Absorption spectra of CdSe–ZnS core–shell quantum dots at highphoton energies: Experiment and modeling

Amlan Mukherjee, Sandip Ghosh n

Department of Condensed Matter Physics and Materials Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India

H I G H L I G H T S

� CdSe–ZnS core–shell Quantum Dotsshow relatively high absorption athigh energies.

� QD potential and shape asymmetrylead to normally disallowed (ND)transitions.

� Such ND transitions can account forthe high absorption at high photonenergies.

G R A P H I C A L A B S T R A C T

We show that to explain the measured high absorption coefficient of CdSe–ZnS core–shell quantum dotsat energies � 1 eV above their effective bandgap, it is necessary to consider the role of normallydisallowed transitions. We provide a simple model under effective mass approximation to account forsuch transitions.

a r t i c l e i n f o

Article history:Received 27 March 2014Received in revised form31 July 2014Accepted 1 August 2014Available online 8 August 2014

Keywords:CdSe–ZnS quantum dotAbsorption spectrum modelingPhotoluminescence excitation spectroscopy

a b s t r a c t

Absorption spectra of CdSe–ZnS core–shell quantum dot (QD) ensembles, with average core diametersranging from 2.6 nm to 7.2 nm have been obtained using both transmission and photoluminescenceexcitation measurements. In agreement with previous reports, the absorption coefficient at energiesC1 eV above the effective bandgap increases monotonically as in bulk solids. A simple effective-massspherical core–shell potential model cannot explain the relatively high absorption at higher energies.The calculated electron and hole radial envelope wavefunctions show asymmetry due to the core–shellstructure. It leads to normally symmetry-disallowed transitions acquiring a weak oscillator strength,with their number and strength increasing with energy. A phenomenological model that invokesnormally disallowed transitions in general is shown to reproduce the absorption spectrum at higherenergies quite well. The oscillator strength scaling factor for such transitions increases with decrease inQD size, consistent with expectations.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Solution synthesized group II–VI semiconductor quantum dots(QDs) have in recent times found application in areas ranging frommulticolored luminescence imaging of biological tissues [1,2]to flat-panel electronic display screens [3,4]. The basis of these

applications is the size-dependent quantum-confinement relatedtunable effective bandgap of these QDs which allows for controlover emission wavelength. A majority of studies on such QDsare therefore concerned with understanding light emission andabsorption properties around the effective bandgap. Howeverfor luminescence imaging applications it is also important toknow the absorption spectrum at higher energies away from theeffective bandgap, since it determines the choice of pump beamcharacteristics such as wavelength and intensity. In anotherpromising medical application called photodynamic therapy, such

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Physica E

http://dx.doi.org/10.1016/j.physe.2014.08.0041386-9477/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author.E-mail address: sangho10@tifr.res.in (S. Ghosh).

Physica E 64 (2014) 234–239

QDs are selectively directed to malignant tissue and excited withlight, thereafter the absorbed energy is transferred to eitheractivate a drug or directly help the kill the malignant tissues [5].For such applications too, estimating the absorption spectrum athigher energies becomes important.

The absorption spectrum of an ensemble of such QDs, with anarrow distribution of core diameters, show clear peaked featuresaround the effective bandgap. These features can be associatedwith inhomogeneously broadened transitions involving quantumconfined electron and hole levels. However with increase inphoton energy the peaked features gradually loose prominenceand eventually the absorption spectrum becomes a featurelessmonotonically increasing function, resembling absorption in bulkmaterial [6–8]. In an early study [9] this absorption was taken tobe a background and simply fitted using a cubic equation. Somestudies have used bulk optical constants to model the measuredabsorption spectrum of the QDs to especially account for this bulklike nature of the absorption spectra at higher photon energies[10]. It was suggested that light extinction at high photon energiescan be explained through the phenomenon of light scattering bysmall absorbing particles but this model cannot explain why theextinction was found to be independent of the refractive index ofthe surrounding medium [11]. It is generally understood that thehigh spectral density of inhomogeneously broadened transitionsin the QDs at higher photon energies is responsible for theabsorption spectra imitating that of bulk solids [12]. However weshall show that transitions that are strongly allowed by selectionrules cannot by themselves account for the relative increase in theabsorption strength at energies C1 eV above the effective band-gap. The main motivation of the present work is to obtain a simpleempirical model for describing the absorption spectra of such QDs,especially the increased bulk-like absorption, at higher photonenergies. In general empirical modeling, for example for determin-ing the size dependence of bandgaps [10], is practically quiteuseful in the absence of exact and detailed knowledge of the shapeand structure of the QDs in an ensemble. We will also discuss thephysical logic behind adopting our model and compare thepresent case with an example from semiconductor quantum well(QW) systems.

