1. Introduction

Boron doping plays an important role in the technology of silica based optical fibers and planar waveguides. Although the attenuation of boron doped silica in the NIR region cannot reach the minimum limit of germanium doped silica, it is often used to design and prepare functional fibers and waveguides with special properties. Boron decreases the refractive index of silica and influences its dispersive characteristics [1]; therefore, it is used to produce optical fiber devices with complex concentration and index profiles [2]. Moreover, it has an enhancing effect on the photosensitivity, which is beneficial to the preparation of the widely applied fiber and waveguide Bragg gratings [3,4]. High mechanical stresses in boron doped waveguides and fibers can be used both for stress compensated low-birefringent planar waveguides on silicon [5] and high-birefringent polarization maintaining optical fibers [6].

The understanding of dopant diffusion is an essential aspect in the preparation of boron doped fibers and waveguides. As a contribution to this area, we have investigated the diffusion behavior of boron in vitreous silica in the temperature range of fiber technology between 1700°C to 2000°C. Similar procedures of sample preparation, high temperature annealing and determination of diffusion coefficients from measured concentration profiles were already used for the investigation of fluorine, phosphorus, germanium, and aluminum/ytterbium diffusion in vitreous silica. These procedures are described in detail in [7–10] and are briefly summarized in the following.

2. Experiments

2.1 Preparation of the specimen

Boron doped quartz glass samples with initial concentrations between 1 and 12 mol% B₂O₃ (y = 0.01 - 0.12) were prepared by Modified Chemical Vapor Deposition (MCVD) from gaseous SiCl₄, BBr₃, and O₂. The initial concentrations cannot be directly measured, they were derived from the deposition conditions [11]. A detailed description of the MCVD process, which is important in optical technology for the preparation of fiber preforms, can be found in [12].

Thin glassy layers with thickness of about 12 μm were deposited on the inner surface of quartz glass carrier tubes with outer and inner diameters of 14 mm and 11.5 mm, respectively, at temperatures of 1600°C. After deposition, the tubes were annealed at temperatures between 1700°C and 2000°C in order to promote boron diffusion from the doped into undoped glass regions. During annealing, the tube is flushed with gaseous oxygen (200 standard cubic centimeters per minute) and a part of the B₂O₃ is removed from the inner surface by vaporization into the flowing oxygen atmosphere within the tube. For the determination of the diffusion coefficient, this effect is rather advantageous because, in addition to the profile form, the total dopant amount also yields information on the diffusion coefficient. In order to regulate the vaporization to a suitable degree, we deposited an undoped SiO₂ layer subsequent to the doped layer on the inner surface of the tube. For comparison, parts of the tube remained unannealed.

Afterwards, the tubes were collapsed to solid rods by short-term heating to 2200°C during two burner passes. The deposited layers form the core of the rod with a diameter of 0.7 mm. The diffusion effect during this collapsing step is small compared with the annealing step. This was taken into account while comparing the annealed and unannealed samples, which undergo the same collapsing procedure. In Fig. 1(a) , a segment of the cross section of the tube with inner layer is shown in the state after annealing. Figure 1(b) shows the cross section of the collapsed tube where the layer has formed the circular core of the rod. These images were taken by backscattered electrons, where the brighter areas give a visual impression of the boron depletion at smaller radii in both the uncollapsed and collapsed layers.

 

Fig. 1 (a) Segment of cross section of the tube with inner layer. (b) Cross section of the collapsed tube.

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Exact radial concentration profiles were measured on thin, polished and carbon-coated slices by Wavelength-Dispersive Electron Probe Microanalyss (WD-EPMA) (see Section 2.2). The radial profile’s shape was measured on each slice along two crossing diameter traces, i.e. four half profiles could be compared for control of accuracy. Examples of measured profiles are shown in Figs. 2(a) and 2(b). In order to avoid breaking the samples with a larger boron concentration (> 6 mol% B₂O₃) as a result of inner stresses, the outer SiO₂ tube material was etched away as much as possible with hydrofluoric acid before cutting the slices. For comparison, refractive index profiles of all specimens were measured nondestructively by a deflection angle method [13] on the rods before the preparation of slices for EPMA measurements.

