Photo-absorption spectra of small hydrogenated silicon clusters using the time-dependent density functional theory
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Photo-absorption spectra of small hydrogenated silicon clusters using the time-dependent densityfunctional theory
Juzar Thingna,
R. Prasad, ∗ and S. Auluck Physics Department, Center for Computation Science and Engineering, National University of Singapore, Singapore Physics Department, Indian Institute of Technology Kanpur, Kanpur UP 208016, India (Dated: November 14, 2018)We present a systematic study of the photo-absorption spectra of various Si n H m clusters (n=1-10, m=1-14) using the time-dependent density functional theory (TDDFT). The method uses a real-time, real-spaceimplementation of TDDFT involving full propagation of the time dependent Kohn-Sham equations. Our resultsfor SiH and Si H show good agreement with the earlier calculations and experimental data. We find that forsmall clusters (n<7) the photo-absorption spectrum is atomic-like while for the larger clusters it shows bulk-likebehaviour. We study the photo-absorption spectra of silicon clusters as a function of hydrogenation. For singlehydrogenation, we find that in general, the absorption optical gap decreases and as the number of silicon atomsincrease the effect of a single hydrogen atom on the optical gap diminishes. For further hydrogenation theoptical gap increases and for the fully hydrogenated clusters the optical gap is larger compared to correspondingpure silicon clusters.PACS number(s): 78.67. -n, 73.22.-f, 71.15 Pd PACS numbers:
I. INTRODUCTION
Recently, there has been a renewed interest in understand-ing the optical properties of clusters because confinement ofelectrons changes the physical properties. Thus by varyingsize of the clusters the optical properties can be tuned accord-ing to the desired application . In particular, optical proper-ties of silicon and hydrogenated silicon clusters have been ofgreat interest due to the observation of photo luminescence(PL) in porous silicon . The structure and properties of sili-con clusters can be tuned by varying the cluster size as well asdoping. An important dopant for silicon clusters is hydrogenand it plays an important role in structural stability. Experi-mental studies also confirm this fact , but in spite of severalinvestigations many issues about hydrogenated silicon clus-ters have not been understood. It is not clear how the structureand optical properties of the cluster evolves with size and asa function of hydrogenation. It is therefore interesting to cal-culate the photo-absorption (PA) spectra and compare it withexperiment.There has been a considerable amount of work experimen-tal as well as theoretical on silicon and hydrogenated siliconclusters. Suto and Lee measured the photo-absorption and flu-orescence of silane in the energy range 8-12 eV using syn-chrotron radiation .The optical absorption of silane and dis-ilane has been measured by Itoh et. al. in the energy range6-12 eV . Cheshnovsky et. al. have measured the photo-electron spectra of charged silicon and germanium clusters .Rinnen and Mandich have measured the photo-dissociationspectra of neutral silicon clusters Si n (n=18-41) . Murakamiand Kanayama investigated the stability of some silicon andhydrogenated silicon clusters using a quadruple ion trap .More recently Antonietti et. al. deduced the photo-absorptionspectra of charged silicon clusters from photo-dissociationof charged xenon-silicon clusters . On the theoretical sideChantranupong et. al. performed a configuration interac-tion (CI) calculation for a large number of low lying states in silane . Rubio et. al. calculated the photo-absorptionspectra of silicon and alkali metal clusters using time de-pendent local density approximation (TDLDA) . Rohlfingand Louie calculated the optical absorption spectra of hydro-gen terminated silicon clusters by solving the Bethe-Salpeterequation . Vasiliev et. al. and Marques et. al. calculatedthe optical absorption spectra of Si n H m clusters using lin-ear response theory within TDLDA. Rao et. al. measuredthe photo luminescence of a dispersion of 1 nm silicon par-ticles obtained from crystalline silicon that is dispersed intonanoparticles through electrochemical etching with HF andH O . They also calculated the photo-absorption spectraof Si H using time dependent density functional theory(TDDFT). Lehtonen and Sundholm calculated the absorptionspectra of three hydrogen terminated silicon clusters usingTDDFT .The earlier studies on the optical properties of silicon andhydrogenated silicon clusters focused on the size dependenceof the