Quantification of Iron (Fe) in Lithium Niobateby Optical Absorption

MARIA VITTORIA CIAMPOLILLO, ANNAMARIA ZALTRON, MARCO BAZZAN,NICOLA ARGIOLAS, and CINZIA SADA*Physics Department and CNISM, University of Padova, Via Marzolo 8 I-35131 Padova, Italy

A quantitative method, based solely on optical absorption, to determine

the total iron (Fe) concentration in Fe : LiNbO3 is proposed. Absorption

spectra of several samples doped by thermal diffusion with different

concentrations and different [Fe2þ]/[Fe3þ] ratios show an isosbestic point

at 342 nm. At this wavelength the absorption is proportional to the total

Fe concentration and does not depend on the oxidation state. Thanks to

the large number of samples covering a wide range of concentrations, in

this work it was possible to estimate an effective absorption cross-section

relating the absorbance of a given sample to its iron content. The main

advantage of the proposed method is in its simplicity and the fact that the

result does not depend on the reduction degree of the sample. As it is

known that the absorbance of Fe : LN at another wavelength (532 nm)

gives information on the amount of Fe2þ present in the sample, our

method makes it possible to characterize both the total Fe amount and its

reduction degree within a single optical absorption measurement.

Index Headings: Optical absorption; Iron; Fe; Lithium niobate; Reduction

degree.

INTRODUCTION

Its combination of excellent electro-optical and nonlinearoptical properties makes lithium niobate (LiNbO3) an attractivehost material for applications in integrated optics, especially inthe field of holographic data storage. In particular, it is wellknown (see Ref. 1 and references therein) that iron-dopedlithium niobate (Fe : LN) crystals present enhanced photo-refractive response. Because the iron content plays a key role inthe photorefractive response of the material, it is mandatory tofind a simple way to quantify the amount of Fe, both whenlithium niobate is intentionally doped and when Fe is present asan unwanted impurity. In order to reach this goal, the opticalproperties of iron can be exploited. As a matter of fact, it isknown2 that Fe in LiNbO3 gives rise to several absorptionbands. In the ultraviolet (UV)–visible region essentially twobands are important: according to the nomenclature introducedby Dischler et al.3 and the assignment done by Clark et al.,4 theC-band, beginning at 400 nm and extending to lowerwavelengths, which is due to charge transfer transitions ofFe2þ and Fe3þ ions and the D-band, partially overlapping the C-band, centered at about 480 nm, due to intervalence transferbetween Fe2þ and Nb. Of course, the D-band is visible only ifFe occurs in the 2þ valence state, and hence it can be exploitedto quantify the Fe2þ amount.

All these bands are polarized mainly perpendicularly to theferroelectric axis; therefore, ordinary polarization is preferredto investigate them. For the sake of clarity the two bands aredepicted in Fig. 1. The first quantitative characterization of the

Fe2þ content was proposed by Kurz et al.:2 using their data theFe2þ absorption cross-section for ordinarily polarized light at477 nm (1/r ¼ 2.16 3 1021 m�2) can be determined.1 In asimilar approach, Berben et al.5 used the ordinary absorptioncoefficient at 532 nm to determine the Fe2þ absoluteconcentration. However, an optical method to estimate thetotal Fe concentration regardless of its oxidation state waslacking up to 2002 when Basun et al.6 made an attempt to usesolely absorption to determine the Fe content. Their methodused a very weak absorption band of Fe3þ and was calibratedusing two samples in a set of only three samples: unfortunatelyit gave inconsistent results when applied to the third sample. Asa consequence the method cannot be applied with a reasonablelevel of confidence.

In this work we present a new method that allows accuratequantification of the total iron concentration in lithium niobatecrystals by measuring their optical absorption at a specificwavelength, which is shown to be independent of the Fereduction degree and proportional to the total concentration.The method exploits the analysis of thermally diffused iron-doped samples to correlate the absorbance to the ironconcentration. This approach has the advantage that the totalFe content can be measured independently with good accuracy.The obtained relation is checked against other bulk iron-dopedsamples obtained by Czochralski growth. The consistency ofthe results indicates that the method is robust and accurate.

