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Kinetic model of proton-lithium exchange in LiNbO3 and LiTaO3 crystals: The role of cation vacancies
SOLID STATE IOWICS
Solid State Ionics 58 ( 1992) 23-32 North-Holland
Kinetic model sf proton-lithium exchange in LiNb03 and LiTaO, crystals: The role of cation vacancies V.A. Ganshin
and Yu.N.
Korkishko
Chair of Chemistry, Moscow Institute ofElectronic Technology Moscow. Russia Received
15 March
199
1;accepted for publication
1 May 1992
The kinetic model of proton exchange in LiNbOs, LiTaO> crystals is proposed. The model is based on the formalism of relative composing units, which enables to find the expressions for gradients of chemical potentials of diffused particles in the sublattice of sites and intersites. The diffusion equations for hydrogen, lithium and vacancies have been obtained. Their solutions with some values of the self-diffusion coefftcients enable for the first time to explain the step-like concentration profile of hydrogen, the upper limit of its concentration, the asymmetry of hydrogen and lithium depth-distributions and some other specific features of proton exchange process.
1. Introduction Throughout recent years, the number of reports on application of proton exchange (PE) in integrated optics technology for producing waveguides in lithium niobate (LN) and lithium tantalate (LT) crystals, has been rapidly growing. Since the time the PE has been discovered ( 1982, ref. [ 1 J ), a substantial body of data on the properties of PE-waveguides and on the regularities of the PE process, especially in LN has been accumulated. However, in our opinion, there is one important point in the description of PE that is not yet clear: its mechanism and kinetics, which differ from the ordinary ion exchange in several aspects. The aim of our present study is to examine this problem and to develop a kinetic model of PE.
2. Principal facts about the PE in LN and LT The following brief account of the main facts will be mostly based on the data for the well-studied PE in LN. The specifics of the PE in LT are discussed in the text. It is known that the processing of LN in melts of many acids [ I-81 results in the formation of H: LiNbO, layers (hereafter denoted as H: LN) on 0167-2738/92/$
05.00 0 1992 Elsevier Science Publishers
the surface of the samples and that the profile of the extraordinary refractive index (e-RI) depth distribution is step-like with a clearly distinguished boundary with coordinate x= h. The increase of e-RI due to the PE in LN is practically independent of the conditions of the process and amount to 0.11-o. 12. The distribution of lattice deformations in the H : LN layers is of a similar step-like character [ 9, lo]. There are data, though not much, on the depth distribution of the concentrations of the exchanging ions in H:LN [ 1l-141 and in H:LT layers [ 151. As an illustration we reproduce in fig. 1 the concentration profiles for lithium and hydrogen as presented in ref. [ 13 1. According to the data of refs. [ 1I- 15 ] one can state that the distribution of hydrogen has a clearly distinguished zero-gradient range with the size of the order of h. On the contrary, the lithium concentration profile is smooth. The maximum content of hydrogen in the doping region in LN may reach 0.72 in relative units [ 161, while after the LN polycrystalline powder is processed, it can be as high as 0.77 [ 17 1. In high-acidity melts [ 18 1, the intersite hydrogen HI takes part in PE [ 191. The term “intersite hydrogen” we use here and below for the particle that is the part of OHgroups with characteristic absorption at 3240-3260 cm-’ in IR-spectra. The known data on PE kinetics do not go beyond
B.V. All rights reserved.
V.A. Ganshin, Yu. Korkishko /Kinetic model ofproton-lithium exchange
24
tric current through the sample if no external field is applied:
1’ IO4
z
1
c G.Jk=O
t -? ,g
2
and (ii) the electrical region:
2 1) I
IO3
/’
-2 ‘ii
(3)
I’ I
neutrality
of the exchanged
/’
cl2
, .a’
2 &.Nk=O.
4
IO1
____---
c-
IO0 0.5
I .2
r.ym
Fig. 1. Depth distributions of concentrations of exchanged ions in H:LN layers (Z-cut), data of SIMS [ 131: 1, 1’ - hydrogen; 2, 2’ - lithium; I, 2 - PE in benzoic acid, 175 ‘C, 90 min; l’, 2’ - the same at 225”C, 35 min.
the “technological” framework and confirm the fact that PE is a stationary diffusion-controlled process. In any case, the optical thickness of H:LN layers grows with time as h=2fi, where D is regarded coefficient.
(4)
k
(1) as an effective
interdiffusion
Formulae ( 1 )-( 4) can also be expressed in terms of relative concentrations: Ck=Nk/NO. In this case the fluxes should be, accordingly, renormalized to N,, i.e. to the total number of all particles in a unit volume, which is taken to be constant. In the case where one has only two kinds of particles (n), (m) with equal charges, eq. (2) can be reduced, with account of eqs. ( 3 ) and (4), to the equation of ion exchange diffusion with respect to the concentration of one of the two particles. If it is, for instance, the incoming one (m ), we have [ 13,22 ] :
ac, at
a
[
1
=Dpn.G 1+JfC,
1’
.ac,, ax
(5)
where f= (D,- D,) ID,. For the incoming particles, one usually takes the following boundary and initial conditions: C,(x,O)=O,
C,n(~,
t)=O,
C,X(O, t)=const. (6)
3. Principal facts about the ion exchange kinetics The PE is a particular case of ion exchange, and its kinetic description is usually based on the NernstPlanck equation for charged particle fluxes [ 13,2022]: Jk=-Dk.(VNk+Nk.qk.V~lkT),
(2)
where Dk is the self-diffusion coefficient for k particles, Nk is their volume concentration, qk is their charge, k is the Boltzmann constant, T is temperature and VP is the potential gradient of the external or the internal (produced by the motion of charged particles) electric field. Since we consider the diffusion along only one direction x, the V operator is simply a/ax. There are two natural conditions under which eq. (2) is usually solved: (i) the absence of a total elec-
The attempts to solve eq. (5) for PE under these conditions, ref. [ 131 gave results that disagree with the experimental data and are contradictory. In the first place, the asymmetry of the depth distributions for concentrations of hydrogen and lithium does not agree with eq. (4), and thus it is necessary to find a realistic substitute for it. As far as we know, no one has ever tried to study the ion exchange and the PE process without imposing the above constraints.
4. The model of PE kinetics and diffusion The model we propose is based on the following principles: (a) The behaviour of particles in the crystal is described according to the formalism of relative composing units (RCU, [ 23,241). The crystal is re-
V.A. Ganshin, Yu. Korkishko /Kinetic model ojproton-lithium exchange
garded as a system of cation (lithium), anion and intersite sublattices. The model makes allowance for the diffusion of lithium and hydrogen along the sites of the cation sublattice together with cation vacancies and, also, along the intersites with corresponding self-diffusion coefficients. We assume, that the anion particles take no part in the diffusion. (b) The cation and intersite sublattices interact by means of the following chemical reactions: LiX’Li.. LI+-
I
+V
L,
and
H,X,PHi+Vt,.