PL and photo-absorption . Some of these studiesignored the influence of oscillator strength’s (electric dipolematrix elements) and hence were not in good agreement withthe experimental data. Calculations based on DFT using LDAand the generalised gradient approximation (GGA) which in-cluded the dipole matrix elements suffered from the drawbackthat is inherent in LDA/GGA i.e. the energy gaps were un-derestimated because these calculations ignored the effect ofexcited states. To overcome this drawback, the configurationinteraction method or the methods based on solving the Bethe-Salpeter equation along with the GW approximation havebeen suggested . These methods require a lot of computertime and hence have been restricted to small clusters. A com-putational technique based on linear response theory withinTDLDA had been proposed by Vasiliev et. al. . This is anatural extension of the LDA ground state density functionalformulation designed to include excited states. This methodis faster than the BS and GW methods and can therefore beused for larger clusters. Vasiliev et. al. have shown the vi-ability of their method by performing calculations on siliconand hydrogenated and oxygenated silicon clusters. Anotherimplementation of TDDFT has been formulated by Castro et.al. which allows the calculation of the excited state energiesand optical absorption spectra. In the present work, we haveused this method .Most of earlier calculations have considered only hydrogenterminated silicon clusters. There seems to be a lack of a sys-tematic study on small hydrogenated silicon clusters. In thispaper we report calculations for the silicon and hydrogenatedsilicon clusters in addition to some hydrogen terminated sili-con clusters. Thus our calculations will show the effect of hy-drogenation on the optical properties of silicon clusters. Ouremphasis is on the smaller clusters so as to bring out the evo-lution of the optical properties as we increase the number ofsilicon and hydrogen atoms in the cluster.The plan of the paper is as follows. In section II, we brieflydiscuss the method and give computational details. In sectionIII we present our results and discussion. In section IV wepresent our conclusions. II. METHOD AND COMPUTATIONAL DETAILS
We have used TDDFT for our calculation of the photo-absorption spectrum . For the sake of completeness weshall summarise the essentials of the method. In TDDFT, thebasic variable is the one electron density n ( r , t ) , which is ob-tained with the help of a fictitious system of non-interactingelectrons, the Kohn-Sham system. The time-dependent Kohn-Sham equations are i ¶¶ t y i ( r , t ) = (cid:2) − (cid:209) / + v KS ( r , t ) (cid:3) y i ( r , t ) (1)where y i ( r , t ) are Kohn-Sham one electron orbitals. In termsthese orbitals n ( r , t ) can be written as n ( r , t ) = occ (cid:229) i | y i ( r , t ) | (2)The Kohn-Sham potential can be written as v KS ( r , t ) = v ext ( r , t ) + v Hartree ( r , t ) + v xc ( r , t ) (3)where the first term is the external potential, the secondHartree potential and the last the exchange and correlationpotential. For obtaining this potential we use adiabatic localdensity approximation (ALDA).In our work we shall use the TDDFT scheme which cal-culates propagation of the time-dependent Kohn-Sham equa-tions in real time. In this scheme the electrons are given somesmall momentum( k ) to excite all the frequencies . This isachieved by transforming the ground state wave function ac-cording to y i ( r , d t ) = e i k z y i ( r , ) (4)and then propagating these wave functions for some finitetime. The spectrum is then obtained from dipole strength function S ( w ) S ( w ) = wp Im a ( w ) (5)where the a ( w ) is the dynamical polarizability and is given by a ( w ) = k Z dte i w t [ d ( t ) − d ( )] (6)where d ( t ) is dipole moment of the system.We can also define a quantity know as the oscillator strengthto express the strength of the transition. f I ( x ) = w I (cid:229) n ∈ x , y , z | < f | n | f I > | (7)where f and f I are the ground and excited state respectively.The oscillator strength is related to the dipole strength func-tion defined earlier using the following relationship S ( w ) = (cid:229) I f I × d ( w − w I ) (8)The sum over the oscillator strength gives the number off ac-tive electrons in the system, (cid:229) f I = N (9)where N is the number of active electrons in the system.The initial structures of the clusters used for our calcula-tions have been obtained by Balamurugan and Prasad ear-lier from their analysis based on the Car-Parrinello moleculardynamics (CPMD) . The resulting structures have been fur-ther optimised using