Electromagnetic excitations in a non-ideal two-sublattice microcavity chain

Physica B: Condensed Matter 571 (2019) 296–300

Contents lists available at ScienceDirect

Physica B: Condensed Matter journal homepage: www.elsevier.com/locate/physb

Electromagnetic excitations in a non-ideal two-sublattice microcavity chain a,b

V.V. Rumyantsev a b

a

a,*

a

, S.A. Fedorov , K.V. Gumennyk , Yu.A. Paladyan

T

A.A.Galkin Institute for Physics & Engineering, 83114, Donetsk, Ukraine Mediterranean Institute of Fundamental Physics, 00047, Marino, Rome, Italy

ARTICLE INFO

ABSTRACT

Keywords: Microcavity lattice Structural defects Quantum dots Polaritonic excitations Virtual crystal approximation

We study the dependence of the dispersion characteristics of quasi-particle excitations in a non-ideal one-dimensional microcavity lattice on concentrations of structural imperfections formed by uneven spacing of cavities. Two configurations are considered where either one of the sublattices contains quantum dots or the whole structure is quantum-dot-free. It is shown that an appropriate incorporation of structural defects into a onedimensional microcavity array can be used to modulate the energy spectrum of electromagnetic excitations (and hence the optical properties) of the crystal.

1. Introduction The advent of optoelectronic devices utilizing various recent advances in photonics such as the harvesting of light by nanophotonic waveguides or quantum information processing has elevated the importance of the correct theoretical understanding of nanocrystalline photonic structures [1,2]. Recently, a considerable interest has been drawn to crystalline structures known as Lieb lattices. Strong confinement of light in photonic Lieb lattices opens up routes to development of new light-trapping schemes [3–5]. Among the problems raised by fabrication of novel nanocomposite materials (used as sources of coherent radiation) and by construction of corresponding devices one encounters the necessity of an adequate description of the so-called polaritonic crystals [6]. The latter constitute a special class of photonic crystals [7] featured by a strong coupling between quantum excitations (excitons) and optical fields. Investigations into polaritonic structures have given rise to polaritonics as a separate branch of photonics. Examples of polaritonic structures are given by spatially periodic systems of coupled microcavities (resonators) [8,9] as well as by the arrays of quantum dots embedded within photonic nanostructures [5,10]. Lately, there has been a growing interest for optical modes in microcavity arrays with embedded quantum dots. For instance, Ref. [11] studies the defect-based resonators in photonic crystals; Ref. [12] demonstrates the attainment of a strong coupling between a quantum dot and a microresonator. Theoretical studies [6,12] are devoted to formation of quantum solitons coupled to lower dispersion branch (LDB) polaritons in microcavity chains. The authors of Refs. [6,12] opine that microcavities may be useful for the purposes of quantum information processing. Recent progress in creation of high-Q *

semiconductor resonators with Bragg mirrors allowed to obtain and study the Bose-Einstein condensation as well as the superfluid properties of LDB-polaritons in quantum wells embedded within semiconductor (CdTe/CdMgTe, GaAs) microcavity structures [13–15]. In these systems polaritons are treated as a thermodynamically quasiequilibrium two-dimensional gas of interacting bosonic particles. Basing on our previously developed concepts of non-ideal one-dimensional polaritonic structures [11] in the present work we study the effect of random uneven spacing of cavities on the dispersion characteristics of electromagnetic excitations in a one-dimensional microcavity lattice. More specifically, we consider the case of polaritonic excitations in a non-ideal two-sublattice tunnel-coupled microcavity chain with embedded quantum dots along with the case of exciton-like excitations in a system of quantum-dot-free cavities. 2. Theoretical model Basing on the approach developed in Refs. [11,16,17] let us consider electromagnetic excitations in a non-ideal two-sublattice microcavity (resonator) chain with variable cavity spacing d = a1 + a2µ . Here a1 , a2µ are variable cavity positions in the first and second sublattices, respectively. Lower indices 1, 2 specify the number of sublattice, whereas upper indices ( , µ ) denote the position of resonator in a cell and assume values 1 or 2. Each of the tunnel-coupled microresonators possesses a single optical mode. In the general case, under the assumption of a small density of excited states of structural elements in ex the resonator and atomic subsystems, the quadratic part Hˆ of Hamiltonian, which describes elementary excitations in a microcavity chain (containing quantum dots or otherwise) within the Heitler-