 

Fig. 2 Radial concentration profiles of the B₂O₃ concentration c, initial (─ • ─ • ─ •), measured (────), calculated (─ ─ ─) ρ*: radius in the collapsed rods. Fitting parameters according to Eqs. (3) and (5): 0: initial τ · F(c) = 0 τ · F(c) = 0; 1: unannealed τ · F(c) = 0.1· exp (0.35· c) µm² τ · F(c) = 0.2· exp (0.58· c) µm²; 2: annealed τ · F(c) = 4.6· exp (0.35· c) µm² τ · F(c) = 2.9· exp (0.58· c) µm².

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In order to determine the diffusion coefficient, the conditions of the annealing procedure must be carefully controlled. Annealing is carried out with an oxy-hydrogen torch which moves slowly along the rotating tube at a velocity of v_B (0.5 cm·min⁻¹). In contrast to a standing torch, the moved torch leads to a better controlled homogeneity of the tube across a length of several centimeters. The torch produces a narrow stationary axial temperature profile which is stable in time and traverses the tube together with the torch. An example of the profile is shown in Fig. 3 . This profile is measured directly on the tube’s surface both by an infrared thermometer, moved relative to the torch position, and by an infrared camera, which provides snapshots of the temperature distribution. In this way, the temperature and the profile shape can be very exactly controlled (in contrast to the use of a closed furnace where only an indirect measurement of temperature, e.g. by themocouples, would be possible).

 

Fig. 3 Axial temperature profile, T_max: temperature in the maximum of the profile, Δz_B: effective axial width calculated according to Eq. (9) with E = 500 kJ·mol⁻¹ (Δz_B = 1.9 cm).

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In detail, however, there is still a certain difference between the temperature at the outer tube surface, which is measured, and the temperature at the inner tube surface, which is responsible for the diffusion effect in the thin doped layer. The temperature difference depends on the torch velocity, the axial width of the temperature profile, the quartz glass type and the wall thickness of the tube. In [14], a method is described to estimate this effect by collapsing experiments. For the conditions of the annealing procedure (low torch velocity, small wall thickness) the temperature difference between outer and inner surface is small (about 10-15 K). All measured temperatures reported here are corrected on this basis to the real temperature within the diffusion layer.

Because of the rotation of the tube and the stabilizing influence of the silica carrier tube, the layers and the interface between layer and tube remain stable despite the relatively low viscosity of the doped layers down to almost 10³ Pa·s.

The collapsing step is of great importance for the exact determination of the diffusion coefficient. It is accompanied by a strong elongation of the radius coordinate by a mean factor of about 70. After collapsing, it is possible to measure exact radial distribution profiles by EPMA, which yields a spatial resolution of about 3-5µm. This resolution would by far not be sufficient for a profile measurement directly on the initial layers, before collapsing. Even if a complicated coordinate transformation must be accepted for the evaluation, only the collapsing step makes the determination of diffusion coefficients possible from the annealed samples.

2.2 Determination of the boron content

The WD-EPMA measurements were carried out using a JXA 8800 L (JEOL) microprobe equipped with multilayer diffractive elements (LDE) for the detection of ultra-soft boron KL_III (α) radiation. This instrument enables the investigation of both element concentration and distribution with high spatial resolution.

Unfortunately, there are a number of difficulties connected with the analysis of light elements and some of them are already described in literature [15]. In particular, the B₂O₃-SiO₂ system exhibits some issues which require special effort concerning sample preparation and experimental strategy [16], as described in the following.

First of all, one has to mind the enormous absorption of boron Kα radiation in silica and the energy of the exciting electrons is crucial for a best possible detection capability. The electron energy must be selected as low as possible, so that a maximum of boron Kα intensity can be detected. On the other hand the energy must be high enough for a sufficient excitation of silicon Kα radiation [15]. Therefore, both experiments and Monte Carlo simulations were carried out to optimize the excitation conditions.

A further difficulty is the peak overlap of boron Kα (183 eV) and oxygen Kα (525 eV) in 3rd order reflection. Because of the broad pulse-height spectrum of boron Kα radiation, the oxygen signal cannot be discriminated completely without a significant loss of boron intensity. Only by replacing the previously used LDE (2d: 94 Å) by an optimized multilayer element (2d: 145 Å) with a much lower high order sensitivity were we able to overcome this problem. Moreover, this LDE provides an increased reflectivity for boron Kα radiation, and together with the optimized excitation conditions, the detection limit could be improved by approximately one order of magnitude to a value of about 0.1 mol% B₂O₃.

