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Orientation dependence of ionization edges in EELS
Ultramicroscopy 86 (2001) 343–353
Orientation dependence of ionization edges in EELS P. Schattschneidera,*, C. He´berta, B. Jouffreyb a
Institut fu¨r Angewandte und Technische Physik, Technische Universita¨t Wien, Wiedner Hauptstrasse 8-10/137, A-1040 Wien, Austria b LMSSMat, CNRS-URA 850, E´cole Centrale Paris, F-92295 Chaˆtenay-Malabry, France Received 11 August 2000; received in revised form 31 October 2000
Abstract Anisotropy in the density of unoccupied states can be detected in the fine structure of ionization edges in angleresolved EELS. It is shown that in a crystal an interference term occurs in the inelastic signal, and how it relates to electron channeling and site selection. The combination of orientation and site selection induces subtle variations in the ELNES. It is shown how this technique can be used to analyze local anisotropy related to the point group of the target atom. A second example shows how to extract non-dipole transitions at small scattering angles. # 2001 Elsevier Science B.V. All rights reserved. PACS: 61.16.Bg; 82.80.Pv Keywords: EELS; ELNES; DOS; Monopole transitions; Anisotropy
1. Introduction In electron energy loss spectrometry of crystals, the choice of a scattering vector allows selection of particular final crystal orbitals for the transitions in question. Accordingly, the fine structure of ionization edges (ELNES) can depend on the orientation of the scattering vector to the lattice. This effect has long been known. It is particularly strong for anisotropic materials such as highly oriented pyrolytic graphite (HOPG) [1] or BN [2]. The notion of orientation dependence is also used in the completely different context of electron *Corresponding author. Tel.: +43-222-58801; fax: +43-222564203. E-mail address: [email protected] (P. Schattschneider).
channeling: slight changes of the orientation of the incident beam influence the inelastically scattered signal of atoms in a solid [3,4] and may thus induce errors in quantification [5]. This technique was coined ELCE (energy loss by channelled electrons). Together with ist EDX counterpart ALCHEMI (atom location by channeling enhanced microanalysis) it has been used to advantage for the study of atomic site location with EDX [6] and EELS [7,8]. The influence of channeling on the fine structure has been studied in lesser detail [9]. In order to detect anisotropy in the electronic structure of a scatterer (not necessarily a crystal) a direction must be selected during the inelastic interaction. In EELS, this is the scattering vector; in X-ray spectroscopy, it is the polarization vector. For channeling, on the other hand at least two coherent incident plane waves are necessary (see later).
0304-3991/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 0 ) 0 0 1 2 5 - X
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It is important not to confuse these two phenomena. They may or may not occur simultaneously in one experiment. In the present paper, we describe both effects in the same theoretical framework. We begin with a description of ELNES in anisotropic systems without reference to channeling, i.e., under kinematic diffraction conditions. Then we introduce the principle of interferometric EELS that allows to describe inelastic channeling effects; finally, we present two new applications of the theory: probing of selected orbitals and extraction of dipole-forbidden transitions.
2. Anisotropy in ELNES under kinematical conditions When a free incident electron (i.e. a plane wave) ionizes an atom in a crystal the cross section reads 2
2
@ skin 4g kf 1 ¼ 2 SðQ; EÞ; @E@O a0 k0 Q4
ð1Þ
where we used the dynamic form factor (DFF) X ^ SðQ; EÞ ¼ jhijeiQR jf ij2 dðE þ Ei Ef Þ: ð2Þ i;f
Here, a0 is the Bohr radius, g ¼ ð1 b2 Þ1=2 the relativistic factor, and k0 ; kf the length of the fast electron’s wave vectors k0 and k before and after interaction, respectively. The kinematics of scattering define Q ¼ k0 kf , the scattering vector in the Fourier-transformed Coulomb interaction potential. The sum is over all occupied initial and all empty final states. Note that in general the electron that ionizes an atom is in a crystal eigenstate and not a plane wave. We shall see that this leads to a significant modification of Eq. (1). Only under kinematical diffraction conditions is Eq. (1) valid. Via the direction and length of the scattering vector Q in the operator the cross section will depend on the position of the spectrometer entrance aperture in the diffraction plane and on the orientation of the incident beam to the crystal. For small Q the exponential operator can be linearized, and according to the orthogonality of jii and j f i the matrix element in the DFF
simplifies to ^
^ f i: hijeiQR j f i8iQhijRj
ð3Þ
This is the dipole approximation. It is valid if Q 5r1 where r is the average radius of the initial (ground state) orbital. Since those orbitals occupy only a fraction of the elementary cell, the dipole approximation is good for scattering angles up to the Bragg angles of the lowest Miller indices. Eq. (3) clearly shows how anisotropy enters the play: it is the projection of the (vectorial) matrix element onto the scattering vector. Thus, choice of the scattering vector selects oriented matrix elements. The final state j f i can be expanded into spherical harmonics cf ¼
1 þ‘0 X X ‘0 ¼0
0
~ D‘f0 m0 u‘0 ðEf ; RÞYm‘ 0 ðRÞ
ð4Þ
m0 ¼‘0
with expansion coefficients Df‘0 m0 . The partial (lm-projected) density of states is X f 2 f jD‘0 m0 j dðe0 Ef Þ: ð5Þ w‘0 m0 ðe0 Þ ¼ f
For atomic orbitals, w is a Dirac function in energy and yields the occupation number of the orbital with quantum numbers ‘0 ; m0 . We choose the z-coordinate parallel to Q. After integration over the radial parts Z r‘0 :¼ R3 ui ðRÞu‘0 ðEf ; RÞ ð6Þ we obtain in dipole approximation 2 X X z^ f 2 rl Dl;m i lm SðQ; EÞ 8Q R f l;m dðE þ Ei Ef Þ:
ð7Þ
We label the states with roman fonts jii in order to signify that the radial parts were already integrated out in Eq. (6) and only the angular part remains. The state i is then s, p, d, f; . . . : As an example, we consider K-edges. Since the operator z^=R applied to an initial s state creates a pz state, the sum over final angular momenta l reduces to l ¼ 1 due to orthogonality. When we transform the remaining three final states from magnetic quantum numbers to Cartesian
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coordinates (ðl; mÞ ! px;y;z ) Eq. (7) reads X SðQ; EÞ 8Q2 r21 jDfpx hpz jpx i þ Dfpy hpz jpy i
3. The effect of electron channeling: interferometric EELS
f
þ
Dfpz hpz jpz ij2 dðE
þ Ei Ef Þ
ð8Þ
and only the last member remains. It is, by Eq. (5), the pz -projected final DOS, and Eq. (8) reads SðQ; EÞ8Q2 r21 wpz ðEf Ei Þ:
345
ð9Þ
In the experiment, the fine structure will mirror the partial density of final states projected onto the direction of Q. r1 is the matrix element of the radial distance jRj taken between states ui and u‘0 . Loosely speaking, the scattering vector is the symmetry axis of the selected final orbitals. When we change the direction of Q either by tilting the specimen relative to the beam, or by changing the scattering angle, different final orbitals are selected. In HOPG or in hexagonal BN, for instance, the probe sees the final s * states when Q is in the basal plane, and the final p * orbitals when it is perpendicular [1,2]. Angle resolved experiments allow thus a simple visual interpretation of ELNES in terms of oriented orbitals. At this point, a comment is in place: K-edges are especially simple. For L-edges, the dipole selection rules allow final s and d states. Eq. (8) contains then transitions to s, dz2 ; dxz , and dyz . Eq. (9) will be replaced by
When electrons channel along atomic columns or planes in a crystal, the apparent cross section increases. The usual explanation is that the true atomic cross section remains unchanged, but the incident intensity is higher at the atomic site under channeling conditions. This reasoning is not rigorously true as we shall see. In a crystal, and particularly under channeling conditions, the incident electron is a Bloch wave, i.e., a coherent superposition of plane waves. We would then expect interference terms to occur in Eq. (1). Indeed, for a coherent superposition of two plane waves C1 jk1 i þ C2 jk2 i impinging on an atom, the inelastic scattering amplitude for a particular transition would read C1 f ðQ1 Þ þ C2 f ðQ2 Þ; where f ðQÞ denotes the generalized oscillator strength (GOS). See the scattering geometry Fig. 1. The phases of the interfering waves can be changed by phaseplates which correspond to the
SðQ; EÞ 8Q2 r20 ws ðEf Ei Þ þ Q2 r22 ½wdz2 ðEf Ei Þ þ wdxz ðEf Ei Þ þ wdyz ðEf Ei Þ :
ð10Þ
This is not simply the product ‘‘matrix element’’ ‘‘partial DOS’’, as often stated. It is not even a sum of such products; rather, it is a weighted sum of partial DOSes, with the matrix elements depending on energy loss, see Eq. (6). Moreover, the weighting coefficients of the d-DOSes are different: the dxy vanishes. Only when we integrate over all directions of Q will all the d-DOSes acquire equal weight. A general description of orientation sensitive ELNES surpassing the dipole approximation can be found in [10,11].
