Method for deriving the diffusion coefficient
The method addresses the underestimation of diffusion coefficients in silicon wafers by using exponential decay and simulation to exclude non-isothermal heating effects, providing accurate results for rapidly diffusing elements and thin wafers.
Patent Information
- Authority / Receiving Office
- JP · JP
- Patent Type
- Applications
- Current Assignee / Owner
- SHIN ETSU HANDOTAI CO LTD
- Filing Date
- 2024-12-18
- Publication Date
- 2026-06-30
AI Technical Summary
Conventional methods for deriving the diffusion coefficient of elements in silicon wafers are inadequate for rapidly diffusing elements or thin wafers, as they underestimate the diffusion coefficient due to non-isothermal conditions during short heat treatments, especially when the concentration of the element decreases below detection limits.
A method involving heat treatment at varying durations to determine the decay constant and diffusion coefficient using the exponential decay relationship, followed by outward diffusion simulation to derive an accurate diffusion coefficient, excluding non-isothermal effects.
Enables accurate derivation of the diffusion coefficient by excluding measurement values from initial non-isothermal heating phases, ensuring precise results even for rapidly diffusing elements and thin wafers.
Smart Images
Figure 2026106703000001_ABST
Abstract
Description
[Technical Field]
[0001] This invention relates to a method for deriving the diffusion coefficient. [Background technology]
[0002] The properties of silicon wafers used as substrates for semiconductor integrated circuits are influenced by impurities in the wafer. These impurities range from light elements such as oxygen, nitrogen, and hydrogen to heavy metals such as iron and nickel. To understand the impact of such impurities on wafer properties during device processing, it is crucial to understand their concentration and diffusion behavior, and understanding diffusion behavior requires the diffusion coefficient.
[0003] A common method for deriving the diffusion coefficient of an element in a silicon wafer is to determine the diffusion coefficient based on the outward diffusion profile of the element under investigation after heat treatment at a predetermined temperature (for example, the technique for determining the diffusion coefficient of oxygen described in Patent Document 1). [Prior art documents] [Patent Documents]
[0004] [Patent Document 1] Japanese Patent Publication No. 2022-20972 [Overview of the project] [Problems that the invention aims to solve]
[0005] As described above, conventionally, the method for deriving the diffusion coefficient of an element in a silicon wafer has generally involved determining the diffusion coefficient based on the outward diffusion profile of the element under investigation after heat treatment at a predetermined temperature.
[0006] However, this prior art is difficult to apply to rapidly diffusing elements or when the wafer containing the element under investigation is thin. The aforementioned prior art uses the outward diffusion profile after heat treatment, but since the concentration of the element under investigation decreases due to outward diffusion heat treatment, the outward diffusion heat treatment time must be set so that the concentration after outward diffusion is above the detection limit of the quantitative method. Therefore, in the case of rapidly diffusing elements or when the wafer containing the element under investigation is thin, the heat treatment becomes short. Furthermore, in actual heat treatment, there is a heating time at the beginning of the heat treatment until the silicon wafer reaches the predetermined heat treatment temperature, so when the heat treatment time is short, the proportion of the heating time to the total heat treatment time becomes high, and the difference between the isothermal heat treatment process assumed in the simulation and the thermal history actually experienced by the silicon wafer becomes large. In such cases, the thermal history becomes lower than the isothermal heat treatment process assumed in the simulation due to the heating time at the beginning of the heat treatment, which is problematic because the derived diffusion coefficient is estimated to be lower than the actual diffusion coefficient.
