Efficient 3D Simulation of Photonic Crystal VCSELs

Transcription

1 Efficient 3D Simulation of Photonic Crystal VCSELs Aditya Kulkarni, Vivek Krishnamurthy, Michael Povolotskyi, and Benjamin Klein Georgia Institute of Technology, 21 Technology Circle, Savannah GA 3147, USA ABSTRACT An efficient modular approach is used to develop components for a 3D simulator for complex semiconductor LED and laser structures. In this approach, only drift transport is simulated in bulk regions, while the active region is simulated with models of varying complexity. The approach is tested using a basic vertical-cavity surface emitting laser (VCSEL) structure, and comparisons are made with experimental data. This approach is advantageous for fast simulation of complex photonic crystal LEDs and VCSELs, for which 2D simulation is inadequate. Keywords: VCSEL, tetrahedral grid, modular approach, box integration, active resistor, Newton s method 1. INTRODUCTION Complex photonic crystal structures have been proposed for use in a variety of light emitting devices in order to exert greater control over the optical modes. For example, etched arrays of holes arranged into a 2d photonic crystal have been integrated into high-power LEDs to increase the light extraction efficiency. 1 Similarly, 2D photonic crystal defect cavities have been etched into the top DBR mirrors of VCSELs to promote singlemode operation at high output powers. 2, 3 Two dimensional cross section simulation techniques 4, 5 are clearly inadequate for the simulation of such complex light emitting devices. The modeling of simpler edge emitting lasers and ordinary LEDs could reasonably neglect regions away from the active regions of the device, as these regions are fairly uniform in their characteristics. This is not possible for photonic crystal structures. Comprehensive 3D modeling of the device structure provides the most generality, and such simulation techniques are being thoroughly investigated by several groups. 6 Several commercially available 3D simulators also use a comprehensive model for the entire 3D device, for example, PICS3D and TCAD Sentaurus. This involves solving a large number of nonlinear discretised equations in the entire device domain. These techniques require significant computational resources and time. In an iterated design or time limited scenario, the usage of a model with lower computational complexity at the price of a tolerable reduction in generality is desirable. In this paper we investigate such an approach. To reduce the complexity of computation, a hybrid model can be used wherein the problem is divided into parts. The parts are created on the basis of functionality separation, i.e. we can approximate the amount of coupling between them to be small. The tasks we describe here are the modeling of the bulk regions of the device, modeling of the active regions of the device and the simulation of the optical modes in the device. Several models of varying complexity can be applied to the active region in particular. The active region of the laser is treated as a set of parallel 1D non linear resistors. The bulk region in 3D is modeled with the drift continuity equation. This approach was inspired by the work of Evstratov 7 in the investigation of blue LEDs, and Osinski 8 in the study of VCSELs. In this way, the total number of device equations is reduced significantly as compared to the full 3d drift-diffusion model of the entire device structure. For now, it is necessary to neglect current spreading in the active region, but this approximation can be mitigated. 2. THEORY A simple VCSEL structure in three dimensions is depicted in Figure 1. This is accompanied by a discretised structure closely approximating the VCSEL structure. The discretized approximation made in Figure 1 is a model wherein the multiple DBR layers are modeled as a single effective material with equivalent conductivity. An oxide aperture is included in the VCSEL structure just above the active region. The active region of the device is modeled as a single plane in the discretised structure. This will be discussed further below in the section describing the effective resistor model of the active region. Physics and Simulation of Optoelectronic Devices XVII, edited by Marek Osinski, Bernd Witzigmann, Fritz Henneberger, Yasuhiko Arakawa, Proc. of SPIE Vol. 7211, SPIE CCC code: X/9/$18 doi: / Proc. of SPIE Vol

