- Email: [email protected]
Aberration characteristics of immersion lenses for LVSEM
Ultramicroscopy 93 (2002) 331–338
Aberration characteristics of immersion lenses for LVSEM Anjam Khursheed* Electrical & Computer Engineering Department, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 28 January 2002; received in revised form 15 May 2002
Abstract This paper investigates the on-axis aberration characteristics of various immersion objective lenses for low voltage scanning electron microscopy (LVSEM). A simple aperture lens model is used to generate smooth axial field distributions. The simulation results show that mixed field electric–magnetic immersion lenses are predicted to have between 1.5 and 2 times smaller aberration limited probe diameters than their pure-field counterparts. At a landing energy of 1 keV; mixed field immersion lenses operating at the vacuum electrical field breakdown limit are predicted to have on-axis aberration coefficients between 50 and 60 mm; yielding an ultimate image resolution of below 1 nm: These aberrations lie in the same range as those for LVSEM systems that employ aberration correctors. r 2002 Elsevier Science B.V. All rights reserved. PACS: 41.85 Keywords: Immersion lenses; Low voltage scanning electron microscopy (LVSEM); On-axis aberrations
1. Introduction The use of immersion lenses is now widespread in low-voltage scanning electron microscopy (LVSEM) [1,2]. Since there are a variety of immersion lenses in use, and because other methods of resolution improvement based upon aberration correction are also being introduced, it is important to critically compare the performance of different immersion lens configurations with one another and to estimate their ultimate resolution limit. Although some previous attempts at reviewing the performance of immersion lenses for LVSEM have been made [3], the simulation results *Corresponding author. Tel.: +65-8742295; fax: +657791103. E-mail address: [email protected] (A. Khursheed).
presented in this study predict significantly lower aberrations. To achieve aberration limited probe diameters of less than 1 nm at the specimen with a landing energy of 1 keV, Zach estimates that the spherical aberration coefficient (Cs ) of an objective lens needs to be below 100 mm; while the chromatic aberration coefficient (Cc ) should be smaller than 84 mm [4]. Aberration correction systems promise to greatly enhance the performance of existing SEMs, and Zach presents a magnetic/electrostatic/ octupole corrector applied to LVSEM that can satisfy the above resolution criterion. The following study uses simple numerical simulation methods to show that these demands can also be met by some immersion objective lens designs. It should be noted however that significant improvement by immersion lenses is only possible for situations
0304-3991/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 2 ) 0 0 2 8 8 - 7
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usually magnetic. Unlike the mixed field immersion lens depicted in Fig. 1c, however, that also employs, overlapping magnetic and electric fields, the focusing magnetic field of the electric retarding field lens does not immerse the specimen in a magnetic field. In the following work, it is assumed that the resolution of immersion lenses is limited by vacuum electrical field breakdown (10 kV=mm) and magnetic saturation (around 1 T for iron). It should also be stated from the outset that only onaxis aberration limitations of immersion lenses are considered. Other effects that are important at high resolution such as the projected source size,
where the specimen can be immersed in moderate to high field strengths. There are at present three broad immersion lens categories: those that immerse the specimen in a magnetic field [5], those that immerse the specimen in an electric field [6], and those that immerse the specimen in a mixed electric–magnetic field combination [7]. These different configurations are depicted in Figs. 1a–c. The electric immersion lens is more popularly known as the retarding field lens, since it slows down the primary beam just before it strikes the specimen. Although it is referred to here as an electric immersion lens, it does employ an auxiliary focusing lens that is
Magnetic Pole pieces for focusing lens
Aperture pole piece
0V
Primary Beam Specimen
Specimen -Vs
Primary Beam
Axis
Axis
Beam voltage
Magnetic field
Focusing field distribution
Landing voltage
(b)
(a)
0 V pole piece aperture
pole piece and specimen at -Vs
Primary Beam Axis
Beam voltage Magnetic Field
Landing voltage
(c) Fig. 1. Axial field distributions for immersion objective lenses: (a) Pure magnetic field; (b) Retarding electric field; (c) Mixed field combination.
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the primary beam/specimen interaction volume, vibrations at the specimen and surface charging, are not taken into account.