Corresponding author. R. Luxembourg st. 72, 83114, Donetsk, Ukraine. E-mail address: [email protected] (K.V. Gumennyk).

https://doi.org/10.1016/j.physb.2019.07.022 Received 21 May 2019; Received in revised form 12 July 2019; Accepted 13 July 2019 Available online 15 July 2019 0921-4526/ © 2019 Elsevier B.V. All rights reserved.

Physica B: Condensed Matter 571 (2019) 296–300

V.V. Rumyantsev, et al.

London approximation [18] and a single-level model has the form: ex Hˆ =

2 , = 1 n, m, ,

Dn

,m

Dn12, m

=

A21 (k , C1, C2) = A21 [a1 (C1)]exp[ik a1 (C1)] + A21 [a2 (C2)]exp

(1)

at n n ,m

Dn21, m

= gn

+ Vn

,m

n ,m

,

, Dn22, m = ˆn

( = 2)

ph n ,m n

= ˆn ,

ˆn

An ( = 1)

indicate par-

,m

= Bˆn

Quantities A12(21) [a1(2) (C1(2) )] are the components of resonant interaction matrix An m , which correspond to the nearest neighbors:

,

A12(21) [a1(2) (C1(2) )] = A12(21) [a1(2) (0)]exp

= 0,

a1(2) (C1(2) )

(2)

a1(2) (0)

a1(2) (0)

. (5)

Below we assume that A12 A21 > > A11 , A22 . The Fourier transform of matrix Vn1m1 of resonant interaction between quantum dots at sites n1 and m1 in the frames of the adopted model has the form V11 (k ) = V11 (k, C1, C2) 2V11 [d (C1, C2)]cos[k d (C1, C2)], V11 where

are bosonic creation and annihilation operators of this excitation in site representation, An m is the matrix of resonant interaction characterizing an overlap of optical fields of resonators at the n -th and m -th lattice sites and therefore defining the jump probability of the corresponding electromagnetic excitation. Vn m is the matrix of resonant interaction between quantum dots at the n -th and m -th sites, gn is the matrix of resonant interaction of quantum dot at the n -th site with electromagnetic field localized at the same site. Values 1, 2 of indices , indicate, correspondingly, the presence or absence of a quantum dot at the corresponding cavity. Let us evaluate the spectrum (k ) of quasiparticle excitations in the system by adopting the virtual crystal approximation [19,20] and the averaged Green's function formalism. Under such an approach the averaged resolvent of the quasi-particle Hamiltonian equals to the resolvent of the averaged Hamiltonian; hence the quantities Dn , m in Eq. Dn , m ) config(1) should be replaced by their values (Dn , m urationally averaged over all feasible variations of cavity positions (the averaging procedure is denoted by angular brackets). Such an operation “restores” the translation invariance of the structure and permits to use the k-representation and the subsequent diagonalization of Hamiltonian through Bogolyubov-Tyablikov transformation [18]. These procedures lead to the following equation for (k ) :

(k )

(4)

{ ik [a2 (C2)]} ,

In expressions (1) and (2) nph is the photonic mode frequency of + electromagnetic excitation localized at the n -th site (cavity), ˆ n , ˆn are bosonic creation and annihilation operators of the photonic mode, at ˆ ˆ+ n is excitation energy of the quantum dot at the n -the site, Bn , Bn

det D (k )

A12 [a1 (C1)]exp[ ik a1 (C1)] + A12 [a2 (C2)]exp {ik [a2 (C2)]},

ˆ +n ˆ m ,

where indices n, m numerate elementary cells, and , ticular sublattices;

Dn11, m =

A12 (k, C1, C2)

[d (C1, C2 )] equals to: V11 [d (C1, C2 )] = V11 [d (0)]exp

d (C1, C2) d (0) d (0)

.