the electronic structure method imple-mented in VASP (Vienna Ab-initio Simulation Package ) . Wedo not find any significant changes in the structures comparedto the CPMD results. In fact the starting CPMD configura-tion gives forces that are around 0.5 eV/Å which eventuallyreduce to around 0.05 eV/Å in VASP. VASP employs den-sity functional theory (DFT) and we have employed the lo-cal density approximation (LDA) for the exchange correlationusing norm conserving pseudo-potentials. The optimisation isdone only by relaxing the ions via the conjugate gradient (CG)method and using a k-point Monkhorst-Pack mesh of 4x4x4.All calculations have been performed in a cubic supercell oflength 20 Å.In this work the photo-absorption spectra of the optimisedstructures was studied using OCTOPUS code where theabove approach is implemented. All calculations are ex-panded in a regular mesh in real space, and the simulationsare performed in real time. The local density approximationwas employed to keep consistency with the geometry optimi-sation process. OCTOPUS uses a uniform grid in real space,which is located inside the sum of n spheres, one around eachatom of the n-atom cluster. For all clusters a minimisation ofenergy with respect to the radius and the grid spacing was car-ried out. We required the radius of each sphere to be 6-8 Å andthe grid spacing 0.28-0.4 Å for optimal energy minimisation.For the TDDFT the propagation in real time has been per-formed with 30,000 time steps with a total simulation timeof around 124 fs. This gives a good resolution of the spec-tra. Throughout the calculations the ions were kept static andin order to approximate the evolution operator the approxi-mated enforced time reversal symmetry (aetrs) method wasemployed. Numerically the exponential of the Hamiltonian,which is used to approximate the evolution operator was eval-uated using a simple Taylor expansion of the exponential. III. RESULTS AND DISCUSSIONS
In order to verify our calculations with the experimentaldata, we chose to compare the two most stable clusters SiH and Si H with the available experimental data as our bench-mark. In optical properties we are interested in transitionsform the occupied levels to the unoccupied levels. The struc-tures in the photo-absorption spectra are identified with thesetransitions. It is therefore interesting to calculate these energydifferences and compare them with the energy differences de-duced from the experimental spectra. Table I shows such acomparison. For our calculations, we have checked that thef-sum rule is satisfied . In Table 1 we have also included theBethe-Salpeter results of Rohlfing and Louie , the TDLDAresults of Vasilev et. al. and the TDLDA results from Oc-topus of Marques et. al. along with the transitions identifi-cation. The uppermost occupied states result from a hybridi-sation of the silicon and hydrogen states while the lowest un-occupied states are primarily silicon states. Thus the energygap is dependent on the the bonding and anti-bonding siliconstates. The transition between these states have been identi-fied as 4s, 4p, 4d states (these refer to the angular momentumcharacter of the final states). We find good agreement withthe experimental data of Itoh et. al. and Suto and Lee andother theoretical calculations. In particular, we get very goodagreement with the results of Marques et. al. . As comparedto the work of Marques et. al. we obtain an additional peakat 8.3 eV for Si H which is observed in other works andthe experiments . Our simulations pick up this additionalpeak because our broadening factor, which is the inverse ofthe time steps in the TDDFT run is quite narrow as comparedto Marques et. al. Fig. 1 shows the structures and photo-absorption spectra of the current work, Vasiliev et.al. andMarques et. al. for SiH and Si H clusters. The calculatedphoto-absorption cross-section (using TDDFT) seems to be ingood agreement with the TDLDA results of Vasiliev et. al. and the TDLDA results of Marques et. al. using Octopus.In order to see the effect of increasing the size of siliconclusters on the optical spectra, we present in Fig. 2 I the opti-mised structures of silicon clusters Si n [n=1-10] along with thephoto-absorption spectra. To see the effect of adding a singleH atom to these clusters, we have also calculated the photo-absorption spectra of Si n H clusters which are also shown inFig. 2 II along with the silicon clusters results. Each figureshows the structure of silicon and hydrogenated silicon clus-ters and the calculated photo-absorption cross-section. Con-sider first the silicon clusters. For small clusters (up to n=7)we find that the photo-absorption spectra is a combination ofmany peaks and looks like that of isolated atoms. However
