Nanostructured CdSe Films in Low Size-Quantization Regime

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Nanostructured CdSe Films in Low Size-Quantization Regime: Temperature Dependence of the Band Gap Energy and Sub-Band Gap Absorption Tails Biljana Pejova*,† and Bahattin Abay‡ †

Institute of Chemistry, Faculty of Natural Sciences and Mathematics, Sts. Cyril and Methodius University, POB 162, 1001 Skopje, Macedonia ‡ € niversitesi, Fen Fak€ultesi, Fizik B€ol€um€u, 25240, Erzurum, Turkey Atat€urk U ABSTRACT:

Temperature dependence of the band gap energy and sub-band gap absorption tails in 3D assemblies of close packed weakly quantized CdSe quantum dots deposited as thin films was studied. The range from cryogenic temperatures (∼10 K) up to 340 K was covered. Excitonic absorption peaks were not observed even at temperatures as low as 11 K, which was attributed to the finite particle size distribution and interdot electronic coupling effects. The temperature coefficient of the band gap energy of the nanostructured films of 9.4
104 eV K1 is higher by a factor of 1.35 than the corresponding value for the bulk CdSe specimen. As compared to the case of films constituted of strongly quantized ZnSe QDs, where the factor αnanocrystal/αbulk was found to be 1.82 (Pejova, B.; Abay, B.; Bineva, I. J. Phys. Chem. C 2011, 115, 37), the present findings imply that this ratio increases upon enhancement of size quantization effects in semiconductor nanocrystals. Analysis of the temperature-dependent optical absorption data within the Bose-Einstein model implies that no phonon confinement effects influence the phonon spectrum in the presently studied material due to the very small size-quantization effects. This situation is opposite to what we have recently found in the case of 3D arrays of strongly quantized ZnSe films. The characteristic Einstein temperature of the presently studied material corresponds to phonon frequency of about 220 cm1, in excellent agreement with the LO mode frequency of bulk CdSe (210214 cm1). It is demonstrated that the Urbach rule is valid in the presently studied nanostructured material in low size-quantization regime. Urbach energies are several times higher than the values characteristic for macrocrystalline materials, due to the relatively high degree of inherent structural disorder in the studied QD solids. At the same time, however, these values are approximately three times smaller than those reported for strongly quantized ZnSe films in our previous study. The dynamical (temperature-dependent) term accounts for only about 22% of the overall Urbach energy values, though this value is higher than the corresponding ratio in the case of strongly quantized ZnSe films.

1. INTRODUCTION Nanostructured, that is, low-dimensional semiconductor materials are of considerable significance to fundamental physical sciences as well as to contemporary high technology. The mentioned systems are characterized with remarkable optical and electronic properties that are notably size-dependent in certain size intervals.18 Size-dependent band gap variation of nanostructured semiconductors has enabled tunability of light absorption and emission properties, along with a considerable enhancement of the oscillator strength for these transitions. All these effects are due to quantum confinement of the charge carrier motions r 2011 American Chemical Society

within a nanocrystal (quantum dot, QD). A characteristic length that characterizes the abrupt occurrence of confinement effects in semiconductor nanocrystals is the value of the Bohr’s excitonic radius.912 Certain aspects related to light absorption and tunability of optical properties in individual QD nanostructures, as well as in systems composed of individual or diluted semiconductor QDs as analogues of 2D quantum wells,5,6 have been Received: May 12, 2011 Revised: August 7, 2011 Published: September 12, 2011 23241

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The Journal of Physical Chemistry C studied a lot.18 However, a number of issues that are by no means of less importance to fundamental sciences and technology have been only partly addressed. This is particularly true when the temperature-dependence of the band gap energy and sub-band gap absorption tails are in question. These two tendencies contain substantial information related to lattice dynamics and statistical thermodynamic properties of nanocrystals, either isolated or close packed, for example, in thin film form. In fact, when the nanocrystals are close packed, forming a 3D assembly (array), they may be regarded as building blocks forming a certain new form of organization of matter, referred to as quantum dot solids (or colloidal crystals; see refs 1320, and the references therein). Individual QDs, in the last case, play the role of basic structural motifs in the formed superstructures. These novel artificial superstructures open certain new opportunities for studying fundamentally new aspects of physical chemistry of solid state- and low-dimensional materials. Actually, when the nanocrystals are deposited in the form of 3D assemblies, a number of collective physical phenomena develop upon interaction of the proximal QDs, while certain properties that are characteristic of individual QDs are retained (see refs 1320, and references therein). It is exactly this duality of properties that makes such novel superstructures very convenient for potential applications in optoelectronic devices, so the continuously increasing interest in this area is self-understood. In this context, studies of optoelectrical and photophysical properties of the mentioned systems are of particular interest from both fundamental and applicative aspects. Generally speaking, the most exploited property of the semiconductor nano- and macrocrystals has been their fundamental absorption edge.57 This parameter has often been characterized by only a single numerical parameter, the band gap value Eg, and it has been shown to provide extensive information on the band structure, disorder/impurity effects, the electron/excitonphonon interactions, as well as the very existence of excitons in the studied material and their role in light absorption.57 A more in-depth physical insight into the mentioned phenomena and a more complete description of the overall complex process of interaction of a solid-state material with electromagnetic radiation, however, could be derived if also the sub-band gap part of the absorption spectrum is considered and analyzed in sufficient details. As recognized for the first time by Urbach21 and Martienssen22 for two particular cases, namely, silver halides and alkali halides, instead of occurring sharply, that is, at a single energy value corresponding to Eg, the light absorption takes place in a more complex manner. A sub-band gap absorption tail, which they have shown to be of exponential type is developed, of the form α(hv,T)
exp[(hv  E0)/EU(T,X)]. The steepness of the so-called Urbach (or Urbach-Martienssen (UM)) absorption tail is determined by the parameter EU, known as Urbach energy. Besides on temperature, it depends on the degree of structural disorder in the studied material, expressed through the parameter X, introduced by Cody et al.23 The validity of this “Urbach-Martienssen rule”5,2426 has been demonstrated for a number of bulk (macrocrystalline) materials, and it has been considered as being a quite universal behavior of semiconductors. However, the sub-band gap absorption of light in nanostructured materials has been studied to a much lesser extent. The last statement is particularly valid in the case of 3D assemblies of quantum dots. Even besides the fact that the UM rule has been considered as a general one in the case of macrocrystalline semiconductors, there are still a number of open questions concerning the mechanisms that actually lead to such

