- Email: [email protected]
New results using capacitance transient studies to investigate deep defect relaxation in hydrogenated amorphous silicon
ELSWIER
Journal of Non-Crystalline
Solids 198-200
(1996) 530-534
New results using capacitance transient studies to investigate deep defect relaxation in hydrogenated amorphous silicon Daewon Kwon Department
*,
Adam Gardner, J. David Cohen
ofPhysics, lJnicersi@ of Oregon, Eugene, OR 97403, USA
Abstract Recent studies of deep defect relaxation processes in a-Si:H using capacitance transient measurements are reported. First, to disprove any significant role of from contact effects, nearly identical transients for samples deposited on pt crystalline Si with a blocking back contact or on n+ crystalline Si substrates with single junction characteristics have been obtained. Similar transients are also obtained for a film deposited on a thin n+ a-Si:H layer over a Cr coated substrate. Second, the effects of incorporating temperature steps during the time evolution of the transients have been investigated. Such transients are compared to the case where the emission of charge is allowed to proceed isothermally. These results verify the existence of a defect distribution whose thermal emission properties evolve in time.
1. Introduction Recent capacitance transient studies of deep defect properties in lightly doped a-Si:H have revealed an unusual relaxation process such that the emission rate of trapped defect electrons varies inversely with their occupation time [ 1,2]. More recently, some other workers have suggested that the unusual behavior may be due to contact effects rather than the properties of defects in the bulk a-Si:H [3]. Others have suggested that the observed behavior might arise from thermalization within a fixed distribution of defects rather than the relaxation of the defects themselves. In this paper we examine such issues by comparing results on samples with a variety of contacts. Also, we present some capacitance transient
* Corresponding author. Tel.: + l-503 346 4774; fax: + l-503 346 3422; e-mail: [email protected]. 0022-3093/96/$15.00 Copyright SSDI 0022-3093(95)00753-9
data using temperature switching during the defect emission to conclusively rule out the thermalization hypothesis.
2. Experimental
details
We deposited 2.9 km thick, 12 Vppm PH, doped a-Si:H films simultaneously on p+ and on n’ crystalline silicon (c-S8 substrates by the glow discharge method. Another set of samples of 1.4 ym thickness with a 1 Vppm doping level were deposited by the glow discharge method at the University of Chicago onto n+a-Si:H/Cr coated substrates. These substrates were prepared by first evaporating Cr onto p+c-Si, followed by the deposition of 100 nm of heavily phosphorus doped a-Si:H (1 at.% PH,). All devices were completed by evaporating semitransparent Pd Schottky contacts of area 0.002 cm2 onto the top surface. These devices were examined in a
0 1996 Elsevier Science B.V. All rights reserved.
D. Kwon
et al. / .Iournal
qf Non-Crystalline
dark annealed state as well as metastable states produced by light soaking followed by partial anneals at elevated temperatures. Capacitance transients, C(t), were obtained using a I kHz measurement frequency. The barrier region was reverse biased at -2 V, and this bias was reduced to 0 V (or in some cases - 1 V) to collapse the depletion region to allow defect filling. The effects of varying the duration of this ‘filling pulse’ on the capacitance transients was studied. To obtain a quantity more directly related to the remaining defect electronic occupation we have usually used the depletion approximation to define No(t) = C*(m) - C2(t), where C(s) represents the asymptotic value of capacitance at long times following the filling pulse. The effects of an abrupt temperature change during these capacitance transients have also been studied. Here, as described previously [4], we employed two constant temperature oil baths with a probe assembly which allowed the oil from either bath to be shunted into a small sample chamber. This allows a 20 K temperature change to be carried out within a few seconds. There is also a simple procedure which allows us to obtain N,(t) in these latter experiments always in the same arbitrary units (independent of the variations in the sample temperature). We simultaneously record the sample temperature during the switch. From the theory of the junction capacitance for amorphous semiconducting devices [5], there is only one parameter, independent of the applied bias or degree of deep charge emission, required to convert the measured capacitance at one temperature to its equivalent value at another temperature (provided the film is spatially uniform). The variation of this single parameter with temperature is easily obtained; for example, by recording the dependence of capacitance with temperature at a fixed dc bias.