(7)
We use here and below the Kriiger-Vink notation: Li5, HL”, and VI, are lithium, hydrogen and vacancies in the cation sites, while others (with index i) are the intersite particles. (c) In general, from the thermodynamical point of view the system “crystal-melt” is open, but there are some characteristics which simplify the description. This system is isothermic and isobaric according to the technological conditions. Then, one can undoubtably neglect the volume changes of the samples. Furthermore, the reactions (7) occur inside a certain elementary cell of crystal volume. They are not related to transport processes, so the equilibrium is reached instantaneously as compared with the characteristic time of the diffusion, and so we can base the description upon the principles and the laws of the local chemical equilibrium. (d) As is known, the classical transport equations present the fluxes as a linear combination of the thermodynamical forces. The most important forces are the gradients of the temperature, stress, chemical and electric potentials. As is cited above, in our case the gradient of temperature is equal to zero and the corresponding force is absent. Let us now study the question concerning the internal mechanical stresses in exchanged layers. It is known [ 9, lo], that in the case of H: LN layers on Z-cut plates all the components of the elastic stress tensor ar equal to zero. In the case of X-and Y-cuts it is also true for the components normal to the surface plane of the samples (along to the transport direction). At the same time all other properties of the exchanged layers on the different cut plates are identical. This enables one to conclude that the gradients of the stresses are of minor influence on the PE process. In addition, we assume that the gradient of the internal electric potential is equal to zero, too. Therefore, one can use the
25
particular form of the transport equation, where only the gradient of the chemical potential is taken into account:
(e) As was already mentioned, an important feature of the PE is that the process is stationary. In the open system this is possible if (i) the thermodynamical forces on the boundary surface are stabilized and (ii) within each elementary volume of the medium the time required to reach chemical equilibrium is shorter than the characteristic time of the diffusion. The latter demand is already discussed and seems to be real. The first demand (i) can be accounted for in boundary conditions. (f) Condition (4) is replaced by the stoichiometric relation for any cross-section of the cation sublattice with N,=N, sites in a unit volume: N,+N,+N,=N,.
(9a)
The indexes k=l, h, v are introduced to mark the particles Li, H and V, respectively. In terms of the relative concentrations c,+c,+c,=1.
(9b)
In a similar manner, by introducing the concentrations of intersite vacancies and occupied intersites one can write down the stoichiometric relation for the intersite sublattice. In addition to (9) there in one more, also important, relation between the concentrations of particles in the system (7): Nlh +Nir = NV .
(10)
The equilibrium in system (7) is characterized by the constant K,. In accordance with the classical concepts one must present K, in the form: K,=---,
aih’al
ah
(11)
‘ail
where a,,, a, are the activities of hydrogen and lithium at the sites, and aih, ai, - at the intersites, respectively. In the framework of the RCU and the statistical description of the crystal according to Einstein, these activities are [ 23 ] :
26
ak
V.A. Ganshin. Yu. Korkishko /Kinetic model ofproton-lithium exchange
=N,/JJ, (for I-G,
aik=Nik/N,,
Xi
(forHi,
),
Li;)
.
(12)
Here N,, is the volume concentration vacancies. The gradients of chemical potentials through activities:
of intersite are expressed
Li; +HG +LiA +H,’ ;
VfiLk= kF Vhak .
(13)
It is now necessary to find the activity of Lizi and VP,. Proceeding from the considerations (c), we use to this end the Gibbs-Duhem relation in the form: >
T Nk’V&=O
(14)
where the summation is performed separately for particles of different sublattices. After substituting ak into (14), we get: VP, = kT. $
I
rectly with the hydrogen and lithium ions in the melt (lithium ions appear in it as a result of PE, or sometime are deliberately introduced into the melt). Let us indicate the following parallel reactions that mediate the PE: (i) in intersite sublattice
. VN,
(ii) in the sublattice LiEi +HL *Liz
(17a)
of sites
+H:i
.
(17b)
The index (m) labels the particles in the melt. If the rate constants of these reactions are known, one can reduce eqs. ( 17) and (7 ) to a nonlinear system of equations of the first order by applying the mass action law and use its solutions to express the nonstationary boundary conditions at x=0. The matter concentration equation that relates Jk in the cation sublattice to the constant number of sites has the form: J,+J,,+J,=O.
and then find after integration, account:
From here, with account of eqs. (8), ( 12), ( 14) and ( 15 ), one obtains the following expressions for Jk of site particles normalized to N,:
.
a, =Q. exp( -N,/N,)
Here Q is an integration constant. Since from the thermodynamical point of view the RCU belongs to the nonsymmetric systems and Ni is a normalizing basis for activities of the site particles, Q is found from the limit a,+1 at N,+N, (C1+1):Q=2.718.., which enables one to express K, in terms of concentrations only:
(15) By using eqs. ( lo)-( N,, =A exp( 1 -NJNi)
(18)
taking eq. ( 13 ) into
12) we obtain now Nii and Nib: ,
J, = -D,Cc’VC,
J,,=-bDh(b-C,,)-‘VC,,
,
J,=-uD,(u+C,)-‘VC,, where
ack -=--.
at
(19)
b= (D,-D,)/(D,-D,)
(DI-&)l(&--Dv). The diffusion the usual one:
equation
and
u=b-
l=
for any for the particles
a.h
is.
(20)
ax
The principles analyze concrete
Nib =AK, ,
,
formulated above enable solutions to the problem.
one to
where A=N,[K,+N,N;’ Let us mention
exp(l-N,/N,)]-’
5. Variants of solutions
.
that Nib and Ni, are independent
of
N,,. (g) Reactions (7) take place in the crystal volume. The situation on the crystal surface is more complicated though, since the particles interact di-
The diffusion equation for hydrogen in sites, C,, (x, t), is obtained from eqs. (19), (20): aCh -=D,b&-&t$j.
at
(21)
21
V.A. Ganshin, Yu. Korkishko /Kinetic model ofproton-lithium exchange
It resembles eq. (5), but has a singularity at Ch= b, provided that 0 < b-c 1. We will be interested in this particular case, which may give anomalous solutions. The unequality 0 D,, or Dh c D, c D,. Of all the initial and boundary conditions (6) only Ch(co, t)=O
(22)
is accepted without remarks. The other require substantiation. Let the process begin at t= 0. For some time after, the exchange will proceed in a nonstationary manner and C,(O, t) will grow. For this stage we have C,,(x, 0) =O, while C,(O, t) can be found by analyzing the system (17). As C,,( 0, t) approaches b, the concentration growth rate should become smaller. This is true, of course, not only of the cross-section x=0 but for any other as well. The natural requirement that Jr, ( 19) should be continuous and limited means that everywhere in the vicinity of C, = b, with increase of t the derivative aC,/ax approaches 0 not slower than C, approaches b. As it follows from (2 1 ), b is the upper limit of C,,. The. stable growth of C, at C,, > b is impossible: the direction of the flux Jh in this case becomes opposite to its usual direction. At large x, where C,+O, eq. (2 1) degenerates into the usual equation of Fick’s second law. The parameter of this asymptotic solution is self-diffusion coefficient D,,. It is apparently impossible to obtain an analytical solution of (2 1). When searching for a numerical solution, it is convenient to introduce the function W=ln (b- C’,) and, since D,, is independent of boundary conditions, the automodel Boltzmann variable Z=x/$ [ 251. By using these new variables, from eq. (2 1) we obtain: w,,+
Zexp(W W’=O
(23)
2bD,
with the derivative to the Z coordinate. Eq. (23) is easily solved numerically, for example, by the RungeKutta method. We used the following boundary conditions: C,(O)=const
and
C,,(m>=O.
