Oxygen-18 tracer diffusion in nearly stoichiometric single crystalline lithium niobate

Solid State Ionics 189 (2011) 1–6

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Solid State Ionics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / s s i

Oxygen-18 tracer diffusion in nearly stoichiometric single crystalline lithium niobate P. Fielitz a,⁎, O. Schneider a, G. Borchardt a, A. Weidenfelder b, H. Fritze b, J. Shi c, K.D. Becker c, S. Ganschow d, R. Bertram d a

Technische Universität Clausthal, Institut für Metallurgie, Robert-Koch-Str. 42, D-38678 Clausthal-Zellerfeld, Germany Technische Universität Clausthal, Institut für Energieforschung und Physikalische Technologien, Am Stollen 19, D-38640 Goslar, Germany Technische Universität Braunschweig, Institut für Physikalische und Theoretische Chemie, Hans-Sommer-Str. 10, D-38106 Braunschweig, Germany d Leibniz-Institut für Kristallzüchtung, Max-Born-Str. 2, D-12489 Berlin, Germany b c

a r t i c l e

i n f o

Article history: Received 20 December 2010 Received in revised form 21 February 2011 Accepted 24 February 2011 Available online 9 April 2011 Keywords: Nearly stoichiometric lithium niobate Oxygen diffusion SIMS

a b s t r a c t Oxygen-18 tracer transport measurements were performed in nearly stoichiometric lithium niobate. To produce such crystals the vapour transport equilibration (VTE) technique was applied using a much larger mass of a lithium-rich two-phase mixture. To avoid Li2O loss the samples were also held in close proximity to the lithium-rich two-phase mixture during the diffusion annealing. This maintains the Li2O activity in the system so that a rigorous defect chemical analysis is applicable. The oxygen-18 tracer diffusivity data were rationalised on the basis of current point defect models for lithium niobate taken from literature. Considering calculated energy values for the different point defect models it can be concluded that the oxygen transport in both Li2O-rich and Li2O-poor lithium niobate occurs via oxygen interstitials rather than via oxygen vacancies. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Commercial LiNbO3 single crystals grown from a melt by the Czochralski technique usually have the congruent melting point composition which has a lithium deficiency of about 1.5% compared to the stoichiometric one. From the viewpoint of optical and other physical properties, the stoichiometric composition is more desirable because defects typical for the congruently melting composition are detrimental to the optical properties of lithium niobate. As stoichiometric LiNbO3 single crystals have much less point defects several methods where developed to produce near-stoichiometric LiNbO3 (see [1] p. 5 for an overview). To understand the physical properties of lithium niobate, the defect structure has been intensively investigated in the past (see chapter 2 in [1] for an overview). The application of lithium niobate at high temperatures requires, however, deeper understanding of the atomic transport of the constituents. While lithium is proved to be the most mobile species [2–8] data for niobium diffusion [9–11] are extremely contradictory: At an overlap temperature around 1300 K the single tracer diffusion data set [11] with an activation energy of 1.07 eV yields Nb diffusivities which are more than two orders of magnitude higher than the Nb–Ta interdiffusion coefficients [9,10] with activation energies of 1.73 eV and 3.05 eV, respectively. For oxygen only two (controversial) data sets (for lithium-deficient LiNbO3) are available [12,13]. While the 18O tracer diffusivity data in [12] seem to be doubtful as the solid–gas

⁎ Corresponding author. Tel.: +49 5323 72 2634. E-mail address: peter.fi[email protected] (P. Fielitz). 0167-2738/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2011.02.023

exchange method with mass spectrometric monitoring of the 18O2 concentration of the gas phase is much less reliable than a state-ofthe-art SIMS profiling method [13], the unambiguous assessment of the Nb diffusion data [9–11] is not possible from the experimental details given in the respective publications. Neglecting the fairly high diffusivities of the Nb tracer experiment because of the unusually low activation energy of 1.07 eV [11] together with the earlier 18O (effective) diffusivities [12] the following ranking of the diffusivities of the three host species in (unspecified) lithium niobate in the temperature range 1073 K ≤ T ≤ 1473 K is to be stated: D Li : D O : DNb ≈ 106 : 10 : 1. On the basis of the resulting tentative assumption that niobium is least mobile even at fairly high temperatures [14] it is obvious that any compositional change at high temperatures is governed by the transport of both lithium and oxygen. It is therefore necessary to investigate also the transport of oxygen in LiNbO3 as a function of composition. To get a practical answer to this question we recently performed oxygen-18 tracer transport measurements in Li-deficient lithium niobate [13]. In the current work oxygen-18 tracer diffusivity data for nearly stoichiometric LiNbO3 is presented.

