2.2 Theory of electron diffraction
2.2.2 Kinematic approach
When the electron plane waveψ(r)interacts with the crystal lattice, the amplitude and the phase of the incoming electron wave is changed. Therefore, the object wave function carries information about the sample characteristics. In kinematic approximation it is assumed that the electrons are scattered only once under conservation of their energy which implies that the intensity of the diffracted beams is negligible with respect to the undiffracted beam. For the description of the crystalline specimen each atom is assumed as scattering center for electrons which emits a secondary wavelet. The wave formed by the periodic arrangement of the atoms can be then expressed as a sum of scattered wavelets coming from individual atoms.
First it is considered the elastic scattering from a single isolated atom and extend afterwards to many atoms. For a single atom, the elastic scattering is caused by the Coulomb potential of the scattering atom. The phenomena can be described using the Schr ¨odinger equation in the stationary case:
∇2ψ(r) + 8π2me
h2 (E+φ(r))ψ(r) = 0, (2.1) whereeandmdenote the electron charge and the mass of the incoming electron,φ(r)is the Coulomb potential of the scattering atom andhis Planck‘s constant. In an electron microscope, operating at 200 keV, the electrons travel at over half the speed of light.
Therefore, relativistic corrections for the acceleration potential E, the mass m of the incident electron and the de Broglie wavelength λ were taking into consideration in Eq. (2.1):
E =U(1 + eU 2m0c2), m=m0
s
(1 + h2
m20c2λ2) (2.2)
λ = h
q
2m0eU(1 + 2meU0c2)
with U being the accelerating voltage, c the speed of light in vacuum and m0 the rest mass of the incident electron.
Eq. (2.1) is an inhomogeneous differential equation with the inhomogeneity : g(r) =−8π2me
h2 φ(r)ψ(r). (2.3)
Using the abbreviation: k2 = 2meh2 E and defining a linear differential operator we can obtain a Green’s function. A solution of the inhomogeneous differential equation (2.1) using the Green’s function is given by:
ψ(r) = ψ0(r) + 2πme h2
Z
Ω
exp(−2πik|r−r0|)
|r−r0| φ(r0)ψ(r0)d3r0 (2.4) called also the integral Schr ¨odinger equation. In this equation, the integral expands over the scattering volumeΩand randr0 are the coordinates of the observation point and of the scattering point, respectively (see Fig. 2.3).
In the absence of a Coulomb potential (φ(r) = 0), the first term of Eq. (2.4) is the solution of the homogeneous differential equation:
ψ0(r) = exp(−2πik0r) (2.5)
and is the incident electron plane wave, with the wave vectork0.
Furthermore, taking into account that the point of observation is at a distance much larger than the size of the scattering center (isolated atom) (Fig. 2.3) and, to first order, the amplitude of the secondary spherical wave can be considered much weaker than the
2.2 Theory of electron diffraction
Figure 2.3: Sketch showing an incident wave with wave vectork0, scattered by a scat-tering center at r’.
amplitude of the transmitted wave, it is possible to expressψ(r)in first Born approxima-tion. Using the first Born approximation for Eq. (2.4) and considering in the following ψ(r)for|r0 |<| r|so that therandr−r0are approximately parallel (Fig. 2.3), we obtain theasymptotic solutionof the inhomogeneous equation (2.1):
ψ(r,q) = exp(−2πik0r) + exp(−2πikr)
r f(q), (2.6)
where the second term shows the spherical wave function associated to the scattered electron with the amplitudef(q)called atomic scattering factor for electrons:
f(q) =σ Z
Ω
exp(−2πiqr0)Φ(r0)d3r0, (2.7) withσbeing the interaction constant, given by:
σ = 2πme
h2 , (2.8)
and the scattering wave vectorqdefined by:
q=k0−k (2.9)
which is related with the scattering angle by: q= λ2 sinθ2 (see Fig. 2.3).
The atomic scattering factor for electronsf(q)is a very important parameter due to the fact that it is a measure of the amplitude of the electron wave scattered by an isolated
atom and| f(q) |2 is proportional to the scattered intensity. Moreover, from Eq. (2.7) it becomes clear that the f(q) is proportional to the Fourier transform of the Coulomb potentialφ(r). An analogous expression can be deduced for the X-ray scattering factor fx(q)as well. Since the values forfx(q)are available in the literature for most elements, using Poisson’s equation, thefx(q)can be converted tof(q)by the Mott formula:
fe(s) = me2 8πε0h2
·Z −fx(s) s2
¸
(2.10) withε0 the permittivity of vacuum, Z the atomic number, ands= q2 = sin(θλB) (θ = 2θB).