Fig. 1. Scattering geometry for two coherent plane waves impinging on an atomic monolayer. The ionization process caused by simultaneous transfer of momenta hQ1 and hQ2 bears the interference effects of the experiment. The phases of the interfering waves can be changed by phase plates which correspond to the diffraction conditions inside a crystal. Cosinusoidal charge distribution in the slab (full line, channeling conditions). Dashed line: antichanneling.
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diffraction conditions inside a real crystal. Channeling along the lattice planes or atomic rows corresponds to a cosinusoidal charge distribution in the lateral plane. By shifting the relative phase by p, the charge distribution becomes sinusoidal. This situation will be referred to as antichanneling in the following. In the two-beam case a situation is meant in which a maximum of electron intensity occurs between atomic planes. This corresponds to a tilt of the incoming electron beam from channeling conditions by twice the Bragg angle. The measured intensity is the amplitude squared Iex ¼ jC1 f ðQ1 Þ þ C2 f ðQ2 Þj2 ¼ jC1 f ðQ1 Þj2 þ jC2 f ðQ2 Þj2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} direct
þ 2R½C1 C2* f ðQ1 Þf ðQ2 Þ * : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð11Þ
mixed
The direct terms correspond to the ionization cross sections as obtained in the usual derivation for two incident plane waves, e.g. [12–14]. The coherent superposition of the two plane waves has created an additional interference term in the ionisation cross section which is not contained in Eq. (1). Note the formal similarity to Youngs double-slit experiment. In an energy loss experiment, one has to sum the scattering amplitude f ðQÞ over all energetically possible final states. Instead of Eq. (11) we write @2s SðQ1 ; Q1 ; EÞ SðQ2 ; Q2 ; EÞ ¼ jC1 j2 þ jC2 j2 4 @E@O Q1 Q42 SðQ1 ; Q2 ; EÞ þ 2R C1 C2* eidðQ1 Q2 Þ : ð12Þ Q21 Q22 We normalized the incident current implicitely to one; therefore, the outgoing intensity equals the cross section. Here we used the mixed dynamic form factor (MDFF), an extension of the DFF X ^ ^ SðQ1 ; Q2 ; EÞ ¼ hijeiQ1 R jf ihf jeiQ2 R jii i;f
dðE þ Ei Ef Þ:
ð13Þ
Eq. (12) is formally equivalent to Eq. (11). The new ‘‘mixed’’ term describes the effect of interference between the coherent plane waves on the
inelastic signal. It replaces the intensity oscillations of a conventional interferometer. The energy spectrum of a fast electron leaving the interferometer in some direction kf will thus contain intensity due to the MDFF, weighted by the complex amplitudes C1 ; C2 of the coherent plane waves, and reveal thereby their interference. The exponential member in Eq. (12) accounts for the position d of the ionized atom by introducing a geometric phase dðQ1 Q2 Þ in the interference term. This means that inelastic cross sections of chemically identical atoms can be different dependent on their location. The apparent increase in cross section under channeling conditions can now be understood easily: the Bloch wave coefficients C1 ; C2 are both positive; the exponential factor in the mixed term causes a cosinusoidal variation of scattered intensity with periodicity 2p=ðQ1 Q2 Þ ¼ 2p=G, and this is exactly the density distribution of the incident probe electron in the lattice under planar channeling conditions. Note, however, that we assumed implicitly that the MDFF is positive. The effect would be inverted when the MDFF were negative. This is in fact the case since in the dipole regime the MDFF is proportional to the scalar product of the two scattering vectors. For large collection angles the expected behavior is retrieved because the integral of the MDFF over all outgoing directions is positive [15]. The value of the integral depends on the collection angle and may even be slightly negative. This fact could explain difficulties in the interpretation of early ELCE experiments, and the ad hoc demand of choosing large Q known as localization enhancement [16]. The situation sketched in Fig. 1 can be approximately realized in a crystal. Bragg diffraction serves as a beam splitter, and the complex amplitudes of the resulting plane waves can be tuned by slightly rocking the beam or changing the depth where ionization takes place, due to the phase of every plane wave which increases linearly with thickness. The difference between Q1 and Q2 is the reciprocal lattice vector g perpendicular to the Bragg diffracting plane. In practice, the situation is more involved because the inelastic signal will also be Bragg diffracted. Moreover, instead of two coherent
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plane waves we have many of them before and after inelastic scattering. The combination of elastic and inelastic interactions leads to the following expression for the differential scattering cross section [17]
2 @ s 4g2 kf X X ¼ 2 TJz ;Jz0 ðtÞAJ AJ*0 @E@O crystal a0 k 0 J 0 J
1 X iuðQJxy QJxy0 Þ Su ðQJxy ; QJxy0 ; EÞ e N u Q2Jxy Q2J 0
ð14Þ
xy
with ðjj 0 Þ
TJz Jz0 ðtÞ ¼ eiðdg
0
0
0
þdgðll Þ Þt=2 ðZðjj Þ þZðll Þ Þt=2
e
cosh dZðt=2Þ sin Dt=2 þ i sinh dZðt=2Þ cos Dt=2 : Dt=2 þ idZt=2 ð15Þ
ðj;j 0 Þ
j
j0
dg ¼ g g are differences of eigenvalues of the Schro¨dinger equation to Bloch waves labelled 0 0 j; j 0 ; Zðl;l Þ ¼ Zl þ Zl are sums of their imaginary 0 parts (describing absorption), D ¼ dgðj;j Þ 0 0 0 dgðl;l Þ ; dZ ¼ Zðj;j Þ Zðl;l Þ . J is a shorthand notation for a running index that covers all Bloch wave indices j; l and all plane wave components g; h j before and after inelastic scattering. AJ ¼ C0* Cgj l Dl0 Dh* are products of Bloch wave coefficients before and after inelastic scattering, u are the positions of the atoms in the elementary cell, Su is the mixed dynamic form factor (MDFF) for the atom at position u; N is the number of atoms in the unit cell, kf ; k0 are the wave numbers of the final (initial) incident plane electron wave. The formal structure of Eq. (14) is easy to understand when we refer back to Eq. (12): AJ correspond to Cg . We retrieve also the same exponential factor and the same MDFFs. The sum over u reduces to one term for a mono-atomic elementary cell. The factor T stems from integration over the thickness of the specimen. It is a thickness- and orientation-dependent term similar to the Pendello¨sung oscillations in dynamical diffraction. Eq. (14) can be evaluated with any program for Bloch wave simulations. As an example, we show the angular intensity distribution of the Al L3 cross section for
Fig. 2. Angular profile of Al L3 intensity for planar channeling and antichanneling conditions. Full line: antichanneling } center of Laue circle (CLC) at ð2 2 0Þ; dashed line: channeling } CLC at ð2 2 0Þ. E0 ¼ 160 kV. Upper lines: intensity traces. Lower lines: interference term.
