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Electronic, elastic, optical and thermal transport properties of penta-PdAs2 monolayer: First-principles study
Solid State Communications 307 (2020) 113802
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Electronic, elastic, optical and thermal transport properties of penta-PdAs2 monolayer: First-principles study Xiao-Long Pan a, Ying-Qin Zhao a, Zhao-Yi Zeng b, *, Xiang-Rong Chen a, **, Qi-Feng Chen c a
Institute of Atomic and Molecular Physics, College of Physics, Sichuan University, Chengdu 610065, China College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing 400047, China c National Key Laboratory for Shock Wave and Detonation Physics Research, Institute of Fluid Physics, CAEP, Mianyang 621900, China b
A R T I C L E I N F O
A B S T R A C T
Communicated by L. Brey
PdAs2 monolayer, a new pentagonal two-dimensional (2D) material, greatly attracts widely research interest due to its extremely high carrier mobility with a direct band gap. We have systematically studied the electronic structure, elastic, optical and thermal transport properties of monolayer PdAs2 from first-principles calculations. Our electronic structure calculations show that the monolayer PdAs2 is a semiconductor with a direct band gap of 0.78 eV. Tensile strain has a good regulating effect on the band gap, which can be adjusted from 0.01 eV to 0.84 eV. The calculated elastic constants of monolayer PdAs2 confirm that the PdAs2 monolayer is mechanically stable. The calculated optical properties reveal that the energy range of the absorption spectrum is 1 eV–15 eV. The absorption range of PdAs2 monolayer is mainly visible and ultraviolet light. Under the tensile strain, the absorption spectrum shows obvious redshift. With the increase of strain, the reflectivity and refractivity increase. The calculated thermal conductivity of the monolayer PdAs2 at 300 K is 1.04 W/(mK), which is lower than that of known pentagonal monolayer. In addition, the effects of phonon free path, group velocity and phonon scattering rate on lattice thermal conductivity are analyzed.
Keywords: PdAs2 Electronic structure Optical property Thermal conductivity First-principles
1. Introduction Since the influential discovery of graphene [1], two-dimensional (2D) materials has attracted wide attentions owing to their peculiar properties [2–6]. Many 2D materials, such as hexagonal boron nitride (h-BN) [7,8], group-IVA (silicene, germanene, and stanene) [9–13] and group-VA layered materials (P, As, Sb, Bi) [14–21] have been exten sively studied in experiment and theory over the past years. As a new carbon allotrope, the penta-graphene also has become a hot topic in the scientific community in recent years due to its unusual negative Pois son’s ratio and ultrahigh ideal strength that can even outperform gra phene [22]. Besides penta-graphene, other pentagonal materials have been re ported gradually. In 2016, Liu et al. [23] found that the penta-graphene, penta-SiC2, and penta-SiN2 exhibit disparate strain dependent thermal conductivity and robust tunability of thermal conductivity, which may inspire intensive research on other derivatives of penta-structures to seek potential materials for emerging nanoelectronic devices. Under acidic conditions, the penta-GeP2 can be useful in photocatalyzed CO2
splitting to CO and photocatalyzed water splitting [24]. Lan et al. [25] reported that the penta-PdX2 (X ¼ S, Se, Te) monolayers has been to promising anisotropic thermoelectric materials with high thermoelectric figure of merit. The planar penta-MN2 (M ¼ Pd, Pt) sheets identified through structure search are dynamically, thermally and mechanically stable, and the penta-PtN2 can withstand a temperature as high as 2000 K, showing its potential as a refractory material [26]. Recently, the planar penta-PdAs2 has been found to be a direct band gap semiconductor with a high carrier mobility (104 cm2V 1s-1 for electrons and 105 cm2V 1s 1 for holes) and a modest band gap (0.80 eV, indicating its suitability for electronic application [27]. It is shown that PdAs2 is dynamically and thermally stable. The electron effective mass climbs up firstly, and then declines largely under tensile load with a turning point at 2% for penta-PdAs2. Like graphene and carbon nano tubes, a PdAs2 sheet can be rolled into various forms of PdAs2 nanotubes as well [28]. However, up to date, the electronic property, thermal transport properties and optical properties of PdAs2 has not been enough research yet, which greatly attracts our research interest. Thus, in this work, we investigate the electronic structure, elastic
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (Z.-Y. Zeng), [email protected] (X.-R. Chen). https://doi.org/10.1016/j.ssc.2019.113802 Received 29 July 2019; Received in revised form 7 November 2019; Accepted 29 November 2019 Available online 2 December 2019 0038-1098/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. (a) The top and side views of the geometric structure of PdAs2 monolayer, and (b) the phonon dispersion spectra of PdAs2 monolayer, where the red balls represent the Pd atoms and the green the As atoms. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
constants, optical properties and lattice thermal conductivity of PdAs2 monolayer. We study the electronic structure under different tensile strain, and obtain the contribution of atomic orbitals to the energy band near the Fermi level according to the density of state. By calculating the imaginary part and the real part of the dielectric constant, we get the energy absorption spectra, energy loss spectra, refractivity and reflec tivity. In addition, we also obtain the lattice thermal conductivity by ShengBTE code [29] which can be successfully predicted the lattice thermal conductivity [30–36].