2 ILIIII1 ULIIU Figure 1. Example VCSEL structure and its discretized approximation The oxide aperture is treated as a non conducting region. A tetrahedral grid is used for the other regions of the device. The tetrahedral grid generator must satisfy the Delaunay criteria in order to be appropriate for use with the box integration method. A tetrahedral grid generator Tetgen 9 was used to generate these three dimensional grids Bulk region The regions far from the p-n junction form the bulk regions of the device. In the VCSEL structure modeled above, these include the DBR layers, the substrate and the oxide region. The bulk region is modeled using the drift current continuity equation. As described below, this relation reduces to a linear relation between electrostatic potentials at adjacent nodes with the aid of approximations such as constant conductivity in the region between two nodes. The continuity equation is,. J = ρ (1) t where J is the current density and ρ is the charge density. We have neglected recombination and generation in the bulk regions of the device for now. We simulate the device only in steady state, so the right hand side of Equation 1 is zero. In addition, we neglect diffusion current in the uniformly doped bulk regions. Therefore, Using J = σ E and E = φ we have. {σ φ} = (2) Here E is the electric field vector and φ is the electrostatic potential. The box integration technique is used to solve Equation 2 in its discretised form. We provide the final expression below. An analogous and detailed derivation can be found in 1 for the Poisson equation. φ n φ m σ i A ti = (3) d mn tet i, p i p j Here tet i represents the i th tetrahedron in the tetrahedral mesh, p m p n represents an edge in the tetrahedron between points m and n, and φ n and φ m represent the potentials at points m and n. d mn represents the distance between points m and n and A ti represents the partial area of the surface normal to p m p n in the tetrahedron i and common to the volumes enclosing the two points m and n. A Dirichlet boundary condition is applied to the nodes on the electrodes where the potential has a fixed value, and a Neumann boundary condition of vanishing normal derivative is applied on all other points of the outer surface of the simulation domain. Proc. of SPIE Vol

3 2.2. Active region The continuity equation in the bulk region is coupled to the active region model through the points on the top and bottom of the active layer. This is essentially boundary matching at the interfaces of the layers. The active region can be modeled with varying complexity depending on the desired speed and accuracy of the simulation. The simplest model, an effective resistor model, is described in detail below. A two dimensional plane coincides with the location of the active region in the device in our 3D tetrahedral mesh. The nodes on the plane representing the active region are duplicated to obtain their counterparts on the other side of the active region. We assume that the current spreading is minimal in the thin active region,so that each point on the active region plane is only coupled to its lower counterpart. In this way, the active region modeling is reduced to a set of parallel 1D equations, based on the model presented by Joyce and Dixon. 11 These should be solved consistently with the bulk region continuity equations. The problem is still nonlinear, but the nonlinearity is reduced significantly and restricted to the active region. This approximation may be lifted later to solve for current spreading in the active region. Bulk region Active region Bulk region 3D view Figure 2. Pictorial representation of the active region in the 1D model The resistors indicated in Figure 2 represent the 1D coupling between a point on the top of the active layer and a point on the bottom. Each of the resistors represents a relation between the current flow through the active region and the potential drop across the active region for that resistor (2D region). The charge carriers in the active region must satisfy the carrier balance equation. Finally, the current density obtained from the potential drop in the active region should satisfy the continuity equation in the adjacent bulk regions. These conditions are represented mathematically below. μ c + μ v + E g qu = (4) Equation 4 relates the external potential drop to the electrochemical potentials at the active region boundaries. μ c and μ v are the electrochemical potentials on the n and p sides of the junction region, E g is the band gap of the device and U is the electrostatic potential drop. The electrochemical potentials are related to the carrier concentrations by the following equation. The linear resistance of the active region is neglected in this equation. ( ) ( ) μ N kt =ln K2 N + K 1 ln +1 + K 3N (5) N eff N eff N eff where N is the carrier concentration in the active region, N eff is the effective density of states in the active region, and K 1,K 2 and K 3 are constants. Equation 5 provides a good approximation to the Fermi-Dirac Integral for parabolic bands. 12 The current density across this resistor is then calculated using the carrier balance equation. In steady state, assuming unity internal efficiency, dn w dt = J in q R st R sp R nr = (6) Proc. of SPIE Vol

