SIMS analysis using ejected CsX+ ions

SIMS analysis using ejected CsX+ ions

Nuclear Instruments and Methods in Physics Research B73 (1993) 214-220 North-Holland NOMIB Beam Interactions with Materials 8 Atoms SIMS analysis u...

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Nuclear Instruments and Methods in Physics Research B73 (1993) 214-220 North-Holland

NOMIB

Beam Interactions with Materials 8 Atoms

SIMS analysis using ejected CsX+ ions Edward W. Thomas School of Physics Georgia Tech, Atlanta, GA 30332, USA

Abbas Torabi Research Institute Georgia Tech, Atlanta, GA 30332, USA

Received 16 July 1992 and in revised form 4 September 1992

A structure consisting of Ge/Pd/GaAs is analyzed with the SIMS technique using a Cs+ probe. Comparison is made between the signals of ejected ions X+ and ejected diatomics CsX+. It is proposed that CsX + is formed by an ion molecule reaction between emerging sputtered neutral X and surface Csf. Depth profiles of Ge, Pd, Ga and As monitored by the species CsX+ are shown to be free of matrix effects at interfaces and the species exhibit approximately equal detection efficiencies. Use of the CsXc signals to study an annealed Ge/Pd/GaAs structure clearly shows the segregation of Ge to the GaAs interface.

1. Introduction

Secondary ion mass spectroscopy (SIMS) is a widely used technique for studying distributions of elemental concentration with depth in materials. Generally electronegative species are analyzed by sputtering with Cs+ and detection of negative ions; electropositive species are analyzed by sputtering with 0: and detection of positive ions. The technique is most useful, and the results least ambiguous, when it is used to record relative distributions with depth of dilute dopants in an otherwise uniform matrix. Disadvantages of the technique include wide variations in relative detection sensitivity between different elements and poorly understood variations in signal levels at interfaces between matrixes. The practices and problems of SIMS are well documented in a number of major reviews (see for example refs. [ 1,211. It has been proposed (see for example refs. [3-61) that there may be some advantages to recording distributions of a material X by sputtering with Cs+ and monitoring the ejected flux of CsX+. The present work studies this possil$lity for profiliag a structure consisting of Ge (1250 A) on Pd (500 A) on n+-GaAs. After annealing to create a GePd alloy the surface layers provide a useful ohmic contact. This structure, particularly in its annealed form, has been extensively studied using oxygen induced positive

Correspondence

to: E.W. Thomas, Georgia Institute of Technology, Atlanta, GA 30332-0430, USA.

0168-583X/93/$06.00

ion SIMS by Palmstrom et al. [7]. They were interested in detecting the presence of a segregated layer of Ge, in annealed samples, at the interface with the underlying GaAs. To circumvent the difficulties associated with oxygen induced segregation and collisionally induced interface broadening they performed the SIMS from the backside of chemically thinned samples. In the present work the same structure is analyzed but from the front surface utilizing a Cs+ probe and detecting a species X both by its monatomic species X+ and also in a molecular combination CsX+. It will be shown that by detection in the form CsX+ rather than X+ the sensitivities for certain species are enhanced, sensitivity variations between species are reduced considerably and unexplained variations of signals at interfaces are largely eliminated. In particular there is no difficulty in detecting and measuring the width of the Ge layer which segregates to the interface in an annealed sample.

2. Experiment A sample of GaAs was grown in a Varian GENII MBE system. On a (100) oriented SI GaAs wafer a 0.1 p,rn undoped buffer layer was grown followed by 0.3 pm of Si doped layer. The growth rate for these layers was measured by a RHEED intensity oscillation technique. After removal from the MBE system the sample was put in a plasma system to grow a thin oxide layer. After etching in HCL and HF, rinsing and drying, the

