J. Phys. Chem. C 2009, 113, 6439–6443
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Structural, Energetic, and Mechanical Properties of ZnSe Nanotubes Biplab Goswami, Sougata Pal, Chanchal Ghosh, and Pranab Sarkar* Department of Chemistry, VisVa-Bharati UniVersity, Santiniketan-731235, India ReceiVed: NoVember 23, 2008; ReVised Manuscript ReceiVed: February 21, 2009
We present results of our theoretical calculations on the structural, energetic, and mechanical properties of ZnSe nanotubes. We have calculated the strain energy, buckling, band gap, and Young modulus for both zigzag and armchair nanotubes and have studied their variation with tube radius. We have also studied the effect of flattening on the band gap values for both zigzag and armchair nanotubes. Our study predicts that the flattening may cause possible semiconductor to metal transition in ZnSe nanotubes. Introduction The existence of tubular forms of matter with nanoscale diameter has opened up an exciting field of research in physics and chemistry.1-4 The exceptional properties of carbon nanotubes increase the interest in one-dimensional structures such as nanowires and nanotubes. The reason for their unique properties which made them different from the corresponding bulk structures is commonly known as the quantum size effect. In addition to the size effect, properties of one-dimensional (1D) nanostructures may also sensitively depend on their structure and morphologies. The particular physical and chemical properties of 1D solids promise new inventions, new materials with new interesting properties. The most common structures in which graphene layers are folded to form carbon nanotubes can now be grown in a relatively controlled fashion. It has been shown that the carbon nanotubes can be either metallic or semiconducting depending upon the tube diameter and chirality.5-7 However, probably the highest potential of nanotubes is in connection with their exceptional mechanical properties. Many theoretical and experimental investigations have been carried out to find the mechanical properties of carbon nanotubes.8-14 Carbon nanotubes possess a range of potentially useful electronic and mechanical properties. However, analogous structures from alternative materials may also offer much greater potential flexibility in properties. Recently, nanotubes made by noncarbon-based inorganic semiconductor nanostructures popularly known as inorganic nanotubes have become the subject of extensive studies.15 The first successful report on inorganic nanotubes (WS2) was made by Tenne et al.16 followed by intense experimental and theoretical research on hollow cylindrical structures that led to the development of numerous inorganic nanotubes with diverse properties.17-28 The physical properties of these nanotubes strongly depend on their chirality as well as on their radius. Their potential applications range from high porous catalytic and ultralight anticorrosive materials to electron field emitters and nontoxic strengthening fibers. The helical structure of nanotubes, with their semiconducting behavior and optical activity, opens up possible applications in nonlinear optics and solar-cell technology. Wide band gap II-IV materials have been studied extensively due to their wide applications in the fields of light-emitting devices, solar cells, sensors, and optical materials. Zinc selenide * To whom correspondence should be addressed. E-mail: pranab.sarkar@ visva-bharati.ac.in.
(ZnSe) with a band gap of 2.7 eV at room temperature has received attention as a promising material for the fabrication of short-wavelength devices such as blue-green lasers, photodetectors, and light-emitting diodes.29-32 There are both theoretical and experimental studies reporting the electronic structure, synthesis, and application of ZnSe quantum dots,33-39 but studies addressing the properties of the tubular form of ZnSe are very scarce. Very recently, Karanikolos et al.40 reported the synthesis of ZnSe nanotubes. To the best of our knowledge, this is the only study on ZnSe nanotubes to date. In this paper we propose to study theoretically the electronic structure of ZnSe nanotubes of different radii and helicities. In the theoretical study in particular, the exploration of new nanostructured materials is of crucial importance in guiding relevant experiments, rational explanation of experimental observations, and also offering new results to be verified experimentally. We will study the variation of the strain energy, buckling, and band gap as functions of both the tube diameter and tube helicity. We calculate the Young modulus and the effect of flattening on the band gap values of ZnSe nanotubes. Theoretical Method In our work we have employed the parametrized densityfunctional tight-binding (DFTB) method of Porezag et al., which has been described in detail elsewhere.41,42 The approximate DFTB method is based on the density-functional theory of Hohenberg and Kohn in the formulation of Kohn and Sham. In this method, the single-particle wave functions Ψi(r) of the Kohn-Sham equations are expanded in a set of atomic-like basis functions Φm, with m being a compound index that describes the atom at which the function is centered and the angular dependence of the function as well as its radial dependence. These functions are obtained from self-consistent densityfunctional calculations on the isolated atoms employing a large set of Slater-type basis functions. The effective Kohn-Sham b) is approximated as a simple superposition of potential Veff(r b) ) ∑jV0j (|r b- b Rj|). the potentials of the neutral atoms Veff(r Furthermore, we make use of a tight-binding approximation, so that 〈φm|V0j |φn〉 is nonvanishing only when φm and/or φn is centered at b Rj. In our calculations only the 3d and 4s electrons of Zn and 4s and 4p electrons of Se are explicitly included while the other electrons are treated within a frozen-core approximation. As the method is parametrized, we have tested its transferability to larger systems by first performing calculations on infinite periodic crystalline structures of ZnSe, and the details are given elsewhere.38,39
10.1021/jp8102854 CCC: $40.75 2009 American Chemical Society Published on Web 03/26/2009
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Figure 1. Strain energies per pair as a function of the radius for ZnSe armchair (×) and zigzag (0) nanotubes.
