- Email: [email protected]
Stress relief in ion-beam deposited ZnO thin films due to post-deposition thermal treatment
Nuclear Instruments and Methods
580
in Physics Research 861 (1991) 580-584
North-Holland
Letter to the Editor
Stress relief in ion-beam deposited ZnO thin films due to post-deposition therm4 treatment * S.A.Isa Lkpartment of Electrical
Engineering
Technology,
South
Carolina
State College,
Orangeburg,
SC 29117,
USA
P.K. Ghash L&artntent
qf
Electrical
and Computer
Engineering.
Syracuse
U&ersity,
Syracuse,
NY 13244, USA
Received 4 February 1991
We have deposited ZnU thin films on glass and silicon substrates using ion-beam sputtering. Afthough the quality of the films obtained by this method depends on the depasitian parameters, even the best films obtained show evidence of internal stress. We demonstrate here by X-ray diffraction and Auger depth profiting that a post-deposition thermal treatment decreases the internal stress and produces a more stoichiometric ZnO film..
Zinc oxide is a piezoelectric material which finds considerable application in surface acoustic wave (SAW) devices for signal processing at ultrasonic frequencies [1,2]. During the past decade significant improvements have occurred in many piezoelectric thin films, largely through a better control of film growth, These developments have been motivated in large measure by the need for improved performance in communication systems. These applications demand materials displaying large electromechanical coupling coefficients over a wide bandwidth. To meet these requirements together with the need for SAW propagation with low insertion loss, one needs to grow highly or&ted ZnO thin films. Our inrtial efforts addressed these properties of ZnO thin fiims deposited by ion-beam sputtering techniques [3]. While jm~rovements in ZnO film properties can be pursued through improved deposition techniques and variation of deposition parameters, the question also arises as to whether film properties can be improved through post-deposition thermal treatment. Nearly all films, by whatever method they are produced, are found to be in a state of internal stress (compressive and/or tensile). ZnO thin films are widely used as the active piezoelectric layer far many SAW devices and the internal stress in these films has been found to affect the acoustic velocity and the perfor* First presented at the 7th 11%Conf. on km Beam Modification of Materials. Knoxvilte, Tennessee, USA, Sept. 9-14, 1991. ~~~~-~~3X/91/~~3.5~
mance of the device as a whole [7]. Our results on annealing experiments show stress relief in the films and increased acoustic velocities. The Auger analyses indicate a substantial improvement in the stoichiometry of the films. Our data on sheet resistance measuremcnts afso show that the sheet resistance increased from 6.02 to f1.89 Mf;t,/n. The corresponding variations in the microstructure of the films have been observed l-31.We have used X-ray diffraction analysis to assess the stress by observing the shift of the (002) diffraction peak. ZnO thin films were deposited by an ion-beam sputtering technique. The sputtering system utilized is the Veeco 341. Micro-etch system combined with a Kaufmann type ion source. The system is evacuated with a diffusion pump and liquid nitrogen cold-trap to an initial base pressure of 3 x IO-’ Torr. A mixture of argon and oxygen (75% Ar + 25% Ua) is used as the sputtering gas. The ions are provided by the ion source where an arc discharge between the cathode and the anode takes place. The positive ions of the plasma arc extracted and accelerated by a special multi-grid system with the resulting homogeneous, collimated and monoenergetic ion beam bombarding the surface of a 99.9% pure ZnO ceramic target used as the sputterhrg source. Sputter depositions are done at a chamber pressure of 2 X IF4 Torr and substrate temperature of 325 ’ C. Anncaiing of the ZnO samples was performed in the diffusion furnace. One set of samples was not annealed and two other sets were annealed at a fur-
0 1991 - Efsevier Science Publishers B.V. AI1 rights reserved
S.A. ka, P.K. Ghosh / Ion-beam deposited ZnO thin films
