- Email: [email protected]
Ta superlattices
~
Solid State Communications, Vol. 72, No. 7, pp. 667-670, 1989. Printed in Great Britain.
0038-1098/8953.00+.00 Pergamon Press plc
ELASTIC A N D STRUCTURAL PROPERTIES OF MoFfa SUPERLATTICES John L. Makous" a n d Charles M. Falco Department of Physics a n d Optical Sciences Center University of Arizona, Tucson, Arizona 85721 (Revised manuscript received 6 September 1989 by A. A. Maradudin)
Previously 1 we reported the fabrication, structure, a n d electrical transport properties of M o / T a superlattices. This present paper reports a more quantitative analysis used to explain a n observed w a v e l e n g t h - d e p e n d e n t strain perpendicular to the layers in terms of the v a n der Merwe 2"3 model for epitaxial superlattices. These results are correlated with a n anomalous decrease in the c~ elastic stiffness constant of M o / T a for 20 A < A < 50 A.
peaked crests, a n d thus might be less well suited for describing metals such as M o / T a than the sinusoidal description of the Frenkel-Kontorowa 6 a n d PeierlsNabarro 7 models, the complexity of the problem does not allow such a free choice of the model potential, while still obtaining an exactly solvable solution to the elastic behavior of crystalline interfaces. Therefore, we have chosen to apply the v a n der Merwe model to M o / T a superlattices in order to obtain a semi-quantitative picture to compare to the experimental results. The advantage is that this model is exactly solvable. A simple cubic lattice is a s s u m e d for both layers, one with lattice constant a a n d the other b. According to the model, an equally-spaced sequence of dislocations occurs with a spacing p to accommodate the misfit between the two lattices. This spacing satisfies the relations
INTRODUCTION Two possibilities exist to accommodate the lattice mismatch in ideal epitaxial superlattices consisting of crystalline layers with different lattice spacings. The layers can b e strained in order to have commensurate interfaces, or this coherency strain can be relieved partially or completely by a cross-grid of misfit dislocations at the interfaces. Which configuration is more favorable in a superlattice of a given combination of materials d e p e n d s on the natural misfit between the lattices, the elastic properties of the layers, a n d the individual layer thicknesses. Recently we reported 1 the fabrication of M o / T a superlattices m o d u l a t e d by integer n u m b e r s of atomic planes of Mo a n d Ta. These superlattices maintain perpendicular structural coherence over m a n y bilayers, a n d exhibit a w a v e l e n g t h - d e p e n d e n t lattice strain perpendicular to the layers. A qualitative explanation was offered in Ref. 1, in which the strain was assumed to exist in the interfacial region only. Here we present a more ~uantitative analysis using the v a n der Merwe modeFC This model is used to calculate the critical wavelength A,, a n d to determine the wavelength dependence of the strain for epitaxial M o / T a superlattices that have the same specifications as those described in Ref. 1. Comparison of the experimental strain to the model shows good agreement. These results are used to explain a previously observed, w a v e l e n g t h - d e p e n d e n t decrease in the c~ elastic constant of M o / T a . I'~
p = Na = (N + 1)b = a b / ( a - b),
where N is a n integer. Uniform strain e can exist in the layers to reduce or eliminate the misfit. The existence of strain in the layers changes the effective lattice spacings of the layers and, therefore, the dislocation spacing p according to Eq. (2). W h e n all of the misfit is accommodated b y strain a n d the interfaces are commensurate, the layers have the same effective (parallel) lattice spacings, a n d p ---) ~. This model assumes that the deformations of the lattices can be characterized by the shear moduli a n d Poisson ratios of each layer. By solving the biharmonic equation a n d assuming suitable b o u n d a r y conditions, v a n d e r Merwe determined the stress components in the layers a n d the energy ED of the dislocation array7 ~ The energy of a cross-grid of dislocations is taken as 2ED u n d e r the assumption that the dislocation density is small enough that the dislocations do not interact with one another. The total energy of the interface can be written E = Es + 2ED, where the total strain energy of the layers Es is given by the well-known expression 8'9
THE VAN DER MERWE MODEL
The model proposed by v a n der Merwe 2~ assumes a parabolic atomic potential across the interface. In terms of the shear stress pzx this is written as s p~ = laU/c,
-c/2 < U < c/2
(2)
(1)
w h e r e p is the interfacial shear modulus, U is the relative tangential displacement of two atoms on either side of the interface, a n d c is the spacing of the reference lattice. Although this potential has sharply
Es = 4{(l+v)/(1-v)}phe2,
"Present address: Naval Research Laboratory, Washington, D.C. 20375-5000 667
(3)
668
ELASTIC AND STRUCTURAL P R O P E R T I E S OF M o / T a SUPERLATTICES
where 2h is the layer thickness, and v is the Poisson ratio. This expression assumes layers of equal thicknesses (i.e. 4h = A) a n d elasticities, but an analogous expression is easily obtained for layers with different properties. Expressions for the layer strains a n d the critical layer thicknesses are obtained by minimizing the total energy with respect to the strain e. By setting 3E/3e = 0, one obtains an implicit equation for the optimal strain e ~ which d e p e n d s on the layer thicknesses, the shear moduli of the layers, a n d the natural misfit between the lattices. 2~ We apply these results to the specific case of the M o / T a superlattices described in Ref. 1, in which the superlattice wavelength A is modulated by alternating integer n atomic planes of Mo a n d Ta, i.e. A = n~(dMo + d~,) = n×(4.57 A).
