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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 14 — Jul. 4, 2011
  • pp: 13225–13244
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A surface-emitting 3D metal-nanocavity laser: proposal and theory

Chien-Yao Lu and Shun Lien Chuang  »View Author Affiliations


Optics Express, Vol. 19, Issue 14, pp. 13225-13244 (2011)
http://dx.doi.org/10.1364/OE.19.013225


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Abstract

A novel three-dimensional (3D) metal-nanocavity (or nano-coin) semiconductor laser suitable for electrical injection is proposed and analyzed. Our design uses metals as both the cavity sidewall and the top/bottom reflectors (i. e., a fully metal encapsulated nanolaser) and maintains the surface-emitting nature. As a result of the large permittivity contrast between the dielectric and metal, the optical energy can be well-confined inside the metal nanocavity. With a proper design and the choice of the HE111 mode, which has the best top surface radiation pattern, a laser with a physical size smaller than 0.01λ03 is achievable at 1.55 μm wavelength with a reasonable semiconductor gain at room temperature. We provide a detailed theoretical model starting from the waveguide analysis to full 3D structure simulations by taking into account both geometry and metal dispersion. We show a systematic procedure for analyzing this class of 3D metal-cavity (or nano-coin) lasers with discussions on the optimization of the performance such as light output power, threshold reduction, and output beam shaping.

© 2011 OSA

1. Introduction and proposal of a metal-nanocavity (nano-coin) semiconductor laser

In this paper, we propose and analyze a new class of electrical-injection metal-nanocavity (nano-coin) lasers with full coverage of metal, which preserves the surface-emitting and substrate-free configurations. Our design uses metal as a cavity wall on all sides, which eases the requirement of precise design of optical feedback structures such as DBR [4

4. C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface emitting microlaser at room temperature,” Appl. Phys. Lett. 96(25), 251101 (2010). [CrossRef]

7

7. C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlasers with hybrid metal-DBR reflectors,” Opt. Lett. (in press). [PubMed]

] or cutoff waveguides [11

11. M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4(6), 395–399 (2010). [CrossRef]

13

13. M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009). [CrossRef] [PubMed]

]. Moreover, with the help of a full metal coverage, the radiation to adjacent devices can be greatly reduced, and, thus a crosstalk-free environment can be achieved. The proposed structure is transferable and stackable due to its substrate-free configuration. With a proper design, the metal-cavity nanolaser can achieve room temperature operation with a physical cavity volume of only 0.01λ0 3. Our proposed three-dimensional metal-nanocavity laser is shown in Fig. 1
Fig. 1 Our proposed surface-emitting three-dimensional (3D) metal-nanocavity (nano-coin) laser. The active region is composed of a bulk In0.53Ga0.47As with a height h in the active region height and a in radius. InP is used as both electron and hole injectors. An InGaAsP layer serves as a contact layer for n-contact. The whole device is encapsulated in silver with SiNx as a current blocker. (a) A 3D view and (b) a cross sectional view of the structure.
, which consists of an In0.53Ga0.47As bulk gain material with InP as carrier injectors. The cavity wall is defined by silver metal with silicon nitride (SiNx) as both a current blocker to prevent any short circuit and a part of optical waveguide shell to reduce optical leakage to the metal. The metal serves a few purposes including the electrical contacts, the cavity wall for optical mode confinement, shielding from crosstalk among lasers, and, more importantly, an efficient heat sink. The emission comes from the top with a thin silver layer as the reflector. Due to the full coverage of metal on the cavity wall, the modes resemble those in a circular waveguide resonator. In order to capture the general features of the cavity modes resulting from the superposition of two counter-propagating fields, the waveguide modes are analyzed and discussed in section 2. In section 3, cavity modes resulting from the lowest few orders of the waveguide modes are analyzed for laser applications. In section 4 we discuss the optimization through a proper design of the SiNx and silver reflector layers. A general discussion of the far-field patterns is given for those designs. In section 5, the rate equations are used to analyze the laser performance. The issues of light output power will also be addressed. We then conclude in section 6.

2. Metal-clad waveguide analysis and design consideration

A simplified perfect electric conductor (PEC) model can be used to capture the guiding properties of circular metallic waveguide. For a homogeneous metallic waveguide, the propagation constant kz should satisfy the following equation,
kρ2+kz2=ω2μεcore=(2πλ0ncore)2,
(1)
where kρ is equal to χmn/b and χmn is the root of Bessel functions or their derivatives and b is the outer radius as shown in Fig. 2. εcore is the permittivity of the homogeneous waveguide core and n core and λ 0 are the refractive index of the core and the free space wavelength. The mode cutoff occurs at kz = 0, or equivalently,
(2πλcncore)=kρ=χmnb.
(2)
The resulting cutoff wavelength (λc) is then equal to 2πbn coremn. For a PEC waveguide with a homogeneous cross-section, the value χmnλc/2πn core b should be 1.

