Lossless propagation in metal-semiconductor-metal plasmonic waveguides using quantum dot active medium K. Sheikhi,1 N. Granpayeh,1,* V. Ahmadi,2 and S. Pahlavan2 1

Center of Excellence in Electromagnetics, Optical Communications Lab., Faculty of Electrical Engineering, K. N. Toosi University of Technology, Tehran 1631714191, Iran 2

Faculty of Electrical and Computer Engineering, Tarbiat Modares University, Tehran 1411713116, Iran *Corresponding author: [email protected] Received 11 September 2014; revised 18 February 2015; accepted 23 February 2015; posted 24 February 2015 (Doc. ID 222919); published 27 March 2015

In this paper, we analyze and simulate the lossless propagation of lightwaves in the active metalsemiconductor-metal plasmonic waveguides (MSMPWs) at the wavelength range of 1540–1560 nm using a quantum dot (QD) active medium. The Maxwell’s equations are solved in the waveguide, and the required gains for achieving lossless propagation are derived. On the other hand, the rate equations in quantum dot active regions are solved by using the Runge–Kutta method, and the achievable optical gain is derived. The analyses results show that the required optical gain for lossless propagation in MSMPWs is achievable using the QD active medium. Also, by adjusting the active medium parameters, the MSMPWs loss can be eliminated in a specific bandwidth, and the propagation length increases obviously. © 2015 Optical Society of America OCIS codes: (230.7370) Waveguides; (240.6680) Surface plasmons; (250.5403) Plasmonics; (250.5590) Quantum-well, -wire and -dot devices. http://dx.doi.org/10.1364/AO.54.002790

1. Introduction

Surface plasmon polariton (SPP)-based waveguides known as surface plasmonic waveguides (SPWs) are the subject of numerous researches in nanotechnologies [1]. These waveguides can confine the electromagnetic waves in the form of SPPs, along the surface of metals. Propagation of the electromagnetic energy in the subwavelength devices significantly overcomes the diffraction limit of light. Therefore, the SPW devices considerably increase the level of integration of optical devices and components [2]. Thin metal film surface plasmonic waveguides [3] were proposed in 1969. Various metallic geometries such as trench [4], gap [5], wedge [6], 1559-128X/15/102790-08$15.00/0 © 2015 Optical Society of America 2790

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slot [7], and mixed SPWs [8] have been proposed and analyzed. Among these structures, metalsemiconductor-metal (MSM) plasmonic waveguides supporting propagation of strongly localized plasmon modes with high group velocity are intensely considered as a way for the design of nanoscale optical devices due to their easy fabrication compared with the other types of plasmonic waveguides. However, the vicinity of the plasmonic modes and the metal layers of MSMPWs causes high absorption losses and consequently limits the waveguide propagation length. One proposal for eliminating the loss of plasmonic waveguides is the use of active medium for compensation of the loss in the waveguides [9–14]. In this paper, we numerically analyze, for the first time, the lossless propagation of the optical wave in a MSM plasmonic waveguide including InAs/InP QDs in the wavelength range of 1540–1560 nm. The

Maxwell’s equations are solved to attain the required gain values for achieving lossless propagation in the waveguide. The rate equations of the quantum dot active region are solved to obtain the achievable gain. Our results indicate that the propagation loss of the MSM plasmonic waveguides can be compensated by using QD active gain medium. The paper is organized as follows. In Section 2, an active MSM plasmonic waveguide is analyzed. The quantum dot active region is simulated in Section 3. In Section 4, the simulation results are presented and discussed. The paper is concluded in Section 5. 2. Active MSM Plasmonic Waveguides

Figure 1(a) shows the cross section of an active MSM plasmonic waveguide. Silver is used as the metal layers of the waveguide. The complex dielectric constant of the silver is characterized by the Drude model with
ε∞ ;ωp ;γ 
4.2;13460 THz;96.17 THz [15]. The gain medium is composed of quantum dots with dielectric constant of 12.25 located in an InP wetting layer with dielectric constant of 9.61 at the wavelength of 1550 nm. By using Comsol Multiphysics mode solver, MSM plasmonic waveguides are analyzed numerically to attain their propagation losses and the required gain values for achieving lossless propagation. In the simulations, transparent and continuity boundary conditions are used, respectively, for the outer and inner boundaries. Figure 1(b) demonstrates the optical intensity distribution for the fundamental guided mode of the MSM plasmonic waveguide. The operating wavelength is 1550 nm. However, the close vicinity of the metal and the dielectric in the waveguide results in high absorption losses. The propagation characteristic of the waveguide is obtained by computing the complex propagation constant, β, of the fundamental guided mode of the waveguide. The propagation loss of the waveguide is derived from the Im
β. Using the mode solver, the calculated propagation constant of the waveguide of Fig. 1(a) is 11.8781 j0.0612 μm−1 , corresponding to the propagation loss

Fig. 1. (a) Cross section of an active MSM plasmonic waveguide, and (b) optical intensity distribution for the fundamental guided mode of the waveguide for an active MSM plasmonic waveguide with W  200 nm and h1  h2  h  100 nm.

