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Shape-generated blue-shift in the photoluminescence spectra of hemispherical nanocrystals
Physica E 4 (1999) 176–179
Shape-generated blue-shift in the photoluminescence spectra of hemispherical nanocrystals W. Jaskolskia; ∗ , J. Planellesb , G. Perisb a Instytut
b Departament
Fizyki UMK, Grudziadzka 5, 87-100 ToruÃn, Poland de CiÂencies Experimentals, Universitat Jaume I, CastellÃo, Spain
Received 30 November 1998; accepted 11 February 1999
Abstract The blue-shift observed in the photoluminescence spectra of CdS quantum dots electrochemically deposited on a graphite substrate [M.A. Anderson, S. Gorger, R.M. Penner, J. Phys. Chem. 101 (1997) 5895] is explained in terms of quantum size and dot shape e ects. The line shape of the observed peak is also theoretically reproduced by assuming a gaussian distribution of the sizes of hemispherical dots with the average radius and standard deviation taken from the experiment. ? 1999 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx; 85.30.Vw; 71.24.+q; 78.66.−w Keywords: Photoluminescence; Blue shift; Quantum dots; Nanocrystals; CdS
Recently a new electrochemical=chemical method of growing CuI and CdS nanocrystals on a graphite substrate has been achieved [1,2]. The microscopic studies of this kind of quantum dots seem to indicate that, to a large extent, their shapes can be approximated by hemispheres [1]. Just as in chemically synthesized spherical nanocrystals [3–5] these dots can vary in size from almost 1 to 10 nm. Very recently a large blue-shift in the photoluminescence (PL) spectra from such CdS dots, in comparison with the CdS bulk emission, has been observed [1]. Since the origin of the observed blue-shift has not been explained in Ref. [1] we attempt to do this in this letter. We study the role of dot shape on the ∗ Corresponding author. E-mail address: [email protected] (W. Jaskolski)
photoluminescence spectra of quantum dots with respect to the bulk and we show that the observed blue-shift is associated with a hemispherical dot shape. In particular, we nd out that the shift can be easily explained by simple geometrical reasoning within the one band e ective mass approximation. Although the mass parameters are not very well known in CdS we assume, following Bryant [6] and Mews et al. [7], an electron e ective mass equal to m∗e = 0:2 and a hole mass m∗h = 0:7. Although the one band approach is not suitable to describe the entire hole energy spectrum in such semiconductor nanocrystals, it is accepted [6] that the lowest electron–hole transition energies can be quite well approximated within this model. We consider a hemispherical dot of radius R = 4:13 nm. It corresponds to the mean value of the
1386-9477/99/$ - see front matter ? 1999 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 0 1 2 - 0
W. JaskÃolski et al. / Physica E 4 (1999) 176–179
radii of the nanocrystals quoted in Ref. [1] under the conditions, for which the quoted blue-shift has been observed. Let us rst consider totally spherical dots of the same radius and check what shift in the PL spectra would be expected if the dots were spherical. For simplicity we assume in nite potential barriers at the dot boundaries for both electrons and holes. In nite barriers have been proved to be appropriate for holes even if multiband approaches are considered [3,8]. For the electrons a reasonable choice for the height of the barrier could be 4.8 eV due to the experimental value of the electron anity in CdS [9]. However, energy positions of the lowest s- and p-type electron states do not change signi cantly if an in nite potential barrier is considered. To simplify our model we do not consider small strain e ects that could be induced by the substrate. Under these assumptions the energies of the lowest electron and hole states of s and p syme = 0:110 eV, Esh = 0:031 metry are respectively: E1s e h eV, E1p = 0:225 eV and E1p = 0:064 eV [10] 1. The band gap energy Eg in the bulk CdS is about 2.5 eV [11,9]. Thus the energy of the lowest e–h (1sh –1se ) transition (corrected by Vss = −0:110 eV, i.e., rst order Coulomb interaction term [12,13]) in spherical nanocrystals of radius 4.13 nm is equal to 2.531 eV. The energy di erence |Eg − E1sh −1se | = 0:031 eV, i.e., wavelengths equal to 6 nm is much less than the observed shift of about 31.4 nm. Spherical symmetry is broken in hemispherical nanocrystals. Due to the high value of the work function in graphite, the graphite surface acts as a hard wall for the carriers in CdS nanocrystals. Then, high or in nite potential barriers for both the holes and electrons can be considered on the graphite side. Following Kovalenko [14] the energy of the lowest state in an in nite well of hemispherical geometry corresponds exactly to the lowest p-state in a spherical well of the same radius. It has to be mentioned that p-type symmetry of the lowest electron and hole states has also been found for pyramidal self-assembled quantum dots [15 –17]. Therefore the energy of the lowest e–h transition of a hemispherical dot corresponds to 1pe –1ph transition of its spherical partner. 1 The results of a three-band model show [10] that the ground valence-band state may be of p-like symmetry. Since it is energeticaly close to the lowest s-type valence state, this will not in uence signi cantly our results and conclusions.