L(ω), refractivity n(ω) and reflectivity R(ω) can be obtained from the real part and imaginary part of the dielectric function [46,47]: �pffiffiffiffiffiffi �hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i1=2 IðωÞ ¼ 2ω ε1 ðωÞ2 þ ε2 ðωÞ2 ε1 ðωÞ (1)
� . pffiffi �hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i1=2 nðωÞ ¼ 1 2 ε1 ðωÞ2 þ ε2 ðωÞ2 þ ε1 ðωÞ
(3)
2. Theoretical methods and computational details
�pffiffiffi � � ω 1�2 � � ffi ffi ffi p RðωÞ ¼ � � � ω þ 1�
(4)
LðωÞ ¼ ε2 ðωÞ
The first-principles calculations are performed by using the Vienna ab initio simulation package (VASP) based on density functional theory (DFT) [37,38]. We use the generalized gradient approximation (GGA) [39] in the Perdew-Burke-Ernzerhof (PBE) parametrization for the exchange-correlation functional and the projector augmented wave (PAW) method [40]. The primitive unit cell of PdAs2 is optimized at a plane-wave cutoff energy of 500 eV using a 11 � 11 � 1 Mon khorst–Pack grid [41] to ensure that the total energy converges to a precision of at least 10 8 eV and that all forces converge to a precision of at least 0.001 eV/Å. A 20 Å vacuum layer along the z-direction is adopted to prevent any interaction between neighboring layers. The Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional [42] and the spin-orbital coupling (SOC) [43] are considered during the band struc ture calculations. On the basis of HSE06 band structure calculation, we calculate the optical properties of PdAs2 monolayer. The interaction between photons and electrons leads to transitions between occupied and non-occupied states, which determine the optical properties of solid materials. In the range of linear response, the macroscopic optical response function of solid is usually measured by the complex dielectric function of light, and expressed by εðωÞ ¼ ε1 ðωÞ þ iε2 ðωÞ Refs. [44,45], where ε1 ðωÞ and ε2 ðωÞ is the real part and the imaginary part of the dielectric function. Other optical constants such as absorption coefficient I(ω), energy loss function
a (Å)
As-As (Å)
Pd-As (Å)
Ecoh (eV)
6.19 6.19 6.19
2.31 2.31 2.31
2.42 2.42 2.42
3.18 3.18 3.55
ε1 ðωÞ2 þ ε2 ðωÞ2
�
(2)
The lattice thermal transport is determined by solving the semiclassical phonon Boltzmann transport equation (BTE) iteratively from ShengBTE code [29]. The second-order (harmonic) force constants are obtained by PHONOPY software package [48], and the third-order (anharmonic) force constants are obtained from ShengBTE code. The second-order (harmonic) and third-order (anharmonic) force constants are both calculated using 3 � 3 � 1 supercell based on the relaxed unit cell. To get a more accurate thermal conductivity, the third-nearest neighbors serve as the interaction cutoff in the third-order (anhar monic) force constants. The lattice thermal conductivity from ShengBTE code is calculated by: kαl ;β ¼
X 1 f0 ðf0 þ 1Þðℏωλ Þ2 vαλ Fβλ kB T 2 ΩN λ
(5)
where Ω is the volume of the unit cell, N is the number of q points uniformly sampled in the Brillouin zone, ωλ is the angular frequency of the phonon mode λ, vλ is the phonon group velocity, f0 is the Bose
–Einstein distribution function, and Fβλ is the projection of the mean free displacement along the β direction. 3. Results and discussion 3.1. Geometry and electronic structure
Table 1 The calculated lattice constant a and bond lengths of Pd-As and As–As, as well as the cohesive energy. Present work Yuan et al. [27] Qian et al. [28]
��
We get PdAs2 monolayer by incising its bulk crystal with space group of Pa3. By geometry optimizations, PdAs2 monolayer has a planar pentagonal structure with space group of P4/mbm. In Fig. 1(a), we illustrate the top and side view of the 2D structure of PdAs2. PdAs2 monolayer is composed by an infinite number of pentagons, and every pentagon contains two Pd atoms and three As atoms. The obtained 2
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Fig. 2. The electronic structures of PdAs2 monolayer calculated by PBE method, with (a) and without (b) spin-orbit coupling (SOC), respectively. The electronic structures of PdAs2 monolayer calculated by HSE06 (c) and HSE06 þ SOC (d) are also presented.
optimized lattice constant a is 6.19 Å, well consistent with other results [28]. The calculated bond lengths and cohesive energy of the PdAs2 monolayer are listed in Table 1. Obviously, these results agree well with previous ones, indicating that the geometry structure of PdAs2 mono layer in our work is reliable. To prove the dynamic stability for the PdAs2 monolayer, we display its phonon dispersion spectra in Fig. 1(b). It can be seen that, there is no imaginary frequency mode, suggesting that PdAs2 monolayer is dynamic stable. From the phonon spectra, there are 18 branches, including three acoustic branches and 15 optical branches. According to phonon spectra, we can see that the largest frequency is 9.61 THz. It is obvious that the frequency has two phonon gaps between 6 THz and 9 THz. In Fig. 2, we illustrate the energy band structure of PdAs2 monolayer obtained from different methods. Fig. 2 (a) and (b) show the electronic structure calculated by PBE and PBE þ SOC, and Fig. 2 (c) and (d) show the electronic structure calculated by HSE06 and HSE06 þ SOC. It is obvious from Fig. 2(a) that PdAs2 monolayer has a direct band gap up to 0.32 eV around the high symmetric point path S, and from Fig. 2(b), the band gap is 0.34 eV, which indicating that SOC has a weak influence on the band gap. As is known that, PBE method underestimates the band gap [49,50], so we also calculate the energy band structures using HSE06 and HSE06 þ SOC methods. Band gaps from HSE06 and HSE06 þ SOC are 0.78 eV and 0.80 eV, respectively, consistent with the band gap 0.80 eV by Yuan et al. [27] and 0.81 eV by Qian et al. [28] Compared to the penta-PdN2 with no band gap and the penta-PtN2 with a narrow 75 meV band gap, the PdAs2 monolayer has a wider band gap. In all, PdAs2 monolayer is a direct band-gap semiconductor with a modest band gap, and the band gaps are insensitive to spin-orbit coupling (SOC). The electronic structures of PdAs2 monolayer under tensile strain are
also calculated from HSE06 method. Fig. 3(a–e) shows the energy band structures with tensile strains of 2%, 4%, 6%, 8% and 10%, respectively. At different tensile strains, the energy band structures of PdAs2 mono layer have obvious difference. As the strain increases, the band gap first decreases and then increases. The band gap reaches a maximum when the strain is 2%. When the strain is more than 2%, the band gap grad ually decreases with the increase of strain, and when the strain reaches 10%, the band gap becomes 0.01 eV. It can be seen from Fig. 3 that, compared with the change of valence band, the change of conduction band is very obvious. When the strain is larger than 2% the conduction band decreases with the increase of strain, which mainly leads to the reduction of band gap. To better intuitively feel the band gap changes, we show the relationship between band gap and strain in Fig. 3(f). In general, the effect of tensile strain on the band gap of PdAs2 monolayer is very obvious, which indicates that PdAs2 monolayer may be a good application prospect in the electronic devices. In Fig. 4, we illustrate the calculated total and partial density of states of monolayer PdAs2. Fig. 4(a) shows the contribution of Pd atom and As atom to the total density of states. It is seen from Fig. 4(a) that, near the Fermi level, the contribution of Pd atom and As atom to the conduction band is almost the same, while the contribution of As atom to the valence band is obviously greater than that of Pd atom. In order to judge the contribution of the orbitals to the energy band near the Fermi level more accurately, we show the contribution of the orbitals of Pd and As atoms to the total density of state in Fig. 4(b), from which we can see that the density of states near the Fermi level is mainly contributed by d orbitals of Pd atom and p orbitals of As atom.
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Fig. 3. The energy band structures of PdAs2 monolayer under different tensile strains: (a) 2% strain, (b) 4% strain, (c) 6% strain, (d) 8% strain, and (e) 10% strain, together with the relationship between band gaps and strains (f).
Fig. 4. The contributions of atoms (a) and orbitals (b) to the total density of states of PdAs2 monolayer.