scattering angles between 0 and 2y220 . The upper lines in Fig. 2 are simulated traces through an energy filtered diffraction pattern under planar (2 2 0) channeling and antichanneling conditions. The dashed line is for channeling – center of Laue circle sitting at (0 0 0), and the full line is for antichanneling – center of Laue circle at (2 2 0). The lower lines are the corresponding interference terms, stemming from the MDFF. In the middle between the incident beam and the Bragg reflection the mixed term is most significant, changing its sign at yBragg 10:5 mrad. Note that in spite of enhanced intensity at the lattice planes (planar channeling!) a reduced L-edge intensity is observed at scattering angles smaller than the Bragg angle. This is because the MDFF is negative in this angular range. The change in contrast seen at 10.5 mrad is the right-hand edge of a defect Kikuchi band subtending the range from 10:5 to 10.5 mrad. (In the standard explanation, the scattered intensity would be proportional to the product of the intensities of the incident and outgoing (reciprocal) Bloch waves at the ionized site [18] and thus generate an excess Kikuchi band.) The mixed term becomes smaller under antichanneling conditions (electrons channel between the (2 2 0) lattice planes, lower full line) but we do not observe contrast reversal as would have been expected from the simple two-beam explanation of Eq. (12). This is because inclusion of 5 beams as
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well as absorption in the simulation complicates the simple 2-beam picture. Specimen thickness plays also an important role in the interpretation of energy filtered diffraction patterns or ELCE experiments. It was shown that the intensity enhancement expected for an infinitely thin crystal under channeling conditions is inverted for a thickness of less than one extinction length [9] when the factor T, Eq. (15) is properly taken into account. It can safely be assumed that numbers of unpublished ELCE experiments are eagerly awaiting a sound interpretation.
4. Interferometric EELS: selection of orbitals in rutile Rutile (TiO2 )is an interesting case for testing the local anisotropy: although the structure is quite isotropic and no orientation effects are visible in kinematical conditions, the O atoms have a point group of low symmetry (2 mm). It thus appears possible to select orbitals of different types, that correspond to different energies and so will be visible as a change in the ELNES of the spectra. At the O K-edge, we probe O p states which hybridize with Ti d states. The projection on the (1 1 1) plane somewhat hides the tetragonal structure of the indicated unit cells (thin lines in Fig. 3). Note that there are two types of oxygen positions in this orientation, those of the unit cell’s basal plane which are aligned in the ð 1 1 0Þ planes, and positions at half height of the unit cell, forming a zig-zag line in this projection. Those two positions are crystallographically equivalent but as they are rotated differently with respect to the crystal axes, the orbitals are also differently oriented. The py and pz shown in Fig. 3 are taken in the local coordinate system of the atom. The two positions are therefore not equivalent with respect to Q1 . The py orbitals of O hybridize with the Ti t2g states to build bonding and antibonding p and p * states. The px and pz orbitals of O hybridize with the Ti eg to build s and s * states [19,20]. Both p * and s * are unoccupied. Transitions to these states correspond to peaks A at 4 eV and B at
Fig. 3. Rutile lattice in 111 zone axis projection. The smaller spheres are the oxygen atoms. Two different oxygen antibonding p orbitals are probed. Under channeling conditions they are local py and form p * bonds with Ti t2g . Under antichanneling conditions they are local pz . Together with px they form s * bonds with the dx2 y2 Ti orbital (taken in the octahedral coordinate system). The six-fold coordination of Ti is also indicated as a shadowed octahedron.
Fig. 4. The px ; py and pz projected density of unoccupied states at the O site in rutile shows the energy difference between py on the one hand and px and pz on the other hand.
7 eV in Fig. 5, with the s * at higher energies as shown in Fig. 4. From Fig. 3 one would intuitively expect the py to be excited under channeling conditions and the pz under antichanneling conditions and thus peak B of the O K-edge to be higher under antichanneling conditions. However, the MDFF is negative under the chosen scattering geometry and thus the effect will exactly be the reverse.
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For experiments, a wedge-shaped border of a cleaved rutile ðTiO2 ) needle was used. In a Philips CM20 Transmission Electron Microscope operated at 160 keV, it was oriented with the ð 1 1 0Þ lattice planes almost parallel to the incident electron beam, approximately 4:58 off the ½1 1 1 crystallographic orientation. At this tilt, the nonsystematic reflections contribute 5% to the direct term in the edge signal, and nothing to the interference term since the conditions are almost kinematic. This is on the level of the expected noise, and differences from switching between channeling and antichanneling should thus remain visible. EELS spectra were taken in diffraction mode using a Gatan 666 PEELS, operated at a dispersion of 0:24 eV=channel. The energy resolution limited by the energy spread of the primary beam was about 1.8 eV. The spectrometer entrance aperture was placed midway between the (0 0 0) and ð 1 1 0Þ Bragg spots. Apart from a small component parallel to the incident wave vector which is due to the energy loss, the scattering vectors Q1 and Q2 point in ð 1 1 0Þ direction in reciprocal space, i.e., perpendicular to the ð 1 1 0Þ lattice planes. The vector Q1 is indicated in Fig. 3. The precise orientation of the crystal was verified using the Kikuchi lines visible in the thicker parts of the wedge. The probe was fully condensed to 40 nm diameter. The illumination and collection half angles were 3 and 1.3 mrad, respectively. The specimen thickness was measured from the (1 1 0) thickness fringes on the wedge. A thin area (t50:5 mean free paths for low-loss scattering) was selected in order to avoid strong plasmon contributions and multiple scattering corrections. All spectra were background subtracted with the same pre-edge window; thus, possible errors induced by the variations in the high-energy tail of the Ti edge should cancel in the difference of the processed spectra. Radiation damage may occur in rutile. It was tested beforehand that during 10 s of irradiation no changes in the Ti to O atomic ratio could be observed. Therefore, an integration time of 5 s was chosen for each spectrum to be on the safe side. This is the reason for the rather noisy spectra, (Fig. 5). It shows the O K ELNES taken at
349
Fig. 5. EELS spectra of the oxygen K-edge in rutile, with peaks A at 4 eV and B at 7 eV, under non-interference conditions (diamonds), and under strong Bragg diffraction interference conditions at a thickness of 25 nm, exciting once the (2 2 0) Bragg spot (thin solid line) and then symmetrically both ð1 1 0Þ and (1 1 0) spots (thin dashed line).
channeling and antichanneling conditions. A significant difference in the oxygen K-edge is seen when normalizing the spectra to the same height in the first peak. When integrating the intensity over 2 eV centered about the two maxima at 4 and 7 eV, the noise is largely reduced, and the predicted variation in peak height ratio is clearly seen. The mixed terms for the two oxygen sites have different sign because of their geometric phase difference g d ¼ p – see Fig. 1. Any action that shifts the relative phases of the incident waves will then lead to complementary intensity variations of the cross section at the two sites, the net effect being a variation of the peak heights B=A. Under planar channeling (thin dashed line) the incident electron has crests at the atoms in the basal plane and hence the dpp * orbitals are probed (see Fig. 3). Tilting the beam by twice the Bragg angle of the (1 1 0) plane to ð1 1 0Þ antichanneling (thin full line) the crests are at the zig-zag line of oxygen atoms and the dps * orbitals are probed. The thick lines in the figure are simulations based on evaluation of Eq. (14), taking a weighted average over the thickness variation of 5 nm within the illuminated area. All spectra, measured and simulated, are normalized to peak A. It should be borne in mind that the specimen thickness plays an important role. For thicker
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Fig. 6. Same as in Fig. 5 at a thickness of 40 nm. Note the inversion of the peak height ratios relative to the thinner specimen.