EXPERIMENTAL

Thin films of metallic Fe were deposited on commercial X-cut samples of pure congruent LiNbO3 (Crystal Tech., 1 mmthick polished on both faces) by means of magnetron sputteringdeposition. The number of Fe atoms per unit surface (i.e., thefluence or areal density) in the deposited film was estimatedfrom the deposition time using a previously determinedcalibration. Iron was incorporated into the matrix by meansof the thermal diffusion process: the samples were annealed inoxygen atmosphere (gas purity 5.0, dry oxygen flux¼90 NL/h)at high temperature in the range of 900 to 1050 8C for severalhours (8–40 h) to promote Fe diffusion into the substrate. Thein-depth iron profiles were measured using secondary ion massspectrometry (SIMS). All the profiles showed a semi-Gaussianshape and their widths at half maximum ranged from 2 to 10lm depending on the process parameters. A typical SIMSspectrum is reported in Fig. 2. The maximum iron concentra-tion was found at the surface of the diffused samples, rangingin the interval from 0.19 to 3.40 3 1020 at/cm3.

As the above-mentioned diffusion treatment is performed inoxygen, it is expected to promote mainly the Fe3þ state. Inorder to favor the partial reduction of Fe3þ to Fe2þ, asubsequent reducing annealing treatment at 500 8C in Ar þH2 (4 wt %) (gas purity 5.0, flux¼110 NL/h) was performed. It

Received 28 May 2010; accepted 25 October 2010.

* Author to whom correspondence should be sent. E-mail: cinzia.sada@unipd.it.

DOI: 10.1366/10-06015

216 Volume 65, Number 2, 2011 APPLIED SPECTROSCOPY0003-7028/11/6502-0216$2.00/0

� 2011 Society for Applied Spectroscopy

was verified with further SIMS profiles that this treatment isable to alter the reduction state of iron without modifying itscompositional profile. By changing the duration of thisreducing annealing (from 0.5 to 6 hours) we were able toobtain a wide range of reduction degrees, which in some cases

reached 100%. Details of the preparation conditions of the iron-

doped samples are reported in the first three columns of Table

I, including iron fluence, annealing conditions, and maximum

iron concentration.

After the diffusion in oxygen and after each reducing post-

treatment, the optical transmittance was measured in the

spectral range 330–850 nm with a Jasco V-670 double-beam

spectrophotometer equipped with a halogen lamp. Spectra were

FIG. 2. Typical SIMS spectrum of the in-depth profiles of the elements in anFe doped sample.

TABLE I. List of investigated samples. Each entry reports the total Fe fluence, the annealing parameters, the measured absorbance, the maximumconcentration of Fe at the surface and the reduction degree expressed in terms of percentage of Fe2þ concentration over the total Fe concentration, and interms of the [Fe2þ]/[Fe3þ] ratio.

Fluence Annealing Absorbance

Maximum Feconcentration 3

1020 at/cm3

Reductiondegree

[Fe2þ]/[Fe] [Fe2þ]/[Fe3þ]

25 6 1 3 1015 at/cm2 40 h at 1000 8C in O2 0.158 6 0.008 0.19 3 0.0325 6 1 3 1015 at/cm2 40 h at 1000 8C in O2 then 6 h at 500 8C in Ar þ H2 0.144 6 0.008 0.19 100 .10027 6 1 3 1015 at/cm2 8 h at 1000 8C in O2 0.156 6 0.008 0.44 1 0.0127 6 1 3 1015 at/cm2 8 h at 1000 8C in O2 then 1.5 h at 500 8C in Ar þ H2 0.142 6 0.008 0.44 75 350 6 2 3 1015 at/cm2 8 h at 900 8C in O2 0.29 6 0.01 2.30 0 050 6 2 3 1015 at/cm2 8 h in O2 at 900 8C then 0.5 h at 500 8C in Ar þ H2 0.30 6 0.01 2.30 50 150 6 2 3 1015 at/cm2 10 h at 900 8C in O2 0.26 6 0.01 1.79 0 054 6 2 3 1015 at/cm2 8 h at 1000 8C in O2 0.29 6 0.01 0.78 1 0.0154 6 2 3 1015 at/cm2 8 h at 1000 8C in O2 then 1.5 h at 500 8C in Ar þ H2 0.29 6 0.01 0.78 21 0.2654 6 2 3 1015 at/cm2 8 h at 1000 8C in O2 then 3 h at 500 8C in Ar þ H2 0.29 6 0.01 0.78 100 . 10066 6 3 3 1015 at/cm2 10 h at 900 8C in O2 0.43 6 0.01 2.30 2 0.0266 6 3 3 1015 at/cm2 10 h at 900 8C in O2 then 1.5 h at 500 8C in Ar þ H2 0.43 6 0.01 2.30 39 0.6466 6 3 3 1015 at/cm2 10 h at 900 8C in O2 then 2 h at 500 8C in Ar þ H2 0.43 6 0.01 2.30 58 275 6 3 3 1015 at/cm2 8 h at 900 8C in O2 0.45 6 0.01 3.40 6 0.0675 6 3 3 1015 at/cm2 8 h at 900 8C in O2 then 1.5 h at 500 8C in Ar þ H2 0.43 6 0.01 3.40 50 180 6 3 3 1015 at/cm2 8 h at 1000 8C in O2 0.43 6 0.01 1.26 0 080 6 3 3 1015 at/cm2 8 h at 1000 8C in O2 then 1.5 h at 500 8C in Ar þ H2 0.45 6 0.01 1.26 25 0.3380 6 3 3 1015 at/cm2 8 h at 1000 8C in O2 then 3 h at 500 8C in Ar þ H2 0.44 6 0.01 1.26 96 24