In the descending order of cutoff wavelengths, the dominant modes are HE11, TM01, HE21, TE01, and EH11. Figure 3(a)
Fig. 3 (a) Cutoff wavelengths of different modes as a function of core radius. The fundamental mode HE11 has the longest cutoff frequency among all the other modes and can be used to design a cavity of a minimal radial dimension. (b) χ mn λ c/2πn core(b + Δ) as function of core radius. χ mn is the root of Bessel functions or their derivatives (HE11: 1.84, TM01: 2.405, HE21: 3.05, TE01: 3.83, and EH11: 3.83). The dashed line represents the prediction by the use of homogeneous waveguide with a receded PEC wall by a skin depth Δ. The actual cutoff wavelength will be close to the prediction when the wavelengths are long enough such that the thin cladding layer becomes negligible.
shows the cutoff wavelengths (λc) as a function of the outer radius (b) calculated with a SiNx shell thicknesses of 50 nm and material (metal and semiconductor) permittivities from [21

21. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

,22

22. S. Adachi, “Refractive indices of III-V compounds: key properties of InGaAsP relevant to device design,” J. Appl. Phys. 53(8), 5863–5869 (1982). [CrossRef]

]. To ensure a propagating behavior inside the cavity within the window of material emission spectrum, the radius with a cutoff wavelength longer than the emission wavelength should be chosen. For example, at around 1550 nm (corresponding to In0.53Ga0.47As emission), the HE11 (EH11) mode can only propagate when the radius (b) is larger than 130 nm (260 nm) as shown in Fig. 3(a). The fundamental mode HE11 has the longest cutoff wavelength for the same size, and, thus can potentially be a good choice in designing an optical cavity with a minimal transverse dimension [23

23. S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic fabry-perot nanolasers,” Opt. Express 18(14), 15039–15053 (2010). [CrossRef] [PubMed]

].

Figure 3(b) plots the value χmnλc/[2πn core(b + Δ)] as a function of the outer radius b, where χmn is the root of Bessel functions or their derivatives (HE11: 1.84, TM01: 2.405, HE21: 3.05, TE01: 3.83, and EH11: 3.83), Δ is the penetration skin depth, and n core is the core (In0.53Ga0.47As) refractive index. The values deviate from unity as the radius decreases. It means that the approximation by a homogeneous circular waveguide with a real dispersive metal replaced by a PEC wall and receding by the penetration depth fails in this region. This discrepancy comes from the non-negligible permittivity transition from semiconductor core to the insulator SiNx shell. Moreover, for hybrid modes which should not occur in a homogeneous waveguide with a PEC wall, the approximation will hold only when the radius is large enough for the hybrid modes to be considered as transverse modes and the cutoff wavelength is long enough to ignore the shell thickness.

In an inhomogeneous waveguide such as a core-shell structure (assuming n core > n shell in this paper) with a real metal surrounding in Fig. 2, there are three propagating modes in the waveguide: surface (plasmonic) mode, core-shell mode, and core (dielectric or fiber-optical) mode. The surface mode exists when the propagation constant k z of the mode is larger than k core = 2πn core0, where λ0 is the free space wavelength, or equivalently, the effective index n eff (defined by k z = (2πn eff0)) is larger than the material indices inside the waveguide (n shell < n core < n eff). In surface mode, the optical energy is concentrated near the metal-dielectric interface and decays exponentially into both regions. As a result of large field penetration into metal, this mode usually suffers from a high propagation loss. The core-shell mode exists when n eff < n shell, and in this case, a significant field leaks out of the core region and propagates (with a real kρ) inside the shell region. The optical power is actually guided in both the core and the shell regions. The core mode represents n shell < n eff < n core, which is the normal propagation mode (or dielectric mode or fiber-optical mode) with power propagation mostly inside the core with an evanescent decay to the shell region, and is expected to be the most lossless among all three modes. The design with the core mode not only helps reduce the sidewall metal loss but also concentrate the energy more into the center region. Figure 4(a)
Fig. 4 (a) The effective index as a function of core radius a and the guiding wavelength of the HE11 mode of a silver-coated circular waveguide with an In0.53Ga0.47As core and a SiNx shell (50 nm) as shown in Fig. 2. (b) The guiding wavelength as a function of core radius a and effective index. The wavelength is plotted only in the window of 1000-2000 nm. The white line represents the cutoff wavelength of the HE11 mode, on which the effective index and the propagation constant Re(kz) equal zero. To have a mode guiding inside the core region, the effective index has to be larger than the refractive index of SiNx (~2.0: dashed line in (a)). This also regulates the choice of cavity radius to be larger than ~70 nm.
shows a two-dimensional (2D) plot of the effective index n eff as a function of radius a with a constant SiNx thickness (s = 50 nm) and the wavelength. Figure 4(b) shows a 2D plot of the guiding wavelength as a function of the core radius (a) and the effective index.

3. Theory of metal-nanocavity lasers: modal analysis

3.1. Cavity structures and resonant modes

A cavity can be formed by shorting both ends of a waveguide with a conductive wall. For a given wavelength, to form a standing wave in the longitudinal direction, a minimum length satisfying the round-trip phase condition (2π) is required. (An exception is the TM0 n mode which lacks the longitudinal magnetic field (Hz = 0) and the azimuthal variation, and, therefore, the boundary conditions can be satisfied at both ends with Ez ≠ 0 at an arbitrary length.) In the PEC model shown in Fig. 5
Fig. 5 (a) The cross-section of a core-shell waveguide with perfect electrical conductor (PEC) surroundings. (b) A cavity formed of shorting both ends of a waveguide in (a) with a cavity length of L.
, the fields of the mode (mnp) in the cavity can be written as a sum of two counter-propagating waves,
Εmnp(ρ,φ,z)=Εmn+(ρ,φ)eikzz+Εmn(ρ,φ)eikzz,
(3a)
Hmnp(ρ,φ,z)=Hmn+(ρ,φ)eikzz+Hmn(ρ,φ)eikzz,
(3b)
where E mnp is the electric field, H mnp is the magnetic field of longitudinal order p, k z = pπ/L (L: the cavity height), Emn+and Emn (Hmn+ and Hmn) are the electric (magnetic) fields of counterpropagating modes along the waveguide. The translation invariant symmetry of the waveguide along the z axis leads to the following relations [24

