Layers dependent dielectric properties of two dimensional hexagonal

Layers dependent dielectric properties of two dimensional hexagonal boron nitridenanosheets Liang Wang, Yayun Pu, Ai Kah Soh, Yuping Shi, and Shuangyi Liu Citation: AIP Advances 6, 125126 (2016); View online: https://doi.org/10.1063/1.4973566 View Table of Contents: http://aip.scitation.org/toc/adv/6/12 Published by the American Institute of Physics

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AIP ADVANCES 6, 125126 (2016)

Layers dependent dielectric properties of two dimensional hexagonal boron nitridenanosheets Liang Wang,1,2 Yayun Pu,1,2 Ai Kah Soh,3 Yuping Shi,4 and Shuangyi Liu1,2,a 1 Chongqing

Institute of Green and Intelligent Technology, Chinese Academy of Sciences, 266 Fangzheng Ave., ShuiTu Technology Development Zone, Beibei District, Chongqing, People’s Republic of China 2 Chongqing Key Laboratory of Multi-Scale Manufacturing Technology, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714, People’s Republic of China 3 School of Engineering, Monash University Malaysia, Bandar Sunway, Selangor, Malaysia 4 Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Hong Kong (Received 24 October 2016; accepted 19 December 2016; published online 30 December 2016)

Two dimensional (2D) boron nitride (h-BN) nanosheets are well known as their tunable electric properties and well compatible with graphene. Studying the dielectric properties carefully is essential for their promised applications. Most previous first principle studies treated 2D h-BN as a strict 2D system, where the contribution of ion polarization is neglected. The results show obvious deviation from experimental values, and the situations are worse with the stacking layer increasing. Thus, in present works, the dielectric properties of 2D h-BN nanosheets are studied with involving the ion contributions appropriately. The evolution of dielectric performance with stacking layers varying is also studied. Obvious layer dependent anisotropic dielectric properties are predicted, which reaches the bulk h-BN level as the thickness approaching 5.8nm (20L). There should be a balance between dielectric properties and the thickness (stacking layers) for the dielectric applications of 2D h-BN nanosheets. © 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4973566] Free standing two dimensional(2D) nanosheets with several atoms/molecules thickness are current thinnest promised functional building blocks for next generation electronic and photonic devices.1 Layered hexagonal boron nitride (h-BN) is excellent chemical and thermodynamic stable insulator, and its 2D counterpart could serve as atomic scale insulators and gate dielectrics for maintaining the Moore’s law and continuous miniaturization of the electronic devices.2–5 Moreover, many studies reveal that the 2D h-BN possesses better compatibility with 2D conductors and semiconductors such as graphene and MoS2 .6–9 Thus, to pushing the applications of 2D h-BN, well understanding of its dielectric properties with varying the staking layers (L) is required urgently. Compared to thoroughly studied bulk h-BN,10,11 the researches on free-standing 2D h-BN nanosheets are in the early stage.12–15 Ab-initio calculation is an effective method for studying the properties of 2D materials, and more predictions to the dielectric behaviors by it are emerging. Beiranvand’s ab-initio calculations on band structures and optical properties of monolayer h-BN show that the electronic dielectric 1 16 17 constant perpendicular and parallel z axis, " 1 ?Z and " kZ , are 2.19 and 1.5 respectively. Li et al. conclude that the dielectric properties of h-BN nanosheets are thickness dependent, and calculated out of plane dielectric constants of 1L, 2L and 3L nanosheets are 2.31, 2.43 and 2.49. Their simulations by applying external electric field are appropriate for large scale models, and inevitably, many details especially ion-electric field coupling are missed. Meanwhile, most reported calculations on

a

Author to whom correspondence should be addressed. Electronic mail: [email protected]

2158-3226/2016/6(12)/125126/6

6, 125126-1

© Author(s) 2016

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dielectric properties did not involve the contribution of ion polarization, which is unneglectable for further understanding the dielectric properties of 2D h-BN nanosheets. Longitudinal-transverse optical phonons (LO-TO) splitting in the vicinity of reciprocal space origin( -point), which is caused by electron-phonon coupling, is crucial impact factor for evaluating ion contribution and predicting the dielectric properties.18–20 The first principal calculations to the 2D h-NB based on S´anchez-Portal’s model conclude that LO-TO splitting is absent for in-plane mode in strict 2D system.21,22 Nevertheless, it is always present for actual 2D materials system, for examples, LO-TO splitting for in-plane E u modes of monolayer MoS2 23,24 and HfS2 25 are found by both ab-initio calculations and experiments. Thus, in this paper, the dielectric properties of 2D h-BN nanosheets are studied based on Density Functional Perturbation Theory (DFPT),26 where ion polarization contributions are included. The general dielectric function can be expressed as18 1 " ↵ (!) = " 1 ↵ + "↵

X

j

2 !LO,j 2 !TO,j

2 !TO,j

!2 + i !