100 6 4 3 1015 at/cm2 8 h at 1000 8C in O2 0.58 6 0.02 1.44 0 0100 6 4 3 1015 at/cm2 8 h at 1000 8C in O2 then 1.5 h at 500 8C in Ar þ H2 0.60 6 0.02 1.44 36 0.56100 6 4 3 1015 at/cm2 8 h at 1000 8C in O2 then 3 h at 500 8C in Ar þ H2 0.59 6 0.02 1.44 50 1108 6 4 3 1015 at/cm2 8 h at 1000 8C in O2 0.59 6 0.02 1.75 1 0.01108 6 4 3 1015 at/cm2 8 h at 1000 8C in O2 then 1.5 h at 500 8C in Ar þ H2 0.58 6 0.02 1.75 26 0.35108 6 4 3 1015 at/cm2 8 h at 1000 8C in O2 then 3 h at 500 8C in Ar þ H2 0.59 6 0.02 1.75 53 1.1108 6 4 3 1015 at/cm2 8 h at 1000 8C in O2 then 3 h at 500 8C in Ar þ H2

then 3h at 500 8C in O2

0.57 6 0.02 1.75 36 0.56

149 6 6 3 1015 at/cm2 2 h at 1050 8C in O2 0.83 6 0.03 2.40 0 0149 6 6 3 1015 at/cm2 2 h at 1050 8C in O2 then 1.5 h at 500 8C in Ar þ H2 0.83 6 0.03 2.40 40 0.67149 6 6 3 1015 at/cm2 4 h at 950 8C in O2 then 1.5 h at 500 8C in Ar þ H2 0.83 6 0.03 3.40 47 0.89149 6 6 3 1015 at/cm2 8 h at 950 8C in O2 then 1.5 h at 500 8C in Ar þ H2 0.82 6 0.03 3.10 47 0.89199 6 8 3 1015 at/cm2 6 h at 1050 8C in O2 1.11 6 0.06 2.00 0 0

FIG. 1. (Color version available online.) Transmittance spectra of pureLiNbO3 and three samples of LiNbO3 doped with (66 6 3) 3 1015 at/cm2 of Fewith different reduction degrees. (Inset) An enlargement of the region wherethe isosbestic point is located.

APPLIED SPECTROSCOPY 217

collected in ordinary polarization with a bandwidth equal to 1nm at a scanning speed 100 nm/min. Both the iron-dopedsamples and a pure untreated sample were measured forcomparison.

RESULTS

Figure 1 shows the transmittance spectra of three iron-dopedsamples containing (66 6 3) 3 1015 at/cm2 each, together withthe spectrum of a pure sample for comparison. After diffusionin oxygen, only the C-band below 400 nm is visible,confirmation of the fact that in an oxidizing atmosphere Feoccurs only in the 3þ valence state. In a fully oxidized sample,the intensity of this band is proportional to Fe3þ concentration.By increasing the duration of the reducing treatment, the D-band starts to grow while the intensity of the C-band decreasesslightly. The C-band contains a contribution due to both Fe2þ

and Fe3þ: because of their different electronic configurations, inprinciple they may contribute differently to the absorption andhave to be treated as two different centers. If the concentrationsare not too high, we may assume that each center behavesindependently of the others, and we may express the internalabsorbance due to Fe atoms AFe as a weighted sum of thecontributions from the two active centers:

AFeðkÞ ¼ r3ðkÞNFe3þ þ r2ðkÞNFe2þ ð1Þ

where NFe3þ and NFe2þ are the areal densities of Fe3þ and Fe2þ,and r3(k) and r2(k) are the absorption cross-sections related tothe contributions of Fe3þ and Fe2þ, respectively.

