Electronic, Optical, and Mechanical Properties of Diamond Nanowires

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Electronic, Optical and Mechanical Properties of Diamond Nanowires Encapsulated in Carbon Nanotubes: A First-Principles View Haidi Wang, Bin Li, and Jinlong Yang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b12864 • Publication Date (Web): 18 Jan 2017 Downloaded from http://pubs.acs.org on January 19, 2017

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The Journal of Physical Chemistry

Electronic, Optical and Mechanical Properties of Diamond Nanowires Encapsulated in Carbon Nanotubes: A First-Principles View Haidi Wang,a Bin Li,a,b,* and Jinlong Yanga,b,* a

Hefei National Laboratory of Physical Science at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China Synergetic Innovation Center of Quantum Information & Quantum Physics, University of

b

Science and Technology of China, Hefei, Anhui 230026, China ABSTRACT In recent years, carbon-based complex nano-structures have been explored due to many of their unique properties and related applications. Here we employ theoretical simulation based on density functional theory to investigate electronic, optical and mechanical properties of a new type of the carbon-based complex nano-structure, i.e. experimentally fabricated one-dimensional complex material of hydrogenated diamond nanowires encapsulated in carbon nanotubes (CNW@CNT). The complex structure CNW@CNT is found to possess metallicity for the outer CNT and wide band gap nature for the inner CNW simultaneously. Under uniaxial strain a specific insulator-to-metal transition occurs for the inner CNW in the complex structure, with

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threshold value much smaller than that for the individual insulator. This effect is interpreted as that the strain induces relative shifting of bands of CNW and CNT and even charge transfer between them, making the valence band of CNW become not fully occupied. The inner CNW in the complex structure has optical absorption only in the ultraviolet waveband. The further examinations on the conductive bands reveal existences of nearly-free-electron states which entirely dominate the conductive bands of the inner CNW, and suggest that the electron-hole separation will happen in the CNW@CNT upon the ultraviolet illumination. The simulation results also reveal higher Young’s modulus of the CNW@CNT and the individual CNW even larger than those of CNT and graphene. We propose a simple parallel spring model to establish the relationship between the Young’s modulus of the complex structure and the one of its component which should be helpful to future predictions for other complex structures. The potential applications of this new type of carbon-based complex structure as a multi-functional integrated nano-material in the future nano-electronics, nano-optoelectronics and nanoelectromechanics are discussed.

1. INTRODUCTION With developments of nanoscience and nanotechnology, a variety of carbon-based nanostructured materials, such as zero-dimensional (0D) fullerenes,1 one-dimensional (1D) nanotubes,2 nanowires,3 nanoribbons,4 and two-dimensional (2D) graphene,5 have been introduced and explored for their excellent properties and potential applications in many fields. Among these carbon-based materials, the 1D carbon nano-materials are suggested to become one of important components of the future nano-electronic and nano-mechanical devices and play an integral part in their design and construction.


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The carbon nanotube (CNT) is the firstly synthesized 1D carbon nano-material. An ideal single-wall CNT can be constructed by cutting a strip with infinite length from the graphene

v v v v sheet and then rolling it up into a cylinder along the 2D lattice vector R = nR1 + mR2 , where R1 v and R2 are primitive lattice vectors of the graphene sheet. Here a two-integers index (n,m) is used to identify different CNT: (n,0)-CNTs and (n,n)-CNTs are called zigzag and armchair nanotubes respectively, and the rest are chiral.6 Generally, there exists the following rule for the single-wall CNT:7 (n,n)-CNTs are all metals; (n,m)-CNTs with n − m = 3 j , where j is a non-zero integer, are semiconductors with very tiny-gap; and all others are semiconductors with large-gap. Many studies have showed that the CNTs have unique mechanical, electronic, catalytic, adsorption and transport properties so as to be interesting for many applications.8–10 A wide range of theoretical simulations and experiments also indicate that external strain can be used to manipulate properties of CNT.11,12 The 1D sp3 hybridization carbon nanowires (CNW) have also attracted many attentions.13 A large number of techniques, including laser-induced chemical vapor deposition,14 high-pressure treatment of catalyst-containing thin films,15 annealing of silicon carbide films,16 and annealing of pressed tablets containing graphite,17 have been developed to synthesize CNW. The first principles simulations conclude that CNWs prefer structures with their principal axis parallel to [100] direction and the surface of (110) appears less structurally favorable.18 However, for the hydrogenated CNWs, H-C interactions at their surface could influence this relative stability.13 The CNWs have been revealed to possess some remarkable properties so that they can be applied in field emission, scanning probe microscopy probes, mass spectrometric analysis of small molecules, high-performance nano-electromechanical switches, electrochemical sensors, and so on.13