4 where R st is the stimulated recombination rate, R sp is the spontaneous recombination rate, R nr is the nonradiative recombination rate (thermal and Auger recombination) and J is the vertical current density for a given area in the active region(representing one active resistor). The recombination terms are related to the carrier density by the following equations: R st = γg w N ph R sp = dbn 2 R nr = d ( AN + CN 3) (7) where g w is the carrier density dependent gain per unit area, N ph is the number of photons in the cavity, and γ is the optical mode confinement factor per unit volume. B is the spontaneous recombination coefficient, A is the coefficient for Shockley-Reed-Hall recombination due to defect states and C is the Auger recombination coefficient. The current density calculated in this manner acts as a boundary condition to the continuity equation in the bulk regions and should equal the current density leaving the p side of the active region interface and entering the n side. Finally we also have the optical part of the problem for which the photon rate equation is solved: dn ph dt = N ph τ c + G w N ph + R sp (8) where τ c is the photon lifetime, and G w is the gain integrated over the active region area. The confinement factor and the photon lifetime are dependent on the optical modes of the laser. These are obtained from an optical solver. A more advanced quasi-equilibrium model that includes a KP calculation of the quantum well bound states and recombination rates, as well as quantum well filling via constant quasi-fermi levels in the active region, is under development. This work will be described in a future publication. 3. OPTICAL MODEL ITa4 /Q c c \ / c c cm.25.3 c ED I JR (a) a> a>,8.6.4, Normalized frequency, f (in cia units) j Normalized frequency, f In c/a units) (b) Figure 3. Calculated transmission and reflection for a plane wave incident on the photonic crystal slab as shown. For photonic crystal structures, a planewave expansion based modal method 13, 14 is used to solve Maxwell s equations for the optical modes throughout the structure. In this method, the 3D structure is first split into various 2D slices which are uniform along the out-of-plane direction. Modes are calculated in each 2D slice and are propagated along out-of-plane direction through the use of scattering matrices. 15, 16 The usage of planewaves for expansion of layer modes enable us to embed a Fourier operator directly into the Maxwell solver, rendering the solver matrix-free. Computationally, O(Nlog(N)) operations are performed and O(2N) storage space is needed if N is the number of grids. The solver has been tested for various structures. An example problem that Proc. of SPIE Vol

5 was simulated is the calculation of scattering by a photonic crystal slab, as shown in Figure 3. Presently the solver is being integrated with the electronic solver for photonic crystal defect based VCSELs and LEDs. The modular approach requires the confinement factors and the photon lifetime, and these will be provided by the optical solver. Simpler optical models may be used for less complex structures. 4. RESULTS The simulation of the VCSEL shown in Figure 1 exploits the symmetry of the VCSEL, requiring only discretisation of one quarter of the VCSEL rather than the entire structure. The quarter section of the VCSEL is shown in the figure below. Since this is not a photonic crystal structure, the optical solver described in the previous section was not used to generate these results. Optical emission peleckode lop DOR., n... remcrrred region Bottom DBR Figure 4. Discretised structure of 1/4 th of the VCSEL shown in Figure 1 The device geometry and dimensions of the simulated VCSEL are specified below: Table1 : Device dimensions for 1/4 th of the VCSEL Length in x and y directions 5 μm Height μm z co-ordinate active region μm z co-ordinate aperture bottom 5. 7 μm z co-ordinate aperture top μm x and y co-ordinate of aperture 3. 5 μm x and y co-ordinate of top electrode 2. 5 μm Bottom electrode Entire bottom surface Conductivity of regions 2 and 3 85 Ω 1 m 1 Conductivity of region Ω 1 m 1 The I-V curves obtained for the laser device are shown below. A linear resistance of Ω was obtained for one quarter of the VCSEL structure (from the I-V curve). Proc. of SPIE Vol