Fig. 2. Spectra of the effective index and propagation length of the MSM plasmonic waveguide of Fig. 1(a).

of 0.5312 dB∕μm. Figure 2 demonstrates the spectra of the effective index, neff  Re
β∕k0 , and the propagation length, Lprop  1∕Im
β, of the waveguide of Fig. 1(a), where k0  2π∕λ, and λ is the free space operating wavelength. For achieving lossless propagation, the imaginary part of the propagation constant must be vanished by providing gain for the wave propagation medium. The dielectric constant of the gain medium is assumed to be ε  εr − jεi , where εr and εi have positive values. The imaginary part, εi , corresponds to the optical gain with a gain coefficient of γ  k0 εi ∕εr1∕2 [16]. Therefore, εi  0 expresses the zero gain value. The zero value for the imaginary part of the propagation constant of the waveguide which results in lossless propagation is achieved by adding incremental value to εi iteratively [17]. According to this method, the required gain for lossless propagation in the MSMPW of Fig. 1(a) is 0.5189 μm−1. Figure 3 illustrates the required gain values for attaining lossless propagation in the MSM plasmonic waveguide of Fig. 1(a) versus gain medium width, W, for different values of height, h. These amounts of required gain are compatible with the current semiconductor technologies and can be achieved by using the quantum dot active region. The loss and the required gain spectra for achieving lossless propagation in the MSM plasmonic waveguide of Fig. 1(a) for different values of W and h are depicted in Fig. 4. When the dimensions

Fig. 3. Required gain for achieving lossless propagation in the MSM plasmonic waveguide of Fig. 1(a) versus gain medium width (W) for different gain medium heights (h). 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS

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of the active region increase, the interaction between the guided mode and the active medium increases, and consequently the gain value decreases. Hence, as shown in Fig. 4, the waveguide of Fig. 1(a) with dimensions of W  200 nm and h  100 nm, requires maximum gain to be lossless. 3. Simulation of Quantum Dot Active Region

For analysis of quantum dot active region, the modeling approaches based on quantum dot-laser diode (QD-LD) and quantum dot-semiconductor optical amplifier (QD-SOA) are used [18,19]. This model contains the rate equations for the propagation equation of different modes of photons and different energy levels of quantum dots. We assume that the dots are spatially isolated, and the exchange of carrier between different dots is only possible through the wetting layer. Furthermore, the dots are grouped to include size differences, material distribution and strain appearing in the self-assembled dots. In the quantum-dot active medium, grouping of the dots plays the role of inhomogeneous broadening of the gain. For each dot group, the optical gain is in turn broadened homogeneously. InAs dots can be fabricated on InP substrate to produce gain in the operational band of the waveguide in the range of 1540–1560 nm [20,21]. The parameters and properties of quantum dots used in the simulations are listed in Table 1. A. Grouping the Dots

Fig. 4. Spectra of the loss (solid line) and the required gain for lossless propagation (dashed line) for the MSM plasmonic waveguide of Fig. 1(a) for: (a) W  200 nm, (b) W  210 nm, and (c) W  220 nm.

Table 1.

Figure 5 demonstrates the schematic view of energy band diagram of self-assembled quantum dots grown through the Stranski–Krastanov mode which is used in our simulation [23]. There are ground, excited, and upper continuum states in each dot. The wetting layer is a common energy level between all dots. By applying current, carriers are injected to the

Parameters and Properties of Quantum Dots Used in Simulations [18,19,22,23]

Parameter and property Current density Input power Quantum dot density Ground state degeneracy Excited state degeneracy Upper continuum state degeneracy Wetting layer degeneracy Temperature Energy gap between excited state and upper levels Energy gap between ground and excited states Relaxation time between wetting layer and upper level Relaxation times between levels Escape time from upper level to wetting layer Recombination rate in the upper continuum, excited, and ground states Recombination rate in the wetting layer Inhomogeneous broadening Homogeneous broadening Ground state resonance energy Ground state energy separation

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Notation

Value

J Pin ND Dg De Du Dw T ΔEue ΔEeg τw→u τu→e  τe→g  τu→g τu→w Ru  Re  Rg Rw ħΓinhom ħΓhom