0.030
0.025
0.020
6
'~ 0.015
0.010
(4)
The lattice spacings of the Mo a n d Ta layers in the direction of growth, i.e. the (110) direction, are denoted by dMo a n d dT,, respectively, with the natural misfit between these spacings being 5%. The elastic' moduli used in our calculations are PMo = 12.7X10~0 N / m 2 a n d p~, = 7.05x10 TM N / m 2. These values are obtained from the Voigt approximation for calculating the shear modulus, ~° where bulk values of the elastic constants of Mo n'~2 a n d Ta ~3'~a'~5are used in the approximation. Whenever a n interfacial value is required for the shear modulus, the average of these is used. The value v = 1 / 3 is taken as the Poisson ratio for both layers. Using the above layer design a n d elasticities in the calculations for the M o / T a superlattices, we find a critical wavelength of 49 /~. For wavelengths longer than * dislocations are introduced to reduce the strain, a n d we can use a n iterative routine on the implicit equation in e~ (= eT,) to calculate the expected strain as a function of A. With the initial value set at em= 0.025, which is the value expected w h e n all of the misfit is accommodated by strain, the solution converges to within +0.1% in approximately six iterations. The results are s h o w n in Fig. 1, which is a plot of the parallel strain in the Ta layer, e , = % , as a function of A as determined by the model. Notice that for A > A~ the strain decreases as A increases, indicating that it becomes more favorable to introduce misfit dislocations to relieve the layer strain. The strain in the Mo layer is obtained easily by the expression eT, = RreMo, where R = PMo/PT, a n d r = hMo/hT,, h~ being the thickness of the i~h layer)
0.005
.......................
0.000 0
20
40
60
80
100
1
A (h) Fig. I. Strain versus superlattice wavelength A for M o / T a as calculated with the van der Merwe model.
2.32 A
o4
o} ¢: o
2.31
°m
T
I T
2.30
l o c Q.
2.29
o. 2.28
STRUCTURAL ANALYSIS As described in Ref. 1 the films were deposited by dc triode sputtering, where the set of samples range from a bulk-like A = 699 ./~ (n = 153) to the monolayer limit A = 4.57 ~ (n = 1) according to Eq. (4). The structures of the films were characterized by several techniques including x-ray diffraction with a BraggBrentano diffractometer. From the linewidths of the Bragg peaks in the spectra the structural coherence length perpendicular to the layers was found to be constant at =250 A for 9 A < A < 115 /~, decreasing to =165 ~ for the n = 1 monolayer sample. This indicates that the structure perpendicular to the layers is coherent over m a n y interfaces. 1 From the Bragg peak positions the average perpendicular lattice spacings were determined using the Bragg law a n d are plotted in Fig. 2 as a function of superlattice wavelength. The large error bar associated with the A = 114 /~ sample is a result of the decrease in the intensity of the Bragg peak as A increases2 In Fig. 2 we see that the perpendicular d-spacing changes with A, indicating a Adependent strain in the layers. These spacings are
Vol. 72, No. 7
. 0
20
.
. 40
.
.
60
a
80
100
120
(h)
Fig. 2. Average perpendicular lattice spacings of M o / T a superlattices as determined from the Bragg peak positions from the 0-20 x-ray spectra.
converted to perpendicular strain ez using the relation (dsr~ - do) e~gs
-
do
(5)
where dB,~ is the perpendicular spacing s h o w n in Fig. 2, and do is the unstrained spacing. The perpendicular strains calculated from the values in Fig. 2 are converted into parallel strains by use of the definition of the Poisson ratio ez = -re,,
Vol. 72, No. 7
a) 0.030
,j
Theory
0.025 0.020
\ \ \
.0.015
\
C
N
0.010 0.005 0.000
I": J.
-0.005
. . . .
