With the rise of artificial magnetism and metamaterials, toroidal resonance has gained much attention for its special properties. In this paper, we propose a novel hybrid graphene-metal metamolecule consisting of a square bracket-like resonator and two asymmetric U-shaped resonators. By applying various Fermi energies to graphene, the amplitude of electromagnetically induced transparency (EIT) can be efficiently manipulated, and the maximum amplitude modulation depth can attain 81% in the microwave region. Numerical simulations and theoretical analysis demonstrate that the dynamic manipulation is mainly induced by the active tuning toroidal resonance through the recombination effect of the conductive graphene. Also, the maximum group delay of 85 ps can be attained and controlled with the increasing Fermi energy. The proposed hybrid graphene-metal metamolecule and dynamically manipulating mode presents a novel modulating strategy of EIT-like analog based on the toroidal response, which has great application for the design of efficient tunable resonators, filters, and sensors.

1.
T.
Xiang
,
T.
Lei
,
J.
Wu
,
J.
Wang
, and
H.
Yang
, “
Dual-Fano resonances based on all-dielectric toroidal metamaterial
,”
Appl. Phys. Express
15
,
032002
(
2022
).
2.
J. D.
Jackson
and
R. F.
Fox
, “
Classical electrodynamics, 3rd ed.
,”
Am. J. Phys.
67
,
841
842
(
1999
).
3.
I.
Zel’Dovich
, “
Electromagnetic interaction with parity violation
,”
Sov. J. Exp. Theor. Phys.
6
,
1184
1186
(
1958
).
4.
D. W.
Watson
,
S. D.
Jenkins
,
J.
Ruostekoski
,
V. A.
Fedotov
, and
N. I.
Zheludev
, “
Toroidal dipole excitations in metamolecules formed by interacting plasmonic nanorods
,”
Phys. Rev. B
93
,
125420
(
2016
).
5.
T.
Kaelberer
,
V. A.
Fedotov
,
N.
Papasimakis
,
D. P.
Tsai
, and
N. I.
Zheludev
, “
Toroidal dipolar response in a metamaterial
,”
Science
330
,
1510
1512
(
2010
).
6.
X.
Luo
,
X.
Li
,
T.
Lang
,
X.
Jing
, and
Z.
Hong
, “
Excitation of high Q toroidal dipole resonance in an all-dielectric metasurface
,”
Opt. Mater. Express
10
,
358
(
2020
).
7.
D.
Kumar
,
S. P.
Ambatipudi
,
S.
Banerjee
,
R.
Kumar
, and
D.
Roy Chowdhury
, “
Multifold coupling enabled high quality factor toroidal resonances in metasurfaces
,”
J. Appl. Phys.
127
,
193103
(
2020
).
8.
Z.
Liu
,
S.
Du
,
A.
Cui
,
Z.
Li
,
Y.
Fan
,
S.
Chen
,
W.
Li
,
J.
Li
, and
C.
Gu
, “
High-quality-factor mid-infrared toroidal excitation in folded 3D metamaterials
,”
Adv. Mater.
29
, 1606298 (
2017
).
9.
X.
Li
,
J.
Yin
,
Z.
Liu
,
Y.
Wang
, and
Z.
Hong
, “
Tailoring the excitation of two kinds of toroidal dipoles in all-dielectric metasurfaces
,”
Optik
201
,
163502
(
2020
).
10.
S.
Han
,
M.
Gupta
,
L.
Cong
,
Y. K.
Srivastava
, and
R.
Singh
, “
Toroidal and magnetic Fano resonances in planar THz metamaterials
,”
J. Appl. Phys.
122
, 113105 (
2017
).
11.
S.
Song
,
S.
Yu
,
H.
Li
, and
T.
Zhao
, “
Ultra-high Q-factor toroidal dipole resonance and magnetic dipole quasi-bound state in the continuum in an all-dielectric hollow metasurface
,”
Laser Phys.
32
,
025403
(
2022
).
12.
Z.
Shen
,
H.
Yang
,
X.
Huang
,
T.
Xiang
,
J.
Wu
,
Y.
Zhou
, and
Z.
Yu
, “
Electromagnetically induced transparency metamaterial with strong toroidal dipole response
,”
Mater. Res. Express
7
,
035802
(
2020
).
13.
T.
Xiang
,
T.
Lei
,
S.
Hu
,
J.
Chen
,
X.
Huang
, and
H.
Yang
, “
Resonance transparency with low-loss in toroidal planar metamaterial
,”
J. Appl. Phys.
123
,
095104
(
2018
).
14.
K.-J.
Boller
,
A.
Imamoğlu
, and
S. E.
Harris
, “
Observation of electromagnetically induced transparency
,”
Phys. Rev. Lett.
66
,
2593
2596
(
1991
).
15.
C.
