Structural stability, electronic, optical and lattice thermal conductivity properties of bulk and monolayer PtS_2

Journal Pre-proof Structural stability, electronic, optical and lattice thermal conductivity properties of bulk and monolayer PtS 2 Hamza A H Mohammed, GM Dongho-Nguimdo, Daniel P Joubert

PII:

S2352-4928(19)30340-X

DOI:

https://doi.org/10.1016/j.mtcomm.2019.100661

Reference:

MTCOMM 100661

To appear in:

Materials Today Communications

Received Date:

29 June 2019

Revised Date:

22 September 2019

Accepted Date:

24 September 2019

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Structural stability, electronic, optical and lattice thermal conductivity properties of bulk and monolayer PtS2 Hamza A H Mohammeda,b,∗ , GM Dongho-Nguimdoc and Daniel P Jouberta a The

National Institute for Theoretical Physics, School of Physics and Mandelstam Institute for Theoretical Physics, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa. b Department of Physics, Shendi University, Shendi, Sudan. c College of Science, Engineering, and Technology, University of South Africa (UNISA), South Africa.

ABSTRACT

Keywords: Bulk and monolayer Stabilities Absorbance Thermal conductivity

In the present study, we used density functional theory to study the cohesive and formation energies, mechanical and dynamical stabilities, optical properties and lattice thermal conductivity of trigonal bulk and monolayer PtS2 . A careful investigation of the electronic band structure and density of states of the bulk and monolayer PtS2 shows that both are indirect band-gap semiconductors. A manybody GW0 approximation on top of density functional theory results was employed to calculate the fundamental gap of the bulk and monolayer PtS2 , while the optical properties were explored with the Tamm-Dancoff approximation of the Bethe-Salpeter equation. Our calculations show that the maximum absorbance, in the visible range, of a monolayer is substantially higher compared to that of single layer in the bulk. Boltzmann transport equations were implemented to examine lattice thermal conductivity over a wide range of temperatures. We found that at 300 K the in-plane and out-of-plane bulk lattice thermal conductivities are 2.30×10−8 and 0.08×10−8 Wm−1 K−1 respectively for a single bulk layer, whereas the in-plane lattice thermal conductivity of a single monolayer is 0.15 × 10−8 Wm−1 K−1 . This shows a substantial decrease in the thermal conductivity in the monolayer compared to the bulk. The low value of the lattice thermal conductivity in the out-of-plane direction for the bulk suggests that bulk PtS2 merits further investigation as a candidate for thermoelectric applications.

1. Introduction

Bulk and monolayer PtS2 crystallise in a CdI2 structure with ̄ space group 𝑃 3𝑚1 (No. 164) [15]. Previous spectroscopic studies at 300 K show that bulk PtS2 is a semiconductor with an indirect band-gap of 0.95 and 0.87 eV parallel and perpendicular to the trilayers, respectively [17, 18]. Electronic structure calculations of infinite periodic bulk PtS2 confirmed that bulk and monolayer PtS2 are indirect bandgap semiconductors [11, 12, 13]. Recently, thermoelectric generators have attracted tremendous attention since they can convert heat energy into electri2 𝜎𝑇 cal energy. The dimensionless figure of merit 𝑍𝑇 = (𝜅𝑆 +𝜅 , 𝑒 𝑙) is used to describe the efficiency of a thermoelectric material. Here, 𝑆 is the Seebeck coefficient, 𝜎 is the electrical conductivity, 𝑇 is the temperature, 𝜅𝑒 is the electronic thermal conductivity and 𝜅𝑙 is the lattice thermal conductivity [19, 20]. The higher the figure of merit of a material, the better its potential as a thermoelectric material. A high figure of merit requires a low total thermal conductivity. In this report, we show that both the bulk and the monolayer PtS2 have a relatively low lattice there thermal conductivity. In this study we investigate the structural, electronic, optical and lattice thermal conductivity of bulk and monolayer PtS2 using DFT and post-DFT techniques. In Section 2, we describe the computational methods. The results and discussions are presented in Section 3, and in Section 4 a summary and conclusion follows.

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The environmental challenges posed by fossil energy have stimulated extensive research on green energy [1]. Transition metal dichalcogenides (TMDs), in layered structures, have diverse properties that extend beyond those of graphene. They exhibit a wide range of unique optical [2, 3], electrical [4, 5], mechanical [6] and thermal properties [7] which make them promising materials for energy generation as well as optoelectronics applications [8]. The general formula for the layered TMDs family materials is MX2 , where M is a transition metal and X a chalcogenide atom (X = S, Se and Te). Among the MX2 compounds, MoX2 , for instance, has been reported to have great potential in the field of medicine, microelectronics, lithium battery and hydrogen storage because of its excellent mechanical stability, adsorption capacity and biocompatibility [9]. The exceptional optical, electronic and mechanical properties of WS2 make it an excellent material for applications in catalysis, energy storage and medical devices [10].