Figure 1 shows that different spectra, pertaining to sampleswith the same total Fe areal density but with different reductiondegrees, cross at a wavelength that was found to be the same(342 nm) for all the samples considered, indicating that theabsorption at this wavelength does not depend on the valencestate of Fe. This fact is of extreme importance because itsuggests that at k* ¼ 342 nm, A(k*) remains the same bychanging Fe3þ into Fe2þ. Taking into account that in sampleswith the same areal density NFe2þ þNFe3þ ¼NFe is constant, thisis possible only if the two cross-sections in Eq. 2 are equal.Therefore, we can regard k* ¼ 342 nm as the wavelength forwhich r3(k*)¼r2(k*)¼rFe where rFe represents an effectivecross-section of iron that does not depend on its valence state.Moreover the presence of this kind of isosbestic point ensuresthat the absorption at this wavelength is due only to the twospecies considered. For these reasons, Eq.1 may be expressedas follows:

AFeðk�Þ ¼ rFeðNFe3þ þ NFe2þÞ ¼ rFeNFe ð2Þ

In order to verify our hypothesis and check whether therelation between absorption at k* and total amount of Fe islinear, we must isolate from the spectra the contribution due toFe atoms, i.e., we must remove the contribution due to multiplereflections at the surfaces of the sample and the contributiondue to the absorption of LN, which at 342 nm is non-negligible(this wavelength is not so far from the UV absorption edge).The transmittance can be fully corrected for reflection lossesusing the following formula, which relates the internaltransmittance Ti of the sample to the measured transmittance T:

T ¼ s2 � Ti

1� r2 � T2i

ð3Þ

where s and r are transmission and reflection coefficients of thematerial. By using this formula we get rid of multiplereflections, but the internal transmission calculated alsocontains the intrinsic LN contribution. A simple way toremove both contributions is to normalize each spectrum to thespectrum of the pure untreated sample of the same thickness.By defining as Td the transmittance of the iron-doped sampleand Tu the transmittance of a pure undoped sample, thenormalized spectrum is equal to

Td

Tu

¼ s2d � Tid

s2u � Tiu

� 1� r2u � T2

iu

1� r2d � T2

id

ð4Þ

where the subscript d refers to the doped sample and thesubscript u refers to the undoped one. The factor (1� r2

u � T2iu)/

(1 � r2d � T2

iu) can be approximated to 1, because the productsr2 � T2 are much smaller than 1. With this approximation weobtain

Td

Tu

’s2

d � Tid

s2u � Tiu

¼ s2d

s2u

� 10�ðAid�AiuÞ ð5Þ

where Aid and Aiu are the internal absorbances of the twosamples. In the exponent �(Aid � Aiu) the contributions ofintrinsic LN and native Fe impurities cancel out, and only thecontribution of the added Fe atoms remains. We thereforedefine the absorbance related to the ratio between the twomeasured trasmittances:

AFe ¼ �logTd

Tu

� �’�log

s2d � Tid

s2u � Tiu

� �¼ �log

s2d

s2u

þ ðAid � AiuÞ

ð6Þ

Fe doping can noticeably change the refractive index of thedoped surface and thus the transmission coefficient. We do nothave exhaustive data about how much the refractive indexchanges by varying the concentration and the reduction degreeof Fe. However, even if Fe doping could induce a highrefractive index variation, for instance 0.05, this would result ina variation of transmission coefficient giving a factor of �logðs2

d=s2uÞ ’ 0.007, which is smaller than the smallest error we

obtained on the experimental values of AFe, reported in Table I.Therefore, the following approximation

AFe ¼ �logTd

Tu

� �’ Aid � Aiu ð7Þ

is allowed.It must be pointed out that Eq. 3 is strictly valid only in the

case of normal incidence of a collimated beam. In ourspectrometer, and many others, the incident light is notcollimated; instead it is focused on the sample. For this reasonwe did not directly use this formula, but we adopted thenormalization, which furthermore allows the cancellation of thecontribution of intrinsic absorption and native Fe impurities.We have shown that in the case of normal incidence ournormalization method is justifiable; we assume that it is alsovalid in the case of non-collimated beams, i.e., that the lightreflections in the measurement chamber are approximately thesame (or their variations contribute negligibly to the finalresult) in the case of the pure crystal and in the case of thedoped one.