2. Experimental details and results

In our experiments we used commercially available solutionsynthesized CdSe–ZnS core–shell QDs. Four QD ensembles werestudied where the average CdSe core diameters do were 7.2 nm,5.1 nm, 3.0 nm and 2.6 nm, with a ZnS shell which was about fourmonolayer thick. The samples were characterized using highresolution transmission electron microscopy (HR-TEM). The QDshad carboxyl molecules coated on the surface to facilitate ligandconjugation and which also helped prevent coagulation in aqu-eous solution. The absorption (Abs) spectra were first obtainedthrough transmission measurements using a conventional setup.

Fig. 1(a)–(d) shows the measured room temperature Absspectrum of the QDs along with their photoluminescence (PL)spectrum. The PL spectrum and the QD size agree well with earlierreports [13]. The Abs spectrum measured with the QDs dispersedin water with 1 μM concentration and in the form of a coated filmon sapphire substrate were similar. Also shown in Fig. 1(a)–(d) arethe simulated Abs spectrum which considered only stronglyallowed transitions, the details of which will be presented in thenext section. Both the measured and the simulated Abs spectrahave peaked features at lower energies which gradually vanishgiving rise to a monotonically increasing bulk-like Abs at higherenergies. The simulated Abs spectrum was scaled to mach thelowest energy peak of the measured Abs spectrum. One notices

that while the measured and the simulated Abs spectrum magni-tudes fairly match at low energies, at higher energies the mea-sured Abs magnitude is larger by factors � 2 or more. Thisdiscrepancy, which is the main thrust of our study, becomes moreand starts to arise closer to the lowest energy Abs peak as the QDsize reduces. It has been suggested that the high Abs strengthmeasured using transmission spectroscopy could be a conse-quence of the probe light getting scattered [11] by the QDs andnot getting detected. A way to test this would be by measuring aphoto-luminescence excitation (PLE) spectrum which is expectedto mimic the true Abs spectrum over a certain energy range andnot be modified significantly by light scattering.

For the PLE measurements light from a 75W Xenon arc lampdispersed by a 0.125 m focal length monochromator (MC1) withbandpass 3.2 nm was used as the pump beam. The luminescenceoutput was dispersed using a 0.5 m focal length monochromator(MC2) with bandpass 0.8 nm and detected with a thermo-electrically cooled Si charge coupled device. For normalizing thePLE data the excitation intensity was separately measured using apyroelectric detector. Fig. 2 shows an example of PLE data on QDswith do ¼ 7:2 nm, measured using the above conventional setup.The PLE signal detectionwas at 1.837 eV and the large signal seen at

Energy (eV)2.0 2.5 3.0 3.5

Abs

. coe

ff.,

PL si

gnal

(arb

. uni

ts)

Wavelength (nm)

x 1/2

x 1/2

PL

Abs Meas

Abs Sim

700 600 500 400 350450550

e-hAe-hBe-hC

x 1/3

Fig. 1. Measured photoluminescence (PL), measured and simulated absorption(Abs) spectra of CdSe–ZnS core–shell QD ensembles with four different averagecore diameters do. The PL magnitudes are scaled to identical peak value, similarlylowest energy Abs peak are scaled to identical value. The simulation is based on amodel which considered only the strongly allowed transitions. The vertical linesindicate the position and relative strength of the strongly allowed transitionsinvolving confined electron levels and confined hole levels corresponding to thethree A, B and C valance bands in CdSe. (a) do¼7.2 nm, (b) do¼5.1 nm, (c) do¼3.0nm and (d) do¼2.6 nm