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ARTICLE INFO

PtX2 also belong to this family of layered TMD materials. The structural, electronic and optical properties of PtX2 have been well investigated [11, 12, 13] and they are predicted to have promising photocatalytic properties [14], but their thermoelectric properties are yet to be well understood. ∗ Corresponding author at: The National Institute for Theoretical Physics, School of Physics and Mandelstam Institute for Theoretical Physics, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa. Tel.: +27 117176854; fax: +27 117176879. [email protected] (H.A.H. Mohammed) ORCID (s): 0000-0002-6763-3844 (H.A.H. Mohammed)

Hamza A H Mohammed et al.: Preprint submitted to Elsevier

2. Methodology and Computational details The density functional theory (DFT) calculations were carried out using the projector augmented wave (PAW) method Page 1 of 11

Structural stability, electronic, optical and thermal properties of bulk and monolayer PtS2 (a)

(c)

(b)

(d)

c

a c b

b

a

Figure 1: The crystal structure of PtS2 bulk and monolayer. (a) unit cell of bulk structure, (b) unit cell of monolayer structure, (c) monolayer top view and (d) monolayer side view. The grey and yellow balls represent the platinum and sulphur atoms.

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and the PBE approximations for the bulk and monolayer, respectively. Figure 2 depicts the EOS fitted to the 3rd-order Birch Murnaghan equation of state [33]. Optimized lattice parameters (𝑎, 𝑏, 𝑐) and equilibrium volume (V0 ) of the bulk and monolayer structures are presented in Table 1. The lattice parameter of bulk PtS2 , 𝑎 = 3.55Å and 𝑐 = 5.03Å obtained by using optB86b-vdW are similar to 𝑎 = 3.54Å and 𝑐 = 5.04Å from the experimental measurements [34, 35, 36]. The obtained lattice parameters for the monolayer are close to the results from other studies. Note that for the PBE bulk results, the lattice parameter 𝑐, perpendicular to the layers, is relatively large compared to the experimental values, while the optB86b-vdW value is close to the experimental values. The in-plane parameters are similar. The optB86b-vdW takes into consideration the long-range interactions which are not captured by the PBE, approximation, which may account for the difference. Cohesive and formation energies were examined at the equilibrium structures. The cohesive energy is defined as the negative of the energy required to decompose a compound into its constituent atoms, infinitely apart at rest,

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[21], as implemented in the Vienna 𝑎𝑏-𝑖𝑛𝑖𝑡𝑖𝑜 Simulation Package (VASP) [22, 23].The generalized gradient approximations (GGA) with the Perdew-Burke-Ernzerhof (PBE) [24] flavour was employed as exchange-correlation functional. We also included the optB86b-vdW [25] dispersion term to take into account the layered nature of the material in the calculations. Kinetic energy cut-off of 520 and 350 eV were used for the optimization of the bulk and monolayer, respectively. Brillouin-zone integrations were performed using a Γ centred 8 × 8 × 6 and 12 × 12 × 1 𝑘-point mesh within the Monkhorst-Pack scheme [26] for electronic structure and density of states calculations for bulk and monolayer, respectively. In order to limit the interaction between monolayers in a supercell, we set a vacuum of 15 Å between the periodically repeated layers. The energy convergence criterion was set to 1×10−8 eV, and the inter-atomic forces were converged to less than 1 × 10−3 eV/Å for both bulk and monolayer systems. We calculated the band structure, total density of states and partial density of states with both the optB86bvdW and PBE for the bulk and monolayer structures. For comparison, we used the hybrid function (HSE03) [27] and GW0 method [28], the time-dependent Hartree-Fock (TDHF) [29] and the meta GGA functional (MBJ) [30] to calculate band gaps. The Bethe-Salpeter equation (BSE) within the Tamm-Dancoff approximation [31] was used to calculate optical properties. Highly converged forces were required for the calculations of the dynamical matrix using the direct force constant approach as implemented in the PHONOPY code [32]. We employed a 4×4×4 and 5×5×1 supercell for bulk and monolayer, respectively. For the calculation of the thermal conductivity, we use the PHONO3PY code [32] to solve the single-mode relaxation-time approximation of the linearised phonon Boltzmann transport equation.

(PtS )

(PtS ) 𝐸coh 2

=

(Pt) (S) 𝐸comp.2 − 𝑁 × (𝑛𝐸atom + 𝑚𝐸atom )

𝑁 × (𝑛 + 𝑚)

Figure 1 shows the bulk and monolayer (a) unit cell of bulk structure, (b) unit cell of monolayer structure, (c) monolayer top view and (d) monolayer side view. The grey and yellow balls represent the Platinum and Sulfide atoms. To determine the equilibrium structure, we calculate the energyvolume E(V) equation of state (EOS) using the optB86bvdW Hamza A H Mohammed et al.: Preprint submitted to Elsevier

