Terzis Department of Physics University of Patras GR26504 Greece Received 21 July 2005 accepted 15 November 2005 published online 11 January 2006 The sizedependent band gap of semiconductor quantum dots is a wellknown and widely studied quantum con6 ID: 26123 Download Pdf

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Terzis Department of Physics University of Patras GR26504 Greece Received 21 July 2005 accepted 15 November 2005 published online 11 January 2006 The sizedependent band gap of semiconductor quantum dots is a wellknown and widely studied quantum con6

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Size-dependent band gap of colloidal quantum dots Sotirios Baskoutas Materials Science Department, University of Patras, GR-26504, Greece Andreas F. Terzis Department of Physics, University of Patras, GR-26504, Greece Received 21 July 2005; accepted 15 November 2005; published online 11 January 2006 The size-dependent band gap of semiconductor quantum dots is a well-known and widely studied quantum conﬁnement effect. In order to understand the size-dependent band gap, different theoretical approaches have been adopted, including the effective-mass approximation with

inﬁnite or ﬁnite conﬁnement potentials, the tight-binding method, the linear combination of atomic orbitals method, and the empirical pseudopotential method. In the present work we calculate the size-dependent band gap of colloidal quantum dots using a recently developed method that predicts accurately the eigenstates and eigenenergies of nanostructures by utilizing the adiabatic theorem of quantum mechanics. We have studied various semiconductor CdS, CdSe, CdTe, PbSe, InP, and InAs quantum dots in different matrices. The theoretical predictions are, in most cases, in good

agreement with the corresponding experimental data. In addition, our results indicate that the height of the ﬁnite-depth well conﬁning potential is independent of the speciﬁc semiconductor of the quantum dot and exclusively depends on the matrix energy-band gap by a simple linear relation. 2006 American Institute of Physics DOI: 10.1063/1.2158502 I. INTRODUCTION Quantum-conﬁned semiconductor structures, including quantum wells, quantum rods, and quantum dots QDs , have been extensively investigated in the past few years. 1–3 One of the most interesting effect of

low-dimensional semiconductor structures is the size-dependent band gap. 4–10 The theoretical investigation of this phenomenon includes several methods, such as the effective-mass approximation EMA 11 the kp method, 12 the tight-binding approach, 13 the linear combina- tion of atomic orbitals method, 14 and the empirical pseudo- potential method. 15 The oldest and least computationally de- manding approach is the EMA, which has relied mostly on inﬁnite-well conﬁning potentials. 11,16 In 1990 Kayanuma and Momiji 17 introduced the ﬁnite-depth square-well effective- mass

approximation FWEMA model. 18,19 Recently, this more reﬁned method has been adopted by other researchers and as has been shown, it considerably improves the model and makes it suitable for quantitative predictions. 18–20 In a very recent publication, Pellegrini et al. 20 systematically in- vestigated the applicability and limitations of the FWEMA model and applied it to several semiconductor QDs embed- ded in different matrices. In their work they have studied QDs by assuming a spherical potential well. The electron and hole energies are estimated numerically by solving appropri- ate

nonlinear algebraic equations. The Coulomb interaction between the electron and hole is treated by ﬁrst-order pertur- bation theory. In the present article we apply our recently developed potential-morphing method 21,22 PMM in order to investi- gate mainly QDs of wide-band-gap semiconductors charac- terized by parabolic bands. As the band structure is para- bolic, we assume that the EMA is appropriate. Moreover, we take that the conﬁning potential of the QD is a ﬁnite-well potential. We systematically investigate the dependence of the conﬁning potential on the QD

material and on the mate- rial of the matrix surrounding the QD. II. THEORY In the EMA the Hamiltonian for the electron-hole system can be written as 23 = eh where is the effective electron hole mass, is the effective dielectric constant, eh is the electron-hole distance in three dimensions, and is the conﬁnement potential of the electron hole . For QDs the potential is as- sumed centrosymmetric; hence, it has a constant value for distances larger than the QD radius and vanishes inside the dot. The Hartree-Fock formulation for two particles electron and hole results in the following

coupled equations: 24 where the and indices refer to the electron and hole, respectively. The self-consistent effective ﬁeld that acts on the electron is given by Electronic mail: terzis@physics.upatras.gr JOURNAL OF APPLIED PHYSICS 99 , 013708 2006 0021-8979/2006/99 /013708/4/$23.00 2006 American Institute of Physics 99 , 013708-1 Downloaded 26 Jan 2006 to 150.140.174.93. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

