I. INTRODUCTION
Er3+-doped LiNbO3 (Er:LN) is a potential substrate material for active optical waveguide devices as it combines the excellent electro-optic, acousto-optic, and nonlinear optical properties of LN with the laser property of Er3+. In particular, it allows broadband amplification and lasing in the telecommunication wavelength region around 1.5 μm. Since the first continuous-wave waveguide laser was reported in 1991,Reference Brinkmann, Sohler and Suche1 a family of Ti (vapor Zn)-diffused Er:LN waveguide lasers (amplifiers) and integrated devices operated in the infrared regime have been demonstrated over the past years.Reference Becker, Oesselke, Pandavenes, Ricken, Rochhausen, Schreiberg, Sohler, Suche, Wessel, Balsamo, Montrosset and Sciancalepore2–Reference Schreiber, Hofmann, Grundkotter, Lee, Suche, Quiring, Ricken and Sohler10 For any practical Ti:Er:LN waveguide device, selective Er3+ doping is a prerequisite for the monolithic integration of active (optically pumped, Er3+ doped) and passive (unpumped) devices on a same substrate to avoid reabsorption in unpumped Er-doped waveguide. This is the reason why the active ions in almost all the Ti:Er:LN devices demonstrated so far were incorporated by the local Er3+ doping instead of the homogeneously bulk doping method.Reference Brinkmann, Sohler and Suche1–Reference Huang and McCaughan6, Reference Das, Ricken and Sohler8–Reference Schreiber, Hofmann, Grundkotter, Lee, Suche, Quiring, Ricken and Sohler10 Local Er3+ doping in an LN crystal is usually realized via thermal diffusion of Er metal at the temperature close to the Curie point of the crystal (1142.3 ± 0.7 °C at the congruent point). The diffusion of Er3+ into a congruent (Li2O content ∼48.4 mol%) LN (CLN) crystal in an argon or O2 atmosphere has been studied in detail by Baumann et al.Reference Baumann, Brinkmann, Dinand, Sohler, Beckers, Buchal, Fleuster, Holzbrecher, Paulus, Muller, Gog, Materlik, Witte, Stolz and von der Osten11 and Caccavale et al.Reference Caccavale, Segato, Mansour, Almeida and Leite12 The results show that the solubility of Er3+ ion in the crystal is limited. In a CLN crystal, the solubility is only ∼2.65 × 10Reference Zhang, Zhuang, Hua and Pun20 ions/cm3 (∼1.43 mol%) at the Curie point of 1142 °C.Reference Baumann, Brinkmann, Dinand, Sohler, Beckers, Buchal, Fleuster, Holzbrecher, Paulus, Muller, Gog, Materlik, Witte, Stolz and von der Osten11 Since the optical gain of a laser or an amplifier usually increases with the increased active ion concentration, the relatively low solubility of Er3+ ions in a CLN limits further demonstration of more efficient devices. The Er3+ solubility depends on not only the diffusion temperature but also the Li2O content in the crystal. Under the same diffusion temperature, the lower the Li2O content in the crystal is, the more the intrinsic defects present in the crystal are, the more the Er ions that can be accepted are, and hence the higher the Er3+ solubility would be. It can be anticipated that if the Er3+ diffusion is carried out in either a Li-poor atmosphere [i.e., Li-poor vapor transport equilibration (VTE) treatment and Er3+ codiffusion] or an off-congruent, Li-deficient LN, the Er3+ solubility would be prompted. In the earlier work, we have demonstrated the feasibility of the former method and the results showed that the solubility can be prompted by as much as 80%.Reference Zhang, Chen and Pun13 Present work aims at demonstrating the feasibility of using the latter method (i.e., Er3+ diffusion in an off-congruent, Li-deficient substrate) to prepare a locally Er3+-doped high solubility LN substrate for integrated optics use. The related work could not yet be found in the earlier literatures as far as we know.
II. EXPERIMENT
Five 1-mm-thick X-cut CLN plates with optical grade surfaces were used as the starting materials. At first, four of them were selected and subjected to the Li-poor VTE treatments at 1100 °C over different durations of 40, 120, 160, and 180 h. The remaining plate did not undergo the VTE process and retains the as-grown (i.e., congruent) state. Accordingly, the five plates have different crystal compositions. After the VTE procedure, the Li2O content at the surface of each VTE-treated plate was evaluated by the measurement on the birefringence at the crystal surface. For comparison purposes, the Li2O content at the surface of the as-grown plate was also evaluated in a similar way. Subsequently, pure Er metal (99.99%) film with a thickness of 30 or 40 nm was locally coated onto part of one surface of each plate, whereas the other part of the same surface was uncoated for reference. Next, the Er3+ diffusion was carried out at 1130 °C for 60 h in an environment of flowing O2 bubbled through room temperature water at a rate of ∼1.5 × 10−3 m3/min. After the diffusion procedure, the Li2O content at the Er3+-diffused surface was characterized again. Table I brings together the preparation parameters and the surface Li2O contents of all the five samples studied. Finally, the Er3+ depth profile and diffusion characteristics were studied by secondary ion mass spectrometry (SIMS).