(1)

where 𝐸coh is the total energy per unit cell 𝑁 is the numS Pt ber of PtS2 formula units per unit cell, 𝐸atom and 𝐸atom are the atomic energies of Pt and S, while 𝑛 and 𝑚 are the number of Pt and S atoms per unit cell, respectively. Obtained cohesive energies for bulk and monolayer, are summarised in Table 1. The formation energy, 𝐸form , is a test for relative phase stabilities for any compound. Formation energy is the negative of the energy required to decompose a compound into its most stable constituent compounds. It can be expressed in terms of cohesive energies as

3. Results and discussions 3.1. Structural parameters

,

(PtS ) 𝐸form2

(PtS ) 𝐸coh 2

[ 𝑛𝐸 (Pt) + 𝑚𝐸 (S) ]

coh . (2) 𝑛+𝑚 The results of the calculated formation energy, for bulk and monolayer, are presented in Table 1. Negative formation energy indicates that the compound is stable with respect to decomposition into its constituent stable components. All our calculated cohesive and formation energies are negative, which confirms that the bulk and monolayer compounds

=

−

coh

Page 2 of 11

Structural stability, electronic, optical and thermal properties of bulk and monolayer PtS2 (b)

(a) -14.005

-25.95

-15.1 PBE

-12.3

-26

PBE -15.2

-14.01 -26.05

-15.3 Ecoh(eV/atom)

Ecoh (eV/atom)

-26.1

-14.015

-26.15 52

-14.02

54

56

58

60

-15.4 140

160

-12.9

optB86b-vdW

-14.025

-12.6

180

200

optB86b-vdW

-14.03

52

53

54 55 3 V0 (A /atom)

56

57

-13.2 100

58

120

140 160 3 V0 (A /atom)

180

200

f

51

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Figure 2: Calculated total cohesive energy versus cell volume PtS2 compound using optB86b-vdW and PBE approximations (a) bulk and (b) monolayer.

𝑐 (Å)

PBE optB86b-vdW Exp[34] Exp[35] Exp[36]

3.51 3.55 3.54 3.54 3.54

5.35 5.03 5.04 5.04 5.04

PBE optB86b-vdW Previous work[37] Previous work[12] Previous work[11] Previous work[38]

3.58 3.58 3.57 3.58 3.52 3.57

– – – – – –

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monolayer

Pr

bulk

𝑉𝑜 (Å3 )

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are energetically stable. The bulk and monolayer cohesive and formation energies per atom, respectively, are similar for each approximation, which is consistent with the understanding that the inter-layer bonding is weak and its contribution to the total energy is relatively small.

3.2. Mechanical stability Elastic constants give the lowest order relationship between stress and strain when a structure is subjected to deformation. For bulk PtS2 , in a trigonal crystal structure, point group 32, there are six independent elastic constants, namely, 𝐶11 , 𝐶12 , 𝐶13 , 𝐶33 , 𝐶44 and 𝐶66 [39]. Our calculated elastic constants are presented in Table 2. Young’s modulus (𝐸), shear modulus (𝐺), and bulk modulus (𝐵), were obtained with Hill’s formula [40]. The Young’s moduli is a function of two variables; bulk and shear moduli- which are proportional to the elastic constants. The relations between the isotropic bulk and shear moduli, and the elastic constants Hamza A H Mohammed et al.: Preprint submitted to Elsevier

𝐸𝑐𝑜ℎ (eV)

𝐸𝑓 𝑜𝑟𝑚 (eV)

56.92 54.87 54.79 54.78 54.76

-4.21 -3.60 – – –

-0.34 -0.45 – – –

– – – – – –

-4.23 -3.43 – – – –

-0.36 -2.04 – – – -0.33

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𝑎 (Å)

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Table 1 Calculated equilibrium lattice parameters (𝑎, 𝑐 and 𝑉0 ), cohesive energy (𝐸𝑐𝑜ℎ ) per atom and formation energy (𝐸𝑓 𝑜𝑟𝑚 ) per atom, of bulk and monolayer structure, compared to experimental and theoretical data.

in the Voigt notation are: 𝐵𝑉

=

1∕9(𝐶11 + 𝐶33 ) + 2∕9(𝐶12 + 𝐶13 ),

(3)

𝐺𝑉

=

1∕15(𝐶11 + 𝐶33 ) − 1∕15(𝐶12 + 𝐶13 ).

(4)

In the Reuss notation, they are written 1∕𝐵𝑅

=

(2𝐶11 + 𝐶33 ) + 2(𝐶12 + 2𝐶13 ),

15∕𝐺𝑅

=

4(𝐶11 + 𝐶33 ) − 4(𝐶12 + 2𝐶13 ) + 3(𝐶44 + 𝐶66 (6) ).

