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Photo-absorption spectra of small hydrogenated silicon clusters using the time-dependent density functional theory
Journal of Physics and Chemistry of Solids 72 (2011) 1096–1100
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Photo-absorption spectra of small hydrogenated silicon clusters using the time-dependent density functional theory Juzar Thingna a,b, R. Prasad b,, S. Auluck b a b
Physics Department, Centre for Computational Science and Engineering, National University of Singapore, Singapore Physics Department, Indian Institute of Technology Kanpur, Kanpur, UP 208016, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 January 2011 Received in revised form 5 May 2011 Accepted 18 June 2011 Available online 28 June 2011
We present a systematic study of the photo-absorption spectra of various SinHm clusters ðn ¼ 1210, m ¼ 1214Þ using the time-dependent density functional theory (TDDFT). The method uses a real-time, real-space implementation of TDDFT involving full propagation of the time dependent Kohn–Sham equations. Our results for SiH4 and Si2H6 show good agreement with the earlier calculations and experimental data. We study the photo-absorption spectra of silicon clusters as a function of hydrogenation. For single hydrogenation, we find that in general, the absorption optical gap decreases showing a significant red shift for small sized clusters and as the number of silicon atoms increases the effect of a single hydrogen atom on the optical gap diminishes. For further hydrogenation the optical gap increases and for the fully hydrogenated clusters the optical gap is larger compared to corresponding pure silicon clusters corresponding to a blue shifted spectra. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction Recently, there has been a renewed interest in understanding the optical properties of clusters because confinement of electrons changes the physical properties. Thus by varying size of the clusters the optical properties can be tuned according to the desired application [1]. In particular, optical properties of silicon and hydrogenated silicon clusters have been of great interest due to the observation of photo-luminescence (PL) in porous silicon [2,3]. The structure and properties of silicon clusters can be tuned by varying the cluster size as well as doping. An important dopant for silicon clusters is hydrogen and it plays an important role in structural stability. Experimental studies also confirm this fact [4], but in spite of several investigations many issues about hydrogenated silicon clusters have not been understood. It is not clear how the structure and optical properties of the cluster evolves with size and as a function of hydrogenation. It is therefore interesting to calculate the photo-absorption (PA) spectra and compare it with experiment. There has been a considerable amount of work experimental as well as theoretical on silicon and hydrogenated silicon clusters. Suto et al. measured the photo-absorption and fluorescence of silane in the energy range 8–12 eV using synchrotron radiation [5]. The optical absorption of silane and disilane has been
Corresponding author. Tel.: þ 91 512 2597065; fax: þ 91 512 2590914.
E-mail address: [email protected] (R. Prasad). 0022-3697/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2011.06.011
measured by Itoh et al. in the energy range 6–12 eV [6]. Cheshnovsky et al. have measured the photo-electron spectra of charged silicon and germanium clusters [7]. Rinnen et al. have measured the photo-dissociation spectra of neutral silicon clusters Sin ðn ¼ 18241Þ [8]. Murakami et al. investigated the stability of some silicon and hydrogenated silicon clusters using a quadruple ion trap [4]. More recently Antonietti et al. deduced the photo-absorption spectra of charged silicon clusters from photodissociation of charged xenon-silicon clusters [9]. On the theoretical side, one of the earliest works was from Rantala et al. where they used a tight binding model to investigate the linear as well as non-linear polarizabilities of silicon clusters [10]. Chantranupong et al. performed a configuration interaction (CI) calculation for a large number of low lying states in silane [11]. Lehtonen et al. [12] have performed calculations at the coupled cluster approximate singles and doubles (CC2) level using a quadruple-z basis sets augmented with diffuse functions and shown the effects of a basis set on the optical spectra of various silanes. Rubio et al. calculated the photo-absorption spectra of silicon and alkali metal clusters using time dependent local density approximation (TDLDA) [13]. Rohlfing et al. calculated the optical absorption spectra of hydrogen terminated silicon clusters by solving the Bethe–Salpeter equation [14]. Vasiliev et al. [15] and Marques et al. [16] calculated the optical absorption spectra of SinHm clusters using linear response theory within TDLDA. Rao et al. measured the photo-luminescence of a dispersion of 1 nm silicon particles obtained from crystalline silicon that is dispersed into nanoparticles through electrochemical etching with HF and H2O2
J. Thingna et al. / Journal of Physics and Chemistry of Solids 72 (2011) 1096–1100
[17]. They also calculated the photo-absorption spectra of Si29H24 using time dependent density functional theory (TDDFT). Lehtonen et al. calculated the absorption spectra of three hydrogen terminated silicon clusters using TDDFT [18] and the spectra of numerous silicon clusters [19]. The earlier studies on the optical properties of silicon and hydrogenated silicon clusters focused on the size dependence of the PL and photo-absorption [3,20–24]. Some of these studies ignored the influence of oscillator strengths (electric dipole matrix elements) and hence were not in good agreement with the experimental data. Calculations based on DFT using LDA and the generalised gradient approximation (GGA) which included the dipole matrix elements suffered from the drawback that is inherent in LDA/GGA i.e. the energy gaps were underestimated because these calculations ignored the effect of excited states. To overcome this drawback, the configuration interaction method or the methods based on solving