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Decay of a quantum dot in two-dimensional metallic photonic crystals
Optics Communications 284 (2011) 2363–2369
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Decay of a quantum dot in two-dimensional metallic photonic crystals A. Hatef, M. Singh ⁎ Department of Physics and Astronomy, The University of Western, Ontario, London, Canada N6A 3K7
a r t i c l e
i n f o
Article history: Received 3 August 2010 Received in revised form 17 October 2010 Accepted 3 December 2010 Available online 18 December 2010 Keywords: Metallic photonic crystal Density of states Plasma energy Quantum dots Photonic band gap Decay rate
a b s t r a c t In this paper we have developed a theory for the decay of a quantum dot doped in a two-dimensional metallic photonic crystal consisting of two different metallic pillars in an air background medium. This crystal structure forms a full two-dimensional photonic band gap when the appropriate pillar sizes are chosen. The advantage of using two metals is that one can easily control the density of states and optical properties of these photonic crystals by changing the plasma energies of two metals rather than one. Using the Schrödinger equation method and the photonic density of states, we calculated the linewidth broadening and the spectral function of radiation due to spontaneous emission for two-level quantum dots doped in the system. Our results show that by changing the plasma energies one can control spontaneous emission of quantum dots doped in the metallic photonic crystal. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In this paper we have studied the quantum optics and band structure of metallic photonic crystals. Photonic crystals are structures characterized by their periodically modulating dielectric constant, which may vary in one, two or three spatial dimensions. Due to multiple reflections at the interfaces between regions with different dielectric constants in the structure, electromagnetic (EM) waves of a certain frequency range cannot propagate through a photonic crystal; this range of frequencies is referred to as the photonic band gap (PBG). Generally, the wavelengths of light which fall within the PBG are on the order of the crystal's lattice constant, while the width of the PBG is proportional to the crystal's dielectric contrast. The PBG of a onedimensional photonic crystal lies in only one direction, while for a two- or three-dimensional photonic crystal it varies for different photon propagation directions. If a certain range of photon energies is prohibited from travelling in a two- or three-dimensional photonic crystal for all Brillouin zone directions then this energy range is referred to as the complete PBG. Essentially, the PBG controls the propagation of EM waves in photonic crystals in the same manner that the electronic band gap controls electrons in semiconductors. It is well-known that in a dielectric photonic crystal, high dielectric contrast is required to have a complete PBG [1]. For example, inverse opal photonic crystals made from ordinary dielectric materials require that the contrast should be greater than eight to have a complete PBG in the optical regime [2]. One way to overcome this barrier is to
⁎ Corresponding author. E-mail address: [email protected] (M. Singh). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.12.012
include metallic components in the structure of photonic crystals. This greatly enhances the dielectric contrast, which leads to the formation of larger PBGs in each Brillouin zone direction and thus increases the likelihood of their overlap for all directions [3–7]. These structures are called metallic photonic crystals (MPCs). MPCs can also provide strong simultaneous coupling between electronic and photonic resonances in the same range of the frequencies, which gives them interesting optical properties [8], and can lead to well-pronounced spectral features [8,9]. One of the promising features of MPCs is their capability of tailoring thermal emission spectra. Theoretical analysis has shown that the existence of the PBG and modification of the photon density of states (DOS) in these structures can lead to significant enhancement and suppression of thermal radiation near and within the PBG, respectively [10–15]. By simultaneously controlling photonic and electronic resonances, MPCs can be used to manipulate light–matter interaction for developing integrated photonic nanodevices. MPCs have been used in the field of telecommunications as antennas [16], all-optical switches [17], biosensors [18], solar cells [19] and other optoelectronic devices [20]. There is a great deal of interest in two-dimensional (2-D) photonic crystals due to their relative ease of fabrication as well as the underlying idea that two-dimensional (2-D) photonic crystals can be used to develop a direct continuation of planar integrated optics [21]. Recently, 2-D MPCs have been fabricated by using various methods [22–24]. For example, Katsarakis et al. [22] reported the fabrication of 2-D MPCs consisting of square and triangular lattices of nickel pillars using deep X-ray lithography. They showed that the photonic band structures of these crystals exhibited high-pass filter characteristics with a cutoff in the far-infrared energy regime. Puscasu et al. [23] used a 2-D metallo-dielectric photonic crystal to investigate narrow-band
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infrared sources and spectroscopic sensors. Zhang et al. [24] fabricated 2-D MPC waveguides using solution-processible gold nanoparticles. For device applications it is critical to accurately calculate higherorder PBGs. Typically, the band structures of 2-D MPCs are determined with numerical approaches such as the plane wave method (PWM) [25], transfer matrix method (TMM) [26], finite-difference timedomain method (FDTD) [27,28] and multiple multipole method (MMP) [29]. For example, Ustyantsev et al. [28] investigated the effect of different dielectric backgrounds on the band structure of MPCs composed of circular metallic pillars in a square lattice. They used the PWM and FDTD methods to determine the photonic dispersion bands and the TMM to determine the reflectivity characteristics. Takayma and Cada [30] reported theoretical results of PBGs based on the TMM for a MPC in the form of a hexagonal lattice consisting of silver pillars in anodic porous alumina. However, it is very complicated to use numerical band structure methods in studying the absorption and emission of light in these structures. On the other hand, analytical expressions are very useful for experimentalists so that they can analyze their data easily. There are also many useful applications for analytical expressions of photonic dispersion relations. For example, these expressions allow for an accurate determination of band gaps in all crystal directions. Determination of very narrow PBGs is important for the design of oversized single-mode photonic cavities or waveguides. They are also very useful for rigorous error analysis and help establish error tolerance in numerical simulation techniques. Finding an analytical formulation of band structures in photonic crystals represents a major mathematical challenge because analytical expressions for the band structure are generally limited to one-dimensional structures [31]. Recently, some effort has been devoted towards developing analytical models [32–36] for 2-D photonic crystals in order to understand and predict the physical properties of these photonic