(5)

According to Hill, the bulk and shear moduli are a average of the Voigt and Reuss expressions, [41]: 𝐵𝐻 = 1∕2(𝐵𝑉 + 𝐵𝑅 ),

(7)

𝐺𝐻 = 1∕2(𝐺𝑉 + 𝐺𝑅 ),

(8)

and the Young’s modulus can be expressed as: 𝑌𝐻 =

9𝐵𝐻 × 𝐺𝐻 . (3𝐵𝐻 + 𝐺𝐻 )

(9) Page 3 of 11

Structural stability, electronic, optical and thermal properties of bulk and monolayer PtS2 Table 2 The calculated elastic constants (𝐶𝑖𝑗 ), Bulk modulus (B) Shear modulus (G),Young’s modulus (Y) and Poisson’s ratio (𝜈), for bulk and monolayer. 𝐶11

𝐶12

𝐶13

𝐶14

𝐶33

𝐶44

𝐵 (GPa)

𝑌 (GPa)

𝐺 (GPa)

𝜈

PBE optB86b-vdW

185.79 130.43

61.73 62.52

20.66 23.41

17.05 16.99

13.87 11.46

62.03 26.49

41.50 31.84

65.39 55.85

26.43 23.12

0.24 0.21

PBE optB86b-vdW Other work[12]

81.86 88.22 120.4

24.25 26.83 32.10

– – –

– – –

– – –

– – –

– – –

74.67 80.06 80.59

– – –

0.29 0.30 0.27

bulk

monolayer

(10)

where 𝐸𝑠 is the strain energy and 𝜀 is the uniaxial strain. The two dimensional expression for 𝑌 can also be written in terms of elastic constants as [12, 42, 43, 44, 45] 2 2 𝑌 = (𝐶11 − 𝐶12 )∕𝐶11 ,

(11)

f

Pr

and the two dimensional Poisson’s ratio can be expressed as

oo

( 2 ) 1 𝜕 𝐸𝑠 , 𝐴0 𝜕𝜀2

pr

𝑌 =

were computed along several high symmetry directions. Figure 3 and Figure 4 show the phonon dispersion relations of PtS2 bulk and monolayer, respectively, calculated at the equilibrium configuration with the PHONOPY code [32]. The phonon frequencies are positive throughout the Brillouin zone, which indicates that the bulk and monolayer compounds are dynamically stable. The phonon frequencies are in the range of 0 − 11.7 THz for bulk and 0 − 10.47 THz for the monolayer, respectively. The reduction in the frequency range for the monolayer is probably due to the reduction in a number of atomic bonds in the monolayer compared to the bulk. The highest frequency of the acoustic modes, defined here as the acoustic cut-off, are approximately 6 THz for both bulk and monolayer. The total and partial density of states respectively for bulk and monolayer are shown in Figure 3(bc) and Figure 4(b-c). It is evident that the Pt atom dominates in the low-frequency acoustic region, and the S atom contributes more to the high-frequency optical mode region.

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To calculate the Young’s modulus and Poisson’s ratio 𝜈 for a monolayer, we used two independent elastic constants, 𝐶11 and 𝐶12 [42]. Since the monolayer is two dimensional, the in-plane Young’s modulus depends on the equilibrium inplane area, 𝐴0 , of the supercell and the strain energy. The Young’s modulus is defined as

𝜈 = 𝐶12 ∕𝐶11 .

(12)

3.4. Electronic properties

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In Table 2, we present the elastic constants, bulk moduli, Young’s moduli, shear moduli and Poisson’s ratios. The obtained results show that in the bulk compound, Young’s modulus is greater than the bulk modulus, which suggests that the compound resists tensile strain more than uniform compression. The value of the Shear modulus indicates how a material responds to shear strain, while Poisson’s ratio (𝜈) supplies information on bonding nature. Our calculated bulk values of 0.24 and 0.21, for PBE and optB86b-vdW, respectively, predict a marginally ionic bonding for PtS2 bulk, while the calculated of 0.29 and 0.30 for the monolayer, suggest a more covalent type of bonding. A comparison of Young’s moduli in Table 2, suggests that the bulk has an in-plane stiffness which is less than for the monolayer.

3.3. Dynamical stability The bulk lattice parameters obtained from the optB86vdW approximation were closer to the experimental values than those from the PBE. Hence, the optB86-vdW was also used for the phonon calculation of the bulk system. For the monolayer, optB86b-vdW and PBE yielded the same lattice parameters. We therefore chosed to use only the PBE for the phonon calculations. The phonon dispersion relations Hamza A H Mohammed et al.: Preprint submitted to Elsevier

In this section, we present the electronic structure calculations for the bulk and monolayer. The band structure of the bulk and the monolayer PtS2 are shown in Figure 5(a) and 6(a), respectively. Bulk PtS2 is an indirect band-gap with a minimum of the conduction band at 𝐾 point and the maximum of the valence band at Γ point. For monolayer PtS2 , the minimum of the conduction band lies between Γ and 𝑀, while the maximum of the valence is between the 𝐾 and Γ points, indicating an indirect band-gap for bulk and monolayer. We also present the total density of states (TDOS) and partial density of states (PDOS) for each atom of PtS2 , bulk and monolayer, in Figures 5(b) and 6(b), respectively. S(p) has the largest contribution at the top of the valence band followed by Pt(d), for both bulk and monolayer. In the conduction band, orbital hybridization between the Pt(d) and the S(p) states occurs for bulk and monolayer. To investigate the intrinsic band-gap we used a number of approximations namely the optB86b-vdW, PBE, HSE03, GW0 , MBJ, BSE and TDHF. In the calculations involving post DFT approximations, we used the relaxed lattice parameters from the optB86b-vdW geometric optimisation in the case of bulk PtS2 and the values for the PBE calculation for the monolayer. The obtained values are summarised in Table 3 with previously calculated and experimental results inPage 4 of 11