The final task is a reliable and accurate quantification. Due to the strong influence of chemical bonding on the boron Kα peak, the careful choice of the standard material is crucial for the specification of exact concentration values [17]. We have carried out a set of measurements of boron doped silica glasses by using different standards and compared the results with wet chemical analysis of the boron content as well as with refractive index measurements. The best agreement of EPMA with the reference methods was achieved by standards with a composition similar to that of the investigated sample. Finally, we have favored a certified DURAN glass [18] as standard for the quantitative measurements of the data presented in this work. For comparison, refractive index profiles of all specimens were measured nondestructively by a deflection angle method [13] on the rods before the preparation of slices for EPMA measurements. A linear relationship between the refractive index and the boron concentration was found with a slope of (5 ± 0.2)·10⁻³ / mol% B₂O₃. This relationship is in close agreement with the relationship reported in [1] and confirms the accuracy of the presented concentration data. In consequence, the relative error of the specified concentrations can be expected to be less than 5% for absolute concentrations between 1 and 10 mol% B₂O₃.

The spatial resolution of the EPMA method is 3-5 µm under optimum conditions. In practice, most measurements have been carried out with a defocused electron beam and a resolution of 20 µm, which is sufficient for the precise determination of the profile shape in all cases investigated.

3. Calculation of the diffusion coefficients

Diffusion coefficients have been determined by a comparison of the measured radial concentration profiles with calculated ones, taking into account the coordinate elongation during collapsing.

The program used for the calculation implements the numerical solution of the radial diffusion in cylindrical coordinates by finite elements according to the diffusion equation

The coordinate ρ in Eq. (3) corresponds to the radius coordinate during annealing (tube with inner layer). The measured profile, on the other hand, corresponds to the radius coordinate after collapsing, ρ*. The coordinate transformation between both states is performed by

For the fitting of each concentration profile, we tentatively used a concentration dependence according to

The calculation starts with a step profile for the initial deposition state (which cannot be directly measured with sufficient spatial resolution), and the calculated curves are successively adjusted to the “unannealed” and “annealed” profiles. Examples are shown in Figs. 2(a) and 2(b) for two different boron concentrations. From Fig. 2 it can be seen that the τ values of the unannealed samples (which comprise only the diffusion effect during collapsing) are much smaller than the τ values of the annealed samples (which comprise the combined diffusion effect of annealing and collapsing). This demonstrates that the tube collapsing provides only a small contribution to the total diffusion effect. The diffusion coefficient during annealing is then determined from the difference of the τ values achieved for the “annealed” and “unannealed” profiles, Δτ.

Alternatively, we can ignore the initial step profile and start the calculation with the measured profile of the unannealed sample at τ = 0. In this case, the diffusion coefficient is directly determined from the τ value achieved for the adjustment of the “annealed” profile (Δτ = τ). Both approaches yield fully agreeing results.

The effect of surface vaporization is considered by setting the boundary concentration at the inner layer surface to zero. Small deviations at the inner part of the profile result from a certain re-incorporation of boron during the collapsing steps similar to in the case of phosphorus diffusion [8]. This effect, however, is of no consequence to profile fitting because the inner region of the collapsed profiles is overestimated due to the strong coordinate elongation in the rod center compared with the layer state in the tube during annealing.

In this way, an adjustment has been made possible by different τ's and k's in all cases here investigated, i.e for all measured radial profiles within the experimental error limits.

In order to determine the diffusion coefficients, we have still to consider the coordinate τ. According to Eq. (2), the quantity τ is related to the diffusion coefficient. From the adjusted values of τ and k, the diffusion coefficient can be calculated by

Strictly speaking, the temperature dependence of the diffusion coefficient must already be known in order to determine Δz_B. This is carried out by an iterative approach, starting with an intuitively reasonable width and improving it by the first step calculation of the diffusion coefficients (see Section 4). In the experiments carried out here, v_B was always 0.5 cm·min⁻¹, Δz_b varied between 1.5 and 2.0 cm depending upon the temperature, and the number of burner passes was changed from 2 (for 2000°C) to 45 (for 1700°C).

Previous investigations of the diffusion of phosphorus [8] and aluminum [10] in silica have shown that the concentration dependence of the diffusion coefficients could not be adjusted by Eq. (5) but rather only by a more general ansatz

The accuracy of the individual diffusion coefficients is represented by uncertainties in the adjustment, and direct error influences of experimental parameters are of secondary importance. In practice, four half profiles of each sample were measured, and each profile was individually adjusted by calculated profiles with good agreement in general. The estimated errors of the different coefficients are between ± 0.07 and ± 0.15 referring to the common (decadic) logarithm.

