Segregation and Twinning in the Rare-Earth ... - ACS Publications

Segregation and Twinning in the Rare-Earth Doped KPb2Cl5 Laser Crystals Matias Vela´zquez,*,† Jean-Francis Marucco,‡ Patrick Mounaix,§ Olivier Pe´rez,| Alban Ferrier,⊥ and Richard Moncorge´⊥

CRYSTAL GROWTH & DESIGN 2009 VOL. 9, NO. 4 1949–1955

Institut de Chimie de la Matie`re Condenseé de Bordeaux (ICMCB), UPR 9048 CNRS, 87 aVenue du Dr. A. Schweitzer, 33608 Pessac cedex, France, Laboratoire d’Etude des Mate´riaux Hors E´quilibre (LEMHE) de l’Institut de Chimie Molećulaire et des Mate´riaux d’Orsay (ICMMO), UMR 8182 CNRS/UniVersite´ Paris Sud Centre d’Orsay, Baˆtiment 410, 15 rue Georges Cle´menceau, 91405 Orsay cedex, France, Centre de Physique Molećulaire Optique et Hertzienne (CPMOH), UMR 5798 CNRS/UniVersite´ de Bordeaux 1, 351 Cours de la Libe´ration, 33405 Talence cedex, France, Laboratoire de Cristallographie et Sciences des Mate´riaux (CRISMAT), UMR 6508 CNRS/ UniVersite´ de Caen, 6 BouleVard du Marećhal Juin, 14050 Caen cedex 04, France, and Centre de Recherche sur les Ions, les Mate´riaux et la Photonique (CIMAP), UMR 6252 CNRS/CEA/ENSICaen/ UniVersite´ de Caen Basse-Normandie, 6 BouleVard du Marećhal Juin, 14050 Caen cedex 04, France ReceiVed NoVember 12, 2008; ReVised Manuscript ReceiVed January 15, 2009

ABSTRACT: We present the Er3+:KPb2Cl5 lattice parameters thermal expansion and twinning characterization, pure KPb2Cl5 crystals room temperature static dielectric constant measurements, and equilibrium segregation coefficients determination of Er3+, Ho3+, and Pr3+ defects in KPb2Cl5. The two main current limitations which still hinder the development of this crystal as a laser material, namely, the Er3+ cations segregation and low solubility limit as well as the crystal twinning, are discussed. Introduction The landmark papers of Bowman and his co-workers, who established first the potential for middle infrared (MIR) laser operation of such host compounds as Er3+:KPb2Cl5 (hereafter called “Er:KPC”) and LaCl3:Pr3+,1-3 have given a strong impetus to the development of large single crystalline MIR laser rods.4-12 With its low highest phonon energy, non-hygroscopicity and ability to dissolve around 1020 Er3+ ions per cm3, Er: KPC stands out as the most promising material for amplification energy storage, with a view to pulsed and quasi-continuous laser operation at 4.6 µm (4I9/2 f 4I11/2 transition). However, 12 years after the pioneering work of Bowman and his co-workers, we have to admit that well-established solid-state laser operations in the >4 µm infrared region are few and far between1-3,13-15 and that the LaCl3:Pr3+ 7.2 µm laser operation has not been reproduced anywhere. A great deal of work in growing and characterizing Er:KPC from the chemical,7 crystallographic, thermomechanical,8,9 and spectroscopical16 viewpoints has already been performed, and we report here our most recent results concerning the crystal growth of this material by the Bridgman-Stockbarger method. We analyze in more detail two of the reasons why, in spite of the successful growth of centimeter-sized and seemingly high-quality single crystals, the development of 4.6 µm solid state laser operation has been hindered until now. The discussion dealing with the difficulties in sufficiently doping this material and avoiding twinning pattern generation lies upon segregation and extensive X-ray diffraction * Corresponding author. E-mail: [email protected]; phone: +33 05 40 00 27 56; fax: +33 05 40 00 27 61. † Institut de Chimie de la Matie`re Condenseé de Bordeaux (ICMCB), UPR 9048 CNRS. ‡ Laboratoire d’Etude des Mate´riaux Hors E´quilibre (LEMHE) de l’Institut de Chimie Molećulaire et des Mate´riaux d’Orsay (ICMMO), UMR 8182 CNRS/ Universite´ Paris Sud Centre d’Orsay. § Centre de Physique Molećulaire Optique et Hertzienne (CPMOH), UMR 5798 CNRS/Universite´ de Bordeaux 1. | Laboratoire de Cristallographie et Sciences des Mate´riaux (CRISMAT), UMR 6508 CNRS/Universite´ de Caen. ⊥ Centre de Recherche sur les Ions, les Mate´riaux et la Photonique (CIMAP), UMR 6252 CNRS/CEA/ENSICaen/Universite´ de Caen Basse-Normandie.