24. S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]

26

26. S. L. Chuang, Physics of Optoelectronic Devices, 1st ed. (Wiley, 1995), Appendix H.

]:
Emn+(ρ,φ)=Emnt(ρ,φ)+Emnz(ρ,φ),
(4a)
Emn(ρ,φ)=Emnt(ρ,φ)Emnz(ρ,φ),
(4b)
Hmn+(ρ,φ)=Hmnt(ρ,φ)+Hmnz(ρ,φ),
(4c)
Hmn(ρ,φ)=Hmnt(ρ,φ)+Hmnz(ρ,φ),
(4d)
where superscripts represent transverse (t) or longitudinal (z) components. The field components inside the cavity can be further explicitly written as,
E-profile:{Ez,mnp~Emnz(ρ,φ)cos(pπLz)Eρ,mnp~Emnρ(ρ,φ)sin(pπLz)Eφ,mnp~Emnφ(ρ,φ)sin(pπLz),
(5a)
H-profile:{Hz,mnp~Hmnz(ρ,φ)sin(pπLz)Hρ,mnp~Hmnρ(ρ,φ)cos(pπLz)Hφ,mnp~Hmnφ(ρ,φ)cos(pπLz).
(5b)
A field node of the waveguide mode will also result in a node in the corresponding cavity mode formed with the same transverse field distributions.

To investigate which cavity mode is the best one for laser application, the full structure simulation with real experimental material parameters is performed. Figure 6(a)
Fig. 6 Calculated resonance wavelengths of the higher order modes by a full 3D structure FDTD simulation. (a) The resonance wavelengths as a function of the core radius a of the six lowest order modes. (b) Mode chart of the structure curves with the same slope corresponding to the same longitudinal mode number (p/2nop)2.
shows the resonance wavelengths of the six lowest cavity modes (TM010, HE111, TM011, HE211, TE011, EH111) calculated by finite-difference time-domain (FDTD) method [27

27. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).

]. The structure is shown in Fig. 1(a) with vertical 160-nm In0.53Ga0.47As, 30-nm n- and p-InP, and 20-nm InGaAsP layer thickness, and a conformal insulator layer of 50-nm SiNx. The material parameters can be found in [21

21. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

,22

22. S. Adachi, “Refractive indices of III-V compounds: key properties of InGaAsP relevant to device design,” J. Appl. Phys. 53(8), 5863–5869 (1982). [CrossRef]

]. According to the waveguide analysis in section 2, all modes except HE111 have a node in either the electric field or the power (Pz) distribution. This node will make a poor overlap between the gain region and the optical field. The dotted line indicates that the resonant wavelength is in the cutoff region of the corresponding waveguide mode. The fundamental mode TM010 has a wavelength far above the rest due to the zeroth order in z direction.

For an inhomogeneous waveguide with a small permittivity variation, perturbation method can be applied to calculate the resonance wavelength shift from that of a homogeneous waveguide [19

19. J. A. Kong, Electromagnetic Wave Theory (EWM, 2008).

,23

23. S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic fabry-perot nanolasers,” Opt. Express 18(14), 15039–15053 (2010). [CrossRef] [PubMed]

]. However, the permittivity contrast between insulator and semiconductor are usually too large to be considered as a perturbation. To properly include the effect of the shell layer, a model with a multilayered structure should be used. As a result of a large negative real part of silver permittivity at near IR region, the optical modes inside the nanolaser surrounded by silver will be similar to those surrounded by a PEC wall. Figure 7
Fig. 7 The cavity resonance wavelength as a function of the core radius a of various cavity heights L from 180 to 320 nm. The SiNx shell thickness is fixed at 50 nm. The resonance wavelength with a solid curve is the core (dielectric) mode with n eff > n shell. Fields concentrate inside the core region. The resonance wavelength with a dashed curve is the shell mode with n eff < n shell. Fields leak out of the core region and the wave can propagate inside the shell region. Lines of waveguide cutoff and transition from core mode to shell mode (n eff = n shell = 2.0) are plotted. A PEC model (Section 3) with the metal wall receding by a skin-depth is used in these calculations.
shows the theoretical calculation of the cavity resonance wavelength of the HE111 mode in Fig. 6, based on the PEC model with the metal wall receding by a skin depth, as a function of the core radius a for various cavity lengths L = 180 to 320 nm. The resonance wavelength gradually becomes shorter and approaches the waveguide cutoff as the geometry shrinks. A transition from core (dielectric) mode (solid lines) to core-shell mode (dashed lines) is indicated by the dispersion curve with the effective index n eff = n shell = 2.0. This transition marks the onset of propagating fields inside the shell region. In the design with real metals, the transition also means more energy penetration into the metal plasma and, thus, results in a higher propagation loss while propagating along the waveguide.

3.2. Simulation results of the cavity properties

To analyze the full nanolaser structure in Fig. 1 with specific material properties, we carried out the simulation using the FDTD method. In our design, In0.53Ga0.47As is used as the active material with p- and n- doped InP layers as cladding for contact. The thickness is fixed at 30 nm for both n- and p-doped regions. Also, an n-doped 20-nm InGaAsP layer is used to support the contact area. Silver is used to encapsulate the whole cavity with a low-index material SiNx in between the semiconductor and metal as a current blocker. At the output, a thin silver is coated to increase the reflectivity and serves as a part of the contact. The material parameters used in the simulation are taken from [21

21. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

,22

22. S. Adachi, “Refractive indices of III-V compounds: key properties of InGaAsP relevant to device design,” J. Appl. Phys. 53(8), 5863–5869 (1982). [CrossRef]

]. Drude model is adopted to fit the metal permittivity for simplicity.