[0007] This invention was made to solve the above problems, and aims to provide a method for more accurately deriving the diffusion coefficient even for elements under investigation that diffuse outward due to the heat treatment of a silicon wafer. [Means for solving the problem]
[0008] The present invention was made to achieve the above objective, and is a method for deriving the diffusion coefficient of elements that diffuse outward due to heat treatment in a silicon wafer cut from a silicon single crystal ingot, Step 1 involves preparing a silicon wafer containing the element to be investigated, subjecting the silicon wafer to heat treatment at a predetermined heat treatment temperature for varying durations, determining the relationship between the concentration of the element to be investigated at the bulk center of the silicon wafer and the heat treatment time, and determining the decay constant a[ / min] obtained by fitting using the following formula (1) within the range of heat treatment times in which the concentration of the element to be investigated decreases exponentially; Based on the thickness of the silicon wafer containing the element under investigation, an outward diffusion simulation based on the diffusion equation is performed for multiple diffusion coefficients. From the relationship between the concentration of the element under investigation at the bulk center of the silicon wafer and the outward diffusion heat treatment time obtained from the simulation, the decay constant a[ / min] and the diffusion coefficient D[cm] in the following equation (1) are obtained. 2 Step 2 is to find the relationship [ / s], The damping constant a[ / min] and the diffusion coefficient D[cm] obtained in step 2 above. 2 Based on the relationship [ / s], the diffusion coefficient D[cm] at the damping constant a[ / min] calculated in step 1 above. 2 Step 3 to derive / s The present invention provides a method for deriving the diffusion coefficient, characterized by having [a specific characteristic]. C = C0 exp(-at) ... (1) (In the formula, C: concentration of the element under investigation at the bulk center of the silicon wafer [atoms / cm³]) 3 ], C0: constant [atoms / cm 3 ], a: damping constant [ / min], t: heat treatment time [min])
[0009] When isothermal treatment is applied to a sample containing an outwardly diffusing element of investigation, the concentration of the element of investigation at the center of the silicon wafer bulk decreases exponentially. Therefore, according to equation (1), the range of heat treatment time in which the concentration changes over time can be determined to be isothermal treatment. Accordingly, by deriving the diffusion coefficient from the relationship between the heat treatment time within the aforementioned range and the concentration of the element of investigation at the center of the silicon wafer bulk, it is possible to exclude measurement values where the heat treatment was performed at a temperature different from the target temperature during the initial heating time of the heat treatment, and thus derive a more accurate diffusion coefficient. [Effects of the Invention]
[0010] In the method for deriving the diffusion coefficient of the present invention, it can be determined that the range of the heat treatment time during which the element concentration changes exponentially with time during heat treatment is an isothermal heat treatment. Therefore, by deriving the diffusion coefficient from the relationship between the heat treatment time within the range of the heat treatment time and the concentration at the bulk center of the silicon wafer of the element to be investigated, it is possible to analyze excluding the measured values that are heat treatments at temperatures different from the target during the temperature rise time at the initial stage of the heat treatment, and a correct diffusion coefficient can be derived.
Brief Description of the Drawings
[0011] [Figure 1] It is a graph showing the relationship between the hydrogen concentration at the bulk center of a silicon wafer and the outward diffusion heat treatment time by heat treatment at 650°C. [Figure 2] It is a graph showing the relationship between the decay constant and the diffusion coefficient of the hydrogen concentration at the bulk center of a silicon wafer with a thickness of 1.2 mm. [Figure 3] It is a graph showing a comparison between the measured values of the relationship between the hydrogen concentration at the bulk center of a silicon wafer and the outward diffusion heat treatment time and the outward diffusion simulation at various diffusion coefficients (the solid line is Example 1). [Figure 4] It is a graph showing the case where the diffusion coefficient is obtained from the outward diffusion profile in the same manner as the conventional method, and shows the cases of (a) heat treatment time of 5 minutes, (b) heat treatment time of 6 minutes, (c) heat treatment time of 7 minutes, and (d) heat treatment time of 8 minutes.
Embodiments for Carrying Out the Invention
[0012] Hereinafter, the present invention will be described in detail, but the present invention is not limited thereto.