2. Experimental Z-cut LiNbO3 wafers (composition: 49.0 mol% Li2O, diameter = 50.8 mm, thickness = 0.5 mm) were purchased from Del Mar Photonics. The chemical analysis by ICP-OES spectrometry (after a microwave assisted conversion with a HNO3/HF mixture into a soluble form) revealed the following impurities (in ppm): Mg (b36), Al(402),

2

P. Fielitz et al. / Solid State Ionics 189 (2011) 1–6

Ca(294), Cr(738), Mn (18), Fe(b 16), K(b 286), Na (b157), and Ti (156). The Li2O content in the lithium niobate samples was determined by the position of the absorption edge. Measurements of the absorption edge were carried out using a UV–vis–NIR optical spectrometer (Perkin Elmer Lambda 900). The wavelength at absorption coefficient of 20 cm−1 has been used to calculate the Li2O content according to the following equation recommended by Wöhlecke et al. [15] λ20 = 320:4−1:829 x−5:485 x

2

ð1Þ

where x is the deviation from the congruent composition in mol%. Thus the Li2O content in the sample is given by Li2 O mol % = 48:38 + x:

ð2Þ

2.1. Preparation of nearly stoichiometric single crystals by VTE processing Lithium niobate produced by the Czochralski technique has a composition near the congruent melting point composition of roughly 48.4 mol% Li2O [1], [16]. To get a specified off-congruent composition vapour transport equilibration (VTE) technique is usually applied [17]. This technique was first described by Holman [18] for preparing uniform lithium niobate crystals of any desired composition within the solid solution phase field. During a VTE process lithium niobate crystal samples are held in close proximity to a much larger mass of lithium niobate powder of a desired composition. After a sufficient time at sufficiently high temperature, the Li/Nb ratio in the crystal equilibrates to that in the powder via a mechanism involving vapour transport and solid-state diffusion. The supplied lithium niobate wafer was cut into two pieces of equal size. One piece was used as reference whereas the other one was used for the VTE process. A two-phase powder batch was prepared using Li2CO3 (Merck, Suprapur) and Nb2O5 (H. C. Starck GmbH, Germany, Ultra pure grade) as starting chemicals. The powder ratio was chosen to establish a net composition of 65 mol% Li2O for a lithium-rich two-phase mixture (Li3NbO4 and LiNbO3) which acts as a Li2O source during the VTE process. The powder mixture was slowly heated (50 K/h) to 1098 K and held at this temperature for 24 h. After cooling to room temperature the mass loss was determined to amount to 99.4% of the total CO2 contents of the starting material. A second heating cycle to 1123 K with a dwell time of 16 h and negligible mass loss confirmed that the Li2CO3 was decomposed entirely. A few grammes of the powder mixture were used to prepare small tablets for the diffusion experiments (see Fig. 1). The powder was pressed into a rubber mould and subsequently compacted in an isostatic press at 2 kbar for 1 h. The so prepared tablets were sintered in a muffle furnace in air for 8 h at 1273 K.