0 1993 - Elsevier Science Publishers B.V. All rights reserved

E. W. Thomas, A. Torabi / SIMS analysis using ejected CsX + ions

sample was loaded into a! e-beam evaporation system for deposition of a 500 A layer of Pd followed by a 1265 A layer of Ge following Marshall [8]. Layer thickness was measured by a quartz resonator. The sample was cleaved into several segments for further treatment and study. The present paper will show results for a sample that was heat treated at 325°C for 30 min in a AET model RM-4 rapid anneal processing system and also results for a sample which was not heat treated. The thermally annealed samples were characterized for their electrical properties using the transmission line method for determining ohmic contact resistance. The TLM structure was made by lithography and lift off techniques. A contact resistivity of 1.32 X 10m6 fi cm* was obtained. SEM evaluation showed a clear and smooth demarkation at the interface between the MBE layer and the subsequent metallic layers without any evidence of metal semiconductor interdiffusion. Detailed results of the electrical measurements will be published elsewhere. SIMS was perfomed in an ATOMIKA ADIDAsystem at a base pressure of 3 x 10e9 Torr. The projectile beam was Cs+, generally at an energy of 12 keV, directed onto the sample at an angle of 45” to the normal. Ejected ions are energy filtered, mass analyzed in a quadrupole, and counted on an electron multiplier. Some care was necessary in setting the energy filter to pass the peak signal of the ejected species of interest. It was observed that ejected atomic species X’ exhibited a broad energy distribution, while the molecular species CsX+ has a narrow, low, energy distribution; setting of the filter pass energy were different for the two species. Mass spectrometer settings were adjusted to provide adequate resolution between adjacent mass peaks. Preliminary analyses showed no significant signals from contaminants. Typical beam currents were 20 nA, spot size 20 urn; the beam was rastered over an area of 200 x 280 urn and the detection system gated to record only ions ejected from the central 30% of the rastered area. For comparison purposes we also performed some analyses with 0: beams at normal incidence to the surface but otherwise similar conditions. No attempt was made to correct for variations of mass spectrometer transmission with ion mass.

3. Results In figs. 1, 2, and 3 we show the raw signals of Ge, Pd, Ga and As detected as positive ions induced by 0: (fig. 1) as positive ions induced by Csf (fig. 2) and positive CsX ions induced by Cs+ (fig. 3). SIMS is of course mass sensitive and we are in fact detecting at the mass 69 isotope of Ga, mass 73 of Ge (except in fig. 1 where mass 70 was used), 75 of As and 105 of Pd.

0

I

215

20

30

DEPTH(ARBlTRARYUNlTS)

Fig. 1. SIMS profiles of Ge, Pd, Ga and As in an unannealed Ge/Pd/GaAs structure using a 12 keV 0: beam and recording monatomic ejected ions Xc. Signals are as recorded. Depth is in arbitrary units. As explained in the text oOneunit of depth is believed to represent approximately 100 A. Note particularly the peaks of the Ge and Pd signals at depth 10.0, the approximate location of the Ge/Pd interface.

Signals are recorded as a function of time and time is in turn related to depth. To facilitate comparison of the various figures we have replaced the time axis with an approximate depth scale. The location where the signal of CsGa+ rises to one half of its value deep in the GaAs matrix has been assigned the value of 17.5 depth units. The total thickness of the0 deposited Ge plus Pd layers are designed to be 1750 A as monitored by weight gain during the deposition process so that one unit of the arbitrary depth scale represents approximately 100 A. The sole exception to this procedure is for the data taken with the 0: probe in fig. 1 where we have assigned the 17.5 unit of depth location to the point where the Ga+ signal rises to one half of its equilibrium value. This is only an approximate depth scale and a more precise assessment would require consideration of the differences in sputtering rates between the different materials and any interfacial compounds that might exist. We would note that the data for ejection of X+ by Cs+ and ejection of CsX+ by Cs+ (figs. 2 and 3) were taken in the same run and therefore under identical conditions; the depth axis scale for the two figures is identical. Despite the fact that the densities of Ge, Pd and GaAs, in their respective layers, are approximately the

E. W Thomas, A. Torabi / SIMS analysis using ejected CsX + ions

216

same the signal levels in the ejected X+ spectra (figs. 1 and 2) vary by more than two orders of magnitude. By contrast the CsX+ signals (fig. 3) are all of the same order of magnitude and some of the difference can be related to our detection at only one isotope; for example we are detecting at the mass 73 isotope of Ge which has an abundance of only 7.8%. While the raw signal levels for Ga and As were reduced when deby an tected as CsX+ the signal of Pd is increased order of magnitude when detected as CsPd+ indicating an enhanced detection sensitivity. It is noted that certain of the signals of monatomic ions X+ show peaks at interfaces. In the profile persuch peaks are seen in the Pd and formed using 0: Ge signals at the Pd/Ge interface and in Pd at the Pd/GaAs interface; smaller peaks in the Pd signals are seen in the profile performed with Csf at both interfaces and in the Ga signal at the Pd/GaAs interface. These peaks cannot in any way be related to an increase in species density. In each case the peak rises above the signal level experienced when the species X is pure and at its maximum density; if anything the species X at an interface is mixed with the neighboring material and density should be decreased. The peaks

