Breakdown of the dipole approximation in core losses.

The validity of the dipole approximations commonly used in the inelastic scattering theory for transmission electron microscopy is reviewed. Both experimental and numerical arguments are presented, emphasizing that the dipole approximations cause significant errors of the order of up to 25% even at small momentum transfer. This behavior is attributed mainly to non-linear contributions to the dynamic form factor due to the overlap of wave functions.

Introduction

Electron energy loss spectroscopy (EELS) can be used for a wide variety of investigations, including material composition, site-selective elemental analysis and determination of the crystal environment, as well as magnetic measurements [1,2]. All these methods have in common that a thorough understanding of the scattering of fast electrons in the specimen is necessary for data interpretation.

Giving analytical solutions to the scattering problem is usually impossible in all but the simplest cases. Therefore, approximations are frequently used. The most popular approximation — especially for small momentum transfer — is the dipole approximation [3,4].

In recent years, the applicability of this approximation has been questioned for high-resolution scanning transmission electron microscopy [5] as well as the angular dependence of plasmons [6]. In this paper, we show that this approximation can introduce significant errors of up to 25% also in conventional core-loss spectra.

Theoretical framework

In first order Born approximation, the double differential scattering cross section (DDSC) for fast electrons in a solid is given by [7]where a0 is the Bohr radius, is the relativistic factor, S is the dynamic form factor (DFF), is the scattering vector, E is the energy transfer, and k0, k are the wave numbers of the incident and outgoing waves, respectively. It is noteworthy that the term stems from the long-ranged Coulomb interaction.

According to Fermi's golden rule, the DFF is commonly written as [8]with the sum running over all occupied initial and unoccupied final states and being the position operator. In the following, we will evaluate this in configuration space. By insertion of identities, the matrix element then becomes where and are the usual wave functions of the sample electron as a function of position .

is frequently written in the Rayleigh expansion as where are the spherical harmonics, are spherical Bessel functions, and the magnitudes r, Q corresponding to have been introduced.

At this point, the first dipole approximation (DA1) is usually employed by only considering the terms for , corresponding to dipole allowed transitions:

A second dipole approximation (DA2) is made by expanding the into a Taylor series around the origin,and retaining only the term of lowest order. With this, the transition operator becomes proportional to , warranting the name “dipole approximation”.

Both approximations DA1+DA2 together result in the DDSC to take the form [3,9]where f(E) sums up all the prefactors and matrix elements . While in its most general form, f(E) still contains , it does not depend on the magnitude of the momentum transfer but only on the direction, which in the present study is merely a parameter that is fixed before the experiments and simulations. In the common case of isotropic scattering with only one transition channel, this direction dependence drops out as well, giving the well-known radially symmetric diffraction pattern.

The last proportionality in Eq. (5) follows from Q2=q2+q2 with the in-plane momentum transfer and the characteristic momentum transfer due to energy loss [3,10]Here, T denotes the relativistic kinetic energy.

Note that q is of the order of (0.75…7.5) nm−1 for energy losses in the range of (100…1000) eV and an acceleration voltage of 200 kV. For comparison, the Compton momentum transfer q, at which the maximum of the Bethe ridge is located [9], is of the order of (50…150) nm−1.

The experimentally accessible intensity of electrons that are scattered inelastically into an energy interval and an angular region is given by the integral over the DDSC, provided the specimen is thin. Otherwise, the intensity redistribution by (elastic) Bragg and (inelastic) multiple scattering has to be taken into account as well.

If the integration regions and are small and thickness effects can largely be neglected, the intensity is essentially given by . Assuming and are constant throughout the measurements, one arrives atshowing the well-known Lorentzian angular distribution of the intensity.

This Lorentzian function is taken as the starting point for many further derivations and its uses range from the quantitative analysis of EELS [3,11] to Kramers–Kronig analysis of low energy losses [12].

Experiments

In Section 2, we have summarized the derivation of the well-known Lorentzian intensity distribution equation (7), using two dipole approximations DA1 and DA2. In the remainder of this work, we will evaluate their validity based on both measurements and simulations.

Since f(E) in Eq. (7) was taken to be the q-independent factor of the intensity, it is useful to divide all spectra by a reference spectrum, such as the one for q = 0, or equivalently Q=q, so this energy-dependent prefactor cancels and one arrives atIt must be emphasized that this simple formula was derived using the dipole approximations DA1 and DA2. Deviations from it in the experimental data or the numerical simulations therefore indicate contributions that are incompatible with DA1, DA2, or both.

Energy filtered selected area diffraction

The most straight-forward, albeit technically rather difficult approach is to keep the energy loss E in Eqs. (7) and (8) constant and to record an energy-filtered selected area diffraction (EFSAD) pattern which includes a large range of different momentum transfers q.

The primary challenge of this method is the enormous dynamic range. As will be shown below, deviations from the Lorentzian-type behavior are increasing slowly with q, but the intensity is strongly peaked at q=0. Therefore, long exposure times or the summation over several exposures is necessary to record the central peak as a reference as well as obtain a reasonable signal to noise ratio (SNR) in the q range of interest. In our case, we summed over 15 images with an exposure time of 40 s each, amounting to a total exposure time of 10 min. Obviously, this places high demands on the stability of both the sample and the instrument. In our case, a TECNAI F20 equipped with a GIF Tridiem with 200 kV acceleration voltage was used for all measurements presented here.