The reference [52] provides f(q) in a parametric form. The relativistically corrected atomic scattering amplitudes are then given by:
f(q) = s
1 + h2 m0c2λ2
X4
n=1
anexp µ
−bnq2 4
¶
. (2.11)
Eq. (2.11) was used for plotting the atomic scattering factors of the atoms present in our crystal structure: Cd, Zn, S and Se. The atomic scattering factors are shown in Fig. 2.4.
It can be clearly seen thatf(q)depends on the type of scattering atom (increasing with atomic number Z). Considering now a unit cell (Chapter 1) of a crystal with N atoms
Figure 2.4: Atomic scattering amplitudes for Cd, Zn,S and Se plotted versus scattering parameterqin the case of an accelerating voltage of 200 kV
characterised by atomic scattering factors fk(q) at the positions ri, the structure factor FS(q)can be obtained by summing up the atomic scattering factors of all N atoms:
FS(q) = XN
i=1
f(i)(q) exp (−2πiq·ri), (2.12) Eq. (2.12) gives the dependence ofFS(q)on the positions and type of atoms inside the unit cell.(The discussion about the structure factors will be continued later, in the section
2.2 Theory of electron diffraction dedicated to the chemical sensitive reflections (2.2.2.1)). Continuing the argumentation, the crystal is a periodic assemble of unit cells, therefore the total scattered wave is ex-pressed as:
ψ(r,q) = exp(−2πikr)
r ·FS(q)G(q) (2.13)
whereG(q)is called the lattice amplitude:
G(q) = X
m,n,o
exp(−2πiq·rc) m, n, o∈N (2.14) and depends on the shape of the crystal. Here, therc =ma1+na2+oa3is the translation vector of the crystal lattice. Then, the interaction between scattered waves emanating from this periodic assemble will lead to the effect of electron diffraction.
In the kinematical approximation, the electron wave diffracted by a real specimen is calculated considering the spatial dimension of the sample in an electron beam di-rection, t,to be very small (<20 nm). However, from each point of the sample surface, elementary waves of the form derived in Eq. (2.13), have different phase at an obser-vation point. It is well known from wave theory that waves can interfere if they are coherent. Summing up the contributions from all points of the exit surface of the thin foil, which interfere at the observation point situated at the distance D from the surface, one obtains, using Fresnel’s zone reconstruction method [49], the total secondary wave corresponding to the diffraction vectorqas:
ψs(q) =iλtFs(q)G(q)
VE exp(−2πik0D), (2.15)
whereVE is the volume of the unit cell andDa vector, normal to the specimen surface.
This expression states that the amplitude of the diffracted beam increases linearly with the specimen thickness. Looking at Eq. (2.14), in the case of a infinite crystal, can be seen that for :
q·rc=n, n∈Z (2.16)
a sum of peak functions corresponding to each diffracted beam is obtained. Comparing now this equation with definition of the reciprocal lattice (see Chapter 1) it becomes clear that the scattering vector q is a reciprocal lattice vector ghkl. In this conditions, Eq. (2.16) can be written as:
ghkl·rc =n (2.17)
known as Laue condition for constructive interference (diffraction) and used to gene-rate a construction first employed by Ewald. Figure 2.5 shows the Ewald construction.
A vector k0= MO with length | k0 |= 1λ is drawn ending at the origin O of the reci-procal lattice. The starting point M of k0 is taken as the center of a sphere of radius 1λ. Diffraction will be observed only if thisEwald sphere intersects one or more pointsgof the reciprocal lattice (for example N). The vectork =MN corresponds to the scattered wave and k0 −k = g is the vector that connects the end points of k0 and k. From Fig. 2.5 it can be deduced that| ghkl |= 2 sinλθB and combined with| ghkl |= d1
hkl (Eq. 1.1) the Eq. (2.17) becomes the well knownBragg conditionfor constructive interference:
λ= 2dsinθB (2.18)
In electron diffraction the radius of the Ewald sphere, 1λ = 600nm−1at 200 keV, is much larger then the distances between the reciprocal lattice points e.g. a1 = 1.76 nm−1 for ZnSe, which is not the case for X-ray diffraction [49]. Therefore, the sphere is flatter and
Figure 2.5: Ewald sphere of radius λ1 in a reciprocal lattice. A Bragg reflection is excited if the sphere intersects a reciprocal lattice point, e.g. N.
it intersects, if the electron beam is oriented along a low-index direction, many more points of the crystal. If the incident beam is parallel to a zone axis (direction common to a set of planes), the diffraction pattern contains Bragg reflections near to the primary beam from the zero-order Laue zone. At larger Bragg angles, circles of reflections occur when the Ewald sphere cuts the first and higher Laue zones.