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behavior. It is generally thought that the effect could be due to smearing of the absorption edge at relatively higher temperatures, at which excitonic absorption bands are not observed, but the formation of excitons (due to the electrostatic interaction between photogenerated charge carriers) is still important for the overall process of light absorption in the semiconductor. The concept of excitons as electronhole bound states is especially important in the case of nanostructured semiconductors. This is so since the charge carriers are confined to a small space region in nanocrystalline solids, which enhances the Coulomb interaction between them (as compared to the bulk counterparts). Due to these facts, however, studies of sub-band gap absorption phenomena in nanostructured semiconductors could shed more light on the very essence of the mechanism of light absorption in QDs, and on the excitonic states developed in such structures. As already implied, however, though the sub-band gap absorption has been studied a lot in the case of macrocrystalline and amorphous materials,2755 emphasizing many aspects as well as the influence of a number of external parameters on the UM absorption tails, the number of studies has been much smaller in the case of nanostructured semiconductors (3D QD assemblies, in particular) and also in the case of thin films. As we have already argued before, however, it is of certain importance for both fundamental aspects of physical chemistry, materials science, and the potential applications of semiconductor quantum dot thin films to follow the evolution of the sub-band gap absorption tails with both temperature and controlled particle size reduction. In a recent study,56 we have investigated these aspects in the case of 3D assemblies of strongly quantized close packed ZnSe QDs deposited in thin film form (i.e., we have considered 3D assemblies composed by strongly quantized nanocrystals). The present paper, which is a logical continuation of our previous work, deals with a system that is in a sense complementary to the previous one. Here, we deal with 3D assemblies of close packed CdSe QDs deposited in thin film form, which are characterized by only very slight size quantization effects (expressed, as we demonstrate later, through a very small blue shift of the band gap energy in the postdeposition annealed samples). The two semiconductors have been chosen as they are characterized with a similar band structure, aside from certain complexities that are not relevant to the fundamental band-to-band transitions from the top (i.e., absolute maximum) of the valence band to the bottom (i.e., absolute minimum) of the conduction band. Both semiconductors, synthesized by the chemical methods developed in our group,5761 are of sphalerite structural type and are characterized by very high phase purity. Such choice of systems, which are complementary, enabled us to follow certain trends and imply more general rules for the behavior of semiconductor QD arrays related to their light-absorption properties. Some of these implications are closely related to the very essence of the mechanism of light absorption in the low-dimensional systems in general. The main focus in the present study is put on the temperature-dependent optical absorption of 3D assemblies of weakly quantized CdSe QDs, deposited in thin film form by our recently developed chemical deposition method. We have investigated the temperature range from cryogenic temperatures (11 K) to about 350 K. To derive the conclusions mentioned before, we have focused on the analysis of the temperature dependence of band gap and Urbach energy in relation to the new physical insights into the structural and optical properties of QD assemblies that could be gained from the analysis of these data. Concerning the relevance of the chosen 23242

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The Journal of Physical Chemistry C semiconductor system (CdSe), it is a member of AIIBVI group of binary semiconductors. Characteristic combination of properties of this material (related mainly to its characteristic band structure) makes it of prime importance in energy conservation problems.6264 Cadmium selenide, especially deposited in thin film form, has found numerous applications, for example, as thin film transistors,65 gas sensing,66,67 acousto-optical devices,68 vidicones,69 photographic photoreceptors,70 and so on. This semiconductor exists in two polymorph modifications: cubic and hexagonal,71 the first one being of sphalerite structural type, whereas the second one is of wurtzite type. Cubic cadmium selenide converts to hexagonal modification at 350400 C.72 Although the number of published papers treating cadmium selenide is rather large,7379 especially the cubic modification of this semiconductor in thin film form is one of the least studied AIIBVI systems. This is due to the fact that it is far from an easy task to obtain crystallographically pure cubic modification of this semiconductor, especially in the form of thin film. For that purpose, usually highly sophisticated techniques were required.7779 Therefore, studies of its physicochemical properties have been in a sense hampered. The crystal structure, along with the particle size and shape can affect optical, electrical and mechanical properties of the nanodots themselves as well as of the materials built up by these “artificial atoms”. As we have previously mentioned, there is a lack of thorough data related to optoelectronic properties of individual CdSe quantum dots with cubic crystal structure and the situation is even more emphasized when 3D assemblies of cubic CdSe QDs are in question. For example, aside from our previous studies of the optical properties of this CdSe polymorph,5961 the first more thorough study of its optical properties has been published very recently.80 It has been often assumed in the literature that optical properties of cubic CdSe should be similar to those of the wurtzite polymorph, in essentially all aspects. Such assumption is based on the similar band gap values for bulk specimen of both polymorphs, as well as on the similarity of bulk effective masses of the charge carriers and bulk optical constants for the two cases. However, such widely used assumption does not seem to be fully justified, as zinc-blende and wurtzite polymophs of CdSe show remarkable differences in the electronic energy band structures near the Γ-point. Thus, zinc-blende polymorph is characterized by 4-fold degenerate valence band level at Γ-point, while in the case of wurtzite polymorph, due to the lower symmetry of the crystalline field, the valence band is split into two 2-fold degenerate levels. As the crystal size reduces and the confinement regime is entered, the hole states in these two polymorphs are described by different Hamiltonians. It is therefore highly desirable from both fundamental and practical aspects of physical chemistry to undertake more thorough studies of the optical and other properties of cubic-phase CdSe QDs and especially of various superstructures built up by these building blocks. Our recently developed chemical route to nanostructured CdSe films,60 however, has enabled us to synthesize thin films of low-dimensional cubic modification of CdSe, therefore, enabling thorough and in-depth studies of the properties of this polymorph. Our current work is related to the previous works of Scholes et al.81 and Crooker et al.,82 devoted to temperature-dependence studies of various properties in nanocrystalline CdSe. In the first of these papers, the spectral line shape origin in colloidal CdSe nanocrystal QDs was studied. Special emphasis was, however, put also on the temperature dependence of optical gap and Stokes shift with temperature. Implementing the technique of

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temperature-dependent three-pulse photon echo peak shift (3PEPS) in combination with theoretical modeling, the authors have derived thorough conclusions concerning the role of exciton phonon coupling, static inhomogeneity, exciton fine structure, and exciton state disorder in the nanocrystal’s line width. They have also looked for Urbach-Marteinssen tails in the absorption and photoluminescence spectra of the studied colloidal nanocrystals, in the context of the excitonphonon coupling strength (which is related to the Urbach energy), but found out that the situation was substantially complicated by trap emission. The CdSe QDs considered in this study were characterized with strong quantum size effects. In the second of the abovementioned papers, the temperature dependence of radiative lifetime of electronhole excitations in colloidal CdSe nanocrystal QDs was addressed. Our study, as mentioned before, is devoted to 3D assemblies of CdSe QDs, in weak size quantization regime. As we demonstrate further, the obtained results imply certain trends going from strongly quantized to weakly quantized nanocrystals. We note in this context, however, that several other studies related to temperature-dependence of various parameters in nanostructured CdSe or heterostructures involving CdSe have also been published in the literature.8387 In a broader context of contemporary physical chemistry, the relevance of studies devoted to the “artificial solids” built up by QDs arises due to the following facts.88101 Namely, except for biophysical chemistry applications (e.g., sensing and imaging), most of the envisioned technologies are not single-particle based and are not operable in solution. Therefore, development of methods for assembling of QDs into various solid-state architectures as well as thorough investigation and understanding of optoelectrical properties thereof are important challenges in physical chemistry nowadays.101 The optoelectronic properties of QD solids may be tuned either by the choice of semiconductor nanocrystals which build up the artificial solid or by tuning the electronic interactions between these “artificial atoms”. It is not surprising that even in this context, CdSe nanocrystals are of exceptional importance. The bulk band gap enegy of this material of about 1.70 eV is much smaller than the transition energies of the most widely used capping ligands. Upon decrease of the nanocrystal size (entering the size-quantization regime) it is possible to achieve a rather large range of band gap blue shifts (essentially in the full visible range) prior to achieving the possibility of mixing the band gap transition and the intraligand transition. It is thus of certain importance to study 3D assemblies of the present material in the context of harnessing the nanoscale tunability on a macroscopic length scale, which could provide a whole new class of materials. In the context of solar energy conversion, on the other hand, thin films constituted of electronically coupled QDs are important as they represent a new class of granular conductors in which it is possible to investigate aspects such as junction formation in nanoscale systems.89,91100 Finally, the “artificial solids” built up from chemically synthesized QDs offer the unique possibility to exert chemical control over the parameters which determine the electronic properties of solids, and to rationally design quantum phase transitions in such solids, which is of exceptional importance to numerous aspects of contemporary physical chemistry.88

