confinement-deconfinement transition in graphene quantum dots p. a. maksym university of leicester...
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Confinement-deconfinement Transition in Graphene Quantum Dots
P. A. Maksym
University of Leicester
1. Introduction to quantum dots.2. Massless electron dynamics.3. How to confine electrons in graphene.4. Experimental consequences.
Collaborators: G. Giavaras and M. Roy
Semiconductor quantum dots• Artificial atoms. Electrons confined on a nm length scale.
• Graphene dots are extremely promising. But
- No technology to grow and cut graphene. Dots of self-assembled type not yet possible.
- Pure electrostatic confinement is difficult. The interesting linear dispersion causes problems.
Self-assembled dot.Confinement fromband offset.
Electrostatic dot.Confinement fromexternal potential.
1D potential barrier
No reflected wave needed!
Transmission coefficient = 1! No confinement!
Klein paradox.
i
r t
V
Single layer graphene in a magnetic field
Wave function decays like
States localise in a magnetic field.
McLure, PR 104, 666 (1956)
Electric and magnetic confinement
• Scalar potential → deconfinement.• Vector potential → confinement.
What happens when both potentials are present?
Model :
Circularly symmetric states:
Radial function satisfies:
Radial functions
Let
Get
Physical meaning:
oscillations, no confinement
no oscillations, states always confined
confinement-deconfinement transition when
In the large r limit
When are the states confined?
Typical quantum states
Character of states depends on s, t, B:s > t: deconfined statess < t: confined statess = t: confinement deconfinement transition (above)
Energy spectrum near transition
Bound state levels emerge from continuum.Continuum slope diverges linearly with system size.Vertical transition in infinite size limit.
Physical reason for transition
Bounded Unbounded
Quantumtunnelling
Confined states only when classical motion is bounded.E cannot confine massless, charged particles.Need E and B.
Confinement in an ideal dot
Confinement occurs when s < t.Confinement-deconfinement transition when s = t.
How can this be used to make a single layer graphene dot?Need to consider the potential in a realistic dot model.
A realistic potential
Real potential does not increase without limit.Problem is to isolate the dot level from the bulk Landau levels.
Real dots: density of states
Dot level
Need a potential with a barrier to isolate the dot state.
Real dot: confinement-deconfinement transition
Character changes:oscillations →smooth decay.
Similar to Klein Paradox.
Possible experiments
• Probe LDOS with STM:
• Attach contacts and study transport:
• Many other geometries possible.
Conclusion
• Confinement in graphene dots is conditional.
• Can be achieved with a combination of E and B.
• Character of states can be manipulated at will.