LED Design, Optimization Using TCAD Modeling

TCAD modeling is widely used in semiconductor manufacturing simulation and analysis routines, from wafer track schedulers to yield management systems. The optoelectronics industry is now applying TCAD to its manufacturing processes, enabling control of process variations and improved yields.

Ricardo Borges, Wei-Choon Ng and Gergö Letay, Synopsys Inc., Mountain View, Calif. -- Semiconductor International, 4/1/2008

Semiconductor manufacturers are continually adapting to rising product complexity and technology development costs, shorter product lifecycles, and increased competition in a global marketplace. To meet these challenges, companies have deployed computer simulation and analysis in many semiconductor manufacturing disciplines.

Technology computer-aided design (TCAD) plays an increasingly important role in the development and optimization of new semiconductor technologies, and has been noted in the latest International Technology Roadmap for Semiconductors (ITRS) as key to reducing technology development costs and time-to-market.¹ Well-established in silicon-based microelectronics, TCAD has recently been successfully applied to various optoelectronic devices, including CMOS image sensors, solar cells, lasers, and light-emitting diodes (LEDs). Some of these efforts address manufacturing issues, such as electrostatic discharge (ESD) effects and yield analysis of vertical-cavity surface-emitting lasers (VCSELs).^2,3

Leveraging nearly three decades of R&D of models and numerical algorithms, including detailed optoelectronic models, TCAD has the power to accurately analyze the impact of process parameters on device characteristics and, thereby, address and control process variability as needed for manufacturing. This article demonstrates the capabilities of TCAD for LEDs in the framework of design for manufacturing (DFM) methodology.

Controlling variations is key

Comprehensive device models and the complete design flow approach permit quantification of process device relationships, allowing optimization of the process flow and yield improvement. The key underlying concept is the establishment of process device relationships, for which TCAD offers two primary advantages over more traditional or purely experimental approaches: the ability to precisely and comprehensively correlate process (independent) variables with parametric (dependent) variables; and the cost-effectiveness of simulations relative to running experimental wafers.

To assess the impact of process variation, it is necessary to control or measure the independent (i.e., process) variables with sufficient precision to minimize noise. This is difficult, if not impossible, to do experimentally thanks to a lack of sufficient control in setting the variables and measurement uncertainties in characterization methods. Furthermore, even if such experimental studies were feasible, they would incur prohibitive costs and time-to-results.

TCAD allows the assessment of the impact of process variations on device performance, but such a methodology requires predictive TCAD models and innovative approaches to reduce the computational cost of simulations in a time-constrained manufacturing environment.

TCAD models for LED simulation

The development of a simulator for LED design and optimization must cover a range of electrical and optical phenomena. The models for semiconductor transport in bulk materials and across heterojunction interfaces are well-known and have been suitably adapted to LED simulation. The electrical transport of holes and electrons requires the combined solution of the Poisson, continuity and thermal equations together with associated mobility, recombination and generation models. Carrier scattering into the quantum wells (QWs) requires special treatment. Commercial simulators typically use a relatively simple scattering model that has to be calibrated with measurement data.⁴ Moreover, QW capture should be treated in conjunction with the gain model, which, through the Kubo-Schwinger formula, is directly related to the spontaneous emission spectrum.

Two main ingredients are necessary to ensure accurate and predictive gain respective to spontaneous emission calculations:

• An accurate ban-structure calculation to obtain the energy dispersion and wave functions of the charge carriers. The most widely used method is the k•p theory. Stress and strain of the material must also be included in the formulation of the k•p Hamiltonian.⁵
• Advanced models for stimulated and spontaneous emission. Comparing a hierarchy of models, including the free carrier theory, the screened Hartree-Fock theory, and the second Born approximation, it was shown that advanced models are necessary to attain accurate results.⁶

