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Ion exchange in zinc selenide crystals: the role of lattice-misfit-induced stresses and dislocations
Journal of Crystal Growth 112 (1991) 781—790 North-Holland
781
Ion exchange in zinc selenide crystals: the role of lattice-misfit-induced stresses and dislocations V.A. Fedorov, V.A. Ganshin and Yu.N. Korkishko Chair of Chemistry, Moscow Institute of Electronic Technology, SU-103498 Moscow, USSR
Received 2 January 1991; manuscript received in final form 15 March 1991
A new method has been applied to form Cd ,,Zn 1 — ~Se solid ion single exchange in plates the melts ofbeen salts.studied. The regularities 2~~ Zn2~in ZnSesolutions: powder and crystal have Cd,,Zn and specific features of ion exchange processes Cd 1_~Se solid solution obeys the model of regular solution. Temperature dependences of equilibrium constant and interaction energy of strained and unstrained Cd~Zn1~Sesolid solutions have been defined. We proposed the model taking into account the role of lattice-misfit-induced stresses and dislocations in the ion exchange equilibrium. The depth profiles of elements in 2the ± and ion Cd2 exchanged ± ions Cd : ZnSe were determined. single crystal structures were analysed and temperature dependences of self-diffusion coefficients of Zn
1. Introduction
powder, and to examination of the regularities and specific features of this process.
Earlier we have reported on the possibility of ion exchange diffusion doping (IEDD) of Il—VI single crystal ZnSe, ZnS, CdS and CdTe by divalent ions (Cd2~, Hg2~, Cu21, Pb2~, Co2~,Ni2~and Mn2~)in melts and solutions of inorganic salts at relatively low temperature [1,2]. The IEDD is a two-stage process and involves: (i) the heterogeneous ion exchange (IE) reaction Me2~+AX ~ A2~+MeX on the melt—crystal boundary and (ii) the simultaneous IE diffusion into the crystal. The IEDD results in the forma—
—
tion of graded surface structures, which, in particular, can be used in integrated optoelectronics [1,2]. To be able to use the IEDD method for processing Il—VI single crystals for manufacturing the optoelectronic elements with a desired depth distribution of the refractive index and a corresponding element distribution inside the crystal, one has to know the diffusion-kinetic parameters and the equilibrium parameters of IE in the crystal—melt system. Our present study is devoted to finding thesesingle parameters for the JE in process 2~ Zn2~in crystal ZnSe plates, ZnSe Cd 0022-0248/91/803.50 © 1991
—
2. Manufacturing of samples and investigation methods For our experiments we have used plates grown from the vapour phase ZnSe single crystals with the zinc-blende (ZB) structure of (111) orientation. The samples underwent, in turn, mechanical, chemical-mechanical and chemical polishing (with Cr0 3—HC1 etchant). We have studied also the IE process in ZnSe fine-disperse (with submicron particles) powder used to obtain single crystals by sublimation. As a doping melt we have used a mixture of potassium, sodium, zinc and cadmium suiphates: Na2SO4—K2S04--ZnSO4—CdSO4. The active ingredients here are only ZnSO4 and CdSO4, while Na2SO4 and K2S04 are added to lower the melting point of mixture. The basic mixture ZnSO4(40.7 mol%)—K2S04(30.2 mol%)—Na2SO4 is a eutectic (T~t 384°C) [3]. We used the melts with a constant and CdSOsum of the molar fractions of ZnSO4 (m) 4 (n), i.e. m + n 0.407 and 0.002 < n =
Elsevier Science Publishers B.V. (North-Holland)
=
782
V.A. Fedorov et at.
/
Ion exchange in ZnSe crystals
<0.04. The exchange of a small quantity of ZnSO4 on CdSO4 does not change substantially the melting temperature of the system. The doping melt is a transparent liquid stable in air. The IE layers were formed by submerging the ZnSe plates or powders into the melt with a stabilized temperature (±1°C) in the range between 400 and 500°C. The rate of heating of the plates was 30°C/mn. The interplane spacings were measured by Xray diffraction method (Rigaku Denki diffractometer D-2, Cu Kct1 radiation). These data were used to determine the composition of the solid phase according to the composition—lattice parameter dependence constructed from Vegard’s law. The method of determining the crystal lattice parameter of the ZB-CdSe is described below. A Philips CM-30 was used to study the surface structure of IE layers by reflection high energy electron diffractometry (RHEED). The depth distribution of the elements was examined by means of Auger technique by successive ion beam sputtering of the surface (Riber LAS 2000). The rate of sputtering, obtained by using standard ZnSe epitaxial films on GaAs plates was 20 A/mm. Signal multiplication was used.
Fig. 1. RHEED pattern of surface of Cd,,Zn1 - ,~SeIE layer.
directly indicates that time the process occurs (from 0.1 to 50 within h), the the IE reaction CdSO4(l)
+
ZnSe(s)
~±
ZnSO4(l)
+
CdSe
(i)
3. Results and their discussion
proceeds to a state of the equilibrium and that the rate of IE diffusion into the crystal depth is much
3.1. Ion exchange equilibrium
lower than that of reaction (1). X-ray phase analysis has shown that within the
By examining the IE in the ZnSe crystal—melt system, one is able to find different thermodynamical characteristics, both of the IE itself and of the solid phase properties. According to the RHEED pattern, after IE the surface of the plates remains single crystalline (fig. 1). Sodium and potassium were not present in the ion-exchanged single crystals and powders as determined by Auger technique. There is therefore no significant effect of diffusion of these ions from the melt to form complex defect centres. It was found that the composition (u parameter) of the solid solution (SS) on the surface Cd~Zn1_~Seis independent of the time of processing and is uniquely determined by the temperature T of IE and the value of n/m. This
whole concentration range (0 ~ u ~ 1), the produced SS Cd5Zn~_~Sehas the ZB structure. In its general form, the equilibrium constant for (1) is: uy2mf1 = Kx K~’~i-, (2) K~= (1 u)y1nf2 s Y1f2 =
—
where u and (1 u) are the mole fractions of CdSe and ZnSe on the crystal surface, Yt and ~Y2 are the activity coefficients of ZnSe and CdSe, and f~and f2 are the activity coefficients of ZnSO4 and CdSO4 in the melt. Since the produced SS and ZnSe themselves have very different crystal lattice parameters, the process of SS formation should be accompanied by the appearance of elastic stresses and/or by —
VA. Fedoroc et a!.
/ Ion exchange in ZnSe
the partial or complete relaxation with creation of a system of misfit dislocations. In the general case, the total excess molar Gibbs energy of the strained surface layer of SS is [41: thS (3) g=geX±gsi~g
:: 0.8
and the activity coefficients of the SS components are:
I
02
ths, i=1,2, (4) 1=lny,~~(+lny,~t+lny~ where g~is the excess molar Gibbs energy of the unstrained SS with the given composition; g~°is the molar elastic energy of the strained surface layer of SS; and g~ is the molar energy of misfit dislocations.
lny
entire particle transforms to a homogeneous binary material. This is suggested by X-ray diffraction experiments. So the elastic and dislocation components of the energy and activity coefficients are zero, and the Gibbs energy for powder g~ g~ and ln y,J) in yex Here and below, the indices p and s refer to the powder and to the single crystal, respectively, As in the IE in single crystals, it was found that the u parameter of the composition of the powder SS is independent of the time of processing and is uniquely T and byofthe values, Fig. 2adetermined shows the by isotherms then/rn IE process (Cd2~~ Zn2~)~for different T in the region where a one-phase SS with ZB structure exists. From (2) for the apparent equilibrium constant we get:
/
7 /
/
I,,
-
0.0
6.0
3.1.1. In Ion order exchange to findin geX the unstrained and the corresponding solid phase
components of the activity coefficients of the SS compounds we have examined the IE in ZnSe powder. Its particles are very small (<1 /.tm) and the diffusion coefficients of the exchanges ions are large (see section 3.2). Therefore, by a sufficiently long treatment (> 50 h in our experiments), the
783
crystals
0.5
0.5
8.0
2.5
2:0
..&.fDz
m
~
0.0
0.2
0.4
0.6
0.8
2.0
zi(,iwt.f~-.)