333
reaches a constant value on either side of the aperture. If the field strength on the right hand side of the lens is E ¼ ðV2 V1 Þ=W ; and if a fieldfree region is assumed on the left-hand side, then the potential along the z-axis, V ðzÞ; is given by [8] E E z V ðzÞ ¼ V1 z R 1 þ tan1 ðz=RÞ : ð1Þ 2 p R
2. Tools and methods An aperture lens model assumes that all electrodes in the immersion lens are infinitely thin, as shown in Fig. 2a, and that the field strength
Note that as the aperture radius R; tends to zero, the potential distribution is linear in z; as expected. W
R
Axis
E or B V1
V1
V2
V2
Aperture lens approximation
Immersion lens
(a)
R = 1 mm 1.20
FEM solution
Potential distribution (volts)
Aperture lens approximation
0.80
Aperture plate 0 V 1 V conductor boundary
0.40
R 0.00 -1.00
(b)
-0.50
0.00
0.50
1.00
Distance along axis (mm)
Fig. 2. The aperture lens approximation: (a) Schematic diagram; (b) Axial potential distribution.
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Fig. 2b shows how the aperture axial potential distribution compares with that of a finite element potential solution that takes into account the aperture electrode thickness. The immersion lens in this case has a 3 mm thick aperture plate, an aperture hole radius of 1 mm; and a working distance of 1 mm: The finite element program used is part of the KEOS package [9]. The aperture electrode is located at z ¼ 0: The two axial potential distributions are similar, except that the aperture electrode distribution is shifted to the left by around 0:3 mm and rises a little less steeply than the finite element solution. The aperture lens model has the merit of providing a smooth axial field distribution that can easily be differentiated, facilitating the accurate calculation of on-axis aberration coefficients. On-axis aberration coefficients were found by running programs in the KEOS package. These programs are based upon using the standard procedure of numerically solving for paraxial ray trajectories, and then calculating on-axis aberration coefficients from evaluating integrals derived by perturbation methods. A variable step method was used in the calculation of the paraxial rays and the evaluation of the perturbation integrals, so that many steps were concentrated in places where the axial field varies steeply, typically around the centre of the aperture lens field distribution. For the retarding field lens case, a focusing magnetic field BðzÞ is assumed, which is approximated by the following Glaser field distribution: BðzÞ ¼
B0 ; ½1 þ ðz=aÞ2 2
ð2Þ
where B0 is the peak of the distribution, and the parameter a controls how sharply the distribution falls on either side of the peak value. Some way of combining the spherical, chromatic and diffraction aberrations is required, if the much more complicated full wave solution technique for calculating the aberration limited probe size is not used. Various formulas have been presented. At one extreme, the standard quadrature formula provides the largest estimate of the probe size [10]. At the other extreme, a root-sum formula by Barth and Kruit based upon the probe containing 50% of the electron current gives the lowest estimate [11].
Zach uses a variation of the quadrature formula to provide an estimate lying approximately half-way between these two extremes [4]. For a source energy spread of 0:15 eV; a landing energy of 1 keV; aberration coefficients of Cs ¼ 100 mm and Cc ¼ 84 mm; the normal quadrature formula gives the minimum aberration limited spot diameter as 1:22 nm; Zach’s formula gives 1:06 nm; while Barth and Kruit’s root-sum formula calculates it to be 0:742 nm: For the purposes of this paper, the more accurate root-sum formula by Barth and Kruit is used. In all subsequent calculations, the electron source is assumed to be a cold field emission electron gun, which has a root-mean-square energy spread of 0:15 eV: This figure has previously been used in the context of simulating high-resolution objective lenses for LVSEM [12]. Also, for a given set of aberrations and source energy spread, the final aperture radius is varied systematically so that the probe diameter is a minimum. For the purposes of this study, it is assumed that such an aperture is available.