The structure of a single-wall nanotube can be conceptualized by wrapping a one-atom-thick layer of graphite called graphene into a seamless cylinder. The key geometric parameter associated with this process is the rolled-up vector b, r which can be expressed as the linear combination of the lattice basis (a b and b b):
b b r ) na b + mb
(1)
It is then possible to associate a particular integer pair (n, m) with each single-wall nanotube. The relation between n and m also defines two different categories of nanotubes: m ) 0, “zigzag” nanotubes; m ) n, “armchair” nanotubes. For each tubular structure of ZnSe having a different radius and helicity, a series of calculations have been carried out in which the initial structures were fully relaxed with respect to both the atomic positions and the tube cell length using the conjugate gradient technique. The periodic boundary conditions along the tube axis were employed with the vacuum region (∼16.0 Å) between neighboring tubes to avoid significant interactions among periodic images, and the length of the repeated unit is about 8.04 Å for zigzag and 4.44 Å for armchair nanotubes. In the total energy calculations, the Brillouin zone of nanotubes has been sampled by 1 × 1 × 8 mesh points in k space within the Monkhorst-Pack scheme. We have performed our calculation for (n, n) and (n, 0) ZnSe nanotubes as a function of n with the diameter (D) ranging from 6.6 to 23 Å, which corresponds to indices (n, 0)-(n, n) from (6, 0)-(3, 3) to (18, 0)-(10, 10), respectively. The optimized bond length of Zn-Se is slightly larger than that of planar sheet and bulk structures. The energy of the flat sheet of ZnSe is found to be lower than that of ZnSe nanotubes but higher relative to that of the bulk zinc blende crystal by 0.8 eV per ZnSe unit. We have also checked the stability of the nanotubes by performing molecular dynamics (MD) calculations, and we found that nanotubes retain their tubular character and hexagonal structures on the surface at temperatures up to 500 K. Results and Discussion One of the important properties of the nanotubes which ensures the possibility of their formation is the strain energy. The strain energy is defined as the energy difference between the energy per ZnSe pair of the nanotubes and the corresponding flat sheet (zero strain). In Figure 1 we present the strain energy of both zigzag and armchair ZnSe nanotubes as a function of the tube radius. From the figure it is seen that, similar to other nanotubes, the strain energy decreases with increasing tube radius. The strain energy of single-wall carbon nanotubes and
Figure 2. Buckling in the ZnS nanotube equilibrium structures vs tube radius: (×) zigzag nanotubes; (0) armchair nanotubes. Buckling is defined as the mean radius of zinc atoms minus the mean radius of the sulfur atoms.
other reported inorganic nanotubes converges approximately as 1/D2 (where D is the diameter), and for ZnSe nanotubes the general feature is nearly the same. From the figure it is evident that the strain energy in ZnSe nanotubes like other inorganic nanotubes is relatively insensitive to the detailed structures of the tube, i.e., independent of the helicity of the tube. This finding is in sharp contrast with that of carbon nanotubes for which tubes with armchair symmetry are more stable than zigzag nanotubes. The values of the strain energy of ZnSe nanotubes are smaller compared to those of carbon nanotubes. The growth of such tubes would certainly depend on other experimental parameters, but clearly the strain energy results support the possibilities of these tubes to exist. Another important property to understand the structural behavior of the tube is buckling, which is defined as the difference of the mean radius of zinc and selenium atoms. The buckling on the tube surface arises because the structural relaxation displaced Zn atoms toward the tube axis and the Se atoms in the opposite direction. In Figure 2, we have shown the buckling of both zigzag and armchair nanotubes as a function of the tube radius. The extent of buckling in ZnSe nanotubes is approximately of the same order as that of ZnS27 nanotubes but is relatively higher than that of BN20 nanotubes. From Figure 2, we see that the buckling decreases with increasing tube radius and the extent of buckling is dependent on the chirality of the tube, being slightly more for zigzag nanotubes compared to armchair nanotubes of similar size. This observation is in sharp contrast to that of BN20 and ZnS27 nanotubes where the buckling is almost insensitive to the tube chirality. The presence of buckling of these nanotubes may have the effect of forming a surface dipole and hence be highly relevant for potential applications of these nanotubes. In Figure 3 we have plotted the values of the band gap for both zigzag and armchair ZnSe nanotubes as a function of the radius of the tube. The values of the band gap of ZnSe nanotubes are higher than those of bulk ZnSe (2.7 eV) and also ZnSe quantum dots studied earlier.38 Thus, the absorption spectra would show a blue shift compared to those of both bulk ZnSe and ZnSe quantum dots of comparable size (zinc blende). Karanikolos et al.40 have also observed a blue shift in the absorption spectra of their synthesized nanotubes compared to the bulk value. Although the nanotubes they synthesized differ both in structure (zinc blende) and size (30 nm diameter and 7 nm wall thickness) from ours, the qualitative behavior is the same. The values of the band gap for both zigzag and armchair nanotubes, in conformity with the quantum confinement effect, decrease with increasing tube radius. However, this feature is
Properties of ZnSe Nanotubes
Figure 3. Band gap of the ZnSe nanotube vs tube radius: (×) zigzag nanotube; (0) armchair nanotube.