Structural stability, electronic, optical and thermal properties of bulk and monolayer PtS2 (a)

(b)

(c)

12

12

Pt S

Frequency

TDOS 10

10

8

8

6

6

4

4

2

2

0 M

0 K

L

0

1 2 3 Total density of states

4 0

1 0.5 1.5 Partial density of states

2

f

Wave vector

A

Γ

6

6

4

4

2

2

Γ

K M Wave vector

0

pr

8

Γ

Pt S

e-

8

TDOS

Pr

10

al

Frequency

10

0

(c)

(b)

(a)

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Figure 3: Phonon dispersion of bulk PtS2 from optB86-vdW approximation. (a) phonon band structure, (b) phonon TDOS and (c) atomic contributions to acoustic and optical modes .

0

1

2 3 4 Total density of states

5 0

1 0.5 1.5 Partial density of states

2

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Figure 4: Phonon dispersion of monolayer PtS2 from PBE approximation. (a) Phonon band structure, (b) phonon TDOS and (c) atomic contributions to acoustic and optical modes.

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cluded for comparison. The measured band-gaps of this material vary from 0.7 to 0.95 eV, a relatively wide range of values. We observer that the optB86b-vdW, MBJ and HSE03 approximations give values within the range of the reported experimental results, while PBE, surprisingly, gives a value above the experimental range. Such unusual behaviour, where GGA DFT overestimates the band-gap of transition metal dichalcogenids has also been reported in previous studies

[50]. Note that all the DFT gap energies are approximate fundamental gap energies [51], while the experimental values are optical gap values. Direct comparisons are strictly not justified, but it is a common practice in the literature to compare these values. Experimental band-gap values for the bulk PtS2 have been reported by an number of groups: F. Parsapour et al. reported a band-gap of 0.87 eV using optical

Table 3 Calculated fundamental and experimental optical band-gaps (in eV) of PtS2 structures.

bulk monolayer

optB86-vdW

PBE

MBJ

GW0

HSE03

Exp.

Other calculations

0.81 1.67

1.06 1.80

0.83 —

1.10 3.90

0.75 3.92

0.87[18] , 0.95[46] , 0.7[47] 1.6[48]

0.48[48] ,1.2[15] ,0.73[37] 1.81[37] ,1.76[12] ,1.94[11] ,1.78[49] ,1.69[13]

Hamza A H Mohammed et al.: Preprint submitted to Elsevier

Page 5 of 11

Structural stability, electronic, optical and thermal properties of bulk and monolayer PtS2

(a)

(b) TDOS Pt(s) Pt(p) Pt(d) S(s) S(p)

4

E - EF (eV/atom)

2

0

-2

-4 M

K

Γ

A

L

H

A|L

M|K

H

0

2

Wave vector

4 Density of states

6

f

Γ

e-

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electronic eigenspectrum. The real part 𝜀1 (𝜔), can then be extracted from the Kramer-Kronig relations and the absorption coefficient 𝛼(𝜔) can be found from the dielectric function according to the expression [54] √ ]1∕2 2𝜔 [√ 2 𝛼𝛼𝛽 (𝜔) = 𝜀1,𝛼𝛽 (𝜔) + 𝜀22,𝛼𝛽 (𝜔) − 𝜀1,𝛼𝛽 (𝜔) . (13) 𝑐 Figure 7(a-c) show the imaginary and real parts of the dielectric functions, and the absorbance respectively, calculated at the Bethe-Salpeter equation (BSE) level of approximation. The calculation of the optical properties of the monolayer was performed for a supercell, where a single layer is included per supercell. For comparison of the bulk and monolayer, we calculate the optical absorbance, the percentage of the radiation absorbed, in the visible photon energy range, 1 − 3 eV, by a single-layer in the bulk and a monolayer. The calculated result for a single-layer of bulk and monolayer were obtained by using the absorbance, 𝐴 = 1 − 𝑒𝑥𝑝(−𝛼(𝜔)Δ𝑧), where 𝛼(𝜔) is the absorption coefficient and Δ𝑧 is the thickness of a slab of the material. For comparison with a monolayer, we set Δ𝑧 equal to the thickness of a single layer in the bulk. For a monolayer, the absorption coefficient scales like the inverse of the length of the supercell perpendicular to the layer for a sufficiently large supercell and the correct value of Δ𝑧 in a periodic supercell calculation is the length of the supercell perpendicular to the layer [55, 56]. Figure 7(c) shows the absorbance of a single bulk layer and a monolayer PtS2 . The maximum absorbance for in-plane polarisation for a monolayer is ≈ 13.15% at 2.5 eV and ≈ 2% at 2.9 eV for a single bulk layer. For out-of-plane polarisation, the maximum absorbance is ≈ 0.09% at 2.6 eV for monolayer and 0.9% at 3 eV for bulk. The absorbance of a monolayer is substantially higher than the absorbance of a single bulk layer. This is consistent with results for graphene and MoS2 [55, 56], where the monolayer absorbance is higher than the absorbance of a single bulk layer. The optical bandgap was estimated by fitting the BSE optical absorption with