Fig. 2. Interdependence of equilibrium compositions of the Cd~Zn 1_~Sesolid solution and of the melt: (a) ion exchange isotherm; (b) apparent equilibrium constant versus the cornposition of the formed solid solution: ( ) single crystal; — -—)powder; (1) T== 420°C;(2) T= 500°C.
=
=
ln K~° ln K~+ ln(’yF/yfl =
+
(5)
According to the symmetric way of normalization, the standard state is taken to be the state of the “pure” substance, i.e. I at u
=
1,
‘y~
=
1 at u
=
=
~
—
,
=
w
ln(f 2/f~).
=
In many cases, binary mixtures of solid and liquid compounds obey the laws of regular solutions [5,6], where the activity coefficients are related to concentrations as 2, ln ~2 (I u) 2 (7) in y1 -k~u where W is the interaction energy of the components. From (5) and (7), for the regular SS we find:
0.
(6)
in K~ ln K~+ ~k~(2u —1) + ln(f2/f1). =
(8)
As one can see from fig. 2a, the maximum change in the composition of the SS occurs at a very insignificant change of the mole fraction of CdSO4 (0.001
784
V.A. Fedorov ci a!.
/ Ion
exchange in ZnSe crystals
phase with the model of simple solutions [7], we have RT ln =
~
j
5,4
f, 2 + ~ ~rkr!(I~I
Wr
j
+
W~
—
~k)’
(9)
a
5,2
k
where r with formal indices i, j, k mark the mole fractions of the components (below we use t for Na 2SO4, / for K2S04, and m, n as they were already in the sum, j ± i,of kthe ± icornand k >j; W,,described); are the interaction energies ponents in a regular two-component melt. After several transformations of (9), for the 4-component solution Na 2S04—K2S04—ZnSO4— RT ln(f2/f1)=A+B(C—2n), (10)
1(lcd)
5,8 2.30
1.35
0.00
~
P~
80
b
7.0
CdSO4, we obtain: where A, B and C are constants:
6.0
A
=
t(~3
5.0
B
=
j~~”12,
C
=
n+m
—
=
W23) + l(w14 — W24),
(11) 0.407.
Here W12, W©© W23, W14 and W24 are the interaction energies for the solutions CdSO4—ZnSO4, Cd504—Na 2SO4~ZnSO4—Na 2S04, CdSO4—K 2~~4 and ZnSO4—K2SO4, respectively. The absence of data on the W12 value prevents one from describing ln(f2/f1). Let us note, however, that the value of 2n in our experiments is more than an order of magnitude smaller than that of C. Consequently, one can consider eq. (10) and the last term in (8) to be constant. Fig. 2b shows the experimental dependence of In K~ on u. One can see that the dependence is linear and, according to (8), the unstrained ~ obeys the model of a regular solution. From the dependence presented, one can easily find J4~and, up to ln(f2/f1), the equilibrium constant of reaction (1). Figs. 3a and 3b show the temperature dependences of the equilibrium constant and of the interaction energy for unstrained SS. One can 1 dependence is pracsee thatlinear the inand K~ can versus T from it, again up to ticaily find ln( f 2/f1), the enthalpy and entropy of reaction (I): L1H~ 20.7 kJ/mol, ~ 14.0 J/mol’ K. Let us note that the energy W~we have found is a fundamental thermodynamical characteristic =
=
/ /
/
670 720 T70 T (t~) of Fig. 3. Temperature dependence of the equilibrium constant the ion-exchange reaction (a) and of the interaction energy of formed Cd~Zn1.~Sesolid solution (b). Solid lines: (1) single crystal, (2) powder; dashed line: dependence calculated according to (33) and (34).
of a SS, which, for example, determines the state diagram of the system [8]. The above method of determining it at a temperatures below the melting point is, probably, the only one there is and is sufficiently accurate (about 3% to 5%). The theoretical values of W found in ref. [9] from the delta lattice parameter model [5] and the strain energy model [101 are equal to 9 and 8.5 kJ/mol, respectively, and are close to our experimental data. 3.1.2. Ion exchange in a strained crystal 3.1.2.1. The elastic component of the excess molar Gibbs energy. According to the linear elasticity
theory is [11], the molar elastic energy of a strained crystal gSt CiIki~IjEkiVm/2 VmEhk/(~, (12) =
=
where C~Jk,are the coefficients of elastic stiffness, are the components of the tensor of elastic deformations, Vm is the molar volume of the SS
V.A. Fedorov ci a!.
/ Ion
with the given composition, and is the elastic deformation in the plane of the sample. The formed layers are graded in depth (along the z-coordinate) and, according to ref. [12], =
~(z)
—
~J IL
a(z) dz,
)1
2
—
Parameter
(A)
2)
C~
=
.
_____________________________________________
a
where L is the thickness of the sample. Since the depth of the IE layer is much less than L, aboi(z) cJ(z). So account of the depth distribution of stresses has no effect upon the strained state of the surface SS. In the general case, depending on the orientation of the plate, Ehk/ has the form:
561 =
Table 1 Values of the material parameters for ZB-ZnSe and ZB-CdSe used in the calculations described in the text
_____________________
0
E
785
exchange in ZnSe crystals
[~( C~+ 2C44s~+ 31~Is
Value ZB-ZnSe 5.6684 ~ 8.59 8)
ZB-CdSe 6.070 6.59
5.06 h) 4.06 “~ 13.04 2.53 0.232
4.69 2.19 8.49 1.36 0.319
1(10’°N/m 2)
C
12 ((1010 N/m2) C~
E 2) N/rn G111 10i0 10N/m2) (10’°N/rn ~ 11, (10 ‘°
bI
Ref. [21]. Ref. [22].
for ZB-CdSe and the corresponding data taken from the literature are presented in table 1.
>< [CIIC2I + (C
11 + C12)C44S~ +
c~)HJ
(13)
1},
where
Q
=
With account of (14) and (15), the concentration dependence of E111 is practically linear: E111 E1(1 u) + E2u E1(1 + eu), (16) =
—
=
with
C11 + 2C12,
S
=
C11
—
C12
—
2C44,
e = (E2
—
Ei)/Er
H=n~n~n~, ~
and n1, n2, n3 are direction cosines that relate the direction normal to the interface of the plate. For (111) orientation we have: E111
=
6QC44/(C11 + 2C12 + 4C~).
