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Dipolar polarization and piezoelectricity of a hexagonal boron nitride sheet decorated with hydrogen and fluorine Mohammad Noor-A-Alam, Hye Jung Kim and Young-Han Shin* In contrast to graphene, a hexagonal boron nitride (h-BN) monolayer is piezoelectric because it is noncentrosymmetric. However, h-BN shows neither in-plane nor out-of-plane dipole moments due to its three-fold symmetry on the plane and the fact that it is completely flat. Here, we show that the controlled adsorption of hydrogen and/or fluorine atoms on both sides of a pristine h-BN sheet induces flatness distortion in a chair form and an out-of plane dipole moment. In contrast, a boat form has no out-of-plane dipole moment due to the alternating boron and nitrogen positions normal to the plane. Consequently, the chair form of surface-modified h-BN shows both in-plane and out-of-plane
Received 20th September 2013, Accepted 6th February 2014
piezoelectric responses; while pristine h-BN and the boat form of decorated h-BN have only in-plane
DOI: 10.1039/c3cp53971g
are comparable to those in known three-dimensional piezoelectric materials. Such an engineered
piezoelectric responses. These in-plane and out-of-plane piezoelectric responses of the modified h-BN piezoelectric two-dimensional boron nitride monolayer can be a candidate material for various nano-
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electromechanical applications.
1. Introduction The extraordinary success of atomically thick graphene1–10 both experimentally and theoretically has revolutionized the whole of nanoscience because of its suitability for future nanotechnology applications. Consequently, hexagonal boron nitride (h-BN) monolayers, two-dimensional honeycomb networks of boron and nitrogen atoms similar to carbons in graphene, have been brought to the forefront of research11–14 because they have remarkably different electronic properties. For example, while graphene is a conductor and hydrogenation of graphene makes it semiconducting,15–17 an h-BN sheet is an insulator and hydrogenation decreases its band gap.18,19 One of the many reasons for these differences comes from the fact that C–C bonds in graphene are purely covalent, whereas B–N bonds in an h-BN sheet have a partial ionic character because of the large electronegativity difference between boron and nitrogen.20,21 Therefore, despite similar lattice parameters and crystal structure, two seemingly disparate materials with different band structures can complement each other in many potential applications to provide interesting functional properties arising from their integration.22–31 One of the biggest challenges in harnessing the power of nanotechnology is achieving dynamic control of mechanical, Department of Physics and EHSRC, University of Ulsan, Ulsan 680-749, Republic of Korea. E-mail: [email protected]; Fax: +82-52-259-1693
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chemical and electronic properties of nanoscale devices. Such control is crucial for many devices on which our modern life heavily relies, including transistors, sensors, actuators, energy harvesters, motors, robots, artificial muscles, and other locomotive devices. Usually, piezoelectric materials are the first choice for dynamic control of material deformation by the application of an external electric field. They are central to a wide variety of applications from pressure sensors to acoustic transducers to high voltage generators.32–37 Moving to the nanoscale, semiconductor and piezoelectric nanomaterials have opened a new burgeoning area of nanotechnology called nanopiezotronics; these materials have exciting and unique properties for a wide range of applications in electronics, optics, optoelectronics, nanogenerators, sensors, resonators, and the biological sciences that may improve quality of life.38–41 However, graphene is unfortunately not intrinsically piezoelectric due to its centrosymmetric crystal structure. Interestingly, Ong and Reed have recently discovered that piezoelectric effects can be engineered into non-piezoelectric graphene by breaking the inversion symmetry through the selective surface adsorption of atoms.37 There has also been a totally different approach to coax piezoelectricity from graphene merely by creating holes of a specific symmetry.42 In contrast to graphene, an h-BN sheet does not exhibit inversion symmetry, and thus is intrinsically piezoelectric.43,44 h-BN sheets are electrical insulators (a band gap of around 4.6 eV) and have profound chemical and thermal stabilities. At the same time, like their C counterparts (graphene), they are
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thermally conductive, mechanically robust, and highly resistant to oxidation.11,43–46 Such unique properties make h-BN sheets a promising nanomaterial in a variety of potential fields such as insulators with high thermal conductivity in electronic devices, ultraviolet-light emitters in optoelectronics, hydrogen accumulators, high strength nano-fillers, and thermally conductive nanocomposites. Being intrinsically piezoelectric, an h-BN monolayer also has the potential to be used as a nano-piezoelectric material. Unfortunately, it does not show any out of plane piezoelectric response due to its complete flatness, which limits its potential applications in nanoelectromechanical devices.34 Because of its three-fold symmetry, it has no in-plane polarization either. Recently, electric polarization has been found in boron nitride nanotubes.48–52 However, we find that both in-plane and out-ofplane piezoelectric response as well as out-of-plane dipolar polarization can be induced in h-BN sheets by merely decorating the surface with atoms. Having made advances11–14 in experimental synthesis of h-BN, it is expected that the same is possible for its chemical functionalization. By hydrogen plasma treatment of h-BN sheets on a substrate, partially one-side hydrogenated few-layered h-BN membranes have been achieved so far.53 Fully hydrogenated and fluorinated layers are expected when free standing h-BN sheets are used. However, the electronic structure and magnetic properties of fully decorated and semi-decorated BN sheets using H and/or F atoms have been studied extensively using first principles calculations, especially to tune the band gap of h-BN sheets.18,19,47 Interestingly, in contrast to endothermic hydrogenation, fluorination and codecoration of h-BN sheet are found to be exothermic mainly because of weaker bonding in F2 than in H2.19 The adatoms’ pattern, which is experimentally
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difficult to control, and is often formed in a random way on a host structure depending on processes largely determines the properties of the functionalized h-BN structures. As a result, many different configurations with different electromechanical properties are expected to be formed under experimental conditions. In this paper, we have conducted an extensive study of the structures and out-of-plane dipolar polarization as well as piezoelectric properties of fully hydrogenated or fluorinated h-BN sheets (labeled as chair and boat for HBNH and FBNF) and co-decorated h-BN sheets using H and F (labeled as chair and boat for FBNH and HBNF). Optimized geometric structures of our considered configurations are shown in Fig. 1. We show that hydrogenation, fluorination, or co-decoration of H and F on an h-BN sheet breaks the flatness of the sheet and results in an out-of-plane electric polarization. Our finding of these uncompensated polar surfaces of chemically modified h-BN monolayers may cast light on their surface chemistry, and electronic properties of their multi-layers. We have also calculated piezoelectric constants e31 and d31, and they are comparable to those of other known piezoelectric materials.
