- Email: [email protected]
Simulation of an improved magnetic–electrostatic detector objective lens for LVSEM
Nuclear Instruments and Methods in Physics Research A 427 (1999) 99}103
Simulation of an improved magnetic}electrostatic detector objective lens for LVSEM G. Knell*, E. Plies Institut fu( r Angewandte Physik, Universita( t Tu( bingen, Auf der Morgenstelle 10, D-72076 Tu( bingen, Germany
Abstract The simulations of the imaging properties of several magnetic}electrostatic detector objective lenses are presented. We have assumed that the magnetic circuit has a radially arranged pole-piece gap. By using this snorkel lens design the specimen is immersed in a strong magnetic "eld. The calculations show that the chromatic aberration coe$cient, which essentially determines the resolution in low-voltage scanning electron microscopy, only decreases with increasing immersion ratio if the radius of the inner pole piece is not too small. Furthermore we determined the collection e$ciency of secondary electrons for a lens variant which is optimized with respect to primary electron optics. 1999 Elsevier Science B.V. All rights reserved. PACS: 41.85 Keywords: Electron optics; Design of lenses; Combined magnetic}electrostatic lenses; Simulation of electron optical properties; Low-voltage scanning electron microscopy; SEM
1. Introduction In recent years numerous lens designs were developed to improve the resolution in low-voltage scanning electron microscopy (LVSEM). A review of the di!erent approaches was given by MuK llerovaH and Lenc [1] and by Tsuno et al. [2]. In all cases where purely magnetic lenses are used, such as single pole-piece lenses below or above the specimen, side pole-gap lenses, snorkel lenses, and
* Corresponding author. Tel.: #49-7071-29-76365; fax: #49-7071-29-5400. E-mail address: [email protected] (G. Knell).
TEM-like objective lenses, the specimen is immersed in a strong magnetic "eld. Thus the lens excitation is comparatively high. A purely electrostatic objective lens in combination with a multipole corrector was proposed by Zach [3]. With this arrangement it is possible to reduce the resolution limit to less than 2 nm at 1 keV primary electron (PE) energy [4]. Combined magnetic-electrostatic lenses were presented in Refs. [2,5}8]. The objective lens described by Frosien et al. [5] consists of a conical magnetic pole-piece lens with an axially arranged pole-piece gap and an integrated electrostatic lens in its lower part. The electrostatic lens is composed of a liner tube on high positive potential (e.g., 8 kV) and a grounded electrode. In this system
0168-9002/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 1 5 4 8 - 4
100
G. Knell, E. Plies / Nuclear Instruments and Methods in Physics Research A 427 (1999) 99}103
the specimen is nearly free of magnetic and electrostatic "elds. The lens is characterised by low values for the spherical and chromatic aberration coe$cients, which decrease with decreasing PE energy. The arrangement of a detector above the lens allows the detection of secondary electrons (SEs) even at small working distances. Additionally, there is no side-extraction "eld which would disturb the PE optics. The use of this magnetic}electrostatic detector objective lens (MEDOL) has proved its worth in low voltage scanning electron microscopy [6] and in critical dimension measurements [7]. Various lens designs, such as combined lenses and purely magnetic lenses, with di!erent geometries were also simulated by Tsuno et al. [2,8]. Preikszas and Rose [9] discussed the optimized axial electromagnetic "eld of compound lenses for various arrangements and constraints. They showed that for low-voltage compound lenses a high magnetic "eld in the region of the specimen strongly reduces the aberration coe$cients. If the specimen is located in a "eld-free region, it does not seem to be possible to achieve a signi"cant improvement of the resolution by optimizing the combined magnetic}electrostatic objective lens by Frosien et al. [5]. Increasing the electrostatic "eld in the specimen plane results in a reduction of spherical and chromatic aberration coe$cients. For strong electrostatic "elds at the specimen plane one gets a design which is similar to that of a cathode lens [2,10]. However, in many applications a strong electrostatic "eld is disturbing. In a recent paper [11] the collection e$ciency for MEDOL variants with a radially arranged pole-piece gap (see Figs. 1 and 2) was presented. Our goal is to get an optimized variant of this lens type with an electric "eld strength lower than 100 V/mm at the specimen plane (working distance wd " 1 mm). The maximum number of ampere turns should be smaller than 1.5 kA-t. The "eld calculations and simulations of trajectories were carried out with the program Electron Optical Systems by Prof E. Kasper. In this program the "eld calculations are based on the boundary element method (see e.g., [12]). It is possible to solve Dirichlet-problems and material-boundaryproblems for a homogeneous magnetic yoke with constant permeability. The sources of the "elds,
Fig. 1. Schematic drawing of a MEDOL variant with radially arranged pole-piece gap (PE"primary electrons, SE"secondary electrons).