the Bethe–Salpeter equation along with the GW approximation have been suggested [14]. These methods require a lot of computer time and hence have been restricted to small clusters. A computational technique based on linear response theory within TDLDA had been proposed by Vasiliev et al. [15]. This is a natural extension of the LDA ground state density functional formulation designed to include excited states. This method is faster than the BS and GW methods and can therefore be used for larger clusters. Vasiliev et al. have shown the viability of their method by performing calculations on silicon and hydrogenated and oxygenated silicon clusters. Another implementation of TDDFT has been formulated by Castro et al. which performs the time propagation of the electronic orbitals in order to calculate the optical absorption spectra. In the present work, we have used this method [25]. Most of the earlier calculations have considered only hydrogen terminated silicon clusters. There seems to be a lack of a systematic study on small hydrogenated silicon clusters. In this paper we report calculations for the silicon and hydrogenated silicon clusters in addition to some hydrogen terminated silicon clusters. Thus our calculations will show the effect of hydrogenation on the optical properties of silicon clusters. Our emphasis is on the smaller clusters so as to bring out the evolution of the optical properties as we increase the number of silicon and hydrogen atoms in the cluster. Our main findings show that for small silicon clusters of comparable size, where quantum confinement effects cannot be used to tune the optical gap, hydrogenation can be used as a viable alternative. The plan of the paper is as follows. In Section 2, we briefly discuss the method and give computational details. In Section 3 we present our results and discussion. In Section 4 we present our conclusions.
2. Method and computational details The initial structures of the clusters used for our calculations have been obtained by Balamurugan et al. [29,30] earlier from their analysis based on the Car–Parrinello molecular dynamics (CPMD) [31]. The resulting structures have been further optimised using the electronic structure method implemented in VASP (Vienna Ab-initio Simulation Package) [32]. We do not find any significant changes in the structures compared to the CPMD results. In fact the starting CPMD configuration gives forces that are around 0.5 eV/A˚ which eventually reduce to around 0.05 eV/A˚ in VASP. VASP employs density functional theory (DFT) and we have employed the local density approximation (LDA) for the exchange correlation using ultra soft pseudo-potentials. The optimisation is done only by relaxing the ions via the conjugate gradient (CG) method and using a k-point Monkhorst–Pack mesh
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of 4 4 4. All calculations have been performed in a cubic ˚ super-cell of length 20 A. In this work we present the photo-absorption spectra [26,27] of the optimised structures using OCTOPUS code [25,33], which employs the LDA with norm-conserving pseudo-potentials. Since OCTOPUS uses a uniform grid in real space, it is essential to carry out a minimisation of energy with respect to the radius and the grid spacing for the ground state calculation before we proceed to TDDFT. In this work we required the radius of each sphere to be 6–8 A˚ and the grid spacing 0.28–0.4 A˚ for optimal energy minimisation [16]. In TDDFT to calculate the absorption spectra we give some small momentum (k) to the electrons in order to excite all the frequencies [28]. This is achieved by transforming the ground state wave function according to
cj ðr, dtÞ ¼ eikz cj ðr,0Þ,
ð1Þ
where z is the direction along which the perturbation is applied. These wave functions are then propagated for 124 fs (with 30,000 time steps giving a resolution of 0:00413 fs) using the approximated enforced time reversal symmetry (aetrs) to approximate the time evolution operator. The choice of the small momentum (k) is made such that it is small enough that we remain in the linear regime, but large enough to avoid numerical errors. The spectrum is then obtained from dipole strength function SðoÞ: SðoÞ ¼
2o
p
ImaðoÞ,
ð2Þ
where the aðoÞ is the dynamical polarizability, which is the Fourier transform of the dipole moment of the system, and is given by Z 1 dteiot ½dðtÞdð0Þ, aðoÞ ¼ ð3Þ
k
where d(t) is dipole moment of the system. Many times a damping function for example: exponential or a polynomial is multiplied to the dipole moment before calculating the Fourier transform. The damping function causes a broadening in the peaks and makes the spectrum ‘‘look’’ more like experimental spectra. Since there is no physics used to determine this damping function, in this work we use a third order polynomial damping function for our calculations. Using the above definition the Thomas–Reiche–Kuhn f-sum rule for the number of electrons (N) of the system is given by the integral: Z 1 N¼ doSðoÞ: ð4Þ 0
The above sum rule and energy conservation when no external field is applied are important checks in order to choose the TDDFT parameters. Although it is numerically impossible to integrate till 1 we ensure that this rule is approximately satisfied in this work [34].
3. Results and discussions In order to verify our calculations with the experimental data, we chose to compare the two most stable clusters SiH4 and Si2H6 with the available experimental data as our benchmark. In optical properties we are interested in transitions form the occupied levels to the unoccupied levels. The structures in the photo-absorption spectra are identified with these transitions. It is therefore interesting to calculate these energy differences and compare them with the energy differences deduced from the experimental spectra. Table 1 shows such a comparison. In Table 1 we have also included the Bethe–Salpeter results of Rohlfing et al. [14], the TDLDA results of
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J. Thingna et al. / Journal of Physics and Chemistry of Solids 72 (2011) 1096–1100
Table 1 Excitation energies in eV for SiH4 and Si2H6 clusters. Cluster
Transition
SiH4
4s 4p 4d 4s 4p 5p 6p
Si2H6
a b
Present work
Marques et al.