3. Simulated aberration predictions A comparison of aberration predictions based upon the aperture lens model for the magnetic, retarding field and mixed field immersion lenses is given in Table 1 for a fixed landing energy of 1 keV: For the magnetic immersion lens, the focal length, f ; Cs ; and Cc are predicted to be, 257, 168 and 187 mm; respectively, giving a probe diameter of 0:87 nm: The lens bore diameter is varied so that the calculated probe diameter is a minimum. An optimum lens bore size exists because while chromatic aberration steadily decreases as the lens bore radius becomes smaller, spherical aberration becomes large once the radius is small relative to the working distance. As the magnetic field distribution approaches a step function, the point at which the primary beam focuses does not change significantly. The working distance in this case is therefore allowed to vary. The optimum lens bore radius is 0:25 mm and the 1 keV primary beam focuses at a point 0:3 mm below the aperture
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Table 1 Simulation of aberrations for aperture lenses at magnetic saturation and electrical breakdown limits for a landing energy of 1 keV Aperture lens model at 1 keV primary landing energy
Magnetic Lens ð1 TÞ
Electric retarding field lens (10 kV=mm)
Mixed field lens
f ðmm) Cs ðmmÞ Cc ðmmÞ dp (nm) (Root-sum method)
257 168 187 0.87
860 713 144 1.2
163 50.45 58.2 0.625
lens electrode (with an axial peak field strength of 1 T). In practice, the required lens bore size to achieve this condition will not be so small, since the zero plate thickness in the aperture lens model underestimates how sharply the field distribution rises. Finite element simulations showed that a 0:5 mm radius hole in a 3 mm thick lens aperture plate provides similar aberrations. Lens bore diameters down to 0:5 mm can be manufactured with standard methods. Although the 0:3 mm working distance for the magnetic immersion lens is considerably smaller than that used for conventional lenses, it is easily achievable with motorised specimen height movement. In the case of the electric retarding field lens the strength of the focusing magnetic field is adjusted to provide a focal point 1 mm below the aperture lens electrode. An 11 keV incoming primary beam is retarded down to a landing energy of 1 keV by biasing the specimen to 10 kV: For a Glaser magnetic field distribution which has a ¼ 1 mm; the position which minimised the aberration coefficients was found to be approximately 1 mm to the left of the centre of the electric field lens distribution, as shown in Fig. 1b. Note that although the magnetic and electric field distributions overlap, the specimen is still free of a magnetic field (it is still an electric immersion lens). The probe diameter is found to be a minimum for an aperture radius of around 1mm. The on-axis aberrations for this radius are calculated to be: f ¼ 860 mm; Cs ¼ 713 mm; and Cc ¼ 144 mm; giving a probe diameter of 1:2 nm: This probe diameter is around 38% higher than that predicted for the immersion magnetic lens. For the mixed field immersion lens, a 10 kV=mm electric field strength is maintained on the specimen side of the aperture, while the
strength of the magnetic field is adjusted to focus the primary beam on to the specimen, located at a working distance of 1 mm: For simplicity, the radius of the lens aperture electrode is specified to be 1 mm; the same as for the electric retarding field lens. The aberration coefficients are found to be, Cs ¼ 50:45 mm and Cc ¼ 58:2 mm with f ¼ 163 mm; giving a probe diameter of 0:625 nm: This probe diameter is around 2 times smaller than the retarding field electric lens, and around 1.4 times smaller than the immersion magnetic lens. The aberration coefficients of this mixed field lens obviously satisfy the criteria set by Zach for achieving an aberration limited probe diameter of less than 1 nm (and less than 0:742 nm using the Bath and Kruit root-sum formula). Figs. 3a–d show simulation predictions for how Cs and Cc vary with landing energy at electric field strengths of 2, 5 and 10 kV=mm and a peak magnetic field of 1 T: The electric field strengths of 2 and 5 kV=mm are of particular interest here, since 10 kV=mm is rarely used in practice. Designing a lens to operate close to the vacuum electrical breakdown value is not easy. The magnitude of the electrical field may be below the breakdown limit in some parts of the lens, while in other parts, typically around corners, it may exceed the limit. Unlike magnetic saturation, electrical breakdown is not a self-limiting process. The simulation results shown in Figs. 3a–d indicate that mixed immersion lenses are predicted to have significantly lower aberration coefficients than pure field immersion lenses (for all the energies examined). The electric retarding field lens has comparable or better aberration coefficients than the magnetic immersion lens at low landing energies: at an electric field strength of 10 kV=mm; the spherical aberration coefficient is
A. Khursheed / Ultramicroscopy 93 (2002) 331–338
336 4000
300
Magnetic (1 T)
Electric (2 kV/mm)
3000
C s (microns)
C s (microns)
200
2000 Electric (5 kV/mm)
Mixed (2 kV/mm)
Mixed (5 kV/mm)
100
Mixed (10 kV/mm)
1000 Electric (10 kV/mm)
Magnetic (1 T)
0
0 0
500
(a)
1000
1500
2000
0
2500
500
(b)
Landing Energy (eV)
1000
1500
2000
2500
Landing Energy (eV) 300
1000 Electric (2 kV/mm)
Magnetic (1 T)
800
Mixed (2 kV/mm)
Electric (5 kV/mm)
400
C c (microns)
C c (microns)
200 600
Mixed (5 kV/mm)
100 Electric (10 kV/mm)
Mixed (10 kV/mm)
Magnetic (1 T)
200
0
0 0
(c)
500
1000
1500
2000
2500
Landing Energy (eV)
0
(d)
500
1000
1500
2000
2500
Landing Energy (eV)
Fig. 3. Simulated immersion lens aberration coefficients as a function of primary beam landing energy: (a) Spherical aberration for immersion magnetic and electric retarding field lenses; (b) Spherical aberration for immersion magnetic and mixed field lenses; (c) Chromatic aberration for immersion magnetic and electric retarding field lenses; (d) Chromatic aberration for immersion magnetic and mixed field lenses.