Fig. 2. Pole-piece region of the simulated MEDOL variants. The dashed curve shows the geometry of the "nal lens design (see Table 1).
which are located at the surfaces of electrodes or iron-yokes, are charges in the "rst case and currents in the second case.
G. Knell, E. Plies / Nuclear Instruments and Methods in Physics Research A 427 (1999) 99}103
2. MEDOL variants with radially arranged pole-piece gap The lens geometry was simpli"ed by rejecting the conical design of the outer pole-piece (Fig. 2). The snorkel of the inner pole-piece also runs parallel to the specimen plane. Provided that the permeability of the iron yoke is high, the "eld calculations of the magnetic lens can be performed by solving the Dirichlet-problem. This is possible because we are not dealing with open magnetic circuits. The whole lens circuit was also simulated with the help of ring currents, to ensure that no saturation e!ects occur. We checked that the magnetic #ux density along the boundary of the magnetic yoke was smaller than the saturation #ux density of the iron. The maximum lens excitation in this case amounts to +1.5 kA-t. Higher currents are probably possible, but we consider only values up to 1.5 kA-t for our "nal lens design (Table 1, see below). It was assumed that the "rst specimen-sided electrode (inner pole-piece), the outer pole-piece and the specimen are at ground potential. The potential of the liner tube U was varied between 0 V and 8 kV. Some authors ignore the specimen while calculating the electrostatic "eld. Since one cannot neglect the "eld strength at the specimen plane in all cases discussed here it was supposed that the specimen is an extended planar conductor. Because of that, one can simulate the electrostatic "eld by a negative mirror symmetry of the electrodes. In
101
order to "nd an optimized variant of this lens type, the radius of the inner pole-piece bore R and the distance between the liner tube and inner pole-piece S were varied. Since the inner pole-piece also # serves as the "rst electrode of the electrostatic lens, both "eld distributions are modi"ed if we change the radius of the bore. As shown in Ref. [13], the axial magnetic #ux density is essentially determined by the radius R of the inner bore. If we decrease R , the maximum of the "eld distribution moves to the lens and the maximum #ux density B in creases (Fig. 3a). The position of the maximum electrostatic "eld moves only slightly towards the inner pole-piece, U increases and the "eld strength at the speci men plane decreases (Fig. 3b). If we change the inner bore radius of the "rst electrode, the curvature of the axial potential, which is a measure for
Table 1 Optical properties of a compound lens with R "1.7 mm, S "3.95 mm (reference plane for S : dotted line in Fig. 2). The # # potential of the liner tube is 8 kV, the pole-pieces and the specimen are on ground potential. The lens excitation NI amounts to 1.1 kA-t for E "200 eV and +1.5 kA-t for the .# other cases. Aberration coe$cients are given with respect to specimen plane E .# (keV)
wd (mm)
C 1 (mm)
C ! (mm)
d (nm)
g (%)
0.2 0.5 1 5 10
1.0 1.4 2.0 4.5 7.0
0.48 0.66 0.96 3.15 7.99
0.51 0.79 1.11 2.38 3.93
7.6 4.7 3.3 1.5 1.2
55 43 36 25 26
Fig. 3. (a) Axial magnetic #ux density distribution B(z) (NI"1 kA-t). (b) Electrostatic "eld strength on the axis U (z) (U "1 kV, S "4 mm). #
102
G. Knell, E. Plies / Nuclear Instruments and Methods in Physics Research A 427 (1999) 99}103
the refracting power of the electrostatic lens, varies strongly in the focusing lens part. The distribution of the diverging part alters only marginally. If R is small, e.g. 0.5 mm, the overlap of magnetic and electrostatic "elds is weak and the "elds are almost sequentially arranged. We calculated the imaging properties for a PE energy E of 1 keV with wd " 2 mm and for .# E "200 eV with wd "1 mm. The behaviour of .# the aberration coe$cients, C and C (Fig. 4), and 1 ! the lens excitation NI strongly depend on the bore radius of the inner pole-piece. At small radii the aberrations grow with increasing potential of the liner tube, and the lens excitation decreases. As already discussed by Tsuno et al. [2], the aberrations are large if the "elds are sequentially arranged. In some cases the electrostatic "eld already focuses the electron beam. For example, the curve in Fig. 4c for C (R "1.5 mm, S "2 mm) "nishes ! # at a potential U of +5 kV. The magnetic excita tion is zero at this point. At large radii the aberrations decrease or increase only slightly with U . For E "200 eV .