Method for the crystal pulling of a doped monocrystalline crystal from silicon

Abstract

A method for the crystal pulling of a monocrystalline crystal from a melt composed of silicon and a dopant according to Czochralski, comprising the following steps in the sequence given: a) performing a time-dependent computer simulation of a predefined crystal-pulling process with a predefined dopant concentration (C) in the melt, with the aim of obtaining a temperature gradient (ꝺc / ꝺn) and a gradient of the dopant (ꝺT / ꝺn) for each time increment (t), where the two gradients are ascertained at the interface between crystal and melt in the direction of the normal of the interface to the melt, b) ascertaining a maximum dopant concentration (Cmax) in the melt and c) pulling a single crystal with a dopant concentration less than the ascertained maximum dopant concentration (Cmax), characterized in that a parameter (CS) which is determined by the formula CS = ꝺT / ꝺn - f CS ꝺc / ꝺn is calculated for each time increment t, where the parameter fCS is determined by the product of the average concentration of the dopant (C) in the melt, the density of the silicon melt, the Avogadro constant, the molar mass of silicon and the slope of the liquidus line, and then an average (formula (I)) of the values obtained for the parameter (CS) is formed, where only those values having a (CS) value of less than zero make a contribution, and a maximum dopant concentration (Cmax) is ascertained by adapting the predefined dopant concentration (C) until the amount of (formula (I)) is equal to zero.

Description

[0001] Process for growing a doped monocrystalline silicon crystal

[0002] The invention relates to a method for growing a doped single crystal of silicon.

[0003] Crystal growth is an important precursor to the production of semiconductor devices. Semiconductor crystal growth processes have been improved to produce single-crystal semiconductor blocks that can, for example, have a diameter of 200 mm to 300 mm and a length of more than two meters.

[0004] Components for controlling large currents (power components) require highly doped silicon crystals, which are further processed into the corresponding components.

[0005] One of the challenges in growing heavily doped silicon using the Czochralski method is increasing the maximum doping concentration or reducing the minimum resistance without reducing the yield of usable crystals.

[0006] State of the art

[0007] DE 10 2019 210 254 A1 describes a method for growing a single crystal of silicon having a resistivity of not more than 20 x 10' 3 Ohm cm, optimizing the first phase of the crystal growth process to reduce the probability of dislocations occurring.

[0008] EP 3 864 197 A1 deals with a method for controlling the concentration of the dopant in the melt with the aim of reducing the occurrence of dislocations.

[0009] DE 10 2021 206 550 A1 describes a method for growing a single crystal with a very low resistance using a magnetic field. The phenomenon of constitutional supercooling is not initially problematic at a relatively low doping level, which is generally used for growing semiconductor crystals.

[0010] However, during crystal growth from a heavily doped melt—where the dopants have a small equilibrium segregation coefficient much smaller than one—constitutional undercooling can occur in the melt at the melt-crystal interface. This causes the ingot to become dislocated and no longer usable for the device process.

[0011] A crystal-growing process can be modified in a variety of ways to produce crystals with moderate dopant levels. For example, crystal rotation, crucible rotation, magnetic field position and strength, and distances between various components can be adjusted, to name just a few.

[0012] If the dopant level is to be increased, i.e. the resistance in the crystal is to be further reduced, it becomes apparent that many processes that can be easily produced with high resistance are offset, i.e. rejects.

[0013] What the cited prior art references have in common is that they do not provide a targeted method for optimizing processes in this regard. Numerous expensive experiments must be conducted, in which it is unclear a priori whether, on the one hand, the corresponding crystal meets the requirements and, on the other hand, whether the proposed process will produce sufficiently stable, suitable crystals.

[0014] The object of this invention is therefore to provide a method that makes it possible to modify crystal pulling processes so that they have a high dopant level and at the same time meet the quality requirements while reducing experimental effort.

[0015] This object is achieved by the methods described in the claims. Brief description of the figures

[0016] Figure 1 shows the factor f cs as a function of the specific resistance for the dopants phosphorus (P), arsenic (As) and antimony (Sb).

[0017] Figure 2 shows the sum of the simulated parameter CS and an offset constant C O f fsetfor an example process A shown as a function of the resistance (dashed curve). The limiting resistance at which the crystal can still be pulled is determined by varying the resistance until the curve is the sum of the simulated parameter CS at the triple point of the simulated crystal and the offset constant C offset becomes zero.

[0018] The dark-colored bar indicates the resistance range where crystal pulling is hopeless. Realistically, there is a small transition area (light-shaded) where crystal pulling is possible with high risk, but is not recommended for economic reasons.

[0019] Figure 3 shows the sum of the simulated parameter CS at the triple point of the respective crystal and an offset constant C offsetfor four example processes (A “dashed”, B “solid”, C “dash dot”, D “dash dot”) each shown as a function of resistance.