A. Mukherjee, S. Ghosh / Physica E 64 (2014) 234–239 235

this energy is the scattered pump beam leaking through. Howeverthere are two additional sharp features at � 1:87 eV and � 1:9 eVmarked by arrows. Their intensities are � 104 times lower than themain scattered pump beam signal. Their linewidths are � 9 meVwhich is much smaller than kBT � 26 meV and are determined bythe overall bandpass of the measurement system. Given that theemission linewidths of single QDs at room temperature [14] are� 35 meV, it is very surprising to find such features in the PLEspectrum. Fine structures very close to the pump beam energy havebeen reported before and their origins have been attributed toexchange-interaction split exciton levels and phonon emission[15,16]. However there are optics related coma aberrations inmonochromators which can give rise to multiple images of theinput slit at the exit slit plane, although with significantly reducedintensity. This can translate into weak replicas of the scatteredpump beam being observed close to the main beam but with verylow intensity. That this may be occurring here is evident from thepresence of such a feature at 1.935 eV in a conventional PL spectrumof the sample shown in the inset of Fig. 2. Here the pump beamenergy (1.968 eV) was chosen to be close to the PL emission peak sothat both the PL and the scattered pump beam are simultaneouslyseen in the spectrum. Use of a double monochromator for MC2 canminimize the scattered pump beam. We have adopted a differentapproach shown schematically as an inset in Fig. 2. Here the pumpbeamwas linearly polarized using a Glan–Taylor polarizer P1 and inthe detection path another polarizer P2 was placed with its pass-axis perpendicular to that of P1. Since the scattered pump beamwillmostly be polarized along the pass-axis of P1, it will be significantlycut off by P2. The QDs are randomly oriented, that apart during thecarrier relaxation process the memory of the incident pumpexcitation polarization is mostly lost and consequently the emissionis expected to be unpolarized. Thus P2 will only reduce the PL signalby a factor little more than 2. This way the strength of the scatteredpump beam relative to the PL signal can be significantly reduced.Fig. 2 also shows results from such a polarized PLE measurementand indeed we no longer see these extra sharp features seen earlier.

Fig. 3(a) and (b) compares the Abs spectrum and the polarizedPLE spectrum of QD samples with average diameter 3.0 nm and2.6 nm. It is true that the PLE spectra of larger QDs deviate fromthe Abs spectra at higher energies due to non-radiative carrier loss

mechanisms [17] however one can see from these plots that in thesmaller QDs, where light scattering is expected to be strong, thereis not much difference between the PLE and the Abs spectrum upto fairly high energies above the effective bandgap. This suggeststhat the relatively stronger measured Abs at higher energies is anintrinsic property of the QDs. We will next describe the modelused to account for this increased Abs at high energies.

3. Absorption modeling and discussion

CdSe is a direct band gap semiconductor with wurtzite crystalstructure. At the Brillouin zone center it has three closely spacedvalence bands (VB) conventionally labeled as A, B and C bands withgaps relative to the conduction band (CB) being EgAoEgBoEgCrespectively. The increased Abs we see here cannot be due totransitions involving CdSe energy bands at higher energies, sincethe next direct inter-band transition after the A, B and C transitionsoccurs beyond 4 eV. Also the increased Abs cannot be due to the ZnSshell since the band gap of hexagonal ZnS is beyond � 3:8 eV. Wewill therefore consider three sets of transitions, one from each of theabove VB to the CB. To calculate the Abs spectra we first need todetermine the energy spectrum of the confined electron and holes inthe QDs. To be exact one may need to account for not only quantumconfinement but also consider aspects such as coulomb interactions,summation of angular momentum quantum numbers associatedwith the confinement potential and the bands and its consequencefor transition selection rules, finite crystal momentum due toconfinement and VB mixing, and also exchange interactions [18].Such a calculation requires several parameters with associateduncertainties, in addition it can be computationally challenging ifthe QD shapes are not simple. These aspects determine the fine-structure of the spectrum of individual QDs but for practicallyrelevant size distribution the effect of these phenomenon is unlikelyto be distinguishable and possibly cannot account for the relativeincrease in Abs at high energies.