In Fig. 8
Fig. 8 (a) Top view of z component of electric and magnetic fields of HE111 mode. (b) Side view of all field components inside the cavity. Note the azimuthal dependence for (b) has been suppressed for simplicity.
, all HE111 field plots of our proposed cavity with a radius a = 190 nm, b = 240 nm, and cavity height L = 240 nm are shown. The top view in (a) with the z-component of both E and H fields at the center of the cavity shows their m = 1 azimuthal distribution (in the form of sin() and cos()). Also, the lack of nodal point along the ρ-direction represents the radial order n = 1. In Fig. 8(b), the side view of the field components follows Eq. (3) in their z distribution. Note in Fig. 8(b) plots, the azimuthal dependence has been suppressed for simplicity. The symmetry follows the model with p = 1 and the field patterns closely resemble those of the PEC waveguide.

Figure 9
Fig. 9 (a) Calculated resonance wavelengths of various cavity heights as a function of radius a using the FDTD method (solid curves) for the real structure (including the complex permittivity of the metal) compared with those using the PEC model (dashed curves) presented in Section 3. (b) Calculated cavity quality factors as a function of radius. The maximum quality factor is around 275 for all cavity heights.
shows the calculated resonance wavelengths and the corresponding quality factors with various heights (220 nm - 300 nm). The thicknesses of SiNx and the thin silver reflector are both set to 50 nm. The effect of varying these values will be discussed in section 4. In Fig. 9(a) the resonance wavelength of various cavity heights as a function of the core radius can be modeled using the structure analyzed in section 3 by receding the metal wall with a penetration (skin) depth to account for the finite permittivity in real metals. The dashed lines in Fig. 9(a) show the result from the model. Compared to the full structure calculation by the FDTD method, the simplified model agrees very well except at the small radius region where the field leaks into SiNx and InGaAsP layers. It starts to dominate especially when the effective index becomes smaller than that of the shielding SiNx layer.

By taking the Fourier transform of the time varying field inside the cavity, the complex resonant frequency (ω res = ω r i) of the cavity with metal dispersions can be obtained. ω r represents the oscillation frequency of the optical field and ω i is the attenuation due to the loss (both material and radiation). The cold cavity quality factor can be expressed as Q = ω r/2ω i [28

28. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1989).

]. Figure 9(b) shows the corresponding ideal cold cavity quality factor Q. A clear drop of Q below 100-nm core radius is caused by the energy leakage to SiNx and InGaAsP layers as described above. The quality factor can be separated into two parts: Qrad from radiation out of the cavity and Qmat from material loss (metal loss). With the calculated field components from the FDTD method, Qrad can be evaluated from the definition [19

19. J. A. Kong, Electromagnetic Wave Theory (EWM, 2008).

,28

28. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1989).

],
Qrad=2πenergy   stored   in   cavityradiated   energy   per   optical   period=ωWPrad         =ωVdrε04[εg(r)+εR(r)]|E(r)|2Sds12Re[E(r)×H*(r)]n^,
(8)
where W is the total electromagnetic energy inside the cavity, Prad is the radiated power outside of the cavity surface, n^ is the unit normal to the surface, εR(r) and εg(r) are, respectively, the real part of the relative permittivity and the relative group permittivity, which is defined as [29

29. S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 45(8), 1014–1023 (2009). [CrossRef]

],
εg(r)=[ωεR(r)]ω.
(9)
The material quality factor Qmat due to absorption can then be obtained from the relation [24

24. S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]

],
Q1=Qmat1+Qrad1.
(10)
The results are shown in Fig. 10
Fig. 10 (a) The radiation quality factor Qrad of cavities as a function of the core radius a for various heights L = 220 to 300 nm. The peaks represent the coupling to a resonance structure. (b) The material quality factor Qmat as a function of radius. The material quality factor has only a minimum change among different cavity heights and radii. A sharp drop at around R = 100 nm depicts the transition from core modes to shell modes.
. The radiation loss has a minimum which originates from minimum energy coupling to a resonant circular slot structure near the output facet [30

30. R. S. Elliott, Antenna Theory and Design (Prentice-Hall, 1981)

]. Due to the coupling at this radius, the radiation loss through the composite waveguide of InGaAsP and SiNx is significantly reduced. According to the result of Qmat, the material loss has a smooth change and reduces as the radius decreases. This reduction comes from the effective increase of shell layer portion inside the cavity, which buffers the field penetration into metal [31

31. A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33(11), 1261–1263 (2008). [CrossRef] [PubMed]

]. One should note the sharp increase of material loss below radius 100 nm. The sharp transition region represents the transition from core (dielectric) mode of the waveguide to its shell mode as discussed before.

In dealing with small lasers, especially with dispersive plasmonic materials like metals, the confinement factor associated with energy should be used to better account for the dispersive and plasmonic effects. The use of the energy confinement factor corrects the improper use of the negative power flow and negative energy resulting from negative permittivity of metals. The energy confinement factor is defined as [29

29. S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 45(8), 1014–1023 (2009). [CrossRef]

],
ΓE=Vadrε04[εg(r)+εR(r)]|E(r)|2Vdrε04[εg(r)+εR(r)]|E(r)|2,
(11)
where Va represents the volume of the active material. In Fig. 11(a)
Fig. 11 The energy confinement factor (a) and extraction efficiency (b) of the cavity as a function of the core radius a with different cavity heights (220 nm – 300 nm). In the inner mode region, ΓE is above 0.6.
, the energy confinement factor is shown for various cavity heights. The energy confinement factor associated with the percentage of active material over the whole cavity increases as both the core radius and height increase. As the radius shrinks, especially when n eff < n SiN x, the energy starts to leak out and results in a reduction in the energy confinement. The overall confinement factors are more than 60% for devices with a radius larger than 100 nm. For a given height, the energy confinement factor approaches a constant and corresponds to the case that the SiNx has a minimal effect. Compared to conventional dielectric lasers, which usually have lower values due to poor dielectric confinement, our proposed laser structure with the full coverage by metal, helps confine more energy inside the cavity. Figure 11(b) shows the extraction efficiency of the cavity, which is a measurement of how much loss contributes to the radiation or light output. It is defined as,

