Ballistic transport, chiral anomaly and radiation from the electron

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 Schwingers 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 , T0 bypassing
the Kubo formula. The first quantized function
obeys
Fradkin, Gitman, Shvarzman, QED with unstable
vacuum

Electric field is switched on at t0


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 Schwingers formula 1

• For small electric field
• Dirac points dominate the pair creation rate.
• Following the Schwingers rate at zero mass
  limit

Schwinger, PR (1962)

• This would lead to electron hole plasma
• An excellent chance to verify
• Schwingers result.
• Creation vs. decay

15
Pair creation rate and Schwingers 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. Conclusions 1

• 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 Schwingers 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