Fig. 2 shows a family of curves C,(Z)
(24) for D,, = 0.5
0.6
0.4
0.2
0
1 0
\._. \.. \.. 2
4
6
E, /urn h-b
Fig. 2. Concentration profiles of hydrogen at lattice sites, calculated at D,=O.5 pm*,h-‘, bc0.6 for different values of the flux J,,(O). l:J,,(O)=O.564 pm.h-‘; 2: 1.264 prn,h-‘; 3: 1.680 krn.h-‘.
um2/h, bE0.6 and different values ofJ,,(O). Each of these curves is a unique solution at all t for the conditions (24 ). As a characteristic depth parameter let us use the bending point Z. of the function C,(Z) and take its x-coordinate to be the layer depth h. By analogy with the usual diffusion distributions, let us take (1) to determine h, considering the C,,-profile and e-RIprofile to be similar. Then one finds the interdiffusion coefficient D in an experiment after measuring the optical depth of the layer, and Z,,=2fi. Let us point out the most important properties of the solutions of eq. (23). It turned out that with very high accuracy one has Z0=2.b-‘.J,,(0), and at C, (0) 4 b the slope of C,(Z) at the bending point, the value of Z, and, thus, D all increase. In all cases of practical importance D is several times greater than D,,. As is seen from fig. 2, one is able to obtain the front of C,(Z) as steep as one desires. Z,, is independent of b. This result seems to be formally contrary to the above noted relation, but means simply that the values of J,,(O), which satisfy (24), linearly depend on 6. Everything we have said above is true provided that 0
%A. Ganshin,
28
ac,=$T at
‘!!s
‘ax [ c, ax
1 .
Yu. Korkishko
/Kinetic
model ofproton-lithium exchange
(25)
In contrast to (2 1 ), it has no singularities except the meaningless case C, = 0 (where the heterogenious process ceases). Two conditions of the problem are evident: C,(x, 0) = 1 and C,( co, t) = 1. C, monotolically increases with x at all t. In the stationary regime, at a certain stable C,, (0, t ) the value C, (0, t) may also remain constant. It and D,,, D,, D, should be selected, in accordance with the supposition that 0
3.6
-
3.4 “,‘, 3.2
%_
0.L&PL 0
2
4
6
a
Z,ym hey2
Fig. 3. Particle distributions throughout the depth calculated for k0.6, D,=O.5 Fm*.h-I, D=5.6 Jh(0) = 1.264 pm.h-‘, ~rn’.h-‘, and.T,(0)=2.167 pm,h-I, C,(O, t)=O.l.
4a, b and c. The difference is significant, but without additional data one cannot say which variant is preferable.
(26) where 0 i C, < 1. Let us recall that u = b - 1 and, since for describing the specific features of the [ HL: ] profile we have taken that 0 < b < 1, we have: - 1 < u < 0. Consequently, (26) has the same singularity as (2 1). It is difficult to formulate a total set of initial and boundary conditions for these particles that undoubtly exist in LN and LT but whose behaviour in the PE has not been studied by direct methods. It is easier to find C,(x, t) from (9b) after one finds C, (x, t) and C, (x, t) . Let us point out that solutions can be found only if IJ,(O) I > (J,,(O) 1. An important property of these solutions is that there exist characteristic maxima and minima in the C, near the front of C,, distribution (fig. 3 ) . The distributions of intersite particles H,’ and Li; can be found by substituting C, (x, t) and Ch(x, t) into ( 16) for a certain given value of K,. We will confine ourselves to simulation of three cases only: K,= 0.001, 1.O and 100. The corresponding Z-distributions [ Li,’ ] and [H,’ 1, together with the vacancy concentration profile (fig. 3 ), are shown in figs.
6. Consequences of the developed model
The model proposed enables one to explain three principal features of the PE according to the character of the particle distribution at the sites. These are: the asymmetry of hydrogen and lithium concentration profiles; the step-like distribution of [ Hc, ] ; the stability of the e-RI of the layers H : LN (due to existence of a natural limit for [ Hri ] C,, = b). An indirect confirmation of the vacancies taking part in the PE comes from two facts: (i) that in the case of PE one observes a displacement of the absorption maximum associated with OH-groups from the frequency 3486 cm-’ to the shortwave region, as it happens when LN is depleted on lithium [ 26 1, and (ii) that with increase of concentration [VL] the LN-lattice expands [ 27 1, which is typical of PE as well. The model creates also a basis for a qualitative understanding of several facts up to now not considered in connection with the PE kinetics and mechanism. Let us briefly consider some of them.
29
%A. Ganshin, Yu. Korkishko /Kinetic model ofproton-lithium exchange
6.2. PE in highly acidic melts
0
2
0
2
6
4
8
Z,
,urn.h-“z
0.2 0.1 0
4
6
8
z, pm.
h- %
Fig. 4. Theoretical distributions of interstitial lithium, of intersite hydrogen and cation vacancies (fig. 3): (a) K,=O.Ol; (b) K,= 1.0; (c) K,= 100.0.
As an example, let us consider the properties of H:LN layers at the Z-cut plates of the crystal after they have been processed in highly acidic melts [ 18 1. Such layers are known to contain a considerable amount of intersite hydrogen, while the shape of their e-RI profile is slightly different from the step-like form (fig. 5). The annealing of the samples at 300400°C will initially vary so that the e-RI profile acquires a pronounced step-like form with increase of the thickness h by approximately 25 to 30% and the disappearance, by that time, of intersite hydrogen. After this I-stage annealing proceeds without any particularities, with transformation of the e-RI profile into a gradient one. During the reserve lithium-proton exchange Li+eH+ that occurs when the samples with H:LN layers are processed, for example, in a melt of lithium nitrate [ 28,291, the resolution of intersite hydrogen, proceeds tens of times faster than when the samples are annealed at the same temperature. Furthermore, in the case of the reverse exchange one observes only an insignificant increase of h, and after a certain period of time the properties of the initial LN are completely restored [ 28,291.
6.1. The position of LiNb,O, and LiTa308 phases 2.32
In several of our papers [ 9,10,15 ] we have shown that the PE in LN and LT results in the appearance of the phases LiNbSOp and LiTa308 in the H: LN, H: LT layers. There are several indirect indications to the fact that these phases are most likely to be found at the front of the hydrogen diffusion. The clearly distinguished vacancy concentration maximum before the [ Hci ]-concentration front (fig. 3) indicates that here the depletion of the cation sublattice is the greatest. From the stoichiometric point of view this is the necessary condition of the phase appearance. The other sufficient condition is that the lattices of the main and of the built-in phases should be crystallographically conjugated. In the case of LN, this condition is favourable [ 91.
2.30 2.28 2.26 2.24 2.22 2.20 0
2
4
6
a
?p
Fig. 5. Transformation of e-RI profiles in H :LN (2) layers formed in highly acidic melt of NH,H*PO, with annealing at 320°C. The technique of our measurements of e-RI is described in ref. [9]: ( 1) initial profile; (2,3,4) after annealing during IO,20 and 150 min, respectively.