2. Computational details Our calculations are based on density functional theory (DFT) using a generalized gradient approximation (GGA) for exchange– correlation potential with a Perdew–Burke–Ernzerhof functional as implemented in the Vienna ab initio Simulation Package (VASP).54–56 Supercells are used to simulate the isolated sheets, and the distance between neighboring sheets is kept larger than 20 Å in order to avoid interactions between them. To cancel out
Fig. 1 Top and side views of optimized configurations: (a) chair HBNH, (b) chair FBNH, (c) chair HBNF, (d) chair FBNF, (e) boat HBNH, (f) boat FBNH, (g) boat HBNF, and (h) boat FBNF. Rectangles represent conventional unit cells.
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PCCP Table 1
Paper Structural data of the systems studied
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Bond lengths (Å) Structure
B–N
B–H
B–F
N–H
N–F
Thickness of the sheets (Å)
h-BN Chair HBNH Boat HBNH Chair FBNH Boat FBNH Chair HBNF Boat HBNF Chair FBNF Boat FBNF
1.45 1.58 1.57/1.62a 1.62 159/1.58 1.58 1.56/1.60 1.62 1.60/1.70
— 1.20 1.21 — — 1.21 1.20 — —
— — — 1.34 1.40 — — 1.35 1.35
— 1.03 1.02 1.03 1.03 — — — —
— — — — — 1.46 1.49 1.44 1.45
— 2.75 2.81 2.91 3.02 3.13 3.23 3.30 3.34
a
Two different bond lengths for the boat conformation.
Table 2 Cohesive energy (Ecoh), formation energies per atom (Oatom and Omolecule from atoms and moleculesa respectively), and band gap (Eg) with a unit of eV
Structure
Oatom
Omolecule
Ecoh
Eg
Chair HBNH 1.04 0.10 4.57 3.3 Boat HBNH 1.05 0.08 4.59 5.0 Chair HBNF 0.76 0.13 4.36 2.0 Boat HBNF 0.74 0.16 4.33 4.4 Chair FBNH 1.62 0.73 5.20 2.0 Boat FBNH 1.75 0.85 5.34 6.4 Chair FBNF 1.31 0.65 4.96 3.3 Boat FBNF 1.25 0.59 4.91 3.4 . nH nF a N where E is the total Omolecule ¼ E Eh-BN EH2 EF2 2 2 energy of the system, Eh-BN is the total energy of the pristine h-BN sheet, EH2 and EF2 are energies of hydrogen and fluorine molecules, respectively, nH and nF are the number of hydrogen and fluorine atoms per unit cell, respectively, while N denotes the total number of atoms in the unit cell.
the artificial electric field due to the net electric dipole moment that arises in polar surface calculations, a dipole correction57 is also used. After generating a G-centered grid with a k-point mesh of 14 14 1, the Brillouin zone integration is carried out. Using a 600 eV energy cutoff for the plane-wave expansion, all the sheets are fully relaxed until the interatomic forces are less than 0.003 eV Å1. Structural optimizations are performed using a conjugated gradient method. The calculated bond lengths, binding energies, and band gaps shown in Tables 1 and 2 are in good agreement with previous theoretical results.16,19
3. Results and discussion Similar to graphene, hydrogenation, fluorination, or co-decoration of h-BN sheets with H and F make B and N atoms sp3-hybridized, which distorts the planar geometry. The thickness of those distorted sheets is defined as the spacing between the upper hydrogen/fluorine plane and the lower hydrogen/fluorine plane, and is the measurement of planar distortion. The stability of the hydrogenated/fluorinated/co-decorated BN sheet is determined by the formation energy per atom Oatom (shown in Table 2), which is defined as Oatom = (E Eh-BN nHEH nFEF)/N, where E is the total energy of the system, Eh-BN is the total energy of the pristine h-BN sheet, EH and EF are energies of an isolated
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spin-polarized hydrogen and fluorine atom, respectively, nH and nF are the number of hydrogen and fluorine atoms per unit cell, respectively, while N denotes the total number of atoms in the unit cell. If the formation energy Oatom is negative, it is thermodynamically possible for the atoms to be combined with the h-BN sheet. Here, we find that the hydrogenation of h-BN is an exothermic process, and thus a stable hydrogenated BN sheet can be obtained by the reaction of the BN sheet with isolated hydrogen atoms. We also observe that, in contrast to graphene,16,17 the boat form is slightly more stable than the chair form for the hydrogenated BN sheet. However, using hydrogen molecules instead of hydrogen atoms to calculate the formation energy (Omolecule as shown in Table 2) of the hydrogenated h-BN, we obtain positive values (0.08 eV per atom for the boat form and 0.10 eV per atom for the chair form). This means that hydrogenation is difficult in the presence of hydrogen molecules. On the other hand, for FBNH, a co-decorated h-BN sheet with H and F atoms where F is at B-sites and H is at N-sites, we find that the formation energy is negative (i.e. exothermic reaction) even in the presence of hydrogen and fluorine molecules (0.73 eV per atom for chair FBNH and 0.85 eV per atom for boat FBNH). Again, the boat form is slightly more stable than the chair form. Since semi-fluorination (F at B sites) that distorts the sheet is exothermic,19 hydrogen can easily bind in that distorted sheet at N sites, resulting in a stable co-decorated BN sheet with H at N sites and F at B sites. The reverse process is also possible where H atoms are attached at B sites and F atoms are at N sites. But, this is energetically unstable because F does not like to bind with N atoms, giving large positive formation energy in the presence of hydrogen and fluorine molecules (0.13 eV per atom for chair HBNF and 0.16 eV per atom for boat HBNF). However, the formation energy of both chair and boat forms of HBNF using H and F atoms is exothermic. Interestingly, totally fluorination, which results in a large distortion of the pristine h-BN sheet, is quite stable even if we take hydrogen and fluorine molecules as a reference for our formation energy (0.65 eV per atom for chair FBNF and 0.59 eV per atom for boat FBNF). We also calculate cohesive energy of our 2D monolayers. The cohesive energy per atom Ecoh is defined as the heat of formation per atom when these atoms are assembled into a crystal structure by the following equation: !, X Ecoh ¼ E N (1) nX EX X¼B;N;H;F
where E is the total energy of the system consisting of N atoms, EX is the total energy of an isolated X (B, N, H, and F) atom, and nX is the number of a specific atom X per unit cell. We find that the cohesive energy is negative in each case. So, assembly of atoms into a crystal structure lowers the energy and might be possible in experiment. To get insights into the electronic properties of our systems, their band structures (calculated by GGA) are shown in Fig. 2. We find that chemical modification of h-BN can be a good way to generate direct band gap covering a wide range from 1.98 eV to 6.39 eV (Table 2). Due to the electronegativity difference
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PCCP Table 3 Electric polarization (P3) perpendicular to the sheets corresponding to the structures in Fig. 1 with a unit: pC m1
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Structure
Chair HBNH Chair HBNF Chair FBNH Chair FBNF
P3 (Berry phase) 48.6 P3 (Charge density) 47.0
Fig. 2 Band structures: the dashed line shows the Fermi energy in each case and the insets show the symmetric points in Brillouin zone.