(14)
-~
xl ‘I-’
The account for the dependence of the elastic stiffness coefficients of the SS composition was made in the form [13]: C, 1a1(1 C (u)=
‘
a1(1
—
u) +
—
u) + a2u
94
‘36
94
96
88
90
29(de9)
(15)
C,11, a1 and C,~2, a2 are elastic stiffness coefficients and crystal lattice parameters of the pure components of SS. Here and below, indices 1 and 2 refer respectively to ZnSe and CdSe. The coefficients C,1 for CdSe have been measured experimentally only for its wurtzite modification [14]. Their calculation for ZB-CdSe has been made according to ref. [15] from the requirement that recalculation C,~ (ZB-CdSe) to ~J (wurtzite CdSe) makes the latter to be the best least-squares fit to date [14]. The calculated data
-
—
98
130
~t9(de,q)
Fig. X-ray diffractograms Cd~Zni.~Se solidCdSO4 solutions 4.obtained at T 460°C ofin the melts with different (n) and ZnSO4 (rn) concentrations: (a) (333) plane; (b) (440)
plane; (1) n/m==l/100; (2) n/rn ==l/250.
786
V.A. Fedorov ci a!.
/
Ion exchange in ZnSe crystals
The crystal lattice parameter of the surface SS, a, and were found by the X-ray diffraction method. The diffractograms were recorded for two
SS with the same composition, and the deformations in the plane of the plate:
planes: for the plane (333) parallel to the plate surface and for the inclined plane (440) (fig. 4). Considering the lattice of the strained crystal to be (001)tetr), oneorientation finds the tetragonal and 1~11)cub adjusting its axes to the of the of plate values the ((tetragonal lattice parameters in the plane of the plate (aH) and in the perpendicular plane (a
a
1):
I
a~
11 + (1
1 (1
=
,
1/~
1/~
(18)
—2/~ \
1/~
0
1/~
1/~
1/~
3~/~d333, a1
=
where 2~(2C 2. F= 2S 44~+3S11), V= HS For a (111)-oriented plate this gives (C 11 + 2C12 2C44)/2(C11 + 2C12 =
where MT is the matrix of transformation to the tetragonal basis and (h, k, l)~is the plane indexation with respect to the standard crystallographic axes of the crystal. With account of (17) and (18) we have =
8/~!6(1/d~40 64/3a~) (19) —
~=(a2—a1)/a1.
=
—e12v/(1
—
c
~ by the Poisson
v),
where t’ is the Poisson coefficient, we obtain an equation from which one can determine the parameter of the crystal lattice for an unstrained
(23)
a=a1(l—u)+a2u=a1(1+~u).
~1=(a11—a)/a,
and
C44). (22)
only in a relatively narrow concentration range (u <0.38) [16]. This dependence is practically linear, which indicates that Vegard’s law holds, and can be extrapolated over the whole range of compositions of ZB-SS: with
1=(a1—a)/a,
+
It is known that most Il—VI solid solutions obey well Vegard’s law. However, data on the cor’°’osition—latticeparameter dependence for the ZB structure of the Cd~Zn1 ~Se S are available
By using the definition of deformations,
and the relation between effect,
(~~)
—1
,
MT111=~ -1/k
a1
7C12 + 2C~~)~
+3V(2C11+6C12+C44)+FI
MT~k
I
+
+V(2C11+10C17C44)+F] x[4C~(C11+C12) +SC44(5C11 + 11C12 + 2C44)~
(17)
(h\ =
v)(a~1— a~)
[4C~C12 + SC44(C11
tragonalbasis:
k
(20a)
P
—
(h, k, ‘)~is the plane indexation in the new te-
(h\
—
+
=
~2
~
—
(20b) 2va + (1 ~)a1 In the general form, the Poisson coefficient is
E
V
h~+k~
~
2va =
This extrapolated dependence practically coincided with the one we have plotted using Vegard’s law. In order to determine a7 by processing in the melt that contains no zinc ions (CdSO4—K2S04— Na2SO4), a layer of pure ZB-CdSe was formed on the surface, and by using (20) one found the lattice parameter of the cubic ZB-CdSe. This way it was found that a2 6.070 A (r~ 0.0708), which =
=
V.A. Fedorov et a!.
/ Ion exchange in ZnSe
is in good agreement with the data presented in ref. [17]: a2 6.077 A. Thus, considering the con=
centration dependence of stiffness coefficients (15) and the fact that Vegard’s law (23) holds, by solving jointly (15), (19), (20), (22) and (23), we found u of the surface Cd~Zn1 ~Se2’3 SS.~±Figs. 2a Zn2~ and 5 show the isotherms of the IE (Cd and the u dependence of elastic deformation for different temperatures. One can see that, within the error, the residual elastic deformation in the working temperature range is independent of T and has a linear dependence on the SS composi~
tion:
vector of MD, and zk is the magnitude of the relaxation: =
~coher
—
t~a/a
+
=
=
(~ ~)u. —
(26)
II
The contribution of dislocations to the total excess molar Gibbs energy is g~~’ = E~~Isp~V~,
(27)
where E’~ is the MD energy (per unit length) [19], Eds
=
GG 2 1b 2~(I v)(G+ G —
(24)
—ou.
=
787
crystals
Considering that =
N
3/4
=
0a
1) [1— ln(~c)], pdiS is the MD surface density [19], 2, pdlS 2(~/b) and G is the shear modulus, =
N
3/4
=
Vmi(1
+
~u)3,
0a~(I+ ~u)
from (12), (16) and (24) we obtain the following
G 11,
formula for the elastic component of the molar excess Gibbs energy of the strained surface layer of the Cd~Zn1~SeSS: t Vmi(1 + ~u)3E 2. (25) g~ 1(l + eu)E~u =
=
(C1,
C12 + c44)/3.
—
The concentration dependence of G111 is practically linear: G 111
=
G1(1
—
u)
+
G2u
=
G1(1
+ gu),
(28)
with In the case of complete coherence between the formed IE layer and the ZnSe substrate, one has COl i~. However, from the data presented (fig. 5) one can see that ~ <<1~, owing to the relaxation of elastic deformations via production of misfit dislocations (MD). The linear density of MD is p’~ i./b [18], where b is the edge component of the Burgers e,, /i~) 3.1.2.2. Dis/ocational component.
=
g
=
(G2 — G1)/G1.
By substitution this into (27), we find the final result for the dislocation contribution: dis
g5
— —
=
G1(l + gu)(ij o)2 _________________ ir(1 v)(2 + gu) —
—
><
(1
—
ln[(~
—
Co)uj
2Vm.
(29)
}U
-
Thus, from (3), (25) and (29), we find the total excess 3.1.2.3. Total excess molar Gibbs energy.
of the Cd~Zn 1.~Se SS: ~
molar Gibbs energy2Vmi(1 of the +~u)3(El(I strained surface +eu)e~ layer g~=W~(l—u)u+u
.3.
+ 0
9.2
0.4
3.6
0.8
3.0
~
Fig. 5. Concentration dependence of the residual elastic deformation in the plane of the plate: (so) T= 420°C; (+) T~
~(1 X
{l
—
—
v)(2
ln[(1 1
460°C; (S) T=500°C. s~=~0.46%.
—
+ gu)
~)U] } J.
(30)
788
V.A. Fedorov Pt a!.
