The diffusivity of erbium in the anatase phase of titanium dioxide (TiO_{2}) has been studied for various temperatures ranging from 800 °C to 1, 000 °C. Samples of TiO_{2}, with a 10 nm thick buried layer containing 0.5 at% erbium, were fabricated by radio-frequency magnetron sputtering and subsequently heat treated. The erbium concentration profiles were measured by secondary ion mass spectrometry, allowing for determination of the temperature-dependent diffusion coefficients. These were found to follow an Arrhenius law with an activation energy of $ ( 2.1 \xb1 0.2 ) $ eV. X-ray diffraction revealed that the TiO_{2} films consisted of polycrystalline grains of size ≈ 100 nm.

The lanthanide rare earth (RE) metal ions, doped into inorganic host materials, have rather unique optical properties due to their active 4*f*-shell electrons, which are shielded from the surrounding environment by the outermost filled 5*s* and 5*p* shells. This leads to relatively well-defined energy levels and transition energies as compared to other solids, and this opens for many applications. Restricting ourselves to the trivalent erbium ion, Er^{3+}, these include upconversion layers for solar cells,^{1} upconversion nanocrystals for imaging,^{2} three-dimensional displays,^{3} and fiber amplifiers.^{4} A huge variety of host materials have been studied, among which TiO_{2} is just one. The most energetic resonance at 639 cm^{−1} of the TiO_{2} phonon spectrum^{5} is rather low, which suppresses multi-phonon relaxation mechanisms and thus makes this host material appealing. Hence, Er^{3+}-doped TiO_{2} has received considerable attention, ranging from basic optical studies in sol-gel^{6} and magnetron-sputtered^{7} materials to upconversion in wave guides,^{8} nanoparticles,^{6,9,10} and thin films.^{11} In addition, the TiO_{2} bandgap energy of 3.3 eV allows for exploiting the interaction between the host material and excited states in RE ions for, e.g., above-bandgap sensitization^{12} and improved photo-catalytic activity.^{13} In many cases, the material processing involves heat treatments at temperatures ranging up to ≈ 1, 000 °C in order to control the structural properties of the host material^{9,10,14,15} and in consequence the optical properties of the Er^{3+} luminescence. However, not much thought has been given to the fact that erbium ions might start to migrate at these elevated temperatures, which could potentially influence the material properties. For this reason, it is important to know the diffusivity of Er ions in TiO_{2}, which is the subject of the present work. The experiments are carried out in two steps. The first part, described below, determines the temperature-dependent diffusivity of Er ions in the anatase phase of TiO_{2}. The second part, to be described later, establishes that the conclusions of the first part are unaffected by the fact that the TiO_{2} was grown in the amorphous phase and only converted to anatase in the very initial stages of the heat treatment.

In order to establish a well-defined starting point for the diffusion experiments, a film of TiO_{2}, with a thin, buried, Er-doped layer of thickness 10 nm, was deposited on a silicon wafer, see Fig. 1(a), using a radio-frequency magnetron sputtering system from AJA Orion ATC. The pure and Er-doped layers were deposited from two separate sputter targets of diameter 2 inches: One pure TiO_{2} target and one target containing a mixture of TiO_{2} and Er_{2}O_{3}, leading to an Er concentration of $ \u2248 0.5 $ at % in a slightly oxygen-rich titania host. The film was deposited at room temperature with a deposition rate of $ 1.3 nm / min $ at a pressure of 3 mT and with a gas flow of Ar (13 sccm) and O_{2} (0.5 sccm). The power on each target was 200 W with a generated target DC bias of −345 V.

This wafer, used in the first part of the experiment and called the “main wafer,” was cut into smaller pieces, which were heat treated at different temperatures ranging from 800 °C to 1,000 °C and different times; 15 minutes or one hour depending on temperature. Two different furnaces were used; a rapid thermal annealing (RTA) furnace primarily for the higher end of the temperature range, and a Vecstar tube furnace, which was convenient for treatments of longer duration. The sample temperature in the RTA furnace was measured by a pyrometer, which was calibrated up against a thermocouple inserted into a bulk piece of silicon. In the tube furnace, the sample temperature was measured by a thermocouple mounted inside a quartz crucible close to the sample position. As we shall see in Fig. 4, the experimental results obtained by the two different heating methods are consistent with each other, which adds great confidence to the accuracy of the temperature readings. A selection of the samples were broken in the middle and the TiO_{2} films subsequently imaged by scanning electron microscopy, see Fig. 1, revealing a columnar structure of the as-grown film, which disappeared upon heat treatment. There are no distinct differences visible between the samples treated at 800 °C and 980 °C.