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14.8 14.1
every case the boat form has no net out-of-plane polarization due to the atomic positions in its unit cell; namely that the upward dipole moments are perfectly canceled out by downward dipole moments. In fact, although the B–N bond is partially ionic, pristine h-BN has no net electric polarization due to its three-fold symmetry. But, hydrogenation or co-decoration of atoms in the sheet distorts the sheet flatness, causing sp3 hybridization and redistributing the charge density due to the electronegativity difference between the atoms. Thus, a dipole moment per unit cell originates in chair forms. We find that polarization in chair FBNH is higher than that in other cases we have tested. F atoms at B sites draw electrons and become negatively charged on one side while, on the other side, H atoms at N sites become positively charged, thus creating a net dipole moment per unit cell that is larger than that in others. In fact, due to its large electronegativity, F atoms at B sites can drag more electrons from B atoms than H atoms can. But, making one side of the sheet partially positive while the other side is negative, chair HBNH still manages a moderate polarization because electrons are dragged by N atoms from H atoms at N-sites, making them partially positive and by H atoms at B-sites from B atoms to become slightly negatively charged. On the other hand, for chair HBNF and chair FBNF cases, the scenario is quite different. In those cases, both sides of the sheets are partially negative, and consequently the polarization is very small. Bader charge analysis60 (shown in Table 4) is used to obtain partial charges. To quantify the real-space-charge rearrangement due to adsorbed atoms on the h-BN sheets, we calculate the polarization in the z-direction as
P3 ¼ between B and N atoms, in contrast to purely covalent bond in graphene, the bond in h-BN gains an ionic character. As a result, h-BN is an insulator with a high direct band gap (4.66 eV) at k-point K. However, keeping both the valence band maximum and the conduction band minimum at k-point G, indicating that band gaps are direct, we find that chemical modification in chair forms reduces the band gap. On the other hand, boat forms can show very high direct gap. Since the GGA calculations usually underestimate the energy band gap of semiconductors and insulators, we expect higher band gap for our systems in experiment. After obtaining the optimized structures as described above, polarizations are computed by using the ‘‘Berry phase’’ technique.58,59 Since our tested systems are monolayers, we have considered a two-dimensional definition for polarization that is dipole moment per unit area (i.e., charge per unit length as the unit), and the values are given in Table 3. We find that for
80.6 77.6
23.8 22.0
ð
rðzÞzdz þ
X
!, Zi zi
A
(2)
i
where r(z) is the electron charge density integrated over the x–y plane, Zi is the ionic charge of ion i, zi is the z coordinate of ion i, and A is the area of the unit cell. The integral and sum are computed over
Table 4 Bader partial charge analysis. HB, HN, FB, and FN stand for H and F at B-site and N-site, respectively
Structure
B
N
HB
HN
FB
FN
Chair HBNH Boat HBNH Chair HBNF Boat HBNF Chair FBNH Boat FBNH Chair FBNF Boat FBNF
+1.98 +2.02 +2.22 +2.03 +2.22 +2.25 +2.22 +2.24
1.79 1.82 1.64 1.24 1.75 1.86 1.12 1.10
0.49 0.60 0.14 0.46 — — —
+0.31 +0.40
— —
— +0.32 +0.44 —
— 0.76 0.83 0.77 0.81
— — 0.45 0.33 — — 0.32 0.32
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the unit cell. We find that polarizations calculated from charge density are close to Berry phase values as shown in Table 3. The piezoelectric effect is understood as the electromechanical interaction between the mechanical and the electrical state in crystalline materials with no inversion symmetry. Using Maxwell relations, the linear piezoelectric effect can be expressed as a first-order coupling between polarization (Pi) or the macroscopic electric field (Ei), and stress (sjk) or the strain (Zjk) tensors, where i, j, k = 1, 2, 3: ! @Pi @sjk ¼ ; (3) eijk ¼ @Zjk @Ei Z E
dijk ¼
@Pi @sjk
¼ E
@Zjk ; @Ei s
(4)
where eijk and dijk are third-rank piezoelectric stress and strain tensors,61 respectively. To limit the number of indices, Voigt notation61 is used throughout the paper. It can be shown that the two sets of piezoelectric constants eij and dij are related each other through the elastic constants Ckj by the transformation X dik Ckj : (5) eij ¼ k
It is much easier to compute the strain tensor in a first-principles calculation than to determine the stress tensor. Therefore, almost all density functional theory studies aimed at investigating piezoelectricity in solids evaluate eij coefficients via the use of the Pind,i = eijZj relationship, where Pind,i is an induced electric polarization in the i-direction and Zj is a strain element in the Voigt notation. After calculating the polarization as a Berry phase of the Bloch states using the modern theory of polarization, piezoelectric improper stress coefficients can be computed as @Pi eij ¼ (6) @Zj
Table 5
Calculated in-plane piezoelectric constants, e11 and d11 a
Structure
e11 (1010 C m1)
d11 (pm V1)
Chair Chair Chair Chair
1.61 1.81 0.84 1.38
0.99 1.27 0.50 0.90
HBNH FBNH HBNF FBNF
a
In-plane piezoelectric and elastic constants shown in Table 7 are related as d11 = e11/(C11 C12) for 3m point group monolayers and C12 = C21. Our calculated in-plane piezoelectric values are comparable with other known piezoelectric nano-materials.33,44,52
uniaxial in-plane strain. For low strains between 2.5% to 2.5%, we find the relationship is linear for each case (as shown in Fig. 3), and the slope of each line gives the improper e31 piezoelectric coefficients for each system. The calculated piezoelectric constants are shown in Table 6. Chair FBNH has a higher improper piezoelectric response than that of the chair HBNH by a factor of two, as expected due to its higher electric polarization. These improper constants are suitable for the interpretation or modeling of experiments involving depolarizing fields and polarization induced interface charges.64 To compare our calculated piezoelectric constants to those of known bulk piezoelectric materials such as wurtzite GaN, the difference in dimensionality must be taken into account because we have expressed the coefficients in two-dimensional units as charge per unit length, while it is charge per unit area in threedimensional units. However, a rough account can be made for the dimensionality by dividing the improper e31 coefficients by the h-BN interlayer separation of 3.3 Å in bulk h-BN crystals. We find e31,3D = 0.11 C m2 for chair HBNH, which is comparable to the piezoelectricity of PVDF (e31 = 0.27 C m2) and its copolymers.65 Similarly, e31,3D = 0.25 C m2 for chair FBNH is quite comparable to the theoretical value of 0.31 C m2 obtained for wurtzite BN,66 and almost half of the experimental value of 0.55 C m2 for wurtzite GaN.67 For 2D h-BN,52 the calculated e11,3D