# (R "2.5 mm, S "2 mm) C is reduced from # ! 0.84 mm (U "0 kV) to 0.29 mm (U "8 kV). How ever, the electrostatic "eld strength in the specimen plane amounts to 321 V/mm and it is therefore large compared to the requested "eld strength of 100 V/mm. Fig. 4c shows a comparison of C for ! di!erent spacings between the liner tube and the inner pole-piece. The lenses with S "2 mm have # smaller aberrations than the variants with S "4 mm. In the "rst approximation, this can be # attributed to the higher electrostatic "eld strength at the specimen plane. The comparison of the aberrations for a "xed potential of the liner tube shows that at high immersion ratios C and C grow with decreasing 1 ! radii of the inner pole-piece. At low immersion ratios or in the purely magnetic case (Fig. 4b and c), the chromatic aberration coe$cient decreases, while the behaviour of the spherical aberration is not obvious. A further improvement of the aberrations was obtained, when the surface of the snorkel, opposite the liner tube, had a conical shape (dashed curve in Fig. 2). Table 1 shows some of the optical properties of such a MEDOL variant with R "1.7 mm. The
Fig. 4. Aberration coe$cients vs potential of the liner tube for di!erent radii of the inner pole-piece bore. (a) C , (b) C (PE 1 ! energy 1 keV, S "2 mm) and (c) C (PE energy 200 eV). # !
electrostatic "eld strength amounts to 100 V/mm at the specimen plane (wd " 1 mm). The spherical and chromatic aberration coe$cients are noticeably smaller than the aberration coe$cients of the MEDOL with an axially arranged pole-piece gap
G. Knell, E. Plies / Nuclear Instruments and Methods in Physics Research A 427 (1999) 99}103
(see Fig. 3 in Ref. [7]). For example, the chromatic aberration coe$cient is reduced by a factor of +2 at a PE energy of 1 keV. In order to estimate the probe diameter d, we used the root-power-sum algorithm according to Barth and Kruit [14], which gives a more realistic value than the common way of a quadratic superposition. The size of the virtual source was neglected, since we assumed a large distance between source and objective lens (magni"cation +0). The energy spread of the electron source was supposed to be $0.35 eV. It should be noted that the calculated probe diameter for a PE energy of 1 keV, shown in Table 1, is almost equal to the measured resolution by Frosien et al. [7], although the aberrations of the lens design presented here are signi"cantly reduced. Table 1 also shows the collection e$ciency of the secondary electrons (SEs) g , which is speci"ed by the number of electrons that hit the detector surface to the number of emitted electrons. It was assumed that the detector is located at z "125 mm (Fig. 1), " the inner bore radius of the detector is 1 mm and the outer radius is 10 mm. Further assumptions are a cosine angular distribution and a Maxwell energy distribution with a most probable energy of 2 eV. For high PE energies we have to increase the working distance, since the magnetic excitation is limited. Thus, the SEs strongly spiral up in the area between the specimen and the beginning of the electrostatic "eld. Several crossovers of the SEs are located in the detector plane. Therefore, most of them miss the annular detector, if its inner radius is larger than 2 mm. A detailed discussion of the determination of g for MEDOLs was given in Ref. [11]. 3. Conclusion In order to design a high-resolution MEDOL with a radially arranged pole-piece gap, attention must be given to the geometrical parameters which essentially determine the "eld distributions: the inner pole-piece bore and the distance between the "rst specimen-sided electrode and the liner tube. In the low-voltage range the chromatic aberration coe$cient decreases with increasing immersion ratio,
103
if the overlapping of the magnetic and electrostatic "elds is not too small. The presented optimized lens variant shows noticeably smaller aberrations than the original MEDOL with an axially arranged pole-piece gap. Constraints for our lens design are a moderate electrostatic "eld strength at the specimen plane of 100 V/mm (wd " 1 mm) and a maximum lens excitation of 1.5 kA-t. We get a chromatic aberration coe$cient of 1.1 mm at a primary electron energy of 1 keV (wd " 2 mm) and of 0.51 mm at 200 eV (wd " 1 mm). The collection e$ciency of the secondary electrons is also satisfactory.