,.r’------__

Il~l,,l,ll,l~llll~IllI)lIll~ 0

!I

10 DEPTH

II

20

2,

30

(ARBITRARY UNITS)

Fig. 2. SIMS profiles of Ge, Pd Ga and As in an unannealed Ge/Pd/GaAs structure using a 12 keV Cs+ beam and recording monatomic ejected ions Xf. Signals are as recorded. Depth is in arbitrary units. As explained in the text oOne unit of depth is believed to represent approximately 100 A. Note particularly the peaks of Pd and Ga signals at about depth 17.5, the approximate location of the Pd/GaAs interface.

so0

l~llll~llll~llll~llll~Illl~’ 5

10

!I

20

25

JO

DEPTH (ARBITRARY UNITS) Fig. 3. SIMS profiles of Ge, Pd, Ga and As in an unannealed Ge/Pd/GaAs structure using a 12 keV Cs+ beam and recording diatomic ejected ions CsX+. Signals are as recorded. Depth is in arbitrary units. As explained in the text one unit of depth is believed to represent approximately 100 A. Note the absence of any peaks in signals at depths 10.0 and 17.5, the approximate location of the interfaces.

are therefore spurious. None of these peaks at all are seen in the CsX+ signals. It is clear, from fig. 3 that by detecting a species in for the form CsX+ the raw data, before correction detection sensitivity, gives a clearer representation. of the actual density of species. Matrix dependent differences in sensitivities have been reduced and certain spurious signals at interfaces have been eliminated. Wittmaack [3] anticipates the value of using CsX+ signals to reduce matrix dependent sensitivity differences. He recommends that a Cs’ incidence angle of 60” or more be used to minimize such differences but his own work indicates that an angle of 45”, as used here, should be quite satisfactory. In each of figs. 1, 2, and 3 there is a clear indication that some Pd signal arises from the region beyond depth unit 17.50 where the material is expected to be GaAs. The tail is very long for the 0: impact and rather shorter for Cs+ impact. The tail is also apparent in the work of Palmstrom et al. [7] when profiling from the front of the sample (as we do here) but is completely absent when profiling from the back of the sample. This tail may therefore be attributed to collisionaly related effects such as radiation enhanced diffusion or knock on. The longer tail for 0: impact than

217

E. W. Thomas, A. Torabi / SIMS analysis using ejected CsX + ions

for Cs+ impact is consistent with the range of 0: in the material being greater than for the heavier Cs+. Use of Cs+ rather than 0: does have some advantage in reducing the collisional distortion but by no means eliminates the problem. The tail observed when detecting X+ is the same as when detecting CsX+. Signals of CsX+ ejected by Cs+ apparently provide a less confusing indication of species density than do signals from the monatomic Xf, whether that is ejected by 0: or by Cs+. To indicate the clarity of the presentation we show in fig. 4 the data of fig. 3 with each signal, at the center of the region where it is designed to be uniform, normalized to the same level. The depth profiles resulting from a sample which had been annealed at 325°C for 30 min are shown in fig. 5 again with each signal level normalized to the same value. This latter figure shows quite clearly that a GePd alloy has been formed close to the surface but that there is an excess of Ge at the interface between the GePd alloy and the underlying GaAs. This is just as was demonstrated by Palmstrom et al. [7], using the back thinning and back analysis technique. In fig. 6 we have shown the difference between the normalized Ge and Pd signals of fig. 5 and this we propose indicates the excess concentration of Ge at the GaAs interface. The deposited thicknesses of Ge and Pd provide 2.29 x 1017 atoms per cm* of area excess Ge over the Pd deposit. For a pure crystalline form of Ge this excess

100

t

I”“I”“I~“‘l”“l”“l”“1

0

I

. 10

IS

20

25

20

DEPl-H (ARBlTRARYUNITS) Fig. 4. SIMS profiles of Ge, Pd and Ga in an unannealed Ge/Pd/GaAs structure. The data are the same as for fig. 3 but with the raw signals now all normalized to the same level.