Figure 1: Structure and photo-absorption spectra of present work(Black solid line), Vasiliev et.al. (Blue dashed line) and Marqueset. al. (Red solid line) for SiH and Si H clusters.Figure 2: I: Structure and photo-absorption spectra of Si n clusters(n=1-10 from top to bottom) ; II: Structure and photo-absorptionspectra of Si n H clusters (n=1-10 from top to bottom). a a Solid line corresponds to the PA spectra of the ground state structure Fig. 4(I), whereas dashed line corresponds to the PA spectra of higher energy statestructure Fig. 4 (II). for n>7, the optical spectra looks bulk-like . This can beunderstood using a simple tight-binding picture. In a largersilicon cluster, the overlap between electronic wave-functionslifts the degeneracy of the energy levels resulting in bunch-ing of energy levels in a narrow energy range. This resultsin broadening of PA spectrum for larger clusters. In Fig 2we see that the main structure in the PA spectra is located ataround 9 eV and a minor structure starts to build up at around15 eV. The same trends are found in the singly hydrogenatedclusters. For smaller clusters (n<7) the PA spectrum changessignificantly as we increase the number of hydrogen atoms,while for larger clusters this change is small.To see the effect of further hydrogenation, we present thePA spectra of hydrogenated silicon clusters where the numberof hydrogen atoms is increased in Fig. 3. We have performed Table I: Excitation energies in eV for SiH and Si H clusters.Transition Present Work Marques et. al. Rohlfing & Louie Vasiliev et. al. Experiment SiH
4s 8.2 8.2 9.0 8.2 8.8 a ,9.0 b
4p 9.2 9.4 10.2 9.2 9.7 a,b
4d 9.8 10 11.2 9.7 10.7 a,b Si H
4s 7.3 7.3 7.6 7.3 7.6 a
4p 8.3 - 9.0 7.8 8.4 a a a calculations for Si H , Si H and Si H , Si H and Si H and Si H and Si H clusters. The different positions of thehydrogen atoms in Si H and Si H makes the two structuresvery different. Hence there is a large difference between thePA spectra. In going from Si H to Si H we find that thenumber of structures in the PA spectra increases and some ofthe new structures have larger peaks. In fact the PA spectrumgoes from atom-like to bulk-like. We see similar trends inSi H and Si H except that the silicon backbone is a rhom-bus. We find that the new structures in Si H PA spectrumhave significantly larger peaks compared to Si H . In thesehydrogenated clusters (Si H , Si H , Si H , Si H , Si H ,Si H and Si H ) which have different silicon backbones,we observe that the bulk-like behaviour seems to start at n=4which is much earlier than in singly hydrogenated and siliconclusters. This effect may be attributed to the large number ofhydrogen atoms.We obtained two very closely related structures of Si Hhaving a difference of about 0.043 eV in their total energy.Both the structures have been shown in Fig. 4 and the cor-responding PA spectra in Figure 2 II. The silicon backbone inboth these clusters remains the same and the only difference isfound in the position of the hydrogen atom. In the ground statestructure (solid line in the PA spectra) hydrogen is bonded totwo silicon atoms while in the other (dashed line in the PAspectra) it is bonded to only one silicon atom. The PA spectraof the two structures is quite different. This shows that the PAspectrum is sensitive to small changes in the structure. Thusthe PA spectrum can be used to identify the structure of theclusters which differ by a very small energy.In Table II we present the optical absorption gaps of the var-ious clusters. We define the optical gaps through the integraloscillator strength rather than as the energy of the first dipoleallowed transition in the absorption spectra. The integral os-cillator strength gives the total number of active electrons inthe system. In this approach the value of the optical absorp-tion gap is determined at a very small but non-zero fractionof the complete oscillator strength . We set this threshold to10 − of the total oscillator strength. This value is chosen be-cause it stands above the value of “numerical noise”and at thesame time it is sufficiently small so as to not suppress the ex-perimentally detectable dipole allowed transitions. This def-inition of absorption gap doesn’t affect the values of opticalgaps for small clusters, since the intensity of first transitionsis well above the selected threshold. Figure 3: Structure and photo-absorption spectra of Si n H m clusters(from top to bottom: Si H , Si H , Si H , Si H , Si H , Si H ,Si H )Figure 4: Structures of ground state (I) and higher energy state (II)Si H clusters.