About 750 g of the lithium-rich two-phase mixture was loaded into a cylindrical platinum crucible of about 3 l volume which was loosely closed by a platinum lid. The lithium niobate wafer (about 2 g) was supported by platinum rings parallel to and slightly above the powder surface (about 5 mm). For the VTE processing the filled and loosely closed platinum crucible was heated in air with a heating rate of 100 K/ h from room temperature to 1373 K. It was kept at 1373 K for 96 h and was then cooled to room temperature with a cooling rate of 45 K/h. The Li2O content in the samples after the VTE processing is found to be 49.9 mol%. Taking into account the uncertainty (±0.1 mol%) of the applied method this is in agreement with the observation that the composition of lithium niobate at the Li2O-rich phase boundary is nearly identical to that of stoichiometric LiNbO3 [1,18]. 2.2. Diffusion annealing of the nearly stoichiometric single crystals To avoid Li2O loss of the VTE processed lithium niobate samples during annealing in 18O2 enriched oxygen gas, a small platinum box was built (see Fig. 1). Two sintered tablets of the lithium-rich twophase mixture used for the VTE processing where positioned in the platinum box as illustrated by the drawing. The lithium niobate sample was placed on a platinum holder between the two tablets and the platinum box was loosely closed by a platinum lid. The filled platinum box was moved into a furnace tube and the furnace tube was subsequently evacuated to about 10−3 mbar total pressure for 1 h before filling it with 185 mbar 18O2 gas (which results in a pressure of about 200 mbar at annealing temperature). As the platinum box was not tightly closed the 18O2 gas could freely diffuse inside the platinum box. The applied oxygen gas was enriched with 97% of the rare 18O isotope (natural abundance 0.02%). In the next step, the furnace was heated with a heating rate of 12 K/min and held for the annealing time, t, at annealing temperature, T, (see Table 1). Thereafter the furnace was cooled to room temperature with a cooling rate of 12 K/min. The Li2O content in the samples after the diffusion anneal was verified and it was found that it had not changed within the uncertainty (0.1 mol%) of the applied method. The resulting depth distribution of the 18O isotope (Fig. 2) was determined by SIMS using a Cameca IMS 3f instrument. Negative 14.5 keV oxygen ions were used as primary beam with 100 nA ion current and a spot size of about 50 μm. The raster-scanned area was 250 × 250 μm2 and the diameter of the analysed zone was 60 μm. Positive secondary ions (O+) were used for the analysis of the samples. Sample charging was prevented by coating the sample surface with a 50 nm thick carbon film. For depth calibration the SIMS crater depth was measured using a surface profiler (Tencor, Alpha Step 500). 2.3. Data evaluation To evaluate the diffusion length, L, from the depth profiles the solution of the diffusion equation for a constant diffusion source (c0 = constant) located at x = 0 was used [19] cðxÞ−c∞ = ½c0 −c∞ erfc


x L

with

pffiffiffiffiffiffi L = 2 Dt

ð3Þ

Table 1 Compilation of the diffusion experiment parameters, where T is the annealing temperature, t the annealing time at T, L the diffusion length measured by SIMS depth profiling, and α the heating and cooling rate of the furnace. The effective additional annealing time, Δt, and the 18O tracer diffusion coefficient, D, were calculated by Eq. (5). Fig. 1. Schematic drawing of the platinum container (l = 55 mm, h = 11 mm) which was used during the diffusion annealing of the lithium niobate samples in 18O2 gas. Two sintered tablets (diameter = 18 mm, height = 10 mm) of the lithium-rich two-phase mixture used for the VTE processing where positioned in the platinum box as illustrated by the drawing. The lithium niobate sample (about 5 × 5 × 0.5 mm3) was placed onto a platinum holder between the two tablets, and the platinum box was then loosely closed by a platinum lid.

T

T

t

αup = αdown

L

Δt

D

°C

K

h

K/min

μm

min

m2/s

775 850 925 1000

1048 1123 1198 1273

85 10 2 2

12 12 12 12

0.29 0.44 0.71 2.34

3.9 4.5 5.1 5.8

6.87E-20 1.33E-18 1.68E-17 1.81E-16

P. Fielitz et al. / Solid State Ionics 189 (2011) 1–6

3

Fig. 2. Typical SIMS depth profiles (grey curves) of a VTE processed lithium niobate sample after 85 h annealing at 775 °C in 200 mbar 18O2 gas. The black solid curve is a fit curve of Eq. (3) which results in an oxygen-18 diffusion length L = 0.29 μm.

where erfc is referred to as the error-function complement. The black solid curve in Fig. 2 shows a least-squares fit of Eq. (3) which results in a diffusion length L = 1.24 μm for the 18O tracer. The fitting of Eq. (3) to the measured 18O+ SIMS depth profile works very well over almost two orders of magnitude. To calculate the diffusion coefficient, D, from the measured diffusion length, L, one has to solve the time integral [20]

3. Results and discussion

t

  L = ∫ D t ′ dt ′ 4 0 2

ð4Þ

which gives simply D × t in the case of infinite heating and cooling rates, where t is the annealing time at constant annealing temperature. For finite heating and cooling rates one can express the solution by D × (t + Δt) where Δt is an effective additional annealing time taking into account diffusion during heating and cooling of the furnace. In the case of constant heating and cooling rates there is a simple approximate solution to calculate Δt so that one has [20]

D=

L2 4ðt + Δt Þ

Fig. 3. Arrhenius diagram of 18O tracer diffusivities in single crystalline lithium niobate. Points with error bars represent measurements in nearly stoichiometric single crystalline lithium niobate (49.9 mol% Li2O, VTE processed, this work). The grey line shows measured 18O tracer diffusivities in single crystalline Li2O-deficient lithium niobate (46.4 mol% Li2O) [13].

with

Δt ≅

1 1 + αup αdown

!

