Download Quantum Faraday effect in graphene within relativistic Dirac model

GIANT
Quantum Faraday effect in
graphene systems
I.V. Fialkovsky1,2, D. Vassilevitch2,3
1 Instituto
de Física, Universidade de São Paulo, Brasil
of Theoretical Physics, St. Petersburg State University, Russia
3 Universidade Federal ABC, São Paulo, Brasil
2 Department
2
Contents
• A reminder on Dirac description of Graphene
• One-loop effective theory:
▫ Polarization operator
▫ Matching conditions
▫ Light propagation
• Conductivity and Faraday effect:
▫ Non-Drudeness of the conductivity
▫ Cyclotron resonance: fitting of existing experiment
▫ Predicting new results
Pictures’ courtesy:
• Novoselov & Geim’s group
• Kuzmenko et al.
Related talks
• Earlier today, by V. Marachevskiy
• Tomorrow by D. Vassilevich
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Tight-binding model
Two sub-lattices, two atoms in a
primitive cell.
Strong covalent  -bonds in the
graphene plane,
Weekly-intersected  -orbitals perpendicular to the plane.
Hamiltonian function of  -electrons hopping to the nearest atom
H  t
a

 
†
n , n  ,
b
 с.с.
n, i ,
t - hopping parameter, n - atom position,  , - spin
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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The Linearity of Spectrum
In vicinity of Dirac (K-)points
energy of quasy-particles has
linear dispersion
E(k )   vF k
vF  106 m s
Fermi velocity
Thus the low-energy quasy-particles are quasy-relativistic Dirack fermions
H
 0
 p  ip
2p
d
y
†  x
vF


2
 0
 DC (2 )

 0

px  ip y
0
0
0
0
0
0
 px  ip y



 px  ip y  

0

0
0
Note 8-fold degeneracy: 2 sub-lattices * 2 valleys * 2 spins
Wallace, 47, Semenoff, Apelquist,Redlich, 80-s
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Action functional
x1
The action of the systems is straightforward (with EM field)
S  S EM  S


S   d x  i   eAx3 0  m  ... 
3
x3
x2
SEM   14  d 4 x F2
   l  l , l  0,1,2
  , 
0
0
1,2
 c  1, vF  (300)1
 vF ,    
1,2
2
0

1,2 2
1
With a second plate – Casimir effect, see the talk by Valeriy Marachevsky
The model is well established and proved by several experiments:
absorption of light, etc.
Gusynin et al, 2004
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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QFT approach
The generating functional of the theory is
2 i ( x )  i eA m   AJ ( )
 i F

3 

Z [ J , ]  D [ A ] e 4


The Casimir energy in this approach is given by
QFT
E
i

Ln Z [0]
TS
To calculate the generating functional we first ‘integrate out’ the fermions

2 i ( x ) S ( A) AJ
 i F
3 eff
Z[J ]  D A e 4
where the effective action is given by
Seff  A  i Ln Det i   eA  m 
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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QFT approach II
In the quadratic approximation the effective action is given by polarization
operator
SEff 
1
2
3
3
lm
d
xd
y
A
(
x
)

( y  x ) Am ( y )
l

 Al
Am
|x3 0
x  ( x 0 , x1 , x 2 )
The full line denotes the fermion propagtor of considered system
1
 i   m  eAex   Sˆ (m, , T , B,...)
and the one-loop polarization operator is defined in a standard way

2
r
2 dq0 d q
ˆ (q , q)%l Sˆ (q  p , q  p)%m
 ( p0 , p)  ie 
tr
S
0
0
0
(2 )3
lm

Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Effective EM theory
Thus in the lowest order approximation we had the following effective action
S   d 4 x   14 F2  12   x3  Al  lm Am 
This gives the modification to the Maxwell equations
  F     x3   A  0
the delta potential is equivalent to the following matching conditions
A |z 0  A |z 0 ,


A


A


 z  z0  z  z0  A
|z0
This lets one to describe the plane waves propagation in the system:
- reflection coefficients (for TE,TM modes, etc.)
- rotation of polarization (if any)
- absorption of light
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Normal Incidence
For our purposes we need to consider the normal incidence only
 ( )   ( , p  0)
when the tensorial structure of PO is quite simple
 0
 ( )  
 0



gij  evn  i ij odd 
0
In the language of condensed matter physics, polarization operator П
corresponds to ac (and dc) conductivity
 ij ( ) 
 ij ( )

 dc  lim  (  i0)
0
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Propagation of light
x1
For linearly polarized waves one has
A  eit
e x eik3 z   rxx e x  rxy e y  e ik3 z , z  0


ik3 z
t
e

t
e
e
,
z0



 xx x xy y
x3
x2
the (cumbersome) calculation of the reflection coefficient gives
2 (i evn  2 )
t xx  2 2
,
2
2
2
  evn  4i evn  (4    odd )
rxx  t xx  1,
2 odd  2
t xy  2 2
2
  evn  4i evn  (4   2 odd
) 2
rxy  t xy
If both P-even and P-odd parts of polarization operator are real, the flux is
conserved
| txx |2  | txy |2  | rxx |2  | rxy |2  1
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Faraday Effect
The transmission coefficient is then given in the first order in
T
t xx  t xy
2
2