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dr dr 3a while the self-consistent effective ﬁeld that acts on the hole takes the form dr

dr 3b The calculation of these spatial integrals is a very time- demanding procedure. In order to overcome this technical difﬁculty we estimate these integrals by means of a three- dimensional fast Fourier transform. 25 The exact methodology of the fast Fourier transform technique is explained in a pre- vious publication. 26 In order to solve the iterative Hartree-Fock equations we apply the PMM Refs. 21 and 22 using as reference system the three-dimensional harmonic oscillator with well-known eigenstates. At this point, we stress that the PMM is capable of ﬁnding eigenvalues and

eigenfunctions of the stationary Schr”dinger equation for any arbitrary potential. This method is based on the quantum adiabatic theorem 27 that concerns the dynamic evolution of quantum-mechanical sys- tems and states that if the Hamiltonian of the system varies slowly with time, then the th eigenstate of the initial Hamil- tonian will be carried on to the th eigenstate of the ﬁnal Hamiltonian. Once the Hartree-Fock iteration scheme converges, the total energy of the exciton is estimated by the following ex- pression: and the corresponding effective band gap is given by eff where is

the bulk band-gap energy. III. RESULTS We start with the results for different semiconductor quantum dots CdS, CdSe, CdTe, and InP all characterized by a wide band gap. Each QD of speciﬁc semiconductor type is surrounded by a matrix of different material. In our study, we have assumed that the conﬁning potential has the same value for both electron and hole Refs. 18 and 20 . The material parameters effective masses and dielec- tric constants utilized in our investigations are reported in Table I. By systematically investigating these nanoscale sys- tems, we have found that the

conﬁning potential can be cho- sen such that it is independent of the type of the QD semi- conductor and depends exclusively on the matrix material, i.e., it depends exclusively on the matrix band-gap energy. This is in contrast to what other studies in the ﬁeld have used, where they take the conﬁnement potential proportional to the difference between the matrix and semiconductor band-gap energies. 18,20 We ﬁrst study the CdS semiconductor QD. The CdS QDs are capped with organic ligands such as oleic acid and 1-thioglycerol. Both matrices have a band-gap energy of

about 5 eV. 20 From Fig. 1, where we plotted the estimated band-gap energies for various conﬁning potentials, we clearly observe that the optimum value for the conﬁnement potential, , is at 400 meV. Then, in Fig. 2 we present the results for three more semiconductor QDs conﬁned in or- ganic matrices such as oleic acid, 1-thioglycerol, and trio- ctylphosphine oxide TOP/TOPO . All matrices have the same band-gap energy of about 5 eV. Figure 2 shows that there is a good agreement between experimental and theoret- ical values. We stress that the chosen value of is the same in

all cases and is equal to the one chosen for the CdS semi- conductor QD =400 meV . This is a strong indication that the conﬁnement potential depends exclusively on the matrix material and not on the QD material, too. In order to further investigate the validity of our method, we systematically investigated CdS QDs of various sizes conﬁned in different matrices. Figure 3 shows the experi- mental and calculated effective band-gap energies for CdS QDs capped in different matrices silicate glass 17 and oleic acid 20 . The best agreement between theoretical and experi- mental results

is found for the conﬁnement potential at TABLE I. Material parameters used in the PPM calculations Ref. 20 QD material CdS 0.18 0.53 5.23 CdSe 0.13 0.3 6.23 CdTe 0.11 0.35 7.1 InP 0.065 0.4 10.6 PbSe 0.07 0.06 25.0 InAs 0.028 0.33 12.3 FIG. 1. Effective band-gap energy for CdS dots as a function of the QD radius. The experimental data Ref. 20 are plotted with symbols. The three curves correspond to theoretical results for different values of the conﬁne- ment potential: =100 meV dotted curve =400 meV solid curve and =1000 meV dashed curve 013708-2 S. Baskoutas and A. F. Terzis J.

Appl. Phys. 99 , 013708 2006 Downloaded 26 Jan 2006 to 150.140.174.93. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

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=560 meV for the silicate glass matrix and at =400 meV for the oleic acid matrix. As the band-gap energy of the silicate glass matrix is around 7 eV Ref. 17 and the band-gap energy of the oleic acid matrix is around 5 eV, 20 we conclude that there is a simple relation between the band- gap energy of the matrix and the conﬁnement potential, i.e., 0.08 Finally, in order to better investigate the validity and

limitations of our model, we compared theoretical results and experimental data for two narrow-band-gap semiconductors, namely, PbSe and InAs. The material parameters used for these semiconductors are also listed in Table I. The results, depicted in Fig. 4, show a good agreement between theoret- ical predictions and experimental data. This agreement is ab- sent in the results obtained from the FWEMA method and from all the other existing models. 20 The reason for the dis- crepancy has been attributed to the breakdown of the EMA for narrow-band-gap semiconductors. The matrix for the InAs QD is

the TOP/TOPO matrix with a band-gap energy of 5 eV and the matrix for the PbSe QD is the phosphate glass with a band-gap energy of 3.5 eV. We have found that the best agreement is achieved for =0.08 , which is the same formula that we extracted for wide-band-gap semiconduc- tors. Hence the conﬁnement potential is =280 meV for the phosphate glass matrix and =400 meV for the TOP/TOPO matrix. IV. CONCLUSIONS We have applied the PMM within the EMA, assuming ﬁnite-depth square-well conﬁning potentials for both elec- trons and holes, in order to systematically investigate the

