The range of validity of EELS microanalysis formulae

Ultramicroscopy 6 (1981) 2 9 7 - 3 0 0 North-Holland Publishing Company

297

LETTER TO THE EDITOR THE RANGE OF VALIDITY OF EELS MICROANALYSIS FORMULAE R.F. EGERTON Department of Physics, University o f Alberta, Edmonton, Canada T6G 2,11 Received 22 April 1981

ments, x a n d y , eq. (1) may be rewritten as:

Electron energy-loss spectrometry (EELS) can be employed for quantitative microanalysis of low- and medium-atomic-number elements through the use of a formula of the type [1 ] :

Nx_oY(a,A) FI~(a,A)"] Ny

l [ I~-(a, A)] NX - oxk(&, A) L i t ( a , / x )

'

o'~k(a,A) LIY(a, A ) J '

provided the same integration range A is used in measuring the ionization edges of both elements (see fig. I). Eqs. (1) and (2) become inaccurate if the integration range A or the collection semi-angle a of the spectrum are made too small, or if the specimen thickness is too large [1,4]. However, an exact treatment in this case would involve a vector deconvolution over scattering angle combined with deconvolution over energy loss, which is not convenient for routine analysis.

(I)

where N x represents the concentration (atoms per unit area) of an element x, I~(a, A) and Ii(a, A) are integrated intensities within the energy-loss spectrum (see fig. 1) and o~(a, A) is the partial cross section for an inner shell (k = K or L, usually) of the element [2,3]. For measuring the ratio N X / N y of two ele-

sTo

i

o

16o

i

200

1

-I

i

~

300

I1"

400

IIP

E(ev)

Fig. 1. Energy-loss spectrum of boron nitride showing the zero-loss peak, plasmon peaks due to n- and o-electrons, a rapid gain increment (X 436) and K-shell ionization edges of boron and nitrogen.

0304-3991/81/0000-0000/$02.50

(2)

© 1981 North-Holland

298

R.F. Egerton / Range of validiO, ofEELS microanalysisformulae Io

Zaluzec [5] has recently described a simple way of experimentally estimating the maximum specimen thickness tm for which eq. (2) is a reasonable approximation. His method involves plotting the ratio I~(a, A)/IY(c~, A) for ionization edges of two elements (in a homogeneous sample) as a function of local specimen thickness (relative to the mean free path Xp for valence-electron scattering), the latter being estimated from the low-energy-loss portion of the spectrum. Here we repeat Zaluzec's measurements for the compound BN, obtaining results which are qualitatively similar but which suggest a higher value of

100

,

F

t ~-

(nm)

\ \\s,

tm/~kp. Single-crystal boron nitride [6] was cleaved using adhesive tape, then mounted on 400-mesh grids for viewing (in CTEM mode at 80 keV) in a JEM ]00B electron microscope fitted with a magnetic spectrometer and serial-detection system [7]. The collection semi-angle for energy-loss spectroscopy, set by a 40 /ira diameter objective aperture, was e = 8.4 mrad. Spectra were recorded using a Tracor T N I 7 ]0 multichannel analyser which can be programmed for quantitative light-element analysis [7]. A typical spectrum is shown in fig. I. In order to estimate specimen thickness, Zaluzec employed the formula [8]:

t/Xp = lp/Io,

(3)

where lp represents the area under the first (main) plasmon peak and Io the area under the zero-loss peak. But because successive plasmon peaks (corresponding to plural scattering) overlap in energy, the choice of an upper integration limit for the measurements o f l p can be somewhat arbitrary. So for this study, use has been made of an alternative formula (which can be derived from Poisson's Law [8]):

t/Xp = loge (It/lo),

(4)

where I t is the total area (including Io) under the spectrum, integrated up to an energy loss ~ 130 eV (above which further contribution from multiple scattering is less than 1% for t/Xp ~< 1.5). In order to measure the K-shell intensities, I~(c~, A) and I~(a, A), due to boron and nitrogen, allowance must be made for the spectral background due to other energy-loss processes (see fig. 1). This was taken to be of the form A e -r, the constants A and r being determined by measuring two areas under the back-

0

I

,

,

I , ,,jl

I 0-I

J

~ I ~,~FI

~.~p 1'0

Fig. 2. R a t i o o f b o r o n to n i t r o g e n K-shell intensities, p l o t t e d as a f u n c t i o n o f specimen thickness. T h e s i g n a l / b a c k g r o u n d ratios at the b o r o n and n i t r o g e n edges are also s h o w n .

ground just preceding each K-edge [7,9]. Fig. 2 shows a plot of the ratio I~(a, A)/I~(a, A) against local thickness, obtained from measurements on 17 different regions of a BN specimen. Areas close to bend-extinction contours were avoided, in order to prevent an additional error in eqs. (1) and (2) which can occur under strongly-diffracting conditions [1,10, 11 ]. Making the reasonable assumption that the boron nitride sample is homogeneous, the ratio NB/N N is independent of thickness and according to eq. (2) the B N ratio IK/IK should also be constant. Therefore, the observed decrease in I~/l~ at higher thickness indicates that the maximum thickness t m for which eq. (2) remains accurate (within experimental error, ~10%) is given by t m / X p ~ 1.0. This is larger than the value tm/kp ~ 0.2 suggested by Zaluzec [5], but is more in accord with previous estimates [2] and with theoretical expectations [4]. Also plotted in fig. 2 is the ratio of K-loss and background intensities, integrated over the same energy region (see fig. 1). This ratio falls with increasing thickness due to multiple-scattering contribution to the background [ 1], an effect which also limits the range of useful sample thickness (for sensitive EELS microanalysis) to t/kp <~ 1. As an alternative to determining elemental ratios, EELS can be employed to measure absolute concen-