characterization. We have also measured the static dielectric function of pure KPC crystals, the knowledge of which allows us to give a first insight into the rare-earth and cation vacancy defects association kinetics. Experimental Section Sample Preparation, Crystal Growth, and Rare-Earth Concentration Profiles. The Bridgman-Stockbarger crystal growth process, which has been given in detail in ref 17, yielded crack- and bubblefree single crystalline rods similar to the ones displayed in Figure 1. The nucleation temperature and length could be obtained over several runs by sticking a thermocouple to the capillary and measuring the height over which the material rapidly solidifies into a polycrystalline bulk in the very first stage of the growth. It indicates that the nucleation temperature is barely the same, ∼380 °C, as that obtained in DSC experiments7 in spite of the setup configuration differences and that, at least in the first stages of the growth, the liquid-solid interface is located at the bottom of the adiabatic zone. The experiments dedicated to the determination of the equilibrium segregation coefficients were carried out in the same way, with ∼5 g charges of ∼2.3 mol % rareearth (ErCl3, HoCl3, PrCl3) doped materials and with a 2.3 mol % ErCl3doped nonstoichiometric mixture. The silica ampoules of 4 mm inner diameter and 1 mm thickness, were translated at rates of ≈0.8 and 0.5 mm h-1, for each composition. The RE3+ concentration profile was established by inductively coupled plasma atomic emission spectroscopy performed on crystal slabs cut out exclusively from crack-free and constant-diameter bulks. The results are summarized in Table 1 and Figure 2. Room and High-Temperature Single Crystal X-ray Diffraction. The twinning occurring in Er:KPC crystals and the temperature dependence of their lattice parameters were characterized by the same procedure as that given in ref 8. The Er3+ concentration in the crystal studied in the Twinning Study section is ≈5.9 × 1019 ions cm-3 (1.29 mol %). X-ray diffraction measurements were performed using Mo KR radiations on a KappaCCD (Bruker-Nonius) four circles diffractometer equipped with a bidimensional detector (CCD: charge coupled device). A Cyberstar gas blower system was used for high temperature investigations. The data collection consists of large scans of the reciprocal space, the procedure of which has been repeated at different temperatures (from 20 to 400 °C). The experimental frames allow for both checking the crystalline quality of the sample and an accurate determination of the unit cell parameters. Above the transition temperature, Tt ) 255 °C,

10.1021/cg801252t CCC: $40.75  2009 American Chemical Society Published on Web 02/25/2009

1950 Crystal Growth & Design, Vol. 9, No. 4, 2009

Vela´zquez et al.

Figure 1. (a) Er:KPC crystal containing a single crystalline domain 8 cm long over the whole cross-section; (b) polished Er:KPC parallelepiped extracted from the bulk displayed in (a).

Figure 2. RE3+ concentration profiles found in RE:KPC crystals. The straight dash line is the best nonlinear least-squares fit to the chemical lever function. Table 1. Equilibrium Segregation Coefficients, Ionic Radii (with 0 C.N. ) 8) and gRECl Energy of Selected Rare-Earth Ions in KPC 3 Crystalsa Er3+ k0 ri (Å) 0 gRECl (kJ mol-1) 3 a

0.49 0.43a 1.004 -4567.0

Ho3+

Pr3+

0.79

1.12

1.015 -4545.4

1.126 -4357.1

Value obtained for the initially substoichiometric mixture.