Shu
,
Q.-G.
Chen
,
J.-S.
Mei
, and
J.-H.
Yin
, “
Dynamically tunable implementation of electromagnetically induced transparency with two coupling graphene-nanostrips in terahertz region
,”
Opt. Commun.
411
,
48
52
(
2018
).
16.
T.-T.
Kim
,
H.-D.
Kim
,
R.
Zhao
,
S. S.
Oh
,
T.
Ha
,
D. S.
Chung
,
Y. H.
Lee
,
B.
Min
, and
S.
Zhang
, “
Electrically tunable slow light using graphene metamaterials
,”
ACS Photonics
5
,
1800
1807
(
2018
).
17.
S.
Zheng
,
M.
Ma
,
Y.
Lv
,
T.
Fu
,
L.
Peng
, and
Q.
Zhao
, “
Dual-band electromagnetically induced transparent metamaterial with slow light effect and energy storage
,”
J. Phys. D: Appl. Phys.
55
,
255103
(
2022
).
18.
Z.
Wang
and
B.
Yu
, “
Optical bistability via dual electromagnetically induced transparency in a coupled quantum-well nanostructure
,”
J. Appl. Phys.
113
,
113101
(
2013
).
19.
F.
He
,
B.
Han
,
X.
Li
,
T.
Lang
,
X.
Jing
, and
Z.
Hong
, “
Analogue of electromagnetically induced transparency with high-Q factor in metal-dielectric metamaterials based on bright-bright mode coupling
,”
Opt. Express
27
,
37590
(
2019
).
20.
Z.-Y.
Shen
,
H.-L.
Yang
,
X.
Liu
,
X.-J.
Huang
,
T.-Y.
Xiang
,
J.
Wu
, and
W.
Chen
, “
Electromagnetically induced transparency in novel dual-band metamaterial excited by toroidal dipolar response
,”
Front. Phys.
15
,
12601
(
2020
).
21.
T.
Fu
,
Z.
Zhou
,
D.
Wang
,
T.
Yang
,
H.
Li
, and
Y.
Chen
, “
Electromagnetically induced transparency based on magnetic toroidal mode of dielectric reverse-symmetric spiral metasurfaces
,”
New J. Phys.
24
,
033024
(
2022
).
22.
B.
Han
,
X.
Li
,
C.
Sui
,
J.
Diao
,
X.
Jing
, and
Z.
Hong
, “
Analog of electromagnetically induced transparency in an E-shaped all-dielectric metasurface based on toroidal dipolar response
,”
Opt. Mater. Express
8
,
2197
(
2018
).
23.
T. C.
Huang
,
B. X.
Wang
, and
C. Y.
Zhao
, “
Tuning toroidal dipole resonances in dielectric metamolecules by an additional electric dipolar response
,”
J. Appl. Phys.
125
,
093102
(
2019
).
24.
X.
Wang
,
H.
Meng
,
S.
Deng
,
C.
Lao
,
Z.
Wei
,
F.
Wang
,
C.
Tan
, and
X.
Huang
, “
Hybrid metal graphene-based tunable plasmon-induced transparency in terahertz metasurface
,”
Nanomaterials
9
,
385
(
2019
).
25.
M.
Chen
and
Z.
Xiao
, “
Metal-graphene hybrid terahertz metamaterial based on dynamically switchable electromagnetically induced transparency effect and its sensing performance
,”
Diam. Relat. Mater.
124
,
108935
(
2022
).
26.
Y.
Sun
,
D.
Liao
,
J.
Xu
,
Y.
Wu
, and
L.
Chen
, “
Active switching of toroidal resonances by using a Dirac semimetal for terahertz communication
,”
Front. Phys.
8
,
602772
(
2020
).
27.
X.
Chen
and
W.
Fan
, “
Study of the interaction between graphene and planar terahertz metamaterial with toroidal dipolar resonance
,”
Opt. Lett.
42
,
2034
(
2017
).
28.
D.
Yan
,
M.
Meng
,
J.
Li
, and
X.
Li
, “
Graphene-assisted narrow bandwidth dual-band tunable terahertz metamaterial absorber
,”
Front. Phys.
8
,
306
(
2020
).
29.
Z.
Song
,
Y.
Deng
,
Y.
Zhou
, and
Z.
Liu
, “
Tunable toroidal dipolar resonance for terahertz wave enabled by a vanadium dioxide metamaterial
,”
IEEE Photonics J.
11
,
4600705
(
2019
).
30.
X.
Jin
,
F.
Wang
,
S.
Huang
,
Z.
Xie
,
L.
Li
,
X.
Han
,
H.
Chen
, and
H.
Zhou
, “
Coherent perfect absorber with independently tunable frequency based on multilayer graphene
,”
Opt. Commun.
446
,
44
50
(
2019
).