E¼0
As the polarization in Berry-phase is a multi-valued quantity, naively computed improper piezoelectric response using a finite difference approach is branch-dependent and should not be compared with experimental results.62–64 So, to find a branchindependent proper piezoelectric response, which is comparable to experiments, we should account for the relation between the improper and proper piezoelectric tensor.63,64 Freely suspended two-dimensional chair forms of HBNH, FBNF, HBNF, and FBNH show 3m point group symmetry. Since coefficients involved with out-of-plane strain are ill-defined due to the vacuum along the direction perpendicular to the sheet, we have only two independent piezoelectric coefficients (e31 and e11) for each case. Although there is no polarization in the plane, the in-plane piezoelectricity still exists (shown in Table 5) – because inplane displacement of boron or nitrogen atoms due to strain or electric field induces dipole moment in the plane. As no polarization in the plane, the finite difference approach for our calculated e11 values gives proper piezoelectric response. However, to calculate the improper e31 piezoelectric coefficients, we evaluated the polarization change perpendicular to the surface as a function of
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Fig. 3 Relationship between a uniaxial in-plane strain and change of induced polarization perpendicular to the sheet. The slope of each line gives the e31 piezoelectric coefficients.
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Table 6
Table 7
Proper and improper e31 values and d31
d31 using proper ep31 and elastic constants
Structure
1 eimp 31 (pC m )
ep31 (pC m1)
d31 (pm V1)
Structure
C11 (N m1)
C21 (N m1)
d31 (pm V1)
Chair Chair Chair Chair
27.60 77.03 7.17 42.80
21.03 3.54 16.61 27.96
0.10 0.05 0.06 0.12
Chair Chair Chair Chair
187.40 174.25 200.41 188.64
25.65 31.52 32.36 35.02
0.10 0.02 0.07 0.13
HBNH FBNH HBNF FBNF
of 0.76 C m2 is very large, and not comparable to our e31,3D values as we expect, because the e11 coefficient is generally larger than the e31 coefficient. Here, we should remember that our studied systems are different from those three dimensional materials. On the other hand, we can compare our calculated e31 coefficients directly with recently discovered engineered piezoelectricity in graphene,37 and find that the e31 in chair FBNH is slightly higher than that in Li absorbed graphene, while chair HBNH is comparable to H and F or F and Li adsorbed graphene e31 coefficients. However, the proper piezoelectric response, corresponding to a set of so-called proper piezoelectric constants, is related to a current flow across a sample in response to a deformation. So, the proper constants should be compared with experiments based on measurements of current across piezoelectric samples and with values computed from linear response theory. Recently, it has been shown that the distinction between the proper and improper piezoelectric tensor is only present if a spontaneous polarization is present. So, corrections to the improper constants must be taken into account for a finite difference calculation of the proper piezoelectric response and for proper ep31, we have ep31 = eimp 31 + P3
(7) 63,68
where P3 is the electric polarization at equilibrium. These proper ep31 constants are shown in Table 6. We also determined the strain piezoelectric coefficients, d31. We use the converse piezoelectric effect, where piezoelectric materials change shape when subject to an external electric field E, and which can be expressed as Zj = dijEi (for a fixed
HBNH FBNH HBNF FBNF
stress), where Zj is the strain tensor and Ei is the applied electric field in the i-direction. Fig. 4 shows how an applied electric field perpendicular to the surface of the sheets induces an equibiaxial strain in the plane of the sheets. For each case, we find an approximately linear relationship between the field and the strain at field amplitudes between 0.3 and 0.3 V Å1, the slope of which gives the desired d31 piezoelectric coefficient. We observe that chair HBNH and FBNF have a d31 value (0.12 pm V1) that is almost twice that of the chair FBNH and HBNF values, as shown in Table 6. This means that the shape of chair HBNH and FBNF can easily be changed by an applied electric field. Comparing the d31 of chair HBNH/FBNF to the known piezoelectric materials, we find that it is a factor of 3 lower than that of wurtzite BN (0.33 pm V1)66 and almost ten times smaller than that of wurtzite GaN (1.0 pm V1).67 However, keeping dimensionality in mind, we see that our calculated values are quite comparable to the engineered piezoelectricity in graphene.37 We also determine the strain piezoelectric coefficients d31 using the relation between stress and strain piezoelectric constants for 2D materials (ill-defined d33) through elastic constants (shown in Table 7 and two-dimensional units are used): d31 ¼
ep31 ðC11 þ C21 Þ
(8)
4. Conclusions By using DFT calculations, we found that an out-of-plane electric polarization could be obtained by selectively adsorbing hydrogen and/or fluorine atoms on the BN sheet in the chair conformation. We also observed the out-of-plane piezoelectric response and compared the response to those of the typical two-dimensional and three-dimensional piezoelectrics. Obviously, the three-fold symmetry on the BN plane gives no in-plane polarization for the chair form, but the in-plane piezoelectric response still exists. So, we expect that the modified h-BN sheet by hydrogen and/or fluorine can widen the area of the h-BN sheet’s application as a piezoelectric material for various nano-electromechanical applications. On the other hand, having an uncompensated polar surface, these chemically modified h-BN monolayers can play an important role to understand surface chemistry, critical thickness and concomitant thickness-dependent properties (band gap, dipole moment, depolarization field, etc.) of polar materials.
Acknowledgements Fig. 4 Relationship between an applied electric field normal to the sheet and an induced equibiaxial strain in the plane of the sheet. The slope of each line gives the d31 piezoelectric coefficients.
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This research was supported by the research fund of University of Ulsan (2010-0126).