Acknowledgements The authors would like to thank Prof. E. Kasper for placing the program Electron Optical Systems at our disposal and R. Wilcox for checking the English manuscript. We also thank Y. Lutsch for productive discussions.
References [1] I. MuK llerovaH , M. Lenc, Ultramicroscopy 41 (1992) 399. [2] K. Tsuno, N. Handa, S. Matsumoto, SPIE Electron-Beam Sources and Charged-Particle Optics, Vol. 2522, 1995, p. 243. [3] J. Zach, Optik 83 (1989) 30. [4] J. Zach, M. Haider, Nucl. Instr. and Meth. A 363 (1995) 316. [5] J. Frosien, E. Plies, K. Anger, J. Vac. Sci. Technol. B7 (1989) 1874. [6] E. Weimer, J.-P. Martin, Proc. 13th Int. Congr. on Electron Microscopy, ICEM, Paris, Vol. 1, 1994, p. 67. [7] J. Frosien, S. Lanio, H.P. Feuerbaum, Nucl. Instr. and Meth. A 363 (1995) 25. [8] K. Tsuno, N. Handa, S. Matsumoto, A. Mogami, SPIE Charged-Particle Optics II, Vol. 2858, 1996, p. 46. [9] D. Preikszas, H. Rose, Optik 100 (1995) 179. [10] S. Beck, E. Plies, B. Schiebel, Nucl. Instr. and Meth. A 363 (1995) 31. [11] G. Knell, E. Plies, Optik 108 (1998) 37. [12] P.W. Hawkes, E. Kasper, Principles of Electron Optics. Vol. 1: Basic Geometrical Optics, Academic Press, London, 1989, Chapter 10. [13] T.-T. Tang, J.-P. Song, Optik 84 (1990) 108. [14] J.E. Barth, P. Kruit, Optik 101 (1996) 101.
Simulation of an improved magnetic}electrostatic detector objective lens for LVSEM G. Knell*, E. Plies Institut fu( r Angewandte Physik, Universita( t Tu( bingen, Auf der Morgenstelle 10, D-72076 Tu( bingen, Germany
Abstract The simulations of the imaging properties of several magnetic}electrostatic detector objective lenses are presented. We have assumed that the magnetic circuit has a radially arranged pole-piece gap. By using this snorkel lens design the specimen is immersed in a strong magnetic "eld. The calculations show that the chromatic aberration coe$cient, which essentially determines the resolution in low-voltage scanning electron microscopy, only decreases with increasing immersion ratio if the radius of the inner pole piece is not too small. Furthermore we determined the collection e$ciency of secondary electrons for a lens variant which is optimized with respect to primary electron optics. 1999 Elsevier Science B.V. All rights reserved. PACS: 41.85 Keywords: Electron optics; Design of lenses; Combined magnetic}electrostatic lenses; Simulation of electron optical properties; Low-voltage scanning electron microscopy; SEM
1. Introduction In recent years numerous lens designs were developed to improve the resolution in low-voltage scanning electron microscopy (LVSEM). A review of the di!erent approaches was given by MuK llerovaH and Lenc [1] and by Tsuno et al. [2]. In all cases where purely magnetic lenses are used, such as single pole-piece lenses below or above the specimen, side pole-gap lenses, snorkel lenses, and
* Corresponding author. Tel.: #49-7071-29-76365; fax: #49-7071-29-5400. E-mail address: [email protected] (G. Knell).