3. Results 3.1. @jkYs
ofcontacts
To disprove the proposal that blocking contacts might in any way be responsible for the temporal
Solids
198-200
11996)
530-534
Fig. 1. Comparison of defect charge transients for two 12 Vppm PH, doped samples, one deposited onto n’ c-Si with an ohmic back contact, and the other deposited onto p+ c-Si with a barrier back contact. For the former (solid lines) the Schottky contact is biased at -2 V and the filling pulse reduces this bias to zero; in the latter case (open symbols) the p+ back contact is held under -2 V reverse bias with a zero bias filling pulse.
shifts in the capacitance transients, we have carried out two tests. First, we compared results on the films deposited simultaneously on p+ and n+ crystalline Si substrates. The n’c-Si device exhibited fairly good single junction rectifying behavior with a forward to reverse conductance ratio at kO.5 V of nearly 50. The p’c-Si substrate device exhibited Z-V characteristics indicative of blocking contacts on both sides. The two devices were exposed to light simultaneously and then annealed at 400 K. This treatment resulted in an activation energy for conduction of 0.67 + 0.02 eV for both. Capacitance transients for both structures were obtained at 330 K. The pfc-Si device transients were recorded both under reverse bias conditions for the substrate barrier and for the top Schottky. (These transients were almost indistinguishable!) Fig. 1 overlays the results for the n+c-Si top Schottky barrier with those obtained for the substrate barrier of the p+c-Si device. The nearly identical transients indicate that, although the barrier contacts were very different, and although the second case requires the depletion charge to be delivered through a forward biased Schottky contact, these contacting details do not affect the experimental results.
532
D. Kwon et al. / Journal @‘Non-Crystalline
The same qualitative transient behavior has also been obtained for the sample deposited on the chromium coated substrate with the n+a-Si:H layer. This device was light soaked and annealed such that the conductivity activation energy was 0.57 eV. (This sample still exhibited a nearly 20000 conductance ratio at +0.3 V in this state.) Fig. 2 shows the capacitance and associated depletion charge transients for this sample. Fig. 3 displays the integrated current during the filling pulse and indicates that, to within 14%, all of the charge is delivered to the collapsed depletion region within 0.2 ms. The time scale for the subsequent thermal emission (see Fig. 2) varies by over a factor of 100 for filling pulses between 1 ms and 10 s. Moreover, for pulses of 10 ms or longer the capacitance values at 10 ms following the restoration of reverse bias are nearly identical: 56 + 1.5 pF, implying that the initial depletion charge is the same to within +6%.
1 04
c n
‘1
“’ ‘1 “““I “I’ ‘_nl 1 Vppm PH3 doped
” “’ 77.
iii”,”
-.-.-
1Oms
i
80
c3
BIAS VOLTAGE 5oI
o-2
IO-’
100
10’
102
(volts)
103
TIME (seconds) Fig. 2. Capacitance and charge transients for the Chicago I Vppm PH, doped sample grown on Cr with a heavily doped n+ a-Si:H layer. The I-V curve indicating excellent single junction behavior is shown in the inset.
Solids 198-200 (1996) 530-534 15
,,,,,
m,
,,,
1
,.,
_- __-------.__-.__
1o-7 1O-6 1 o-5
1 o-4
,,
VppmPHsdoped 330K
10-S
1 O-2
,,,,,,,
250
: : 200 0 1
El50 ,__.^“,,’
,P
-100
t[r
.50
a I 0
1 o-1
, go
TIME (seconds) Fig. 3. Current and total charge delivered during filling pulse to Schottky barrier region under the same conditions used in Fig. 2. This implies all of the defect states are filled within less than 100 p”s.
3.2. Efsects of temperature
switching
Recently [4,6] we reported results of capacitance transient measurements in which the temperature was abruptly altered during the course of the thermal emission. Such experiments are valuable for two reasons. First, they help discriminate against alternative mechanisms that might be responsible for the anomalous transient behavior we observe. Second, in an isothermal experiment thermal emission and configurational changes occur in direct competition. This means that, if relaxation is relatively slow only the shallower state emission will be observed whereas, if relaxation is relatively fast only the deeper state properties will be seen (see Fig. 4). This means that configurational changes are generally quite difficult to observe in capacitance transient experiments unless conditions are exactly correct. This may account for our need to carefully select the conductivities of our samples to obtain the clearest evidence of the relaxation behavior. However, by switching the temperature during the emission phase we can alter the relative rates of relaxation and emission mid-stream and thus observe behaviors that more clearly indicate configurational switching. In Fig. 5 we show the direct comparison of three such capacitance transients: two isothermal transients using long (10 s) filling pulses at 3 10 K and 330 K, respectively, and one in which the pulse filling and first 1 s of thermal emission occurred at 310 K before the temperature was switched to 330
D. Kwon et al. /Journal
c?fNon-Cryrtalline Solids 198-200 (19961 530-534 4. Discussion
CONFIGURATIONAL
1000/T Fig. 4. Schematic of possible relaxation process between two defect configurations. The solid lines indicate the variation of the emission rates with temperature for each configuration and the dashed line represents the rate of the configurational relaxation process. The open circles at (A) and (B) indicate that, in general, the behavior of only a single configuration will be observable in a capacitance transient measurement; however, during a temperature switch (arrows) the configurational change may be more easily visible.