Rohlfing et al.
Vasiliev et al.
Experiment
8.2 9.2 9.8 7.3 8.6
8.2 9.4 10 7.3 8.7
9.0 10.2 11.2 7.6 9.0
8.2 9.2 9.7 7.3 7.8
10.6
10.6
8.8a, 9.0b 9.7a,b 10.7a,b 7.6a 8.4a 9.5a 9.9a
The experimental data is taken from the works of Itoh et al. [6]. The experimental data is taken from the works of Suto et al. [5].
Fig. 1. Structure and photo-absorption spectra of present work (black solid line) and Marques et al. [16] (black dashed line) for SiH4 and Si2H6 clusters.
Vasilev et al. [15] and the Octopus results of Marques et al. [16] along with the transitions identification. The uppermost occupied states result from a hybridisation of the silicon and hydrogen states while the lowest unoccupied states are primarily silicon states. Thus the energy gap is dependent on the bonding and anti-bonding silicon states. The transition between these states has been identified as 4s, 4p, 4d states (these refer to the angular momentum character of the final states). We find good agreement with the experimental data of Itoh et al. [6] and Suto et al. [5] and other theoretical calculations. In particular, we get good agreement with the results of Marques et al. [16]. As compared to the work of Marques et al. (Fig. 1) the discrepancies seen in the two spectra are because of the different choices of the damping function used to evaluate the Fourier transform of the dipole moment. In this case we have used an exponential damping function in order to replicate the data of Marques et al. It is worth noting that this damping function is just a numerical artefact and has no ‘‘real’’ physics. In order to see the effect of the number of silicon atoms (n) on the optical spectra, we present in Fig. 2(I) the optimised structures of silicon clusters Sin ½n ¼ 1210 along with the photoabsorption spectra. For small clusters (up to n ¼ 7) we find that the photo-absorption spectra is a combination of many peaks and looks like that of isolated atoms. However for n 4 7, the optical spectra looks bulk-like [35]. This can be understood using a simple tight-binding picture. In a larger silicon cluster, the overlap between electronic wave-functions lifts the degeneracy of the energy levels resulting in bunching of energy levels in a narrow
energy range. This results in broadening of PA spectrum for larger clusters. We see that the main structure in the PA spectra is located at around 9 eV and a minor structure starts to build up at around 15 eV. The same trends are found in the singly hydrogenated clusters (Fig. 2(II)). For smaller singly hydrogenated clusters (n o7) the PA spectrum changes significantly as we increase the number of hydrogen atoms, while for larger clusters this change is small. In order to quantify the difference between singly hydrogenated and silicon clusters we plot the optical gap as a function of the number of silicon atoms (n) for various Sin (solid line) and SinH (dashed line) clusters ðn ¼ 1210Þ in Fig. 3. We define the optical gaps through the integral oscillator strength rather than as the energy of the first dipole allowed transition in the absorption spectra. The integral oscillator strength gives the total number of active electrons in the system. In this approach the value of the optical absorption gap is determined at a very small but non-zero fraction of the complete oscillator strength [36]. We set this threshold to 10 4 of the total oscillator strength. This value is chosen because it stands above the value of ‘‘numerical noise’’ and at the same time it is sufficiently small so as to not suppress the experimentally detectable dipole allowed transitions. This definition of absorption gap does not affect the values of optical gaps for small clusters, since the intensity of first transitions is well above the selected threshold. It can be clearly seen that the addition of a single hydrogen atom reduces the optical gap as compared to the silicon clusters causing the PA spectra to be red shifted. As the number of silicon atoms in the cluster increases we find that the difference between the optical gap of Sin and SinH gradually decreases with the increase in silicon atoms and hence in the bulk limit a single hydrogen atom should not distort the optical gap and thus in that limit the optical gap of the silicon will be the same as the optical gap of the singly hydrogenated silicon. The plot of the HOMO–LUMO gap for these clusters is very similar to that obtained by Balamurugan et al. [29] and therefore is not given here. However these gaps are listed in Table 2. We note that the HOMO–LUMO gap does not show this trend [29]. This is because the optical gap is quite different from the HOMO–LUMO gap as it involves self-energy correction [23] which perhaps increases with n and provides this trend to the optical gap. To study the effect of further hydrogenation, we present the PA spectra of intermediate as well as fully hydrogenated silicon clusters (Fig. 4). We have performed calculations for Si3H3, Si4H4 and Si5H6, Si5H12, Si6H7 and Si6H14 clusters. In all these clusters we find similar trends of atomic like behaviour to a bulk like transition. Although in these clusters the bulk like behaviour of the spectra is observed much earlier (n ¼ 4) than the singly hydrogenated clusters. This effect is attributed to the large number of hydrogen atoms. To see the effect of further hydrogenation, we list the optical gaps and HOMO–LUMO gaps of all the clusters in Table 2. We see that addition of a single hydrogen atom reduces the optical gap, but with further hydrogenation the optical gap increases. This is because as the number of hydrogen
J. Thingna et al. / Journal of Physics and Chemistry of Solids 72 (2011) 1096–1100
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Fig. 2. I: Structure and photo-absorption spectra of Sin clusters (n ¼ 1210 from top to bottom); II: Structure and photo-absorption spectra of SinH clusters (n ¼ 1210 from top to bottom).