TEM-like objective lenses, the specimen is immersed in a strong magnetic "eld. Thus the lens excitation is comparatively high. A purely electrostatic objective lens in combination with a multipole corrector was proposed by Zach [3]. With this arrangement it is possible to reduce the resolution limit to less than 2 nm at 1 keV primary electron (PE) energy [4]. Combined magnetic-electrostatic lenses were presented in Refs. [2,5}8]. The objective lens described by Frosien et al. [5] consists of a conical magnetic pole-piece lens with an axially arranged pole-piece gap and an integrated electrostatic lens in its lower part. The electrostatic lens is composed of a liner tube on high positive potential (e.g., 8 kV) and a grounded electrode. In this system
0168-9002/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 1 5 4 8 - 4
100
G. Knell, E. Plies / Nuclear Instruments and Methods in Physics Research A 427 (1999) 99}103
the specimen is nearly free of magnetic and electrostatic "elds. The lens is characterised by low values for the spherical and chromatic aberration coe$cients, which decrease with decreasing PE energy. The arrangement of a detector above the lens allows the detection of secondary electrons (SEs) even at small working distances. Additionally, there is no side-extraction "eld which would disturb the PE optics. The use of this magnetic}electrostatic detector objective lens (MEDOL) has proved its worth in low voltage scanning electron microscopy [6] and in critical dimension measurements [7]. Various lens designs, such as combined lenses and purely magnetic lenses, with di!erent geometries were also simulated by Tsuno et al. [2,8]. Preikszas and Rose [9] discussed the optimized axial electromagnetic "eld of compound lenses for various arrangements and constraints. They showed that for low-voltage compound lenses a high magnetic "eld in the region of the specimen strongly reduces the aberration coe$cients. If the specimen is located in a "eld-free region, it does not seem to be possible to achieve a signi"cant improvement of the resolution by optimizing the combined magnetic}electrostatic objective lens by Frosien et al. [5]. Increasing the electrostatic "eld in the specimen plane results in a reduction of spherical and chromatic aberration coe$cients. For strong electrostatic "elds at the specimen plane one gets a design which is similar to that of a cathode lens [2,10]. However, in many applications a strong electrostatic "eld is disturbing. In a recent paper [11] the collection e$ciency for MEDOL variants with a radially arranged pole-piece gap (see Figs. 1 and 2) was presented. Our goal is to get an optimized variant of this lens type with an electric "eld strength lower than 100 V/mm at the specimen plane (working distance wd " 1 mm). The maximum number of ampere turns should be smaller than 1.5 kA-t. The "eld calculations and simulations of trajectories were carried out with the program Electron Optical Systems by Prof E. Kasper. In this program the "eld calculations are based on the boundary element method (see e.g., [12]). It is possible to solve Dirichlet-problems and material-boundaryproblems for a homogeneous magnetic yoke with constant permeability. The sources of the "elds,
Fig. 1. Schematic drawing of a MEDOL variant with radially arranged pole-piece gap (PE"primary electrons, SE"secondary electrons).
Fig. 2. Pole-piece region of the simulated MEDOL variants. The dashed curve shows the geometry of the "nal lens design (see Table 1).
which are located at the surfaces of electrodes or iron-yokes, are charges in the "rst case and currents in the second case.
G. Knell, E. Plies / Nuclear Instruments and Methods in Physics Research A 427 (1999) 99}103
2. MEDOL variants with radially arranged pole-piece gap The lens geometry was simpli"ed by rejecting the conical design of the outer pole-piece (Fig. 2). The snorkel of the inner pole-piece also runs parallel to the specimen plane. Provided that the permeability of the iron yoke is high, the "eld calculations of the magnetic lens can be performed by solving the Dirichlet-problem. This is possible because we are not dealing with open magnetic circuits. The whole lens circuit was also simulated with the help of ring currents, to ensure that no saturation e!ects occur. We checked that the magnetic #ux density along the boundary of the magnetic yoke was smaller than the saturation #ux density of the iron. The maximum lens excitation in this case amounts to +1.5 kA-t. Higher currents are probably possible, but we consider only values up to 1.5 kA-t for our "nal lens design (Table 1, see below). It was assumed that the "rst specimen-sided electrode (inner pole-piece), the outer pole-piece and the specimen are at ground potential. The potential of the liner tube U was varied between 0 V and 8 kV. Some authors ignore the specimen while calculating the electrostatic "eld. Since one cannot neglect the "eld strength at the specimen plane in all cases discussed here it was supposed that the specimen is an extended planar conductor. Because of that, one can simulate the electrostatic "eld by a negative mirror symmetry of the electrodes. In