0
669
ELASTIC AND STRUCTURAL PROPERTIES OF Mo/Ta SUPERLATTICES
,
20
. . . .
,
40
. . . .
I
60
. . . .
J . . . .
80
,
. . . .
100
120
a (X) b) 0.080 0.070 0.060 0.050
6
'~ 0.040
mental behavior agrees semi-quantitatively with the theory. The data seem to suggest a critical wavelength A~ = 28 /~. The decrease in strain for A > A~ is in qualitative agreement with the theory. For A < 18 ,/~ the strain decreases in contrast to what is expected from the theory. Since Mo a n d Ta form solid solutions in all proportions, '6 we can expect some interdiffusion to occur at the interfaces. Therefore, a possible explanation for the decrease in the strain is that the role of interdiffusion at the interfaces becomes significant for these smaller A samples. Assuming that for the A = 4.57 A a n d 9,14 A samples the lattice has relaxed toward some "unstrained" alloy value, we take as do their average d-spacing to generate the data s h o w n in Fig. 3(a). However, s a t e l l i t e p e a k s do exist in the x-ray spectrum of the A = 9.14 A sample, indicating that there is composition modulation in that sample.' These results are consistent with the previous findings of Bennett, w h o determined from xray data an upper limit of 8 /~ for the interdiffusion width of M o / T a . '7 In Fig. 4 we plot the ratio of the first order satellite peaks, [-/P, as a function of A after correcting for the Lorentz, polarization, a n d Debye-Waller factors in the x-ray intensities. TM Since in M o / T a the lattice with the larger lattice spacing (Ta) has a smaller scattering factor, the ratio I-/P should increase as A decreases. In Fig. 4 this expected behavior occurs for A > 14/~. However, at A = 14 /~ there is a s u d d e n increase in this ratio followed b y a decrease at A = 9 A, possibly indicating that some kind of structural transition is occuring. However, caution must be taken w h e n trying to determine information about structure parallel to the interfaces from spectra obtained with a scattering vector perpendicular to the layers, as this can lead to ambiguous results# 9 If we assume that in the superlattice the scattering factor varies between f(1 + 11) a n d lattice spacing between do(1 + e), and we assume that the strain amplitude ¢ is small, then we can write a n expression for the ratio of the first order satellites as 19
Theory
0.030
\
0.020
25 0.010
O
i
. m
0.000
0
20
40
60
80
100
120
a (X) Fig. 3. Wavelength d e p e n d e n c e of the experimental strain, where the theoretical curve from Fig. 1 is included for comparison. In (a) the strain was determined from the Bragg peak positions. In (b) the values were d e t e r m i n e d from the high angle x-ray satellite peak intensities.
rv
20
>, CO c-
15
c10
..=
T i
(D
co where v = 1 / 3 is a s s u m e d for b o t h layers. These strains are plotted as a function of A in Fig. 3(a), where we used do = 2.291 /~. This value is the average ~erpendicular d-spacings of the A = 4.57 A a n d 9.14 samples s h o w n in Fig. 2. The explanation for this choice is presented below. Also s h o w n in Fig. 3(a) with the experimental strains is the theoretical curve from Fig. 1. Ignoring for n o w the relaxation of the strain for A < 18 .~, we see in Fig. 3(a) that the experi-
0
.
0
.
.
.
i
20
. . . .
,
40
. . . .
I , i J r l l , l , l l l ,
60
80
100
'
120
A(A) Fig. 4. Wavelength d e p e n d e n c e of the first order satellite intensity ratio I-/I ÷ as defined by Eq. (6).
670
ELASTIC AND STRUCTURAL PROPERTIES I-/I + = [(A/d
- 1)e + ll]/[(A/d
+ 1 ) e - n]-
(6)
For M o / T a 11 = 0.29, 20 and we take do = 2.291 /~ for the reasons described above. By considering the elastic nature of each layer, we can convert the coherency strains ¢ obtained from Eq. (6) into parallel strains era. The A dependence of the strains determined from the satellite intensity ratios of Fig. 4 are plotted in Fig. 3(b), where the solid line is the theoretical curve from Fig. 2. Except for the A = 9 ~ sample we see that the experimental data agree well with the theory. The data in Fig. 3(b) also agree with the results in Fig. 3(a), both suggesting a critical wavelength Ac = 28 A, and both showing the expected decreasing strain with further increasing A. The results in Fig 3(b) are in agreement with those suggested by the data in Figs. 3(a) and 4, namely that interdiffusion effects dominate the structural behavior for A < 14 ~. Hence, we expect that the assumptions by which Eq. (6) were determined are invalid for the A = 9 /~ sample, consistent with its anomalous value in Fig. 3(b). The above analysis neglects impurities and nonideal defects that can be introduced during the deposition process. More detailed information on the structure parallel to the interfaces and on the dislocation density is needed for a more conclusive understanding of the structural behavior of these superlattices. Previously 1 we correlated an anomalous dip in the ca lattice stiffness of these samples obtained by Brillouin light scattering 4 with the strain in the superlattices. The analysis presented above confirms this correlation. The light scattering results indicated a =6% decrease in the c~ stiffness constant for 20 /~ < A < 50 /~, which correlated with the A-dependent strain. I In the context of the analysis presented here, we reiterate the explanation for this effect as follows.
