Conventional Optics from Unconventional Electronics in ZnO Quantum

May 5, 2010 - Coulomb interactions lower the bright (electron-S−hole-S) exciton below the orbitally forbidden (electron-S−hole-P) exciton to recov...
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J. Phys. Chem. C 2010, 114, 9301–9307

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Conventional Optics from Unconventional Electronics in ZnO Quantum Dots Sotirios Baskoutas† and Gabriel Bester* Max-Planck-Institut fu¨r Festko¨rperforschung, D-70569 Stuttgart, Germany ReceiVed: March 3, 2010; ReVised Manuscript ReceiVed: April 12, 2010

We study the electronic and optical properties of ZnO quantum dots within the atomistic empirical pseudopotential framework. The highest occupied molecular orbital (HOMO) is found to be of orbital P character for structures larger than 2.6 nm in diameter. We identify the origin of this unconventional situation in the electronic character of the HOMO state, originating from an even mixture of the A- and B-bands of the Wurtzite band structure. This situation, however, does not lead to an orbitally dark exciton ground state, as one might expect. Coulomb interactions lower the bright (electron-S-hole-S) exciton below the orbitally forbidden (electron-S-hole-P) exciton to recover the conventional situation of an orbitally allowed, but spinforbidden, exciton ground state and a Stoke’s shift originating from electron-hole exchange interactions. Introduction Zinc oxide (ZnO) is a material with a great variety of technological applications, such as surface acoustic wave devices, varistors, piezoelectric transducers, optical waveguides, transparent conductive oxides, chemical and gas sensors, spin functional devices, and UV-light emitters.1 Its wide band gap (3.445 eV) makes ZnO a promising material for photonic applications in the UV or blue spectral range, while the high exciton binding energy (around 60 meV) allows efficient excitonic emission, even at room temperature. Additionally, ZnO doped with transition metals is a promising candidate for spintronics applications.2 Finally, ZnO is an environmentally friendly material, which is desirable, especially for bioapplications such as bioimaging and cancer detection. Although bulk ZnO has been investigated for many years, in the past few years ZnO nanostructures became the subject of renewed interest due to the important modifications that appear at the nanoscale in the optical properties3,4 and also due to the increasing demands for green materials. The most recent developments are toward the fabrication of zero-dimensional ZnO clusters, or quantum dots (QDs), through different methods such as thermal vapor transport and condensation,5 polyol methods,6 thermolysis,7 precipitation,8 sol-gel,9 and microemulsion.10 Optical properties11,12 and spin dynamics13 have already been investigated. The hopes for future applications span a wide range of areas, such as ultraviolet photovoltaic devices,14,15 laser diodes,16 ultraviolet photodetectors,17 and catalysts.18 From a theoretical standpoint, the subject has mainly been addressed at the effective mass or envelope function level. These approaches may be appropriate for larger structures,19 but are questionable for small colloidal quantum dots. Furthermore, the electronic structure of bulk ZnO is rather exotic, with a negative spin-orbit splitting leading to almost degenerate Wurtzite A and B valence bands. A situation that is in contrast to the exhaustively studied CdSe quantum dots, where we can expect to discover new physics. In this contribution we derive a new empirical pseudopotential for ZnO that enables us to calculate the electronic and the optical properties of Wurtzite quantum dots of realistic sizes at an * To whom correspondence should be addressed: E-mail: g.bester@ fkf.mpg.de. † Permanent address: Department of Materials Science, University of Patras, 26504 Patras, Greece.