3.2. Elastic properties
Table 2 The calculated elastic constants (C11, C12 and C66), Young’s modulus Y and Poisson’s ratio ν (all in N/m). Present work Qian et al. [28]
C11
C12
C66
Y
ν
93.0 92.6
33.1 32.4
23.7 23.7
81.2 81.2
0.36
We calculate the elastic constants of monolayer PdAs2 by elastic matrix method. For the 2D materials, it is accepted that the elastic constants matrix is a six order square matrix containing four non-zero elastic constants, C11, C12, C22 and C66. For monolayer PdAs2, the elastic constant C11 is equal to C22, so there are three independent elastic constants (C11, C12, and C66). In Table 2, we list the calculated elastic constants. Our results are consistent well with previous work. Young’s modulus is the same along the x and y directions. Young’s modulus Y and Poisson’s ratio ν of the 2D material, as important criterion of me chanical stability, is calculated by Ref. [51]: 4
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3.3. Optical properties We obtained the real and imaginary parts of the dielectric function of monolayer PdAs2 by HSE06 method, as are displayed in Fig. 6. Since the monolayer PdAs2 has no anisotropy in x and y directions, we only analyze the real and imaginary parts of the dielectric function in x di rection. When the incident photon has zero energy, the static dielectric constant is obtained from the real part of the dielectric function. Fig. 6 (a) shows the real part of the dielectric constant under three different strains, where black represents no strain, red represents 3% strain, and blue represents 6% strain. We can see from Fig. 6(a) that the static dielectric constant is 5.24 for no strain, 5.94 for 3% strain and 9.37 for 6% strain. Obviously, as the strain increases, the static dielectric con stant increases. As the energy of the incoming photon increases, the real part increases to a peak and then decreases rapidly, remaining almost flat at high energies. In the absence of strain, a sharp decrease in the real part between 1 eV and 5 eV, which corresponds to a sharp increase in the absorption coefficient because the absorption of photon energy by electrons in the band transition increases. In the equilibrium state, the maximum peak value of the dielectric function is 6.96, corresponding to the incident photon energy of 1.03 eV. Under strain conditions, the peak of the dielectric function appears in advance and corresponds to a larger value, 7.78 for 3% at 0.78 eV and 10.83 for 6% at 0.41 eV. The imagi nary part of the dielectric function reflects transitions between energy bands near Fermi energy levels, which determine the linear response of the material to light under small wave vectors. From Fig. 6(b), in the absence of strain, we can see that the imaginary part of monolayer PdAs2 has four major peaks at 1.32 eV, 1.94 eV, 2.76 eV, 3.31 eV, respectively. With the increase of strain, the lower the energy of the incident photon corresponding to the first peak of the imaginary part is, an obvious redshift occurs. When the natural frequency of the incident wave is consistent with the natural frequency of the atoms in the crystal, light will be absorbed. When the light travels through the material, its intensity will be more or less absorbed. We research the absorption spectrum of light based on imaginary parts of the dielectric function of monolayer PdAs2, as is displayed in Fig. 7(a). It is seen that when there’s no strain, the energy range of the absorption spectrum is 1 eV–15 eV. Absorption spectra have the maximum absorption peak at 5.1 � 105 cm 1eV. The absorption range of PdAs2 monolayer is very wide, including ultraviolet, visible, and infrared regions. And the absorption range of PdAs2 monolayer is mainly visible and ultraviolet light. When the strain is 3% and 6%, the maximum absorption peak is 4:7 � 105 cm 1 eV and 43 � 104 cm 1 eV, which indicates that the light absorption coefficient of PdAs2 monolayer decreases with the increase of strain. From the absorption spectrum, the absorption edge of monolayer PdAs2 under three different strains is 0.30 eV at no strain, 0.21 eV at 3% strain and 0.07 at 6% strain, which in dicates that there is an obvious redshift in the absorption spectrum under tensile strain. To show why the absorption coefficient decreases
Fig. 5. The relationship between strains and pressures for PdAs2 monolayer, where the black and red curves are for the biaxial and uniaxial, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Y10 ¼ Y01 ¼ C11 C12
C212
��
� C22 ; v ¼ C12 C11
(6)
Moreover, sheer modulus G is simple equal to C66. Young’s modulus, sheer modulus G and Poisson’s ratio ν in this work is 81.2 N/m, 23.7 N/ m and 0.36, respectively. These results are also listed in Table 2. Our results are completely consistent with previous results. For a 2D monolayer, according to Born–Huang criteria [52], its mechanically stability can be judged by the following formula: C11 C12
C212 > 0; C66 > 0
(7)
From Table 2, the elastic constants of monolayer PdAs2 satisfy the above criteria very well, confirming that the PdAs2 monolayer is me chanically stable. Elastic limit of the monolayer PdAs2 can be estimated by stress-strain cure. We calculated the biaxial and uniaxial stress-strain cures, as is illustrated in Fig. 5. During the calculations of the uniaxial stress-strain cures, the x-axis tensile strain is just considered since the lattice con stants of monolayer PdAs2 in the x and y directions are the same, which implies that PdAs2 monolayer has no anisotropy. This is consentient with the equality of elastic constants C11 and C22, so the monolayer PdAs2 has no anisotropy in x and y direction. It can be seen from Fig. 5 that the monolayer PdAs2 bears the biggest stress is 5.91 N/m for the biaxial strain and 6.46 N/m for the uniaxial strain corresponding to 12% strain and 16% strain, respectively. Compared with penta-PdN2 (17.5%) and penta-PtN2(15%) [26], the monolayer PdAs2 has weak tensile resistance.
Fig. 6. The real part (a) and imaginary part (b) of the dielectric function of PdAs2monolayer. 5
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Fig. 7. Optical parameters of PdAs2 monolayer under different tensile strain: (a) absorption coefficient, (b) energy loss spectrum, (c) reflectivity, and (d) refractivity.
Fig. 8. (a) Lattice thermal conductivity varies with temperature and (b) the heat capacity varies with temperature for PdAs2 monolayer.
under tensile strain, let’s analyze the energy loss spectrum, reflectivity and refractivity, as are showed in Fig. 7(b–d). From energy loss spectra, we can see that the energy loss is about the same for all three strains, and the energy loss is mainly distributed between 2 eV and 13 eV. There is a rapid increase in the energy loss spectrum before it reaches its maximum, which corresponds to a rapid decrease in the refractivity and reflectivity. Between 0 eV and 5 eV, the energy loss spectrum has a relatively small value, while the reflectivity and refractive index have a large value, which indicates that most of the energy of the incident photon is reflected and refracted and only a small part is absorbed by the material. Obviously, as the strain increases, the reflectivity increases gradually. When there is no incident light, the reflectivity of the three strains is 0.15, 0.18 and 0.26 for different strains, respectively. Like the reflectivity, the refractive index also increases with the increase of strain, when there is no incident light. The refractivity of the three strains is 2.29, 2.44 and 3.06, respectively. In general, as the strain in creases, the refractive index and reflectivity increase, implying that when the incident light shines on the material, the larger the strain, the
more light will be reflected and refracted. The reason for the decrease of absorption coefficient under strain is that the reflectivity and refractivity increase with the increase of strain. 3.4. Lattice thermal conductivity We obtained lattice thermal conductivity of monolayer PdAs2 based on solving the Boltzmann transport equation using the iterative solution method. The variation of lattice thermal conductivity with temperature is shown Fig. 8(a). The lattice thermal conductivity decreases with the increase of temperature, and the decreasing rate decreases gradually. The lattice thermal conductivity is 1.04 W/(mK) at 300 K, which is lower than that of the penta-PdX2 (X ¼ S, Se, Te) monolayers [25]. The heat capacity increases gradually with the rise of temperature, and the rate is proportional to T3, which remains basically unchanged after 600 K, as is shown in Fig. 8(b). Further, the lattice thermal conductivity varying with the mean free path is shown in Fig. 9(a). We show the phonon mean free path and 6
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Fig. 9. (a) The relationship between the cumulative lattice thermal conductivity and the free path at 300 K, 500 K and 900 K; (b) The relationship between effective phonon mean free path and the temperature for PdAs2 monolayer.
