CHAPTER 2. PHOTOABSORPTION AND PHOTOELECTRIC PROCESS IN Si QUANTUM DOTS

Transcription

1 CHAPTER PHOTOABSORPTION AND PHOTOELECTRIC PROCESS IN S QUANTUM DOTS.1 INTRODUCTION At the nanometer sze o a matera comparabe to the characterstc de-broge waveength o the eectron, ts contnuous band structure s repaced by dscrete energy eves because o the quantum connement. Quantum wes and quantum wres are obtaned by the one-dmensona and the two-dmensona connement, respectvey. Among a the nanostructures the quantum dots are o partcuar nterest due to the threedmensona quantum connement eects. In terature semconductor quantum dots are aso reerred to as semconductor nanocrystas or nanopartces. The energy eves o semconductor quantum dots can be montored by controng the shape o the dot, the sze o the dot, the dot matera, and the connement potenta. In a semconductor nanocrysta embedded n a deectrc matrx the photoexcted charge carrers ocated n the potenta we are conned n a drecton and due to the nte sze o the quantum dot both the conducton and the vaence bands energy eves are uy quantzed. The absorpton spectra o a quantum dot show a seres o dscrete transtons whch are derent rom the spectra o the buk semconductor. Due to ther dscrete energy spectra quantum dots are aso termed as artca atoms. The eectronc transtons n the dscrete energy eves can be used to taor the optca propertes o the quantum dots. The observaton o the quantum connement n two dmensons was gven by Dnge et a. [], whereas the evdence o three-dmensona quantum connement was notced by Ekmov and Onushchenko [41] n nanocrystas o the cadmum sude and cadmum seende dspersed n a gass matrx. Subsequenty, severa expermenta [9-11] and theoretca [1,14-17] studes on the quantum connement were reported n the terature. The quantum connement eect n semconductor QDs has been expoted to desgn and deveop severa optoeectronc devces. Many potenta appcatons o quantum dots are reported e.g. photovotac ces [43-45], ght emttng dodes [46-47], snge eectron transstors [49-5], quantum dot asers [51]. Aso, there are 54

2 severa boogca appcatons o quantum dots such as n ceuar magng and abeng [5-54]. Due to ts nontoxc and natura abundance, scon s a most wdey used matera n makng derent devces o semconductor eectroncs, optoeectroncs, photovotacs [55]. The nanostructure scon quantum dots are potenta canddate or abrcatng the devces o the next generaton photovotacs ke tandem soar ces, detectors, and snge eectron transstors. Severa expermenta studes have been reported n the terature usng both the bottom up and the top down approaches or abrcatng scon quantum dots. The bottom up approaches ncude gas evaporaton [1-], sputterng [3-5], reactve evaporaton [6], chemca vapor deposton [7-3], on mpantaton [33-35], aser abaton [36-37], souton synthess [38] etc. The top down approaches ncude mechanca mng [39], eectrochemca etchng [4] etc. In severa studes abrcaton o the scon nano partces o derent szes are reported and photoumnescence optca spectra are measured to study the quantum connement eects [9-17,56-57]. The detaed revew on the recent deveopments n studes on photoumnescence rom scon nanostructures s gven by Saar [57]. Most o the expermenta studes suggest the orgn o the umnescence n quantum dots as nterband transtons or the exctonc recombnaton. Both type o charge carrers (eectron and hoe) are nvoved n nterband transtons and take pace between the vaence and conducton band. On njectng an eectron to a quantum dot, ntraband transtons n the conducton band w be mportant as these transtons are mosty n the nrared regon. In an ntraband transton ony one type o charge carrer s nvoved and aows one to separatey study the dynamcs o the eectrons and hoes. The studes on the ntraband transtons n nanostructures are requred n many potenta appcatons n optoeectroncs and nanoeectronc devces such as the quantum dot nrared photodetector [163] and quantum cascade asers [154]. Severa studes o ntersubband transtons n the quantum we and the quantum dots have been reported n the terature [ ]. West and Egash [15] reported the rst observaton o ntersubband transtons n square quantum we by nrared spectroscopy. Sauvage and Boucaud [164] reported the pane poarzed ntraband absorpton n InAs/GaAs seassembed quantum dots. 55