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Figure 5. Young modulus of the ZnSe nanotube vs tube radius: (×) zigzag nanotube; (0) armchair nanotube.
Y)
Figure 4. DOS for (a) (4, 4) armchair and (b) (7, 0) zigzag nanotubes.
in sharp contrast with those of other inorganic nanotubes such as BN,20 AlN,21 GaN,23 and ZnS27 nanotubes. In most of the cases, the band gap increases with increasing tube radius. For ZnS nanotubes with zigzag symmetry, the band gap decreases with increasing tube radius while it passes through a maximum for armchair nanotubes. The calculated band gap values are relatively higher for armchair ZnSe nanotubes than the zigzag nanotubes of comparable radius, and this feature is the same as those of other inorganic nanotubes. The higher values of the band gap for armchair nanotubes compared to zigzag nanotubes of comparable radius may be explained following the ideas of Blase et al.43 and Reich et al.44 These authors have pointed out that the curvature-induced σ-π hybridizations have the most pronounced effect on zigzag nanotubes where there is a downshift of conduction bands, whereas the electronic band structure is less affected in armchair nanotubes. We have shown the density of states (DOS) of (4, 4) armchair and (7, 0) zigzag nanotubes in Figure 4. It is clearly seen that the position of the valence band in nanotubes of both symmetries appears almost at the same energy range but the conduction band of the zigzag tube appears at lower energy compared to that of the armchair nanotube. There are a large number of both experimental and theoretical studiesaddressingthemechanicalpropertiesofcarbonnanotubes.8-14 These experimental and theoretical studies confirm the expectation that nanotubes have exceptional stiffness and could therefore be used in the synthesis of highly resistant composite materials. In what follows, we study the Young modulus of ZnSe nanotubes. The conventional definition of the Young modulus is
( )
1 ∂2E V0 ∂ε2
(2) ε)0
where V0 is the equilibrium volume, E is the strain energy, and ε is the axial strain. In the case of the single-wall nanotube, the volume V0 is defined as the volume of a hollow cylinder which is given by V0 ) 2πLR δR, where L is the length, R is the radius, and δR is the shell thickness. Several authors have adopted different conventions for the value of δR. Instead of adopting this ad hoc convention for δR, we have followed the approach of Hernandez et al.20 These authors in their study on the mechanical properties of C and BxCyNz composite nanotubes used a different magnitude to characterize the stiffness of a single-wall nanotube, which is independent of any shell thickness and is defined as
Ys )
( )
1 ∂2E S0 ∂ε2
(3) ε)0
Here, S0 is the surface area of a single nanotube at equilibrium. Therefore, the Young modulus Y of the nanotubes can be written in terms of the stiffness of the nanotubes Ys as Y ) Ys/δR. Using this different approach, we calculated the values of the stiffness (Ys) for different tube radii with different chiralities, and these values are shown in Figure 5. We see from the figure that, for both zigzag and armchair nanotubes, the stiffness of the tube increases with the tube radius and the values of Ys are a little higher for armchair nanotubes compared to zigzag nanotubes. The rate of increase in stiffness is relatively higher at lower tube radius, and it reaches almost a constant value at higher radius. For higher radius, the Young modulus of the nanotubes is nearly equal to that of a flat graphite sheet. The nature of the curve is very much consistent with the expectation that the tube of smaller radius has the higher curvature and will have the weaker bonds, which are responsible for the lower value of the Young modulus. Both experimental and theoretical studies have predicted that nanotubes are extremely flexible45,46 and at the same time have a high Young modulus. Chesnokov et al.45 recently reported that the cross section of carbon nanotubes may be flattened. Ma´Rio et al.46 applying the first principle calculation studied the effect of flattening on the carbon nanotubes and showed that flattening induces a semiconductor-metal transition. The flattening of the nanotubes may be experimentally realized by applying force by atomic force microscopy (AFM) or scanning tunneling microscopy (STM) on the nanotubes. Now, to study the effect of flattening, we calculated the band gap values of (14, 0) zigzag and (7, 7) armchair ZnSe nanotubes with specific
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Figure 6. Relaxed cross sections of (14, 0) ZnSe nanotubes with different degrees of flatness. The flatness is characterized by the quantity D (please see the text), which assumes the values (a) 0.10, (b) 0.20, and (c) 0.30.