To obtain the energy spectrum we adopt the effective-massenvelope wavefunction approximation and solve the Schrödingerequation for a spherically symmetric core–shell potential [13,19].The full wavefunction is expressed as a product of the unit-cell-periodic part and an envelope wavefunction. The radial depen-dence of the confinement potential is given by

VðrÞ ¼0 for 0rrrrcVo for rcorrrs1 for r4rs

8><>: ð1Þ

where rc is the radius of the CdSe core and rs is the outer radius ofthe ZnS shell (rs4rc) and Vo is the height of potential barrier

Energy (eV)1.8 1.9 2.0 2.1 2.2

PLE

Sign

al (a

rb. u

nits

)

Energy (eV)1.82 1.87 1.92 1.97

PL (a

rb. u

nits

)

Convnl measPol meas

MC2

CC

D

Xe Lmp

P2P1

MC1

Sample

Fig. 2. Photoluminescence excitation (PLE) spectrum of the QD ensemble withdo ¼ 7:2 nm at room temperature. The inset plot is the photoluminescence (PL)spectrum of the sample with excitation at 1.968 eV. Signals detected in bothconventional and polarized measurement modes are shown. A schematic of thepolarized PLE measurement mode is shown. P1 and P2 are crossed polarizers, MC1and MC2 are the excitation end and detection end monochromators.

Energy (eV)Energy (eV)2.50 3.00 3.502.50 3.00 3.50

Sign

al (a

rb. u

nits

)

PLE

Abs x 1/2

Fig. 3. The measured room temperature absorption (Abs) and photoluminescenceexcitation (PLE) spectrum of QD ensembles having average core diameter do(a) 3.0 nm and (b) 2.6 nm. The PLE signal detection energy was � 25 meV belowthe start of the plotted PLE spectrum.

A. Mukherjee, S. Ghosh / Physica E 64 (2014) 234–239236

between the core and the shell region. For electrons Vo ¼ fΔEgA,where ΔEgA is the difference between fundamental bandgap ofCdSe and ZnS, while f is the CdSe/ZnS CB offset ratio. For holes inthe A band of CdSe Vo ¼ ð1� f ÞΔEgA and for holes in the other twobands Vo is determined by considering the energy position of thesebands relative to the A band. The spherical symmetry allows us towrite the envelope wavefunction as a product Ylmðθ;ϕÞχnlðrÞ,where Ylmðθ;ϕÞ are the spherical harmonics [20] containing theangular dependencies and χnlðrÞ the solution of the radial part ofthe Schrödinger equation

1χnl

ddr

r2dχnl

dr

� ��2mnr2

ℏ2 ½VðrÞ�Enl� ¼ lðlþ1Þ ð2Þ

where mn is the carrier effective mass. Each confined state isdefined by three eigen numbers l, m, n and the correspondingeigenvalues Enl are the required confinement energies. To obtainEnl and χnlðrÞ one begins by writing the solutions in the core andthe shell regions as

χðrÞ ¼

¼ a jlðk1rÞ if 0rrorc¼ b ilðk2rÞþc κlðk2rÞ if rcorors; 0rErVo

¼ d jlðk2rÞþe ηlðk2rÞ if rcorors; VooEo1¼ 0 if rZrs

8>>><>>>:

ð3Þ

k1 ¼ffiffiffiffiffiffiffiffiffiffiffiffi2mn

CE

ℏ2

s; k2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mn

S jVo�Ejℏ2

s

Here jlðk1rÞ;ηlðk2rÞ are lth order spherical Bessel functions of 1stand 2nd kind, while ilðk2rÞ; κlðk2rÞ are lth order modified sphericalBessel functions of 1st and 2nd kind. [20] mn

C; S are the carriereffective masses in the core and the shell regions. The constantsa�e need to be determined using boundary conditions whichimpose continuity of χðrÞ and 1=mn d=dr½χr� at r¼ rc and vanishingof χðrÞ at r¼ rs. Application of these boundary conditions leads tothe following transcendental equations where 0 represents firstderivative with respect to r

jlðk1rcÞ ¼k1mn

S

k2mn

Cj0lðk1rcÞ

ilðk2rcÞκlðk2rsÞ� ilðk2rsÞκlðk2rcÞi0lðk2rcÞκlðk2rsÞ� ilðk2rsÞκ0lðk2rcÞ

" #0rErVojlðk1rcÞ

¼ k1mn

S

k2mn

Cj0lðk1rcÞ

jlðk2rcÞηlðk2rsÞ� jlðk2rsÞηlðk2rcÞj0lðk2rcÞηlðk2rsÞ� jlðk2rsÞη0lðk2rcÞ

" #VooEo1

ð4Þ

We solved the above equations numerically for electrons andholes, using a FORTRAN program with Bessel function subroutinesfrom Ref. [21] and appropriate parameter values [22]. The nth rootof Eq. (4) for a given l yields Enl. For a given l, m takes integervalues � l to þ l which have the same energy resulting in adegeneracy of 2lþ1. The transition energies Ei(d) in the QD withcore diameter d¼ 2rc are then given by

