Ballistic transport, c hiral anomaly and radiation from the electron – hole plasma in graphene

1. Ballistic transport, chiral anomaly and radiation from the electron –hole plasma in graphene Hsien-Chong Kao NTNU, Taiwan , Collaborators: Baruch Rosenstein(NCTU) Meir Lewkowicz (Ariel UC) , 2, April , 2011

2. Outline • Tight binding model of the graphene sheet. Dirac points and quasi – Ohmic “resistivity” without either impurities or carriers. • Linear response and the chiral anomaly. Role of electrons far from the Dirac points. • Beyond linear response and the Schwinger’s pair creation rate. • Radiation emitted from graphene. • 5. Conclusion

3. 1. Tight binding nearest neighbour model Single graphene sheet as seen by STM E. Andrei et al, Nature Nano 3, 491(08) There are two sublattices and consequently Hamiltonian is an off diagonal matrix.

4. In momentum space: Spectrum: Wallace, PR71, 622 (1949) Fermi surface: two in-equivalent points around which the spectrum becomes “ultra - relativistic”

5. 2. The minimal DC conductivity of the absolutely clean graphene Fradkin, PRB33, 3257 (1986) Lee, PRL71, 1887 (1993) Ludwig et al, PRB (1994) Ando et al, J. P. S. Jap. 71, 1318 (02) Gusynin, Sharapov, PRB73, 245411 (06) Peres et al, PRB73, 125411 (06)… Theory 1: Theory 2: Ziegler, PRB75, 233407 (07); Beneventano et al, arXiv 0901.0396 (09) … Regularization dependent Geim et al, Nature Mat. 6, 183 (07)

6. Recent advance – suspended graphene (SG). Graphene on substrate (NSG) exhibits a network of positive and negative puddles and therefore does not probe directly the Dirac point. In suspended graphene (SG), conductivity mismatch drops to 1.7 instead of 3 at zero temperature. E. Andrei et al, Nature Nano 3, 491(08)

7. Transparency at optical frequencies. Optical frequencies conductivity agrees with and was measured to accuracy of 1%. Geim, Novoselov et al, Science 102 10451 (08) This value remain the same in the high frequency limit. The only time scale for pure graphene is Hard to imagine why DC value is different from this.

8. 3. The dynamical approach to ballistic transport in graphene. The basic picture of the quasi – Ohmic resistivity in pure graphene is the creation of the electron – hole pairs by electric field. We use a method which has a natural “regularizations” and can be applied directly to DC at Dirac point , T=0 bypassing the Kubo formula. The first quantized function obeys Fradkin, Gitman, Shvarzman, “QED with unstable vacuum” Electric field is switched on at t=0

9. Linear response to DC field Expanding the electric current to first order, Lewkowicz, B.R., PRL102, 106802 (09); Rosenstein, et al, PRB81, 041416 (10); Kao et al, PRB82, 035406 (10). Two terms: The first is divergent, but vanishes upon integration over BZ. The second, upon integration over BZ,approaches the “dynamical” value . Frequency independent for

10. Universality and chiral anomaly • Two gapless points on BZ ~ massless fermions “species doubling”. • Hamiltonian staggered fermions in the lattice gauge theory. • Nielsen – Ninomiya theorem: two Dirac points and • correct “matching” of two massless “species”. • The finite part of the conductivity is • dominated by Dirac points • The “divergent” part is not • dominated by Dirac points. • Feasibility of effective Dirac model hinges • on using chirally invariant regularization. Non-invariant regularization (mass, … ) leads to an arbitrary result.

11. 3. Beyond linear response (DC). Numerical result of the tight binding model: Electric field in microscopic units Crossover time from the linear response into a linear dependence Consistent with 3rd order perturbation Rosenstein, et al, PRB81, 041416 (10); Kao et al, PRB82, 035406 (10).

12. Beyond linear response (AC). • New phenomena appears: • Inductive part of conductivity • 3rd harmonics generation • Perturbation fails for Rosenstein, et al, PRB81, 041416 (10); Kao et al, PRB82, 035406 (10).

13. Bloch oscillation For Bloch oscillation sets in. Gives excellent fit. Floquet theory must be applied to obtain the result.

14. Pair creation rate and Schwinger’s formula 1 • For small electric field: • Dirac points dominate the pair creation rate. • Following the Schwinger’s rate at zero mass limit Schwinger, PR (1962) • This would lead to electron –hole plasma: • An excellent chance to verify • Schwinger’s result. • Creation vs. decay

15. Pair creation rate and Schwinger’s formula 2 Are these fields and ballistic times feasible?

16. 4. Radiation emitted from graphene 1 The one photon emission process is dominant due to an extra factor of , with the phase space remaining the same.

17. Radiation emitted from graphene 2 From Golden rule, photon emission rate: Landau-Zener creation rate: Transition amplitude:

18. Radiation emitted from graphene 3 Matrix element: Spectral emittance per volume in the momentum space: In the perpendicular direction:

19. Radiation emitted from graphene 4 Emittance at various Emittance at various

20. Radiation emitted from graphene 5 Window of frequency to observe the Schwinger effect: Radiated power per unit area: The radiant flux from a flake of , for is 12 photons per second.

21. Radiation emitted from graphene 6 Angular dependence of radiation intensity at : In plane polarization. Out of plane polarization. At small , of the same order. At , in plane term dominant.

22. Radiation emitted from graphene 7 Angular dependence of un-polarized intensity at : Radiation is suppressed at .

23. 5. Conclusions1 • DC conductivity is equal to its “dynamical” • value . • It is independent of frequency (at zero temperature) all the way up to UV. • The experimental and theoretical values are now in better agreement with .

24. Conclusions 2 4. (i) linear response regime. (ii) linear rise in conductivity and fast Schwinger’s pair creation phase sets in leading to creation of electron – hole plasma. (iii) Bloch regime.

25. Conclusions 3 5. 6.

26. Conclusions 4 7. “Radiation friction” is not significant until for Equilibrium is reached at