atomistic level, including all relevant effects such as multiband coupling and spin orbit interaction. We find that the highest occupied molecule orbital (HOMO) state for structures larger than 2.6 nm diameter have P-like orbital character. This unusual situation was predicted earlier for CdSe, CdTe, and CdS for sufficiently small structures by continuum theories (envelope function approximation, k.p.-method20-22) but was later found to be erroneous based on atomistic calculations.23-25 So, this type of orbitally forbidden HOMO state we find in ZnO nanocrystals seems to be yet unobserved. We find the reason for this behavior in the mixed Bloch A- and B-band character of the HOMO state to be a situation related to the peculiar bulk band structure of ZnO. Performing calculations of the Coulomb and exchange integrals, we find, however, that the exciton derived from an electron in the lowest unoccupied molecular orbital (LUMO) and a hole in the HOMO does not represent energetically the lowest state. Coulomb interactions favor energetically the optically bright exciton, where both electron and hole occupy an S-like state. Hence, the orbitally forbidden HOMO-LUMO exciton is promoted to an excited state and the conventional situation (e.g., CdSe), where the lowest exciton state is derived from conduction band S and valence band S states, is recovered. The lowest exciton state is therefore a spinforbidden (but not orbitally forbidden) dark state and the Stokes shift is given by the electron-hole exchange splitting. We derive scaling laws for the experimentally relevant quantities. Method We follow the empirical pseudopotential method26 and derive new empirical pseudopotentials VR for Zn and O. The Hamiltonian has the form

ˆ ) - 1 ∇2 + H 2

∑ [VR(r - Rn) + VˆSO R ]

(1)

nR

ˆ SO where n is an atomic index, R specifies the atom type, and V R is the nonlocal spin-orbit operator, including one parameter λ for each atom type. A review of the method employed in this work can be found in ref 27. The screened atomic pseudopotentials VR (with R ) Zn, O) are centered at each atomic position and their superposition generates the crystal potential. The

10.1021/jp101921g  2010 American Chemical Society Published on Web 05/05/2010

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TABLE 1: Compiled Reference Bulk Properties and Empirical Pseudopotential Results (EPM Results) for ZnO Using the Parameters from Table 3 property

experiments

theory

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ε(Γ7V–Γ3V) ε(Γ1c–Γ7V) ε(Γ3c–Γ1c) ε(Γ6c–Γ1c) ε(H3c–Γ1c) ε(K2c–Γ1c) ε(M1c–Γ1c) m⊥e m|e mA⊥ mA| mB⊥ mB| mC⊥ mC| ∆so ∆cr

30

5.0 3.4449,34 3.435,35 3.437636

TABLE 2: Structural Parameters for ZnO 3.249 Å52 5.205 Å52

6.14110 a.u. 9.8373 a.u. 0.38253 1.6019

TABLE 3: Empirical Pseudopotential Parameters, Described in Eq 2 (a1, 2, 3, 4) and Spin-Orbit Interaction Parameter λa Zn O a

32

33

3.8, 5.517, 8.92, 5.0 3.4437,38 3.45831 3.666,39 4.39431 13.21,40 7.303,39 7.320,39 7.389731 6.44.,41 5.00,40 7.10531 7.25,41 6.75,42 5.99,40 6.83131 5.03,42 5.11,43 5.01731 0.177,37,38 0.21,45 0.130,46 0.21131 0.21,45 0.137,46 0.22531 0.351,46 4.31,47 2.589931 1.98,47 3.06,39 1.09131 0.30,39 0.55,39 0.227,46 0.581331 2.979,46 3.227,46 4.330,46 3.06,39 0.845431 0.288,46 0.537,46 1.12,39 0.176931 0.169,46 0.330,46 0.26,39 0.27,39 0.207131 –0.003537 0.0391,37 0.039231

0.26544 0.26544 0.5936 0.5936 0.5936 0.5936 0.5536 0.3136 –0.0035,48 –0.004734 0.0404,49 0.040834

a c u c/a

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λ (spin-orbit)

a1

a2

a3

a4

0.00533 0.0

–8.561564 –28.419299

0.0062352 4.2557852

–0.001589 –0.0100096

0.5521333 0.95926425

EPM

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3.454 3.458 3.032 8.590 4.655 5.361 3.502 0.213 0.239 0.712 2.120 0.515 0.572 1.075 0.250 –0.0035 0.040

5.0 3.444934 3.66639 7.30339 5.0040 5.9940 5.0342 0.26544 0.26544 0.5936 0.5936 0.5936 0.5936 0.5536 0.3136 –0.0035 0.040

parameters given in Table 2 (see ref 51 for more information about the bulk structure). Our computed values for in-plane and perpendicular components of electron and hole effective masses ⊥ | at the Γ point (m⊥e , m|e, mA, B, C and mA, B, C), the energy gaps at high symmetry points, the values for the crystal field splitting, as well as the spin-orbit splitting, are given in Table 1 and are shown to be in good agreement with experimental results. we project To visualize the quantum dot wave functions ψQD i them onto the bulk ZnO Wurtzite wave functions with band bulk : index n and wave vector k: ψnk

|ψiQD〉 )