R T2 ΔH

ð5Þ

where α up is the heating rate, α down the cooling rate, T the constant annealing temperature, t is the annealing time at T, R the universal gas constant and ΔH the activation enthalpy of diffusion. To estimate Δt by Eq. (5) one has first to estimate the activation energy of diffusion. This was done by calculating in a first step all the diffusion coefficients with Δt = 0 which resulted in an activation enthalpy ΔH1 = 390 kJ/mol so that the effective additional annealing time, as compiled in Table 1, could be estimated. Comparing Δt and t one finds that the additional annealing time is virtually negligible. Nevertheless, in a second step all the diffusion coefficients were recalculated taking into account the estimated Δt values. This resulted in an activation enthalpy ΔH2 = 387 kJ/mol, so that one finally gets the following Arrhenius relation  
 2 ð387F 3ÞkJ = mol + 0:49 m exp − D = 1:4 −0:36 RT s

ð6Þ

for the 18O tracer diffusivity in nearly stoichiometric single crystalline lithium niobate. The measured activation enthalpy in common units is consequently (387 ± 3) kJ/mol or (4.01 ± 0.03) eV/at.

In Fig. 3 the measured oxygen-18 tracer diffusion coefficients in nearly stoichiometric single crystalline lithium niobate (49.9 mol% Li2O, VTE processed, this work) are compared with measurements in single crystalline Li2O-deficient lithium niobate (46.4 mol% Li2O) [13]. As we can see the oxygen-18 tracer diffusivity does not significantly depend on the Li2O composition of the crystals. In the following subsections the data for the oxygen-18 tracer diffusivity in lithium niobate with slight Li2O excess and in Li2O-deficient lithium niobate were rationalised on the basis of current point defect models taken from literature [21], [26]. 3.1. LiNbO3 with a slight Li2O excess The accuracy of the applied Li2O concentration measurement method is about ±0.1 mol% Li2O. That is, from this measurement we cannot decide whether our diffusion experiments in nearly stoichiometric single crystalline lithium niobate were performed in crystals with slight Li2O deficiency or in crystals with slight Li2O excess. However, the LiNbO3 crystals were VTE processed with a lithium-rich two-phase mixture and also the diffusion annealing was performed in a platinum box which contained a lithium-rich two-phase mixture (see Fig. 1). It is, therefore, conceivable to assume that there is a slight Li2O excess in the crystals. The lithium-rich two-phase mixture maintains the Li2O activity corresponding to the equilibrium Li2 OðsÞ + LiNbO3 ðsÞ⇔Li3 NbO4 ðsÞ

ð7Þ

with the equilibrium constant KVTEð + Þ =

1 aLi2 O

= exp −

ΔG∘VTEð+Þ kB T

! ð8Þ

where aLi2O is the activity of Li2O in the system (the platinum box (see Fig. 1)), ΔG∘VTE(+) is the standard Gibbs free energy of reaction (7), kB is the Boltzman constant, and T the temperature. For an all-solid-state reaction like Eq. (7) with a standard Gibbs energy, ΔG∘VTE(+), the standard