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Dipolar polarization and piezoelectricity of a hexagonal boron nitride sheet decorated with hydrogen and fluorine Mohammad Noor-A-Alam, Hye Jung Kim and Young-Han Shin* In contrast to graphene, a hexagonal boron nitride (h-BN) monolayer is piezoelectric because it is noncentrosymmetric. However, h-BN shows neither in-plane nor out-of-plane dipole moments due to its three-fold symmetry on the plane and the fact that it is completely flat. Here, we show that the controlled adsorption of hydrogen and/or fluorine atoms on both sides of a pristine h-BN sheet induces flatness distortion in a chair form and an out-of plane dipole moment. In contrast, a boat form has no out-of-plane dipole moment due to the alternating boron and nitrogen positions normal to the plane. Consequently, the chair form of surface-modified h-BN shows both in-plane and out-of-plane
Received 20th September 2013, Accepted 6th February 2014
piezoelectric responses; while pristine h-BN and the boat form of decorated h-BN have only in-plane
DOI: 10.1039/c3cp53971g
are comparable to those in known three-dimensional piezoelectric materials. Such an engineered
piezoelectric responses. These in-plane and out-of-plane piezoelectric responses of the modified h-BN piezoelectric two-dimensional boron nitride monolayer can be a candidate material for various nano-
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electromechanical applications.
1. Introduction The extraordinary success of atomically thick graphene1–10 both experimentally and theoretically has revolutionized the whole of nanoscience because of its suitability for future nanotechnology applications. Consequently, hexagonal boron nitride (h-BN) monolayers, two-dimensional honeycomb networks of boron and nitrogen atoms similar to carbons in graphene, have been brought to the forefront of research11–14 because they have remarkably different electronic properties. For example, while graphene is a conductor and hydrogenation of graphene makes it semiconducting,15–17 an h-BN sheet is an insulator and hydrogenation decreases its band gap.18,19 One of the many reasons for these differences comes from the fact that C–C bonds in graphene are purely covalent, whereas B–N bonds in an h-BN sheet have a partial ionic character because of the large electronegativity difference between boron and nitrogen.20,21 Therefore, despite similar lattice parameters and crystal structure, two seemingly disparate materials with different band structures can complement each other in many potential applications to provide interesting functional properties arising from their integration.22–31 One of the biggest challenges in harnessing the power of nanotechnology is achieving dynamic control of mechanical, Department of Physics and EHSRC, University of Ulsan, Ulsan 680-749, Republic of Korea. E-mail: [email protected]; Fax: +82-52-259-1693
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chemical and electronic properties of nanoscale devices. Such control is crucial for many devices on which our modern life heavily relies, including transistors, sensors, actuators, energy harvesters, motors, robots, artificial muscles, and other locomotive devices. Usually, piezoelectric materials are the first choice for dynamic control of material deformation by the application of an external electric field. They are central to a wide variety of applications from pressure sensors to acoustic transducers to high voltage generators.32–37 Moving to the nanoscale, semiconductor and piezoelectric nanomaterials have opened a new burgeoning area of nanotechnology called nanopiezotronics; these materials have exciting and unique properties for a wide range of applications in electronics, optics, optoelectronics, nanogenerators, sensors, resonators, and the biological sciences that may improve quality of life.38–41 However, graphene is unfortunately not intrinsically piezoelectric due to its centrosymmetric crystal structure. Interestingly, Ong and Reed have recently discovered that piezoelectric effects can be engineered into non-piezoelectric graphene by breaking the inversion symmetry through the selective surface adsorption of atoms.37 There has also been a totally different approach to coax piezoelectricity from graphene merely by creating holes of a specific symmetry.42 In contrast to graphene, an h-BN sheet does not exhibit inversion symmetry, and thus is intrinsically piezoelectric.43,44 h-BN sheets are electrical insulators (a band gap of around 4.6 eV) and have profound chemical and thermal stabilities. At the same time, like their C counterparts (graphene), they are
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thermally conductive, mechanically robust, and highly resistant to oxidation.11,43–46 Such unique properties make h-BN sheets a promising nanomaterial in a variety of potential fields such as insulators with high thermal conductivity in electronic devices, ultraviolet-light emitters in optoelectronics, hydrogen accumulators, high strength nano-fillers, and thermally conductive nanocomposites. Being intrinsically piezoelectric, an h-BN monolayer also has the potential to be used as a nano-piezoelectric material. Unfortunately, it does not show any out of plane piezoelectric response due to its complete flatness, which limits its potential applications in nanoelectromechanical devices.34 Because of its three-fold symmetry, it has no in-plane polarization either. Recently, electric polarization has been found in boron nitride nanotubes.48–52 However, we find that both in-plane and out-ofplane piezoelectric response as well as out-of-plane dipolar polarization can be induced in h-BN sheets by merely decorating the surface with atoms. Having made advances11–14 in experimental synthesis of h-BN, it is expected that the same is possible for its chemical functionalization. By hydrogen plasma treatment of h-BN sheets on a substrate, partially one-side hydrogenated few-layered h-BN membranes have been achieved so far.53 Fully hydrogenated and fluorinated layers are expected when free standing h-BN sheets are used. However, the electronic structure and magnetic properties of fully decorated and semi-decorated BN sheets using H and/or F atoms have been studied extensively using first principles calculations, especially to tune the band gap of h-BN sheets.18,19,47 Interestingly, in contrast to endothermic hydrogenation, fluorination and codecoration of h-BN sheet are found to be exothermic mainly because of weaker bonding in F2 than in H2.19 The adatoms’ pattern, which is experimentally
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difficult to control, and is often formed in a random way on a host structure depending on processes largely determines the properties of the functionalized h-BN structures. As a result, many different configurations with different electromechanical properties are expected to be formed under experimental conditions. In this paper, we have conducted an extensive study of the structures and out-of-plane dipolar polarization as well as piezoelectric properties of fully hydrogenated or fluorinated h-BN sheets (labeled as chair and boat for HBNH and FBNF) and co-decorated h-BN sheets using H and F (labeled as chair and boat for FBNH and HBNF). Optimized geometric structures of our considered configurations are shown in Fig. 1. We show that hydrogenation, fluorination, or co-decoration of H and F on an h-BN sheet breaks the flatness of the sheet and results in an out-of-plane electric polarization. Our finding of these uncompensated polar surfaces of chemically modified h-BN monolayers may cast light on their surface chemistry, and electronic properties of their multi-layers. We have also calculated piezoelectric constants e31 and d31, and they are comparable to those of other known piezoelectric materials.