K. We observe that, in this case, the initial part of the transient follows the 310 K isothermal curve then, shortly after the temperature switch, the charge is emitted more quickly than the isothermal 330 K case. After a few seconds this emission slows down so that, at long times, the curve nearly joins up with the isothermal 330 K curve.
1 Vppm PHsdoped
10-a
10-Z
10-1
100
10'
533
102
103
104
TIME (seconds) Fig. 5. Results of isothermal transients at 3 10 K (dashed line) and 330 K (solid line) along with a transient in which the filling and first I s of the emission proceeds at 310 K before the temperature is abruptly switched to 330 K. The temperature profile during the recording of this last transient is indicated at the bottom.
The nearly identical transients for the sample devices with and without blocking back contacts, illustrated in Fig. I, makes a strong case that the observed behavior must arise from bulk defect properties. Also. we have demonstrated that we can observe essentially the same behavior on a sample device identical in structure to that reported on by workers attempting to identify such effects with a contact artifact [3]. Indeed, these latter data, exhibited in Figs. 2 and 3, provide an extremely simple but powerful argument for defect relaxation. The 10 ms and 10 s pulse cases exhibit identical (to within the measurement error) values of integrated current during the filling pulse. This means that the charge delivered into the deep defects is the same. These cases also exhibit nearly identical capacitance values following the filling pulse. One can show that, based only on the validity of Poisson’s equation (and also that the device capacitance value is not pinned at the geometric value of the sample thickness), this implies the first moment of charge within the depletion region is the same. That is, for the 10 ms and the 10 s pulse cases we hate the same number of defect electrons delivered with the same spatial distrihution. However, the emission time varies by over an order of magnitude. Such a situation clearly requires that the defect emission properties themselves have been changed: i.e., some configurational relaxation has taken place. Finally, the temperature switching data of Fig. 5 provides additional insight into the switching process. These data are consistent with those presented in earlier reports [4,6] but, in this case, have been obtained on a sample with good single junction characteristics. In all cases we observe that, if the temperature is raised during the emission phase, the depletion charge is lost more quickly than for the isothermal case at higher temperature. Such behavior is quite inconsistent with the expected behavior of thermalization within a fixed density of states. In Fig. 5 it is also interesting that, after this initial faster release, the emission nearly ceases for awhile, then resumes so that it approaches the 330 K isothermal case. A possible explanation is suggested by Fig. 4. Initially, as the temperature rises, the emission rate increases before a configuration change is fast enough
534
D. Kwon et al. /Journal
of Non-Crystalline
to occur. Hence the loss of charge is accelerated. However, after the crossing point is reached the emission rate drops to the lower curve. Then we must wait until a much later time before appreciable charge loss is resumed. This explanation is undoubtedly too simple as it incorporates only a single configurational jump. However, for a sufficiently small temperature jump it may offer a qualitatively correct picture.
Acknowledgements The authors would like to thank Professor Hellmut Fritzsche for providing some of the samples for
Solids 198-200
(1996) 530-534
this study. Research at Oregon NSF Grant DMR9208334.
was supported
by
References [l] J.D. Cohen, T.M. Leen and R.J. Rasmussen, Phys. Rev. Lett. 69 (1992) 3358. [2] M.W. Carlen, Y. Xu and R.S. Crandall, Phys. Rev. B51 (1995) 2173. [3] W.B. Jackson and N.M. Johnson, Phys. Rev. B52 (1995) 2233. [4] A. Gardner and J.D. Cohen, Mater. Res. Sot. Symp. Proc. 336 (1994) 207. [5] J.D. Cohen and D.V. Lang, Phys. Rev. 825 (1982) 5321. [6] J.D. Cohen and T.M. Leen, Phys. Rev. Lett. 73 (1994) 366.