OF Mo/Ta SUPERLATTICES
Vol. 72, No.
At large A the interfaces are incoherent, and the interiors of the layers are relaxed. As A decreases, the strain in the layers increases as the interfaces become more coherent. Since the Ta layers have the larger lattice spacing, they are expected to contract in the plane of the interface in order to match with the Mo layers. Consequently, perpendicular to the layers the Ta layers are expected to expand. As a result, the perpendicular interatomic potential of the Ta layers weakens, and their stiffness in this direction is expected to decrease. The Mo layers behave in an opposite manner. In superlattices the ca stiffnesses of the constituents add in parallel to generate the resulting c44 stiffness of the multilayer3 ~ Therefore, the layer with the smaller ca constant will dominate the ca behavior of the superlattice. In this case it is Ta with the smaller c44 and, hence, as the coherency of the interfaces and the strain in the layers increases, the resultant ca stiffness of the superlattice is expected to soften. Below approximately 18 /~ we have shown that interdiffusion causes the lattice to relax, and the observed rehardening of c~4 is expected. This explains the correlation of the c44 elastic constant with the layer strain. In summary, for M o / T a superlattices we have described a wavelength dependent strain in terms of the van der Merwe model for epitaxial superlattices. The experimental strain determined in two different ways from x-ray diffraction data agrees semiquantitatively with this model. These results suggest a critical wavelength A = 28 A, above which interfacial misfit dislocations will be introduced to relieve the strain. Our results also indicate interdiffusion effects to be significant for A < 18 A. We are able to explain the wavelength dependence of the c44 stiffness constant in terms of the observed strain in the superlattices. Acknowledgements - This work was supported by the Office of Naval Research under Contract N00014-88-K0298.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
J.L. Makous and C. M. Falco, Solid State Commun. 68, 375 (1988). J . H . van der Merwe and W. A. Jesser, J. Appl. Phys. 63, 1509 (1988). W . A . ]esser and J. H. van der Merwe, J. Appl. Phys. 63, 1928 (1988). J.A. Bell, W. R. Bennett, R. Zanoni, G. I. Stegeman, C. M. Falco, and F. Nizzoli, Phys. Rev. B 35, 4127 (1987). J . H . van der Merwe, J. Appl. Phys. 34, 117, 123 (1963). J. Frenkel and T. Kontorowa, Phys. Z. Sowjetunion 13, 1 (1938). R.E. Peierls, Proc. Phys. Soc. London 52, 34 (1940); F. R. N. Nabarro, Proc. Phys. Soc. London 52, 90 (1940). J.W. Matthews, ed., Epitaxial Growth (Academic Press, New York, 1975). W . A . Jesser and D. Kuhlmann-Wilsdorf, Phys. Status Solidi 19, 95 (1967). E. Schreiber, O. L. Anderson, and N. Soga, in Elastic Constants and Their Measurements (McGraw-Hill, New York, 1973), pp. 30-31.
ll.
J. M. Dickinson and P. E. Armstrong, J. Appl. Phys. 38, 602 (1967). 12. D. I. Bolef and J. D. Klerk, J. Appl. Phys. 33, 2311 (1962). 13. F . H . Featherton and J. R. Neighbors, Phys. Rev. 130, 1324 (1963). 14. D.I. Bolef, J. Appl. Phys. 32, 100 (1961). 15. N. Soga, J. Appl. Phys. 37, 3416 (1966). 16. G. A. Geach and D. Summers-Smith, J. Inst. Metals 80, 143 (1951-52). 17. W.R. Bennett, Ph.D. dissertation, University of Arizona, 1985. 18. H . P . Klug and L. E. Alexander, in X-Ray Diffraction Procedures 2nd ed. (Wiley, New York, 1974). 19. D.B. McWhan, in Synthetic Modulated Structures, eds. L. L. Chang and B. C. Giessen (Academic Press, New York, 1985), chap. 2. 20. B.D. Cullity, Elements of X-Ray Diffraction, 2nd ed. (Addison-Wesley, Reading, 1978). 21. M. Grimsditch, Phys. Rev. B 31, 6818 (1985).