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spectroscopy [18]; H. Tributsch and O. Gorochov measured a band-gap of 0.95 eV, by means of photoelectrochemical mechanisms [46]; and diffuse-reflectance measurements by F. Hulliger give a band-gap of 0.7 eV [47]. The band-gap of the bulk PtS2 has also been calculated by many authors: Y. Zhao et al. used the optB86b-vdW to predict a bandgap of 0.48 eV [48]; a band-gap of 1.2 eV was obtained by G. Guo and W. Liang [15] using the linear muffin-tin orbital method combined with the atomic sphere approximation method [52]; and the PBE calculations by H.L. Zhuang and R.G. Hennig give a 0.73 eV band-gap [37]. The only experimental band-gap of the monolayer PtS2 was reported by Y. Zhao et al [48]. They obtained, using the Fouriertransform infra red spectrometry, a band-gap of 1.6 eV. Theoretical monolayer gap values have also been reported. Bandgaps of 1.76 and 1.94 eV were predicted by J. Du et al. [12] using the PBE, with the spin orbit coupling included; and S. Ahmed [11] using the PBE, respectively. P. Miró et al. [49] used the dispersion corrected PBE exchange-correlation functional (PBE-BJ-D3) [24, 53] to obtain a band-gap of 1.78 eV, whereas the local density approximation calculations by Z. Huang give an estimation of 1.69 eV [13]. The difference between our band-gap results which we calculated with the optB86b-vdW and PBE approximations and the results of other calculations performed with the same approximations, can come from differences between DFT implementations and input variables including lattice parameters, 𝑘-points and energy cut-offs, after convergence test for all.

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Figure 5: Bulk PtS2 calculated electronic structure using optB86b-vdW. (a) Band structure along high symmetry directions, (b) TDOS and PDOS for each atom and orbital.

3.5. Optical properties When a material is exposed to electromagnetic radiation, an interaction will occur between the photons and electrons. An investigation of this interaction can reveal the energy structure of the material. In this paper, we explore the optical properties by studying the behaviour of the dielectric matrix. The imaginary part 𝜀2 (𝜔) of the dielectric function, can be calculated from excitation oscillator strengths and the Hamza A H Mohammed et al.: Preprint submitted to Elsevier

Page 6 of 11

Structural stability, electronic, optical and thermal properties of bulk and monolayer PtS2

(a)

(b) TDOS Pt (s) Pt (p) Pt(d) S (s) S(p)

4

E - EF (eV)

2

0

-2

-4 M

L Γ A Wave Vector

K

H

A|LM|K 0

2

4 Density of states

6

8

f

Γ

(a)

(b)

60

(c)

Bulk ε2(ω)xy

Bulk ε1(ω)xy

Bulk ε2(ω)z Mono ε2(ω)xy

Bulk ε1(ω)z Mono ε1(ω)xy

40

Mono ε2(ω)z

Mono ε1(ω)z

30

Bulkxy

12

Bulkz Monoxy Monoz

20

30

Pr

10

20 0

10

-10

0.5

1 2 1.5 Photon energy (eV)

2.5

3

-20

0

0.5

1 2 1.5 Photon energy (eV)

al

0

Absorbance %

e-

10

ε1(ω)

ε2(ω)

40

14

pr

50

50

0

oo

Figure 6: Monolayer PtS2 calculated electronic structure using PBE (a) Band structure along high symmetry directions, (b) TDOS and PDOS for each atom and orbital.

8

6

4 2

2.5

3

0

0

0.5

1 2 1.5 Photon energy (eV)

2.5

3

ur n

Figure 7: GW0 BSE results (a) imaginary part 𝜀2 (𝜔), (b) real part 𝜀1 (𝜔) of dielectric function and (c) absorbance for bulk and monolayer of PtS2 .

Jo

the Tauc plot [57]. The BSE approximations is expected to give a good estimation of the optical properties. In order to estimate the optical band-gap, we fitted an optical absorption spectrum from the BSE calculations by the Tauc plot [57]. For the bulk PtS2 , the optical band-gap for the in-plane and out-of-plane polarisations are 1.09 and 1.36 eV, respectively. The difference between the two values shows significant optical anisotropy. The value for in-plane polarisation gap is consistent with the experimental value. For the monolayer, optical band-gaps of 1.95 and 2.30 eV were predicted for the in-plane and out-plane polarisation, respectively. The in-plane optical gap is once again consistent with the experimental values as given in Table 4.