/
Ion exchange in ZnSe crystals
In a manner similar to (5), for the apparent equilibrium constant of the IE reaction in ZnSe single crystal, we obtain
also close to linear (fig. 6). This allows one to determine the value of the interaction energy of the IE surface SS by transforming the dependences (33) in the (34) view by least-squares
ln K
method. Fig. 3b shows the temperature dependences of J4~of a regular IE surface SS found experimentally from the data of fig. 2b and calculated according to (33) and (34). These dependences are very close: the maximum deviation is 4%, which is of the same magnitude as the error in W (about 3% to 5%). 1 depenThe experimental in K~ versus T dence is linear (fig. 3a) and, up to ln(f 2/f1), the enthalpy and entropy of IE reaction (1) for single crystal are 20.0 kJ/mol and ~ 16.2
In K5’
=
+
In y~ in —
‘v~•
(31)
By using RT ln y~= g5 — u(Bg5/Bu), RT in y~= g5 + (1 — u)(Bg5/Bu),
with account of the symmetric normalization (6), we have RT~IIn y RT(ln y~ In .,~) =
—
8g =
g5(u
=
1)
—
(
(32)
5/Bu).
From (31) and (32), we find the theoretical dependence In K~’:
‘
=
=
J/mol K. The difference between the experimental and the calculated values of the equilibrium constant is constant for all T, ~ In K~’ In K~’~ , leo — ln K5 0.15 (fig. 6), and can be attributed to the difference in the standard chemical potentials of the SS components for a single crystal and =
in K In K~+ ~ In Y. (33) Fig. 2b shows the experimental dependence In K~ versus u. This dependence and the calculated dependences according to (33) and the data of table I are presented in fig. 6 for several temperatures. The experimental dependences are linear and, in a manner similar to (8), can be presented in the form =
In K~’ in K~’+ (2u =
—
1) W5/RT,
(34)
where J4’~is the interaction energy of a regular IE surface SS. The calculated dependences (33) are
:
(,~Xx 6.0
V
=
a powder, which is not accounted for by this model. 3.2. Jon exchange diffusion
As a result of the reaction (1), on the surface of the crystal there forms the Cd,,Zn1.~Se SS and inside the plate there appears a gradient of the chemical potential, which is a motive force for IE diffusion (lED) of zinc and cadmium inside the crystal. The equation that describes the lED process
~=~(D12~), has the form [20]:
/
Bu B where
//
+
4.0
0.0
0.2
0.4
0.6
0.8
1.0
D12=
Fig. 6. Concentration dependence of the apparent equilibrium constant: (—) experimental dependence; (— — —) dependence calculated according (2) T= 500°C. to (33): (1) T = 420°C:
=
Bu\
D1D2[1+u(1—u)P] D1(1—u)+D2u
B In
U~OW!.Jr.)
(35)
~
B in
B(1—u)~Bu
~2
,
(36a)
(36b)
2~ D1 and and Cd21D2ions, are respectively, self-diffusionincoefficients ZnSe and of CdSe. Zn
~( V.A. Fedorov ci a!.
1.0 1.8
0.6
u(m~o~fr) .4
0.2
/
0.0 .0
0.2
0.2
0.3
0.1,
789
Ion exchange in ZnSe crystals
Z (‘~~‘~
mental and calculated profiles.profiles The fact coincide that for the different experimental sampies at a constant temperature and the same values of self-diffusion coefficients is an additional the solutions that were the closest to the experiindication to the validity of the model we have described. We have also analysed the depth distributions of the elements in the layers obtained in the same melt but at different temperatures. In a manner similar to that described above, by solving (37) and comparing the results obtained with the cxperimental profiles we found the values of D1 and D2. The results obtained (fig. 7b) have enabled us
~ (c.’n¼) 2 io~’
dence of the self-diffusion coefficients of Zn2’ and Cd2’3 ions in the form: to find the parameters of the temperature depenD D 0 exp(— Q/kT). 2~we got D 2/s, Q 1.54 eV, For Zn 0 0.040 cm and for Cd2~,D 2/s, Q 1.73 eV. 0 0.058 cm =
=
=
=
=
1.35
1.sJ
1.45
2.50
T4
(jc~,~
Fig. 7. Characteristics of the diffusion stage of the process. (a) Distribution profiles of cadmium (A) and zinc (B) in the samples obtained in melts with different concentration of the zinc and cadmium sulphates by 3 h processing at 420°C: (1) n/rn = 1/200; (2) n/rn = 1/10. (b) Temperature dependence of self-diffusion coefficients of cadmium (A) and zinc (B) ions.
Acknowledgments The authors would like to thank Dr.V.Sh. Ivanov for help with the Auger measurements and E.M. Kirov for many useful discussions.
References From (32)—(34), we find P 214’ç/RT, and the equation of lED in the case where a regular ~ forms reads as: =
—
[1] A.O. Aleksanyan, V.A. Ganshin, Yu.N. Korkishko and [2]
Bu
B / D 1D2 [i — 2W~u(1 u)/RT] Bu) D1(l — r~)+ D2u —
Bt
V.Z. Petrova, Zh. Tekh. Fiz. 59 (1989) 351. A.O. Aleksanyan, V.A. Ganshin, Yu.N. Korkishko and
=
Bz
.
(37)
We have examined the profiles of the element distribution in Cd : ZnSe layers obtained in melts with different n and m values (fig. 7a). It was assumed that the magnitude of the Auger signal is proportional to the element concentration. Eq. (37) was solved by a finite-difference technique for different D1 and D2. The concentration on the surface was considered constant and equal to the value measured experimentally. We selected
V.A. Fedorov, Phys. Status Solidi (b) 161 (1990) 629. [3J N.V. Hahlova and N.S. Dornbrovskaya, Zh. Neorg. Khim 4 (1959) 920. 14] P.K. Bhattacharya and S. Srinivasa, J. AppI. Phys. 54 (1983) 5090. [5] [6] [7] [8] [9] [10J [11] [12]
GB. Stringfellow, J. Crystal Growth 27 (1974) 21. A. Laugier, Rev. Physique AppI. 8 (1973) 259. A.S. Jordan, J. Electrochem. Soc 119 (1972) 123. V.A. Fedorov, V.A. Ganshin and Yu.N. Korkishko, to be published. D.W. K.isker. J. Crystal Growth 98 (1989) 127. P.A. Fedders and MW. Muller, J. Phys. Chem. Solids 45 (1984) 685. L D Landau and EM. Lifshits, Teoriya Uprugosti (Nauka, Moscow, 1965). A. Brandenburg. J. Lightwave Technol. 14 (1986) 1580.
790
V.A. Fedorov et at.
/
Ion exchange in ZnSe crystals
[13] A.K. Gin and GB. Mitra, Phys. Status Solidi (b) 134 (1986) KIl. [14] D. Berlincourt, H. Jaffe and L.R. Shiozawa, Phys. Rev. 129 (1963) 1009. {15J R.M. Martin, Phys. Rev. B6 (1972) 4546. [16] K.V. Shalimova, A.F. Botnev, V.A. Dmitriev. N.Z. Kognovitskaya and V.V. Starostin. Kristallografiya 14 (1969) 629. [17] N. Samarth, H. Luo, J.K. Furdyna, S.B. Qadri, YR. Lee, AK. Ramdas and N. Otsuka, AppI. Phys. Letters 54 (1989) 2680.
[181 S.N.G. Chu. A.T. Macrander. K.E. Strege and W.D. Johnstori, Jr., J. AppI. Phys. 57 (1985) 249. [19] J.W. Matthews, in: Epitaxial Growth, Part B, Ed. J.W. Matthews (Academic Press, New York, 1975) ch. 8. [20] R.M. Barrer, R.F. Bartholomew and L.V.C. Rees, J. Phys. Chem. Solids 24 (1963) 309. [21] S. Larch. RE. Schrader and CF. Stocker. Phys. Rev. 108 (1957) 587. [22] B.H. Lee. J. AppI. Phys. 41(1970) 2984.