2. EXPERIMENTAL DETAILS 2.1. Chemical Deposition of 3D Assemblies of CdSe QDs Close Packed in Thin Film Form. We have recently developed a 23243

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novel chemical route to nanostructured CdSe thin films,60 composed by 3D assemblies of close packed QDs of this material. Our method has certain advantages over other literature approaches to thin films of this rather important semiconductor. First of all, it enabled synthesis of the studied material with very high phase purity (i.e., of only the cubic modification of CdSe, avoiding completely the presence of any impurities of the hexagonal modification). Synthesis of the pure cubic modification of CdSe (of sphalerite structural type) appeared to be a significant problem in most methods which have been reported in the literature.7779 Thus, to the best of our knowledge, this aim seemed to be practically unachievable by other chemical methods. The method developed by our group also enables presence of Cd(OH)2 in the as-deposited material to be practically completely avoided. This component, if present in significant concentrations in the deposited films, could potentially significantly degrade its photoelectrical properties. Glass or polyester substrates, with dimensions of a standard microscope glass, were used to deposit the films. To achieve good adhesion of the films to the substrate surface, which is of course of crucial importance for further investigations of their optical and photoelectrical properties, the substrates were subjected to predeposition treatment, which has been explained in details in our previous publications.102,103 The essence of our chemical deposition method for synthesis of 3D assemblies of CdSe QDs in thin film form is to carry out a highly controllable precipitation reaction in aqueous medium. Sodium selenosulfate was used as precursor of selenide anions, while the concentration of metal (Cd2+) cations was controlled by their complexation with ammonia, introduced in the reactor as ammonia buffer solution. Using ammonia buffer solution allowed us, at the same time, to control the pH of the reaction system. Due to its inherent instability, sodium selenosulfate solution was prepared freshly, just before the deposition process, as explained elsewhere.60 Though we have carried out a classical optimization of experimental deposition conditions, it was actually carried out only after a careful analysis of the thermodynamical and kinetical aspects of the reaction system. Coherent light-scattering experiments were performed in order to study the mechanisms of cadmium selenide crystal growth.104 For more details concerning the description of the chemical and thermodynamical aspects of the synthesis, the reader is referred to our previous publications.59,60 The as-deposited CdSe nanostructured films were further subjected to postdeposition thermal annealing treatment (at 300 C in air atmosphere), for a total duration of 60 min. 2.2. Structural Characterization and Variable-Temperature Optical Measurements. Identification and characterization of the basic structural properties of the 3D assemblies of CdSe QDs, deposited either in thin film form or as bulk precipitates was performed by the X-ray diffraction method (XRD) for polycrystalline samples with monochromatic Cu Kα radiation. X-ray diffractograms were recorded on a Philips PW 1710 diffractometer. Average crystal size in the nanostructured CdSe films was calculated on the basis of intrinsic broadening of the diffraction maxima using the following form of the Debye Scherrer equation:105 Ædæ ¼

4 0:9λ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 β2  β2 cos θ m s

ð1Þ

In eq 1, βm is the raw value of the full width at half-maximum intensity (FWHMI) of the peak (which contains both instrumental and intrinsic broadening factors), while βs is a standard FWHMI value referring to a correspondent peak in the case of macrocrystalline and strain-free Al2O3 sample. To determine correctly all of the parameters in eq 1, we have fitted the diffraction peaks using the pseudo-Voigt profiles (i.e., with linear combinations of Gaussian and Cauchy (Lorentzian) functions) by the least-squares based methodology. In other words, each peak was fitted by a function of the form Ið2θÞ ¼ γ

pffiffiffiffiffiffiffiffiffiffiffi 2 4 ln 2 ½1 þ 4ð2θÞ2 1 þ ð1  γÞ pffiffiffi exp½4 ln 2ð2θÞ2  Hπ H π

ð2Þ where the parameters H and γ were determined by least-squares fitting techniques. Optical absorption spectra of the 3D assemblies of CdSe QDs deposited as thin films were recorded in the spectral range from 190 to 1100 nm, in temperature interval from 10 to 340 K. The measurements were performed on a Perkin-Elmer Lambda 2S UV-vis-NIR spectrophotometer, using a Leybold-Heraeus cryogenic system. Closed liquid He system was employed for lowtemperature measurements, controlling the sample temperatures with a precision of (0.2 K. For temperature control, a computercontrolled Leybold-Heraeus Variotemp HR1 temperature regulation system was used. Analysis and processing or the spectra was performed using the MS Excel and Microcal Origin software packages.106,107 To convert the spectral dependence of transmission coefficient T(hv) to the spectral dependence of absorption coefficient α(hv) of the films, their thicknesses were determined using either gravimetrical or interferometrical methods.

3. RESULTS AND DISCUSSION 3.1. Temperature-Dependent Optical Absorption of the 3D QD Assemblies in Low Quantization Regime. As outlined

in the experimental section, our synthetic method provides a way to produce three-dimensional nanocrystal assemblies of the cubic CdSe modification (the polymorph with sphalerite structural type, i.e., with sufficient phase purity), Figure 1. As-deposited films (as well as bulk precipitates) are characterized with an average crystal radius of 2.65 nm. Postdeposition thermal treatment at 300 C leads to crystal growth and coalescence processes, finally resulting in an average crystal radius of 12 nm.59,60 No chemical changes or phase transformations in the films or precipitates have been detected upon such postdeposition thermal treatment. Having in mind that Bohr’s excitonic radius for macrocrystalline CdSe is 5.6 nm,108 after thermal treatment, thus, though the material is still nanostructured, the regime of low (weak) size-quantization has been entered. This is reflected in the electronic band structure of the solid semiconductor, and further manifested in its optical absorption properties. For example, upon thermal treatment of the as-deposited CdSe films, the band gap energy reduces from 2.08 to 1.77 eV.5961 The last value is only very slightly higher than the one characteristic for a bulk specimen (1.74 eV).109 In the present study, thus, we deal with a QD assembly system in regime of a very weak size quantization. This is in contrast with the system to which we have paid a special attention in our recent study, namely, the thin films composed by 3D assemblies of strongly quantized 23244

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Table 1. Basic Structural Parameters of As-Deposited and Thermally Annealed CdSe Thin Films Derived on the Basis of XRD Dataa

a

Figure 1. Recorded X-ray diffractograms for the as-deposited and thermally treated (60 min at 300 C) CdSe bulk precipitates and thin films, compared to the standard.