Fig. 10. (a) The phonon group velocity of three acoustical branches ZA, TA and LA, respectively. (b) The phonon scattering rate of three acoustical branches and partial optical branches for PdAs2 monolayer.
cumulative lattice thermal conductivity at three temperatures respec tively. The cumulative lattice thermal conductivity increases with the increase of the phonon average free path and remains unchanged after the free path is large enough. The cumulative lattice thermal conduc tivity at 300 K, 400 K and 500 K is 1.04 W/(mK), 0.65 W/(mK) and 0.37 W/(mK), respectively. At room temperature, the phonon average free path only contributes to the lattice thermal conductivity between 1 nm and 410 nm, and this distance decreases with the increase of tempera ture. In order to better study the influence of phonon mean free path on lattice thermal conductivity, we fitted the data through the following formula [53,54]: �� � �n � kl ðLÞ ¼ Kmax 1 þ Λeff L (8)
are 18 vibration modes including three acoustic branches and 15 optical branches, The three acoustic modes are the longitudinal (LA) mode, the transverse (TA), and the flexural mode (ZA). We analyzed the contri bution of different vibration modes to the lattice thermal conductivity at 300 K, concluding that the contribution of three acoustic modes (ZA, TA, LA) is greater than 15 optical modes, accounting for 61%. For the monolayer PdAs2 at 300 K, the contributions of the three acoustic modes (ZA, TA, LA) to the total lattice thermal conductivity is 32%, 16% and 13%, respectively. Therefore, in the following study, we mainly consider the influence of three acoustic branches on the lattice thermal conduc tivity. From the lattice thermal conductivity formula, we know that the group velocity is an important parameter to determine the lattice ther mal conductivity. The group velocity with respect to the phonon vi bration frequency is showed in Fig. 10(a). The ZA mode has a minimum group velocity consistent with its maximum contribution to the lattice thermal conductivity. Obviously, the group velocity which is lower than those of penta-PdX2 (X ¼ S, Se, Te) [25] monolayers mainly varies from 0 to 2.5 Km/s, which implies that the small group velocity is one of the reasons for the low lattice thermal conductivity. In addition, we calcu late the phonon scattering rate ωλ of monolayer PdAs2 with three acoustic branches and partial optical branches at room temperature (Fig. 10(b)). With the increase of frequency, the scattering rate of the three acoustic branches gradually increases, and finally stays between 0.1 ps 1 and 1 ps 1. The scattering rates of the optic modes are only in the high-frequency regions, and the scattering rate mainly ranges from 0.1 ps 1 and 1 ps 1. Obviously, the acoustic mode is only in the low frequency region while the optical mode is in the high frequency region,
where Λeff is effective phonon mean free path, L is the length, and kl is the lattice thermal conductivity after convergence. Λeff and n are ob tained after fitting the curve. At room temperature we get that Λeff and n are 11.42 and 2. Effective phonon mean free path at other temperature is displayed in Fig. 9(b). It is seen that effective phonon mean free path is decreases with increasing temperature, corresponding to the change of thermal conductivity with temperature. From the diagram we can see that there is a small effective phonon mean free path, suggesting that monolayer PdAs2 may have a low lattice thermal conductivity. To explore the reason why the lattice thermal conductivity of monolayer PdAs2 is lower than that of other pentagonal twodimensional materials, we analyze the phonon group velocity and scattering rate. From the phonon scattering spectra, we know that there 7
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suggesting that the acoustic mode contributes a lot to the thermal con ductivity. The phonon scattering rate of PdAs2 monolayer is mainly distributed between 0.1 ps 1 and 1 ps 1, which is relatively high in the pentagonal two-dimensional material. Thus higher phonon scattering rate ωλ of Pd monolayer is one of the reasons for the low thermal conductivity.
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4. Conclusions We have systematically studied the electronic structure, elastic, op tical and thermal transport properties of monolayer PdAs2 from firstprinciples calculations. By phonon dispersion spectrum calculation, we conclude that PdAs2 monolayer is dynamic stable. Our electronic structure calculations show that monolayer PdAs2 is a semiconductor with a direct band gap of 0.78 eV. Tensile strain has a good regulating effect on the band gap, which can be adjusted from 0.01 eV to 0.84 eV. The adjustable band gap implies that monolayer PdAs2 has a good application prospect in the field of electronic devices. The partial density of states of the monolayer PdAs2 shows that the valence band maximum is mainly due to the p orbitals of As atoms and the conducting band minimum due to the p orbitals of Pd atoms and the p orbitals of As atoms. We also calculate the elastic constants of mono layer PdAs2, confirming that the PdAs2 monolayer is mechanically sta ble. The monolayer PdAs2 bears the biggest stress is 5.91 N/m for biaxial strain and 6.46 N/m for uniaxial strain corresponding to 12% strain and 16% strain. The calculation of the optical properties reveals that the energy range of the absorption spectra is 1 eV–15 eV. Absorption spectra have the maximum absorption peak at 5.1 � 105cm 1eV. The absorption range of PdAs2 monolayer is very wide, including ultraviolet, visible, and infrared regions. The absorption range of PdAs2 monolayer is mainly infrared region. Under the tensile strain, the absorption spectra show obvious redshift. With the increase of strain, the reflectivity and refractivity increase. In addition, we have calculated the lattice thermal conductivity of monolayer PdAs2. Our results show that, at room temperature, the thermal conductivity of monolayer PdAs2 is 1.04 W/(mK), which is lower than that of known pentagonal monolayer. The main reasons of low thermal conductivity are low effective free path, low phonon group velocity and high phonon scattering rates. In general, PdAs2 monolayer has a promising application prospect in the field of semiconductor de vices due to its unique electronic, optical and thermal transport properties. Acknowledgment The authors are grateful for the supports of the NSAF (Grant No. U1830101), the Science Challenge Project (Grant No. TZ2016001), and the National Natural Science Foundation of China (Grant No. 11504035). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.ssc.2019.113802. References [1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I. V. Grigorieva, A.A. Firsov, Science 306 (2004) 666–669. [2] A. Bablich, S. Kataria, M. Lemme, Electronics 5 (2016) 13. [3] A.N. Gandi, U. Schwingenschl€ ogl, Europhys. Lett. 113 (2016) 36002. [4] S.J. Kim, K. Choi, B. Lee, Y. Kim, B.H. Hong, Annu. Rev. Mater. Res. 45 (2015) 63–84.
