Tunable Optical Bistability and Tristability in Nonlinear Graphene

Subscriber access provided by University of Florida | Smathers Libraries

Article

Tunable Optical Bistability and Tristability in Nonlinear Graphene-Wrapped Nanospheres Kai Zhang, Yang Huang, Andrey Miroshnichenko, and Lei Gao J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 05 May 2017 Downloaded from http://pubs.acs.org on May 6, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Tunable Optical Bistability and Tristability in Nonlinear Graphene-Wrapped Nanospheres Kai Zhang1, Yang. Huang1, #, Andrey E. Miroshnichenko2, and Lei Gao1, 3,* 1. College of Physics, Optoelectronics and Energy of Soochow University, Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China 2. Nonlinear Physics Centre, The Australian National University, Canberra ACT 0200, Australia 3. Jiangsu Key Laboratory of Thin Films, Soochow University, Suzhou 215006, China.

1

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 22

[ABSTRACT] We develop the nonlinear electromagnetic theory (NET) including the self-consistent mean-field approach to investigate the optical multi-stability of graphene-wrapped dielectric nanoparticles. We demonstrate the optical bistability (OB) of the graphene-wrapped nanoparticle in both near-field and far-field spectra due to electric dipolar modes for small sizes, as predicted in the quasistatic limit (QL). For small sizes, two OB regions can be observed when the magnetic dipolar modes arise under the strong field. On the other hand, for large sizes, one observes the optical tristability (OT) and even optical multi-stability arising from the contributions of higher-order magnetic modes. Further more, both the optical stable region and the switching threshold values can be tuned by changing either the Fermi level or the size of the nanoparticles. Our results promise the graphene-wrapped dielectric nanoparticle a candidate of multi-state optical switching, optical memories and relevant optoelectronic devices.

2


Page 3 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


[INTRODUCTION] Nonlinear optical effects, characterized by the interaction between light and the nonlinear optical materials, play an important role in modern photonic functionalities.1 In general, optical nonlinearities are inherently weak, and are superlinearly dependent on the electromagnetic field. One possible way of enhancing the nonlinear optical effects is to introduce metal plasmonic nanostructures,2 which support (local) surface plasmon resonance in the metal/dielectric interface. It is known that surface plasmon, arising from coherent oscillations of conduction electrons near the surface of noble-metal structures,2 results in strong confinement and enhancement of local electromagnetic field. Correspondingly, the enhancement of local electromagnetic fields shall boost the nonlinear optical effects of the metal/dielectric planar or two (or three)-dimensional composite microstructures. Besides, hybrid3-4 or two-level5 systems, which support exciton-phonon/plasmon interaction or systematic self-action, can also result in enhanced nonlinearity and even optical bistability. As one remarkable feature of nonlinearity, optical bistability provides a new way of manipulating the light by light, where a nonlinear optical system exhibits two distinguished stable excited states for a single input intensity.6 Optical structures with such properties can be the candidate for all-optical switching, optical transistor and optical memory.7-8 Proposals were put forward to decrease the intensity threshold for the bistability by exploiting the field enhancement produced by the local surface plasmon resonance of the metal plasmonic composite media.9-11 However, due to intrinsic Ohmic losses,2 plasmonic field enhancement is fundamentally limited even in noble metals, which constrains the functionality of some of metamaterials and transformation optical devices. On the other hand, Graphene, with extremely large electron mobility12 and large Kerr nonlinear coefficients, has attracted 3



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 22

significant interest in the plasmonic community.13-15 In this connection, there have been some theoretical works on the optical bistabiltiy in one-dimensional (1D) graphene planar structures involving the Kerr nonlinearity without/with the excitation of the nonlinear surface plasmon polarizations.16-19 In addition, the condition for the emergence of the plasmonic bistability in two-dimensional (2D) graphene nanoribbons with Kerr nonlinearity was theoretically examined under plane-wave excitation.20 Moreover, the nonlinear plasmonic modes in the graphene-coated dielectric nanowire were investigated analytically, and the propagation constants of different plasmonic modes can be tuned by the incident fields.21 In this paper, we would like to study an analogous Kerr nonlinearity but in three dimensional

(3D)

graphene-coated

nanospheres

theoretically.