crystals without the need for heavy and time consuming calculations. Some efforts have also been placed towards obtaining analytical expressions for photonic dispersion relations in 2-D MPCs. For example, Pokrovsky et al. [37] presented an analytical theory for low-energy EM waves in MPCs with a small volume fraction of metal. Wang et al. [38] developed an analytical expression for the effective velocity of EM waves at low energies for 2-D dielectric photonic crystals and MPCs. In this paper, we have considered 2-D MPCs made from rectangular metallic pillars arranged periodically in a 2-D plane, where air is taken as the background medium. The advantage of choosing two metals lies in the fact that one can easily control the size and location of the crystal's PBG by manipulating the plasma frequencies of two metals rather than one. This structure has another advantage in that one can obtain analytical expressions for the band structure, DOS, linewidth and absorption coefficients. These expressions can be very useful for experimental studies and device design involving metallic photonic crystals. We have investigated the effect of modifying the plasma energy of the metals in the MPC on the spontaneous emission in these structures. To study spontaneous emission, we consider the situation where a twolevel quantum dot is doped in our MPC. The quantum dot is interacting with the photonic states of the crystal via the electron–photon interaction. Analytical expressions of the linewidth and spectral function of the quantum dot due to spontaneous emission have been calculated by using the Schrödinger equation method. Numerical simulations have been performed on the band structure and DOS in all 2-D crystal directions. We consider a MPC consisting of aluminum (Al) with zinc (Zn) in an air background. This structure gives a complete 2-D PBG. The DOS of photons has also been calculated in all crystal directions. It is found that DOS has singularities near the band edges. This agrees with the findings of other researchers in the literature [28]. Finally we have calculated the effect of the plasma energy on the spectral function for spontaneous emission, which contains information about the linewidths of the energy levels of the quantum dot. It is found
that manipulating the normalized plasma energy of the metal changes the width of the absorption peak. This means that by changing the plasma energy one can control spontaneous emission in these structures. 2. Two dimensional metallic photonic crystals As is schematically depicted in Fig. 1, we consider a 2-D metallic photonic crystal (MPC) made from two rectangular metallic pillars B and C with dielectric constants b and c, infinite in the z-direction. In each unit cell the opening domain (A) has a square shape, and is empty (i.e. an air space) with a dielectric constant a. The structure is homogeneous in the z-direction and periodic in the x and y directions with period L; therefore the unit cell for this crystal is a square. The metallic pillars used in the MPC are conductors in which electrons are free to move. When an EM field is propagating in a metal, the conductivity of the metal is energy dependant. According to the Lorentz model, the refractive index of a metal is obtained as [39]
ns ðεk Þ =
1−
!1 = 2 ε2p ; εk ðεk + iγÞ
ð1Þ
where εp is called the plasma energy and is obtained as 2
εp =
nℏ2 e2 : m0
ð2Þ
The plasma energy is the natural energy of oscillation for the electron gas in the metal pillars. In the above equations, n is the electron concentration, e is the electron charge, m is the mass of an electron and γ is the relaxation energy of the electrons in a pillar. Kady et al. [6] utilized the energy dependency of the real and imaginary parts of the dielectric constant for noble metals (i.e. Al, Ag, Au, and Cu). They showed that the imaginary part of dielectric constant (i.e. loss) is negligible in the range of optical and ultraviolet frequencies. Here the value of the relaxation energy is on the order of few meV and can be ignored without any significant variation in our calculations [40]. 3. Spontaneous emission decay rate The 2-D MPC is doped with a two-level quantum dot. Energy levels of the quantum dot are denoted as |a〉 and |b〉, where the former is the ground state (see Fig. 2). We consider that the quantum dot decays
Fig. 1. Schematic of the dielectric function in the 2-D separable rectangular MPC. The large white square region has a dielectric constant a = 2n21, while the small dark square and rectangular regions have b = 2n22 and c = n21 + n22, respectively. The parameters a and b give the thicknesses of the layers and L = a + b is the lattice constant in both the x and y directions.
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where η is the separation constant, n21 and n22 are related to a, b and c as a = 2n21, b = 2n22 and c = n21 + n22. Generally, the band structure of a square lattice is plotted in symmetric directions in the Brillouin zone. The centre of the Brillouin zone is located at (Kx = 0, Ky = 0) and is denoted as Γ. The other symmetric points are called X and M and are at (Kx = π/L, Ky = 0) and (Kx = 0, Ky = π/L), respectively. The band structure is then plotted along three major directions Γ − X, Γ − M and X-M. In Eq. (3), z(εk) is called the coupling constant and is obtained as Fig. 2. A schematic diagram of a two-level quantum dot where the levels are denoted by |a 〉 and |b 〉. When the quantum dot is in the excited state it decays to the ground state spontaneously.
spontaneously from the excited state to the ground state via the electron–photon (EP) interaction. The Hamiltonian of the system is written as: 2
3 1 dεk þ z 6 εba σba + 2 + ∫C 2π εk p ðεk Þpðεk Þ 7 6 7 H=6 ⋅ h i 7 4 5 dεk þ − þ −∫C zðεk Þ pðεk Þσba + σba p ðεk Þ 2π
ð3Þ
− z Here σ+ ba = |b〉〈a|, σba = |a〉〈b| and σba = |b〉〈b| − |a〉〈a|. The p(εk) and p (εk) operators denote the annihilation and creation of photons, respectively. The first and second terms correspond to the Hamiltonians of the quantum dot and photons in the photonic crystal, respectively. The third term describes the coupling between a quantum dot and photons. The integration contour C consists of two intervals: − ∞ b εk ≤ εv and εc ≤εk b ∞. Here εv and εv are the lower and the upper band edges of PBG, respectively, such that the PBG of the MPC lies between εv and εc. Note that there is no integration between energies εv and εc. In Eq. (3) εk is photon energy which is obtained from the band structure of 2-D MPC. The band structure is calculated by using the separable model [32–36]. We consider transverse electric (TE) plane waves propagating in the x–y plane (i.e. kz = 0) on 2-D crystal. The length and width of pillar B are taken the same, and are denoted as b. Similarly the length and width of pillar C is taken as a and b, respectively. The periodicity of the crystal along x and y directions is given as L = a + b. The dispersion relation is obtained as +
cosðKx LÞ = Fx ðεk ; ηÞ
ð4aÞ
cos Ky L = Fy ðεk ; ηÞ;
ð4bÞ
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! ffi ! ε n 2 ε n 2 k 1 k 2 −η2 a cos −η2 b ℏc ℏc ε n 2 ε n 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! k 1 ε n 2 + k 2 −2η k 1 ℏc ℏcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin 2 r ffi − rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a −η ε n 2 ε n 2 ℏc k 1 k 2 2 −η2 −η2 ℏc ℏc rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! ε n 2 k 2 × sin −η2 b ð5aÞ ℏc
Fx ðεk ; ηÞ = cos
! ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εk n 1 2 εk n2 2 Fy ðεk ; ηÞ = cos + η2 a cos + η2 b ℏc ℏc ε n 2 ε n 2 ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 k 1 k 2 + + 2η εk n1 2 ℏc ℏc 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos a + η − r ε n 2 ℏc εk n1 2 k 2 2 + η2 + η2 ℏc ℏc ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε n 2 k 2 × cos + η2 b ð5bÞ ℏc
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 3 pffiffiffiffiffiffi 3π ℏ c Dðεk Þ zðεk Þ = γ0 dx dy dz ε2k
ð6Þ
μ 2 ε3ba γ0 = 3π0 ℏ4 c3
where γ0 is the linewidth for an energy level of the quantum dot when it is located in a vacuum. The coupling constant controls the coupling between photons and electrons and depends on the photon density of states function D(εk) (DOS) in the 2-D MPC. This function is calculated from the dispersion relation (Eqs. (4a) and (4b)) as follows. We consider that dx, dy and dz are the dimensions of 2-D MPC along x, y and z direction, respectively. By using the concept of the DOS the summation over and Kx and Ky can be replaced by integration over photon energy εk as ∑ ∑ = ∫Dðεk Þdεk ; Kx
ð7Þ
Ky
where D(εk) is obtained as Dðεk Þ =
dx dy dk⊥ k ; π ⊥ dεk
ð8Þ
where k⊥ =
1 L
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi 2
½arccos Fx ðεk ; ηÞ2 + arccos Fy ðεk ; ηÞ ;
ð9Þ
with the help of the above equation the DOS is then evaluated as 2 3 dx dy 6 Gx ðεÞξx ðεk Þ Gy ðεÞξy ðεk Þ 7 Dðεk Þ = 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5: πL 1−Fy2 ðεk Þ 1−Fx2 ðεk Þ ξ ðεk Þ = ξ 1 ðεk Þ + ξ2 ðεk Þ + ξ3 ðεk Þ⋅ n2d n2 εk a ξ ð ε Þ = −cos ð k a Þsin ð k a Þ 1 k 1 2 ðℏcÞ2 k2 ! an21 εk −cosðk2 aÞsinðk1 aÞ k1 ðℏcÞ2 n2d n2 εk a ð ε Þ = −cos ð k a Þsin ð k a Þ ξ 1 k 1 2 ðℏcÞ2 k2 ! 2 an1 εk −cosðk2 aÞsinðk1 aÞ k1 ðℏcÞ2
ξ 2 ðεk Þ = sinðk1 aÞsinðk2 bÞ ×
½
2 n2d n2 εk k1 n1 εk − 2 3 2 ðℏcÞ k2 ðℏcÞ k2 k1
n21 εk k2 n2d n2 εk − ðℏcÞ2 k31 ðℏcÞ2 k1 k2 2n2d n2 εk a ξ3 ðεk Þ = sinðk1 aÞcosðk2 aÞ ðℏcÞ2 k2 +
+ sinðk2 aÞcosðk1 aÞ
! an21 εk ⋅ k1 ðℏcÞ2
ð10aÞ
ð10bÞ
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where + and − stand for y and x, respectively. Other terms are defined as
n2d
1 + ε2p ε−2 k = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1−ε2p = ε2k
k1− k2−
Δba
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ε n 2 ε n 2 k 1 k 1 = −η2 ; k1 + = + η2 ℏc ℏc rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ε n 2 ε n 2 k 2 k 2 = −η2 ; k2 + = + η2 ⋅ ℏc ℏc
ð10cÞ
ðH−εk Þjψk 〉 = 0
ð11Þ
The eigenstate of system can be written as a single particle eigenket in the form þ
dεk þ f ðε Þp ðεk Þj0〉; 2π k k
ð12Þ
where the vacuum state of the system is denoted as |0 〉. Putting Eq. (12) into Eq. (11) the Schrödinger equation takes the form pffiffiffiffiffiffiffiffiffiffiffiffi ðε′ −εk Þfk ðεk Þ = C Dðεk Þgk dεk pffiffiffiffiffiffiffiffiffiffiffiffi ðεba −εk Þgk = C∫ Dðεk Þfk ðεk Þ; C 2π
ð13Þ
C=
!1 = 2 ð14Þ
When the eigenvalue εk lies outside the PBG the solution of the first equation can be written as pffiffiffiffiffiffiffiffiffiffiffiffi
C Dðεk Þ fk ðεk Þ = 2πδ ε′ −εk + limþ gk : s→0 ε′ −εk −is
ð15Þ
Putting fk(εk) into Eq. (13-b) we obtain ðεba −εk Þgk = C
pffiffiffiffiffiffiffiffiffiffiffiffi dεk C 2 Dðεk Þ Dðεk Þ + limþ ∫ g : C 2π ε′ −ε −is k s→0 k
ð16Þ
! C 2 Dðεk Þ ; ε′ −εk
ð20Þ
and P stands for the principal part. The expression for the linewidth is calculated as ! 3π2 ℏ3 c3 Dðεab Þ: dx dy dz ε2ab
ð21Þ
Note that the linewidth has a very large value when the resonance energy εba lies near the photonic band edges but outside the PBG. Putting the expression of DOS into the above expression, we find 2 3 ! Gy ðεk Þξy ðεk Þ 7 3πℏ3 c3 6 Gx ðεk Þξx ðεk Þ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5: dz ε2ab Ly 1−Fy2 ðεk Þ Lx 1−Fx2 ðεk Þ
Γba = γ0
ð22Þ
The probability of finding the quantum dot in the excited state is then found as 2
Iðεk Þ = jgk j :
ð27Þ
This quantity is called the spectral density of the radiation due to spontaneous emission and is found as Iðεk Þ =
where C is a constant, given as 3π2 ℏ3 c3 γ0 dx dy dz ε2ba
dεk = P∫ C 2π
Γba = γ0
To find the decay rate of the quantum dot we use the Schrödinger equation method. The Schrödinger equation is written as
jψk 〉 = gk σ j0〉 + ∫C
where
εk −εba
C 2 Dðεab Þ :
2 + Δ2ba + C 4 D2 ðεab Þ
ð24Þ
Note that in empty space, I(εk) reduces into the standard Wigner– Weisskopf expression for a natural radiation line width. 4. Results and discussions For our numerical simulations of the 2-D MPC we choose a dielectric-metal composite. The structure is homogeneous in the zdirection and periodic in the x and y directions. We consider that the crystal is made from square and rectangular metallic pillars embedded in a background dielectric material. The rectangular pillars are connected to the sides of square pillars. In our simulation the background dielectric material is air. The dielectric functions in Eqs. (5a) and (5b) can be written as follows: 2
a = 2n1 = 1 Let us define a quantity called self-energy Ξba(εk) as 2 dεk C Dðεk Þ ⋅ Ξba ðεk Þ = limþ ∫ C 2π ε′ −ε −is s→0 k
b = 2n22 = 1− ð17Þ
Putting this into Eq. (13-b) we get the expression for gk as pffiffiffiffiffiffiffiffiffiffiffiffi C Dðεk Þ ⋅ gk ðεk Þ = εba −εk −Ξba ðεk Þ
2
ð25Þ
ε2 ε2pc 1− 2 ε
;
C, where the εbp and εcp are the plasma energy of pillars B and ! respectively. Substituting the values of n21 ð18Þ
Note that Ξba is a complex quantity. The real part of the self-energy gives the energy shift and the imaginary part gives the decay rate or linewidth: Ξba = Δba + iΓba ;
2
c = n1 + n2 =
ε2pb
ð19Þ
=
1 1 and n22 = 2 2
1−
ε2pb ε2
into
Eq. (25) we get the following relation between the plasma frequencies of the metallic pillars. εpb εpc = pffiffiffi 2
ð26Þ
This shows that our calculation is valid for any metallic pillars which satisfy this relation. Several options are available for different metals whose plasma energies satisfy Eq. (26). For instance, options for pillars B and C include aluminum (Al) and zinc (Zn), with plasma
A. Hatef, M. Singh / Optics Communications 284 (2011) 2363–2369
energies εp(Al) = 15.1 eV [41] and εp(Zn) = 10.1 eV [42] or copper (Cu) and platinum (Pt) with plasma energies of εp(Cu) = 7.4 eV and εp(Pt) = 5.1 eV [43]. In our simulations, εpb is chosen to satisfy the relation εpb L = λ in order to study the effect of changingpplasma energy, and ffiffiffi 2πℏc pffiffiffi ε 1 2 Lεk we take n21 = 0.5 and n22 = 0:5− 2 , where εn = = 2 k is 2πℏc εpb λ εn defined as the normalized energy. Since this separable model has not been used for MPCs, it is important to see whether this model produces a band gap in these crystals. To visualize and analyze the direction and propagation behavior of incident EM waves inside the MPC, we employ constantenergy contour diagrams in wave vector space (K) whose gradient vectors give the group velocities of the photonic modes. To plot constant-energy contours of the crystal we eliminate η in Eqs. (4a) and (4b) using MAPLE code for plotting a parametric system of real nonlinear equations. Fig. 3(a) and (b) illustrate several constantenergy contours, which are the projections of the photonic bands in the (Kx, Ky) plane, for various normalized energy in the reduced Brillouin zone (− 1 b KxL/π b 1, − 1 b KyL/π b 1). For the given parameters (a = 0.5L and b = 0.5L) and λ = 1, our calculations show there is a low-energy complete PBG from zero up to a normalized cut-off energy (εn = 0.9), and below that the normal surface is an exact
Fig. 3. Normal surface in the (Kx, Ky) plane of the 2-D MPC at different normalized energies. The normal surfaces plotted in the reduced Brillouin zone exhibit a fourfold symmetry. The parameters are n21 = 0.5, n22 = 0.5 − 1/ε2n, a = 0.5L and b = 0.5L.