8
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Solid State Communications journal homepage: http://www.elsevier.com/locate/ssc
Electronic, elastic, optical and thermal transport properties of penta-PdAs2 monolayer: First-principles study Xiao-Long Pan a, Ying-Qin Zhao a, Zhao-Yi Zeng b, *, Xiang-Rong Chen a, **, Qi-Feng Chen c a
Institute of Atomic and Molecular Physics, College of Physics, Sichuan University, Chengdu 610065, China College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing 400047, China c National Key Laboratory for Shock Wave and Detonation Physics Research, Institute of Fluid Physics, CAEP, Mianyang 621900, China b
A R T I C L E I N F O
A B S T R A C T
Communicated by L. Brey
PdAs2 monolayer, a new pentagonal two-dimensional (2D) material, greatly attracts widely research interest due to its extremely high carrier mobility with a direct band gap. We have systematically studied the electronic structure, elastic, optical and thermal transport properties of monolayer PdAs2 from first-principles calculations. Our electronic structure calculations show that the monolayer PdAs2 is a semiconductor with a direct band gap of 0.78 eV. Tensile strain has a good regulating effect on the band gap, which can be adjusted from 0.01 eV to 0.84 eV. The calculated elastic constants of monolayer PdAs2 confirm that the PdAs2 monolayer is mechanically stable. The calculated optical properties reveal that the energy range of the absorption spectrum is 1 eV–15 eV. The absorption range of PdAs2 monolayer is mainly visible and ultraviolet light. Under the tensile strain, the absorption spectrum shows obvious redshift. With the increase of strain, the reflectivity and refractivity increase. The calculated thermal conductivity of the monolayer PdAs2 at 300 K is 1.04 W/(mK), which is lower than that of known pentagonal monolayer. In addition, the effects of phonon free path, group velocity and phonon scattering rate on lattice thermal conductivity are analyzed.
Keywords: PdAs2 Electronic structure Optical property Thermal conductivity First-principles
1. Introduction Since the influential discovery of graphene [1], two-dimensional (2D) materials has attracted wide attentions owing to their peculiar properties [2–6]. Many 2D materials, such as hexagonal boron nitride (h-BN) [7,8], group-IVA (silicene, germanene, and stanene) [9–13] and group-VA layered materials (P, As, Sb, Bi) [14–21] have been exten sively studied in experiment and theory over the past years. As a new carbon allotrope, the penta-graphene also has become a hot topic in the scientific community in recent years due to its unusual negative Pois son’s ratio and ultrahigh ideal strength that can even outperform gra phene [22]. Besides penta-graphene, other pentagonal materials have been re ported gradually. In 2016, Liu et al. [23] found that the penta-graphene, penta-SiC2, and penta-SiN2 exhibit disparate strain dependent thermal conductivity and robust tunability of thermal conductivity, which may inspire intensive research on other derivatives of penta-structures to seek potential materials for emerging nanoelectronic devices. Under acidic conditions, the penta-GeP2 can be useful in photocatalyzed CO2
splitting to CO and photocatalyzed water splitting [24]. Lan et al. [25] reported that the penta-PdX2 (X ¼ S, Se, Te) monolayers has been to promising anisotropic thermoelectric materials with high thermoelectric figure of merit. The planar penta-MN2 (M ¼ Pd, Pt) sheets identified through structure search are dynamically, thermally and mechanically stable, and the penta-PtN2 can withstand a temperature as high as 2000 K, showing its potential as a refractory material [26]. Recently, the planar penta-PdAs2 has been found to be a direct band gap semiconductor with a high carrier mobility (104 cm2V 1s-1 for electrons and 105 cm2V 1s 1 for holes) and a modest band gap (0.80 eV, indicating its suitability for electronic application [27]. It is shown that PdAs2 is dynamically and thermally stable. The electron effective mass climbs up firstly, and then declines largely under tensile load with a turning point at 2% for penta-PdAs2. Like graphene and carbon nano tubes, a PdAs2 sheet can be rolled into various forms of PdAs2 nanotubes as well [28]. However, up to date, the electronic property, thermal transport properties and optical properties of PdAs2 has not been enough research yet, which greatly attracts our research interest. Thus, in this work, we investigate the electronic structure, elastic
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (Z.-Y. Zeng), [email protected] (X.-R. Chen). https://doi.org/10.1016/j.ssc.2019.113802 Received 29 July 2019; Received in revised form 7 November 2019; Accepted 29 November 2019 Available online 2 December 2019 0038-1098/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. (a) The top and side views of the geometric structure of PdAs2 monolayer, and (b) the phonon dispersion spectra of PdAs2 monolayer, where the red balls represent the Pd atoms and the green the As atoms. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
constants, optical properties and lattice thermal conductivity of PdAs2 monolayer. We study the electronic structure under different tensile strain, and obtain the contribution of atomic orbitals to the energy band near the Fermi level according to the density of state. By calculating the imaginary part and the real part of the dielectric constant, we get the energy absorption spectra, energy loss spectra, refractivity and reflec tivity. In addition, we also obtain the lattice thermal conductivity by ShengBTE code [29] which can be successfully predicted the lattice thermal conductivity [30–36].