[0013] The present invention has been made to achieve the above object, and is a method for deriving the diffusion coefficient of an element that diffuses outward by heat treatment in a silicon wafer cut out from a silicon single crystal ingot, Prepare a silicon wafer containing the element to be investigated, perform heat treatment on the silicon wafer at a predetermined heat treatment temperature with varying time, determine the relationship between the concentration of the element to be investigated at the bulk center of the silicon wafer and the heat treatment time, and within the range of the heat treatment time in which the concentration of the element to be investigated decreases exponentially, obtain the decay constant a [ / min] determined by fitting according to the following formula (1). Step 1 Based on the thickness of the silicon wafer containing the element to be investigated, perform external diffusion simulations based on the diffusion equation at multiple diffusion coefficients. From the relationship between the concentration of the element to be investigated at the bulk center of the silicon wafer obtained from the simulation and the external diffusion heat treatment time, obtain the relationship between the decay constant a [ / min] and the diffusion coefficient D [cm 2 / s] in the following formula (1). Step 2 Based on the relationship between the decay constant a [ / min] and the diffusion coefficient D [cm 2 / s] obtained in Step 2, derive the diffusion coefficient D [cm 2 / s] at the decay constant a [ / min] calculated in Step 1. Step 3 Provided is a method for deriving a diffusion coefficient, characterized by having the above steps. C = C0exp(-at) ···(1) (In the formula, C: Concentration of the element to be investigated at the bulk center of the silicon wafer [atoms / cm 3 , C0: Constant [atoms / cm<00000?10>, a: Decay constant [ / min], t: Heat treatment time [min])
[0014] Hereinafter, Steps 1 to 3 will be described step by step. The order of Step 1 and Step 2 is not particularly limited, and Step 1 may be performed after Step 2.
[0015] [Step 1] Step 1 of the method of the present invention is a step of obtaining the decay constant a [ / min] of the decrease in the concentration at the bulk center by heat treatment at a desired temperature for the element to be investigated. <s
[0016] It seems there is a formatting issue in the original text where "?10" is shown in ID=19, which might be a typo. I've translated it as is for now. If it's incorrect, please provide the correct information.In step 1, silicon wafers containing the elements to be investigated are prepared. More specifically, three or more silicon wafers containing the elements to be investigated at a level that allows for quantitative analysis are prepared. Examples of elements to be investigated include hydrogen, oxygen, and nitrogen.
[0017] Next, the prepared silicon wafers are subjected to heat treatment at predetermined heat treatment temperatures for varying durations to determine the relationship between the concentration of the target element at the bulk center of the silicon wafer and the heat treatment time. More specifically, the procedure is as follows: First, the prepared silicon wafers are subjected to outward diffusion heat treatment at the temperature for which the diffusion coefficient is to be determined, with different heat treatment times for each wafer. After heat treatment, the concentration of the target element at the bulk center of the silicon wafer is measured. Here, the heat treatment time for the outward diffusion heat treatment is set so that there are at least three points in which the target element decreases exponentially, for the purpose of determining the isothermal heat treatment range described later. Furthermore, the atmosphere for the outward diffusion heat treatment should preferably be an inert gas atmosphere such as nitrogen or argon, which does not contain the target element, so as not to affect outward diffusion. In addition, the concentration of the target element at the bulk center of the silicon wafer is used because it is the position with the highest concentration in the depth direction in the outward diffusion profile, making quantification easy.
[0018] Next, the decay constant a[ / min] is determined by fitting using the following equation (1) within the range of heat treatment time in which the concentration of the element under investigation decreases exponentially. C = C0 exp(-at) ... (1) Here, C[atoms / cm 3 ] is the concentration of the element under investigation at the bulk center of the silicon wafer, and C0 is a constant [atoms / cm³]. 3 ], where t is the heat treatment time [min].
[0019] More specifically, first, the range of heat treatment time in which isothermal treatment occurs is determined from the relationship between the obtained outward diffusion heat treatment time and the concentration of the element under investigation at the bulk center of the silicon wafer. Here, if isothermal treatment occurs, the concentration at the bulk center decreases exponentially, so the range of isothermal treatment can be determined to be the range in which there is a linear decrease on a semi-logarithmic plot. Then, within the range of heat treatment time, the decay constant a[ / min] is calculated by fitting using the following equation (1).