EiðdÞ ¼ ECdSegj þEenlðdÞþEhjnlðdÞ; j� A;B;C ð5Þ

where EgjCdSe are the energy gaps corresponding to the three

relevant VB of CdSe. The overlap integral of the electron and holeenvelope wavefunctions determines the relative oscillator strength(ROS) of the transitions. For transitions where ne¼nh and le¼ lh theROS� 1 and such transitions are strongly allowed (SA). Therelative contribution of these transitions to Abs then depends onlyon the degeneracy 2lþ1. The Abs spectrum αSAðE; dÞ of a single QDof core diameter d as a function of photon energy E due toinhomogeneously broadened SA transitions of width ΔE at ener-gies Ei(d) then becomes

αSAðE; dÞp∑ið2liþ1Þ GðE; ESAi ðdÞ;ΔEÞ; where

Gðx; xo;ΔxÞ ¼ 2Δx

ffiffiffiffiffiffiffiffiffiln 2π

rexp �4 ln 2

x�xoΔx

� �2� �

: ð6Þ

Here Gðx; xo;ΔxÞ represents a normalized Gaussian distributionwith mean xo and full-width at half-maximum (FWHM) Δx. Theinhomogeneous line width ΔE for a single QD arises essentiallydue to spectral diffusion [14] and was taken to be 35 meV. Nextthe Abs spectrum α(E) for an ensemble of such QDs with randomdistribution of core diameters around a mean diameter do andFWHM Δd is obtained by summing αSAðE; dÞ with a weight factordetermined by a Gaussian distribution as

αðEÞ ¼Λ∑dGðd; do;ΔdÞ αSAðE; dÞ ð7Þ

where Λ is a single scaling factor. We calculated α(E) using theabove formulation and matched the lowest energy peak of the Absspectrum by adjusting Λ and Δd. The estimated values of Δd forthe four samples starting from the largest diameter QDs were2.6 nm, 1.6 nm, 1.0 nm and 0.7 nm, which on an average represents� 33% of the mean diameters. The resultant simulated Abs spectraare shown in Fig. 1(a)–(d) where we have also shown individualtransitions, involving all three hole bands A, B, C for a QD corediameter equal to do, as lines whose length proportional to theirrelative contribution to the Abs spectrum. As is evident thissimulated Abs spectrum does not match with the experimentaldata shown alongside. We have tried to understand the inade-quacy of the above formulation as follows.

Let us consider some typical radial envelope wavefuncions χðrÞthat result from the above calculation. Fig. 4(a) shows two pairs ofelectron and hole χðrÞ for the QD with do ¼ 5:1 nm, which haveidentical l but different n quantum numbers (i) le ¼ 0, ne ¼ 2,lhA ¼ 0, nhA ¼ 3 and (ii) le ¼ 2, ne ¼ 1, lhA ¼ 2, nhA ¼ 2. One observesthat the different extent of leakage of the electron and holewavefunctions into the shell region makes these pairs looseperfect anti-symmetry with respect to each other. This suggeststhat the overlap integral of the electron and hole wavefunctionsthat determine ROS may be non-zero although neanh. Indeed theROS for the transition between states in the first (second) pair is0.029 (0.03) and occur at 3.567 eV (3.318 eV). Fig. 4(b) shows thecalculated ROS of all transitions for which neanh in this QD withdo ¼ 5:1 nm as a function of energy. It indicates that such normallydisallowed (ND) transitions can occur, although their strengthwould be weak. Note that the ROS magnitudes of these NDtransitions tend to increase with energy. This is because theenvelope wavefunctions of the higher energy states leak moreinto the shell and therefore are likely to be more asymmetric. That

Radial dist. (nm)0 1 2 3 4

χ(r )

