Ultralow lattice thermal conductivity and anisotropic thermoelectric performance of AA stacked SnSe bilayer

Journal Pre-proofs Full Length Article Ultralow lattice thermal conductivity and anisotropic thermoelectric performance of AA stacked SnSe bilayer Shagun Nag, Anuradha Saini, Ranber Singh, Ranjan Kumar PII: DOI: Reference:

S0169-4332(20)30396-2 https://doi.org/10.1016/j.apsusc.2020.145640 APSUSC 145640

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Applied Surface Science

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Please cite this article as: S. Nag, A. Saini, R. Singh, R. Kumar, Ultralow lattice thermal conductivity and anisotropic thermoelectric performance of AA stacked SnSe bilayer, Applied Surface Science (2020), doi: https:// doi.org/10.1016/j.apsusc.2020.145640

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Ultralow lattice thermal conductivity and anisotropic thermoelectric performance of AA stacked SnSe bilayer Shagun Nag Department of Physics, Panjab University, Chandigarh-160014

Anuradha Saini Department of Physics, Panjab University, Chandigarh-160014

Ranber Singh Department of Physics, Sri Guru Gobind Singh College, Chandigarh-160019

Ranjan Kumar Department of Physics, Panjab University, Chandigarh-160014 and Physics department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

Abstract The low-temperature bulk phase of SnSe has a layered structure with AB-type of stacking of two-atom-thick SnSe monolayers. However, in the case of bilayers the AA-stacked SnSe bilayer structure is most stable. Similar to bulk SnSe, AA-stacked SnSe bilayer has ultralow lattice thermal conductivity of 0.90 W m−1 K −1 at room temperature which is quite low as compared to other two dimensional materials. Our results show that n-type SnSe bilayer has good thermoelectric performance as compared to that of the p-type bilayer. Keywords: Relaxation Time, Two-dimensional materials

1. Introduction Harvesting the waste heat produced in thermal power generation and other thermal energy based systems using thermoelectric materials is an interesting non-conventional source of energy. However, most thermoelectric materials produced are highly toxic, expensive and/or have low thermoelectric performance.

Preprint submitted to Elsevier

February 5, 2020

Recently, layered SnSe chalcogenide materials have been observed to have excellent thermoelectric performance [1, 2] which is attributed to the ultralow lattice thermal conductivity [1] and huge increase in charge carrier concentration and conductivity at high temperatures [3]. SnSe is nontoxic and economically competent with other conventional thermoelectric materials. The discovery of high thermoelectric performance of layered SnSe has led to an extensive investigation of similar layered chalcogenide materials, both theoretically and experimentally [1, 4]. Layered bulk SnSe is a narrow band gap semiconductor that undergoes a phase transition from Pnma to Cmcm phase at high temperature in the range 600-800 K. Both Pnma and Cmcm phases have distorted rock-salt Fm3m structures of SnSe. The two-atom-thick SnSe monolayer has zigzag and armchair like projections of atoms within the plane of the monolayer. The distinctive structure along the zigzag and armchair directions results in its anisotropic properties. This bulk phase of SnSe has very low intrinsic thermal conductivity in the range of 0.3-0.8 W m−1 K −1 at room temperature [5]. The two-dimensional monolayers of group IV-VI chalcogenides have also been predicted to have better thermoelectric properties as compared to their bulk counterparts [4]. The thermoelectric performance of above mentioned layered materials can be further improved by doping [6, 7], nanostructuring [8, 9], or strain [10, 11]. It is also still an open question how does the properties of these layered chalcogenides evolve when constructed by stacking the monolayers. There are different ways of stacking the monolayers [12, 13]. The bilayer structures of group IV-VI chalcogenides in different stackings have been investigated [12]. It has been found that AA stacking of bilayer of these IV-VI chalcogenides is most stable. The structural and electronic properties of these bilayer structures have been investigated. However, electronic and thermal transport properties of these bilayer structures are not yet investigated. In this paper, we investigate the electronic and lattice thermal transport properties of AA-stacked SnSe bilayer. We find that n-type SnSe bilayer has better thermoelectric performance as compared to p-type bilayer. Rest of the 2

paper is organized as follows; in section 2, we present the computational details, the results are presented and discussed in detail in section 3 and finally the results are summarized in section 4.