In addition, for nitride-based QWs, spontaneous and piezo polarization occur at interfaces, resulting in a tilted energy band of the QWs.⁷ Because of the tilt, the electron and hole wave functions are shifted in such a way that their overlap integral is reduced, decreasing the optical emission. However, the spatial separation of electron and holes builds up an electrical field in the opposite direction of the polarization field, resulting in a screening of the latter one. In the k•p approach, the polarization can be taken into account by introducing interface charges and self-consistently solving the Poisson and Schrödinger equations. Research demonstrates the successful calibration of indium gallium nitride (InGaN) gain curves, including piezoelectric screening, using the TCAD-based Sentaurus Device simulator.⁸

Creating optical models

Because of the relatively large dimensions (a few hundred microns across) of LEDs, ray optics approaches such as raytracing are typically employed and accepted as valid optical solutions. Using a spectrally resolved raytracing method allows spontaneous emission spectra to be embedded in the simulations and then evolved as the rays traverse regions of stimulated gain, absorption and luminescence.

There are four possible spectral evolution processes that give rise to a modified spontaneous emission spectrum: amplification by stimulated emission; absorption with subsequent electron-hole pair generation; reemission; and spectral conversion. The first three processes constitute a novel photon-recycling model that includes amplified spontaneous emission (ASE). Because brighter LEDs require higher-current injection, the active region (QW) is pumped with enough carriers for stimulated emission to occur. Stimulated emission will amplify the spontaneous emission spectrum, giving rise to ASE. Although ASE has been extensively studied in resonating structures, such as lasers, optical amplifiers, resonant-cavity LED (RCLED) and super-luminescent LED (SLED), the inclusion of ASE in the non-resonant structures typical in modern white LED design has not been fully explored, and is now enabled by this novel photon-recycling model. The photons’ production/absorption and carrier recombination/generation are carefully balanced to ensure particle conservation when a net photon-recycling recombination rate is fed back into the continuity equations. This enables the important physical effects in white LED structures to be captured in a fully coupled and self-consistent electro-opto-thermal simulation.

Today, the most popular and cost-effective way to produce white-light LEDs is through spectral conversion. Typically, a phosphorus coating or filling material down converts blue light to yellow or green, which, together with the blue emission, add up to “warm” white light. The spectral conversion can be accounted for by using spectrally resolved raytracing and specifying appropriate absorption and emission profiles.⁹

LED manufacturers apply increasingly complex surface treatments to enhance optical out-coupling and shape the far-field pattern. Where geometric optics of the raytracer are violated, the conventional Fresnel equations used to calculate refraction angle, reflection and transmission coefficients can be replaced in many cases by other adequate algorithms. For instance, scattering at rough surfaces can be captured by suitable diffuse reflective models, such as the Phong model,¹⁰ or the reflectance coefficients of thin coatings can be calculated by the transfer matrix method.¹¹

For some advanced LED concepts (e.g., SLEDs, which are edge-emitting devices that exploit stimulated emission of the cavity), the raytracer approach is not valid. However, even where raytracing fails, suitable TCAD approximations exist.¹² All of these peculiarities of LED physics are addressed by the Sentaurus Device simulator.¹³

Process compact models

TCAD tools employ meshes and numerical methods to solve the physical models and partial differential equations described herein. These tools are highly flexible, adapting to a wide variety of process technologies and device structures, and predictive, because the physical model parameters can be calibrated to specific technologies. However, solution times are typically too long to meet the near instantaneous demands of manufacturing environments. Moreover, to achieve a fast, efficient design and manufacturing process, it is inevitable that the properties of the device and its behavior with respect to parameter variations are passed on to engineers who may not be device experts — process engineers, yield analysts, etc. It is therefore essential to provide an easy-to-use but accurate device model where changes in input parameters, such as layer thickness and composition sidewall angles or the phosphorus concentration in the spectral converter, are immediately translated to realistic device performance characteristics.

