Tests of two alternative methods for measuring specimen thickness in a transmission electron microscope

Pergamon

Micron, Vol. 26, No. 1, pp. 1-5, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0968-4328/95 $9.50 + 0.00

0968-4328(94)00039-5

RESEARCH PAPERS

Tests of Two Alternative Methods for Measuring Specimen Thickness in a Transmission Electron Microscope Y.-Y. YANG* and R. F. E G E R T O N t

*Department of Materials Science, State Universityof New York, Stony Brook, NY 11794, U.S.A. tPhysics Department, Universityof Alberta, Edmonton, Canada T6G 2Jl (Received 17 May 1994; Accepted 8 August 1994)

Abstract--Using thin-film standards of AI, Ag, In, Sn, Sb, and Te whose mass-thickness was determined by weighing, we investigated the accuracy of two methods of thickness measurement, both of which employ electron energy-loss spectroscopy (EELS) in a transmission electron microscope (TEM). A method based on the Kramers-Kronig sum rule was found to provide an accuracy of better than 15%, given suitable treatment of the low-energy and surface losses. The log-ratio method, based on measurement of the total and zero-loss integrals and a parameterized expression for the total-inelastic mean free path, yielded more variable accuracy: the discrepancy with weighed values was less than 5*/o for A1 but amounted to more than 50% in the case of Se and Te. Key words: Electron energy-loss spectroscopy (EELS), thickness measurement.

INTRODUCTION

Em~ f E(df/dE) dE/ f (df/dE)dE When carrying out transmission electron microscopy (TEM), it is sometimes desirable to know the local thickness of the specimen, especially in the case of analytical microscopy or high-resolution imaging. Now that many TEMs are equipped with an electron spectrometer, electron energy-loss spectroscopy (EELS) provides a practical and convenient method of thickness measurement. The simplest procedure for obtaining the thickness t is based on the log-ratio formula:

t/2=ln(It/Io).

where E denotes energy loss and f represents oscillator strength. Experimental measurements (Malis et al., 1988), using thin films or thinned foils of Be, A1, Fe, Cu, Hf, Au, NiO, SnO 2 and stainless steel, gave the following empirical formula for Era: E m= 7.6 Z 0"36

(4)

where Z is the (mean) atomic number of the specimen. Equation 4 is not expected to be more accurate than about 20%, since the mean free path may be sensitive to the physical state (e.g. density) of an element, as well as its atomic number. In addition, eqn 2 is exact only if the value of fl is limited (e.g. by an objective aperture) to the dipole region of scattering (fl < 10 mrad at Eo = 100 keV). The log-ratio method, including eqn 4, has been applied to thin-film standards of C, AI, Fe, CU, Ag and Au and found accurate to within 15% in all cases, provided the collection semi-angle (for 100 keV incident electrons) was no larger than 21 mrad (Crozier, 1990)i It has also been tested on small (< 240 nm) vanadium spheres, where eqn 4 was found to be accurate to better than 10% (Bonney, 1990). A second method for the determination of thickness is based on Kramers-Kronig analysis of the energy-loss spectrum J(E), which enables real and imaginary parts of the energy-dependent permittivity to be derived (Egerton, 1986). Iterative processing, including deconvolution to remove plural scattering and successive corrections for surface-mode scattering, leads to estimates of thickness t which converge to a limit, provided the energy resolution

(1)

I t and I o are integrals of the entire spectrum and of the zero-loss peak, respectively. Since the spectral intensity decreases fairly rapidly with increasing energy loss, the spectrum need only be measured up to an energy loss of a few hundred eV in order to measure I t. Equation 1 can be used by itself to measure relative thickness, in units of the total-inelastic mean free path 2. But it is often more useful to obtain an absolute thickness, even if only approximate; this requires knowledge of 2, which is a function of the specimen composition, the electron incident energy E o and the collection semi-angle of the electron spectrometer. For Eo>20 keV, electron scattering theory suggests the following formula (Malis et al., 1988) for mean free path in nm: /~= 106 F(Eo/Em)/In(2flEo/Em)

(3)

(2)

where F=(l+Eo/lO22)/(l+Eo/511) 2 is a relativistic factor (0.73 for E 0 = 120 keV), E o is expressed in keV and fl in miUiradians. The parameter E m is an average energy loss, expressed in eV and given by: 1

2

Y.-Y. Yang and R. F. Egerton

is sufficient (Wehenkel, 1975). A simplified version of this process uses the following approximation (Egerton and Cheng, 1987):

t ~ (2C/D) (aoFEo/Io) f E - 1j (E)/ln(fl/OE) dE-- At.