ηext=Qrad1Q1.
(12)

A special care has to be taken to get a decent light output. Observing from Fig. 11(b), the design with a maximum radiation Qrad can lead to a poor extraction efficiency even though the condition of lasing can be achieved more easily like photonic crystal lasers [1

1. H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305(5689), 1444–1447 (2004). [CrossRef] [PubMed]

,2

2. K. Nozaki, S. Kita, and T. Baba, “Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser,” Opt. Express 15(12), 7506–7514 (2007). [CrossRef] [PubMed]

]. The physical sizes are summarized in Fig. 12(a)
Fig. 12 (a) The physical cavity volume and (b) the threshold material gain for different cavity heights L = 220 to 300 nm. The cavity volume approaches the diffraction limit ~(λ 0/2n)3. The threshold can be as low as 600 cm−1 for a≥ 250 nm and cavity heights L≥ 240 nm.
. Due to the nature of standing wave formation in HE111 mode, the size should be around the minimum standing wave volume or the diffraction limit (λ 0/2n)3, where n represents the effective index of the optical mode. With the knowledge of cavity Q, confinement factor ΓE, and the corresponding resonance frequency ω, the threshold material gain gth can be formulated as [29

29. S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 45(8), 1014–1023 (2009). [CrossRef]

],
gth=2πngQΓEλ,
(13)
where n g is the group index of the active region and λ is the wavelength in vacuum. The values of gth shown in Fig. 12(b) are around 600 cm−1 – 1,200 cm−1 for cavity heights (240 - 300 nm) and a≥ 250 nm and should be achievable at room temperature within the emission spectrum of In0.53Ga0.47As materials.

4. Design optimization of metal-nanocavity lasers

4.1. Effects of shell insulator layer

The shell insulator layer plays a very important role in the design of electrical injection lasers. It is not only an electrical buffer layer but also an effective optical buffer layer for reducing metal loss. Recently, a proper design of the insulator thickness was proven to be useful for reducing the laser threshold by optimizing the material loss from metal and its optical confinement factor [31

31. A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33(11), 1261–1263 (2008). [CrossRef] [PubMed]

]. The theoretical results of major physical parameters for a cavity with a cavity height L = 240 nm and various SiNx thicknesses from 20 nm – 80 nm are shown in Fig. 13
Fig. 13 The physical properties for different SiNx shell thicknesses s = 20 to 80 nm. (a) The resonance wavelength. (b) The quality factor of the cavity Q. (c) The computed radiation quality factor Qrad. (d) The material quality factor Qmat. (e) The energy confinement factor ΓE. (f) The threshold material gain gth.
. Since the cavity boundary is defined by metal, it is more illustrative to plot the result as a function of outer radius b (i.e. the radius of the metal wall or the sum of the core radius a and the SiNx shell thickness s). Since the boundary remains the same for each b, a smaller change in the resonance wavelength is expected in larger radius case, where the SiNx layer occupies only a small portion of the cavity and the field inside SiNx layer is reduced. In Fig. 13(a), the resonance wavelength decreases with increasing SiNx layer thickness from 20 nm to 80 nm. This is caused by the increase of the average material index inside the metal cavity, or more physically, the decrease of effective wavelength. The cavity quality factors Q in Fig. 13(b) are similar in all cases except in larger radius region. The decrease of quality factor in large radius region as b increases comes mainly from their material loss, or field penetration into metal as can be seen from Figs. 13(c) and 13(d). For longer wavelengths, the effect of SiNx layer as a buffer layer for reducing loss becomes smaller, as a result, cavities with the same outer radii will experience more loss if SiNx becomes relatively thinner compared with the wavelengths. Besides, a thicker SiNx can help reduce field into the metal or minimize the field leaking out of the cavity as radiation loss.

For the confinement factor ΓE in Fig. 13(e), reducing the portion of SiNx inside the cavity will result in the increase of ΓE for small radius. For a large radius, the dependence of confinement factor on the SiNx thickness becomes less obvious except for 20 nm and 30 nm cases in which the wavelengths are longer than the other cases and the radial electric field starts to dominate over the other components (k ρ increases), and, thus experiences more field penetration into the metal. Because of the large permittivity difference between Ag and SiNx, the field inside the SiNx builds up as the thickness becomes thinner and this results in the reduction of ΓE. Figure 13(f) shows the corresponding threshold material gain. The minimum threshold material gain for each case has only a small dependence on SiNx thicknesses.

The effect of insulator layer thickness has only a small change on the resonance wavelength. The optimization should be focused on the confinement factor and material loss issues. In general, the confinement factor will increase with thinner insulator layers for b < 250 nm with a sacrifice of reducing the material quality factor.