Ga -

J

Fig. 5. SIMS profiles of Ge, Pd and Ga in a sample of Ge/Pd/GaAs after annealing at 325°C for 30 min. The signals are recorded as CsX+ and we assert that they represent the variation with depth of the species X. All three signals have been normalized to the same level for display purposes. Note the excess of Ge at the interfacial region. Depth is in arbitrary units; as explained in the text oneOunit of depth is believed to represent approximately 100 A.

would correspond to 505 .& thickness. From the SIMS analysis displayed on fig. 6 we would estimate the thickness of the Ge layer, measured at half maximum, to be 482 A, suggesting that essentially all the excess Ge is concentrated in this layer. It is not possible to reliably estimate the thickness of the Ge deposit in the work of Palmstrom et al. [7], since their analysis with detection of X+ shows at the interface large variations in the Ge signals that clearly do not represent variations in the Ge density. In reviewing the data for sputtered X+ one should bear in mind that this is only a small fraction of the ejected X species and that most of the ejection is as neutrals. This can be confirmed from our own records of monatomic ions shown in figs. 1 and 2. Knowing the depth of the Ge and Pd layers from weight gain during the growth of the target and with the known ion beam current and analyzed area we can arrive at a sputtering coefficient which in turn gives the rate of ejection of X in all its charge states. Using the estimated instrumental transmission factors for a quadrupole machine determined by Clegg et al. [9], of about 3 X 10e4 and our known ion count rates we estimate that only one tenth of one percent of the ejected X is in the form of ions

E. W. Thpmas, A. Torabi / SIMS analysis using ejected CsX + ions

218

DEPTH

(ARBITRARY

UNITS)

Fig. 6. Difference between the Ge and Pd signals of fig. 5 representing the excess signal of Ge over the signal of Pd. Depth is in arbitrary units; as explained in the text oneOunit of depth is believed to represent approximately 100 A. The apparent indication of excess Ge at depth of 5.0 units and less is not necessarily reliable; it occurs during the period when all signals are rising due to the implantation of Cs.

X+. Thus essentially tral.

4. Mechanisms

all the ejected

species

X is neu-

for formation of CsX +

It has been proposed [3-61 that the CsX+ is formed by a combination or recombination of sputtered neutral X with a Cs+ ion but there has been no detailed discussion of a mechanism. We propose that the CsX+ is formed by an ion molecule reaction between neutral X emerging from the surface and Cs+ residing on the surface; the reaction equation would be of the form: X + (Cs + + surface)

+ CsX+ + surface.

(1)

The surface acts as the third body to allow for conservation of momentum and energy in the reaction. The Cs arises at the surface as a result of implantation from the projectile beam. Presumably implanted Cs revealed on the surface by erosion will be ionically bonded and in the form of positive ions. According to Williams and Baker [lo] the fractional surface concentration of the beam probe species, under equilibrium conditions, will be approximately S-i where S is the sputtering coeffi-

cient of the matrix in which it resides. The atom X is ejected by the collision cascade; we recall that the majority of the ejected X atoms are expected to be in a neutral state. Collisions of the type given in eq. (1) are well known in the study of gas phase atomic collisions. Borrowing from a simple classical model proposed by Bates et al. [ll], we suggest that the emerging X collides with a surface Cs+ which recoils to collide with a surface atom and thereby has its direction and velocity changed to be approximately the same as that of the X which initiated the events. Provided the final velocities of X and Cs’ lie within a volume of velocity space defined by the mutual affinity of the two species they will emerge as a molecule. Cross sections for such processes have been estimated on a clasical basis [ll] and for light species in gas phase reactions are of the order lo- i6 ems’ and lower in the energy region of 10 to 100 eV. The present model is completely consistent with a similar proposal of Wittmaack [3] who also suggests that the CsX+ is formed by a combination, or recombination, of emerging neutral X and surface Cs+. He shows evidence that CsX+ signals vary with incidence angle in a manner that is similar to that of total sputtering coefficient. Sputtering of course ejects primarily neutral X and it is this species emerging across the surface which initiates the reaction proposed in eq. (1). Wittmaack [3] does not, however, progose a specific model. The present SIMS observations do not permit a detailed test of the proposed model. We can however examine the general systematics implied by the model and relate them to the observed behavior. Moreover we should examine whether signal levels are consistent with the anticipated cross sections for a process such as that described in eq. (1). We model the formation of the CsX+ in terms of a flux of particles X emerging through a layer of the target (Cs++ surface) and undergoing the proposed reaction with a specific cross section rr. Using conventional definition of cross sections the number of CsX’ ions created per unit time will be given by: N(CsX+)

=N(X)alnT.

