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 0.01λ0 3 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 Optical Society of America
1. Introduction and proposal of a metal-nanocavity (nano-coin) semiconductor laser
There are a lot of advantages of making ultrasmall lasers such as thresholdless current with almost zero power consumption. Compact nanocavities such as photonic crystals or distributed Bragg reflectors (DBR) have been used to tightly confine the optical field in a tiny volume [1,2]. Because at least a few pairs of these feedback structures are required to achieve a high quality factor, the sizes are usually several times larger than the wavelength. In the past few years, plasmonic structures have been extensively studied for photonic applications, especially for manipulating optical fields within a subwavelength scale. Metals exhibit a large negative permittivity at the optical frequency due to the interaction of electromagnetic fields with free electrons. Nanolasers with a metal cavity can reduce the optical volume below the diffraction limit with the help of surface plasmon resonance . Also, with the use of the metal as an optical isolator, it is possible to have a crosstalk free environment for dense photonic integration. The challenges for practical applications are how to realize continuous-wave operation by electrical injection at room temperature, light collection, performance, and integration. A few types of metal-cavity micro/nanolasers have been analyzed theoretically or demonstrated experimentally [4–14]. Designs with metal as parts of the cavity wall and feedback structures to compress the optical mode have been demonstrated by optical pumping from low to high temperature [8,9,11]. A subwavelength metal-clad disk laser  with a metal reflector under the cavity, similar to that of the nanopan laser  has been analyzed by considering the TEmnp modes. Room temperature operation of metal-cavity lasers by optical pumping was demonstrated with an optimized plug-in structure for both mode confinement and metal loss reduction . In 2007, Hill et. al. demonstrated the first metal-cavity nanolaser under CW current injection at 78 K with the idea of cutoff to improve the mode confinement ; and in 2009, demonstrated a Fabry-Perot type laser which operated under pulsed current injection at 298 K . Due to the strong distortion of optical fields in certain directions, those metal-cavity nanolasers all suffer from low output power and poor beam shape. With the full coverage of metal, light can only emit through the substrate, which makes the collection inefficient [11–13]. Vertical-cavity surface-emitting lasers (VCSELs) have been studied extensively during the past few decades, and significant improvements have been made such as small active volumes (for example, a 1.5 μm diameter micropillar with a volume of ~8 λ0 3), high-power arrays, and high speed applications [15–17]. To further reduce the size of VCSELs, the high reflection spectral window of DBR will become extremely sensitive to the modal dispersion, and the scattering loss from the sidewall irregularity will be important. Recently, metal-cavity surface-emitting micro/nanolasers with DBR as part of the feedback structure have been proposed and realized under continuous-wave current injection at room temperature [4–7]. With the surface-emitting configuration, the performance of metal-cavity lasers has been significantly improved, including microwatt output power, ultra-narrow linewidth, circular beam shape, and extremely low thermal impedance. From an application point of view, a compatible platform for advanced stacking with integrated circuits is also an important issue . The metal-cavity surface-emitting nanolasers are substrate-free with the transferability to other platforms. A further improvement with hybrid metal-DBR mirrors of these metal-cavity lasers is demonstrated experimentally [6,7]. Even though several metal-cavity lasers were demonstrated, the fundamental limit of the smallest realizable metal-cavity nanolaser is still under investigation.