3.6. Lattice thermal conductivity In this work we used the single-mode relaxation-time approximation to the Boltzmann equation as implemented Hamza A H Mohammed et al.: Preprint submitted to Elsevier

in the PHONO3PY package [32]. Details of the implantation are discussed in reference [32]. In the single-mode relaxation-time approximation the lattice thermal conductivity can be expressed as 𝜅=

1 ∑ 𝐶 v ⊗ v 𝜆 𝜏𝜆 , 𝑍Ω 𝜆 𝜆 𝜆

(14)

where 𝑍 and Ω are the unit cell number and unit cell volume respectively. 𝜆 represents of the phonon mode. 𝐶𝜆 is the heat capacity 𝐶𝜆 = 𝑘 𝐵

( ℏ𝜔 )2 𝜆

𝑘𝐵 𝑇

𝑒(ℏ𝜔𝜆 ∕𝑘𝐵 𝑇 ) , [𝑒(ℏ𝜔𝜆 ∕𝑘𝐵 𝑇 ) − 1]2

(15)

where 𝜔𝜆 = 𝜔(q) is the phonon frequency, 𝑇 is the temperature, and 𝑘𝐵 and ℏ indicate the reduced Boltzmann constant Page 7 of 11

Structural stability, electronic, optical and thermal properties of bulk and monolayer PtS2 Table 4 BSE and TDHF estimated and experimental optical band-gaps in (eV) for PtS2 structures. BSE

PtS2

Exp.

in-plane

out-of-plane

in-plane

out-of-plane

1.09 1.95

1.36 3.50

1.07 2.37

1.19 3.05

3.7. Phonon lifetime

The phonon lifetime included in this study is a measure of the time between phonon-phonon scattering events and it is determined from third-order force constants [59, 32]. The phonon lifetime of bulk and monolayer PtS2 at 300 K are shown in Figure 9 as a function of frequency. The lifetime ranges are from 0 to 32 ps for PtS2 bulk, while for monolayer is from 0 to 27 ps. The acoustic mode lifetimes in the lowfrequency range, are relatively long compared to the optical modes in the high-frequency range. This is consistent with relatively strong scattering the the higher frequency optical modes. The lifetime distributions of the bulk and monolayer for in-plane almost are similar, except in out-of-plane which have a value of 10 ps for bulk, while it is 0 ps for monolayer. The phonon lifetime distribution can partly explain the difference between the acoustic and optical phonon mode contributions to the total lattice thermal conductivity [60].

Jo

ur n

al

Pr

𝜏𝜆 is the relaxation time or phonon lifetime. Figure 8 (a) and (b) show the temperature and frequency dependence of the lattice thermal conductivity for a single bulk layer and a monolayer. The normal unit for thermal conductivity is Wm−1 K−1 per unit cross-sectional area, which gives the thermal conductivity of a volume of material of a cross-sectional area of 1 m2 and 1 m thick, with heat conducted perpendicular to the cross-sectional area. For a single layer, the cross-sectional area for out-of-plane thermal conductivity is taken as 1 m2 , but the thickness is that of a single layer. The cross-sectional area for single layer in-plane thermal conductivity is that of the cross-sectional area of a single layer perpendicular to the plane of the layer. The ’thickness’ is taken as 1 m. For single layers we use units of Wm−1 K−1 per cross-sectional area (not per unit cross-sectional area), where the cross-sectional area and thickness depends on the direction in which the heat flow is measured. With this definition, the units for thermal conductivity are consistent for bulk and single layers. As shown in Figure 8(a), the average in-plane lattice thermal conductivity for a single layer at 300 K is 2.30×10−8 Wm−1 K−1 and 0.15×10−8 Wm−1 K−1 for bulk and monolayer, respectively. The out-of-plane thermal conductivity is 0.08×10−8 Wm−1 K−1 for a single layer of the bulk , whereas it is zero for the monolayer. The in-plane lattice thermal conductivity for a monolayer is ∼6% of that of a single bulk layer. This is a significant decrease and points to a possible technique for decreasing lattice thermal conductivity by decreasing the thickness to a few layers. Further, the highest in-plane value of lattice thermal conductivity found to be in the low-temperature range with 8.66×10−8 Wm−1 K−1 and 0.61×10−8 Wm−1 K−1 at 60 K for a single bulk layer and a monolayer, respectively. The highest out-of-plane thermal conductivity for a single bulk layer is 0.34×10−8 Wm−1 K−1 at 10 K. We observe that the lattice thermal conductivity is highly anisotropic, with the in-plane thermal conductivity much higher than the out-of-plane thermal conductivity for both bulk and monolayer. To understand the contribution ratio of phonons of different frequencies to the lattice thermal conductivity, we examine the contribution from the acoustic and optical phonons branches. The cumulative lattice thermal conductivity against frequency at 300 K is presented in Figure 8(b). From the acoustic mode, the approximate in-plane contribution to the

f

(16)

oo

v𝜆 = ∇q 𝜔(q, 𝑗),

thermal conductivity at 300 K is 96% and 97% for the bulk and monolayer, respectively; and reduces to 62% and 0% out-of-plane direction. This is consistent with the conventional understanding that the acoustic modes dominate thermal conductivity [58]. For reference, in standard units, the average bulk lattice thermal conductivity at 300 K is 46.01 Wm−1 K−1 and 1.65 Wm−1 K−1 , in-plane and out-of-plane, respectively. The low out-of-plane thermal conductivity makes the bulk system a candidate for further investigation as a potential active component in the thermoelectric device.

