Fabrication Process Stochastic Model for Yield Estimation in Microwave Semiconductor Devices Production
Fabrication Process Stochastic Model for Yield Estimation in Microwave Semiconductor Devices...
Paszkiewicz, Bartlomiej K.;Paszkiewicz, Bogdan;Dziedzic, Andrzej
2022-04-29 00:00:00
Hindawi Journal of Engineering Volume 2022, Article ID 5561059, 7 pages https://doi.org/10.1155/2022/5561059 Research Article Fabrication Process Stochastic Model for Yield Estimation in Microwave Semiconductor Devices Production Bartlomiej K. Paszkiewicz , Bogdan Paszkiewicz , and Andrzej Dziedzic Wroclaw University of Science and Technology, Faculty of Electronics, Photonics and Microsystems, Wroclaw, Poland Correspondence should be addressed to Bartlomiej K. Paszkiewicz; bartlomiej.paszkiewicz@pwr.edu.pl Received 10 February 2021; Accepted 16 February 2022; Published 29 April 2022 Academic Editor: Amiya K. Jana Copyright © 2022 Bartlomiej K. Paszkiewicz et al. �is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. �is paper presents a new methodology for simulation of production processes in order to determine device parametric yield. �e elaborated methodology is focused on capturing stochastic relations between every parameter of the subsequent processes that are impossible to determine directly. �e current state-of-the-art together with the gaps in the knowledge regarding yield modelling is presented. A novel approach to the important issue of e…ective yield modelling that allows the overcoming of current challenges is presented. �e methodology’s usefulness is validated with the example of the fabrication process of AlGaN/GaN HEMT (high electron mobility transistor) for application in high-frequency electronics. Fabrication of AlGaN/GaN HEMTs is a complex process due to the large number of technological stages required, most of which are still the subject of ongoing research. Most importantly, the approach presented in this paper could be easily applied to the modelling of any complex production process in every case where it is necessary to examine relations between the ”nal product parameters distribution and the values of the involved process parameters. �e yield analysis has to be performed comprehensively 1. Introduction in every stage of device design and fabrication. Typical �e characteristic feature of the semiconductor devices methods of yield analysis impede introducing the necessary manufacturing industry is multistage, in•exible processes, optimizing changes, due to the large amount of data re- which demands sophisticated and high-cost equipment and quired, which can be collected only after ”nishing the fabrication process deployment. �e methods usually consist utilizes expensive materials. Although demanding, the in- dustry is important economically. �e size of the global market of two stages. In the beginning, results of technological for semiconductor devices in 2016 reached 339 billion USD processes are collected, and later, the various statistical with further projected growth dynamics in 2017 of 11.5%. An models are ”tted to the data [1–3]. Unfortunately, the scope additional trend that impacts its operation is a constant de- of possible changes that can be implemented on the base of crease in capital margins. �us, every possible means of cost the formulated conclusions is rather limited, because of the reduction should be explored and applied for developing and aforementioned in•exibility of the manufacturing cycle. �e protecting the competitive advantage of a company. Essential visible need for faster knowledge implementation regarding for reaching this goal is the incorporation into the design and the predicted device yields in the design of the semicon- fabrication process of semiconductor devices as well as analysis ductor, and their fabrication process constitutes the main of both parametric and functional yields. �is allows for de- research motivation of this paper. In the presented paper, the elaborated model of the semiconductor device fabrication cisions optimization in numerous areas: device design, tech- nology node selection, and the choice of fabrication process process is presented elaborated on the experience of aca- parameters values, which should result in the increased eco- demic research, and development laboratory involved in the nomic e…ectiveness of the developed solution both in terms of design and fabrication of modern devices based on com- production chain optimization and ”nal product margin. pound semiconductors. �e model described below enables 2 Journal of Engineering the simulation of the whole process starting from separated y � f A , D , (2) n n c o procedures. It was applied to analyze the parametric yield of AlGaN/GaN HEMT dedicated for high-power RF switching where A is the size of the chip critical area and D is the unit c 0 at C (4–8 GHz) and X (8–12 GHz) bands. AlGaN and GaN defect density. *e range of f functions family is wide and are nitride alloys of metals from the third group of the includes different statistical approaches to the described periodic table (respectively, aluminium and gallium). *ey process. Model calibration is performed based on the pro- are widely used in the semiconductor industry, due