3 Theoretcay, severa studes on the S quantum dot embedded n SO matrx are reported n the terature [76,136,167,169-17,59-64]. See et a. [76] used the densty unctona theory and the Hartree treatment to cacuate scon nanocrysta energy states as a uncton o the number o eectrons n the quantum dot. de Sousa et a. [169-17] nvestgated the ntraband absorpton n S/SO quantum dot usng a trdmensona quantum mechanca mode. Buutay [59] used an atomstc pseudopotenta approach to study the photon absorpton n scon nanocrystas. Aan and Deerue [167] studed the ecent ntraband optca transtons n n-type scon nanocrystas usng tght-bndng cacuatons. Krap et a. [166] used the photo-nduced nrared absorpton spectroscopy o porous scon to study the sze quantzaton and ntersubeve transtons n the conducton band o the S nano-crystates. Not much work has been reported n the terature on the photoeectrc process n S quantum dot. In ths chapter, we present a study on the photoabsorpton and the photoeectrc process o sngy charged sphercay symmetrc S semconductor quantum dot embedded n amorphous SO matrx, usng the eectve mass approxmaton (EMA) [65]. We consder the barrer heght at the nterace o the dot matera and the matrx matera as nnte as we as nte. We aso ncude the eect o se energy assocated wth the poarzaton due to the chargng o the dot [144]. Resuts are obtaned or the photoabsorpton coecents, and the photoeectrc cross-secton and ther varaton wth dot sze. We compare our resuts wth the expermenta data o Krap et a. [166,6] n the photon energy range 14 mev-175 mev.. OPTICAL TRANSITIONS IN SEMICONDUCTOR QUANTUM DOTS We consder a sngy charged spherca S QD embedded n a matrx. The quantum conned waveunctons o an eectron n the dot are descrbed by ψ ( r ) = U ( r ) φ ( r ), b where Ub ( r ) represents the Boch uncton at zone center or band b and φ ( r ) the enveope uncton. The probabty o an optca transton rom an nta conned state to a na state by the absorpton o one-photon (OPA) s governed by the square o the optca matrx eement o the momentum operator p between these states [144], ψ p ψ U p U φ φ + U U φ p φ, (.1) b b b b 56

4 The ntraband transtons n the dot occur between the conned eectron states n the same band ( b = b ) wth the same Boch uncton ( U = U ) and a derent enveope uncton ( φ φ ). Ths gves U p U = and U U = 1, and ψ p ψ = φ p φ. b b b b b b nt raband Thus or ntraband transtons the optca matrx eement s determned by the enveope part o the waveuncton. The party seecton rue and conservaton o anguar momentum ( = ± 1) appy to the enveope uncton [144]. The nterband transtons n the dot occur between states o derent bands (vaence band b and conducton band reduces to the rst term n equaton (.1). ψ p ψ = U p U φ φ nt erband b b b ). The optca matrx eement or nterband transtons, (.) The seecton rues are determned by the eement φ φ whch requre that the enveope uncton o the states and have the same party and the same orbta anguar momentum. Thus the nterband optca transtons can occur ony between states havng the same quantum numbers..3 WAVE FUNCTIONS AND ENERGY LEVELS The Schrödnger equaton or the enveope uncton φ or an eectron n a spherca quantum dot o scon o radus R s gven by, ħ 1 ([ m] ) + Ve ( r) φ( r) = Eφ( r) 57 (.3) where V ( r) = V ( r) + (.4) e con where [ m] s the eectve mass tensor [17,66]. Foowng See et a. [76,15,63-64], Barber [66], and Conbeer et a. [44] we repace the eectve mass tensor by the scaar eectve mass whch s taken as harmonc mean o the transverse and the ongtudna masses as, m 1 S 3(1/ m L + / m T ) =, wth ml =.9m and mt =.19m. m s the mass o ree eectron. The above representaton or the sotropc conducton band eectve mass (wth each conducton band vaey gvng rse to two od degeneracy n transverse drecton) has been ound sutabe or descrbng the spherca S nanocrystas o radus arger than 1.5 nm. Comparson between the EMA and the tght bndng approxmaton shows that the EMA s sutabe [15] to descrbe S quantum dots.