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P. Fielitz et al. / Solid State Ionics 189 (2011) 1–6

reaction entropy is generally very small. Therefore ΔG∘VTE(+) ≃ΔH∘VTE(+) is a reasonable approximation. Typical Gibbs energy values for such reactions are about −(0.2… 0.4)eV [22–25]. Donnerberg et al. [26] proposed three Li2O solution reactions. These authors took no account of temperature since there were no temperature dependent terms in their calculations. The smallest calculated energy per Li2O, ΔEcalc, is realised by the following solution reaction •

hhLiNbO3 ii + Li2 O⇔2LiI + OI″



 ••

1 1 1=3 ∘ ∘ ∘ VO = 4 exp − ΔGFrenkel − ΔGLi2 O sol + ΔGVTEðþÞ kB T 3 3

ð9Þ

ð17Þ

where the excess of Li2O is incorporated as interstitial ions and 〈〈LiNbO3〉〉 represents the whole LiNbO3 crystal. The equilibrium constant of the Li2O solution reaction (9) is

and hence for the formation enthalpy, ΔHVf O, of Frenkel oxygen vacancies

KLi2 O sol

ΔEcalc = 4:71 eV

concentration of oxygen interstitials, and ΔG∘Frenkel is the standard Gibbs free energy of reaction (15). We see that at equilibrium the concentration of oxygen vacancies is inversely proportional to the concentration of oxygen interstitials which are created by the Li2O solution reaction (9). Combining Eqs. (12) and (16) we get for the concentration of oxygen vacancies

with

! • 2 ΔG∘Li2 O sol LiI ⋅½OI″  = = exp − aLi2 O kB T

ð10Þ

where aLi2O is the activity of Li2O, [Li•I] is the concentration of lithium ∘ interstitials, [O″I ] the concentration of oxygen interstitials, and ΔGLi 2O sol is the standard Gibbs free energy of reaction (9). The electroneutrality condition is in this case •



LiI = 2 O″I

ð11Þ

Considering Eqs (8), (10) and (11) we get for the concentration of oxygen interstitials in a nearly stoichiometric crystal with a slight Li2O excess ″ 3 ΔG∘Li2 O sol −ΔG∘VTEðþÞ 1 exp − OI = 4 kB T

!

O

ð12Þ



 d ln O″I 1
∘ ∘ = ΔHLi2 O sol −ΔHVTEðþÞ dð1 = T Þ 3

ð13Þ

Applying the calculated energy per Li2O of the solution reacI tion (9) allows us to estimate the formation enthalpy, ΔHO f , of oxygen interstitials O

ΔHf I =

 1
∘ 4:71 eV−ΔHVTEðþÞ ≈ 1:7 eV 3

ð14Þ

For a first order approximation the standard enthalpy ΔH∘VTE(+) of reaction (7) is estimated to be of the order of some tenth of an eV, e.g. − 0.4 eV (see explanations given for Eq. (7)). A formation enthalpy value of about 1.7 eV is consistent with the measured activation enthalpy, ΔH = ΔHf + ΔHm = 4.0 eV, of the oxygen transport so that it is plausible to assume that the oxygen transport occurs via oxygen interstitials with a migration enthalpy of about 2.3 eV. However, if the oxygen interstitials are relatively immobile the oxygen transport could also be supported by oxygen vacancies which are created by the (oxygen) Frenkel equilibrium according to reaction ••

hhLiNbO3 ii⇔VO + O″I

with

ΔEcalc = 6:84 eV

ð15Þ

where VO•• are oxygen vacancies, O″I are oxygen interstitial ions, and ΔEcalc is the calculated energy [26] for the (oxygen) Frenkel reaction (15). The equilibrium constant is 

••

ΔG∘Frenkel KFrenkel = VO ⋅ O″I = exp − kB T

••

d ln VO 1 1 ∘ ∘ ∘ = ΔHFrenkel − ΔHLi2 O sol + ΔHVTEðþÞ 3 3 dð1 = T Þ

ð18Þ

Applying the calculated energies of reactions (9) and (15) we can estimate V

ΔHf O =

  4:71 1 ∘ eV + ΔHVTEðþÞ ≈ 5:14eV 6:84− 3 3

ð19Þ

with ΔH∘VTE(+) ≈ − 0.4 eV like in Eq. (14). Now the estimated formation enthalpy, ΔHVf O, of Frenkel oxygen vacancies becomes even larger than the measured activation enthalpy, ΔH = ΔHf + ΔHm = 4.0 eV, of the oxygen transport. It is therefore plausible to assume that the oxygen transport is not supported by (Frenkel) oxygen vacancies. 3.2. LiNbO3 with a slight Li2O deficiency

I We can now calculate from Eq. (12) the formation enthalpy, ΔHO f , of oxygen interstitials

