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

8

3

0.005

10 !

1

10

R

m

D 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 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.

Thank you

Homogenization: local and nonlocal approaches

nloc

2

20

0

2 20 0

2 2

2 2 20

0 0

ˆ 0 0

0 0

12

1 2 ( , )

12

1 2 ( , )

2 Im( ) / ( )

sin( ) /( , )

2(cos( ) cos( ))

xx

yy

xx

x z

yy

x z

z zx z

z z z z

z x

kzf k k

k

f k k

ck d

k d k d k kf k k

k d k d k k

k k k