Modeling Process-Dependent Thermal Silicon Dioxide (SiO2) Films on Silicon

This study attempts to model correctly the process dependence of thermal SiO2 film physical structures and their associated densities, as well as high frequency dielectric constants, so as to provide a foundation for a ULSI process-dependent device reliability simulator. By exploring the characteristic signature of ellipsometric data reduced using a one-layer film model, and comparing it to a twolayer model, we establish a process-dependent, two-layer model for thermal SiO2 films. Internal consistency in this model is demonstrated using three intrinsicstress-related phenomena in thermal SiO2 films on Si. Both the interfacial layer and bulk film are characterized quantitatively for 38 samples, dry-oxidized at four temperatures, leading to three empirical equations describing interlayer thickness, bulk layer density, and bulk layer optical frequency dielectric constant, as functions of oxidation temperature. The interfacial layer refractive index is taken to be independent of oxidation time, and found to be independent of oxidation temperature. The oxidation-temperature-dependent index of refraction of bulk SiO2 films obtained using the proposed model agrees well with independent data on thick oxides, for which the interlayer effect is minimal. It is also found that interlayer thickness has a relatively weak dependence on oxidation temperature, which supports the strain energy model for interlayer formation. Application of the thermal SiO2 film model to Si-device dielectric characterization using fixed index ellipsometry is also discussed, based on recent, new understanding of the ellipsometry equation. INTRODUCTION Semiconductor device scaling has followed the set of ideal scaling laws proposed by Dennard and co-workers in 1974 [1], increasing system performance and functionality with higher layout density and lower power-speed product. Projections indicate that by the turn of the century memory and microcomputer chips may have 100 million transistors with 0.2 mm channels and 50  gate oxides [2]. On the other hand, chip reliability has improved over the last two decades, driven by customer demand and stiff competition. Projections indicate that by the year 2000 VLSI chips will have failure rates of less than 10 FIT [2]. Both factors have pushed VLSI technologies close to fundamental reliability limits. One of these limits on device scaling is attributed to hot carrier effects caused by less-than-ideal scaling of power supply voltage. Hot carriers generated in high-field channel regions [3] are emitted into the insulator layers, inducing threshold voltage shifts [4,5]. They are also emitted into the substrate, forming substrate currents which can trigger latchup in CMOS [6]. In deep-submicron MOSFETs, substrate current and gate current are measured at drain biases as low as 0.7 V and 1.75 V, respectively [7]. While in the 70's and 80's VLSI reliability engineering focus was primarily on predicting system time to failure, today the focus needs to be on understanding the failure mechanisms at the microstructure level, and controlling those process variables which ultimately affect system failure rate [2]. Our broader intent is to create a ULSI process-dependent device reliability simulator, targeted in part for silicon-based devices, to achieve device built-in reliability by using optimal combinations of input process variables. Due to a lack of much fundamental knowledge needed in our proposed simulator, the present work was undertaken to model correctly the physical structures of thermal SiO2 films and their associated high frequency dielectric constants, as a function of processing temperature. For Si-device gate dielectric thicknesses where the interfacial Si-rich layer (SiOx, x<2) is non-negligible, correct understanding of bulk SiO2 film and interlayer SiOx film thicknesses, as well as their respective dielectric constants and densities, is essential for device physics studies. For instance, accurate CV data interpretation, proper account of the image charge potential well at the Si/SiO2 interface, dielectric breakdown strength analysis, and the a-SiO2 network ring structure statistics, all depend on knowledge of the physical structure of the oxide film throughout its full extent. I. Experiment descriptions: A matrix of 24 device-quality Boron-doped (resistivity 11 ~ 16 W×cm) <100> 5" silicon wafers was RCA cleaned, then dry-oxidized at several temperatures: 800 ûC (group I samples #1 to #8); 900 ûC (group II samples #9 to #16); and 1000 ûC (group III samples #17 to #24). Wafers were processed at the IBM General Technology Division facility in Essex Junction, VT. The ellipsometric data of a fourth group of samples (denoted as IBM samples #1 to #14) were also used in this study. Most samples in this