Ab initio study of the vibrational properties of single-walled silicon nanotubes

Physica E 44 (2012) 1441–1445

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Ab initio study of the vibrational properties of single-walled silicon nanotubes Diana Bogdan a, Radu Isai b, Adrian Calborean a,b, Cristian Morari a,n a b

National Institute for Research and Development of Isotopic and Molecular Technologies, 65-103 Donath, 400293 Cluj-Napoca, Romania Babes- -Bolyai University, 1 Mihail Kog˘alniceanu, 400084 Cluj-Napoca, Romania

a r t i c l e i n f o

abstract

Article history: Received 4 October 2011 Received in revised form 19 January 2012 Accepted 7 March 2012 Available online 14 March 2012

We investigate the vibrational properties of pentagonal and hexagonal single walled silicon nanotubes by using DFT and the frozen phonons method. The phononic band structure and the vibrational density of states are reported. The stability of each structure is discussed based on these results. We investigate the influence of isotopic substitutions upon the vibrational density of states for the two types of nanotubes. Finally, we show that the model of vibrational Hamiltonian in a diatomic chain can be used to give a qualitative description of the relation between the amount of isotopically substituted layers in the nanotube and their influence upon the vibrational density of states. & 2012 Elsevier B.V. All rights reserved.

1. Introduction The discovery of carbon nanotubes (CNTs) [1] attracted much attention in the last two decades due to their intriguing physical properties and because they are believed to be important building blocks of the next generation of electronic devices. This discovery also biased the research on silicon quasi-one-dimensional systems to the pursuit of tube-like structures [2]. The fist sub-micron silicon nanowires (SiNWs) were synthesized since 1964 by Wagner and Ellis [3], using the vapor–liquid– solid (VLS) mechanism. Later, Westwater et al. [4], synthesized SiNWs of about 1 mm long and as thin as 10 nm. Morales and Lieber [5] used laser ablation for generating nanoparticles for the VLS mechanism, and reported single crystal SiNWs with diameters as small as 6 nm and more than 1 mm long. Zhang et al. [6] obtained uniformly smooth curving crystalline SiNWs with diameters ranging from 3 to 43 nm and length up to a few hundred microns and Holmes et al. [7] reported SiNWs with diameter ranging from 4 to 5 nm and several micrometers in length. Among the most promising applications of the SiNWs, we note their thermal properties. Recent experiments [8,9] addressing the thermoelectrical properties of the SiNWs showed high values for the ZT-figure of merit originating mainly from phonon effects. Thus, the variation of the nanowire size and impurity doping levels affect the thermal conductivity k. Another important aspect is the large difference between mean free path lengths of electrons and phonons, the latter being smaller than the

n

Corresponding author. E-mail address: [email protected] (C. Morari).

1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2012.03.008

SiNW’s dimensions. Vo et al. [10], based on Boltzmann transport equation and ab initio electronic structure calculation, computed the thermoelectric figure of merit of SiNWs with diameter up to 3 nm. Their results showed an important ZT dependence on the carrier concentration, wire diameter, wire growth direction, surface structure and type of dopant. The thermal properties can be further enhanced up to two orders of magnitude compared to bulk samples, with the help of isotopic substitutions [11]. A decrease of the thermal conductivity of SiNWs was evidenced even for small percentage of randomly doped isotopes [11]. In the case of isotopic superlattice, the mismatch of phonon density of states spectra of different mass layers can reduce the conductivity significantly, considering superlattice period lengths smaller than a critical value. In contrast to the extensive studies on SiNWs, the research on silicon nanotubes (SiNTs) is still in the primary stage. One reason is that silicon prefers sp3 hybridization instead of sp2 hybridization, which is more suitable for hollow structure, carbon nanotube like [12]. The silicon tendency for sp3 hybridization originates from the relatively small energy (5.66 eV) between s and p orbitals, compared to carbon (10.60 eV), and from the poor p2p overlap for Si¼Si, which is an order of magnitude smaller than the corresponding value in C¼C [13]. The first models addressing the existence of silicon tubular structures considered CNT-like structures with silicon atoms instead of carbon. Fagan et al. [12], based on density functional theory (DFT) calculation, compared the electronic and structural properties of a hypothetical SiNT (with sp2 bonds) with a CNT. They conclude that the cohesive energies for the studied nanotubes are only 82% of the bulk cohesive energy, compared to 99% that corresponds to CNTs. Zhang et al. [14] showed that the