781
Ion exchange in zinc selenide crystals: the role of lattice-misfit-induced stresses and dislocations V.A. Fedorov, V.A. Ganshin and Yu.N. Korkishko Chair of Chemistry, Moscow Institute of Electronic Technology, SU-103498 Moscow, USSR
Received 2 January 1991; manuscript received in final form 15 March 1991
A new method has been applied to form Cd ,,Zn 1 — ~Se solid ion single exchange in plates the melts ofbeen salts.studied. The regularities 2~~ Zn2~in ZnSesolutions: powder and crystal have Cd,,Zn and specific features of ion exchange processes Cd 1_~Se solid solution obeys the model of regular solution. Temperature dependences of equilibrium constant and interaction energy of strained and unstrained Cd~Zn1~Sesolid solutions have been defined. We proposed the model taking into account the role of lattice-misfit-induced stresses and dislocations in the ion exchange equilibrium. The depth profiles of elements in 2the ± and ion Cd2 exchanged ± ions Cd : ZnSe were determined. single crystal structures were analysed and temperature dependences of self-diffusion coefficients of Zn
1. Introduction
powder, and to examination of the regularities and specific features of this process.
Earlier we have reported on the possibility of ion exchange diffusion doping (IEDD) of Il—VI single crystal ZnSe, ZnS, CdS and CdTe by divalent ions (Cd2~, Hg2~, Cu21, Pb2~, Co2~,Ni2~and Mn2~)in melts and solutions of inorganic salts at relatively low temperature [1,2]. The IEDD is a two-stage process and involves: (i) the heterogeneous ion exchange (IE) reaction Me2~+AX ~ A2~+MeX on the melt—crystal boundary and (ii) the simultaneous IE diffusion into the crystal. The IEDD results in the forma—
—
tion of graded surface structures, which, in particular, can be used in integrated optoelectronics [1,2]. To be able to use the IEDD method for processing Il—VI single crystals for manufacturing the optoelectronic elements with a desired depth distribution of the refractive index and a corresponding element distribution inside the crystal, one has to know the diffusion-kinetic parameters and the equilibrium parameters of IE in the crystal—melt system. Our present study is devoted to finding thesesingle parameters for the JE in process 2~ Zn2~in crystal ZnSe plates, ZnSe Cd 0022-0248/91/803.50 © 1991
—
2. Manufacturing of samples and investigation methods For our experiments we have used plates grown from the vapour phase ZnSe single crystals with the zinc-blende (ZB) structure of (111) orientation. The samples underwent, in turn, mechanical, chemical-mechanical and chemical polishing (with Cr0 3—HC1 etchant). We have studied also the IE process in ZnSe fine-disperse (with submicron particles) powder used to obtain single crystals by sublimation. As a doping melt we have used a mixture of potassium, sodium, zinc and cadmium suiphates: Na2SO4—K2S04--ZnSO4—CdSO4. The active ingredients here are only ZnSO4 and CdSO4, while Na2SO4 and K2S04 are added to lower the melting point of mixture. The basic mixture ZnSO4(40.7 mol%)—K2S04(30.2 mol%)—Na2SO4 is a eutectic (T~t 384°C) [3]. We used the melts with a constant and CdSOsum of the molar fractions of ZnSO4 (m) 4 (n), i.e. m + n 0.407 and 0.002 < n =
Elsevier Science Publishers B.V. (North-Holland)
=
782
V.A. Fedorov et at.
/
Ion exchange in ZnSe crystals
<0.04. The exchange of a small quantity of ZnSO4 on CdSO4 does not change substantially the melting temperature of the system. The doping melt is a transparent liquid stable in air. The IE layers were formed by submerging the ZnSe plates or powders into the melt with a stabilized temperature (±1°C) in the range between 400 and 500°C. The rate of heating of the plates was 30°C/mn. The interplane spacings were measured by Xray diffraction method (Rigaku Denki diffractometer D-2, Cu Kct1 radiation). These data were used to determine the composition of the solid phase according to the composition—lattice parameter dependence constructed from Vegard’s law. The method of determining the crystal lattice parameter of the ZB-CdSe is described below. A Philips CM-30 was used to study the surface structure of IE layers by reflection high energy electron diffractometry (RHEED). The depth distribution of the elements was examined by means of Auger technique by successive ion beam sputtering of the surface (Riber LAS 2000). The rate of sputtering, obtained by using standard ZnSe epitaxial films on GaAs plates was 20 A/mm. Signal multiplication was used.
Fig. 1. RHEED pattern of surface of Cd,,Zn1 - ,~SeIE layer.
directly indicates that time the process occurs (from 0.1 to 50 within h), the the IE reaction CdSO4(l)
+
ZnSe(s)
~±
ZnSO4(l)
+
CdSe
(i)
3. Results and their discussion
proceeds to a state of the equilibrium and that the rate of IE diffusion into the crystal depth is much
3.1. Ion exchange equilibrium
lower than that of reaction (1). X-ray phase analysis has shown that within the
By examining the IE in the ZnSe crystal—melt system, one is able to find different thermodynamical characteristics, both of the IE itself and of the solid phase properties. According to the RHEED pattern, after IE the surface of the plates remains single crystalline (fig. 1). Sodium and potassium were not present in the ion-exchanged single crystals and powders as determined by Auger technique. There is therefore no significant effect of diffusion of these ions from the melt to form complex defect centres. It was found that the composition (u parameter) of the solid solution (SS) on the surface Cd~Zn1_~Seis independent of the time of processing and is uniquely determined by the temperature T of IE and the value of n/m. This
whole concentration range (0 ~ u ~ 1), the produced SS Cd5Zn~_~Sehas the ZB structure. In its general form, the equilibrium constant for (1) is: uy2mf1 = Kx K~’~i-, (2) K~= (1 u)y1nf2 s Y1f2 =
—
where u and (1 u) are the mole fractions of CdSe and ZnSe on the crystal surface, Yt and ~Y2 are the activity coefficients of ZnSe and CdSe, and f~and f2 are the activity coefficients of ZnSO4 and CdSO4 in the melt. Since the produced SS and ZnSe themselves have very different crystal lattice parameters, the process of SS formation should be accompanied by the appearance of elastic stresses and/or by —
VA. Fedoroc et a!.
/ Ion exchange in ZnSe
the partial or complete relaxation with creation of a system of misfit dislocations. In the general case, the total excess molar Gibbs energy of the strained surface layer of SS is [41: thS (3) g=geX±gsi~g
:: 0.8
and the activity coefficients of the SS components are:
I
02
ths, i=1,2, (4) 1=lny,~~(+lny,~t+lny~ where g~is the excess molar Gibbs energy of the unstrained SS with the given composition; g~°is the molar elastic energy of the strained surface layer of SS; and g~ is the molar energy of misfit dislocations.
lny
entire particle transforms to a homogeneous binary material. This is suggested by X-ray diffraction experiments. So the elastic and dislocation components of the energy and activity coefficients are zero, and the Gibbs energy for powder g~ g~ and ln y,J) in yex Here and below, the indices p and s refer to the powder and to the single crystal, respectively, As in the IE in single crystals, it was found that the u parameter of the composition of the powder SS is independent of the time of processing and is uniquely T and byofthe values, Fig. 2adetermined shows the by isotherms then/rn IE process (Cd2~~ Zn2~)~for different T in the region where a one-phase SS with ZB structure exists. From (2) for the apparent equilibrium constant we get:
/
7 /
/
I,,
-
0.0
6.0
3.1.1. In Ion order exchange to findin geX the unstrained and the corresponding solid phase
components of the activity coefficients of the SS compounds we have examined the IE in ZnSe powder. Its particles are very small (<1 /.tm) and the diffusion coefficients of the exchanges ions are large (see section 3.2). Therefore, by a sufficiently long treatment (> 50 h in our experiments), the
783
crystals
0.5
0.5
8.0
2.5
2:0
..&.fDz
m
~
0.0
0.2
0.4
0.6
0.8
2.0
zi(,iwt.f~-.)