(5)

Here a o = 0.0529 nm, C = 1 + 0.3 ln(It/Io) is a correction for plural scattering, D = 1 - n-2 where n is the optical refractive index of the specimen (D = 1 for a metal, semimetal or semiconductor of high refractive index) and 0E= (E/Eo) (Eo + 511 keV)/(E o + 1022 keV) is the characteristic angle of inelastic scattering at an energy loss E; A t e 8 nm is a correction for surface scattering. The integral in eqn 5 is over the inelastic part of the spectrum, excluding the zero-loss peak, and up to an energy loss beyond which further contributions to the integral are negligible. Evaluated by a 40-line FORTRAN program, eqn 5 was found to give about 10% accuracy (in the thickness range 10-150 nm) when tested on thin-film standards of AI, Cr, Cu, Ni, Ag, and Au (Egerton and Cheng, 1987). Because of the 1/E weighting factor in eqn 5, the integral converges within 200 eV, so a large range of energy loss need not be recorded. To our knowledge eqn 5 has not been extensively tested and has not been applied to semi-metals or insulators. In this paper, we present results for several elements of medium atomic number, including the semi-metals Sb and Te, comparing our measured thicknesses with those determined by a gravimetric method. We also compare with thicknesses calculated from the log-ratio formula, since eqn 2 has not been previously tested for semi-metals. As a check on our procedure, we include measurements on aluminum films, whose thickness has previously been measured successfully by spectroscopic (log-ratio and Kramers-Kronig) methods.

MATERIALS A N D M E T H O D S

Thin-film standards of silver, indium, tin, antimony and tellurium were prepared by vacuum evaporation from tantalum boats. Aluminum films were made by evaporation from a tungsten wire. Evaporation rates were in the range 0.4-2.0 nm/sec and film thicknesses in the range 30-80 nm. The evaporation rate and film thickness were monitored by a 6 MHz quartz crystal (connected to an Inficon oscillator) placed close to the substrate, which was a cleaved sodium chloride crystal. Except for AI, the substrate was cooled by liquid nitrogen in order to produce a more uniform (fine-grain) film. In the case of Te and Sb, the as-deposited film was probably amorphous but upon warming to room temperature, most of the specimens became crystalline with a large grain size (of the order of 1 ~m). A more precise indication of thickness was obtained by placing a small square of 30 lam-thick aluminum foil close to the NaCI substrate. Using a sensitive balance (Mettler UM3, 0.1 lag readability) to weigh the foil before and after deposition, a direct measurement of the mass thickness