nace temperature of 5000°C for about 3 h. One of these two sets of samples was annealed in pure oxygen ambient and the other in dry nitrogen. We performed X-ray diffraction analysis (28 scan) of the films for stress analysis and the stoichiomet~ of the films was assessed through resistivity measurements and Auger electron spectroscopy. Our efforts on ZnO thin film development mainly addressed the requirements of surface acoustic wave devices. We have grown ZnO films on glass and silicon substrates. Most of our annealing experiments were performed on ZnO films on silicon substrftes with an intermediate layer of 90, (about 1000 A thick and thermally grown). Three sets of ZnO samples were prepared for this investigation all under the same deposition conditions. The internal stress in films deposited by sputtering can be quite appreciable due to the nature of the deposition procedure. This stress can be decomposed into two main components, thermal stress, and intrinsic stress (a = rr,, + oint). The thermal component of the internal stress is a result of mismatch between the thermal expansion coefficients of the film and that of the substrate. Its value can usually be approximated as
[41 where K is a factor that is dependent on both the Young’s modulus of the ZnO film and the difference between the average substrate temperature used in depositing the ZnO film and the ambient temperature under which the thermal stress is measured. (Y, and as are the average coefficients of thermal expansion for the ZnO film and the substrate material, respectively. The cause of the intrinsic stress in the films can often be interpreted in terms of several models. Windischmann [6] proposed a model for predicting the intrinsic stress resulting from ion-beam film deposition as depending on the physical properties of the target, the projectile energy, atomic number, mass, and flux of the ion-beam. In our case, operating at low pressures, the sputtered ZnO molecules and reflected argon/ oxygen atoms impinge on the condensing film with high energies. This is because at lower pressures, the mean-free-path is long and high energy particies arrive at the substrate unattenuated by gas scattering. These high energy incident atoms drive the atoms of the film closer together, imbedding themselves in the film. This mechanism, known as “atomic peening”‘, is believed to cause compression in the films [5]. These energetic particles can penetrate the surface of the film, randomly displacing the atoms from their equilibrium positions through a series of primary and recoil collisions, producing a volumetric distortion. Since the ratio of our deposition temperature to the melting temperature of ZnO is less than 0.2, we can assume that mass
581
10
-.. -
I
i
Unannealed ZnO film Annealed ZnO film with oxygen Annealed ZnO film m dry mtrogen.
.....
ok?-
06
-
04
-
-1 i
a9
Fig. 1. X-ray diffraction results of ZnO film samples.
transport and defect mobility is sufficiently low to freeze the volumetric distortion in place. Fig. 1 is the X-ray diffraction results of an unannealed sample (dashed line) and the sample annealed in the presence of oxygen (solid line). There is no dramatic difference in the X-ray diffraction results of the sample annealed in oxygen and that annealed just in pure nitrogen (dotted line) except the slight increase in the peak intensity of the sample annealed in oxygen ambient (see fig. 1). The (002) diffraction peak of the unannealed sample is shifted to lower 20 values (about 33.8”) than the powder diffraction value of stoichiometric ZnO (34.4” 1. This is because of the large inplane compressive stress which produces d spacing dilation in the growth direction. Comparing the results in fig. 1, observe that the (002) diffraction peak of the annealed films is higher in intensity, narrower in width, and has moved in position (about 0.3”) closer to the ZnO powder value. The increase in the (002) peak intensity and the decrease in half-width of the annealed films indicate evidence of stress relief and that a larger fraction of the material approaches the ideal c-axis normal orientation. We have been able to demonstrate during our preliminary work and in agreement with the work of Hickernell [7] that the internal stress in the piezoelectric layer of the SAW device influences its acoustic velocity. Annealed films with evidence of stress relief gave rise to higher acoustic velocities (up to 4.6% increase was obtained~.