the diffraction pattern can be described within an orthorhombic cell. The conditions limiting the possible reflections are in agreement with the Pmcn space group (h0l: l ) 2n, hk0: h + k ) 2n). Below the transition temperature, a monoclinic cell must be considered, with a weak distortion from orthorhombic lattice (cell parameters at 20 °C: a ) 8.85 Å, b ) 7.93 Å, c ) 12.49 Å and β ≈ 90.1°). Owing to this weak deviation from the orthorhombic lattice, most of the reflections of the twin domains are perfectly overlapped. Consequently, the diffraction patterns seem to be unaffected by the twins. However, an accurate analysis of the reflections at high θ-values would show split reflections. The coupling between KR1 and KR2 radiations and the twin gives rise to a quadruplet of reflections, typical of a nonmerohedral twin. Data collection was optimized to reach the best completeness for a monoclinic space, up to high enough θ-value (>35°) in spite of the presence of both the conditioning apparatus and the high-temperature gas flow that generate shadow zones. Plots of reciprocal lattice planes were assembled from the different scans. They provide an overall view, useful for twin law analysis, of the reciprocal space. The EvalCCD software18 was used to extract reflections from the collected frames,

merge, and rescale them as a function of the exposure time. Data were corrected from absorption using the Sadabs program19 developed for scaling and correction of area detector data. The structure was studied using the program Jana2006:20 twin laws have been introduced for the monoclinic phase, and the ratio of the possible twin domains was refined. Static Dielectric Constant Measurements. A THz transmission spectroscopy utilizing ultrafast photocommuters and a Ti:sapphire laser delivering 650 fs pulses allowed us to measure the static dielectric “constant”, ε ) εrε0, of the pure KPb2Cl5 crystal. This Fourier-transform based spectroscopical method consists in determining the sample parameters through its complex transfer function T(ω). The extraction of the complex dielectric function could be achieved on samples of various thicknesses 2.32 mm, 250 and 70 µm, giving a signal-to-noise ratio of 106 over a wavelength range going from 30 to 150 µm. The real part of the dielectric function was found to be equal to εr ) (17.5 ( 1.0). We shall need it for the last part of the Results and Discussion section.

Results and Discussion Synthesis, Crystal Growth, and Rare-Earth Segregation. The use of rare-earth metals or rare-earth oxides to dope pure KPb2Cl5 can hardly be successful, even under chlorine gas atmosphere. In the former situation, the Ellingham diagram indicates that reduction of the lead cations to metallic lead should readily occur (as in the case of contact with aluminum7). In the latter case, the synthesis eventually produces water vapor

Rare-Earth Doped KPb2Cl5 Laser Crystals

Crystal Growth & Design, Vol. 9, No. 4, 2009 1951

that must be evacuated from the reactor (for instance, a mass action law analysis of a hypothetical reaction between KPb2Cl5 and some RE2O3 oxide under HCl gas straightfor• point defect solubility limit is wardly shows that the REPb -3/4 3/2 proportional to pH2O · pHCl). Thus, it is highly desirable that the chloride source originates from the solid state, in the most common form of a rare-earth anhydrous trichloride. However, trying to dope already synthesized pure KPb2Cl5 with RECl3 alone also leads to several pitfalls. Indeed, the easiest formulation of this equilibrium (in terms of Kro¨ger-Vink symbols): • × RECl3 S REPb + V'K + 3ClCl

(1)

makes it clear that, if the site conservation rule is to be checked in the KPb2Cl5 lattice, we must add a PbCl2 molar equivalent of the RECl3 amount of powder to the KPb2Cl5/RECl3 mixture, according to • × × + V'K + PbPb + 5ClCl RECl3 + PbCl2 S REPb

(2)

If we seek to balance the number of crystallographic sites in eq 1 without adding a PbCl2 molar equivalent as in eq 2, in the following way:

1 v • × 2RECl3 S 2REPb + V'K + 5ClCl + Cl2 2

(3)

then it is the electroneutrality condition that is no longer verified (and, incidentally, a chlorine gas release that must be evacuated from the reactor if the RE3+-content in KPb2Cl5 is to be increased). Equations 1-3 show that the major limitation on the RE3+ cations solubility in KPb2Cl5 arises from the virtually perfect stoichiometry of this compound. The intrinsic potassium, lead, and chloride vacancies concentrations are extremely low, probably due to low Schottky defects formation equilibrium constants. The KPb2Cl5 narrow range of stoichiometry was ascertained during the refinement procedure of the crystal structure by X-ray diffraction carried out on several raw single crystals.8 It was also expected from the low mutual solid solubility of KCl and PbCl2, as deduced from the thermodynamic analysis of the PbCl2/KCl pseudobinary system.21 Another option to eq 2 would be its corollary, consisting of trying to introduce the correct amount of potassium vacancies from the start, in the initial KCl/PbCl2/RECl3 powder mixture, according to

