Size-dependent band gap of colloidal quantum dots

Size-dependent band gap of colloidal quantum dots Sotirios Baskoutas and Andreas F. Terzis Citation: J. Appl. Phys. 99, 013708 (2006); doi: 10.1063/1.2158502 View online: http://dx.doi.org/10.1063/1.2158502 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v99/i1 Published by the American Institute of Physics.

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JOURNAL OF APPLIED PHYSICS 99, 013708 共2006兲

Size-dependent band gap of colloidal quantum dots Sotirios Baskoutas Materials Science Department, University of Patras, GR-26504, Greece

Andreas F. Terzisa兲 Department of Physics, University of Patras, GR-26504, Greece

共Received 21 July 2005; accepted 15 November 2005; published online 11 January 2006兲 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 共CdS, CdSe, CdTe, PbSe, InP, and InAs兲 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. 关DOI: 10.1063/1.2158502兴 I. INTRODUCTION

Quantum-confined semiconductor structures, including quantum wells, quantum rods, and quantum dots 共QDs兲, 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 共EMA兲,11 the kp method,12 the tight-binding approach,13 the linear combination of atomic orbitals method,14 and the empirical pseudopotential method.15 The oldest and least computationally demanding approach is the EMA, which has relied mostly on infinite-well confining potentials.11,16 In 1990 Kayanuma and Momiji17 introduced the finite-depth square-well effectivemass approximation 共FWEMA兲 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 investigated the applicability and limitations of the FWEMA model and applied it to several semiconductor QDs embedded 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 appropriate nonlinear algebraic equations. The Coulomb interaction between the electron and hole is treated by first-order perturbation theory. In the present article we apply our recently developed potential-morphing method21,22 共PMM兲 in order to investigate mainly QDs of wide-band-gap semiconductors characterized by parabolic bands. As the band structure is parabolic, we assume that the EMA is appropriate. Moreover, we a兲

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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 material of the matrix surrounding the QD.

II. THEORY

In the EMA the Hamiltonian for the electron-hole system can be written as23

H=−

ប2 2 ប2 2 e2 1 e h ⵜ − ⵜ + V 共r 兲 + V 共r 兲 − , 0 e 0 h e h ␧ reh 2m*e 2m*h

共1兲

where m*e 共m*h兲 is the effective electron 共hole兲 mass, ␧ is the effective dielectric constant, reh is the electron-hole distance in three dimensions, and Ve0共re兲关Vh0共rh兲兴 is the confinement potential of the electron 共hole兲. For QDs the potential is assumed centrosymmetric; hence, it has a constant value Ve0关Vh0兴 for distances larger than the QD radius and vanishes inside the dot. The Hartree-Fock formulation for two particles 共electron and hole兲 results in the following coupled equations: 24

冋

p2i 2m*i

册

+ Ui共ri兲 ⌽i共ri兲 = ˜Ei⌽i共ri兲,

i = e,h,

共2兲

where the e and h indices refer to the electron and hole, respectively. The self-consistent effective field Ui共ri兲 that acts on the electron is given by

99, 013708-1

© 2006 American Institute of Physics


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S. Baskoutas and A. F. Terzis

冋

1 e2 2␧

Ue共re兲⌽e共re兲 = Ve0共re兲 − 2

+

1e 2␧

冕

drh

冕

drh

册

TABLE I. Material parameters used in the PPM calculations 共Ref. 20兲.

兩⌽h共rh兲兩2 ⌽e共re兲兩re − rh兩

⌽*h共rh兲⌽e共rh兲兩re − rh兩

⌽h共re兲,

共3a兲

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

冋

Uh共rh兲⌽h共rh兲 = Vh0共rh兲 − +

1 e2 2␧

冕

1 e2 2␧

dre

冕

dre

册

兩⌽e共re兲兩2 ⌽h共rh兲兩re − rh兩

⌽*e 共re兲⌽h共re兲 ⌽e共rh兲. 兩rh − re兩

共3b兲

The calculation of these spatial integrals is a very timedemanding procedure. In order to overcome this technical difficulty we estimate these integrals by means of a threedimensional fast Fourier transform.25 The exact methodology of the fast Fourier transform technique is explained in a previous publication.26 In order to solve the iterative Hartree-Fock equations we apply the PMM 共Refs. 21 and 22兲 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 Schrödinger equation for any arbitrary potential. This method is based on the quantum adiabatic theorem27 that concerns the dynamic evolution of quantum-mechanical systems and states that if the Hamiltonian of the system varies slowly with time, then the nth eigenstate of the initial Hamiltonian will be carried on to the nth eigenstate of the final Hamiltonian. Once the Hartree-Fock iteration scheme converges, the total energy of the exciton is estimated by the following expression: E共X兲 = ˜Ee + ˜Eh ,