30
V.A. Ganshin, Yu. Korkishko /Kinetic model ofproton-lithium exchange
In our opinion, the best way to explain these facts is to suppose that the distribution of intersite particles and vacancies in H : LN layers before annealing or reverse exchange qualitatively resemble those shown in fig. 4a (K, < 1)) and to regard 3-RI as being linearly dependent of the concentrations [ HLX,] and
[Kl. As it follows from the experiment, the annealing allows to observe the processes with higher activation energy than does the PE itself, for example, the H; generation (7). Its activation with annealing results in the increasing of [vl,i] and in the lowering of [ HEi 1. Similar processes for lithium proceed even more intensively. When K, increases, the equilibrium in the system (7) is shifted in such a way that [Hi ] and [ V;i ] become closer everywhere where both Hz, and Vii are present. On the contrary, with lowering of K, the [Li; ] maximum shifts into the zone where there is practically no Hz, (see fig. 4). The conditions of particle diffusion also change with temperature. First of all, this happens owing to the absence of external drains and sources at the crystal boundary. Thermodynamically the sample is now an isolated system and all diffusion processes have the general feature of tending to level off chemical potential gradients of the particles. The calculation of these gradients shows that among the site particles, the least noticeable changes occur in the [ HL: ]-distribution where C,, is close to b. On the contrary, the redistribution of cation vacancies should be the most substantial one, particularly in filling the “well” of [V;, ] near the front of [H?, 1. The fluxes of vacancies are directed into the “well” both from the right and from the left. In addition, in this zone, where the value of [Vi,] is initially low, one also has the best conditions for generation of vacancies. The levelling of the [V;, ]-distribution here may, in principle, continue until it reaches its limit value 1u 1 for the annealing temperature. Owing to the constraint [V,, ] = 1u 1, the region of the initial maximum of [Vii ] will broaden both into the depth of the crystal and toward the “well”. Probably, some vacancies may still remain in the layer after annealing but their distribution should now be practically uniform throughout the depth. The main time-phases of this I-stage of annealing in the case, when K, increases with temperature, are shown in fig. 6a, b, c and d.
b t J’
b
t L?
b I c:
x Fig. 6. Evolution of depth distributions of particles in the process of annealing : (a) initial state; (b) H, generation and covering the vacancy “well”; (c) H, diffusion, and (d) cooling.
The second stage of annealing is a usual diffusion process and we will not discuss it. In the reverse exchange, the hydrogen from the near-surface regions is replaced by Li+ ions from the melt. In contrast to annealing, the fluxes of HI and HE, are now directed towards each other. This alone is enough to explain the fact that in the reverse exchange the thickness of the initial layer practically does not change. 6.3. Anomalies of deformation profiles As we have mentioned above, the deformation profile in the layers after PE is step-like, as is the eRI profile. This is true in the majority of cases but
31
V.A. Ganshin, Yu. Korkishko /Kinetic model ofproton-lithium exchange
_1
(4
,(-,
a
-
-b
b
b J--__--_--_L
b
X Fig. 7. Evolution
of the hydrogen
fluctuation:
there are several exceptions. For instance, when examining the PE in X-and Y-cut LT plates in the stearic acid, we have found that PE is associated with a successive discrete accumulation of deformations in H: LT layer [ 15 1. In the framework of the model these facts can be qualitatively understood. Consider a layer with a step-like [Hz ] and deformation distributions. Let us suppose, that in a certain region, owing to some fluctuation, the concentration [Hci ] rises by the amount b (fig. 7a). As it follows from eq. (21), the fluxes of Hzi to the left and to the right of the fluctuation maximum are directed towards each other. Owing to proton diffusion through the boundary, the region to the left of this maximum will be gradually filled with HL”, particles. The source of hydrogen on the right side is limited: this is the falling off section of the fluctuation profile. The diffusion “uphill” will further cause the contraction of this region. The development of
X the growth of a new concentration
step (a-+d).
this process (figs. 7b-d) results in the building up of a new step in H,1-concentration and in the deformations associated with it. The PE according to reaction (17a) will be dominating at the boundary x=0. Since at the maximum of the [ HLX,] we have Jh=O, the moment the fluctuation appears, the advance of the main concentration front will stop , but the PE will continue [ 151, and is limited only by lithium diffusion from the crystal depth to the surface. In principle, within this model the finite value of [ HL7 ] in a region of the built-up step is limited only by the b-parameter. But actually, an additional constraint may be imposed by crystallophysical factors (such as the elastic stresses), which require special examination. The same also applies to the number of built-up steps. In our opinion, a good example to this case is the multistep deformation accumulation after the PE we have discovered at the LN Z-cut (the
V.A. Ganshin, Yu. Korkishko /Kinetic model ofproton-lithium exchange
32
References
0
0.45
300
600
- 40
12”
.
0.30
O.T5
0
. \“; 0
12
3
Fig. 8. Rocking curve (a) and C-parameter deformation profile (b) in the H: LN(Z) layer formed in highly acidic melt NH,HZPOI (240°C 150 min).
PE processing in NH,H,PO, melt). This is confirmed by the characteristic rocking curve with respect to the (00.12) plane of the sample and by the C-parameter deformation distribution (fig. 8 >.
7. Conclusion We have developed a kinetic model of the PE process in LN and LT crystals based on the following principles: the account of the gradients of chemical potentials in particle diffusion in the framework of the formalism of relative composing units; the account of the special role played by the vacancies of the cation sublattice of the crystals; and the assumption about the value of the b parameter. The model proposed enables one to explain for the first time several important features of the PE process in lithium niobate and tantalate crystals.
