Nicholas A. Allsop. Tel: +49 30 80623186; fax: +49 30 80623199; e-mail: firstname.lastname@example.org
High Resolution Electron Microscopy (HREM) is often used to characterize objects supported by amorphous substrates, usually amorphous carbon. HREM is currently undergoing step change in performance due to aberration correctors. This paper examines the aberration corrected imaging of objects supported by amorphous substrates. In particular, we show that a substantial increase in the ratio of the object contrast to the substrate contrast can be achieved by utilizing the strong variation of phase contrast with height, which is present when the spherical aberration has been adjusted to a small value. This variation is examined using the familiar Weak Phase Object Approximation model from which it is determined that the contrast ratio achieves a maximum at a small nonzero value of the spherical aberration. This result is confirmed by multislice modelling which allows for deviations from the Weak Phase Object Approximation and delocalization effects. One important practical result of this study is the need to place the object of interest on the correct side of the amorphous carbon substrate.
High Resolution Electron Microscopy is a commonly used technique used for the analysis of small nanosized objects such as semiconductor nanoparticles and nanotubes. It allows the atomic structure and morphology of the objects to be observed simultaneously with an accuracy approaching 1 Å. However, these nanosized objects must be mechanically supported in the electron microscope, usually by a thin sheet of amorphous carbon and therefore the ability to characterize the object of interest is compromised by the noise of the substrate contrast.
Recently, the development of aberration correctors has lead to a step change in the performance of HREM (Haider et al., 1998). Several authors have examined the impact of spherical aberration (Cs) correction on the imaging of perfect objects (including Den Dekker et al., 1999; Lentzen et al., 2002; Chang et al., 2003). Here, we investigate how spherical aberration will affect the imaging of objects supported by amorphous substrates.
When imaging small faceted particles using HREM there are three possible scenarios regarding the orientation of the particle, as shown in Fig. 1. (i) The particle has no specific crystalline orientation with respect to the beam direction; in this case we observe weak phase contrast and mass thickness contrast. Alternatively the particle may be aligned along some crystal axis giving strong phase contrast and the edge of the particle can then either be; (ii) faceted along the beam direction or (iii) faceted at some angle.
If the facet is aligned, then determining the particle width should be straight forward, as the final atomic plane will be several atoms thick in projection and will show up clearly in HREM. However, if the particle facet is inclined at some angle, then the apparent size of the particle will depend upon our ability to resolve the atomic columns and distinguish them against the amorphous background. This also applies to roughened surfaces or non-faceted particles.
In order to minimize the contrast from the substrate it should be as thin as possible and composed of a light element. The most commonly used support films are composed of amorphous carbon and are between 10 and 20 nm thick. At this thickness the amorphous carbon can still be regarded as a weak phase object (WPO) even though an equivalent thickness of diamond or graphite would be strongly scattering. This is because it is not the total projected potential that dictates a deviation from the weak phase object approximation (WPOA) but rather the maximum variation in the projected potential (Buseck et al., 1988).
Contrast ratios at low spherical aberration
If both the particle and the substrate are well approximated by the weak phase object approximation, it follows that the contrast from each will be additive. For small particles (< 5 nm) on 10–20 nm of amorphous carbon the contrast from the substrate will be the dominant factor when considering the noise in the image. Therefore the signal to noise ratio will be approximately the particle contrast to substrate contrast, henceforth referred to as object contrast to substrate contrast as the principle is the same for all thin objects.
The contrast of a weak phase object is related to the area under the Contrast Transfer Function (CTF) given in Equation 1.
Where U is the reciprocal spacing, z is the focus, λ the electron wavelength, t the substrate thickness and Δ is the focal spread (a microscope parameter). The exponential term is the chromatic dampening envelope, which is a result of the energy spread of the beam. It is reasonable to neglect the temporal dampening envelope because this has a reduced effect at low Cs values. Other mechanical and electrical instabilities have been neglected but their effects could be approximated by slightly increasing the focal spread.
However, the CTF changes sharply with defocus/height when Cs has been adjusted to a few tens on microns or less. Figure 2 shows the variation of the CTF when Cs is zero. At Gaussian focus there will be zero phase contrast, however, we can see that there is strong phase contrast with just 10 nm of underfocus. This means that when using a 10 nm support film, one side of the film can be showing strong phase contrast and the other side will have zero phase contrast under the WPOA. Figure 3 plots the area under the CTF curve, over a small range of defocus for three low Cs values and this gives us an indication of the amount of phase contrast. The reader is referred to the work of Lentzen et al. for a full 2-D contour plot of phase contrast as a function of defocus and Cs (Lentzen et al., 2002).
Figure 3 also shows why, if we are working at small positive Cs values, the object of interest should be placed on the underside of the support film. This will then allow the object to be imaged under strong phase contrast, whilst suppressing the phase contrast coming from the support film. Figure 4 quantifies this by plotting the contrast ratio of a 1 nm object placed underneath a 10 nm support film.
The contrast ratios here were calculated by firstly numerically integrating the modulus of the CTF for each height and then calculating the ratio by adding the substrate contrast contributions in quadrature.