2. Computational Details We first optimize the structure of AA-stacked SnSe bilayer using the DFT as implemented in the Quantum Espresso package with plane wave basis [14]. The exchange-correlation functional of electronic interactions is represented by generalized gradient approximation of Perdew, Burke and Ernzerhof [15]. The optimized energy cutoff for the plane wave basis expansion is set to 90 Ry and the Brillouin zone (BZ) is sampled using the Monkhorst-Pack [16] k-point mesh of 15 × 15 × 1. A vacuum region of 15˚ A normal to the bilayer structure is used to avoid the interactions between the periodic images of the unit cell. Binding energy is calculated using the expression, Eb = (E(SnSe)n − E(SnSe)up − E(SnSe)sub )/4n

(1)

where, E(SnSe)up , E(SnSe)sub and E(SnSe)n are the total energies of upper layer SnSe, lower layer SnSe, and bilayer SnSe, respectively. We then compute the electronic band structure of the SnSe bilayer. From the computed band structure, the effective masses of charge carriers (electron and hole) are evaluated using the relation given as m∗ =

~2 ∂ 2 E(k)

(2)

∂k2

where ~ is the reduced Planck constant and ∂ 2 E(k)/∂k 2 is the second derivative of band energy E(k) w.r.t k. The ∂ 2 E(k)/∂k 2 is evaluated for electrons and holes by polynomial fitting of band energy versus k-points at conduction band minimum (CBM) and valence band maximum (VBM), respectively. We further evaluate the relaxation times of charge carriers using the deformation potential theory [17]. This theory has been extensively used to calculate the relaxation times of charge carriers and the thermoelectric properties of materials [18, 19, 3

20, 21, 22]. Within this theory, the relaxation time (τ ) of charge carriers is given by the relation τ=

~3 C2D kB T md Eβ2

(3)

where C2D is a two-dimensional elastic constant, kB is the Boltzmann constant, Eβ is the deformation potential constant of charge carriers along the β direction, md is the effective mass of charge carriers and T is the temperature in Kelvin. √ md is calculated as md = mx my , where mx and my are effective masses along the armchair and zigzag directions. Eβ for the i-th band is calculated as Eβ = ∆Ei /[∆l/l0 ], where ∆Ei is change in the i-th band energy with the strain (∆l/l0 ) along the β direction. Here, l0 is the equilibrium lattice constant and l is the deformed lattice constant along the β direction. For electrons (holes), the Eβ is evaluated for CBM (VBM) along the armchair and zigzag directions. C2D is evaluated by fitting the total energy versus uniaxial strain curve of SnSe bilayer structure to a quadratic polynomial. The electronic transport properties are calculated using the semi-classical Boltzmann transport theory as implemented in the BoltzTraP code [23]. For this purpose the electronic structure is calculated by sampling the BZ using a dense k-point mesh of 30×30×1. BoltzTraP code calculates electrical conductivity (σ/τ ) and electronic thermal conductivity (κe /τ ) within the constant relaxation time (τ ) approximation. Using the τ calculated within the deformation potential theory using the equation (3), the σ and κe are calculated at different temperatures. The lattice thermal conductivity (κl ) is calculated using the theoretical model developed by Slack [24]. The κl for a two dimensional crystal is given as [20], κl = B

¯ θD δ 3 M 2

γ2n 3 T

(4)

