Research Article Electronic and Optical Properties of Small ...Research Article Electronic and...

Research ArticleElectronic and Optical Properties of SmallHydrogenated Silicon Quantum Dots Using Time-DependentDensity Functional Theory

Muhammad Mus-rsquoab Anas and Geri Gopir

School of Applied Physics Faculty of Science and Technology Universiti Kebangsaan Malaysia 43600 Bangi Selangor Malaysia

Correspondence should be addressed to Muhammad Mus-rsquoab Anas mus physicsyahoocom

Received 29 April 2015 Revised 31 August 2015 Accepted 31 August 2015

Academic Editor Bin Dong

Copyright copy 2015 M M Anas and G Gopir This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents a systematic study of the absorption spectrum of various sizes of small hydrogenated silicon quantum dots ofquasi-spherical symmetry using the time-dependent density functional theory (TDDFT) In this study real-time and real-spaceimplementation of TDDFT involving full propagation of the time-dependent Kohn-Sham equations were used The experimentalresults for SiH

4and Si

5H12showed good agreement with other earlier calculations and experimental data Then these calculations

were extended to study larger hydrogenated silicon quantum dots with diameter up to 16 nm It was found that for smallquantum dots the absorption spectrum is atomic-like while for relatively larger (16 nm) structure it shows bulk-like behavior withcontinuous plateau with noticeable peak This paper also studied the absorption coefficient of silicon quantum dots as a functionof their size Precisely the dependence of dot size on the absorption threshold is elucidated It was found that the silicon quantumdots exhibit direct transition of electron from HOMO to LUMO states hence this theoretical contribution can be very valuable indiscerning the microscopic processes for the future realization of optoelectronic devices

1 Introduction

Current nanostructures including clusters biomoleculesand molecular nanodevices have become the core of manyfundamental and technological research projects The studyof static and dynamic electron-electron correlation makes itpossible to characterize the electronic structural and bond-ing properties in nanostructure regime Additional to that itsnecessity is also related to the optical electronic and time-resolved spectroscopies Since the structural and electronicproperties of nanoparticle in particular silicon quantumdots are absolutely sensitive to the changes of atomic con-figuration impurities and doping effect [1] these challengesurged researchers to understand the phenomena involvedspecifically from quantum mechanics perspective One canobtain the electronic structure information from the opticalanalysis in particular the optical response of the quantumdots will provide informative view of its dependency on

their size and geometrical structure This is an importantfeature since the determination of the structure is in generala very sensitive task either for prototype construction orfor sophisticated relaxation of total energy minimizationFurthermore the knowledge of the geometrical structurefor modeling is highly required from solid state physics asa basis for understanding many of the properties of thenanostructure material

Relatively large quantum dots consisting of thousands ofatoms need very high computational cost in order to solvetheir many-body problem wave functions Here the impor-tance of parallel processing combined with perfect com-putational strategy urged researchers to seek more efficientalternatives to overcome these challenges Real-space gridsare a powerful alternative tool for the simulation of electronicsystems One of the main advantages of this approach is theflexibility and simplicity of working directly in real spacewhere the different fields are discretized on a grid combined

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2015 Article ID 481087 9 pageshttpdxdoiorg1011552015481087

2 Journal of Nanomaterials

with competitive numerical performance and great potentialfor parallelization In this study this paper is divided intotwo main parts In the first section the electronic propertiesof ground state hydrogenated silicon quantum dots usingdensity functional theory by three different methods willbe briefly reviewed This comparison step is taken to act asa benchmark for further study of the optical spectra usingthe time-dependent density functional theory (TDDFT) [23] Secondly the basic framework of TDDFT is used forthe calculation of the quantum dot optical spectra Thisframework is applied in real space by the first principlescode of OCTOPUS [4] that allows the study of electron-iondynamics of many-electron systems under the presence ofarbitrary external perturbation

In particular this paper will investigate the effect of quan-tum confinement on the energy gap of hydrogen passivatedsilicon quantum dots with various sizes In this aspect thisstudy is focused to compute the energy gap and opticalabsorption spectra as a function of dot size in order toexamine the behavior of optical gap under quantum confine-ment effect This study is also focused to get brief insight atthe microscopic level of the electron transition and bondingcharacteristic of the dots identified from the distribution ofthe highest occupied molecular orbitals (HOMO) and thelowest unoccupied molecular orbitals (LUMO) isosurface

Since quite a number of computational studies on siliconquantum dots were reported earlier calculation results werecompared with previous studies as a benchmark of thisextended study conducted Serious selection of computa-tional method and parameters is very important in orderto achieve ideal counter and balance between efficiency andcomputational cost Before the calculation starts impor-tant related background study for comparison purpose wasconducted These results were compared with Zdetsis etal [8] where the same orbital basis-set representation wasused but it was different in parameters Models for variousprototype structures were done especially for the struc-ture below 18 nm Those models also covered the missingstructure reported from Hirao and Uda [6] DFT calculationwith the same exchange-correlation functional Those possi-ble prototypes were discovered during structural construc-tion while preserving quasi-spherically structured quantumdots

Computational works done by Onida and Andreoni [12]Xue Jiang et al [13] and Vasiliev [14] were reported by manythese reports covered small quantum dots (below 08 nm)which also considered as small clusters were used as earlybenchmark for the calculation conducted in this study Sincelarger quantumdots (sim16 nm) fell into quantumconfinementregime of electron movement it was decided to extendthis study and analyze computationally atomistic ab initiomethod to understand quantum behavior of electron underconfinement region where all electron and orbital were takeninto account Using the computational facilities availablecombined with good calculation approach this study is capa-ble of analyzing time-dependent optical properties of siliconquantum dots under 16 nm in diameter corresponding to 160atoms (Si

87H76)

