ballistic transport,hiral anomaly and radiation from the electron hole plasma in graphene ballistic...

Ballistic transport,Ballistic transport, chiral hiral anomaly and radiation from the anomaly and radiation from the

electron electron – hole plasma in hole plasma in graphenegraphene

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

There are two sublattices and consequently Hamiltonian is an off diagonal matrix.

3,12

aa ;3,1

2

aa 21

213

212

211

a2a3

1

,2aa3

1

,aa3

1

sites r,

ˆ .ˆˆ H .BA r r c ca a

Single graphene sheet as seen by STM

E. Andrei et al, Nature Nano 3,

491(08)

In momentum space:

exp 2exp cos23 2 3

y yik xak ak ak

h k e i i

k h k Spectrum:

*

B

††

Z

0ˆ H0

ˆˆ

ˆˆ A

BB

A

h k kk k

h k ka

a

aa

Fermi surface: two in-equivalent points around which the spectrum becomes “ultra - relativistic”

3;

2 300g g

a cv k v

Wallace, PR71, 622 (1949)

2. The minimal DC conductivity of the absolutely clean graphene

Theory 1:

Geim et al, Nature Mat. 6,

183 (07)

2

1

4(missing )

e

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)…

Regularization dependent

Ziegler, PRB75, 233407 (07); Beneventano et al, arXiv 0901.0396

(09) …

2

2 2

e

h

Theory 2:

2

exp 4e

h

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)2

0 22.2e

h

Transparency at optical frequencies.

Optical frequencies conductivity agrees with and was measured to accuracy of 1%.

Geim, Novoselov et al, Science

102 10451 (08)

2

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 .

15/ 0.24 10 sec.

Electric field is switched on at t=0

tyyxx A

c

ekpkp

,

3. The dynamical approach to ballistic transport in graphene.

1 1 1*

2 2 2

0;

0ik r

t

h pe i

h p

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

0FE

Fradkin, Gitman, Shvarzman, “QED

with unstable vacuum”

0,tA c E t t ��

The basic picture of the quasi – Ohmic resistivity in pure graphene is the creation of the electron – hole pairs by electric field.

Linear response to DC field

The first is divergent, but vanishes upon integration over BZ.The second, upon integration over BZ, approaches the “dynamical” value .

Expanding the electric current to first order,

1* *1 2 *

. . 2

0 J

' 0B Z

heE

h

2

2 2

e

h

Lewkowicz, B.R., PRL102, 106802

(09);Rosenstein, et al,

PRB81, 041416 (10);Kao et al, PRB82,

035406 (10).22 2 * *

2 2. .

' ' 2sin

2 2B Z y

e d h h h ht tdk

Two terms:

Frequency independent for

1/ ; / 0.24fst t

• 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”.

Universality and chiral anomaly

K

K

K

• 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.

3. Beyond linear response (DC).

Numerical result of the tight binding model:

E

E 0 2 6 E

E 0 2 7

E

E 0 2 8

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

t im e in u nits o f t

cond

uctiv

ity 2

pair

crea

tion

rate

Electric field in microscopic units

Crossover time from the linear response into a linear dependence

100 / 10 V/mE ea

Consistent with 3rd order perturbation

1/2

0/ = /nl g E Et eEv t

4

4

2

31 .

64 nl

t

t

Rosenstein, et al,PRB81, 041416 (10);Kao et al, PRB82, 035406 (10).

New phenomena appears:

• Inductive part of conductivity

• 3rd harmonics generation

• Perturbation fails for

Beyond linear response (AC).

22

4 20

2

2

30

2

3 40

6

63 9Re 1 ;

128

27Im

256

1

51

4

2

E

E

E

E

E

t

t

E

Rosenstein, et al,PRB81, 041416 (10);Kao et al, PRB82, 035406 (10).

20 0/ and / . /tE EE E

For

Bloch oscillation sets in.

Gives excellent fit.

Floquet theory must be applied to obtain the result.

Bloch oscillation

1/2

20 0

sin( ) 3

.4

3t

J t E t E

E E E

1

0

, 8

3 ,B B

Et t t

Et

3

3 3

10

50

0

0

4

0.5 2.3 , 1.5 10 V/m

, /4= 3.6 / ;

, /4

m, 10 m,

/ ~ 10 1mA, 10 ~

/ ~ 5 10 5 = 7.2 .0 A, 10

bal

B bal

B b

g

al

t t

E E I t t

E E I t t

L W E

t L v

t

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

12 90 ~ 2/ 2E E

3/2

20

.8

g

E

Ev

dN

dt

p

Pair creation rate and Schwinger’s formula 2

4

0.5 m, 1.5 m,

10 V/m.

L W

E

2000~nl balt t t

Are these fields and ballistic times feasible?