L(ω), refractivity n(ω) and reflectivity R(ω) can be obtained from the real part and imaginary part of the dielectric function [46,47]: �pffiffiffiffiffiffi �hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i1=2 IðωÞ ¼ 2ω ε1 ðωÞ2 þ ε2 ðωÞ2 ε1 ðωÞ (1)
� . pffiffi �hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i1=2 nðωÞ ¼ 1 2 ε1 ðωÞ2 þ ε2 ðωÞ2 þ ε1 ðωÞ
(3)
2. Theoretical methods and computational details
�pffiffiffi � � ω 1�2 � � ffi ffi ffi p RðωÞ ¼ � � � ω þ 1�
(4)
LðωÞ ¼ ε2 ðωÞ
The first-principles calculations are performed by using the Vienna ab initio simulation package (VASP) based on density functional theory (DFT) [37,38]. We use the generalized gradient approximation (GGA) [39] in the Perdew-Burke-Ernzerhof (PBE) parametrization for the exchange-correlation functional and the projector augmented wave (PAW) method [40]. The primitive unit cell of PdAs2 is optimized at a plane-wave cutoff energy of 500 eV using a 11 � 11 � 1 Mon khorst–Pack grid [41] to ensure that the total energy converges to a precision of at least 10 8 eV and that all forces converge to a precision of at least 0.001 eV/Å. A 20 Å vacuum layer along the z-direction is adopted to prevent any interaction between neighboring layers. The Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional [42] and the spin-orbital coupling (SOC) [43] are considered during the band struc ture calculations. On the basis of HSE06 band structure calculation, we calculate the optical properties of PdAs2 monolayer. The interaction between photons and electrons leads to transitions between occupied and non-occupied states, which determine the optical properties of solid materials. In the range of linear response, the macroscopic optical response function of solid is usually measured by the complex dielectric function of light, and expressed by εðωÞ ¼ ε1 ðωÞ þ iε2 ðωÞ Refs. [44,45], where ε1 ðωÞ and ε2 ðωÞ is the real part and the imaginary part of the dielectric function. Other optical constants such as absorption coefficient I(ω), energy loss function
a (Å)
As-As (Å)
Pd-As (Å)
Ecoh (eV)
6.19 6.19 6.19
2.31 2.31 2.31
2.42 2.42 2.42
3.18 3.18 3.55
ε1 ðωÞ2 þ ε2 ðωÞ2
�
(2)
The lattice thermal transport is determined by solving the semiclassical phonon Boltzmann transport equation (BTE) iteratively from ShengBTE code [29]. The second-order (harmonic) force constants are obtained by PHONOPY software package [48], and the third-order (anharmonic) force constants are obtained from ShengBTE code. The second-order (harmonic) and third-order (anharmonic) force constants are both calculated using 3 � 3 � 1 supercell based on the relaxed unit cell. To get a more accurate thermal conductivity, the third-nearest neighbors serve as the interaction cutoff in the third-order (anhar monic) force constants. The lattice thermal conductivity from ShengBTE code is calculated by: kαl ;β ¼
X 1 f0 ðf0 þ 1Þðℏωλ Þ2 vαλ Fβλ kB T 2 ΩN λ
(5)
where Ω is the volume of the unit cell, N is the number of q points uniformly sampled in the Brillouin zone, ωλ is the angular frequency of the phonon mode λ, vλ is the phonon group velocity, f0 is the Bose
–Einstein distribution function, and Fβλ is the projection of the mean free displacement along the β direction. 3. Results and discussion 3.1. Geometry and electronic structure
Table 1 The calculated lattice constant a and bond lengths of Pd-As and As–As, as well as the cohesive energy. Present work Yuan et al. [27] Qian et al. [28]
��
We get PdAs2 monolayer by incising its bulk crystal with space group of Pa3. By geometry optimizations, PdAs2 monolayer has a planar pentagonal structure with space group of P4/mbm. In Fig. 1(a), we illustrate the top and side view of the 2D structure of PdAs2. PdAs2 monolayer is composed by an infinite number of pentagons, and every pentagon contains two Pd atoms and three As atoms. The obtained 2
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Fig. 2. The electronic structures of PdAs2 monolayer calculated by PBE method, with (a) and without (b) spin-orbit coupling (SOC), respectively. The electronic structures of PdAs2 monolayer calculated by HSE06 (c) and HSE06 þ SOC (d) are also presented.
optimized lattice constant a is 6.19 Å, well consistent with other results [28]. The calculated bond lengths and cohesive energy of the PdAs2 monolayer are listed in Table 1. Obviously, these results agree well with previous ones, indicating that the geometry structure of PdAs2 mono layer in our work is reliable. To prove the dynamic stability for the PdAs2 monolayer, we display its phonon dispersion spectra in Fig. 1(b). It can be seen that, there is no imaginary frequency mode, suggesting that PdAs2 monolayer is dynamic stable. From the phonon spectra, there are 18 branches, including three acoustic branches and 15 optical branches. According to phonon spectra, we can see that the largest frequency is 9.61 THz. It is obvious that the frequency has two phonon gaps between 6 THz and 9 THz. In Fig. 2, we illustrate the energy band structure of PdAs2 monolayer obtained from different methods. Fig. 2 (a) and (b) show the electronic structure calculated by PBE and PBE þ SOC, and Fig. 2 (c) and (d) show the electronic structure calculated by HSE06 and HSE06 þ SOC. It is obvious from Fig. 2(a) that PdAs2 monolayer has a direct band gap up to 0.32 eV around the high symmetric point path S, and from Fig. 2(b), the band gap is 0.34 eV, which indicating that SOC has a weak influence on the band gap. As is known that, PBE method underestimates the band gap [49,50], so we also calculate the energy band structures using HSE06 and HSE06 þ SOC methods. Band gaps from HSE06 and HSE06 þ SOC are 0.78 eV and 0.80 eV, respectively, consistent with the band gap 0.80 eV by Yuan et al. [27] and 0.81 eV by Qian et al. [28] Compared to the penta-PdN2 with no band gap and the penta-PtN2 with a narrow 75 meV band gap, the PdAs2 monolayer has a wider band gap. In all, PdAs2 monolayer is a direct band-gap semiconductor with a modest band gap, and the band gaps are insensitive to spin-orbit coupling (SOC). The electronic structures of PdAs2 monolayer under tensile strain are
also calculated from HSE06 method. Fig. 3(a–e) shows the energy band structures with tensile strains of 2%, 4%, 6%, 8% and 10%, respectively. At different tensile strains, the energy band structures of PdAs2 mono layer have obvious difference. As the strain increases, the band gap first decreases and then increases. The band gap reaches a maximum when the strain is 2%. When the strain is more than 2%, the band gap grad ually decreases with the increase of strain, and when the strain reaches 10%, the band gap becomes 0.01 eV. It can be seen from Fig. 3 that, compared with the change of valence band, the change of conduction band is very obvious. When the strain is larger than 2% the conduction band decreases with the increase of strain, which mainly leads to the reduction of band gap. To better intuitively feel the band gap changes, we show the relationship between band gap and strain in Fig. 3(f). In general, the effect of tensile strain on the band gap of PdAs2 monolayer is very obvious, which indicates that PdAs2 monolayer may be a good application prospect in the electronic devices. In Fig. 4, we illustrate the calculated total and partial density of states of monolayer PdAs2. Fig. 4(a) shows the contribution of Pd atom and As atom to the total density of states. It is seen from Fig. 4(a) that, near the Fermi level, the contribution of Pd atom and As atom to the conduction band is almost the same, while the contribution of As atom to the valence band is obviously greater than that of Pd atom. In order to judge the contribution of the orbitals to the energy band near the Fermi level more accurately, we show the contribution of the orbitals of Pd and As atoms to the total density of state in Fig. 4(b), from which we can see that the density of states near the Fermi level is mainly contributed by d orbitals of Pd atom and p orbitals of As atom.
3
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Fig. 3. The energy band structures of PdAs2 monolayer under different tensile strains: (a) 2% strain, (b) 4% strain, (c) 6% strain, (d) 8% strain, and (e) 10% strain, together with the relationship between band gaps and strains (f).
Fig. 4. The contributions of atoms (a) and orbitals (b) to the total density of states of PdAs2 monolayer.