2 Theoretical Background

Time-dependent density functional theory (TDDFT) [2 3]can be used to obtain the optical spectra from relaxedgeometries TDDFT is an exact reformulation of time-dependent quantum mechanics in which the fundamentalvariable is no longer the many-body wave function but thetime-dependent density TDDFT is an extension of DFTwith the time-dependent domain to describe what happenswhen a time-dependent perturbation is applied For thesake of completeness the essentials of this method canbe summarized explicitly In TDDFT the basic variable isthe one electron density 119899(119903 119905) = sum119873

119894=1|120593

119894(r 119905)|2 which is

obtained with the help of a fictitious system of noninteractingelectrons the Kohn-Sham systemThe interacting system canbe represented with the time-dependent Kohn-Sham orbitals120593

119894(r 119905) using the time-dependent Kohn-Sham equation

119894

120575

120575119905

120593

119894(r 119905) = [minusnabla

2

2

+ 119881KS (r 119905)] 120593119894 (r 119905) (1)

The Kohn-Sham potential 119881KS(r 119905) is defined as

119881KS (r 119905) = 119881ext (r 119905) + 119881119867 (r 119905) + 119881xc (r 119905) (2)

where 119881ext(r 119905) is the external potential 119881119867(r 119905) is Hartree

potential and119881xc(r 119905) is the exchange and correlation poten-tial In this study the adiabatic local density approximation(ALDA) for the whole simulation was used where it ismapped from homogenous electron gas (HEG)

119881

ALDAxc (r 119905) = 119881

HEGxc (119899)1003816

1003816

1003816

1003816

1003816119899=119899(r119905) (3)

Since the exchange and correlation functional is the heartof density functional theory calculation it is very importantthat this crucial part be reviewed before proceeding into thefurther calculation There are numbers of publications thatexplore the reliability of various exchange-correlation func-tionals used to study electronic properties of silicon quantumdots As reported by Zdetsis et al [8] result of calculatedenergy gap using semiempirical hybrid exchange-correlationfunctional (B3LYP) overestimated the experimental valueespecially for the dots with diameter below 18 nm whileit is accurate for larger structure around 18ndash20 nm Eventhough this hybrid functional was optimized by chemistto overcome the weaknesses of coulomb potential in localdensity approximation (LDA) for molecular structure studyhere the reliability of LDA functional was demonstrated toexplore the electronic and optical properties of small sizedsilicon quantum dots as presented in this paper This isalso confirmed by earlier computational research reported byHirao and Uda [6] where the LDA functional is also usedto study hydrogenated silicon quantum dots For the sake ofreducing computational cost while maintaining its accuracyit was found that LDA results were acceptable to studyquantum dot sizes silicon semiconductor as benchmarkedwith reported experimental results

Next the absorption spectrum was calculated usingthe explicit propagation of the time-dependent Kohn-Sham

Journal of Nanomaterials 3

equations Throughout this approach the system was firstexcited from its ground state by applying a delta electric field119864

0120575(119905)e119898 The unit vector e

119898determines the polarization

direction of the electric field and 1198640is its magnitude

which must be small enough if one is interested in linearresponse The reaction of the noninteracting Kohn-Shamsystem caused by the perturbation can be readily computedThe mechanism described that each ground state Kohn-Sham orbital 120593GS

119894(r) instantaneously phase-shifted where

120593

119894(r 119905 = 0+) = 1198901198941198640e119898 sdotr120593GS

119894(r) The Kohn-Sham equations

are then propagated forward in real time and by then thetime-dependent density 119899(r 119905) can be computedThe induceddipole moment variation is an explicit functional of thedensity

120575D119898 (119905) = 120575 ⟨R⟩ (119905) = int1198893119903 [119899 (r 119905) minus 119899 (r 119905 = 0)] r (4)

The super index 119898 indicates that the perturbation hasbeen applied along the 119898th Cartesian direction Then thecomponent of the dynamical dipole polarizability tensor120572(120596)is directly related to the Fourier transform of the induceddipole moment function

120572

119898119899=

120575119863

119898

119899(120596)

119864

0

(5)

The spatially averaged absorption cross section is triviallyobtained from the imaginary part of the dynamical polariz-ability

120590 (120596) =

4120587120596

119888

Im [120572 (120596)] (6)

where 120572 as the function of 120596 is the spatial average or trace ofthe 120572119898119899

tensor

120572 (120596) =

1

3

Tr [120572 (120596)] (7)

It is well known that the simpler approach of taking thedifferences of eigenvalues between Kohn-Sham orbitals givespeaks at lower frequencies in disagreement with the experi-mental spectra [15] On the other hand TDDFT within theALDA exchange-correlation functional typically producesthe optical linear response of the molecular compound thatis in good agreement with experimental results with accuracybelow 02 eV

3 Computational Detail

Hydrogen passivated silicon quantum dots (Si-QDs) withselected sizes up to 16 nm were constructed by repeat-ing tetrahedral silicon geometry from its crystal geometri-cal structure (bulk structural properties) until the desiredrepetition was achieved The prototype quantum dots areconstructed in the order of quasi-spherical shape Nextthe quantum dots structures were passivated with hydrogenatoms on the surface All the ground state relaxed structureswere obtained after performing geometry optimization using

the quasi-Newtonmethod to get an ideal relaxation structureThe quasi-Newton method used is the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method [16ndash19] all in Cartesiancoordinates This optimized structure is then used to cal-culate the electronic and optical properties in particularthe energy gap and absorption spectrum linear responseAll the computational approaches were successfully done inreal space and implemented using OCTOPUS [4] The coreelectrons with strong coulombic effect were treated usingthe Troullier and Martins pseudo potentials [20] throughoutthis simulation For the ground state calculation the com-putational parameters used in our previous work were usedexactly [5] to solve for the Kohn-Sham eigenvalues Thenthe time-dependent calculations of the perturbed systemwere conducted Subsequently the time reversal symmetrypropagator in the algorithms was used to approximate theevolution operator

