Nano Science (Lec6 Quantum Dots)

Nano Science (Lec6 Quantum Dots) - MAE 287/EE 257 1 In...

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MAE 287/EE 257 1
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In natural materials, the material structures are defined by structures of atoms, molecules, and solids. However, we can modify the material structures further by controlling the sizes down to nanoscale at various dimensions to create 0D (quantum dot), 1D (quantum wire) and 2D (quantum well) nano materials. 2
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A quantum dot (QD) is a nanostructure that confines electrons and holes in all three spatial directions. Small QDs can be as small as 2 to 50 nm , corresponding to 10 to 100 atoms in diameter and a total of 100 to 10 7 atoms within the quantum dot volume. The QDs can have multiple-layer structures. 3
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Ge Islands on Si(001) T. Kamins, HP Labs Ge Islands on Si(001) Gio Ribeiro, HP Labs
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1,2,3,. .. n with ) L n ( 2 2 2 m E n The quantized energy levels of a particle- in-a-box model are shown. The horizontal lines drawn across the box show the allowed energies of a particle in this box; the height of each line above the box floor corresponds to the value of the energy (a higher line means a higher energy). 6
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The corresponding wave functions, L x 0 for ) ( L 2 ) ( x L n Sin x n String harmonics: The strings are held at two ends. The quantum levels are represented by n = 1,2,3,…,∞. Waves only exist when there is an integral multiple of the half-wavelengths in the string. 7
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1D Schrodinger’s Equation ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V x x m 8 3D Schrodinger’s Equation 2 2 2 2 2 2 2 2 2 ( , , ) ( , , ) ( , , ) ( , , ) 2 ψ x y z V x y z ψ x y z Eψ x y z m with x y z              
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Lx Lz Ly Solution of the wave function, For a particle confined in a nano box, within the well V=0, ) , , ( 0 ) , , ( 2 2 2 z y x E z y x m x y z ( , , ) ( ) ( ) ( ) L L L y x z ny nx nz x y z A Sin Sin Sin 9
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Solution for particle energy, E n x n y n z 1 1 1 1 2 1 2 1 1 1 1 2 There are three quantum numbers corresponds to three dimensional space. When different quantum numbers correspond to the same energy, the energy level is degenerated; Otherwise it is non-degenerated. 1,2,3,. .. n , n , n with ) L n L n L n ( 2 z y x 2 z 2 z 2 y 2 y 2 x 2 x 2 2 m E n 10
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The hydrogen atom represents the simplest possible atom since it consists of only one proton and one electron, but it is a three-dimensional quantum mechanical problem. The potential, V ( r ), is due to the electrostatic force between the positively charged proton and the negatively charged electron. 0 () 4 e Vr r  e: electron charge. 0 : vacuum dielectric constant.
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The energy of electron in hydrogen can be obtained as, 2 2 0 24 e E mr       4 2 2 2 2 0 1 13.6 with n=1,2,3,. ..
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Nano Science (Lec6 Quantum Dots) - MAE 287/EE 257 1 In...

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