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Now, the rst order Coulomb interaction term, Vpp , has to be estimated due to the electron–hole p–p state in the spherical dot [13] 2. For radius of R = 4:13 nm the energy of this transition is nally equal to 2.673 eV (yielding a blue-shift of 32.0 nm), which ts unexpectedly well to the observed peak at 2.669 eV (31.4 nm shift). The blue shift of luminescence is then much more pronounced in electrochemically deposited CdS nanocrystals than in chemically synthesized spherical dots of the same radius. It means that this shift originates not only from the quantum size of dot but also signi cantly from its particular shape. Next, using the experimentally reported distribution of quantum dots sizes in a sample similar to the one for which the photoluminescence blue-shift was observed in Ref. [1], we have reproduced also the shape of the observed band-gap emission peak (at 464 nm). The quoted distribution of sizes is characterized by a mean radius R0 = 4:13 nm and standard deviation 2 = 6:63 nm (see Fig. 3b of Ref. [1]). We assume that the oscillator strengths of the fundamental transition are the same for all the dots in the sample and that the emission shape is caused by the dot sizes’ distribution. Although the radii distribution is assumed to be gaussian, the distribution of wavelengths corresponding to the fundamental transition is not. This is shown in Fig. 1 (dotted line). To compare the predicted line shape with the experimental one we have extracted the observed peak (at 464 nm) from the tail of 620 nm broad maximum (assigned in Ref. [1] to emission from trap states located in the gap). This yields the experimental shape corresponding to the pure band-gap transition (plotted as solid line in Fig. 1). The 2 Due to the same symmetry of the ground state charge density in a hemispherical dot and of the lowest p-type state density in a spherical one, the rst order Coulomb correction, Vpp , will change only slightly when going from spherical to hemispherical geometry. To estimate this change we have performed exact numerical calculation of the Coulomb energy term for two model cases of (i) 2D square [2R × 2R] box and (ii) rectangular half-square [2R × R] box. The ratio S of the calculated Coulomb √ corrections in these two cases appears to be of the order of 2, i.e., of the order of square root of volume ratio of these two boxes. Since the Coulomb correction scales inversly to the box size [12,15 – √ 17], S would be of the order of 3 2 ¡ 1:2 in the 3D case and would not essentially change the absolute di erence between the calculated and experimental transition energies. For simplicity we neglect this change.
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W. JaskÃolski et al. / Physica E 4 (1999) 176–179
Fig. 1. Comparison of the experimental (solid line) and theoretical (dashed line) photoluminescence spectra corresponding to the band-gap transition in hemispherical CdS nanocrystals. The experimental line shape has been extracted from the tail of a broad peak assigned to emission from trap states located in the gap [1].
