Depth profiles of the interfacial strains of Si0.7Ge0.3/Si using three-beam Bragg-surface diffraction

Interfacial strains are important factors affecting the structural and physical properties of crystalline multilayers and heterojunctions, and the performance of the devices made of multilayers used, for example, in nanowires, optoelectronic components, and many other applications. Currently existing strain measurement methods, such as grazing incidence X-ray diffraction (GIXD), cross-section transmission electron microscope, TEM, and coherent diffractive imaging, CDI, are limited by either the nanometer spatial resolution, penetration depth, or a destructive nature. Here we report a new non-destructive method of direct mapping the interfacial strain of [001] Si0.7Ge0.3/Si along the depth up to ~287 nm below the interface using three-beam Bragg-surface X-ray diffraction (BSD), where one wide-angle symmetric Bragg reflection and a surface reflection are simultaneously involved. Our method combining with the dynamical diffraction theory simulation can uniquely provide unit cell dimensions layer by layer, and is applicable to thicker samples.

⃑ ⃑⃑ = 1 ( ℕ 1 cos 2 cos + ℕ 2 sin 2 cos 2 sin ℕ 1 sin 2 + ℕ 2 cos 2 cos ) − ⃑⃑ The three symbols, θ B , and ̅ are the Bragg angle, miscut angle with respect to atomic layers and the refractive index. The surface diffraction can be aligned by adjusting the theta angle to the value, θ B , and by rotating the azimuth, ϕ, around the reciprocal lattice vector, g ⃑⃑ (see, Figure 1b). As the results, the reciprocal lattice vector, ⃑ , of the surface reflection, L, may be rewritten as below The three values, 0 , 0 and 0 , represent the x-, y-and z-components of the incident wavevector, ⃑ ⃑⃑ . We may replace ⃑ ⃑⃑ and ⃑ of Supplementary equation (4) by Supplementary equations (2) and (3) where E ⃑ ⃑⃑ , D ⃑⃑⃑ , H ⃑⃑⃑ and B ⃑ ⃑⃑ are the electric field, electric displacement, magnetic field , and magnetic induction, respectively. The two symbols, ρ t and J ⃑ t , are the given net charge and current density, respectively. In X-ray frequency, the conductivity is about zero, thus the two values, ρ t and J ⃑ t are both set to zero. Also the magnetic field, H ⃑⃑⃑ , is approximately equal to magnetic induction, B ⃑ ⃑⃑ due to the magnetic permeability, μ, being about unity. Considering all these conditions mentioned, the Maxwell`s equations can be rewritten as below: The Supplementary equation (7) may be deduced as: By considering the periodic nature of the crystal, the assumed solution to Supplementary equation (8) is a Bloch function, namely And the electric susceptibility,χ, can be also expressed as a Fourier series as The two symbols, and k ⃑⃑ G , represent time and the wavevector of the G th reflection, respectively. The two vectors, ⃑ and r ⃑, stand for the positions of the electric displacement and susceptibility, respectively. The coefficient,χ h m , is the electric susceptibility of the h m th reflection.
By substituting Supplementary equations (9) and (10) into (8), the relation among the Fourier components and wavevectors lead to the following so-called fundamental equation of wavefield:

III. Linearization of Eigenvalue problem of the fundamental equation in single Cartesian coordinates [1]
The fundamental equation usually can be solve as an eigenvalue equation, but with quadratic terms of eigenvalues involved. This fact makes numerical calculation difficult. According to Ref. [1] , the use of single Cartesian coordinates can transform the quadratic eigenvalue problem into a linear form, which is easy to solve numerically. Following Ref. [1], a Cartesian coordinates system is chosen where the z-axis, pointing towards the empty space, is normal to the crystal entrance surface. In an N-beam case, in order to describe the diffracted E-fields, , and wavevectors, , the m' reflection with a single Cartesian coordinates system, the N  N diagonal matrices, A, B, and C S below are utilized to represent the x-, y-and z-component of Also is changed as a 3N  1 matrix as: where F ≡ ∑ χ h m −h n n , and χ h m −h n is the electric susceptibility of the (h m − h n ) th reflection. Using G 2 ≡ K 2 (I + F), with I as a unit matrix, the fundamental equation can be expressed as : To transform Supplementary equation (14) into a linear eigenvalue problem, two new variables, E v and E w , are introduced: Finally, the fundamental equation in a linear form is obtained as: , and (E j , E j , E j ) is the x-, y-and z-component of the E-field of the j th mode for the m th reflection. The diffracted E-field is then the superposition of the 4N modes with the proportional coefficient, , which is determined though the boundary conditions given in Supplement IV: where ⃑⃑ is the wavevector inside the crystal of the j th mode for the m th reflection, and ⃑ ⃑⃑ is the wavevector outside the crystal of the m th reflection, and ⃑ is the position vector.