Fig. 7. Diffraction mode spectra of Si L23 -edge at Q ¼ ð2 0 0Þ after removal of the thermal diffuse background, plural scattering deconvolution and background subtraction. Upper spectrum for kinematic conditions, lower one for (4 0 0) planar channeling.
specimens, the peak ratio shows the opposite tendency, see Fig. 6. Details can be found in [9]. By its ability to probe the local anisotropy at the ionized site, angle resolved interferometric EELS aims at the point group symmetry of the target atom. Knowledge of the specimen thickness as well as the precise scattering geometry is important for a correct interpretation.
geometry described above we have Q1 ¼ G=2 and Q2 ¼ G=2. Restricting ourselves to the three most important transition probabilities (l ¼ 0; 1; 2), we find for the DFF DFF ¼ SðQ1 ; Q1 ; EÞ8
3 X
Pl ð1Þfl ðQ1 ; Q1 ; EÞ ð17Þ
l¼0
and for the MDFF 5. Interferometric EELS: non-dipole transitions Non-dipole transitions can be seen for large scattering angles [21]. The disadvantage is often the low intensity according to the Lorentzian angular profile of inelastic scattering. Interferometric EELS circumvents this problem as we shall see. The formula for the direct and mixed terms contain the DFF and the MDFF. Now that the transition with l ¼ 1 dominates (dipole transition), the cross section should by and large resemble that of the dipole approximation, irrespective of channeling. Evaluation of Eq. (13) for atomic orbitals yields [10]
1 X Q1 Q2 SðQ1 ; Q2 ; EÞ ¼ Pl fl ðQ1 ; Q2 ; EÞ; Q1 Q2 l¼0 ð16Þ where fl is the probability for a l-pole transition. Pl is the Legendre polynomial. In the scattering
MDFF ¼ SðQ1 ; Q2 ; EÞ8
3 X
Pl ð1Þfl ðQ1 ; Q2 ; EÞ:
l¼0
ð18Þ
A closer inspection of the two equations shows that in the MDFF the odd transitions – l odd – contribute with positive sign, and the even transitions – l even – with negative sign whereas the DFF contains only positive contributions. From Eqs. (18) and (17) the relative strength of dipoleforbidden transitions can be calculated immediately: f0 þ f2 1 þ SðQ1 ; Q2 ; EÞ=SðQ1 ; Q1 ; EÞ : 8 1 SðQ1 ; Q2 ; EÞ=SðQ1 ; Q1 ; EÞ f1
ð19Þ
This enables extraction of non-dipole transitions from a measurement of the cross section at yBragg corresponding to Q1 ¼ G=2. Under channeling conditions, the ELNES is a linear combination of DFF and MDFF; under kinematic conditions only the DFF contributes. So, both the DFF and
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the MDFF as a function of energy could be obtained from a linear combination of these two experimental ELNES spectra when the coefficients are known from a simulation of the profiles calculated via Eq. (14), such as in Fig. 2.
Fig. 8. Si L-edge at yBragg ¼ 10:5 mrad. Also shown is the ratio of Si L-edges for kinematic to channeling conditions.
351
In practice, it is easier to impose a normalization condition. For the Si L23 edge, this condition derives from the fact that at 120 eV the non-dipole terms are small (they contribute 7%Þ. Consequently, the cross sections at this energy are almost identical for channeling and kinematical spectra. This can be used for normalizing the spectra and extracting the ratio MDFF=DFF. Before doing so it is necessary to remove the contributions from coupling of inelastic with thermal diffuse background scattering and the plural scattering contributions. The method described by Batson and Silcox was applied [22]. For background subtraction the same pre-edge windows were used. Fig. 7 shows the edge region after background subtraction and for both channeling and kinematic conditions. The former is ’ 30% lower in intensity than the latter. As in the rutile case, this is because the MDFF is negative – enhanced intensity at the 400 lattice planes reduces the cross section. The fine structure is by and large the same in both spectra.
Fig. 9. Extracted relative non-dipole transition probabilities, compared with the atomic simulations and a solid-state model. Next to the edge the error bars of the experimental results are high according to data processing.
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To judge the differences better we show the ratio of the two spectra in Fig. 8. Any deviation from unity signifies presence of non-dipole terms or errors in the background subtraction. Non-dipole transitions occur in the first few eV above threshold. In fact, there is little variation for losses higher than 10 eV beyond the edge – proving that deconvolution and background subtraction were successful. Fig. 9 shows the relative non-dipole transition probabilities ðf0 þ f2 Þ=f1 obtained from the two experimental ELNES Fig. 7 via Eq. (19). The experimental values are compared with an atomic model and a simulation based on the TELNES package of WIEN97 [11,23]. From the atomic model we see that the MDFF is dominated by monopole transitions. Quadrupole transitions f2 are negligible [10]. Furthermore, since dipole transitions to final s states are 510%; what we see in Fig. 9 is essentially the ratio of p!p to p!d transitions. The tendency to increased monopole transitions in the vicinity of the edge can be seen in both the atomic and the solid-state model. The leftmost data point is not reliable because of strong noise amplification near threshold. Although the error bars for a single data point at 2 eV from the edge onset are still high, the local maximum of final p states at 3 eV and the minimum at 6–7 eV are significant experimental features. The 3 eV maximum is well reproduced; the origin of the disagreement at 5–6 eV is not known. A possible explanation is that the core hole significantly changes the final DOS in Si. This could quench more d states into the center of the band at 5 eV than the simulation predicts, causing a reduction of the ratio of p to d states.
6. Conclusion The two experiments show that subtle variations in the fine structure of ionization edges are related to channeling of the probe electron. The concept of interferometric EELS facilitates the understanding of such surprising effects as the occurrence of defect Kikuchi bands for particular
scattering geometries, or contrast inversion of the inelastic signal as a function of specimen thickness. The symmetry properties of the mixed dynamic form factor can be exploited to differentiate between dipole and non-dipole transitions. The dependence of the fine structure on the channeling conditions in combination with the choice of a scattering vector can be used to obtain directional information on particular unoccupied states, even for crystals with a single type of atoms and bonding. Acknowledgements This work was supported by the Austrian Science Fund under Project No. P14038-PHY. The financial support of parts of this work by the Austrian O¨AD (Amade´e III.5) and the French CNRS (PICS No. 913) is gratefully acknowledged. References [1] R.D. Leapman, P.L. Fejes, J. Silcox, Phys. Rev. B 28 (1983) 2361. [2] C. Souche, B. Jouffrey, G. Hug, M. Nelhiebel, Micron 29 (1998) 419. [3] J. Gjonnes, Acta Crystallogr. 20 (1966) 240. [4] P. Schattschneider, B. Jouffrey, M. Nelhiebel, Phys. Rev. B 54 (1996) 3861. [5] M. Schenner, M. Nelhiebel, P. Schattschneider, Ultramicroscopy 65 (1996) 95. [6] J.C.H. Spence, J. Taft, J. Microsc. 130 (1983) 147. [7] J. Taft, O.L. Krivanek, Nucl. Instr. and Meth. 194 (1982) 153. [8] J. Taft, O.L. Krivanek, Phys. Rev. Lett. 48 (1982) 560. [9] M. Nelhiebel, P. Schattschneider, B. Jouffrey, Phys. Rev. Lett. 85 (2000) 1847. [10] M. Nelhiebel, N. Luchier, P. Schorsch, P. Schattschneider, B. Jouffrey, Philos. Mag. B 79 (1999). [11] C. Souche, P.-H. Louf, P. Blaha, M. Nelhiebel, J. Luitz, P. Schattschneider, K. Schwarz, B. Jouffrey, Ultramicroscopy 83 (2000) 9. [12] H.A. Bethe, Ann. Phys. Lpz. 5 (1930) 325. [13] R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 2nd Edition, Plenum Press, New York, 1996. [14] P. Schattschneider, Fundamentals of Inelastic Electron Scattering, Springer, Wien, 1986. [15] W. Schu¨lke, S. Mourikis, Acta Crystallogr. A 42 (1986) 86. [16] O. Krivanek, M. Disko, J. Taft, J. Spence, Ultramicroscopy 9 (1982) 249.
P. Schattschneider et al. / Ultramicroscopy 86 (2001) 343–353 [17] P. Schattschneider, M. Nelhiebel, M. Schenner, W. Grogger, F. Hofer, J. Microsc. 183 (1996) 18. [18] L. Reimer, Adv. Electron. Electr. Phys. 81 (1991) 43. [19] P.I. Sorantin, K. Schwarz, Inorg. Chem. 31 (1992) 567. [20] P. Sorantin, Elektronische Eigenschaften vonSubstanzen in der Rutilstruktur, Ph.D. Thesis, Technische Universita¨t Wien, Wien, 1990.