See ref 27.

targets

bulk QD ∑ |ψbulk nk 〉〈ψnk |ψi 〉

(4)

n,k

surface passivation is approximated by a high-band gap artificial material, as practiced successfully previously.28 The empirical pseudopotentials used in this work have four free parameters (a1,2,3,4) and are defined in reciprocal space by the analytic form

VR(q) )

bbcin )

a1(q2 - a2)

1/2 1 8 EAC(Γ7ν) ) ( (∆so + ∆cr)2 - ∆so∆cr 2 3 1 EB(Γ9ν) ) (∆so + ∆cr) 2

]

(3)

During the fit, we put a larger emphasis (larger weight) on the most relevant quantities close to the conduction band minimum (CBM) and valence band maximum (VBM), while we relaxed the constraints (smaller weights) for the band structure parameters remote from CBM and VBM. We used the experimental bulk Wurtzite structure in the pseudopotential generation (and later in the quantum dot calculations) with the

i i* cn,k ∑ cn,k

(5)

k

(2)

2

a3ea4q - 1

The parameters are optimized to reproduce the known bulk properties of ZnO we listed in Table 1. The target properties include specific points of the electronic Brillouin zone (ε), the effective masses (perpendicular m⊥, and parallel m| to the c-axis) of electron me and hole mA, mB, and mC for the top three valence bands A, B, and C. We indicate the values of the spin-orbit ∆so and crystal-field ∆cr splittings following the conventional quasicubic model of Hopfield50

[

i bulk QD The expansion coefficients cn,k ) 〈ψnk |ψi 〉 are used to determine the bulk band character

of the dot wave function i. By chopping the quantum dot wave functions into bulk-cells, and projecting each of them onto bulk states, we obtain space-resolved bccni(x, y, z)s. Summing them over bands leads to the envelope functions

envi(x, y, z) )

∑ bccin(x, y, z)

(6)

n

The calculation of excitonic states requires the two center Coulomb and exchange integrals

〈ejhi |V|hi'ej'〉 )

∫ ∫ ψ*(r j e)ψ*(r i h)V(re′rh)ψi′(re)ψj'(rh)dredrh (7)

For the screened Coulomb interaction ν(re, rh), we used54 the phenomenological isotropic and uniform model proposed by Resta.55 The required parameters are given in ref 51, while a review of the method56 can be found in ref 27. The optical dipole matrix elements are calculated within the dipole approximation and the oscillator strength was calculated using Fermi’s golden rule.

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Figure 2. Geometry of our smallest Zn87O81 cluster.

Figure 1. (a) Symmetries of the conduction band minimum (Γ6, 1, 7, c) and valence band maximum (Γ5, 7, 8, 9, V) for bulk ZnO with zinc-blende (Td point group) and Wurtzite structures (C6V point group). (b) Topmost three valence bands around Γ, neglecting spin-orbit coupling and (c) including spin-orbit coupling.

Band Structure of Bulk Wurtzite ZnO. Before we discuss the results on nanostructure, we briefly describe the valence band of bulk ZnO because this will be relevant to our discussion. The valence band structure of ZnO is qualitatively given in Figure 1a for the zinc-blende structure (Td, hypothetical structure only used for illustration purposes) and the Wurtzite structure (C6V). The characteristic features of Wurtzite is the crystal field, splitting the Γ5V (Td) band into a Γ5V and Γ1V (C6V). The characteristic feature of ZnO is an inVerted spin-orbit splitting. In the zinc-blende structure, the Γ8V(J ) 3/2) band is below the Γ7V(J ) 1/2) band, which is unconventional. In the Wurtzite structure, this translates into a Γ7V band being above the Γ9V band. The quantitative results using our new pseudopotentials are given in Figure 1b for the case neglecting spin-orbit interaction and in Figure 1c including it (our subsequent calculations will include spin-orbit interactions). The origin of this negative spin-orbit has been discussed in the literature initially for cuprous halides57 and has been attributed to the spin-orbit effect of the cation d states.58 Spin orbit has only a small direct effect on the light oxygen anion p states forming the VBM, but a much larger effect on the zinc cation d band, splitting it into a higher energy J ) 5/2 and a lower energy J ) 3/2 state. This specific order (J ) 5/2 states being above the J ) 3/2 states) adds a negative term to the spin-orbit splitting of the valence band at the Γ-point. In the case of ZnO, the order is just reversed, as shown in Figure 1c. The reversed order is not a unique situation, and it can be expected following