Table 2 Optical absorption (OA) gaps, -(HOMO-LUMO) (HL) gaps, binding energy (BE) per atom and the H adsorption energy (AE) per atom of various clusters. Cluster OA gap (eV) HL gap (eV) BE/atom (eV/atom) AE/atom (eV/atom)
Fig. 3. Optical absorption gap of various Sin (solid line filled circle) and SinH (dashed line empty circle) clusters as a function of the number of silicon atoms (n).
atoms increases in the cluster, the HOMO–LUMO gap (Table 2) of the cluster increases and thus increasing the optical gap assuming that the self-energy correction increases or remains more or less constant. Hence for the fully hydrogenated clusters the optical gap is larger than the un-hydrogenated counterparts. In the bulk limit the hydrogenated system will show a blue shifted optical gap which is consistent with the experimental observations [37]. This result is quite different to the quantum confinement result of Vasiliev et al. [15] where they observe the increase in the optical gap due to confinement effects. Here the increase in the optical gap is solely due to hydrogenation since the cluster size is nearly the same for all the clusters. In order to manipulate the PA spectrum as discussed above the stability of the various clusters is very important. A valuable measure of the stability of these clusters is the binding energy and the hydrogen adsorption energy. Table 2 gives the binding energy per atom for the silicon clusters (defined as ðE½Sin nE½SiÞ=nÞ, the hydrogenated silicon clusters (defined as ðE½Sin Hm nE½SimE½HÞ=
Si SiH SiH4 Si2 Si2H Si2H6 Si3 Si3H Si3H3 Si4 Si4H Si4H4 Si4H8 Si5 Si5H Si5H6 Si5H12 Si6 Si6H Si6H7 Si6H14 Si7 Si7H Si8 Si8H Si9 Si9H Si10 Si10H
4.7 3.8 8.2 5.3 4.4 7.3 4.9 4.3 4.6 5.4 4.5 5.5 5.6 4.9 4.5 5.2 6.4 5.4 5.1 4.3 6.3 5.3 5.1 5.5 5.3 6.1 5.6 6.0 5.6
0.66 0.30 8.10 2.69 2.68 6.60 0.96 2.37 1.77 1.20 1.62 2.08 4.70 2.00 1.97 3.02 5.19 2.09 1.70 2.86 5.20 2.13 2.52 1.47 1.07 1.85 1.23 2.14 2.50
0.00 2.59 – 2.65 3.62 – 4.11 4.16 4.22 4.10 4.49 4.36 4.41 4.98 4.76 4.58 4.38 5.19 4.94 4.54 4.37 5.34 4.91 5.23 5.11 5.31 5.20 5.52 5.32
– 2.59 – – 1.85 – – 1.08 2.17 – 1.22 2.32 3.06 – 0.60 2.31 2.92 – 0.49 2.15 2.82 – 0.24 – 0.47 – 0.42 – 0.30
ðn þmÞ) and the H adsorption energy per atom for the hydrogenated silicon clusters (defined as ðE½Sin þ mE½HE½Sin Hm Þ=ðn þ mÞÞ. For the silicon clusters the values presented in Table 2 are very close to the values reported by Grossman et al. [38]. In case of the singly hydrogenated clusters the binding energy per atom values and the adsorption energy values given in Table 2 are quite close to values reported by Tiznado et al. [39] and Balamurugan et al. [29]. It can be clearly seen that adding hydrogen to a cluster of silicon atoms
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J. Thingna et al. / Journal of Physics and Chemistry of Solids 72 (2011) 1096–1100
References
Fig. 4. Structure and photo-absorption spectra of SinHm clusters (from top to bottom: Si3H3, Si4H4, Si4H8, Si5H6, Si5H12, Si6H7, Si6H14).
decreases the binding energy, thus stabilising the clusters. Hence these structures are quite likely to be obtained in actual experiments.
4. Conclusions The photo-absorption spectra of silicon and hydrogenated silicon clusters have been calculated using TDDFT. Our calculations show good agreement for the benchmarks of SiH4 and Si2H6 with the earlier theoretical calculations and experimental results. In the singly hydrogenated clusters we find bulk-like behaviour for larger clusters (n 4 7) while for the smaller clusters we find the PA spectra is composed of numerous peaks as in atoms. The addition of a single hydrogen atom to the cluster leads to a decrease in the optical gap of the cluster. This effect diminishes as the number of silicon atoms increases, thus in the bulk limit the addition of a single hydrogen atom will not affect the absorption spectra of the cluster. For the other hydrogenated clusters the bulk-like behaviour appears for smaller number of silicon atoms in the cluster. The optical gap of such hydrogenated clusters shows an optical gap larger than the corresponding gap for the pure silicon clusters. Thus hydrogenation turns out to be an alternate way to manipulate the optical gaps for silicon nano-clusters.