ΔHf I = −kB

V

ΔHf O = −kB

 ð16Þ

where [VO••] is the concentration of oxygen vacancies, [O″I ] is the

As discussed above it is conceivable to assume that our nearly stoichiometric LiNbO3 crystals have a slight Li2O excess during the diffusion annealing. Nevertheless, for completeness, we will also discuss the alternative, though hypothetical, case that the nearly stoichiometric LiNbO3 crystals are slightly Li2O-deficient (less than −0.1 mol% Li2O). This would mean that the crystal is not yet in equilibrium with the lithium-rich two-phase mixture in the platinum box (see Fig. 1) so that a rigorous defect chemical analysis as presented above is not possible. Approximately, however, we could assume the same diffusion mechanism in both cases (slightly and strongly Li2O deficient) which would enable us to compare the absolute values of the presented oxygen diffusion coefficients in Fig. 3. In the early literature charge compensation for lithium-deficient niobate was alternatively thought to be achieved by the following defect reaction ••

′ + VO hhLiNbO3 ii⇔Li2 O + 2V Li

with

ΔEcalc = 5:82 eV

ð20Þ

where V ′Li are Li vacancies, VO•• are oxygen vacancies and 〈〈LiNbO3〉〉 represents the whole crystal where the deficit of Li is achieved by Li2O outdiffusion. ΔEcalc is the calculated energy per Li2O [26]. If we now assume that oxygen vacancies would be mobile enough to support the oxygen transport we would expect (if the nearly stoichiometric crystal is slightly Li2O-deficient) a higher oxygen diffusivity in the strongly Li2O-deficient lithium niobate sample according to the straightforward estimation Dstrong Dslight

! expected

••

VO strong ½Li2 Oout ≈ 3 mol % strong ≈ 30 ð21Þ = ••

= N ≈ 0:1 mol % VO slight ½Li2 Oout slight

where the subscript “slight” indicates the (hypothetical) nearly stoichiometric case with slight Li2O deficiency ([Li2O]out slight ≈ 0.1mol %) and the subscript “strong” indicates the non-stoichiometric case with strong Li2O deficiency ([Li2O]out strong ≈ 3 mol %). The experimental data (see Fig. 3), however, do not agree with this (hypothetical) ratio, which indicates that reaction (20) cannot be the dominant defect

P. Fielitz et al. / Solid State Ionics 189 (2011) 1–6

reaction for the necessary charge compensation in Li2O-deficient niobate. The conclusion that oxygen vacancies are not majority point defects in lithium niobate is in accordance with structure investigations by X-ray and neutron diffraction (see [1] p. 12 for an overview) and with theoretical calculations [26].

5

and the corresponding electroneutrality condition for Li2O deficient single crystals h i

4• ′ 4 NbLi = V Li

ð26Þ

we get for the concentration of lithium vacancies in equilibrium

3.3. LiNbO3 with a strong Li2O deficiency

∘

The LiNbO3 crystals of this work were VTE processed with a lithium-rich two-phase mixture (reaction (7)) so that we measured the oxygen diffusivity in nearly stoichiometric LiNbO3 crystals. We will now discuss the case of oxygen self diffusion in crystals with a strong Li2O deficiency. The best way to realise a defined strong Li2O deficiency during the diffusion experiment is the same experimental approach as describe above, however, this time applying a lithiumpoor two-phase mixture during the VTE process and the subsequent diffusion annealing. Such experimental work has not been done so far. However, in our earlier work [13] we investigated 18O tracer diffusion in Li2O-deficient single crystals whose composition (46.4 mol% Li2O) guaranteed a high temperature defect population in very close proximity to the one valid for the true LiNb3O8/LiNbO3 equilibrium. As the crystals were consequently practically saturated with respect to the Nb2O5 excess, i.e. corresponding to the Li2O deficit, there was only a negligible gradient of the chemical potential of Li2O, at least in the near surface region of the crystals where the oxygen diffusion and SIMS profiling occurred. This picture was supported by a slightly milky aspect of the crystals in some cases, depending on the quenching rate, which indicates LiNb3O8 precipitates to be formed [13]. It is therefore justified to discuss our earlier data for Li2O-deficient single crystals as if the material had really undergone a preliminary VTE process on the Li2O-poor side of the lithium niobate composition. This assumption will allow to estimate formation enthalpies for oxygen vacancies and for oxygen interstitials from calculated energy values [26] and will yield clues to decide whether the oxygen transport occurs via oxygen vacancies or via oxygen interstitials in Li2O deficient lithium niobate. In the following we will therefore present the formal theoretical treatment for the LiNb3O8/LiNbO3 equilibrium case although our experimental data [13] do not completely correspond to this case. The lithium-poor two-phase mixture will maintain the Li2O activity corresponding to the equilibrium Li2 OðsÞ + LiNb3 O8 ðsÞ⇔3LiNbO3 ðsÞ