better for landing energies below 200 eV; while for the chromatic aberration coefficient, it is better for landing energies below 1500 eV: However, this advantage is dramatically reduced as the electric field strength is decreased. An important result of these simulations is that for the mixed field immersion lens operating at the moderate electric field strength of 5 kV=mm; Cs is less than 100 mm and Cc is less than 84 mm;
satisfying Zach’s criteria for high resolution LVSEM. Figs. 4a–c show the simulation results for the aberration limited probe diameter as the landing energy is varied. For an electric field strength of 10 kV=mm and landing energies below 300 eV; electric retarding field lenses are predicted to provide smaller aberration limited probe diameters than magnetic immersion lenses, while above
2.0
2.0
1.6
1.6
1.2
Electric (10 kV/mm)
Probe Diameter (nm)
Probe Diameter (nm)
A. Khursheed / Ultramicroscopy 93 (2002) 331–338
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Electric (5 kV/mm) 1.2
0.8
0.8
Magnetic (1 T)
Magnetic (1 T) Mixed
Mixed 0.4
0.4 0
(a)
500
1000
1500
2000
2500
0
500
(b)
Landing Energy (eV)
1000
1500
2000
2500
Landing Energy (eV)
2.4
Probe Diameter (nm)
2.0
1.6
Electric (2 kV/mm)
1.2
0.8
Magnetic (1 T) Mixed
0.4 0
(c)
500
1000
1500
2000
2500
Landing Energy (eV)
Fig. 4. Simulated probe diameters as a function of landing energy for immersion lenses, electric field strength of (a) 10 kV=mm; (b) 5 kV=mm; (c) 2 kV=mm:
300 eV; magnetic immersion lenses are expected to perform better. Mixed field immersion lenses are consistently predicted to provide smaller probe diameters than their pure-field counterparts, ranging from 1.5 to 2 factor of improvement in the probe diameter. At a landing energy of 200 eV and a maximum electric field strength of 10 kV=mm; a mixed field lens is predicted to provide on-axis aberrations as low as Cs ¼ 16:21 mm and Cc ¼ 17:41 mm with a focal length of f ¼ 84 mm; giving an aberration limited probe size of 0:89 nm:
It is also apparent that as the electrical field strength decreases, the landing energy at which the magnetic immersion lens and electric retarding field lens are predicted to give a comparable aberration limited probe size also drops. At an electric field strength of 5 kV=mm; the crossing point drops to just under 200 eV; while for 2 kV=mm; the crossing point falls to less than 100 eV (not shown due to reasons of scale). These simulation results indicate that for most practical situations, where an electric field strength of
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5 kV=mm or less is used, the electric retarding field lens is only advantageous for very low landing energies (less than 200 eV). The best form of immersion lens in all cases is the mixed field immersion lens.
displays a remarkable capacity for effective communication. More importantly, he has used this talent to give others the opportunity of publishing their innovative research findings. Peter has brought the international community of electron optics much closer together, and we do well to honour him on this occasion.
4. Conclusions This paper has carried out a study to estimate on-axis aberrations of immersion objective lenses for LVSEM. A simple aperture lens model was used to generate smooth axial field distributions. Of the different immersion lenses investigated, simulation results showed that mixed field electric– magnetic immersion lenses are predicted to provide the lowest aberrations for a wide range of landing energies. Mixed field immersion lenses are predicted to have on-axis aberration coefficients in the same range as LVSEM systems that employ aberration correctors.