35 KA∕cm2 7.15 × 1013 W∕cm2 9 × 1017 cm−3 1 6 20 8.5 × 1017 cm−3 295 K 70 meV 70 meV 1 ps 10 ps 10 ps 1 ns−1 2.5 ns−1 40 meV 20 meV

ℏω0cv;g ℏΔωcv;g

806 meV 1.5 meV

g
t; ωm ; cvlj 

2πe2 jPcv j2
N l;j ∕Dl N D Gj − 1 cnr ε0 m20 ℏωjcv;l × B
ωm − ωjcv;l Dl N D Gj ;

Fig. 5. Energy band diagram of self-assembled quantum dots. N w is the carrier density of the wetting layer, and N u;j , N e;j , and N g;j denote the carrier density of the upper continuum, excited, and ground state at the jth dot group, respectively [23].

wetting layer and transmit from the wetting layer to the upper continuum state. Exchange of carriers between the states is possible after corresponding time constants. Let the number of groups be 2M  1, according to the resonant frequency and the frequency separation of groups, Δωcv;g , the angular frequency of the ground state of the jth group will be [18]: ωjcv;g  ω0cv;g −
M − jΔωcv;g ;

(1)

where j varies from 0 to 2M, and ω0cv;g is the central angular frequency. The index c and v denote the conduction and valence bands, respectively. For inhomogeneous broadening, the statistical distribution is assumed to be a Gaussian function as Gj  G
ωjcv;g − ω0cv;g   
ωjcv;g − ω0cv;g 2 1 ;  p





exp − 2ξ2 2π ξ

(4)

where l is the ground, excited or upper continuum state, e is the electron charge, c is the speed of light in vacuum, nr is the refractive index of the active medium, ε0 is the permittivity of vacuum, m0 is the free electron mass, Pcv is the matrix element, N l;j is the density of carriers in state l at the jth dot group, Dl is the degeneracy of the corresponding state, N D is quantum dot density, and ℏωjcv;l is the resonance energy of state l in dot group j, where ℏ is reduced Planck constant. A Lorentzian function for homogeneous broadening is supposed as B
ωm − ωjcv;l  


ωm −

Γhom ∕2π j ωcv;l 2 −
Γhom ∕22

;

(5)

where Γhom is the FWHM of homogeneous broadening in state l. The rate equation of the wetting layer is written as [18]: h i dN w
z; t  −T w N w
z; t − N w − Rw N w
z; t; dt

(6)

where  Nw 

X N u; j
z; t J  T −1 w ; τu→w ed j

(7)

and ξ

Γinhom ; 2.35ℏ

Tw 

X N u;j
z; t j

   N u;j
z; t Gj τ−1  1− w→u ; (8) 2Dw τu→w 2Du N D Gj

(2) where Γinhom is the inhomogeneous broadening and ħ is the reduced Planck’s constant. The same method is applied for excited and upper continuum states
l  e; u as ωjcv;l  ω0cv;l −
M − jΔωcv;l :

(3)

For the upper continuum and excited states, the statistical distribution is again a Gaussian function with different full-width at half-maximum (FWHM). With assumption of a full correlation between the statistical distributions of the ground, excited, and upper continuum state resonance energies, the relations for dots grouping have been written. B.

where Rw is the recombination rate in the wetting layer, J is the current density, d is the thickness of active medium. Parameters τu→w , τw→u , Dw , and Du are defined in Table 1. The rate equation of the upper continuum state is written as dN u;j
z; t  −T ju N u;j
z; t − N u;j  − Ru N u;j
z; t; dt

(9)

where  −1 N u;j  N w
z; tGj τ−1 w→u  N g;j
z; tτg→u  j −1  N e;j
z; tτ−1 e→u
T u  ;

(10)

T ju  T wu  T jgu  T jeu ;

(11)

  N w
z; t −1 N w
z; t −1  τ  1− τu→w ; 2Du N D w→u 2Dw

(12)

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Carrier Rate Equations

Based on the density matrix formalism, optical gain of the energy state l for an input photon of angular frequency ωm is [18]:

T wu

  N g;j
z; t −1 N g;j
z; t −1 τg→u  1 − ; τ 2Du N D Gj 2Dg N D Gj u→g

(13)

  N e;j
z; t −1 N e;j
z; t −1  τ  1− τ ; 2Du N D Gj e→u 2De N D Gj u→e

(14)

T jgu 

T jeu

and Sm is the photon density, and Rg is the carrier recombination rate in the ground state. The optical gain is calculated by: 2M1 X

Cmj
2Pj − 1B
ωm − ωjcv;g ;