was obtained. The weight before deposition was close to 30 mg; the increase in weight was in the range 90-370 #g. Repeated weighing of a foil (without deposition) gave readings which varied by less than 3 lag. Allowing for the fact that the foil must be weighed both before and after film deposition, this corresponds to an accuracy of 2-5% in the measurement of mass thickness. Repeated measurements on thin glass substrates gave poor reproducibility, possibly due to electrostatic charging of the glass. Small areas of film were detached from the NaC1 substrates by floatation onto the surface of distilled water. They were picked up on 400-mesh electron-microscope grids and washed in a second water bath before examination in a Philips CM12 transmission electron microscope, fitted with a Gatan model 666 parallel-recording spectrometer. For energy-loss spectroscopy, the microscope was operated at 120 kV in image mode, with a screen magnification in the range 5000-10,000. A 5 mm spectrometer entrance aperture (SEA) defined the region of analysis; the spectrometer acceptance angle fl, within the range 2-20 mrad, was defined by a TEM objective aperture. Spectra were acquired into an EDAX 9900 multichannel analyzer, where energy-loss software was used to measure the integral I o of the zero-loss peak and the area I t under the low-loss region. Before such measurement, the instrumental (detector) background was removed from each spectrum by subtracting a 'bias spectrum' recorded under the same conditions but with the microscope screen lowered to exclude the electron beam. By using the middle portion of the photodiode array to record the low-loss region, the root-mean square variation in channel gain was reduced to below 1%, so a 'gain correction' (to allow for the varying sensitivity of individual detector channels) was considered unnecessary. The gravimetric method of determining elemental mass-thickness is accurate only if there is negligible oxidation of the film. Our electron microscope specimens were therefore checked for oxygen content by closely examining the energy-loss spectra at an energy corresponding to the oxygen K-ionization edge (535 eV). Oxidation was found to be appreciable (up to 20%) only in the case of aluminum; for other elements, the oxygen content was estimated to be in the range 0.1% to 10%, and would not greatly affect our measurements of mean free path.

RESULTS FROM THE KRAMERS-KRONIG METHOD

Because J(E) in eqn 5 denotes inelastic intensity, the zero-loss must be removed from the data before integration. This can be done by programming the dataanalyzing computer to detect the first minimum in intensity and employ linear interpolation to estimate the inelastic intensity below this energy before computing the integral (Egerton and Cheng, 1987). However, other forms of interpolation are possible, such as a parabolic energy dependence (dotted curve in Fig. 1). Alternatively,

Measurement of TEM Specimen Thickness

the inelastic intensity can simply be truncated and set to zero below the minimum, on the assumption that the tail of the zero-loss component (above the minimum) is equivalent in area to the missing inelastic intensity. Because of the E - ~ weighting in eqn 5, these alternative procedures can give rise to appreciable differences in measured thickness, as illustrated in Fig. 2.

I:1 o

f

/

,z. / //..

.: ":

Z,... ""

Fig. 1. Part of an energy-loss spectrum recorded with a Gatan 666 spectrometer, showing the zero-loss and plasmon peaks. The dashed, dotted and vertical solid lines represent three different approximations for the inelastic intensity below the minimum value. 100

I

el qD

Method

I

AI Ag In Sn Sb Te

vV DD

qo

[]

¢1

o []

[]

It appears that the linear-interpolation method (triangular data points in Fig. 2) overestimates the thickness, whereas the parabolic interpolation (inverted triangles) gives values within 10 nm of those obtained by weighing. Although abrupt truncation gives consistently low values (open squares), this procedure yields thicknesses (filled circles) which are within 10 nm of weighed values if the surface correction term is omitted from eqn 5. The success of this latter procedure may be due to the fact that most of the surface-loss intensity occurs at energy losses below the bulk plasmon-peak, so that elimination of the [ow energy losses is roughly equivalent to exclusion of the surfaceloss intensity. Overall, there is a tendency (see Fig. 2) for the measured thickness to decrease as the semi-angle fl of collection of the energy-loss spectrum is increased, an effect which can be explained in terms of a falloff in oscillator strength at higher scattering angles (Egerton, 1986). Values average over several spectra recorded at //-values in the range 3.8 to 10 mrad are shown in Table 1. Table 1. Film thickness (in rim) measured by EELS, Compared to weighed values (W). L R represents the log-ratio method, with ,1. taken from eqn 2. KK represents the Kramers-Kronig methodi with various interpolations at low energy loss, including abrupt truncation (KK-cut) with no surface scattering correction. Error bars represent standard deviations based on the analysis of spectra recorded a t collection semiangles between 3.8 and 20 mrad. The last column gives the physical density used to derive (W) thickness from weight d{fference

Energy Loss (eV)