[0020] The respective limit resistances at which a crystal can still be pulled were symbolized by the downward-pointing arrows on the abscissa axis.

[0021] Detailed description of the invention

[0022] As already mentioned, the phenomenon of constitutional supercooling is initially not problematic at a relatively low doping level, which is generally used for growing semiconductor crystals. Only increasing the desired dopant concentration can lead to constitutional supercooling and thus to a high probability of dislocations and, consequently, to the total failure of the respective crystal.

[0023] Among other things, the inventors have successfully attempted to use a significantly modified computer simulation method to narrow down and solve the problem in question.

[0024] The physical effect of constitutional supercooling is described, for example, in the dissertation by L. Stockmeier (ISBN 978-3-8396-1345-0, “Heavily n-type doped silicon and the dislocation formation during its growth by the Czochralski method”).

[0025] The underlying mathematical model for the numerical modeling of this problem, together with suitable software, are described in the necessary details in Mukaiyama et al. [Y. Mukaiyama, V. Artemyev, K. Sueoka, J. Crystal Growth, 597 (2022) 126844, https: / / doi.org / 10.1016 / j crysgro.2022.126844].

[0026] The inventors have recognized that significant adjustments are necessary to achieve significant yield improvements. These significant adjustments are explained below.

[0027] A value CS calculated at the interface between melt and crystal is defined as: where — is the partial derivative of temperature and — is the partial derivative of con concentration of the dopant along the normal to the crystal-melt interface in the melt.

[0028] The factor f csrepresents a function of the specific resistance and the material properties of the dopant and can be determined using material data constants and the “density / resistivity” diagram of the dopant (SEMI Standard MF723- 0307, ​​2012 Practice for conversion between resistivity and dopant or carrier density for boron-doped, phosphorus-doped, and arsenic-doped silicon).

[0029] In Figure 1, this factor cs as a function of the specific resistance for different dopants such as phosphorus, arsenic, or antimony. It is easy to determine this factor for other dopants as well, and thus the inventive method is not limited to these dopants.

[0030] The numerical differences in f csfor different dopants are due to differences in the ionized fraction of the dopant and in the segregation coefficients. The material properties can explain the difference for phosphorus (P), arsenic (As), and Sb as dopants.

[0031] In the following, the modified boundary conditions for heat transport and dopant transport for modeling are explained.

[0032] For the 3D simulations performed, the temperature at the interface between crystal and melt was assumed to be constant.

[0033] The inventors have found that the concentration boundary conditions (demixing boundary conditions) from Mukaiyama et al. [Y. Mukaiyama, V. Artemyev, K. Sueoka, J. Crystal Growth, 597 (2022) 126844,]] are valid for a positive crystallization rate. The inventors have further recognized that in the case of a negative crystallization rate (=melting), it is preferable to replace it with the following equation:

[0034] Where c is the mass concentration, p meU the density of the melt, D meit the diffusion coefficient of the dopant in the melt, c rrv . the concentration of dopants in the crystal and n the normal to the interface at the melting or growth front.

[0035] The parameter c crys is a value known from current calculations when the crystallization rate is positive and it is a function of the crystal length.

[0036] In the present invention, its value was approximated by the average dopant concentration in the crystal. It has been found that the exact values ​​of c crys are not necessary for the calculation. Furthermore, they are difficult to implement in the software.

[0037] An experimental shape of the interface between melt and crystal can be determined by cutting a rectangular slice (a so-called plank) out of a piece of crystal under investigation using two longitudinal cuts made along the center of the piece. This plank can then be examined for resistance fluctuations. The added dopant incorporates itself irregularly into the crystal. The dopant preferably contains boron or phosphorus. This leads to a locally inhomogeneous resistance distribution of the silicon, which is called "striation." Although great efforts are made to avoid striation to avoid negative effects during the device process, striation is always measurable on the doped rod.Since the dopant is incorporated from the melt into the crystal along the melt / crystal interface, the original shape of the crystal-melt interface can be determined in the form of growth stripes by analyzing the measured resistance distribution. Unsteady 3D simulations are required for the simulation, and there are two ways to perform them:

[0038] Either the shape of the interface between crystal and melt is adjusted by adjusting the Stefan boundary condition to match reality, or the shape of the interface is fixed, accepting that the Stefan condition is not necessarily fulfilled.

[0039] Both approaches may be unsuitable for calculations under certain conditions. In the first case, the interface may deviate from the experimental one, affecting the flow and species calculations. In the second case, the calculated crystallization rate would be incorrect, and accordingly, the species distribution at the interface would also be incorrect, as it follows the demixing boundary conditions.