(1 - x)KCl + (2 - x)PbCl2 + xRECl3 S (1 - x)KK× + × • × + xREPb + 5ClCl xV'K + (2 - x)PbPb

(4)

and leaving us with a correctly mass-, charge-, and site-balanced equation. This led us to determine the Er3+ cations equilibrium segregation coefficient in a nonstoichiometric K1-x(VK)xPb2-xRExCl5 solid solution. Indeed, as the temperature increases, the K1-x(VK)xPb2-xRExCl5 solid solution comes in contact with the liquid phase, and one has to analyze the equilibrium segregation coefficient k0. In the crystal growth experiments dedicated to the segregation coefficients measurements, we deliberately chose a smaller diameter to avoid as much as possible radial temperature differences and convection movements in the liquid phase. The early stage of the 0.5 mm h-1 concentration profiles (0 < f e 0.41, see in Figure 2) were fitted to the chemical lever rule:

i nRE 3+k0 nRE3+(f ) ) 1 - f (1 - k0)

(5)

with f ) z/L, z being the axial distance and L the total length of the solidified ingot. As a part of the rare-earth ions is trapped in the bulk material rapidly solidified upon nucleation in the thin capillary, and another part is incorporated in the crystal which has a nonconstant diameter different from that of the bulk crystal, the initial i 3+ had to be fitted simultarare-earth cation concentration nRE neously with k0, which could not be extracted by setting f ) 0 in eq 5. In Table 1, it is seen that k0 is greater for Pr3+ than for Er3+ cations, that of Ho3+ cations being in between the latter two values. This trend is in agreement with the decreasing ionic radius across this series of cations22 as already stressed by Isaenko et al.10 Nevertheless, as we are dealing with an heterovalent substitution, we would like to highlight that not only size effects alone may not account for the elastic strain energy that remains stored in the crystal upon doping (there is also a Coulombic contribution to it), but also this elastic energy does not constitute the sole driving force for rare-earth segregation. As a matter of fact, according to ref 23, the KPC elastic moduli are E1 ≈ 29.9 GPa, E2 ≈ 24.9 GPa, E3 ≈ 27.0 GPa, G44 ≈ 11.0 GPa, G55 ≈ 11.1 GPa, and G66 ≈ 14.3 GPa. In comparison with those of a great majority of oxide and fluoride crystals, such values are at least 1 order of magnitude lower. Consequently, one may expect other types of energies to be relevant. The segregation free energy can be looked upon as a difference of excess Gibbs free energy changes related to the liquid and the solid states, each of which may in turn be split into an enthalpic part and an excess entropic one.24 As these enthalpy changes (containing the contribution from the above-mentioned strain energy) and excess entropy are relative to the pure RECl3 trichlorides and KPb2Cl5 0 as a reference state, one is led to appraise gRECl (298 K) ) 3 0 RECl3 Ulat. (298 K) - 298Sf,RECl3 (298 K), in which Ulat. symbolizes the lattice energy, readily obtained by means of a Born-Haber cycle, and S0f the standard formation entropy.25 It is found that 0 : the more stable the solidk0 increases consistently with gRECl 3 state RECl3, the more thermodynamically unfavored its dissolution in the solid phase, the greater the segregation. Contrary to our expectations, the Er3+ cations k0 value in the initially substoichiometric solid solution proved to be smaller than that in the KPC/ErCl3 mixture, leading in fact to an unwanted slight increase in segregation free energy of ≈320 J mol-1. This reveals the necessity of characterizing and understanding the RECl3 dissolution thermodynamics in the liquid state as well. Finding some chemical trick that would cancel the segregation free energy (≈-1.77 kJ mol-1) and, by the same token, increase the Er3+ cations solubility limit in the solid state upon cooling would undoubtedly help increase the very small intracavity round trip gain of lasers based on such crystals and operating around 4.5 µm. Condon et al. have found random growth directions of the Er:KPC single crystals from one growth run to the other,26 and our X-ray diffraction experiments led to similar results in these crystals as well as in those of Tl3PbCl5, where the crystals facets turned out to change upon regrowing them in strictly identical conditions.27 These observations, which reflect the threedimensional nature of the chemical bonding and the relatively high density in these systems, demonstrate that the growth habit is not unique. Somewhat surprisingly, this occurs under conditions which are not so far from equilibrium. As a matter of fact, owing to the moderate undercooling observed in both systems,