¯ is the average mass of an atom in the crystal (in atomic mass unit), where M θD is the average Debye temperature of acoustic phonons, δ 3 is the volume per atom (δ in ˚ A), n is the number of atoms in the primitive unit cell, γ is the high temperature acoustic phonon mode Gruneisen parameter, and B=3.1× 10−6 is 4

a constant. θD and γ are calculated using the thermo pw package [25] interfaced with Quantum Espresso. Finally, the thermoelectric performance is computed in terms of dimensionless figure of merit (zT ), defined as zT =

S 2 σT κe +κl ,

where S

is the Seebeck coefficient computed using the BoltTrap code discussed above. (a)

(b)

Se

Armchair direction

Sn

(c) b

a

Zigzag direction

Figure 1: Optimized crystal structure of SnSe bilayer. (a) top view, (b) side view perpendicular to zigzag direction and (c) side view perpendicular to the armchair direction. The crystalline structure of SnSe along the zigzag and armchair directions is distinctive which leads to its anisotropic properties.

3. Results and Discussion Taking 2D monolayer SnSe as a basic building block, we constructed four types of stacking configurations of bilayer: AA-, AB-, AC- and AD-stacking[12]. First we optimize the four different stackings. The optimized lattice constants and structural parameters (l1 , l2 , θ1 and θ2 ) of these stacking configurations are given in ESI (see Supporting Information, Table S1) Next we calculate the binding energies of four different stacking orders which are also presented in ESI (see Supporting Information, Table S1). The lowest binding energy of -39.3 meV is achieved in AA-stacking which suggest that AA-stacking is most stable structure among four stacking structures. The optimized structure of AA-stacked SnSe bilayer is shown in figure 1. Figure 1 shows the top view and side view structures of optimized AA-stacked

5

SnSe bilayer along armchair and zigzag directions. Clearly, the crystalline structure along the zigzag and armchair directions is quite distinctive. The optimized lattice constants are 4.29 ˚ A and 4.33 ˚ A along zigzag and armchair directions. To study the thermal stability of the bilayer, we performed ab initio molecular dynamics (MD) simulations at some finite temperature. A 3×3×1 supercell is constructed to simulate the bilayer for minimizing the constraint induced by periodicity. The simulated results at 800 K are given in figure. 2. We found that geometry of the bilayer remain almost invarient after heating for 10 pico-seconds at 800 K. Therefore, bilayer is thermally stable at 800 K.

-12690

-12695

Energy (eV)

-12700

-12705

-12710

-12715

-12720

-12725

-12730 0

2

4

6

8

10

Time (ps)

Figure 2: Energy fluctuation with respect to time in ab initio molecular dynamics simulations at 800 K. Total energies remain almost invariant after heating for 10 pico-seconds.

We further calculate the electronic band structure of AA-stacked SnSe bilayer. Figure 3 shows the band structure and corresponding electronic density of states (EDOS) of above mentioned structure. It has an indirect band gap of 0.88 eV with VBM at the Γ-point and CBM along Y − Γ path. This band gap is larger than that of bulk SnSe (0.39 eV ) [1]. Such a high band gap of SnSe bilayer will be capable of overcoming the high temperature bipolar conduction problem which degrade the thermoelectric performance [26]. Using the band structure shown in figure 3, we compute the effective mass

6

Figure 3: Electronic band structure of AA-stacked SnSe bilayer and the corresponding electronic density of states (EDOS). It has indirect band gap of 0.88 eV with VBM at Γ-point and CBM along the Y − Γ path.

Figure 4:

The VBM and CBM energies as a function of uniaxial strain along zigzag and

armchair directions. The slopes of these CBM (VBM) curves give the values of Eβ for electrons (holes). The results are given in table 1. The values of Eβ for holes are same along both directions, while Eβ for electrons are different along armchair and zigzag directions.