T
S.A. Isa, P.K. Ghosh / Ion-beam deposited ZnO thin films
Table 1 Some of the parameters measured from unannealed [ZnO(UN)], annealed in dry nitrogen [ZnOCANII, and annealed with oxygen [ZnOfAO)] ZnO film samples Film
ZnOGJN) ZnO(AN) ZnO(AO)
Sheet resistance
‘3% Auger peak on the surface
[Mfi,‘ol
Oxygen
Zinc
6.02 6.54 6.89
22 26 28
78 74 72
those annealed
in the presence of oxygen except for the slightly higher average percent peak height of zinc
(74% versus 72%) in the films annealed in dry nitrogen. Observe the variations in the peak heights of zinc and oxygen, especially close to the film surface. Upon annealing, the peak height of zinc dropped from 78 to about 72%. The low resistivity of unannealed films indicates that they are nonstoichiometric and that there is excess zinc in the deposited films. Two possibilities exist [8,9]: that the excess zinc was interstitial and/or there were oxygen vacancies. We are assuming that the presence of oxygen in the annealing process also helps to fill some of the oxygen vacancies resulting in a more stoichiometric film. It should be noted that the percentages quoted here are the Auger peak of each element normalized to the total Auger signal and they are not weight percentages. The Auger intensity for each element is also dependent on the concentration of that element in the material. These figures are not absolute values since neither the Auger scattering efficiency nor the system resolution are accurately accounted for (the system was initially calibrated for Ta,O,). We have chosen 500” C as an annealing temperature. ZnO has a melting temperature (under high pressure) close to 2000 o C. ZnO also sublimates (at atmospheric pressure) around 1950 o C. However, using the existing models for structural changes in annealed polycrystalline materials, we anticipated that at temperatures between 300 and 6OO”C, stress relief would take place. This was confined by our X-ray results. Between 600 and lOOO”C, rec~stallization and major grain growth was anticipated. Our experimental data indicates that for samples annealed even for 6 hours at annealing temperatures up to 425”C, there were no noticeable changes in their diffraction pattern from those of unannealed samples. Only samples annealed for 6 hours and between annealing temperatures of 450 ’ C and 475 ’ C show slight increase in peak intensity and an upward shift of the diffraction peak of about 0.1’. Therefore, using annealing temperatures below 500 OC involves longer annealing times and very small change in the internal stress of the films. Considering the very thin nature of the films (about 0.76 Fm>,
583
higher annealing temperatures can result in the development of microfracture which would result in defect boundaries and considerable acoustic scattering. Consider also that one of the goals of ZnO technology is to integrate it with the existing silicon technology in order to create monolithic devices. One of our objectives, therefore, is that ZnO fabrication procedures must not be detrimental to any active circuitry. These circuitries which are normally metallized prior to film deposition and annealing places an upper limit on the processing temperatures. Certainly an upper limit is 577°C [lo] above which silicon and aluminum will form an eutetic alloy, thus rapidly destroying active devices. In conclusion, the internal stress in the ion-beam deposited ZnO thin film can be decomposed into two main components, thermal stress and intrinsic stress. The thermal component is as a result of mismatch between the thermal expansion ~efficients of the ZnO film and that of the substrate as the sample heats and cools from deposition temperatures. The intrinsic stress can be accounted for as a result of energetic particles bombarding the condensing film and can often be explained via an “atomic peening” model. The results of X-ray diffraction indicate that the (002) diffraction peak of the annealed films is higher in intensity, narrower in width, and has moved about 0.3’ in position closer to the powder diffraction value. These are indications of stress relief and that a larger fraction of the material approaches the ideal c-axis orientation. The results of Auger analysis also indicate that a more stoichiometric ZnO fifm results after undergoing the annealing process. The results show that the Auger peak of zinc was reduced from 78% to about 72% upon annealing, resulting in higher sheet resistance and a more stoichiometric ZnO film. We have chosen 500°C as the annealing temperature to avoid the development of microfracture in the films that could result in defect boundaries and considerable acoustic scattering. The temperature limitation of silicon technology is also a very important consideration for using ZnO thin films for SAW devices integrated with active circuitry on silicon. In this regard, we are currently exploring possibilities of lower annealing temperatures.
Acknowledgement
The authors would like to thank Lois Walsh of RADC/RBRE Griffis AFB, Rome, New York for performing the Auger analysis of the films.
References 111B.T. Klmri-Yakub and G.S. Kino, Appl. Phys. Lett. 25 (1974) 188.
584
S.A. Isa, P.K. Ghosh / Ion-beam deposited ZnO thin films
[2] N. Chubachi, Proc. IEEE 64 (1976) 772.
[3] S.A. Isa, P.K, Ghosh and P.G. Kornreich, Mater. Res. Sot. Symp. Proc. 128 (1989) 725. [4] J. Anderson, in: Use of Thin Films in Physical Investigation (Academic Press, New York, 1966) p. 261. 151 D.W. Hoffman and J.A. Thornton, Thin Solid Films 40 (19771 355. [6] H. Windischmann, J. Appl. Phys. 62 (1987) 1800.
[7] F.J. Hickernell, Proc. IEEE Ultrson. Symp. (1977) p. 309. [8] G. Heiland, E. Mollwo and F. Stockmann, in: Solid State Physics, eds. F. Seitz and D. Turnbull, vol. 8 Academic Press, New York, 1959) p. 191. [9] P.H. Kasai, Phys. Rev. 130 (1963) 989. 1101 D.J. Hamilton and W.G. Howard, Basic Integrated Circuit Engineering (McGraw-Hill, 1975).