group (IV) were dry-oxidized at 1050 ûC on <100> silicon wafers; a few thin oxides were grown at 900 ûC with 10 minute anneals at 1050 ûC, as described in [8]. A Rudolph research model 436 manual ellipsometer at the Measurement Standards Lab of IBM Essex Junction was used in this study (operated at wavelength 6328 ), which is of research grade with polarizer and analyzer resolutions of 0.01û, and specially calibrated [8]. All ellipsometric data were reduced using a recently developed, robust, graphical algorithm [9]. This algorithm uses n-Si=3.8737 [8] and k-Si=0.018 for the real and imaginary components of the Si complex refractive index, and n-air=1.0. We assume all media except the Si substrate are non-absorbing at 6328 . The mathematical formulations for the oneand two-layer models used in this work are found in the Appendix. A Digital Instruments NanoScope III atomic force microscope was also used to measure the standard deviation of oxide surface roughness. II. Experiment results and analysis 2.1 Evidence of an optically-different interlayer between thermal SiO2 film and Si : Figure 1 shows the oxide thickness-dependent refractive index assuming a onelayer model (no interfacial layer), as reported previously [10,11,12,13]. Worsterror curves were calculated using a one-layer model with refractive index of 1.465 and based on the measurement error combination (D, y, and angle of incidence) which leads to the largest refractive index deviation from 1.465, as a function of film thickness. Error magnitudes are based on the resolution of the research grade ellipsometer, namely 0.01û for D, y , and AOI (angle of incidence). Each measurement error for D, y, and AOI was assigned 3 possible states (-0.01û, 0û, 0.01û). Thus, each pair of points on the worst error curves was generated after 27 runs of all possible error combinations. It was found that the upper worst-error curve refers to the case where all three measurement errors are negative while the lower worst-error curve refers to the case where all three measurement errors are positive, for oxide thickness less than 970 . Neither event is highly probable due to the often random nature of ellipsometric measurement (D and y), and AOI error due to laser beam deviation [8]. Data points falling outside the worst-error envelop are considered reliable. Thickness-dependent refractive index based on the one-layer model for the IBM samples is shown in Figure 2. The unique "U" shape centered around oxide thickness 1400  (half of the first ellipsometric cycle at wavelength 6328 ) is noteworthy. Note also that the minimum value is less than 1.460, commonly believed to be the index of refraction for fully relaxed thermal a-SiO2 [14]. Should the oxide have a discrete interlayer of different optical property than that of the bulk oxide film, as suggested previously [15,16], then a one-layer model interpretation of the actual two-layer structure will estimate film thickness and refractive index incorrectly. An optically-distinct interlayer creates a unique shape in the graph of relativeerror versus film thickness, as shown in the simulation experiment of Figure 3. Here we presume a two-layer structure, where the interlayer thickness (d2) is 10  with a refractive index of 2.8, and the oxide layer thickness is d1 with a refractive index of 1.465. The state of polarization (D and y) was generated as a function of d1. The one-layer model was used to interpret this state of polarization, which yielded the n0 (refractive index) and d0 (thickness) for the hypothetical one-layer oxide. The signature of the interlayer should thus be a "U"-shaped (centered around oxide thickness 1400 ), thickness-dependent refractive index, with the minimum less than the refractive index for fully-relaxed a-SiO2 (i.e., 1.460). The data in both Figure 1 (half of the "U" shape) and Figure 2 (the whole "U" shape) support the existence of this optically-different interlayer. Its formation can be attributed to the Si/SiO2 interface roughness [17], an off-stoichiometric SiOx boundary layer [17], and a structurally-distinct region of near-interfacial SiO2 [17]. 