Created to bridge this gap, process compact models (PCMs) are analytic, multi-dimensional surface-response models containing the non-linear parameter interactions of the entire manufacturing process, as well as the relevant range and accuracy information about the variables. While real experiments are cost-intensive and time-consuming, and TCAD simulations can take hours, a PCM can be evaluated almost instantaneously, making it well-suited for advanced statistical data analysis.

PCMs are extracted from systematic TCAD simulations for a specific technology and over a range of process conditions. They typically include internal parameters in the statistical analysis that are not accessible to measurements, such as the optical extraction efficiency, QW carrier densities, or temperature and current distributions. In this way, the design engineer gains insight into the importance of particular parameters in the context of manufacturing. Although TCAD experiments allow controlling process parameters largely independent of the feasibility and cost constraints of the manufacturing domain, manufacturing and cost constraints can be built into the PCM. Additionally, in a simulation domain, there is perfect control over environment and boundary conditions, and simulations are reproducible.

A typical PCM extraction flow consists of the following steps

1. Definition of a calibrated TCAD flow
2. Analysis of process parameter sensitivity
3. Simulations of process/design splits
4. PCM generation

In Step 1, process and device simulation flows are defined, while Step 2 studies the sensitivity of output device characteristics to changes in the input process parameters. This helps with selection of the process parameters to be included in the model and the appropriate ranges (i.e., the definition of the process splits). Step 3 is the execution of the process or design split simulations, which can be done in parallel across distributed machines on a network. Finally, in Step 4, PCMs are generated by fitting the extracted data to suitable polynomial or neural network-based functions.

PCMs are analogous to circuit compact models (SPICE), but when applied to semiconductor technology development and manufacturing they generate a set of responses (device characteristics) for a given set of parameters (process parameters, Fig. 1).

1. For any given set of process parameters, PCMs generate a set of responses.

The multi-dimensional data PCMs generate can be visualized in pairs using scatter plots or directly in many dimensions using parallel coordinate plots. Parallel coordinate plots, originally proposed by Inselberg,² are an effective way to display multi-dimensional data in one representation. Each variable is assigned its own vertical axis. With all the axes drawn parallel to one another, each observation (or sample) is then represented as a series of line segments intersecting each vertical axis at the corresponding values for the variables. This way of visualizing multi-dimensional data greatly enhances human pattern recognition.

TCAD provides process and device engineers with a systematic way to optimize process parameters to meet certain device performance characteristics. It is likely that the parameters leading to the highest achievable performance reside in a highly sensitive (narrow) process window, whereas a less sensitive (broader) process window still achieves fairly good performance (Fig. 2).

2. The parameter value leading to the highest achievable performance typically resides in a highly sensitive (narrow) process window, whereas a less sensitive (broader) process window often achieves fairly good performance.

This situation leads to an interesting trade-off decision: Should one run the process in a region of higher manufacturability at the expense of some performance? PCMs are very useful for assessing these kinds of trade-offs.

Applying PCMs to blue LED design/optimization

To demonstrate the use of PCMs for LEDs, let’s look at a generic InGaN LED with a truncated pyramid structure emitting in the blue. The 2-D model describes a typical InGaN-on-silicon carbide (SiC) LED. To enhance the out-coupling efficiency, a truncated pyramid structure is realized as the socket. Standard drift diffusion with Shockley-Read-Hall (SRH), radiative and Auger recombination, together with doping-dependent mobility and thermionic emission at hetero interfaces, were used. For the optical modeling, the housing was included because it has a strong influence on the optical far-field pattern. Doping-dependent absorption coefficients are used to account for free carrier absorption (FCA) in highly doped regions.

The active region contains four undoped InGaN QWs sandwiched between undoped GaN barriers emitting at 470 nm. An aluminum GaN (AlGaN) blocking layer on the p-side prevents electrons from reaching the p-side to reduce carrier recombination outside the active region.