ZnSe QDs56 and also with the temperature-dependent studies in refs 81 and 82. For the purpose of the present study, certain structural parameters were deduced by further analysis of the X-ray diffraction data. First of all, we have performed a refinement of the unit cell parameters of the QD solids deposited as thin films before and after the thermal treatment. This was done on the basis of linear regression analysis of the dependence between sin2 θhkl and (h2 + k2 + l2), which, according to the basic relations that could be derived between the structural and XRD parameters for the cubic crystalline system has the form105 sin2 θhkl ¼ Aðh2 þ k2 þ l2 Þ

ð3Þ

In eq 3, θhkl is the angle corresponding to the diffraction (“reflection”) from crystallographic planes with Miller indices h, k and l, while A = λ2/4a2, which is obtained as the slope of the linear fit to the experimental data. Alternatively, the lattice constant can be calculated from the XRD data for each peak by the formula pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ h2 þ k2 þ l2 ð4Þ a¼ 3 2 sin θhkl and further averaged over all relevant peaks in the X-ray diffractogram. From the computed lattice constant values, we have further estimated the isotropic average lattice strain defined as:110 Æεæ ¼

jΔaj abulk

ð5Þ

or, alternatively as111 Δθhkl ¼  ε 3 tanðθhkl Þ

ð6Þ

Æaæ (Å)

ÆRæ (nm)

δ (nm2)

Æεæ

Eg (eV)

as-deposited

6.0746

2.7

0.040

0.0028

2.08

annealed at 300 C

6.0842

12.0

0.002

0.0012

1.77

See text for details.

Figure 2. Spectral dependencies of the absorption coefficient in the case of a thin film composed by close packed 3D assemblies of CdSe QDs in the very weak size-quantization regime recorded at, from right to left, 11, 100, 180, 260, and 340 K.

parameters which have been deduced by the outlined approach are given in Table 1, where also the band gap energies computed from the optical spectroscopy data at room temperature are given. It is clearly seen from Table 1 that while the lattice constant (a) in the case of as-deposited films is notably smaller than the bulk value for this material, the discrepancy almost vanishes upon thermal treatment at 300 C. This is followed by lattice strain relaxation and notable dislocation density decrease upon thermal annealing. In comparison to the results concerning the strongly quantized ZnSe nanocrystal arrays which we have recently reported,56 the current results for a 3D QD assembly system in the very weak quantization regime lead to a conclusion that all the parameters that characterize the degree of structural disorder in the chemically synthesized CdSe QDs in thin film form are notably smaller. This is reflected, as we discuss below, in their temperature-dependent optical absorption as well. To the best of our knowledge, this is one of the first experimental studies devoted to temperature dependence of the band gap energy of thin films composed of close packed semiconductor quantum dots in the range from cryogenic to room temperatures, in both strong and weak size-quantization regime. Variable temperature optical absorption spectroscopy was used to investigate the optical properties of the close-packed CdSe nanocrystals deposited as continual thin films. The spectral dependencies of absorption coefficient at various temperatures (starting from 11 K up to 340 K) were constructed on the basis of experimentally measured spectral dependencies of transmission coefficient in the studied spectral region, that is, the functions T(hv) using the relation

(with subsequent averaging over all relevant peaks in the X-ray diffractograms). The ε values computed by both approaches were found to be in excellent agreement. The relevant structural

αðhvÞ ¼ 23245

1 1 ln d ðhvÞ

ð7Þ

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demonstrated when colloidal QDs have been dispersed in solutions or in cases of diluted QD ensembles (achieving the dilution, e.g., by an appropriate capping ligand).1320 In the present case, however, we deal with thin films composed by close-packed CdSe QDs (in the very weak size-quantization regime), which have been synthesized by a wet chemical method. As discussed elsewhere, the nanoparticles synthesized by this approach are not monodispersed, but are instead characterized by a relatively wide size distribution. In the following discussion, we will denote the QD size probability distribution function (PDF) by F(R). When the QD size range approaches the Bohr excitonic radius of the material in question, the spectral dependence of the absorption coefficient (i.e., the function α(hv)) becomes a rather complex function of the particle size distribution. In the simplest case, one can represent the dependence of absorption coefficient on the incident photon energy for a QD with a radius R in the very weak size-quantization regime accounting only for the continuum DOS contribution, which has the following form for direct band-to-band transitions:136,137 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αðhv, RÞ ¼ A0 hv  Eg ðRÞ ð8Þ

Figure 3. Simplified band structure of semiconductors with zinc-blende structural type, emphasizing the complexity of the valence band: ΔSO, spinorbit splitting energy; hh, heavy hole band; lh, light hole band; so, split-off band; J = L + S.

where A0 is a constant. Equation 8 actually represents a limiting case in which the electronhole electrostatic interaction is regarded as being completely screened during the photon absorption. Within this approximation, the spectral dependence of absorption coefficient will be given by the following function: Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FðRÞ½A0 hv  Eg ðRÞdR Z ð9Þ αðhvÞ ¼ FðRÞdR

where d is the film thickness (measured either gravimetrically or interferometrically in the present case). Typical α = f(hv) plots in the region of fundamental band-to-band electronic transitions for several temperatures within the mentioned range are given in Figure 2. The assignment of higher-order transitions in zincblende polymorph of the studied material has been has been shown to be a rather difficult task,1618,112132 and therefore, in the present study we have focused our attention on the lowestenergy direct band-to-band Γ8v f Γ6c transition from the absolute maximum of the valence band to the absolute minimum of the conduction band.24,114,133,134 The band structure of bulk CdSe is schematically presented in Figure 3. Concerning the assignment of higher-order transitions in the presently studied QD solids detected by optical spectroscopy at room temperature, the reader is referred to our previous publication.135 As can be seen from Figure 2, the absorption onset of the films is blue-shifted upon lowering of the temperature. This indicated that the temperature coefficient of the band gap energy is negative, a point that will be discussed quantitatively further in this paper. A notable qualitative characteristic of the α = f(hv) functions shown in Figure 2 is the absence of clearly defined excitonic peaks, even at temperatures as low as 11 K. Appearance of more or less clearly differentiated excitonic peaks would be expected in cases of individual QDs, or diluted QD assemblies, even at rather high temperatures, due to the evolution of bulk energy bands into series of discretized electronic energy levels as a consequence of the 3D confinement of charge carrier motions within a QD. This property of individual QDs has actually been

QD size probability distribution function F(R) has most frequently been described by a Gaussian or log-normal function. On the other hand, the size-dependence of the band gap energy Eg(R) may be represented by some analytical model function, such as the one arising from the model proposed by Brus, the hyperbolic band model or Nosaka’s model.58 Regardless of the actual form of F(R) or Eg(R), or on the exact approximation that has been used for α(hv,R), however, it becomes obvious from eq 9 that the finite particle size distribution leads to smearing of the excitionic absorption peaks, even if they should be visible in single (individual) QDs. Such smearing has frequently been encountered in QD thin films synthesized by chemical methods (see ref 56 and references therein). In addition to this reason, however, one should also keep in mind that the synthesized CdSe nanocrystals are close packed when forming the continual thin film (with no organic or capping layer between the nanocrystals). Bringing the isolated quantized nanodots in contact enhances the probability of interdot electronic coupling and the overall effects thereof. Perhaps the most important consequence of the interdot coupling in this context is the formation of collective electronic states,1618,20,138,139 which are delocalized over a finite number of nanocrystals which constitute the nanocrystal ensemble. Despite the continuing debate in the literature concerning its origin, such delocalization of charge carrier states seems to occur mostly due to tunneling effects. An overall consequence of these coupling effects would be an appearance of structureless absorption spectra in the studied nanocrystal assemblies. 23246