agreement between the theoretical and experimental shape is remarkable. We have checked also the dependence of the photoluminescence peak position on the geometry of quantum dots. In particular we have calculated the energy of the lowest transition for: (i) a rectangular three-dimensional box of the base 2R × 2R and the volume of the hemispherical dot and (ii) a cylindrical box of the same radius and volume as the hemispherical box. The results are: (i) 2.973 eV and (ii) 2.641 eV. These values di er from the experimental value (2.669 eV) much more than the result obtained for hemispherical dot (2.673 eV), suggesting that the dots are most likely hemispherical 3. Finally, we have estimated a possible eccentricity of the hemispherical shapes of the dots in question. Provided we already know that the eccentricity is small, we can estimate its e ect on the transition energy by assuming that the energy E of a particle con ned in a volume V is proportional to V −2=3 . Since the volume of a slightly elliptically deformed hemispherical top having a basic radius R and height h is Vnh = 12 34 R2 h and the volume of the correspond3 Comparison of the transition energy for a cone or pyramidal shape, although interesting itself, would require numerical solution on 3D grid that is above the scope of this paper.
ing hemispherical top is Vh = 12 34 R3 , the ratio Vh =Vnh equals to (R=h)2=3 . Making an assumption that the experimental quantum dot is a deformed hemispherical top, i.e., Enh = Eexp and using the above calculated energy Eh = 2:673 eV for the exactly hemispherical dot of radius R = 4:13 nm, we get h = 4:14 nm. It shows that the deviation of the calculated energy from the experimental value could not be explained by any considerable eccentricity. In summary, it has been shown that not only dot size but, additionally, the hemispherical dot shape is relevant for the explanation of the blue-shift observed in the photoluminescence spectra of CdS nanocrystals deposited by Anderson et al. [1] on a graphite surface. We have shown that the observed blue-shift can be easily explained within the one band e ective mass approximation if one remembers that the lowest electron and hole states in hemispherical dots are of the p-type symmetry. The experimental line-shape has also been very well reproduced under the assumption of a gaussian distribution of dot sizes in the sample. Acknowledgements WJ acknowledges Generalitat Valenciana for grant INV98-CB-II-35. Continuous support from the Universitat Jaume I, project CICYT PB97-0397, fundacio Caixa de Castello, project P1B 97-23, from the Polish Scienti c Research Council (KBN) and from MS-C Fund II is gratefully acknowledged. References [1] M.A. Anderson, S. Gorger, R.M. Penner, J. Phys. Chem. 101 (1997) 5895. [2] G.S. Hsiao, M.A. Anderson, D. Harris, R.M. Penner, J. Am. Chem. Soc. 119 (1997) 1439. [3] D.J. Norris, M.G. Bawendi, Phys. Rev. B 53 (1996) 16 338. [4] A. Mews, A.V. Kadavanich, U. Banun, A.P. Alivisatos, Phys. Rev. B 53 (1996) R13 242. [5] U. Banin, J.C. Lee, A. Guzelian, A.V. Kadavanich, A.P. Alivisatos, W. Jaskolski, G.W. Bryant, Al.L. Efros, M. Rosen, J. Chem. Phys. 109 (1998) 2306. [6] G.W. Bryant, Phys. Rev. B 52 (1995) R16 997. [7] A. Mews, A. Eychmuller, M. Giersig, D. Schooss, H. Weller, J. Phys. Chem. 98 (1994) 934. [8] H.H. Grunberg, Phys. Rev. B 55 (1997) 2293. [9] A.H. Nethercot, Phys. Rev. Lett. 33 (1974) 1088.
W. JaskÃolski et al. / Physica E 4 (1999) 176–179 [10] G.B. Grigoryan, E.M. Kazaryan, Al.L. Efros, T.V. Yazeva, Sov. Phys. Solid State 32 (1990) 1031. [11] Landolt-Borstein, in: O. Madelung (Ed.), Numerical Data and Functional Relationships in Science and Technology, Springer, Berlin, 1982. [12] L.E. Brus, J. Chem. Phys. 80 (1984) 4403. [13] Y. Kayanuma, Phys. Rev. B 38 (1988) 9797.
179
[14] A.F. Kovalenko, E.N. Sovyak, M.F. Golovko, Phys. Stat. Sol. (b) 155 (1989) 549. [15] M. Grundman, O. Stier, D. Bimberg, Phys. Rev. B 52 (1995) 11 969. [16] C. Pryor, Phys. Rev. B 57 (1998) 7190. [17] J. Kim, L.-W. Wang, A. Zunger, Phys. Rev. B 57 (1998) R9408.