Actually,

in

such

graphene-wrapped nanoparticles, second harmonic generations,22 localized plasmons,23 and superscattering phenomena24 were investigated. On the experimental side, graphene wrapped spheres may be achieved by using electrostatic self-assembly25-27 and in an emulsification process,28 where the flexible core-diameters range from tens nanometers to several micrometers, More recently, in the quasistatic limit, we theoretically study the effective third-order

nonlinear

response

and

optical

bistability

of

the

three-dimensional

nanocomposites, containing nonlinear graphene wrapped dielectric nanoparticles embedded in the host medium.29 Here, we shall investigate the optical bistable (tristable) behavior in the near-field and far-field spectra from graphene-wrapped spheres by generalizing full-wave Mie scattering theory to nonlinear theory.30 In conjunction with the self-consistent mean-field method,11, 30-31 we demonstrate the existence one optical bistability (OB) double OB, and optical tristability (OT) for large sizes and for large incident fields. In addition, the existence of large and tunable Kerr nonlinearity in the graphene atomically thin layer is 4


Page 5 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


helpful to decrease the threshold values of both OB and OT either by varying the particle size or changing the Fermi energies. Our results may be useful in the design of multi-state optical switching, which has potential applications in optical communications and computing.

[THEORETICAL MODEL AND METHODS]

Figure 1. Geometric model of a graphene-wrapped dielectric particle embedded in the dielectric host. The incident plane wave is polarized in x-direction and propagates along z-direction.

Let’s consider the structure consisting of the monolayer graphene-wrapped dielectric ε

spheres of radius a and relative permittivity

, embedded in a host medium with relative

permittivity ε h ，as shown in Figure 1. This kind of structures can be now fabricated experimentally.25-28

A. Full-wave Mie theory for linear graphene-coated nanosphere We apply Mie scattering theory to solve the scattering problem from a graphene-coated nanosphere. The incident electric fields upon the graphene-wrapped dielectric nanoparticle have the form Ein = xˆE 0 eikz ⋅ e −iωt ,

where

k = k0 ε h

denotes the wave vector in the host medium, and 5


(1) k0 = ω ε 0 µ 0

is the one in


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 22

vacuum. In comparison with the size of the dielectric nanoparticle, the monolayer graphene is only one atom thick and it can be considered as an extremely thin conducting shell with linear conductivity σg .32 Hence, we adopt the following boundary conditions, nˆ × ( H i + H s − H c ) = J

Here, Hi , Ei , Hs , Es , Hc , and

Ec

and

nˆ ⋅ ( Ei + Es − Ec ) = 0 .

(2)

are the incident, scattered, and internal magnetic and electric J = σ g Et

fields of the dielectric sphere, respectively. In addition,

represents the surface

current density induced by the tangential component of the electric field E t . According to Mie scattering theory,23 the general solutions for the local electromagnetic field can be written as follows,

(

∞

Ec = ∑ En cn M (o1)n − id n N(e1)n n =1

1

1

)

) ( = E + E = ∑ E ( M ( ) − iN( ) ) + ∑ E ( ia N( ) − b M ( ) )   = H + H = − k (ωµ )  ∑ E ( M ( ) + iN ( ) ) − ∑ E ( ib N( ) + a M ( ) )    ∞

H c = − k1 (ωµ )−1 ∑ En d n M (e1)n + icn N(o1)n 1

n =1 ∞

Eout H out

where

i

s

n =1

n

1 o1n

1 e1n

∞

−1

i

1

s

n =1

n

∞

n =1

1 e1n

n

n

1 o1n

3 e1n

n

∞

n =1

n

 jn  1 En = i n E 0 ( 2n + 1) / n ( n + 1) , Mσ(1),(3) ( kr )Yσ 1n (θ ,ϕ )  , N = ∇ × M . 1n = ∇ × r k  hn 

(3)

3 o1n

n

3 e1n

Here

n

M

3 o1n

and

N

are

the vector spherical harmonics, and the upper indices (1) and (3) indicate the use of the spherical Bessel function jn ( x) and the first-order spherical Hankel function hn ( x) , respectively, and Ye (θ ,ϕ ) = Pn(1) (cosθ ) 1n o

where

Pn(1)

cos (ϕ ), sin

are the associated Legendre functions. In addition,

(4) k1 = k0 ε

is the wave number

inside the sphere. an , bn , cn and d n are the five unknown coefficients to be determined. Applying boundary conditions in eq 2, we arrive at the following coefficients,15