5 In equaton (.4) Σ s the se energy assocated wth the chargng o the quantum dot wth an eectron. Ths charged partce (eectron) w poarze the deectrc and nduce poarzaton charges. The se energy represents the nteracton energy between ths charged partce and the poarzaton charges. The se-energy Σ s gven as [144], Σ = e ε n ε πε R ε [ ε + ε 8 ε n and n n η ] η (.5) ε are the deectrc constants o the dot matera and the surroundng matrx matera, respectvey, and ε s the permttvty o the ree space. η ε ε + ε =, or 1 n η <, the se energy s gven by [144,67], e Σ = + ε εn 4πε R δ Σ, (.6) where, e εn ε δσ πε εnr εn + ε We consder the barrer potenta V con n equaton (.4) as o nnte heght, and o nte heght..3.1 THE INFINITE BARRIER For the nnte barrer the connement potenta s, V ( r ) = or r < R e and Ve ( r ) = or r R, (.7) R s the dot radus. The waveuncton φ( r ) ξ ( r) Y m ( rˆ ) = n where ξ ( r) s the rada waveuncton and n equaton (.3) wth V ( r ) o equaton (.7) gves, χnr ξ n = Aj ( ) R e m Y s a spherca harmonc. The souton o 58

6 where, A s the normazaton constant and χ n s the n th zero o the spherca Besse uncton j (z). The energes o the quantum eves n the dot are gven by, E n = ħ m R χ S n (.8) The energy o the dscrete eves n the dot decreases wth the ncrease o dot sze as nverse square o radus. Some o the energy eves o a dot and vaues o the zeros o the spherca Besse unctons χn are gven n Tabe.1. Tabe.1: Vaues o χn or some ower states. Leve χ n 1s p d s p THE FINITE BARRIER For the nte barrer we represent the eectve potenta as, V ( r ) = or r < R con V ( r) = V or r R (.9) con V s barrer potenta at the nterace o the dot and the surroundng matrx. The souton o equaton (.3) usng equatons (.5-.6) and (.9) gves, φ( r ) ξ ( r) Y m ( rˆ ), (.1) = n wth the rada part o the wave uncton ξ ( r) gven by, ξn( r) = Aj ( αr) or r < R and n ξn( r) = Bk ( βr) or r R (.11) 59

7 A and B are the normazaton constants and j and k are the spherca Besse and the moded spherca Besse unctons, respectvey. The moded spherca Besse uncton k ( z) wth z = β r s expressed as [68], 1 k( z) = e ( +, s)( z), wth are, π z s z s= 1 ( + s)! ( +, s) =. The parameters α and β s! Γ( s + 1) α and ms ( En ) = ħ V ms β = γ r α ħ wth the eectve mass rato, m γ r = (.1) m SO S From the contnuty o waveunctons at the dot nterace, the normazaton coecent B s obtaned as, B Aj ( α R), and rom the normazaton o the waveuncton the k ( β R) = coecent A s obtaned as, A T T 1/ = ( 1 + ) (.13) R j ( αr) wth T1 = [ j ( αr)] r dr, and T = [ k ( β r)] r dr k ( βr) R Appyng the boundary condton on the dervatves o waveuncton at the dot nterace at r = R one obtans the oowng equaton, 1 j ( α R) j ( α R) k = ( β R) (.14) γ k ( β R) r where j ( r) = j 1( r) j ( r) and k ( r) = k 1( r) k ( r) r r Usng the above reaton n equaton (.14), one obtans the oowng equaton, j 1 α R k 1 β R + γ rα β γ r j ( α R) k( β R) R ( ) ( ) ( 1) = + ( 1) The parameter α s obtaned by sovng the above transcendenta equaton numercay. (.15) 6