A summary of Li-poor vapor transport equilibration (VTE) condition, Li2O content, Er3+ diffusion condition, Er3+ profile type, 1/e Er3+ diffusion depth, Er3+ diffusivity, surface Er3+ concentration, and Er3+ solubility for the diffusion in X-cut Li-poor VTE-treated LiNbO3 plates.
Erfc, complementary error function.
The Li-poor VTE treatment was carried out in a Li-poor atmosphere created by a closed Nb-rich two-phase (LiNbO3 + LiNb3O8) crucible at elevated temperature. The two-phase crucible was prepared by sintering homogeneous Nb-rich mixture of Li2CO3 and Nb2O5 powder, which have the same purity of 99.99%. The molar ratio was 40 mol% (Li2CO3): 60 mol% (Nb2O5). The mixture was pressurized and molded into a crucible model of 6 cm in inner diameter and 4 cm in height. A precalcination at 1000 °C for 10 h and an additional calcination at 1100 °C for 1 h produced an Nb-rich two-phase crucible. The solid phase reaction involved is 40 mol% Li2CO3 + 60 mol% Nb2O5 = 40 mol% CO2↑ + 60 mol% LiNbO3 + 20 mol% LiNb3O8. The detailed Li-poor VTE procedure is described as follows. At first, the plate to be treated was wrapped with Pt wires and placed on a Pt foil, which was placed in the crucible before the crystal to be treated. Then, the crucible was sealed by a lid, made of the same Nb-rich two-phase powder, and heated up to a target temperature and dwelled for a scheduled duration. Due to the Li concentration gradient inside and outside the crystal, the Li ions inside the crystal diffused away from the crystal surface at the elevated temperature and the crystal evaporated the Li2O gas, which sunk in the two-phase powder due to the solid phase reaction Li2O↑ + LiNb3O8 = 3LiNbO3. As a result, the crystal became Li-deficient more and more due to the continuous Li2O evaporation.
The ordinary and extraordinary refractive indices of the crystals studied were measured at the wavelengths of 632.8 and 1553 nm using a Metricon 2010 prism coupler (Metricon Corp., Pennington, NJ), which has a working principle of measurement on the critical angle of the total reflection. Note that the refractive index measured by this method should be the value at the crystal surface because the total reflection phenomenon takes place there. It is convenient to choose a transverse magnetic (TM) or transverse electric (TE) polarization scheme to measure the ordinary or extraordinary index, depending on the cut of the plate to be measured. All the measurements were carried out at room temperature (24.5 ± 0.1 °C).
SIMS analysis was accomplished by a time-of-flight second ion mass spectrometry (ToF SIMS V; ION-TOF GmbH, Münster, Germany.). A Cs+ beam (∼45-μm in diameter) of 42 nA at 3 keV was used to sputter a crater of 120 × 120 μm2, and a pulsed bismuth ion beam of 1 pA at 25 keV was used to analyze the secondary ions 6Li, 93Nb, 16O, and 166Er as a function of time. Positive secondary ions were detected. Ions from a central area of 20.5 × 20.5–22.5 × 22.5 μm2 inside the crater were detected. During the analysis, a low energy pulsed electron gun was used to neutralize the positive charges and hence degrade the surface charge accumulation. For the same purposes, before the SIMS analysis, a 30-nm-thick Ag film was coated onto the diffused surface of each sample to be analyzed. The trace and the depth of each erosion crater were measured by a Tencor Alpha Step 200 profilometer (KLA-Tencor Corp., Milpitas, CA). The depth resolution was mainly determined by the roughness of the crater under analysis and is better than 5 nm in our case.
III. RESULTS AND DISCUSSION
A. Li2O content
The Li2O content in an LN crystal can be determined by many optical and nonoptical methods.Reference Wöhlecke, Corradi and Betzler14 Here, the optical method of birefringence measurement was used to quantitatively evaluate the Li2O contents at the surface of the studied crystals. From the measured ordinary and extraordinary refractive indices, one can evaluate the corresponding Li2O content by utilizing the Li2O content-dependent Sellmeier equation.Reference Schlarb and Betzler15 The measurement error of the refractive index is within 10−3, which yields a Li2O content uncertainty of 0.1 mol%. The Li2O contents evaluated after the VTE process and Er3+ diffusion procedures are collected in Table I. Regarding the Li2O content evaluation after the Er3+ diffusion procedure, there is one point to be clarified. Earlier study has shown that the Er3+ dopant has a weak effect on the refractive index.Reference Zhang, Yang, Hua, Liu, Cui, Sun, Xu and Pun16 Therefore, after the Er3+ doping procedure, the measured birefringence value can be approximately used to evaluate the Li2O content and the presence of Er3+ has a small effect on the result.