101
order to "nd an optimized variant of this lens type, the radius of the inner pole-piece bore R and the distance between the liner tube and inner pole-piece S were varied. Since the inner pole-piece also # serves as the "rst electrode of the electrostatic lens, both "eld distributions are modi"ed if we change the radius of the bore. As shown in Ref. [13], the axial magnetic #ux density is essentially determined by the radius R of the inner bore. If we decrease R , the maximum of the "eld distribution moves to the lens and the maximum #ux density B in creases (Fig. 3a). The position of the maximum electrostatic "eld moves only slightly towards the inner pole-piece, U increases and the "eld strength at the speci men plane decreases (Fig. 3b). If we change the inner bore radius of the "rst electrode, the curvature of the axial potential, which is a measure for
Table 1 Optical properties of a compound lens with R "1.7 mm, S "3.95 mm (reference plane for S : dotted line in Fig. 2). The # # potential of the liner tube is 8 kV, the pole-pieces and the specimen are on ground potential. The lens excitation NI amounts to 1.1 kA-t for E "200 eV and +1.5 kA-t for the .# other cases. Aberration coe$cients are given with respect to specimen plane E .# (keV)
wd (mm)
C 1 (mm)
C ! (mm)
d (nm)
g (%)
0.2 0.5 1 5 10
1.0 1.4 2.0 4.5 7.0
0.48 0.66 0.96 3.15 7.99
0.51 0.79 1.11 2.38 3.93
7.6 4.7 3.3 1.5 1.2
55 43 36 25 26
Fig. 3. (a) Axial magnetic #ux density distribution B(z) (NI"1 kA-t). (b) Electrostatic "eld strength on the axis U (z) (U "1 kV, S "4 mm). #
102
G. Knell, E. Plies / Nuclear Instruments and Methods in Physics Research A 427 (1999) 99}103
the refracting power of the electrostatic lens, varies strongly in the focusing lens part. The distribution of the diverging part alters only marginally. If R is small, e.g. 0.5 mm, the overlap of magnetic and electrostatic "elds is weak and the "elds are almost sequentially arranged. We calculated the imaging properties for a PE energy E of 1 keV with wd " 2 mm and for .# E "200 eV with wd "1 mm. The behaviour of .# the aberration coe$cients, C and C (Fig. 4), and 1 ! the lens excitation NI strongly depend on the bore radius of the inner pole-piece. At small radii the aberrations grow with increasing potential of the liner tube, and the lens excitation decreases. As already discussed by Tsuno et al. [2], the aberrations are large if the "elds are sequentially arranged. In some cases the electrostatic "eld already focuses the electron beam. For example, the curve in Fig. 4c for C (R "1.5 mm, S "2 mm) "nishes ! # at a potential U of +5 kV. The magnetic excita tion is zero at this point. At large radii the aberrations decrease or increase only slightly with U . For E "200 eV .# (R "2.5 mm, S "2 mm) C is reduced from # ! 0.84 mm (U "0 kV) to 0.29 mm (U "8 kV). How ever, the electrostatic "eld strength in the specimen plane amounts to 321 V/mm and it is therefore large compared to the requested "eld strength of 100 V/mm. Fig. 4c shows a comparison of C for ! di!erent spacings between the liner tube and the inner pole-piece. The lenses with S "2 mm have # smaller aberrations than the variants with S "4 mm. In the "rst approximation, this can be # attributed to the higher electrostatic "eld strength at the specimen plane. The comparison of the aberrations for a "xed potential of the liner tube shows that at high immersion ratios C and C grow with decreasing 1 ! radii of the inner pole-piece. At low immersion ratios or in the purely magnetic case (Fig. 4b and c), the chromatic aberration coe$cient decreases, while the behaviour of the spherical aberration is not obvious. A further improvement of the aberrations was obtained, when the surface of the snorkel, opposite the liner tube, had a conical shape (dashed curve in Fig. 2). Table 1 shows some of the optical properties of such a MEDOL variant with R "1.7 mm. The
Fig. 4. Aberration coe$cients vs potential of the liner tube for di!erent radii of the inner pole-piece bore. (a) C , (b) C (PE 1 ! energy 1 keV, S "2 mm) and (c) C (PE energy 200 eV). # !
