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
Outline
1.
Tight binding model of the graphene sheet. Dirac points and
quasi – Ohmic “resistivity” without either impurities or
carriers.
2.
Linear response and the chiral anomaly. Role of electrons far
from the Dirac points.
3.
Beyond linear response and the Schwinger’s pair creation
rate.
4.
Radiation emitted from graphene.
5. Conclusion
1. Tight binding nearest neighbour model
a1 



a
a
1, 3 ; a 2  1, 3
2
2

There are two sublattices and
consequently Hamiltonian is an off
diagonal matrix.
ˆ 
H

sites r,
aˆ A  r  aˆB  r     c.c.
Single graphene sheet
as seen by STM
E. Andrei et al, Nature
Nano 3, 491(08)
1
a1  a 2 ,
3
1
 2  a1  2a 2 ,
3
1
 3   2a1  a 2 
3
1 
In momentum
space:
h  k    aˆ A  k  
 0
†
†
ˆ
H    aˆ A  k  aˆ B  k    *


ˆ
h
k
0
BZ
  
  aB  k  
h  k     e
Spectrum:
ik 
  ak y 
 ak y   ak x 
  exp  i
  2exp  i
 cos  2 

 2 3 
  3
 k   h k 
Wallace, PR71, 622 (1949)
Fermi surface: two in-equivalent
points around which the spectrum
becomes “ultra - relativistic”
3 a
c
  vg k ;vg 

2
300
2. The minimal DC conductivity of the absolutely
clean graphene
Theory 1:
Theory 2:
2
4e
1 
(missing  )
 h
2 
 e2
2 h
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)…
Ziegler, PRB75, 233407 (07);
Beneventano et al, arXiv 0901.0396 (09)
…
Regularization dependent
 exp
 e2 
 4 
h
Geim et al,
Nature Mat. 6,
183 (07)
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.
e2
 0  2.2   2
h
E. Andrei et al, Nature
Nano 3, 491(08)
Transparency at optical frequencies.
Optical frequencies conductivity agrees with
was measured to accuracy of 1%.
2
and
Geim, Novoselov
et al, Science 102
10451 (08)
This value remain the same in the high frequency limit.
15
/


0.24

10
sec.
The only time scale for pure graphene is
Hard to imagine why DC value is different from this .
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 EF  0 , 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
e
ik r
h  p    1 
 1 
 1   0
 
  ; i  t     h* p
0   2 
 2
 2   
At  c  0,  E  t    t  
px  k x , p y  k y 
e
At
c
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:
J
 
e
B. Z .
*
1
 0 h    1 
2  *
 E



 h ' 0   2 
*
2
2
*
*

e
 d   h h ' h ' h 
 2
    t 2  
sin


2 B. Z .  dk y 
2 2



2


t 


The first is divergent, but vanishes upon integration over BZ.
The second, upon integration over BZ, approaches the
“dynamical” value .    e
2
2
2 h
Frequency independent for
  1/ t ;t  /   0.24fs
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.
K
K
K
Non-invariant regularization (mass, … ) leads to an arbitrary
result.
3. Beyond linear response (DC).
Numerical result of the tight binding model:
Electric field in microscopic units
conductivity
2
E
2.0
E0
E
2 6
E0
2 7
pair creation rate
2.5
E
2 8
E0
1.5
1.0
0.5
0.0
0
20
40
60
80
100
time in units of t
120
E0   / ea  1010 V/m
Crossover time from the linear
response into a linear dependence
/ eEvg =  E / E0 
tnl 
Consistent with 3rd order perturbation

3  t4
   2 1   4
 64  tnl

 .

Rosenstein, et al,
PRB81, 041416 (10);
Kao et al, PRB82, 035406 (10).
1/2
t
Beyond linear response (AC).
New phenomena appears:
• Inductive part of conductivity
• 3rd harmonics generation
2
2
  63
9t   E  
Re    2 1  

 ;
4
2 
64   E0  
  128

27t  E 
Im  
 
256 3  E0 
 3
1  E 

 
512 4  E0 
2
Rosenstein, et al,
PRB81, 041416 (10);
Kao et al, PRB82,
035406 (10).
2
• Perturbation fails for
 2  E / E0 and t   /  E / E0  .
Bloch oscillation
1
8  E 
For t  t B , t B 
  t ,
3  E0 
Bloch oscillation sets in.
E
J (t )
 3 2  
E
 E0 
1/2
 3t E 
sin 
.
 4t E0 
Gives excellent fit.
Floquet theory must be applied to obtain the result.
L  0.5 m, tbal  2.3 103 t , W  1.5 m, E0  1010 V/m
 E / E0  ~ 103 , I  1mA, tB /4= 3.6 103 t ~ tbal  L / vg ;
 E / E0  ~ 5 105 , I  50 A, tB /4= 7.2 104 t tbal .
Pair creation rate and Schwinger’s formula 1
For small electric field: E / E0 ~ 212  29
• 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:
dNp
dt

8

2
 E 
 
vg  E0 
3/2
.
• An excellent chance to verify
Schwinger’s result.
• Creation vs. decay
Pair creation rate and Schwinger’s formula 2
 3 
J  t    2 
E 
 2 
3/2
1/2
 evg 

 t.


