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C.M. Expt. 5 Cadmium Selenide (CdSe) Quantum Dot/Quantum Mechanical Models Developed by Karen S. Quaal and Jason Hofstein, Department of Chemistry and Biochemistry, Siena College, Loudonville, NY. [email protected] Several quantum mechanical models were used to predict the size of the Quantum Dots based on the wavelengths observed for the first exciton. The results were compared to the TEM values reported in the literature1. Approximations2 1-Dimensional PIB: E 2 2 (2n 1) 2mL2 where L = length of box in meters, me = 9.11 x 10-31 kg, and n = 1. 3 2 2 (2n 1) E 2mL2 3-Dimensional PIB: where L = length of box in meters, me = 9.11 x 10-31 kg, and n = 1. Particle in a spherical well: E 2 2 (2n 1) 2ma 2 where a = radius of sphere in meters, me = 9.11 x 10-31 kg, and n = 1. m *e m *h Replace me with for an exciton: * where me* = 0.13me and mh* = 0.45me. m e m *h Show that for an exciton = 0.1009/me or 9.18 x 10-32 kg. Higher order Approximations Weak Confinement Approximation: adot is small, but greater than ab 2 E1s1s a b Ry * adot Where ab is the Bohr radius (CdSe) = 4.9 nm and Ry* is the exciton Rydberg for CdSe = 0.016 eV. Show that this is an INVALID approximation. Strong Confinement Approximation: adot << ab 2 a a E1s1s Eg 2 b Ry * 1.786 b Ry * 0.248 Ry * a dot adot where ab is the Bohr radius (CdSe) = 4.9 nm and Ry* is the exciton Rydberg for CdSe = 0.016 eV. Show that this is a VALID approximation, even though the conditions for the approximation are not met. Application of Spectroscopic Data to Quantum Mechanical Models Several quantum mechanical models were used to predict the size of the Q.D. The best agreement with TEM values was found with the strong confinement model. E1s1s = Eg + π2 (ab/adot)2 Ry* - 1.786 (ab/adot) Ry* - 0.248 Ry* Where E1S1S = Energy calculated from UV/VIS spectrum Eg= bang gap (CdSe= 1.84 eV) ab= exciton Bohr radius (CdSe= 4.9 nm) adot= radius of the Q.D Ry* = Rydberg constant (CdSe= 0.016 eV) Table 1: CdSe Spectral Data Temp ºC Lambda Max (nm) 128 453 138 473 148 501 158 510 168 521 178 522 188 530 198 538 208 546 218 555 Energy (J) 4.38E-19 4.20E-19 3.97E-19 3.90E-19 3.81E-19 3.81E-19 3.75E-19 3.69E-19 3.64E-19 3.58E-19 Radius (nm) 1.97 2.11 2.33 2.41 2.51 2.52 2.61 2.70 2.79 2.91 References: 1. Yu, W., Qu L., Guo W., and Peng X., Experimental Determination of Extinction Coefficient of CdTe, CdSe, and CdS Nanocrystals. Chem. Mater. 2003, 15 (14) 28542860. 2. Gaponenko, S.V., Optical Properties of Semiconductor Nanocrystals, Cambridge, U.K., Cambridge University Press, 1998.