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square which means no EM modes can propagate through the crystal, and hence the MPC behaves like a perfect reflector. However, a highenergy complete PBG appears in the normalized energy region between (1.15, 1.30) where the normal surfaces on this range of energy are all exactly square (see Fig. 3). Above the normalized cut-off energy between (0.93, 1.01) the contours are in the form of circles, meaning that the MPC behaves like a isotropic homogenous medium. By increasing the normalized energy the contours are no longer circular, showing the effect of the periodic structure of MPC. For example, at the normalized energy of εn = 1.03, a partial PBG occurs at KxL/π = 1. However, at the normalized energy of εn = 1.06 a partial PBG occurs at KxL/π = 1 and KyL/π = 1. We also studied the effect of an increase in the plasma energy on the PBG; our calculations show that by increasing the value of plasma energy the PBG shifts towards the smaller energies. For example for 5% (λ = 1.05) and 10% (λ = 1.10) increases in plasma energy, the normalized full PBG lies between (1.10, 1.28) and (1.00, 1.27), respectively. Using the relation between normalized energy and real energy one can easily calculate the range of the energies of the PBG. For example for εpb = 15.1 eV the PBG for λ = 1.0 lies in the ultraviolet region of the electromagnetic spectrum between (12.28 eV, 13.89 eV). Fig. 4 shows the numerical result of the band structure for TE waves with the electric field vector lying perpendicular to the plane of propagation (Kx, Ky plane) along the paths connected by the symmetry wave vector points of Γ, X and M. These points are located at (Kx = 0, Ky = 0), (Kx = π/L, Ky = 0) and (Kx = π/L, Ky = π/L), respectively. To calculate the band structure of rectangular MPC we have developed MAPLE code for solving the system of real nonlinear equations given in Eqs. (4a) and (4b) using the function FSOLVE. To plot the band structure along the Γ − X direction, we set Ky = 0 in Eq. (4b) and divide the normalized energy into 600 intervals ([εni, εni + 1], i = 0, 1, 2...600). Substituting a given value of normalized energy (εn) we are able to determine the values of η, and using these values with their correspondent normalized energy in Eq. (4a), we obtain the corresponding value of Kx between (0, π/L). In this direction we have the largest PBG, which lies between (1.03, 1.3). In the X–M direction we put Kx = π/L in Eq. (4a) and perform the same calculation to find all the possible values of Ky between (0, π/L). In this direction the PBG lies between (1.15, 1.3). For the M − Γ direction we divide the (K = 0, K = π/L) interval into 40 intervals ([Ki, Ki + 1], i = 0, 1, 2...40) and use the same Ki instead of Kx and Ky in Eqs. (4a) and (4b). We simultaneously solve these equations for η and normalized energy. In this direction the first PBG occurs at the normalized energy between (1.15, 1.4). Our
Fig. 4. Band structure of the 2-D MPC with parameters n21 = 0.5, n22 = 0.5 − 1/ε2n, a = 0.5L and b = 0.5L. The vertical axis is the normalized energy (εn) and the horizontal axis is normalized wave vector (K/L). Special points Γ, M, and X correspond to K = 0, K = (π/L)i, K = (π/L)i + (π/L)j and K = (π/L)j respectively. The second complete PBG occupies the normalized energy region between (1.15, 1.30).
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Fig. 5. Plot ! normalized density of states D(εn) of the MPC versus the normalized pffiffiffiof the 2Lεk energy with parameters n21 = 0.5, n22 = 0.5 − 1/ε2n, a = 0.5L and b = 0.5L. 2πℏc
results are approximately consistent with the results found by Ustyantsev et al. [28], especially in first and second bands, where they used the FDTD method combined with auxiliary differential equations. Their photonic crystal was composed of circular metallic rods arranged in a square lattice and embedded in an air background. The existence of PBGs in each direction is confirmed in the DOS calculation. We show that the DOS vanishes for energies lying within the PBG. To calculate the spectral density of the radiation due to spontaneous emission, one needs to find the total DOS by adding up all the values of DOS in each direction for a given energy. To calculate this parameter we used Eqs. (10a) to (10c) and the developed MAPLE code which uses the FSOLVE function. We divide the (K = 0, K = π/L) interval into 40 intervals ([Ki, Ki + 1], i = 0, 1, 2...40). Substituting each Ki instead of Kx and Ky in Eqs. (5a) and (5b) for each direction gives all the possible solutions for η and normalized energy. We substituted these solutions in Eqs. (10a) to (10c) and calculate the total DOS value. The results of the calculation for λ = 1.0 are shown in Fig. 5, where we show that the DOS vanishes in the region of the PBG. Note that at the band edges the value of DOS approaches infinity. To show the detailed features of the of DOS behavior, we omit values greater than 150. The normalized spectral function I(ε)/I(0) has also been calculated as a function of energy, where the results are shown in Fig. 6. The peak of the spectrum is located at zero detuning. We know that the width of
Fig. 7. Plot of the normalized spectral function as a function of the photon energy detuning for a fixed resonance energy (εba = 10.8 eV) and different normalized plasma energies (λ = 1.0, 1.05 and 1.10).
the peak at half maximum gives the linewidth. It can be proved as follows: The height of the maximum is I(0) = 2I0/Γ. Then the half maximum has a value of I0/Γ. Putting this into Eq. (24) we get I0 Γ= 2 = I0 : Γ ðεk −εba Þ2 + ½Γ= 22
ð27Þ
Solving this equation we get ε = εba Γ = 2:
ð28Þ
This proves that the energy width (ε+ − ε−) of the half maxima is equal to Γ. To calculate the linewidth for different excitation energies (εba) we use Eq. (22), where the height of the 2-D MPC and the plasma energy is 200 nm and εp(Al) = 15.1 eV, respectively. We choose three excitation energies which are far away from (εba = 10.7 eV), close to (εba = 11.75 eV) and very close to (εba = 12.1 eV) the lower-energy PBG. The values of the linewidth for these three energies are 3.1γ0, 2.8γ0 and 12.1γ0, respectively. The results from our simulation of the normalized spectral function are shown in Fig. 7. Note that the dotted curve has the greatest width because of the maximum linewidth near the edge of the PBG. We also studied the normalized spectral function's variation due to modification of the plasma energy. Fig. 7 shows the behavior of the spectral function for a fixed energy, εba = 10.8 eV, where the solid, dashed and dotted curves correspond to λ = 1.00, 1.05 and 1.10, respectively. The values of the linewidth energies are 5.70γ0 (λ = 1.0), 3.99γ0 (λ = 1.05) and 7.09γ0 (λ = 1.10). Note that when the plasma energy is changed, the linewidth is also changed. This occurs due to the linewidth dependency on the DOS, as pointed out by Eq. (22). 5. Conclusions
Fig. 6. Plot of the normalized spectral function as a function of photon energy detuning for different resonance energies (εba) and a fixed normalized plasma energy (λ = 1.0).
We have investigated the decay rate of a two-level quantum dot doped in the 2-D metallic photonic crystal, and have calculated the linewidth and spectral function due to spontaneous emission using our analytical expression for the photonic density of states. The linewidth and spectral function of the quantum dot due to spontaneous emission were calculated by using the Schrödinger equation method. Finally, we showed that by changing the plasma energy one can control the spontaneous emission in the metallic photonic crystal.