Im  evn
 1
2
while the rotation of the angle of the polarization and elipticity of
transmitted wave are defined as
 
R
 Re  imp
2
 Im  imp
2
IVF, D. Vassilevitch, 09
 O( 2 )
 O( 2 )
Earlier predictions on the issue:
Mikhailov, Volkov 1985, O’Connell, Wallace, 1982
See also works by Morimoto, Hatsugai, Aoki
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Polarization Operator
Just to remind you
 lm ( ) 

2
dq
d
q
2
0
ˆ (q , q)%l Sˆ (q  , q  p)%m
 ie 
tr
S
0
0
(2 )3

The full line denotes the fermion propagator of considered system
 i   m  eAex   Sˆ (m, , T , B,...)
1
We will be interested in the case of zero temperature but with external
magnetic field and non-vanishing chemical potential.
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Propagator, B>0
In presence of external magentic field perpendicular to the graphene plane
one has for the propagator
Sˆ (q0 , q; B)  eq
2
Sn (q0   , q)
(1)

2
2
(
q



i

sgn
q
)

M
n 0
0
0
n

/|eB|
n

 2q 2 
 2q 2  
 2q 2 
1
Sn  2( q0 0  m)  P Ln 
  P Ln 1 
   4vF qLn 1 

|
eB
|
|
eB
|
|
eB
|







where Ln are the Laguerre polynomials, and
P  (1  i 1 2 B ) / 2,  B  sgn B,
M n  2nvF2 | eB | m2
Initially the propagator in this case was calculated in the field of the heavy
ions collisions.
Chodos, Evering, Owen, 1990
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Polarization Operator
In the normal incidence case, the internal moment integration can be
resolved analytically due to the orthogonality of the Laguerre polynomials


0
dx e x x a Lan ( x ) Lam ( x ) 
  n  a  1
 nm
n!
which lead to the following expressions for P-odd and P-even components


 evn ( )  2 N B vF2 | eB |   dq0  I n,n 1  I n 1,n 
n  0 
 odd ( ) 
where
I k ,n
2 B vF2 | eB |



  dq  I
n  0 
0
n , n 1
 I n 1,n 
q0 (  q0 )  m2 

1

2 i  (q0  i sgn(q0   ))2  M k2  (q0    i sgn(q0     )) 2  M n2 
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Relativistic Hall effect
In the limit   0, m  0 we obtain the relativistic Hall conductivity
 xy  lim  odd
 0
e2

N  B 1  2n0 
4 h
 2 
n0   2

 2vF | eB | 
For graphene N=4, and thus
Novoselov et al, 2005
Experimental: Novoselov et al, 2005; Kim et al, 2005
Theory: Zheng, Ando 2002; Gusynin, Sharapov, 2005, Peres et al., 2006
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Non-Drudeness of conductivity
The next step is in resolving the frequency integration which leads to
 odd ( ) 
i

8
2 B vF2 | eB |


G
n 0
n

Gn 
 dq  I
0
n , n 1
 I n 1,n  

 g1 ( M n ,  ' M n 1 )

g1 ( M n ,  ' M n 1 )


  (   )

  2i  ( M n   ' M n 1 ) 
 , '    ( M n   ' M n 1 )
Surprisingly, G has NO POLES in the complex plane!! Only a tricky
combination of logarithmic divergences:
g1 : log   i    M n1   i    M n i    M n1 i    M n 
g2 : log   i    M n1   i    M n i    M n1 i    M n 
However, its behavior on real axe only is almost undistinguishable from
the Drude conductivity, see below!
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Non-Drudeness of conductivity
A similar results holds for the P-even part: NO POLES

 evn ( )  2 N B vF2 | eB |  H n
n 0

Hn 
 dq  I
0
n , n 1
 I n 1,n 

Unlike P-odd part the P-even is neither odd, nor even as a function of
chemical potential for non vanishing mass gap.
Actually, polarization operator in 2+1 is power-counting divergent. It reveals
itself in the even part
Hn ; 
n 
1
3/2

O
(
n
)
2
4vF | eB | n
However, at our level of consideration it does not effect the results since
T  1
Im  evn
2
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Small chemical potential
Small chemical potentials corresponds to suspended graphene layers. In this
limit we have
Gn ;
 0
   M
2
2   M n21  M n2  2  3i  2 2 
2
n 1
 