Solid State Communications, Vol. 72, No. 7, pp. 667-670, 1989. Printed in Great Britain.
0038-1098/8953.00+.00 Pergamon Press plc
ELASTIC A N D STRUCTURAL PROPERTIES OF MoFfa SUPERLATTICES John L. Makous" a n d Charles M. Falco Department of Physics a n d Optical Sciences Center University of Arizona, Tucson, Arizona 85721 (Revised manuscript received 6 September 1989 by A. A. Maradudin)
Previously 1 we reported the fabrication, structure, a n d electrical transport properties of M o / T a superlattices. This present paper reports a more quantitative analysis used to explain a n observed w a v e l e n g t h - d e p e n d e n t strain perpendicular to the layers in terms of the v a n der Merwe 2"3 model for epitaxial superlattices. These results are correlated with a n anomalous decrease in the c~ elastic stiffness constant of M o / T a for 20 A < A < 50 A.
peaked crests, a n d thus might be less well suited for describing metals such as M o / T a than the sinusoidal description of the Frenkel-Kontorowa 6 a n d PeierlsNabarro 7 models, the complexity of the problem does not allow such a free choice of the model potential, while still obtaining an exactly solvable solution to the elastic behavior of crystalline interfaces. Therefore, we have chosen to apply the v a n der Merwe model to M o / T a superlattices in order to obtain a semi-quantitative picture to compare to the experimental results. The advantage is that this model is exactly solvable. A simple cubic lattice is a s s u m e d for both layers, one with lattice constant a a n d the other b. According to the model, an equally-spaced sequence of dislocations occurs with a spacing p to accommodate the misfit between the two lattices. This spacing satisfies the relations
INTRODUCTION Two possibilities exist to accommodate the lattice mismatch in ideal epitaxial superlattices consisting of crystalline layers with different lattice spacings. The layers can b e strained in order to have commensurate interfaces, or this coherency strain can be relieved partially or completely by a cross-grid of misfit dislocations at the interfaces. Which configuration is more favorable in a superlattice of a given combination of materials d e p e n d s on the natural misfit between the lattices, the elastic properties of the layers, a n d the individual layer thicknesses. Recently we reported 1 the fabrication of M o / T a superlattices m o d u l a t e d by integer n u m b e r s of atomic planes of Mo a n d Ta. These superlattices maintain perpendicular structural coherence over m a n y bilayers, a n d exhibit a w a v e l e n g t h - d e p e n d e n t lattice strain perpendicular to the layers. A qualitative explanation was offered in Ref. 1, in which the strain was assumed to exist in the interfacial region only. Here we present a more ~uantitative analysis using the v a n der Merwe modeFC This model is used to calculate the critical wavelength A,, a n d to determine the wavelength dependence of the strain for epitaxial M o / T a superlattices that have the same specifications as those described in Ref. 1. Comparison of the experimental strain to the model shows good agreement. These results are used to explain a previously observed, w a v e l e n g t h - d e p e n d e n t decrease in the c~ elastic constant of M o / T a . I'~
p = Na = (N + 1)b = a b / ( a - b),
where N is a n integer. Uniform strain e can exist in the layers to reduce or eliminate the misfit. The existence of strain in the layers changes the effective lattice spacings of the layers and, therefore, the dislocation spacing p according to Eq. (2). W h e n all of the misfit is accommodated b y strain a n d the interfaces are commensurate, the layers have the same effective (parallel) lattice spacings, a n d p ---) ~. This model assumes that the deformations of the lattices can be characterized by the shear moduli a n d Poisson ratios of each layer. By solving the biharmonic equation a n d assuming suitable b o u n d a r y conditions, v a n d e r Merwe determined the stress components in the layers a n d the energy ED of the dislocation array7 ~ The energy of a cross-grid of dislocations is taken as 2ED u n d e r the assumption that the dislocation density is small enough that the dislocations do not interact with one another. The total energy of the interface can be written E = Es + 2ED, where the total strain energy of the layers Es is given by the well-known expression 8'9
THE VAN DER MERWE MODEL
The model proposed by v a n der Merwe 2~ assumes a parabolic atomic potential across the interface. In terms of the shear stress pzx this is written as s p~ = laU/c,
-c/2 < U < c/2
(2)
(1)
w h e r e p is the interfacial shear modulus, U is the relative tangential displacement of two atoms on either side of the interface, a n d c is the spacing of the reference lattice. Although this potential has sharply
Es = 4{(l+v)/(1-v)}phe2,
"Present address: Naval Research Laboratory, Washington, D.C. 20375-5000 667
(3)
668
ELASTIC AND STRUCTURAL P R O P E R T I E S OF M o / T a SUPERLATTICES
where 2h is the layer thickness, and v is the Poisson ratio. This expression assumes layers of equal thicknesses (i.e. 4h = A) a n d elasticities, but an analogous expression is easily obtained for layers with different properties. Expressions for the layer strains a n d the critical layer thicknesses are obtained by minimizing the total energy with respect to the strain e. By setting 3E/3e = 0, one obtains an implicit equation for the optimal strain e ~ which d e p e n d s on the layer thicknesses, the shear moduli of the layers, a n d the natural misfit between the lattices. 2~ We apply these results to the specific case of the M o / T a superlattices described in Ref. 1, in which the superlattice wavelength A is modulated by alternating integer n atomic planes of Mo a n d Ta, i.e. A = n~(dMo + d~,) = n×(4.57 A).