IV. Boundary Conditions
For a plane parallel crystal slab (M = 1, where M is the total number of the simulated layers), the coefficient, , can be solved by applying the boundary conditions, i.e., the continuity of the tangential components of the electric field, ⃑⃑ , and magnetic field, ⃑ ⃑⃑ , and of the normal components of the electric displacement, ⃑ ⃑⃑ , and magnetic induction, ⃑⃑ , at the top (l = 1) and bottom (l = 2) boundaries. These lead to the following equations:  , where the symbol, t, is the thickness of the parallel crystal slab. For a multi atomic layers (Supplement V), t (1) ( ) and t (2) ( ) are the top and bottom of the n th atomic layer, and t ( ) is the thickness of the n th atomic layer.

V. Multi-layer dynamical theory of X-ray diffraction form crystalline [2]
Because  lead to the following matrix equations: , (1) (1) ) 2 ×4 , ̅ (M) ) 2 ×4 (1) ) ( (1) ) + ( The symbol, 0 ̅ , is a 2N×4N null matrix. The Supplementary equation (29) can be solved as a linear matrix equation as: for ℚ , (4N×M)×4N for both ℂ and , and 4N×4N for the null matrix . The unknown coefficients, ℂ can be solved as: ℂ = ℚ − (31). Because the size of the matrix, ℚ, is so huge that its inverse matrix cannot be estimated in the general software due to the problem, "out of memory". However, using the method given in Ref. [3], the inversed matrix of ℚ can be solved. Then, the elements, ̂( ) , of the matrix ℂ are determined:

VI. Construction of fitting function, ( ) ( ⃑⃑⃑ ( , ))
A fitting function, ℱ ( ) ( ⃑⃑ (tth, beta)), is to simulate the intensity distributions of the surface diffraction, m, at the n th layer, which is monitored by a detector. Because of the shallow diffraction angle of the surface reflection, m, with respect to the crystal surface, part of the diffracted beam is absorbed by the crystal. The acceptance of the detector depends therefore on the diffraction geometry, crystal orientation, crystal depth, beam divergence, and the miscut angle between the surface and atomic layers.
For better handling the simulated intensity, we use the measured tth-and beta-scans to construct a fitting function of the n th layer. Consequently, the surface wavevector (l = 1), ⃑ ⃑⃑ ( ) ( ) (=( ( ) , ( ) , − (1) ( ) )), should be firstly transferred to our laboratory tth-beta coordinate system by rotating the crystal a Bragg angle, , around the y-axis, so that the incident beam is along the x-axis. The surface wavevector,− ⃑ ⃑⃑ (1) ( ) , denoted as ( ( ) , ( ) , ( ) ), in the laboratory coordinate system, then takes the form as:

VIII. Evaluation the uncertainties of the tensor and lattice parameters
In the tth-and beta-scans, the slit width of the detector may cause uncertainties in measuring the  tensor and lattice parameters. The Bragg`s law can be written in relation to the lattice parameters and  tensor of the n th layer as:

IX. Strain
The three unit lattice vectors, ⃑, ⃑ ⃑ and ⃑, (also referred to in Supplement Ⅰ) may be rewritten in term of tensile strains, ε, and shear strains, τ, in a cubic system as a first-order approximation: a ⃑⃑ = ( a x a y a z) = a 0 ((1 + ) ), where the three symbols, , and , are the x-, y-and z-direction tensile strains, and , are the y-and z-direction shear strains of a-axis. Similarly, , represent the x-and z-direction shear strains of b-axis, and , stand for the x-, y-direction shear strains of c-axis. The cubic lattice constant, a 0 , is estimated by Vegard's law: a 0 = xa si + ya Ge (39), where x and y are the composition of the Si and Ge, respectively, and the lattice constant of the cubic system for Si, a si~5 .43075 Å (see Ref. [4]) and Ge, a Ge~5 .6583 Å (Ref. [5]). From the equation (39) and Figure 1c