[0040] To avoid this difficulty, the inventors realized that both the calculations have to be carried out with the calculated interface and at the same time artificial heat sources have to be introduced in order to obtain both the prescribed interface shape and the correct crystallization rate.

[0041] This method is also advantageous because it allows fluctuations of the crystallization rate and dopant species around the correct averaged values.

[0042] This procedure can be implemented, for example, using a modified version of the CGSim software from STR (https: / / str-soft.com).

[0043] Four exemplary cases were selected for the calculations.

[0044] The first two cases are from two different processes for 200 mm crystal diameters for arsenic (Process A) and red phosphorus (Process B). The other two processes, both doped with arsenic and with a crystal diameter of 300 mm, are referred to below as Process C and Process D. The difference between Processes C and D lies in the use of different crystal pullers and different pull speeds, as well as different crystal and crucible rotations.

[0045] The inventors have found that the statistically averaged lowest possible resistances for the processes with 200 mm 0.9 x 10' 3 Ohm cm for process A and 1.7 x 10'3 Ohm cm for process B. For the 300 mm processes, these resistances are 1.75 x 10' 3 Ohm cm (process C) and 2.1 x 10' 3 Ohm cm (process D).

[0046] All of these experimental values ​​apply to a specific position in the crystal and the calculation cases were selected according to this one corresponding position.

[0047] It should be noted that the resistance values ​​determined represent average values ​​of many real experiments and that the results of each experiment vary from process to process and some experiments are unsuccessful, e.g. due to dislocations.

[0048] So far, the concrete causal relationship between the occurrence of dislocations and the level of dopant concentration is not known.

[0049] Nevertheless, the inventive method offers means to at least prevent the problem.

[0050] The inventors assume that CS may be related to these experimental results.

[0051] The parameter CS defined above can be calculated using the Tiller criterion as follows:

[0052] CS = G L (1 - fc) • C o • (- Q

[0053] VV k - D L where C o - Concentration in the melt, k - Segregation coefficient, D L - diffusion coefficient in the melt, V - crystallization rate, - temperature- gradient in the melt and mi - slope of the liquidus line.

[0054] The parameter can be obtained, for example, from the corresponding 2D simulation be taken over.

[0055] According to the theory, whenever the condition CS < 0 is met, constitutional supercooling would occur and the correspondingly drawn rod would become unusable.

[0056] However, this turned out to be incorrect. This means that a simple analysis using the well-known "tiller criterion" is not possible and therefore useless.

[0057] The inventors concluded that further measures are necessary.

[0058] The inventors have recognized that the numerical grid used in all 3D simulations must adequately resolve the concentration boundary layer.

[0059] In the simulations, it is preferred that the distance of the nearest control volume to the crystal-melt interface is a maximum of 0.125 mm.

[0060] The complete numerical grids in these examples varied from about 3 million control volumes to 3.8 million control volumes.

[0061] The result of the 3D simulations is the instantaneous distribution of CS values ​​at each position at the crystal-melt interface. These points are located on the crystal rotating above the melt.

[0062] For further calculations, CS values ​​calculated at points equidistant from the center of the crystal interface were averaged.

[0063] This results in a radial distribution of the CS values ​​as a function of time. When examining the determined CS values ​​at different radii, the inventors noticed that the smaller values ​​appear to occur more often in the triple point region, while in the center of the crystal, the CS values ​​are either not in the negative range or are significantly less likely to be in the negative range. This would indicate that constitutional supercooling originates at the triple point.

[0064] However, experimental findings by Mukaiyama et al. [Y. Mukaiyama, V. Artemyev, K. Sueoka, J. Crystal Growth, 597 (2022) 126844.] show, in contrast, that the consequences of constitutional supercooling tend to start in the center of the crystal.

[0065] Despite this apparent contradiction, the inventors have further applied their result and arrived at a method that uses simulation to predict the possible suitability of a crystal pulling process for low resistance.

[0066] With this method it is possible to modify crystal pulling parameters so that crystals with lower resistances can be pulled without loss.

[0067] The inventive method for crystal pulling a monocrystalline crystal from a melt and a dopant according to the Czochralski method comprises the following steps:

[0068] (a) performing a time-dependent computer simulation of a predefined crystal pulling process with a predefined dopant concentration C in the melt, with the aim of obtaining a temperature gradient dT / dn and a dopant gradient dc / dn for each time period t, both gradients being determined at the interface between crystal and melt in the direction of the normal of the interface to the melt,

[0069] (b) determining a maximum dopant concentration Cmax in the melt and (c) pulling a single crystal with a dopant concentration lower than the determined maximum dopant concentration Cmax,

[0070] The inventive method also includes the calculation of a value CS for each journal t, which is determined by the following formula:

[0071] CS = dT / dn — f cs dc / dn where the parameter f csis determined by the product of the average concentration of the dopant in the melt, the density of the silicon melt, the Avogadro constant, the molar mass of silicon and the slope of the liquidus line.