ð22Þ

with the equilibrium constant KVTEð−Þ =

1 aLi2 O

∘

= exp −

ΔGVTEð−Þ kB T

! ð23Þ

where aLi2O is the activity of Li2O in the system, and ΔG∘VTE(−) is the standard Gibbs free energy of reaction (22). Currently the Li site vacancy model is preferred for Li2O deficient single crystals (see [1] p. 9 for an overview) in which an Nb antisite is compensated by four VLi′ so that we have the defect reaction ′ + NbLi hhLiNbO3 ii⇔3Li2 O + 4V Li

4•

with

ΔEcalc = 4:56 eV ð24Þ

where the Li2O deficiency is discussed in terms of Li2O-outdiffusion and ΔEcalc is the calculated energy per Li2O [26]. Considering the equilibrium constant of reaction (24)

KLi2 O out =

h

4 3 ′ aLi2 O ⋅ V Li ⋅

4• NbLi

i

= exp −

ΔG∘Li2 O out kB T

!



1 ΔGLi2 O out −3 = 5 1=5 ′ = aLi 4 exp − V Li 2O 5 kB T

ð27Þ

where ΔGLi∘2O out is the standard Gibbs free energy of reaction (24) and aLi2O is the activity of Li2O. Defect reaction (20) can formally also be considered as a Schottky reaction [26] so that we have for the Schottky reaction the following equilibrium constant   2 ••

ΔG∘Schottky ′ ⋅ VO = exp − KSchottky = aLi2 O ⋅ V Li kB T

ð28Þ

where ΔG∘Schottky is the standard Gibbs free energy of reaction (20). Combining Eqs. (23), (27) and (28) we get for the concentration of Schottky oxygen vacancies in equilibrium ∘

∘

∘

••

ΔGSchottky − 25 ΔGLi2 O out − 15 ΔGVTEð−Þ −2 = 5 VO = 4 exp − kB T

! ð29Þ

and hence for the formation enthalpy, ΔHVf O, of Schottky oxygen vacancies V ΔHf O

••

−kB d ln VO 2 1 ∘ ∘ ∘ = ΔHSchottky − ΔHLi2 O out − ΔHVTEð−Þ = 5 5 dð1 = T Þ

ð30Þ

Applying the calculated energies of reactions (20) and (24) we can estimate V

ΔHf O =

  2 1 ∘ 5:82− 4:56 eV− ΔHVTEð−Þ ≈ 4:1eV 5 5

ð31Þ

∘ ∘ ≈ − 0.4 eV (here ΔHVTE(−) was estimated in Taking again ΔHVTE(−) ∘ the same way as ΔHVTE(+) in Eq. (14)) it can be seen that the estimated value for the formation enthalpy of Schottky oxygen vacancies is too high and hence inconsistent with the measured activation enthalpy, ΔH = ΔHf + ΔHm = 3.5 eV, of oxygen diffusion in strongly Li2Odeficient lithium niobate [13]. The concentration of Schottky oxygen vacancies is (according to the equilibrium constant (28)) suppressed by the concentration of lithium vacancies which are created by defect reaction (24) in Li2O deficient lithium niobate. That is, if the concentration of oxygen vacancies is generally suppressed in Li2O deficient lithium niobate, we can expect in equilibrium an increase of the concentration of oxygen interstitial ions created by the Frenkel reaction (15) (see equilibrium constant (16). From this point of view it is plausible to assume that the oxygen transport should be supported rather by oxygen interstitials than by oxygen vacancies. Combining Eqs. (16) and (29) we get for the concentration of Frenkel oxygen interstitials

″

OI = 4

+ 2=5

"  # , 2 1 ∘ ∘ ∘ ∘ kB T exp − ΔGFrenkel −ΔGSchottky + ΔGLi out + ΔGVTEð−Þ 5 5

ð32Þ I and hence for the formation enthalpy, ΔHO f , of Frenkel oxygen interstitials

O

ð25Þ

!