Figure 3. (a) Single particle electron and (b) hole states as a function of the dot diameter. The energy is given relative to the VBM of bulk ZnO. (c) Optical gap (including Coulomb interactions) is represented by a black line and single-particle gap is represented as a gray line. The red stars,62 the blue diamonds,63 and the triangles64 correspond to experimental results. (d) Stokes shift of the lowest exciton transition.

the simple argument above for materials with heavy cations and light anions. It has been measured in the case of CuCl57 and predicted theoretically by density functional theory (although the method tends to overestimate the effect of d states due to the typically too large p-d couplings) for CdO, HgO, and HgS.59 For ZnO, the splitting between the A and B bands is very small, around -3.5 meV (see Table 1), and the material could be described as having vanishing effective spin-orbit splitting, similar to InN,60 with a splitting of +9 meV. Electronic Properties Before we apply the methodology to quantum dots, we tested the passivation of the structure on ZnO quantum wells. Indeed, the passivation of the dangling bonds requires careful testing and monitoring of the appearance of surface states. We tested five different passivating potentials with similar large band gaps of 5.7 eV, corresponding approximately to organic capping

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Figure 4. Envelope functions of the first four electron states, e0, e1, e2, and e3, for a QD with diameter of 3.1 nm.

Figure 5. Envelope functions (eq 6) of the first three hole states, h0, h1, and h2, for QDs of varying diameter.

groups but different effective masses. We find that potentials with electron and hole effective masses similar to the ZnO effective masses remove the surface states most effectively from the gap. A good surface passivation, however, seems easier to achieve for ZnO than for CdSe or GaN, and the electronic states are rather insensitive to the passivant as long as it removes unwanted states from the energetic area of interest. Results for quantum wells are presented in Supporting Information. We define five different ZnO colloidal quantum dots of diameter d ) 1.7, 2.1, 2.6, 3.1, 3.6 nm with the crystal parameters from Table 2. The numbers of atoms for the respective structures are Zn87O81, Zn192O198, Zn381O372, Zn669O678, and Zn1071O1050. A pictorial representation of our smallest structure is given in Figure 2. Smaller clusters are usually experimentally not available. Furthermore, they would be significantly deformed by surface reconstructions61 and would not be meaningfully addressed by the present approach. The diameter dependence of the electron and hole eigenvalues are given in Figure 3a,b. For the electron states, we can recognize the atomic shells with S(e0), P(e1, 2, 3), and D(e4, 5, 6, 7, 8) levels split by the crystal field. No such shells are present for the holes. To visualize the wave functions, we project the fast oscillating atomic wave functions onto the ZnO Bloch states according to eq 6. This leads to the envelope function of the dot, which is more convenient to visualize than the fast oscillating real wave functions. The results for the first four electron states e0, 1, 2, 3 are given in Figure 4. The first state is S-like, while the next three states are P-like. The state e1 is elongated along the c-axis, while e2, 3 are inplane. This fact will have consequences for the optical polarization of the transitions. Figure 5 shows the envelope functions, eq 6, of the first three hole states, h0, 1, 2, for QDs with five different diameters. For the smaller diameter we see three states with S-like envelope functions, whereby h2 has some deformation along the c-axis. When the diameter of the dot is increased, we notice that the state h0 (HOMO) changes from an S-like to a P-like state. The transition occurs at around 2.6 nm diameter. We now revisit the diameter dependence of the hole-state eigenvalues from Figure 3b and plot them now relative to the first hole state with S-character (which now has an energy of 0 eV) in Figure 6.