pr

and Planck constant, respectively. v𝜆 is phonon group velocity

0.87[18] , 0.95[46] , 0.7[47] 1.6[48]

e-

Bulk Monolayer

TDHF

Hamza A H Mohammed et al.: Preprint submitted to Elsevier

3.8. Group velocity The group velocity depends on the dispersion as shown in Equation (16). In this work, we calculated the average of directional group velocities for bulk and monolayer PtS2 . The obtained results are shown in Figure 10. In the lowfrequency range, the in-plane group velocities are higher than out-of-plane, while at high-frequency the in-plane and outof-plane are small compared to the low-frequency values. The anisotropy in group velocities has a great impact on the lattice thermal conductivity [61]. The combined effect of the frequency dependence of the phonon lifetimes and group velocities are consistent with the anisotropy in the thermal conductivity as well as the large contribution to the total thermal conductivity form the low frequency phonons.

Page 8 of 11

Structural stability, electronic, optical and thermal properties of bulk and monolayer PtS2

(a)

-1 -1

(Wm K )

-8

10

Lattice thermal conductivity κL per layer

Bulk 0.3

Out-of-plane

0.2 6 0.1 4

0

0

100

200

300

400

500

In-plane 2

2.4 2.1

100

200 300 Temperture (K)

400

In-plane

1.8

Bulk

0.08

1.5

0.06

1.2

0.04

0.9

Out-of-plane

0.02

0.6

0

0

10

5

15

0.3 0

0

Bulk Mono

500

0

10

5 Frequency (THz)

15

oo

0

Cumulative lattice thermal conductivity κL per layer

0.4 Bulk Mono

8

(b)

-1 -1

(Wm K )

f

-8

10

pr

Figure 8: Lattice thermal conductivity per layer, (a) lattice thermal conductivity against temperature, (b) cumulative lattice thermal conductivity against frequency .

e-

30

20

Pr

average τγ (ps)

25

15

10

5

al

0 0

2

4

6 Frequency (THz)

8

10

12

ur n

Figure 9: Phonon lifetime as a function of frequency at 300 K. Bulk (left), monolayer (right).

35

35 30

Jo

30

vxy vz

25 vg (˚ A.THz)

vg (˚ A.THz)

25 20 15

20

10

10

5

5

0

vxy vz

15

0 0

2

4

6 Frequency (THz)

8

10

12

0

2

4

6 Frequency (THz)

8

10

12

Figure 10: Average directional phonon group velocities as a function of frequency. Bulk (left) and monolayer (right).

Hamza A H Mohammed et al.: Preprint submitted to Elsevier

Page 9 of 11

Structural stability, electronic, optical and thermal properties of bulk and monolayer PtS2

oo

pr

Pr

Acknowledgements

e-

We have calculated the structural, mechanical, dynamical, electronic and optical properties as well as the lattice thermal conductivity, phonon lifetime and group velocity of bulk and monolayer PtS2 from a first-principles DFT approach. Imaginary and real parts of the dielectric function and the absorbance for single-layer were examined in bulk and monolayer. Our calculations were found more comparable to the available experimental (for structural for bulk and band-gap values for bulk and monolayer PtS2 ) and previous theoretical work. We showed that the maximum absorbance, in the visible range, of a monolayer is substantially higher compared to a single layer in the bulk. Furthermore, we showed that the lattice thermal conductivity for single-layer in bulk and monolayer PtS2 is highly anisotropic with the in-plane thermal conductivity much higher than the out-ofplane thermal conductivity for both bulk and monolayer. At 300 K the in-plane and out-of-plane lattice thermal conductivities for a single bulk layer are 2.30 and 0.08 Wm−1 K−1 , respectively, whereas the in-plane lattice thermal conductivity of single-layer in the monolayer is 0.15 Wm−1 K−1 . This shows a substantial decrease in the thermal conductivity in the monolayer compared to the bulk. The low value of the lattice thermal conductivity in the out-of-plane direction for the bulk suggests that bulk PtS2 merits further investigation as a candidate for thermoelectric applications.

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f

4. Conclusion

al

We would like to acknowledge the financial support received from Shendi University, Sudan. We also wish to acknowledge the Center for High-Performance Computing (CHPC), Cape Town, South Africa, for providing us with computing facilities.

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