lecture 2 magnetic field: classical mechanics magnetism: landau levels aharonov-bohm effect...
TRANSCRIPT
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Lecture 2
Magnetic Field:
Classical MechanicsMagnetism: Landau levelsAharonov-Bohm effectMagneto-translations
Josep Planelles
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Newton’s Law
Lagrange
equation
Conservative systems
Classical Mechanics: an overview
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No magnetic monopoles
Time-independent field
Velocity-dependent potentials
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Define:
Velocity-dependent potentials: cont.
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Velocity-dependent potentials: cont.
kinematic momentum:
canonical momentum:
Hamiltonian:
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Conservative systems: V(x,y,z) & L = T - V
kinematic momentum:
canonical momentum:
Hamiltonian:
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Gauge
We may select :
;
Coulomb Gauge :
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Hamiltonian (coulomb gauge)
;
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Axial magnetic field B & coulomb gauge
;
gauge
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Hamiltonian: axial magnetic field B & coulomb gauge
;
Cyclotron frequency
;
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Electron in a magnetic field
• Landau levels
• No crossings
Rosas et al. AJP 68 (2000) 835
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Confined electron pierced by a magnetic field
Spherical confinement, axial symmetry
Competition: quadratic vs. linear term
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Electron energy levels,
nLM, R = 3 nm InAs NC
Electron in a spherical QD pierced by a magnetic field
Electron energy levels,
nLM, R = 12 nm InAs
NC
Landau levels
Electron energy levels, nLM, GaAs/InAs/GaAs QDQW
AB crossings
J. Planelles, J. Díaz, J. Climente and W. Jaskólski, PRB 65 (2002) 245302
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Electron in a QR pierced by a magnetic field
Electron energy levels (n=1, M) (n=2,M), of a InAs QR
Dimensions: r = 10, R=60, h=2J. Planelles ,W. Jaskólski, and I. Aliaga, PRB 65 (2001) 033306
AB crossings
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Peierls/Berry phase
R. Peierls, Z. Phys. 80 (1933) 763;M. Graf & P. Volg, PRB 51 (1995) 4940M.V. Berry, Proc. R. Soc. A 392 (1984) 45.R.Resta, JPCM 12 (2000) R107;
A phase in is introduced in the wf by the magnetic field:
Choose:
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Peierls/Berry phase cont
Peierls subtitution:
Hamiltonian substitution:
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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.
• Berry’s phase (gauge dependent) • Gauge independentAdl
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Semiconductor Quantum Rings
Litographic rings
GaAs/AlGaAsA.Fuhrer et al., Nature
413 (2001) 822;
M. Bayer et al.,
Phys. Rev. Lett.
90 (2003) 186801.
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
self-assembled rings
InAs
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Example 1: Quantum ring 1D
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Electron in a potential vector but no magnetic field
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Aharonov-Bohm Effect
• Periodic symmetry changes of the energy levels
•Energetic oscillations
• Persistent currents
E
Φ/ Φ0
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Some 2D calculations: off-centering
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FIR absorption of one electron in QD and QR
J. Planelles and J. I. Climente, Collect. Czech. Chem. Commun. 70 (2005) 605 arXiv: cond-mat/0412552;
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Fractional Aharonov-Bohm Effect
1 electron2 electrons coulomb
interaction
J.I. Climente , J. Planelles and F. Rajadell,J. Phys. Condens. Matter 17 (2005) 1573
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Optic Aharonov-Bohm effect
The Aharonov-Bohm effect leads to changes in electron and hole
symmetry at different B values:
J. Climente, J. Planelles and W. Jaskólski, Phys. Rev. B 68 (2003) 075307
Observed in stacked type II QDs: Kuskovsky et al . Phys. Rev. B 76 (2007) 035342; Sellers et al PRL 100, 136405 (2008).
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Translations and magneto-translations I
Switching the magnetic field on: No
translational symmetry
Though, can we get Bloch functions?
Simultaneous
eigenfunctions:
Bloch functions
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Translations and magneto-translations II
Commute …. but
and we can get Bloch functionsCommute …. if
Physical meaning: Translation + Lorentz Strength compensation
Two-fold periodicity: magnetic and spatial cells
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Translations and magneto-translations III
Two-fold periodicity: magnetic and spatial cells
J.L. Movilla and J. Planelles, PRB 83 (2011) 014410
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Thanks for your attention!
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SUMMARY
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Magnetic field: summary
No magnetic monopoles:
vector potential
No conservative field: velocity-dependent potential:
Lagrangian: kinematic momentum
Canonical momentum:
Hamiltonian:
Coulomb gauge:
Hamiltonian operator:
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Magnetic field: summary (cont.)
axial symmetry
Spatial confinement
dominates at strong confinement (atomic scale)
Relevant at soft confinement (nanoscale and bulk)
Aharonov-Bhom oscillations in non-simple topologies
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Magnetic field: summary (cont.)
Magneto-translations and Super-lattices
Fractal spectrum (Hofstadter butterfly)
Periodicity and homogeneous magnetic field
B-dependent (super)-lattice constant
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MORE
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Example 2: Flux outside only (QR 1D)
A = 0 at the position of system
B does not make any influence on it
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Example 3: irregular 1D system
x = distance along the circuit
L= circuit length
With magnetic flux:
F 2 F (flux units)
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Example 4: non homogeneous B, 1D system
Only flux dependent