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In recent years, peoples have begun to pay attention to the complex nano-structures or hybrid nano-structures consisting of different nano-materials with homogeneous or similar elemental composition which are combined via weak or strong interaction. For example, the ZnO nanowires with specific core-shell structures formed by amorphization of the surface or introducing interfacial Zn-doping layer were fabricated, and good improvements in their optical and electrical properties were observed.19,20 As for the carbon-based complex nano-structures, the carbon nanopeapod in which a C60 molecular chain is enwrapped by a CNT21 and carbon nanobud in which the C60 molecules are covalently attached onto sidewall of a single-wall CNT22 had been synthesized experimentally. And the complex structure of fullerene molecules adsorbed on the graphene had been extensively studied.23–27 Theoretical simulation also predicted the carbon nanobud in which the C60 molecules are covalently attached onto surface of a graphene monolayer28 and the carbon sheet-tube frameworks constructed by tailored graphene sheets and armchair single-wall CNTs.29 It had been shown that these carbon-based complex structures and their derivatives not only can inherit some advantages of fullerene and CNT or graphene, but also may be utilized as a new building block in the electronic, optoelectronic and photoelectrochemical nano-devices.24–28,30–36 In recent years, a new type of carbon-based complex nano-structure, which consists of a 1D sp3 CNW encapsulated inside a 1D sp2 CNT (CNW@CNT), was synthesized via fusion reaction of diamondoid derivatives diamantane-4,9-dicarboxylic acid.37 In this paper, we employ the firstprinciples calculations to explore the electronic, optical and mechanical properties of this new complex nano-structure. The calculations validate that the diamond nanowires encapsulated in (8,8)-CNT is the most stable structure in our examined systems. It is found that for the complex structure CNW@(8,8)-CNT, both metallicity of the outer armchair CNT and wide-gap of the


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inner CNW have been maintained. Under the compressive strain, an insulator-to-metal transition can occur for the inner CNW as a consequence of the charge transfer between the CNW and CNT via electron tunneling, with a threshold value much smaller than that for the individual CNW. It is also found that the optical absorption of CNW in the complex structure occur in the ultraviolet waveband, and the complex structure has hybrid nearly free electron states which entirely dominate the conductive bands (CBs) of CNW, which will result in the electron-hole separation when the complex structure is illuminated by the ultraviolet light. The individual CNWs have higher Young’s modulus even comparable to the diamond bulk, and Young’s modulus of the complex structure is appreciably larger than that of CNT or graphene. A simple parallel spring model is proposed to deduce the Young’s modulus of any complex structure based on the ones of each components. These properties of the complex structure CNW@CNT suggest its many potential applications in the fields of nano-electronics, nano-optoelectronics, and nano-electromechanics. 2. COMPUTATIONAL METHODS AND MODELS Our theoretical calculations employ Vienna ab-initio simulation package (VASP)38,39 approach within the Perdew-Burke-Ernzerhof (PBE)40 exchange-correlation functional of generalized gradient approximation (GGA). Projector-augmented wave (PAW)41 pseudopotentials are adopted to describe the electron-ion interaction, and van der Waals interactions are taken into account by choosing DFT-D2 method.42 The kinetic energy cutoff for the plane-wave basis set is chosen to be 500 eV. The energy convergence criteria for electronic SCF iterations is 10-5 eV and force convergence criteria in ionic step iterations is set to be 10-2 eV/Å. Axes of the nanotube and nanowire is along z direction, and vacuum slabs of 15 Å are added along both x and y directions to eliminate interaction of the neighboring nanotubes. We adopt variable cell