The crystal structure of the film was investigated by X-ray diffraction using the Cu $ K \alpha $ line at 1.54056 Å before and after the heat treatments. Spectra were obtained for samples heat treated at 800, 900, 960, 980, and 1,000 °C, see a selection in Fig. 2. Prior to the heat treatment the TiO_{2} film was amorphous and showed no diffraction peaks. All other samples showed the anatase phase with no systematic variations in the spectra and hence in crystallinity between 800 and 980 °C (consistent with the images of Fig. 1(c,d)), but the spectrum obtained from the sample heat treated at 1,000 °C indicated a beginning decomposition of the film with reduced crystallinity and build-up of material strain. Specifically, between 800 and 980 °C the most intense peak showed a relative standard deviation in intensity of 5 %, a peak position of $ 2 \theta = 25.324 \xb0 \xb1 0.005 \xb0 $, and a $ 2 \theta $ full width at half maximum of $ 0.222 \xb0 \xb1 0.006 \xb0 $. In comparison, for the sample heat treated at 1,000 °C the most intense peak was reduced by 86 % in intensity and increased by 0.3 % in peak position. A diffraction peak from the (3 2 0) plane of the silicon wafer (not shown) presented a $ 2 \theta $ full width at half maximum of $0.13\xb0$, which we may take as an estimate of the instrument resolution. Subtracting this from the observed peak width from the TiO_{2} film, we estimate by Scherrer’s equation a mean crystallite size of $ \u2248 100 $ nm. We note that it requires heat treatment at temperatures exceeding 1,000 °C for observing rutile as the dominant phase,^{9,10} which is consistent with its absence in the present investigation, see Fig. 2.

The samples were then investigated by secondary ion mass spectrometry (SIMS) using a Tof-SIMS 5 instrument. Examples of raw data from the measurements are shown in Fig. 3 for the as-grown sample and for a sample heat treated at 980 °C. The depth scale of the SIMS data was calibrated by measuring the depth of the sputtered crater by a Veeco Dektak 150 surface profiler and assuming a constant sputter rate. For the as-grown sample the measured SIMS profile, $ y Er ( ASG ) ( x ) $, is theoretically given by a convolution of the true concentration profile, $ \rho Er ( ASG ) ( x ) $, with the SIMS instrument response function *M*(*x*), i.e. $ y Er ( ASG ) ( x ) = \u222b M ( x \u2212 x \u2032 ) \rho Er ( ASG ) ( x \u2032 ) d x \u2032 $. The true concentration profile, $ \rho Er ( ASG ) ( x ) $, is nominally a 10 nm wide top-hat distribution, and *M*(*x*) takes into account the finite spatial resolution of the instrument, which is partly due to the fact that Er^{3+} ions are kicked further into the material during the sputtering process, leading to an asymmetric SIMS profile as clearly seen by the as-grown case in Fig. 3. However, if the concentration profile *ρ* evolves according to Fick’s second law, $ \u2202 \rho \u2202 t = D \u2202 2 \rho \u2202 x 2 $ with temperature-dependent diffusion coefficient *D* during the heat treatment at temperature *T* and of duration *t*, then it follows (see the appendix) that the resulting SIMS profile should theoretically be given as the convolution, $ y Er ( T ) ( x ) = \u222b 1 2 \pi \sigma exp ( \u2212 ( x \u2212 x \u2032 ) 2 2 \sigma 2 ) y Er ( ASG ) ( x \u2032 ) d x \u2032 $, between the as-grown SIMS profile and a Gaussian function of variance $ \sigma 2 = 2 D t $. In practice, we model the SIMS profile of heat treated samples as:

for the following reasons: First, there is finite precision to the calibration of the depth scale, which may lead to an artificial shift in mean depth of the SIMS profiles. Since the different samples are derived from the same wafer, these depths should nominally be equal, and possible variations are compensated by the scaling parameter *η* in the fitting model. Next, the amplitude *A* absorbs possible variations in the sputter yield between each measurement. Finally, a background parameter *B* is added, since a small background contribution is observable in the data. The physical interpretation of this will be discussed later. The experimental SIMS profiles, which essentially consist of histograms of counting data with Poissonian statistics, are fitted by the model of Eq. (1), where the free parameters *A*, *B*, *η*, and *σ* are determined by a likelihood ratio estimation.^{16} An example of this fitting procedure is shown in Fig. 3.