Figure 6. Single particle energies of the first six hole states h0,1,2,3,4,5 vs quantum dot diameter. The energies are relative to the first hole state with S-orbital character. The dominant orbital character of the states is indicated by the letters S and P.

It becomes clear that state h3 with P-like envelope at a small diameter becomes h0 at large diameters. The S-like and P-like hole state exhibit a size dependence that is different enough to lead to crossings. The marking result is that the first hole state h0, or HOMO state, is for larger structures of P-character. To understand this behavior, we project the single particle quantum dot hole states h0, 1, 2 on bulk ZnO bands, according to eq 5. The results for the three hole states are given in Figure 7, where the contribution of the A-, B-, and C-bands are given in percent. The regions where the states have orbital S-like and P-like character are marked. The h0 state has dominant A-band character below 2.6 nm diameter while it has an evenly mixed A- and B- band character above this critical value, which corresponds to the transition between S-like and P-like orbital character. A similar behavior is observed for the states h1 and h2 at 2.6 and 2.1 nm, respectively, where they have P-like character. At these radii, the Bloch function character is evenly mixed. We conclude that the existence of a HOMO state with P-character is linked to a mixed Bloch character of its wave function. It strikingly demonstrates that the energetics are not

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J. Phys. Chem. C, Vol. 114, No. 20, 2010 9305 discussed earlier for CdSe, CdTe, and CdS,20,22-24,65 where spin-orbit interactions was responsible for S and P state ordering. Optical Properties

Figure 7. Analysis of the Bloch function character of the first three hole states h0,1,2 for five different dots. The band labels A, B, and C refer to the Wurtzite bulk bands (Figure 1c). The letters S and P refer to regions of the dot diameter where the states are S-like and P-like (Figure 6).

governed by the orbital character of the envelope function alone, but by the atomistic full wave function derived from a mixed Γ7V and Γ9V Bloch function in this case. This mixed character is a direct consequence of the almost degenerate nature of the A-(Γ7V) and B-(Γ9V) bands in ZnO, as described previously (see Figure 1), a situation that is rather unique (only InN shows a similar valence band structure) and much in contrast to most of the heavily investigated quantum dot materials (CdSe, CdS, PbSe, InP, InAs, GaP, etc.). To conclude, we have to mention that we have studied the ordering of the states for the same QD diameters and neglecting the spin-orbit interaction. The ordering remains the same and the eigenvalues are shifted by less than one meV. The situation described here is, therefore, very different than the issues

We calculate the Coulomb and exchange integrals according to eq 7 to obtain the excitonic properties, and we calculate the optical dipole matrix elements to obtain the optical properties. In Figure 8 we illustrate the effects of adding Coulomb and exchange interaction terms to the uncorrelated electron-hole pair. In the left panels of Figure 8a and b, we give the difference in the single particle energies for the electron and hole states shown in Figure 3a,b. Figure 8a shows the results for a dot with diameter of 2.1 nm, that is, when the HOMO state has orbital S-like character. Figure 8b shows the results for a dot with 3.6 nm diameter, that is, after it changes from an S-like to a P-like orbital character (see Figure 6). For the small dot, the lowest electron-hole pair energy corresponds to a bright state (solid red line) where both electron and hole states have S-like character. For the larger dot, the lowest energy state corresponds to a hole with P-like character and electron with S-like character and is therefore optically dark (dashed black line). The modification through the inclusion of Coulomb and exchange interactions can be seen in the right panels of Figure 8a and b. In the case of the smaller dot, the lower two (bright) states are shifted down through direct Coulomb attraction by 352 meV, while the third (dark, P-like) state is lowered only by 337 meV. In the exciton picture (right panels), the lowest eight states (counting spin) originate from electron and hole states with orbital S-character. The exchange interaction (illustrated by the letter K) splits these states into optically spin-forbidden states and spin-allowed states. The spin-forbidden states are energetically lower and represent the exciton ground state. In the case of the larger dot (Figure 8b), the situation is unexpectedly very similar, despite the reverse order of dark and bright states at the electron-hole pair level (neglecting Coulomb and exchange). The reason for this restored situation is the larger Coulomb interaction felt by the bright states (225 meV) and than felt by the dark state (196 meV). Hence, in the exciton case the lowest eight states (counting spin) originate from electron and hole states with orbital S-character, much like in

Figure 8. Left panels of (a) and (b) show the electron-hole pair energy, neglecting Coulomb and exchange interactions. The right panels of (a) and (b) show the exciton energy (including Coulomb and exchange integrals). Red lines show optically bright states, while dashed black lines are the optically dark states.