[ 11 J.L. Jackel, C.E. Rice and J.J. Veselka, Appl. Phys. Lett. 4 I (1982) 607. [2] M. De Micheli, J. Botineau, P. Sibillot, D.B. Ostrowsky and M. Papuchon, Opt. Commun. 42 (1982) 101. [3] D.E.Clark,A.C.G.Nutt,K.K. Wong,P.J.LaybournandM. De La Rue, J. Appl. Phys. 54 (1983) 6218. [4] C. Canali, A. Camera, P. Mazzoldi and R.M. De La Rue, SPIE517(1984)119. [ 5] SM. Al-Shukri, J. Duffy, R.M. De La Rue, M.N. Armenise, C.Canali and A. Camera, SPIE 578 (1985) 2. [6]A.L.Dawar,S.M.Al-Shukri,R.M.DeLaRue,A.C.G.Nutt and G. Stewart, Appl. Opt. 25 (1986) 1495. [7] A.L. Dawar, S.M. Al-Shukri, R.M. DeLaRue, A.C.G. Nutt and G. Stewart, Opt. Commun. 61 (1987) 100. [ 81 M. Goodwin and C. Stewart, Electron. Lett. 19 ( 1983) 222. [9] V.A. Ganshin, V.Sh. Ivanov, Yu.N. Korkishko and V.Z. Petrova, J. Techn. Phys. (USSR) 56 (1986) 1354. [ lo] V.A. Ganshin, Yu.N. Korkishko and T.M. Morozova, Phys. Status Solidi (a) 110 (1988) 397. 111 R. Richter, T. Bremer, P. Hertel and E. Kraetzig, Phys. Status Solidi (a) 114 (1989) 765. 12) C. Canali, A. Camera, G. Della Mea, R.M. De La Rue, A.C.G. Nutt and J.R. Tobin, SPlE 460 (1984) 49. [ 131 S.T. Vohra, A.R. Mickelson and S.E. Asher, J. Appl. Phys. 66 (1989) 5161. [ 141 K.K. Wong, GEC J. Research 3 (1985) 243. [ 151 V.A. Ganshin, Yu.N. Korkishko and T.M. Morozova, Phys. Status Solidi (a) 114 (1989) 457. [ 161 J.L. Jackel, C.E. Rice and J.J. Veselka, Ferroelectr. 50 (1983) 165. [ 171 C.E. Rice and J.J. Jackel, Mat. Res. Bull. 19 (1984) 591. [ 181 Yu.N. Korkishko and V.A. Ganshin, J. Techn. Phys. (USSR) 58 (1988) 692. [ 191 M. De Micheli, D.B. Ostrowksy, J.P. Barety, C. Canali, A. Camera, G. Mazzi and M. Papuchon, J. Lightwave Technol. LT-4 (1986) 743. [ 201 F. Helfferich and M.S. Plesset, J. Chem. Phys. 28 (1958) 418. [ 2 1 ] R.M. Barrer and L.V. Rees, J. Phys. Chem. Solids 25 (1964) 1035. [ 221 C.D.W. Wilkinson and R. Walker, Electron. Lett. 14 (1978) 599. [23] V.N. Chebotin, Fizicheskaya Khimiya Tverdoga Tela (Nauka, Moscow, 1982). [ 241 F.A. Kroger, The Chemistry of Imperfect Crystals (NorthHolland, Amsterdam, 1964). [ 251 J. Crank, The Mathematics of Diffusion (Clarendon Press, Oxford, 1957 ) [ 261 L. Kovacs, V. Szalay and R. Capelletti, Solid State Commun. 52 (1984) 1029. [27] SC. Abrahams and P. Marsh, Acta Cryst. B42 (1986) 61. [28] V.A. Ganshin and Yu.N. Korkishko J. Techn. Phys. (USSR) 58 (1988) 1168. [ 291 V.A. Ganshin and Yu.N. Korkishko J. Techn. Phys. (USSR) 60 (1990) 153.
Solid State Ionics 58 ( 1992) 23-32 North-Holland
Kinetic model sf proton-lithium exchange in LiNb03 and LiTaO, crystals: The role of cation vacancies V.A. Ganshin
and Yu.N.
Korkishko
Chair of Chemistry, Moscow Institute ofElectronic Technology Moscow. Russia Received
15 March
199
1;accepted for publication
1 May 1992
The kinetic model of proton exchange in LiNbOs, LiTaO> crystals is proposed. The model is based on the formalism of relative composing units, which enables to find the expressions for gradients of chemical potentials of diffused particles in the sublattice of sites and intersites. The diffusion equations for hydrogen, lithium and vacancies have been obtained. Their solutions with some values of the self-diffusion coefftcients enable for the first time to explain the step-like concentration profile of hydrogen, the upper limit of its concentration, the asymmetry of hydrogen and lithium depth-distributions and some other specific features of proton exchange process.
1. Introduction Throughout recent years, the number of reports on application of proton exchange (PE) in integrated optics technology for producing waveguides in lithium niobate (LN) and lithium tantalate (LT) crystals, has been rapidly growing. Since the time the PE has been discovered ( 1982, ref. [ 1 J ), a substantial body of data on the properties of PE-waveguides and on the regularities of the PE process, especially in LN has been accumulated. However, in our opinion, there is one important point in the description of PE that is not yet clear: its mechanism and kinetics, which differ from the ordinary ion exchange in several aspects. The aim of our present study is to examine this problem and to develop a kinetic model of PE.
2. Principal facts about the PE in LN and LT The following brief account of the main facts will be mostly based on the data for the well-studied PE in LN. The specifics of the PE in LT are discussed in the text. It is known that the processing of LN in melts of many acids [ I-81 results in the formation of H: LiNbO, layers (hereafter denoted as H: LN) on 0167-2738/92/$
05.00 0 1992 Elsevier Science Publishers
the surface of the samples and that the profile of the extraordinary refractive index (e-RI) depth distribution is step-like with a clearly distinguished boundary with coordinate x= h. The increase of e-RI due to the PE in LN is practically independent of the conditions of the process and amount to 0.11-o. 12. The distribution of lattice deformations in the H : LN layers is of a similar step-like character [ 9, lo]. There are data, though not much, on the depth distribution of the concentrations of the exchanging ions in H:LN [ 1l-141 and in H:LT layers [ 151. As an illustration we reproduce in fig. 1 the concentration profiles for lithium and hydrogen as presented in ref. [ 13 1. According to the data of refs. [ 1I- 15 ] one can state that the distribution of hydrogen has a clearly distinguished zero-gradient range with the size of the order of h. On the contrary, the lithium concentration profile is smooth. The maximum content of hydrogen in the doping region in LN may reach 0.72 in relative units [ 161, while after the LN polycrystalline powder is processed, it can be as high as 0.77 [ 17 1. In high-acidity melts [ 18 1, the intersite hydrogen HI takes part in PE [ 191. The term “intersite hydrogen” we use here and below for the particle that is the part of OHgroups with characteristic absorption at 3240-3260 cm-’ in IR-spectra. The known data on PE kinetics do not go beyond
B.V. All rights reserved.
V.A. Ganshin, Yu. Korkishko /Kinetic model ofproton-lithium exchange
24
tric current through the sample if no external field is applied:
1’ IO4
z
1
c G.Jk=O
t -? ,g
2
and (ii) the electrical region:
2 1) I
IO3
/’
-2 ‘ii
(3)
I’ I
neutrality
of the exchanged
/’
cl2
, .a’
2 &.Nk=O.
4
IO1
____---
c-
IO0 0.5
I .2
r.ym
Fig. 1. Depth distributions of concentrations of exchanged ions in H:LN layers (Z-cut), data of SIMS [ 131: 1, 1’ - hydrogen; 2, 2’ - lithium; I, 2 - PE in benzoic acid, 175 ‘C, 90 min; l’, 2’ - the same at 225”C, 35 min.
the “technological” framework and confirm the fact that PE is a stationary diffusion-controlled process. In any case, the optical thickness of H:LN layers grows with time as h=2fi, where D is regarded coefficient.
(4)
k
(1) as an effective
interdiffusion
Formulae ( 1 )-( 4) can also be expressed in terms of relative concentrations: Ck=Nk/NO. In this case the fluxes should be, accordingly, renormalized to N,, i.e. to the total number of all particles in a unit volume, which is taken to be constant. In the case where one has only two kinds of particles (n), (m) with equal charges, eq. (2) can be reduced, with account of eqs. ( 3 ) and (4), to the equation of ion exchange diffusion with respect to the concentration of one of the two particles. If it is, for instance, the incoming one (m ), we have [ 13,22 ] :
ac, at
a
[
1
=Dpn.G 1+JfC,
1’
.ac,, ax
(5)
where f= (D,- D,) ID,. For the incoming particles, one usually takes the following boundary and initial conditions: C,(x,O)=O,
C,n(~,
t)=O,
C,X(O, t)=const. (6)
3. Principal facts about the ion exchange kinetics The PE is a particular case of ion exchange, and its kinetic description is usually based on the NernstPlanck equation for charged particle fluxes [ 13,2022]: Jk=-Dk.(VNk+Nk.qk.V~lkT),
(2)
where Dk is the self-diffusion coefficient for k particles, Nk is their volume concentration, qk is their charge, k is the Boltzmann constant, T is temperature and VP is the potential gradient of the external or the internal (produced by the motion of charged particles) electric field. Since we consider the diffusion along only one direction x, the V operator is simply a/ax. There are two natural conditions under which eq. (2) is usually solved: (i) the absence of a total elec-
The attempts to solve eq. (5) for PE under these conditions, ref. [ 131 gave results that disagree with the experimental data and are contradictory. In the first place, the asymmetry of the depth distributions for concentrations of hydrogen and lithium does not agree with eq. (4), and thus it is necessary to find a realistic substitute for it. As far as we know, no one has ever tried to study the ion exchange and the PE process without imposing the above constraints.