2.2 Establishment and verification of a two-layer thermal SiO2 film model : Ideally, any assumptions made in establishing the film model should be based on experimental observations. At the same time, a model based on these same assumptions should not lead to predictions which contradict other experimental observations. There are three distinct intrinsic-stress-related phenomena in thermal SiO2 films on Si [18,19] which lead to assumptions used in our two-layer model: (1) Intrinsic stress at the interface is oxidation temperature independent; (2) Intrinsic stress decreases quickly with increasing oxide thickness for oxides grown for various times at the same temperature, finally becoming nearly constant in thick oxides (> 300 ); (3) The magnitude of this constant stress in thick oxides is oxidationtemperature-dependent: the higher the oxidation temperature, the lower this constant stress. Using index of refraction as an estimator of oxide density and intrinsic stress magnitude, we make the following assumptions: Assumption I: Based on phenomenon (2) above, we assume the bulk oxide film has the same refractive index (n1) among samples in each group (notice all samples in this study are grouped according to oxidation temperature). That is, n1 is oxidation-time-independent at a fixed oxidation temperature; Assumption II: We further assume both the interlayer index of refraction n2 and thickness d2 is also oxidation-time-independent at a fixed oxidation temperature; Assumption III: For simplicity, we assume both the interlayer and bulk film are uniform; that is, there is no refractive index gradient within each layer. The d2 (interlayer thickness) standard deviation, or variance estimator, is examined as a function of n1 (bulk oxide refractive index) and n2 (interlayer refractive index) for all four groups of samples. The d2 variance estimation procedure for a given group of samples is described as follows, using the three assumptions made above: [i] Fix n2; [ii] For a given (variable, 1.3<n1<2) n1, calculate d1 and d2 for every sample in the group using the ellipsometric data measured with the research grade ellipsometer, then plot the whole sample group d2 variance estimator versus n1; [iii] Repeat steps [i] and [ii] for a different n2 (1.3<n2<4) Notice the d2 variance thus formed for each sample group is a highly non-linear function of n1 and n2, using the two-layer model (see Appendix). Based on this procedure, we expect the following outcomes: A. The value of n2 obtained should be the same for all four groups of samples based on intrinsic-stress phenomenon (1), above; B. The value of n1 obtained should be oxidation-temperature-dependent: the higher the oxidation temperature, the lower the n1, based on intrinsic-stress phenomenon (3); C. On the plane of n1 and n2 which is of physical significance (1.3<n1<2 and 1.3<n2<4) the resulting d2 variance minimum should be unique, and the magnitude of the minimum should not be more than one mono-layer thickness of a-SiO2 (3.3 ). Figure 4 shows the d2 variance estimation process for group I (800 ûC), where only the d2 variance minimum portion was plotted and the d2 mean (scaled by 10) was plotted only at the variance estimation curve minimum. Evidently there is a best set of n1 and n2 which minimizes the d2 variance down to 1.45 . Thus, the solution of n1 and n2 is unique to group I. Incidentally, both d2 mean and variance are minimized at the same n1 value. That the d2 variance minimum (1.45 ) is less than one mono-layer thickness of a-SiO2 (3.3 ) supports the Assumption II that interlayer thickness is essentially the same for sample group I. The d2 variance minimums for the rest sample groups are found to be between 1.4  to 1.5 . The standard deviation of oxide surface roughness of sample #1 was found to be 1.8  in a total scanning distance of 8000  using the NanoScope III atomic force microscope. It is likely the roughness of the Si-substrate-to-oxide interlayer interface, as well as the interlayer-to-bulk-film interface, are also of the same order (1.8 ). This may explain why the d2 variance can not be minimized to zero for any combinations of n1 and n2 which are of physical significance. Figure 5 shows the d2 mean for the four groups of samples, using the d2 standard deviation to mark the upper and lower limit of the error bars. The interlayer refractive index n2 which minimizes the d2 standard deviation within each sample group was found to be 2.95 for all four groups of samples. This result is expected based on the growth-temperature-independent interface stress data of [18,19]. Though the d2 means obtained in this study differ from Taft and Cordes' reported values at wavelength 5461  (d2=7~8  for 900 ûC and 4  for 1200 ûC, n2=2.8) [15], and Aspnes and Theeten's reported values at 5461  (d2=7±2 , n2=3.2±0.5) [16], the d2 means of this study are close to those found in MOS solar cell open circuit voltage experiments (the oxide non-stoichiometric transition thickness was found to be 13~14 ) [20]. The relatively weak dependence of interlayer thickness on oxidation temperature in Figure 5 supports the strain energy argument for the interlayer: a large lattice mismatch at the Si/SiO2 interface can favor an intermediate layer so as to reduce strain energy, and the interlayer thus formed should be a very slow function of oxidation temperature [21]. Having verified the internal consistency in this two-layer, thermal SiO2 film model, we turn our attention to the bulk oxide film refractive index (n1) for the four groups of samples. This was found to be a near-linear function of oxidation temperature, as shown in Figure 6 (n2 is 2.95). Independent data for thick oxides (1000~1400 ) based on a one-layer model [22,23] are also plotted in Figure 6. According to the prediction of Figure 3 (that for oxide thickness around 1400  the interlayer effect is a minimum when the one-layer model is used to interpret a twolayer structure), it is expected all these data agree with each other. 