Acknowledgements I greatly value this opportunity to pay tribute to Peter Hawkes. Peter’s contributions to electron optics are immense. In my opinion, his greatest contributions lie in the area of disseminating information on electron optics. Whether it is through his many roles as author, translator, editor, reviewer, or keynote speaker, Peter Hawkes
References [1] K. Tsuno, in: J. Orloff (Ed.), Magnetic Lenses for Electron Microscopy in the Handbook of Charged Particle Optics, CRC Press LLC, Boca Raton, FL, 1997. [2] B. Lencov!a, in: J. Orloff (Ed.), Electrostatic Lenses in the Handbook of Charged Particle Optics, CRC Press LLC, Boca Raton, FL, 1997. [3] K. Tsuno, N. Handa, S. Matsumoto, SPIE 2522 (1995) 243. [4] J. Zach, Proceedings of EUREM 12, Brno, Czech Republic III (2000) 169. [5] M. Sato, H. Todokoro, K. Kageyama, SPIE Charge Particle Opt. 2014 (1993) 17. [6] L. Frank, I. Mullerov!a, J. Electron. Microsc. 48 (3) (1999) 205. [7] Y.W. Yau, R.F. Pease, A.A. Iranmanesh, K.J. Polasko, J. Vac. Sci. Technol. 19 (4) (1981) 1048. [8] P.W. Hawkes, E. Kasper, Principles of Electron Optics, Vol. 2, Academic Press, 1989, p. 632. [9] A. Khursheed, KEOS, Electrical Engineering Department, The National University of Singapore, 10 Kent Ridge Crescent, Singapore, 1995. [10] L. Reimer, Scanning Electron Microscopy, 2nd Edition, Springer, 1998, p. 32. [11] J.E. Bath, P. Kruit, Optik 101 (3) (1996) 101. [12] Z. Shao, Rev. Sci. Instrum. 59 (9) (1988) 1985.
Aberration characteristics of immersion lenses for LVSEM Anjam Khursheed* Electrical & Computer Engineering Department, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 28 January 2002; received in revised form 15 May 2002
Abstract This paper investigates the on-axis aberration characteristics of various immersion objective lenses for low voltage scanning electron microscopy (LVSEM). A simple aperture lens model is used to generate smooth axial field distributions. The simulation results show that mixed field electric–magnetic immersion lenses are predicted to have between 1.5 and 2 times smaller aberration limited probe diameters than their pure-field counterparts. At a landing energy of 1 keV; mixed field immersion lenses operating at the vacuum electrical field breakdown limit are predicted to have on-axis aberration coefficients between 50 and 60 mm; yielding an ultimate image resolution of below 1 nm: These aberrations lie in the same range as those for LVSEM systems that employ aberration correctors. r 2002 Elsevier Science B.V. All rights reserved. PACS: 41.85 Keywords: Immersion lenses; Low voltage scanning electron microscopy (LVSEM); On-axis aberrations
1. Introduction The use of immersion lenses is now widespread in low-voltage scanning electron microscopy (LVSEM) [1,2]. Since there are a variety of immersion lenses in use, and because other methods of resolution improvement based upon aberration correction are also being introduced, it is important to critically compare the performance of different immersion lens configurations with one another and to estimate their ultimate resolution limit. Although some previous attempts at reviewing the performance of immersion lenses for LVSEM have been made [3], the simulation results *Corresponding author. Tel.: +65-8742295; fax: +657791103. E-mail address: [email protected] (A. Khursheed).
presented in this study predict significantly lower aberrations. To achieve aberration limited probe diameters of less than 1 nm at the specimen with a landing energy of 1 keV, Zach estimates that the spherical aberration coefficient (Cs ) of an objective lens needs to be below 100 mm; while the chromatic aberration coefficient (Cc ) should be smaller than 84 mm [4]. Aberration correction systems promise to greatly enhance the performance of existing SEMs, and Zach presents a magnetic/electrostatic/ octupole corrector applied to LVSEM that can satisfy the above resolution criterion. The following study uses simple numerical simulation methods to show that these demands can also be met by some immersion objective lens designs. It should be noted however that significant improvement by immersion lenses is only possible for situations
0304-3991/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 2 ) 0 0 2 8 8 - 7
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usually magnetic. Unlike the mixed field immersion lens depicted in Fig. 1c, however, that also employs, overlapping magnetic and electric fields, the focusing magnetic field of the electric retarding field lens does not immerse the specimen in a magnetic field. In the following work, it is assumed that the resolution of immersion lenses is limited by vacuum electrical field breakdown (10 kV=mm) and magnetic saturation (around 1 T for iron). It should also be stated from the outset that only onaxis aberration limitations of immersion lenses are considered. Other effects that are important at high resolution such as the projected source size,
where the specimen can be immersed in moderate to high field strengths. There are at present three broad immersion lens categories: those that immerse the specimen in a magnetic field [5], those that immerse the specimen in an electric field [6], and those that immerse the specimen in a mixed electric–magnetic field combination [7]. These different configurations are depicted in Figs. 1a–c. The electric immersion lens is more popularly known as the retarding field lens, since it slows down the primary beam just before it strikes the specimen. Although it is referred to here as an electric immersion lens, it does employ an auxiliary focusing lens that is
Magnetic Pole pieces for focusing lens
Aperture pole piece
0V
Primary Beam Specimen
Specimen -Vs
Primary Beam
Axis
Axis
Beam voltage
Magnetic field
Focusing field distribution
Landing voltage
(b)
(a)
0 V pole piece aperture
pole piece and specimen at -Vs
Primary Beam Axis
Beam voltage Magnetic Field
Landing voltage
(c) Fig. 1. Axial field distributions for immersion objective lenses: (a) Pure magnetic field; (b) Retarding electric field; (c) Mixed field combination.