∆Tu ∼ (10-40) K,7,28 the driving force for crystal growth amounts to half the thermal energy at the most at the growth temperature, and it seems sufficient to induce kinetic facetting of the crystals. However, such a randomness in growth direction has virtually no impact on the axial rare-earth segregation profiles for at least one simple reason: the room temperature KPC elastic shear moduli are closely distributed around a mean value of ≈12.1 GPa. Consequently, the segregation anisotropy that would be likely to result from the shear moduli anisotropy is thermally averaged, all the more so since the Gij’s tend toward 0 as the temperature approaches the melting point. Twinning Study. In the pure KPb2Cl5 crystal structure refinements as a function of temperature that we recently published,8 we never had to use a twinning matrix to achieve good agreement factors. Although our crystals were slightly smaller than those of Merkulov et al.’s study (100 × 150 × 150 µm3 versus 480 × 290 × 200 µm3 in ref 29), we repeated our measurements several times on different samples: the absence of twinning at room temperature was checked on different crystals manipulated by two independent crystallographers using two distinct diffractometers located in two distant laboratories.30 Nevertheless, it did happen that, after having mistakenly quenched a Er:KPC crystal at its intrinsic cooling rate, the subsequent diffraction data refinement procedure required the introduction of a twinning matrix,

( )

1 M1 ) 0 k

Vela´zquez et al.

0 0 1¯ 0 0 1¯

(with k ) 2c/a sin (π/2 - β) ≈ -0.0054 < 0) to reach an agreement factor of ∼3% with a twin ratio of ∼50%. This twinning matrix corresponds to a (100) mirror times an inversion (equivalent to a 2-fold axis parallel to b a) and a tiny ka b translation (|ka b| ≈ 0.048 Å). Thus, it describes twin domains I and III (in Figure 3) that are equivalent to those of Merkulov and his cobIII, b cIII) with b cIII ) -c b + ka b. workers, (a b, b b, b c) and (a bIII, b The second type of twinning matrix,

( )

1¯ M2 ) 0 0

0 k′ 1¯ 0 0 1

with k′ ) 2a/c sin (π/2 - β) ≈ -0.0027 < 0), leads to domain II. It consists of a (001) mirror plane times an inversion (equivalent to a 2-fold axis parallel to b c) and a tiny k′c b translation (|k′c b| ≈ 0.034 Å). Since domain II can be switched back to domain I or transformed into two different domains by the application of M1 or M2, and setting aside strain energy minimization complications in the transition regions from domains with M1-type twins to domains with M2-type twins,

Figure 3. Drawing of the twin domain pairs (I, II) and (I, III) of the P21/c phase with the twin mirror planes (100) and (001). The circle and bar in each domain is a guide to visualize both the 180° inversion by 2-fold axes parallel to b a or b c and the tiny rotation of the optical indicatrix with respect to b b.


Figure 4. Section of an oriented monoclinic Er:KPC crystal observed under polarized light optical microscopy showing the twinning due to the phase transition Pmcn f P21/c in a crystal cooled at its intrinsic cooling rate.