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conjugate gradient algorithm to relax structures of the examined systems. The reciprocal space is meshed at 1×1×18 for all systems using Monkhorst-Pack meshes. In our calculations, four types of armchair CNTs [(m,m)-CNT, m=7-10] are chosen to encapsulate the diamond CNW with (110) surface. The reasons why we choose these nanotubes are as follows: firstly, we find that in order to encapsulate the CNW obtained in the experiment,37 the CNT must has an enough large diameter and the (7,7)-CNT may become the smallest nanotube among all of possible candidates; secondly, the armchair CNTs of above families have small lattice mismatch of about 2.4% with the diamond CNW with (110) face, but the zigzag CNTs have larger lattice mismatch of about 17.2%. As an example, the schematic of CNW obtained in the experiment and its encapsulated structure CNW@(8,8)-CNT are shown in Figure 1. Here we adopt single-walled CNTs which are different from the double-walled cases in the experiment, because the single-walled CNTs are more representative system for general study of the encapsulated structure.

Figure 1. (a) Atomic structure of CNW. (b) Side view and (c) vertical view of the CNW@(8,8)CNT. Ball-and-stick model is used for CNW, and stick model is used for CNT. The white, green and blue represents H atoms, C atoms in CNW and the C atoms in CNT, respectively. 3. RESULTS AND DISCUSSIONS


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In the beginning, we pay attention to stability of the complex structures CNW@CNT. All structures are fully relaxed. As a typical example, the optimized structure of CNW@(8,8)-CNT is given in Figure 1(b) and (c). From its vertical view along the tube axis [Figure 1(c)], one can clearly find that cross section of the complex structure CNW@(8,8)-CNT becomes ellipse instead of primitive circle in CNT. It is also discovered that all cross sections of these complex structures are ellipse, and the length difference between the major axis dA and minor axis dB (see Figure 1) gradually decreases with the increased radius of CNT (see Table 1). To evaluate the relative stability of different complex structures, we calculated the formation energy43 per unit cell Δ and per hydrogen atom Δ , which is defined as:

Δ = − −

Δ = ( − − )/

(1) (2)

where is total energy of the complex structures, and and are total energy of the individual CNW and CNT respectively. All complex structures have eight hydrogen atoms, i.e. = 8. The results are summarized in Table 1. The negative values of Δ or Δ indicate that the encapsulation is an exothermic process. Obviously, one may deduce that the CNW@(8,8)-CNT is the most stable among these complex structures. In the CNW@(8,8)-CNT, the calculated C-C bond lengths of CNW are 1.541-1.556 Å, which are consistent with C-C bond in bulk diamond. The C-H bond length is 1.120 Å. The C-C bond length of CNW@(8,8)-CNT changes little compared to that of CNW, which may be attributed to suitable diameter of CNT for encapsulating CNW. Table 1. Formation energy per unit cell and per H atom of four complex structures CNW@(m,m)-CNT (m = 7-10) along with major axis dA and minor axis dB of their ellipse cross section perpendicular to the tube axis.


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System

CNW@(7,7)-CNT

CNW@ (8,8)-CNT

CNW@ (9,9)-CNT

CNW@(10,10)-CNT

Δ /(eV/cell) /

0.272

-0.730

-0.481

-0.251

/

0.034

-0.091

-0.060

-0.031

dA/Å

5.753

10.184

11.172

12.323

13.621

dB/Å

4.309

9.141

10.563

12.075

13.555

Δ /(eV/H)

CNW

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To deeply understand this stability behavior, we have performed a fitting of the relationship between the total energy and averaged diameter of CNT@CNT by using the Lennard-Jones (612) potential formula.44 The fitted result is shown in Figure 2(a). Combining the results in Table 1 with Figure 2(a), we come to the conclusion that the process of the CNW encapsulated in CNT with too small diameter is endothermic, resulting in an unstable encapsulated structure. When the diameter of CNT increases, the formation energy decreases in value and finally trends to zero. The relationship between the formation energy and CNT diameter accords with the LennardJones potential formula on the whole, indicating that the CNW-CNT interaction in the complex structure belongs to weak van der Waals type. In order to validate this view point, we also calculate electron-localized functions (ELF)45 of the complex structure CNW@(8,8)-CNT. The contour plot of 2D projection ELF on (001) surface of CNT is shown in Figure 2(b). The strong covalent bonding between C atoms and their neighboring H atoms in CNW and the adjacent C atoms in (8,8)-CNT can be deduced from the ELF plot. However, no bonding between the CNW and (8,8)-CNT is found.