6


Page 7 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


an = bn = cn = dn =

ψ n ( x )ψ n' ( mx ) − mψ n' ( x )ψ n ( mx ) − iσ gαψ n' ( x )ψ n' ( mx ) ξ n ( x )ψ n' ( mx ) − mξ n' ( x )ψ n ( mx ) − iσ gαξ n' ( x )ψ n' ( mx ) ψ n ( mx )ψ n' ( x ) − mψ n' ( mx )ψ n ( x ) + iσ gαψ n ( x )ψ n ( mx ) ψ n ( mx ) ξ n' ( x ) − mψ n' ( mx ) ξ n ( x ) + iσ gαξ n ( x )ψ n ( mx ) mψ n ( x ) ξ n' ( x ) − mψ n' ( x ) ξ n ( x )

ψ n ( mx ) ξ n' ( x ) − mψ n' ( mx ) ξ n ( x ) + iσ gαξ n ( x )ψ n ( mx ) mψ n' ( x ) ξ n ( x ) − mψ n ( x ) ξ n' ( x )

ξ n ( x )ψ n' ( mx ) − mξ n' ( x )ψ n ( mx ) − iσ gαξ n' ( x )ψ n' ( mx )

where ψ n ( x) = xjn ( x) , ξn ( x) = xhn ( x) , α = µ0 / ε 0ε h . refractive index, and

x = ka

(5)

m = k1 / k = ε / ε h

is defined as relative

is the size parameter. Note that when σ g = 0 , all the coefficients

shown in eq 7 reduce to those in the case of a bare spherical dielectric particle. The distribution of the local electric field in the graphene thin layer can be obtained with eq 3 with r = a , and the spatial average of the square of the linear local electric field within the graphene, which will be generalized in the following subsection, can be written as, E

2

2 lin, g

with

1 N= 4π

2π π ∞

∫

0

= N E0 ,

(6)

2

 2n + 1  (1) (1) 2 ∫ ∑  n ( n + 1)  cn M o1n − idn N e1n sin θ dθ dϕ .  0 n =1 

In addition, we define the linear scattering cross section efficiency as, Qsca = 2 ( ka )

−2

∑ ( 2n + 1) ( an ∞

2

n =1

+ bn

2

).

(7)

B. Nonlinear theory for nonlinear graphene-coated nanosphere In general, the surface conductivity of the graphene is field-dependent and nonlinear. Here, we introduce the simplified version within the random-phase approximation,16, 21 2

σ% g = σ g + σ 3 E ,

(8)

in which the linear term σ g = σ int ra + σ int er , and σ int ra , σ int ra are the intraband and interband terms which have the following forms,16

7



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 22

(

)

−1 −1 −1 σ intra = ie 2 kBT π h 2 ( ω + i / τ )   EF ( kBT ) + 2ln exp  − EF ( kBT )  + 1  ,



−1

σ inter = ie 2 ( 4π h ) ln

In above formula,





( ) + (ω + iτ ) h

(9)

2 EF − ω + iτ −1 h 2 EF

e , h , kB , EF , τ

−1

and

k1 = k0 ε

are electron charge, reduced Plank

constant, Boltzmann constant, Fermi energy, electron-phonon relaxation time and temperature respectively. The Fermi energy

EF = hυ F π n2D

can be electrically controlled by

an applied gate voltage due to the strong dependence of the carrier density n2D on the gate voltage, and the relaxation time is determined by the carrier mobility

mc

as τ = mc EF / eυ F2 ,

wherein υF is the Fermi velocity of electrons. For photon energy

hω

far less than the Fermi energy

EF ,

the interband transitions in

graphene is negligible compared with the intraband part. Therefore, in the THz range graphene is well described by the Drude-like surface conductivity σ int ra . And since EF

kBT

in the room temperature ( T = 300K ), we have the simplified form of graphene linear conductivity σ g and Kerr nonlinear surface conductivity σ 3 ,16, 20 σg =

ie 2 EF

π h 2 (ω + i / τ )

, σ 3 = −i

9e 4υ F2 8π EF h 2ω 3

.

(10)

As one knows, when the Kerr nonlinear conductivity of the coated graphene is taken into account, it is quite difficult to determine the local electromagnetic fields inside the graphene layer. In general, the local fields within the atomically thin graphene layer cannot be solved exactly. To give a simple and efficient way, we alternatively introduce the self-consistent mean-field approximation,11 in which the nonlinear local field atomically thin graphene layer is replaced by the mean one σ% g ≈ σ g + σ 3 E

2 non , g

.