to their duction data analyses and allows for accurate functional unique combination of features: high-temperature con- yield prediction of the semiconductor devices and integrated ductivity, wide and straight bandgap as well as the occur- circuits (ICs). Recent research in this area is focused on rence of piezoelectric effects. including mathematical models of learning curves and *e main objective of the performed research was to predictors of stochastic variability in the function, f [1, 4, 5]. establish the statistical distribution of transistor drain cur- Although it is commonly utilized in various analyses, rent in production batches. Particularly important was the functional yield due to its specificity has limited applica- determination of the impact of an early stage technological bility. Functional yield is the scalar measure, describing only process–namely the selection of AlGaN/GaN HEMT-type the percentage of acceptable elements after the single pro- heterostructures grown by the MOCVD (metal-organic cess, losing details of the technology complexity of semi- chemical vapor deposition) technique on the sapphire conductor devices fabrication technology, thus providing no substrate. *e knowledge of the statistical distribution of information about distributions of the values of their op- transistor drain current in production batches is essential for erating parameters. Such information is required to optimize cost reduction and yield increase. device design and fabrication process in regard to their It is obtained by eliminating wafers with deposited het- performance and manufacturing cost. erostructures whose parameters do not guarantee the fabri- Measure describing device parameters distribution is the cation of devices with expected parameters. Moreover, the parametric yield that is represented by the probability density developed approach enables the comprehensive description functions set of random variables {X }, where any X rep- i i of available knowledge, including all involved areas: applied resents some output parameter measured either between technology, device design, element design, characteristics of processes or of the finished device. In the case of AlGaN/GaN specific equipment, process parameters, applied materials HEMT, parameters yield analysis includes distributions of and reagents, measurement methods, environmental influ- maximal drain current (I ), pitch-of voltage U , cut-off DSS p ence, as well as the staff’s knowledge, experience, and their frequency f , MAG (maximal available gain), and other individual learning curves. An additional advantage is the parameters. Contemporary methods of semiconductor de- possibility of incorporating this analysis early, during the vices and IC parametric yield analysis in most cases utilize design phase or even during the selection of the technology. elements of big data methods combined with statistic re- gression models [3, 5, 6]. Complex GLMMs (generalized linear mixed models) are applied, allowing for the description 2. Yield Analysis Methods of the nonlinear dependencies between variables. Such an approach, although effective in a production environment, is In the literature concerning semiconductor manufacturing, hard to apply into research and development of device design few kinds of yield are specified along with different phases and technology, due to the limited volume of experimental during production when being measured [4]. *e main data available and the high cost of their acquisition. Ap- division lies between functional and parametric yield. proach taken in such cases is based on TCAD methods and Former called catastrophic yield is considered fundamental simulation of respective processes [7–9]. It enables to acquire and most widely used, due to the advanced mathematic data regarding specific relations that influence device op- theory of its description. In basic terms, the functional yield eration and its parameters distribution without conducting for a given manufacturing process is defined as the ratio of numerous experiments. *ough, the limitation of this good elements at the output stage of the process to the methodology is challenged by a precise description of rela- number of elements at the input. In the case of AlGaN/GaN tions between parameters in the scope of the whole devices high-electron-mobility transistors (HEMTs) fabrication, a fabrication process. Consequently, due to the complex list of main processes includes cleaning procedures, epitaxy, characteristic of involved relations, there is impossible to lithography, etching, and metallization deposition, and their develop tools that use purely analytical formulas and capture electrolytic thickening, passivation, to the separation and the stochastic characteristic of modeled fabrication process. packaging of the whole chips. To conclude, the functional In the frame of the conducted research, the approach was yield is a scalar measure of process efficiency. From a single proposed that enables the unification of all aforementioned yield of respective processes, denoted y , one can determine methods: big data analysis, TCAD simulations, and heuristic the yield, Y, of the whole process as observations about