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Acknowledgements The authors are grateful to NSERC of Canada for financial support in the form of a research grant and Mr. J. D. Cox for helpful discussions and for proofreading the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Decay of a quantum dot in two-dimensional metallic photonic crystals A. Hatef, M. Singh ⁎ Department of Physics and Astronomy, The University of Western, Ontario, London, Canada N6A 3K7
a r t i c l e
i n f o
Article history: Received 3 August 2010 Received in revised form 17 October 2010 Accepted 3 December 2010 Available online 18 December 2010 Keywords: Metallic photonic crystal Density of states Plasma energy Quantum dots Photonic band gap Decay rate
a b s t r a c t In this paper we have developed a theory for the decay of a quantum dot doped in a two-dimensional metallic photonic crystal consisting of two different metallic pillars in an air background medium. This crystal structure forms a full two-dimensional photonic band gap when the appropriate pillar sizes are chosen. The advantage of using two metals is that one can easily control the density of states and optical properties of these photonic crystals by changing the plasma energies of two metals rather than one. Using the Schrödinger equation method and the photonic density of states, we calculated the linewidth broadening and the spectral function of radiation due to spontaneous emission for two-level quantum dots doped in the system. Our results show that by changing the plasma energies one can control spontaneous emission of quantum dots doped in the metallic photonic crystal. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In this paper we have studied the quantum optics and band structure of metallic photonic crystals. Photonic crystals are structures characterized by their periodically modulating dielectric constant, which may vary in one, two or three spatial dimensions. Due to multiple reflections at the interfaces between regions with different dielectric constants in the structure, electromagnetic (EM) waves of a certain frequency range cannot propagate through a photonic crystal; this range of frequencies is referred to as the photonic band gap (PBG). Generally, the wavelengths of light which fall within the PBG are on the order of the crystal's lattice constant, while the width of the PBG is proportional to the crystal's dielectric contrast. The PBG of a onedimensional photonic crystal lies in only one direction, while for a two- or three-dimensional photonic crystal it varies for different photon propagation directions. If a certain range of photon energies is prohibited from travelling in a two- or three-dimensional photonic crystal for all Brillouin zone directions then this energy range is referred to as the complete PBG. Essentially, the PBG controls the propagation of EM waves in photonic crystals in the same manner that the electronic band gap controls electrons in semiconductors. It is well-known that in a dielectric photonic crystal, high dielectric contrast is required to have a complete PBG [1]. For example, inverse opal photonic crystals made from ordinary dielectric materials require that the contrast should be greater than eight to have a complete PBG in the optical regime [2]. One way to overcome this barrier is to
⁎ Corresponding author. E-mail address: [email protected] (M. Singh). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.12.012
include metallic components in the structure of photonic crystals. This greatly enhances the dielectric contrast, which leads to the formation of larger PBGs in each Brillouin zone direction and thus increases the likelihood of their overlap for all directions [3–7]. These structures are called metallic photonic crystals (MPCs). MPCs can also provide strong simultaneous coupling between electronic and photonic resonances in the same range of the frequencies, which gives them interesting optical properties [8], and can lead to well-pronounced spectral features [8,9]. One of the promising features of MPCs is their capability of tailoring thermal emission spectra. Theoretical analysis has shown that the existence of the PBG and modification of the photon density of states (DOS) in these structures can lead to significant enhancement and suppression of thermal radiation near and within the PBG, respectively [10–15]. By simultaneously controlling photonic and electronic resonances, MPCs can be used to manipulate light–matter interaction for developing integrated photonic nanodevices. MPCs have been used in the field of telecommunications as antennas [16], all-optical switches [17], biosensors [18], solar cells [19] and other optoelectronic devices [20]. There is a great deal of interest in two-dimensional (2-D) photonic crystals due to their relative ease of fabrication as well as the underlying idea that two-dimensional (2-D) photonic crystals can be used to develop a direct continuation of planar integrated optics [21]. Recently, 2-D MPCs have been fabricated by using various methods [22–24]. For example, Katsarakis et al. [22] reported the fabrication of 2-D MPCs consisting of square and triangular lattices of nickel pillars using deep X-ray lithography. They showed that the photonic band structures of these crystals exhibited high-pass filter characteristics with a cutoff in the far-infrared energy regime. Puscasu et al. [23] used a 2-D metallo-dielectric photonic crystal to investigate narrow-band
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infrared sources and spectroscopic sensors. Zhang et al. [24] fabricated 2-D MPC waveguides using solution-processible gold nanoparticles. For device applications it is critical to accurately calculate higherorder PBGs. Typically, the band structures of 2-D MPCs are determined with numerical approaches such as the plane wave method (PWM) [25], transfer matrix method (TMM) [26], finite-difference timedomain method (FDTD) [27,28] and multiple multipole method (MMP) [29]. For example, Ustyantsev et al. [28] investigated the effect of different dielectric backgrounds on the band structure of MPCs composed of circular metallic pillars in a square lattice. They used the PWM and FDTD methods to determine the photonic dispersion bands and the TMM to determine the reflectivity characteristics. Takayma and Cada [30] reported theoretical results of PBGs based on the TMM for a MPC in the form of a hexagonal lattice consisting of silver pillars in anodic porous alumina. However, it is very complicated to use numerical band structure methods in studying the absorption and emission of light in these structures. On the other hand, analytical expressions are very useful for experimentalists so that they can analyze their data easily. There are also many useful applications for analytical expressions of photonic dispersion relations. For example, these expressions allow for an accurate determination of band gaps in all crystal directions. Determination of very narrow PBGs is important for the design of oversized single-mode photonic cavities or waveguides. They are also very useful for rigorous error analysis and help establish error tolerance in numerical simulation techniques. Finding an analytical formulation of band structures in photonic crystals represents a major mathematical challenge because analytical expressions for the band structure are generally limited to one-dimensional structures [31]. Recently, some effort has been devoted towards developing analytical models [32–36] for 2-D photonic crystals in order to understand and