100

3

[-I

W

LR

34.9 65.8 46.1 34.7 43.4 50.1

35.6±0.3 57.0±2.0 34.0±0.7 26.0±1.0 27.5±0.7 33.0±2.0

KK-linear KK-parabolic 43±2 77±4 54±4 41±3 46±2 57±3

37!1 70±4 48±4 34±2 38±2 50±3

KK-cut

p

35.0±2.0 2.7 66.0±2.010.5 46.0±3.0 7.3 35.0±1.0 7.3 37.4±0.5 6.7 45.0±2.0 6.0

w 0 lOOn

I

I

i

,

l

,

0 A

eJ

100

i:I

RESULTS FROM THE LOG-RATIO METHOD qD

i:1

Oo

0

o

I

Ao

,ti o

I

100

,

I

100~

I

v V

o

A^

v

.M

[]

&

[.-,

0

I

[]

5

i 10

n 15

Beta (mrad)

OI 20

0

I 5

I 10

15

Beta (mrad)

Fig. 2. Thickness of Al, Ag, In, Sn, Sb and Te films computed according to eqn 5 and plotted as a function of the collection sere±angle, defined by an objective aperture. Inverted triangles denote linear interpolation and upright triangles parabolic interpolation below the first minimum in intensity. Open squares represent abrupt truncation below the minimum; solid circles correspond to abrupt truncation and At = 0 in eqn 5. Horizontal lines show the thickness determined by weighing.

20

Table 1 also shows thicknesses measured by the log-ratio (LR) method, taking the integral up to the first minimum at the zero-loss intensity I o and eqn 4 for the inelastic mean free path 2. Except for aluminum, the LR thicknesses are consistently lower than the weighed values, as illustrated also in Fig. 3 which dononstrates that the measured thickness is effectively independent of the collection angle employed. These results suggest that eqn 4 gives values of 2 for the medium-Z elements which are too low, by factors ranging from 1.15 (for Ag) to 1.58 (for Sb). We therefore used eqnl to deduce 2, taking t as the weighed value of thickness, with the results shown in Fig. 4. By leastsquares fitting of these experimental points to smooth curves generated from eqn 2, we obtained values of E~ which best match the experimental data. These fitted values are listed in Table 2, and might be used to predict 2 (for an appropriate value of fl) for log-ratio thickness measurements of these elements, hopefully with better accuracy than by use of eqn 4.

4

Y.-Y. Yang and R. F. Egerton

We do not know why eqn 4 appears to be less accurate for these medium-Z elements; Malis et al. (1987) found that this equation gave 2 to within 20% of measured values for Be, AI, Fe, Cu, Zr, Ag, Hf, and Au. Equation 2 is itself an approximation; a more precise formula would include a denominator term of D = 1 - n -2. But D is 100

100 el Itl m

e-,

el

eo

-v V

60

60

00

40

40

80

.m

v

I

o

I

I

I

1

I

I

I

el

60

eo

VV V

,tO O 0

0

0

~

~-0 I

o

A

I:1 I:l ,.M

I

I

el

80 60

.2

40

t-,

20 0

V

v

0

0

0

40

Vv

v

~o

O0

0 I

o

100

I

I

I

5

10

15

-Vv

V

40 -00 0 20 0 i t t 0 5 10 15 Beta (mrad)

20

Beta ( m r a d )

20

Fig. 3. Results from three methods of thickness measurement: horizontal lines represent values obtained by weighing, triangles are from the Kramers-Kronig method with parabolic interpolation and correction for surface scattering, and the circles are from the log-ratio method.

•--,~ 160

140

140

12o

a

AI Ag In Sn Sb Te

19.6 25.0 20.4 21.2 17.0 18.3

19.1 30.4 30.8 31.1 31.3 31.5

Region

2 (nm)

Centre of a large grain At a bend contour Bend--contour intersection Grain boundary Fine-grain polycrystalline Amorphous

101 116 126 120 123 101

gives the mean free path deduced via eqn 1 from spectra recorded from different areas of a Te specimen containing both crystalline and anorphous regions. It appears that the effective mean free path increases in more strongly diffracting areas; the maximum variation in mean free path is about 25%0.