On the other hand, eq 7 can be modified as, 8


E

2 non , g

E

2

within

. Therefore eq 9 turns to (11)

Page 9 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry 2 = N% E 0 ,

2

E

non, g

1 N% = 4π

with

2π π ∞

∫

0

2

2

n

(12)

2n + 1  (1) (1) ∫ ∑ ( −1)  n ( n + 1)  c%n M o1n − id%n N e1n sin θ dθ dϕ .   0 n =1

Here we mention that c%n and

d%n

have the same form as cn and d n but with σ g in cn and d n replaced by the field-dependent conductivity σ% g ≈ σ g + σ 3

E

2 non, g

. Hence, N% in above equation is dependent on

and eq 14 is a self-consistent equation for relation between local field

E

2 non, g

E

2 non, g

E

2 non, g

. As a consequence, one can obtain the

and external field

E0

2

by directly solving eq 12 in a

self-consistent manner, and hence one may achieve the optical bistable behavior for the near field. Moreover, the nonlinear scattering efficiency

Q%sca

of the graphene-wrapped nanoparticle

can also be generalized as, ∞

2 2 −2 Q% sca = 2 ( ka ) ∑ ( 2n + 1)  a%n + b%n ,   n =1

Again, a%n and

b%n

(13)

have the same form as an and bn but with σ g in an and bn replaced by

the field-dependent conductivity σ% g ≈ σ g + σ 3

E

2

. non, g

[RESULTS AND DISSCUSSIONS] We are now in a position to provide some numerical results. At first, let’s investigate the scattering properties of the monolayer graphene-wrapped dielectric sphere in the linear case, by neglecting the field-dependent term of the surface conductivity in eq 8. In the THz frequencies, the imaginary part of the surface conductivity of the graphene is positive and indicates a “metallic” type nature which can support local surface plasmon.32 Therefore, this graphene layer acts as a very thin “metallic” shell and leads to the resonant peaks in the scattering efficiency spectra, as shown in Figure 2. 9



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 22

Figure 2. Linear scattering efficiency of the monolayer graphene-wrapped dielectric sphere as a function of the incident wavelength, with (a) different sphere radii at Fermi energy E F = 0.3eV ; and (b) with different Fermi energies at a fixed sphere radius a = 100nm . Other parameters are ε = ε h = 2.25 and τ = 0.1ps , respectively.

Unlike metallic nanoparticle with small size, the graphene-wrapped dielectric nanoparticle has tunable surface plasmon resonant frequency associated with the particle size in its linear scattering efficiency spectra. Figure 2a indicates that when the radius of the nanoparticle is increased, the surface plasmon resonant frequencies are red-shifted, accompanied with the enhanced peaks. In the present model, although the radius of the nanoparticle is much smaller than the incident wavelength, with the plasmonic nature of the graphene layer in the THz frequencies, we can still achieve the size-dependent resonances. For a metallic sphere, however, the surface plasmonic resonances are almost independent on the size when the incident wavelength is much longer. In addition, besides the radius of the nanoparticle, different Fermi levels of graphene can also modify the surface conductivity, hence lead to the tunable surface plasmon resonances as well. Figure 2b shows the dependence of the resonant peak on the Fermi level, it clearly indicates that higher Fermi level will generally result in the blue-shift of the surface plasmon resonant wavelength and 10


Page 11 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


enhanced peak. Our results qualitatively coincide with those in the graphene ribbons.13

Figure 3. (a) Dependence of Qsca on incident wavelength with a = 100nm , E F = 0.3eV , ε = ε h = 2.25 and τ = 0.1ps . And correspondent distributions of the linear electric fields inside and outside the sphere for (b) a = 100nm , λ = 17.3µ m ; (c) a = 100nm , λ = 22 µ m .

To observe strong nonlinear effects in the graphene described by a field-dependent surface conductivity (i.e., eq 9), strong field intensity is required due to small Kerr-nonlinearity σ 3 . As one knows, localized surface plasmon resonance would enhance the local field so that it is expected to boost the naturally weak nonlinear effects in this graphene-wrapped nanoparticle. Along this line, we would like to take one step forward to investigate the spatial local fields near the resonant wavelength. In Figure 3, we plot the Qsca with a fixed radius

a = 100nm .

It is obvious that the field-enhancement in the surface resonant

frequency λ = 17.3µ m is larger than that of λ = 22µ m . Note that the scattering efficiency depicted by eq 7 depends on the Mie coefficients for TM ( an ) and TE ( bn ) modes, while the resonance here totally results from the electric dipole resonance for the chosen small size, 11



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 22

which can be proved by comparing the dotted line with the solid line in Figure 3.

Figure 4. The average local field E non,g as a function of the external applied field E0 for (a) same Fermi energy EF = 0.3eV with different radii; (b) same radius a = 100nm with different Fermi energy. (c) and (d) are the corresponding nonlinear far-field scattering efficiency versus E0 . Other parameters are ε = ε h = 2.25 , τ = 0.1ps and λ = 20 µ m .