共4兲

and the corresponding effective band gap is given by Eeff g 共X兲 = Eg + E共X兲,

QD material

me* / me

m*h / me

␧

CdS CdSe CdTe InP PbSe InAs

0.18 0.13 0.11 0.065 0.07 0.028

0.53 0.3 0.35 0.4 0.06 0.33

5.23 6.23 7.1 10.6 25.0 12.3

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, V0, is at 400 meV. Then, in Fig. 2 we present the results for three more semiconductor QDs confined in organic matrices such as oleic acid, 1-thioglycerol, and trioctylphosphine oxide 共TOP/TOPO兲. All matrices have the same band-gap energy of about 5 eV. Figure 2 shows that there is a good agreement between experimental and theoretical values. We stress that the chosen value of V0 is the same in all cases and is equal to the one chosen for the CdS semiconductor QD 共V0 = 400 meV兲. 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 experimental and calculated effective band-gap energies for CdS QDs capped in different matrices 共silicate glass17 and oleic acid 20兲. The best agreement between theoretical and experimental results is found for the confinement potential at V0

共5兲

where Eg is the bulk band-gap energy.

III. RESULTS

We start with the results for different semiconductor quantum dots 共CdS, CdSe, CdTe, and InP兲 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 关Ve0 = Vh0 = V0 共Refs. 18 and 20兲兴. The material parameters 共effective masses and dielectric constants兲 utilized in our investigations are reported in Table I. By systematically investigating these nanoscale systems, we have found that the confining potential can be chosen such that it is independent of the type of the QD semiconductor 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

FIG. 1. Effective band-gap energy for CdS dots as a function of the QD radius. The experimental data 共Ref. 20兲 are plotted with symbols. The three curves correspond to theoretical results for different values of the confinement potential: V0 = 100 meV 共dotted curve兲, V0 = 400 meV 共solid curve兲, and V0 = 1000 meV 共dashed curve兲.


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FIG. 3. Comparison between experimental and calculated effective bandgap 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 V0 = 560 meV for the silicate glass matrix and V0 = 400 meV for the oleic acid matrix.

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 theoretical predictions and experimental data. This agreement is absent in the results obtained from the FWEMA method and from all the other existing models.20 The reason for the discrepancy 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 V0 = 0.08Eg, which is the same formula that we extracted for wide-band-gap semiconductors. Hence the confinement potential is V0 = 280 meV for the phosphate glass matrix and V0 = 400 meV for the TOP/TOPO matrix.

IV. CONCLUSIONS FIG. 2. Comparison between experimental and calculated effective bandgap energies for three semiconductor Qds 共CdSe, CdTe, and InP兲 as functions of the dot radius. The experimental data are plotted with symbols and the theoretical predictions with solid curves. The confinement potential is V0 = 400 meV in all cases.

= 560 meV for the silicate glass matrix and at V0 = 400 meV for the oleic acid matrix. As the band-gap energy of the silicate glass matrix is around 7 eV 共Ref. 17兲 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 bandgap energy of the matrix and the confinement potential, i.e., V0 ⬇ 0.08Eg. Finally, in order to better investigate the validity and

We have applied the PMM within the EMA, assuming finite-depth square-well confining potentials for both electrons and holes, in order to systematically investigate the size-dependent band gap of semiconductor quantum dots embedded in various matrices. We have found that our results are sensitive to the value of the confinement potential. Actually, we have shown that the confining potential solely depends on the material of the matrix and not on the material of the dot. Moreover, a simple expression was found that connects the confinement potential and band-gap energy of the matrix. The validity and limitations of our method were investigated assuming several semiconductor QDs embedded in various matrices. In closing we can state that we have