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armchair SiNT are more stable due to efficient overlapping of pz orbitals and delocalized p bonds. They also showed that all the small diameter SiNTs are metallic regardless of their chiralities. Recently, Bai et al. [15] proposed geometric structures for square, pentagonal and hexagonal single-walled silicon nanotubes. These structures maintain the fourfold coordination of silicon, but at different angles compared to the tetrahedral structure of cubic diamond silicon. Using ab initio calculations, Bai et al. argue that such structures have a zero band gap. The structure and stability issues of SiNTs were also tackled by Lee et al. [16,17]. Thermophysical properties of SiNTs were investigated by Wang et al. [18], showing that the thermal conductivity of SiNTs is significantly lower compared to that of bulk silicon; similar results were obtained by Chen et al. [19]. Recently, Ishai and Patolski [20] reported new routes to control and engineer the SiNTs surface chemistry. They functionalized the inner and outer surfaces of SiNTs with organic molecular layers containing different functional groups and hydrophobicity/hydrophilicity chemical nature, via covalent binding, in order to give nanotubular structures with dual chemical properties. Fahad et al. [21] introduced the concept of SiNT field effect transistor. Chen et al. [22] proposed linear and branched crystalline SiNTs via porous anodic aluminum oxide (AAO) self-catalyzed growth, and the connection of crystalline SiNTs and gold nanowires (AuNWs). A large variety of AuNW/SiNT and SiNT/AuNW/SiNT heteronanostructures with both linear and branched topologies have been achieved, paving the way for the rational design and fabrication of SiNT-based nanocircuits, nanodevices, and multifunctional nanosystems. The metallic character and the thermal properties of SiNTs suggest that a strong enhancement of the ZT-figure of merit can be expected in the case of SiNTs. This makes them ideal candidates for the technological applications, as previously suggested [8,9]. Our aim is to give a detailed description of the vibrational properties of pentagonal and hexagonal SiNTs. We report here the phonon band structure and the vibrational density of states (VbDOSs). We also determine the influence of the isotopic substitutions upon the vibrational density of states and propose a simple model to rationalize the VbDOS features resulting from our ab initio calculations.

supercell used in calculation. Too short supercell can lead to wrong results; precisely, the resulting vibrational bands have negative frequencies. By gradually increasing the supercell in the Z direction we found that, for the 5SiNT, a number of 19 layers produces converged results with no negative phononic bands. In the case of 6SiNT, we get converged results for a number of 23 layers. Our geometrical models are given in Fig. 1; the numerical parameters of the models are listed in Table 1. We note that the interatomic, as well as the interlayer distances are slightly larger (i.e. about 2%) compared with those reported by Bai et al. [15]. This is in qualitative agreement with the well known overestimation of the bondlengths in GGA. For the vibrational density of states, we followed the straightforward definition of VbDOS given in Ref. [26] XX w 2 I g ð oÞ ¼ 9Ai ðoÞ9 ð1Þ wAg i w

where 9Ai ðoÞ9 is the ith component of the vibrational eigenvector (i¼ 1, 2, 3) of the mode with vibrational frequency o (for o 40). Here, g labels a group of atoms, indexed by w. For graphic representations we use a smearing function of Lorentzian shape, having a width of 5 cm  1 and a discretization step of 1 cm  1. Supercell approach and G point calculation were used to compute the VbDOS. We check the quality of the pseudopotentials used in our calculations by comparing the results obtained for models systems with all-electrons calculation performed with Gaussian [28]. The same GGA functional (PBE) was used for all-electrons and pseudopotential calculations; a 6-31G(d) basis set was used for all-electron calculations. As a geometric models we used two structures: Si5n H10 (we call it pentagonal model) and Si6n H1 2 (hexagonal model). In both cases we took n¼ 5, representing minimal models for the two types of nanotubes under discussion (see Fig. 2, left). Overall, the bondlengths computed for the geometrical models by using pseudopotential methods tend to be 0.5 up to 0.8%