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the primary beam/specimen interaction volume, vibrations at the specimen and surface charging, are not taken into account.
333
reaches a constant value on either side of the aperture. If the field strength on the right hand side of the lens is E ¼ ðV2 V1 Þ=W ; and if a fieldfree region is assumed on the left-hand side, then the potential along the z-axis, V ðzÞ; is given by [8] E E z V ðzÞ ¼ V1 z R 1 þ tan1 ðz=RÞ : ð1Þ 2 p R
2. Tools and methods An aperture lens model assumes that all electrodes in the immersion lens are infinitely thin, as shown in Fig. 2a, and that the field strength
Note that as the aperture radius R; tends to zero, the potential distribution is linear in z; as expected. W
R
Axis
E or B V1
V1
V2
V2
Aperture lens approximation
Immersion lens
(a)
R = 1 mm 1.20
FEM solution
Potential distribution (volts)
Aperture lens approximation
0.80
Aperture plate 0 V 1 V conductor boundary
0.40
R 0.00 -1.00
(b)
-0.50
0.00
0.50
1.00
Distance along axis (mm)
Fig. 2. The aperture lens approximation: (a) Schematic diagram; (b) Axial potential distribution.
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Fig. 2b shows how the aperture axial potential distribution compares with that of a finite element potential solution that takes into account the aperture electrode thickness. The immersion lens in this case has a 3 mm thick aperture plate, an aperture hole radius of 1 mm; and a working distance of 1 mm: The finite element program used is part of the KEOS package [9]. The aperture electrode is located at z ¼ 0: The two axial potential distributions are similar, except that the aperture electrode distribution is shifted to the left by around 0:3 mm and rises a little less steeply than the finite element solution. The aperture lens model has the merit of providing a smooth axial field distribution that can easily be differentiated, facilitating the accurate calculation of on-axis aberration coefficients. On-axis aberration coefficients were found by running programs in the KEOS package. These programs are based upon using the standard procedure of numerically solving for paraxial ray trajectories, and then calculating on-axis aberration coefficients from evaluating integrals derived by perturbation methods. A variable step method was used in the calculation of the paraxial rays and the evaluation of the perturbation integrals, so that many steps were concentrated in places where the axial field varies steeply, typically around the centre of the aperture lens field distribution. For the retarding field lens case, a focusing magnetic field BðzÞ is assumed, which is approximated by the following Glaser field distribution: BðzÞ ¼
B0 ; ½1 þ ðz=aÞ2 2
ð2Þ
where B0 is the peak of the distribution, and the parameter a controls how sharply the distribution falls on either side of the peak value. Some way of combining the spherical, chromatic and diffraction aberrations is required, if the much more complicated full wave solution technique for calculating the aberration limited probe size is not used. Various formulas have been presented. At one extreme, the standard quadrature formula provides the largest estimate of the probe size [10]. At the other extreme, a root-sum formula by Barth and Kruit based upon the probe containing 50% of the electron current gives the lowest estimate [11].
Zach uses a variation of the quadrature formula to provide an estimate lying approximately half-way between these two extremes [4]. For a source energy spread of 0:15 eV; a landing energy of 1 keV; aberration coefficients of Cs ¼ 100 mm and Cc ¼ 84 mm; the normal quadrature formula gives the minimum aberration limited spot diameter as 1:22 nm; Zach’s formula gives 1:06 nm; while Barth and Kruit’s root-sum formula calculates it to be 0:742 nm: For the purposes of this paper, the more accurate root-sum formula by Barth and Kruit is used. In all subsequent calculations, the electron source is assumed to be a cold field emission electron gun, which has a root-mean-square energy spread of 0:15 eV: This figure has previously been used in the context of simulating high-resolution objective lenses for LVSEM [12]. Also, for a given set of aberrations and source energy spread, the final aperture radius is varied systematically so that the probe diameter is a minimum. For the purposes of this study, it is assumed that such an aperture is available.