3.2. Elastic properties
Table 2 The calculated elastic constants (C11, C12 and C66), Young’s modulus Y and Poisson’s ratio ν (all in N/m). Present work Qian et al. [28]
C11
C12
C66
Y
ν
93.0 92.6
33.1 32.4
23.7 23.7
81.2 81.2
0.36
We calculate the elastic constants of monolayer PdAs2 by elastic matrix method. For the 2D materials, it is accepted that the elastic constants matrix is a six order square matrix containing four non-zero elastic constants, C11, C12, C22 and C66. For monolayer PdAs2, the elastic constant C11 is equal to C22, so there are three independent elastic constants (C11, C12, and C66). In Table 2, we list the calculated elastic constants. Our results are consistent well with previous work. Young’s modulus is the same along the x and y directions. Young’s modulus Y and Poisson’s ratio ν of the 2D material, as important criterion of me chanical stability, is calculated by Ref. [51]: 4
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3.3. Optical properties We obtained the real and imaginary parts of the dielectric function of monolayer PdAs2 by HSE06 method, as are displayed in Fig. 6. Since the monolayer PdAs2 has no anisotropy in x and y directions, we only analyze the real and imaginary parts of the dielectric function in x di rection. When the incident photon has zero energy, the static dielectric constant is obtained from the real part of the dielectric function. Fig. 6 (a) shows the real part of the dielectric constant under three different strains, where black represents no strain, red represents 3% strain, and blue represents 6% strain. We can see from Fig. 6(a) that the static dielectric constant is 5.24 for no strain, 5.94 for 3% strain and 9.37 for 6% strain. Obviously, as the strain increases, the static dielectric con stant increases. As the energy of the incoming photon increases, the real part increases to a peak and then decreases rapidly, remaining almost flat at high energies. In the absence of strain, a sharp decrease in the real part between 1 eV and 5 eV, which corresponds to a sharp increase in the absorption coefficient because the absorption of photon energy by electrons in the band transition increases. In the equilibrium state, the maximum peak value of the dielectric function is 6.96, corresponding to the incident photon energy of 1.03 eV. Under strain conditions, the peak of the dielectric function appears in advance and corresponds to a larger value, 7.78 for 3% at 0.78 eV and 10.83 for 6% at 0.41 eV. The imagi nary part of the dielectric function reflects transitions between energy bands near Fermi energy levels, which determine the linear response of the material to light under small wave vectors. From Fig. 6(b), in the absence of strain, we can see that the imaginary part of monolayer PdAs2 has four major peaks at 1.32 eV, 1.94 eV, 2.76 eV, 3.31 eV, respectively. With the increase of strain, the lower the energy of the incident photon corresponding to the first peak of the imaginary part is, an obvious redshift occurs. When the natural frequency of the incident wave is consistent with the natural frequency of the atoms in the crystal, light will be absorbed. When the light travels through the material, its intensity will be more or less absorbed. We research the absorption spectrum of light based on imaginary parts of the dielectric function of monolayer PdAs2, as is displayed in Fig. 7(a). It is seen that when there’s no strain, the energy range of the absorption spectrum is 1 eV–15 eV. Absorption spectra have the maximum absorption peak at 5.1 � 105 cm 1eV. The absorption range of PdAs2 monolayer is very wide, including ultraviolet, visible, and infrared regions. And the absorption range of PdAs2 monolayer is mainly visible and ultraviolet light. When the strain is 3% and 6%, the maximum absorption peak is 4:7 � 105 cm 1 eV and 43 � 104 cm 1 eV, which indicates that the light absorption coefficient of PdAs2 monolayer decreases with the increase of strain. From the absorption spectrum, the absorption edge of monolayer PdAs2 under three different strains is 0.30 eV at no strain, 0.21 eV at 3% strain and 0.07 at 6% strain, which in dicates that there is an obvious redshift in the absorption spectrum under tensile strain. To show why the absorption coefficient decreases
Fig. 5. The relationship between strains and pressures for PdAs2 monolayer, where the black and red curves are for the biaxial and uniaxial, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Y10 ¼ Y01 ¼ C11 C12
C212
��
� C22 ; v ¼ C12 C11
(6)
Moreover, sheer modulus G is simple equal to C66. Young’s modulus, sheer modulus G and Poisson’s ratio ν in this work is 81.2 N/m, 23.7 N/ m and 0.36, respectively. These results are also listed in Table 2. Our results are completely consistent with previous results. For a 2D monolayer, according to Born–Huang criteria [52], its mechanically stability can be judged by the following formula: C11 C12
C212 > 0; C66 > 0
(7)
From Table 2, the elastic constants of monolayer PdAs2 satisfy the above criteria very well, confirming that the PdAs2 monolayer is me chanically stable. Elastic limit of the monolayer PdAs2 can be estimated by stress-strain cure. We calculated the biaxial and uniaxial stress-strain cures, as is illustrated in Fig. 5. During the calculations of the uniaxial stress-strain cures, the x-axis tensile strain is just considered since the lattice con stants of monolayer PdAs2 in the x and y directions are the same, which implies that PdAs2 monolayer has no anisotropy. This is consentient with the equality of elastic constants C11 and C22, so the monolayer PdAs2 has no anisotropy in x and y direction. It can be seen from Fig. 5 that the monolayer PdAs2 bears the biggest stress is 5.91 N/m for the biaxial strain and 6.46 N/m for the uniaxial strain corresponding to 12% strain and 16% strain, respectively. Compared with penta-PdN2 (17.5%) and penta-PtN2(15%) [26], the monolayer PdAs2 has weak tensile resistance.
Fig. 6. The real part (a) and imaginary part (b) of the dielectric function of PdAs2monolayer. 5
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Fig. 7. Optical parameters of PdAs2 monolayer under different tensile strain: (a) absorption coefficient, (b) energy loss spectrum, (c) reflectivity, and (d) refractivity.
Fig. 8. (a) Lattice thermal conductivity varies with temperature and (b) the heat capacity varies with temperature for PdAs2 monolayer.
under tensile strain, let’s analyze the energy loss spectrum, reflectivity and refractivity, as are showed in Fig. 7(b–d). From energy loss spectra, we can see that the energy loss is about the same for all three strains, and the energy loss is mainly distributed between 2 eV and 13 eV. There is a rapid increase in the energy loss spectrum before it reaches its maximum, which corresponds to a rapid decrease in the refractivity and reflectivity. Between 0 eV and 5 eV, the energy loss spectrum has a relatively small value, while the reflectivity and refractive index have a large value, which indicates that most of the energy of the incident photon is reflected and refracted and only a small part is absorbed by the material. Obviously, as the strain increases, the reflectivity increases gradually. When there is no incident light, the reflectivity of the three strains is 0.15, 0.18 and 0.26 for different strains, respectively. Like the reflectivity, the refractive index also increases with the increase of strain, when there is no incident light. The refractivity of the three strains is 2.29, 2.44 and 3.06, respectively. In general, as the strain in creases, the refractive index and reflectivity increase, implying that when the incident light shines on the material, the larger the strain, the
more light will be reflected and refracted. The reason for the decrease of absorption coefficient under strain is that the reflectivity and refractivity increase with the increase of strain. 3.4. Lattice thermal conductivity We obtained lattice thermal conductivity of monolayer PdAs2 based on solving the Boltzmann transport equation using the iterative solution method. The variation of lattice thermal conductivity with temperature is shown Fig. 8(a). The lattice thermal conductivity decreases with the increase of temperature, and the decreasing rate decreases gradually. The lattice thermal conductivity is 1.04 W/(mK) at 300 K, which is lower than that of the penta-PdX2 (X ¼ S, Se, Te) monolayers [25]. The heat capacity increases gradually with the rise of temperature, and the rate is proportional to T3, which remains basically unchanged after 600 K, as is shown in Fig. 8(b). Further, the lattice thermal conductivity varying with the mean free path is shown in Fig. 9(a). We show the phonon mean free path and 6
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Fig. 9. (a) The relationship between the cumulative lattice thermal conductivity and the free path at 300 K, 500 K and 900 K; (b) The relationship between effective phonon mean free path and the temperature for PdAs2 monolayer.