2. Computational details We use the SIESTA package [23,24] for the DFT calculations. To describe the electronic structure of the 3s electrons, we employ a DZP basis set, while for the 3p electrons we use a TZP basis set. The energy shift used to build the basis set was 150 meV. We use Troullier–Martins norm conserving pseudopotentials [25] and the GGA (PBE version) [27] as exchange–correlation functional. In order to represent the grid related quantities we use an energy grid of 250 Ry. We relax the preliminary geometry of the system (i.e. atomic positions and cell length) by using the conjugate gradient method. The final gradients in the system were less ˚ while the stress components were less than than 10  3 eV/A, 5  10  6 eV/A˚ 3. The vibrational analysis was performed by using the frozen-phonons technique, as implemented in SIESTA. We build the geometric structures for pentagonal (5SiNT) and hexagonal (6SiNT) silicon single-walled nanotubes starting from the results of Bai et al. [15] and using the supercell periodic approach for our study. The supercell geometry was taken to be tetragonal, with the dimension L  L  Lz , where the Z direction is ˚ LZ is defined as the axial direction of the nanotubes with L¼25 A; ˚ ˚ 46.4 A (5SiNT) and 56.2 A (6SiNT). These values are obtained after a full relaxation of the atomic positions and of the cell length. The frozen-phonon technique requires carefully chosen size of the

Fig. 1. A 3D perspective representation of the geometric structures for 5SiNT and 6SiNT. We indicate the relevant geometric parameters. As an intuitive example, for 6SiNT we indicate by green circles the isotopic substitutions for three layers of the nanotube. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 1 Geometric parameters of the SiNTs investigated in this work: d0 denotes the interatomic distance for the atoms located in the same plane of the nanotube; a0 represents the interplanar distance, while N is the number of layers used to build the supercells (see also Fig. 1). The values in parentheses indicate the results of Bai et al. [15]. Model

˚ d0 (A)

˚ a0 (A)

N

5SiNT 6SiNT

2.43 2.42

2.44 (2.39) 2.44 (2.40)

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VbDOS [arb. units]

Hexagonal

Pentagonal

0

100

200

300

400

500

600

Frequency [cm-1] Fig. 2. Left: geometric models used for pseudopotential tests. Right: comparison of the VbDOS for the two models presented in the right side. Black lines: pseudopotential calculations (Siesta); red lines: all electrons calculation (Gaussian). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

500

500

6SiNT

Frequency [cm-1]

longer compared to all-electrons results. For example the length of Si–H bond was 1.52 A˚ in the case of pseudopotential calculations and 1.51 A˚ for the all-electrons calculations. The average distances between two Si layers in the pentagonal structure were 2.43 A˚ (pseudopotential) versus 2.42 A˚ (all electrons). For the hexagonal structure these distances are 2.42 A˚ (pseudopotential) versus 2.41 A˚ (all electrons). The vibrational density of states projected over the Si atoms for the two model-structures is represented in the right side of Fig. 2 [29]. Only the values for frequencies up to 600 cm  1 are represented, since VbDOS has negligible values beyond this value. By inspecting Fig. 2 we remark that typical differences between the peaks in the two curves are of 10 cm  1 or less. The most important differences between the results produced with the two approaches occur at the extreme values of the frequency range. The differences between the smallest vibrational frequency computed using the two strategies are about 35 cm  1 (for hexagonal model) and 45 cm  1 (for pentagonal model). On the other hand, pseudopotential calculations predict the existence of VbDOS peaks between 550 and 600 cm  1—for both models. They are not reproduced by all-electrons calculations. Nevertheless, the amplitude of these peaks is relatively small. By taking into account the excellent agreement between geometric parameters and the good agreement between the vibrational properties for the two realistic models investigated, we conclude that the pseudopotential approach leads to reliable results for the systems under consideration.