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[21] J. Auerhammer, P. Rez, Phys. Rev. B 40 (1989) 2024. [22] P.E. Batson, J. Silcox, Phys. Rev. B 27 (1983) 5224. [23] P. Blaha, K. Schwarz, J. Luitz, WIEN97, Vienna University of Technology 1997 (Updated Unix version of P. Blaha, K. Schwarz, P. Sorantin, and S.B. Trickey, Comput. Phys. Commun. 59 (1990) 399).
Orientation dependence of ionization edges in EELS P. Schattschneidera,*, C. He´berta, B. Jouffreyb a
Institut fu¨r Angewandte und Technische Physik, Technische Universita¨t Wien, Wiedner Hauptstrasse 8-10/137, A-1040 Wien, Austria b LMSSMat, CNRS-URA 850, E´cole Centrale Paris, F-92295 Chaˆtenay-Malabry, France Received 11 August 2000; received in revised form 31 October 2000
Abstract Anisotropy in the density of unoccupied states can be detected in the fine structure of ionization edges in angleresolved EELS. It is shown that in a crystal an interference term occurs in the inelastic signal, and how it relates to electron channeling and site selection. The combination of orientation and site selection induces subtle variations in the ELNES. It is shown how this technique can be used to analyze local anisotropy related to the point group of the target atom. A second example shows how to extract non-dipole transitions at small scattering angles. # 2001 Elsevier Science B.V. All rights reserved. PACS: 61.16.Bg; 82.80.Pv Keywords: EELS; ELNES; DOS; Monopole transitions; Anisotropy
1. Introduction In electron energy loss spectrometry of crystals, the choice of a scattering vector allows selection of particular final crystal orbitals for the transitions in question. Accordingly, the fine structure of ionization edges (ELNES) can depend on the orientation of the scattering vector to the lattice. This effect has long been known. It is particularly strong for anisotropic materials such as highly oriented pyrolytic graphite (HOPG) [1] or BN [2]. The notion of orientation dependence is also used in the completely different context of electron *Corresponding author. Tel.: +43-222-58801; fax: +43-222564203. E-mail address: [email protected] (P. Schattschneider).
channeling: slight changes of the orientation of the incident beam influence the inelastically scattered signal of atoms in a solid [3,4] and may thus induce errors in quantification [5]. This technique was coined ELCE (energy loss by channelled electrons). Together with ist EDX counterpart ALCHEMI (atom location by channeling enhanced microanalysis) it has been used to advantage for the study of atomic site location with EDX [6] and EELS [7,8]. The influence of channeling on the fine structure has been studied in lesser detail [9]. In order to detect anisotropy in the electronic structure of a scatterer (not necessarily a crystal) a direction must be selected during the inelastic interaction. In EELS, this is the scattering vector; in X-ray spectroscopy, it is the polarization vector. For channeling, on the other hand at least two coherent incident plane waves are necessary (see later).
0304-3991/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 0 ) 0 0 1 2 5 - X
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It is important not to confuse these two phenomena. They may or may not occur simultaneously in one experiment. In the present paper, we describe both effects in the same theoretical framework. We begin with a description of ELNES in anisotropic systems without reference to channeling, i.e., under kinematic diffraction conditions. Then we introduce the principle of interferometric EELS that allows to describe inelastic channeling effects; finally, we present two new applications of the theory: probing of selected orbitals and extraction of dipole-forbidden transitions.
2. Anisotropy in ELNES under kinematical conditions When a free incident electron (i.e. a plane wave) ionizes an atom in a crystal the cross section reads 2
2
@ skin 4g kf 1 ¼ 2 SðQ; EÞ; @E@O a0 k0 Q4
ð1Þ
where we used the dynamic form factor (DFF) X ^ SðQ; EÞ ¼ jhijeiQR jf ij2 dðE þ Ei Ef Þ: ð2Þ i;f
Here, a0 is the Bohr radius, g ¼ ð1 b2 Þ1=2 the relativistic factor, and k0 ; kf the length of the fast electron’s wave vectors k0 and k before and after interaction, respectively. The kinematics of scattering define Q ¼ k0 kf , the scattering vector in the Fourier-transformed Coulomb interaction potential. The sum is over all occupied initial and all empty final states. Note that in general the electron that ionizes an atom is in a crystal eigenstate and not a plane wave. We shall see that this leads to a significant modification of Eq. (1). Only under kinematical diffraction conditions is Eq. (1) valid. Via the direction and length of the scattering vector Q in the operator the cross section will depend on the position of the spectrometer entrance aperture in the diffraction plane and on the orientation of the incident beam to the crystal. For small Q the exponential operator can be linearized, and according to the orthogonality of jii and j f i the matrix element in the DFF
simplifies to ^
^ f i: hijeiQR j f i8iQhijRj
ð3Þ
This is the dipole approximation. It is valid if Q 5r1 where r is the average radius of the initial (ground state) orbital. Since those orbitals occupy only a fraction of the elementary cell, the dipole approximation is good for scattering angles up to the Bragg angles of the lowest Miller indices. Eq. (3) clearly shows how anisotropy enters the play: it is the projection of the (vectorial) matrix element onto the scattering vector. Thus, choice of the scattering vector selects oriented matrix elements. The final state j f i can be expanded into spherical harmonics cf ¼
1 þ‘0 X X ‘0 ¼0
0
~ D‘f0 m0 u‘0 ðEf ; RÞYm‘ 0 ðRÞ
ð4Þ
m0 ¼‘0
with expansion coefficients Df‘0 m0 . The partial (lm-projected) density of states is X f 2 f jD‘0 m0 j dðe0 Ef Þ: ð5Þ w‘0 m0 ðe0 Þ ¼ f
For atomic orbitals, w is a Dirac function in energy and yields the occupation number of the orbital with quantum numbers ‘0 ; m0 . We choose the z-coordinate parallel to Q. After integration over the radial parts Z r‘0 :¼ R3 ui ðRÞu‘0 ðEf ; RÞ ð6Þ we obtain in dipole approximation 2 X X z^ f 2 rl Dl;m i lm SðQ; EÞ 8Q R f l;m dðE þ Ei Ef Þ:
ð7Þ
We label the states with roman fonts jii in order to signify that the radial parts were already integrated out in Eq. (6) and only the angular part remains. The state i is then s, p, d, f; . . . : As an example, we consider K-edges. Since the operator z^=R applied to an initial s state creates a pz state, the sum over final angular momenta l reduces to l ¼ 1 due to orthogonality. When we transform the remaining three final states from magnetic quantum numbers to Cartesian
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coordinates (ðl; mÞ ! px;y;z ) Eq. (7) reads X SðQ; EÞ 8Q2 r21 jDfpx hpz jpx i þ Dfpy hpz jpy i
3. The effect of electron channeling: interferometric EELS
f
þ
Dfpz hpz jpz ij2 dðE
þ Ei Ef Þ
ð8Þ
and only the last member remains. It is, by Eq. (5), the pz -projected final DOS, and Eq. (8) reads SðQ; EÞ8Q2 r21 wpz ðEf Ei Þ:
345
ð9Þ
In the experiment, the fine structure will mirror the partial density of final states projected onto the direction of Q. r1 is the matrix element of the radial distance jRj taken between states ui and u‘0 . Loosely speaking, the scattering vector is the symmetry axis of the selected final orbitals. When we change the direction of Q either by tilting the specimen relative to the beam, or by changing the scattering angle, different final orbitals are selected. In HOPG or in hexagonal BN, for instance, the probe sees the final s * states when Q is in the basal plane, and the final p * orbitals when it is perpendicular [1,2]. Angle resolved experiments allow thus a simple visual interpretation of ELNES in terms of oriented orbitals. At this point, a comment is in place: K-edges are especially simple. For L-edges, the dipole selection rules allow final s and d states. Eq. (8) contains then transitions to s, dz2 ; dxz , and dyz . Eq. (9) will be replaced by
When electrons channel along atomic columns or planes in a crystal, the apparent cross section increases. The usual explanation is that the true atomic cross section remains unchanged, but the incident intensity is higher at the atomic site under channeling conditions. This reasoning is not rigorously true as we shall see. In a crystal, and particularly under channeling conditions, the incident electron is a Bloch wave, i.e., a coherent superposition of plane waves. We would then expect interference terms to occur in Eq. (1). Indeed, for a coherent superposition of two plane waves C1 jk1 i þ C2 jk2 i impinging on an atom, the inelastic scattering amplitude for a particular transition would read C1 f ðQ1 Þ þ C2 f ðQ2 Þ; where f ðQÞ denotes the generalized oscillator strength (GOS). See the scattering geometry Fig. 1. The phases of the interfering waves can be changed by phaseplates which correspond to the
SðQ; EÞ 8Q2 r20 ws ðEf Ei Þ þ Q2 r22 ½wdz2 ðEf Ei Þ þ wdxz ðEf Ei Þ þ wdyz ðEf Ei Þ :
ð10Þ
This is not simply the product ‘‘matrix element’’ ‘‘partial DOS’’, as often stated. It is not even a sum of such products; rather, it is a weighted sum of partial DOSes, with the matrix elements depending on energy loss, see Eq. (6). Moreover, the weighting coefficients of the d-DOSes are different: the dxy vanishes. Only when we integrate over all directions of Q will all the d-DOSes acquire equal weight. A general description of orientation sensitive ELNES surpassing the dipole approximation can be found in [10,11].