For the as-grown congruent plate, as expected, the Li2O content, evaluated from the measured birefringence, is similar to 48.4 ± 0.1 mol%. After the VTE procedure of 40, 120 h, 160, and 180 h, the surface Li2O content decreases to 48.2 ± 0.1, 48.0 ± 0.1, 47.7 ± 0.1, and 47.3 ± 0.1 mol%, respectively. It is reasonable that the Li2O content decreases with the prolonged VTE. Note that after the 180 h VTE process, the decrease is only similar to 1.0 mol%. However, in the case of Li-rich VTE treatment, the composition at the crystal surface can reach the Li-rich boundary after a short VTE duration, which is not longer than 24 h.Reference Jundt, Fejer, Norwood and Bordui17 In other words, after a relatively much short period, the Li-rich VTE can give rise to ∼1.6 mol% Li2O content change. It is thus concluded that the Li-poor VTE is much slower than the Li-rich VTE, which is consistent with the previously reported result.Reference Bordui, Norwood, Jundt and Fejer18 This is because the Li+ diffusivity in an LN crystal decreases with the decreased Li2O content.Reference Jundt, Fejer, Norwood and Bordui17 Very slow Li-poor VTE procedure implies that the Li2O content in each treated crystal can be considered as approximately homogeneous.
The subsequent Er3+ diffusion procedure gave rise to further lowering of the surface Li2O content. This implies that the Li+ out-diffusion phenomenon took place still in the Er3+ in-diffusion procedure though the suppression measure that the diffusion was carried out in the wet O2 atmosphere has been taken. One can see from Table I that the Er3+ in-diffusion procedure led to the further surface Li2O content loss of 0.4, 0.5, 0.5, 0.3, and 0.3 mol% for the plate that has undergone the VTE treatment of 0, 40, 120, 160, and 180 h, respectively. It appears that the lower the original surface Li2O content is and the slightly smaller the further loss induced by the Er3+ in-diffusion procedure is. As mentioned above, this is because the Li+ diffusivity decreases with the decreased Li2O content.Reference Jundt, Fejer, Norwood and Bordui17 It is worthwhile to mention here that the quantification for Li+ out-diffusion in the Er3+ in-diffusion procedure was not reported previously as far as we know.
B. Er3+ profile and diffusivity
Figures 1(a)–1(e) show depth profiles of 166Er SIMS signals (thick black curves) detected from the five samples studied. As a representative, in Fig. 1(f), we show the substrate constituent 7Li, 93Nb, and 16O profiles measured from the sample no. 4. Since the concentration profiles of the substrate constituents 6Li, 93Nb, and 16O are homogeneous inside the crystals, their SIMS signals should be depth independent. Indeed, one can see from Fig. 1(f) that all the substrate signals show constancy with the depth during the analysis (so does for the other samples), implying that the charge compensation by the electron gun is effective. Nevertheless, the Er3+ profiles shown in Figs. 1(a)–1(e) need to be further justified by checking if the detecting area 20.5 × 20.5–22.5 × 22.5 μm2 is affirmatively smaller than the flat area at the bottom of the corresponding crater. The trace profiling has revealed that all the erosion craters have a flat bottom area larger than 50 × 70 μm2, which is considerably larger than the maximum detecting area 22.5 × 22.5 μm2. Thus, the experimental data shown in Figs. 1(a)–1(e) reflect the practical scenario of the constituent profiles and all results inferred are convincing.
(a)–(e): Depth profiles (black thick curves) of 166Er secondary ion mass spectrometry (SIMS) signal normalized to Nb signal for X-cut plates that has undergone the Li-poor vapor transport equilibration (VTE) treatments of 0, 40, 120, 160, and 180 h. The red/green plot overlapped on each measured Er3+ profile curve represents the best complementary error function (Erfc)/Gauss fit to the measured profile, and the fitted expressions are indicated. (f) Depth profiles of 6Li, 93Nb, and 16O SIMS signals detected from the plate that has undergone the Li-poor VTE treatment of 160 h.
Obviously, it is reasonable to consider the Er3+ SIMS intensity relative to a matrix ion intensity, so that the possible influence arising from the fluctuation of the primary ion (Cs+) beam may be degraded somewhat. Each measured Er3+ profile in Figs. 1(a)–1(e) is the result after the normalization to the corresponding Nb signal.