All linear response calculations were done using OCTO-PUS [4] implementing the Perdew-Zunger [21] parameter-ization of the local density approximation (LDA) for theexchange-correlation potentialThewave functions represen-tation in real space was mapped onto a uniform grid with aspacing of 0175 A and spheres of radius 4 A around everyatom The system was perturbed by short electrical pulses001 Aminus1 A time step of 00017 femtoseconds ensured thestability of the time propagation and a total propagation timeof 10 femtoseconds allowed a resolution of about 001 eV inthe resulting energy spectrum with almost 6000 steps Theresults obtained are summarized in Table 2 and also Figure 2where the absorption spectrum was plotted for the wholeperturbation

4 Results and Discussion

41 Ground State Properties At the beginning of this simula-tion the accuracy of ground states calculation was examinedby comparing the Kohn-Sham energy gaps obtained usingdifferent types ofwave function representation and exchange-correlation function This comparison acts as a benchmarkfor the simulation reliability of the selected parameter usedTable 1 shows the comparison of Kohn-Sham energy gaps(119864119892 in eV) obtained from the ground state calculations

of geometry optimized hydrogenated silicon quantum dotsThree different wave function representations were usedto solve the Kohn-Sham equations namely the numericalatomic orbital (NAO) plane-waves (PWs) and real-space(RS) basis set All the reported data were analyzed using thesame computational approach (DFT) but in different wavefunction representation (PWs NAO and RS basis set) andexchange-correlation potential Then the approximated 119864

119892

values were checked This calculation result using real-spacemethod for the ground state eigenvalues is found to be ingood agreement with othersrsquo theoretical report

Before the absorption coefficient was calculated attentionwas given first to the spatial distribution for the groundstate of the HOMO and LUMO states The results of the 3-dimensional spatial distribution for both states in differentdot size are visualized in Figure 1 For larger sizes it is noted


SiH4 Si5H12 Si17H36 Si29H36 Si35H36

HO

MO

LUM

O

Si47H60 Si66H72 Si71H84 Si84H76

HO

MO

LUM

O

Figure 1 Spatial distribution of HOMO and LUMO plot

Table 1 Comparison of ground state 119864119892(eV) calculated using different basis-set representation and exchange-correlation functional (lowast

corresponding to Si66H64)

Quantum dots NAO [5] (LDA)(without discontinuity correction)

PWs [6](LDA)

PWs [7](LDA)

RS [8](B3LYP)

RS (LDA)(present study)

SiH4 900 mdash 790 mdash 851

Si5H12 558 mdash 560 76 555

Si17H36 480 mdash 450 572 428

Si29H36 430 332 420 515 395

Si35H36 356 mdash 370 504 366

Si47H60 314 mdash mdash 464 314

Si66H72 272 295lowast 28 mdash 273


Si84H76 251 mdash 25 mdash 254


005

115

225

335

4

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)

Absorption spectrum SiH4

(a)

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)

Absorption spectrum Si5H12

(b)

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(c)

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18Energy (eV)

3 64 50

051

152

Stre

ngth

func

tion

(1e

V)


(d)

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18Energy (eV)

25 35 45

02040608

Stre

ngth

func

tion

(1e

V)


(e)

05

101520253035

0 2 4 6 8 10 12 14 16 18

3 50

4

02040608

Energy (eV)

1

Stre

ngth

func

tion

(1e

V)


(f)

05

1015202530354045

0 2 4 6 8 10 12 14 16 18

3 50

4

1

2

Energy (eV)

Stre

ngth

func

tion

(1e

V)


(g)

0

10

20

30

40

50

0 2 4 6 8 10 12 14 16 18Energy (eV)

3 50

4

1

2

Stre

ngth

func

tion

(1e

V)


(h)

Figure 2 Continued


0

10

20

30

40

50

60

0

01234

2 4 6 8 10 12 14 16 18Energy (eV)

25 5535 45

Stre

ngth

func

tion

(1e

V)


(i)

Figure 2 Optical absorption spectrum of hydrogenated silicon quantum dots

Table 2 Comparison of results for optical properties [9ndash11]

Quantum dots Optical properties

Diameter (nm) Empirical formula Optical threshold (eV) Firstnoticeable peak (eV)

Secondnoticeable peak (eV)

mdash SiH4 81 80 [9] 80 [11] 854 87 [9] 82 [11] 945 96 [9] 91 [11]

06 Si5H12 55 57 [10] 56 [11] 67 65 [10] 64 [11] 812

08 Si17H36 40 547 672

10 Si29H36 35 42 48

11 Si35H36 343 37 44

12 Si47H60 34 41 505

14 Si66H74 33 372 493

15 Si71H84 32 381 598

16 Si87H76 30 478 672

that the HOMO states are principally distributed along thedirection of the bond axis and just a little on the surfacestates whereas the LUMO states are located the other wayaround It is also clearly seen that the surface distributionof the HOMO states contributes significantly to the valueof the electronic property of energy gap In contrast thephenomena of phonon-assisted transition in bulk siliconmake this element an ineffective purpose for optoelectronicdevicesTheir efficiency was reduced by phonon effect duringelectron transition in Brillouin zone However from spatialdistribution of electron wave function shown in Figure 1 itwas found that the probability of electron position in realspace between HOMO and LUMO eigenstates was occupiedin the same space This will provide a direct transitionof electron which can make silicon quantum dots a goodcandidate for future optoelectronic device

Results in this study show that as the size increased thenature of the localization of the HOMO and LUMO statesdiffers from the smaller sizes and this obviously reflects theeffect of the structural configuration on the electronic prop-erties It is clearly seen that hydrogen is bonded with siliconatoms even when its valance is oneThis phenomenon can beelucidated from the electron spatial distribution isosurface ofthe HOMO and LUMO states for Si