4. The model of PE kinetics and diffusion The model we propose is based on the following principles: (a) The behaviour of particles in the crystal is described according to the formalism of relative composing units (RCU, [ 23,241). The crystal is re-
V.A. Ganshin, Yu. Korkishko /Kinetic model ojproton-lithium exchange
garded as a system of cation (lithium), anion and intersite sublattices. The model makes allowance for the diffusion of lithium and hydrogen along the sites of the cation sublattice together with cation vacancies and, also, along the intersites with corresponding self-diffusion coefficients. We assume, that the anion particles take no part in the diffusion. (b) The cation and intersite sublattices interact by means of the following chemical reactions: LiX’Li.. LI+-
I
+V
L,
and
H,X,PHi+Vt,.
(7)
We use here and below the Kriiger-Vink notation: Li5, HL”, and VI, are lithium, hydrogen and vacancies in the cation sites, while others (with index i) are the intersite particles. (c) In general, from the thermodynamical point of view the system “crystal-melt” is open, but there are some characteristics which simplify the description. This system is isothermic and isobaric according to the technological conditions. Then, one can undoubtably neglect the volume changes of the samples. Furthermore, the reactions (7) occur inside a certain elementary cell of crystal volume. They are not related to transport processes, so the equilibrium is reached instantaneously as compared with the characteristic time of the diffusion, and so we can base the description upon the principles and the laws of the local chemical equilibrium. (d) As is known, the classical transport equations present the fluxes as a linear combination of the thermodynamical forces. The most important forces are the gradients of the temperature, stress, chemical and electric potentials. As is cited above, in our case the gradient of temperature is equal to zero and the corresponding force is absent. Let us now study the question concerning the internal mechanical stresses in exchanged layers. It is known [ 9, lo], that in the case of H: LN layers on Z-cut plates all the components of the elastic stress tensor ar equal to zero. In the case of X-and Y-cuts it is also true for the components normal to the surface plane of the samples (along to the transport direction). At the same time all other properties of the exchanged layers on the different cut plates are identical. This enables one to conclude that the gradients of the stresses are of minor influence on the PE process. In addition, we assume that the gradient of the internal electric potential is equal to zero, too. Therefore, one can use the
25
particular form of the transport equation, where only the gradient of the chemical potential is taken into account:
(e) As was already mentioned, an important feature of the PE is that the process is stationary. In the open system this is possible if (i) the thermodynamical forces on the boundary surface are stabilized and (ii) within each elementary volume of the medium the time required to reach chemical equilibrium is shorter than the characteristic time of the diffusion. The latter demand is already discussed and seems to be real. The first demand (i) can be accounted for in boundary conditions. (f) Condition (4) is replaced by the stoichiometric relation for any cross-section of the cation sublattice with N,=N, sites in a unit volume: N,+N,+N,=N,.
(9a)
The indexes k=l, h, v are introduced to mark the particles Li, H and V, respectively. In terms of the relative concentrations c,+c,+c,=1.
(9b)
In a similar manner, by introducing the concentrations of intersite vacancies and occupied intersites one can write down the stoichiometric relation for the intersite sublattice. In addition to (9) there in one more, also important, relation between the concentrations of particles in the system (7): Nlh +Nir = NV .
(10)
The equilibrium in system (7) is characterized by the constant K,. In accordance with the classical concepts one must present K, in the form: K,=---,
aih’al
ah
(11)
‘ail
where a,,, a, are the activities of hydrogen and lithium at the sites, and aih, ai, - at the intersites, respectively. In the framework of the RCU and the statistical description of the crystal according to Einstein, these activities are [ 23 ] :
26
ak
V.A. Ganshin. Yu. Korkishko /Kinetic model ofproton-lithium exchange
=N,/JJ, (for I-G,
aik=Nik/N,,
Xi
(forHi,
),
Li;)
.
(12)
Here N,, is the volume concentration vacancies. The gradients of chemical potentials through activities:
of intersite are expressed
Li; +HG +LiA +H,’ ;
VfiLk= kF Vhak .
(13)
It is now necessary to find the activity of Lizi and VP,. Proceeding from the considerations (c), we use to this end the Gibbs-Duhem relation in the form: >
T Nk’V&=O
(14)
where the summation is performed separately for particles of different sublattices. After substituting ak into (14), we get: VP, = kT. $
I
rectly with the hydrogen and lithium ions in the melt (lithium ions appear in it as a result of PE, or sometime are deliberately introduced into the melt). Let us indicate the following parallel reactions that mediate the PE: (i) in intersite sublattice
. VN,
(ii) in the sublattice LiEi +HL *Liz
(17a)
of sites
+H:i
.
(17b)
The index (m) labels the particles in the melt. If the rate constants of these reactions are known, one can reduce eqs. ( 17) and (7 ) to a nonlinear system of equations of the first order by applying the mass action law and use its solutions to express the nonstationary boundary conditions at x=0. The matter concentration equation that relates Jk in the cation sublattice to the constant number of sites has the form: J,+J,,+J,=O.
and then find after integration, account:
From here, with account of eqs. (8), ( 12), ( 14) and ( 15 ), one obtains the following expressions for Jk of site particles normalized to N,:
.
a, =Q. exp( -N,/N,)
Here Q is an integration constant. Since from the thermodynamical point of view the RCU belongs to the nonsymmetric systems and Ni is a normalizing basis for activities of the site particles, Q is found from the limit a,+1 at N,+N, (C1+1):Q=2.718.., which enables one to express K, in terms of concentrations only:
(15) By using eqs. ( lo)-( N,, =A exp( 1 -NJNi)
(18)
taking eq. ( 13 ) into
12) we obtain now Nii and Nib: ,
J, = -D,Cc’VC,
J,,=-bDh(b-C,,)-‘VC,,
,
J,=-uD,(u+C,)-‘VC,, where
ack -=--.
at
(19)
b= (D,-D,)/(D,-D,)
(DI-&)l(&--Dv). The diffusion the usual one:
equation
and
u=b-
l=
for any for the particles
a.h
is.
(20)
ax
The principles analyze concrete
Nib =AK, ,
,
formulated above enable solutions to the problem.
one to
where A=N,[K,+N,N;’ Let us mention
exp(l-N,/N,)]-’
5. Variants of solutions
.
that Nib and Ni, are independent
of
N,,. (g) Reactions (7) take place in the crystal volume. The situation on the crystal surface is more complicated though, since the particles interact di-
The diffusion equation for hydrogen in sites, C,, (x, t), is obtained from eqs. (19), (20): aCh -=D,b&-&t$j.
at
(21)
21
V.A. Ganshin, Yu. Korkishko /Kinetic model ofproton-lithium exchange
It resembles eq. (5), but has a singularity at Ch= b, provided that 0 < b-c 1. We will be interested in this particular case, which may give anomalous solutions. The unequality 0 D,, or Dh c D, c D,. Of all the initial and boundary conditions (6) only Ch(co, t)=O
(22)
is accepted without remarks. The other require substantiation. Let the process begin at t= 0. For some time after, the exchange will proceed in a nonstationary manner and C,(O, t) will grow. For this stage we have C,,(x, 0) =O, while C,(O, t) can be found by analyzing the system (17). As C,,( 0, t) approaches b, the concentration growth rate should become smaller. This is true, of course, not only of the cross-section x=0 but for any other as well. The natural requirement that Jr, ( 19) should be continuous and limited means that everywhere in the vicinity of C, = b, with increase of t the derivative aC,/ax approaches 0 not slower than C, approaches b. As it follows from (2 1 ), b is the upper limit of C,,. The. stable growth of C, at C,, > b is impossible: the direction of the flux Jh in this case becomes opposite to its usual direction. At large x, where C,+O, eq. (2 1) degenerates into the usual equation of Fick’s second law. The parameter of this asymptotic solution is self-diffusion coefficient D,,. It is apparently impossible to obtain an analytical solution of (2 1). When searching for a numerical solution, it is convenient to introduce the function W=ln (b- C’,) and, since D,, is independent of boundary conditions, the automodel Boltzmann variable Z=x/$ [ 251. By using these new variables, from eq. (2 1) we obtain: w,,+
Zexp(W W’=O
(23)
2bD,
with the derivative to the Z coordinate. Eq. (23) is easily solved numerically, for example, by the RungeKutta method. We used the following boundary conditions: C,(O)=const
and
C,,(m>=O.
Fig. 2 shows a family of curves C,(Z)
(24) for D,, = 0.5
0.6
0.4
0.2
0
1 0
\._. \.. \.. 2
4
6
E, /urn h-b
Fig. 2. Concentration profiles of hydrogen at lattice sites, calculated at D,=O.5 pm*,h-‘, bc0.6 for different values of the flux J,,(O). l:J,,(O)=O.564 pm.h-‘; 2: 1.264 prn,h-‘; 3: 1.680 krn.h-‘.
um2/h, bE0.6 and different values ofJ,,(O). Each of these curves is a unique solution at all t for the conditions (24 ). As a characteristic depth parameter let us use the bending point Z. of the function C,(Z) and take its x-coordinate to be the layer depth h. By analogy with the usual diffusion distributions, let us take (1) to determine h, considering the C,,-profile and e-RIprofile to be similar. Then one finds the interdiffusion coefficient D in an experiment after measuring the optical depth of the layer, and Z,,=2fi. Let us point out the most important properties of the solutions of eq. (23). It turned out that with very high accuracy one has Z0=2.b-‘.J,,(0), and at C, (0) 4 b the slope of C,(Z) at the bending point, the value of Z, and, thus, D all increase. In all cases of practical importance D is several times greater than D,,. As is seen from fig. 2, one is able to obtain the front of C,(Z) as steep as one desires. Z,, is independent of b. This result seems to be formally contrary to the above noted relation, but means simply that the values of J,,(O), which satisfy (24), linearly depend on 6. Everything we have said above is true provided that 0
%A. Ganshin,
28
ac,=$T at
‘!!s
‘ax [ c, ax
1 .
Yu. Korkishko
/Kinetic
model ofproton-lithium exchange
(25)
In contrast to (2 1 ), it has no singularities except the meaningless case C, = 0 (where the heterogenious process ceases). Two conditions of the problem are evident: C,(x, 0) = 1 and C,( co, t) = 1. C, monotolically increases with x at all t. In the stationary regime, at a certain stable C,, (0, t ) the value C, (0, t) may also remain constant. It and D,,, D,, D, should be selected, in accordance with the supposition that 0
3.6
-
3.4 “,‘, 3.2
%_
0.L&PL 0
2
4
6
a
Z,ym hey2
Fig. 3. Particle distributions throughout the depth calculated for k0.6, D,=O.5 Fm*.h-I, D=5.6 Jh(0) = 1.264 pm.h-‘, ~rn’.h-‘, and.T,(0)=2.167 pm,h-I, C,(O, t)=O.l.
4a, b and c. The difference is significant, but without additional data one cannot say which variant is preferable.
(26) where 0 i C, < 1. Let us recall that u = b - 1 and, since for describing the specific features of the [ HL: ] profile we have taken that 0 < b < 1, we have: - 1 < u < 0. Consequently, (26) has the same singularity as (2 1). It is difficult to formulate a total set of initial and boundary conditions for these particles that undoubtly exist in LN and LT but whose behaviour in the PE has not been studied by direct methods. It is easier to find C,(x, t) from (9b) after one finds C, (x, t) and C, (x, t) . Let us point out that solutions can be found only if IJ,(O) I > (J,,(O) 1. An important property of these solutions is that there exist characteristic maxima and minima in the C, near the front of C,, distribution (fig. 3 ) . The distributions of intersite particles H,’ and Li; can be found by substituting C, (x, t) and Ch(x, t) into ( 16) for a certain given value of K,. We will confine ourselves to simulation of three cases only: K,= 0.001, 1.O and 100. The corresponding Z-distributions [ Li,’ ] and [H,’ 1, together with the vacancy concentration profile (fig. 3 ), are shown in figs.
6. Consequences of the developed model
The model proposed enables one to explain three principal features of the PE according to the character of the particle distribution at the sites. These are: the asymmetry of hydrogen and lithium concentration profiles; the step-like distribution of [ Hc, ] ; the stability of the e-RI of the layers H : LN (due to existence of a natural limit for [ Hri ] C,, = b). An indirect confirmation of the vacancies taking part in the PE comes from two facts: (i) that in the case of PE one observes a displacement of the absorption maximum associated with OH-groups from the frequency 3486 cm-’ to the shortwave region, as it happens when LN is depleted on lithium [ 26 1, and (ii) that with increase of concentration [VL] the LN-lattice expands [ 27 1, which is typical of PE as well. The model creates also a basis for a qualitative understanding of several facts up to now not considered in connection with the PE kinetics and mechanism. Let us briefly consider some of them.
29
%A. Ganshin, Yu. Korkishko /Kinetic model ofproton-lithium exchange
6.2. PE in highly acidic melts
0
2
0
2
6
4
8
Z,
,urn.h-“z
0.2 0.1 0
4
6
8
z, pm.
h- %
Fig. 4. Theoretical distributions of interstitial lithium, of intersite hydrogen and cation vacancies (fig. 3): (a) K,=O.Ol; (b) K,= 1.0; (c) K,= 100.0.