0.030
0.025
0.020
6
'~ 0.015
0.010
(4)
The lattice spacings of the Mo a n d Ta layers in the direction of growth, i.e. the (110) direction, are denoted by dMo a n d dT,, respectively, with the natural misfit between these spacings being 5%. The elastic' moduli used in our calculations are PMo = 12.7X10~0 N / m 2 a n d p~, = 7.05x10 TM N / m 2. These values are obtained from the Voigt approximation for calculating the shear modulus, ~° where bulk values of the elastic constants of Mo n'~2 a n d Ta ~3'~a'~5are used in the approximation. Whenever a n interfacial value is required for the shear modulus, the average of these is used. The value v = 1 / 3 is taken as the Poisson ratio for both layers. Using the above layer design a n d elasticities in the calculations for the M o / T a superlattices, we find a critical wavelength of 49 /~. For wavelengths longer than * dislocations are introduced to reduce the strain, a n d we can use a n iterative routine on the implicit equation in e~ (= eT,) to calculate the expected strain as a function of A. With the initial value set at em= 0.025, which is the value expected w h e n all of the misfit is accommodated by strain, the solution converges to within +0.1% in approximately six iterations. The results are s h o w n in Fig. 1, which is a plot of the parallel strain in the Ta layer, e , = % , as a function of A as determined by the model. Notice that for A > A~ the strain decreases as A increases, indicating that it becomes more favorable to introduce misfit dislocations to relieve the layer strain. The strain in the Mo layer is obtained easily by the expression eT, = RreMo, where R = PMo/PT, a n d r = hMo/hT,, h~ being the thickness of the i~h layer)
0.005
.......................
0.000 0
20
40
60
80
100
1
A (h) Fig. I. Strain versus superlattice wavelength A for M o / T a as calculated with the van der Merwe model.
2.32 A
o4
o} ¢: o
2.31
°m
T
I T
2.30
l o c Q.
2.29
o. 2.28
STRUCTURAL ANALYSIS As described in Ref. 1 the films were deposited by dc triode sputtering, where the set of samples range from a bulk-like A = 699 ./~ (n = 153) to the monolayer limit A = 4.57 ~ (n = 1) according to Eq. (4). The structures of the films were characterized by several techniques including x-ray diffraction with a BraggBrentano diffractometer. From the linewidths of the Bragg peaks in the spectra the structural coherence length perpendicular to the layers was found to be constant at =250 A for 9 A < A < 115 /~, decreasing to =165 ~ for the n = 1 monolayer sample. This indicates that the structure perpendicular to the layers is coherent over m a n y interfaces. 1 From the Bragg peak positions the average perpendicular lattice spacings were determined using the Bragg law a n d are plotted in Fig. 2 as a function of superlattice wavelength. The large error bar associated with the A = 114 /~ sample is a result of the decrease in the intensity of the Bragg peak as A increases2 In Fig. 2 we see that the perpendicular d-spacing changes with A, indicating a Adependent strain in the layers. These spacings are
Vol. 72, No. 7
. 0
20
.
. 40
.
.
60
a
80
100
120
(h)
Fig. 2. Average perpendicular lattice spacings of M o / T a superlattices as determined from the Bragg peak positions from the 0-20 x-ray spectra.
converted to perpendicular strain ez using the relation (dsr~ - do) e~gs
-
do
(5)
where dB,~ is the perpendicular spacing s h o w n in Fig. 2, and do is the unstrained spacing. The perpendicular strains calculated from the values in Fig. 2 are converted into parallel strains by use of the definition of the Poisson ratio ez = -re,,
Vol. 72, No. 7
a) 0.030
,j
Theory
0.025 0.020
\ \ \
.0.015
\
C
N
0.010 0.005 0.000
I": J.
-0.005
. . . .
0
669
ELASTIC AND STRUCTURAL PROPERTIES OF Mo/Ta SUPERLATTICES
,
20
. . . .