(m∗ ) of electron (hole) at CBM (VBM) along the zigzag and armchair directions as discussed in the computational details section. The results are tabulated in table 1. m∗ of holes is larger than m∗ of electrons. This is due to the fact that hole moves in an otherwise filled band, whereas electron moves in an otherwise empty band. The deformation potential constant Eβ of electron (hole) is evaluated by calculating the change in CBM (VBM) under the uniaxial strain along the zigzag and armchair directions. The relaxed structural parameters (l1 , l2 , θ1 and θ2 ) of the bilayer at different uniaxial strain values are given in ESI (see Supporting Information, Table S2). The CBM and VBM energies as

7

Figure 5: The change in total energy (E-E0 ) of a lattice as a function of uniaxial strain along (a) zigzag and (b) armchair directions. Here, E0 and E are the total energies of equilibrium lattice and deformed lattice, respectively. A fitting of a quadratic polynomial to a curve of EE0 versus strain give value of C2D . The results are given in ESI (see Supporting Information, Table S1). Table 1: Calculated deformation potential (Eβ ), elastic constant (C2D ), effective mass (m∗ ), and relaxation time (τ ) at 300 K, 500 K, and 800 K, along the zigzag and armchair directions of SnSe bilayer.

C2D

Eβ

m∗

τ (f s)

Direction

(N/m)

Carrier

(eV)

(m0 )

300 K

500 K

800 K

Zigzag

0.009

Electron

0.04

0.20

28.5

17.1

10.7

Hole

0.07

6.48

0.37

0.22

0.14

Electron

0.10

0.26

41.3

24.8

15.5

Hole

0.07

4.33

0.32

0.19

0.12

Armchair

0.008

a function of uniaxial strain along zigzag and armchair directions are given in figure 4. The calculated values of Eβ are given in table 1. Further, we calculate the elastic constant C2D from change in total energy of lattice versus uniaxial strain curve of SnSe bilayer structure along the zigzag and armchair directions. The change in total energy of lattice versus uniaxial strain curve of SnSe bilayer structure along the zigzag and and armchair directions are given in figure 5. The calculated values of C2D are given in figure 4. Using these calculated m∗ , Eβ and C2D , we calculate the relaxation time of electron and hole along the zigzag and armchair directions using the equation (3). Table 1 shows that SnSe

8

Figure 6: Electronic Charge conductivity (σ) as a function of carrier concentration (n) for the SnSe bilayer along (a) zigzag and (b) armchair direction at 300 K, 500 K and 800 K. Negative values of n represent n-type charge carrier concentration, whereas positive values of n represent p-type charge carrier concentration. Clearly, for n-type SnSe bilayer the σ is larger as compared to that of p-type SnSe bilayer. The σ decrease with an increase in temperature. This is due to the increase in the scattering of charge carrier with an increase in temperature.

80

120 300 K

(a)

500 K

70

300 K

(b)

1.2

500 K

800 K

90 0.6

50

0.8

-1

-1

(Wm K )

0.8

800 K

1.0

60

40

e

0.6

60 0.4

30 0.4 20

30

0 -30

0.2

0.2

10

0.0 -20

-10

0 n (10

14

10

20

30

0 -30

0.0 -20

-10

0

-2

cm )

n (10

14

10

20

30

-2

cm )

Figure 7: Electronic charge thermal conductivity (κe ) as a function of n for the SnSe bilayer along (a) zigzag and (b) armchair direction at 300 K, 500 K, and 800 K. The κe shows a complicated behaviour with an increase in temperature. The κe for n-type SnSe bilayer is larger as compared to that of p-type SnSe bilayer.

bilayer has anisotropic properties along zigzag and armchair directions. The electronic charge conductivity and thermal transport properties of above mentioned bilayer are given in figures 6-8. BoltzTrap code calculates σ and κe as σ/τ and κe /τ within constant τ approximation. Using τ as computed above using the deformation potential theory, we compute σ and κe at different temperatures. Figure 6 shows σ along the zigzag and armchair directions as a function of charge carrier concentration (n) at 300 K, 500 K and 800 K temperatures. Negative values of n represent n-type charge carrier concentration, whereas positive values of n represent p-type charge carrier concentration.