predict the physical properties of these photonic crystals without the need for heavy and time consuming calculations. Some efforts have also been placed towards obtaining analytical expressions for photonic dispersion relations in 2-D MPCs. For example, Pokrovsky et al. [37] presented an analytical theory for low-energy EM waves in MPCs with a small volume fraction of metal. Wang et al. [38] developed an analytical expression for the effective velocity of EM waves at low energies for 2-D dielectric photonic crystals and MPCs. In this paper, we have considered 2-D MPCs made from rectangular metallic pillars arranged periodically in a 2-D plane, where air is taken as the background medium. The advantage of choosing two metals lies in the fact that one can easily control the size and location of the crystal's PBG by manipulating the plasma frequencies of two metals rather than one. This structure has another advantage in that one can obtain analytical expressions for the band structure, DOS, linewidth and absorption coefficients. These expressions can be very useful for experimental studies and device design involving metallic photonic crystals. We have investigated the effect of modifying the plasma energy of the metals in the MPC on the spontaneous emission in these structures. To study spontaneous emission, we consider the situation where a twolevel quantum dot is doped in our MPC. The quantum dot is interacting with the photonic states of the crystal via the electron–photon interaction. Analytical expressions of the linewidth and spectral function of the quantum dot due to spontaneous emission have been calculated by using the Schrödinger equation method. Numerical simulations have been performed on the band structure and DOS in all 2-D crystal directions. We consider a MPC consisting of aluminum (Al) with zinc (Zn) in an air background. This structure gives a complete 2-D PBG. The DOS of photons has also been calculated in all crystal directions. It is found that DOS has singularities near the band edges. This agrees with the findings of other researchers in the literature [28]. Finally we have calculated the effect of the plasma energy on the spectral function for spontaneous emission, which contains information about the linewidths of the energy levels of the quantum dot. It is found
that manipulating the normalized plasma energy of the metal changes the width of the absorption peak. This means that by changing the plasma energy one can control spontaneous emission in these structures. 2. Two dimensional metallic photonic crystals As is schematically depicted in Fig. 1, we consider a 2-D metallic photonic crystal (MPC) made from two rectangular metallic pillars B and C with dielectric constants b and c, infinite in the z-direction. In each unit cell the opening domain (A) has a square shape, and is empty (i.e. an air space) with a dielectric constant a. The structure is homogeneous in the z-direction and periodic in the x and y directions with period L; therefore the unit cell for this crystal is a square. The metallic pillars used in the MPC are conductors in which electrons are free to move. When an EM field is propagating in a metal, the conductivity of the metal is energy dependant. According to the Lorentz model, the refractive index of a metal is obtained as [39]
ns ðεk Þ =
1−
!1 = 2 ε2p ; εk ðεk + iγÞ
ð1Þ
where εp is called the plasma energy and is obtained as 2
εp =
nℏ2 e2 : m0
ð2Þ
The plasma energy is the natural energy of oscillation for the electron gas in the metal pillars. In the above equations, n is the electron concentration, e is the electron charge, m is the mass of an electron and γ is the relaxation energy of the electrons in a pillar. Kady et al. [6] utilized the energy dependency of the real and imaginary parts of the dielectric constant for noble metals (i.e. Al, Ag, Au, and Cu). They showed that the imaginary part of dielectric constant (i.e. loss) is negligible in the range of optical and ultraviolet frequencies. Here the value of the relaxation energy is on the order of few meV and can be ignored without any significant variation in our calculations [40]. 3. Spontaneous emission decay rate The 2-D MPC is doped with a two-level quantum dot. Energy levels of the quantum dot are denoted as |a〉 and |b〉, where the former is the ground state (see Fig. 2). We consider that the quantum dot decays
Fig. 1. Schematic of the dielectric function in the 2-D separable rectangular MPC. The large white square region has a dielectric constant a = 2n21, while the small dark square and rectangular regions have b = 2n22 and c = n21 + n22, respectively. The parameters a and b give the thicknesses of the layers and L = a + b is the lattice constant in both the x and y directions.
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where η is the separation constant, n21 and n22 are related to a, b and c as a = 2n21, b = 2n22 and c = n21 + n22. Generally, the band structure of a square lattice is plotted in symmetric directions in the Brillouin zone. The centre of the Brillouin zone is located at (Kx = 0, Ky = 0) and is denoted as Γ. The other symmetric points are called X and M and are at (Kx = π/L, Ky = 0) and (Kx = 0, Ky = π/L), respectively. The band structure is then plotted along three major directions Γ − X, Γ − M and X-M. In Eq. (3), z(εk) is called the coupling constant and is obtained as Fig. 2. A schematic diagram of a two-level quantum dot where the levels are denoted by |a 〉 and |b 〉. When the quantum dot is in the excited state it decays to the ground state spontaneously.
spontaneously from the excited state to the ground state via the electron–photon (EP) interaction. The Hamiltonian of the system is written as: 2
3 1 dεk þ z 6 εba σba + 2 + ∫C 2π εk p ðεk Þpðεk Þ 7 6 7 H=6 ⋅ h i 7 4 5 dεk þ − þ −∫C zðεk Þ pðεk Þσba + σba p ðεk Þ 2π
ð3Þ
− z Here σ+ ba = |b〉〈a|, σba = |a〉〈b| and σba = |b〉〈b| − |a〉〈a|. The p(εk) and p (εk) operators denote the annihilation and creation of photons, respectively. The first and second terms correspond to the Hamiltonians of the quantum dot and photons in the photonic crystal, respectively. The third term describes the coupling between a quantum dot and photons. The integration contour C consists of two intervals: − ∞ b εk ≤ εv and εc ≤εk b ∞. Here εv and εv are the lower and the upper band edges of PBG, respectively, such that the PBG of the MPC lies between εv and εc. Note that there is no integration between energies εv and εc. In Eq. (3) εk is photon energy which is obtained from the band structure of 2-D MPC. The band structure is calculated by using the separable model [32–36]. We consider transverse electric (TE) plane waves propagating in the x–y plane (i.e. kz = 0) on 2-D crystal. The length and width of pillar B are taken the same, and are denoted as b. Similarly the length and width of pillar C is taken as a and b, respectively. The periodicity of the crystal along x and y directions is given as L = a + b. The dispersion relation is obtained as +
cosðKx LÞ = Fx ðεk ; ηÞ
ð4aÞ
cos Ky L = Fy ðεk ; ηÞ;
ð4bÞ
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! ffi ! ε n 2 ε n 2 k 1 k 2 −η2 a cos −η2 b ℏc ℏc ε n 2 ε n 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! k 1 ε n 2 + k 2 −2η k 1 ℏc ℏcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin 2 r ffi − rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a −η ε n 2 ε n 2 ℏc k 1 k 2 2 −η2 −η2 ℏc ℏc rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! ε n 2 k 2 × sin −η2 b ð5aÞ ℏc
Fx ðεk ; ηÞ = cos
! ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εk n 1 2 εk n2 2 Fy ðεk ; ηÞ = cos + η2 a cos + η2 b ℏc ℏc ε n 2 ε n 2 ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 k 1 k 2 + + 2η εk n1 2 ℏc ℏc 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos a + η − r ε n 2 ℏc εk n1 2 k 2 2 + η2 + η2 ℏc ℏc ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε n 2 k 2 × cos + η2 b ð5bÞ ℏc
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 3 pffiffiffiffiffiffi 3π ℏ c Dðεk Þ zðεk Þ = γ0 dx dy dz ε2k
ð6Þ
μ 2 ε3ba γ0 = 3π0 ℏ4 c3
where γ0 is the linewidth for an energy level of the quantum dot when it is located in a vacuum. The coupling constant controls the coupling between photons and electrons and depends on the photon density of states function D(εk) (DOS) in the 2-D MPC. This function is calculated from the dispersion relation (Eqs. (4a) and (4b)) as follows. We consider that dx, dy and dz are the dimensions of 2-D MPC along x, y and z direction, respectively. By using the concept of the DOS the summation over and Kx and Ky can be replaced by integration over photon energy εk as ∑ ∑ = ∫Dðεk Þdεk ; Kx
ð7Þ
Ky
where D(εk) is obtained as Dðεk Þ =
dx dy dk⊥ k ; π ⊥ dεk
ð8Þ
where k⊥ =
1 L
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi 2
½arccos Fx ðεk ; ηÞ2 + arccos Fy ðεk ; ηÞ ;
ð9Þ
with the help of the above equation the DOS is then evaluated as 2 3 dx dy 6 Gx ðεÞξx ðεk Þ Gy ðεÞξy ðεk Þ 7 Dðεk Þ = 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5: πL 1−Fy2 ðεk Þ 1−Fx2 ðεk Þ ξ ðεk Þ = ξ 1 ðεk Þ + ξ2 ðεk Þ + ξ3 ðεk Þ⋅ n2d n2 εk a ξ ð ε Þ = −cos ð k a Þsin ð k a Þ 1 k 1 2 ðℏcÞ2 k2 ! an21 εk −cosðk2 aÞsinðk1 aÞ k1 ðℏcÞ2 n2d n2 εk a ð ε Þ = −cos ð k a Þsin ð k a Þ ξ 1 k 1 2 ðℏcÞ2 k2 ! 2 an1 εk −cosðk2 aÞsinðk1 aÞ k1 ðℏcÞ2
ξ 2 ðεk Þ = sinðk1 aÞsinðk2 bÞ ×
½
2 n2d n2 εk k1 n1 εk − 2 3 2 ðℏcÞ k2 ðℏcÞ k2 k1
n21 εk k2 n2d n2 εk − ðℏcÞ2 k31 ðℏcÞ2 k1 k2 2n2d n2 εk a ξ3 ðεk Þ = sinðk1 aÞcosðk2 aÞ ðℏcÞ2 k2 +
+ sinðk2 aÞcosðk1 aÞ
! an21 εk ⋅ k1 ðℏcÞ2
ð10aÞ
ð10bÞ
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where + and − stand for y and x, respectively. Other terms are defined as
n2d
1 + ε2p ε−2 k = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1−ε2p = ε2k
k1− k2−
Δba
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ε n 2 ε n 2 k 1 k 1 = −η2 ; k1 + = + η2 ℏc ℏc rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ε n 2 ε n 2 k 2 k 2 = −η2 ; k2 + = + η2 ⋅ ℏc ℏc
ð10cÞ
ðH−εk Þjψk 〉 = 0
ð11Þ
The eigenstate of system can be written as a single particle eigenket in the form þ
dεk þ f ðε Þp ðεk Þj0〉; 2π k k
ð12Þ
where the vacuum state of the system is denoted as |0 〉. Putting Eq. (12) into Eq. (11) the Schrödinger equation takes the form pffiffiffiffiffiffiffiffiffiffiffiffi ðε′ −εk Þfk ðεk Þ = C Dðεk Þgk dεk pffiffiffiffiffiffiffiffiffiffiffiffi ðεba −εk Þgk = C∫ Dðεk Þfk ðεk Þ; C 2π
ð13Þ
C=
!1 = 2 ð14Þ
When the eigenvalue εk lies outside the PBG the solution of the first equation can be written as pffiffiffiffiffiffiffiffiffiffiffiffi
C Dðεk Þ fk ðεk Þ = 2πδ ε′ −εk + limþ gk : s→0 ε′ −εk −is
ð15Þ
Putting fk(εk) into Eq. (13-b) we obtain ðεba −εk Þgk = C
pffiffiffiffiffiffiffiffiffiffiffiffi dεk C 2 Dðεk Þ Dðεk Þ + limþ ∫ g : C 2π ε′ −ε −is k s→0 k
ð16Þ
! C 2 Dðεk Þ ; ε′ −εk
ð20Þ
and P stands for the principal part. The expression for the linewidth is calculated as ! 3π2 ℏ3 c3 Dðεab Þ: dx dy dz ε2ab
ð21Þ
Note that the linewidth has a very large value when the resonance energy εba lies near the photonic band edges but outside the PBG. Putting the expression of DOS into the above expression, we find 2 3 ! Gy ðεk Þξy ðεk Þ 7 3πℏ3 c3 6 Gx ðεk Þξx ðεk Þ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5: dz ε2ab Ly 1−Fy2 ðεk Þ Lx 1−Fx2 ðεk Þ
Γba = γ0
ð22Þ
The probability of finding the quantum dot in the excited state is then found as 2
Iðεk Þ = jgk j :
ð27Þ
This quantity is called the spectral density of the radiation due to spontaneous emission and is found as Iðεk Þ =
where C is a constant, given as 3π2 ℏ3 c3 γ0 dx dy dz ε2ba
dεk = P∫ C 2π
Γba = γ0
To find the decay rate of the quantum dot we use the Schrödinger equation method. The Schrödinger equation is written as
jψk 〉 = gk σ j0〉 + ∫C
where
εk −εba
C 2 Dðεab Þ :
2 + Δ2ba + C 4 D2 ðεab Þ
ð24Þ
Note that in empty space, I(εk) reduces into the standard Wigner– Weisskopf expression for a natural radiation line width. 4. Results and discussions For our numerical simulations of the 2-D MPC we choose a dielectric-metal composite. The structure is homogeneous in the zdirection and periodic in the x and y directions. We consider that the crystal is made from square and rectangular metallic pillars embedded in a background dielectric material. The rectangular pillars are connected to the sides of square pillars. In our simulation the background dielectric material is air. The dielectric functions in Eqs. (5a) and (5b) can be written as follows: 2
a = 2n1 = 1 Let us define a quantity called self-energy Ξba(εk) as 2 dεk C Dðεk Þ ⋅ Ξba ðεk Þ = limþ ∫ C 2π ε′ −ε −is s→0 k
b = 2n22 = 1− ð17Þ
Putting this into Eq. (13-b) we get the expression for gk as pffiffiffiffiffiffiffiffiffiffiffiffi C Dðεk Þ ⋅ gk ðεk Þ = εba −εk −Ξba ðεk Þ
2
ð25Þ
ε2 ε2pc 1− 2 ε
;
C, where the εbp and εcp are the plasma energy of pillars B and ! respectively. Substituting the values of n21 ð18Þ
Note that Ξba is a complex quantity. The real part of the self-energy gives the energy shift and the imaginary part gives the decay rate or linewidth: Ξba = Δba + iΓba ;
2
c = n1 + n2 =
ε2pb
ð19Þ
=
1 1 and n22 = 2 2
1−
ε2pb ε2
into
Eq. (25) we get the following relation between the plasma frequencies of the metallic pillars. εpb εpc = pffiffiffi 2
ð26Þ
This shows that our calculation is valid for any metallic pillars which satisfy this relation. Several options are available for different metals whose plasma energies satisfy Eq. (26). For instance, options for pillars B and C include aluminum (Al) and zinc (Zn), with plasma
A. Hatef, M. Singh / Optics Communications 284 (2011) 2363–2369
energies εp(Al) = 15.1 eV [41] and εp(Zn) = 10.1 eV [42] or copper (Cu) and platinum (Pt) with plasma energies of εp(Cu) = 7.4 eV and εp(Pt) = 5.1 eV [43]. In our simulations, εpb is chosen to satisfy the relation εpb L = λ in order to study the effect of changingpplasma energy, and ffiffiffi 2πℏc pffiffiffi ε 1 2 Lεk we take n21 = 0.5 and n22 = 0:5− 2 , where εn = = 2 k is 2πℏc εpb λ εn defined as the normalized energy. Since this separable model has not been used for MPCs, it is important to see whether this model produces a band gap in these crystals. To visualize and analyze the direction and propagation behavior of incident EM waves inside the MPC, we employ constantenergy contour diagrams in wave vector space (K) whose gradient vectors give the group velocities of the photonic modes. To plot constant-energy contours of the crystal we eliminate η in Eqs. (4a) and (4b) using MAPLE code for plotting a parametric system of real nonlinear equations. Fig. 3(a) and (b) illustrate several constantenergy contours, which are the projections of the photonic bands in the (Kx, Ky) plane, for various normalized energy in the reduced Brillouin zone (− 1 b KxL/π b 1, − 1 b KyL/π b 1). For the given parameters (a = 0.5L and b = 0.5L) and λ = 1, our calculations show there is a low-energy complete PBG from zero up to a normalized cut-off energy (εn = 0.9), and below that the normal surface is an exact
Fig. 3. Normal surface in the (Kx, Ky) plane of the 2-D MPC at different normalized energies. The normal surfaces plotted in the reduced Brillouin zone exhibit a fourfold symmetry. The parameters are n21 = 0.5, n22 = 0.5 − 1/ε2n, a = 0.5L and b = 0.5L.