ΔHf I = −kB

d ln½OI″ 2 1 ∘ ∘ ∘ ∘ = ΔHFrenkel −ΔHSchottky + ΔHLi out + ΔHVTEð−Þ dð1 = T Þ 5 5

ð33Þ

6

P. Fielitz et al. / Solid State Ionics 189 (2011) 1–6

Applying the calculated energies of reactions (15), (20) and (24) we now can estimate O

ΔHf I =

  2 1 ∘ 6:84−5:82 + 4:56 eV + ΔHVTEð−Þ ≈ 2:67 eV 5 5

ð34Þ

where ΔH∘VTE(−) ≈ − 0.4 eV is taken as above. This time the estimated formation enthalpy is not too high and hence consistent with the measured activation enthalpy, ΔH = ΔHf + ΔHm = 3.5 eV, of oxygen diffusion in strongly Li2O-deficient lithium niobate [13]. The estimated formation enthalpy values (31) and (34) support the assumption that the oxygen transport occurs in the Li2O deficient case also rather via oxygen interstitials than via oxygen vacancies. At this point, the following facts should be remembered: i. The interpretation of the experimental results, which was given above, relies on our experience that very often the theoretical energy values [26] give at best a correct ranking of the respective defect formation energies — albeit that the real values are most probably significantly different. ii. In the “all-interstitial” picture of oxygen transport oxygen vacancy mediated migration becomes generally less probable: Our earlier assumption that aliovalent dopants like Al introduced •• defects (see Al ″Nb and VO ) at constant concentrations in the 100 ppm range and that the experimentally determined activation enthalpy of 3.5 eV corresponded to the migration enthalpy only [13] has therefore to be discarded as a dominant diffusion mechanism even in Li2O-deficient LiNbO3. iii. The data obtained by Jorgenson and Bartlett [12] do not fit either of the proposed mechanisms. iv. For an unequivocal proof of the conclusions drawn above it would, however, be necessary to measure the oxygen diffusivity in correctly VTE processed Li2O-deficient crystals. 4. Summary Oxygen-18 tracer diffusion experiments were performed in nearly stoichiometric LiNbO3 crystals. A VTE process with a lithium-rich twophase mixture was applied to produce nearly stoichiometric crystals. To avoid Li2O loss during the diffusion annealing a special platinum box was built (Fig. 1) which contained two sintered tablets of the lithium-rich two-phase mixture used for the VTE processing. In this way the activity of Li2O was also well defined during the diffusion annealing. In Fig. 3 the measured oxygen-18 tracer diffusion coefficients in nearly stoichiometric single crystalline lithium niobate (49.9 mol% Li2O, VTE processed, this work) are compared with measurements in single crystalline Li2O-deficient lithium niobate (46.4 mol% Li2O) [13]. There is no significant dependency of the oxygen diffusivity on the Li2O composition of the crystals. Older oxygen diffusivity data of Jorgensen and Bartlett [12], measured by a different method, are no more seriously considered because they do not fit any defect model. The data for the oxygen-18 tracer diffusivity in lithium niobate with slight Li2O excess and in Li2O-deficient lithium niobate were

rationalised on the basis of current point defect models taken from literature [21], [26]. Considering calculated energy values [26] for the different point defect models it can be concluded that the oxygen transport in lithium niobate should occur via oxygen interstitials rather than via oxygen vacancies. In the case of a potential Li2O excess it is reasonable to assume that Li2O is dissolved interstitially in lithium niobate so that the assumption that the oxygen transport is supported by oxygen interstitials is straightforward. In the case of Li2O deficiency it is well confirmed in the literature that Li2O loss from the crystal results in the creation of lithium vacancies by defect reaction (24) so that the concentration of (Schottky) oxygen vacancies is generally very low at equilibrium. A decrease of the oxygen vacancy concentration results, however, at equilibrium in an increase of the (Frenkel) oxygen interstitial concentration so that from this point of view an oxygen transport via oxygen interstitials seems to be more probable than via oxygen vacancies also for Li2O deficient lithium niobate.

Acknowledgements We are indebted to Dr. R.A. Jackson for his valuable comments on defect energy calculation issues. The remarks of an anonymous reviewer helped to improve the clarity of the presentation. Financial support from Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged.

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