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Figure 2. (a) Lennard-Jones (6-12) potential fitting of the relationship between Δ and the averaged diameter of CNT@CNT. (b) 2D contour plots of ELF on (001) surface of CNW@(8,8)CNT. Next, we examine electronic structure of the complex structure. The band structures of three systems: CNW, (8,8)-CNT and the most stable complex structure CNW@(8,8)-CNT are shown in Figure 3. Obviously, CNW is an insulator with a wide direct band gap of 4.298 eV [Figure 3(a)], because CNW can be seen as a generalized alkane chain with the cross section extended by C-C bonding and all C-C bonds are sp3-sp3 σ type that have no contribution to the conductivity. The (8,8)-CNT is metallic [Figure 3(b)], consistent with the general argument about the electronic properties of all the armchair CNTs.46 Furthermore, we obtain the band structure [Figure 3(c)] and density of states (DOS) [Figure 3(d)] of the complex structure CNW@(8,8)CNT. The projected DOS (PDOS) of CNW shows that it has no contribution to total DOS of the complex structure near the Fermi level (EF), and the complex structure should maintain conductivity of the armchair CNTs which dominates the DOS at the EF of the complex structure. We also perform a test of hybrid functional calculation by adopting HSE06 functional,47–49 which


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gives qualitatively consistent results with the PBE calculation, although the HSE06-derived band gap of CNW turns to be 5.210 eV.

Figure 3. (a)-(c) band structure of CNW, (8,8)-CNT and CNW@(8,8)-CNT. (d) DOS of CNW@(8,8)-CNT. The EF is set to zero. As shown above, the outer CNT and inner CNW in the complex structure have different bonding and electronic structures in despite of similar elemental composition. It should be interesting to examine response of the electronic structure of the complex structure upon the external strain. We have calculated the band structures of the complex structure CNW@(8,8)CNT under different uniaxial strains εz along the tube axis ranging from -9% to 9%, which are shown in Figure 4. The complex structure basically keeps metallic during the variation of uniaxial strain εz. It is noted that the highest occupied state at the Γ point (denoted as SΓW), which is actually valence band (VB) edge of CNW as shown by partial DOS plot in Figure 3(d), markedly moves relative to the EF: under the tensile strain, the SΓW is away from the EF and reaches about -1.0 eV (relative to the EF) when εz = 9%; under the compressive strain, the SΓW approaches the EF and even exceeds the latter when |εz| > 5%, which means that it becomes unoccupied and the CNW turns from the insulator with wide band gap to metal. Meanwhile, the


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Dirac point of CNT begins to fall below the EF when the CNW becomes metal, and under other situations it is always fixed at the EF. This behavior suggests that the insulator-to-metal transition of CNW is related to specific interaction between the CNW and CNT in the complex structure.

Figure 4. (a)-(h) Band structures of CNW@(8,8)-CNT under different uniaxial strains εz = 3%, 5%, 7%, 9%, -3%, -5%, -7%, and -9%, respectively. The EF is set to zero. In order to explore mechanism of the strain-induced insulator-to-metal transition of CNW in the complex structure, we firstly examine the CNW-CNT interaction in the complex structure after the insulator-to-metal transition. Figure 5(a) shows differential charge density of the complex structure under the uniaxial strain εz = -9% relative to the CNW and CNT. There exist obviously regions of electron accumulation around the CNT, corresponding to the shifting down


