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.