Energy (eV)2.0 2.5 3.0 3.5

RO

S x

100

0

1

2

3

4

0.0

1.0

(i)

ehA

e-hBe-hC

e-hA(ii)

rc rs

0.0

-1.0

Fig. 4. (a) Radial envelope wavefunctions for confined electron (thin line) andA-type holes (thick line) in a QD with do ¼ 5:1 nm. The states are characterized by(i) le ¼ 0;ne ¼ 2, lhA ¼ 0;nhA ¼ 3 and (ii) le ¼ 2;ne ¼ 1, lhA ¼ 2;nhA ¼ 2. The shadedpart represents the shell region. (b) Relative oscillator strength (ROS) of transitionsin this QD for which Δle;h ¼ 0 but Δne;ha0, as a function of the transition energy.The dashed line indicates position of the first strongly allowed transition.

A. Mukherjee, S. Ghosh / Physica E 64 (2014) 234–239 237

apart the QD shapes are unlikely to be perfectly spherical andtherefore distort the spherical symmetry of the confinementpotential. Strictly speaking this negates the option of using Yl;m

as solutions but in an approximate analysis for small distortionsone may consider its consequence to be the appearance ofadditional ND transitions that violate the le¼ lh selection rule. Athigher energies the confined level spacings reduce, therefore thedensity of such ND transitions will be higher and they can add upto increase α(E). Such ND transitions, we suggest, could be a sourceof the increased Abs seen at high photon energies. In practice it isdifficult to quantify arbitrary deviations from spherical symmetryand even if that can be done it would be computationally quitedifficult to solve the Schrödinger equation to determine the energyspectrum in each case. We therefore propose a simple model inwhich we consider ND transitions mentioned above and assign atransition energy [ENDi ðdÞ] dependent oscillator strength βENDi ðdÞ toeach, where β is a constant. The latter because we saw thatthe oscillator strength of the ND transitions tend to increasewith energy. We therefore write such contribution to the Absspectrum as

αNDðE; dÞpβ∑ið2lminþ1Þ ENDi ðdÞ GðE; ENDi ðdÞ;ΔEÞ: ð8Þ

Here the index i runs over all the transitions that satisfy eitherneanh, le ¼ lh or ne ¼ nh; lea lh. We avoid transitions that violateboth l and n conservation since ROS then would become theproduct of two very small numbers o1 and therefore insignif-icant. The degeneracy is taken as 2lminþ1, where lmin is the lowerof le and lh for that ND transition. Thereafter we sum over the QDsize distribution as was done through Eq. (7) and write the totalAbs as

αðEÞ ¼Λ∑dGðd; do;ΔdÞ ½αSAðE; dÞþαNDðE; dÞ� ð9Þ

We used this formulation that includes the ND transitions tosimulate the Abs spectra. We kept all the parameters the same asin Eq. (7) and only varied β to fit the measured Abs spectra.Fig. 5(a)–(d) compares the measured and the simulated Absspectra after including the ND transitions. It is evident that thereis now a much better agreement between the measured and thesimulated Abs spectra at high photon energies. Also shown inFig. 5(a)–(d) are the contribution to Abs by the ND transitionsalone, which is comparable to that of the SA transitions at highenergies. This might seem odd since ROS of ND transitions aremuch smaller. To understand this note that the index i in Eq. (6)for SA transitions only runs over the ones satisfying ne¼nh andle¼ lh, while in Eq. (8) for ND transitions i runs over neanh butle¼ lh and also ne¼nh but lea lh. Thus the set of possible NDtransitions becomes quite large making their contribution to theAbs coefficient significant in spite of a smaller ROS. Including NDtransition contributions to Abs in the larger QDs seems like addinga simple function which monotonically increases with energy.However note that for do ¼ 2:6 nm such a simple function cannotachieve the good fit we get, which gives us confidence in ourmodel. Such ND transition contribution can explain the origin ofthe smooth fitted background in Ref. [9] which was arbitrarilychosen to be a cubic polynomial.