[0020] [Process 2] Next, we have the damping constant a[ / min] and the diffusion coefficient D[cm 2 Step 2, which determines the relationship between the decay constant a[ / min] and the diffusion coefficient D[cm], is described below. In Step 2, based on the thickness of the silicon wafer containing the element under investigation, an outward diffusion simulation based on the diffusion equation is performed for multiple diffusion coefficients. From the relationship between the concentration of the element under investigation at the bulk center of the silicon wafer and the outward diffusion heat treatment time obtained from the simulation, the decay constant a[ / min] and the diffusion coefficient D[cm] in the following equation (1) are determined. 2 The relationship [ / s] is determined. In this way, based on the thickness of the silicon wafer containing the element under investigation, a diffusion simulation is performed to calculate the time change in the concentration of the element under investigation at the bulk center of the silicon wafer at various diffusion coefficients. Here, the diffusion simulation is performed using the finite volume method according to Fick's second law, the boundary conditions are Dirichlet boundary conditions, and the boundary is 0 atoms / cm 3 The calculation is performed as follows. In the outward diffusion simulation results for various calculated diffusion coefficients, the fitting is performed using equation (1), and the damping constant a [ / min] and the diffusion coefficient D [cm 2 Find the relationship between / s and s.
[0021] As described above, the order of steps 1 and 2 is not particularly limited, and step 1 may be performed after step 2.
[0022] [Process 3] Finally, in step 3, the damping constant a[ / min] and diffusion coefficient D[cm] obtained in step 2 are used. 2Based on the relationship [ / s], the diffusion coefficient D[cm] at the damping constant a[ / min] calculated in step 1 above. 2 Derive / s].
[0023] This invention is particularly effective when heat treatment is performed for a short time, but the diffusion coefficient can also be derived using the same method when heat treatment is performed for a long time. Cases where heat treatment is performed for a short time include when rapidly diffusing elements such as hydrogen are being investigated, when the wafer thickness containing the element under investigation is thin, and when the concentration of the element under investigation contained in the wafer used for the investigation is close to the detection limit. [Examples]
[0024] The present invention will be described in detail below with reference to examples and comparative examples, but these are not intended to limit the present invention.
[0025] [Example 1] The element under investigation was hydrogen in a silicon wafer. A silicon wafer (thickness 1.2 mm, p-type, resistivity 10 Ω·cm, oxygen concentration 10 ppm) was prepared by introducing hydrogen through a heat treatment at 1100°C for 60 seconds under a hydrogen atmosphere. The silicon wafer was then subjected to an outward diffusion heat treatment at 650°C under a nitrogen atmosphere for 0 min to 8 min. Subsequently, the hydrogen concentration at the bulk center of the silicon wafer was quantified using the SIMS raster change method, and the relationship between the outward diffusion heat treatment time and the hydrogen concentration at the bulk center was obtained as shown in Figure 1. At this time, a linear decrease, i.e., an exponential decrease, was observed in the semi-logarithmic plot from 5 min to 8 min. When fitting was performed using equation (1), the decay constant a was calculated to be 0.72 [ / min] (Step 1).
[0026] Next, outward diffusion simulations based on the diffusion equation were performed at a thickness of 1.2 mm with various diffusion coefficients, and the time evolution of the hydrogen concentration at the bulk center obtained was fitted to equation (1) to determine the decay constant a[ / min] and the diffusion coefficient D[cm]. 2 When we derive [ / s], we obtain the relationship shown in Figure 2 (Step 2). D and a are D[cm2 / s]=2.51×10 -5 a[ / min]-5.00×10 -7 The following relationship was obtained.
[0027] Since a nearly linear relationship was obtained in step 2, the relationship was derived from fitting a linear function, and substituting the decay constant of 0.72 [ / min] calculated from the measured values, the diffusion coefficient of hydrogen at 650°C is 1.77 × 10⁻⁶. -5 cm 2 We were able to derive / s (Step 3).