In what follows, we’d like to study the optical switching effects in such nonlinear nanosphere near the surface plasmon resonant wavelengths for small size. We plot the average of the local field

E non, g ≡

E

2 non, g

within the graphene-wrapped nanoparticle as a

function of the applied external field for several sizes and different Fermi levels in Figure 4a and Figure 4b. From Figures 4a and 4b, it shows that the average of the local-field within the nanoparticles firstly increases with increasing the applied field, and it jumps discontinuously 12


Page 13 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


to the upper branch when the applied field reaches the switching-up threshold field. Further increasing the external incident field leads to monotonic increase of the local-field. On the other hand, the continuous decrease of the applied field will lead to the discontinuous jump of the local field from the upper branch to the lower branch at the switching-down threshold field [see Figure 4b]. This OB behavior in the near field shows a potential application in nano-switches and nano-memories.33 From Figures 4a, one observes that larger size of the nanoparticle will generally result in a lower bistable switching-up threshold field accompanied by a narrower bistable region. This can be understood as follows: the magnitude of the linear local field within the graphene thin layer is indeed enhanced as increasing the radii of the nanoparticle [see the insert of Figure 4a], resulting in the decrease of the threshold field. In addition, the results given by the quasistatic approximation29 are shown in Figure 4a (dotted line) as well to make comparison. The results within the quasistatic approximation agree well with those by applying our nonlinear full-wave theory, indicating that both the quasistatic approximation and the full-wave theory in the study of nonlinear OB are accurate within the dimension of the nano-device. Figure 4b illustrates the influence of Fermi level on bistable near-field. In this case, the threshold values differ because the Fermi energy influences the nonlinear coefficient of graphene conductivity [see eq 10], which in turn affects the average local field. In detail, increasing the Fermi energy results in the small magnitude of the Kerr-nonlinearity. As a consequence, much larger threshold field is needed to stimulate the optical bistable behavior. Next, the dependence of the far field scattering property of this graphene-wrapped nanoparticle is investigated. By substituting eq 12 into field-dependent scattering efficiency in eq 13, one yields the nonlinear scattering efficiency as the function of the applied external 13



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 22

field, as shown in Figures 4 c and 4d. Again, nonlinear scattering efficiency reveals optical bistable behavior unlike conventional hysteretic loops, the upper branch of these bistable curves exhibit very abrupt variation compared with the lower one, and the maximal scattering efficiency before dropping at switching-down threshold is extremely high. Hence, from Figure 4, we conclude that this graphene-wrapped dielectric nanoparticle provides us one possible way to realize nonlinear optical switch devices and nonlinear optical sensors, whose switching threshold field can be tunable by changing the Fermi level or the spherical sizes.

Figure 5. Switching threshold fields as a function of the sphere size for two Fermi energies.

Then, we investigate the switching threshold fields against the particles’ size, as shown in Figure 5. It is evident that both switching-up and switching-down threshold fields decrease monotonically, and the bistable region becomes narrow with increasing the particle sizes (see symbols). As a consequence, there are the critical sizes, above which the bistable properties vanish. Take

E F = 0.3eV

for example, the critical size

a

is about

135nm

. In

addition, it’s shown that larger Fermi energy results in the broader bistable region for a given particle size, and larger critical sizes. For comparison, analytical results for the switching 14


Page 15 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


threshold fields derived from the quasistatic limit29 are also shown by the lines in Figure 5. Again, good agreement between the present nonlinear Mie theory and the quasistatic one is found for the sizes.

Figure 6. Contributions from electric dipole (dotted line) and magnetic dipole (dash-dotted line) to the bistable behavior with the increasing of applied field E0 .

To one’s interest, as we further increase the magnitude of the applied field, the other hysteretic region appears. Note that for small size

a = 100nm ,

the dipolar approximation ( n = 1 )

is enough to estimate the optical bistable properties. From Figure 6, we conclude that the former bistable behavior origins totally from the electric dipole resonance, whereas the latter one is due to the magnetic dipole excitation. This may be understood as follows: with the increasing of the applied field, the field-dependent surface conductivity of the graphene layer is large, and the equivalent permittivity of the graphene-wrapped sphere is high. In general, high permittivity will excite the magnetic dipole mode, and boost the nonlinearity of graphene, resulting in the latter optical bistability. Here, we would like to mention that, the magnetic dipole excitation can only be predicted by our present theory, while it cannot be found within our previous theory based on the quasistatic limit and self-consistent 15



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 22

mean-field approximation.29 Since the sizes we adopted above are much smaller than the incident wavelengths, the first order term ( n = 1 ) is enough to predict the optical bistability even for large applied fields. In what follows, we would like to how the higher-order terms affect the nonlinear properties of the proposed graphene-wrapped nanoparticle for large sizes. In Figure 6, we show the average local field a = 1µ m .