What we are suggesting here is that the confinement potentialstructure and asymmetry in these QDs leads to transitions that arenot possible in a perfect shell-less hard-sphere potential model.These transitions modify the Abs spectrum by helping to smearfine-structure and increasing α(E) at high energies thereby makingit more like three-dimensional bulk solid. This idea has similaritywith the important work of Miller et al. [23] where they showedthat when an electric field is applied to a two-dimensional QWsystem, the resultant asymmetry of the potential leads to ND

transitions. In the high field limit, it makes the QW Abs spectrumacquire features that resemble Franz–Keldysh oscillations in bulkthree-dimensional solid. In their case field induced tunneling ofcarriers through the QW barriers made the system behave like abulk three-dimensional solid however in our case the system stillremains quasi zero-dimensional.

It is instructive to see how β varies with do. To make it adimensionless parameter for comparison with ROS, we haveplotted the product βEo as an inset in Fig. 5. Eo is the energyposition of the first peak in the Abs spectrum. First note that theβEo value for large QDs is much smaller than 1, which agrees withprevious calculated ROS values. For example in Fig. 4(b) fordo ¼ 5:1 nm the ROS is seen to go up to 0.03 and here we getβEo � 0:05, which is comparable in magnitude. Thus again a smallaverage ROS value is obtained for ND transitions from fitting ourmodel, although their contribution to Abs becomes comparable tothat of SA transitions at high energies. One also finds that βEoincreases significantly as do becomes smaller, which can be under-stood as follows. In the smaller QDs the relative leakage of thewavefunction into the shell region will be larger and also amonolayer scale distortion of the QD shape will have greaterrelative impact on the confinement potential symmetry. For thesereasons the occurrence of ND transitions and its role in determin-ing α(E) will be much more significant in the smaller QDs. Thevariation of this average effective ROS of the ND transitions βEo

Energy (eV)2.0 2.5 3.0 3.5

Abs

. coe

ff. (

arb.

uni

ts)

Wavelength (nm)

x 1/3

x 1/2

Meas

700 600 500 400 350450550

do (nm)0 2 4 6 8 10

β Eo

100

10-1

10-2

Sim ND

Sim Tot

x 1/2

Fig. 5. Measured and simulated room temperature absorption spectra of CdSe–ZnScore–shell QD ensembles for different average core diameters do. The simulatedtotal absorption considered both strongly allowed and normally disallowed (ND)transitions. The contribution of ND transitions is shown separately. The inset showshow βEo , which gives a measure of the average oscillator strength of the NDtransitions, varies with do. (a) do¼7.2 nm, (b) do¼5.1 nm, (c) do¼3.0 nm and(d) do¼2.6 nm.

A. Mukherjee, S. Ghosh / Physica E 64 (2014) 234–239238

with do is such that one can fit to it a equation of the typeβEo ¼ a=d3o where a¼8.259, which interestingly approximates toβEo � ðdo=2Þ�3. The line in the inset figure represents a fit usingthis equation. Thus for a given do one can find βEo and then usingthe above model it becomes possible to predict the shape of theAbs spectrum at high energies for such CdSe–ZnS core–shell QDensembles.

4. Conclusions

In conclusion we have shown through two independent mea-surements that CdSe–ZnS core–shell QDs have relatively highbulk-like absorption at high energies above the effective bandgapthat cannot be explained by considering only strongly allowedinter-band transitions between confined levels in the QDs. Weshowed that due to the core–shell structure it is possible to haveinter-band transitions in these QDs that would be symmetryforbidden in a simple hard sphere confinement potential. Therecan also be additional normally disallowed transitions due toabberations in the spherical shape. Such transitions increase innumber at high energies and add up to increase the absorptioncoefficient. We have provided a simple phenomenological modelrequiring just one extra parameter that can reproduce thisobserved increase in the absorption coefficient at high photonenergies.

Acknowledgments

The authors acknowledge useful discussion with G.H. Döhler.They thank R.D. Bapat, S.C. Purandare for HR-TEM measurementsand V. Sugunakar for help with instrumentation.

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jj þ2=mn? . For wurtzite ZnS: fundamental band gap

EgA ¼ 3:81 eV; mhAjj ¼ 1:4m○ , mhA? ¼ 0:49m○; me ¼ 0:28m○; a¼ 0:382 nm,c¼ 0:626 nm. The CdSe hole effective masses are from Ref. [24], otherparameter values are directly obtained or derived from Ref. [25]. CdSe/ZnSband offset ratio f ¼ΔECB=ΔEgA ¼ 0:7 [26].

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