4.2. Effects of the output metal reflector layer

The metals on the output (or top) facet provides both optical feedback and mode confinement of the cavity. Designs with a thin metal usually suffer from both radiation loss through the metal and ohmic loss dissipated inside the metal [24

24. S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]

,32

32. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]

]. However, designs with a thick metal will regulate the output power. Unlike the other metal surrounding of the cavity, which usually has little constraint on the thickness, the metal reflector on the output facet plays an important role as a cavity wall and an output coupler. Figures 14(a)
Fig. 14 The cavity properties with different metal reflector thicknesses from 20 nm to 80 nm. (a) The resonance wavelength. (b) The quality factor of the cavity Q. (c) The computed radiation quality factor Qrad. (d) The material quality factor Qmat. (e) The energy confinement factor ΓE. (f) The threshold material gain gth.
and 14(b) show the resonance wavelengths and their corresponding cavity quality factors of a 240-nm height cavity with different silver reflector thicknesses. The cavity resonance wavelength, which is determined by the boundary conditions, has little dependence on the metal thickness, especially when the thickness is close to or thicker than its skin depth (for example 15~30 nm for silver at this optical frequency). The quality factor in Fig. 14(b) shows a more obvious dependence on the metal thickness. The separation of Q into Qrad and Qmat is plotted in Figs. 14(c) and 14(d). The material quality factor shows a strong dependence on the metal thickness. As a result of a large permittivity difference between semiconductor and metal and/or air and metal, the optical field tends to accumulate near the interface. As the metal becomes thinner, fields from both interfaces start to couple and result in a strong field distribution across the metal which in turn contributes to the high material loss for thin metals [24

24. S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]

,32

32. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]

]. In Fig. 14(c), the radiation Qrad is shown for various metal thicknesses. Also, as shown in Fig. 14(e), the energy confinement factor has an almost unchanged behavior due to the quick decaying tail inside the metal, which usually contains a small portion of the total stored energy in the cavity.

In general, the radiation loss is not the main path of loss out of the cavity. However, in order to get a reasonable output power, the thickness should not be too large. A thicker metal will hinder the light penetration through, or equivalently worsen the extraction efficiency, and has only a minimal increase in the reflectivity. The threshold material gains for different metal thicknesses are shown in Fig. 14(f). When the thickness is over 40 nm, the required threshold material gain reduces to below 1,000 cm−1 and is achievable for bulk materials at room temperature. According to the calculated result, a design with a metal thickness of around 40 nm to 60 nm should meet the requirement.

4.3 Far field radiation pattern

The radiation pattern depends strongly on the cavity mode. For a laser with dimensions comparable or even smaller than the wavelength, the emission pattern usually is divergent since the fields are strongly localized in the cavity with large wave vectors. In the waveguide case with an open end, the HE11 mode preserves the most useful radiation pattern compared to others since it has the smallest kρ (~χ11/b). A cavity mode design associated with HE11 mode will also prevent a null at the center of the circular waveguide. The radiation pattern I(θ,ϕ) can be defined as [19

19. J. A. Kong, Electromagnetic Wave Theory (EWM, 2008).

,30

30. R. S. Elliott, Antenna Theory and Design (Prentice-Hall, 1981)

],
I(θ,φ)=P(r,θ,φ)r^Max[P(r,θ,φ)r^],
(14)
where P(r,θ,ϕ) is the Poynting vector in spherical coordinates and Max[·] represents the maximum value over the hemisphere with a constant radius r in the far field. In Fig. 15
Fig. 15 Computed far field radiation pattern of Hz and Eρ components. Note the azimuthal dependence (e ± ) has been suppressed.
, calculated radiation fields of two dominant components (Hz, Eρ) of the HE111 mode with cavity parameter L = 240 nm, R = 190 nm, R out = 240 nm, and metal thickness = 50 nm are shown. The radiation comes from top metal reflector facet with a high uniform spreading toward the hemispherical space. In the Eρ plot, a clear guided wave along the radial direction can be observed in the structure composed of Ag/InGaAsP/SiNx/Ag hybrid waveguides. These fields do not contribute to the emission into the hemispherical region and have a minimum effect on the radiation pattern.

The calculated radiation patterns as a function of θ are shown in Fig. 16
Fig. 16 Far-field radiation patterns of cavity with different radii, a, from 130 to 370 nm. The azimuthal dependence has been assumed to be e ± . The radiation pattern narrows as the aperture size (radius) increases.
for cavities with different radii. Note the azimuthal dependence has been assumed to be in e ± iϕ convention. The radiation pattern narrows as the radius becomes larger. The divergence comes from large radial wave component kρ at a small radius. A small tail near the interface (θ ~90°) comes from the surface plasma propagation along the Ag/air interface. Compared with nanolasers of other designs such as nanopatchs [8

8. K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18(9), 8790–8799 (2010). [CrossRef] [PubMed]

,14

14. C. Manolatou and F. Rana, “Subwavelength nanopatch cavities for semiconductor plasmon lasers,” IEEE J. Quantum Electron. 44(5), 435–447 (2008). [CrossRef]

] in which the output comes from the interference of fields from the opening sides, our design of the HE111 mode with a direct facet output has the advantage of less sensitive radiation pattern dependence on the geometry.

5. Threshold analysis and the light output power (L-I curve)

To investigate the possibility of achieving an injection laser and the performance of our proposed metal-cavity structure, we use the rate equations to perform the analysis [23

23. S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic fabry-perot nanolasers,” Opt. Express 18(14), 15039–15053 (2010). [CrossRef] [PubMed]

,29

29. S. W. Chang and S. L. Chuang, “Fundamental formulation for plasmonic nanolasers,” IEEE J. Quantum Electron. 45(8), 1014–1023 (2009). [CrossRef]

],
nt=ηIqVaRnr(n)Rsp,cont(n)Rsp(n)Rst(n)S,
(15a)
St=Sτp+ΓERsp(n)+ΓERst(n)S,
(15b)
Rnr(n)=An+Cn3,
(15c)
Rsp,cont(n)=1τradVac,v,kfc,k(1fv,k),
(15d)
Rst=υgg(n).
(15e)
Here n (cm−3) is the carrier density, S (cm−3) is the photon density, η is the injection efficiency, q is the charge of electron, Va (cm3) is the volume of active region, Rnr (cm−3s−1) is the non-radiative recombination rate, Rsp,cont (cm−3s−1) and Rsp (cm−3s−1) are the spontaneous emission rate into continuum modes and cavity mode, respectively, RstS (cm−3s−1) is the stimulated emission rate, and g(n) (cm−1) is material gain. The non-radiative recombination rate can be modeled as a sum of surface recombination rate (An) plus Auger recombination rate (Cn 3). The coefficient A is given by νsAs, where νs is the surface recombination velocity and is set to be 2.0 × 102 ms−1 [33