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Figure 9. (a) Oscillator stengths for the transition |0〉 to |X〉 in a 3.1 nm diameter ZnO quantum dot. (b) Single particle eigenvalues labeled from 0 to 3 for electron states and 0 to 40 for hole states, with energy given in eV. The numbers on top of the absorption peaks in (a) refer to the dominant single particle levels involved in the transition. Transitions polarized along the c-axis are shown in red, while the transitions polarized in-plane are shown in black.

the case for smaller dots. The exchange interaction has a smaller magnitude than in the case of the smaller dot but, qualitatively, the results are the same, with an orbitally allowed but spin forbidden exciton ground state. The optical gap for the full range of diameters is given in Figure 3c. The direct Coulomb interactions (the difference between the gray and black curves in Figure 3) varies between 413 meV for a diameter of 1.6 nm to 219 meV for 3.6 nm diameter, following a nearly linear behavior, best fitted by ∆Coul ) 0.6364 - 0.1644d + 0.0136d2, in eV and nm. The dependence of the optical gap can be fitted by Eg ) 3.235 + 1.602d-1.17, in eV and nm, within the range studied. In Figure 3c we collected the experimental value from literature and noticed a very good agreement. The Stokes shift, defined as the energetic difference between the lowest exciton state and the first bright state, is given in Figure 3d. It can be approximately fitted (blue dashes line) by the power law: ∆Stokes ) 35.2844d-0.71 - 11.3773, in meV and nm. The best fit (red line), however, is achieved by the exponential ∆Stokes ) 46.4 exp(-0.754d). Finally, we calculate the oscillator strength for the transition from the ground state |0〉 to the exciton state |X〉 in Figure 9. This corresponds to a measurement of the absorption or photoluminescence excitation (PLE); however, dynamical effects of carrier relaxation may change the relative intensities in the experimental spectra, so a one to one correspondence of the intensities might not be given. To include the electron states up to the orbital P-like states, we must include hole states up to the same single-particle excitation energy, which corresponds to 40 hole states. The results for the oscillator strength are shown in Figure 9a for a QD with a diameter of 3.1 nm. The pair of numbers above the peaks indicate which single particle state is dominantly involved in the respective transitions. Transitions polarized along the c-axis are shown in red, while the transitions polarized in-plane are shown in black. The energy eigenvalues of the single-particle states are given in Figure 9b. Summary In the present work we derive a new empirical pseudopotential for ZnO that enables us to calculate the electronic and optical properties of nanostructures with several thousand atoms. This allows us to make contact with the experimentally realized