5H12(Figure 1) It can be

seen clearly too that the HOMO exhibits a bonding characterwhere the electron cloud is predominantly located in theintermediate regions where the bonds are formed On theother hand the antibonding character of LUMO state is con-firmed by the significant distribution of charge density at thevicinity of the atoms For the largest quantum dot examinedin this study size around 16 nm diameter the LUMO clearlyshows its antibonding nature correspondingly The HOMOdisplays a region of high density inside the QD as anticipatedfor 1s type (sphere) wave functions It is inferred that fromthe isosurface plots of HOMO and LUMO the formationof the energy gap is originated from the localization of theelectron cloud at the surface The localization of both theHOMO and LUMO orbitals on hydrogen atoms illustratedthat the gap is prominent and can be identified by hydrogenbondsmore clearly seen for smaller size quantumdots In thenanostructures case if a strong electronegativity difference ispresent between Si and the passivating atoms the HOMOtends to concentrate (while preserving its delocalization)on the weakened surface SindashSi bonds Hence the surfacestates have different contribution for the HOMO and LUMOstates Also the LUMO states indicate that the electronsare more localized near the hydrogen atoms This situationis anticipated because there will be small charge transfer


from the silicon atoms to hydrogen This may be attributeddue to the slightly higher electronegativity of H atom ascompared to Si atom As a result the H atom prefers tobond with the silicon atom which possesses a larger numberof electrons For silicon quantum dots this can be at thecost of giving up the sp3-hybridization Therefore the sp3-character of diamond-like bulk materials may not be presentin these small dots As a matter of fact the Pauling scaleof electronegativity defined that silicon electronegativity is19 while that of H is 22 If the bond considered is onlybetween both elements (SindashH) then it can be considered asnonpolar because the electronegativity difference betweenthese two elements is 03 But in this case of quantum dotsstudy the electron cloud from the core of the silicon atomsand the neighboring atom silicon-silicon charges contributeto increments of electronegativity differences which can becategorized as polar covalent bonding interaction This is aclear explanation that the interaction between hydrogen andsilicon leads not only to passivation of the dangling bondsbut also to alteration of the electronic properties of the siliconquantum dots

42 Optical Properties As the optical response of nanometerstructures depends crucially on the particle size due toquantum effects their optical properties can be tuned bychanging their size opening the road towards potentialapplications in optical nanodevices Therefore it is importantto have a reliable scheme to address the calculation ofresponse functions In the case of nonlinear phenomena thepresent exchange-correlation functional might not be highlyaccurate but they nevertheless yield relevant informationabout the systems Again all of the results reported havebeen obtained using the OCTOPUS code based on real-spacemethod of wave function representation

In order to interpret the absorption spectrum of varioussized silicon quantum dots let us take the absorption edge asthe energy threshold opening gap of absorption spectra It iswell known that the optical gap is different from the energygap calculated from differences energy level of HOMO andLUMO where 119864gap = 119864HOMO minus 119864LUMO because the firstallowed excitation is often too weak to be detected For ameaningful comparison with experiment the same empiricalcriterion employed in interpreting experimental data whichis related to the detection of the absorption threshold wasused and the first two absorption peaks were detected [9 10]The conformity of these time-dependent calculations waschecked by comparing these results with the experimentalresults for SiH

4(silane) and Si

5H12 It is observed that the first

peak of silane optical threshold resulting in this simulation is810 eV and the first peak is 854 eV Meanwhile for Si

5H12

the optical threshold is 545 eV the first allowed transitionpeakwas observed at 67 eV as shown in Figures 2(a) and 2(b)On the other hand the experimental value reported that theonset of absorption spectrum for silane was 8 eV and the firstpeak observed was around 87 eV [9] Experimental opticalthreshold and the first peak observed for Si

5H12were 57 eV

and 653 eV respectively [10] These experimental results arein good agreement with our calculated results with smallerrors between 01 and 02 eV This benchmark indicates that

this conducted study is in a right way as its results areclose to the reported experimental values Subsequently thissimulation is extended to study the absorption spectra oflarger quasi-spherical silicon quantum dots with diameter upto 16 nm

For the structure with the largest diameter (Si87H76) the

first noticeable transition peak was detected close to 50 eV(see Figure 2(i))Thepresence of a very small numerical noisewas also noticed on the left side of the optical thresholdobserved for Si

35H36

structure Those noise peaks occupyvery small oscillator strengthwhich can be neglected from thecharacteristic of prominent optical absorption A distributionof an absorption peak in the optical absorption versus photonenergy curve denotes that increments of photon absorptionbegin from 4 to 11 eV with noticeable peaks which infers thatthere is an important absorption of photons at this regionfor Si nanostructures which is different from bulk Si Otherintriguing characteristics that also can be noticed are that thehighest absorption peak of nanosized structure was shiftedtowards higher energy level where the highest absorptionpeak for silicon quantum dots was located around 11 eV Thisdiffers from the experimental determinations of bulk siliconoptical properties where the main peak (highest peak) occur-rence of the photon absorption is at energy around 43 eV [22]that indicates the electron confinement phenomena muchhigher in nanosized structure The calculated optical gaps asidentified from these absorption spectra are shown in Table 2and compared with experimental and previously reportedvalues

The obtained calculation results infer that the surfaceeffects caused by the single bonded passivators participate inthe optical transitions It is obvious to understand that withthe increment of dot size both the energy gap and opticalgap are reduced because of deconfinement effect Accordingto the exciton de Broglie wavelength for silicon the effectivediameter of silicon quantum dots where the confinementeffectwill affect its electronic properties should be below 5nm[23]This specific tendency of tuneable sizes that produce thevariation of energy gap makes the silicon a new candidatefor optoelectronic devices Despite its excellent character oftuneable wavelengths from ultraviolet to infrared siliconquantumdotswere biofriendly element andnontoxic elementto use in biological living