(2)

Here N(X) is the number of X atoms emerging through the target (surface + Cs’) per unit time; + is the number density of the target; 1 is the thickness of that target traversed by the emerging neutral X; (+ is the cross section for the reaction. The flux of X atoms is related to the sputtering coefficient of the matrix S and the incident beam flux I, of probing Cs+ by: N(X)

= I,SC,/C,.

(3)

C, and C, are respectively the concentration of the species of interest X and of the matrix; obviously if one is actually profiling the matrix itself then their ratio is unity. Combining eqs. (2) and (3), allowing also for an

E. W. Thomas, A. Torabi / SIMS analysis using ejected CsX * ions

instrumental transmission function f, we arrive at a predicted signal in counts per second of Signal(CsX+)

=fZ,S(C,/C,)c~lnr.

(4)

Now nr is the volume concentration of Cs in the near surface region. Under equilibrium sputtering conditions we may estimate this, following Williams and Baker [lo], as CM/S which simplifies eq. (4) to Signal(CsX+)

= fZ,C,al.

(5)

Let us now use the observed signal levels to estimate what would be the necessary cross section for the process described by eq. (1). Let us assume, following Dumke et al. [12], that the emerging X which undergoes the ion molecule reaction comes from only the fir$ few monolayers and estimate the path length 1 as 6 A. Let us make the estimate for a pure matrix region such as the Ge and insert for C, the conventional matrix density. As the instrumental transmission factor f we will again take 3 X lop4 from the work of Clegg et al. [9]. With the signal levels indicated on fig. 3 the cross section cr would be of the order 2 x lo-l9 cm2. There is no available data on the reaction described by eq. (1) with which this estimate may be compared. We recall, however, that the classical predictions by Bates et al. [ll], of cross sections for similar reactions in a low density gas phase environment give cross sections between 10-r’ and lo-l6 cm2 for energies in the region of 10 eV. Cross sections per atom for the denser situation represented by a solid are likely to be smaller due to shadowing of one atom by a neighbor. Within the spirit of the rough approximations used here we would conclude only that the cross section of the proposed ion molecule reaction is expected to be quite adequate to provide the observed signal levels. Attempts to make a more detailed comparison between observed signals and cross sections are not warranted. Let us now for comparison review the equation that is normally used to predict the flux of monatomic X+ ejected from a solid. Accepted models propose [2] that initial formation of the ion X+ may be related to the breaking of the bonds that holds X to the solid matrix; then as the X+ ejects there is the opportunity for electron transfer between the solid and the emerging X+ that might result in significant neutralization of X+ and thereby loss of ion signal. Both processes will be related to the details of the chemical bonding and electronic structure in the solid. A general expression [2] that gives the fraction of emerging X that are in the X+ state is Fraction(X+)

=A exp - (I-

4)/B,

(6) where Z is the ionization potential of the species X and 4 is the work function of the target from which it is ejected. The factors A and B are both constants for the particular matrix from which X is ejected. Using

219

eq. (3) to give the total ejected flux of X and allowing again for instrumental transmission function f one arrives at a signal count rate expression of Signal

= fZ,S(C,/C,)A

exp - (I - 4)/B.

(7)

Contrast now eq. (5), predicting CsX+ formation, with eq. (7), predicting X+ formation. Both equations show a linear dependence of signal on C,, the concentration of the species of interest. For CsX+ formation, signal levels are further influenced by only the escape depth 1 and the cross section c. There is evidence [12] that most sputter ejected particles emerge only from the first one or two monolayers; thus no great variation in 1 is expected with matrix nor with ejected species. Extensive compilations [13] of data on gas phase collisions of the type exemplified by eq. (1) show that for a fixed ion species the reaction rates (and therefore the cross sections) vary by less than an order of magnitude for a wide variety of species X. Thus we anticipate little variation of CsX’ signal level with matrix. By contrast extensive studies of monatomic ion ejection [l] show that the ion fraction varies by many orders of magnitude for a single species in different matrices or different species in a single matrix. Variation comes through the different ionization potentials Z of the species X, through difference in the effective value of the work function term 4 and through differences of the constants A and B from one matrix to another. These simplistic models suggest that the determination of concentration of X by monitoring the signal of CsX+ will be a more reliable indicator of density that the use of Xf. Signal of X+ is well known to be greatly dependent on matrix as well as on the characteristics of the species X. Signal of CsX+ is likely to be far less dependent on both matrix and on the nature of the species X.