31.
C.
Argyropoulos
, “
Enhanced transmission modulation based on dielectric metasurfaces loaded with graphene
,”
Opt. Express
23
,
23787
(
2015
).
32.
G.-D.
Liu
,
X.
Zhai
,
S.-X.
Xia
,
Q.
Lin
,
C.-J.
Zhao
, and
L.-L.
Wang
, “
Toroidal resonance based optical modulator employing hybrid graphene-dielectric metasurface
,”
Opt. Express
25
,
26045
(
2017
).
33.
A.
Bhattacharya
,
K. M.
Devi
,
T.
Nguyen
, and
G.
Kumar
, “
Actively tunable toroidal excitations in graphene based terahertz metamaterials
,”
Opt. Commun.
459
,
124919
(
2020
).
34.
X.
Liu
,
Z.
Liu
,
M.
Hua
,
L.
Wang
,
K.
Wang
,
W.
Zhang
,
Y.
Ning
,
Y.
Shi
,
X.
Wang
, and
F.
Yang
, “
Tunable terahertz metamaterials based on anapole excitation with graphene for reconfigurable sensors
,”
ACS Appl. Nano Mater.
3
,
2129
2133
(
2020
).
35.
G.
Sun
,
S.
Peng
,
X.
Zhang
, and
Y.
Zhu
, “
Switchable electromagnetically induced transparency with toroidal mode in a graphene-loaded all-dielectric metasurface
,”
Nanomaterials
10
,
1064
(
2020
).
36.
Y.
Xu
,
X.
Wang
,
X.
Chen
, and
L.
Zhang
, “
Structure-based tunable metamaterials for electromagnetically induced transparency windows in low terahertz frequency
,”
J. Appl. Phys.
127
,
034501
(
2020
).
37.
W. B.
Lu
,
J. W.
Wang
,
J.
Zhang
,
Z. G.
Liu
,
H.
Chen
,
W. J.
Song
, and
Z. H.
Jiang
, “
Flexible and optically transparent microwave absorber with wide bandwidth based on graphene
,”
Carbon
152
,
70
76
(
2019
).
38.
M.
Chen
,
Z.
Xiao
,
X.
Lu
,
F.
Lv
, and
Y.
Zhou
, “
Simulation of dynamically tunable and switchable electromagnetically induced transparency analogue based on metal-graphene hybrid metamaterial
,”
Carbon
159
,
273
282
(
2020
).
39.
S.
Xiao
,
T.
Wang
,
T.
Liu
,
X.
Yan
,
Z.
Li
, and
C.
Xu
, “
Active modulation of electromagnetically induced transparency analogue in terahertz hybrid metal-graphene metamaterials
,”
Carbon
126
,
271
278
(
2018
).
40.
T.
Chen
,
T.
Xiang
,
J.
Wang
,
T.
Lei
, and
F.
Lu
, “
Double E-shaped toroidal metasurface with high Q-factor Fano resonance and electromagnetically induced transparency
,”
AIP Adv.
11
,
095011
(
2021
).
41.
Y.
Fan
,
F.
Zhang
,
N. H.
Shen
,
Q.
Fu
,
Z.
Wei
,
H.
Li
, and
C. M.
Soukoulis
, “
Achieving a high-Q response in metamaterials by manipulating the toroidal excitations
,”
Phys. Rev. A
97
,
033816
(
2018
).
42.
V. A.
Fedotov
,
A. V.
Rogacheva
,
V.
Savinov
,
D. P.
Tsai
, and
N. I.
Zheludev
, “
Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials
,”
Sci. Rep.
3
,
1
5
(
2013
).
43.
D.
Yi
,
X.-C.
Wei
, and
Y.-L.
Xu
, “
Tunable microwave absorber based on patterned graphene
,”
IEEE Trans. Microwave Theory Tech.
65
,
2819
2826
(
2017
).
44.
B.
Wu
,
H. M.
Tuncer
,
M.
Naeem
,
B.
Yang
,
M. T.
Cole
,
W. I.
Milne
, and
Y.
Hao
, “
Experimental demonstration of a transparent graphene millimetre wave absorber with 28% fractional bandwidth at 140 GHz
,”
Sci. Rep.
4
,
4130
(
2014
).
45.
X.
Wang
and
S. A.
Tretyakov
, “
Toward ultimate control of terahertz wave absorption in graphene
,”
IEEE Trans. Antennas Propag.
67
,
2452
2461
(
2019
).
46.
X.
Xu
,
C.
Zhang
,
G.
Lv
,
J.
Jiang
, and
X.
He
, “
Actively tunable and switchable electromagnetically induced transparency in hybrid metal-graphene metamaterials
,”
Mater. Res. Express
8
,
025802
(
2021
).

Supplementary Material

You do not currently have access to this content.