the maximum number of domains that could fill the space roughly amounts to ≈π/(β - π/2) ∼ 1800. In Figure 4, one can see that some of the twin domains sharpen as they approach another perpendicular twin domain. This is likely to be due to the overlap between type II domains and M2-transformed type III domains, that gives rise to regions in which local stress relaxation and minimization of elastic strain energy induce lattice distorsions, responsible for high optical losses. Indeed, the resulting losses could be estimated at 50% of the 632.8 nm transmitted beam of a He-Ne laser, whatever the crystal orientation. Such high intracavity losses are all the more detrimental to laser operation since these crystals display a very low intracavity round trip gain around 4.5 µm. The two twin walls parallel to (100) and (001) planes evoked above reproduce macroscopically the mirror planes of the hightemperature phase that are lost during the phase transition intervening upon cooling at Tt ) 255 °C. It turns out that they are the only permissible and prominent domain walls in the coordinate system of the orthorhombic phase (x ) 0, z ) 0) for a species undergoing such a phase transition.31 As P21/c is a subgroup of Pmcn, group theoretical considerations show that fast temperature cycles around Tt should frequently lead to a twinned microstructure with two distinct orientational states below Tt, and always result in an untwinned microstructure above Tt.32,33 With regard to the stress involved in the twinning formation and patterns, if twinning occurs at all, it has both mechanical and thermal contributions. Our X-ray diffraction investigations suggest that a stress threshold exists below which the crystals do not twin over distances up to a few millimeters. Broadly speaking, the stress levels depend on the lattice thermal expansion. Figure 5 displays the thermal dependence of the pure KPC and Er:KPC crystals lattice parameters. In comparison with the situation in pure KPC crystals,8 Er3+doping seems to lower a little bit the thermal contraction (Figure 5a,c) and to smooth the effect of the phase transition (Figure 5d). While the a (respectively b) parameter in both types of crystals shows a linear dependency on temperature from 293 to 643 (respectively 513) K, the c parameter break of slope at ∼473 K provide clear evidence for the phase transition, as does

Crystal Growth & Design, Vol. 9, No. 4, 2009 1953

the evolution of the monoclinic angle at temperatures higher than 533 K (Figure 5a-c). Mean lattice expansion coefficients are 1/a(293 K) da/dT ≈ 36.8 × 10-6 K-1 (293-603 K), 1/b(293 K) db/dT ≈ 37.4 × 10-6 K-1 (293-603 K) and 1/c(293 K) dc/dT ≈ 39.6 × 10-6 K-1 (293-468 K). Worthy of note is the patent decrease in c parameter upon cooling from 583 to 468 K, 1/c(468 K) dc/dT≈ 80.7 × 10-6 K-1, which, in spite of its lower magnitude than in pure KPC crystals, entails that in this temperature range the thermal stress may be substantial, and suggests that a very slow cooling under an extremely flat temperature profile should be applied to the crystal to prevent generation of dislocations in too large a concentration. Below 473 K, the lattice volume contraction in both pure and Er:KPC crystals becomes approximately isotropic with a substantial average ∼(35-40) × 10-6 K-1. The relative difference in unit cell volume between 293 K and the Pmcn phase at 643 K is ∼ -4.4% (Figure 5d). Point Defects Association Kinetics. The discussion of the previous section suggests that laser performances may prove to be difficult to reproduce from one author to the other, but also for a fixed author using different samples. In this section, the static dielectric constant determination allows us to address another aspect of the elaboration-structure relationships that might affect the local structure-spectroscopic properties relationship, namely, the complicated issue of point defects association kinetics. Let us first envisage the case of the (Er•Pb(2) + V′K) associated defect, the existence of which was claimed to be the majority in Er:KPC crystals.34 With the static dielectric constant measured, ε ) 17.5ε0, and the closest distance of approach of the two point defects, dmin ) 4.3 Å, the capture radius defined by Waite35,36 equals rc ≈ 14.1 Å at 705.45 K, the melting point of the pure KPb2Cl5 crystal. On the other hand, for an initial Er3+ ion concentration of [Er•Pb(2)]i ≈ 1.955 × 1019 cm-3, the BjerrumFuoss theory refined by Reiss37 leads to a distribution of nearest neighbors that displays two inflection points at b ) 18.5 and c )22.1 Å, respectively. The former may tentatively be assigned to the distance above which the point defects can be looked upon as isolated and distributed in a random way with an average separation distance of c ) 22.1 Å. The latter value is approximately equal to the radius of a sphere that would exhibit the molecular volume of the solute (here ErCl3 dissolved into KPb2Cl5 in the form of point defects), rm ≈ 23 Å. The fraction of associated defects, as given by Reiss’ theory, amounts to ≈0.86 in the case discussed here. Consequently, it seems reasonable to state that the initial average separation distance (rm,c) between the two point defects is larger than the capture radius (rc). The kinetics of the irreversible association reaction occurring under such circumstances has been detailed theoretically by Waite35,36,38 who established that the dependency on time of the point defects concentrations follows a pseudosecond-order behavior. The analytical integration of his equation taking into account the lattice electroneutrality condition applied • ] ) [V′K], leads to the present extrinsic doping situation, [ErPb(2) • to the following type of isolated defect concentration ErPb(2) decay:

{

• [ErPb(2) ](t) ) 1/

1 • [ErPb(2) ]i

+

[

4πDrcs 2src t+ s+1 s+1

2sr2c πD(s + 1)

2

(

ln 1 +

πDt -

)] }

s+1 √πDt rc

(6)

In eq 6, s stands for an adimensional coefficient that quantifies the probability of the association reaction, once the two point


Vela´zquez et al.