4. Results

A total of 16 diffusion experiments have been carried out at temperatures between 1700 and 2000°C with maximum concentrations after annealing between 0.5 and 8 mol% B₂O₃. A first-step adjustment of all profiles was carried out on the basis of Eq. (5). The individual fitting parameters τ and k for each measured diffusion profile were used to calculate diffusion coefficients according to Eqs. (6), (7), and (9) for the respective maximum concentration of each “annealed” profile, with an assumed activation energy of 500 kJ∙mol⁻¹. For the overall adjustment of the calculated diffusion coefficients we have used the relationship

 

Fig. 4 Determined diffusion coefficients in comparison with Eqs. (12)-(15) (broken line) and Eqs. (17)-(20) (solid line); one error bar is given as example.

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In an iteration step, the adjustment of all profiles was repeated on the basis of Eq. (10) with x = 1.03, leading to improved values of τ and k for each diffusion experiment. Diffusion coefficients were calculated with the improved values of the activation energies by Eqs. (6), (7) and (9). In the result, small changes of the 16 diffusion coefficient were achieved, but practically all near the limit of the experimental error. Because of the steepness of the temperature profile (see Fig. 3), the activation energy has only small effect on the profile width and the resulting diffusion time. The final values (as result of the iteration) are shown in Fig. 4 as common logarithms. The diffusion coefficients increase with the boron content of the glasses in a practically linear relationship between lgD and the concentration c, with

At 0.050, the MAD is well within the limits of the experimental error.

In Eqs. (12)-(15), the concentration influences the factor of 1/T (which refers to the activation energy) more than the constant term (which refers to the preexponential). Therefore, we have repeated the regression without considering B₀ (B₀ = 0)

In Fig. 4, both Eqs. (12) and (17) are displayed, showing the agreement with the experimental results. Within the limits of experimental error, both functions are equivalent. For reasons of simplification, we based the formulation of the results upon Eqs. (17)-(20) and obtained

(Note that each experiment yields in principle not a single diffusion coefficient but a nearly straight line according to Eq. (6), which is valid for the concentration region of the respective experiment. Depicting several line bundles in Fig. 4, however, looks cluttered. Therefore, we have preferred to show only one diffusion coefficient for each experiment, at the relevant concentration of the profile maximum. For the mathematical evaluation there is no difference.)

The error of lgD, calculated from Eq. (21) can be estimated to be ± 0.06 (according to the order of the MAD, this means a relative error in D of about 15%). The individual uncertainties of E and lgD₀, however, are essentially larger and can be estimated to be ± 25 kJ∙mol⁻¹ and ± 0.6, respectively. This is the result of the limited temperature range of the investigations. Bear in mind, however, that the errors of E and lgD₀ are interdependent in effect on the diffusion coefficient. (A change in the activation energy is largely compensated by a change in lgD₀).

In Fig. 5 , the dependence of the adjusted diffusion functions on the temperature is shown for selected concentrations as an Arrhenius plot.

 

Fig. 5 Arrhenius plots of boron diffusion in vitreous silica according to Eqs. (12)-(15) (broken line) and Eqs. (17)-(20) (solid line).

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5. Discussion

5.1 Comparison with literature data

Diffusion investigations in B₂O₃-SiO₂ glass at high temperatures > 1500°C by other autors are not known to the best of our knowledge. In the temperature range between 1000°C and 1250°C, however, diffusion coefficients of boron in amorphous SiO₂ have been determined in thin layers, with respect to the importance of boron as a semiconductor dopant in thin film device technology [20–28]. The concentration dependence of the diffusion coefficient has been observed, although the evaluation of the data has been founded in all cases on the concentration-independent diffusion formalism. Therefore, an exact relationship of the reported diffusion coefficients to a defined concentration is simply not possible. Moreover, a certain dependence of the diffusion coefficient on the preparation method [24], on the ambient [26,27], and on codopants [27,28] has been observed. A detailed review of the different effects is not within the scope of our present report. Nevertheless, if we extrapolate the high temperature data derived here to lower temperatures, then we achieve the result that all reported diffusion coefficients for nominally pure B₂O₃-SiO₂ layers and inert ambients are well covered by the extrapolated D values for concentrations between 0 and about 10 mol%. This is shown in Fig. 6 for some examples. The comparison with the reported data supports the assumption that the boron diffusion – both in high purity glasses and amorphous layers – is distinguished by a simple and consistent behavior over a large range down to temperatures as low as 1000°C.