size-dependent band gap of semiconductor quantum dots em- bedded in various matrices. We have found that our results are sensitive to the value of the conﬁnement potential. Actu- ally, we have shown that the conﬁning potential solely de- pends on the material of the matrix and not on the material of the dot. Moreover, a simple expression was found that con- nects the conﬁnement potential and band-gap energy of the matrix. The validity and limitations of our method were in- vestigated assuming several semiconductor QDs embedded in various matrices. In closing we can state

that we have FIG. 2. Comparison between experimental and calculated effective band- gap energies for three semiconductor Qds CdSe, CdTe, and InP as func- tions of the dot radius. The experimental data are plotted with symbols and the theoretical predictions with solid curves. The conﬁnement potential is =400 meV in all cases. FIG. 3. Comparison between experimental and calculated effective band- gap energies for CdS semiconductor Qds as functions of the QD radius for dots capped in two different matrices. The experimental data are plotted with symbols and the theoretical predictions

with curves. The cycles and solid curve correspond to the oleic acid matrix and the squares and dotted curve to the silicate glass matrix. The conﬁnement potential is =560 meV for the silicate glass matrix and =400 meV for the oleic acid matrix. 013708-3 S. Baskoutas and A. F. Terzis J. Appl. Phys. 99 , 013708 2006 Downloaded 26 Jan 2006 to 150.140.174.93. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

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found a generic method with high predictability for various semiconductor QDs conﬁned in a matrix of known band-gap

energy. ACKNOWLEDGMENTS The authors would like to thank Dr. E. Paspalakis for fruitful discussions and helpful comments on the manuscript. One of the authors S.B. also thanks the Research Commit- tee of the University of Patras, Greece, for the ﬁnancial sup- port under the projects Karatheodoris B393. The authors also acknowledge the ﬁnancial support by the Archimedes- EPEAEK II Research Programme” cofunded by the Euro- pean Social Fund and National Resources. J. H. Davies, The Physics of Low Dimensional Semiconductors Cam- bridge University Press, Cambridge, 2000 L. Jacak, P.

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, 16168 2003 X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadavanish, and A. P. Alivisatos, Nature London 404 ,59 2000 10 L. L. Li, J. Hu, W. Yang, and A. P. Alivisatos, Nano Lett. ,349 2001 11 L. E. Brus, J. Chem. Phys. 80 ,4403 1984 12 H. Fu, L. W. Wang, and A. Zunger, Phys. Rev. B 57 , 9971 1998 13 P. E. Lippens and M. Lannoo, Phys. Rev. B 39 , 10935 1989 14 C. Delerue, G. Allan, and M. Lannoo, Phys. Rev. B 48 , 11024 1993 15 L. W. Wang and A. Zunger, Phys. Rev. B 53 ,9579 1996 16 Al. L. Efros and A. L. Efros, Sov. Phys. Semicond. 16 ,772 1982 17 Y. Kayanuma and H. Momiji, Phys.

Rev. B 41 , 10261 1990 18 K. K. Nanda, F. E. Kruis, and H. Fissan, Nano Lett. , 605 2001 19 K. K. Nanda, F. E. Kruis, and H. Fissan, J. Appl. Phys. 95 ,5035 2004 20 G. Pellegrini, G. Mattei, and P. Mazzoldi, J. Appl. Phys. 97 , 073706 2005 21 M. Rieth, W. Schommers, and S. Baskoutas, Int. J. Mod. Phys. B 16 ,4081 2002 22 S. Baskoutas, W. Schommers, A. F. Terzis, M. Rieth, V. Kapaklis, and C. Politis, Phys. Lett. A 308 , 219 2003 23 U. Woggon, Optical Properties of Semiconductor Quantum Dots Springer, Berlin, Heidelberg, 1997 24 N. Ashcroft and N. Mermin, Solid State Physics Holt-Saunders, New

York, 1976 ; R. G. Parr and W. Yang, Density-Functional Theory ofAtoms and Molecules Oxford University Press, Oxford, 1989 25 D. Sullivan and D. S. Citrin, J. Appl. Phys. 89 , 3841 2001 26 S. Baskoutas, Chem. Phys. Lett. 404 ,107 2005 ; Phys. Lett. A 341 , 303 2005 27 A. Messiah, Quantum Mechanics North-Holland, Amsterdam, 1961 ,Vol. II; D. Grifﬁths, Introduction to Quantum Mechanics Prentice-Hall, Lon- don, 2000 FIG. 4. Comparison between experimental and calculated effective band- gap energies for InAs and PbSe narrow-gap semiconductor Qds as functions of the QD radius in two

different matrices. The matrix for the InAs is the TOP/TOPO matrix with band-gap energy at 5 eV and the matrix for the PbSe is the phosphate glass with band-gap energy at 3.5 eV. Hence the conﬁnement potential is =280 meV for the phosphate glass matrix and =400 meV for the TOP/TOPO matrix, which is in both cases 8% of band-gap energy of the matrix. The experimental data are plotted with sym- bols and the theoretical predictions with solid curves. 013708-4 S. Baskoutas and A. F. Terzis J. Appl. Phys. 99 , 013708 2006 Downloaded 26 Jan 2006 to 150.140.174.93. Redistribution subject to AIP

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