Fig. 1. Scattering geometry for two coherent plane waves impinging on an atomic monolayer. The ionization process caused by simultaneous transfer of momenta hQ1 and hQ2 bears the interference effects of the experiment. The phases of the interfering waves can be changed by phase plates which correspond to the diffraction conditions inside a crystal. Cosinusoidal charge distribution in the slab (full line, channeling conditions). Dashed line: antichanneling.
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diffraction conditions inside a real crystal. Channeling along the lattice planes or atomic rows corresponds to a cosinusoidal charge distribution in the lateral plane. By shifting the relative phase by p, the charge distribution becomes sinusoidal. This situation will be referred to as antichanneling in the following. In the two-beam case a situation is meant in which a maximum of electron intensity occurs between atomic planes. This corresponds to a tilt of the incoming electron beam from channeling conditions by twice the Bragg angle. The measured intensity is the amplitude squared Iex ¼ jC1 f ðQ1 Þ þ C2 f ðQ2 Þj2 ¼ jC1 f ðQ1 Þj2 þ jC2 f ðQ2 Þj2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} direct
þ 2R½C1 C2* f ðQ1 Þf ðQ2 Þ * : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð11Þ
mixed
The direct terms correspond to the ionization cross sections as obtained in the usual derivation for two incident plane waves, e.g. [12–14]. The coherent superposition of the two plane waves has created an additional interference term in the ionisation cross section which is not contained in Eq. (1). Note the formal similarity to Youngs double-slit experiment. In an energy loss experiment, one has to sum the scattering amplitude f ðQÞ over all energetically possible final states. Instead of Eq. (11) we write @2s SðQ1 ; Q1 ; EÞ SðQ2 ; Q2 ; EÞ ¼ jC1 j2 þ jC2 j2 4 @E@O Q1 Q42 SðQ1 ; Q2 ; EÞ þ 2R C1 C2* eidðQ1 Q2 Þ : ð12Þ Q21 Q22 We normalized the incident current implicitely to one; therefore, the outgoing intensity equals the cross section. Here we used the mixed dynamic form factor (MDFF), an extension of the DFF X ^ ^ SðQ1 ; Q2 ; EÞ ¼ hijeiQ1 R jf ihf jeiQ2 R jii i;f
dðE þ Ei Ef Þ:
ð13Þ
Eq. (12) is formally equivalent to Eq. (11). The new ‘‘mixed’’ term describes the effect of interference between the coherent plane waves on the
inelastic signal. It replaces the intensity oscillations of a conventional interferometer. The energy spectrum of a fast electron leaving the interferometer in some direction kf will thus contain intensity due to the MDFF, weighted by the complex amplitudes C1 ; C2 of the coherent plane waves, and reveal thereby their interference. The exponential member in Eq. (12) accounts for the position d of the ionized atom by introducing a geometric phase dðQ1 Q2 Þ in the interference term. This means that inelastic cross sections of chemically identical atoms can be different dependent on their location. The apparent increase in cross section under channeling conditions can now be understood easily: the Bloch wave coefficients C1 ; C2 are both positive; the exponential factor in the mixed term causes a cosinusoidal variation of scattered intensity with periodicity 2p=ðQ1 Q2 Þ ¼ 2p=G, and this is exactly the density distribution of the incident probe electron in the lattice under planar channeling conditions. Note, however, that we assumed implicitly that the MDFF is positive. The effect would be inverted when the MDFF were negative. This is in fact the case since in the dipole regime the MDFF is proportional to the scalar product of the two scattering vectors. For large collection angles the expected behavior is retrieved because the integral of the MDFF over all outgoing directions is positive [15]. The value of the integral depends on the collection angle and may even be slightly negative. This fact could explain difficulties in the interpretation of early ELCE experiments, and the ad hoc demand of choosing large Q known as localization enhancement [16]. The situation sketched in Fig. 1 can be approximately realized in a crystal. Bragg diffraction serves as a beam splitter, and the complex amplitudes of the resulting plane waves can be tuned by slightly rocking the beam or changing the depth where ionization takes place, due to the phase of every plane wave which increases linearly with thickness. The difference between Q1 and Q2 is the reciprocal lattice vector g perpendicular to the Bragg diffracting plane. In practice, the situation is more involved because the inelastic signal will also be Bragg diffracted. Moreover, instead of two coherent
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plane waves we have many of them before and after inelastic scattering. The combination of elastic and inelastic interactions leads to the following expression for the differential scattering cross section [17]
2 @ s 4g2 kf X X ¼ 2 TJz ;Jz0 ðtÞAJ AJ*0 @E@O crystal a0 k 0 J 0 J
1 X iuðQJxy QJxy0 Þ Su ðQJxy ; QJxy0 ; EÞ e N u Q2Jxy Q2J 0
ð14Þ
xy
with ðjj 0 Þ
TJz Jz0 ðtÞ ¼ eiðdg
0
0
0
þdgðll Þ Þt=2 ðZðjj Þ þZðll Þ Þt=2
e
cosh dZðt=2Þ sin Dt=2 þ i sinh dZðt=2Þ cos Dt=2 : Dt=2 þ idZt=2 ð15Þ
ðj;j 0 Þ
j
j0
dg ¼ g g are differences of eigenvalues of the Schro¨dinger equation to Bloch waves labelled 0 0 j; j 0 ; Zðl;l Þ ¼ Zl þ Zl are sums of their imaginary 0 parts (describing absorption), D ¼ dgðj;j Þ 0 0 0 dgðl;l Þ ; dZ ¼ Zðj;j Þ Zðl;l Þ . J is a shorthand notation for a running index that covers all Bloch wave indices j; l and all plane wave components g; h j before and after inelastic scattering. AJ ¼ C0* Cgj l Dl0 Dh* are products of Bloch wave coefficients before and after inelastic scattering, u are the positions of the atoms in the elementary cell, Su is the mixed dynamic form factor (MDFF) for the atom at position u; N is the number of atoms in the unit cell, kf ; k0 are the wave numbers of the final (initial) incident plane electron wave. The formal structure of Eq. (14) is easy to understand when we refer back to Eq. (12): AJ correspond to Cg . We retrieve also the same exponential factor and the same MDFFs. The sum over u reduces to one term for a mono-atomic elementary cell. The factor T stems from integration over the thickness of the specimen. It is a thickness- and orientation-dependent term similar to the Pendello¨sung oscillations in dynamical diffraction. Eq. (14) can be evaluated with any program for Bloch wave simulations. As an example, we show the angular intensity distribution of the Al L3 cross section for
Fig. 2. Angular profile of Al L3 intensity for planar channeling and antichanneling conditions. Full line: antichanneling } center of Laue circle (CLC) at ð2 2 0Þ; dashed line: channeling } CLC at ð2 2 0Þ. E0 ¼ 160 kV. Upper lines: intensity traces. Lower lines: interference term.