As an example system, we have chosen Si. The sample was oriented in a systematic row condition including the (1 1 1) diffraction spot to have a well-defined, simple situation for later simulations. The Si L2,3 edge at E=99.2 eV (q=0.72 nm−1, q=51 nm−1) was used for analysis. Background subtraction was performed using the three-window method.1 Subsequently, intensity traces perpendicular to the systematic row were obtained by averaging over a small wedge with its center at the (0 0 0) spot to obtain a better signal to noise ratio at larger momentum transfers q. These traces were then normalized to the maximum intensity at q=0.

Fig. 1 shows the square root of such a normalized trace multiplied by Q2. In the ideal dipolar case, the data points should all lie on the straight line . It is clearly evident that for , this is indeed the case to good accuracy. Above that value, however, there appear significant deviations, indicating that the dipole approximations no longer hold in that region. As is derived below, this plot directly visualizes the weighted, averaged wave function overlap (see Eq. (11)).

Fig. 1

Trace of an approximately 12 nm thick Si sample measured perpendicular to a systematic row including the (1 1 1) reflex after background subtraction using the Si L2,3 edge. The dashed line shows the ideal dipolar behavior according to Eq. (8). Note that in the experiment, data is only accessible for .

Angle-resolved electron energy loss spectroscopy

Instead of recording energy filtered diffraction patterns as described in Section 3.1 — with all its problems regarding dynamic range and stability — one can also record several conventional spectra at different momentum transfers.

Since one has to use a finite-sized objective aperture to get reasonable count rates, the angular resolution is greatly decreased by this method. On the other hand, this also reduces the stability requirement greatly. Furthermore, the dynamic range is decreased dramatically because only a small region of the diffraction plane is recorded during each measurement.

For this measurement method, Eq. (8) is still valid in principle. The major difference compared to before is that now q is a parameter, and the energy loss E — which in turn enters in q — is the variable.

As before, Si was chosen as an example system to elucidate the procedure. The method was also applied to other materials, however, as is shown in Fig. 4. The 35 nm thick Si sample was tilted into a systematic row including the diffraction spot before measuring spectra at the L2,3 edge (E=99.2 eV, q=0.72 nm−1, q=51 nm−1). To set the momentum transfer, the diffraction pattern was shifted with respect to the detector perpendicular to the systematic row. To achieve reasonable exposure times, the convergence semi-angle of the beam was chosen to be similar to the collection semi-angle of the spectrometer. For Si, the values were used. A sketch of all relevant parameters is shown in Fig. 2

Fig. 4

Relative deviation from the Lorentzian for (a) Si L2,3 (), (b) Gd (), (c) Co (), (d) NiO (, Ni L2,3: boxes, O K: disks). The horizontal error bars are derived from the finite aperture sizes, the vertical error bars are derived from the quality of the Lorentzian fit in the background. The gray vertical lines indicate the characteristic energy loss q (for NiO: qNi (top), qO (bottom)). Note that the Ni spectra show a small effect superimposed on much noise, which averages out in the calculations of the effect.

Fig. 2

Sketch of the measurement geometry in the diffraction plane. The diffraction pattern was shifted with respect to the detector (gray disk) to measure spectra at different momentum transfers q. specifies the convergence semi-angle and specifies the collection semi-angle.

After the acquisition of the spectra, a background subtraction was performed using a conventional power-law fit to obtain the pure edge spectra. To be able to compare this data with Eq. (8), all the pure edge spectra were then divided by the one corresponding to q=0. Fig. 3 shows an example of these ratio spectra.

Fig. 3

Si L2,3 ratio spectrum (top) and relative deviation spectrum (bottom) for q=19.6 nm−1. The specimen was 35 nm thick, oriented in a systematic row including the (2 2 0) diffraction spot, and were used. The dashed line in the upper graph shows the dipolar prediction using Eq. (8).

To compare these ratio spectra with Eq. (8), the latter was fitted to the data in the smooth post-edge region (see Fig. 3). To obtain quantitative information about the actual deviations, we subsequently computed the relative deviation of the experimental data from the fit curve.2 Fig. 3 shows an example of such a relative deviation spectrum.

It is evident from Fig. 3 that in this mode, similarly astonishing deviations are observed as with EFSAD. It is also quite remarkable that the relative deviation exhibits a general shape similar to the original pure edge spectrum, indicating that the energy dependence is indeed fairly independent of the momentum transfer in the isotropic systems investigated here, but the angular dependence is mis-judged by Eq. (7). Note that this cannot be attributed to the Bethe ridge, as the momentum transfers are much smaller than q and the Bethe ridge would cause positive deviations due to an increased scattering cross section instead of the observed negative deviations. It should also be emphasized that despite the fact that the collection angles were of the order of the characteristic energy loss, the resulting integration over the Lorentzian scattering distribution merely gives a constant factor, which is automatically compensated for by the fitting routine.