2.3 Further evaluations of the two-layer thermal SiO2 film model : A simulation experiment in Figure 7 further verifies the validity of the results in Figures 5 and 6 obtained by the technique of d2 variance estimation. Three sets of thickness-dependent refractive index curves are depicted in Figure 7. Each set consists of three hypothetical two-layer simulations, assuming interlayers of constant thickness (d2) and refractive index (n2=2.8), and with bulk films of different refractive index n1. Interlayer thickness differs between the three sets of simulations. The state of polarization (D and y) is generated as a function of bulk film thickness d1, using the two-layer model. The one-layer model is then used to interpret this state of polarization in terms of the thickness-dependent refractive index curves plotted in Figure 7, similar to the procedure used in Figure 3. The simulation curves in Figure 7 show little dependence on n2 (varied from 2.6 to 3.0), compared with the effects caused by d2 changes. However, the state of polarization (D and y) generated using a two-layer model in which n2 is varied from 2.6 to 3.0, does indicate shifts though much smaller than that caused by the d2 changes, but large enough (more than 0.01û) to be detected. Thus, the range of n2 (2.8±0.2) should not be interpreted as an uncertainty range. Four features can be generalized from Figure 7: (1) The thickness-dependent refractive index (two-layer structure interpreted by a one-layer model) is relatively insensitive to interlayer refractive index n2 (compared to its sensitivity to d2 and n1), for the range n2=2.6 ~ 3.0 and the axis scale used in Figure 7; (2) Differences in bulk film refractive index n1 cause marked distinctions in the thickness-dependent refractive index curves using the one-layer model interpretation; (3) All thickness-dependent refractive index curves merge together for oxides thinner than 150 , should the two-layer structures these curves represent have the same interfacial layer (n2, d2), but different n1; (4) For oxide thickness less than 150 , the thickness-dependent refractive index curves are distinct only when the two-layer structures these curves represent have interfacial layer (d2) thicknesses differing by more than around one monolayer of a-SiO2. The relative insensitivity of the d2 variance estimator and the minimum of n1 versus n2 found in Figure 4, agree with feature (1). According to feature (2), distinctions for oxides thicker than 150  in Figure 1 imply each sample group should have a distinct bulk film refractive index n1. Based on features (3) and (4), the convergent behavior for oxide thickness less than 150  in Figure 1 implies the interlayer thickness should not differ by more than one a-SiO2 monolayer for all 38 samples studied. Both conclusions agree with the experimental results in Figures 5 and 6. Consequently, we compare thickness-dependent refractive index simulations of two-layer structures as interpreted by a one-layer model, with the experimental curves of Figure 1, as shown in Figure 8 (the values of n2, d2, and n1 used in the simulations are from results of Figures 5 and 6). Note that the simulations in Figure 8 are NOT obtained by direct fitting of the experimental curves of Figure 1. Instead, they are obtained using the d2 variance estimations as shown in Figure 4. The fine match between simulations and experimental curves shown in Figure 8 in turn supports the validity of the d2 variance estimation procedure using the assumptions inherent in the two-layer thermal SiO2 film model. It is worth noting that we have assumed the interlayer is also non-absorbing at 6328  in our ellipsometric data reductions. The assumption that k2 » 0 is indeed a valid one, since by setting k2=0.009 (half of the extinction coefficient of Si at 6328 ) in the ellipsometric program using the n2, d1, d2, and n1 ranges in this study, the change in the state of polarization (D and y) generated is less than 0.01û in most cases, which is beyond the resolution of the research grade ellipsometer used. 2.4 Empirical equations in the two-layer thermal SiO2 film model: Fitting the data of Figure 5 yields the following empirical equation for interlayer thickness d2 () as a function of oxidation temperature T (ûC): d2 = -11.035 + 6.1146 ́10-2 T 3.8181 ́10-5 T2 (1) Film density can be extracted from refractive index, using the Lorentz-Lorenz relation given by: