2D Semiconductor Quantum Dots

 Part 1: Fock-Darwin Energy Levels  ☝

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The motion of a conduction-band electron confined in a 2D parabolic well in an external perpendicular magnetic field, in the effective- mass approximation, is described by the following Hamiltonian 

\[H =\frac{p^2}{2 m^*} + \frac{1}{2} m^* \omega^2_0 r^2\] \[p \rightarrow p - e A\] \[H =\frac{1}{2 m^*} (p - e A)^2 + \frac{1}{2} m^* \omega^2_0 r^2\]

• where $m^*$ is the effective mass
• r is the position
• p is the momentum
• A is the vector potential ofthe magnetic field B
• B = rot A, in the symmetric gauge: \[A = \frac{1}{2} r \times B = \frac{1}{2} \begin{pmatrix} x\\ y\\ z \end{pmatrix} \times \begin{pmatrix} 0\\ 0\\ B \end{pmatrix} = \frac{B}{2} \begin{pmatrix} y\\ -x\\ 0 \end{pmatrix}\]         

\[H =\frac{p^2}{2 m^*}+ \frac{(e A)^2}{2 m^*}-\frac{e}{2 m^*} (p\, A- A \,p)+ \frac{1}{2} m^* \omega^2_0 r^2\]

On the one hand

\[\left\{\begin{matrix} p. A = \begin{pmatrix} -i\hbar\partial_x\\ -i\hbar\partial_y\\ -i\hbar\partial_z \end{pmatrix} . \begin{pmatrix} \frac{B}{2}y\\ -\frac{B}{2}x\\ 0 \end{pmatrix} = \frac{-i\hbar B }{2 } (\partial_x y- \partial_y x) =0\\ A. p = \begin{pmatrix} \frac{B}{2}y\\ -\frac{B}{2}x\\ 0 \end{pmatrix} . \begin{pmatrix} -i\hbar\partial_x\\ -i\hbar\partial_y\\ -i\hbar\partial_z \end{pmatrix} = \frac{-i\hbar B }{2 } (y \partial_x - x \partial_y ) = \frac{B }{2 } (y p_x - x p_y )= -\frac{B }{2 } l_z \end{matrix}\right.\]

• $l_z$ is the projection of the angular momentum onto the field direction

On the other hand $A^2= \left \|  \frac{B}{2} \begin{pmatrix} y\\ -x\\ 0 \end{pmatrix} \right \|^2=\frac{B^2}{4} r^2$

So we can rewrite the previous Hamiltonian H :

\[H =\frac{p^2}{2 m^*}+ \frac{1}{2} m^*  \Omega^2 \,r^2-\frac{\omega_c}{4 }\, l_z\] 

Where $ \omega_c =\frac{e\, B}{m^*} $ is the cyclotron frequency and $\Omega = \sqrt{\omega^2_0 +\frac{\omega_c^2}{4} }$.

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