Dynamical theory of electron diffraction: Bloch-wave approach

Figure 2.6 shows the structure factor of the (002) and (004) beams, used here for the TEM analysis of ternary ZnS_xSe_1−x and respectively Cd_xZn_1−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^∂ (^F_F^s(ghkl,x)

s(g_hkl,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(g_hkl)through:

S(g_hkl) := ∂

∂x

µF_s(g_hkl, x) F_s(g_hkl,0)

¶

= ∂

∂xF_s^N(g_hkl, x), (2.24) withF_s^N(~ghkl, x)the normalised structure amplitude. In Fig. 2.7 the normalised structure amplitude of the ternary Cd_xZn_1−xSe and Zn_xS_1−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.

2.2 Theory of electron diffraction time independent wave equation of a waveψwith wavenumberk(known as Helmholtz equation):

∆ψ+ 4π²k²ψ = 0, (2.25)

where

eU^∗ = h²k²

2m₀ (2.26)

establishes the relationship between the wavevector kand the relativistically corrected acceleration potential of the electrons, before reaching the specimen. In the crystal, elec-trons are also accelerated by the atoms. Then, taking into account that the crystal poten-tial V(r) is usually very small (∼ 10V) in comparison with the acceleration potential of incoming electronsU or considering the mean inner potentialV₀, U^∗ can be expressed as:

U^∗ =U^∗_acc+γV =U^∗_acc+γ(V(r)−V₀) +γV₀, (2.27) withV0the mean inner crystal potential. Introducing the Eq. (2.27) and (2.26) in Eq. (2.25), the general equation for a relativistic electron scattered by the crystal is obtained:

∆ψ+ 4π²k²₀ψ =−4π²U(r)ψ, (2.28) called the governingequation. This establishes the relationship between the waveψ(r) inside the crystal and the crystal potential, where

k₀ =

r2m₀e(U^∗_acc+γV₀)

h² (2.29)

is the wave vector of the incident electron beam within the crystal, and U(r) = 2m0γe

h² (V(r)−V₀) (2.30)

is the new definition (used from now on) for the crystal potential. Then, the general solution for ψ within the crystal, which reflects the lattice periodicity, is written as a product of a wave witharbitrary wave vectork, and a functionC(r)with the periodicity of the lattice. This type of function is calledBloch wave:

ψ(r) =C(r) exp(2πik·r) = X

g

Cgexp(2πi(k+g)r) (2.31) where a Fourier expansion for C(r)with Cg being the Bloch wave coefficients was in-troduced. Since the potentialU(r) is a superposition of all atomic potentials, it has the same periodicity as the lattice and can, therefore, be expanded in a Fourier series:

U(r) = X

h

Uhexp(2πih·r) (2.32)

Taking now into account that the coefficientsCg, for a perfect crystal, are dependent on the wave vector k and independent of position, the substitution ofψ(r) and U(r)into equation (2.28) leads to:

X

g

n

[k²₀ −(k+g)²]C_g+X

h6=g

U_g−hC_ho

exp(2πi(k+g)r) = 0 (2.33)

which must be valid for any positionrin the crystal. This equation can be written also as:

[k₀²−(k+g)²]Cg+X

h6=g

Ug−hCh = 0 (2.34)

This set of equations, one for eachg, relate the vectorkof the Bloch wave to the energy of the incident electron (throughk₀), therefore are calleddispersion relations.

In order to solve the equation (2.34) for the multibeam case, it has to be observed that the scattering problem can be separated into two contributions: one is coming from the first term of this equation, describing the geometry of the scattering and can be repre-sented as a diagonal matrix, and the second contribution is coming from the interaction with the crystal potential and gives rise to off-diagonal matrix elements. Furthermore, for convenience, the transmitted beam is selected as the first in the entry in the column vector of Bloch wave coefficients. Then, the matrix form of this equation will be:

n

[k²₀−(k+h)²]δ_hg+X

h

U_g−h o

C_h = 0 (2.35)

or

M ·C= 0 (2.36)

and the non-trivial solutions will be obtained when the determinant of the matrixM is equal to zero. The main contribution to this determinant is coming from the product of the elements from the main diagonal, and this term is proportional to k^2N, where N (up to 500 is enough for numerical calculations) is the number of beams taken into account. The characteristic equation obtained is then a polynomial equation of 2N order with 2N roots. Therefore, in the N-beam case, the total number of allowed wave vectors in the crystal is 2N. To distinguish each wave vector a superscript j is assigned, k^(j) (j = 1..2N). Hence the general wave solution inside the crystal is a linear combination of 2N Bloch waves:

ψ(r) = X

j

α^(j)X

g

C_g^(j)exp(2πi(k^(j)+g)r) (2.37) each with its own amplitudeα_j calledexcitation amplitudeof the Bloch wave. This gene-ral solution contains many unknown parameters as: α^(j),Cg^(j) andk^(j). In consequence additional information is needed in order to determine them.