(25)

2πe2 N D jPcv j2 G
ωjcv;g − ω0cv;g Δωcv;g ; ε0 cnr m20 ℏω0cv;g

(26)

g
t; ωm  

j1

and Ru is the recombination rate of the carriers in the upper continuum state, τg→u and τe→u are the escape times from the ground and excited state to the upper continuum level, respectively. Parameters τu→e, τu→g , Dg , and De are defined in Table 1. The rate equation of the excited state is written as h i dN e;j
z; t  −T je N e;j
z; t − N e;j − Re N e;j
z; t; (15) dt

where Cmj  and Pj 

N g;j
t ; 2Dg N D Gj

(27)

where j −1 −1 N e;j  N u;j
z; tτ−1 u→e  N g;j
z; tτg→e 
T e  ;

C.

Calculation of Escape Times

(16)

Escape times τg→u , τg→e , and τe→u used in the rate equations are calculated by [19]:

T je  T jge  T jue ;

(17)

τg→u 
Dg ∕Du τu→g exp
ΔEug ∕kB T;

(28)

T jge

  N g;j
z; t −1 N g;j
z; t −1 τ ;  τ  1− 2De N D Gj g→e 2Dg N D Gj e→g

(18)

τg→e 
Dg ∕De τe→g exp
ΔEeg ∕kB T;

(29)

τe→u 
De ∕Du τu→e exp
ΔEue ∕kB T;

(30)

T jue

  N u;j
z; t −1 N u;j
z; t −1 τ ;  τ  1− 2De N D Gj u→e 2Du N D Gj e→u

(19)

and Re is the carrier recombination rate in the excited state, τg→e is the escape time from the ground state to the excited state. Parameter τe→g is defined in Table 1. The rate equation for the ground state is written as h i dN g;j
z; t  −Sm g
t; ωm ; cvgj − T jg N g;j
z; t − N g;j dt (20) − Rg N g;j
z; t; where j −1 −1 N g;j  N u;j
z; tτ−1 u→g  N e;j
z; tτe→g 
T g  ;

T jeg

T jg  T jeg  T jug ;

(22)

  N e;j
z; t −1 N e;j
z; t −1 τ ;  τ  1− 2Dg N D Gj e→g 2De N D Gj g→e

(23)

  N u;j
z; t −1 N u;j
z; t −1 τu→g  1 − ; τ 2Dg N D Gj 2Du N D Gj g→u

(24)

T jug 

2794

(21)

APPLIED OPTICS / Vol. 54, No. 10 / 1 April 2015

where kB is the Boltzmann constant, T is temperature, and ΔEug  ΔEue  ΔEeg . Parameters ΔEue and ΔEeg are defined in Table 1. 4. Simulation Results and Discussion

As mentioned before, we have assumed 2M  1 rate equations for each energy state of upper, excited, and ground and one rate equation for the wetting layer. Thus, we have a set of 6M  4 coupled differential equations. Supposing M to be in the order of 100, therefore, a set of 604 differential equations must be solved which is a burdensome and time-consuming task. Density of carriers is not constant along the length of waveguide. Therefore, in order to solve the differential equations, transfer matrix method (TMM) is employed [23]. Thus, we divide the length of the active medium to an appropriate number of sections. For the waveguide length of 100 μm, 40 sections are acceptable [23]. Along each section, carrier and photon density is supposed to be constant in each energy state. Therefore, there are a set of 604 coupled differential equations in each section. Density of carriers is assumed to be z-independent, hence the optical gain is also z-independent in each section [23]: dSm  Sm g
ωm  − αloss : dz

(31)

Thus, we have: Sm;out  Sm;in fg
ωm  − αloss Lsection g;

(32)

where Sm;in and Sm;out are the input and output photon densities of a section, respectively. Lsection is the length of each section, and αloss is the internal loss of the waveguide shown in Fig. 4. For each section of the waveguide, the input photon density is actually the output photon density of the previous section. The set of differential equations can be solved by Runge–Kutta numerical method. In each time step, the set of z-independent rate equations are simulated from t  0 to tmax  Lsection nr ∕c. The accuracy of our numerical method represented in this paper has been validated in the previous work [23]. There is a direct relationship between the light intensity and photon density in the active medium of the waveguide. The speed of light in the active medium is c∕nr. Thus, in 1 s photons pass through the length of L  c∕nr of the waveguide. Let A be the area of the cross section of the active medium of the waveguide. Therefore, the photons passing through the length of L of the waveguide are the photons which exist in the volume V  AL of the active medium. Thus, number of photons, N, is derived by the multiplication of the photon density and volume V as N  Sm V  Sm AL 