Fig. 2. Interdependence of equilibrium compositions of the Cd~Zn 1_~Sesolid solution and of the melt: (a) ion exchange isotherm; (b) apparent equilibrium constant versus the cornposition of the formed solid solution: ( ) single crystal; — -—)powder; (1) T== 420°C;(2) T= 500°C.
=
=
ln K~° ln K~+ ln(’yF/yfl =
+
(5)
According to the symmetric way of normalization, the standard state is taken to be the state of the “pure” substance, i.e. I at u
=
1,
‘y~
=
1 at u
=
=
~
—
,
=
w
ln(f 2/f~).
=
In many cases, binary mixtures of solid and liquid compounds obey the laws of regular solutions [5,6], where the activity coefficients are related to concentrations as 2, ln ~2 (I u) 2 (7) in y1 -k~u where W is the interaction energy of the components. From (5) and (7), for the regular SS we find:
0.
(6)
in K~ ln K~+ ~k~(2u —1) + ln(f2/f1). =
(8)
As one can see from fig. 2a, the maximum change in the composition of the SS occurs at a very insignificant change of the mole fraction of CdSO4 (0.001
784
V.A. Fedorov ci a!.
/ Ion
exchange in ZnSe crystals
phase with the model of simple solutions [7], we have RT ln =
~
j
5,4
f, 2 + ~ ~rkr!(I~I
Wr
j
+
W~
—
~k)’
(9)
a
5,2
k
where r with formal indices i, j, k mark the mole fractions of the components (below we use t for Na 2SO4, / for K2S04, and m, n as they were already in the sum, j ± i,of kthe ± icornand k >j; W,,described); are the interaction energies ponents in a regular two-component melt. After several transformations of (9), for the 4-component solution Na 2S04—K2S04—ZnSO4— RT ln(f2/f1)=A+B(C—2n), (10)
1(lcd)
5,8 2.30
1.35
0.00
~
P~
80
b
7.0
CdSO4, we obtain: where A, B and C are constants:
6.0
A
=
t(~3
5.0
B
=
j~~”12,
C
=
n+m
—
=
W23) + l(w14 — W24),
(11) 0.407.
Here W12, W©© W23, W14 and W24 are the interaction energies for the solutions CdSO4—ZnSO4, Cd504—Na 2SO4~ZnSO4—Na 2S04, CdSO4—K 2~~4 and ZnSO4—K2SO4, respectively. The absence of data on the W12 value prevents one from describing ln(f2/f1). Let us note, however, that the value of 2n in our experiments is more than an order of magnitude smaller than that of C. Consequently, one can consider eq. (10) and the last term in (8) to be constant. Fig. 2b shows the experimental dependence of In K~ on u. One can see that the dependence is linear and, according to (8), the unstrained ~ obeys the model of a regular solution. From the dependence presented, one can easily find J4~and, up to ln(f2/f1), the equilibrium constant of reaction (1). Figs. 3a and 3b show the temperature dependences of the equilibrium constant and of the interaction energy for unstrained SS. One can 1 dependence is pracsee thatlinear the inand K~ can versus T from it, again up to ticaily find ln( f 2/f1), the enthalpy and entropy of reaction (I): L1H~ 20.7 kJ/mol, ~ 14.0 J/mol’ K. Let us note that the energy W~we have found is a fundamental thermodynamical characteristic =
=
/ /
/
670 720 T70 T (t~) of Fig. 3. Temperature dependence of the equilibrium constant the ion-exchange reaction (a) and of the interaction energy of formed Cd~Zn1.~Sesolid solution (b). Solid lines: (1) single crystal, (2) powder; dashed line: dependence calculated according to (33) and (34).
of a SS, which, for example, determines the state diagram of the system [8]. The above method of determining it at a temperatures below the melting point is, probably, the only one there is and is sufficiently accurate (about 3% to 5%). The theoretical values of W found in ref. [9] from the delta lattice parameter model [5] and the strain energy model [101 are equal to 9 and 8.5 kJ/mol, respectively, and are close to our experimental data. 3.1.2. Ion exchange in a strained crystal 3.1.2.1. The elastic component of the excess molar Gibbs energy. According to the linear elasticity
theory is [11], the molar elastic energy of a strained crystal gSt CiIki~IjEkiVm/2 VmEhk/(~, (12) =
=
where C~Jk,are the coefficients of elastic stiffness, are the components of the tensor of elastic deformations, Vm is the molar volume of the SS
V.A. Fedorov ci a!.
/ Ion
with the given composition, and is the elastic deformation in the plane of the sample. The formed layers are graded in depth (along the z-coordinate) and, according to ref. [12], =
~(z)
—
~J IL
a(z) dz,
)1
2
—
Parameter
(A)
2)
C~
=
.
_____________________________________________
a
where L is the thickness of the sample. Since the depth of the IE layer is much less than L, aboi(z) cJ(z). So account of the depth distribution of stresses has no effect upon the strained state of the surface SS. In the general case, depending on the orientation of the plate, Ehk/ has the form:
561 =
Table 1 Values of the material parameters for ZB-ZnSe and ZB-CdSe used in the calculations described in the text
_____________________
0
E
785
exchange in ZnSe crystals
[~( C~+ 2C44s~+ 31~Is
Value ZB-ZnSe 5.6684 ~ 8.59 8)
ZB-CdSe 6.070 6.59
5.06 h) 4.06 “~ 13.04 2.53 0.232
4.69 2.19 8.49 1.36 0.319
1(10’°N/m 2)
C
12 ((1010 N/m2) C~
E 2) N/rn G111 10i0 10N/m2) (10’°N/rn ~ 11, (10 ‘°
bI
Ref. [21]. Ref. [22].
for ZB-CdSe and the corresponding data taken from the literature are presented in table 1.
>< [CIIC2I + (C
11 + C12)C44S~ +
c~)HJ
(13)
1},
where
Q
=
With account of (14) and (15), the concentration dependence of E111 is practically linear: E111 E1(1 u) + E2u E1(1 + eu), (16) =
—
=
with
C11 + 2C12,
S
=
C11
—
C12
—
2C44,
e = (E2
—
Ei)/Er
H=n~n~n~, ~
and n1, n2, n3 are direction cosines that relate the direction normal to the interface of the plate. For (111) orientation we have: E111
=
6QC44/(C11 + 2C12 + 4C~).