to induce a semiconductor to metal transition can be obtained as the derivative of the strain energy relative to the distortion. The reduction of the band gap may be due to the hybridization effect (as discussed earlier) that arises from the increase in the curvature on a hexagonal curved surface because of flattening. Conclusions
Figure 7. Energy as a function of the flatness (D) of the ZnSe nanotube: (×) zigzag (14, 0) nanotube; (0) armchair (7, 7) nanotube.
Figure 8. Band gap values as a function of the flatness (D) of the ZnSe nanotube: (×) zigzag (14, 0) nanotube; (0) armchair (7, 7) nanotube.
degrees of flatness. The relaxed cross sections of (14, 0) singlewall ZnSe nanotubes with different degrees of flatness are shown in Figure 6. When the nanotube becomes flattened, its circular curvature becomes an elliptical shape. In an ellipse there are two axes; one is a major axis, and the other is a minor axis. Now, to characterize the flatness, we defined a dimensionless quantity D ) (D1 - D2)/D0, where D1 and D2 are the major and minor axes of the ellipse, respectively, and D0 is the diameter of the tube at the equilibrium position. In Figures 7 and 8 we have shown the variation of the total energy and band gap with the flatness for both (14, 0) zigzag and (7, 7) armchair nanotubes. The band gap value decreases with increasing flatness of the tube while the energy increases. The decrease in band gap values with increasing flatness implies a possible semiconductor to metal transition for higher values of D. For low values of flatness, the band gap decreases slowly with increasing flatness coefficient, but the band gap decreases rapidly with a linear relationship for higher values of flatness, and it decreases below 0.5 eV from 3.5 eV. The force necessary
In summary, our calculation shows the possibility of formation of single-wall ZnSe nanotubes because of their lower strain energy, the value of which decreases with increasing tube radius and is independent of the tube helicity. A certain degree of buckling is present in ZnSe nanotubes and is due to different hybridizations of Zn and Se atoms on the hexagonal curved surface. The band gap values of the tube consistent with the quantum confinement effect decreases with increasing tube radius. Band gap values of the ZnSe nanotubes are relatively higher compared to those of bulk ZnSe and ZnSe quantum dots. We have also studied the mechanical properties of single-wall ZnSe nanotubes. We see that the Young modulus of singlewall ZnSe nanotubes is slightly higher than that of carbon nanotubes and depends on the tube radius and also on the chirality of the tube. For its high stiffness it can have potential use in nanoelectronics. The challenge remains of establishing a database of mechanical properties of ZnSe nanotubes such as the concentration and type of defects, temperature, chemical environment, presence of chemical functionality, and so on. We have studied the effect of flattening of their cross section on the band gap of single-wall ZnSe nanotubes. Our result predicts a possibility of semiconductor-metal transition in ZnSe nanotubes as a result of curvature-induced hybridization effects. Acknowledgment. Financial support from CSIR, Government of India [Grant 01(2148)-EMR-II/2003], and UGC(SAP) through research grants is gratefully acknowledged. S.P. is grateful to CSIR for the award of a senior research fellowship. References and Notes (1) Nalwa, H. S., Ed. Handbook of Nanostructures Materials and Nanotechnology; Academic Press: New York, 1999. (2) Hu, J.; Odom, T. W.; Lieber, C. M. Acc. Chem. Res. 1999, 32, 435. (3) Ouyang, M.; Huang, J. L.; Cheung, C. L.; Lieber, C. M. Science 2001, 292, 702. (4) Iijima, S. Nature (London) 1991, 354, 56. (5) Collins, P. G.; Arnold, M. S.; Avouris, P. Science 2001, 292, 706. (6) Mintmire, J. W.; Dunlap, B. I.; White, C. T. Phys. ReV. Lett. 1992, 68, 631. (7) Hamada, N.; Sawada, S. I.; Oshiyama, A. Phys. ReV. Lett. 1992, 68, 1579. (8) Belova, E.; Chernozatonskii, L. A. Phys. ReV. B 2007, 75, 073412. (9) Iijima, S.; Brabec, C. J.; Maiti, A.; Bemholc, J. J. Chem. Phys. 1996, 104, 2089.
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