,

(1)

where ↵ and are Cartesian coordinate components, " 1 ↵ is the electronic permittivity, !LO,j and th !TO,j are the j longitudinal and transverse infrared active(IR) phonon modes, and is the damping parameter and set as 3% of !TO,j .27,28 Phonon frequency and LO-TO splitting are calculated by means of Linear Response26 with non-analytical term correction.18 The electronic permittivity is determined by26 * + 16⇡e XN/2 @ 1 "↵ = ↵ (2) n r↵ n , n=1 Nc ⌦ @E

where Nc and N are the numbers of cells in crystal and electrons in the cell, ⌦ is the volume of cell, r↵ is the ↵ component of vector ~r , E is the component of perturbation electric field and n is eigenfunction of Kohn-Sam equation. Note that a vacuum layer is induced into the cell for sheet model calculation. The effective electronic permittivity (" eff ) is evaluated by subtracting this effect by eff

"↵ =

↵

+

c(" 1 ↵

↵

c0

)

,

(3)

eff

according to Eq. 2. " ↵ and " 1 ↵ are the effective electronic permittivity and calculated electronic permittivity from vacuum sheet model, c and c0 are the lattice constant in z direction of h-BN sheet in calculation model and the thickness of the nanosheet, respectively. The calculations are performed by Quantum ESPRESSO code.29 According to convergence testing, Norm-conserving pseudopotential30 of local density approximation (LDA)31 is chosen and energy cutoffs are 140Ry and 560Ry for the wave function and charge density, respectively. The k-point grid for self consistent field (SCF), lattice-dynamical matrices and phonon density of states (PhDOS) calculations are 16 ⇥ 16 ⇥ 1,16 ⇥ 16 ⇥ 1 and 500 ⇥ 500 ⇥ 1 respectively. For h-BN nanosheet, a vacuum TABLE I. Calculated and experimental lattice parameter (in Å), dielectric constant and frequency of IR optical mode (in cm-1 ) of h-BN 1L, 2L sheet and bulk. 1L h-BN

c/a 1 "?z 1 "kz 0 "?z 0 "kz A(TO) A(LO) E(TO) E(LO) a

present

Calc.

2.5013 5.37 1.82 6.28 1.83 810 813 1388 1501

2.50533

h-BN monolayer on Ni(111).

2L h-BN exp.

2.1916 1.516

81422

79038a

130122

137138a

Bulk h-BN

present

present

calc.27

exp.

2.500 5.30 1.85 6.57 1.89 805 814 1388 1545

5.827/2.500 5.27 3.61 7.21 5.09 687 815 1390 1626

6.378/2.491 4.87 2.95 6.71 3.35 746 819 1372 1610

6.66/2.50636,37 4.9510 4.110 6.8510 5.0610

137010 161010

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FIG. 1. Phonon dispersion of stacked 2L h-BN sheet and phonon density of state.

with width of 20 Å is set between the layers for ensuring no interaction between them.16,17,32,33 The structure is relaxed completely with Van der Waals correction (dft-d2) before phonon calculations with the force threshold of 10-6 Ry/bohr.34–37 The accuracy test is performed through calculating the anisotropic dielectric constants of bulk h-BN depending on present method and indicates that more 0 accurate results can be obtained by, where the results for out of plane (" 1 kz and " kz ) are very close to 0 experimental results and that for in plane (" 1 ?z and " ?z ) are no larger than 6.5%(Table I). Through the DFPT calculations, the electronic permittivity of the monolayer(1L) are " 1 kz = 5.37 and " 1 = 1.82. In comparing with bulk h-BN (Table I), bigger in plane and smaller out of plane ?z electronic permittivity are reasonable due to quantum confinement effect that electrons are restricted

FIG. 2. Total dielectric function of 2L stacked h-BN sheet, dielectrics of bulk h-BN from calculation are presented for comparison. = 0.3% (damping parameter) for all. (a) and (b): dielectric function in and out of plane.

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in 2D plane, and such results furtherly prove that 1L h-BN is less effective for electric field screening along z direction.17 The 1L h-BN possesses six lattice vibrations, three acoustic and three optical modes. It is well known that only infrared active (IR) optical modes contribute to the dielectric function.39 The as-calculated frequencies of IR A2 00(out of plane) and E 0(in plane) modes of 1L h-BN are 810.35 cm-1 and 1388 cm-1 , and both in and out of plane LO-TO splitting, 113.08 cm-1 and 2.99 cm-1 , are observed38 (Table I). The bigger in plane splitting indicates stronger in plane electron-phonon coupling, which leads to bigger enhancement of in plane dielectric constant (" 0?z ). For further understanding the evolution of dielectric properties with the dimension reduction along z axis, the study on 2L h-BN sheet is performed because it is a minimum unit which has same primitive cell with the bulk counterparts and is the direct bridge to the pristine 2D system. The optical vibration modes increased to 9 for 2L h-BN, and IR active A2u and E u modes have effect on dielectric functions along z and xy directions respectively. The phonon dispersion of 2L h-BN sheet as well as phonon density of state (PhDOS) are shown in Figure 1. The k.p theorem is used to sort the phonon branches according to the continuity of their eigenvectors.32,40 By comparing with bulk h-BN, the characteristic frequency of A2u (TO) increases ⇠62 cm-1 , E u (LO) decreases ⇠68 cm-1 , while A2u (LO) and E u (TO) are almost the same. These cause the LO/TO splitting reduction as well as the ion dipole contribution to dielectric function. The DOS of 2L h-BN nanosheet presents more 2D characters, such as the step is less ambiguous in the range of 0-350 cm-1 and 1400-1600 cm-1 , however it is still not a strict 2D system.41–43 The total dielectric functions of 2L h-BN nanosheet are displayed in Figure 2. The dieleckz tric constant is reaching its extremum with the applied frequency approaching !TO (" ?z Re = 17.34, " Re

