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Novel hyperbolic metamaterials based on multilayer graphene
structures.
I.V. Iorsh, I.V. Shadrivov, P.A. Belov, and Yu.S. Kivshar
Benasque, 03.03-08.0.3, 2013
Isotropic media:
Disp. equation:
Isofrequency surface:
D E
22 2 2
2x y zk k kc
Hyperbolic medium:
Disp. equation :
Isofrequency surface :
||
|| 0
xx yy
zz
D E
222 2
||
/kk
c
Anisotropic media:
Disp. equation :
Isofrequency surface:
0 0
0 0
0 0
xx
yy
zz
D E
22 2 2
2
yx z
xx yy zz
kk k
c
Hyperbolic medium
Spontaneous emission
21| | d | | ( , , )
3eg
ege g r
e
Transition rate (Fermi Golden Rule):
For atom in vacuum:
LDOS
2 2 3
0 0 30
1( )
3
d nd
c
Note: Fermi Golden Rule is not an exact result, but rather a first approximation solution of the integro-differential equation obtained from time-dependant perturbation theory
/( ) tP t e
Purcell factor
E.M. Purcell(1912-1997)
3 23 / 4f Q V
Purcell worked with RF range and small metallicCavities: enhancement of the order of 2010
Infinite density of states
2isofrequencysurface
1
(2 ) ( )k
ds
E k
isofrequency surface unbound
=
Narimanov et al, Appl. Phys. B: 100, 215–218 (2010)
Realizations of hyperbolic mediaWire medium
J. Sun et al. Appl. Phys. Lett. 98, 101901 (2011)
Graphite (for UV)
Magnetized plasma (for RF)
Layered metal dielectric nanostructure – the simplest realization of hyperbolic media Within the effective media approximation the layered metal
dielectric nanostructure can be described as a hyperbolic media
2
0 0
0 0 ;( )
0 0
( )
Me Me D D
Me D
Me D M
pMe
e D
Me D D Me
d d
d d
d d
d d
i
||,
Me D
Purcell factor in layered structures. Theory.
•Extremum is observed at the bulk plasmon frequency .
3
RD
Im (0,0, )R G
0
0
3
3 2 20 0 0
3 3
3 2 2 3 2 20 0
|| ||
||
|| || || ||
|| || 00 0
3Im (0,0, ) Re ( )
4
3 3Re( ) Im( )
4 4
TM
TM T
k
k
M
dG r
k k
d dr r
k
k k
k
k k k k
k k kk k
T. Tumkur, G. Zhu, P. Black, Yu. A. Barnakov, C. E. Bonner, and M. A. Noginov, APL 99, 151115, (2011)
O. Kidway, S.V. Zhukovsky, J.E. Sipe, OL, 36,13,(2011)
Purcell factor in layered structures. Experiment.
Spontaneous emission enhancement in THz range
100
1
N
R D
R
A s
ns
But what if to utilize Purcell effect?
Efficiency is very low
/RAD RAD R
From the other hand, THz frequency range lies well below the characteristic bulk plasmon frequencies in the conventional metal-dielectric multilayers, which significantly limits the achievable values of the Purcell factors.
Graphene multilayer structure ashyperbolic metamaterial
1.Hyperbolic isofrequency contours in metal-dielectric nanostructuresarise due to near field Bloch waves
2.Near field Bloch waves – essentially areThe coupled surface plasmon polaritons
3. Graphene sheet supports surface plasmon modes which can be coupled if we organise an array of graphene sheets.
Multilayer graphene structure should behaveAs a hyperbolic metamaterial
Isofrequency contours
0
0
: cos( ) cos( )
cos( ) co
2sin( )
2: sin )s( ()
z z
z
z
z z
i kk
k
i kT
TE KD k D D
KD k D DM kk
Purcell factor (analytics)32
0
0
2
0 0
31: exp ;
2 2 Im( ) | 2 Im( ) |
31:
8( )
4
| I ) |
4
2 m(
TM
TM
ck dcR
ck D
cR
ck D k d
‖
‖ ‖
Largest Purcell factors correspond to:
0
41
ck D
Limitations of the local approach
||
||local approach: ( , ) ( )
works only for: k /F Fk v
k
||
0
|
0
|
Fk
dk dk
2 | Im( ) |coth( / (2 ))F
F
d vv
To be done: separating the far-field and near-field input to the Purcell factor
Vogel, Welsch, “Quantum optics”:
23 *
2Im( ( , )) ( , , ) ( , , ) Im ( , , )
Im (0,0, )
ik jks s G r s G r s G r rdc
G
To separate the far field and near field:2
2*
23
*32
Im( ( , )) (0, , ) (0, , )
Im( ( , )) (0, , ) (0, , )
RAD ik jk
ik jk
s s G s G sc
s s G s G s
d
dc
Application of perpendicular magnetic field
Perpendicular magnetic field couples the TE and TM polarized Bloch waves:
2 2 22
1,2
0
0
sin ( )( )cos( ) ,
2 42
cos( ) sin( ),
2cos( ) sin( ).
H z
z zz
zz z
k dK d
kik d k d
c k
kik d k d
c k
A B A B
A
B Coupling term
Conclusion
Multilayered graphene structures could be used as a new realization of hyperbolic metamaterials for THz range to boost the terahertz transitions in semiconductor devices.