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of Dirac point of CNT. But the situation is complicated for the CNW because around the CNW there coexist regions of electron accumulation and depletion. We find that the mechanism of electron transfer in this complex structure under compressive strain includes two effects: 1) under the uniaxial strain εz = -9%, the originally occupied SΓW has been lifted beyond Dirac point of CNT which is originally at the EF, so the electrons at the SΓW and its nearby occupied states transfer to the occupied states near the Dirac point of CNT, which should be achieved through the electron tunneling process between the CNT and CNW. This electron transfer results in the electron depletion between the C atoms across the z-direction in the CNW, which corresponds to the location of the SΓW [see Figure 5(c) and (e)], and electron accumulation around the CNW; 2) after the above electron transfer process, the regions between the C atoms across the z-direction in the CNW become positively charged and the CNT becomes negative charged. Under this potential field, the electrons in the complex structure are redistributed, bringing the electron accumulation around the C atoms in the CNW and electron depletion near the H atoms. The first effect should be the main mechanism of the insulator-to-metal transition of CNW in the complex structure under the compressive strain, and therefore we should probe into why the SΓW is shifting upward relative to the Dirac point of CNT under the compressive strain. Figure 5(b) shows absolute eigen-energy of the state SΓW and the lowest unoccupied state at the Γ point which is localized in the CNT [see Figure 3(c) and (d)] (denoted as SΓT) under different uniaxial compressive strains εz. It is found that the state SΓT changes a little with the varied strain, whereas the state SΓW is evidently increased when the compressive strain is enhanced, indicating that shifting of the SΓW relative to the Dirac point of CNT is mainly ascribed to the strain-induced energy shifting of the SΓW itself. The wavefunction of the SΓW is displayed in Figures 5(c) and (e), from which one can find that the SΓW is contributed by C pz


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orbitals, with characteristics of bonding orbital between the nearest neighboring (NN) C atoms across the z-direction and antibonding orbital between the next nearest neighboring (NNN) C atoms across the z-direction. Although the former orbital interaction occurs between the NN C atoms, the lobes of pz orbitals of the NN C atoms do not point towards each other so as to result in a hybridization of σ and π bonds, and it is difficult to evaluate the change of this orbital interaction under the strain along the z-direction which is not parallel to the direction of the NN C-C bond. By contrast, the antibonding orbital interaction between the NNN C atoms of the SΓW involves the pz orbitals of which the lobes point towards each other, meaning a σ bond, and it is obvious that under the compressive strain this orbital interaction will be enhanced. The enhanced antibonding orbital interaction can make the SΓW shift upward at the energy scale. On the other hand, the SΓT has characteristics of antibonding orbital between the NN C atoms across the zdirection and bonding orbital between the NNN C atoms across the z-direction, as shown in Figure 5(d) and (f). Because these two orbital interactions of the SΓT are of pure π bond type and different bonding/antibonding properties, the strain-induced enhancements of these two orbital interactions will bring contrary energy shifting of the SΓT. Then, their joint effect is a slower energy shifting than that of SΓW, as shown in Figure 5(b). So the differences of the bonding and electronic structures between the CNT and CNW in the complex structure bring the dissimilar strain-induced energy shifting of their frontier orbitals, which ultimately results in the insulatorto-metal transition of CNW.


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Figure 5. (a) The contour plot of differential charge density of the complex structure CNW@(8,8)-CNT under the uniaxial strain εz = -9% relative to the CNT and CNW (isodensity: 2.0×10-4 Å-3). The yellow and cyan represent the electron accumulation and depletion respectively. (b) Absolute eigen-energy of the eigenstates SΓW (red) and SΓT (black) under different uniaxial compressive strains εz. (c) Top view along the z-direction and (e) side view of the wavefunction of the eigenstate SΓW without strain (isovalue: 2.1×10-6 Å-1.5). (d) Top view along the z-direction and (f) side view of the wavefunction of the eigenstate SΓT without strain (isovalue: ±1.1×10-6 Å-1.5). The CNT in (e) and CNW in (f) are not shown for better display.