Furthermore, a diffracted beam can occur even when the Bragg condition is not e-xactly satisfied. The actual intensity of the diffracted beam will depend on the distance of the reciprocal lattice point from the Ewald sphere. This distance is measured by a vectors(excitation error), in reciprocal space such that:
q=g+s (2.19)
The length of the vector s can be computed taking into account the equation of a sphere, in this case, the equation of the Ewald sphere:
(g+s)(2k+ g+s) = 0 (2.20)
If the angle between the k + g and the specimen normal is α, then we have the excitation error corresponding to the vectorgwritten as:
sg ≡ −g(2k+g)
2|k+g|cosα (2.21)
2.2 Theory of electron diffraction 2.2.2.1 Chemical sensitivity
The results obtained in the preceding paragraph will now be used for the definition of chemical sensitivity in kinematic approach. Therefore the discussion will be concentrate on structure factors. The computation of the structure factors FS(q) in the case of a binary material (AB) with zincblende configuration requires the knowledge about the position of the atoms in the non-primitive unit cell. In this case the Eq. (2.12) becomes:
Fs(ghkl) = 4{fA(ghkl, x) +fB(ghkl, x) exp(2πi(h+k+l)/4), (2.22) wherefA(ghkl)andfB(ghkl, x), are the atomic structure amplitudes for metal and respec-tively non-metal components. For the zincblende configurations the following selection rules are obtained for calculating the structure factors and furthermore the intensity of the diffracted beam:
|Fs |2=
0 : (h, k, l)mixed 16(fA2 +fB2) : (h, k, l)all odd 16(fA+fB)2 : (h, k, l)all even and
h+k+l = 4n
16(fA−fB)2 : (h, k, l)all even and h+k+l = 4(n+ 1/2)
(2.23)
It can be seen that the structure factors for zincblende materials exhibit sets of (hkl) for which Fs is 0. This explains why some of the diffracted beams that occur for the unit cell with only one atom disappear in the zincblende structure. For ternary compounds
Figure 2.6:Structure factors of the (002) (lower curve) and (004) (upper curve) beams in CdxZn1−xSe and ZnSxSe1−x, plotted versus the Cd and respectively S concentration x.
The curves were computed assuming an acceleration voltage of 200 kV.
AxB1−xC we assume a random distribution of e.g. two different sorts of metal or non-metal atoms (e.g. Cd and Zn in CdxZn1−xSe or Se and S in ZnSx Se1−x in the metal (non-metal) sublattice. In this case the structure factor can be obtained, in good appro-ximation, by a linear interpolation of structure factors of the binary materials involved.
Figure 2.6 shows the structure factor of the (002) and (004) beams, used here for the TEM analysis of ternary ZnSxSe1−x and respectively CdxZn1−xSe. It is clearly seen that the structure amplitude of the (002) beam is smaller then that for (004) beam. In order
Figure 2.7: The normalised structure factors ∂x∂ (FFs(ghkl,x)
s(ghkl,0))which determine the chemical sensitivity as indicated in Eq. (2.24) for (002) and (004) reflections.
to have a measure how the structure amplitude changes with the composition of the material, one can define the chemical sensitivityS(ghkl)through:
S(ghkl) := ∂
∂x
µFs(ghkl, x) Fs(ghkl,0)
¶
= ∂
∂xFsN(ghkl, x), (2.24) withFsN(~ghkl, x)the normalised structure amplitude. In Fig. 2.7 the normalised structure amplitude of the ternary CdxZn1−xSe and ZnxS1−xSe for the (002) and (004) are shown.
The slope of the (002) beam is larger than for (004), which reveals the higher chemical sensitivity of (002) in zincblende structure.
In conclusion, kinematic theory leads to the Bragg condition and a description of the influence of structure of a unit cell and of the external size of a crystal on the diffracted amplitude in terms of structure and lattice amplitude. The observed diffraction pa-ttern is equivalent to the points of intersection of the Ewald sphere of a radius λ1(thin specimen) with the reciprocal lattice. Moreover, the diffracted beams in an electron diffraction experiment are solutions of the Schr¨ı¿12inger equation.