E non,g

as a function of the external applied field E0 for large size such as

For large sizes, according to our calculations, the optical bistable region due to the

electric dipole disappears, which can also be predicted from Figure 4a, where the bistable region becomes narrow. Actually, for large sizes, the contributions from the electric dipole or even electric multipole are quite small, and the hysteretic curves mainly result from the magnetic multipole contributions excited by high incident fields.

Figure 7. Three typical behaviors in nonlinear composites for a = 1µ m , and λ = 40 µ m , with considering the nth order of incident TM wave: (a) the first order; (b) the first and second orders; and (c) the first, second 16


Page 17 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and third orders.

Taking a close look at Figure 7, it is found that once the higher-order terms are taken into account, the nonlinear curves will become more complicated. For instance, magnetic-dipole mode together with magnetic quadrupole term ( n = 2 ) contribute to the optical tristability (OT),9 which has one more stable branch compared to the OB [see Figures 7a and 7b]. OT curve in Figure 7b reveals the following nonlinear process: when the applied external field

E0

starts to increase over the first upper threshold field, the discontinuous

jump of the local field takes place from the lower branch to the middle branch; as E0 further increases up to the second upper threshold field, we find the other discontinuous jump from the middle branch to the upper branch. On the contrary, if one decreases E0 the average local field will first jump to the middle branch before jump to the lower branch. The difference between the OB and OT is that for a given

E0 ,

the average field has three real

roots within one electric field domain, hence the desired OB. However, if it has five real roots in one incident field region, hence the desired OT. Different from previous OT predicted in nonlinear plasmonic system of nonspherical particles, for which one electric dipolar mode along one’s direction interacts with the other electric dipole mode along the other direction,11 here the OT are associated with magnetic dipolar mode with magnetic quadrupolar one. Besides the dipole and quadrupolar terms, once the octupole term ( n = 3 ) is included, another bistable curve exists at much higher E0 region in Figure 7c. Therefore, one might achieve more functionality about optical switching in this proposed nano- or micro-devices.

17



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 22

Figure 8. Dependence of the average local fields on the applied field for (a) different sizes with E F = 0.3eV ; and (b) different Fermi energy with a = 1µ m . Other parameters are ε = ε h = 2.25 , τ = 0.1ps and λ = 40 µ m .

In the end, we investigate the influence of the particle size and Fermi energy on the multi-stable curves in the near-field for large sizes. As shown in Figure 8a, the multi-stable region is found to be strongly dependent on the size of nanoparticles, and it is possible to achieve low switching threshold with larger sizes. In contrast, one might obtain broader OT and OB regions when the Fermi energy is increased. For further increasing the radius of sphere, the octupole modes may shift to the dipole and quadrupole region, resulting in optical multistability. Therefore, this graphene-wrapped nanoparticle can provide more freedoms to control its multi-state optical switching, and may find many potential applications in optical communications and computing.34

[CONCLUSIONS] In conclusion, we have proposed the design and operation of graphene-wrapped dielectric nano-switches based on the nonlinear Mie theory and self-consistent mean-field approach. We find optical bistable behavior for the near-field and far-field scattering efficiency in such coated nanoparticle system at terahertz frequencies, and the switching thresholds are highly 18


Page 19 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


dependent on the Fermi energy of graphene besides the particle size. Our design provides a new degree of freedom to control the local field and scattering field with the input one. For small sizes and weak-field case, our theory is in good agreement with the quasistatic one.26 When the applied field is large, the magnetic dipolar mode can be excited, and additional optical bistability is found at large-field region even for small sizes. In this connection, one observe double OBs in both near-field and far-field spectra as a function of the incident field. Furthermore, for large sizes, the higher-order terms should be considered, and one observes multi-bistability in its near field spectra. In other words, tristability occurs once we further take into account the quadrupole term, and there is another bistable region under higher input field if the octupole term is considered. The influence of the particle size and Fermi energy on the multi-stable curves is also studied. As the possible detection of the optical bistability/tristability in nonlinear graphene-wrapped nanospheres, we would like to mention the experimental observations of the optical bistable behavior in nanometer-size spherical CdS coated with silver35, and in buckled dome microcavities36 by detecting the transmitted intensity with an incident intensity. Since works on the experimental fabrication of graphene-wrapped nanospheres25-27 have been reported, one may take one step forward to observe the optical bistability/tristability by detecting the nonlinear far-field scattering efficiency with the increase (or decrease) the input intensity. Our results offer insights into the role of the interaction between the Kerr-nonlinearity and plasmonic graphene-wrapped particles, which may pave the way for experimental investigations including the design in optoelectronic devices, like two/three state switching, memory, optical transistors and so on.