,
40
. . . .
I
60
. . . .
J . . . .
80
,
. . . .
100
120
a (X) b) 0.080 0.070 0.060 0.050
6
'~ 0.040
mental behavior agrees semi-quantitatively with the theory. The data seem to suggest a critical wavelength A~ = 28 /~. The decrease in strain for A > A~ is in qualitative agreement with the theory. For A < 18 ,/~ the strain decreases in contrast to what is expected from the theory. Since Mo a n d Ta form solid solutions in all proportions, '6 we can expect some interdiffusion to occur at the interfaces. Therefore, a possible explanation for the decrease in the strain is that the role of interdiffusion at the interfaces becomes significant for these smaller A samples. Assuming that for the A = 4.57 A a n d 9,14 A samples the lattice has relaxed toward some "unstrained" alloy value, we take as do their average d-spacing to generate the data s h o w n in Fig. 3(a). However, s a t e l l i t e p e a k s do exist in the x-ray spectrum of the A = 9.14 A sample, indicating that there is composition modulation in that sample.' These results are consistent with the previous findings of Bennett, w h o determined from xray data an upper limit of 8 /~ for the interdiffusion width of M o / T a . '7 In Fig. 4 we plot the ratio of the first order satellite peaks, [-/P, as a function of A after correcting for the Lorentz, polarization, a n d Debye-Waller factors in the x-ray intensities. TM Since in M o / T a the lattice with the larger lattice spacing (Ta) has a smaller scattering factor, the ratio I-/P should increase as A decreases. In Fig. 4 this expected behavior occurs for A > 14/~. However, at A = 14 /~ there is a s u d d e n increase in this ratio followed b y a decrease at A = 9 A, possibly indicating that some kind of structural transition is occuring. However, caution must be taken w h e n trying to determine information about structure parallel to the interfaces from spectra obtained with a scattering vector perpendicular to the layers, as this can lead to ambiguous results# 9 If we assume that in the superlattice the scattering factor varies between f(1 + 11) a n d lattice spacing between do(1 + e), and we assume that the strain amplitude ¢ is small, then we can write a n expression for the ratio of the first order satellites as 19
Theory
0.030
\
0.020
25 0.010
O
i
. m
0.000
0
20
40
60
80
100
120
a (X) Fig. 3. Wavelength d e p e n d e n c e of the experimental strain, where the theoretical curve from Fig. 1 is included for comparison. In (a) the strain was determined from the Bragg peak positions. In (b) the values were d e t e r m i n e d from the high angle x-ray satellite peak intensities.
rv
20
>, CO c-
15
c10
..=
T i
(D
co where v = 1 / 3 is a s s u m e d for b o t h layers. These strains are plotted as a function of A in Fig. 3(a), where we used do = 2.291 /~. This value is the average ~erpendicular d-spacings of the A = 4.57 A a n d 9.14 samples s h o w n in Fig. 2. The explanation for this choice is presented below. Also s h o w n in Fig. 3(a) with the experimental strains is the theoretical curve from Fig. 1. Ignoring for n o w the relaxation of the strain for A < 18 .~, we see in Fig. 3(a) that the experi-
0
.
0
.
.
.
i
20
. . . .
,
40
. . . .
I , i J r l l , l , l l l ,
60
80
100
'
120
A(A) Fig. 4. Wavelength d e p e n d e n c e of the first order satellite intensity ratio I-/I ÷ as defined by Eq. (6).
670
ELASTIC AND STRUCTURAL PROPERTIES I-/I + = [(A/d
- 1)e + ll]/[(A/d
+ 1 ) e - n]-
(6)
For M o / T a 11 = 0.29, 20 and we take do = 2.291 /~ for the reasons described above. By considering the elastic nature of each layer, we can convert the coherency strains ¢ obtained from Eq. (6) into parallel strains era. The A dependence of the strains determined from the satellite intensity ratios of Fig. 4 are plotted in Fig. 3(b), where the solid line is the theoretical curve from Fig. 2. Except for the A = 9 ~ sample we see that the experimental data agree well with the theory. The data in Fig. 3(b) also agree with the results in Fig. 3(a), both suggesting a critical wavelength Ac = 28 A, and both showing the expected decreasing strain with further increasing A. The results in Fig 3(b) are in agreement with those suggested by the data in Figs. 3(a) and 4, namely that interdiffusion effects dominate the structural behavior for A < 14 ~. Hence, we expect that the assumptions by which Eq. (6) were determined are invalid for the A = 9 /~ sample, consistent with its anomalous value in Fig. 3(b). The above analysis neglects impurities and nonideal defects that can be introduced during the deposition process. More detailed information on the structure parallel to the interfaces and on the dislocation density is needed for a more conclusive understanding of the structural behavior of these superlattices. Previously 1 we correlated an anomalous dip in the ca lattice stiffness of these samples obtained by Brillouin light scattering 4 with the strain in the superlattices. The analysis presented above confirms this correlation. The light scattering results indicated a =6% decrease in the c~ stiffness constant for 20 /~ < A < 50 /~, which correlated with the A-dependent strain. I In the context of the analysis presented here, we reiterate the explanation for this effect as follows.