As an example, let us consider the properties of H:LN layers at the Z-cut plates of the crystal after they have been processed in highly acidic melts [ 18 1. Such layers are known to contain a considerable amount of intersite hydrogen, while the shape of their e-RI profile is slightly different from the step-like form (fig. 5). The annealing of the samples at 300400°C will initially vary so that the e-RI profile acquires a pronounced step-like form with increase of the thickness h by approximately 25 to 30% and the disappearance, by that time, of intersite hydrogen. After this I-stage annealing proceeds without any particularities, with transformation of the e-RI profile into a gradient one. During the reserve lithium-proton exchange Li+eH+ that occurs when the samples with H:LN layers are processed, for example, in a melt of lithium nitrate [ 28,291, the resolution of intersite hydrogen, proceeds tens of times faster than when the samples are annealed at the same temperature. Furthermore, in the case of the reverse exchange one observes only an insignificant increase of h, and after a certain period of time the properties of the initial LN are completely restored [ 28,291.
6.1. The position of LiNb,O, and LiTa308 phases 2.32
In several of our papers [ 9,10,15 ] we have shown that the PE in LN and LT results in the appearance of the phases LiNbSOp and LiTa308 in the H: LN, H: LT layers. There are several indirect indications to the fact that these phases are most likely to be found at the front of the hydrogen diffusion. The clearly distinguished vacancy concentration maximum before the [ Hci ]-concentration front (fig. 3) indicates that here the depletion of the cation sublattice is the greatest. From the stoichiometric point of view this is the necessary condition of the phase appearance. The other sufficient condition is that the lattices of the main and of the built-in phases should be crystallographically conjugated. In the case of LN, this condition is favourable [ 91.
2.30 2.28 2.26 2.24 2.22 2.20 0
2
4
6
a
?p
Fig. 5. Transformation of e-RI profiles in H :LN (2) layers formed in highly acidic melt of NH,H*PO, with annealing at 320°C. The technique of our measurements of e-RI is described in ref. [9]: ( 1) initial profile; (2,3,4) after annealing during IO,20 and 150 min, respectively.
30
V.A. Ganshin, Yu. Korkishko /Kinetic model ofproton-lithium exchange
In our opinion, the best way to explain these facts is to suppose that the distribution of intersite particles and vacancies in H : LN layers before annealing or reverse exchange qualitatively resemble those shown in fig. 4a (K, < 1)) and to regard 3-RI as being linearly dependent of the concentrations [ HLX,] and
[Kl. As it follows from the experiment, the annealing allows to observe the processes with higher activation energy than does the PE itself, for example, the H; generation (7). Its activation with annealing results in the increasing of [vl,i] and in the lowering of [ HEi 1. Similar processes for lithium proceed even more intensively. When K, increases, the equilibrium in the system (7) is shifted in such a way that [Hi ] and [ V;i ] become closer everywhere where both Hz, and Vii are present. On the contrary, with lowering of K, the [Li; ] maximum shifts into the zone where there is practically no Hz, (see fig. 4). The conditions of particle diffusion also change with temperature. First of all, this happens owing to the absence of external drains and sources at the crystal boundary. Thermodynamically the sample is now an isolated system and all diffusion processes have the general feature of tending to level off chemical potential gradients of the particles. The calculation of these gradients shows that among the site particles, the least noticeable changes occur in the [ HL: ]-distribution where C,, is close to b. On the contrary, the redistribution of cation vacancies should be the most substantial one, particularly in filling the “well” of [V;, ] near the front of [H?, 1. The fluxes of vacancies are directed into the “well” both from the right and from the left. In addition, in this zone, where the value of [Vi,] is initially low, one also has the best conditions for generation of vacancies. The levelling of the [V;, ]-distribution here may, in principle, continue until it reaches its limit value 1u 1 for the annealing temperature. Owing to the constraint [V,, ] = 1u 1, the region of the initial maximum of [Vii ] will broaden both into the depth of the crystal and toward the “well”. Probably, some vacancies may still remain in the layer after annealing but their distribution should now be practically uniform throughout the depth. The main time-phases of this I-stage of annealing in the case, when K, increases with temperature, are shown in fig. 6a, b, c and d.
b t J’
b
t L?
b I c:
x Fig. 6. Evolution of depth distributions of particles in the process of annealing : (a) initial state; (b) H, generation and covering the vacancy “well”; (c) H, diffusion, and (d) cooling.
The second stage of annealing is a usual diffusion process and we will not discuss it. In the reverse exchange, the hydrogen from the near-surface regions is replaced by Li+ ions from the melt. In contrast to annealing, the fluxes of HI and HE, are now directed towards each other. This alone is enough to explain the fact that in the reverse exchange the thickness of the initial layer practically does not change. 6.3. Anomalies of deformation profiles As we have mentioned above, the deformation profile in the layers after PE is step-like, as is the eRI profile. This is true in the majority of cases but
31
V.A. Ganshin, Yu. Korkishko /Kinetic model ofproton-lithium exchange
_1
(4
,(-,
a
-
-b
b
b J--__--_--_L
b
X Fig. 7. Evolution
of the hydrogen
fluctuation:
there are several exceptions. For instance, when examining the PE in X-and Y-cut LT plates in the stearic acid, we have found that PE is associated with a successive discrete accumulation of deformations in H: LT layer [ 15 1. In the framework of the model these facts can be qualitatively understood. Consider a layer with a step-like [Hz ] and deformation distributions. Let us suppose, that in a certain region, owing to some fluctuation, the concentration [Hci ] rises by the amount b (fig. 7a). As it follows from eq. (21), the fluxes of Hzi to the left and to the right of the fluctuation maximum are directed towards each other. Owing to proton diffusion through the boundary, the region to the left of this maximum will be gradually filled with HL”, particles. The source of hydrogen on the right side is limited: this is the falling off section of the fluctuation profile. The diffusion “uphill” will further cause the contraction of this region. The development of
X the growth of a new concentration
step (a-+d).
this process (figs. 7b-d) results in the building up of a new step in H,1-concentration and in the deformations associated with it. The PE according to reaction (17a) will be dominating at the boundary x=0. Since at the maximum of the [ HLX,] we have Jh=O, the moment the fluctuation appears, the advance of the main concentration front will stop , but the PE will continue [ 151, and is limited only by lithium diffusion from the crystal depth to the surface. In principle, within this model the finite value of [ HL7 ] in a region of the built-up step is limited only by the b-parameter. But actually, an additional constraint may be imposed by crystallophysical factors (such as the elastic stresses), which require special examination. The same also applies to the number of built-up steps. In our opinion, a good example to this case is the multistep deformation accumulation after the PE we have discovered at the LN Z-cut (the
V.A. Ganshin, Yu. Korkishko /Kinetic model ofproton-lithium exchange
32
References
0
0.45
300
600
- 40
12”
.
0.30
O.T5
0
. \“; 0
12
3
Fig. 8. Rocking curve (a) and C-parameter deformation profile (b) in the H: LN(Z) layer formed in highly acidic melt NH,HZPOI (240°C 150 min).
PE processing in NH,H,PO, melt). This is confirmed by the characteristic rocking curve with respect to the (00.12) plane of the sample and by the C-parameter deformation distribution (fig. 8 >.
7. Conclusion We have developed a kinetic model of the PE process in LN and LT crystals based on the following principles: the account of the gradients of chemical potentials in particle diffusion in the framework of the formalism of relative composing units; the account of the special role played by the vacancies of the cation sublattice of the crystals; and the assumption about the value of the b parameter. The model proposed enables one to explain for the first time several important features of the PE process in lithium niobate and tantalate crystals.
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