2. Computational details Our calculations are based on density functional theory (DFT) using a generalized gradient approximation (GGA) for exchange– correlation potential with a Perdew–Burke–Ernzerhof functional as implemented in the Vienna ab initio Simulation Package (VASP).54–56 Supercells are used to simulate the isolated sheets, and the distance between neighboring sheets is kept larger than 20 Å in order to avoid interactions between them. To cancel out
Fig. 1 Top and side views of optimized configurations: (a) chair HBNH, (b) chair FBNH, (c) chair HBNF, (d) chair FBNF, (e) boat HBNH, (f) boat FBNH, (g) boat HBNF, and (h) boat FBNF. Rectangles represent conventional unit cells.
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PCCP Table 1
Paper Structural data of the systems studied
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Bond lengths (Å) Structure
B–N
B–H
B–F
N–H
N–F
Thickness of the sheets (Å)
h-BN Chair HBNH Boat HBNH Chair FBNH Boat FBNH Chair HBNF Boat HBNF Chair FBNF Boat FBNF
1.45 1.58 1.57/1.62a 1.62 159/1.58 1.58 1.56/1.60 1.62 1.60/1.70
— 1.20 1.21 — — 1.21 1.20 — —
— — — 1.34 1.40 — — 1.35 1.35
— 1.03 1.02 1.03 1.03 — — — —
— — — — — 1.46 1.49 1.44 1.45
— 2.75 2.81 2.91 3.02 3.13 3.23 3.30 3.34
a
Two different bond lengths for the boat conformation.
Table 2 Cohesive energy (Ecoh), formation energies per atom (Oatom and Omolecule from atoms and moleculesa respectively), and band gap (Eg) with a unit of eV
Structure
Oatom
Omolecule
Ecoh
Eg
Chair HBNH 1.04 0.10 4.57 3.3 Boat HBNH 1.05 0.08 4.59 5.0 Chair HBNF 0.76 0.13 4.36 2.0 Boat HBNF 0.74 0.16 4.33 4.4 Chair FBNH 1.62 0.73 5.20 2.0 Boat FBNH 1.75 0.85 5.34 6.4 Chair FBNF 1.31 0.65 4.96 3.3 Boat FBNF 1.25 0.59 4.91 3.4 . nH nF a N where E is the total Omolecule ¼ E Eh-BN EH2 EF2 2 2 energy of the system, Eh-BN is the total energy of the pristine h-BN sheet, EH2 and EF2 are energies of hydrogen and fluorine molecules, respectively, nH and nF are the number of hydrogen and fluorine atoms per unit cell, respectively, while N denotes the total number of atoms in the unit cell.
the artificial electric field due to the net electric dipole moment that arises in polar surface calculations, a dipole correction57 is also used. After generating a G-centered grid with a k-point mesh of 14 14 1, the Brillouin zone integration is carried out. Using a 600 eV energy cutoff for the plane-wave expansion, all the sheets are fully relaxed until the interatomic forces are less than 0.003 eV Å1. Structural optimizations are performed using a conjugated gradient method. The calculated bond lengths, binding energies, and band gaps shown in Tables 1 and 2 are in good agreement with previous theoretical results.16,19
3. Results and discussion Similar to graphene, hydrogenation, fluorination, or co-decoration of h-BN sheets with H and F make B and N atoms sp3-hybridized, which distorts the planar geometry. The thickness of those distorted sheets is defined as the spacing between the upper hydrogen/fluorine plane and the lower hydrogen/fluorine plane, and is the measurement of planar distortion. The stability of the hydrogenated/fluorinated/co-decorated BN sheet is determined by the formation energy per atom Oatom (shown in Table 2), which is defined as Oatom = (E Eh-BN nHEH nFEF)/N, where E is the total energy of the system, Eh-BN is the total energy of the pristine h-BN sheet, EH and EF are energies of an isolated
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spin-polarized hydrogen and fluorine atom, respectively, nH and nF are the number of hydrogen and fluorine atoms per unit cell, respectively, while N denotes the total number of atoms in the unit cell. If the formation energy Oatom is negative, it is thermodynamically possible for the atoms to be combined with the h-BN sheet. Here, we find that the hydrogenation of h-BN is an exothermic process, and thus a stable hydrogenated BN sheet can be obtained by the reaction of the BN sheet with isolated hydrogen atoms. We also observe that, in contrast to graphene,16,17 the boat form is slightly more stable than the chair form for the hydrogenated BN sheet. However, using hydrogen molecules instead of hydrogen atoms to calculate the formation energy (Omolecule as shown in Table 2) of the hydrogenated h-BN, we obtain positive values (0.08 eV per atom for the boat form and 0.10 eV per atom for the chair form). This means that hydrogenation is difficult in the presence of hydrogen molecules. On the other hand, for FBNH, a co-decorated h-BN sheet with H and F atoms where F is at B-sites and H is at N-sites, we find that the formation energy is negative (i.e. exothermic reaction) even in the presence of hydrogen and fluorine molecules (0.73 eV per atom for chair FBNH and 0.85 eV per atom for boat FBNH). Again, the boat form is slightly more stable than the chair form. Since semi-fluorination (F at B sites) that distorts the sheet is exothermic,19 hydrogen can easily bind in that distorted sheet at N sites, resulting in a stable co-decorated BN sheet with H at N sites and F at B sites. The reverse process is also possible where H atoms are attached at B sites and F atoms are at N sites. But, this is energetically unstable because F does not like to bind with N atoms, giving large positive formation energy in the presence of hydrogen and fluorine molecules (0.13 eV per atom for chair HBNF and 0.16 eV per atom for boat HBNF). However, the formation energy of both chair and boat forms of HBNF using H and F atoms is exothermic. Interestingly, totally fluorination, which results in a large distortion of the pristine h-BN sheet, is quite stable even if we take hydrogen and fluorine molecules as a reference for our formation energy (0.65 eV per atom for chair FBNF and 0.59 eV per atom for boat FBNF). We also calculate cohesive energy of our 2D monolayers. The cohesive energy per atom Ecoh is defined as the heat of formation per atom when these atoms are assembled into a crystal structure by the following equation: !, X Ecoh ¼ E N (1) nX EX X¼B;N;H;F
where E is the total energy of the system consisting of N atoms, EX is the total energy of an isolated X (B, N, H, and F) atom, and nX is the number of a specific atom X per unit cell. We find that the cohesive energy is negative in each case. So, assembly of atoms into a crystal structure lowers the energy and might be possible in experiment. To get insights into the electronic properties of our systems, their band structures (calculated by GGA) are shown in Fig. 2. We find that chemical modification of h-BN can be a good way to generate direct band gap covering a wide range from 1.98 eV to 6.39 eV (Table 2). Due to the electronegativity difference
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PCCP Table 3 Electric polarization (P3) perpendicular to the sheets corresponding to the structures in Fig. 1 with a unit: pC m1
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Structure
Chair HBNH Chair HBNF Chair FBNH Chair FBNF
P3 (Berry phase) 48.6 P3 (Charge density) 47.0
Fig. 2 Band structures: the dashed line shows the Fermi energy in each case and the insets show the symmetric points in Brillouin zone.