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Though the nonobservance of excitonic peaks in semiconductor nanocrystalline films at room temperature could be understood in terms of the size distribution (and also the interdot electron coupling when nanocrystals are small), to the best of our knowledge, this is one of the first experimental studies (along with our previous paper devoted to ultrasmall strongly quantized ZnSe QDs in thin film form) which demonstrates that the excitonic transitions are not manifested at all in the optical absorption spectra of thin films composed of close-packed QDs even at a temperature as low as 11 K. 3.2. Temperature Dependence of the Band Gap Energy. In the absence of clearly defined transitions to excitonic states in the studied CdSe nanocrystalline assemblies, we have further focused our analysis on interband transition energies. These were obtained by analysis of the semiconductor absorption functions and their temperature dependencies. To provide a thorough physical insight into the electronic transitions in the studied nanocrystalline semiconductor films, the experimental absorption spectra will here be interpreted in the framework of band theory of solid state,58,2426 but also accounting for the low-dimensional character of the studied systems. The dimensionality reduce could be effectively treated as a spatial confinement imposed on the charge-carrier motions.14 Assuming, to a first approximation, a spherical symmetry of the energy bands of the studied semiconductor compound, the semiconductor absorption function could be written in the form ðαðhvÞhvÞn ¼ const:ðhv  Eg Þ

ð10Þ

where Eg denotes the band gap energy, while n depends on the type of electronic transitions. We assume in the foregoing analysis that the type of band structure is preserved even upon dimensionality reduce, a situation that has frequently been encountered in the semiconductor systems with varying dimensionality. Because a bulk specimen of cubic polymorph of CdSe is a direct-type of semiconductor (in which the valence band absolute maximum and conduction band absolute minimum correspond to same point of k-space), the interband transitions are not accompanied with change of electronic momentum. The exponent n in eq 10 can therefore take two values, 2 or 2/3, the first case corresponding to dipole-allowed, and the second one to dipole-forbidden electronic transitions. In line with our expectations, analyses of the semiconductor absorption functions constructed from the measured spectral dependencies of α at various temperatures revealed that best interpolations could be obtained setting the exponent n to 2. Note that we have made a thorough analyses of the experimental data, using all physically acceptable values for n, including those for indirect band gap semiconductors (n = 1/2) as well as for the amorphous specimen (n = 1).140 We have therefore concluded on the basis of these analyses that the “band-to-band” transitions in the studied nanocrystalline assemblies are direct and at the same time dipole-allowed. The corresponding band gap energies of the nanocrystalline films at various temperatures were further on determined by detailed analysis of the (α(hv)hv)2 = f(hv) dependencies on the basis of a combined interpolation-extrapolation procedure. Strictly speaking, the semiconductor absorption function of the form in eq 10 is based on the parabolic approximation for the dispersion relation. This approximation is, however, valid only in the neighborhood of first Brillouin zone borders. One thus expects that certain deviations from linearity should occur as one moves away from the zone borders. To account for this error, we have

actually carried out linear least-squares fits of (α(hv)hv)2 = f(hv) dependencies by successive inclusion or elimination of a number of neighboring points in relevant energy range (wide enough), keeping in parallel an eye on the value of R2. After that, extrapolating the linear dependencies to (α(hv)hv) = 0, band gap energy for each temperature was calculated on the basis of the obtained correlation equations. The band gap value for a bulk specimen of cubic CdSe polymorph is 1.74 eV.109 The room temperature band gap value of our specimen is thus characterized by only a very slight band gap blue-shift of 0.03 eV. Our experimentally measured band gap blue shift is actually in excellent agreement with the value of this quantity predicted by the Brus’ model,141143 that is, by using the formula ! h2 1 1 1:8e2 þ ΔEg ðRÞ ¼  mh 8m0 R 2 me 4πε0 εr R 4π2 e4 m0

 0:248 2ð4πε0 εr

Þ2 h 2

1 1 þ   me mh

!

ð11Þ

where R is the nanocrystal radius, h is the Planck’s constant, m0 is the electron mass, while me* and mh* are the electron and hole relative effective masses, respectively (i.e., the effective masses expressed in units of electron mass m0), e is the electron charge, ε0 is the permittivity of vacuum, and εr is the relative dielectric constant of the semiconductor. In the present study, we therefore deal with an assembly of semiconductor nanocrystals in a very weak size quantization regime, that is, we consider a complementary situation to the one addressed in our previous paper.56 The weak size-quantization effects arise due to the fact that the average nanocrystal radius in CdSe thin films synthesized by our chemical method and subsequently annealed at 300 C for 60 min is 12 nm, as compared to the excitonic Bohr’s radius in the case of macrocrystalline (bulk) CdSe of 5.6 nm.144 As it has been mentioned before, to the best of our knowledge, this is one of the first studies devoted to temperature dependence of the band gap energy (and also of the sub-band gap absorption tails) in semiconductor nanocrystal arrays. The dependence of band gap energy on temperature for the studied nanocrystalline assemblies deposited in thin film form, constructed on the basis of previously outlined procedure is presented in Figure 4. We have first fitted the experimental data to the empirical Varshni-type function, which has the form145 Eg ðTÞ ¼ Eg ð0Þ  α

T2 T þ T0

ð12Þ

The value for the temperature coefficient of the band gap energy (i.e., Varshni coefficient α in 12), which we have obtained from the least-squares nonlinear fit to the experimental data, is 9.4
104 eV K1. Compared to the literature data for this quantity in the case of bulk cubic CdSe specimen of 6.96
104 eV K1,146 our present value is somewhat higher (actually, by a factor of 1.35, defined as αnanocrystal/αbulk). In our previous study, where we have considered 3D assemblies of strongly quantized ZnSe QDs deposited in thin film form, we have also found out that the Varshni coefficient is higher than the value characteristic for a bulk cubic ZnSe polymorph.56 However, in the case of strongly quantized ZnSe, the factor αnanocrystal/αbulk was found to be 1.82. This implies that the αnanocrystal/αbulk ratio could increase upon 23247

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Table 2. Values of Parameters Obtained by Least-Squares Fitting of the Experimental Eg = f(T) Data to the Varshni, Bose-Einstein, and Four-Parameter Functionsa Varshni

Eg(0)/eV

α/(eV K1)

T0/K

Bose-Einstein

1.903 Eg(0)/eV

9.4
104 aB/meV

701.6 ΘE/K

1.900

77.4

317.2

Eg(0)/eV

α p/(eV K1)

Θp/K

1.901

4.3
104

265.6

3.0

Eg(0)/eV

α p/(eV K1)

Θp/K

p

3.229

6.5
104

384.3

2.0

4-parameter 4-parameter (ZnSe) a

p

See text for details.