Relations (1)–(3) show that in the general case the dispersion law (k ) of elementary quasi-particle excitations in the considered microcavity system is determined by both quantities A (k ) and V (k ) . 3.1. Exciton-like excitations in a non-ideal one-dimensional microcavity lattice

Let us first consider electromagnetic excitations (the so-called exciton-like excitations [16]) in a two-sublattice one-dimensional microcavity lattice (see Fig. 1) without quantum dots (where Dn12, m = Dn21, m = 0 ). In such a case within the nearest-neighbor approximation the corresponding spectrum (k ) follows from relations (1)–(3): ph 1

(k , C1, C2) A21 (k , C1, C2)

A12 (k, C1, C2) ph 2

(k, C1, C2 )

= 0.

(6)

Numerical evaluation of quantities, which define the spectrum peculiarities shall be performed for the values of resonant photonic modes localized at lattice sites 1ph = 2 × 311 THz and 2ph = 2 × 331 THz . Similarly to Ref. [11] it is assumed that A12 [a1 (0)]/2 = 3,5·1014 Hz, A12 [a2 (0)]/2 = 1,2·1014 Hz , g1/ = 5 1012 Hz, V11/2 = 1·1013 Hz a2 (0) = a2(1) = 3 10 7 m , d a1 (0) = a1(1) = 1 10 7 m . where (0) = a1 (0) + a2 (0) . Numerically computed surfaces in Fig. 2 describe the dispersion dependence of frequencies ± (k , C1, C2) of the studied collective excitations. We remind that k ranges within the first Brillouin < k < d (C , C ) . Fig. 3 depicts the concentration zone d (C1, C2) 1 2 dependence of the corresponding band gap width (C1, C2 ) min[ + (C1, C2) (C1, C2)].

(3)

rm )]. Let us note that the where D (k ) = m Dn m exp[ik (rn wave number k , which characterizes the eigenstates of electromagnetic excitations in the crystal ranges within the first Brillouin zone of the d = a1µ + a2 = virtual lattice with period 1(2) C1(1) a1(1) + C1(2) a1(2) + C2(1) a2(1) + C2(2) a2(2) , where C1(2) are concentrations of the corresponding cavity spacings a1 , a2µ . These concentrations are subject to constraints C1(1) + C1(2) = 1, C2(1) + C2(2) = 1, and hence, C1(2) = 1 C1(1) C1, C2(2) = 1 C2(1) C2 . d = a1 Therefore, (C1) + a2 (C2) d (C1, C2) , a1 (C1) = a1(1) +(a1(2) a1(1) ) C1, where a2 (C2) = a2(1) + (a2(2) a2(1) ) C2 .

k

3.2. Polaritonic excitations in a one-dimensional two-sublattice non-ideal microcavity lattice Basing on the general theory developed in Section 1 let us proceed to consider quasi-particle (polaritonic) excitations in a two-sublattice one-dimensional microcavity lattice with same-type quantum dots embedded in one of the sublattice (e.g. in the first one, i.e. = = 1).

3. Results and discussion To fix our ideas let us study quasiparticle excitations in a two-sublattice one-dimensional virtual microcavity lattice. A resonator position is defined by the equality rn = rn + r . Hence positions of cavities in the first and second sublattices in the zeroth elementary cell (rn = 0 = 0 ) are r01 = 0 defined, correspondingly, by relations: and r02 = a1 (C1) = d (C1, C2) a2 (C2) . Quantities A (k ) A (k, C1, C2) in Eq. (6) are the Fourier transforms of matrix An m , which describes an interaction between cavities resulting from the overlap of their optical rm )]. Within the frames of the fields A (k ) = m An m exp[ik (rn adopted model the Fourier transforms A (k, C1, C2 ) assume the following form:

Fig. 1. Schematic of the non-ideal two-sublattice one-dimensional microcavity array with quantum dots embedded in the first sublattice. 297

Physica B: Condensed Matter 571 (2019) 296–300

V.V. Rumyantsev, et al.

Fig. 2. Concentration dependence

± (k ,

C1, C2) for: a) C1 = 0,3 and b) C2 = 0,4 .