We checked the validity of Eq. 2 by plotting the absorbance

218 Volume 65, Number 2, 2011

(determined using Eq. 7) as a function of the iron fluence,which in our case is known from independent measurements.In Fig. 3 we report the values of AFe(k*) as a function of theiron fluence: it clearly emerges that a linear dependence holds.The linear fit of the data gives rFe (342 nm)¼ (5.69 6 0.06 3

10�18 cm2/at, with a correlation coefficient higher than 0.99.The error on rFe (342 nm) is calculated using NFe as anindependent variable and the errors on AFe values as theweights for the fit; it is worth noting that the fitting with AFe as

the independent variable and the errors on NFe as the weightsgave a result compatible within errors. The result of the linearfit in Fig. 3 is a first a posteriori confirmation that theassumptions underlying Eq. 2 are adequate.

In order to further investigate this aspect we estimated themean iron-to-iron distance expected in the iron-diffusedsamples. Because in our samples the concentration is depthdependent, the mean distance between Fe ions also changeswith depth. Moreover, as the Fe areal density is the integral ofthe diffusion profile, two samples with the same fluence canhave significantly different ranges of Fe concentrations andconsequently different ranges of Fe–Fe distances. However, wecan refer our data to concentration at the surface, which is avalue that can be used to compare different samples even ifthey do not have the same diffusion profiles. By consideringthat the maximum concentrations at the surface (reported inTable I) lie in a range from 0.2 3 1020 at/cm3 to 3.4 3 1020 at/cm3, the minimum mean distances among nearest neighbor Featoms lie in the range from 3.7 nm to 1.4 nm. Theseconcentrations are much higher than those used for photo-refractive applications (typically 2 3 1019 at/cm3 or less). Toensure that there is no effect of enhancement or inhibition ofabsorption due to the nearness of Fe atoms, we checked theeventual dependence of the absorbance divided by the fluence(which corresponds to the effective Fe cross-section) on thesurface concentration for the different samples: no correlationcould be found (see Fig. 4, left panel); therefore, we can safelyassume that our result is also valid also for lower Feconcentrations.

In order to verify whether our results depend on thereduction degree of Fe, we also evaluated the correlationbetween absorbance divided by the fluence and the reductiondegree: the latter was calculated from the optical absorption at

FIG. 3. (Color version available online.) Absorbance values at 342 nm plottedas a function of total Fe concentration of samples doped by thermal diffusion,fitted with a line passing through the origin.

FIG. 4. The absorbance divided by the fluence is reported as a function of (left) the maximum iron concentration and (right) the iron reduction degree.

APPLIED SPECTROSCOPY 219

532 nm according to Ref. 5 and its values for each sample arereported in Table I, expressed as percentage of Fe2þ over thetotal Fe amount. Again, no correlation between the effectivecross-section and reduction degree was noticed (see Fig. 4,right panel), confirming our former observation.

Once rFe(k*) has been determined, it is possible to estimatethe Fe content in unknown samples. This method presents thegreat advantages of being nondestructive and reproducible. Aspreviously stated, we underline that the effective cross-sectioncalculated here may be conveniently used to quantify theunknown Fe concentration in bulk doped samples. This is atypical problem in the crystal growth procedure because it isgenerally known that the nominal dopant concentration (i.e.,that added to the starting melt) may differ significantly from thereal concentration in the crystal due to the fact that thesegregation coefficient is usually different from one.7 More-over, the exploitation of this sort of isosbestic point has thegreat advantage that it is a simple method and the result doesnot depend on the reduction degree of the sample. It shouldalso be underlined that because the measurements areperformed in ordinary polarization, the result does not dependon the stoichiometry, i.e., on the [Li]/[Nb] ratio. In fact, theordinary refractive index (see Ref. 7, Chap. 8.2) and theintrinsic absorption vary negligibly with the ratio [Li]/[Nb]. Asa consequence, a possible variation of [Li]/[Nb] along thegrowth axis, or a different stoichiometry from one sample toanother, do not affect the determination of Fe concentration.