Acknowledgement We are grateful to D. Balamurugan for providing the co-ordinates of some clusters. We thank Professor M.K. Harbola for helpful discussions.
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Photo-absorption spectra of small hydrogenated silicon clusters using the time-dependent density functional theory Juzar Thingna a,b, R. Prasad b,, S. Auluck b a b
Physics Department, Centre for Computational Science and Engineering, National University of Singapore, Singapore Physics Department, Indian Institute of Technology Kanpur, Kanpur, UP 208016, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 January 2011 Received in revised form 5 May 2011 Accepted 18 June 2011 Available online 28 June 2011
We present a systematic study of the photo-absorption spectra of various SinHm clusters ðn ¼ 1210, m ¼ 1214Þ using the time-dependent density functional theory (TDDFT). The method uses a real-time, real-space implementation of TDDFT involving full propagation of the time dependent Kohn–Sham equations. Our results for SiH4 and Si2H6 show good agreement with the earlier calculations and experimental data. We study the photo-absorption spectra of silicon clusters as a function of hydrogenation. For single hydrogenation, we find that in general, the absorption optical gap decreases showing a significant red shift for small sized clusters and as the number of silicon atoms increases the effect of a single hydrogen atom on the optical gap diminishes. For further hydrogenation the optical gap increases and for the fully hydrogenated clusters the optical gap is larger compared to corresponding pure silicon clusters corresponding to a blue shifted spectra. & 2011 Elsevier Ltd. All rights reserved.
1. Introduction Recently, there has been a renewed interest in understanding the optical properties of clusters because confinement of electrons changes the physical properties. Thus by varying size of the clusters the optical properties can be tuned according to the desired application [1]. In particular, optical properties of silicon and hydrogenated silicon clusters have been of great interest due to the observation of photo-luminescence (PL) in porous silicon [2,3]. The structure and properties of silicon clusters can be tuned by varying the cluster size as well as doping. An important dopant for silicon clusters is hydrogen and it plays an important role in structural stability. Experimental studies also confirm this fact [4], but in spite of several investigations many issues about hydrogenated silicon clusters have not been understood. It is not clear how the structure and optical properties of the cluster evolves with size and as a function of hydrogenation. It is therefore interesting to calculate the photo-absorption (PA) spectra and compare it with experiment. There has been a considerable amount of work experimental as well as theoretical on silicon and hydrogenated silicon clusters. Suto et al. measured the photo-absorption and fluorescence of silane in the energy range 8–12 eV using synchrotron radiation [5]. The optical absorption of silane and disilane has been
Corresponding author. Tel.: þ 91 512 2597065; fax: þ 91 512 2590914.
E-mail address: [email protected] (R. Prasad). 0022-3697/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2011.06.011
measured by Itoh et al. in the energy range 6–12 eV [6]. Cheshnovsky et al. have measured the photo-electron spectra of charged silicon and germanium clusters [7]. Rinnen et al. have measured the photo-dissociation spectra of neutral silicon clusters Sin ðn ¼ 18241Þ [8]. Murakami et al. investigated the stability of some silicon and hydrogenated silicon clusters using a quadruple ion trap [4]. More recently Antonietti et al. deduced the photo-absorption spectra of charged silicon clusters from photodissociation of charged xenon-silicon clusters [9]. On the theoretical side, one of the earliest works was from Rantala et al. where they used a tight binding model to investigate the linear as well as non-linear polarizabilities of silicon clusters [10]. Chantranupong et al. performed a configuration interaction (CI) calculation for a large number of low lying states in silane [11]. Lehtonen et al. [12] have performed calculations at the coupled cluster approximate singles and doubles (CC2) level using a quadruple-z basis sets augmented with diffuse functions and shown the effects of a basis set on the optical spectra of various silanes. Rubio et al. calculated the photo-absorption spectra of silicon and alkali metal clusters using time dependent local density approximation (TDLDA) [13]. Rohlfing et al. calculated the optical absorption spectra of hydrogen terminated silicon clusters by solving the Bethe–Salpeter equation [14]. Vasiliev et al. [15] and Marques et al. [16] calculated the optical absorption spectra of SinHm clusters using linear response theory within TDLDA. Rao et al. measured the photo-luminescence of a dispersion of 1 nm silicon particles obtained from crystalline silicon that is dispersed into nanoparticles through electrochemical etching with HF and H2O2
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[17]. They also calculated the photo-absorption spectra of Si29H24 using time dependent density functional theory (TDDFT). Lehtonen et al. calculated the absorption spectra of three hydrogen terminated silicon clusters using TDDFT [18] and the spectra of numerous silicon clusters [19]. The earlier studies on the optical properties of silicon and hydrogenated silicon clusters focused on the size dependence of the PL and photo-absorption [3,20–24]. Some of these studies ignored the influence of oscillator strengths (electric dipole matrix elements) and hence were not in good agreement with the experimental data. Calculations based on DFT using LDA and the generalised gradient approximation (GGA) which included the dipole matrix elements suffered from the drawback that is inherent in LDA/GGA i.e. the energy gaps were underestimated because these calculations ignored the effect of excited states. To overcome this drawback, the configuration interaction method or the methods based on solving the Bethe–Salpeter