The

12(23) (C1,

band gap widths C2) min[ 2(3) (C1, C2)

of

1(2) (C1,

k

polaritonic spectrum C2)] are plotted as

functions of concentrations of structural defects in Fig. 5a and b.

3.3. Effective mass and the density of states of the polaritonic quasi-particles The

nonmonotonic character of concentration dependence C1, C2) of the spectrum of exciton-like and polaritonic excitations in the considered defect-containing structures has its manifestations in the quasiparticles' effective masses: 1,2,3 (k ,

(1,2,3) meff (C1, C2)

Fig. 3. Concentration dependence of the band gap width

V11 (k )

(k )

0

0

(k )

g1

0

0

0

ph 1

g1

0

0

0 (k )

A21 (k,)

= 0.

A12 (k ) ph 2

1,2,3 (k , C1, k2

1

C2) k=0

(8)

where the upper index numbers dispersion branches. From examining the corresponding plots shown in Fig. 6a–b one may conclude that the carefully designed introduction of structural defects can be an effective tool of controlling the group velocity of wave packets in a one-dimensional microcavity lattice. Considerable interest lies in how peculiarities of the spectrum are manifested in the quasi-particle density of states 1,2,3 ( , C1, C2) . In the case of a non-ideal one-dimensional microcavity array functions 1,2,3 ( , C1, C2 ) are computed as follows:

(C1, C2) .

In that case (within the nearest neighbor approximation) the relations (1)–(3) yield the following equation for the elementary excitation spectrum (k ) : at 1

2

(k )

1,2,3 (

(7) Since composition of quantum dots is not being varied the parameter of resonant interaction g1 between a quantum dot and electromagnetic field localized at a same site is always the same. Calculation of quantities, which define the spectrum shape of polaritonic excitations was performed for the above data and excitation frequency of quantum dots 2at = 2 202 THz . Also, similarly to Ref. [11] we put V11/2 = 1·1013 Hz , g1/ = 5 1012 Hz. Fig. 4 shows surfaces, which describe the dispersion dependence of polaritonic frequencies 1,2,3 (k , C1, C2 ) in the two-sublattice microcavity array with quantum dots embedded in one of the sublattice (surfaces are numbered bottomup). The wave number k ranges as always within the first Brillouin zone < k < d (C , C ) (shaded region in plane (k, C1(2) ) in Fig. 4a–d). d (C1, C2) 1 2 Let us note that the presence of local minima at k 0 in the dispersion surface 3 (k, C1, C2) in Fig. 4a–d (as well as in Fig. 2) indicates the possibility of existence (for certain defect concentrations) of BoseEinstein polaritonic condensate for non-zero k 's (in addition to BoseEinstein condensation at zero k 's at the corresponding minima in surfaces 1,2 (k, C1, C2) ).

, C1, C2) =

d (C1, C2) 2

1 i

d 1,2,3 (k , C1, C2) dk ki

(9)

In (9) the sum is evaluated over the roots ki of equation C1, C2) = falling within the first Brillouin zone. Fig. 7 shows concentration dependence of the density of states of the considered electromagnetic excitations 1,2,3 ( , C1, C2) , where the lower index numerates dispersion branches. The curves 1.2.3 ( ) were computed for concentration values C1 = 0.1, C2 = 0.2 and g1/ = 1014 Hz. 1,2,3 (k ,

4. Conclusion The present study of the spectrum of elementary quasiparticle excitations (along with some of its manifestations) in a binary one-dimensional lattice of coupled microcavities allows to conclude that introduction of structural defects leads to a significant alteration of optical properties of the crystal and hence may be used as an effective tool to attain the desired energy structure of electromagnetic excitations. We have performed numerical computation of the dispersion characteristics of two types of quasiparticles: a) exciton-like excitations in a non-ideal two-sublattice chain of tunnel-coupled microcavities 298

Physica B: Condensed Matter 571 (2019) 296–300

V.V. Rumyantsev, et al.

Fig. 4. Dependence of polariton dispersion 1,2,3 (k , C1, C2) on structural defect concentration plotted for various values of parameter g g1/ responsible for the resonant interaction between a quantum dot and electromagnetic field localized at a same site (arrows indicate the effect of changing g on the width of the so-called “bottle neck”.

