39-4 Two- and Three-Dimensional Electron Traps

39-4 Two- and Three-Dimensional Electron Traps#

Prompts

  • For a rectangular corral (2D infinite well) with widths \(L_x\) and \(L_y\), the matter wave must fit into each dimension. What are the quantum numbers \(n_x\) and \(n_y\)? Write the energy \(E_{n_x,n_y}\) and the wave function \(\psi_{n_x,n_y}(x,y)\).

  • For a square corral (\(L_x = L_y = L\)), the ground state is \((1,1)\). What are the next few energy levels? Which states are degenerate (same energy)?

  • Extend to a 3D rectangular box with \(L_x\), \(L_y\), \(L_z\). Write \(E_{n_x,n_y,n_z}\). How does the energy add from each dimension?

  • What is degeneracy? Why do \((1,2)\) and \((2,1)\) have the same energy in a square corral?

  • Nanocrystallites and quantum dots are real electron traps. How does nanocrystallite size affect the threshold wavelength for absorption and the observed color?

Lecture Notes#

Overview#

  • A two-dimensional infinite potential well (rectangular corral) confines an electron in the \(xy\) plane. The wave is quantized separately in \(x\) and \(y\); quantum numbers \(n_x\) and \(n_y\); energy \(E = E_{n_x} + E_{n_y}\).

  • A three-dimensional rectangular box adds \(n_z\) and \(L_z\); \(E = E_{n_x} + E_{n_y} + E_{n_z}\).

  • Degeneracy: different states \((n_x, n_y)\) can have the same energy (e.g., \((1,2)\) and \((2,1)\) in a square corral).

  • Applications: Nanocrystallites, quantum dots, and quantum corrals are real electron traps; their size determines energy levels and optical properties.

Rectangular corral (2D)#

A rectangular corral is a two-dimensional infinite potential well: \(U = 0\) for \(0 < x < L_x\) and \(0 < y < L_y\); \(U \to \infty\) elsewhere. The electron is confined to the rectangle (e.g., on a surface).

The matter wave must fit into each dimension. Quantization is separate:

(394)#\[ \psi_{n_x,n_y}(x,y) = \sqrt{\frac{2}{L_x}}\sin\left(\frac{n_x\pi x}{L_x}\right) \sqrt{\frac{2}{L_y}}\sin\left(\frac{n_y\pi y}{L_y}\right) \]
(395)#\[ E_{n_x,n_y} = \frac{h^2}{8m}\left(\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2}\right), \quad n_x, n_y = 1, 2, 3, \ldots \]

The energy is the sum of the 1D energies for confinement in \(x\) and \(y\) separately.

Square corral and degeneracy#

For a square corral (\(L_x = L_y = L\)):

(396)#\[ E_{n_x,n_y} = \frac{h^2}{8mL^2}(n_x^2 + n_y^2) \]

\((n_x, n_y)\)

Energy (multiple of \(h^2/8mL^2\))

(1,1)

2 (ground state)

(1,2), (2,1)

5 (degenerate)

(2,2)

8

(1,3), (3,1)

10 (degenerate)

(2,3), (3,2)

13 (degenerate)

Degeneracy occurs when different combinations \((n_x, n_y)\) give the same \(n_x^2 + n_y^2\). For example, \((1,2)\) and \((2,1)\) are distinct states (different wave functions) but have the same energy.

Note

For a square corral, \((4,1)\) and \((1,4)\) have energy 17; \((3,3)\) has energy 18. So \((4,1)\) and \((1,4)\) lie below \((3,3)\) in the energy-level diagram.

Rectangular box (3D)#

A 3D infinite potential well (rectangular box) with widths \(L_x\), \(L_y\), \(L_z\):

(397)#\[ E_{n_x,n_y,n_z} = \frac{h^2}{8m}\left(\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2}\right) \]

\(n_x\), \(n_y\), \(n_z = 1, 2, 3, \ldots\). Quantum jumps and photon absorption/emission follow the same rules as in 1D; \(\Delta E = hf = hc/\lambda\).

Applications: nanocrystallites, quantum dots, quantum corrals#

Nanocrystallites: Semiconductor powder with nanometer-sized granules. Each granule acts as a potential well. Smaller granules \(\Rightarrow\) larger level spacing \(\Rightarrow\) higher threshold energy \(E_t\) for absorption \(\Rightarrow\) shorter threshold wavelength \(\lambda_t\). Light with \(\lambda > \lambda_t\) is scattered; the scattered spectrum determines the observed color. Smaller granules can appear yellow/green; larger ones red.

Quantum dots: Semiconductor layers between insulators; electrons trapped in the central layer. Thin insulating layer allows tunneling to add/remove electrons—an “artificial atom” with controllable electron count.

Quantum corrals: Atoms arranged in a circle on a surface (e.g., by STM manipulation). Electrons on the surface are confined; the resulting standing matter waves produce visible ripples in STM images.

Summary#

  • 2D corral: \(E_{n_x,n_y} = (h^2/8m)(n_x^2/L_x^2 + n_y^2/L_y^2)\); wave function \(\psi = \psi_{n_x}(x)\psi_{n_y}(y)\).

  • 3D box: \(E_{n_x,n_y,n_z} = (h^2/8m)(n_x^2/L_x^2 + n_y^2/L_y^2 + n_z^2/L_z^2)\).

  • Degeneracy: different \((n_x,n_y)\) can yield same energy.

  • Applications: Nanocrystallite size \(\Rightarrow\) color; quantum dots; quantum corrals.