For convenience, we consider a one-dimensional reference frame fixed at the diffused surface with x-axis pointing to the diffusion depth direction of the X-cut plate. Because the Li+ out-diffusion accompanied the Er3+ in-diffusion procedure, the one-dimensional (planar) Er3+ diffusion process, according to the diffusion theory,Reference Crank19 can be described by a standard Fick-type diffusion equation with a Li2O content-dependent diffusion coefficient:
![$${{\partial C_{{\rm{Er}}}^{\rm{X}} (x,t)} \over {\partial t}} = D_{{\rm{Er}}}^{\rm{X}} [C({\rm{Li}}_{\rm{2}} {\rm{O}})]{{\partial ^2 C_{{\rm{Er}}}^{\rm{X}} (x,t)} \over {\partial x^2 }},$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160921011417762-0106:S0884291411001579:S0884291411001579_eqn1.gif?pub-status=live){kind=small}
where 
denotes the Er3+ concentration at the depth position x after diffusion time t and ![$D_{{\rm{Er}}}^{\rm{X}} [C({\rm{Li}}_{\rm{2}} {\rm{O}})]$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160921011417762-0106:S0884291411001579:S0884291411001579_inline2.gif?pub-status=live){kind=small}
is the diffusion coefficient, which is assumed to be independent of the Er3+ concentration 
but dependent of the Li2O content, named C(Li2O). Since the C(Li2O) is time dependent, the ![$D_{{\rm{Er}}}^{\rm{X}} [C({\rm{Li}}_{\rm{2}} {\rm{O}})]$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160921011417762-0106:S0884291411001579:S0884291411001579_inline4.gif?pub-status=live){kind=small}
is thus actually dependent of the diffusion time t. The diffusion equation in this case has mathematical solutions similar to the case that the diffusion coefficient is a constant.Reference Crank19 This argument is well supported by the present experimental results as demonstrated in the following and the numerical results on Ti4+ in-diffusion in an Li-enriched atmosphere,Reference Zhang, Zhuang, Hua and Pun20 which is described by a diffusion equation similar to Eq. (1).
The mathematical solutions to Eq. (1) can be briefly described as follows. For a given Er metal film thickness and under a given diffusion temperature, there exists a diffusion time t 1, named as the depletion time of Er3+ diffusion reservoir. When t < t 1, the Er3+ diffusion reservoir is not exhausted. In this case, the solution to Eq. (1) is a complementary error function (Erfc) with a constant surface concentration 
, which can be regarded as the solid solubility under the diffusion condition adopted. When the diffusion time is longer than the depletion time t 1, the Er3+ reservoir is exhausted. As the diffusion is prolonged from t 1, the Er3+ ions diffuse continuously into the bulk, the surface Er3+ concentration degrades gradually and the Er3+ profile transforms gradually from a Erfc to a Gaussian function with a surface concentration below the solid solubility 
.
Next, we discuss the Er3+ profile characteristics of the studied samples and compare with the theoretically predicted mathematical solutions described above. Figure 1(a) clearly shows that a certain number of Er3+ ions did not diffuse into the bulk of sample no. 1 and residue within the ∼0.1-μm-thick shallow layer near the crystal surface, implying that the Er3+ diffusion reservoir was not exhausted. Deeper than 0.1 μm, the Er3+ profile, as expected, can be well fitted by an Erfc:
![$$I_{{\rm{Er}}}^{\rm{X}} (x) = I_{{\rm{Er}}}^{\rm{X}} (x_0){\rm{Erfc}}[(x - x_0)/d_x ] + I_{\rm{N}}^{\rm{X}} ,$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160921011417762-0106:S0884291411001579:S0884291411001579_eqn2.gif?pub-status=live){kind=small}
where 
represents the yield of the secondary ion Er3+, dx denotes the diffusion depth, 
is the background noise, and x 0 represents the initial coordinate of the Er3+-diffused layer. The red plot overlapped on the measured Er3+ profile in Fig. 1(a) denotes the best fit to the experimental data. The good Erfc fit to the experimental data shows that the experiment and the theory are consistent and verifies that the diffusion equation with a time-dependent diffusion coefficient has a solution similar to the case that the diffusion coefficient is a constant.