[0028] [Comparative Example 1] Figure 3 shows the results of comparing the measured relationship between the bulk center hydrogen concentration and the outward diffusion heat treatment time obtained in Example 1 with outward diffusion simulations at various diffusion coefficients. No diffusion coefficient exhibited behavior consistent with the measured values, making it impossible to estimate the correct diffusion coefficient. This is likely because the diffusion simulation assumes isothermal treatment at 650°C, but the actual outward diffusion heat treatment includes a heating time between 0 and 5 minutes, thus not being isothermal treatment.
[0029] [Comparative Example 2] Based on the relationship between the bulk center hydrogen concentration and the outward diffusion heat treatment time obtained in Example 1, the diffusion coefficient was derived using a conventional method. In the conventional method, the outward diffusion profile obtained from measurement is compared with the outward diffusion profile obtained from outward diffusion simulation, and the diffusion coefficient that best matches is derived. In Example 1, since the bulk center hydrogen concentration was determined, the outward diffusion profile was calculated using diffusion simulation with various diffusion coefficients for each heat treatment time, and a diffusion coefficient was derived that matched the bulk center value with the measured value.
[0030] As a result, as shown in Figure 4, when the heat treatment time is (a) 5 min, the result is approximately 3.0 × 10 -6 cm 2 / s, (b) For 6 min, approximately 5.2 × 10 -6 cm 2 / s, (c)7min is approximately 7.4 × 10 -6cm 2 / s, (d)8min is approximately 8.6 × 10 -6 cm 2 The estimated value was / s, and it varied depending on the heat treatment time. This is thought to be due to the effect of the heating time at the beginning of the heat treatment, and the difference from the true value becomes larger as the proportion of heating time in the total heat treatment is higher. In fact, the value derived in this comparative example becomes closer to the value derived in the example as the heat treatment time increases. If the heat treatment time can be made long enough that the heating time does not have an effect, the diffusion coefficient can be derived using the conventional method, but if diffusion is fast and the heat treatment time needs to be short in order to quantify the concentration, the conventional method cannot determine the correct diffusion coefficient.
[0031] It should be noted that the present invention is not limited to the embodiments described above. The embodiments described above are illustrative, and any configuration that is substantially identical to the technical idea described in the claims of the present invention and achieves similar effects is included within the technical scope of the present invention.
Claims
[Claim 1] A method for deriving the diffusion coefficient of elements that diffuse outward due to heat treatment in a silicon wafer cut from a silicon single crystal ingot, Step 1 involves preparing a silicon wafer containing the element to be investigated, subjecting the silicon wafer to heat treatment at a predetermined heat treatment temperature for varying durations, determining the relationship between the concentration of the element to be investigated at the bulk center of the silicon wafer and the heat treatment time, and determining the decay constant a [ / min] obtained by fitting according to the following formula (1) within the range of heat treatment times in which the concentration of the element to be investigated decreases exponentially, Based on the thickness of the silicon wafer containing the element under investigation, an outward diffusion simulation based on the diffusion equation is performed for multiple diffusion coefficients. From the relationship between the concentration of the element under investigation at the bulk center of the silicon wafer and the outward diffusion heat treatment time obtained from the simulation, the decay constant a [ / min] and the diffusion coefficient D [cm] in the following equation (1) are obtained. ２ Step 2 involves determining the relationship [ / s], The damping constant a [ / min] and the diffusion coefficient D [cm] obtained in step 2 above. ２ Based on the relationship [ / s], the diffusion coefficient D [cm] at the damping constant a [ / min] calculated in step 1 is used. ２ Step 3 to derive / s and A method for deriving the diffusion coefficient, characterized by having [a certain characteristic]. C=C ０ exp(-at) ・・・(1) (In the formula, C: concentration of the element under investigation at the bulk center of the silicon wafer [atoms / cm³]) ３ ], C ０ : Constant [atoms / cm] ３ ], a: damping constant [ / min], t: heat treatment time [min])