8 We consder the connement potenta [76] V o heght 3.1 ev. The ncuson o the se energy sghty ncreases the barrer heght. The eectve mass o the eectron n the S dot and the SO matrx are taken as m S =.7m and m SO =.5m, respectvey. Ths eads to the ratoγ r = The deectrc constants [76] o S and SO are taken as ε n = 11.7 and ε = 3. 8, respectvey. The eectve Bohr radus a ε a = or the S dot n ms s obtaned as Ht ms a =.9nm and the reduced Hartree energy Ht = s obtaned as ε n Ht = 53.67meV..4 PHOTOABSORPTION PROCESS In the study o the optca propertes o materas, the photoabsorpton and the photoumnescence are the basc processes. In the absorpton process a monochromatc radaton o ntensty I s ncdent on the sampe surace and I t s the transmtted radaton ntensty. Insde the nanocrystas the photons are absorbed and the nteracton o the eectromagnetc ed wth the eectron nduces the transton o eectron rom a nta state to a hgher energy state. Such transtons can be treated usng semcassca approach. The eectromagnetc waveengths are much arger compared to the sze o the nanocrysta, and the eectrc dpoe approxmaton woud be vad. The eectrc ed nsde the nanocrysta E n s derent rom the eectrc ed E n the medum surroundng the nanocrysta. The two eds are reated as En = FE, where F s caed the oca ed actor. For spherca nanocrystas the oca ed actor s [144], F = ε /( ε + ε ) (.16) where 3 n respectvey. ε n and ε are the deectrc constants o the nanocrysta and the host matera, Insde the crysta, the oca eectrc ed causes eectronc transton and poarzes the medum by nduced dpoes. The expectaton vaue o the nduced dpoe moment (1) tr( ) µ = ρ µ, where ρ s the densty matrx and µ the dpoe moment. The poarzaton P = N µ where N s the nanocrysta densty. 61

9 E and Consderng a nanocrysta wth ony two eectronc eves ( and wth energes E, respectvey), and usng ρ rom equaton (1.94), the poarzaton n the t nanocrysta s obtaned as, P eµ ( ρ ρ ) e ω = + Ω, where Ω s the voume o nanocrysta. Usng equaton (1.94) o chapter 1, the poarzaton s, µ µ. E( ω) e t N ω () () ( ρ ρ ) ε ħ ω ω γ P = where γ s the newdth. ( ) (.17) P = ε χ ω ω (1) ( ) E( ) (1) wth χ, the near susceptbty, gven by, N χ ( ω) = ( ρ ρ ) ħ µ µ (1) () () ε ω ω γ ( ) The poarzaton can aso be expressed as, P( ω) = ε ( ε 1) E( ω). n (.18) The near deectrc constant ε (1) ( ω ) s reated to the near susceptbty as, ( ) = 1 + ( ) (1) (1) ε ω χ ω The near reractve ndex s reated to the near deectrc constant n 1 1 (1) (1) (1) (1) ( ω) = ε ( ω) 1 + χ ( ω) = 1 + ( χ + χ ). (1) ε as where (1) χ and (1) χ are rea and magnary parts o the near susceptbty. Here n( ω) = n ( ω) + n ( ω), where n and n are the rea and magnary parts o reractve ndex. The ntensty o the eectromagnetc wave n the medum vares as I = Ie α z, where α s the absorpton coecent o the matera system. It s dened as, ω (1) α( ω) = n ( ω) = ( ω c) χ ( ω) c Usng the vaue o gven by [144,69], (1) χ, the photo nduced absorpton coecent o the nanocrysta s πωe α ω ψ ε ψ δ ħ ω (.19) ( ) = pf. ˆ r ( E E ) cnrε Ω wth the deta uncton as [144], 6

10 ħγ δ ( E E ħω) = π ω γ and [( E E ħ ) + ( ħ ) ] are popuaton n the states and, respectvey, p s the voume racton o the nanocrystas n the medum. n r s the reractve ndex, c s the veocty o ght, ε s the permttvty o ree space. ˆε s the unt poarzaton vector o the ncdent neary poarzed photon, and ω s the requency o the ncdent photon..5 PHOTOELECTRIC PROCESS In ths secton we consder the process o the ejecton o eectron rom the dot matera under the photon mpact.e. the photoeectrc process, ħ ω + D D + e ( k ) (.) D and D denote the neutra dot and the charged dot (n the nta quantum state ), respectvey. k s the wave vector o the eectron ejected rom the dot and ω s the requency o the ncdent photon. The transton rate or the photoeectrc process (equaton.) s gven as, π Wk = µ ke ρ ( ε k ) ħ µ = e ψ ˆ. ε r ψ k k (.1) The eectrc ed nsde the dot E FE =, where E s the ed sde the dot. The ntensty o ncdent ght beam ε I = n c E. ρ( ε k ) s the densty o states gven by, L m k ρ ( ε k ) = d Ω π ħ where 3 dω = snθ dθ dφ k k k k k (.) For the deta uncton normazed pane wave, ρ( ε k ) s obtaned by repacng L by π. The wave vector o the ejected eectron k s gven by, k m = [ ħω Φ] ħ (.3) 63

11 Φ s the energy requred to eject the eectron rom the dot and s dened as Φ = χa E, where χ a s the eectron anty o the buk S matera. E s the bound-state energy o the nta state o the dot n the conducton band. ψ and ψ k are the waveunctons o the bound and the ejected eectron, respectvey. We represent the ejected eectron wth a deta-uncton normazed pane wave. The consderaton o the ejected eectron as a pane wave s reasonabe as ater the ejecton o the eectron rom the dot, a neutrazed dot s et [69]. 1 ψ k ( r ) = e 3 ( π ) k. r The transton rate W can be expressed as, Wk (.4) = σ I (.5) The derenta cross-secton or the photoeectrc process (equaton.) s obtaned as, dσ 4π = F m kα sω M dω k ħ wth. M = e k r ˆ. ε r ψ k k (.6) (.7) Consderng the dot n the nta ns state ψ = ξns ( r) Y ( r) and ncdent photon to be neary poarzed, ˆ 3 k1s 1 ( ) 1( ) ξns ( ) 3π M = Y k j kr r r dr (.8) j ( kr ) s the spherca Besse uncton. The tota photoeectrc cross-secton ˆ σ T or the ejecton o a photoeectron by photon mpact rom an nta bound state o the dot s obtaned by ntegratng equaton (.6) over a the ejected eectron ange as, dσ σ T = snθkdθkdφk (.9) dω k.6 RESULTS AND DISCUSSION Fgure.1 shows the varaton o energy o conned energy states wth dot radus n the conducton band or nte and nnte barrer potentas. The ground-state s desgnated as 1s or ( n = 1, = ) and the excted states as 1p or ( n = 1, = 1), s 64

12 or ( n =, = ), and p or ( n =, = 1). It s seen that or a the states the energy decreases wth ncrease o radus. We notce that the energy vaues o conned states are hgher or nnte barrer potenta as compared to nte barrer potenta. At arge rad both the potenta yed equa vaues o energy or conned states. Fgure.1: The varaton o the energy or the ground (1s) and excted (1p, s, p) states. Fgure. shows the varaton o ISL transton energy o the dot wth the radus o the dot or the owest two subeves ( E1 E1 ) n the conducton band or nte and p nnte barrers. It s observed that ths ISL energy derence at a rad s hgher or nnte barrer as compared to the nte barrer. Ths woud mean that the peak photoabsorpton energy o the dot at a gven radus w be hgher n the case o the nnte barrer as compared to the nte barrer. It s notced that the ISL energy derence n the conducton band or nte and nnte barrers decreases wth ncreases o dot radus. s 65

13 Fgure.: The Varaton o energy derence ( E1 p E1 s ) as a uncton o dot radus R or nte and nnte barrer heght. Krap et a. [166,6] have observed expermentay our photoabsorpton peaks at 14.6 mev, mev, 16.9 mev, and 17.8 mev n the energy range between 14 mev 175 mev. From the Fgure we estmate the radus o the dot correspondng to these our peaks as (R = Å, 8.9 Å, 7.7 Å, and 6. Å) or nte and (R = 34.6 Å, Å, 9.96 Å, and 8.9 Å) or nnte barrer heghts. Fgures.3 and.4 show the varaton o rada part o 1s, 1p, s, and p states n the conducton band o the sngy-charged spherca quantum dot o radus R = 3 Å or nnte and nte barrers, respectvey. 66