100

'~ 120

el ~

Equation 4

Table 3. Mean free paths in different regions of a tellurium specimen, measured using eqn 2 with fl = 9.3 mrad

100

ct

Experiment

believed to be close to unity for the materials we have investigated, as evidenced by our successful use of the Kramers-Kronig method with D = 1 in eqn 5. One reason why eqn 4 may not provide highly accurate values of 2 is that the mean free path may vary with the physical state of the specimen, and possibly with its orientation if the specimen is crystalline. An indication of the extent of these effects is given by Table 3, which

.~, 100

,-. 100

Element

0

eo

20

Table 2. E,, values (in eV) as deduced from experimental data, via eqns 1 and 2, and as given by the parameterized formula, eqn 4

DISCUSSION AND CONCLUSIONS

80

100 M

"~ so

"~

60

I

I

I

I

I

I

5

10

15

140 ,t 120

"~ 120

100

100

,o

N

a

8o

A~ 160 ~'~ 140 120

~

160

~

el 14o

'~ 120

100 I~

eo

,.~ 100

80 0

J

I

5

10

15

Beta ( m r a d )

20

~'

eo

Beta ( m r a d )

Fig. 4. Total-inelastic mean free paths as a function of collection semiangle. Circles with error bars are experimental results (where no error bars are shown, they lie within the circle); the curves represent least-square fitting to eqn 2 with E m as an adjustable parameter.

20

We have applied electron-spectroscopic methods of thickness determination to thin films of atomic number 47-52 and thicknesses in the range 30-70 nm, as determined by weighing. We show that the accuracy of the Kramers-Kronig method of thickness determination is sensitive to the form of interpolation used to separate the zero-loss intensity. Our best results, corresponding to 10%o accuracy, were obtained with a parabolic interpolation (and 8 nm surface-scattering correction) or by completely excluding energy losses below the first minimum in the energy-loss spectrum and omitting the surface-scattering correction. Our results using the log-ratio method appear to show that 20% accuracy in thickness may not be possible for some elements if the formula E m - 7 . 6 Z T M is used to predict the mean free path. Our measured values of Era are up to 46% smaller than given by eqn 4, corresponding to a mean free path 58% larger than predicted. It is not expected that ). should be a completely smooth function of atomic number; atomic calculations (Inokuti et al., 1981)

Measurement of TEM Specimen Thickness

show oscillations in mean free path corresponding to the structure of the outermost atomic shell, but this would not account for the deviations in the range Z=47-52. However, atomic models do not take into account solidstate effects such as changes in density and diffraction conditions, which may produce a variation in 2 of up to 25%. Acknowledgements--This work was funded by the U.S. National Science Foundation under their instrumentation program, and by the Natural Sciences and Engineering Research Council of Canada.

REFERENCES Bonney, L., 1990. Measurement of the inelastic mean free path by EELS analyses of submicron particles. Proc. Xllth International Congress

5

for Electron Microscopy, pp. 74-75. San Francisco Press, San Francisco. Crozier, P. A., 1990. Measurement of inelastic electron scattering crosssections by electron energy-loss spectroscopy. Phil. Mag. B, 61, 311-336. Egerton, R. F., 1986. Electron Energy-Loss Spectroscopy in the Electron Microscope. Plenum, New York. Egerton, R. F. and Cheng, S. C., 1987. Measurement of local thickness by electron energy-loss spectroscopy. Ultramicroscopy, 21,231-244. Inokuti, M., Dehmer, J. L., Bauer, T. and Hanson, J. D., 1981. Oscillator-strength moments, stopping powers and total inelasticscattering cross sections for all elements through strontium. Phys. Rev. A, 23, 95 109. Malis, T., Cheng, S. C. and Egerton, R. F., 1988. EELS log-ratio technique for specimen-thickness measurement in the TEM. J. Electron Microsc. Tech., 8, 193-200. Wehenkel, C., 1975. Mise au point d'une nouvelle m6thode d'analyse quantitative des spectres de pertes d'6nergie d'6lectrons rapides difus6s dans la direction du faiseau incident: application a l'6tudes des m&aux nobles. Le Journal de Physique, 36, 199 213.