A range of diffusion coefficients have been plotted in Fig. 4 as a function of inverse temperature, using a logarithmic vertical axis. Clearly, the data follows largely a straight line and thus conforms to an Arrhenius law. The statistical uncertainty of each fitted diffusion coefficient *D* is typically 3 %. However, the fitted values of *η* are found in a range between 1.03 and 1.13, which indicate the uncertainty in the length scale of the SIMS profiles. In relative units, the diffusion coefficients $ D = \sigma 2 / 2 t $ must inherit twice the relative uncertainty due to fluctuations in the determination of the length scale. In order to make a conservative estimate of all statistical and uncontrolled systematic errors, we take a relative uncertainty of each diffusion coefficient to be 30 %, which appears consistent with the spread of data points in Fig. 4. Note in particular that measurements of *D* for $ T = 900 \xb0 $C were performed on three independent samples with a spread consistent with the proposed error bars. The data in Fig. 4 is fitted by the relation,

with a fitted activation energy of $ Q = ( 2.1 \xb1 0.2 ) $ eV and a fitted diffusion coefficient $ D 0 = ( 1.85 \xb1 0.18 ) \xd7 10 \u2212 16 cm 2 / s $ at the temperature *T*_{0} = 1,194.4 K, which is chosen such that the fitted values of *Q* and *D*_{0} become uncorrelated. With these fit values and uncertainties, the standard deviation of the fit model is calculated and shown as dotted curves in Fig. 4. Extrapolated to infinite temperature, the diffusion coefficient becomes $ D \u221e = ( 1.3 \xb1 2.6 ) \xd7 10 \u2212 7 cm 2 / s $. To exemplify the typical diffusion length $ L = 2 D t $ predicted by the experimental results, one may choose one-hour heat treatments at 800 °C, 900 °C, and 1,000 °C, which lead to diffusion lengths *L* equal to $ ( 3.7 \xb1 0.4 ) $ nm, $ ( 9.6 \xb1 0.5 ) $ nm, and $ ( 22 \xb1 2 ) $ nm, respectively.

While the above results constitute the main conclusion of the manuscript, it still remains to prove that the observed diffusivities really relate to the anatase phase of TiO_{2}. We observe from Fig. 2 that a phase transition occurs, from amorphous to anatase, during the heat treatment, and in principle the diffusion could originate from processes during both phases. For this reason we deposited yet another wafer, called the “control wafer,” which was nominally identical to the “main wafer” sketched in Fig. 1. The wafer was cut into smaller pieces and heat treated for various times at 800 °C and at 980 °C, representing the lowest and highest temperatures in Figs. 2 and 4 without observable film degradation. The samples were investigated by X-ray diffraction as seen in Fig. 5. The results show that the phase transition from the amorphous phase to the anatase phase occurs on a time scale faster than 15 seconds and that there is no further development in the film crystallinity on longer time scales. Evidently, this holds true for heat treatment at both 800 °C and 980 °C. Next, the samples were investigated by SIMS, the result of which are also presented in Fig. 5. To enable a better comparison, the various data curves have been stretched along the horizontal scale in order to obtain an equal mean depth of the SIMS profiles. This corresponds to circumventing the slight uncertainty in the depth calibration, which was previously parametrized by the scaling parameter *η* in the fitting procedure of Fig. 3. The observed count signals on the vertical axis are also scaled to facilitate a convenient comparison of the shape of the SIMS profile peaks. Similar graphs from the 980 °C heat treated sample from the main wafer have also been added for comparison in Fig. 5. The variations in depths are caused by the fact that the control wafer turned out to have a thinner TiO_{2} top layer than the main wafer. Based on our observations, we conclude the following: (i) It is possible that some diffusion occurs during the amorphous phase or during the process of crystallization into the anatase phase, despite the fact that it can last at most 15 seconds. This can be seen as a small, but significant change in the background level (depths smaller than ≈ 40 nm) for the control wafer at 15 seconds heat treatment. The effect is most evident at 980 °C heat treatment, where the background level reaches ≈ 2 % and ≈ 3 % of the maximum peak height of the SIMS profile for 15 and 60 seconds treatment, respectively. (ii) Apart from these changes in the background level, the shape of the SIMS profile peak changes negligibly within the measurement accuracy. Since the main results of this manuscript, presented in Fig. 4, are determined solely from such peaks, our conclusions cannot be affected by the initial amorphous phase or crystallization process into the anatase phase. The experimental parameters, *D*_{0} and *Q*, are thus properties of anatase TiO_{2}. To underline this conclusion, the observed peak shape for 15 minutes heat treatment of the main wafer at 980 °C is evidently different from that of the ASG profile, whereas nothing detectable occurs during the first 15 seconds of heating at 980 °C for the control wafer. (iii) The background level, which were parametrized by *B* in the fitting procedure of Fig. 3, increases to ≈ 10 % of the SIMS profile peak height during the entire 15 minutes heat treatment at 980 °C. In comparison to the ≈ 2 % background level at 15 seconds of heat treatment, this background level must largely (and possibly exclusively) be a result of diffusion in the anatase phase, and we attribute this effect to diffusion of a small amount of Er^{3+} with large diffusivity at grain boundaries. In addition to the above considerations, we also remind that the typically observed diffusion lengths (exemplified by the thick line of length 13.2 nm in Fig. 3) are significantly smaller than the ≈ 100 nm average grain size as determined by X-ray diffraction. Hence, most Er ions will always reside at a distance far from the grain boundaries, which will then not have any significant, systematic impact on the main results regarding diffusion within the anatase grains.