In sample no. 2, the Er3+ concentration [see Fig. 1(b)] reveals a profile characteristic similar to that in the sample no. 1. The difference with each other is that fewer Er3+ ions residue on the crystal surface and the reservoir was almost exhausted. The situation is slightly different for sample no. 3. One can see from Fig. 1(c) that, although the profile retains the Erfc in nature, there is a slight misfit between the measured and the Erfc fitted curves near the crystal surface, and the measured ion yield is smaller than the predicted there. We refer to the Er3+ concentration profile in the sample no. 3 as the “Erfc-like.” Such a misfit implies that the diffusion time (60 h) adopted is slightly longer than the depletion time t 1; the Er3+ reservoir has been exhausted; and the surface Er3+ concentration is slightly below the solid solubility as a result of the diffusion of some surface Er3+ ions into the bulk. Due to the same reason, the misfit between the measured and the Erfc fitted curves near the crystal surface becomes more evident for the samples nos. 4 and 5. As mentioned above, with the decrease of the surface concentration, the Er3+ profile broadens and transforms gradually from an Erfc type to a Gaussian type. On the basis of this consideration, the measured Er3+ profiles in samples nos. 4 and 5 were also fitted by a Gaussian trial function:
![$$I_{{\rm{Er}}}^{\rm{X}} (x) = I_{{\rm{Er}}}^{\rm{X}} (0)\;{\rm exp} [ - (x/d_x)^2 ] + I_{\rm{N}}^{\rm{X}} .$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160921011417762-0106:S0884291411001579:S0884291411001579_eqn3.gif?pub-status=live){kind=small}
The green plots in Figs. 1(d) and 1(e) denote the best Gaussian fits, and the fitted expressions are indicated. One can see from Fig. 1(d) that the Er3+ profile in sample no. 4 can be well fitted by neither the Erfc nor the Gaussian function. It seems that the profile is closer to the Gaussian type, indicating that the Er3+ profile in sample no. 4 is just near the transition regime from the Erfc type to the Gaussian type. The complete transformation to the Gaussian type profile took place for the sample no. 5. One can see from Fig. 1(e) that the Er3+ profile in sample no. 5 can be well fitted by the Gaussian trial function while the Erfc fit leads to a large misfit near the diffused surface.
The fitted parameter values of 
and 
[see the fitted expressions indicated in Figs. 1(a)–1(e)] show that the fraction of the background noise is within 5%. Accordingly, the Er3+ concentration profile can be approximated as:
![$$C_{{\rm{Er}}}^{\rm{X}} (x) = C_{{\rm{Er}}}^{\rm{X}} (x_0){\rm{Erfc}}[(x - x_0)/d_x ]\quad ,$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160921011417762-0106:S0884291411001579:S0884291411001579_eqn4.gif?pub-status=live){kind=small}
for the sample nos. 1, 2, and 3 and
![$$C_{{\rm{Er}}}^{\rm{X}} (x) = C_{{\rm{Er}}}^{\rm{X}} (0)\exp [ - (x/d_x)^2 ]\quad ,$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160921011417762-0106:S0884291411001579:S0884291411001579_eqn5.gif?pub-status=live){kind=small}
for the sample nos. 4 and 5. In the preceding two expressions, 
or 
represents the surface Er3+ concentration. Note that the fitting parameter dx in the two expressions has different physical implications. For the Gaussian profile, it denotes the diffusion depth at 1/e surface concentration, whereas for the Erfc profile, it no longer denotes the 1/e diffusion depth (simple numerical calculation shows that the dx value is about 1.575 times the x value corresponding to the 1/e intensity). To quantitatively characterize and compare the profile characteristics of the five samples studied, it is essential to unitedly consider the 1/e diffusion depth. Table I collects the values of this parameter, which is similar to 2.8 ± 0.1, 2.9 ± 0.1, 3.1 ± 0.1, 3.3 ± 0.1, and 3.7 ± 0.1 μm for the sample nos. 1–5 in order. From the obtained 1/e diffusion depth data, the Er3+ diffusivity 
can be readily evaluated according to the relation 
. The results are listed in Table I. Note that these diffusivity data are actually the averaged results over the 60 h diffusion time as the 
is time dependent as mentioned above. For the diffusion into the initial congruent crystal, the diffusivity is similar to 3.3 ± 0.2 × 10−2 μm2/h, which is similar to the result reported by Sohler and coworkers,Reference Baumann, Brinkmann, Dinand, Sohler, Beckers, Buchal, Fleuster, Holzbrecher, Paulus, Muller, Gog, Materlik, Witte, Stolz and von der Osten11 ∼3 × 10−2 μm2/h under the same diffusion temperature of 1130 °C in the argon atmosphere with the consideration of 1/e diffusion depth and is comparable to the data reported by Caccavale et al.,Reference Caccavale, Segato, Mansour, Almeida and Leite12 and ∼1.8 × 10−2 μm2/h at the 1150 °C diffusion temperature in the dry O2 atmosphere. Continue to discuss the diffusivities of other samples studied in this work. One can see from Table I that the diffusivity is similar to (3.5 ± 0.2), (4.0 ± 0.3), (4.5 ± 0.3), and (5.7 ± 0.3) × 10−2 μm2/h for the sample nos. 2–5 in order and increases with the decrease of the surface Li2O content. As the Li2O content in the crystal decreases from 48.0 mol% (sample no. 1) to 47.0 mol% (sample no. 5), the Er3+ diffusivity is nearly doubled.