Fig. 5. Dependences

12(23) (C1,

C2 ) of the band gap widths on structural defect concentrations: a)

(resonators) with no atomic subsystem and b) polaritonic excitations in a microcavity chain with an atomic subsystem (a system of quantum dots) embedded in one of the sublattices. The obtained results regarding the dependence of the effective mass of electromagnetic excitations on structural defect concentrations demonstrate the possibility of controlling the group velocity of excitations, which is responsible for signaling rates in optoelectronic devices. Our results contribute to the

12 (C1,

C2) , b)

23 (C1,

C2) .

modeling of the new class of functional porous materials, namely the so-called polaritonic systems (microcavity arrays with embedded quantum dots, e.g. in porous materials) where controlling of propagation of electromagnetic excitations is accomplished by an appropriate introduction of structural defects.

299

Physica B: Condensed Matter 571 (2019) 296–300

V.V. Rumyantsev, et al.

a)

b)

c)

1,2,3 (C1, C2) of polaritonic excitations on defect concentrations in a non-ideal one-dimensional microcavity lattice. Fig. 6. Dependence of the effective mass meff

Fig. 7. Dependence of the quasiparticle density of states in a one-dimensional microcavity lattice on structural defect concentrations in three dispersion branches plotted for g1/ = 1014 Hz .

Conflicts of interest

[9] K.J. Vahala, Nature 424 (2003) 839–846. [10] P. Tighineanu, A.S. Sørensen, S. Stobbe, P. Lodahl, The mesoscopic nature of quantum dots in photon emission, in: P. Michler (Ed.), Quantum Dots for Quantum Information Technologies. Nano-Optics and Nanophotonics, Springer, Cham, 2017. [11] V.V. Rumyantsev, S.A. Fedorov, K.V. Gumennyk, et al., Superlattice Microstruct. 120 (2018) 642. [12] A.P. Alodjants, I.O. Barinov, S.M. Arakelian, J. Phys. B At. Mol. Opt. Phys. 43 (2010) 095502. [13] J. Kasprzak, M. Richard, S. Kundermann, et al., Nature 443 (7110) (2006) 409. [14] R. Balili, V. Hartwell, D. Snoke, et al., Science 316 (2007) 1007. [15] A. Amo, J. Lefrère, S. Pigeon, et al., Nat. Phys. 5 (2009) 805. [16] V.V. Rumyantsev, S.A. Fedorov, K.V. Gumennyk, et al., Nature Sci. Rep. 4 (2014) 6945. [17] V.V. Rumyantsev, S.A. Fedorov, K.V. Gumennyk, et al., Physica B 461 (2015) 32–37. [18] V.M. Agranovich, Theory of Excitons, Nauka, Moscow, 1968. [19] J.M. Ziman, Models of Disorder: the Theoretical Physics of Homogeneously Disordered Systems, Cambridge University Press, Cambridge, 1979. [20] V.F. Los, Theor. Math. Phys. 73 (1) (1987) 1076.

None. References [1] [2] [3] [4] [5] [6] [7] [8]

I. Söllner, S. Mahmoodian, S. Lindskov Hansen, et al., Nat. Nanotechnol. 10 (2015) 775. P. Lodahl, Q. Sci. Technol. 3 (2018) 013001. D. Guzman-Silva, C. Mejía-Cortés, M.A. Bandres, et al., New J. Phys. 16 (2014) 063061. Rodrigo A. Vicencio, Camilo Cantillano, Luis Morales-Inostroza, et al., Phys. Rev. Lett. 114 (2015) 245503. Shiqi Xia, Ajith Ramachandran, Shiqiang Xia, et al., Phys. Rev. Lett. 121 (2018) 263902. E.S. Sedov, A.P. Alodjants, S.M. Arakelian, et al., Phys. Rev. A. 89 (2014) 033828. J.D. Joannopoulos, S.G. Johnso, J.N. Winn, R.D. Meade, Photonic Crystals. Molding the Flow of Light, second ed., Princeton University Press, Princeton, 2008. M.A. Kaliteevskii, Tech. Phys. Lett. 23 (2) (1997) 120.

300