scattering angles between 0 and 2y220 . The upper lines in Fig. 2 are simulated traces through an energy filtered diffraction pattern under planar (2 2 0) channeling and antichanneling conditions. The dashed line is for channeling – center of Laue circle sitting at (0 0 0), and the full line is for antichanneling – center of Laue circle at (2 2 0). The lower lines are the corresponding interference terms, stemming from the MDFF. In the middle between the incident beam and the Bragg reflection the mixed term is most significant, changing its sign at yBragg 10:5 mrad. Note that in spite of enhanced intensity at the lattice planes (planar channeling!) a reduced L-edge intensity is observed at scattering angles smaller than the Bragg angle. This is because the MDFF is negative in this angular range. The change in contrast seen at 10.5 mrad is the right-hand edge of a defect Kikuchi band subtending the range from 10:5 to 10.5 mrad. (In the standard explanation, the scattered intensity would be proportional to the product of the intensities of the incident and outgoing (reciprocal) Bloch waves at the ionized site [18] and thus generate an excess Kikuchi band.) The mixed term becomes smaller under antichanneling conditions (electrons channel between the (2 2 0) lattice planes, lower full line) but we do not observe contrast reversal as would have been expected from the simple two-beam explanation of Eq. (12). This is because inclusion of 5 beams as
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well as absorption in the simulation complicates the simple 2-beam picture. Specimen thickness plays also an important role in the interpretation of energy filtered diffraction patterns or ELCE experiments. It was shown that the intensity enhancement expected for an infinitely thin crystal under channeling conditions is inverted for a thickness of less than one extinction length [9] when the factor T, Eq. (15) is properly taken into account. It can safely be assumed that numbers of unpublished ELCE experiments are eagerly awaiting a sound interpretation.
4. Interferometric EELS: selection of orbitals in rutile Rutile (TiO2 )is an interesting case for testing the local anisotropy: although the structure is quite isotropic and no orientation effects are visible in kinematical conditions, the O atoms have a point group of low symmetry (2 mm). It thus appears possible to select orbitals of different types, that correspond to different energies and so will be visible as a change in the ELNES of the spectra. At the O K-edge, we probe O p states which hybridize with Ti d states. The projection on the (1 1 1) plane somewhat hides the tetragonal structure of the indicated unit cells (thin lines in Fig. 3). Note that there are two types of oxygen positions in this orientation, those of the unit cell’s basal plane which are aligned in the ð 1 1 0Þ planes, and positions at half height of the unit cell, forming a zig-zag line in this projection. Those two positions are crystallographically equivalent but as they are rotated differently with respect to the crystal axes, the orbitals are also differently oriented. The py and pz shown in Fig. 3 are taken in the local coordinate system of the atom. The two positions are therefore not equivalent with respect to Q1 . The py orbitals of O hybridize with the Ti t2g states to build bonding and antibonding p and p * states. The px and pz orbitals of O hybridize with the Ti eg to build s and s * states [19,20]. Both p * and s * are unoccupied. Transitions to these states correspond to peaks A at 4 eV and B at
Fig. 3. Rutile lattice in 111 zone axis projection. The smaller spheres are the oxygen atoms. Two different oxygen antibonding p orbitals are probed. Under channeling conditions they are local py and form p * bonds with Ti t2g . Under antichanneling conditions they are local pz . Together with px they form s * bonds with the dx2 y2 Ti orbital (taken in the octahedral coordinate system). The six-fold coordination of Ti is also indicated as a shadowed octahedron.
Fig. 4. The px ; py and pz projected density of unoccupied states at the O site in rutile shows the energy difference between py on the one hand and px and pz on the other hand.
7 eV in Fig. 5, with the s * at higher energies as shown in Fig. 4. From Fig. 3 one would intuitively expect the py to be excited under channeling conditions and the pz under antichanneling conditions and thus peak B of the O K-edge to be higher under antichanneling conditions. However, the MDFF is negative under the chosen scattering geometry and thus the effect will exactly be the reverse.
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For experiments, a wedge-shaped border of a cleaved rutile ðTiO2 ) needle was used. In a Philips CM20 Transmission Electron Microscope operated at 160 keV, it was oriented with the ð 1 1 0Þ lattice planes almost parallel to the incident electron beam, approximately 4:58 off the ½1 1 1 crystallographic orientation. At this tilt, the nonsystematic reflections contribute 5% to the direct term in the edge signal, and nothing to the interference term since the conditions are almost kinematic. This is on the level of the expected noise, and differences from switching between channeling and antichanneling should thus remain visible. EELS spectra were taken in diffraction mode using a Gatan 666 PEELS, operated at a dispersion of 0:24 eV=channel. The energy resolution limited by the energy spread of the primary beam was about 1.8 eV. The spectrometer entrance aperture was placed midway between the (0 0 0) and ð 1 1 0Þ Bragg spots. Apart from a small component parallel to the incident wave vector which is due to the energy loss, the scattering vectors Q1 and Q2 point in ð 1 1 0Þ direction in reciprocal space, i.e., perpendicular to the ð 1 1 0Þ lattice planes. The vector Q1 is indicated in Fig. 3. The precise orientation of the crystal was verified using the Kikuchi lines visible in the thicker parts of the wedge. The probe was fully condensed to 40 nm diameter. The illumination and collection half angles were 3 and 1.3 mrad, respectively. The specimen thickness was measured from the (1 1 0) thickness fringes on the wedge. A thin area (t50:5 mean free paths for low-loss scattering) was selected in order to avoid strong plasmon contributions and multiple scattering corrections. All spectra were background subtracted with the same pre-edge window; thus, possible errors induced by the variations in the high-energy tail of the Ti edge should cancel in the difference of the processed spectra. Radiation damage may occur in rutile. It was tested beforehand that during 10 s of irradiation no changes in the Ti to O atomic ratio could be observed. Therefore, an integration time of 5 s was chosen for each spectrum to be on the safe side. This is the reason for the rather noisy spectra, (Fig. 5). It shows the O K ELNES taken at
349
Fig. 5. EELS spectra of the oxygen K-edge in rutile, with peaks A at 4 eV and B at 7 eV, under non-interference conditions (diamonds), and under strong Bragg diffraction interference conditions at a thickness of 25 nm, exciting once the (2 2 0) Bragg spot (thin solid line) and then symmetrically both ð1 1 0Þ and (1 1 0) spots (thin dashed line).
channeling and antichanneling conditions. A significant difference in the oxygen K-edge is seen when normalizing the spectra to the same height in the first peak. When integrating the intensity over 2 eV centered about the two maxima at 4 and 7 eV, the noise is largely reduced, and the predicted variation in peak height ratio is clearly seen. The mixed terms for the two oxygen sites have different sign because of their geometric phase difference g d ¼ p – see Fig. 1. Any action that shifts the relative phases of the incident waves will then lead to complementary intensity variations of the cross section at the two sites, the net effect being a variation of the peak heights B=A. Under planar channeling (thin dashed line) the incident electron has crests at the atoms in the basal plane and hence the dpp * orbitals are probed (see Fig. 3). Tilting the beam by twice the Bragg angle of the (1 1 0) plane to ð1 1 0Þ antichanneling (thin full line) the crests are at the zig-zag line of oxygen atoms and the dps * orbitals are probed. The thick lines in the figure are simulations based on evaluation of Eq. (14), taking a weighted average over the thickness variation of 5 nm within the illuminated area. All spectra, measured and simulated, are normalized to peak A. It should be borne in mind that the specimen thickness plays an important role. For thicker
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Fig. 6. Same as in Fig. 5 at a thickness of 40 nm. Note the inversion of the peak height ratios relative to the thinner specimen.