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As a test, the electronic structures of the CNW under the uniaxial strain are also calculated. It is found that the CNW turns to metal until strength of the compressive strain reaches about 20% (not shown). This requested value of strain is much larger than the one when the CNW lies in the complex structure, and is difficult to be achieved at the normal situation. So the encapsulation inside the CNT remarkably decreases the threshold of strain required to realize the insulator-tometal transition of the CNW. This mechanism of insulator-to-metal should be also applicable for other insulators with wide band gaps which is difficult to be metalized by using normal modulation methods. In fact, similar strain-assisted mechanism had been used to weaken Schottky barrier and then improve carrier injection efficiency in the system of layers materials contacting with bulk metal substrate.50–52 In our case, the wide band gap CNW is so thin that the weakening of Schottky barrier evolves into the metallization of CNW, similar to the case of monolayer MoS2 contacting with Sc and Ti substrate.53 We have also tried to explore variances of the band structure of the complex structure CNW@(8,8)-CNT under the uniaxial strain along the major and minor axes (i.e. dA and dB as defined before) of the cross-section of the complex structure. The maximal strength of the applied strain is chosen to make dA or dB reduced by about 10%. It is also found that the SΓW can be shifted relative to the EF under this type of strain (not shown), but the energy shift is not large enough to induce the insulator-to-metal transition of CNW. We have also examined optical property of the complex structure CNW@(8,8)-CNT. Figure 6 presents our calculated optical absorption spectrum of CNW@(8,8)-CNT, i.e. the absorption coefficient α(E) as a function of photon energy E, which is deduced from the frequencydependent dielectric function () = () + () computed by VASP as follows54:


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() =

"# % % ! $ &#$ '#$

Page 16 of 34

(3)

It is found that that the complex structure can absorb light with wave band from near infrared to ultraviolet (UV), although there exist some differences between the detailed absorption behaviors in the directions parallel and vertical to the tube axis. The optical absorption spectra of CNW and CNT in the complex structure and their individual forms are also shown in Figure 6. Obviously, the absorption spectrum of CNW@(8,8)-CNT can be almost considered as simple summation of those of CNW and CNT except for some small deviations. And one can note that the CNW in the complex structure has absorption only in the UV region, which is also the optical property of its individual form originating from its wide band gap.

Figure 6. The calculated optical absorptions for the complex structure CNW@(8,8)-CNT, the CNW and CNT in the complex structure and their individual forms with the polarization vector (a) parallel (α//) and (b) perpendicular (α⊥)to z-axis. In order to explore influence of the outer CNT on the UV photoresponse property of the inner CNW, we have checked characteristics of VBs and conduction bands (CBs) of CNW@(8,8)-


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CNT and the individual CNW/CNT. The VBs of CNW@(8,8)-CNT are localized at either CNW or CNT without obvious orbital hybridization between the CNW and CNT, which is consistent with the weak van der Waals type CNW-CNT interaction as revealed before. However, the situation is different for their conduction bands: we find that some conduction bands of CNW@(8,8)-CNT [labeled in Figure 7(a)] have larger components far from the CNT or localized between the CNT and CNW, showing that these bands are of nearly-free-electron (NFE) states. Figure 7(b) displays the wavefunctions of these NFE states at Γ point which show different atom-like orbital characteristics and have the same number of angular nodes as the corresponding s, p, d and f atomic orbitals. Our further calculations find that all the conduction bands of the individual CNW in the examined energy range are NFE states, consistent with negative or very low electron affinity of CNW.55–57 Previous study showed that there also exist some NFE states with substantial probability density within and outside of the CNT shell among the conduction bands of the individual CNT with similar energy region.58 The hybridizations between the NFE states of the CNW and CNT produce the NFE states of the complex structure CNW@(8,8)-CNT showed in Figure 7(b). It can be reasonably surmised that this specific electronic structure of CNW@(8,8)-CNT can result in the following photoresponse property [Figure 7(c)]: if the complex structure CNW@(8,8)-CNT is illuminated by UV light with appropriate waveband, the electrons in the VBs of the inner CNW will be excited to the NFE states of CNW@(8,8)-CNT, and there appear holes in the VBs of CNW simultaneously; subsequently, the excited electrons in the NFE states have two kinds of relaxation possibilities, i.e. recombination with the holes in the VBs of CNW and the relaxation to the lower CBs of CNT; since the transition probability of the electron between the two energy levels is larger for the smaller energy spacing, the latter relaxation process will dominate, resulting in that most of


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the excited electrons is relaxed to the CBs of CNT and most of the holes in the VB of CNW are kept without recombination; so the electron-hole separation can be realized in this complex structure illuminated by UV light. This electron-hole separation property may be exploited for some applications which will be discussed later.