In Fig. 5 we plot the optical gap as a function of the num-ber of silicon atoms (n) for various Si n (solid Line) and Si n H(dashed Line) clusters (n=1-10). It shows a general trend thatthe addition of a single hydrogen atom reduces the optical gapas compared to the silicon clusters. As the number of siliconatoms in the cluster increase we find that the effect of a sin-gle hydrogen atom reduces the optical gap and the differencebetween the optical gap of Si n and Si n H gradually decreaseswith the increase in silicon atoms. In the bulk limit a singlehydrogen atom should not distort the optical gap and thus inthat limit the optical gap of the silicon will be the same as theoptical gap of the singly hydrogenated silicon.To see the effect of further hydrogenation, we list the op-
Figure 5: Optical absorption gap of various Si n (solid line filled cir-cle) and Si n H (dashed line empty circle) clusters as a function of thenumber of silicon atoms (n). tical gaps of all the clusters in Table II. We see that additionof a single hydrogen atom reduces the optical gap, in general,but with further hydrogenation the optical gap increases. Forthe fully hydrogenated clusters the optical gap is larger thanthe unhydrogenated clusters. Thus in the bulk limit the hy-drogenated system will show a larger gap which is consistentwith the experimental observations. IV. CONCLUSIONS
The photo-absorption spectra of silicon and hydrogenatedsilicon clusters have been calculated using TDDFT. Our cal-culations show very good agreement for the benchmarks ofSiH and Si H with the earlier theoretical calculations andexperimental results. We find that the changes in the photo-absorption spectra correlates well with the changes in thestructure of the cluster. In the singly hydrogenated clusterswe find bulk-like behaviour for larger clusters (n>7) while forthe smaller clusters we find the PA spectra is composed ofnumerous peaks as in atoms. We find that the PA spectra oftwo structures of Si H which are close in energy to be quitedifferent. For the other hydrogenated clusters the bulk-likebehaviour appears for smaller n which can be attributed tothe larger number of hydrogen atoms.The addition of a sin-gle hydrogen to the silicon cluster shows a decrease in theabsorption optical gap. The optical gap decreases graduallywith the number of silicon atoms in the cluster giving the bulklimit in which an addition of a single hydrogen to the siliconshould not effect the optical gap of silicon. With further hy-drogenation the optical gap increases and for the fully clustersthe optical gap is larger than the corresponding gap for puresilicon clusters.