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 [4–7] or cutoff waveguides [11–13]. 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 , 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
In a metal-clad inhomogeneous circular waveguide, the normal modes are either hybrid mode (HEmn or EHmn) for azimuthal order m ≥ 1 or transverse mode (TE0 n or TM0 n) for m = 0, where n represents the order of radial distribution [19,20]. Figure 2 shows such a circular waveguide with an inhomogeneous radial distribution (In0.53Ga0.47As semiconductor core with a SiNx shell and a silver metal wall) and its corresponding mode patterns (in the basis of cos(mϕ) and sin(mϕ)) calculated by the finite element method (FEM). To have a large modal gain, a mode with good overlap between optical field and core gain material should be considered. Figure 2 lists our calculated field distribution (electric field norm: |E|; magnetic field norm: |H|) of the first 5 lowest order modes (HE11, TM01, HE21, TE01, EH11) at 1550 nm with an core radius of 250 nm and a shell thickness of 50 nm. For the transverse mode TM01 (TE01), both the magnetic (electric) field and its resulting time-averaged power along the waveguide direction (Pz) have a node at the center due to the nature of Bessel functions and their lack of longitudinal magnetic (electric) field components. Since the periphery of the waveguide usually degrades during the fabrication, configurations with a node at the center will result in an inefficient interaction between the gain materials and the optical fields when propagating back and forth along the z-direction. Optical modes with m > 1 are not suitable here because of their whispering-gallery behaviors, which usually experience more perturbation from the periphery than the lower order modes and have a power node at the center.
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,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,
In the descending order of cutoff wavelengths, the dominant modes are HE11, TM01, HE21, TE01, and EH11. Figure 3(a) 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 [21,22]. 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 .
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 core/λ0, where λ0 is the free space wavelength, or equivalently, the effective index n eff (defined by k z = (2πn eff /λ0)) 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) 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 , the fields of the mode (mnp) in the cavity can be written as a sum of two counter-propagating waves,24–26]:
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) shows the resonance wavelengths of the six lowest cavity modes (TM010, HE111, TM011, HE211, TE011, EH111) calculated by finite-difference time-domain (FDTD) method . 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 [21,22]. 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 a PEC homogeneous cavity (simplified model for understanding), kz is quantized to pπ/L, and Eq. (1) leads to the modal characteristic equation,Fig. 6(b) with the slope representing (p/2nop)2, where nop is the effective index (instead of n core) seen by the optical field. In Fig. 6(b), the outer radius (core radius a plus shell thickness s) is labeled as b.Although the resonant wavelength of HE111 mode is slightly larger than those of the other modes (except TM010) and will result in a slightly larger cavity size, the corresponding waveguide mode (HE11) has the longest cutoff wavelength as shown in Fig. 3(a) and allows for a wider range of detuning. The nonzero slope of TM010 mode comes from the inhomogeneous permittivity inside the cavity and the asymmetry in z direction. Moreover, HE111 mode is the most suitable mode for achieving a high gain and optical field overlap. In the following analysis, we will focus on the design for the HE111 mode.
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 [19,23]. 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 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 [21,22]. Drude model is adopted to fit the metal permittivity for simplicity.
In Fig. 8 , 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(mϕ) and cos(mϕ)). 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 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 – iω 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 . 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 [19,28],29],24],Fig. 10 . The radiation loss has a minimum which originates from minimum energy coupling to a resonant circular slot structure near the output facet . 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 . 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 ,Fig. 11(a) , 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,
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 [1,2]. The physical sizes are summarized in Fig. 12(a) . 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 ,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 . 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 . 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 [24,32]. 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) 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 [24,32]. 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 [19,30],Fig. 15 , 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 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 [8,14] 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)33] and As is the surface area of the active region. The Auger coefficient is set to be 4.67 × 1027 cm6s−1 . 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)  and fc, k and fv, k represent the occupation numbers at wave vector k in conduction and valance bands.
Figure 17 shows the carrier-dependent material gain spectra of the bulk In0.53Ga0.47As semiconductor calculated by Luttinger-Kohn model at room temperature . 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) 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 , 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.
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
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
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.,
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,
The non-trivial solution for coefficients A, B, C, D exists when the determinant is zero, i.e.,
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
This work was sponsored by the DARPA NACHOS Program under Grant No. W911NF-07-1-0314. We thank many insightful discussions with Dr. Shu-Wei Chang formerly at UIUC and Professor Dieter Bimberg at Technical University of Berlin.
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