9

Figure 8:

Seebeck coefficient (S) as a function of n for the SnSe bilayer along (a) zigzag

and (b) armchair directions at 300 K, 500 K, and 800 K. The S has a discontinuity at n=0. Conventionally, S has negative values for n-type carrier concentration and positive values for p-type carrier concentration.

Clearly, for n-type SnSe bilayer the σ is significantly larger as compared to that of p-type SnSe bilayer. This is due to the higher electron mobilities resulting from smaller effective mass of electrons as compared to holes. The σ decreases with an increase in temperature. This is due to the increase in the scattering of charge carriers with an increase in temperature. The σ along armchair direction is about twice larger as compared to that along zigzag direction.

Figure 9:

Calculated power factor (S 2 σ) as a function of n for the SnSe bilayer along (a)

zigzag and (b) armchair direction at 300 K, 500 K, and 800 K. For n-type SnSe bilayer the S 2 σ is larger as compared to that of p-type SnSe bilayer.

Figure 7 shows κe as a function of n along zigzag and armchair directions at 300 K, 500 K and 800 K. The κe has a complicated behaviour with an increase in temperature. There is no clear monotonic change with an increase in tem-

10

Figure 10: Lattice thermal conductivity (κl ) at different temperatures for the SnSe bilayer. It decreases nonlinearly with an increase in temperature.

perature. The κe for n-type SnSe bilayer is larger as compared to that of p-type SnSe bilayer. This is due to larger effective mass and smaller relaxation time of p-type charge carriers as compared to that of n-type charge carriers in SnSe bilayer structure. The σ and κe are related to each other by the WiedemannFranz Law [27], given as κe = LσT , where L is a Lorenz number having value 1.5 × 10−8 W Ωk −2 for non degenerate semiconductors [28]. Consequently, an increase (decrease) in σ will also cause an increase (decrease) in κe . Figure 8 shows the Seebeck coefficient (S) as a function of n for the SnSe bilayer along zigzag and armchair directions at 300 K, 500 K, and 800 K. Conventionally, S has negative values for n-type carrier concentration and positive values for p-type carrier concentration. The S is calculated with the expression,   R∞ ) dEg(E)(E − µ) −∂f (E,µ,T ∂E −∞   S= (5) R∞ ) T −∞ dEg(E) ∂f (E,µ,T ∂E Here, in Eq. 5; E, g(E), f (E, µ, T ), µ, and T are energy, transport function, fermi function, chemical potential and temperature respectively. The transport function is g(E) = N (E)v 2 (E)τ (E) 11

(6)

0.9

0.20

0.9

300 K

(a)

0.8

800 K

0.7

0.12 300 K

(b)

0.8

500 K

0.16

0.6

500 K 800 K

0.10

0.7 0.6

0.08

zT

0.12 0.5

0.5

0.4

0.4

0.06 0.08 0.3

0.3

0.2

0.04

0.1

0.02 0.1

0.0 -10

0.04

0.2

0.00 -8

-6

-4

-2 n (10

0 14

2

4

6

8

10

0.0 -10

-2

0.00 -8

-6

-4

-2 n (10

cm )

0 14

2

4

6

8

10

-2

cm )

Figure 11: Calculated figures of merit (zT) as a function of the carrier concentration (n) for the SnSe bilayer along the (a) zigzag and (b) armchair direction at 300 K, 500 K, and 800 K. The zT for n-type SnSe bilayer is larger as compared to that of p-type SnSe bilayer.