580
in Physics Research 861 (1991) 580-584
North-Holland
Letter to the Editor
Stress relief in ion-beam deposited ZnO thin films due to post-deposition therm4 treatment * S.A.Isa Lkpartment of Electrical
Engineering
Technology,
South
Carolina
State College,
Orangeburg,
SC 29117,
USA
P.K. Ghash L&artntent
qf
Electrical
and Computer
Engineering.
Syracuse
U&ersity,
Syracuse,
NY 13244, USA
Received 4 February 1991
We have deposited ZnU thin films on glass and silicon substrates using ion-beam sputtering. Afthough the quality of the films obtained by this method depends on the depasitian parameters, even the best films obtained show evidence of internal stress. We demonstrate here by X-ray diffraction and Auger depth profiting that a post-deposition thermal treatment decreases the internal stress and produces a more stoichiometric ZnO film..
Zinc oxide is a piezoelectric material which finds considerable application in surface acoustic wave (SAW) devices for signal processing at ultrasonic frequencies [1,2]. During the past decade significant improvements have occurred in many piezoelectric thin films, largely through a better control of film growth, These developments have been motivated in large measure by the need for improved performance in communication systems. These applications demand materials displaying large electromechanical coupling coefficients over a wide bandwidth. To meet these requirements together with the need for SAW propagation with low insertion loss, one needs to grow highly or&ted ZnO thin films. Our inrtial efforts addressed these properties of ZnO thin fiims deposited by ion-beam sputtering techniques [3]. While jm~rovements in ZnO film properties can be pursued through improved deposition techniques and variation of deposition parameters, the question also arises as to whether film properties can be improved through post-deposition thermal treatment. Nearly all films, by whatever method they are produced, are found to be in a state of internal stress (compressive and/or tensile). ZnO thin films are widely used as the active piezoelectric layer far many SAW devices and the internal stress in these films has been found to affect the acoustic velocity and the perfor* First presented at the 7th 11%Conf. on km Beam Modification of Materials. Knoxvilte, Tennessee, USA, Sept. 9-14, 1991. ~~~~-~~3X/91/~~3.5~
mance of the device as a whole [7]. Our results on annealing experiments show stress relief in the films and increased acoustic velocities. The Auger analyses indicate a substantial improvement in the stoichiometry of the films. Our data on sheet resistance measuremcnts afso show that the sheet resistance increased from 6.02 to f1.89 Mf;t,/n. The corresponding variations in the microstructure of the films have been observed l-31.We have used X-ray diffraction analysis to assess the stress by observing the shift of the (002) diffraction peak. ZnO thin films were deposited by an ion-beam sputtering technique. The sputtering system utilized is the Veeco 341. Micro-etch system combined with a Kaufmann type ion source. The system is evacuated with a diffusion pump and liquid nitrogen cold-trap to an initial base pressure of 3 x IO-’ Torr. A mixture of argon and oxygen (75% Ar + 25% Ua) is used as the sputtering gas. The ions are provided by the ion source where an arc discharge between the cathode and the anode takes place. The positive ions of the plasma arc extracted and accelerated by a special multi-grid system with the resulting homogeneous, collimated and monoenergetic ion beam bombarding the surface of a 99.9% pure ZnO ceramic target used as the sputterhrg source. Sputter depositions are done at a chamber pressure of 2 X IF4 Torr and substrate temperature of 325 ’ C. Anncaiing of the ZnO samples was performed in the diffusion furnace. One set of samples was not annealed and two other sets were annealed at a fur-
0 1991 - Efsevier Science Publishers B.V. AI1 rights reserved
S.A. ka, P.K. Ghosh / Ion-beam deposited ZnO thin films