equation along with the GW approximation have been suggested [14]. These methods require a lot of computer time and hence have been restricted to small clusters. A computational technique based on linear response theory within TDLDA had been proposed by Vasiliev et al. [15]. This is a natural extension of the LDA ground state density functional formulation designed to include excited states. This method is faster than the BS and GW methods and can therefore be used for larger clusters. Vasiliev et al. have shown the viability of their method by performing calculations on silicon and hydrogenated and oxygenated silicon clusters. Another implementation of TDDFT has been formulated by Castro et al. which performs the time propagation of the electronic orbitals in order to calculate the optical absorption spectra. In the present work, we have used this method [25]. Most of the earlier calculations have considered only hydrogen terminated silicon clusters. There seems to be a lack of a systematic study on small hydrogenated silicon clusters. In this paper we report calculations for the silicon and hydrogenated silicon clusters in addition to some hydrogen terminated silicon clusters. Thus our calculations will show the effect of hydrogenation on the optical properties of silicon clusters. Our emphasis is on the smaller clusters so as to bring out the evolution of the optical properties as we increase the number of silicon and hydrogen atoms in the cluster. Our main findings show that for small silicon clusters of comparable size, where quantum confinement effects cannot be used to tune the optical gap, hydrogenation can be used as a viable alternative. The plan of the paper is as follows. In Section 2, we briefly discuss the method and give computational details. In Section 3 we present our results and discussion. In Section 4 we present our conclusions.
2. Method and computational details The initial structures of the clusters used for our calculations have been obtained by Balamurugan et al. [29,30] earlier from their analysis based on the Car–Parrinello molecular dynamics (CPMD) [31]. The resulting structures have been further optimised using the electronic structure method implemented in VASP (Vienna Ab-initio Simulation Package) [32]. We do not find any significant changes in the structures compared to the CPMD results. In fact the starting CPMD configuration gives forces that are around 0.5 eV/A˚ which eventually reduce to around 0.05 eV/A˚ in VASP. VASP employs density functional theory (DFT) and we have employed the local density approximation (LDA) for the exchange correlation using ultra soft pseudo-potentials. The optimisation is done only by relaxing the ions via the conjugate gradient (CG) method and using a k-point Monkhorst–Pack mesh
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of 4 4 4. All calculations have been performed in a cubic ˚ super-cell of length 20 A. In this work we present the photo-absorption spectra [26,27] of the optimised structures using OCTOPUS code [25,33], which employs the LDA with norm-conserving pseudo-potentials. Since OCTOPUS uses a uniform grid in real space, it is essential to carry out a minimisation of energy with respect to the radius and the grid spacing for the ground state calculation before we proceed to TDDFT. In this work we required the radius of each sphere to be 6–8 A˚ and the grid spacing 0.28–0.4 A˚ for optimal energy minimisation [16]. In TDDFT to calculate the absorption spectra we give some small momentum (k) to the electrons in order to excite all the frequencies [28]. This is achieved by transforming the ground state wave function according to
cj ðr, dtÞ ¼ eikz cj ðr,0Þ,
ð1Þ
where z is the direction along which the perturbation is applied. These wave functions are then propagated for 124 fs (with 30,000 time steps giving a resolution of 0:00413 fs) using the approximated enforced time reversal symmetry (aetrs) to approximate the time evolution operator. The choice of the small momentum (k) is made such that it is small enough that we remain in the linear regime, but large enough to avoid numerical errors. The spectrum is then obtained from dipole strength function SðoÞ: SðoÞ ¼
2o
p
ImaðoÞ,
ð2Þ
where the aðoÞ is the dynamical polarizability, which is the Fourier transform of the dipole moment of the system, and is given by Z 1 dteiot ½dðtÞdð0Þ, aðoÞ ¼ ð3Þ
k
where d(t) is dipole moment of the system. Many times a damping function for example: exponential or a polynomial is multiplied to the dipole moment before calculating the Fourier transform. The damping function causes a broadening in the peaks and makes the spectrum ‘‘look’’ more like experimental spectra. Since there is no physics used to determine this damping function, in this work we use a third order polynomial damping function for our calculations. Using the above definition the Thomas–Reiche–Kuhn f-sum rule for the number of electrons (N) of the system is given by the integral: Z 1 N¼ doSðoÞ: ð4Þ 0
The above sum rule and energy conservation when no external field is applied are important checks in order to choose the TDDFT parameters. Although it is numerically impossible to integrate till 1 we ensure that this rule is approximately satisfied in this work [34].
3. Results and discussions In order to verify our calculations with the experimental data, we chose to compare the two most stable clusters SiH4 and Si2H6 with the available experimental data as our benchmark. In optical properties we are interested in transitions form the occupied levels to the unoccupied levels. The structures in the photo-absorption spectra are identified with these transitions. It is therefore interesting to calculate these energy differences and compare them with the energy differences deduced from the experimental spectra. Table 1 shows such a comparison. In Table 1 we have also included the Bethe–Salpeter results of Rohlfing et al. [14], the TDLDA results of
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Table 1 Excitation energies in eV for SiH4 and Si2H6 clusters. Cluster
Transition
SiH4
4s 4p 4d 4s 4p 5p 6p
Si2H6
a b
Present work
Marques et al.
