PEEM

Ultramicroscopy 199 (2019) 46–49

Contents lists available at ScienceDirect

Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic

Measuring chromatic aberration in LEEM/PEEM

T

R.M. Tromp IBM T.J. Watson Research Center, 1101 Kitchawan Road, P.O. Box 218, Yorktown Heights, NY 10598, United States

A R T I C LE I N FO

A B S T R A C T

Keywords: Cathode lens Electron mirror Chromatic aberration Aberration correction

Measurement of chromatic aberration in a Low Energy Electron Microscope (LEEM) or Photo Electron Emission Microscope (PEEM) is necessary for quantitative image interpretation, and for accurate correction of chromatic aberration in an aberration-corrected instrument. While methods have been developed for measuring the spherical aberration coefficient, C3, measuring the chromatic aberration coefficient, Cc, remains a more difficult task. Here a novel method is introduced to simplify such measurements. The viability and accuracy is demonstrated using detailed electron-optical ray-tracing calculations. Experimental results show that the method is easily reduced to practice.

1. Introduction With the advent of mirror-based correction of spherical and chromatic aberrations (C3 and Cc) in cathode lens instruments [1–6] (LEEM/ PEEM), the task of measuring these aberration coefficients has become a topic of interest. But even without aberration correction, quantitative interpretation of LEEM/PEEM images requires accurate knowledge of the electron-energy-dependent values of C3 and Cc of the objective lens, as these instrumental parameters together with defocus, ΔC1, determine the contrast transfer function [7–9], assuming that astigmatism and higher order/rank aberrations are negligible. Measuring C3 is a relatively straightforward task that can be accomplished in LEEM with the real-space micro-spot Low Energy Electron Diffraction (RS-μLEED) method [10]. For PEEM, Scholl et al. have utilized the closely related Hartmann method, measuring image shift as a function of the location of a small contrast aperture in the backfocal plane of the objective lens [11]. However, measuring Cc is not quite as simple. In LEEM, the landing energy of the electron beam is given by the sum of the difference of the potentials of the electron emitter in the gun and of the sample, and the difference in their work functions. In a standard LEEM experiment the landing energy is varied by changing the sample potential, while keeping the gun potential fixed. This has the advantage that the potential of the electron beam in the illumination and projection columns is constant. As we change the landing energy in this fashion, the objective lens excitation must be adjusted since the chromatic aberration coefficient of the uniform electrostatic field between the biased sample and the grounded front plane of the objective lens is strongly dependent on the landing energy, E0 : Cc = −L E/E0 , where E0 is the energy with which the electrons leave the sample, E is the energy gained between the sample and the objective lens, and L is the distance

E-mail address: [email protected]. https://doi.org/10.1016/j.ultramic.2019.01.009 Received 4 December 2018; Accepted 21 January 2019 Available online 10 February 2019 0304-3991/ © 2019 Published by Elsevier B.V.

between sample and objective lens [12,13]. Chromatic aberration is nothing but a change of focal length with electron energy. Thus, as the energy of the electrons inside the objective lens itself is constant (i.e. E + E0 = constant), measuring objective lens excitation required to keep the image focused as a function of E0 ignores the chromatic aberration of the magnetic or electrostatic objective lens [14]. Similarly, as the energy at the electron mirror in an aberrationcorrected system remains constant in this standard LEEM experiment (E + E0 constant as E0 is changed), the focal length of the mirror remains constant, the chromatic aberration coefficient of the mirror remains invisible and the aberration coefficient of the combined system cannot be measured. This problem can be overcome by not changing the sample potential but the gun potential, varying the energy of the electrons in the illumination and projection columns, including the objective lens and the mirror. While this solves the problem conceptually, it has a severe drawback. In a LEEM instrument the electrons pass through the objective lens twice, first on their way to the sample, and once again after they return from the sample. To separate the electron paths of the illumination and projection systems, the electrons are deflected by a magnetic prism array (MPA), typically over an angle of either 60° or 90°, depending on the design of the instrument. Of course, the exact deflection angle depends on the energy of the electron beam passing through the MPA. As we change the potential of the electron gun and thereby the energy of the electrons passing through the MPA between gun and sample, this deflection angle also changes, and the alignment of the system must be corrected for each setting of the gun energy. This makes the measurement tedious, time-consuming, and not conducive for routine use. In the following I will introduce a new method for measuring chromatic aberrations (i.e. defocus as function of E0) at fixed gun potential, while still accurately accounting