We checked this method on samples taken from two boulesgrown in our laboratory using the Czochralski technique withnominal concentrations of 0.01 mol % and 0.005 mol % and anunknown degree of reduction. Two slabs, perpendicular to thegrowth axis, were cut from the top and from the bottom of theboules to check whether the concentration of Fe is equal to thenominal one and whether it varies along the growth axis. Eachvalue of AFe was divided by the thickness of each sample toobtain the linear absorption coefficient. According to theresults, reported in Table II, the actual concentrations differfrom the nominal ones: this is reasonable taking into accountthe large uncertainty in the weighing of the small amount ofFe2O3 to be added to the melt. Moreover, the results show thatFe is more concentrated in the bottom of each boule, indicating

that the distribution coefficient of Fe is smaller than one, asalready reported by many authors (see Ref. 7, Chap. 2.3).

The minimum difference of Fe concentration that can bemeasured with a standard spectrometer depends on the concen-tration and thickness of the sample: by way of example, on asample doped with 0.05 mol %, 0.5 mm thick, it is about 131016

at/cm3. Therefore the method is capable of very good sensitivity.

CONCLUSION

In this work we presented a quantitative method to determinethe total Fe concentration in Fe : LiNbO3, regardless of theoxidation state, using only nondestructive optical measure-ments. As a matter of fact, the absorption spectra of severalsamples doped by thermal diffusion with different concentra-tions and different [Fe2þ]/Fe3þ] ratios show an isosbestic pointat 342 nm. At this wavelength the absorption is proportional tothe total Fe concentration and does not depend on the oxidationstate. By exploiting the linear relation between the absorbanceand the iron areal concentration, it was possible to estimate theeffective cross-section, which can then be used to quantify theFe concentration in bulk doped samples. The main advantage ofthe proposed method lies in its simplicity and in the fact that theresult does not depend on the reduction degree of the sample.

ACKNOWLEDGMENT

This work has been supported by the University of Padova, Progetto diAteneo CPDA073231/07, ‘‘Doping and microstructuration of lithium niobatecrystals for holographic recording in integrated optics’’.

1. T. Volk and M. Wohlecke, Lithium Niobate – Defects, Photorefraction andFerroelectric Switching (Springer, Heidelberg Berlin, 2008).

2. H. Kurz, E. Kratzig, W. Keune, H. Engelmann, U. Gonser, B. Dischler, andA. Rauber, Appl. Phys. 12, 355 (1977).

3. B. Dischler, J. R. Herrington, and A. Rauber, Solid State Commun. 14, 1233(1974).

4. M. G. Clark, F. J. DiSalvo, A. M. Glass, and G. E. Peterson, J. Chem. Phys59, 6209 (1973).

5. D. Berben, K. Buse, S. Wevering, P. Herth, M. Imlau, and Th. Woike, J.Appl. Phys. 87, 1034 (2000).

6. S. A. Basun, D. R. Evans, T. J. Bunning, S. Guha, J. O. Barnes, G. Cook,and R. S. Meltzer, J. Appl. Phys. 92, 7051 (2002).

7. K. K. Wong, Ed., Properties of Lithium Niobate (INSPEC, London, 2002).

TABLE II. Results of the absorption measurements on bulk doped samples. Each entry reports the nominal concentration, the original position in theboule, the absorption coefficient, and the actual Fe concentration. Errors take into account uncertainties on transmittance, thickness, and absorptioncross-section.

Nominal concentration Position in the boule Absorption coefficient of Fe (cm�1) Actual Fe concentration

1.9 3 1018 at/cm3 Top 9.8 6 0.2 (1.72 6 0.04) 3 1018 at/cm3

Bottom 10.1 6 0.2 (1.78 6 0.05) 3 1018 at/cm3

0.93 3 1018 at/cm3 Top 6.5 6 0.1 (1.14 6 0.02) 3 1018 at/cm3

Bottom 7.0 6 0.1 (1.22 6 0.02) 3 1018 at/cm3

220 Volume 65, Number 2, 2011