respective processes that are based on the experience of the research team, as well as available literature Y � y . (1) data. *is approach fits into methodologies reported in n�1 various publications applying big data methodologies in- In general, the single functional yield, y , is determined cluding machine learning methods [10, 11]. However, it by some function f with parameters A , D : diverges from this method by focusing on including n c 0 Journal of Engineering 3 conclusions based on analysis of small scale targeted ex- Fabrication Process Parameters periments. �e approach is based on developing a frame- Pearson Correlations work that will be capable of capturing all of the involved relations together with correlations between them and ef- Normal Distribution ”ciently generating possible realizations of such a stochastic Standard Deviation σ model. It ”ts into the wide scope of Monte Carlo methods. �e next performed step is to analyze acquired data with the Mean μ Dependent Variable aim of statistical inference about the in•uence of respective Relation e.g: "exp (n/kt)" processes variables on operating parameters distributions in Discrete Distribution HEMT, during its designing and manufacturing. Discontinues in CDF function 3. The Developed Model Overview Figure 1: Proposed scheme of variables representation and cor- relations between them. In order to analyze the parametric yield of AlGaN/GaN HEMTs, a novel approach was applied. It is based on created from scratch universal simulation software framework that Additionally, feedback coupling was predicted between operates on the textual representation of the manufacturing the software part, responsible for samples generation, and process. �e representation includes all important techno- the model that can be used for self-calibration as well as for logical processes parameters, results of interprocess mea- performing statistical Bayesian analysis [12]. �e prototype surements as well as ultimate device operation parameters. of the software framework was developed using Python During the development, a number of assumptions were language and is fully functional allowing for model de- made, in order to maximize the •exibility of generated scription loading and transformation into memory objects, predictions. It was necessary, considering the iterative na- generating a stochastic realization of the process, as well as ture of conducted research. �ey result in an incremental saving and data acquisition. Applied mathematical functions increase of available knowledge regarding AlGaN/GaN are from the commonly acclaimed scienti”c computing li- HEMTs manufacturing and design. �e aforementioned brary SciPi (0.18.1). Essential, for the model application, is an reasons force the initial requirement that in the model all e¦cient algorithm for model realization generation. It re- involved parameters can be random variables with arbitrary quires using a method capable of generating samples from continuous or discrete distribution, and for a given random arbitrary cross-correlated distributions both continuous and variable, it should be su¦cient to state solely the arbitrary discrete. cumulative distribution function. Furthermore, every input variable can be freely cross-correlated with each other. 4. Model Structure Regarding the dependent variables derived from model parameters described previously, it will be possible to in- �e model is the description of subsequent manufacturing clude them in the model as arbitrary nonlinear explicit process steps by a number of representative variables, X n,m functions, whose coe¦cient can be also stochastic variables. whose values are realizations of certain stochastic processes. �e dependent variables in the model usually are inter- For example, the expected epitaxy temperature is 800 C; process measurements results or ultimate device parameters. however, the actual temperature of the process is subjected Introduced above described requirement of the possibility to to variability due to mechanical and electrical factors. �e utilize an arbitrary distribution creates challenges regarding distribution of obtained temperatures among the number of the computing complexity of random samples generated by processes can be described with Gaussian distribution. �e the Monte Carlo method. �ough simpli”cation was same regards for all of the other parameters. After every step, implemented, every continuous distribution that is non- a number of resultant output parameters, X are gen- n+1,k normal will be discretized. Possible errors and inaccuracies erated using arbitrary nonlinear functions of X For ex- n,m. created by this approach can be neglected, because of the ample, in the case of the epitaxy process, the resulting sheet possibility of freely increasing discretization density. It is resistance is calculated from the process temperature de- possible, due to applied search algorithms of the compu- scribed above, together with •ows, and composition of tation complexity of the class O (log n). In Figure 1, the reagents. �e function relating all involved parameters is scheme of the proposed model of variables representation nonlinear, and not all input parameters have Gaussian