3/2 1/2

2

3

2.gevJ t E t

4. Radiation emitted from graphene 1

22

†conf2

g z z

e eH iv

cV

czA

m

σ A

The one photon emission process is dominant due to an extra factor of , with the phase space

remaining the same.

QED

Landau-Zener creation rate:

Radiation emitted from graphene 2

2

' ' ',2

, ', , 'nn z nn gW k F v p pt N t N t p pp p k

From Golden rule, photon emission rate:

Transition amplitude:

2exp/ 2/y y g xN t eEt p hp v eE p p

' (2)0' , ' '2

† 2, '

2 2

20

1 1

' ,2

' / .

sin ,cos ,e 0; cos cos ,sin ,e sin .

*

,

g zi v p p tg ik

nn n n

z

z

z

z z

evEF i e e

L

v u

z

E LL

z

p p

p p

p p k

p σ e p

e e

F

F

Spectral emittance per volume in the momentum space:


2

2

'4

2

,2

,z nn ggk

e vt N t N t v p

p p k

p

k p k M F

Matrix element:

In the perpendicular direction:

21

, '

21 2, '

2 1 cos 2 ,

2 1 cos

'

cos ' .2

p p

p p

F

F

20 1/22 2

4 /

2 2

2 22

2

0, /

, / 4 / 2

/ ex 4p 2

nlnlp t

g nlnlt

nl

e v tt t p t p

p

c

t

M

2

' 4nn

F


2 2

2 2

2/21

4 2

2/22 2 2

03

2 4

2For 0, /

For / 2 0, / / 2 .

lim 0

,

, /

, ,

, ,

, 2 / .

nl

nl

t

nl

tnl nl

nl

e tt c e

t

et t c e I t

c e

t

t

t t

M

M

M

Emittance at various Emittance at various t

Radiated power per unit area:


2

' '30, ., ,,nn nn zkt t

c

kL M

2 4min max min1/ , 2 / For 10 V/m, 3 6TH. .. znl nlt t Et

4 3 31 2 2

5/2 4 2 3

4 3 32 2 2 2

5/2 4 3

2

2

2

For , sin cos ;2 3 4

, ,

, cos co, s sin .2 3 4

nlnl nl

nl nl

e v E t tt t

t t

t t

t

tc

e v Et

c t

L

L

1/ 30/ 0 1,gv v c

Window of frequency to observe the Schwinger effect:

The radiant flux from a flake of , for is 12 photons per second.

m m 4 /10 mVE 214.7 10 W ~


In plane polarization. Out of plane polarization.At small , of the same order.

At , in plane term dominant.

Angular dependence of radiation intensity at : nlt t

~ 90


Angular dependence of un-polarized intensity at : nlt t

Radiation is suppressed at . ~ 90 , ~ 90

5 .Conclusions 1

1. DC conductivity is equal to its “dynamical”

value .

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

2 2

e

h

2

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.

1/2

0/ ,nl E Et t t

1

08 / / ,3nl Bt t t E E t

,Bt t

30

100

50

0.5 1.5 10 V/m

/4 / ;

, /

m, m,

/ ~ 10 , ~

/ ~ 5 1 4 .0

B bal

B al

g

b

E E t

L W E

t L v

tE E t

Conclusions 3

5.

6.

2 2

2 2

2/21

4 2

2/22 2 2

03

2 4

2For 0, /

For / 2 0, / / 2 .

lim 0

,

, /

, ,

, ,

, 2 / .

nl

nl

t

nl

tnl nl

nl

e tt c e

t

et t c e I t

c e

t

t

t t

M

M

M

2min max min1/ , 2 / T. ~ .Hznl nlt t t

4 3 31 2 2

5/2 4 2 3

4 3 32 2 2 2

5/2 4 3

2

2

2

For , sin cos ;2 3 4

, ,

, cos co, s sin .2 3 4

nlnl nl

nl nl

e v E t tt t

t t

t t

t

tc

e v Et

c t

L

L

Conclusions 4

7. “Radiation friction” is not significant until for

Equilibrium is reached at

3/2/~ ,400p p nlt t EN t 410 V/m.E

0050 .nlt t

ballistic transport,hiral anomaly and radiation from the electron hole plasma in graphene ballistic...

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