Are these fields and
ballistic times
feasible?
L  0.5 m,W  1.5 m,
E  104 V/m.
tnl ~ tbal  2000t
4. Radiation emitted from graphene 1
2
2


e 
e




† 
H   vg σ   i   A  
Az   Vconf  z  
 z 
c  2m 
c 





The one photon emission process is
dominant due to an extra factor of
 QED , with the phase space
remaining the same.
Radiation emitted from graphene 2
From Golden rule, photon emission rate:
Wnn '  p, p ', k , k z , t  
 
2
 
Fnn '
2
N p  t  N  p '  t    vg  p  p '   
Landau-Zener creation rate:
2

Np  t     p y    eEt /  p y  exp   hvg / 2eE  px 
Transition amplitude:
 
Fnn '
E0 evg i vg  p  p ' t    ik z z
(2)
i
e
F
e

*
z

z





 p  p ' k  ,
p ,p ' 
n
n'
2
 2L
z
Fp ,p'  v†  p ' σ  e  u  p  , E02   /  L2 Lz  .
e     sin  , cos   , ez   0; e    cos   cos  ,sin   , e z   sin  .
1
1
2
2
Radiation emitted from graphene 3
Fp ,p '  2 1  cos  2     '  ,
1
Matrix element:
2
Fp ,p '  2 cos 2  1  cos  2     '  .
1
2
Spectral emittance per volume in the momentum space:
M
 
k, kz , t  
e 2 vg2
F

 2 
 
nn '
4
2
N p  t  N p k  t    vg  p  p  k    
p
In the perpendicular direction:
M  0,  / c, t  
e2 vg2tnl2

4

0
p  t / tnl
 F
 
nn '
2
4
  t / 4  p
2 2
nl

2 1/2
 tnl / 2  p 
 exp  2  2tnl2 / 4  p 2  
Radiation emitted from graphene 4
For t
 1 ,
2e 2t  2tnl2 /2
M  0,  / c, t   4 2 e
,
 tnl
For t  t / 2, M  0,  / c, t  
2
nl
e2

e
3
2
 2tnl
/2
I 0   2tnl2 / 2  .
lim M  0,  / c, t   2e 2 /  4tnl  .
 
Emittance at various 
Emittance at various
t
Radiation emitted from graphene 5
Window of frequency to observe the Schwinger effect:
min  1/ tnl , max  2t / tnl2 . For E  104 V/m, min  3.6THz.
Radiated power per unit area:
v  vg / c  1/ 300
For t
1,

2
 0
c3
Lnn '  ,  , t   
Mnn '   k , k z , t .

e4v3 E 2  t 3
t
2
2
tnl , L  ,  , t   5/2 4 2  3 sin  
cos   ;
2  c  3tnl
4 tnl

1
L
 2
 t3

e4v3 E 2
t
2
2
2
sin   .
 ,  , t   5/2 4 2 cos   3 cos  
2  c
4 tnl
 3tnl

The radiant flux from a flake of  m   m , for E  10 4 V/m
is 4.7 1021 W ~ 12 photons per second.
Radiation emitted from graphene 6
Angular dependence of radiation intensity at t  tnl :
In plane polarization.
At small  , of the same order.
Out of plane polarization.
At  ~ 90 , in plane term dominant.
Radiation emitted from graphene 7
Angular dependence of un-polarized intensity at t  tnl :
Radiation is suppressed at  ~ 90,  ~ 90 .
5. Conclusions 1
1. DC conductivity is equal to its “dynamical”
2

e
value  
.
2
2 h
2. It is independent of frequency (at zero
temperature) all the way up to UV.
3. The experimental and theoretical values are
now in better agreement with  2 .
Conclusions 2
4. (i) t  tnl   E / E0 1/2 t , linear response regime.


(ii) tnl  t  tB  8 / 3  E / E0  t ,
1
linear
rise in conductivity and fast Schwinger’s
pair creation phase sets in leading to
creation of electron – hole plasma.
(iii) t  tB , Bloch regime.
L  0.5 m, W  1.5 m, E0  1010 V/m
 E / E0  ~ 103 ,
 E / E0  ~ 5 105 ,
t B /4 ~ tbal  L / vg ;
t B /4
tbal .
Conclusions 3
5. For t
2e 2t  2tnl2 /2
M  0,  / c, t   4 2 e
,
 tnl
 1 ,
For t  t / 2, M  0,  / c, t  
2
nl
e2

e
3
2
 2tnl
/2
I 0   2tnl2 / 2  .
lim M  0,  / c, t   2e 2 /  4tnl  .
 
min  1/ tnl , max  2t / tnl2 . min ~ THz.
6.
For t

e4v3 E 2  t 3
t
2
2
tnl , L  ,  , t   5/2 4 2  3 sin  
cos   ;
2  c  3tnl
4 tnl

1
L
 2
 t3

e4v3 E 2
t
2
2
2
sin   .
 ,  , t   5/2 4 2 cos   3 cos  
2  c
4 tnl
 3tnl

Conclusions 4
7. “Radiation friction” is not significant until
t  t p ~ N p / E 3/2  400tnl , for E  104 V/m.
Equilibrium is reached at t  5000tnl .