(14)
-~
xl ‘I-’
The account for the dependence of the elastic stiffness coefficients of the SS composition was made in the form [13]: C, 1a1(1 C (u)=
‘
a1(1
—
u) +
—
u) + a2u
94
‘36
94
96
88
90
29(de9)
(15)
C,11, a1 and C,~2, a2 are elastic stiffness coefficients and crystal lattice parameters of the pure components of SS. Here and below, indices 1 and 2 refer respectively to ZnSe and CdSe. The coefficients C,1 for CdSe have been measured experimentally only for its wurtzite modification [14]. Their calculation for ZB-CdSe has been made according to ref. [15] from the requirement that recalculation C,~ (ZB-CdSe) to ~J (wurtzite CdSe) makes the latter to be the best least-squares fit to date [14]. The calculated data
-
—
98
130
~t9(de,q)
Fig. X-ray diffractograms Cd~Zni.~Se solidCdSO4 solutions 4.obtained at T 460°C ofin the melts with different (n) and ZnSO4 (rn) concentrations: (a) (333) plane; (b) (440)
plane; (1) n/m==l/100; (2) n/rn ==l/250.
786
V.A. Fedorov ci a!.
/
Ion exchange in ZnSe crystals
The crystal lattice parameter of the surface SS, a, and were found by the X-ray diffraction method. The diffractograms were recorded for two
SS with the same composition, and the deformations in the plane of the plate:
planes: for the plane (333) parallel to the plate surface and for the inclined plane (440) (fig. 4). Considering the lattice of the strained crystal to be (001)tetr), oneorientation finds the tetragonal and 1~11)cub adjusting its axes to the of the of plate values the ((tetragonal lattice parameters in the plane of the plate (aH) and in the perpendicular plane (a
a
1):
I
a~
11 + (1
1 (1
=
,
1/~
1/~
(18)
—2/~ \
1/~
0
1/~
1/~
1/~
3~/~d333, a1
=
where 2~(2C 2. F= 2S 44~+3S11), V= HS For a (111)-oriented plate this gives (C 11 + 2C12 2C44)/2(C11 + 2C12 =
where MT is the matrix of transformation to the tetragonal basis and (h, k, l)~is the plane indexation with respect to the standard crystallographic axes of the crystal. With account of (17) and (18) we have =
8/~!6(1/d~40 64/3a~) (19) —
~=(a2—a1)/a1.
=
—e12v/(1
—
c
~ by the Poisson
v),
where t’ is the Poisson coefficient, we obtain an equation from which one can determine the parameter of the crystal lattice for an unstrained
(23)
a=a1(l—u)+a2u=a1(1+~u).
~1=(a11—a)/a,
and
C44). (22)
only in a relatively narrow concentration range (u <0.38) [16]. This dependence is practically linear, which indicates that Vegard’s law holds, and can be extrapolated over the whole range of compositions of ZB-SS: with
1=(a1—a)/a,
+
It is known that most Il—VI solid solutions obey well Vegard’s law. However, data on the cor’°’osition—latticeparameter dependence for the ZB structure of the Cd~Zn1 ~Se S are available
By using the definition of deformations,
and the relation between effect,
(~~)
—1
,
MT111=~ -1/k
a1
7C12 + 2C~~)~
+3V(2C11+6C12+C44)+FI
MT~k
I
+
+V(2C11+10C17C44)+F] x[4C~(C11+C12) +SC44(5C11 + 11C12 + 2C44)~
(17)
(h\ =
v)(a~1— a~)
[4C~C12 + SC44(C11
tragonalbasis:
k
(20a)
P
—
(h, k, ‘)~is the plane indexation in the new te-
(h\
—
+
=
~2
~
—
(20b) 2va + (1 ~)a1 In the general form, the Poisson coefficient is
E
V
h~+k~
~
2va =
This extrapolated dependence practically coincided with the one we have plotted using Vegard’s law. In order to determine a7 by processing in the melt that contains no zinc ions (CdSO4—K2S04— Na2SO4), a layer of pure ZB-CdSe was formed on the surface, and by using (20) one found the lattice parameter of the cubic ZB-CdSe. This way it was found that a2 6.070 A (r~ 0.0708), which =
=
V.A. Fedorov et a!.
/ Ion exchange in ZnSe
is in good agreement with the data presented in ref. [17]: a2 6.077 A. Thus, considering the con=
centration dependence of stiffness coefficients (15) and the fact that Vegard’s law (23) holds, by solving jointly (15), (19), (20), (22) and (23), we found u of the surface Cd~Zn1 ~Se2’3 SS.~±Figs. 2a Zn2~ and 5 show the isotherms of the IE (Cd and the u dependence of elastic deformation for different temperatures. One can see that, within the error, the residual elastic deformation in the working temperature range is independent of T and has a linear dependence on the SS composi~
tion:
vector of MD, and zk is the magnitude of the relaxation: =
~coher
—
t~a/a
+
=
=
(~ ~)u. —
(26)
II
The contribution of dislocations to the total excess molar Gibbs energy is g~~’ = E~~Isp~V~,
(27)
where E’~ is the MD energy (per unit length) [19], Eds
=
GG 2 1b 2~(I v)(G+ G —
(24)
—ou.
=
787
crystals
Considering that =
N
3/4
=
0a
1) [1— ln(~c)], pdiS is the MD surface density [19], 2, pdlS 2(~/b) and G is the shear modulus, =
N
3/4
=
Vmi(1
+
~u)3,
0a~(I+ ~u)
from (12), (16) and (24) we obtain the following
G 11,
formula for the elastic component of the molar excess Gibbs energy of the strained surface layer of the Cd~Zn1~SeSS: t Vmi(1 + ~u)3E 2. (25) g~ 1(l + eu)E~u =
=
(C1,
C12 + c44)/3.
—
The concentration dependence of G111 is practically linear: G 111
=
G1(1
—
u)
+
G2u
=
G1(1
+ gu),
(28)
with In the case of complete coherence between the formed IE layer and the ZnSe substrate, one has COl i~. However, from the data presented (fig. 5) one can see that ~ <<1~, owing to the relaxation of elastic deformations via production of misfit dislocations (MD). The linear density of MD is p’~ i./b [18], where b is the edge component of the Burgers e,, /i~) 3.1.2.2. Dis/ocational component.
=
g
=
(G2 — G1)/G1.
By substitution this into (27), we find the final result for the dislocation contribution: dis
g5
— —
=
G1(l + gu)(ij o)2 _________________ ir(1 v)(2 + gu) —
—
><
(1
—
ln[(~
—
Co)uj
2Vm.
(29)
}U
-
Thus, from (3), (25) and (29), we find the total excess 3.1.2.3. Total excess molar Gibbs energy.
of the Cd~Zn 1.~Se SS: ~
molar Gibbs energy2Vmi(1 of the +~u)3(El(I strained surface +eu)e~ layer g~=W~(l—u)u+u
.3.
+ 0
9.2
0.4
3.6
0.8
3.0
~
Fig. 5. Concentration dependence of the residual elastic deformation in the plane of the plate: (so) T= 420°C; (+) T~
~(1 X
{l
—
—
v)(2
ln[(1 1
460°C; (S) T=500°C. s~=~0.46%.
—
+ gu)
~)U] } J.
(30)
788
V.A. Fedorov Pt a!.