400

400

300

300

200

200

100

100

0

0

5SiNT 400

400

300

300

200

200

100

100

0

0

Arb. units 3. Results The phonon band structure proves to be a very effective tool for the study of stability of periodic structures. If all the vibrational modes of a crystal are associated with a positive eigenvalue of the dynamical matrix, all the collective displacements of the atoms, or change of unit cells, increase its energy the crystal is stable against all possible infinitesimal collective changes of its structure. An energy barrier separates this phase from other possible energetically favored phases. The full phonon band structure provides an unique and complete characterization of the (meta)stability of a phase. The phonon dispersion curves obtained for 5SiNT and 6SiNT are given in the right panels of Fig. 3, while the corresponding VbDOS are presented in the left panels. We note that no negative frequencies are present in the phononic band structure. This is a clear indication for the stability of both structures. The main

Γ

X

Fig. 3. Phononic band structure and the vibrational density of states for the 5SiNT and 6SiNT. Breathing modes are indicated by arrows (see Fig. 4 for the geometric representations of the breathing modes).

features of VbDOS for 5- and 6SiNTs are in qualitative agreement with the VbDOS for Si nanowires reported by Peelaers et al. [30]. In particular, two local maxima of the VbDOS are found at 90 cm  1 (acoustic modes) and 344 cm  1 (optical modes) for 5SiNT, and at 83 cm  1 and 312 cm  1 for 6SiNT, respectively. The breathing modes are located between about 230 cm  1 and 315 cm  1 for 5SiNT, while in the case of 6SiNT they are shifted towards lower frequencies (between 160 cm  1 and 250 cm  1). A graphical representation of the breathing modes is given in Fig. 4. Next, we project the vibrational eigenvectors onto the Z-axis (i.e. parallel with the nanotube). The VbDOS of the resulting Z-projected eigenvectors is then constructed. The results of our

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Fig. 4. Graphical representation for the breathing modes indicated by arrows in Fig. 3 for 5SiNT (bottom) and 6SiNT (top).

Transverse

VbDOS [arb. units]

6SiNT

Longitudinal Transverse

5SiNT

Longitudinal 0

100

200

300

400

500

Frequency [cm-1] Fig. 5. Comparison between the total, transverse and longitudinal VbDOS for 5SiNT (bottom) and 6SiNT (top). The effect of the isotopic substitution upon the total and longitudinal VbDOS is also presented: black curves—results for nanotubes including only 28Si; red curves—results for nanotubes isotopically doped with 30Si (see text for details). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

analysis are presented in Fig. 5. We note that transverse modes are dominant in the region 0–70 cm  1, while the longitudinal ones are dominant at large frequencies (i.e. above 425 cm  1). The two local maxima of the VbDOS mentioned above are dominated by transverse modes. In the case of 5SiNT we note the presence of a gap in the transverse VbDOS, between 190 and 210 cm  1; no such gap is present for 6SiNT. Finally, we investigate the effect of isotopic substitution upon the VbDOS of the studied SiNTs. We consider that the layers of the SiNT’s are formed either by 28Si atoms (i.e. the most abundant isotope of the silicon [31]), either by 30Si atoms (see also Fig. 1 for an intuitive representation). We choose the 30Si isotope because it has the largest mass among the stable isotopes of the silicon and about 3% natural abundance [31]. In the case of 29Si dopant we get similar but less pronounced trends (results not presented here). In the presence of isotopic substitutions, the VbDOS is changed. Both systems exhibit a depletion of the VbDOS peaks in the region