the Eg(T) dependence (in particular for a precise determination of the temperature coefficient of the band gap energy), a further physical insight into the mechanisms responsible for generation of the observed trend could not be obtained solely on the basis of 12. This is certainly due to the empirical character of the function of the form 12. At the same time, aside from the overall goodnessof-fit, Varshni-type function actually shows some deviations from the experimentally observed trend in the region of very low temperatures. Such behavior has been demonstrated as well in a series of other studies2935 and is due to the ad hoc character of this function. To be able to derive a more in-depth insight into the fundamental mechanisms behind the temperature dependence of the band gap energy in the presently studied nanocrystalline films in weak size-quantization regime, we undertook additional analyses of the Eg(T) data. These analyses are based on fitting with functions which have a more exact theoretical background. The first of these functions is of a Bose-Einstein (B-E) type, derived on the basis of explicit consideration of the interaction of the electronic subsystem with the “phonon bath” within the crystal. It has the following form2935 Eg ðTÞ ¼ Eg ð0Þ 

2aB  ΘE exp 1 T 

ð13Þ

In eq 13, aB is a parameter characterizing the magnitude of the electronphonon interaction (coupling), and ΘE is the average temperature of phonons which interact with the electrons (in other words, it is actually the Einstein’s characteristic temperature of the solid state system). Assuming that a Debye phonon spectrum with characteristic Debye’s temperature ΘD holds for an Einstein oscillator with temperature ΘE, the following relation between the two characteristic temperatures holds Figure 4. Temperature dependence of the band gap energy of the synthesized nanostructured CdSe films, fitted to (a) Varshni-type function, (b) Bose-Einstein function, and (c) the four-parameter model function.

enhancement of size quantization effects in semiconductor nanocrystals. It is therefore rather tempting to check out further examples of size-quantized nanocrystals, in various size-quantization regimes, and check out this implication, or even establish a quantitative relation between αnanocrystal/αbulk and the degree of size-quantization. Table 2 summarizes the values of all parameters obtained by fitting the experimental Eg(T) data to a Varshni-type model function. Although Varshni function has been found to be rather useful for quantitative characterization of

ΘD ¼ 4ΘE =3

ð14Þ

The fit to the experimental data by the B-E function of the form 13 is shown in Figure 4b. Table 2, on the other hand, summarizes the fitting parameters and some basic fit statistics data. Converting the obtained Einstein’s temperature value of 317 K to the corresponding Debye’s temperature gives a value of 423 K for the last quantity. Both of these values are somewhat higher than those which have usually been cited for a bulk specimen of this material.147153 This finding, along with the previously discussed higher value for the temperature coefficient of the band gap energy in the present case in comparison to the corresponding bulk value, are in agreement with some other recent findings in the literature concerning these two quantities. For example, 23248

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Wei et al. have recently reported higher linear coefficient of the band gap energy and higher Debye temperature/average phonon energy in nanostructured hybrid organicinorganic compound based on CdSe, in comparison to the corresponding bulk values.148 However, it appears that phonon confinement effects influence the phonon spectrum to a much lesser extent in the presently studied material, due to the smaller size-quantization effects, as opposed to what we have recently found in the case of 3D arrays of strongly quantized ZnSe films deposited in thin film form. A further proof of the previous statement is the fact that the characteristic Einstein temperature obtained by fitting the experimental data to B-E model function of the form of eq 13 corresponds to phonon frequency of about 220 cm1, in excellent agreement with the LO mode frequency of bulk CdSe, which is reported (see refs 154156 and references therein) to fall in the range of 210214 cm1. Recent accumulated evidence has favored the usage of some other analytical model functions to describe the nonlinear part of the Eg versus T dependence at low and intermediate temperatures (T < ΘD), the curvature of which, in a sense, serves as a material’s specific fingerprint.146,157,158 This curvature is in close relation to the center of gravity (Æεæ) and the width (expressed as a root-mean-square distance, Δε) of the spectrum of relevant phonon modes (which make substantial contribution to the observed Eg(T) dependence). The ratio Δε/Æεæ can be used to distinguish between the large, small and intermediate dispersion regimes. In the case of large dispersion (Δε/Æεæ ≈ 0.577), the contribution of low-energy phonons to Eg(T) is comparable to that of the high-energy phonons.146,157,158 The cryogenic region is in such case characterized by a strong curvature of the Eg(T) dependence which apparently tends to a quadratic asymptote. In such cases, it can be roughly fitted by the Varshni’s formula. When the ratio Δε/Æεæ is smaller than 1/3, the regime of small dispersion is entered, in which the contribution of low-energy phonons to the apparent Eg(T) function is very small as compared to the contribution of the high-energy phonons. The Eg(T) dependence in the cryogenic region is in such a case very weak and tends to a plateau behavior (“saturation”). Then it can be fairly well described by a Bose-Einstein type model function. Despite these two limiting cases, however, in most of the conventional semiconductor materials, the dispersion ratio lies between 1/3 and 1/31/2, that is, one deals with the regime of intermediate dispersion. Adequate description of this most frequently encountered dispersion regime imposes the necessity to use a more flexible fitting function. The most frequently used function in this context is the four-parameter model discussed in details in refs 146, 157, and 158. It has the following analytical form: 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 !p u αp Θp 6u p 2T 7  15 ð15Þ Eg ðTÞ ¼ Eg ð0Þ  4t1 þ Θp 2 In eq 15, the parameter α (α = S(∞)) corresponds to the limiting magnitude of the slope of the Eg(T) (entropy) in the limit of very high temperatures (T f ∞), that is, S(T) = dEg(T)/dT, while Θp corresponds approximately to the average phonon temperatures, that is, Θp = Æεæ/kB. The fractional exponent p is related to the material-specific degree of phonon dispersion Δε/Æεæ by Δε 1 ≈ pffiffiffiffiffiffiffiffiffiffiffiffi Æεæ p2  1

ð16Þ

that is, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 Δε p≈ 1 þ Æεæ

ð17Þ

Expressed through the value of p, thus, the regimes of large and small dispersion correspond to p < 2 and p > 3.3, respectively. We have carried out a nonlinear least-squares fit of our experimental data for 3D assemblies of CdSe QDs in weak size-quantization regime with the 4-parameter model function of the form, and the results are summarized in Table 2. For comparison purposes, we have also performed this analysis for the recently studied 3D assemblies of strongly quantized ZnSe QDs in thin film form; the corresponding results have also been tabulated in Table 2. First of all, it can be seen that in the case of both nanostructured materials, the value of p corresponds to the regime of intermediate dispersion, though in the case of strongly quantized ZnSe QDs it is practically at the upper limit toward the regime of large dispersion. The contribution of low-energy phonons to the overall shape of the Eg(T) dependence thus seems to be much more significant in the case of the strongly quantized material. The limiting magnitude of the slope of the temperature dependence of the band gap energy is smaller than the “temperature coefficient” of the band gap as determined from the Varshni function. However, the ratio αp/α varies from 0.4574 for CdSe to 0.7927 in the case of ZnSe, which implies that the larger size quantization effects reflect in the lower temperatures at which the limiting slope is almost reached. 3.3. Urbach-Martienssen Absorption Tails in the Optical Spectra of Thin Films Composed by 3D Assemblies of CdSe QDs in Weak Size-Quantization Regime and Their Temperature Dependence. The issue concerning the existence and quantitative characteristics of the sub-band gap absorption tails in the optical absorption of macrocrystalline and amorphous semiconductors and isolators has been thoroughly addressed, mostly in relation to the quality of a particular material. Considering nanostructured materials, however, the situation seems to be quite opposite. Thus, our recently published study56 was one of the first attempts in the literature to address this fundamentally important aspect related to interaction of nanomaterials with electromagnetic radiation. While the existence of UM tails has been shown to be of quite universal nature in macrocrystalline semiconductors and isolators, the validity of the so-called Urbach rule has not even been tested in the case of nanostructured materials until recently. In practically all known cases, the absorption “tails” are of an exponential form and may be described analytically by a function of the form   E  E0 ð18Þ αðE, TÞ ¼ α0 3 exp EU ðT, XÞ In eq 18, α0 and E0 are constants (E0 nearly coinciding with the energy of the lowest free exciton state at zero lattice temperature), while EU, which is known as Urbach energy, determines the steepness of the Urbach tail. The last parameter appears to be a rather important characteristic of a given semiconducting material. All parameters appearing in eq 18 can be determined from the dependencies of ln(α) on E measured at series of different temperatures, as we demonstrate later. As can be seen from eq 18, EU is a function of temperature and the degree of structural disorder of the studied material, expressed through the parameter X in 18. X is defined as the ratio of the 23249