C. Surface Er3+ concentration and solubility
The surface Er3+ concentration of the samples studied can be determined from each measured profile according to the law of mass conservation. This is done more conveniently by applying the law of mass conservation to the corresponding Eq. (4) or (5):
![$$C_{{\rm{Er}}}^{\rm{X}} (x_0)\int\limits_{x_0 }^{ + \infty } {{\rm{\! Erfc}}[(x - x_0)/d_x ]} dx \ \ \= C_{\rm{s}} \tau ^{\rm{X}} {\rm{ \ for\ the\ Erfc\ profile,}}$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160921011417762-0106:S0884291411001579:S0884291411001579_eqn6.gif?pub-status=live)
![$$C_{{\rm{Er}}}^{\rm{X}} (0)\int\limits_0^{ + \infty } {\rm\!exp [ - (x/d_x)^2 ]} dx \ \ \= C_{\rm{s}} \tau ^{\rm{X}} {\rm{ \ for\ the\ Gaussian\ profile,}}$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160921011417762-0106:S0884291411001579:S0884291411001579_eqn7.gif?pub-status=live)
where C s = ρN A/M with ρ, N A, and M representing mass density of erbium metal (9.06 g/cm3), Avogadro’s number (6.02 × 1023 atoms/mol), and molecular weight of the Er metal (166 g/mol), respectively. One can see from Figs. 1(a) and 1(b) that the total Er3+ ion number contained in the unit area of the metal film can be divided into two parts. One part contains the Er3+ ions residual on the unexhausted diffusion reservoir within the depth range of 0–x 0. The other part includes those Er3+ ions diffused into the bulk of the crystal and follows the Erfc profile described by Eq. (4). The τ X in Eq. (6) denotes the Er metal film thickness corresponding to the second part of Er3+ ions. It can be determined from the totally coated Er film thickness and the ratio of the area under the Erfc profile part to that under the entire Er3+ profile: τ X = (totally coated Er thickness) × (area under Erfc profile)/(total area under entire Er3+ profile). Table I brings together the obtained τ X values for the studied samples. Note that the τ X values for the sample nos. 3–5 are all the initial Er film thickness of 40 nm because all the Er3+ diffusion reservoirs of these samples were exhausted under the diffusion condition adopted. The evaluated surface Er3+ concentration is collected in Table I. It is similar to (3.7 ± 0.4), (4.9 ± 0.5), (4.8 ± 0.4), (4.5 ± 0.5), and (4.0 ± 0.4) × 1020 ions/cm3 for the sample nos. 1–5, respectively.
Based upon the obtained surface Er3+ concentrations and the profile characteristics, the solubility is further evaluated quantitatively or semiquantitatively. As described above, the surface concentration of an Erfc profile may be regarded as the solubility. This is the case for the sample nos. 1 and 2, in which all the Er3+ profiles are an Erfc type. So, the solubility in the two samples could be determined quantitatively. For the sample nos. 3–5, however, as the Er3+ profiles in these samples are all not well fitted by an Erfc, the solubility in these three samples cannot be determined accurately but only semiquantitatively. Because the Erfc-predicted ion yield is larger than the measured near the crystal surface, the solubility in these three samples must be above the obtained surface Er3+ concentration. Since the Er3+ profile in the sample no. 3 retains still the Erfc type in nature, the solubility in this sample should be slightly above the obtained surface Er3+ concentration. Because the Er3+ profile in the sample no. 4 and 5 is either Gauss-like or complete Gaussian and has a large mismatch with the Erfc fit near the crystal surface, the solubility in the two samples should be considerably higher than the obtained surface Er3+ concentration, i.e., > 4.5 × 1020 and 4.0 × 1020 atoms/cm3, respectively. We are able to further refine the solubility range of the two samples. The refinement is done in the following way. Note that each Erfc fitting curve in Figs. 1(d) and 1(e) has an 1/e depth value similar to the measured or Gauss-fitted plot. Assume that such an Erfc curve just corresponds to the circumstance that the diffusion reservoir was exhausted, and there are no Er3+ ions residual on the crystal surface in this case. Then, the surface concentration of the Erfc curve, which corresponds to the solubility, can be evaluated by considering that the area under the Erfc curve equals to that under the corresponding Gaussian one (i.e., the law of mass conservation). The evaluation shows that the solubility should be higher than 4.5 × 1020 atoms/cm3 for the sample no. 4 and 4.7 × 1020 atoms/cm3 for the sample no. 5. We should admit that such evaluation yields rather conservative results. This is because the 1/e diffusion depth of the practical Erfc profile just corresponding to the circumstance that the diffusion reservoir was exhausted must be considerably smaller than that adopted in the evaluation. Required by the law of mass conservation, an Erfc profile with a larger 1/e depth corresponds to a lower surface concentration, whereas that with a smaller 1/e depth corresponds to a higher surface concentration. Therefore, the practical solubility in the sample nos. 4 and 5 must be considerably higher than 4.5 × 1020 and 4.7 × 1020 atoms/cm3, respectively. Table I brings together the quantitatively or semiquantitatively evaluated solubility results for all the five samples studied. For convenience, all the data have been converted into the corresponding molar percentages by simply doing the normalization to the Nb5+ concentration in crystal, ∼1.85 × 1022 atoms/cm3. The results are also listed in Table I.