nace temperature of 5000°C for about 3 h. One of these two sets of samples was annealed in pure oxygen ambient and the other in dry nitrogen. We performed X-ray diffraction analysis (28 scan) of the films for stress analysis and the stoichiomet~ of the films was assessed through resistivity measurements and Auger electron spectroscopy. Our efforts on ZnO thin film development mainly addressed the requirements of surface acoustic wave devices. We have grown ZnO films on glass and silicon substrates. Most of our annealing experiments were performed on ZnO films on silicon substrftes with an intermediate layer of 90, (about 1000 A thick and thermally grown). Three sets of ZnO samples were prepared for this investigation all under the same deposition conditions. The internal stress in films deposited by sputtering can be quite appreciable due to the nature of the deposition procedure. This stress can be decomposed into two main components, thermal stress, and intrinsic stress (a = rr,, + oint). The thermal component of the internal stress is a result of mismatch between the thermal expansion coefficients of the film and that of the substrate. Its value can usually be approximated as
[41 where K is a factor that is dependent on both the Young’s modulus of the ZnO film and the difference between the average substrate temperature used in depositing the ZnO film and the ambient temperature under which the thermal stress is measured. (Y, and as are the average coefficients of thermal expansion for the ZnO film and the substrate material, respectively. The cause of the intrinsic stress in the films can often be interpreted in terms of several models. Windischmann [6] proposed a model for predicting the intrinsic stress resulting from ion-beam film deposition as depending on the physical properties of the target, the projectile energy, atomic number, mass, and flux of the ion-beam. In our case, operating at low pressures, the sputtered ZnO molecules and reflected argon/ oxygen atoms impinge on the condensing film with high energies. This is because at lower pressures, the mean-free-path is long and high energy particies arrive at the substrate unattenuated by gas scattering. These high energy incident atoms drive the atoms of the film closer together, imbedding themselves in the film. This mechanism, known as “atomic peening”‘, is believed to cause compression in the films [5]. These energetic particles can penetrate the surface of the film, randomly displacing the atoms from their equilibrium positions through a series of primary and recoil collisions, producing a volumetric distortion. Since the ratio of our deposition temperature to the melting temperature of ZnO is less than 0.2, we can assume that mass
581
10
-.. -
I
i
Unannealed ZnO film Annealed ZnO film with oxygen Annealed ZnO film m dry mtrogen.
.....
ok?-
06
-
04
-
-1 i
a9
Fig. 1. X-ray diffraction results of ZnO film samples.
transport and defect mobility is sufficiently low to freeze the volumetric distortion in place. Fig. 1 is the X-ray diffraction results of an unannealed sample (dashed line) and the sample annealed in the presence of oxygen (solid line). There is no dramatic difference in the X-ray diffraction results of the sample annealed in oxygen and that annealed just in pure nitrogen (dotted line) except the slight increase in the peak intensity of the sample annealed in oxygen ambient (see fig. 1). The (002) diffraction peak of the unannealed sample is shifted to lower 20 values (about 33.8”) than the powder diffraction value of stoichiometric ZnO (34.4” 1. This is because of the large inplane compressive stress which produces d spacing dilation in the growth direction. Comparing the results in fig. 1, observe that the (002) diffraction peak of the annealed films is higher in intensity, narrower in width, and has moved in position (about 0.3”) closer to the ZnO powder value. The increase in the (002) peak intensity and the decrease in half-width of the annealed films indicate evidence of stress relief and that a larger fraction of the material approaches the ideal c-axis normal orientation. We have been able to demonstrate during our preliminary work and in agreement with the work of Hickernell [7] that the internal stress in the piezoelectric layer of the SAW device influences its acoustic velocity. Annealed films with evidence of stress relief gave rise to higher acoustic velocities (up to 4.6% increase was obtained~.