Shape-generated blue-shift in the photoluminescence spectra of hemispherical nanocrystals W. Jaskolskia; ∗ , J. Planellesb , G. Perisb a Instytut
b Departament
Fizyki UMK, Grudziadzka 5, 87-100 ToruÃn, Poland de CiÂencies Experimentals, Universitat Jaume I, CastellÃo, Spain
Received 30 November 1998; accepted 11 February 1999
Abstract The blue-shift observed in the photoluminescence spectra of CdS quantum dots electrochemically deposited on a graphite substrate [M.A. Anderson, S. Gorger, R.M. Penner, J. Phys. Chem. 101 (1997) 5895] is explained in terms of quantum size and dot shape e ects. The line shape of the observed peak is also theoretically reproduced by assuming a gaussian distribution of the sizes of hemispherical dots with the average radius and standard deviation taken from the experiment. ? 1999 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx; 85.30.Vw; 71.24.+q; 78.66.−w Keywords: Photoluminescence; Blue shift; Quantum dots; Nanocrystals; CdS
Recently a new electrochemical=chemical method of growing CuI and CdS nanocrystals on a graphite substrate has been achieved [1,2]. The microscopic studies of this kind of quantum dots seem to indicate that, to a large extent, their shapes can be approximated by hemispheres [1]. Just as in chemically synthesized spherical nanocrystals [3–5] these dots can vary in size from almost 1 to 10 nm. Very recently a large blue-shift in the photoluminescence (PL) spectra from such CdS dots, in comparison with the CdS bulk emission, has been observed [1]. Since the origin of the observed blue-shift has not been explained in Ref. [1] we attempt to do this in this letter. We study the role of dot shape on the ∗ Corresponding author. E-mail address: [email protected] (W. Jaskolski)
photoluminescence spectra of quantum dots with respect to the bulk and we show that the observed blue-shift is associated with a hemispherical dot shape. In particular, we nd out that the shift can be easily explained by simple geometrical reasoning within the one band e ective mass approximation. Although the mass parameters are not very well known in CdS we assume, following Bryant [6] and Mews et al. [7], an electron e ective mass equal to m∗e = 0:2 and a hole mass m∗h = 0:7. Although the one band approach is not suitable to describe the entire hole energy spectrum in such semiconductor nanocrystals, it is accepted [6] that the lowest electron–hole transition energies can be quite well approximated within this model. We consider a hemispherical dot of radius R = 4:13 nm. It corresponds to the mean value of the
1386-9477/99/$ - see front matter ? 1999 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 0 1 2 - 0
W. JaskÃolski et al. / Physica E 4 (1999) 176–179
radii of the nanocrystals quoted in Ref. [1] under the conditions, for which the quoted blue-shift has been observed. Let us rst consider totally spherical dots of the same radius and check what shift in the PL spectra would be expected if the dots were spherical. For simplicity we assume in nite potential barriers at the dot boundaries for both electrons and holes. In nite barriers have been proved to be appropriate for holes even if multiband approaches are considered [3,8]. For the electrons a reasonable choice for the height of the barrier could be 4.8 eV due to the experimental value of the electron anity in CdS [9]. However, energy positions of the lowest s- and p-type electron states do not change signi cantly if an in nite potential barrier is considered. To simplify our model we do not consider small strain e ects that could be induced by the substrate. Under these assumptions the energies of the lowest electron and hole states of s and p syme = 0:110 eV, Esh = 0:031 metry are respectively: E1s e h eV, E1p = 0:225 eV and E1p = 0:064 eV [10] 1. The band gap energy Eg in the bulk CdS is about 2.5 eV [11,9]. Thus the energy of the lowest e–h (1sh –1se ) transition (corrected by Vss = −0:110 eV, i.e., rst order Coulomb interaction term [12,13]) in spherical nanocrystals of radius 4.13 nm is equal to 2.531 eV. The energy di erence |Eg − E1sh −1se | = 0:031 eV, i.e., wavelengths equal to 6 nm is much less than the observed shift of about 31.4 nm. Spherical symmetry is broken in hemispherical nanocrystals. Due to the high value of the work function in graphite, the graphite surface acts as a hard wall for the carriers in CdS nanocrystals. Then, high or in nite potential barriers for both the holes and electrons can be considered on the graphite side. Following Kovalenko [14] the energy of the lowest state in an in nite well of hemispherical geometry corresponds exactly to the lowest p-state in a spherical well of the same radius. It has to be mentioned that p-type symmetry of the lowest electron and hole states has also been found for pyramidal self-assembled quantum dots [15 –17]. Therefore the energy of the lowest e–h transition of a hemispherical dot corresponds to 1pe –1ph transition of its spherical partner. 1 The results of a three-band model show [10] that the ground valence-band state may be of p-like symmetry. Since it is energeticaly close to the lowest s-type valence state, this will not in uence signi cantly our results and conclusions.