It is noticed that for all the silicon quantum dots con-sidered their optical gap is higher than the correspondinglyground state Kohn-Sham energy gap It is inferred that thisoccurrence was caused by the allowed excitation transitionstate of electron fromHOMO to LUMO state with reasonableoscillator strength The absorption spectra for smaller dotsillustrate a combination of many peaks and resemble isolatedatoms While on a larger Si-QD the overlap between theelectronic wave functions removes the nondegeneracy ofthe energetic states that arises in the grouping of energystates in a narrow energy section This induces broadeningof width and increment of the absorption oscillator strengthfor larger nanostructures A red shift of the optical absorptionspectrum is found to appear with the increment in the dotdiameter of silicon quantum dots In larger dots (16 nm) themajor feature (maximum peak) of the absorption spectra is


located around 98 eV and the declination feature begins toappear after 11 eV

5 Conclusion

Theelectronic structure and absorption coefficients of hydro-genated silicon quantum dots have been calculated usingtime-dependent density functional theory Calculations inthis study showed a very good and significant agreement forthe benchmark small silicon quantumdots of SiH

4and Si

5H12

with earlier theoretical calculations and experimental resultsIt is found that the changes in the photoabsorption spectracorrelate well with the changes in the size of the quantumdots For relatively large hydrogenated silicon quantum dots(16 nm) it shows bulk-like behavior with smooth noticeablepeak while for the small structure the absorption spectra arecomposed of numerous peaks as in isolated atoms It is foundthat the optical gap for the large structure with diameteraround 1ndash16 nm is suitable for optoelectronic applicationwhere its first peak of absorption is observed around 37to 5 eV which lies under ultraviolet region The optical gapdecreases gradually with the increase in number of siliconatoms in the quantum dots The absorption spectra displaysignificant red shifts with respect to the increment size of dotThis phenomenon is due to deconfinement effect of electronstrapped in quantumdotsThe calculated spectra present vari-eties of features that could be used for structure identificationIt is noticed that the TDDFT computed optical thresholddiffers significantly from energy gap simply computed fromdifference of ground state energy level of HOMO-LUMOwhich can provide much better approximation of energygap using TDDFT calculation The significant absorptionthreshold obtained can be used to classify the characterand structure precisely up to 01 nm in difference It isalso confirmed that LDA functional in TDDFT calculationmethod predicts good approximation for the optical gap oflow-dimensional structures where the obtained results arevery consistent with the absorption spectrum reported by theexperiment

Conflict of Interests

The authors declared that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the Malaysian Ministry ofScience Technology and Innovation (MOSTI) for ScienceFund Funding 06-01-02-SF0988 and Malaysian Ministry ofEducation Scholarship (MyBrain15) to Muhammad Mus-rsquoabAnas

References

[1] J Zeng and H Yu ldquoA first-principle study of B- and P-dopedsilicon quantum dotsrdquo Journal of Nanomaterials vol 2012Article ID 147169 7 pages 2012

[2] E Runge and E K U Gross ldquoDensity-functional theory fortime-dependent systemsrdquo Physical Review Letters vol 52 no12 pp 997ndash1000 1984

[3] E K U Gross andW Kohn ldquoLocal density-functional theory offrequency-dependent linear responserdquo Physical Review Lettersvol 55 no 26 pp 2850ndash2852 1985

[4] M A LMarques A Castro G F Bertsch andA Rubio ldquoOcto-pus a first-principles tool for excited electronndashion dynamicsrdquoComputer Physics Communications vol 151 no 1 pp 60ndash782003

[5] M M Anas A P Othman and G Gopir ldquoFirst-principlestudy of quantum confinement effect on small sized siliconquantum dots using density-functional theoryrdquo AIP ConferenceProceedings vol 1614 pp 104ndash109 2014

[6] M Hirao and T Uda ldquoElectronic structure and optical proper-ties of hydrogenated silicon clustersrdquo Surface Science vol 306no 1-2 pp 87ndash92 1994

[7] A J Williamson J C Grossman R Q Hood A Puzder andG Galli ldquoQuantumMonte Carlo calculations of nanostructureoptical gaps application to silicon quantum dotsrdquo PhysicalReview Letters vol 89 no 19 Article ID 196803 4 pages 2002

[8] A D Zdetsis C S Garoufalis and S Grimme ldquoReal spaceAb initio calculations of excitation energies in small siliconquantum dotsrdquo in Quantum Dots Fundamentals Applicationsand Frontiers vol 190 of NATO Science Series pp 317ndash332Springer Dordrecht The Netherlands 2005

[9] U Itoh Y Yasutake H Onuki N Washida and T IbukildquoVacuum ultraviolet absorption cross sections of SiH

4 GeH

4

Si2H6 and Si

3H8rdquo The Journal of Chemical Physics vol 85 no

9 pp 4867ndash4872 1986[10] F Feher Molekulspektroskopische Untersuchungen auf dem

Gebiet der Silane und der Heterocyclischen Sulfane Forschungs-berichtdes Landes Nordrhein-Westfalen WestdeutscherCologne German 1977

[11] JThingna R Prasad and S Auluck ldquoPhoto-absorption spectraof small hydrogenated silicon clusters using the time-dependentdensity functional theoryrdquo Journal of Physics amp Chemistry ofSolids vol 72 no 9 pp 1096ndash1100 2011

[12] G Onida and W Andreoni ldquoEffect of size and geometry onthe electronic properties of small hydrogenated silicon clustersrdquoChemical Physics Letters vol 243 no 1-2 pp 183ndash189 1995

[13] X Xue Jiang J Zhao and X Jiang ldquoTuning the electronic andoptical properties of hydrogen-terminated Si nanocluster byuniaxial compressionrdquo Journal of Nanoparticle Research vol 14article 818 10 pages 2012