3. Simulated aberration predictions A comparison of aberration predictions based upon the aperture lens model for the magnetic, retarding field and mixed field immersion lenses is given in Table 1 for a fixed landing energy of 1 keV: For the magnetic immersion lens, the focal length, f ; Cs ; and Cc are predicted to be, 257, 168 and 187 mm; respectively, giving a probe diameter of 0:87 nm: The lens bore diameter is varied so that the calculated probe diameter is a minimum. An optimum lens bore size exists because while chromatic aberration steadily decreases as the lens bore radius becomes smaller, spherical aberration becomes large once the radius is small relative to the working distance. As the magnetic field distribution approaches a step function, the point at which the primary beam focuses does not change significantly. The working distance in this case is therefore allowed to vary. The optimum lens bore radius is 0:25 mm and the 1 keV primary beam focuses at a point 0:3 mm below the aperture
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Table 1 Simulation of aberrations for aperture lenses at magnetic saturation and electrical breakdown limits for a landing energy of 1 keV Aperture lens model at 1 keV primary landing energy
Magnetic Lens ð1 TÞ
Electric retarding field lens (10 kV=mm)
Mixed field lens
f ðmm) Cs ðmmÞ Cc ðmmÞ dp (nm) (Root-sum method)
257 168 187 0.87
860 713 144 1.2
163 50.45 58.2 0.625
lens electrode (with an axial peak field strength of 1 T). In practice, the required lens bore size to achieve this condition will not be so small, since the zero plate thickness in the aperture lens model underestimates how sharply the field distribution rises. Finite element simulations showed that a 0:5 mm radius hole in a 3 mm thick lens aperture plate provides similar aberrations. Lens bore diameters down to 0:5 mm can be manufactured with standard methods. Although the 0:3 mm working distance for the magnetic immersion lens is considerably smaller than that used for conventional lenses, it is easily achievable with motorised specimen height movement. In the case of the electric retarding field lens the strength of the focusing magnetic field is adjusted to provide a focal point 1 mm below the aperture lens electrode. An 11 keV incoming primary beam is retarded down to a landing energy of 1 keV by biasing the specimen to 10 kV: For a Glaser magnetic field distribution which has a ¼ 1 mm; the position which minimised the aberration coefficients was found to be approximately 1 mm to the left of the centre of the electric field lens distribution, as shown in Fig. 1b. Note that although the magnetic and electric field distributions overlap, the specimen is still free of a magnetic field (it is still an electric immersion lens). The probe diameter is found to be a minimum for an aperture radius of around 1mm. The on-axis aberrations for this radius are calculated to be: f ¼ 860 mm; Cs ¼ 713 mm; and Cc ¼ 144 mm; giving a probe diameter of 1:2 nm: This probe diameter is around 38% higher than that predicted for the immersion magnetic lens. For the mixed field immersion lens, a 10 kV=mm electric field strength is maintained on the specimen side of the aperture, while the
strength of the magnetic field is adjusted to focus the primary beam on to the specimen, located at a working distance of 1 mm: For simplicity, the radius of the lens aperture electrode is specified to be 1 mm; the same as for the electric retarding field lens. The aberration coefficients are found to be, Cs ¼ 50:45 mm and Cc ¼ 58:2 mm with f ¼ 163 mm; giving a probe diameter of 0:625 nm: This probe diameter is around 2 times smaller than the retarding field electric lens, and around 1.4 times smaller than the immersion magnetic lens. The aberration coefficients of this mixed field lens obviously satisfy the criteria set by Zach for achieving an aberration limited probe diameter of less than 1 nm (and less than 0:742 nm using the Bath and Kruit root-sum formula). Figs. 3a–d show simulation predictions for how Cs and Cc vary with landing energy at electric field strengths of 2, 5 and 10 kV=mm and a peak magnetic field of 1 T: The electric field strengths of 2 and 5 kV=mm are of particular interest here, since 10 kV=mm is rarely used in practice. Designing a lens to operate close to the vacuum electrical breakdown value is not easy. The magnitude of the electrical field may be below the breakdown limit in some parts of the lens, while in other parts, typically around corners, it may exceed the limit. Unlike magnetic saturation, electrical breakdown is not a self-limiting process. The simulation results shown in Figs. 3a–d indicate that mixed immersion lenses are predicted to have significantly lower aberration coefficients than pure field immersion lenses (for all the energies examined). The electric retarding field lens has comparable or better aberration coefficients than the magnetic immersion lens at low landing energies: at an electric field strength of 10 kV=mm; the spherical aberration coefficient is