Here, in Eq.6; N (E) is the density of states, v(E) is Fermi velocity and τ (E) is the scattering time. The seebeck coefficient changes dramatically near the Fermi level because of the term ∂f /∂E in the Eq.5, which behaves as Dirac delta function. Further, density of states of electrons and holes varies discontinuously near the n=0 because the material changes from n-type to p-type and in equation5, S depends strongly on density of states of charge carriers. Thus, S has a discontinuity at n=0. As the temperature decreases, S increases because of bipolar conduction effect or the excitation of carriers with both positive and negative charges suggesting an increasing inverse Hall coefficient [26]. Further with the increase in the carrier concentration, S becomes almost constant in both n-type and p-type regions. Since σ has higher values at large values of n, while S has large values near n=0, therefore a trade-off between the σ and S is required in order to have high power factor which is needed for good thermoelectric performance of a material. The combined effect of σ and S are computed as a power factor defined as S 2 σ. The results are presented in figure 9. Figure 9 shows that the power factor of SnSe bilayer increases with an increase in temperature. At a fixed temperature, the power factor of n-type SnSe bilayer is significantly larger than that of p-type bilayer which suggests that n-type doping is more favourable for high power factor. The intrinsic lattice thermal conductivity of SnSe bilayer at different temper-

12

atures is estimated using the Slack’s model [24]. The temperature dependence of the lattice thermal conductivity is shown in Fig.10. The bilayer has lattice thermal conductivity equal to 0.90W m−1 K −1 at 300 K which is significantly low as compared to that of other two-dimensional materials like MoSe2 [29], graphene [30], phosphorene [31] etc. The low intrinsic thermal conductivity is combined effect of heavy constituent elements and the low atomic coordination. The average mode Gruneisen parameter of SnSe bilayer is 3.6 as computed using the thermo pw package [25]. It quantifies the extent of anharmonicity of SnSe bilayer structure. Finally, the figure of merit defined as zT =

S 2 σT κe +κl

is computed as a function

of n along the zigzag and armchair directions. The results are shown in figure 11. Since, doping generally benefits σ while it deteriorate S, zT first increases and then decreases with carrier concentration, therefore leading to optimal zT . Further, with the rise in temperature, zT value of the bilayer increases. The bilayer has peak zT =0.72, 0.55, 0.33 along zigzag and zT = 0.78, 0.64, 0.42 along armchair directions at T=800 K, 500 K, 300 K respectively. Also, the thermoelectric performance of n-type SnSe bilayer is found to be more promising than p-type bilayer, as shown in Fig.11 which suggests that AA-stacked bilayer is promising thermoelectric material in n-type applications.

4. Conclusion The structural, electronic and thermal transport properties of AA-stacked SnSe bilayer are investigated using the DFT combined with Boltzmann transport theory. AA-stacked SnSe bilayer is an indirect band gap semiconductor with a band gap of 0.88 eV . It has significant anisotropy in its properties along armchair and zigzag directions. It has low lattice thermal conductivity in comparison to other two-dimensional materials like M oS2 , graphene and phosphorene etc. The thermoelectric performance of n-type SnSe bilayer is better as compared to that of p-type SnSe bilayer structure. The peak values of zT for n-type SnSe bilayer are 0.72, 0.55, 0.33 along the zigzag direction and 0.78,

13

0.64, 0.42 along the armchair direction at 800 K, 500 K, 300 K, respectively. Thus bilayer SnSe is a promising thermoelectric material that could be used in nanoscale thermoelectric devices such as thermocouples, thermoelectric generators etc. We also expect that the advent of nanotechnology will make it easier to use the nanostructured SnSe in thermoelectric devices for practical applications.

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Highligts: • The thermoelectric properties of bilayer SnSe are still not explored yet experimentally and theoretically. • The lattice thermal conductivity of bilayer SnSe is ultralow (0.90Wm -1K-1) at room temperature. • Temperature dependent relaxation time has been used to calculate the thermoelectric properties of bilayer SnSe. • The peak value of ZT is 0.72 (0.78) along zigzag (armchair) direction.

Author contributions The idea behind this work was given by Ranber Singh. Shagun Nag has performed the electronic and thermoelectric study calculations. Relaxation time calculations were performed by Anuradha Saini. Ranjan Kumar has performed the

molecular dynamics calculations and help in drafting the

manuscript. All authors read and approved the final manuscript.