illustrating the correlations between them is presented. distributions; as a result, obtaining analytical distribution is �e entire novel part of the created methodology is the impossible, although by applying Monte Carlo simulation development of a method to textually represent model one can obtain stochastic dependence between variables. parameters. A special XML structure was developed to Consequently, obtained resultant parameters can become satisfy this need. Due to this, the main risk for the imple- input parameters for the next process steps. �anks to this, it mentation of the modern design support and knowledge is possible to analyze a complex multistage production management tools was mitigated, which is extensive IT process with a number of branches. knowledge required in order to e…ectively use them. In the �e number of simulation steps is unbounded and allows developed form, the model can be used and modi”ed by the for the description of the complete manufacturing process. person without special training. Ultimate results of interest are distributions of dependent 4 Journal of Engineering variables in the last process step, which represent device Model components operation parameters. �eir distribution together with the possibility to calculate correlation with input parameters Normal Distribution Discrete Distribution represents a directly parametric yield of fabrication process Variables Variables and dependence of yield on certain process variable values. Information is presented in the form of a vectors table with exact values obtained in every subsequent Monte Carlo it- eration. Such table number representation allows for the Pearson Correlations description of stochastic relations of every parameter of the subsequent processes that are beyond the analytical com- position of normal and discrete distributions. �e •ow of the algorithm is present in Figure 2. �e proposed methodology allows for observing the dependence between parameters in every moment of the production process. It is a useful tool for parametric yield analysis of devices or integrated circuits. Especially, it en- Non-linear functions ables us to understand the impact of given input process parameters on the distribution of ultimate product parameters. 5. Random Samples Generation from NoRTA Algorithm Arbitrary Distribution for random samples generation An essential aspect that needs to be addressed before the hands-on application of described above model is the e¦- cient generation of random samples from arbitrary both continuous and discrete distributions. �erefore, to generate random variables vector, F realizations with covariance Essential yield matrix ΣX, the NorTA method was applied. Expansion of determining parameters this name is “Normal to Anything” which illustrates the principle of its operation. �e advantage of the method is the Figure 2: Flow chart presenting the steps used in modelling ability to generate samples from arbitrary distributions process. combined both continuous and discrete while preserving the computation e¦ciency. �e only requirement, in order to e…ectively apply the method, is the existence of an inverse discrete to the methods based on the optimization of the cumulative distribution function [13]. nonlinear stochastic functions. In the developed framework, �e NorTA method consists of a few steps. In the be- the former was applied, even though more computationally ginning, a vector of random variables realization is drawn complex, enabling the uni”ed approach to the issue of from multivariate normal distribution N(0, Σ ) with a determining covariance’s matrix Σ . Moreover, for a given certain matrix covariance Σ that needs to be previously Z set of model random variables, the covariance matrix is determined. Furthermore, every generated sample is determined only once for any pair of variables, which does transformed by the normal cumulative distribution function not in•uence considerably total computation time [14, 15]. Φ . After this operation, the variables vector of uniform (0,1) distribution U(0, 1) is obtained. �e received relation 6. Model Application and Results Φ (N ) is a Gaussian copula and preserves initial co- 0,1 (0,Σ ) variance relations within the input vector. �e last step is the �e described model was applied to modelling of the transformation of the samples vector by the vector of inverse AlGaN/GaN HEMT fabrication process. �e developed − 1 distributions of desired random variables F . Samples model enables the investigation of arbitrary relations be- obtained in that way are the realization of that random tween various parameters of the fabrication process and was variables vector with cumulative distribution functions, F, aimed at the optimization of the research plan of AlGaN/ and cross-correlation matrix Σ . GaN HEMT dedicated to microwave applications. �e essential question in e…ective NorTA algorithm AlGaN/GaN HEMTs are essential for both civilian and application is the determination of the covariance matrix Σ military markets since they constitute an important part of elements that will result in the generation of the samples radars, telecommunication power ampli”ers as well as vectors with desired correlations. It is