V. ACKNOWLEDGEMENT
We are grateful to D Balamurugan for providing the co-ordinates of some clusters. We thank Professor M. K. Harbola
Table II: Optical gaps of various clusters.Cluster Absorption Optical Gap (eV)Si 4.7SiH 3.8SiH H 4.4Si H H 4.3Si H H 4.5Si H H H (gs) 4.5Si H (he) 5.4Si H H H 5.1Si H H H 5.1Si H 5.3Si H 5.6Si H 5.6 for helpful discussions. ∗ Electronic Address:[email protected] See for example, reviews by A.D. Zdetsis, Rev Ad. Mater. Sci. ,56 (2006); J.R. Chelikowsky, L. Kronik, and I. Vasiliev, J. Phys.:Condens Matter , R1517 (2003). L.T. Canham, Appl. Phys. Lett. , 1046 (1990). C. Delerue, G. Allan and M. Lannoo, Phys. Rev. B. , 11024(1993). H. Murakami and T. Kanayama, Appl. Phys. Lett. , 2341(1995). M. Suto and L.C. Lee, J. Chem. Phys. , 1160 (1986). U. Itoh, Y. Yasutake, H. Onuki, N. Washida and T. Ibuki J. Chem.Phys. , 4867 (1986). O. Cheshnovsky, S.H. Yang, C.L. Pettiette, M.J. Craycraft, Y. Liuand R.E. Smalley, Chem. Phys. Lett. , 119 (1987). K.D. Rinnen and M.L. Mandich, Phys. Rev. Lett. , 1823 (1992). J.M. Antonietti, F. Conus, A. Chatelain and S. Fredigo, Phys. Rev.B. , 035420 (2003). L. Chantranupong, G. Hirsch, R. J. Beunker and M. Dillon,Chem. Phys. , 167 (1993). A. Rubio, J.A. Alonso, X. Blase, L.C. Balbas and S.G. Louie,Phys. Rev. lett. , 247 (1996). M. Rohlfing and S.G. Louie, Phys. Rev. Lett. , 3320 (1998). I. Vasilev, S. Ogut and J. Chelikowsky,Phys. Rev. Lett. , 1813(2001). M.A.L. Marques, Alberto Castro, Angel Rubio, J. Chem. Phys. , 3006 (2001). S. Rao, J. Sutin, R. Clegg, E. Gratton, M.H. Nayfeh, S. Habbal,A. Tsolakidis, and R.M. Martin, Phys. Rev. B. , 205139 (2004). O. Lehtonen and D. Sundholm,Phys. Rev. B , 085424 (2005). T. Takagahara and K. Takeda, Phys. Rev. B. , 15578 (1992). N.A. Hill and K.B. Whaley, Phys. Rev. Lett. , 1130 (1995); ,3039 (1996). C. Delerue, M. Lanoo and G. Allan, ibid, , 3038(1996). L.W. Wang and A. Zunger, J. Chem. Phys. , 2394 (1994). S. Ogut, J.R. Chelikowsky and S.G. Louie, Phys. Rev. Lett. , 1770 (1997). B. Delly and E.F. Steigmeier, Phys. Rev. B. , 1397 (1993). A. Castro, H. Appel, Michael Olliveria, C.A. Rozzi, X. Andrade,F. Lorenzen, M.A.L. Marques, E.K.U Gross and A. Rubio, Phys.Stat. Sol. B. , 2465-2488 (2006). M.A.L. Marques and E.K.U. Gross, Ann. Rev. Phys. Chem. ,427 (2004). Time-dependent density functional theory, M.A.L. Marques, C.Ullrich, F. Nogueira, A. Rubio, K. Burke, and E.K.U. Gross(Eds.), Lecture Notes in Physics, Vol. 706, Springer, Berlin,(2006). K. Yabana and G.F. Bertsch, Phys. Rev. B. , 4484 (1996). D. Balamurugan and R. Prasad, Phys. Rev. B. , 205406 (2001). D. Balamurugan, Ph.D. thesis(unpublished), Indian Institute ofTechnology Kanpur (2004). R. Car and M. Parrinello, Phys. Rev. Lett. G. Kresse and J. Hafner, Phys. Rev. B. , RC558 (1993); G.Kresse and J. Haffner, Phys. Rev. B. , 13115 (1993), G. Kresseand J. Furthmuller, Comput. Mat. Sci. , 15 (1996). 27. G. Kresseand J. Furthmuller, Phys. Rev. B. , 11169 (1996). 28. G.Kresse, and J. Joubert, Phys. Rev. B. , 1758 (1999). See alsohttp://cms.mpi.univie.ac/at/vasp. M.A.L. Marques, A. Castro, G.F. Bertsch andA. Rubio, Comp. Phys. Comm. F. Wooten, Optical Properties of Solids (Academic, New York,1972). B. Adolph, J. Furthmuller and F. Bechstedt, Phys. Rev. B. ,125108 (2001). I. Vasilev, S. Ogut and J. Chelikowsky,Phys. Rev. B. , R8477(1999). G.D. Cody, in Semiconductors and Semimetals ed J.I. Pankove(Acad Press NY 1984) Vol21B