Many different parameters, such as QW composition, doping and thickness, have an important influence on the device behavior. Some of them can be fixed or constrained through priori calculations, such as the QW thickness and composition, if a certain wavelength is desired. Others are more complex and show strong interaction on different levels. For example, on one hand, the top p-spreading layer’s doping and thickness influence the lateral spreading and, hence, lateral injection into the active region. On the other hand, it has a strong influence optically through FCA on the out-coupling efficiency, which is a complex function on the whole geometry, including packaging.

To identify the most important parameters of a device, a so-called parameter screening is performed. First, the list of potentially important parameters is compiled and a meaningful range for each parameter is defined. Second, the list of result variables has to be identified; Table 1 shows the list of input parameters and result variables for the investigated LED. TCAD simulations are then performed by individually varying each parameter while keeping the other parameters at their default values.

After parameter screening, the two parameters with the greatest influence are chosen to further improve device performance. These are quantum barrier doping (N_qb) and front contact width (w_cont). A full-factorial design of experience has been chosen to resolve the cross-interaction. A second-order Hermite polynomial fit function has been chosen for PCM construction. The PCM is then evaluated for 1000 randomly chosen and uniformly distributed values of the input parameters.

To illustrate the optimization process, the wall-plug efficiency (WPE) is maximized while keeping the wavelength in the range of 469-471 nm. By successively applying constraints to the model, the number of successful experiments — that is, process conditions satisfying the design constraints — decreases from the initial value of 1000. First, a lower limit is applied to the WPE, then manufacturability constraints are applied (Fig. 2). An unconstraint sensitivity analysis is performed for each candidate experiment and the variance is evaluated. The experiment with the lowest variance and highest WPE will result in the best combination of high performance and manufacturability. Therefore, a normal distribution during manufacturing is assumed, with a standard deviation of 0.2 and 0.02 for lg(N_qb) and contact width, respectively. Tolerating a variance of 10% in wavelength and <5% in WPE, a moderate barrier doping level around 5 × 10¹⁷ cm^-3 and a contact width of 25 µm are most promising. The process window in Figure 3 shows all result variables overlaid in one contour plot defined by barrier doping and contact width. Parameter combinations inside the remaining white area satisfy the specifications.

3. When multiple device specifications are combined, the acceptable process window (white region) is defined.

Process recentering and reduction of process variability are key steps in continuous yield improvement. This can be reflected in the PCM evaluation by reducing the standard deviation of input parameters. Now the question arises: Where will the process engineer invest time in improving the process to reduce standard deviation and increase yield?

Barrier doping and QW mole fraction are assumed to have a normal distribution with standard deviations of 0.2 and 0.02, respectively. For simplicity, we assume that the process improvement would result in half of the standard deviation for both process steps. Suppose that the specifications for a particular device class are constrained by WPE >9% and 465<w<475 nm. For the evaluation, 1000 experiments for the fluctuation of barrier doping and contact width are performed. The underlying PCM is constructed in a manner similar to that described in the previous section, but using barrier doping and QW mole fraction as input parameters. Counting the number of experiments fulfilling the constraints enables the yield to be easily determined. In our example case (Table 2), the yield can be improved from 23.3% to 29.7% if the standard deviation of contact width, respectively barrier doping, can be cut in half. If both processes were improved, the yield would more than double.

Conclusions

In silicon-based microelectronics, use of TCAD models is well-established for both the design phase and manufacturing process. Now, the optoelectronics industry, which has typically restricted simulation to the device development process, is following suit in using TCAD to resolve manufacturing issues. This is caused by the technology’s ability to accurately analyze the impact of process parameters on device characteristics, thereby enabling control of manufacturing process variations and increasing the device maker’s ability to maximize yields.

Performing design variations on a hypothetical blue InGaN LED illustrated how the concept of process compact models can assist the design process in exploiting parameter screening, multi-dimensional optimization and sensitivity analysis. In addition, the influence of overall yield was analyzed to select potential manufacturing steps for target-oriented enhancement.