Journal of Non-Crystalline
Solids 198-200
(1996) 530-534
New results using capacitance transient studies to investigate deep defect relaxation in hydrogenated amorphous silicon Daewon Kwon Department
*,
Adam Gardner, J. David Cohen
ofPhysics, lJnicersi@ of Oregon, Eugene, OR 97403, USA
Abstract Recent studies of deep defect relaxation processes in a-Si:H using capacitance transient measurements are reported. First, to disprove any significant role of from contact effects, nearly identical transients for samples deposited on pt crystalline Si with a blocking back contact or on n+ crystalline Si substrates with single junction characteristics have been obtained. Similar transients are also obtained for a film deposited on a thin n+ a-Si:H layer over a Cr coated substrate. Second, the effects of incorporating temperature steps during the time evolution of the transients have been investigated. Such transients are compared to the case where the emission of charge is allowed to proceed isothermally. These results verify the existence of a defect distribution whose thermal emission properties evolve in time.
1. Introduction Recent capacitance transient studies of deep defect properties in lightly doped a-Si:H have revealed an unusual relaxation process such that the emission rate of trapped defect electrons varies inversely with their occupation time [ 1,2]. More recently, some other workers have suggested that the unusual behavior may be due to contact effects rather than the properties of defects in the bulk a-Si:H [3]. Others have suggested that the observed behavior might arise from thermalization within a fixed distribution of defects rather than the relaxation of the defects themselves. In this paper we examine such issues by comparing results on samples with a variety of contacts. Also, we present some capacitance transient
* Corresponding author. Tel.: + l-503 346 4774; fax: + l-503 346 3422; e-mail: [email protected]. 0022-3093/96/$15.00 Copyright SSDI 0022-3093(95)00753-9
data using temperature switching during the defect emission to conclusively rule out the thermalization hypothesis.
2. Experimental
details
We deposited 2.9 km thick, 12 Vppm PH, doped a-Si:H films simultaneously on p+ and on n’ crystalline silicon (c-S8 substrates by the glow discharge method. Another set of samples of 1.4 ym thickness with a 1 Vppm doping level were deposited by the glow discharge method at the University of Chicago onto n+a-Si:H/Cr coated substrates. These substrates were prepared by first evaporating Cr onto p+c-Si, followed by the deposition of 100 nm of heavily phosphorus doped a-Si:H (1 at.% PH,). All devices were completed by evaporating semitransparent Pd Schottky contacts of area 0.002 cm2 onto the top surface. These devices were examined in a
0 1996 Elsevier Science B.V. All rights reserved.
D. Kwon
et al. / .Iournal
qf Non-Crystalline
dark annealed state as well as metastable states produced by light soaking followed by partial anneals at elevated temperatures. Capacitance transients, C(t), were obtained using a I kHz measurement frequency. The barrier region was reverse biased at -2 V, and this bias was reduced to 0 V (or in some cases - 1 V) to collapse the depletion region to allow defect filling. The effects of varying the duration of this ‘filling pulse’ on the capacitance transients was studied. To obtain a quantity more directly related to the remaining defect electronic occupation we have usually used the depletion approximation to define No(t) = C*(m) - C2(t), where C(s) represents the asymptotic value of capacitance at long times following the filling pulse. The effects of an abrupt temperature change during these capacitance transients have also been studied. Here, as described previously [4], we employed two constant temperature oil baths with a probe assembly which allowed the oil from either bath to be shunted into a small sample chamber. This allows a 20 K temperature change to be carried out within a few seconds. There is also a simple procedure which allows us to obtain N,(t) in these latter experiments always in the same arbitrary units (independent of the variations in the sample temperature). We simultaneously record the sample temperature during the switch. From the theory of the junction capacitance for amorphous semiconducting devices [5], there is only one parameter, independent of the applied bias or degree of deep charge emission, required to convert the measured capacitance at one temperature to its equivalent value at another temperature (provided the film is spatially uniform). The variation of this single parameter with temperature is easily obtained; for example, by recording the dependence of capacitance with temperature at a fixed dc bias.