19



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 22

[AUTHOR INFORMATION] Corresponding Author *E-mail: [email protected]

[ACKNOWLEDGMENTS] This work was supported by the National Natural Science Foundation of China (Grant No. 11374223), the National Science of Jiangsu Province (Grant No. BK20161210), the Qing Lan project, “333” project (Grant No. BRA2015353), and PAPD of Jiangsu Higher Education Institutions.

[REFERENCES] 1. Boyd, R. W. Nonlinear Optics; Academic Press: Burlington, 2008; 2. Kauranen, M.; Zayats, A. V. Nonlinear Plasmonics. Nat Photon 2012, 6, 737-748. 3. Li, J.-B.; Kim, N. C.; Cheng, M. T.; Zhou, L.; Hao, Z. H.; Wang, Q. Q. Optical Bistability and Nonlinearity of Coherently Coupled Exciton-Plasmon Systems. Opt Express 2012, 20, 1856-1861. 4. Li, J.-B.; Liang, S.; Xiao, S.; He, M.-D.; Kim, N.-C.; Chen, L.-Q.; Wu, G.-H.; Peng, Y.-X.; Luo, X.-Y.; Guo, Z.-P. Four-Wave Mixing Signal Enhancement and Optical Bistability of a Hybrid Metal Nanoparticle-Quantum Dot Molecule in a Nanomechanical Resonator. Opt Express 2016, 24, 2360-2369. 5. Malyshev, A. V. Condition for Resonant Optical Bistability. Phys Rev A 2012, 86. 6. Gibbs, H. Optical Bistability: Controlling Light with Light; Academic: New York, 1985. 7. Assanto, G.; Wang, Z.; Hagan, D. J.; VanStryland, E. W. All-Optical Modulation Via Nonlinear Cascading in Type Ii Second-Harmonic Generation. Appl Phys Lett 1995, 67, 2120-2122. 8. Mazurenko, D. A.; Kerst, R.; Dijkhuis, J. I.; Akimov, A. V.; Golubev, V. G.; Kurdyukov, D. A.; Pevtsov, A. B.; Sel'kin, A. V. Ultrafast Optical Switching in Three-Dimensional Photonic Crystals. Phys Rev Lett 2003, 91. 9. Haus, J. W.; Inguva, R.; Bowden, C. M. Effective-Medium Theory of Nonlinear Ellipsoidal Composites. Phys Rev A 1989, 40, 5729-5734. 10. Bergman, D. J.; Levy, O.; Stroud, D. Theory of Optical Bistability in a Weakly Nonlinear Composite Medium. Phys Rev B 1994, 49, 129-134. 11. Gao, L.; Gu, L.; Li, Z. Optical Bistability and Tristability in Nonlinear Metal/Dielectric Composite Media of Nonspherical Particles. Phys Rev E 2003, 68, 066601. 12. Tassin, P.; Koschny, T.; Soukoulis, C. M. Graphene for Terahertz Applications. Science 2013, 341, 620-621. 13. García de Abajo, F. J. Graphene Plasmonics: Challenges and Opportunities. Acs Photonics 2014, 1, 135-152. 14. Low, T.; Avouris, P. Graphene Plasmonics for Terahertz to Mid-Infrared Applications. Acs Nano 2014, 8, 1086-1101. 20