Where C(z) is the contrast at a single height given in Equation 1, t is the substrate thickness and f is the focus setting at the object plane. The substrate contrast contributions are added in quadrature because they come from an amorphous, approximately random object. Comparisons with simulated images of amorphous carbon, taking the standard deviation in the image as a measure of contrast, show that this is the correct method.
Figure 4 shows that there is an optimum focus value at which the contrast ratio is maximized. This focus value changes with changing Cs but more interestingly so does the maximum contrast ratio. Figure 5 plots the maximum contrast ratio as a function of Cs for several substrate thicknesses.
We can clearly see that the contrast ratio goes through a maximum, the position of which varies with substrate thickness. Note the break in the axis followed by the contrast ratio for the uncorrected microscope with a Cs value of 0.5 mm and a 10 nm substrate. This value however, represents the maximum contrast ratio for the uncorrected microscope if we allow the focus setting to vary from Scherzer defocus, which would of course reduce the resolution achievable and complicate image interpretation. It is shown later that optimizing the contrast ratio for a Cs adjusted microscope does not have such an adverse effect on the resolution. We calculated that the object/substrate contrast ratio for a 10 nm substrate can be increased by 79%, when compared to the uncorrected microscope at the Scherzer setting, if the Cs is adjustable. However, the curves do not indicate a sharply varying contrast ratio and any low Cs value will give a considerable improvement over the uncorrected microscopes.
Contrast ratios with real objects
So far the contrast of our weak phase objects has been represented by the area under the contrast transfer function. However, this is an oversimplification because the contrast of a weak phase object is proportional to the area under the product of the contrast transfer function and the object function, the object function being the Fourier transform of the projected potential. This means that the optimum contrast ratios will be dependent on the object and substrate under investigation. In order to examine the effects of introducing an object function we have repeated the calculations using the scattering factor (object function) for a single carbon atom (Fig. 6). This was chosen as it should be very similar to the object function for amorphous carbon, they would be identical if amorphous carbon had a truly random structure (Boothroyd, 2000). For a discussion on the optimum imaging of crystals using Cs correction and taking into account object functions, the reader is referred to den Dekker et al. (1999).
Recalculating the contrast ratios now provides a quantitative measure (under the approximations used) for imaging carbon supported by amorphous carbon (Fig. 7).
We can see that the contrast ratios achievable with an adjustable Cs are now even higher, and the gain between the optimum setting and the unadjusted Scherzer setting is now 85%.
In practice the majority of objects imaged on top of amorphous carbon will not be made of carbon. However, as we are most interested in defining the edges and corners of small objects, the single atom scattering factor is still a more appropriate test than using the object function of a perfect crystal. Using the scattering factors of heavier atoms would result in shifting the results slightly towards the values we calculated without the object functions. This is because the normalized scattering factor varies more slowly in reciprocal space with increasing atomic number, or equally, the projected potential in real space has a smaller FWHM.
The optimum Cs setting does not affect the achievable resolution or delocalization of the corrected microscope, which is still limited by the information limit, when considering a 10 nm thick substrate and modern corrected microscope with an information limit of 7.5 nm−1. (see Lentzen et al., 2002 for plots of delocalization vs. Cs and defocus.) The contrast transfer function for the microscope is shown in Fig. 8, which compares the Scherzer setting for the uncorrected microscope and the settings for the optimum contrast ratio with a 10 nm substrate.
The variation of the calculated optimum Cs settings as a function of substrate thickness is plotted in Fig. 9. We can see that taking into account the scattering factor of carbon has little effect on the optimum setting. Figure 9 also shows the optimum setting if we wish to maintain the object at the Scherzer defocus. In this case the value of Cs appears to vary quadratically with the relationship:
Where t is the thickness in nanometers and the Cs is given in microns.
The curve for the optimum Cs when the focus is not fixed does not follow a simple quadratic relationship but can be approximated the polynomial relationship:
With a focus value approximated by:
Which represents a deviation from the Scherzer defocus value of less than 10%. Maintaining the object at the Scherzer defocus does give a small reduction in the obtainable contrast ratio, however, Fig. 10 shows that the optimum Cs, Scherzer focus setting, still gives a large improvement when compared to the uncorrected Cs Scherzer setting.
When selecting the parameters for an HREM experiment then the limits of the microscope as well as the theoretical resolution limits must be taken into account. Figure 9 also shows the point resolution at the optimum contrast ratio setting, which is defined by the first crossover in the contrast transfer function:
Plus the delocalization limit adapted from the definition of Lichte (Lichte, 1991; Lentzen et al., 2002) where the delocalization would equal the maximum spatial frequency:
We can see that if the carbon thickness is high or the information limit of the microscope is very good then the optimum setting with respect to the contrast ratio may no longer be appropriate as the resolution of the microscope will be compromised. The delocalization limit is lower in the case of optimizing the parameters for the contrast ratio, however, this is not always true in aberration corrected microscopy (Lentzen et al., 2002). When the delocalization limit is reached there will also be a small shift in the optimum contrast ratio because the analysis used so far does not include a delocalization term, however, once the delocalization limit is reached then delocalization should be a priority in choosing the parameters. Once again, because the contrast ratios do not change rapidly with Cs other factors may dominate the design of the HREM experiment and the contrast ratio will still be substantially improved when compared to the uncorrected microscope.