 

Fig. 6 Coefficients of boron diffusion in amorphous SiO₂ according to [18–20], in comparison with the present study for 0 mol% B₂O₃ (extrapolated) and 10 mol% B₂O₃ according to Eqs. (12)-(15) (broken line) and Eqs. (17)-(20) (solid line).

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The comparison of the diffusion coefficients of boron with the diffusivities determined previously for phosphorus and aluminum in vitreous silica reveals that the values for low concentrations resemble closely [8,10], especially by comparison of boron and aluminum. In the case of aluminum and phosphorus, however, the concentration influence is much more distinct than for boron, and the concentration dependence is strongly nonlinear. This is expressed by x values of 0.41 (Al) and 0.32 (P). The comparison of the diffusion coefficients of boron with the diffusivities determined for germanium [9] in vitreous silica shows a certain similarity. We have a nearly linear relationship between the logarithm of the diffusion coefficient and the molar concentration of the dopant oxide. Again, the diffusion coefficients at low concentrations are similar. However, the activation energy in the case of germanium is distinctly higher. In Fig. 7 , a comparison of the diffusion coefficients at 1800°C is shown.

 

Fig. 7 Diffusion coefficients in binary glasses at 1800°C ●B in B₂O₃-SiO₂ (see this work), ○ Ge in GeO₂-SiO₂ [9], (c: concentration of B₂O₃ and GeO₂).

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5.2 Correlation of diffusion coefficient and viscosity

It would be of general interest for a deeper understanding of the mechanism of the diffusion to correlate it with other transport processes such as viscous flow. The viscosity of glasses in the yB₂O₃ (1-y)SiO₂ system has already been determined by several authors: in the y range between 1 and 0.42 (T = 1030°C…1460°C) [29], between 0.98 and 0.39 (T = 250°C…1200°C) [30], and between 0.25 and 0 (T = 1300°C…1800°C) [31]. In general, Arrhenius-like behavior between viscosity and temperature has been observed for y ≤ 0.4, with a strong dependence on the boron concentration.

We have already re-evaluated all reported viscosity data in [29]- [31] and fitted them in order to describe the temperature and concentration-dependent sintering behavior of porous B₂O₃-SiO₂ layers [11]. These fit functions, Eqs. (2)-(4) in [11], have been used here to calculate viscosity values for the exact conditions (concentration, temperature) of each diffusion experiment. In Fig. 8 , the calculated viscosities are displayed depending on all measured diffusion coefficients. Irrespective of the individual concentrations and temperatures, we find a strongly linear correlation between the logarithms of diffusion coefficient D and viscosity η in the form

 

Fig. 8 Correlation between diffusion coefficient D and viscosity η in the glass system B₂O₃-SiO₂.

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The fit functions used represent the measured viscosity data with a maximum uncertainty of about 0.1 in lgη in the concentration range of interest. This is sufficient for the attempt of a correlation between diffusion and viscosity. Moreover, the comparison between viscosity and diffusion involves a partial extrapolation to higher temperatures, beyond the experimental range of the viscosity data. The uncertainty in this respect is believed to be low because the temperature dependence in both cases is well described by Arrhenius relations.

A linear relationship between viscosity and diffusion is often discussed and interpreted in the literature by the Stokes-Einstein relationship

6. Conclusions

The diffusion of boron in binary B₂O₃-SiO₂ glasses has been investigated. Similar to in the case of germanium, boron diffusion reveals a pronounced concentration dependence, which covers about one order of magnitude in the investigated concentration range from 0.5 to 8 mol% B₂O₃. The diffusion coefficients between 1700°C and 2000°C can be described by an Arrhenius function with the constant preexponential factor D₀ = 10^0.38cm²s⁻¹ and the concentration-dependent activation energy E = (449-4.8·c^1.06) kJ·mol⁻¹ which is decreased with increasing B₂O₃ molar concentration c. The extrapolated diffusion coefficients agree fairly well with those determined previously for boron in amorphous SiO₂ layers at temperatures between 1000°C and 1250°C. A strong correlation between diffusion and viscosity in B₂O₃-SiO₂ glasses has been found.