177
Now, the rst order Coulomb interaction term, Vpp , has to be estimated due to the electron–hole p–p state in the spherical dot [13] 2. For radius of R = 4:13 nm the energy of this transition is nally equal to 2.673 eV (yielding a blue-shift of 32.0 nm), which ts unexpectedly well to the observed peak at 2.669 eV (31.4 nm shift). The blue shift of luminescence is then much more pronounced in electrochemically deposited CdS nanocrystals than in chemically synthesized spherical dots of the same radius. It means that this shift originates not only from the quantum size of dot but also signi cantly from its particular shape. Next, using the experimentally reported distribution of quantum dots sizes in a sample similar to the one for which the photoluminescence blue-shift was observed in Ref. [1], we have reproduced also the shape of the observed band-gap emission peak (at 464 nm). The quoted distribution of sizes is characterized by a mean radius R0 = 4:13 nm and standard deviation 2 = 6:63 nm (see Fig. 3b of Ref. [1]). We assume that the oscillator strengths of the fundamental transition are the same for all the dots in the sample and that the emission shape is caused by the dot sizes’ distribution. Although the radii distribution is assumed to be gaussian, the distribution of wavelengths corresponding to the fundamental transition is not. This is shown in Fig. 1 (dotted line). To compare the predicted line shape with the experimental one we have extracted the observed peak (at 464 nm) from the tail of 620 nm broad maximum (assigned in Ref. [1] to emission from trap states located in the gap). This yields the experimental shape corresponding to the pure band-gap transition (plotted as solid line in Fig. 1). The 2 Due to the same symmetry of the ground state charge density in a hemispherical dot and of the lowest p-type state density in a spherical one, the rst order Coulomb correction, Vpp , will change only slightly when going from spherical to hemispherical geometry. To estimate this change we have performed exact numerical calculation of the Coulomb energy term for two model cases of (i) 2D square [2R × 2R] box and (ii) rectangular half-square [2R × R] box. The ratio S of the calculated Coulomb √ corrections in these two cases appears to be of the order of 2, i.e., of the order of square root of volume ratio of these two boxes. Since the Coulomb correction scales inversly to the box size [12,15 – √ 17], S would be of the order of 3 2 ¡ 1:2 in the 3D case and would not essentially change the absolute di erence between the calculated and experimental transition energies. For simplicity we neglect this change.
178
W. JaskÃolski et al. / Physica E 4 (1999) 176–179
Fig. 1. Comparison of the experimental (solid line) and theoretical (dashed line) photoluminescence spectra corresponding to the band-gap transition in hemispherical CdS nanocrystals. The experimental line shape has been extracted from the tail of a broad peak assigned to emission from trap states located in the gap [1].