a computationally power transforming devices (inverters and converters) [16]. intensive task. Di…erent approaches are applied from the However, the fabrication of such devices is highly de- analytical solutions that unfortunately narrow possible manding in terms of process complexity in comparison with combinations of input parameters distribution to the pair of silicon technology. Usually, the process involves several continuous-continues, continuous-discrete, or discrete- dozen of steps. In general, the ”rst group of processes Journal of Engineering 5 constitutes an epitaxial deposition of AlGaN/GaN HEMT type heterostructures by the MOVPE technique. Any epitaxy process parameters modifications, such as temperatures, pressures, and reagents content and composition, result in a variation of electric heterostructure parameters and as a result strongly influence the main electrical parameters of these structures, namely two-dimensional electron gas (2DEG) concentration, pitch-off voltage, and surface re- sistivity. All of the aforementioned heterostructures’ elec- trical parameters could be measured before the fabrication of devices by nondestructive methods and enabled to select the Figure 3: Construction of X-band AlGaN/GaN HEMT. substrates with required parameters. Subsequently, after the growth of the heterostructure, the fabrication processes of the devices start from the Meza structures of the definition of parameters of obtained transistors and validate the influence the device performed by lithography and reactive ion etching of all involved technological processes on the transistor (RIE), using Cl parameters. /BL -based plasma. Subsequently, process 2 3 quality control is performed via a range of microscopic Listed previously parameters were included in the measurements, using scanning microscopy (SEM) and comprehensive fabrication model of AlGaN/GaN HEMT. atomic force microscopy (AFM). Furthermore, ohmic *e model was applied to the prediction of the parametric contacts are fabricated. *ey are formed by multilayer (Ti/ yield of saturate drain current I . *is parameter is one of dss Au/Mo/Au) metallization and must be annealed at a high the most important enabling the determination of transistor temperature. *en, in order to control the drain current in applicability into the power circuits for microwave band, the transistor channel, Schottky contact has to be fabricated specifying the maximum power that can be obtained using a and placed in the area between the drain and the source single device. It is affected by a number of factors, related to contacts. *ere are two possibilities for its fabrication. *e AlGaN heterostructure, HEMT design, and fabrication selection of the appropriate method depends on desired gate processes parameters. *e elaborated model was written in length. For gates longer than 1μm, the photolithography the form proposed above. *e input parameters distribution technique (PhL) could be used, whereas, for gates of the was chosen on the basis of the previously analyzed results of length between 100÷ 500 nm, the electron beam lithography the research on respective technological processes within the (EBL) technique has to be applied. At our laboratory, two research group. In Table 1, the list of model parameters with different metallizations were used for Schottky contact: Ru/ their distributions is presented. Au or Ni/Au fabricated by lift-off process of metallization *ese parameters consist of complete input into the applying PhL or EBL techniques, respectively. At the next framework. In Figure 4, a simulation of saturate drain stage, the passivation process, using polyimide materials, is current, I , and distribution for the batch of one thousand dss conducted, and the transistor is ready. However, HEMTs are presented. manufacturing of consumer available product requires a few *e figure illustrates important results for production more processes, such as the thickening of all metallization, planning regarding the values of saturate drain currents. *e cutting substrate for chips, bonding, packaging, and distribution differs from usually expected in such case- encapsulation. s—Gaussian distribution. *e simulation allows for sizes At this stage, the range of electric DC and microwave allocation of accurate quality bins. measurements are conducted on the wafer using the spe- Validation of obtained results is performed using two cialized probes as, at this stage, it is possible to measure final approaches. First, obtained results of operation parameter device parameters. One of the most important is the saturate distributions should be consistent with measurements re- drain current, I , that determines, among others, the sults that allow for analysis of prediction accuracy, as well as, dss suitability of the transistor to switch high RF powers. In further model calibration. *e obtained simulation results of Figure 3, the transistor structure operating at X-band the distribution of saturate drain current are in accordance (8–12 GHz) is shown. with measurement data from a small batch of devices fab- They were designed and are applied under the frame of ricated during the preproduction phase at WUST. *e the project to develop novel