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square which means no EM modes can propagate through the crystal, and hence the MPC behaves like a perfect reflector. However, a highenergy complete PBG appears in the normalized energy region between (1.15, 1.30) where the normal surfaces on this range of energy are all exactly square (see Fig. 3). Above the normalized cut-off energy between (0.93, 1.01) the contours are in the form of circles, meaning that the MPC behaves like a isotropic homogenous medium. By increasing the normalized energy the contours are no longer circular, showing the effect of the periodic structure of MPC. For example, at the normalized energy of εn = 1.03, a partial PBG occurs at KxL/π = 1. However, at the normalized energy of εn = 1.06 a partial PBG occurs at KxL/π = 1 and KyL/π = 1. We also studied the effect of an increase in the plasma energy on the PBG; our calculations show that by increasing the value of plasma energy the PBG shifts towards the smaller energies. For example for 5% (λ = 1.05) and 10% (λ = 1.10) increases in plasma energy, the normalized full PBG lies between (1.10, 1.28) and (1.00, 1.27), respectively. Using the relation between normalized energy and real energy one can easily calculate the range of the energies of the PBG. For example for εpb = 15.1 eV the PBG for λ = 1.0 lies in the ultraviolet region of the electromagnetic spectrum between (12.28 eV, 13.89 eV). Fig. 4 shows the numerical result of the band structure for TE waves with the electric field vector lying perpendicular to the plane of propagation (Kx, Ky plane) along the paths connected by the symmetry wave vector points of Γ, X and M. These points are located at (Kx = 0, Ky = 0), (Kx = π/L, Ky = 0) and (Kx = π/L, Ky = π/L), respectively. To calculate the band structure of rectangular MPC we have developed MAPLE code for solving the system of real nonlinear equations given in Eqs. (4a) and (4b) using the function FSOLVE. To plot the band structure along the Γ − X direction, we set Ky = 0 in Eq. (4b) and divide the normalized energy into 600 intervals ([εni, εni + 1], i = 0, 1, 2...600). Substituting a given value of normalized energy (εn) we are able to determine the values of η, and using these values with their correspondent normalized energy in Eq. (4a), we obtain the corresponding value of Kx between (0, π/L). In this direction we have the largest PBG, which lies between (1.03, 1.3). In the X–M direction we put Kx = π/L in Eq. (4a) and perform the same calculation to find all the possible values of Ky between (0, π/L). In this direction the PBG lies between (1.15, 1.3). For the M − Γ direction we divide the (K = 0, K = π/L) interval into 40 intervals ([Ki, Ki + 1], i = 0, 1, 2...40) and use the same Ki instead of Kx and Ky in Eqs. (4a) and (4b). We simultaneously solve these equations for η and normalized energy. In this direction the first PBG occurs at the normalized energy between (1.15, 1.4). Our
Fig. 4. Band structure of the 2-D MPC with parameters n21 = 0.5, n22 = 0.5 − 1/ε2n, a = 0.5L and b = 0.5L. The vertical axis is the normalized energy (εn) and the horizontal axis is normalized wave vector (K/L). Special points Γ, M, and X correspond to K = 0, K = (π/L)i, K = (π/L)i + (π/L)j and K = (π/L)j respectively. The second complete PBG occupies the normalized energy region between (1.15, 1.30).
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A. Hatef, M. Singh / Optics Communications 284 (2011) 2363–2369
Fig. 5. Plot ! normalized density of states D(εn) of the MPC versus the normalized pffiffiffiof the 2Lεk energy with parameters n21 = 0.5, n22 = 0.5 − 1/ε2n, a = 0.5L and b = 0.5L. 2πℏc
results are approximately consistent with the results found by Ustyantsev et al. [28], especially in first and second bands, where they used the FDTD method combined with auxiliary differential equations. Their photonic crystal was composed of circular metallic rods arranged in a square lattice and embedded in an air background. The existence of PBGs in each direction is confirmed in the DOS calculation. We show that the DOS vanishes for energies lying within the PBG. To calculate the spectral density of the radiation due to spontaneous emission, one needs to find the total DOS by adding up all the values of DOS in each direction for a given energy. To calculate this parameter we used Eqs. (10a) to (10c) and the developed MAPLE code which uses the FSOLVE function. We divide the (K = 0, K = π/L) interval into 40 intervals ([Ki, Ki + 1], i = 0, 1, 2...40). Substituting each Ki instead of Kx and Ky in Eqs. (5a) and (5b) for each direction gives all the possible solutions for η and normalized energy. We substituted these solutions in Eqs. (10a) to (10c) and calculate the total DOS value. The results of the calculation for λ = 1.0 are shown in Fig. 5, where we show that the DOS vanishes in the region of the PBG. Note that at the band edges the value of DOS approaches infinity. To show the detailed features of the of DOS behavior, we omit values greater than 150. The normalized spectral function I(ε)/I(0) has also been calculated as a function of energy, where the results are shown in Fig. 6. The peak of the spectrum is located at zero detuning. We know that the width of
Fig. 7. Plot of the normalized spectral function as a function of the photon energy detuning for a fixed resonance energy (εba = 10.8 eV) and different normalized plasma energies (λ = 1.0, 1.05 and 1.10).
the peak at half maximum gives the linewidth. It can be proved as follows: The height of the maximum is I(0) = 2I0/Γ. Then the half maximum has a value of I0/Γ. Putting this into Eq. (24) we get I0 Γ= 2 = I0 : Γ ðεk −εba Þ2 + ½Γ= 22
ð27Þ
Solving this equation we get ε = εba Γ = 2:
ð28Þ
This proves that the energy width (ε+ − ε−) of the half maxima is equal to Γ. To calculate the linewidth for different excitation energies (εba) we use Eq. (22), where the height of the 2-D MPC and the plasma energy is 200 nm and εp(Al) = 15.1 eV, respectively. We choose three excitation energies which are far away from (εba = 10.7 eV), close to (εba = 11.75 eV) and very close to (εba = 12.1 eV) the lower-energy PBG. The values of the linewidth for these three energies are 3.1γ0, 2.8γ0 and 12.1γ0, respectively. The results from our simulation of the normalized spectral function are shown in Fig. 7. Note that the dotted curve has the greatest width because of the maximum linewidth near the edge of the PBG. We also studied the normalized spectral function's variation due to modification of the plasma energy. Fig. 7 shows the behavior of the spectral function for a fixed energy, εba = 10.8 eV, where the solid, dashed and dotted curves correspond to λ = 1.00, 1.05 and 1.10, respectively. The values of the linewidth energies are 5.70γ0 (λ = 1.0), 3.99γ0 (λ = 1.05) and 7.09γ0 (λ = 1.10). Note that when the plasma energy is changed, the linewidth is also changed. This occurs due to the linewidth dependency on the DOS, as pointed out by Eq. (22). 5. Conclusions
Fig. 6. Plot of the normalized spectral function as a function of photon energy detuning for different resonance energies (εba) and a fixed normalized plasma energy (λ = 1.0).
We have investigated the decay rate of a two-level quantum dot doped in the 2-D metallic photonic crystal, and have calculated the linewidth and spectral function due to spontaneous emission using our analytical expression for the photonic density of states. The linewidth and spectral function of the quantum dot due to spontaneous emission were calculated by using the Schrödinger equation method. Finally, we showed that by changing the plasma energy one can control the spontaneous emission in the metallic photonic crystal.
A. Hatef, M. Singh / Optics Communications 284 (2011) 2363–2369
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