Since the total energy of the incident electron is constant, the 2N Bloch wave must have the same total energy. The Bloch waves have different kinetic and potential energy, as the correspondingk^(j)wave vectors are different from each other. Both the incident beam and the wave inside the crystal have to be solutions of Schr ¨odinger’s equation.

This means that their wave functions and first order derivatives must be continuous across the entrance plane of the crystal (see Fig. 2.8). This continuity condition implies that the tangential component of the wavevector has to be conserved, therefore only the normal component can change (Fig. 2.8 b). Defining the unit vectornas the surface normal vector, the vectorsk^(j) can be written as:

k^(j) =k₀+γ^(j)n (2.38)

2.2 Theory of electron diffraction

Figure 2.8: a) Wave function not continuousk_k 6=k_0k b) Continuity conditionk_k =k_0k

with γ^(j) being the coefficients of the normal components. The Eq. (2.38) can be rewri-tten as:

k²₀ −(k^(j)+g)² = 2k0sg−2n(k0 +g)γ^(j)−(γ^(j))² (2.39) where the expression for the excitation error deduced in Eq. (2.21) was used. In the following the coefficients γ^(j) are assumed to be small compared withk₀, therefore the quadratic terms are ignored. In this way only N of the wave vectorsk^(j) remain. This is calledhigh energy approximationand for the acceleration voltages used in typical TEM experiments this is a good approximation.

This is illustrated in Fig. 2.9, which shows the Ewald sphere construction, with the reciprocal lattice point inside the Ewald sphere. When the sample surface is normal

Figure 2.9: Ewald sphere showing the positive and negative excitation errors for a re-ciprocal lattice point located inside the sphere.

to the incident beam with the wave vector k₀, the equation (2.21) has two solutions for the excitation vector s_g, where one has a large magnitude, corresponding to the beam travelling in opposite directions. Since the excitation error corresponding to the reflected beam increases rapidly with increasing acceleration voltage, the probability of Bragg scattering decreases with decreasing wavelength. Therefore, only electrons

scattered in the forward direction are considered. This is the idea of the high energy approximation.

Using the following notations n·g = gn and n·k0 = kn for the case of k0 nearly antiparallel ton(g_n≪k_n) the equation (2.34) becomes :

2k0sgC_g^(j)+X

h6=g

Ug−hC_h^(j)= 2knγ^(j)C_g^(j), (2.40) aneigenvalueequation. This can be rewritten in a matrix form:

A·C^(j) = 2k_nγ^(j)C^(j). (2.41) For the N beam case, the matrix Ais of the typeN ×N and has N eigenvalues and N eigenvectors. A solution of this equation determines all vectorsk^(j)and all Bloch wave coefficientsCg^(j). The diagonal of the matrixA contains information about the orienta-tion of the crystal, through the excitaorienta-tion errorss_g, and the off diagonal elements con-tain information about the interaction between the beams. At this point it is important to note some of the properties of the eingenvalues and eigenvectors. For a centrosy-metric crystal, e.g.,U_g = U_−gand therefore the eigenvalues must be real. It can be also shown that the eigenvectorsC^(j)are complex and form a unitary matrix. Applying the appropriate boundary conditions at the crystal entrance plane, the Bloch wave excita-tion amplitudesα^(j)can be determined. The starting point is the characteristic equation (2.37) written in a matrix form for the high energy approximation case:

ψ(r) =X

g

n X

j

α^(j)C_g^(j)exp(2πiγ^(j)z) o

exp(2πi(k₀+g)r) = X

g

ψ_g(z) exp(2πi(k₀+g)r) (2.42) where eachψ_gcorresponds to a diffracted beam. Furthermore,ψ_gcan be converted into a matrix formulation using:

ψ_i(z) =X

j

C_ijX

l

ε_jl(z)α_l=C·

ε

·

α

(2.43) where

C_g^j =C_ij (2.44)

are forming the matrixCof Bloch wave coefficients and the matrixεhas elements εjl(z) = δjlexp(2πiγ^(l)z) (2.45) Considering the entrance specimen plane atz = 0, ε_jl(0)is reduced to the unity matrix I and the wave function only contains the undiffracted beam:

ψi(0) =

(1 for (undiffracted beam)

0 else (diffracted beams) (2.46)

In this conditions,

ψ(0) =C·α = 1 (2.47)

and

α=C⁻¹·ψ(0) (2.48)

2.3 Image formation in an electron microscope

Im Dokument Transmission electron microscopy investigationsof the CdSe based quantum structures (Seite 46-51)