OF Mo/Ta SUPERLATTICES
Vol. 72, No.
At large A the interfaces are incoherent, and the interiors of the layers are relaxed. As A decreases, the strain in the layers increases as the interfaces become more coherent. Since the Ta layers have the larger lattice spacing, they are expected to contract in the plane of the interface in order to match with the Mo layers. Consequently, perpendicular to the layers the Ta layers are expected to expand. As a result, the perpendicular interatomic potential of the Ta layers weakens, and their stiffness in this direction is expected to decrease. The Mo layers behave in an opposite manner. In superlattices the ca stiffnesses of the constituents add in parallel to generate the resulting c44 stiffness of the multilayer3 ~ Therefore, the layer with the smaller ca constant will dominate the ca behavior of the superlattice. In this case it is Ta with the smaller c44 and, hence, as the coherency of the interfaces and the strain in the layers increases, the resultant ca stiffness of the superlattice is expected to soften. Below approximately 18 /~ we have shown that interdiffusion causes the lattice to relax, and the observed rehardening of c~4 is expected. This explains the correlation of the c44 elastic constant with the layer strain. In summary, for M o / T a superlattices we have described a wavelength dependent strain in terms of the van der Merwe model for epitaxial superlattices. The experimental strain determined in two different ways from x-ray diffraction data agrees semiquantitatively with this model. These results suggest a critical wavelength A = 28 A, above which interfacial misfit dislocations will be introduced to relieve the strain. Our results also indicate interdiffusion effects to be significant for A < 18 A. We are able to explain the wavelength dependence of the c44 stiffness constant in terms of the observed strain in the superlattices. Acknowledgements - This work was supported by the Office of Naval Research under Contract N00014-88-K0298.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
J.L. Makous and C. M. Falco, Solid State Commun. 68, 375 (1988). J . H . van der Merwe and W. A. Jesser, J. Appl. Phys. 63, 1509 (1988). W . A . ]esser and J. H. van der Merwe, J. Appl. Phys. 63, 1928 (1988). J.A. Bell, W. R. Bennett, R. Zanoni, G. I. Stegeman, C. M. Falco, and F. Nizzoli, Phys. Rev. B 35, 4127 (1987). J . H . van der Merwe, J. Appl. Phys. 34, 117, 123 (1963). J. Frenkel and T. Kontorowa, Phys. Z. Sowjetunion 13, 1 (1938). R.E. Peierls, Proc. Phys. Soc. London 52, 34 (1940); F. R. N. Nabarro, Proc. Phys. Soc. London 52, 90 (1940). J.W. Matthews, ed., Epitaxial Growth (Academic Press, New York, 1975). W . A . Jesser and D. Kuhlmann-Wilsdorf, Phys. Status Solidi 19, 95 (1967). E. Schreiber, O. L. Anderson, and N. Soga, in Elastic Constants and Their Measurements (McGraw-Hill, New York, 1973), pp. 30-31.
ll.
J. M. Dickinson and P. E. Armstrong, J. Appl. Phys. 38, 602 (1967). 12. D. I. Bolef and J. D. Klerk, J. Appl. Phys. 33, 2311 (1962). 13. F . H . Featherton and J. R. Neighbors, Phys. Rev. 130, 1324 (1963). 14. D.I. Bolef, J. Appl. Phys. 32, 100 (1961). 15. N. Soga, J. Appl. Phys. 37, 3416 (1966). 16. G. A. Geach and D. Summers-Smith, J. Inst. Metals 80, 143 (1951-52). 17. W.R. Bennett, Ph.D. dissertation, University of Arizona, 1985. 18. H . P . Klug and L. E. Alexander, in X-Ray Diffraction Procedures 2nd ed. (Wiley, New York, 1974). 19. D.B. McWhan, in Synthetic Modulated Structures, eds. L. L. Chang and B. C. Giessen (Academic Press, New York, 1985), chap. 2. 20. B.D. Cullity, Elements of X-Ray Diffraction, 2nd ed. (Addison-Wesley, Reading, 1978). 21. M. Grimsditch, Phys. Rev. B 31, 6818 (1985).