FIG. 3. (a) The frequency of A mode(vibration along z direction) and E mode (vibration in xy plane) which haspinfrared activity as a function of the thickness of h-BN sheet. TO: transverse optical mode; LO: longitudinal ! 2 is q optical mode. p 2 2 2 used for estimating the contribution of LO-TO splitting to dielectric function, defined as ! = !LO,j !TO,j . (b) and (c) schematic of A2u mode and E u mode of 2L h-BN sheet, respectively, 2 ⇥ 2 ⇥ 1 supercell. The nearby atoms configurations of B/N atom are depicted in Figure 3 b-ii and Figure 3 b-ii. (d) The evolution of electronic permittivity and static dielectric constant of h-BN sheet with thickness.

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= 2.34) according to Eq. 1. By comparing with bulk h-BN, in plane dielectric of 2L nanosheets is comparable, but out of plane reduced by 44% at ! = 0, which demonstrates an obvious dimension effect on the dielectric constant. Figure 3a shows the frequency of each IR mode approaches the bulk level by different ways. Those of A(LO) and E(TO) modes are almost constant. The frequency of E(LO) mode increases with thickness and approaches the bulk level (1613 cm-1 ) at around 10L, while that of A(TO) mode behaves oppositely and decreases to the bulk level (743 cm-1 ) at thickness around 18 L. The ion p 2 contribution, ! , enhances gradually with thickness increasing. Figure 3b and Figure 3c show the A2u and E u vibration modes of 2L nanosheet with the atoms configuration and movements. There are 3 nearest heterogeneous atoms with opposite charge around each N/B atom in 1L h-BN. With more layers being stacked, the environment of B/N atoms are not the same any more, the atom b is involved in 2L nanosheet and both atom b, d should be involved for 2D h-BN nanosheets with more than 2 layers (Figure 3b-ii and Figure 3c-ii). These result in the harden of E(LO) mode and soften of A(TO) mode as well as increasing ion contribution with stacking layers,21 and the present calculations prove the situation clearly (Figure 3a). As shown in Figure 3d, " 1 ?z of the nanosheets is a little higher than that of bulk, it approaches to bulk’s level at ⇠6L(⇠1.4 nm). " 1 and increases to bulk’s level kz is affected much by dimension change p 0 2 at ⇠20L (5.8 nm). " ?z varies as thickness increases as the trend of ! of IR E mode (Figure 3a), it approaches bulk’s level at ⇠15 L(4.4 nm). " 0kz possesses similar way with " 1 kz and approaches bulk’s level at ⇠20L(⇠5.8nm). Generally, for the nanosheets with layer lower than 3, the dielectric properties present obvious quantum effect, 3L to tens layers exhibit transitional performance, and when the layers over 20L, the dielectric properties approach the bulk level. In conclusion, the dielectric properties of 2D h-BN nanosheets are studied by first principle calculations. The calculations are well converged and tested, by which, more accurate prediction to the dielectric properties of 2D h-BN nanosheets can be achieved. Anisotropic dielectric properties are observed with obvious dimensional effects. " 1 ?z reduces with the thickness increasing of the sheets, but 0 present the opposite behaviors with different evolution rates. The specific behaviors " 0?z , " 1 and " kz kz of " 1 ?z is due to quantum confinement effect. Moreover, the nanosheets possess better high frequency in plane dielectric properties. Contribution of ion polarization to dielectric increases as harden of E(LO) and soften of A(TO) when thickness of nanosheet increases. The dielectric properties of 2D nanosheets reach the level of bulk h-BN as the thickness approaching 5.8 nm (20L). Thus, there should be a balance between dielectric properties and the thickness (stacking layers) for the dielectric applications of 2D h-BN nanosheets. ACKNOWLEDGMENTS

This work was supported by the program of Hundreds talents of Chinese Academy of Sciences (Grant No. R52A261Z10), the Fundamental and advanced technology Research Funds of Chongqing (Grant No. cstc2015jcyjbx0103), and Nature Science Foundation of China(NSFC)–GuangDong Government unite Foundation Supercomputing scientific applied research projects (2nd session) No. nsfc2015-475. 1 H.

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