Fig. 7. Diffraction mode spectra of Si L23 -edge at Q ¼ ð2 0 0Þ after removal of the thermal diffuse background, plural scattering deconvolution and background subtraction. Upper spectrum for kinematic conditions, lower one for (4 0 0) planar channeling.
specimens, the peak ratio shows the opposite tendency, see Fig. 6. Details can be found in [9]. By its ability to probe the local anisotropy at the ionized site, angle resolved interferometric EELS aims at the point group symmetry of the target atom. Knowledge of the specimen thickness as well as the precise scattering geometry is important for a correct interpretation.
geometry described above we have Q1 ¼ G=2 and Q2 ¼ G=2. Restricting ourselves to the three most important transition probabilities (l ¼ 0; 1; 2), we find for the DFF DFF ¼ SðQ1 ; Q1 ; EÞ8
3 X
Pl ð1Þfl ðQ1 ; Q1 ; EÞ ð17Þ
l¼0
and for the MDFF 5. Interferometric EELS: non-dipole transitions Non-dipole transitions can be seen for large scattering angles [21]. The disadvantage is often the low intensity according to the Lorentzian angular profile of inelastic scattering. Interferometric EELS circumvents this problem as we shall see. The formula for the direct and mixed terms contain the DFF and the MDFF. Now that the transition with l ¼ 1 dominates (dipole transition), the cross section should by and large resemble that of the dipole approximation, irrespective of channeling. Evaluation of Eq. (13) for atomic orbitals yields [10]
1 X Q1 Q2 SðQ1 ; Q2 ; EÞ ¼ Pl fl ðQ1 ; Q2 ; EÞ; Q1 Q2 l¼0 ð16Þ where fl is the probability for a l-pole transition. Pl is the Legendre polynomial. In the scattering
MDFF ¼ SðQ1 ; Q2 ; EÞ8
3 X
Pl ð1Þfl ðQ1 ; Q2 ; EÞ:
l¼0
ð18Þ
A closer inspection of the two equations shows that in the MDFF the odd transitions – l odd – contribute with positive sign, and the even transitions – l even – with negative sign whereas the DFF contains only positive contributions. From Eqs. (18) and (17) the relative strength of dipoleforbidden transitions can be calculated immediately: f0 þ f2 1 þ SðQ1 ; Q2 ; EÞ=SðQ1 ; Q1 ; EÞ : 8 1 SðQ1 ; Q2 ; EÞ=SðQ1 ; Q1 ; EÞ f1
ð19Þ
This enables extraction of non-dipole transitions from a measurement of the cross section at yBragg corresponding to Q1 ¼ G=2. Under channeling conditions, the ELNES is a linear combination of DFF and MDFF; under kinematic conditions only the DFF contributes. So, both the DFF and
P. Schattschneider et al. / Ultramicroscopy 86 (2001) 343–353
the MDFF as a function of energy could be obtained from a linear combination of these two experimental ELNES spectra when the coefficients are known from a simulation of the profiles calculated via Eq. (14), such as in Fig. 2.
Fig. 8. Si L-edge at yBragg ¼ 10:5 mrad. Also shown is the ratio of Si L-edges for kinematic to channeling conditions.
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In practice, it is easier to impose a normalization condition. For the Si L23 edge, this condition derives from the fact that at 120 eV the non-dipole terms are small (they contribute 7%Þ. Consequently, the cross sections at this energy are almost identical for channeling and kinematical spectra. This can be used for normalizing the spectra and extracting the ratio MDFF=DFF. Before doing so it is necessary to remove the contributions from coupling of inelastic with thermal diffuse background scattering and the plural scattering contributions. The method described by Batson and Silcox was applied [22]. For background subtraction the same pre-edge windows were used. Fig. 7 shows the edge region after background subtraction and for both channeling and kinematic conditions. The former is ’ 30% lower in intensity than the latter. As in the rutile case, this is because the MDFF is negative – enhanced intensity at the 400 lattice planes reduces the cross section. The fine structure is by and large the same in both spectra.
Fig. 9. Extracted relative non-dipole transition probabilities, compared with the atomic simulations and a solid-state model. Next to the edge the error bars of the experimental results are high according to data processing.
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To judge the differences better we show the ratio of the two spectra in Fig. 8. Any deviation from unity signifies presence of non-dipole terms or errors in the background subtraction. Non-dipole transitions occur in the first few eV above threshold. In fact, there is little variation for losses higher than 10 eV beyond the edge – proving that deconvolution and background subtraction were successful. Fig. 9 shows the relative non-dipole transition probabilities ðf0 þ f2 Þ=f1 obtained from the two experimental ELNES Fig. 7 via Eq. (19). The experimental values are compared with an atomic model and a simulation based on the TELNES package of WIEN97 [11,23]. From the atomic model we see that the MDFF is dominated by monopole transitions. Quadrupole transitions f2 are negligible [10]. Furthermore, since dipole transitions to final s states are 510%; what we see in Fig. 9 is essentially the ratio of p!p to p!d transitions. The tendency to increased monopole transitions in the vicinity of the edge can be seen in both the atomic and the solid-state model. The leftmost data point is not reliable because of strong noise amplification near threshold. Although the error bars for a single data point at 2 eV from the edge onset are still high, the local maximum of final p states at 3 eV and the minimum at 6–7 eV are significant experimental features. The 3 eV maximum is well reproduced; the origin of the disagreement at 5–6 eV is not known. A possible explanation is that the core hole significantly changes the final DOS in Si. This could quench more d states into the center of the band at 5 eV than the simulation predicts, causing a reduction of the ratio of p to d states.
6. Conclusion The two experiments show that subtle variations in the fine structure of ionization edges are related to channeling of the probe electron. The concept of interferometric EELS facilitates the understanding of such surprising effects as the occurrence of defect Kikuchi bands for particular
scattering geometries, or contrast inversion of the inelastic signal as a function of specimen thickness. The symmetry properties of the mixed dynamic form factor can be exploited to differentiate between dipole and non-dipole transitions. The dependence of the fine structure on the channeling conditions in combination with the choice of a scattering vector can be used to obtain directional information on particular unoccupied states, even for crystals with a single type of atoms and bonding. Acknowledgements This work was supported by the Austrian Science Fund under Project No. P14038-PHY. The financial support of parts of this work by the Austrian O¨AD (Amade´e III.5) and the French CNRS (PICS No. 913) is gratefully acknowledged. References [1] R.D. Leapman, P.L. Fejes, J. Silcox, Phys. Rev. B 28 (1983) 2361. [2] C. Souche, B. Jouffrey, G. Hug, M. Nelhiebel, Micron 29 (1998) 419. [3] J. Gjonnes, Acta Crystallogr. 20 (1966) 240. [4] P. Schattschneider, B. Jouffrey, M. Nelhiebel, Phys. Rev. B 54 (1996) 3861. [5] M. Schenner, M. Nelhiebel, P. Schattschneider, Ultramicroscopy 65 (1996) 95. [6] J.C.H. Spence, J. Taft, J. Microsc. 130 (1983) 147. [7] J. Taft, O.L. Krivanek, Nucl. Instr. and Meth. 194 (1982) 153. [8] J. Taft, O.L. Krivanek, Phys. Rev. Lett. 48 (1982) 560. [9] M. Nelhiebel, P. Schattschneider, B. Jouffrey, Phys. Rev. Lett. 85 (2000) 1847. [10] M. Nelhiebel, N. Luchier, P. Schorsch, P. Schattschneider, B. Jouffrey, Philos. Mag. B 79 (1999). [11] C. Souche, P.-H. Louf, P. Blaha, M. Nelhiebel, J. Luitz, P. Schattschneider, K. Schwarz, B. Jouffrey, Ultramicroscopy 83 (2000) 9. [12] H.A. Bethe, Ann. Phys. Lpz. 5 (1930) 325. [13] R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 2nd Edition, Plenum Press, New York, 1996. [14] P. Schattschneider, Fundamentals of Inelastic Electron Scattering, Springer, Wien, 1986. [15] W. Schu¨lke, S. Mourikis, Acta Crystallogr. A 42 (1986) 86. [16] O. Krivanek, M. Disko, J. Taft, J. Spence, Ultramicroscopy 9 (1982) 249.
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