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14.8 14.1
every case the boat form has no net out-of-plane polarization due to the atomic positions in its unit cell; namely that the upward dipole moments are perfectly canceled out by downward dipole moments. In fact, although the B–N bond is partially ionic, pristine h-BN has no net electric polarization due to its three-fold symmetry. But, hydrogenation or co-decoration of atoms in the sheet distorts the sheet flatness, causing sp3 hybridization and redistributing the charge density due to the electronegativity difference between the atoms. Thus, a dipole moment per unit cell originates in chair forms. We find that polarization in chair FBNH is higher than that in other cases we have tested. F atoms at B sites draw electrons and become negatively charged on one side while, on the other side, H atoms at N sites become positively charged, thus creating a net dipole moment per unit cell that is larger than that in others. In fact, due to its large electronegativity, F atoms at B sites can drag more electrons from B atoms than H atoms can. But, making one side of the sheet partially positive while the other side is negative, chair HBNH still manages a moderate polarization because electrons are dragged by N atoms from H atoms at N-sites, making them partially positive and by H atoms at B-sites from B atoms to become slightly negatively charged. On the other hand, for chair HBNF and chair FBNF cases, the scenario is quite different. In those cases, both sides of the sheets are partially negative, and consequently the polarization is very small. Bader charge analysis60 (shown in Table 4) is used to obtain partial charges. To quantify the real-space-charge rearrangement due to adsorbed atoms on the h-BN sheets, we calculate the polarization in the z-direction as
P3 ¼ between B and N atoms, in contrast to purely covalent bond in graphene, the bond in h-BN gains an ionic character. As a result, h-BN is an insulator with a high direct band gap (4.66 eV) at k-point K. However, keeping both the valence band maximum and the conduction band minimum at k-point G, indicating that band gaps are direct, we find that chemical modification in chair forms reduces the band gap. On the other hand, boat forms can show very high direct gap. Since the GGA calculations usually underestimate the energy band gap of semiconductors and insulators, we expect higher band gap for our systems in experiment. After obtaining the optimized structures as described above, polarizations are computed by using the ‘‘Berry phase’’ technique.58,59 Since our tested systems are monolayers, we have considered a two-dimensional definition for polarization that is dipole moment per unit area (i.e., charge per unit length as the unit), and the values are given in Table 3. We find that for
80.6 77.6
23.8 22.0
ð
rðzÞzdz þ
X
!, Zi zi
A
(2)
i
where r(z) is the electron charge density integrated over the x–y plane, Zi is the ionic charge of ion i, zi is the z coordinate of ion i, and A is the area of the unit cell. The integral and sum are computed over
Table 4 Bader partial charge analysis. HB, HN, FB, and FN stand for H and F at B-site and N-site, respectively
Structure
B
N
HB
HN
FB
FN
Chair HBNH Boat HBNH Chair HBNF Boat HBNF Chair FBNH Boat FBNH Chair FBNF Boat FBNF
+1.98 +2.02 +2.22 +2.03 +2.22 +2.25 +2.22 +2.24
1.79 1.82 1.64 1.24 1.75 1.86 1.12 1.10
0.49 0.60 0.14 0.46 — — —
+0.31 +0.40
— —
— +0.32 +0.44 —
— 0.76 0.83 0.77 0.81
— — 0.45 0.33 — — 0.32 0.32
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the unit cell. We find that polarizations calculated from charge density are close to Berry phase values as shown in Table 3. The piezoelectric effect is understood as the electromechanical interaction between the mechanical and the electrical state in crystalline materials with no inversion symmetry. Using Maxwell relations, the linear piezoelectric effect can be expressed as a first-order coupling between polarization (Pi) or the macroscopic electric field (Ei), and stress (sjk) or the strain (Zjk) tensors, where i, j, k = 1, 2, 3: ! @Pi @sjk ¼ ; (3) eijk ¼ @Zjk @Ei Z E
dijk ¼
@Pi @sjk
¼ E
@Zjk ; @Ei s
(4)
where eijk and dijk are third-rank piezoelectric stress and strain tensors,61 respectively. To limit the number of indices, Voigt notation61 is used throughout the paper. It can be shown that the two sets of piezoelectric constants eij and dij are related each other through the elastic constants Ckj by the transformation X dik Ckj : (5) eij ¼ k
It is much easier to compute the strain tensor in a first-principles calculation than to determine the stress tensor. Therefore, almost all density functional theory studies aimed at investigating piezoelectricity in solids evaluate eij coefficients via the use of the Pind,i = eijZj relationship, where Pind,i is an induced electric polarization in the i-direction and Zj is a strain element in the Voigt notation. After calculating the polarization as a Berry phase of the Bloch states using the modern theory of polarization, piezoelectric improper stress coefficients can be computed as @Pi eij ¼ (6) @Zj
Table 5
Calculated in-plane piezoelectric constants, e11 and d11 a
Structure
e11 (1010 C m1)
d11 (pm V1)
Chair Chair Chair Chair
1.61 1.81 0.84 1.38
0.99 1.27 0.50 0.90
HBNH FBNH HBNF FBNF
a
In-plane piezoelectric and elastic constants shown in Table 7 are related as d11 = e11/(C11 C12) for 3m point group monolayers and C12 = C21. Our calculated in-plane piezoelectric values are comparable with other known piezoelectric nano-materials.33,44,52
uniaxial in-plane strain. For low strains between 2.5% to 2.5%, we find the relationship is linear for each case (as shown in Fig. 3), and the slope of each line gives the improper e31 piezoelectric coefficients for each system. The calculated piezoelectric constants are shown in Table 6. Chair FBNH has a higher improper piezoelectric response than that of the chair HBNH by a factor of two, as expected due to its higher electric polarization. These improper constants are suitable for the interpretation or modeling of experiments involving depolarizing fields and polarization induced interface charges.64 To compare our calculated piezoelectric constants to those of known bulk piezoelectric materials such as wurtzite GaN, the difference in dimensionality must be taken into account because we have expressed the coefficients in two-dimensional units as charge per unit length, while it is charge per unit area in threedimensional units. However, a rough account can be made for the dimensionality by dividing the improper e31 coefficients by the h-BN interlayer separation of 3.3 Å in bulk h-BN crystals. We find e31,3D = 0.11 C m2 for chair HBNH, which is comparable to the piezoelectricity of PVDF (e31 = 0.27 C m2) and its copolymers.65 Similarly, e31,3D = 0.25 C m2 for chair FBNH is quite comparable to the theoretical value of 0.31 C m2 obtained for wurtzite BN,66 and almost half of the experimental value of 0.55 C m2 for wurtzite GaN.67 For 2D h-BN,52 the calculated e11,3D