3. Results 3.1. @jkYs
ofcontacts
To disprove the proposal that blocking contacts might in any way be responsible for the temporal
Solids
198-200
11996)
530-534
Fig. 1. Comparison of defect charge transients for two 12 Vppm PH, doped samples, one deposited onto n’ c-Si with an ohmic back contact, and the other deposited onto p+ c-Si with a barrier back contact. For the former (solid lines) the Schottky contact is biased at -2 V and the filling pulse reduces this bias to zero; in the latter case (open symbols) the p+ back contact is held under -2 V reverse bias with a zero bias filling pulse.
shifts in the capacitance transients, we have carried out two tests. First, we compared results on the films deposited simultaneously on p+ and n+ crystalline Si substrates. The n’c-Si device exhibited fairly good single junction rectifying behavior with a forward to reverse conductance ratio at kO.5 V of nearly 50. The p’c-Si substrate device exhibited Z-V characteristics indicative of blocking contacts on both sides. The two devices were exposed to light simultaneously and then annealed at 400 K. This treatment resulted in an activation energy for conduction of 0.67 + 0.02 eV for both. Capacitance transients for both structures were obtained at 330 K. The pfc-Si device transients were recorded both under reverse bias conditions for the substrate barrier and for the top Schottky. (These transients were almost indistinguishable!) Fig. 1 overlays the results for the n+c-Si top Schottky barrier with those obtained for the substrate barrier of the p+c-Si device. The nearly identical transients indicate that, although the barrier contacts were very different, and although the second case requires the depletion charge to be delivered through a forward biased Schottky contact, these contacting details do not affect the experimental results.
532
D. Kwon et al. / Journal @‘Non-Crystalline
The same qualitative transient behavior has also been obtained for the sample deposited on the chromium coated substrate with the n+a-Si:H layer. This device was light soaked and annealed such that the conductivity activation energy was 0.57 eV. (This sample still exhibited a nearly 20000 conductance ratio at +0.3 V in this state.) Fig. 2 shows the capacitance and associated depletion charge transients for this sample. Fig. 3 displays the integrated current during the filling pulse and indicates that, to within 14%, all of the charge is delivered to the collapsed depletion region within 0.2 ms. The time scale for the subsequent thermal emission (see Fig. 2) varies by over a factor of 100 for filling pulses between 1 ms and 10 s. Moreover, for pulses of 10 ms or longer the capacitance values at 10 ms following the restoration of reverse bias are nearly identical: 56 + 1.5 pF, implying that the initial depletion charge is the same to within +6%.
1 04
c n
‘1
“’ ‘1 “““I “I’ ‘_nl 1 Vppm PH3 doped
” “’ 77.
iii”,”
-.-.-
1Oms
i
80
c3
BIAS VOLTAGE 5oI
o-2
IO-’
100
10’
102
(volts)
103
TIME (seconds) Fig. 2. Capacitance and charge transients for the Chicago I Vppm PH, doped sample grown on Cr with a heavily doped n+ a-Si:H layer. The I-V curve indicating excellent single junction behavior is shown in the inset.
Solids 198-200 (1996) 530-534 15
,,,,,
m,
,,,
1
,.,
_- __-------.__-.__
1o-7 1O-6 1 o-5
1 o-4
,,
VppmPHsdoped 330K
10-S
1 O-2
,,,,,,,
250
: : 200 0 1
El50 ,__.^“,,’
,P
-100
t[r
.50
a I 0
1 o-1
, go
TIME (seconds) Fig. 3. Current and total charge delivered during filling pulse to Schottky barrier region under the same conditions used in Fig. 2. This implies all of the defect states are filled within less than 100 p”s.
3.2. Efsects of temperature
switching
Recently [4,6] we reported results of capacitance transient measurements in which the temperature was abruptly altered during the course of the thermal emission. Such experiments are valuable for two reasons. First, they help discriminate against alternative mechanisms that might be responsible for the anomalous transient behavior we observe. Second, in an isothermal experiment thermal emission and configurational changes occur in direct competition. This means that, if relaxation is relatively slow only the shallower state emission will be observed whereas, if relaxation is relatively fast only the deeper state properties will be seen (see Fig. 4). This means that configurational changes are generally quite difficult to observe in capacitance transient experiments unless conditions are exactly correct. This may account for our need to carefully select the conductivities of our samples to obtain the clearest evidence of the relaxation behavior. However, by switching the temperature during the emission phase we can alter the relative rates of relaxation and emission mid-stream and thus observe behaviors that more clearly indicate configurational switching. In Fig. 5 we show the direct comparison of three such capacitance transients: two isothermal transients using long (10 s) filling pulses at 3 10 K and 330 K, respectively, and one in which the pulse filling and first 1 s of thermal emission occurred at 310 K before the temperature was switched to 330
D. Kwon et al. /Journal
c?fNon-Cryrtalline Solids 198-200 (19961 530-534 4. Discussion
CONFIGURATIONAL
1000/T Fig. 4. Schematic of possible relaxation process between two defect configurations. The solid lines indicate the variation of the emission rates with temperature for each configuration and the dashed line represents the rate of the configurational relaxation process. The open circles at (A) and (B) indicate that, in general, the behavior of only a single configuration will be observable in a capacitance transient measurement; however, during a temperature switch (arrows) the configurational change may be more easily visible.
