Electronic and magnetic properties of carbon tori under external fields

1. Electronic and magnetic properties of carbon tori under external fields Student : C. C. Tsai Professor: M. F. Lin 2005-08-05

2. Outline 1. Introduction 2. Tight-binding model 3. Electronic properties of carbon tori a. Magnetic field b. Electric field 4. Magnetization of carbon tori 5. Conclusion

3. 1. Introduction Roll-up R 2r Ry aC-C≡b=1.42 Å |a1|=|a2|=2.46Å

4. Zero-dimensional carbon tori Nature 385, 780 (1997) Scanning force micrograph (SFM) image Height: 1.0-1.2 nm Diameter: 300-500 nm J. Phys. Chem. B 103, 7551 (1999) Scanning electron microscope (SEM) image Radius: 300-400 nm

5. 2. Tight-binding model • Unit cell of graphite sheet • Tight-binding function A B • Wave equation 

6. Hamiltonian matrix elements v on-site energy v nearest-neighbor interaction B3 g0 B2 g0 A g0 B1

7. Boundary condition and curvature effect

8. B Z R a X Y • Hamiltonian matrix of carbon tori with magnetic field

9. Zeeman effect

10. q • In electric field

11. Energy gap: energy difference between HOS and LUS HOS: highest occupied state LUS: lowest unoccupied state • Density of states

12. 3. Electronic properties of carbon tori EF Low electronic states of armchair carbon tori • Occupied states are symmetric to the unoccupied states about EF=0. • Armchair carbon tori own double degeneracyat the low energy region. • Each state is characterized by a specific L.

13. L L a. magnetic field • B||: shift of L; B⊥: coupling of different L’s • Change the energy spacing, destroy the state degeneracy

14. Wave function (B⊥) • At a=0o, the quantization of the wave function (Yh) remains unchanged; that is, each Yh is well described by one L. • At a=90o, the wave function has the components of L,L±1,L±2 for f≦f0. L

15. Density of states of carbon tori • The peak height corresponds to the state degeneracy, and the distance between two neighboring peaks is the energy spacing between states. • At f/f0>0, peak height is reduced to half its size with increasing a, except at a=90o for armchair tori. Eg=0 • For chiral angle≠30o

16. Energy gaps of carbon tori • Eg is symmetric about f0/2 at a=0, and it exhibits a periodic AB oscillation with a period f0. • SMTs happen frequently at small a. • For chiral angle≠30o

17. b. electric field • At aE=0 and E≦0.02 g0/Å, e(E||) is much smaller than tij. • Transverse electric field could drastically modulate the state energies. • There are more low- and extreme-energy states. • Energy modulation is largely enhanced by B.

18. double degeneracy fourfold degeneracy • Density of states • State degeneracy will destroyed by electric field.

19. Energy gaps with Zeeman effect • There exists SMTs when E increases. • SMT happens more frequently as aE increases. • Magnetic field could enhance the modulation of energy gap.

20. 4. Magnetization of carbon tori

21. Magnetization of carbon tori with Zeeman effect • At a=0o, M depends on f linearly and exhibits two pairs of special jump structures. • Armchair carbon tori changes from diamagnetism into paramagnetism when a increases.

22. Dependence on radius of magnetization • The jump height linearly decrease with the increasing radius. • At f=0, most of armchair carbon tori are diamagnetic except for few systems with zero Eg. Dependence on radius of magnetization • At a=90o, armchair carbon tori exhibit paramagnetism. • The f-dependent Mis not sensitive to radius at a=90°.

23. Dependence on radius of magnetization • There exist a critical angle ac=30o in determining magnetism.

24. Dependence on chirality • Zigzag tori are diamagnetism at small f. • The critical angle ac=30° is also present in carbon tori near amrchair configuration.

25. Dependence on T • At a=0o, the special jump structures become peak structures when T>0. • Magnetization hardly depend on temperature at a=90o.

26. 5. Conclusion • 0D carbon tori are studied for their electronic and magnetic properties. • The magnetic field might destroy state degeneracy, lead to semiconductor-metal transitions, modulate energy spacings, cause Aharonov-Bohm (AB) oscillations. • The electric field in carbon tori could induce similar effects except the AB oscillations. • Magnetism and strength of magnetic response are mainly determined by the geometric structures and the magnitude and the direction of magnetic field.