1. 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

2. 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 from band offset. Electrostatic dot. Confinement from external potential.

3. 1D potential barrier No reflected wave needed! Transmission coefficient = 1! No confinement! Klein paradox. i rt V

4. Single layer graphene in a magnetic field Wave function decays like States localise in a magnetic field. McLure, PR 104, 666 (1956)

5. Electric and magnetic confinement Scalar potential → deconfinement. Vector potential → confinement. What happens when both potentials are present? Model ： Circularly symmetric states: Radial function satisfies:

6. 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?

7. Typical quantum states Character of states depends on s, t, B : s > t : deconfined states s < t : confined states s = t : confinement deconfinement transition (above)

8. Energy spectrum near transition Bound state levels emerge from continuum. Continuum slope diverges linearly with system size. Vertical transition in infinite size limit.

9. Physical reason for transition BoundedUnbounded Quantum tunnelling Confined states only when classical motion is bounded. E cannot confine massless, charged particles. Need E and B.

10. 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.

11. A realistic potential Real potential does not increase without limit. Problem is to isolate the dot level from the bulk Landau levels.

12. Real dots: density of states Dot level Need a potential with a barrier to isolate the dot state.

13. Real dot: confinement-deconfinement transition Character changes: oscillations → smooth decay. Similar to Klein Paradox.

14. Possible experiments Probe LDOS with STM: Attach contacts and study transport: Many other geometries possible.

15. 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.