Fig. 4 shows the relative effect for several materials at the respective edge onsets. To avoid numerical artifacts, the relative deviation was averaged over a 1 eV range around the actual edge onset.

Simulations

In order to understand the origin of the deviations described in Section 3 and in order to assess the validity of the dipole approximations DA1+DA2, extensive calculations using both analytical and numerical methods are required. As before, we will limit the description to Si for the sake of brevity and clarity.

In this case, it can be shown using Wien2k [13] and Telnes.3 [14] that monopole and quadrupole transitions are much smaller in the angular range investigated [15]. It should be emphasized, however, that this is not the case in general. The leading term of quadrupole transitions, for example, has a Q4/Q4 behavior, i.e. it is approximately constant. With increasing scattering angle and momentum transfer, it will thus eventually become dominant as the importance of monopole and dipole transitions decreases. Similar arguments hold for higher multipole terms, although they generally have very small magnitudes. The effects at such large scattering angles can be found elsewhere (see, e.g., [16]). Hence, we will focus on the dipole approximation DA2.

For core losses, usually only one sample electron is involved. For the sake of brevity and clarity, we will assume that this electron's states (i) can be described by spherical harmonics, (ii) are determined by a single orbital angular momentum quantum number for each state (l, l), and (iii) can be described using a single radial wave function for each state (u(R), u(R)). More general derivations can be performed analogously, but are omitted here. Note that employing LS-coupling for the initial state does not change the form of the equations below [8].

With these assumptions, and combining Eqs. (2) and (3), the DFF becomeswhere the are functions only of the energy loss anddenotes the weighted, averaged wave function overlap.

The last term in Eq. (9) depends only on the direction of and the energy loss E. Since the former was constant during the experiments and only the magnitude Q changed, this last term can be ignored when the intensities are divided by I(qE). Inserting the expression for the DFF into Eq. (1) finally gives the expressionfor the DDSC. In the absence of elastic scattering, this is proportional to the experimentally accessible intensity.

As stated by Saldin and co-workers [17-19], the initial and final state wave functions are decisive for the validity of the dipole approximation DA2. In a similar way as described by them, it is possible to estimate the actual deviations caused by the higher order terms in the expansion of the spherical Bessel functions given in Eq. (4).

Using simple Slater-type orbitals (STO) [20], can be evaluated analytically. Fig. 5 shows that the result agrees well with the experimental data. Furthermore, it is possible to include the effects of elastic scattering [21], resulting in a slightly modified curve.

Fig. 5

as calculated by Slater-type orbitals using one beam (no elastic scattering) and three beams (with elastic scattering). The curves are superimposed on the experimental data from Fig. 1. The dashed line shows the ideal dipolar behavior.

Simulating the deviations of angle-resolved spectra from the Lorentzian-like behavior predicted by DA2 is more complicated. Because Eq. (8) has to be fitted to the experimental data relatively far beyond the edge, we have to properly treat the energy dependence of both the and the wave functions that enter in . We used Wien2k for these simulations, the results of which are shown in Fig. 6.

Fig. 6

Wien2k simulations of the relative deviations from the Lorentzian, superimposed on the experimental data for Si as depicted in Fig. 4. These calculations include a proper treatment of the energy dependence, but no elastic scattering.

Again, the numerical results are in remarkably good agreement with the experimental data, given that Wien2k has not been designed with excited states in mind and does not treat elastic scattering. The errors of the latter can be extrapolated from the calculations described in Section 4.1 and are generally smaller than the experimental error bars in the thickness and orientation ranges used. It should be emphasized, though, that the removal of elastic contributions produces larger deviations instead of smaller ones as can readily be seen from Fig. 5.

Conclusion and outlook

In this work, we have demonstrated that the dipole approximation equation (4), which is commonly used for simplification, does not strictly hold, even for small angle scattering and core excitations. Taking only into account dipole-allowed transitions — i.e., DA1 in Eq. (3) — on the other hand, does not cause significant deviations in the angular range investigated.

In the expansion of the spherical Bessel functions, at least the second order terms must be included to avoid errors of the order of up to 25%. If very accurate results are required, one additionally has to take into account contributions from several diffraction spots, the full spherical Bessel function, and possibly even other multipole transitions. These produce errors of the order of (1…3)%.

These results are of paramount importance for all formulas used in the analysis of off-axis experiments that are derived from the DDSC and generally use some form of dipole approximation [10,22,23]. While the on-axis error is small due to the peaked shape of the Lorentzian, all off-axis experiments — which record data only at non-zero q where the deviations are large — are affected. Most notable among these are site-specific methods such as quantitative EELS or energy loss by channeled electrons (ELCE) [24]. Furthermore, this can be an additional source of error for energy loss magnetic chiral dichroism (EMCD), where the signal of several off-axis detector positions has to be compared [8,25].

It should be emphasized that these findings open unique possibilities as well, as they will allow to directly study the overlap of wave functions, thus providing an original method for studying the electronic structure of solids.