6 E = a) S Voltaqe (V) aoi.1.1 le-5 le-6 le-7 le-8 le 9.1 Voltaqe (V) 1 Figure 5. I-V curves on the linear and logarithmic scale for the quarter VCSEL structure E C- a Current (ma) le-lo le 12 1 e-9 is-os I 8-7 Ie-6 I e-5.i.1 1 Current (ma).1 IC Figure 6. Optical power vs current on the linear and logarithmic scale for the entire VCSEL structure The values of the current are for the quarter VCSEL structure shown. This is in contrast to the optical power obtained below which is for the whole device. A simple scaling factor of 4 is used. The current density distribution and charge concentration in the active region is plotted below. The plot is a 2D plot in the plane of the active region. The current density is concentrated at the aperture. The carrier density shows significant spatial hole burning above threshold. The results obtained from the simulator can be compared to previous simulation and experimental results for a similar device; in particular, the simulation and experimental data from Liu et al. 17 Scattering loss in the mirrors and carrier recombination outside of the active region are not modeled in the current simulation. This explains the higher simulated output power (1. 8 mw at. 5*4=2. ma) as compared to the output power obtained in experiments. The lack of scattering loss in the mirrors also accounts for the low threshold current obtained in the simulation (. 63*4=. 252 ma) as compared to the experimental value of. 6 to. 8 ma. The threshold current will also increase if Auger recombination is considered in the simulation. 5. CONCLUSIONS Photonic crystal LED and VCSEL simulation requires a 3D model to capture the essential features of the problem. A full fledged semiconductor modelling of the entire device is computationally intensive and slow, and a simpler approach is desired. In this paper, we present such a model. The device is separated into bulk and active regions, and simplified models appropriate for each region are applied. Three dimensional models can be used to model the bulk regions of the device using bulk parameters such as conductivity. Since current flow is mainly in the vertical direction in the active region of the device, the active region can be modeled by resistors which carry current in the vertical directions and whose resistance is related to the potential drop using a non linear functional depending on the electrochecmical potentials in the heterostructure region. As shown in the results, such a model provides a reasonable comparison with experimental data even in its primitive form. The model can be further improved by simulating the active region more accurately. The existing simulator has low computation complexity with simulation times ranging from 124 seconds for a 45 node problem to 12 seconds for a 25 node problem on a 3.GHz, 1 GB sequential Linux machine. Proc. of SPIE Vol

7 L)= V J (ka'r,1} 18 IS U= V xl 18, N (urn2).45.4, ,5 Figure 7. Current density and carrier concentration as function of position in the active region REFERENCES 1. Ryu, H., Hwang, J., Lee, Y., and Lee, Y., Enhancement of Light Extraction from two-dimensional Photonic Crystal Slab Structures, IEEE Journal of Quantum Electronics 8(2), (25). 2. Yokouchi, N., Danner, A., and Choquette, K., Two-dimensional Photonic Crystal Confined Vertical-Cavity Surface-Emitting Lasers, IEEE Journal of Selected Topics in Quantum Electronics 9(5), (23). 3. Danner, A., Kim, T., and Choquette, K., Single Fundamental Mode Photonic Crystal Vertical Cavity Laser with Improved Output Power, Electronics Letters 41(6), (25). 4. Grupen, M. and Hess, K., Simulation of Carrier Transport and Non-linearities in Quantum-Well Laser Diodes, IEEE Journal of Quantum Electronics 34(1), (1998). 5. Witzigmann, B., Oyafuso, F., and Hess, K., Quasi-Three-dimensional Simulation of Carrier Dynamics in Quantum-Well DFB Lasers, Optoelectronics, IEEE Proceedings 145(6), (1998). 6. Witzigmann, B., Bregy, A., Michel, F., Odermatt, S., Santschi, R., and Streiff, M., A Full Threedimensional Microscopic Simulation for Vertical-Cavity Surface-Emitting Lasers, Numerical Simulation of Optoelectronic Devices, (25). 7. Evstratov, I., Mymrin, V., Karpov, S., and Makarov, Y., Current Crowding Effects on Blue LED Operation, Physica Status Solidi,Applied research 3(6), (26). 8. Osinski, M., Smagley, V., Lu, M., Smolyakov, G., Eliseev, P., Riley, B., Shen, P., and Simonis, G., Self- Consistent Calculation of Current Self-Distribution Effect in GaAs-AlGaAs Oxide-Confined VCSELs, IEEE Journal of Selected Topics In Quantum Electronics 9(5), (23). 9. Tetgen, A Quality Tetrahedral Mesh Generator and Three-Dimensional Delaunay Triangulator. "http: //tetgen.berlios.de/". 1. The Poisson Equation. " 11. Joyce, W. and Dixon, R., Electrical Characterisation of Heterostructure Lasers, Journal of Applied Physics 49(7), (1978). Proc. of SPIE Vol

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