk and nanowire structures were studied. Calculations were performed within density functional 16
Journal Pre-proof framework by using the pseudo-potential method. Based on the calculations of the density of states, the bulk structure shows a metallic behavior and the band structure calculations confirm this result. For the nanowire structure, due to lowering the size, which is accompanied by increasing surface effects and quantum limitations, the compound loses its metallic property and becomes a semiconductor. In the first diameter of the nanowire, the band gap was estimated to be 0.8 eV; however, by increasing the diameter of the nanowire, the band gap decreased towards corresponding bulk behavior. The investigation of the phonon modes shows the existence of a frequency gap for the bulk structure, but for the nanowire structure, this gap is not observed. Moreover, the study of thermal properties indicates that the vibrational free energies at zero temperature and zero pressure for the titanium carbide compound in both bulk and first diameter of nanowire structures are 0.011 and 0.094 Ry/cell, respectively. The free energies for the bulk and nanowire structures, decreases at 280 K and 230 K, respectively, and at 1000 K, they reach -0.02 Ry/cell and -0.24 Ry/cell for the bulk and nanowire structures, respectively. The phonon heat capacity behavior for the calculated structures is according to a Debye model. Since the temperature dependence of Debye temperature in nanowire structure of TiC is much more than that of the bulk structure, the Debye stiffness is greater in bulk structure than in nanowire structure. The Debye temperature in the structure of the nanowire shows completely different behavior from the bulk structure. The reason for this different behavior can be attributed to surface effects and quantum limitation. The amount of entropy increases by increasing temperature in both structures. The entropy values at room temperature for bulk and nanowire structures are 3.17 kB and 36.68 kB,
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Journal Pre-proof respectively. Since nanowire has more vibrational states than bulk does, nanowire exhibits larger entropy than bulk structure.
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Affiliation
-----------------------------------------------------------------------------------------------------------Peiman Amiri , Assiatant professor of shahid chamran university of ahvaz,Ahvaz, Iran Hamdollah Salehi, Associated professor of shahid chamran university of ahvaz,Ahvaz, Iran. Yasamin Loveimi Motlagh, Graduated student of shahid chamran university of ahvaz,Ahvaz, Iran It should be mentioned that the authors declare no conflict of interest.