electrostatic "eld strength amounts to 100 V/mm at the specimen plane (wd " 1 mm). The spherical and chromatic aberration coe$cients are noticeably smaller than the aberration coe$cients of the MEDOL with an axially arranged pole-piece gap
G. Knell, E. Plies / Nuclear Instruments and Methods in Physics Research A 427 (1999) 99}103
(see Fig. 3 in Ref. [7]). For example, the chromatic aberration coe$cient is reduced by a factor of +2 at a PE energy of 1 keV. In order to estimate the probe diameter d, we used the root-power-sum algorithm according to Barth and Kruit [14], which gives a more realistic value than the common way of a quadratic superposition. The size of the virtual source was neglected, since we assumed a large distance between source and objective lens (magni"cation +0). The energy spread of the electron source was supposed to be $0.35 eV. It should be noted that the calculated probe diameter for a PE energy of 1 keV, shown in Table 1, is almost equal to the measured resolution by Frosien et al. [7], although the aberrations of the lens design presented here are signi"cantly reduced. Table 1 also shows the collection e$ciency of the secondary electrons (SEs) g , which is speci"ed by the number of electrons that hit the detector surface to the number of emitted electrons. It was assumed that the detector is located at z "125 mm (Fig. 1), " the inner bore radius of the detector is 1 mm and the outer radius is 10 mm. Further assumptions are a cosine angular distribution and a Maxwell energy distribution with a most probable energy of 2 eV. For high PE energies we have to increase the working distance, since the magnetic excitation is limited. Thus, the SEs strongly spiral up in the area between the specimen and the beginning of the electrostatic "eld. Several crossovers of the SEs are located in the detector plane. Therefore, most of them miss the annular detector, if its inner radius is larger than 2 mm. A detailed discussion of the determination of g for MEDOLs was given in Ref. [11]. 3. Conclusion In order to design a high-resolution MEDOL with a radially arranged pole-piece gap, attention must be given to the geometrical parameters which essentially determine the "eld distributions: the inner pole-piece bore and the distance between the "rst specimen-sided electrode and the liner tube. In the low-voltage range the chromatic aberration coe$cient decreases with increasing immersion ratio,
103
if the overlapping of the magnetic and electrostatic "elds is not too small. The presented optimized lens variant shows noticeably smaller aberrations than the original MEDOL with an axially arranged pole-piece gap. Constraints for our lens design are a moderate electrostatic "eld strength at the specimen plane of 100 V/mm (wd " 1 mm) and a maximum lens excitation of 1.5 kA-t. We get a chromatic aberration coe$cient of 1.1 mm at a primary electron energy of 1 keV (wd " 2 mm) and of 0.51 mm at 200 eV (wd " 1 mm). The collection e$ciency of the secondary electrons is also satisfactory.
Acknowledgements The authors would like to thank Prof. E. Kasper for placing the program Electron Optical Systems at our disposal and R. Wilcox for checking the English manuscript. We also thank Y. Lutsch for productive discussions.
References [1] I. MuK llerovaH , M. Lenc, Ultramicroscopy 41 (1992) 399. [2] K. Tsuno, N. Handa, S. Matsumoto, SPIE Electron-Beam Sources and Charged-Particle Optics, Vol. 2522, 1995, p. 243. [3] J. Zach, Optik 83 (1989) 30. [4] J. Zach, M. Haider, Nucl. Instr. and Meth. A 363 (1995) 316. [5] J. Frosien, E. Plies, K. Anger, J. Vac. Sci. Technol. B7 (1989) 1874. [6] E. Weimer, J.-P. Martin, Proc. 13th Int. Congr. on Electron Microscopy, ICEM, Paris, Vol. 1, 1994, p. 67. [7] J. Frosien, S. Lanio, H.P. Feuerbaum, Nucl. Instr. and Meth. A 363 (1995) 25. [8] K. Tsuno, N. Handa, S. Matsumoto, A. Mogami, SPIE Charged-Particle Optics II, Vol. 2858, 1996, p. 46. [9] D. Preikszas, H. Rose, Optik 100 (1995) 179. [10] S. Beck, E. Plies, B. Schiebel, Nucl. Instr. and Meth. A 363 (1995) 31. [11] G. Knell, E. Plies, Optik 108 (1998) 37. [12] P.W. Hawkes, E. Kasper, Principles of Electron Optics. Vol. 1: Basic Geometrical Optics, Academic Press, London, 1989, Chapter 10. [13] T.-T. Tang, J.-P. Song, Optik 84 (1990) 108. [14] J.E. Barth, P. Kruit, Optik 101 (1996) 101.