Contrast ratios from image simulations
In order to test the contrast ratio predictions, and extend their validity beyond the weak phase object approximation, image simulations of amorphous carbon were performed. The amorphous carbon model was provided by RMIT University, Melbourne and is the result of an environment dependent potential simulation (Peterson, 2003), image simulations were performed by the Java Electron Microscopy Simulation (JEMS) software (Stadelmann, 1987) using the multislice method. The contrast was measured by taking the standard deviations of the pixel intensities in the resulting images. The microscope parameters used where those of an uncorrected JEOL 2010F and are listed in Table 1.
Figure 11 shows the contrast ratios from the image simulations alongside the calculated values. There is an excellent agreement between the calculated and simulated values of the contrast for the uncorrected microscope at the Scherzer setting. The Cs adjusted contrast ratios are in broad agreement, but there are reasons why they may differ slightly. The multislice method is expected to yield more accurate results because it is not using the approximations of weak phase object theory, includes delocalization and is also using a more realistic model of amorphous carbon. (It has also been pointed out that the simulations may use a different definition for the focal spread, when compared to the theoretical calculations presented earlier. However, this will only have a negligible effect in the values calculated.)
Image simulations of a model cluster
Image simulations where also performed on a hypothetical CdSe pyramidal cluster shown in Fig. 12. This object was chosen because when viewed in projection it has a central column of 5 atoms which will be clearly observed in a corrected microscope but the edges of the pyramid are only 1 atom thick. The structure also closely resembles experimentally relevant clusters such as Cd32Se14(SePh)36 (Soloviev et al., 2001)
Figure 13 shows the simulated images of the model cluster upon 10 nm of amorphous carbon. The cluster is almost invisible in the uncorrected 200 kV microscope (although not in the higher voltage machines) but the Cs adjusted images show the centre of the cluster clearly, and ‘just about’ resolve the atomic columns consisting of only two atoms. Imaging the cluster with a negative Cs and the cluster on top of the substrate, reverses atom contrast but maintains resolution. Under the weak phase object approximation the two are exactly inverted. However, it has recently been shown that white atom contrast is preferred when considering the full nonlinear imaging theory (Jia et al., 2004; Lentzen, 2004).
A more quantitative approach to determining the detectability of the atoms is to look at the signal to noise ratio. The amorphous background gives a Gaussian distribution of intensity values and therefore we can use the standard deviation (σ) as a convenient measure of the detectability. The threshold value is somewhat arbitrary and we have chosen the commonly used threshold for detection of 3 σ. Multislice simulations of single atoms show that for the Cs adjusted microscope this requirement is fulfilled by any atom above Xe in the periodic table. Single Cd and Se atoms produce a maximum intensity variation equivalent to 2.67 σ and 2.10 σ, respectively. The results do confirm our ability to detect a column consisting of one Cd atom and one Se atom, which produces a maximum contrast variation of 4.0 σ.
It should be noted that the detection of atomic columns in the Cs adjusted microscope is enhanced both by the overall increase in the ratio of object : subject contrast and the increased resolution. An increase in the point resolution of a microscope means a sharper point spread function and an increase in the maximum intensity variation at the centre of the columns. The contrast of the carbon is also seen to increase in the simulation although not as quickly. This is because the object function of the amorphous carbon, which to a first approximation is the scattering factor of carbon, varies more quickly in reciprocal space as compared to the scattering factor of the heavier atoms of interest. Therefore an increase in the resolution of the microscope produces a greater increase in the contrast of the heavier atoms. In real space this is because the electrons are much closer to the atomic core in the heavier atoms giving a smaller FWHM of the projected potential and therefore decreasing the point spread function of the microscope will have a greater effect. This is a very interesting point and should be the subject of further study using more advanced imaging theory. For now, it should be noted that the results are dependent on the parameters of the microscope in our model and if the chromatic damping envelope is pushed even further out, through the use of electron monochromators or chromatic aberration (Cc) reduction, then resolving single atoms of lower atomic number will be possible. Of course the simplest way to improve the detectability is to reduce the thickness of the amorphous carbon substrate.
The development of Cs correctors will lead to great improvements in our ability to detect and resolve small objects supported by amorphous substrates. This is in part due to the increase in resolution and, as shown in this paper, in part due to our ability to position the substrate in a focal range where the phase contrast is suppressed. A substantial increase in the ratio of the object contrast to the substrate contrast can be achieved, provided that the object is placed on the correct side of the substrate (underneath for positive Cs). The contrast ratio passes through a maximum, which is located at a Cs value of 34 µm for a 10 nm substrate in the microscope we have modeled. It should be noted that for uncorrected microscopes there is only a 2–4% difference in the contrast ratios between placing the object above and below the substrate.
The sharp changes in contrast with specimen height, which have been used to our advantage in imaging supported particles, may also have implications here for the imaging of unsupported objects in Cs corrected microscopes. If objects are to be imaged at a low Cs value near to Gaussian focus, then any small amounts of surface contamination may have a large effect on the image.