Page 21 of 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


15. Ciattoni, A.; Rizza, C. Graphene-Nonlinearity Unleashing at Lasing Threshold in Graphene-Assisted Cavities. Phys Rev A 2015, 91, 053833. 16. Peres, N. M. R.; Bludov, Y. V.; Santos, J. E.; Jauho, A. P.; Vasilevskiy, M. I. Optical Bistability of Graphene in the Terahertz Range. Phys Rev B 2014, 90. 17. Xiang, Y. J.; Dai, X. Y.; Guo, J.; Wen, S. C.; Tang, D. Y. Tunable Optical Bistability at the Graphene-Covered Nonlinear Interface. Appl Phys Lett 2014, 104. 18. Dai, X. Y.; Jiang, L. Y.; Xiang, Y. J. Low Threshold Optical Bistability at Terahertz Frequencies with Graphene Surface Plasmons. Sci Rep 2015, 5. 19. Sanderson, M.; Ang, Y. S.; Gong, S.; Zhao, T.; Hu, M.; Zhong, R. B.; Chen, X. X.; Zhang, P.; Zhang, C.; Liu, S. G. Optical Bistability Induced by Nonlinear Surface Plasmon Polaritons in Graphene in Terahertz Regime. Appl Phys Lett 2015, 107. 20. Christensen, T.; Yan, W.; Jauho, A. P.; Wubs, M.; Mortensen, N. A. Kerr Nonlinearity and Plasmonic Bistability in Graphene Nanoribbons. Phys Rev B 2015, 92. 21. Li, R. J.; Lin, X.; Lin, S. S.; Zhang, X. M.; Li, E. P.; Chen, H. S. Graphene Induced Mode Bifurcation at Low Input Power. Carbon 2016, 98, 463-467. 22. Smirnova, D. A.; Shadrivov, I. V.; Miroshnichenko, A. E.; Smirnov, A. I.; Kivshar, Y. S. Second-Harmonic Generation by a Graphene Nanoparticle. Phys Rev B 2014, 90, 035412. 23. Christensen, T.; Jauho, A.-P.; Wubs, M.; Mortensen, N. A. Localized Plasmons in Graphene-Coated Nanospheres. Phys Rev B 2015, 91, 125414. 24. Rujiang, L.; Xiao, L.; Shisheng, L.; Xu, L.; Hongsheng, C. Atomically Thin Spherical Shell-Shaped Superscatterers Based on a Bohr Model. Nanotechnology 2015, 26, 505201. 25. Lee, J. S.; You, K. H.; Park, C. B. Highly Photoactive, Low Bandgap Tio2 Nanoparticles Wrapped by Graphene. Adv Mater 2012, 24, 1084-1088. 26. Lee, J. S.; Kim, S. I.; Yoon, J. C.; Jang, J. H. Chemical Vapor Deposition of Mesoporous Graphene Nanoballs for Supercapacitor. Acs Nano 2013, 7, 6047-6055. 27. Han, M.; Yin, X.; Kong, L.; Li, M.; Duan, W.; Zhang, L.; Cheng, L. Graphene-Wrapped Zno Hollow Spheres with Enhanced Electromagnetic Wave Absorption Properties. Journal of Materials Chemistry A 2014, 2, 16403-16409. 28. Yoo, D. Y., et al. Graphene Oxide Nanosheet Wrapped White-Emissive Conjugated Polymer Nanoparticles. Acs Nano 2014, 8, 4248-4256. 29. Huang, Y.; Miroshnichenko, A. E.; Gao, L. Low-Threshold Optical Bistability of Graphene-Wrapped Dielectric Composite. Sci Rep 2016, 6. 30. Chen, H. L.; Gao, D. L.; Gao, L. Effective Nonlinear Optical Properties and Optical Bistability in Composite Media Containing Spherical Particles with Different Sizes. Opt Express 2016, 24, 5334-5345. 31. Chen, H.; Zhang, Y.; Zhang, B.; Gao, L. Optical Bistability in a Nonlinear-Shell-Coated Metallic Nanoparticle. Sci Rep 2016, 6, 21741. 32. Vakil, A.; Engheta, N. Transformation Optics Using Graphene. Science 2011, 332, 1291-1294. 33. Argyropoulos, C.; Chen, P.-Y.; Monticone, F.; D’Aguanno, G.; Alù, A. Nonlinear Plasmonic Cloaks to Realize Giant All-Optical Scattering Switching. Phys Rev Lett 2012, 108, 263905. 34. Amin, M.; Farhat, M.; Bagci, H. A Nonlinear Plasmonic Resonator for Three-State All-Optical Switching. Opt Express 2014, 22, 6966-6975. 35. Neuendorf, R.; Quinten, M.; Kreibig, U. Optical Bistability of Small Heterogeneous Clusters. J Chem Phys 1996, 104, 6348-6354. 36. Bitarafan, M. H.; Ramp, H.; Potts, C.; Allen, T. W.; Davis, J. P.; DeCorby, R. G. Bistability in Buckled Dome Microcavities. Opt Lett 2015, 40, 5375-5378.

21



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

“TOC Graphic”

22


Page 22 of 22

Tunable Optical Bistability and Tristability in Nonlinear Graphene

Recommend Documents