[14] I Vasiliev ldquoOptical excitations in small hydrogenated siliconclusters comparison of theory and experimentrdquo Physica StatusSolidi vol 239 no 1 pp 19ndash25 2003

[15] A Castro M A L Marques J A Alonso G F BertschK Yabana and A Rubio ldquoCan optical spectroscopy directlyelucidate the ground state of C20rdquo Journal of Chemical Physicsvol 116 no 5 pp 1930ndash1933 2002

[16] C G Broyden ldquoThe convergence of a class of double-rankminimization algorithms 1 General considerationsrdquo Journal ofthe Institute of Mathematics and Its Applications vol 6 no 1 pp76ndash90 1970

[17] R Fletcher ldquoNew approach to variable metric algorithmsrdquoComputer Journal vol 13 no 3 pp 317ndash322 1970

[18] D Goldfarb ldquoA family of variable-metric methods derived byvariational meansrdquo Mathematics of Computation vol 24 pp23ndash26 1970


[19] D F Shanno ldquoConditioning of quasi-Newton methods forfunction minimizationrdquo Mathematics of Computation vol 24no 111 pp 647ndash656 1970

[20] N Troullier and J L Martins ldquoEfficient pseudopotentials forplane-wave calculationsrdquo Physical Review B vol 43 no 3 pp1993ndash2006 1991

[21] J P Perdew and A Zunger ldquoSelf-interaction correction todensity-functional approximations for many-electron systemsrdquoPhysical Review B vol 23 no 10 pp 5048ndash5079 1981

[22] H R Philipp and E A Taft ldquoOptical constants of silicon in theregion 1 to 10 evrdquo Physical Review vol 120 no 1 pp 37ndash38 1960

[23] C S Solanki Solar Photovoltaics Fundamentals Technologiesand Applications Prentice-Hall of India New Delhi India2009

Submit your manuscripts athttpwwwhindawicom

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with competitive numerical performance and great potentialfor parallelization In this study this paper is divided intotwo main parts In the first section the electronic propertiesof ground state hydrogenated silicon quantum dots usingdensity functional theory by three different methods willbe briefly reviewed This comparison step is taken to act asa benchmark for further study of the optical spectra usingthe time-dependent density functional theory (TDDFT) [23] Secondly the basic framework of TDDFT is used forthe calculation of the quantum dot optical spectra Thisframework is applied in real space by the first principlescode of OCTOPUS [4] that allows the study of electron-iondynamics of many-electron systems under the presence ofarbitrary external perturbation

In particular this paper will investigate the effect of quan-tum confinement on the energy gap of hydrogen passivatedsilicon quantum dots with various sizes In this aspect thisstudy is focused to compute the energy gap and opticalabsorption spectra as a function of dot size in order toexamine the behavior of optical gap under quantum confine-ment effect This study is also focused to get brief insight atthe microscopic level of the electron transition and bondingcharacteristic of the dots identified from the distribution ofthe highest occupied molecular orbitals (HOMO) and thelowest unoccupied molecular orbitals (LUMO) isosurface

Since quite a number of computational studies on siliconquantum dots were reported earlier calculation results werecompared with previous studies as a benchmark of thisextended study conducted Serious selection of computa-tional method and parameters is very important in orderto achieve ideal counter and balance between efficiency andcomputational cost Before the calculation starts impor-tant related background study for comparison purpose wasconducted These results were compared with Zdetsis etal [8] where the same orbital basis-set representation wasused but it was different in parameters Models for variousprototype structures were done especially for the struc-ture below 18 nm Those models also covered the missingstructure reported from Hirao and Uda [6] DFT calculationwith the same exchange-correlation functional Those possi-ble prototypes were discovered during structural construc-tion while preserving quasi-spherically structured quantumdots

Computational works done by Onida and Andreoni [12]Xue Jiang et al [13] and Vasiliev [14] were reported by manythese reports covered small quantum dots (below 08 nm)which also considered as small clusters were used as earlybenchmark for the calculation conducted in this study Sincelarger quantumdots (sim16 nm) fell into quantumconfinementregime of electron movement it was decided to extendthis study and analyze computationally atomistic ab initiomethod to understand quantum behavior of electron underconfinement region where all electron and orbital were takeninto account Using the computational facilities availablecombined with good calculation approach this study is capa-ble of analyzing time-dependent optical properties of siliconquantum dots under 16 nm in diameter corresponding to 160atoms (Si

87H76)

2 Theoretical Background

Time-dependent density functional theory (TDDFT) [2 3]can be used to obtain the optical spectra from relaxedgeometries TDDFT is an exact reformulation of time-dependent quantum mechanics in which the fundamentalvariable is no longer the many-body wave function but thetime-dependent density TDDFT is an extension of DFTwith the time-dependent domain to describe what happenswhen a time-dependent perturbation is applied For thesake of completeness the essentials of this method canbe summarized explicitly In TDDFT the basic variable isthe one electron density 119899(119903 119905) = sum119873

119894=1|120593

119894(r 119905)|2 which is

obtained with the help of a fictitious system of noninteractingelectrons the Kohn-Sham systemThe interacting system canbe represented with the time-dependent Kohn-Sham orbitals120593

119894(r 119905) using the time-dependent Kohn-Sham equation

119894

120575

120575119905

120593

119894(r 119905) = [minusnabla

2

2

+ 119881KS (r 119905)] 120593119894 (r 119905) (1)

The Kohn-Sham potential 119881KS(r 119905) is defined as

119881KS (r 119905) = 119881ext (r 119905) + 119881119867 (r 119905) + 119881xc (r 119905) (2)

where 119881ext(r 119905) is the external potential 119881119867(r 119905) is Hartree

potential and119881xc(r 119905) is the exchange and correlation poten-tial In this study the adiabatic local density approximation(ALDA) for the whole simulation was used where it ismapped from homogenous electron gas (HEG)