Ultramicroscopy 199 (2019) 46–49

R.M. Tromp

for the aberrations of the objective lens and electron mirror. 2. A new method As we change electron energy from E to E+dE, the normalized electron energy changes to:

E + dE dE =1+ E E

(1a)

The defocus due to the small energy change dE is given by dE dC1 = Cc E . We may further write down the following approximate identity:

E + dE E ⎛ dE ≪ 1⎞ ≅ E E − dE ⎝ E ⎠

(1b)

Eq. (1b) appears to be superfluous as it merely states a mathematically approximate identity for small values of dE/E. However, it also suggests a different way of thinking about this. The left-hand side of the equation states that the electron energy changes by a small amount dE (numerator), relative to a reference energy E (denominator). The righthand side says that this is mathematically equivalent to the electron energy remaining constant (numerator constant), while the reference energy changes by a small amount –dE (denominator). When we want to measure the chromatic aberrations of a system, we usually keep the sample potential constant while changing the gun potential. Thus, the fixed sample potential functions as the reference energy (left-hand denominator), while the gun potential changes (left-hand numerator). The right-hand side of Eq. (1b) suggests that we could also take another approach: leave the gun potential fixed (right-hand numerator fixed), but change the reference, i.e. sample potential (right-hand denominator changes from E to E–dE). We must keep in mind that all lens settings of the imaging side of the microscope are coupled to the reference potential. In the usual measurement of Cc (changing gun potential) all imaging lenses are coupled to a fixed reference potential (i.e. the sample potential), and therefore remain fixed as we change the gun potential. In other words, all focal lengths of the imaging lenses stay fixed for an electron at the reference potential. As we consider the right-hand side of Eq. (1b) this condition (fixed focal lengths for an electron at the reference potential) must still be fulfilled. But as we change the reference potential, the imaging lens settings must change with it so as to keep the focal lengths at the reference potential constant. The focal length of an electrostatic lens remains constant as the ratio of the applied potentials to the electron energy (V/E) is held constant. For a magnetic lens the focal length remains constant as the ratio of the square of the lens current to the electron energy (I2/E) is held constant. Thus, as long as a change in E (i.e. sample potential) by an amount −dE is accompanied by a change in either V (electrostatic lens potential) or I (magnetic lens current), so that

dV dE = (electrostatic lenses and electron mirror elements ) V E

Fig. 1. Geometry used in ray tracing calculations. The figure shows a straightened optical path. In (a) the two prism arrays, MPA1 and MPA2, are coupled by an electrostatic transfer lens. Since the MPA's function like achromatic 1:1 transfer lenses, we can simplify the geometry further as shown in (b). Here we have removed both MPA's, and placed an image at the center of the transfer lens. The reference image used in the ray tracing calculations is placed at the center of M3.

3. Ray tracing simulations The chromatic aberrations of the system are evaluated theoretically by changing the landing energy of the electrons, and calculating the excitation of the objective lens required to keep the image focused in a fixed image plane [15]. As we wish to explore the effects of aberration correction, we take as the reference image plane the center of the final magnetic lens, M3, in front of the electron mirror, after the electrons have returned from the mirror. A simplified diagram is shown in Fig. 1a. The calculation can be further simplified by realizing that the MPA's behave like 1:1 transfer lenses, both for the diffraction and image planes, so that we replace the two MPAs plus the transfer lens between the two MPAs with just the transfer lens (Fig. 1b). For a more complete description, see ref. [1]. What we are first interested in is the change in objective lens excitation, as we keep the potential between sample and objective lens constant while we change the landing energy E0. Thus, the electron energy along the optical path changes as we change E0 at fixed sample bias. As this calculation includes a reflection from the electron mirror, we set the excitation of the electron mirror such that the system is corrected for E0 = 10 eV. We then change E0 from 5 to 50 eV, and calculate the excitation of the magnetic objective lens required to keep the image focused in the reference plane, after reflection from the mirror. The result is shown in Fig. 2, small black circles. At E0 = 10 eV, the lens excitation is 711.802 A.turn. The right-hand axis shows the defocus (set to zero at 10 eV), which changes in proportion to the change in lens excitation. The figure shows an upside-down parabolalike curve, similar to an optical achromat [16]. Changing E0 while keeping the sample bias constant corresponds to changing the gun energy at fixed sample potential, and at fixed lens excitations (except for the objective lens which we adjust to keep the image in focus). Theoretically, the defocus ΔC1, relative to the energy Ec for which the instrument is corrected, is given by [16]:

(2a)

and

dI 1 dE = (magnetic lenses ) I 2 E

(2b)

the imaging system behaves as if it is operating at fixed settings, i.e. all focal lengths remain fixed relative to the reference energy. Therefore according to Eq. (1b) we can ‘mimic’ an increase in gun energy by keeping the gun energy constant, and decreasing the lens excitations in the imaging system according to these simple relations. Lens excitations are easily adjusted as we change sample potential at fixed gun potential, while the instrument alignment remains unchanged. In the next section detailed ray tracing calculations will show that this strategy yields exactly the desired result. 47

Ultramicroscopy 199 (2019) 46–49

R.M. Tromp

defocus relative to the changing reference energy, we must subtract this background change from the red data points. When we do so, we obtain the open blue circles in Fig. 2, which coincide with the corresponding black circles to better than 1 part in 105 in absolute lens excitation. Changing lens excitations in proportion to the change in sample bias at fixed gun energy is therefore exactly equivalent to changing gun energy at fixed lens excitations and fixed sample bias. We can now use this scheme to measure defocus as a function of landing energy, keeping the gun energy fixed, in a standard LEEM experiment. To do so, we change landing energy in the usual manner by changing sample bias, while at the same time changing all image forming and transfer lenses (including the electron mirror) in proportion, as given by Eq. (2). The excitations of the prism arrays are fixed, as the images at the center of the MPA's are achromatic to start with. Changing MPA excitations would also affect instrument alignment, which is exactly what we want to avoid. We then measure the objective lens excitation required to keep the image in focus. This will give us the red data points. Then we subtract the ‘background’ change in the objective lens excitation to obtain the blue data points, which coincide with the black data points, i.e. the result that we are after. This procedure works irrespective of the aberration settings of the electron mirror, or even if the system does not have an electron mirror. The shape of the defocus curve depends on the setting of the electron mirror (if there is one), but the procedure does not depend on the mirror settings. In fact, we can now easily measure an achromat curve as shown in Fig. 2, as a function of the mirror settings, and quickly establish the electron energy for which the system is corrected, i.e. the maximum at which lens excitation and defocus do not change with a small change in landing energy. Tracking the lens and mirror excitations to changes in landing energy can easily be implemented with a small macro that runs in the background while the instrument is being operated. This macro only runs while these measurements are in progress, not during routine instrument operation.

Fig. 2. Objective lens excitation (or defocus at M = 1) as a function of E0, calculated using ray tracing. Black circles: changing gun energy at fixed sample bias. Red circles: fixed gun energy, changing sample bias, as well as changing all image forming lenses and electron mirror according to Eq. (2). Open blue circles: after correcting objective lens excitation with dI/I = dE0/2E background term (see text). Blue and small black circles agree to better than 1:105. For lines through data see text. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

ΔC1 = −L

( Ec −

E0 )

Ec E0

2

− Cccmir (Ec − E0)2 / E 2

(3)

With L = 1.5 mm, Cccmir = 4.7 m, E = 15,000, and Ec = 10, the black line through the black circles is given by Eq. (3). Cccmir , the third rank chromatic aberration coefficient of the electron mirror, is usually ignored, but must be included. Eq. (3) gives an accurate and quantitative description of the ray tracing results, as shown in Fig. 2. Alternatively, we can change E0 by changing the sample potential at fixed gun potential, as in a standard LEEM experiment. To mimic the effects of a changing gun potential (even as we keep it fixed) we now change the excitations of the magnetic and electrostatic lenses, including the electron mirror electrodes, using the simple relations given dV dE dE in the introduction (Eq. (2)), i.e. V = E = − E 0 for electrostatic lenses, dI