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Figure 5. Plot of ln(α) vs hv just below the absorption edge for the investigated CdSe nanostructured films in the studied temperature region, demonstrating the validity of Urbach rule.

mean square deviation of atomic positions caused by the actual structural disorder, ÆUx2æ, to the zero-point uncertainty in the atomic positions, ÆU2æ0, that is X ¼

ÆUx2 æ ÆU 2 æ0

ð19Þ

Extending the original formulation of Urbach and Martienssen, Cody et al.159 have expressed the temperature dependence of EU by    Ep Ep X þ coth EU ðT, XÞ ¼ ð20Þ 2σ0 3 2kB T where Ep is the phonon energy, σ0 is a material-dependent parameter (usually of the order of unity), and kB is the Boltzmann constant. Accumulated experimental data, along with some theoretical models, have implied that besides the thermal component to X (giving rise to a temperature-dependent width of the absorption tail), there exists an additional component to the disorder, which has a nonthermal character (i.e., it is a temperature-independent term of the Urbach energy). The last component is particularly clearly manifested in the case of amorphous semiconductors. The Urbach energy of a particular material can thus be split into two contributions, one of which does and the other does not depend on the sample’s temperature: EU ðT, XÞ ¼ EU ðTÞ þ EU ðXÞ

ð21Þ

Cody and co-workers,159 and also Yang et al.160 have proposed the following explicit forms of eq 21:   ΘE 1 þ X 1 þ EU ðT, XÞ ¼ ð22Þ 2 expðΘE =TÞ  1 σ0 3 EU ðT, XÞ ¼ ½EU ðTÞ þ EU ðXÞ   1 ¼A þ B expðΘE =TÞ  1

ð23Þ

The Yang’s eq 23, which may be treated as a generalized form of eq 22, could be derived by modeling the temperature dependence of EU by the concept of Einstein oscillator, taking into account the contributions of dynamical thermal, as well as static

Figure 6. Plot of the calculated Urbach energies vs temperature, together with a fit to the function proposed by Yang et al.

structural and compositional disorders in the sample. Thus, the first term in eq 23 represents the contribution of electron/ excitonphonon interaction like the DebyeWaller factor and the second term is due to the mean-square deviation of atomic positions from a perfectly ordered lattice, caused by the inherent structural disorder. Although a number of theoretical mechanisms have been proposed to explain these absorption phenomena, usually two effects have been given the main emphasis. Both of these effects are closely related to the interaction of excitons with the optical phonons.24 The first one is due to the momentary localization of the excitons in the random, fluctuating field of the optical phonons, and the second one is the ionization of their electric field. Figure 5 shows a series of plots of ln(α) versus hv just below the absorption edge for the investigated nanostructured CdSe films for several temperatures in the studied range. Extrapolations from the plots at various temperatures, confirming that they all converge to the point (E0, α0), are also superimposed on the basic ln(α) versus hv dependencies. This firmly confirms the validity of Urbach rule in the case of the presently studied nanostructured material. The corresponding Urbach energies calculated from the slopes d(ln(α))/d(hv) for the measured data at each temperature have been plotted versus T in Figure 6. A trend conforming to a model function of a general form eq 23 is clearly manifested. Nonlinear least-squares fit with a function of the form of eq 23 to our experimentally determined data points was excellent, judging by all statistical parameters (the corresponding function fitted to the experimental data is also shown in Figure 6 as a continuous line). The parameters A, B, and ΘE, along with their standard errors are given in Table 3. If one compares the absolute values of Urbach energies determined for the presently studied systems to the corresponding values characteristic for other systems where nanocrystalline character is not pronounced at all, it is evident that in the case of our nanostructured films, these values are several times larger.27,28 On the other hand, the present values are approximately three times smaller than those which we have recently reported in the case of strongly quantized ZnSe QDs deposited in thin film form.56 Even a qualitative approach to the problem shows that the degree of nanocrystallinity, expressed, for example, through the magnitude of size-quantization effects, is directly related to the value of Urbach energy of a given nanostructured material. Expressed through quantitative merits of the degree of structural 23250

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Table 3. Values of Parameters Obtained by Least-Squares Fitting of the Experimental EU = f(T) Data to the Function Proposed by Yang et al.a

a

A (meV)

B (meV)

ΘE (K)

88

395

347

See text for details.

disorder (e.g., the dislocation density, average lattice strain, etc.), it is obvious that, in accordance with the model of Cody, EU values are directly proportional to X. The degree of structural disorder in 3D assemblies of QDs is high. The higher the size quantization effects (which are often accompanied by higher ε and δ values), the higher the value of X in 23, as revealed by a direct comparison of the nanostructured systems addressed in our present and previous studies. It is worth noting, however, that such high values for the Urbach energies (i.e., Urbach tails with large widths) have been already found for particular materials, mostly with amorphous or partially crystalline character.4655 Nanostructured thin films, therefore, behave similarly as amorphous materials in this respect. As can be concluded by a more careful inspection of the values given in Table 3, the dynamical term in eq 23 accounts for about 22% of the overall EU values. Though this ratio is larger in the presently studied system than in the case of strongly quantized ZnSe, it is still the intrinsic degree of structural disorder (i.e., the nonthermal contribution) that governs the overall Urbach energy value in the studied nanocrystalline arrays of CdSe QDs. It is worth noting in this context, however, that besides on the degree of structural disorder, a particular EU value for a given material also depends on other material-related parameters (eq 16) such as σ0 and the phonon energy Ep. Even for one and the same material, the Urbach energy varies even in the case of crystals characterized with the same average size, but obtained by different chemical methods.3744 To the best of our knowledge, this is the first experimental study in which the validity of Urbach-Martienssen rule in the case of 3D assemblies of semiconductor QDs in the low sizequantization regime has been demonstrated, and the Urbach energy values have been directly compared to the corresponding ones in various classes of similar systems. According to the model proposed by Cody,159 a simple linear relation should exist between the band gap and the Urbach energy. Expressed in a quantitative manner,   EU ðT, XÞ 1 ð24Þ Eg ðT, XÞ ¼ Eg ð0, 0Þ  ÆU 2 æ0 D EU ð0, 0Þ (In eq 24, Eg(0,0) and EU(0,0) are the band gap energy and Urbach energy at temperature of absolute zero for a material lacking any structural disorder (X = 0), while D is a second-order deformation potential; correspondingly, Eg(T,X) and EU(T,X) are the band gap energy and Urbach energy at finite temperature for a material with a finite degree of structural disorder.) Such linear Eg vs EU relationship has been shown to be valid in cases of various materials, with degree of structural disorder varying in wide ranges.159 Its applicability to the case of nanostructured films, however, has not been tested until recently.56 We have demonstrated its validity in the case of 3D assemblies of strongly quantized ZnSe QDs deposited in thin film form in our previous study.56 Figure 7 shows a plot of the measured Eg versus the calculated EU values in the case of presently studied weakly

Figure 7. Plot of the measured Eg vs the calculated EU values for the studied material, together with a linear least-squares fit.