military radar systems, with second way to validate the model of obtained results was to solid-state devices that could replace the traveling wave tubes perform analysis without stochastic variability. It was (TWT). *e C band transistors have the gate length, L , equal achieved by the reduction of involved random variables to to 1000 nm, with the width of W � 10 ×125µm � 250µm their expected values. *ey are equal to the exact sizes and fabricated by photolithography, whereas the X-band tran- compositions of respective HEMT elements determined sistors have the gate length of the range from 100 nm to during transistor design. *e distribution of three param- 500 nm with W � 2 ×125µm � 250 mm fabricated by elec- eters that are strongly coupled with I , 2DEG electron g dss tron beam lithography. mobility, µ , AlGaN layer thickness, and 2 DEG sheet *e developed model was applied to combine results of electron concentration, n , is presented assuming first that the number of discrete research on subsequent technological I has no stochastic variability and can be determined dss steps in a coherent way that allow the prediction of operating directly. *e value of I in the form of a binary measure is dss lel3 –2 ns[cm ] lel3 –2 ns[cm ] 6 Journal of Engineering Table 1: �e applied model parameters and their distributions are presented. Parameter Distribution type Heterostructure sheet resistance R [Ω/□] Normal (400, 10) −3 Heterostructure 2DEG electron concentration n (m ) Normal (1.15e13, 0.05e13) AlGaN layer thickness d (nm) Normal (19, 1) Al Ohmic contact resistance R (Ω ) Normal (1e-3, 2e-4) Gate length L (μm) Normal (1, 0.2) −1 Max carrier velocity V (cms ) Normal (0.75, 0.05e7) 2 −1 1 Electron mobility μ [cm V s ] Nonlinear function of R and ns 1/(R n (1.6e-19)) s s s Pinch-o… voltage V (V) Nonlinear function of d , n p Al s Gate source distance (μm) Discrete (based on measurements) 180 0.014 0.012 0.010 0.008 1600 390 0.006 0.004 0.002 1200 375 0 0.000 1.35 15 1.30 16 300 350 400 450 500 1.25 1.20 1.15 1.10 I [mA] 1.05 20 dss 1.00 21 0.95 22 Figure 4: Distribution of I for 1000 AlGaN/GaN HEMTs fab- dss Figure 6: Relationship between saturate drain current, electron ricated at WUST. mobility, layer thickness, and electron concentration of AlGaN/ GaN HEMT with stochastic complexity being considered. the need for application of the elaborated methodology, because it allows for optimization of selected parameters with concern to the variation of the others as well as allows for the proper estimation of the parametric yield of fabri- cated devices. 1.35 15 1.30 1.25 17 1.20 18 7. Conclusions 1.15 19 1.10 20 1.05 21 1.00 22 0.95 23 �e production process of advanced devices is of high complexity. During such process, virtually each individual Figure 5: Relationship between saturate drain current, electron mobility, layer thickness, and electron concentration of AlGaN/ technological step is under constant research and devel- GaN HEMT, when stochastic dependencies are omitted. opment and, as a result, requires a nonconventional ap- proach to the question of the expected yield modelling. Described complexity results from a signi”cant number of calculated. �e threshold of I that quali”es the HEMT as independent variables, available interoperation measure- dss approved was set arbitrarily to 375 mA. Obtained results are ments, and the indeterminable in•uence of external factors. presented (Figure 5). Dependence between the aforementioned elements is often �ere can be an unambiguously determined area, where nonlinear. �e additional challenge of the conducted re- approved devices can be found. In that case, for speci”c search is the environment of the R&D laboratory, which values of n ,d and μ saturate drain current can be character limits the availability of a su¦cient number of s AlGaN exactly calculated. Obtained values correspond to the values production data from various technological cycles. For the need of yield analysis, there are only available separate el- calculated using classical device models of HEMTs. Al- though measurement results clearly show that such de- ements of knowledge that usually focus solely on a single scription is incomplete and that is required to examine real technological process and relations between its input and statistical distributions of respective parameters. In Figure 6, outputs. �e full picture is further shadowed by a relatively the true dependence is presented in the real conditions, small number of experiments that hinder the statistical assuming parameters stochastic variation. inference and work on the cutting-edge of the current Distributions obtained with that approach correspond to technological capability. Despite the listed challenges, thanks the measurement results in which the range of acceptable n , to the •exibility of the developed model, there was possible d , and μ parameters is fuzzy. �e ”gure clearly shows to include the knowledge and experience of the research AlGaN d [nm] AlGaN d [nm] AlGaN From 1000 HEMTs Probability Density Function 2 –1 –1 μ[cmv V s ] I [mA] dss 2 –1 –1 μ[cm V s ] I [mA] dss Journal of Engineering 7 [6] C.-F. Chien, Y.-J. Chen, and J.-Z. 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