E¼0
As the polarization in Berry-phase is a multi-valued quantity, naively computed improper piezoelectric response using a finite difference approach is branch-dependent and should not be compared with experimental results.62–64 So, to find a branchindependent proper piezoelectric response, which is comparable to experiments, we should account for the relation between the improper and proper piezoelectric tensor.63,64 Freely suspended two-dimensional chair forms of HBNH, FBNF, HBNF, and FBNH show 3m point group symmetry. Since coefficients involved with out-of-plane strain are ill-defined due to the vacuum along the direction perpendicular to the sheet, we have only two independent piezoelectric coefficients (e31 and e11) for each case. Although there is no polarization in the plane, the in-plane piezoelectricity still exists (shown in Table 5) – because inplane displacement of boron or nitrogen atoms due to strain or electric field induces dipole moment in the plane. As no polarization in the plane, the finite difference approach for our calculated e11 values gives proper piezoelectric response. However, to calculate the improper e31 piezoelectric coefficients, we evaluated the polarization change perpendicular to the surface as a function of
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Fig. 3 Relationship between a uniaxial in-plane strain and change of induced polarization perpendicular to the sheet. The slope of each line gives the e31 piezoelectric coefficients.
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Table 6
Table 7
Proper and improper e31 values and d31
d31 using proper ep31 and elastic constants
Structure
1 eimp 31 (pC m )
ep31 (pC m1)
d31 (pm V1)
Structure
C11 (N m1)
C21 (N m1)
d31 (pm V1)
Chair Chair Chair Chair
27.60 77.03 7.17 42.80
21.03 3.54 16.61 27.96
0.10 0.05 0.06 0.12
Chair Chair Chair Chair
187.40 174.25 200.41 188.64
25.65 31.52 32.36 35.02
0.10 0.02 0.07 0.13
HBNH FBNH HBNF FBNF
of 0.76 C m2 is very large, and not comparable to our e31,3D values as we expect, because the e11 coefficient is generally larger than the e31 coefficient. Here, we should remember that our studied systems are different from those three dimensional materials. On the other hand, we can compare our calculated e31 coefficients directly with recently discovered engineered piezoelectricity in graphene,37 and find that the e31 in chair FBNH is slightly higher than that in Li absorbed graphene, while chair HBNH is comparable to H and F or F and Li adsorbed graphene e31 coefficients. However, the proper piezoelectric response, corresponding to a set of so-called proper piezoelectric constants, is related to a current flow across a sample in response to a deformation. So, the proper constants should be compared with experiments based on measurements of current across piezoelectric samples and with values computed from linear response theory. Recently, it has been shown that the distinction between the proper and improper piezoelectric tensor is only present if a spontaneous polarization is present. So, corrections to the improper constants must be taken into account for a finite difference calculation of the proper piezoelectric response and for proper ep31, we have ep31 = eimp 31 + P3
(7) 63,68
where P3 is the electric polarization at equilibrium. These proper ep31 constants are shown in Table 6. We also determined the strain piezoelectric coefficients, d31. We use the converse piezoelectric effect, where piezoelectric materials change shape when subject to an external electric field E, and which can be expressed as Zj = dijEi (for a fixed
HBNH FBNH HBNF FBNF
stress), where Zj is the strain tensor and Ei is the applied electric field in the i-direction. Fig. 4 shows how an applied electric field perpendicular to the surface of the sheets induces an equibiaxial strain in the plane of the sheets. For each case, we find an approximately linear relationship between the field and the strain at field amplitudes between 0.3 and 0.3 V Å1, the slope of which gives the desired d31 piezoelectric coefficient. We observe that chair HBNH and FBNF have a d31 value (0.12 pm V1) that is almost twice that of the chair FBNH and HBNF values, as shown in Table 6. This means that the shape of chair HBNH and FBNF can easily be changed by an applied electric field. Comparing the d31 of chair HBNH/FBNF to the known piezoelectric materials, we find that it is a factor of 3 lower than that of wurtzite BN (0.33 pm V1)66 and almost ten times smaller than that of wurtzite GaN (1.0 pm V1).67 However, keeping dimensionality in mind, we see that our calculated values are quite comparable to the engineered piezoelectricity in graphene.37 We also determine the strain piezoelectric coefficients d31 using the relation between stress and strain piezoelectric constants for 2D materials (ill-defined d33) through elastic constants (shown in Table 7 and two-dimensional units are used): d31 ¼
ep31 ðC11 þ C21 Þ
(8)
4. Conclusions By using DFT calculations, we found that an out-of-plane electric polarization could be obtained by selectively adsorbing hydrogen and/or fluorine atoms on the BN sheet in the chair conformation. We also observed the out-of-plane piezoelectric response and compared the response to those of the typical two-dimensional and three-dimensional piezoelectrics. Obviously, the three-fold symmetry on the BN plane gives no in-plane polarization for the chair form, but the in-plane piezoelectric response still exists. So, we expect that the modified h-BN sheet by hydrogen and/or fluorine can widen the area of the h-BN sheet’s application as a piezoelectric material for various nano-electromechanical applications. On the other hand, having an uncompensated polar surface, these chemically modified h-BN monolayers can play an important role to understand surface chemistry, critical thickness and concomitant thickness-dependent properties (band gap, dipole moment, depolarization field, etc.) of polar materials.
Acknowledgements Fig. 4 Relationship between an applied electric field normal to the sheet and an induced equibiaxial strain in the plane of the sheet. The slope of each line gives the d31 piezoelectric coefficients.
6580 | Phys. Chem. Chem. Phys., 2014, 16, 6575--6582
This research was supported by the research fund of University of Ulsan (2010-0126).
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