14 Fgure.3: The varaton o rada waveunctons or nnte barrer as a uncton o rada dstance r or the dot radus R= 3 Å. Fgure.4: The varaton o rada waveunctons or nte barrer as a uncton o rada dstance r or the dot radus R= 3 Å. 67

15 In Fgure.5 we present our resuts o the photoabsorpton coecentα ; (obtaned by usng equaton.19) or the two cases o nte ( ) and nnte (.) barrer heghts. We take the newdth γ =1 mev [7]. The varaton o the photoabsorpton coecent wth the ncdent photon energy or ISL transtons n the conducton band rom nta 1s state o the conducton band s shown n Fgure.5 at our vaues o dot rad. In ths Fgure we compare our resuts o the photoabsorpton coecent wth the expermenta data o Krap et a. [166,6] or the ISL transtons between the two owest subeves 1s and 1p n conducton band. Usng the photo nduced nrared absorpton spectroscopy, Krap et a. [166] measured the derenta transmsson ( T / T) rom whch the photo nduced absorpton coecent α s derved usng the reaton T T = α. d 1.. e α d Fgure.5: The varaton o the photoabsorpton coecent α or the (1s-1p) transton as a uncton o the ncdent photon energy at derent vaues o the dot radus R., Fnte barrer heght a(r = 6. Å), b(r = 7.7 Å), c(r = 8.9 Å), d(r = Å);.. Innte barrer heght e(r = 8.9 Å), (R = 9.96 Å), g(r = Å), h(r = 34.6 Å);, Expermenta data o Krap et a. [166,6]. 68

16 The thckness d can be correated wth the dameter D o the dot. We nd that our cacuated vaues o the photoabsorpton coecent α agree very we wth the expermenta data at a our rad o the dot. Our resuts everywhere are wth n a actor o two agreements wth the expermenta vaues. Fgure.6 shows the varaton o the photoabsorpton coecent α or the (s-p) transton wth the ncdent photon energy usng the same rad as n Fgure.5. It s ound that the absorpton peak postons or the (s-p) transton are obtaned at energes o mev, mev, 93. mev, and mev, respectvey. These postons are reatvey at ab two tme s energy vaues as compared to (1s-1p) transton. The magntude o the absorpton coecent or the (s-p) transton s hgher as compared to the (1s-1p) transton. Aso t s observed the peak poston o the photoabsorpton coecent α shows a red sht wth ncreasng dot radus. Ths s because the ISL energy derence between the conned states decreases wth the ncrease n the dot radus. Fgure.6: The varaton o the photoabsorpton coecent α or the (s-p) transton as a uncton o ncdent photon energy at derent vaues o the dot radus R. Captons are same as n Fgure.5. 69

17 Fgure.7 shows the varaton o the photoeectrc cross-secton (equaton.9) wth the ncdent photon energy or the dot rom ts nta 1s state usng the same rad as n Fgure.5. It s ound that the photoeectrc cross-secton decreases on ncreasng the ncdent photon energy at a xed dot radus. On ncreasng the dot radus the cross-sectons decreases sharpy wth n the ncdent photon energy range (4.35 ev to 4.55 ev). It s observed that the photoeectrc cross-secton obtaned usng the nnte barrers are hgher compared to the nte barrer. Fgure.7: The varaton o the tota photoeectrc cross-secton same as n Fgure.5. σ T rom the 1s state. Captons are Fgure.8 shows the varaton o photoeectrc cross-secton o the dot rom ts nta s state at the same dot rad. The nature o varaton wth the ncdent photon energy and dot radus n the photoeectrc cross-sectons s smar to the nta 1s state. 7

18 Fgure.8: The varaton o the tota photoeectrc cross-secton n Fgure.5. σ T rom the s state. Captons are same as.7 CONCLUSIONS We concude that the present theoretca study provdes a good agreement wth the avaabe expermenta resuts or optca transtons n conducton band. Snce these transtons are many n the nrared regon ths study w be useu n the deveopment o nrared devces usng quantum dots (e.g. nrared detectors). Further, the study on the photoeectrc process n S quantum dot w be useu n the deveopment o photovotac devces (soar ce wth an array o S quantum dots). 71