300–400 cm  1. The importance of these changes depends on the number of isotopically substituted layers. Precisely, if up to five layers of 30Si atoms are included in nanowire’s structures, we found that VbDOS remains practically unchanged (pictures not shown here). For a number of seven layers of isotopically substituted Si atoms (i.e. about 38% from the total number of atoms) we notice a change of about 20% in the amplitude of the largest peak in the VbDOS of the 5SiNT (see Fig. 5). For the 6SiNT we see a similar diminution of VbDOS if 11 layers of 30Si isotope are present (i.e. about 47% from the total number of atoms). In order to rationalize this behavior we recall the solutions of the vibrational Hamiltonian for a simple diatomic lattice model [32]. In the center of the Brillouin zone (BZ) (k  0) the two solutions are o2  2Cð1=M 1 þ 1=M 2 Þ (optical branch) and o2  Ck2 a2 =2ðM1 þ M2 Þ (acoustical branch), where M1 and M2 are the masses of the two atoms connected by force constant C; a is the lattice parameter. At the edge of the Brillouin zone k  7 p=a and the solutions are o2  2C=M 1 and o2  2C=M 2 , respectively. Therefore, the frequency gap at k  7 p=a is dependent of the masses of the two atoms, M1 and M2. When the mass of one atom increases, the major VbDOS changes happen at the Brillouin zone’s center for optical branch and at the BZ edge for acoustic branch. The model of vibrational Hamiltonian for diatomic lattice model allows us to explain the results presented in Fig. 5. The vibration of the two types of layers (i.e. those formed by 28Si atoms and those formed by the 30Si atoms) can be approximated with the vibration of two ‘‘particles’’ having the masses M1 and M2, respectively. We note that by increasing the number of layers formed by 30Si atoms, the M 1 =M2 ratio changes. Specifically, M1 (i.e. the mass of the layer formed by 28Si atoms) decreases because more layers are subtracted from it to be substituted with isotopes. On the other hand, the total mass of the group of layers formed by 30Si (i.e. M2) is increasing. As a consequence, the ratio M1 =M 2 is gradually decreasing toward unity when the number of 30 Si layers is going from about 20% to about 40% of the total number of atoms. From Fig. 3, we see that the optical and acoustic branch at the BZ center and edge, respectively, are in the frequency range 300–400 cm  1. This shows that the VbDOS features presented in Fig. 5 are perfectly consistent with the particle-chain model proposed by us. We conclude that the effect of isotopic substitution upon the vibrational density of states of silicon nanotubes is maximum when the number of isotopically substituted layers is equal (or slightly

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smaller) compared to the number of 28Si layers in the nanowire. The simple model investigated here is based only on the fact that the force constants between neighboring layers of silicon is the same, while their mass is modified by isotopic substitutions. Therefore, we think that the conclusion presented above can be applied to other types of silicon nanotubes.

4. Conclusion We studied the phonon spectrum for two types of single walled silicon nanotubes, by performing ab initio calculations (DFT). Our results indicate that the two structures are stable (i.e. no imaginary frequencies are present in the phonon spectra). We show that the vibrational density of states for 5- and 6SiNT display similar features with those of the more complex silicon nanowires [30]. We also investigate the effect of the controlled isotopic substitutions upon the vibrational density of states. We found that, for both types of nanotubes, a relative important depletion of the VbDOS occurs for frequencies ranging from 300 to 400 cm  1. We show that the simple model of vibrational Hamiltonian for diatomic 1D lattice can be used to explain this behavior. Therefore, our conclusion is that the changes of VbDOS are most important when the number of isotopically substituted layers is close to the number of 28Si layers.

Acknowledgments We acknowledge financial support from the Core Program, project PN 09-440101, contract no. 44N/2009. Thanks are due to NIRDIMT Cluj-Napoca Data Center for providing computer facilities. References [1] [2] [3] [4]

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