Geometrical Phase Analysis for nanoscale strain calculations

Geometrical phase analysis (GPA) is a method for obtaining quantitative information from HRTEM images such as measuring and mapping displacements fields and strain fields. This method is based on the Fourier analysis of a HRTEM lattice in which a strong Bragg reflection is selected and a Fourier transform is performed. The resulting image is a complex one from which the phase component give us information about the local displacements of the atomic planes. This method has been developed by M. Hÿtch et al.¹⁸

40

in the late ‘90s and is widely used since then for analysing high resolution lattice fringe images.

The basic principle of GPA is shown in Figure 2.12.¹⁹ The positions of lattice fringes or image features can be measured and then compared with a regular lattice and this gives the displacement field. The local spacing of the lattice can be obtained by sub- tracting neighbouring displacements.

Figure 2.8: 1D illustration of displacement measurements. The displacements un, are with re- spect to a reference of fixed spacing d. The local deviation of lattice spacing (dn-d) is given by the (un+1-un) or the derivative of the displacement field.¹⁹

Displacements can be measured directly from the HRTEM images by calculating the local Fourier components of the lattice fringes.²⁰ The intensity of the image I(r) is given by the expression:

( ) _g( )exp{2 }

g

I ^r =

∑

where g is the reciprocal lattice vector of the undistorted lattice. The Fourier compo- nent Ig(r) becomes a function of position in the image. In Fourier space the deviations in the lattice is contained in the diffuse intensity around the sharp spot in the reciprocal space. In order to obtain the local Fourier components, a mask must be placed around the desired periodicity (spot) and take the inverse Fourier transform. The local Fourier components have an amplitude Ag(r) and a phase Pg(r) which are expressed as follows:

( ) ( )exp{ ( )}

g g g

I r =A r iP r (2.9)

A(r) describes the local contrast of the fringes, while the Pg(r) describes their po- sition. Pg(r) is connected to the displacement u(r) by the expression:

( ) 2 ( )

Pg r = − πg u r⋅ (2.10) If we consider a set of perfect lattice fringes of intensity Bg(r):

41

( ) 2 cos{2 }

g g g

B r = A πg r⋅ +P (2.11)

the amplitude and the phase are fixed and they are equal to the amplitude and phase of the Fourier components that correspond to g. If a displacement field in whichr→ −r u, then the Bg(r) is given by the following:

( ) 2 cos{2 2 }

g g g

B r → A πg r⋅ − πg u⋅ +P (2.12)

In Equation (2.12) an extra phase component is added with the use of Equation (2.10). The image Bg(r) is the Bragg filtered image that is formed by placing a mask around a spot in the Fourier transform.

A 2D displacement field can be obtained by using two phase components Pg1(r) and Pg2(r) of a set of nonparallel lattice fringes which is given by the following Equa- tion:¹⁸

1 1 2 2

( ) 1[ ( ) ( ) ]

2 ^{g g}

u r = − P r a +P r a (2.13)

in which the a1, a2 are the basis vectors of the lattice in the real space. The relationship between the real and reciprocal space lattices is gi·aj=δij. After calculating the displace- ment field, the local deformation can also be calculated. If the lattice distortion e is writ- ten as a matrix:

x x

xx xy

yx yy y y

u u

e e x y

e e e u u

x y

∂ ∂

 

 ∂ ∂ 

   

= =∂ ∂ 

 ∂ ∂ 

 

(2.14)

then this matrix can be separated in a symmetric term ε and an antisymmetric term ω:

1( )

2 e eT

ε = + (2.15) and 1

( )

2 e eT

ω= − (2.16)

In Equations (2.15) and (2.16) T is the transpose. The strain is given by ε and the rigid lattice rotation by ω.

In practice, GPA routines are implemented in the most widely used software for TEM image manipulation, Digital Micrograph. In order to obtain the strain filed of an area of interest (e.g. an interface) certain steps must be followed. The process is illus- trated in Figure 2.9 using as an example an interface between AlN and r-plane sapphire substrate.

Beginning from a HRTEM image (a) we take the Fourier transform (b) which is called Power Spectrum. After placing a mask around the desired reflection we apply the GPA routines and three images are obtained as shown in Figure 2.9 (c)-to-(e). The first one is the amplitude image, the second is the phase image and third one is the Bragg fil-

42

tered image. In the Bragg image the arrows show extra half planes and the misfit dislo- cations that are associated with those extra half planes are visible in the phase image as well as alternating colours. Using the Phase image, after selecting an area as the refer- ence area, a routine for obtaining the strain field is applied (Figure 2.9(f)). The colour variation in (f) shows the difference in the lattice parameters as we move from the one material to the other. The strain field of the dislocations is also visible in the strain field (red dots in the interfacial area).

Figure 2.9: Starting from a HRTEM image of an AlN/sapphire interface (a) a mask is placed around a spot in the FFT of the image (b). The FFT is obtained by the indicated area in the HRTEM image. After the application of the GPA routine on the Power Spectrum, three images are obtained: the amplitude image (c), the phase image (d) and the Bragg filtered image (e). In the phase image and after a reference has been chosen, the routine for the strain filed routine is applied and the strain field is obtained (f). The shade and colour variations in amplitude and phase images correspond to change in the spacing of the planes that were used for the inverse FFT. In the Bragg filtered image, arrows show the points of extra half planes. The positions of the misfit dislocations are visible in the phase image as alternating colours.

43