Hyperbolic Functions. The Hyperbolic Sine, Hyperbolic Cosine & Hyperbolic Tangent.
Novel hyperbolic metamaterials based on multilayer graphene structures.
description
Transcript of Novel hyperbolic metamaterials based on multilayer graphene structures.
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 00 00 0
xx
yy
zz
D E
22 2 2
2yx z
xx yy zz
kk kc
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 ndc
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
dsE 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 00 0 ;
( )0 0
( )
Me Me D D
Me D
Me D M
pMe
e D
Me D D Me
d dd dd 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
kM
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
1001
N
R D
R
A sns
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
ki kT
TE KD k D D
KD k D DM kk
Purcell factor
8
3
0.005
10 !
1
10
R
mD m
D
s
R
p
Phys. Rev. B 87, 075416 (2013)
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 dcRck D
cRck D k d
‖
‖ ‖
Largest Purcell factors correspond to:
0
4 1ck 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 ))FF
d vv
To be done: separating the far-field and near-field input to the Purcell factor
Vogel, Welsch, “Quantum optics”:2
3 *2 Im( ( , )) ( , , ) ( , , ) Im ( , , )
Im (0,0, )
ik jks s G r s G r s G r rdcG
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 42cos( ) sin( ),
2cos( ) sin( ).
H z
z zz
zz z
k dK d
kik d k dc k
kik d k dc 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.
Thank you
Homogenization: local and nonlocal approaches
nloc
2
20
0
2 20 0
2 2
2 2 20
0 0ˆ 0 0
0 0
121 2 ( , )
121 2 ( , )
2 Im( ) / ( )
sin( ) /( , )
2(cos( ) cos( ))
xx
yy
xx
x z
yyx z
z zx z
z z z z
z x
kz f k kk
f k kck d
k d k d k kf k k
k d k d k k
k k k