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better for landing energies below 200 eV; while for the chromatic aberration coefficient, it is better for landing energies below 1500 eV: However, this advantage is dramatically reduced as the electric field strength is decreased. An important result of these simulations is that for the mixed field immersion lens operating at the moderate electric field strength of 5 kV=mm; Cs is less than 100 mm and Cc is less than 84 mm;
satisfying Zach’s criteria for high resolution LVSEM. Figs. 4a–c show the simulation results for the aberration limited probe diameter as the landing energy is varied. For an electric field strength of 10 kV=mm and landing energies below 300 eV; electric retarding field lenses are predicted to provide smaller aberration limited probe diameters than magnetic immersion lenses, while above
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300 eV; magnetic immersion lenses are expected to perform better. Mixed field immersion lenses are consistently predicted to provide smaller probe diameters than their pure-field counterparts, ranging from 1.5 to 2 factor of improvement in the probe diameter. At a landing energy of 200 eV and a maximum electric field strength of 10 kV=mm; a mixed field lens is predicted to provide on-axis aberrations as low as Cs ¼ 16:21 mm and Cc ¼ 17:41 mm with a focal length of f ¼ 84 mm; giving an aberration limited probe size of 0:89 nm:
It is also apparent that as the electrical field strength decreases, the landing energy at which the magnetic immersion lens and electric retarding field lens are predicted to give a comparable aberration limited probe size also drops. At an electric field strength of 5 kV=mm; the crossing point drops to just under 200 eV; while for 2 kV=mm; the crossing point falls to less than 100 eV (not shown due to reasons of scale). These simulation results indicate that for most practical situations, where an electric field strength of
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5 kV=mm or less is used, the electric retarding field lens is only advantageous for very low landing energies (less than 200 eV). The best form of immersion lens in all cases is the mixed field immersion lens.
displays a remarkable capacity for effective communication. More importantly, he has used this talent to give others the opportunity of publishing their innovative research findings. Peter has brought the international community of electron optics much closer together, and we do well to honour him on this occasion.
4. Conclusions This paper has carried out a study to estimate on-axis aberrations of immersion objective lenses for LVSEM. A simple aperture lens model was used to generate smooth axial field distributions. Of the different immersion lenses investigated, simulation results showed that mixed field electric– magnetic immersion lenses are predicted to provide the lowest aberrations for a wide range of landing energies. Mixed field immersion lenses are predicted to have on-axis aberration coefficients in the same range as LVSEM systems that employ aberration correctors.
Acknowledgements I greatly value this opportunity to pay tribute to Peter Hawkes. Peter’s contributions to electron optics are immense. In my opinion, his greatest contributions lie in the area of disseminating information on electron optics. Whether it is through his many roles as author, translator, editor, reviewer, or keynote speaker, Peter Hawkes
References [1] K. Tsuno, in: J. Orloff (Ed.), Magnetic Lenses for Electron Microscopy in the Handbook of Charged Particle Optics, CRC Press LLC, Boca Raton, FL, 1997. [2] B. Lencov!a, in: J. Orloff (Ed.), Electrostatic Lenses in the Handbook of Charged Particle Optics, CRC Press LLC, Boca Raton, FL, 1997. [3] K. Tsuno, N. Handa, S. Matsumoto, SPIE 2522 (1995) 243. [4] J. Zach, Proceedings of EUREM 12, Brno, Czech Republic III (2000) 169. [5] M. Sato, H. Todokoro, K. Kageyama, SPIE Charge Particle Opt. 2014 (1993) 17. [6] L. Frank, I. Mullerov!a, J. Electron. Microsc. 48 (3) (1999) 205. [7] Y.W. Yau, R.F. Pease, A.A. Iranmanesh, K.J. Polasko, J. Vac. Sci. Technol. 19 (4) (1981) 1048. [8] P.W. Hawkes, E. Kasper, Principles of Electron Optics, Vol. 2, Academic Press, 1989, p. 632. [9] A. Khursheed, KEOS, Electrical Engineering Department, The National University of Singapore, 10 Kent Ridge Crescent, Singapore, 1995. [10] L. Reimer, Scanning Electron Microscopy, 2nd Edition, Springer, 1998, p. 32. [11] J.E. Bath, P. Kruit, Optik 101 (3) (1996) 101. [12] Z. Shao, Rev. Sci. Instrum. 59 (9) (1988) 1985.