In conclusion, the diffusion coefficient of Er^{3+} ions in anatase TiO_{2} has been measured and found to follow an Arrhenius law with an activation energy of $ ( 2.1 \xb1 0.2 ) $ eV. Furthermore, the results show that the diffusion occurs significantly faster at grain boundaries and that diffusion of Er^{3+} in the amorphous phase has played a negligible role in our experimental procedure.

This work was supported by the Innovation Fund Denmark under the SunTune project.

### APPENDIX

Here we derive the relation between the diffusion equation and the experimentally measured SIMS profiles. Note first, that with $ \sigma 2 = 2 D t $, the Gaussian function $ g ( x , t ) = 1 2 \pi \sigma exp ( \u2212 x 2 2 \sigma 2 ) $ fulfills the diffusion equation $ \u2202 g \u2202 t = D \u2202 2 g \u2202 x 2 $. Furthermore, $ g ( x , t ) \u2192 \delta ( x ) $ when $ t \u2192 0 $, where $ \delta ( x ) $ is Dirac’s delta function. Provided that the temperature, and hence the diffusivity *D*, is constant, it follows that the time-evolution of the Er concentration profile can be written during the heat treatment as: $ \rho Er ( x \u2032 , t ) = \u222b \rho Er ( ASG ) ( x \u2033 ) g ( x \u2032 \u2212 x \u2033 , t ) d x \u2033 $. The formula is trivially true at *t* = 0, due to the *δ*-function inside the integral, and at later times the equality follows by linearity. By equating *t* with the duration of heat treatment, the resulting concentration profile is thus: $ \rho Er ( T ) ( x \u2032 ) = \u222b \rho Er ( ASG ) ( x \u2033 ) g ( x \u2032 \u2212 x \u2033 ) d x \u2033 = \u222b \rho Er ( ASG ) ( x \u2032 \u2212 x \u2033 ) g ( x \u2033 ) d x \u2033 $, where the time variable *t* was omitted for simplicity in the first step and the second equality follows by substitution. Now, the measured SIMS profile $ y Er ( T ) $ is a convolution between the true concentration profile $ \rho Er ( T ) $ and the instrument response function *M*, resulting in:

In the second line, the above expression for $ \rho Er ( T ) ( x \u2032 ) $ is inserted, and the third line exploits the fact that the order of integration can be reversed (for sufficiently rapidly decreasing functions) and also involves a substitution of variables, *y* = $ x \u2032 \u2212 x \u2033 $. The fourth line presents another substitution, written formally as $ x \u2032 $ = $ x \u2212 x \u2033 $ in order to end with the expression stated in the main text. The above derivations are valid in the absence of interfaces, which was ensured in the experiment by placing the initial Er-doped layer far from both the silicon substrate and the sample surface compared to the relevant diffusion lengths.