Rohlfing et al.
Vasiliev et al.
Experiment
8.2 9.2 9.8 7.3 8.6
8.2 9.4 10 7.3 8.7
9.0 10.2 11.2 7.6 9.0
8.2 9.2 9.7 7.3 7.8
10.6
10.6
8.8a, 9.0b 9.7a,b 10.7a,b 7.6a 8.4a 9.5a 9.9a
The experimental data is taken from the works of Itoh et al. [6]. The experimental data is taken from the works of Suto et al. [5].
Fig. 1. Structure and photo-absorption spectra of present work (black solid line) and Marques et al. [16] (black dashed line) for SiH4 and Si2H6 clusters.
Vasilev et al. [15] and the Octopus results of Marques et al. [16] along with the transitions identification. The uppermost occupied states result from a hybridisation of the silicon and hydrogen states while the lowest unoccupied states are primarily silicon states. Thus the energy gap is dependent on the bonding and anti-bonding silicon states. The transition between these states has been identified as 4s, 4p, 4d states (these refer to the angular momentum character of the final states). We find good agreement with the experimental data of Itoh et al. [6] and Suto et al. [5] and other theoretical calculations. In particular, we get good agreement with the results of Marques et al. [16]. As compared to the work of Marques et al. (Fig. 1) the discrepancies seen in the two spectra are because of the different choices of the damping function used to evaluate the Fourier transform of the dipole moment. In this case we have used an exponential damping function in order to replicate the data of Marques et al. It is worth noting that this damping function is just a numerical artefact and has no ‘‘real’’ physics. In order to see the effect of the number of silicon atoms (n) on the optical spectra, we present in Fig. 2(I) the optimised structures of silicon clusters Sin ½n ¼ 1210 along with the photoabsorption spectra. For small clusters (up to n ¼ 7) we find that the photo-absorption spectra is a combination of many peaks and looks like that of isolated atoms. However for n 4 7, the optical spectra looks bulk-like [35]. This can be understood using a simple tight-binding picture. In a larger silicon cluster, the overlap between electronic wave-functions lifts the degeneracy of the energy levels resulting in bunching of energy levels in a narrow
energy range. This results in broadening of PA spectrum for larger clusters. We see that the main structure in the PA spectra is located at around 9 eV and a minor structure starts to build up at around 15 eV. The same trends are found in the singly hydrogenated clusters (Fig. 2(II)). For smaller singly hydrogenated clusters (n o7) the PA spectrum changes significantly as we increase the number of hydrogen atoms, while for larger clusters this change is small. In order to quantify the difference between singly hydrogenated and silicon clusters we plot the optical gap as a function of the number of silicon atoms (n) for various Sin (solid line) and SinH (dashed line) clusters ðn ¼ 1210Þ in Fig. 3. We define the optical gaps through the integral oscillator strength rather than as the energy of the first dipole allowed transition in the absorption spectra. The integral oscillator strength gives the total number of active electrons in the system. In this approach the value of the optical absorption gap is determined at a very small but non-zero fraction of the complete oscillator strength [36]. We set this threshold to 10 4 of the total oscillator strength. This value is chosen because it stands above the value of ‘‘numerical noise’’ and at the same time it is sufficiently small so as to not suppress the experimentally detectable dipole allowed transitions. This definition of absorption gap does not affect the values of optical gaps for small clusters, since the intensity of first transitions is well above the selected threshold. It can be clearly seen that the addition of a single hydrogen atom reduces the optical gap as compared to the silicon clusters causing the PA spectra to be red shifted. As the number of silicon atoms in the cluster increases we find that the difference between the optical gap of Sin and SinH gradually decreases with the increase in silicon atoms and hence in the bulk limit a single hydrogen atom should not distort the optical gap and thus in that limit the optical gap of the silicon will be the same as the optical gap of the singly hydrogenated silicon. The plot of the HOMO–LUMO gap for these clusters is very similar to that obtained by Balamurugan et al. [29] and therefore is not given here. However these gaps are listed in Table 2. We note that the HOMO–LUMO gap does not show this trend [29]. This is because the optical gap is quite different from the HOMO–LUMO gap as it involves self-energy correction [23] which perhaps increases with n and provides this trend to the optical gap. To study the effect of further hydrogenation, we present the PA spectra of intermediate as well as fully hydrogenated silicon clusters (Fig. 4). We have performed calculations for Si3H3, Si4H4 and Si5H6, Si5H12, Si6H7 and Si6H14 clusters. In all these clusters we find similar trends of atomic like behaviour to a bulk like transition. Although in these clusters the bulk like behaviour of the spectra is observed much earlier (n ¼ 4) than the singly hydrogenated clusters. This effect is attributed to the large number of hydrogen atoms. To see the effect of further hydrogenation, we list the optical gaps and HOMO–LUMO gaps of all the clusters in Table 2. We see that addition of a single hydrogen atom reduces the optical gap, but with further hydrogenation the optical gap increases. This is because as the number of hydrogen
J. Thingna et al. / Journal of Physics and Chemistry of Solids 72 (2011) 1096–1100
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Fig. 2. I: Structure and photo-absorption spectra of Sin clusters (n ¼ 1210 from top to bottom); II: Structure and photo-absorption spectra of SinH clusters (n ¼ 1210 from top to bottom).