agreement between the theoretical and experimental shape is remarkable. We have checked also the dependence of the photoluminescence peak position on the geometry of quantum dots. In particular we have calculated the energy of the lowest transition for: (i) a rectangular three-dimensional box of the base 2R × 2R and the volume of the hemispherical dot and (ii) a cylindrical box of the same radius and volume as the hemispherical box. The results are: (i) 2.973 eV and (ii) 2.641 eV. These values di er from the experimental value (2.669 eV) much more than the result obtained for hemispherical dot (2.673 eV), suggesting that the dots are most likely hemispherical 3. Finally, we have estimated a possible eccentricity of the hemispherical shapes of the dots in question. Provided we already know that the eccentricity is small, we can estimate its e ect on the transition energy by assuming that the energy E of a particle con ned in a volume V is proportional to V −2=3 . Since the volume of a slightly elliptically deformed hemispherical top having a basic radius R and height h is Vnh = 12 34 R2 h and the volume of the correspond3 Comparison of the transition energy for a cone or pyramidal shape, although interesting itself, would require numerical solution on 3D grid that is above the scope of this paper.
ing hemispherical top is Vh = 12 34 R3 , the ratio Vh =Vnh equals to (R=h)2=3 . Making an assumption that the experimental quantum dot is a deformed hemispherical top, i.e., Enh = Eexp and using the above calculated energy Eh = 2:673 eV for the exactly hemispherical dot of radius R = 4:13 nm, we get h = 4:14 nm. It shows that the deviation of the calculated energy from the experimental value could not be explained by any considerable eccentricity. In summary, it has been shown that not only dot size but, additionally, the hemispherical dot shape is relevant for the explanation of the blue-shift observed in the photoluminescence spectra of CdS nanocrystals deposited by Anderson et al. [1] on a graphite surface. We have shown that the observed blue-shift can be easily explained within the one band e ective mass approximation if one remembers that the lowest electron and hole states in hemispherical dots are of the p-type symmetry. The experimental line-shape has also been very well reproduced under the assumption of a gaussian distribution of dot sizes in the sample. Acknowledgements WJ acknowledges Generalitat Valenciana for grant INV98-CB-II-35. Continuous support from the Universitat Jaume I, project CICYT PB97-0397, fundacio Caixa de Castello, project P1B 97-23, from the Polish Scienti c Research Council (KBN) and from MS-C Fund II is gratefully acknowledged. References [1] M.A. Anderson, S. Gorger, R.M. Penner, J. Phys. Chem. 101 (1997) 5895. [2] G.S. Hsiao, M.A. Anderson, D. Harris, R.M. Penner, J. Am. Chem. Soc. 119 (1997) 1439. [3] D.J. Norris, M.G. Bawendi, Phys. Rev. B 53 (1996) 16 338. [4] A. Mews, A.V. Kadavanich, U. Banun, A.P. Alivisatos, Phys. Rev. B 53 (1996) R13 242. [5] U. Banin, J.C. Lee, A. Guzelian, A.V. Kadavanich, A.P. Alivisatos, W. Jaskolski, G.W. Bryant, Al.L. Efros, M. Rosen, J. Chem. Phys. 109 (1998) 2306. [6] G.W. Bryant, Phys. Rev. B 52 (1995) R16 997. [7] A. Mews, A. Eychmuller, M. Giersig, D. Schooss, H. Weller, J. Phys. Chem. 98 (1994) 934. [8] H.H. Grunberg, Phys. Rev. B 55 (1997) 2293. [9] A.H. Nethercot, Phys. Rev. Lett. 33 (1974) 1088.
W. JaskÃolski et al. / Physica E 4 (1999) 176–179 [10] G.B. Grigoryan, E.M. Kazaryan, Al.L. Efros, T.V. Yazeva, Sov. Phys. Solid State 32 (1990) 1031. [11] Landolt-Borstein, in: O. Madelung (Ed.), Numerical Data and Functional Relationships in Science and Technology, Springer, Berlin, 1982. [12] L.E. Brus, J. Chem. Phys. 80 (1984) 4403. [13] Y. Kayanuma, Phys. Rev. B 38 (1988) 9797.
179
[14] A.F. Kovalenko, E.N. Sovyak, M.F. Golovko, Phys. Stat. Sol. (b) 155 (1989) 549. [15] M. Grundman, O. Stier, D. Bimberg, Phys. Rev. B 52 (1995) 11 969. [16] C. Pryor, Phys. Rev. B 57 (1998) 7190. [17] J. Kim, L.-W. Wang, A. Zunger, Phys. Rev. B 57 (1998) R9408.