K. We observe that, in this case, the initial part of the transient follows the 310 K isothermal curve then, shortly after the temperature switch, the charge is emitted more quickly than the isothermal 330 K case. After a few seconds this emission slows down so that, at long times, the curve nearly joins up with the isothermal 330 K curve.
1 Vppm PHsdoped
10-a
10-Z
10-1
100
10'
533
102
103
104
TIME (seconds) Fig. 5. Results of isothermal transients at 3 10 K (dashed line) and 330 K (solid line) along with a transient in which the filling and first I s of the emission proceeds at 310 K before the temperature is abruptly switched to 330 K. The temperature profile during the recording of this last transient is indicated at the bottom.
The nearly identical transients for the sample devices with and without blocking back contacts, illustrated in Fig. I, makes a strong case that the observed behavior must arise from bulk defect properties. Also. we have demonstrated that we can observe essentially the same behavior on a sample device identical in structure to that reported on by workers attempting to identify such effects with a contact artifact [3]. Indeed, these latter data, exhibited in Figs. 2 and 3, provide an extremely simple but powerful argument for defect relaxation. The 10 ms and 10 s pulse cases exhibit identical (to within the measurement error) values of integrated current during the filling pulse. This means that the charge delivered into the deep defects is the same. These cases also exhibit nearly identical capacitance values following the filling pulse. One can show that, based only on the validity of Poisson’s equation (and also that the device capacitance value is not pinned at the geometric value of the sample thickness), this implies the first moment of charge within the depletion region is the same. That is, for the 10 ms and the 10 s pulse cases we hate the same number of defect electrons delivered with the same spatial distrihution. However, the emission time varies by over an order of magnitude. Such a situation clearly requires that the defect emission properties themselves have been changed: i.e., some configurational relaxation has taken place. Finally, the temperature switching data of Fig. 5 provides additional insight into the switching process. These data are consistent with those presented in earlier reports [4,6] but, in this case, have been obtained on a sample with good single junction characteristics. In all cases we observe that, if the temperature is raised during the emission phase, the depletion charge is lost more quickly than for the isothermal case at higher temperature. Such behavior is quite inconsistent with the expected behavior of thermalization within a fixed density of states. In Fig. 5 it is also interesting that, after this initial faster release, the emission nearly ceases for awhile, then resumes so that it approaches the 330 K isothermal case. A possible explanation is suggested by Fig. 4. Initially, as the temperature rises, the emission rate increases before a configuration change is fast enough
534
D. Kwon et al. /Journal
of Non-Crystalline
to occur. Hence the loss of charge is accelerated. However, after the crossing point is reached the emission rate drops to the lower curve. Then we must wait until a much later time before appreciable charge loss is resumed. This explanation is undoubtedly too simple as it incorporates only a single configurational jump. However, for a sufficiently small temperature jump it may offer a qualitatively correct picture.
Acknowledgements The authors would like to thank Professor Hellmut Fritzsche for providing some of the samples for
Solids 198-200
(1996) 530-534
this study. Research at Oregon NSF Grant DMR9208334.
was supported
by
References [l] J.D. Cohen, T.M. Leen and R.J. Rasmussen, Phys. Rev. Lett. 69 (1992) 3358. [2] M.W. Carlen, Y. Xu and R.S. Crandall, Phys. Rev. B51 (1995) 2173. [3] W.B. Jackson and N.M. Johnson, Phys. Rev. B52 (1995) 2233. [4] A. Gardner and J.D. Cohen, Mater. Res. Sot. Symp. Proc. 336 (1994) 207. [5] J.D. Cohen and D.V. Lang, Phys. Rev. 825 (1982) 5321. [6] J.D. Cohen and T.M. Leen, Phys. Rev. Lett. 73 (1994) 366.