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One of the authors 共S.B.兲 also thanks the Research Committee of the University of Patras, Greece, for the financial support under the projects Karatheodoris B393. The authors also acknowledge the financial support by the ArchimedesEPEAEK II Research Programme” cofunded by the European Social Fund and National Resources. J. H. Davies, The Physics of Low Dimensional Semiconductors 共Cambridge University Press, Cambridge, 2000兲. 2 L. Jacak, P. Hawrylak, and A. Wojs, Quantum Dots 共Springer, Berlin, Heidelberg, 1998兲. 3 Trends in Quantum Dots Research, edited by P. A. Ling 共Nova, New York, 2005兲. 4 T. Voddmeyer, D. J. Katsikas, M. Giersig, I. G. Popovic, K. Diesner, A. Chemseddine, A. Eychmuller, and H. Weller, J. Appl. Phys. 90, 265 共2001兲. 5 V. N. Soloviev, A. Eichhoefer, D. Fenske, and U. Banin, J. Am. Chem. Soc. 122, 2673 共2000兲. 6 A. A. Guzelian et al., J. Phys. Chem. 100, 7212 共1996兲. 7 U. Banin, Y. W. Cao, D. Katz, and O. Millo, Nature 共London兲 400, 542 共1999兲. 8 H. Yu, J. Li, R. A. Loomis, P. C. Gibbons, L. W. Wang, and W. E. Buhro, J. Am. Chem. Soc. 125, 16168 共2003兲. 9 X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadavanish, and A. P. Alivisatos, Nature 共London兲 404, 59 共2000兲. 10 L. L. Li, J. Hu, W. Yang, and A. P. Alivisatos, Nano Lett. 1, 349 共2001兲. 11 L. E. Brus, J. Chem. Phys. 80, 4403 共1984兲. 12 H. Fu, L. W. Wang, and A. Zunger, Phys. Rev. B 57, 9971 共1998兲. 13 P. E. Lippens and M. Lannoo, Phys. Rev. B 39, 10935 共1989兲. 14 C. Delerue, G. Allan, and M. Lannoo, Phys. Rev. B 48, 11024 共1993兲. 15 L. W. Wang and A. Zunger, Phys. Rev. B 53, 9579 共1996兲. 16 Al. L. Efros and A. L. Efros, Sov. Phys. Semicond. 16, 772 共1982兲. 17 Y. Kayanuma and H. Momiji, Phys. Rev. B 41, 10261 共1990兲. 18 K. K. Nanda, F. E. Kruis, and H. Fissan, Nano Lett. 1, 605 共2001兲. 19 K. K. Nanda, F. E. Kruis, and H. Fissan, J. Appl. Phys. 95, 5035 共2004兲. 20 G. Pellegrini, G. Mattei, and P. Mazzoldi, J. Appl. Phys. 97, 073706 共2005兲. 21 M. Rieth, W. Schommers, and S. Baskoutas, Int. J. Mod. Phys. B 16, 4081 共2002兲. 22 S. Baskoutas, W. Schommers, A. F. Terzis, M. Rieth, V. Kapaklis, and C. Politis, Phys. Lett. A 308, 219 共2003兲. 23 U. Woggon, Optical Properties of Semiconductor Quantum Dots 共Springer, Berlin, Heidelberg, 1997兲. 24 N. Ashcroft and N. Mermin, Solid State Physics 共Holt-Saunders, New York, 1976兲; R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules 共Oxford University Press, Oxford, 1989兲. 25 D. Sullivan and D. S. Citrin, J. Appl. Phys. 89, 3841 共2001兲. 26 S. Baskoutas, Chem. Phys. Lett. 404, 107 共2005兲; Phys. Lett. A 341, 303 共2005兲. 27 A. Messiah, Quantum Mechanics 共North-Holland, Amsterdam, 1961兲, Vol. II; D. Griffiths, Introduction to Quantum Mechanics 共Prentice-Hall, London, 2000兲. 1

FIG. 4. Comparison between experimental and calculated effective bandgap 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 V0 = 280 meV for the phosphate glass matrix and V0 = 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 symbols and the theoretical predictions with solid curves.

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.