1 dE

4. Experimental results A set of experiments implementing the above procedure was performed in the IBM aberration-corrected LEEM system. Accurately measuring defocus over a wide energy range is not necessarily a simple task, as image intensity can depend strongly on E0. Of course, a minimum of sample preparation is preferable if one is to repeat the measurements for a number of different settings of the electron mirror, and one would like the sample to be stable (i.e. not contaminate) during the experiments. After trying a number of different samples, graphene on Si (with native oxide) proved to be the most suitable sample, reusable many times, and relatively easy to prepare. Graphene was grown on a polycrystalline Cu-foil in a CVD system. Such graphene on Cu can be purchased commercially at modest cost. The graphene layer, with different layer thicknesses in different areas, was then transferred to a Si wafer using standard methods [17]. Areas with different layer thicknesses give different contrast in LEEM at different energies, making it easy to image with good contrast. Exposing a region of interest to >50 eV electrons for a few seconds at the beginning of the experiment is found to remove contaminations, resulting in high quality LEEM images suitable for our purposes. We measured the objective lens current required to keep the image in focus for a number of different nominal mirror settings (Ccmir = 2.5, 5, 7.5, 10, and 15 m, respectively, with C3mir = −1000 m). A term 1 dE dI = 2 I E 0 was then added to the measured data to correct for the changing ‘background’ excitation of the objective lens, and the change in lens current was converted to defocus [18]. Fig. 3 shows defocus vs. E0 for these five different mirror settings. The data points show the results of the measurements. As the chromatic aberration of the uniform field increases with decreasing energy,

1 dE

and I = 2 E = − 2 E 0 for magnetic lenses. With these lens excitations changing in proportion to the change in landing energy E0, we again calculate the change in objective lens excitation required to keep the reference image in focus. The results are given by the red circles in Fig. 2. Obviously, the red and black data points do not coincide, so we are getting different results for these two calculations. However, recall that we need to keep the focal lengths of all lenses constant relative to the reference (i.e. sample) potential. This includes dI 1 dE adjusting the objective lens by an amount I = − 2 E 0 as we change E0. The change in E0 then gives rise to a defocus that depends on the chromatic aberration of the complete system. To measure this defocus we adjust the objective lens current to refocus the image. In the red data points we plot the change in objective lens current relative to the initial setting at E0 = 10 eV, ignoring that at each value of E0 we must correct dI 1 dE for the ‘background’ change in lens current I = − 2 E 0 required to follow the changing reference potential. That is, we must calculate the change in objective lens excitation relative to the ‘background’ change dI 1 dE of I = − 2 E 0 as we change E0. (If an electrostatic objective lens is used, dV

dE

the background change is given by V = − E 0 .) The red line through the red circles is obtained by adding this background change in objective lens current to the black line. In other words, to obtain the 48

Ultramicroscopy 199 (2019) 46–49

R.M. Tromp

normalized energy distribution of secondary electrons is approximated by [20]:

f (E0) = 6φ2

E0 (E0 + φ) 4

(4)

where φ is the work function of the material. In that case, the energydependent defocus Eq. (3) must be weighed with the electron energy distribution Eq. (4), and minimized with respect to Ec to find the electron mirror setting that optimally corrects chromatic aberration. For a typical work function of 4–5 eV, the optimum value of Ec is then 3–4 eV, i.e. about a factor 2 higher than the peak in the secondary electron energy distribution. In conclusion, using both theoretical ray tracing and experimental measurements I have demonstrated a simplified method to determine and measure defocus curves in an aberration-corrected LEEM/PEEM system at fixed electron gun potential. The key to this method is that a small change in gun energy can be replaced by a small opposite change in column energy, and by adjusting the excitations of the electron lens and mirror elements in the imaging system in response to this change in column energy. Theory shows that this somewhat non-intuitive method of changing column energy is identical to changing gun energy, to 5 significant places in the excitation of the objective lens. Experiments show that high quality defocus curves can be readily acquired in this fashion in a matter of minutes, turning a tedious and time-consuming measurement into something that can be done quickly and routinely.

Fig. 3. Measured defocus vs E0 (colored symbols) for various nominal settings of the chromatic aberration coefficient of the electron mirror. Solid lines are fits using Eq. (3) (see text).

Acknowledgments I thank Dr. Satoshi Oida for fabrication of the Si/SiO2/graphene sample, and Dr. Jim Hannon for stimulating discussions.

correction at lower energy requires a higher mirror chromatic aberration coefficient. The solid lines in Fig. 3 are fits using Eq. (3), again using L = 1.5 mm, Cccmir = 4.7 m, E = 15,000. Other than the value of Ec (and a constant defocus offset) there are no adjustable parameters. The overall defocus offset has been subtracted from the data in Fig. 3, so that ΔC1 = 0 for E0 = Ec for each mirror setting. Obviously, the fits are in excellent, quantitative agreement with the measured data, with Ec = 56.9, 31.4, 13.1, 6.4 and 2.3 eV for the nominal mirror settings of 2.5, 5, 7.5, 10 and 15 m, respectively.

Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ultramic.2019.01.009. References [1] R.M. Tromp, J.B. Hannon, A.W. Ellis, W. Wan, A. Berghaus, O. Schaff, Ultramicroscopy 110 (2010) 852–861. [2] R.M. Tromp, J.B. Hannon, W. Wan, A. Berghaus, O. Schaff, Ultramicroscopy 127 (2013) 25–39. [3] T.h. Schmidt, U. Groh, R. Fink, E. Umbach, O. Schaff, W. Engel, B. Richter, H. Kuhlenbeck, R. Schlögl, H.J. Freund, A.M. Bradshaw, D. Preikszas, P. Hartel, R. Spehr, H. Rose, G. Lilienkamp, E. Bauer, G. Benner, Surf. Rev. Lett. 9 (2002) 223. [4] T.h. Schmidt, H. Marchetto, P.L. Lévesque, U. Groh, F. Maier, D. Preikszas, P. Hartel, R. Spehr, G. Lilienkamp, W. Engel, R. Fink, E. Bauer, H. Rose, E. Umbach, H.J. Freund, Ultramicroscopy 110 (2010) 1358. [5] R. Könenkamp, R.C. Word, G.F. Rempfer, T. Dixon, L. Almaraz, T. Jones, Ultramicroscopy 110 (2010) 899–902. [6] J.Feng W.Wan, H.A. Padmore, Nucl. Instrum. Method Phys. Res. A564 (2006) 537. [7] A.B. Pang, Th. Müller, M.S. Altman, E. Bauer, J. Phys. 21 (2009) 314006. [8] S.M. Kennedy, N.E. Schofield, D.M. Paganin, D.E. Jesson, Surf. Rev. Lett. 16 (2009) 855. [9] S.M. Schramm, A.B. Pang, M.S. Altman, R.M. Tromp, Ultramicroscopy 115 (2012) 88–108. [10] R.M. Tromp, Ultramicroscopy 130 (2013) 2–6. [11] A. Scholl, M.A. Marcus, A. Doran, J.R. Nasiatka, A.T. Young, A.A. MacDowell, R. Streubel, N. Kent, J. Feng, W. Wan, H.A. Padmore, Ultramicroscopy 188 (2018) 77–84. [12] E. Bauer, Ultramicroscopy 17 (1985) 51. [13] R.M. Tromp, W. Wan, S.M. Schramm, Ultramicroscopy 119 (2012) 33–39. [14] R.M. Tromp, Ultramicroscopy 111 (2011) 273–281. [15] MIRDA Software Package from Munro's Electron Beam Software Ltd. (London, UK). [16] R.M. Tromp, Ultramicroscopy 159 (2015) 497–502. [17] X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E.l. Tutuc, S.K. Banerjee, L. Colombo, R.S. Ruoff, Science 324 (2009) 1312–1314. [18] R.M. Tromp, M.S. Altman, Ultramicroscopy 183 (2017) 2–7. [19] D. Geelen, A. Thete, O. Schaff, A. Kaiser, S.J. van der Molen, R.M. Tromp, Ultramicroscopy 159 (2015) 482–487. [20] J. Feng, H. Padmore, D.H. Wei, S. Anders, Y. Wu, A. Scholl, D. Robin, Rev. Scient. Instrum. 73 (2002) 1514.

5. Discussion and conclusions In the above we have seen how the simplified procedure to measure defocus curves in a corrected LEEM system can be used in practice, with excellent results. In a PEEM system that is not equipped with an electron gun the situation is more complicated [5–11]. If the PEEM is equipped with an energy filter, one could select a small energy window of interest and then scan this energy window, provided that a sufficiently large range of electron energies is available in the photoemission spectrum. But again, one would have to worry about how this is going to affect the alignment of the system through the prism array(s). Recently, we have shown how incorporation of a compact electron source inside the sample holder, in combination with an electrontransparent sample (such as one or more graphene layers suspended on a TEM grid), can be used to obtain transmission images with good contrast and spatial resolution. This so-called eV-TEM [19] geometry can also be used in a PEEM instrument, in the same manner as in the LEEM experiments described above. Once again, keeping the electron source (now behind the sample) at a fixed bias relative to ground, the transmission energy E0 can be varied by changing the sample potential. The procedure to measure the defocus of objective lens plus electron mirror is then exactly the same as described above, and used to measure the data in Fig. 3. While in LEEM we typically deal only with a very narrow energy range, many synchrotron-PEEM experiments image with the secondary electrons over a wider energy range from 0 to ∼20 eV or so. The

49