Fig. 10. (a) The phonon group velocity of three acoustical branches ZA, TA and LA, respectively. (b) The phonon scattering rate of three acoustical branches and partial optical branches for PdAs2 monolayer.
cumulative lattice thermal conductivity at three temperatures respec tively. The cumulative lattice thermal conductivity increases with the increase of the phonon average free path and remains unchanged after the free path is large enough. The cumulative lattice thermal conduc tivity at 300 K, 400 K and 500 K is 1.04 W/(mK), 0.65 W/(mK) and 0.37 W/(mK), respectively. At room temperature, the phonon average free path only contributes to the lattice thermal conductivity between 1 nm and 410 nm, and this distance decreases with the increase of tempera ture. In order to better study the influence of phonon mean free path on lattice thermal conductivity, we fitted the data through the following formula [53,54]: �� � �n � kl ðLÞ ¼ Kmax 1 þ Λeff L (8)
are 18 vibration modes including three acoustic branches and 15 optical branches, The three acoustic modes are the longitudinal (LA) mode, the transverse (TA), and the flexural mode (ZA). We analyzed the contri bution of different vibration modes to the lattice thermal conductivity at 300 K, concluding that the contribution of three acoustic modes (ZA, TA, LA) is greater than 15 optical modes, accounting for 61%. For the monolayer PdAs2 at 300 K, the contributions of the three acoustic modes (ZA, TA, LA) to the total lattice thermal conductivity is 32%, 16% and 13%, respectively. Therefore, in the following study, we mainly consider the influence of three acoustic branches on the lattice thermal conduc tivity. From the lattice thermal conductivity formula, we know that the group velocity is an important parameter to determine the lattice ther mal conductivity. The group velocity with respect to the phonon vi bration frequency is showed in Fig. 10(a). The ZA mode has a minimum group velocity consistent with its maximum contribution to the lattice thermal conductivity. Obviously, the group velocity which is lower than those of penta-PdX2 (X ¼ S, Se, Te) [25] monolayers mainly varies from 0 to 2.5 Km/s, which implies that the small group velocity is one of the reasons for the low lattice thermal conductivity. In addition, we calcu late the phonon scattering rate ωλ of monolayer PdAs2 with three acoustic branches and partial optical branches at room temperature (Fig. 10(b)). With the increase of frequency, the scattering rate of the three acoustic branches gradually increases, and finally stays between 0.1 ps 1 and 1 ps 1. The scattering rates of the optic modes are only in the high-frequency regions, and the scattering rate mainly ranges from 0.1 ps 1 and 1 ps 1. Obviously, the acoustic mode is only in the low frequency region while the optical mode is in the high frequency region,
where Λeff is effective phonon mean free path, L is the length, and kl is the lattice thermal conductivity after convergence. Λeff and n are ob tained after fitting the curve. At room temperature we get that Λeff and n are 11.42 and 2. Effective phonon mean free path at other temperature is displayed in Fig. 9(b). It is seen that effective phonon mean free path is decreases with increasing temperature, corresponding to the change of thermal conductivity with temperature. From the diagram we can see that there is a small effective phonon mean free path, suggesting that monolayer PdAs2 may have a low lattice thermal conductivity. To explore the reason why the lattice thermal conductivity of monolayer PdAs2 is lower than that of other pentagonal twodimensional materials, we analyze the phonon group velocity and scattering rate. From the phonon scattering spectra, we know that there 7
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suggesting that the acoustic mode contributes a lot to the thermal con ductivity. The phonon scattering rate of PdAs2 monolayer is mainly distributed between 0.1 ps 1 and 1 ps 1, which is relatively high in the pentagonal two-dimensional material. Thus higher phonon scattering rate ωλ of Pd monolayer is one of the reasons for the low thermal conductivity.
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4. Conclusions We have systematically studied the electronic structure, elastic, op tical and thermal transport properties of monolayer PdAs2 from firstprinciples calculations. By phonon dispersion spectrum calculation, we conclude that PdAs2 monolayer is dynamic stable. Our electronic structure calculations show that monolayer PdAs2 is a semiconductor with a direct band gap of 0.78 eV. Tensile strain has a good regulating effect on the band gap, which can be adjusted from 0.01 eV to 0.84 eV. The adjustable band gap implies that monolayer PdAs2 has a good application prospect in the field of electronic devices. The partial density of states of the monolayer PdAs2 shows that the valence band maximum is mainly due to the p orbitals of As atoms and the conducting band minimum due to the p orbitals of Pd atoms and the p orbitals of As atoms. We also calculate the elastic constants of mono layer PdAs2, confirming that the PdAs2 monolayer is mechanically sta ble. The monolayer PdAs2 bears the biggest stress is 5.91 N/m for biaxial strain and 6.46 N/m for uniaxial strain corresponding to 12% strain and 16% strain. The calculation of the optical properties reveals that the energy range of the absorption spectra is 1 eV–15 eV. Absorption spectra have the maximum absorption peak at 5.1 � 105cm 1eV. The absorption range of PdAs2 monolayer is very wide, including ultraviolet, visible, and infrared regions. The absorption range of PdAs2 monolayer is mainly infrared region. Under the tensile strain, the absorption spectra show obvious redshift. With the increase of strain, the reflectivity and refractivity increase. In addition, we have calculated the lattice thermal conductivity of monolayer PdAs2. Our results show that, at room temperature, the thermal conductivity of monolayer PdAs2 is 1.04 W/(mK), which is lower than that of known pentagonal monolayer. The main reasons of low thermal conductivity are low effective free path, low phonon group velocity and high phonon scattering rates. In general, PdAs2 monolayer has a promising application prospect in the field of semiconductor de vices due to its unique electronic, optical and thermal transport properties. Acknowledgment The authors are grateful for the supports of the NSAF (Grant No. U1830101), the Science Challenge Project (Grant No. TZ2016001), and the National Natural Science Foundation of China (Grant No. 11504035). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.ssc.2019.113802. References [1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I. V. Grigorieva, A.A. Firsov, Science 306 (2004) 666–669. [2] A. Bablich, S. Kataria, M. Lemme, Electronics 5 (2016) 13. [3] A.N. Gandi, U. Schwingenschl€ ogl, Europhys. Lett. 113 (2016) 36002. [4] S.J. Kim, K. Choi, B. Lee, Y. Kim, B.H. Hong, Annu. Rev. Mater. Res. 45 (2015) 63–84.
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