Before the discussion on crystal composition effect on the solubility, it is essential to compare our solubility data regarding the diffusion into an initial congruent substrate (i.e., sample no. 1) with the previous results reported by Sohler and coworkersReference Baumann, Brinkmann, Dinand, Sohler, Beckers, Buchal, Fleuster, Holzbrecher, Paulus, Muller, Gog, Materlik, Witte, Stolz and von der Osten11 and Caccavale et al.Reference Caccavale, Segato, Mansour, Almeida and Leite12 As shown in Table I, the solubility in our case is 2.0 ± 0.2 mol%. It is definitely higher than the result reported by Sohler and coworkers,Reference Baumann, Brinkmann, Dinand, Sohler, Beckers, Buchal, Fleuster, Holzbrecher, Paulus, Muller, Gog, Materlik, Witte, Stolz and von der Osten11 1.3 ± 0.2 mol%, and the data reported by Caccavale et al.,Reference Caccavale, Segato, Mansour, Almeida and Leite12 ∼1.6 ± 0.2 mol%, under the same diffusion temperature of 1130 °C. Different diffusion atmospheres of wet O2 (present work), dry O2 (work of Caccavale et al.), and Ar (work of Sohler and coworkers) may be responsible for the difference in the results of solubility between ours, Caccavale et al., and Sohler and coworkers. Different diffusion atmospheres correspond to different extents of Li+ out-diffusion.
Subsequently, let us pay attention to the crystal composition effect on the solubility. It can be concluded from the data shown in Table I that the solubility increases with the decrease of the crystal composition. Table I shows the increase of the solubility of the sample nos. 2–4 relative to that of the sample no. 1. The increase is by 0.6, >0.6, >0.4, and >0.5 mol% and is by 30, >30, >20, and >25%, respectively. Note that the increase is larger than the usual surface concentration (Gaussian profile) of a practical Ti:Er:LN waveguide device fabricated on an initial congruent material, ∼0.5 mol%.Reference Baumann, Bosso, Brinkmann, Corsini, Dinand, Greiner, Schäfer, Söchtig, Sohler, Suche and Wessel21 It is worthwhile to mention that the maximum solubility increase in the case of the simultaneous work of Er3+ diffusion and Li-poor VTE treatment is similar to 80%.Reference Zhang, Chen and Pun13 Here, it is impossible to give a quantitative comparison of the present data with the result of that work as we do not know the exact solubility values of the sample no. 3–5 studied in this work, among which the sample no. 5 has the lowest crystal composition and should, thus, have the maximum solubility.
It is no doubt that the solubility increase is due to the reduction of the Li2O content in the crystal arising from both the initial Li-poor VTE process and the Li out-diffusion accompanied in the subsequent Er3+ diffusion procedure. As the Li2O content in the crystal lowers, more intrinsic defects, like the antisite (NbLi)4+ (Nb5+ on Li+ site) and the Li vacancy (VLi) required by charge equilibrium, are generated in the crystal; the crystal may accept more Er3+ ions, and hence the Er3+ solubility increases. We can see from Table I that the sample nos. 2 and 3 have the similar solubility values. This is obviously because both have the close Li2O contents.