Figure 5. (a-c) Thermal dependence of the pure KPC and Er:KPC lattice parameters; (d) KPC and Er:KPC “X-ray” volumic mass.

defects are separated by a distance lower than rc, over the probability that the two point defects diffuse away from each other without reacting. It takes values really smaller than 1 when, for example, a kinetic barrier to the reaction appears (for example, when the two point defects exhibit the same charge sign). When the association reaction happens almost systematically, s . 1. D symbolizes the sum of the diffusion coefficients of both species. If we define the lifetime τr of the isolated defect • • as the time at which its concentration equals [ErPb(2) ]i/e, ErPb(2) -13 2 -1 it appears that for s ) 1 and D ) 10 cm s (very slow association kinetics), τr ≈ 0.8 s. This time is to be compared with the intrinsic cooling time τc ) FCpL2/Mκ of the crystal around the melting point. With F ≈ 4.55 g cm-3, Cp ≈ 56.5 R, L ≈ 6 mm, M ≈ 630.74 g mol-1, and κ ≈ 4.6 W m-1 K-1, one gets τc ≈ 26.4 s. Thus, in spite of the very low values of s and D chosen for the purpose of our discussion, we have τc . τr and the association reaction is kinetically favored. The case of the (EuK•• + V′′Pb(2)) associated defect39 would bring about a different concentration decay. With dmin ) 4.2 Å and |z+| ) |z-| ) 2, the capture radius now amounts to rc ≈ 54.1 Å. Furthermore, the Reiss distribution function no longer displays inflection points nor extrema. Its integration from dmin to 8.5 Å indicates a virtually complete association. As rc > rm,

it appears that the kinetics is first order,40 with [Eu••K] undergoing a simple exponential decay and

τr ≈

1 4πfa[EuK••]ircD

(7)

Now D represents the diffusion coefficient of the fastest of the two species, EuK•• and V′′Pb(2). With D ) 10-13 cm2 s-1, eq 7 gives τr ≈ 75.2 ms, again a much shorter time than τc, so that the association reaction between the point defects is kinetically favored. It is patent that the amount of associated defects as well as their association kinetics depends on the initial Er3+ • concentration. For a [ErPb(2) ]i ≈ 8.44 × 1019 cm-3, the kinetics would probably be first order with, in the pessimistic case (D ) 10-13 cm2 s-1), τr ≈ 66.7 ms , τc. Owing to the axial segregation phenomena inherent to the crystal growth method, one understands that the nature and amount of a particular rareearth defect may vary with spatial coordinates. Conclusions In summary, the Er:KPC lattice parameters thermal expansion, pure KPC room temperature static dielectric constant and


equilibrium segregation coefficients of Er3+, Ho3+, and Pr3+ point defects in KPb2Cl5 have been characterized and discussed in relation to the two main current limitations that hinder the development of this crystal as a laser material: the Er3+ cations segregation and low solubility limit as well as the crystal twinning. In particular, we have given evidence that fast thermal variations can make the crystals cross a stress threshold above which they twin according to the theory of Sapriel. The resulting twin walls, consisting of macroscopic (100) and (001) mirror planes, entail both double refraction and depolarization of the transmitted beam in the laser cavity. Acknowledgment. M.V. thanks Dr. Y. Tsur and Prof. J. Margerie for very fruitful discussions, and Dr. S. Pećhev for additional crystallographic orientation and twinning characterizations of our crystals. The static dielectric constant measurement would not have been successful without the expert craftmanship of Vivien Meńard, who polished the KPC crystals down to a 70 µm thickness.

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Segregation and Twinning in the Rare-Earth ... - ACS Publications

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