1. Magnetic Field: Classical Mechanics Aharonov-Bohm effect
Lecture 2
Magnetic Field: Classical Mechanics Aharonov-Bohm effect
Josep Planelles

2. The Hamiltonian of a particle in a magnetic field
Where does this Hamiltonian come from?

3. Classical Mechanics: Conservative systems Newton’s Law
Lagrange equation
Hamiltonian

4. Velocity-dependent potentials
Time-independent field
No magnetic monopoles
Lagrange function in this case?
Guess:
Is equivalent to ?

6. Lagrangian
kinematic momentum:
canonical momentum:
Hamiltonian

7. Gauge ;
We may select c :
Coulomb Gauge :

8. Hamiltonian (coulomb gauge);

9. Axial magnetic field B & coulomb gauge;
gauge

10. Confined electron pierced by a magnetic field
Spherical confinement, axial symmetry
Competition: quadratic vs. linear term

11. Electron energy levels, nLM, GaAs/InAs/GaAs QDQW
Electron in a spherical QD pierced by a magnetic field
Electron energy levels, nLM, R = 3 nm InAs NC
Electron energy levels, nLM, R = 12 nm InAs NC
Landau levels
Electron energy levels, nLM, GaAs/InAs/GaAs QDQW
AB crossings

12. Electron in a QR pierced by a magnetic field
AB crossings

13. Aharonov-Bohm Effect
Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485
Classical mechanics: equations of motion can always be expressed in term of field alone.
Quantum mechanics: canonical formalism. Potentials cannot be eliminated.
An electron can be influenced by the potentials even if no fields act upon it.
In their celebrated 1959 paper[1] Aharonov and Bohm pointed out that while the fundamental equations of motion in classical mechanics can always be expressed in term of field alone, in quantum mechanics the canonical formalism is necessary, and as a result, the potentials cannot be eliminated from the basic equations. They proposed several experiments and show that an electron can be influenced by the potentials even if no fields acts upon it. More precisely, in a field-free multiple-connected region of space, the physical properties of a system depends on the potentials through the gauge-invariant quantity H Adl, where A represents the potential vector.

14. Semiconductor Quantum Rings
Litographic rings GaAs/AlGaAs
A.Fuhrer et al., Nature
413 (2001) 822;
M. Bayer et al.,
Phys. Rev. Lett.
90 (2003)
self-assembled rings InAs
Los primeros QRs fueron fabricados en los 80 por técnicas litográficas. Mesoscópicos. Dispersión.
Primeros anillos nanoscópicos recientes: litográficos por Fuhrer y Bayer; auto-ordenados por Jorge García. Éstos crecen por una ligera variación en la síntesis de QDs auto-ordenados.
J.M.García et al., Appl. Phys. Lett. 71 (1997) 2014
T. Raz et al., Appl. Phys. Lett 82 (2003) 1706
B.C. Lee, C.P. Lee, Nanotech. 15 (2004) 848

15. Toy-example : Quantum ring 1D

16. Quantum ring 1D