Figure 7. (a) The band structure of the complex structure CNW@(8,8)-CNT. The sizes of blue dots on the bands illustrate the contribution from vacuum (not localized at the CNW and CNT). (b) The wavefunction schematics of the seven NFE states at Γ point of CNW@(8,8)-CNT (isovalue: ±2.0×10-7 for s-type NFE state and ±2.0×10-6 Å-1.5 for others), and the insert in the


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upper right corner shows their located bands in the band structure. (c) The schematic of the electron-hole separation in the complex structure CNW@(8,8)-CNT upon the UV illumination. We have also examined effects of external electric field on the electronic and optical properties of the complex structure CNW@CNT. The external electric field is applied along the major axis dA or minor axis dB which are perpendicular to the tube axis. It is found that under the external field ranging from 0 to 0.2 V/Å, the outer CNT keeps metallic and the inner CNW is still insulator. But a remarkable phenomenon is observed: when the electric field is increased, the NFE states of the complex structure downshift in energy and approach the EF (not shown). Since the CBs of CNW are dominated by these NFE states, this phenomenon results in decrease of the band gap of CNW. It is ascribed to that the NFE states are not localized on the atomic cores and loosely bound, so they could be strongly affected by external potentials. Similar effect had been discussed in previous study.58 It is also noted that the decrease of band gap of CNW depends on the direction of applied electric field: under the field of 0.2 V/Å, the band gap turns to 3.365 eV and 2.569 eV for the field along the major axis dA and minor axis dB respectively. This anisotropy should be related to the structural anisotropy (ellipse) of CNW@CNT in the cross section and specific spatial distribution of NFE states. At last, we turn to the mechanical properties of the complex structures CNW@CNT. As a quantity related to mechanical behavior of the material, Young’s modulus is a measure of stiffness of an elastic isotropic material. In classical mechanics, Young’s modulus is defined as ratio of stress along an axis to strain along that axis. For nano-mechanical systems, Young’s modulus Y is expressed as follows:59 + % ,-

(=)

*

+# %

(4)


19


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where ε corresponds to the uniaxial strain (ε =

/'/* /*

Page 20 of 34

, with r and r0 being lattice constant in the

deformation direction under a strain and at equilibrium state respectively), ES is strain energy calculated by subtracting the total energy at equilibrium state from the total energy of the strained system, i.e. 0 = |# − |#23 , and volume 43 = 53 6. However, the definition of the cross-sectional area S is ambiguous in previous studies. In order to uniform the standard in the current work, we apply the following definitions of S:

a) For CNW: we assume it is an elliptical cylinder, so 6 = 78 8 . Here, d1 and d2 are

lengths of minor and major axis of the cross section respectively; b) For CNTs: we use the definition of S0 applied in a previous study,59 i.e. 6 = 7893, where c0 (3.4 Å) is interlayer distance of graphite and d is diameter of CNT; c) Finally for CNW@CNTs: an effective area S0 is obtained by 63 = 6 + 6 . The above methodology is validated by calculating the Young’s modulus of four armchair CNTs: (m,m)-CNT, m=7-10. A quadratic dependence of the strain energy upon the strain along the tube axis is obtained in the range of -1.5% < εxx < 1.5%. Table 2 shows the calculated Young’s modulus of CNTs, which are in good agreement with previous studies.59,60 Table 2. Young’s modulus and diameter for different armchair CNT. System

(7,7)-CNT

(8,8)-CNT

(9,9)-CNT

(10,10)-CNT

Y /TPa

0.996

1.015

0.992

0.980

Diameter /Å

9.552

10.951

12.262

13.644

References59,60 /TPa

0.8-1.13

Then the same methodology is used to calculate the Young’s modulus of the complex structures CNW@(m,m)-CNT (m=7-10). As an example, Figure 8 presents the plots of strain


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energy and equivalent stress with the varied strain for the CNW@(8,8)-CNT, and the quadratic dependence of the strain energy and linear dependence of the equivalent stress upon the strain is evident. The calculated Young’s modulus values via DFT model (denoted as YDFT) of the four complex structures are listed in Table 3. On the whole, the Young’s modulus of various complex structures are approximately equal to 1.0 TPa. In addition, CNW has a higher Young’s modulus than the CNTs and complex structures.