5. Discussion of profiles Accepting the above models and conclusions let us re-examine the data of figs. 1, 2, and 3. Number densities of Ge, Pd, Ga and As in the as grown material are the same to within a factor of two. We observe that the indications of density given by monitoring CsX+ in fig. 3, after correction for isotopic abundance, are the same within approximately that factor. By contrast the indication of density given by monitoring X+ in figs. 1 and 2 differ by two or more orders of magnitude. Signal levels of X+ show sharp oscillations at interfaces that cannot be due to density variations; the CsX+ signals show not such oscillations and give reasonably sharp changes at the interfaces as one would expect. The mode1 suggests that the CsX+ signal is a reliable indicator of X profile. We have used this result to

220

E. W Thomas, A. Torabi / SIMS analysis using ejected CsX + ions

estimate the profile of Ge segregated to the interface of the annealed sample as indicated in fig 6. The width of the region is consistent with the calculated excess Ge. PalmstrQm et al. [7], suggest that the variation of X+ signal levels at the interfaces are “signatures” of the formation of compounds. Specifically a stoichiometry of Pd,Ge [14] is suggested at the Ge/Pd interface and Pd,GaAs at the Pd/GaAs interface [15]. Presumably these compounds exhibit values of the constants A, B and 4 in eq. (6) which are different from those of the pure materials to either side. In comparing figs. 2 and 3 at the Pd/GaAs interface one would conclude that both the Ga+ and Pd+ signals are enhanced at the interface while the signal of As+ is depressed relative to the true density. One might argue that the profile of the compound Pd,GaAs at the interface could be achieved by normalizing together the signals of CsGa+ and Ga+ at distances remote from the interface and then evaluating the difference between them in the interface region and regarding this as a measure of the compound profile. The apparent location of interfaces indicated by signals of ejected X+ differ from those indicated by signals of CsX+. Compare the data of figs. 2 and 3, both taken in the same experimental run and where the indicated depth scales are therefore the same. Let us define the apparent location of the interface between Pd and the underlying GaAs as being the depth where the Ge or As signals have risen to half the level measured deep in the pure GaAs. From fig. 3 based on detection of the diatomic CsX+ both the Ga and As signals rise to half the matrix level at the same depth of 17.5 units and the depth profile of both Ga and As are the same implying stoichiometric GaAs. By contrast the signals of X+ in fig. 2, show the Ga+ signal rising to half height at 15.7 depth units while the As+ signal rises to half height at 18.0 depth units: a difference of over 200 A. This would imply that the GaAs is nonstoichiometric at the interfac: with Ga being in excess of As for a depth of 200 A. We would be inclined to reject such a conclusion on the grounds of signal distortion due to matrix related effects. The Pd+ signals at the same location rise above those for the pure Pd at the center of the Pd layer; this rise clearly cannot represent a rise of Pd density and must be related to changes in the unknown constants of eq. (7). These constants in turn will also influence signals of the Ga+ and As+ rendering them unreliable measures of the Ga and As density. By contrast our model of CsX+ production would imply that the location of the interface through this signal should be reliable and indeed these signals do imply a stoichiometric GaAs compound with the same interface location both Ga and for As.

6. Conclusion Analysis of a Ge/Pd/GaAs structure has been performed with a Cs+ probe and comparisons made between the X+ and CsX+ signals from the various elements present. It is proposed that the CsX+ is formed by an ion molecule reaction involving escaping neutral X and implanted Cs+ revealed on the surface. Within this model it is expected that the CsX+ signals will give a more reliable indication of X density and that “artifacts” associated with the matrix will be reduced. Observations are consistent with the model.

Acknowledgements

This work was supported through the programs of the Georgia Tech Microelectronics Research Center. One of us (A.T.) also acknowledges support from M/A-COM.

References

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