[0072] Subsequently, a temporal average CS is calculated from the obtained values ​​of the CS values, whereby only those values ​​CS that are less than zero contribute. The temporal average can be calculated using the formula dT, W CS (T) < 0, where To and Ti are the absolute temporal beginning and represent endpoints.

[0073] Preferably, averaging over 30 minutes of physical time or more is used.

[0074] The mean value CS is therefore a function of the radius and the concentration of the dopant C in the melt. Furthermore, it is now independent of time.

[0075] For the further process, the mean value CS is selected at a radial position of the crystal. Preferably, the CS is used at the triple point of the crystal, i.e., at the nominal radius of the crystal.

[0076] The determination of a maximum dopant concentration Cmax is carried out by adjusting the dopant concentration C until the value of CS is equal to zero, preferably until the value of CS - C offset is equal to zero, where the constant C offset is preferably determined experimentally. Determining the constant C ()ffsel

[0077] If the constant C offset preset to zero, one obtains a lower limit for the dopant Cmax, below which it is always possible to grow a crystal without loss of yield.

[0078] In some cases, however, this lower limit may appear too conservative, meaning that crystal pulling with a dopant concentration above this limit is still possible.

[0079] For such cases, a C offset preferably determined with the aid of experimental data as follows.

[0080] For a crystal pulling process, an empirical limit is determined, for example, by pulling one or more crystals while successively increasing the dopant concentration until the crystal reproducibly exhibits dislocations. This empirical limit is used to obtain the constant Coffset.

[0081] This constant C offsetcan now be used, for example, to compare many different crystal pulling processes and select the one with the potential to avoid dopant-induced dislocations. The advantage of this method is that no further expensive experiments are required.

[0082] Particularly preferred for determining the constant C offset several different processes of a process class are tested, with an average constant C offset is determined by arithmetically averaging the individual values.

[0083] The inventors recognized that it can be useful to define process classes by grouping processes with similar process properties into one process class. These properties include, for example, the nominal diameter of the crystal to be pulled or the use of a magnetic field during crystal pulling. The process class is particularly preferably defined by processes with the same nominal crystal diameter.

[0084] Figure 2 shows an example of the sum of the simulated parameter CS and an offset constant C offset for an example process A shown as a function of the resistance (dashed curve). The limiting resistance at which the crystal can still be pulled is determined by varying the resistance until the curve is the sum of the simulated parameter CS at the triple point of the simulated crystal and the offset constant C offset becomes zero.

[0085] The dark bar shown here is intended to indicate the resistance range where crystal pulling is hopeless. Realistically, there is a small transition area (lightly shaded) where crystal pulling is possible with high risk, but is not recommended for economic reasons.

[0086] Figure 3 shows the sum of the simulated parameter CS at the triple point of the respective crystal and an offset constant C offset for four example processes (A “dashed”, B “solid”, C “dash dot”, D “dash dot”) each shown as a function of resistance.

[0087] The respective limit resistances at which a crystal can still be pulled were symbolized by the downward-pointing arrows on the abscissa axis.

Claims

Claims 1 . A method for growing a monocrystalline crystal from a melt of silicon and a dopant according to Czochralski, comprising the steps in the given order: (a) performing a time-dependent computer simulation of a predefined crystal pulling process with a predefined dopant concentration C in the melt, with the aim of obtaining a temperature gradient dT / dn and a dopant gradient dc / dn for each time period t, both gradients being determined at the interface between crystal and melt in the direction of the normal of the interface to the melt, (b) determining a maximum dopant concentration Cmax in the melt and (c) pulling a single crystal with dopant concentration less than the determined maximum dopant concentration Cmax, characterized in that for each period t a quantity CS is calculated which is determined by the formula dc / dn, where the parameter fcs is determined by the product of the average concentration of the dopant C in the melt, the density of the silicon melt, the Avogadro constant, the molar mass of silicon and the slope of the liquidus line, and then to form an average value CS from the obtained values of the quantity CS, whereby only those values contribute whose value CS is less than zero and to determine a maximum dopant concentration Cmax by adjusting the predefined dopant concentration C until the value of CS is equal to zero.

2. Method according to claim 1, characterized in that the quantity CS is determined at the triple point of the crystal.

3. Method according to claim 1 or 2, characterized in that Determine a maximum dopant concentration Cmax by adjusting the predefined dopant concentration C until the value of CS - C offset is zero, where C offset is determined empirically by measuring one or more crystals are grown, whereby the dopant concentration is successively increased until the crystal shows reproducible dislocations.

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