/
Ion exchange in ZnSe crystals
In a manner similar to (5), for the apparent equilibrium constant of the IE reaction in ZnSe single crystal, we obtain
also close to linear (fig. 6). This allows one to determine the value of the interaction energy of the IE surface SS by transforming the dependences (33) in the (34) view by least-squares
ln K
method. Fig. 3b shows the temperature dependences of J4~of a regular IE surface SS found experimentally from the data of fig. 2b and calculated according to (33) and (34). These dependences are very close: the maximum deviation is 4%, which is of the same magnitude as the error in W (about 3% to 5%). 1 depenThe experimental in K~ versus T dence is linear (fig. 3a) and, up to ln(f 2/f1), the enthalpy and entropy of IE reaction (1) for single crystal are 20.0 kJ/mol and ~ 16.2
In K5’
=
+
In y~ in —
‘v~•
(31)
By using RT ln y~= g5 — u(Bg5/Bu), RT in y~= g5 + (1 — u)(Bg5/Bu),
with account of the symmetric normalization (6), we have RT~IIn y RT(ln y~ In .,~) =
—
8g =
g5(u
=
1)
—
(
(32)
5/Bu).
From (31) and (32), we find the theoretical dependence In K~’:
‘
=
=
J/mol K. The difference between the experimental and the calculated values of the equilibrium constant is constant for all T, ~ In K~’ In K~’~ , leo — ln K5 0.15 (fig. 6), and can be attributed to the difference in the standard chemical potentials of the SS components for a single crystal and =
in K In K~+ ~ In Y. (33) Fig. 2b shows the experimental dependence In K~ versus u. This dependence and the calculated dependences according to (33) and the data of table I are presented in fig. 6 for several temperatures. The experimental dependences are linear and, in a manner similar to (8), can be presented in the form =
In K~’ in K~’+ (2u =
—
1) W5/RT,
(34)
where J4’~is the interaction energy of a regular IE surface SS. The calculated dependences (33) are
:
(,~Xx 6.0
V
=
a powder, which is not accounted for by this model. 3.2. Jon exchange diffusion
As a result of the reaction (1), on the surface of the crystal there forms the Cd,,Zn1.~Se SS and inside the plate there appears a gradient of the chemical potential, which is a motive force for IE diffusion (lED) of zinc and cadmium inside the crystal. The equation that describes the lED process
~=~(D12~), has the form [20]:
/
Bu B where
//
+
4.0
0.0
0.2
0.4
0.6
0.8
1.0
D12=
Fig. 6. Concentration dependence of the apparent equilibrium constant: (—) experimental dependence; (— — —) dependence calculated according (2) T= 500°C. to (33): (1) T = 420°C:
=
Bu\
D1D2[1+u(1—u)P] D1(1—u)+D2u
B In
U~OW!.Jr.)
(35)
~
B in
B(1—u)~Bu
~2
,
(36a)
(36b)
2~ D1 and and Cd21D2ions, are respectively, self-diffusionincoefficients ZnSe and of CdSe. Zn
~( V.A. Fedorov ci a!.
1.0 1.8
0.6
u(m~o~fr) .4
0.2
/
0.0 .0
0.2
0.2
0.3
0.1,
789
Ion exchange in ZnSe crystals
Z (‘~~‘~
mental and calculated profiles.profiles The fact coincide that for the different experimental sampies at a constant temperature and the same values of self-diffusion coefficients is an additional the solutions that were the closest to the experiindication to the validity of the model we have described. We have also analysed the depth distributions of the elements in the layers obtained in the same melt but at different temperatures. In a manner similar to that described above, by solving (37) and comparing the results obtained with the cxperimental profiles we found the values of D1 and D2. The results obtained (fig. 7b) have enabled us
~ (c.’n¼) 2 io~’
dence of the self-diffusion coefficients of Zn2’ and Cd2’3 ions in the form: to find the parameters of the temperature depenD D 0 exp(— Q/kT). 2~we got D 2/s, Q 1.54 eV, For Zn 0 0.040 cm and for Cd2~,D 2/s, Q 1.73 eV. 0 0.058 cm =
=
=
=
=
1.35
1.sJ
1.45
2.50
T4
(jc~,~
Fig. 7. Characteristics of the diffusion stage of the process. (a) Distribution profiles of cadmium (A) and zinc (B) in the samples obtained in melts with different concentration of the zinc and cadmium sulphates by 3 h processing at 420°C: (1) n/rn = 1/200; (2) n/rn = 1/10. (b) Temperature dependence of self-diffusion coefficients of cadmium (A) and zinc (B) ions.
Acknowledgments The authors would like to thank Dr.V.Sh. Ivanov for help with the Auger measurements and E.M. Kirov for many useful discussions.
References From (32)—(34), we find P 214’ç/RT, and the equation of lED in the case where a regular ~ forms reads as: =
—
[1] A.O. Aleksanyan, V.A. Ganshin, Yu.N. Korkishko and [2]
Bu
B / D 1D2 [i — 2W~u(1 u)/RT] Bu) D1(l — r~)+ D2u —
Bt
V.Z. Petrova, Zh. Tekh. Fiz. 59 (1989) 351. A.O. Aleksanyan, V.A. Ganshin, Yu.N. Korkishko and
=
Bz
.
(37)
We have examined the profiles of the element distribution in Cd : ZnSe layers obtained in melts with different n and m values (fig. 7a). It was assumed that the magnitude of the Auger signal is proportional to the element concentration. Eq. (37) was solved by a finite-difference technique for different D1 and D2. The concentration on the surface was considered constant and equal to the value measured experimentally. We selected
V.A. Fedorov, Phys. Status Solidi (b) 161 (1990) 629. [3J N.V. Hahlova and N.S. Dornbrovskaya, Zh. Neorg. Khim 4 (1959) 920. 14] P.K. Bhattacharya and S. Srinivasa, J. AppI. Phys. 54 (1983) 5090. [5] [6] [7] [8] [9] [10J [11] [12]
GB. Stringfellow, J. Crystal Growth 27 (1974) 21. A. Laugier, Rev. Physique AppI. 8 (1973) 259. A.S. Jordan, J. Electrochem. Soc 119 (1972) 123. V.A. Fedorov, V.A. Ganshin and Yu.N. Korkishko, to be published. D.W. K.isker. J. Crystal Growth 98 (1989) 127. P.A. Fedders and MW. Muller, J. Phys. Chem. Solids 45 (1984) 685. L D Landau and EM. Lifshits, Teoriya Uprugosti (Nauka, Moscow, 1965). A. Brandenburg. J. Lightwave Technol. 14 (1986) 1580.
790
V.A. Fedorov et at.
/
Ion exchange in ZnSe crystals
[13] A.K. Gin and GB. Mitra, Phys. Status Solidi (b) 134 (1986) KIl. [14] D. Berlincourt, H. Jaffe and L.R. Shiozawa, Phys. Rev. 129 (1963) 1009. {15J R.M. Martin, Phys. Rev. B6 (1972) 4546. [16] K.V. Shalimova, A.F. Botnev, V.A. Dmitriev. N.Z. Kognovitskaya and V.V. Starostin. Kristallografiya 14 (1969) 629. [17] N. Samarth, H. Luo, J.K. Furdyna, S.B. Qadri, YR. Lee, AK. Ramdas and N. Otsuka, AppI. Phys. Letters 54 (1989) 2680.
[181 S.N.G. Chu. A.T. Macrander. K.E. Strege and W.D. Johnstori, Jr., J. AppI. Phys. 57 (1985) 249. [19] J.W. Matthews, in: Epitaxial Growth, Part B, Ed. J.W. Matthews (Academic Press, New York, 1975) ch. 8. [20] R.M. Barrer, R.F. Bartholomew and L.V.C. Rees, J. Phys. Chem. Solids 24 (1963) 309. [21] S. Larch. RE. Schrader and CF. Stocker. Phys. Rev. 108 (1957) 587. [22] B.H. Lee. J. AppI. Phys. 41(1970) 2984.