T
S.A. Isa, P.K. Ghosh / Ion-beam deposited ZnO thin films
Table 1 Some of the parameters measured from unannealed [ZnO(UN)], annealed in dry nitrogen [ZnOCANII, and annealed with oxygen [ZnOfAO)] ZnO film samples Film
ZnOGJN) ZnO(AN) ZnO(AO)
Sheet resistance
‘3% Auger peak on the surface
[Mfi,‘ol
Oxygen
Zinc
6.02 6.54 6.89
22 26 28
78 74 72
those annealed
in the presence of oxygen except for the slightly higher average percent peak height of zinc
(74% versus 72%) in the films annealed in dry nitrogen. Observe the variations in the peak heights of zinc and oxygen, especially close to the film surface. Upon annealing, the peak height of zinc dropped from 78 to about 72%. The low resistivity of unannealed films indicates that they are nonstoichiometric and that there is excess zinc in the deposited films. Two possibilities exist [8,9]: that the excess zinc was interstitial and/or there were oxygen vacancies. We are assuming that the presence of oxygen in the annealing process also helps to fill some of the oxygen vacancies resulting in a more stoichiometric film. It should be noted that the percentages quoted here are the Auger peak of each element normalized to the total Auger signal and they are not weight percentages. The Auger intensity for each element is also dependent on the concentration of that element in the material. These figures are not absolute values since neither the Auger scattering efficiency nor the system resolution are accurately accounted for (the system was initially calibrated for Ta,O,). We have chosen 500” C as an annealing temperature. ZnO has a melting temperature (under high pressure) close to 2000 o C. ZnO also sublimates (at atmospheric pressure) around 1950 o C. However, using the existing models for structural changes in annealed polycrystalline materials, we anticipated that at temperatures between 300 and 6OO”C, stress relief would take place. This was confined by our X-ray results. Between 600 and lOOO”C, rec~stallization and major grain growth was anticipated. Our experimental data indicates that for samples annealed even for 6 hours at annealing temperatures up to 425”C, there were no noticeable changes in their diffraction pattern from those of unannealed samples. Only samples annealed for 6 hours and between annealing temperatures of 450 ’ C and 475 ’ C show slight increase in peak intensity and an upward shift of the diffraction peak of about 0.1’. Therefore, using annealing temperatures below 500 OC involves longer annealing times and very small change in the internal stress of the films. Considering the very thin nature of the films (about 0.76 Fm>,
583
higher annealing temperatures can result in the development of microfracture which would result in defect boundaries and considerable acoustic scattering. Consider also that one of the goals of ZnO technology is to integrate it with the existing silicon technology in order to create monolithic devices. One of our objectives, therefore, is that ZnO fabrication procedures must not be detrimental to any active circuitry. These circuitries which are normally metallized prior to film deposition and annealing places an upper limit on the processing temperatures. Certainly an upper limit is 577°C [lo] above which silicon and aluminum will form an eutetic alloy, thus rapidly destroying active devices. In conclusion, the internal stress in the ion-beam deposited ZnO thin film can be decomposed into two main components, thermal stress and intrinsic stress. The thermal component is as a result of mismatch between the thermal expansion ~efficients of the ZnO film and that of the substrate as the sample heats and cools from deposition temperatures. The intrinsic stress can be accounted for as a result of energetic particles bombarding the condensing film and can often be explained via an “atomic peening” model. The results of X-ray diffraction indicate that the (002) diffraction peak of the annealed films is higher in intensity, narrower in width, and has moved about 0.3’ in position closer to the powder diffraction value. These are indications of stress relief and that a larger fraction of the material approaches the ideal c-axis orientation. The results of Auger analysis also indicate that a more stoichiometric ZnO fifm results after undergoing the annealing process. The results show that the Auger peak of zinc was reduced from 78% to about 72% upon annealing, resulting in higher sheet resistance and a more stoichiometric ZnO film. We have chosen 500°C as the annealing temperature to avoid the development of microfracture in the films that could result in defect boundaries and considerable acoustic scattering. The temperature limitation of silicon technology is also a very important consideration for using ZnO thin films for SAW devices integrated with active circuitry on silicon. In this regard, we are currently exploring possibilities of lower annealing temperatures.
Acknowledgement
The authors would like to thank Lois Walsh of RADC/RBRE Griffis AFB, Rome, New York for performing the Auger analysis of the films.
References 111B.T. Klmri-Yakub and G.S. Kino, Appl. Phys. Lett. 25 (1974) 188.
584
S.A. Isa, P.K. Ghosh / Ion-beam deposited ZnO thin films
[2] N. Chubachi, Proc. IEEE 64 (1976) 772.
[3] S.A. Isa, P.K, Ghosh and P.G. Kornreich, Mater. Res. Sot. Symp. Proc. 128 (1989) 725. [4] J. Anderson, in: Use of Thin Films in Physical Investigation (Academic Press, New York, 1966) p. 261. 151 D.W. Hoffman and J.A. Thornton, Thin Solid Films 40 (19771 355. [6] H. Windischmann, J. Appl. Phys. 62 (1987) 1800.
[7] F.J. Hickernell, Proc. IEEE Ultrson. Symp. (1977) p. 309. [8] G. Heiland, E. Mollwo and F. Stockmann, in: Solid State Physics, eds. F. Seitz and D. Turnbull, vol. 8 Academic Press, New York, 1959) p. 191. [9] P.H. Kasai, Phys. Rev. 130 (1963) 989. 1101 D.J. Hamilton and W.G. Howard, Basic Integrated Circuit Engineering (McGraw-Hill, 1975).