2
M
2
n
    i 
2
M
2
n 1
   i 
2
 M n2

the maximum/minimum of rotation angle is of the order of
; 
2 M 1
 2
and are achieved, correspondingly, at
 ; M1  
In idealistic case, the chemical potential as well as Г are identically zero.
However, in a realistic situation there is an interplay between them due to
unavoidable presence of impurities.
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Suspended graphene
Thus, possible experimental curves should have the following form
 , rad
T
 , meV
 , meV
Just 1Å of graphene at B=7T rotates polarization of light by about 4.5
degrees and absorbs 25%!
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Epitaxial graphene
Large Fermi energy shifts are usually invoked by the interaction with a
substrate. For big chemical potential one can show

2 B vF2 | eB |

n0 
Re
 G
n  n0 
n
T  1
8 vF2 | eB |

n0 
Im
 H
n  n0 
n
i.e. in cyclotron resonance the main contribution is coming from around
 2 
n0   2

2
v
|
eB
|
 F

For numerical investigation it is sufficient consider  : 0.3 n0
M 12
This holds already for  :
2 
No nicer formulas can be obtained.
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Data fitting
Aims:
- Yet again to check the Dirac model
- Reveal possible field dependence of the parameters
(mass, scattering rate, chemical potential)
The fit was performed around the following initial values
|  | 340 meV   5 meV   0
The Diraс model (nicely) fits the experimental curves:
- mass gap doesn’t influence the result (in a reasonable interval)
- Gamma’s dependence on the magnetic filed is about +/- 10-15%. So,
it cannot be resolved at this stage.
- The fitted value of chemical potential is essentially lower then the Fermi
energy shift measured at zero field, and grows with magnetic field
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Data fitting
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Data fitting
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Chemical potential
Chemical potential (as a parameter entering the Dirac model) grows with
magnetic field:
while experimentally measured Fermi energy shift at zero magnetic field
|  | 340 meV
Self-consistent perturbation theory seems to be a possible solution for the
apparent discrepancy.
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
27
vs Drude
As promised we can compare our results with the Drude ones
 xx 
1/   i
 c2  (  i /  ) 2
2D
 xy  
c
 c2  (  i /  ) 2
2D
Thus, the predictions for the real frequencies are rather indistinguishable,
while analytical properties in the whole complex plane are completely
different!
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
28
Conclusions
After reminding of the Dirac description of graphene we
• revealed the non-Drudeness of the conductivity
• predicted a new regime to observe ‘Giant Faraday Rotation in Graphene’
• showed that model nicely describes the experimental curves in the
cyclotron resonance regime
Attn.: tomorrow at 18.15h “Quantum Field Theory in Graphene”
by D. Vassilevich
Fialkovsky I.V., Vassilevitch D.V.
Parity-odd effects and polarization rotation in graphene
J. Phys. A: Math. Theor. 42 (2009) 442001, arXiv: 0902.2570 [hep-th]
Bordag M., Fialkovsky I. V., Gitman D. M., Vassilevich D. V.,
Polarization rotation and Casimir effect in suspended graphene films,
arXiv:1003.3380 [hep-th]
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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Faraday in Graphene,
IVF, QFEXT'11
Thank you!
30
QFT approach II
No caso mais simples
i
S  S (m) 
p m
o laço dos férmions foi calculado muitos anos atrás e muitas vezes
j l

 n
%
%

p
p 
mn
m
jl
jkl
%k l
 ( p )  2  j   par ( p )  g  2   i imp ( p ) p
%
vF
p 




 mj  diag(1, vF , vF ),
%
p j  l j p l
onde as funções tem seguinte forma explicita
% ( %
% 2m )  / 2 p
%
 par ( p )   2mp
p 2  4m 2 ) arctanh( p
% 2m) p
% 1
 imp ( p )  2m arctanh( p
Semenoff, 1984, Redlich, 1984, Appelquist, 1986, Dunne, 1996-1999, etc.
Faraday effect in Graphene, I. Fialkovksy @QFEXT’11
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QFT approach II

%
%
%l
pj%
pl 
pj%
pl %
p jul  u j p
u jul 2   n
jl
%  B l
 ( p )  2   g  2  A   2 

p
2


%
vF
p 
p
pu 
%
pu 
%

 %
 
mn

m
j
Onde a velocidade da media pode ser escolhida como
u  (1,0,0)
As funções escalares A,B se expressam através dos alguns dois componentes
p2
A  2  00   tr
p

1  p2  %
p2
B 2 2



00
tr 
vF p  p 2

Faraday effect in Graphene, I. Fialkovksy @QFEXT’11