R.F. Egerton / Range of validity ofEELS microanalysisformulae 50

100

I

60

/

i

" t (rim) o

I atorn5 rn"2 /

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,"

~""

./o

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40 a

a

"

All

•

20

NB N N A(ev) a

o

o'.2

~

o'.4

•

&

o'.8 '

1:o

o A o

50 70 O0

1:2

1:~

~Xp

Fig. 3. Measured concentration of boron and nitrogen (per

unit specimen area) as a function of thickness, for three values of energy window 4.

trations, using eq. (1). To find out tile range of validity of this equation, tile separate values o f N B and N N, predicted by eq. (1), can be plotted against t/kp, as in fig. 3. (For tile present measurements, the required values of o~(c~, A) and oR(a, A) were calculated using a hydrogenic program [2] .) The resulting points should ideally lie on a straight line, any deviation from tiffs indicating failure of eq. (1). For a 8.4 mrad and A = 70 eV, it appears that eq. (1) remains accurate (to within 15%) up to t/kp ~ 1 for boron and t/kp ~ 1.2 for the nitrogen K-edge. Fig. 3 also includes a few points obtained using other values of the integration range A. In accordance with expectations [1], a larger value of A increases the range of validity of eqs. (1) and (2) (i.e. increases tin). A smaller value of A can either increase or diminish tin, depending on the energy and width of the plasmonloss peaks relative to A. Increasing the value of c~ should tend to increase tin, due to the reduced importance of the obliquity factor in the angular convolution [1,4]. From the slope of fig. 3 at small thickness, and knowing the physical density p of the sample, the value of kp can be calculated. For an incident energy Eo = 80 keV and collection semi-angle a = 8.4 mrad, the resulting experimental value for crystalline boron nitride (p = 2.25 g cm -3) is Xo = 91 nm, as compared

299

to 85 nm calculated using a free electron model [8]. Knowing kp, the thickness axis in figs. 2 and 3 can then be calibrated in absolute units. Although the value of kp depends on a and on E0, its variation [8] is less than a factor of 2 for 3 mrad < a < 20 mrad and 60 ~< Eo ~< 100 keV. Amongst other sample materials which exhibit distinct plasmon losses, the values of kp do not change by more than a factor ~2, owing to the fact that valence-electron densities do not vary greatly. For specimens whose valence-electron losses are dominated by singleelectron excitation, the appropriate mean free path can be approximated by (noi) -l , where n is the number of atoms per unit volume and oi is an inelastic cross section corresponding to the average atomic number of the sample. In this case, the inelastic mean free path appears to vary by no more than a factor ~3 between different materials. The conclusions from the present study are therefore that under favourable EELS conditions the simple microanalysis formulae remain valid (within 15%) up to a sample thickness equal to the mean free path for inelastic scattering, which is ~91 nm (at an incident energy = 80 keV, collection semi-angle = 8.4 mrad) for boron nitride and probably within the range 5 0 - 1 5 0 nm (at incident energies of 6 0 - 1 0 0 keV) for most other materials. A similar limitation on sample thickness is provided by nmltiple-scattering contributions to the background, which seriously degrade the signal/background ratio (and hence elemental detectability) as t/k o approaches unity. These considerations obviously favour the use of a highvoltage microscope in the case of samples of thickness > 1 0 0 nm, since the valence-electron mean free path can be increased by a factor ~ 4 by raising the incident-electron energy from 60 keV to 1 MeV [12]. The author is grateful to Dr. N. Zaluzec for helpful discussions and for his critical reading of the manuscript.

References I1] R.F. Egerton, Ultramicroscopy 3 (1978) 243. [2] R.F. Egerton, Ultramicroscopy 4 (1979) 169. [3] R.D. Leapman, P. Rez and D.F. Mayers, J. Chem. Phys. 72 (1980) 1232.

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R.F. Egerton /Range o f validity ofEELS microanalysis formulae

[41 A.P. Stephens, Ultramicroscopy 5 (1980) 343. [5] N.J. Zaluzec, in: 38th Ann. Proc. Electron Microscopy Soc. Am., Ed. G.W. Bailey (Claitor's, Baton Rouge, 1980) p. 112. [6] The author is grateful to Dr. Gordon Parkinson for supplying the specimen material. 171 R.F. Egerton, in: 38th Ann. Proc. EMSA (1980) p. 130. [81 H. Raether, Springer Tracts in Modern Physics 38 (Springer, Berlin, 1965) p. 84.

[9] R.F. Egerton, in: Scanning Electron Microscopy/1980/l, Ed. O. Johari (SEM, Chicago, 1980) p. 41. [10] C.J. Rossot, w and M.J. Whelan, Phys. Status Solidi (a) 45 (1978) 277. [111 N.J. Zaluzec, J. Hren and R.W. Carpenter, in: 38th Ann. Proc. EMSA (1980) p. 114. [12] B. Jouffrey, Y. Kihn, J.Ph. P~rez, J. Sevely and G. Zanchi, in: Proc. 5th Intern. Conf. on High Voltage Electron Microscopy, Kyoto, 1977.