It has been proved that majority of Er3+ ions in an LN crystal are located on or in the close vicinity of the regular Li+ sites [(ErLi)2+]Reference Baumann, Brinkmann, Dinand, Sohler, Beckers, Buchal, Fleuster, Holzbrecher, Paulus, Muller, Gog, Materlik, Witte, Stolz and von der Osten11, Reference Kovacs, Rebouta, Soares and da Silva22, Reference Kovacs, Rebouta, Soares, da Silva, Hage-Ali, Stoquert, Siffert, Sanz-Garcia, Corradi, Szaller and Polgar23 and a few percents occupy the Nb5+ sites [(ErNb)2−].Reference Gill, McCaughan and Wright24, Reference Dierolf and Koerdt25 Based upon this fact and the requirement by electric neutrality [every (ErLi)2+ requires two Li vacancies], we can roughly predict the Er3+ solubility increase from the known Li2O content reduction by assuming that all the incorporated Er3+ ions fill the Li+ sites and then compare with the experimental data. The predicted results are given in the last row of Table I. The predicted solubility increase relative to the sample no. 1 is similar to 0.20, 0.33, 0.40, and 0.67 mol% for the sample nos. 2–5, respectively. It appears that the predicted data are smaller than the experimental results for the sample nos. 2 and 3. This should be also true for the sample nos. 4 and 5 though we do not know their accurate solubility values. The difference between the predicted and experimental data gives a hint that more Er3+ ions occupy the Nb5+ sites. In other words, the Er3+ ions are incorporated on both Li+ and Nb5+ lattice sites. It was suggested that the Er3+ diffusion takes place through the easiest path via vacant cation sites and the structural octahedral vacancies.Reference Baumann, Brinkmann, Dinand, Sohler, Beckers, Buchal, Fleuster, Holzbrecher, Paulus, Muller, Gog, Materlik, Witte, Stolz and von der Osten11 For the case that the Er3+ ions are incorporated on the Li+ site, the defect reaction can be modeled as: LiLi + (NbLi)4+ + 4VLi + 2Er3+ → Li+ + Nb5+ + 4VLi’ + 2(ErLi)2+, which assumes that two Er3+ ions are incorporated on regular Li+ lattice sites with one substituting a regular Li+ site and another occupying the Li+ site near an antisite (NbLi)4+, which forces the Nb5+ back to the regular Nb5+ site and hence cancels the antisite. It is also possible that the two Er3+ ions directly fill two Li+ vacancies, which, due to the requirement by charge equilibrium, similarly leads to that an Li+ ion on the regular lattice site is evicted out and an antisite (NbLi)4+ is canceled. Whether the Er3+ ion is incorporated through the former or the later way, four new Li+ vacancies (VLi’) are required for charge neutrality. Meanwhile, one Li+ ion is liberated and part of them diffuse out of the crystal, leading to the reduction of the Li2O content. It is also possible that two of such Li+ ions occupy an Nb5+ lattice together with one Er3+ ion, forming a Li(Li2Er)O3 molecule, which retains still the LN phase.Reference Gill, McCaughan and Wright24
It is well known that the Curie temperature (T c) of the LN crystal (1142.3 ± 0.7 °C at the congruent point) is closely related to the Li2O content inside the crystal ([Li2O]) and both follow the linear relation T c = 39.064 [Li2O] − 746.73.Reference Jundt, Fejer, Norwood and Bordui17 By utilizing this expression, one can judge that all the five crystals studied in this work have a Curie temperature below the diffusion temperature adopted, 1130 °C. This implies that the ferroelectric domain in all these crystals orients no longer conformably but randomly. Moreover, as the crystal composition decreases, the electro-optic effect of the crystal degrades. These two problems would be solved by carrying out further Li compensation (via Li-rich VTE process) and then the uniform ferroelectric poling treatment on the whole crystal plate.
In addition, we should note that the high Er3+ doping concentration inevitably results in the increases of (i) the clustered Er3+ site concentration, (ii) the cooperative upconversion probability, which depends mainly on the clustering site concentration, and (iii) the concentration quenching effect on the 1.5-μm fluorescence intensity, which is associated with the multiphonon relaxation process and the upconversion effect. It is anticipated that there should have a trade-off between the Er3+ concentration and the concentration quenching effect, which is intended to be the future work.
IV. CONCLUSIONS
We have demonstrated the diffusivity and solubility increases of Er3+ in an off-congruent, Li-deficient LN crystal. Present study allows to conclude that:
(1) due to Li out-diffusion, the Er3+ diffusion procedure results in 0.3–0.5 mol% Li2O content loss at the crystal surface, depending on the initial Li2O content in the crystal;
(2) as the Li2O content at the crystal surface decreases from 48.0 to 47.0 mol%, the Er3+ diffusivity is nearly doubled that increases from (3.3 ± 0.2) × 10−2 to (5.7 ± 0.3) × 10−2 μm2/h;
(3) the Er3+ solubility increases also with the decrease of the crystal composition. As the Li2O content decreases only from 48.0 to 47.7 mol%, the solubility increases from 2.0 ± 0.2 to 2.6 ± 0.2 mol%. The increase is similar to 0.6 mol%, which is larger than the usual surface concentration (Gaussian profile) of a practical Ti:Er:LN waveguide device fabricated on an initial congruent material, ∼0.5 mol%. It is expected from the experimental data that as the crystal composition is further decreased, the solubility is increased further.
(4) The difference between the predicted and experimental solubility increase values gives a hint that the Er3+ ions occupy also the Nb5+ site, besides the Li+ site.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China under Project No. 50872089, by the Key Program for Research on Fundamental to Application and Leading Technology, Tianjin Science and Technology Commission of China under Project No. 11JCZDJC15500, and by Specialized Research Fund for the Doctoral Program of Higher Education of China under Project No. 20100032110052.