Figure 8. Plots of (a) the strain energy and (b) the equivalent stress versus strain for CNW@(8,8)-CNT, along with the quadratic fitting for the former and linear fitting for the latter. Table 3. The equilibrium length (L0), cross section area (S), stiffness coefficient (k), Young’s modulus obtained via DFT model (YDFT) and Young’s modulus obtained via parallel spring model (YPS) for the CNW, four CNTs and four complex structures CNW@CNT. The k values of four CNW@CNT are calculated by sum of kCNW and kCNT, and the corresponding YPS are calculated by using Equation (5) with the known L0, S and k obtained before. L0 /Å

S /Å2

k /GPa·Å

YDFT /TPa

YPS /TPa

CNW

2.532

19.460

8.685

1.130

/

(7,7)-CNT

2.468

102.029

41.175

0.996

/

(8,8)-CNT

2.470

116.913

48.043

1.015

/


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(9,9)-CNT

2.469

130.975

52.623

0.992

/

(10,10)-CNT

2.469

145.737

57.846

0.980

/

CNW@(7,7)-CNT

2.475

121.489

49.860

1.063

1.016

CNW@(8,8)-CNT

2.478

136.373

56.728

1.037

1.031

CNW@(9,9)-CNT

2.477

150.435

61.308

1.062

1.009

CNW@(10,10)-CNT

2.477

165.197

66.531

1.040

0.998

The higher Young’s modulus of CNW deserves our further exploration. In Figure 9(a), we present a comparison between the Young’s modulus of different carbon-based materials. Here, we also examine the Young’s modulus of the CNW with cross section consisting of two sixmembered rings (denoted as CNW-2), CNW with cross section consisting of four six-membered rings (denoted as CNW-4), graphene sheet along armchair direction (denoted as AG), graphene sheet along zigzag direction (denoted as ZG) and diamond with different faces. For 2D graphene sheet, the obtained values for equivalent Young’s modulus is 0.990 TPa (AG) and 0.989 TPa (ZG), which are in an acceptable agreement with previous reports.61,62 The calculated Young’s modulus of diamond are 1.179, 1.239 and 1.080 TPa along [110], [111] and [100] directions (denoted as D-110, D-111 and D-100) respectively, which is basically consistent with previous work.63 As for the 1D cases, the Y values of CNW, CNW-2 and CNW-4 are 1.130, 1.112 and 1.059 TPa respectively, which are analogous to those of the 3D diamond, and are larger than those of the 2D graphene. Among the three 1D cases, the Y value decreases with increased diameter of nanowire, which can be interpreted as follows: the geometrical relaxation of the surface atoms of nanowire can result in the shorter C-C bonds across the cross section on the surface than those in the interior of nanowire; the shorter C-C bonds across the cross section have higher bond energy so as to harden the nanowire and make it difficult to be stretched or compressed; for the nanowire with smaller diameter, this effect is more evident due to higher


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ratio between the number of C-C bonds on the surface and that in the bulk. From the above results, we can find that the CNWs and CNW@CNT examined here are stiffer than the CNTs and graphene, although the Y values of the complex structures CNW@CNT are lower than those of the unencapsulated CNWs.

Figure 9. (a) The Young’s modulus of three 1D carbon nanowires (CNW, CNW-2, CNW-4), the complex structure CNW@(8,8)-CNT, 2D graphene sheet along two directions (AG, ZG), and 3D diamond with different directions (D-110, D-111, D-100). (b) Single spring used to model the CNW or CNT. (c) Parallel spring system used to model the complex structure CNW@CNT. It is expected to find a convenient way to deduce the Young’s modulus of the complex structures with different combinations of CNW and CNT from the known values of CNW and CNT, because the DFT calculations for all of their combinations will be very time-consuming. In these complex structures, there only exists weak interaction between the CNW and CNT as discussed before. So we built a simple parallel spring (PS) model based on Hooke’s law64,65 to examine the Young’s modulus of this type of complex structure [see Figures 9(b) and (c)]. We can deduce that the relationship between stiffness coefficient k and Young’s modulus Y is: (=:

;*
< ( @ =

=

?3, @ (: + : ) 6 @ ;*,ABC@ABD

Electronic, Optical, and Mechanical Properties of Diamond Nanowires

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