33. E. Yablonovitch, R. Bhat, C. E. Zah, T. J. Gmitter, and M. A. Koza, “Nearly ideal InP/In0.53Ga0.47As heterojunction regrowth on chemically prepared In0.53Ga0.47As surfaces,” Appl. Phys. Lett. 60(3), 371–373 (1992). [CrossRef]

] and As is the surface area of the active region. The Auger coefficient is set to be 4.67 × 1027 cm6s−1 [34

34. Y. Zou, J. S. Osinski, P. Grodzinski, P. Dapkus, W. C. Rideout, W. F. Sharfin, J. Schlafer, and F. D. Crawford, “Experimental study of Auger recombination, gain, and temperature sensitivity of 1.5 μm compressively strained semiconductor lasers,” IEEE J. Quantum Electron. 29(6), 1565–1575 (1993). [CrossRef]

]. The spontaneous emission rate in to continuum modes Rsp,cont is calculated by the formula in Eq. (15d), where τrad is an effective radiation lifetime (set to be 5 μs) [23

23. S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic fabry-perot nanolasers,” Opt. Express 18(14), 15039–15053 (2010). [CrossRef] [PubMed]

] and fc, k and fv, k represent the occupation numbers at wave vector k in conduction and valance bands.

Figure 17
Fig. 17 Material gain spectra of bulk In0.53Ga0.47As at different injected carrier densities at room temperature.
shows the carrier-dependent material gain spectra of the bulk In0.53Ga0.47As semiconductor calculated by Luttinger-Kohn model at room temperature [35

35. S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009), Chap. 4.

]. Compared with the required threshold material gain for various structures discussed in the context (~700 cm−1 - 1000 cm−1), In0.53Ga0.47As is suitable for achieving room temperature operation at reasonable carrier densities. In Fig. 18(a)
Fig. 18 (a) The L-I curve at room temperature of a device with a = 190 nm, SiNx shell thickness = 50 nm, b = 240 nm, metal reflector thickness = 50 nm, and cavity height = 240 nm. (b) The transition rates of the corresponding device. The stimulated emission rate starts to take over after 0.12 mA. (Inset) The turning point from stimulated absorption to stimulated emission (red).
the light output power as a function of the injection current (L-I curve) of a device with a cavity parameter L = 240 nm, a = 190 nm, SiNx shell thickness of 50 nm, b = 240 nm, and metal thickness = 50 nm is shown. According to the calculation, the output power is predicted to be 22 μW at 1.0 mA current injection with a threshold below 0.1 mA. In Fig. 18(b), the transition rates are shown at low current injection range. When the current passes 0.12 mA, the stimulated emission rate starts to take over and the injected power contributes mostly to the lasing mode. The inset shows a turning point of stimulate emission rate (red curve) from stimulated absorption to stimulate emission at ~30 μA. The turning point also represents the injection level where the transparent carrier density injection condition is reached (i.e., the net material gain is zero).

To illustrate the effect of metal reflectors, a calculation with various metal reflector thicknesses is performed. As shown in Fig. 19
Fig. 19 The light output power as a function of the injected current (L-I curves) of devices with different metal reflector thicknesses from 30 nm to 80 nm. A transition point when the stimulated emission rate equals the non-radiative recombination rate is plotted as the green symbols. Also, the transition point when the stimulated emission rate equals the spontaneous emission is plotted as the orange symbols.
, with a thinner metal reflector (30 nm), the output can be as high as 100 μW at the expense of a higher threshold current due to its lower quality factor. Further increase of the metal thickness will block the output power but reduce the threshold. After 50 nm, the turning point depicting the stimulate emission rate equal to the non-radiative recombination rate, which remains pinned with increasing current injection. Therefore, further increase of metal thickness will not change the threshold but simply reduce the output power. For an optimized design of power and low threshold, a nominal metal thickness should be around 40 to 60 nm. In real situation, not only the thickness of the metal but also the contact resistance will affect the performance. Optimization by inserting a low bandgap material such as InGaAsP in between the p-InP and silver interface can efficiently reduce the contact resistance; and, therefore, minimize the heating effect.

6. Conclusions

We have proposed and presented a new class of three-dimensional metal-nanocavity (nano-coin) semiconductor lasers, which will potentially operate at room temperature with electrical injection and emit more than micro-Watt light output power. A model with a multilayered core-shell structure with a perfect metal conductor coating is used to predict the resonance wavelengths and mode patterns inside the cavity. A numerical simulation using the FDTD method taking into account the lossy metal plasma is also performed to confirm our model. Our design is based on the fundamental waveguide mode (HE11) with both ends bounded by metals. The conditions for making a laser cavity have been analyzed, including waveguide cutoff wavelength, optical field, optical energy confinement factor, and output power optimizations. The effects of insulator layers and metal reflectors are discussed and optimized. We show how the insulator thickness affects the material loss and the corresponding shift in the resonance wavelength. In addition, an optimized thickness of the top metal reflector has been proposed, which optimizes the output power and threshold current. The theoretical treatments are universal and suitable for all metal-cavity nanolasers. The laser device performance and operation principles are presented through the analysis of rate equations. Our proposed surface-emitting 3D metal-nanocavity semiconducator laser and theoretical model will be useful for future size shrinkage of electrical-driven plasmonic nanolasers.