Baskoutas and Bester colloidal quantum dot structures with a few nm diameter. The excellent agreement with available experimental data validates our approach. Our main findings are (i) that the highest occupied dot state (HOMO) has orbital P-character for structures larger than 2.6 nm in diameter. This yet unobserved situation originates from an evenly mixed Bloch function character (from the Aand B-bands of the Wurtzite band structure) of the quantum dot state. It is directly linked to the unusual situation in bulk Wurtzite ZnO where the A- and B-bands are nearly degenerate. (ii) Although this situation fulfills the premises for an optically orbital-forbidden exciton ground state, few body Coulomb interactions restore the more conventional situation of a spinforbidden ground state. Indeed, the optically allowed electronS-hole-S exciton is lowered below the HOMO-LUMO state. We present scaling laws for exciton energies and Stoke shifts, as well as a prediction for forthcoming optical absorption or photoluminescence excitation experiments. Supporting Information Available: The form of the new empirical pseudopotential for ZnO, the band structure of ZnO, and calculation details. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes ¨ zgu¨r, U.; Alinov, Y. I.; Liu, C.; Teke, A.; Reshchikov, M. A.; (1) O Dog˘an, S.; Avrutin, V.; Cho, S.-J.; Morkoc¸, H. J. Appl. Phys. 2005, 98, 041301. (2) Ghosh, S.; Sih, V.; Lau, W. H.; Awschalom, D. D.; Bae, S. Y.; Wang, S.; Vaidya, S.; Chapline, G. Appl. Phys. Lett. 2005, 86, 232507. (3) Djurisˇic´, A. B.; Leung, Y. H. Small 2006, 2, 944. (4) Li, J.; Wang, L.-W. Phys. ReV. B 2005, 72, 125325. (5) Lao, J.; Huang, J.; Wang, D.; Ren, Z. Nano Lett. 2003, 3, 235. (6) Bouropoulos, N.; Tsiaoussis, I.; Poulopoulos, P.; Roditis, P.; Baskoutas, S. Mater. Lett. 2008, 62, 3533. (7) Salavati-Niasari, M.; Davar, F.; Fereshteh, Z. Chem. Eng. J. 2009, 146, 498. (8) Seelig, E. W.; Tang, B.; Yamilov, A.; Cao, H.; Chang, R. Mater. Chem. Phys. 2003, 29, 257. (9) Lin, K.; Cheng, H.; Hsu, H.; Lin, L.; Hsieh, W. Chem. Phys. Lett. 2005, 409, 208. (10) Gao, L.; Ji, Y.; Xu, H.; Simon, P.; Wu, Z. J. Am. Chem. Soc. 2002, 124, 14864. (11) Germeau, A.; Roest, A. L.; Vanmaekelbergh, D.; Allan, G.; Delerue, C.; Meulenkamp, E. A. Phys. ReV. Lett. 2003, 90, 097401. (12) Sun, D.; Sue, H.-J.; Miyatake, N. J. Phys. Chem. C 2008, 112, 16002. (13) Jansen, N.; Whitaker, K.; Gamelin, D. R.; Bratschitsch, R. Nano Lett. 2008, 8, 1991. (14) Schrier, J.; Demchenko, D. O.; Wang, L. W.; Alivisatos, A. P. Nano Lett. 2007, 7, 2377. (15) Li, F.; Cho, S.; Son, D.; Kim, T.; Lee, S.-K.; Cho, Y.-H.; Jin, S. Appl. Phys. Lett. 2009, 94, 111906. (16) Lin, C.; Zapien, J.; Yao, Y.; Meng, X.; Lee, C.; Fan, S.; Lifshitz, Y.; Lee, S. AdV. Mater. 2003, 15, 838. (17) Jun, J. H.; Seong, H.; Cho, K.; Moon, B.-M.; Kim, S. Ceram. Int. 2009, 35, 2797. (18) Strunk, J.; Ko¨hler, K.; Xia, X.; Muhler, M. Surf. Sci. 2009, 603, 1776. (19) Alim, K. Appl. Phys. Lett. 2005, 86, 053103. (20) Richard, T.; Lefebvre, P.; Mathieu, H.; Alle`gre, J. Phys. ReV. B 1996, 53, 7287. (21) Yu, Z.; Li, J.; O’Connor, D.; Wang, L.-W.; Barbara, P. J. Phys. Chem. B 2003, 107, 5670. (22) Li, J.; Xia, J.-B. Phys. ReV. B 2000, 62, 12613–12616. (23) Demchenko, D.; Wang, L.-W. Phys. ReV. B 2006, 73, 155326. (24) Fu, H.; Wang, L.-W.; Zunger, A. Phys. ReV. B 1998, 57, 9971. (25) The difficulty in obtaining reliable results for the valence band states in quantum dots using the envelope function approximation was demonstrated for CdSe,66,67 where two envelope function descriptions, with slightly different sets of (Luttinger) parameters, lead to different hole ground states. One with S-like66 and one with P-like envelope functions.67 (26) Wang, L.-W.; Zunger, A. Phys. ReV. B 1995, 51, 17398. (27) Bester, G. J. Phys.: Condens. Matter 2009, 21, 023202. (28) Califano, M.; Bester, G.; Zunger, A. Nano Lett. 2003, 3, 1197. (29) Powel, R.; Spicer, W.; McMenamin, J. Phys. ReV. B 1972, 6, 3056. (30) Ro¨ssler, U. Phys. ReV. 1969, 184, 733.

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