119881

ALDAxc (r 119905) = 119881

HEGxc (119899)1003816

1003816

1003816

1003816

1003816119899=119899(r119905) (3)

Since the exchange and correlation functional is the heartof density functional theory calculation it is very importantthat this crucial part be reviewed before proceeding into thefurther calculation There are numbers of publications thatexplore the reliability of various exchange-correlation func-tionals used to study electronic properties of silicon quantumdots As reported by Zdetsis et al [8] result of calculatedenergy gap using semiempirical hybrid exchange-correlationfunctional (B3LYP) overestimated the experimental valueespecially for the dots with diameter below 18 nm whileit is accurate for larger structure around 18ndash20 nm Eventhough this hybrid functional was optimized by chemistto overcome the weaknesses of coulomb potential in localdensity approximation (LDA) for molecular structure studyhere the reliability of LDA functional was demonstrated toexplore the electronic and optical properties of small sizedsilicon quantum dots as presented in this paper This isalso confirmed by earlier computational research reported byHirao and Uda [6] where the LDA functional is also usedto study hydrogenated silicon quantum dots For the sake ofreducing computational cost while maintaining its accuracyit was found that LDA results were acceptable to studyquantum dot sizes silicon semiconductor as benchmarkedwith reported experimental results

Next the absorption spectrum was calculated usingthe explicit propagation of the time-dependent Kohn-Sham








120593

119894(r 119905 = 0+) = 1198901198941198640e119898 sdotr120593GS



120575D119898 (119905) = 120575 ⟨R⟩ (119905) = int1198893119903 [119899 (r 119905) minus 119899 (r 119905 = 0)] r (4)


120572

119898119899=

120575119863

119898

119899(120596)

119864

0

(5)


120590 (120596) =

4120587120596

119888

Im [120572 (120596)] (6)


tensor

120572 (120596) =

1

3

Tr [120572 (120596)] (7)









119892





HO

MO

LUM

O

Si47H60 Si66H72 Si71H84 Si84H76

HO

MO

LUM

O





PWs [6](LDA)

PWs [7](LDA)

RS [8](B3LYP)



Si5H12 558 mdash 560 76 555

Si17H36 480 mdash 450 572 428

Si29H36 430 332 420 515 395

Si35H36 356 mdash 370 504 366






005

115

225

335

4

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(a)

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(b)

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(c)

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18Energy (eV)

3 64 50

051

152

Stre

ngth

func

tion

(1e

V)


(d)

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18Energy (eV)

25 35 45

02040608

Stre

ngth

func

tion

(1e

V)


(e)

05

101520253035

0 2 4 6 8 10 12 14 16 18

3 50

4

02040608

Energy (eV)

1

Stre

ngth

func

tion

(1e

V)


(f)

05

1015202530354045

0 2 4 6 8 10 12 14 16 18

3 50

4

1

2

Energy (eV)

Stre

ngth

func

tion

(1e

V)


(g)

0

10

20

30

40

50

0 2 4 6 8 10 12 14 16 18Energy (eV)

3 50

4

1

2

Stre

ngth

func

tion

(1e

V)


(h)

Figure 2 Continued


0

10

20

30

40

50

60

0

01234

2 4 6 8 10 12 14 16 18Energy (eV)

25 5535 45

Stre

ngth

func

tion

(1e

V)


(i)






mdash SiH4 81 80 [9] 80 [11] 854 87 [9] 82 [11] 945 96 [9] 91 [11]

06 Si5H12 55 57 [10] 56 [11] 67 65 [10] 64 [11] 812

08 Si17H36 40 547 672

10 Si29H36 35 42 48

11 Si35H36 343 37 44

12 Si47H60 34 41 505

14 Si66H74 33 372 493

15 Si71H84 32 381 598

16 Si87H76 30 478 672









4(silane) and Si



5H12


5H12were 57 eV





35H36






5 Conclusion


4and Si

5H12




Acknowledgments


References










4 GeH

4

Si2H6 and Si

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










120593

119894(r 119905 = 0+) = 1198901198941198640e119898 sdotr120593GS



120575D119898 (119905) = 120575 ⟨R⟩ (119905) = int1198893119903 [119899 (r 119905) minus 119899 (r 119905 = 0)] r (4)


120572

119898119899=

120575119863

119898

119899(120596)

119864

0

(5)


120590 (120596) =

4120587120596

119888

Im [120572 (120596)] (6)


tensor

120572 (120596) =

1

3

Tr [120572 (120596)] (7)









119892





HO

MO

LUM

O

Si47H60 Si66H72 Si71H84 Si84H76

HO

MO

LUM

O





PWs [6](LDA)

PWs [7](LDA)

RS [8](B3LYP)



Si5H12 558 mdash 560 76 555

Si17H36 480 mdash 450 572 428

Si29H36 430 332 420 515 395

Si35H36 356 mdash 370 504 366






005

115

225

335

4

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(a)

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(b)

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(c)

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18Energy (eV)

3 64 50

051

152

Stre

ngth

func

tion

(1e

V)


(d)

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18Energy (eV)

25 35 45

02040608

Stre

ngth

func

tion

(1e

V)


(e)

05

101520253035

0 2 4 6 8 10 12 14 16 18

3 50

4

02040608

Energy (eV)

1

Stre

ngth

func

tion

(1e

V)


(f)

05

1015202530354045

0 2 4 6 8 10 12 14 16 18

3 50

4

1

2

Energy (eV)

Stre

ngth

func

tion

(1e

V)


(g)

0

10

20

30

40

50

0 2 4 6 8 10 12 14 16 18Energy (eV)

3 50

4

1

2

Stre

ngth

func

tion

(1e

V)


(h)