Appendix: theory of the HE11p mode in a circular core-shell-metal waveguide

The field expressions of HE11 mode in a PEC-surrounded waveguide shown in Fig. 5 are given by [19

19. J. A. Kong, Electromagnetic Wave Theory (EWM, 2008).

,24

24. S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]

]

Core   Region:  Ezc=AF1(ρ)sin(mφ)eikzz,
(16a)
Hzc=BF2(ρ)cos(mφ)eikzz,
(16b)
Eρc=1kρc2[AikzF'1(ρ)BiωμmρF2(ρ)]sin(mφ)eikzz,
(16c)
Eφc=1kρc2[AikzmρF1(ρ)BiωμkρcF'2(ρ)]cos(mφ)eikzz,
(16d)
Hρc=1kρc2[AiωεcoremρF1(ρ)+BikzkρcF'2(ρ)]cos(mφ)eikzz,
(16e)
Hφc=1kρc2[AiωεcorekρcF'1(ρ)BikzmρF2(ρ)]sin(mφ)eikzz,
(16f)
Shell   Region:  Ezs=CF3(ρ)sin(mφ)eikzz,
(17a)
Hzs=DF4(ρ)cos(mφ)eikzz,
(17b)
Eρs=1kρs2[CikzF'3(ρ)DiωμmρF4(ρ)]sin(mφ)eikzz,
(17c)
Eφs=1kρs2[CikzmρF3(ρ)DiωμkρsF'4(ρ)]cos(mφ)eikzz,
(17d)
Hρs=1kρs2[CiωεsmρF3(ρ)+DikzkρsF'4(ρ)]cos(mφ)eikzz,
(17e)
Hφs=1kρs2[CiωεskρsF'3(ρ)DikzmρF4(ρ)]sin(mφ)eikzz,
(17f)

where

{kρs2+kz2=ω2μεs,kρc2+kz2=ω2μεcore.
(18)

In the equations above, s represents shell region and c is the core region, kρ is the corresponding radial wavevector, kz is the propagation constant, and ω is the angular frequency. We have to choose proper Fi(ρ) to satisfy proper conditions. First, inside the core (ρ < a), the wave solution should be a regular function (no singularity), we find

F1(ρ)=Jm(kρcρ),
(19a)
F2(ρ)=Jm(kρcρ).
(19b)

Second, the wave solutions in the shell layer consists of two outgoing and incoming cylindrical decaying waves, which satisfy the boundary conditions that Ez and Eϕ vanish at the outer wall (ρ = b), i.e.,

F3(ρ)=Km(kρsρ)Im(kρsb)Im(kρsρ)Km(kρsb),
(20a)
F4(ρ)=Km(kρsρ)I'm(kρsb)Im(kρsρ)K'm(kρsb),
(20b)

where Jm is the Bessel function of the first kind and Im and Km are the modified Bessel function of the first and second kinds, respectively. Matching the boundary conditions of tangential field components at the core/shell interface, ρ = a, leads to,

AF1(a)=CF3(a),
(21a)
BF2(a)=DF4(a),
(21b)
1kρc2[BkzmaF2(a)+AωεcorekρcF'1(a)]=1kρs2[DkzmaF4(a)+CωεskρsF'3(a)],
(21c)
1kρc2[AkzmaF1(a)BωμkρcF'2(a)]=1kρs2[CkzmaF3(a)DωμkρsF'4(a)].
(21d)

The non-trivial solution for coefficients A, B, C, D exists when the determinant is zero, i.e.,

|F1(a)0F3(a)00F2(a)0F4(a)ωεcoreF'1(a)kρckzmakρc2F2(a)ωεsF'3(a)kρskzmakρs2F4(a)kzmakρc2F1(a)ωμF'2(a)kρckzmakρs2F3(a)ωμF'4(a)kρs|=0.
(22)

The resonant wavelength of the cavity associated with certain modes (mnp) can be obtained by solving equations Eqs. (18), (21), and (22) simultaneously. For modes with no azimuthal variation (m= 0), the characteristic equation Eq. (22) can be further simplified to

(εskρsF1F'3εcorekρcF'1F3)=0  for  TM0n,
(23a)

and

(1kρsF2F'41kρcF'2F4)=0  for  TE0n.
(23b)

Acknowledgments

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S. W. Lee, S. L. Chuang, and C. S. Lee, “Normal modes in an overmoded circular waveguide coated with lossy material,” IEEE Trans. Microw. Theory Tech. 34(7), 773–785 (1986). [CrossRef]

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E. Yablonovitch, R. Bhat, C. E. Zah, T. J. Gmitter, and M. A. Koza, “Nearly ideal InP/In0.53Ga0.47As heterojunction regrowth on chemically prepared In0.53Ga0.47As surfaces,” Appl. Phys. Lett. 60(3), 371–373 (1992). [CrossRef]

34.

Y. Zou, J. S. Osinski, P. Grodzinski, P. Dapkus, W. C. Rideout, W. F. Sharfin, J. Schlafer, and F. D. Crawford, “Experimental study of Auger recombination, gain, and temperature sensitivity of 1.5 μm compressively strained semiconductor lasers,” IEEE J. Quantum Electron. 29(6), 1565–1575 (1993). [CrossRef]

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OCIS Codes
(140.3410) Lasers and laser optics : Laser resonators
(230.7370) Optical devices : Waveguides
(260.3910) Physical optics : Metal optics
(140.3945) Lasers and laser optics : Microcavities
(250.5960) Optoelectronics : Semiconductor lasers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: April 18, 2011
Revised Manuscript: June 9, 2011
Manuscript Accepted: June 11, 2011
Published: June 23, 2011

Citation
Chien-Yao Lu and Shun Lien Chuang, "A surface-emitting 3D metal-nanocavity laser: proposal and theory," Opt. Express 19, 13225-13244 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13225


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References

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