Sm Ac : nr

(33)

By multiplying Eq. (33) by the photon energy, ħω, the power delivered by photons to the area A of the active medium is defined as P  Nℏω 

Sm Acℏω nr


W:

(34)

Therefore, the light intensity or the power density in the active medium of the waveguide is: Intensity  I 

P Sm cℏω  A nr


W∕m2 :

(35)

This equation explicitly expresses the relation between the light intensity and the photon density in the active medium of the waveguide. In an optical amplifier, optical gain value depends on the light intensity, and light intensity depends on optical gain value at every point in the active region. Therefore, analysis of the active waveguide presented in this paper consists of two simulation processes. First, the Maxwell’s equations are numerically solved to attain the light intensity in the waveguide, and, next, the rate equations of the quantum dot active region are solved to calculate the optical gain value. The two processes are coupled and cannot be analyzed separately. In order to overcome this difficulty, we have devised an iterative analysis process:

• Assuming zero-gain in the active medium of the waveguide, Maxwell’s equations are solved using Comsol Multiphysics mode solver in order to calculate the light intensity at all points of the quantum dot active region. • The calculated intensities are used for calculation of optical gain using the rate equations governing carrier transfer in the quantum dots. By solving the set of rate equations using Runge–Kutta numerical method, carrier density in each energy state (wetting layer, upper continuum, excited, and ground state) is derived. Then, optical gain can be calculated using Eq. (25). • Optical gain is then fed into the Maxwell’s equations to recalculate the new value of light intensity in the waveguide. • This process is iterated until sufficient convergence is achieved. Iteration ends when the gain change is less than 1%. This is usually achievable with five iterations. Figure 6 demonstrates the optical gain of QD active medium of the MSM plasmonic waveguide of Fig. 1(a). The required gain for achieving lossless propagation depicted in Fig. 4 is also shown in this figure. At the wavelength of 1560 nm the acquired optical gain is equal to the required gain. Thus, the loss will be compensated with the optical gain of the QD active region. For shorter wavelengths, the obtained gain of the QD active region is higher than the required gain values. This subject is inevitable because by increasing the wavelength, the loss of the waveguide and the optical gain of the active medium do not vary with the same rate. Therefore, QD parameters are tuned in order to attain a lossless propagation over the required wavelength range. Figure 6 depicts the MSM plasmonic waveguide as being lossless at the operating wavelength of 1560 nm and having a little gain for the lower wavelengths. Lossless MSM plasmonic waveguides can be utilized in the field of planar and on-chip optical devices, photonic integrated circuits (PICs), and sensors due to their long propagation length. Figure 7(a) illustrates the propagation of light through a lossy MSM plasmonic waveguide

Fig. 6. Required gain for achieving lossless propagation in the MSM plasmonic waveguide of Fig. 1(a) and the acquired optical gain from the QD active region. 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS

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amount of required gain for achieving lossless propagation can be acquired by the use of InAs/InP QD active medium in the MSM plasmonic waveguides. With lossless MSM plasmonic waveguides, some novel applications such as planar and on-chip optical devices, photonic integrated circuits (PICs), and nanoscale sensors can be proposed in the field of nanophotonics. References Fig. 7. Light propagation through a MSM plasmonic waveguide: (a) with loss and (b) without loss.

Fig. 8. Spectra of the average value of light intensity of the active region in the z-direction for the MSM plasmonic waveguide of Fig. 1(a).

with metal and semiconductor width of 100 and 300 nm, respectively. The wavelength is 1550 nm, and silver is used as the waveguide metal. Semiconductor layer of the waveguide is composed of three different layers with widths of 100 nm and dielectric constants similar to that of Fig. 1(a). The propagation of light through the lossless waveguide is shown in Fig. 7(b). The average value of light intensity of the active region in the z-direction for the MSM plasmonic waveguide of Fig. 1(a) is shown in Fig. 8. The figure shows light intensity attenuation of 3 dB at z  6 μm when no gain medium is used in the waveguide. Figure 8 demonstrates that when QD active medium is used in the waveguide, the required gain value for achieving lossless propagation is acquired according to Fig. 6, and attenuation of the light intensity is compensated. 5. Conclusion

We numerically analyzed the gain requirements for lossless propagation in subwavelength active metalsemiconductor-metal (MSM) plasmonic waveguides for the wavelength range of 1540–1560 nm. For achieving gain in the structure, active region with InAs quantum dots were used. By solving iteratively the Maxwell’s equations and the rate equations of quantum dots in the active region, the required gain for lossless propagation was acquired. We studied the effect of wavelength changes on the QD gain. The 2796

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