Table 2 Optical absorption (OA) gaps, -(HOMO-LUMO) (HL) gaps, binding energy (BE) per atom and the H adsorption energy (AE) per atom of various clusters. Cluster OA gap (eV) HL gap (eV) BE/atom (eV/atom) AE/atom (eV/atom)
Fig. 3. Optical absorption gap of various Sin (solid line filled circle) and SinH (dashed line empty circle) clusters as a function of the number of silicon atoms (n).
atoms increases in the cluster, the HOMO–LUMO gap (Table 2) of the cluster increases and thus increasing the optical gap assuming that the self-energy correction increases or remains more or less constant. Hence for the fully hydrogenated clusters the optical gap is larger than the un-hydrogenated counterparts. In the bulk limit the hydrogenated system will show a blue shifted optical gap which is consistent with the experimental observations [37]. This result is quite different to the quantum confinement result of Vasiliev et al. [15] where they observe the increase in the optical gap due to confinement effects. Here the increase in the optical gap is solely due to hydrogenation since the cluster size is nearly the same for all the clusters. In order to manipulate the PA spectrum as discussed above the stability of the various clusters is very important. A valuable measure of the stability of these clusters is the binding energy and the hydrogen adsorption energy. Table 2 gives the binding energy per atom for the silicon clusters (defined as ðE½Sin nE½SiÞ=nÞ, the hydrogenated silicon clusters (defined as ðE½Sin Hm nE½SimE½HÞ=
Si SiH SiH4 Si2 Si2H Si2H6 Si3 Si3H Si3H3 Si4 Si4H Si4H4 Si4H8 Si5 Si5H Si5H6 Si5H12 Si6 Si6H Si6H7 Si6H14 Si7 Si7H Si8 Si8H Si9 Si9H Si10 Si10H
4.7 3.8 8.2 5.3 4.4 7.3 4.9 4.3 4.6 5.4 4.5 5.5 5.6 4.9 4.5 5.2 6.4 5.4 5.1 4.3 6.3 5.3 5.1 5.5 5.3 6.1 5.6 6.0 5.6
0.66 0.30 8.10 2.69 2.68 6.60 0.96 2.37 1.77 1.20 1.62 2.08 4.70 2.00 1.97 3.02 5.19 2.09 1.70 2.86 5.20 2.13 2.52 1.47 1.07 1.85 1.23 2.14 2.50
0.00 2.59 – 2.65 3.62 – 4.11 4.16 4.22 4.10 4.49 4.36 4.41 4.98 4.76 4.58 4.38 5.19 4.94 4.54 4.37 5.34 4.91 5.23 5.11 5.31 5.20 5.52 5.32
– 2.59 – – 1.85 – – 1.08 2.17 – 1.22 2.32 3.06 – 0.60 2.31 2.92 – 0.49 2.15 2.82 – 0.24 – 0.47 – 0.42 – 0.30
ðn þmÞ) and the H adsorption energy per atom for the hydrogenated silicon clusters (defined as ðE½Sin þ mE½HE½Sin Hm Þ=ðn þ mÞÞ. For the silicon clusters the values presented in Table 2 are very close to the values reported by Grossman et al. [38]. In case of the singly hydrogenated clusters the binding energy per atom values and the adsorption energy values given in Table 2 are quite close to values reported by Tiznado et al. [39] and Balamurugan et al. [29]. It can be clearly seen that adding hydrogen to a cluster of silicon atoms
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References
Fig. 4. Structure and photo-absorption spectra of SinHm clusters (from top to bottom: Si3H3, Si4H4, Si4H8, Si5H6, Si5H12, Si6H7, Si6H14).
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4. Conclusions The photo-absorption spectra of silicon and hydrogenated silicon clusters have been calculated using TDDFT. Our calculations show good agreement for the benchmarks of SiH4 and Si2H6 with the earlier theoretical calculations and experimental results. In the singly hydrogenated clusters we find bulk-like behaviour for larger clusters (n 4 7) while for the smaller clusters we find the PA spectra is composed of numerous peaks as in atoms. The addition of a single hydrogen atom to the cluster leads to a decrease in the optical gap of the cluster. This effect diminishes as the number of silicon atoms increases, thus in the bulk limit the addition of a single hydrogen atom will not affect the absorption spectra of the cluster. For the other hydrogenated clusters the bulk-like behaviour appears for smaller number of silicon atoms in the cluster. The optical gap of such hydrogenated clusters shows an optical gap larger than the corresponding gap for the pure silicon clusters. Thus hydrogenation turns out to be an alternate way to manipulate the optical gaps for silicon nano-clusters.
Acknowledgement We are grateful to D. Balamurugan for providing the co-ordinates of some clusters. We thank Professor M.K. Harbola for helpful discussions.
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