Figure 8. Plot of σ vs T constructed on the basis of temperaturedependent optical spectroscopy data.

quantized, but yet nanocrystalline, CdSe thin films. The linearleast-squares fit to the experimental data is also shown in this figure. As can be seen from the plot, and also from the statistical parameters characterizing the goodness-of-fit, the observed dependence is fairly linear, which further supports the model of Cody with this respect as well. The dependence of absorption coefficient on the incident photon energy in the “Urbach region” can be expressed through another variant of eq 18, that is, including the steepness parameter σ:2935   σðTÞ ðhv  E0 Þ αðhv, TÞ ¼ α0 exp ð25Þ kB T Comparing to eq 18 shows that EU = kBT/σ. The usefulness of such alternative variant to the function 25 lies in the fact that σ can be expressed as2935     2kB T Æεæ σðTÞ ¼ σ 0 tanh ð26Þ Æεæ 2kB T where σ0 is temperature-independent but material-dependent parameter, which is inversely proportional to the strength of the coupling between electrons/excitons and phonons, while Æεæ (=hvp) 23251

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is the average energy of phonons that contribute to the UM tails. It therefore follows that, from the data obtained by experimental studies of temperature dependence of UM tails, valuable information concerning the electron/excitonphonon interactions in semiconductors can be obtained. This is particularly relevant in the case of nanostructured materials, where simply due to the spatially confined motion of the charge carriers the electrostatic interaction between them could be enhanced. Figure 8 shows a plot of the steepness parameter σ versus T, constructed from our experimental optical absorption data. The fitting model function of the form of eq 26 deduced from a nonlinear least-squares fit to our experimental data is also shown in Figure 8. It is evident that the fit is excellent. The fitting parameters σ0 and hvp are 0.109 and 713.2 cm1 (88.5 meV) correspondingly, the last value being significantly larger than the reported position of fundamental LO phonon modes in the Raman spectra of CdSe (see refs 154156 and references therein). Such observation thus seems to be in line with our conclusion outlined before in this paper, that the regime of small dispersion in the studied nanostructured semiconductor implies the relevance of higher-energy phonons to the processes of absorption of electromagnetic radiation. However, note that high values of average phonon frequencies that contribute to the UM tails have been reported in various systems which are not characterized by nanocrystalline nature.161167 Thus, in the case of IIVI type of semiconductors of zinc-blende structural type, higher values of hvp than the LO phonon energies have been attributed to higher crystal symmetry.166,167 In cases of some ternary compounds, mixed chalcogenides, and so on, on the other hand, structural disorders of various types (e.g., cation cation disorder, cation vacancies and interstitials, even deviations from ideal stoichiometry)161165 have been pointed out as the main cause of such observations. The last arguments could be valid for the presently studied system as well, having in mind the relatively high level of structural disorder in 3D arrays of QDs.

4. CONCLUSIONS In this work, we have studied the temperature dependence of the band gap energy and of the sub-band gap absorption tails in the case of chemically deposited thin films constituted of close packed cadmium selenide quantum dots in very weak size-quantization regime (i.e., with very slight size-quantization effects). We have covered the temperature range from 11 up to 340 K. This is one of the first such studies in the literature related to the QD solids and is complementary to our recent work which addresses the related problems in the case of quantum dot solids composed of strongly quantized ZnSe QDs.56 The most important insights from the study may be summarized as follows: • Clearly defined excitonic absorption peaks have not been observed even at temperatures as low as 11 K. This is attributed to the particle size distribution and the interdot electronic coupling effects between proximal QDs. • Analysis of the temperature dependence of the band gap energy by Varshni-type function has led to a value of 9.4
104 eV K1 for the temperature coefficient of the band gap energy in the presently studied system, which is somewhat higher (by a factor of 1.35, defined as αnanocrystal/αbulk) compared to the corresponding value for bulk cubic CdSe specimen (6.96
104 eV K1). In the case of films composed of strongly quantized ZnSe QDs, studied in our previous paper,56 the factor αnanocrystal/αbulk was found to

•

•

•

•

•

be 1.82. This implies that the αnanocrystal/αbulk ratio increases upon enhancement of size quantization effects in semiconductor nanocrystals. Analysis of the Eg(T) dependence by fitting to a BoseEinstein-type function, implies that no phonon confinement effects influence the phonon spectrum in the presently studied material to a significant degree, due to the very small size-quantization effects. This situation is opposite to what we have recently found in the case of 3D arrays of strongly quantized ZnSe films deposited in thin film form. The characteristic Einstein temperature for the presently studied nanostructured material corresponds to phonon frequency of about 220 cm1, which is in excellent agreement with the LO mode frequency of bulk CdSe (reported to fall in the range of 210214 cm1). The four-parameter model for the temperature dependence of Eg allowed us to conclude that the value of characteristic parameter p corresponds to the regime of intermediate dispersion in the case of both CdSe and ZnSe, though in the case of strongly quantized ZnSe QDs it is practically at the upper limit toward the regime of large dispersion. The contribution of low-energy phonons to the overall shape of the Eg(T) is, however, much more significant in the case of the strongly quantized material. In both studied materials, the limiting magnitude of the slope of the temperature dependence of the band gap energy is smaller than the “temperature coefficient” of the band gap as determined from the Varshni function. However, the ration αp/α varies from 0.4574 for CdSe to 0.7927 in the case of ZnSe, which implies that the larger size quantization effects reflect in the lower temperatures at which the limiting slope is almost reached. The Urbach rule was found to hold for the 3D assemblies of weakly quantized CdSe nanocrystals deposited in thin film form. Due to the relatively high degree of structural disorder, in the studied QD solids, the values of Urbach energies are several times higher than those usually found in macrocrystalline semiconductors. In comparison to the recently studied 3D assemblies of strongly quantized ZnSe QDs, however, the present Urbach energy values are approximately three times smaller, in line with the smaller degree of inherent structural disorder in CdSe NC arrays. Temperature dependence of Urbach energy could be excellently fitted to a Cody or Yang-type functions. The dynamical thermal (temperature-dependent) contribution to EU was found to account for about 22% in the overall Urbach energy values in the investigated 3D QD arrays, which is much less than in the case of macrocrystalline semiconductors, but a somewhat higher value than that observed in the case of ZnSe QD arrays (15%). An excellent linear correlation was found between the Urbach energies and band gap energies of the studied materials, in accordance with the predictions of Cody’s model.

’ AUTHOR INFORMATION Corresponding Author

*Tel.: +389-2-3117-055. Fax: +389-2-3226-865. E-mail: biljana@ iunona.pmf.ukim.edu.mk.

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