Figure 2 Continued


0

10

20

30

40

50

60

0

01234

2 4 6 8 10 12 14 16 18Energy (eV)

25 5535 45

Stre

ngth

func

tion

(1e

V)


(i)






mdash SiH4 81 80 [9] 80 [11] 854 87 [9] 82 [11] 945 96 [9] 91 [11]

06 Si5H12 55 57 [10] 56 [11] 67 65 [10] 64 [11] 812

08 Si17H36 40 547 672

10 Si29H36 35 42 48

11 Si35H36 343 37 44

12 Si47H60 34 41 505

14 Si66H74 33 372 493

15 Si71H84 32 381 598

16 Si87H76 30 478 672









4(silane) and Si



5H12


5H12were 57 eV





35H36






5 Conclusion


4and Si

5H12




Acknowledgments


References










4 GeH

4

Si2H6 and Si

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





HO

MO

LUM

O

Si47H60 Si66H72 Si71H84 Si84H76

HO

MO

LUM

O





PWs [6](LDA)

PWs [7](LDA)

RS [8](B3LYP)



Si5H12 558 mdash 560 76 555

Si17H36 480 mdash 450 572 428

Si29H36 430 332 420 515 395

Si35H36 356 mdash 370 504 366






005

115

225

335

4

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(a)

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(b)

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(c)

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18Energy (eV)

3 64 50

051

152

Stre

ngth

func

tion

(1e

V)


(d)

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18Energy (eV)

25 35 45

02040608

Stre

ngth

func

tion

(1e

V)


(e)

05

101520253035

0 2 4 6 8 10 12 14 16 18

3 50

4

02040608

Energy (eV)

1

Stre

ngth

func

tion

(1e

V)


(f)

05

1015202530354045

0 2 4 6 8 10 12 14 16 18

3 50

4

1

2

Energy (eV)

Stre

ngth

func

tion

(1e

V)


(g)

0

10

20

30

40

50

0 2 4 6 8 10 12 14 16 18Energy (eV)

3 50

4

1

2

Stre

ngth

func

tion

(1e

V)


(h)

Figure 2 Continued


0

10

20

30

40

50

60

0

01234

2 4 6 8 10 12 14 16 18Energy (eV)

25 5535 45

Stre

ngth

func

tion

(1e

V)


(i)






mdash SiH4 81 80 [9] 80 [11] 854 87 [9] 82 [11] 945 96 [9] 91 [11]

06 Si5H12 55 57 [10] 56 [11] 67 65 [10] 64 [11] 812

08 Si17H36 40 547 672

10 Si29H36 35 42 48

11 Si35H36 343 37 44

12 Si47H60 34 41 505

14 Si66H74 33 372 493

15 Si71H84 32 381 598

16 Si87H76 30 478 672









4(silane) and Si



5H12


5H12were 57 eV





35H36






5 Conclusion


4and Si

5H12




Acknowledgments


References










4 GeH

4

Si2H6 and Si

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




005

115

225

335

4

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(a)

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(b)

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14 16 18Energy (eV)

Stre

ngth

func

tion

(1e

V)


(c)

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18Energy (eV)

3 64 50

051

152

Stre

ngth

func

tion

(1e

V)


(d)

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18Energy (eV)

25 35 45

02040608

Stre

ngth

func

tion

(1e

V)


(e)

05

101520253035

0 2 4 6 8 10 12 14 16 18

3 50

4

02040608

Energy (eV)

1

Stre

ngth

func

tion

(1e

V)


(f)

05

1015202530354045

0 2 4 6 8 10 12 14 16 18

3 50

4

1

2

Energy (eV)

Stre

ngth

func

tion

(1e

V)


(g)

0

10

20

30

40

50

0 2 4 6 8 10 12 14 16 18Energy (eV)

3 50

4

1

2

Stre

ngth

func

tion

(1e

V)


(h)

Figure 2 Continued


0

10

20

30

40

50

60

0

01234

2 4 6 8 10 12 14 16 18Energy (eV)

25 5535 45

Stre

ngth

func

tion

(1e

V)


(i)






mdash SiH4 81 80 [9] 80 [11] 854 87 [9] 82 [11] 945 96 [9] 91 [11]

06 Si5H12 55 57 [10] 56 [11] 67 65 [10] 64 [11] 812

08 Si17H36 40 547 672

10 Si29H36 35 42 48

11 Si35H36 343 37 44

12 Si47H60 34 41 505

14 Si66H74 33 372 493

15 Si71H84 32 381 598

16 Si87H76 30 478 672









4(silane) and Si



5H12


5H12were 57 eV





35H36






5 Conclusion


4and Si

5H12




Acknowledgments


References










4 GeH

4

Si2H6 and Si

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0

10

20

30

40

50

60

0

01234

2 4 6 8 10 12 14 16 18Energy (eV)

25 5535 45

Stre

ngth

func

tion

(1e

V)


(i)






mdash SiH4 81 80 [9] 80 [11] 854 87 [9] 82 [11] 945 96 [9] 91 [11]

06 Si5H12 55 57 [10] 56 [11] 67 65 [10] 64 [11] 812

08 Si17H36 40 547 672

10 Si29H36 35 42 48

11 Si35H36 343 37 44

12 Si47H60 34 41 505

14 Si66H74 33 372 493

15 Si71H84 32 381 598

16 Si87H76 30 478 672









4(silane) and Si



5H12


5H12were 57 eV





35H36






5 Conclusion


4and Si

5H12




Acknowledgments


References










4 GeH

4

Si2H6 and Si

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







4(silane) and Si



5H12


5H12were 57 eV





35H36






5 Conclusion


4and Si

5H12




Acknowledgments


References










4 GeH

4

Si2H6 and Si

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





5 Conclusion


4and Si

5H12




Acknowledgments


References










4 GeH

4

Si2H6 and Si

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials
















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



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