Off-axis electron holography

This section follows from [62,63]. The conventional TEM records the variation in the intensity of electrons but the phase information is lost. Off-axis electron holography is a technique that allows to recover the phase shift of the electrons that are passing through an electron transparent specimen in the TEM. The analysis of the phase can provide a high-spatial resolution information about local variations in electrostatic potential and magnetic flux density within and around the specimen.

When an electron with charge q = -e enters a magnetic fieldB⃗ and an electric field E⃗, it experiences the Lorentz force F⃗, which depends on the velocity of electron ⃗v. F⃗ can be expressed as,

F⃗ =−e(E⃗ +⃗v×B⃗) (4.15) To first approximation, the presence of an in-plane magnetic field B in a TEM sample of thicknesst results in the small-angle deflection of an incident electron by an angle

ϑ= eλ

h B_⊥t (4.16)

4.5 TEM modes

Figure 4.8: (a) Simplified schematic ray diagram of off-axis electron holography in TEM using single biprism. S_1,2 are the virtual sources of the electron hologram.

Taken from [62]. (b) Electron hologram of the specimen and(c) its Fourier trans-form showing the side bands.

where λ is the (relativistic) wavelength of the electrons and h is the Planck’s con-stant. For electrons that have been accelerated by 300 kV and pass through a specimen of thickness 100 nm that supports an in-plane magnetic induction of 1 T, the deflection angle is 47.6 µrad. For comparison, typical crystallographic Bragg angles in electron diffraction in the TEM are in the range of a few mrads. For co-herent TEM imaging, the electron wave function in the image plane can be written as

ψ_i(⃗r) =A_i(⃗r)exp[iϕ_i(⃗r)] (4.17) where A and ϕ refers to amplitude and phase and subscript i refers to the image plane. The recorded intensity distribution is then given by expression

I_i(⃗r) = |A_i(⃗r)|² (4.18) The phase contrast technique uses this deflection of the electrons by the magnetiza-tion of the sample. The off-axis electron holography records the phase of the electron

wave directly, based on the interference of the primary wave of interest with a ref-erence wave (Fig. 4.8). It uses an electrostatic biprism to interfere an electron wave that passes through the sample (object wave) with another part of same electron wave that passes only through the vacuum (reference wave). A positive voltage is applied to a biprism to generate this interference pattern. The intensity distribution of the off-axis electron hologram can be written as

I_hol(⃗r) = |ψ_i(⃗r) +exp[2πi⃗q_c.⃗r]|² = 1 +A²_i(⃗r) + 2A_i(⃗r) cos [2π ⃗q_c.⃗r+ϕ_i(⃗r)] (4.19) where ψ_i(⃗r) is the electron wave function in image plane i with amplitude A_i and phase ϕ_i, ⃗r is a two-dimensional vector in the plane of the sample and the tilt of the reference wave is specified by a two-dimensional reciprocal space vector ⃗q =

⃗

q_c. Eq.4.19 represents the three separate contributions to the intensity of an off-axis electron hologram: image intensity A²_i(⃗r) and an additional set of cosinusoidal fringes, whose local phase shifts and amplitudes are exactly equivalent to the phase and amplitude of the electron wave function in the image plane, respectively.

The complex Fourier transform of a hologram gives the amplitude and the phase information and can be written as

F T [I_hol(⃗r)] = δ(⃗q) +F T [A²_i(⃗r)] centerband +δ((⃗q) + (q⃗c))⊗F T [Ai(⃗r)exp[iϕi(⃗r)]] −1 sideband

+δ((⃗q)−(q⃗_c))⊗F T [A_i(⃗r)exp[−iϕ_i(⃗r)]] +1 sideband (4.20) It depicts 3 bands (fig. 4.8 c): the center band represents the conventional image and contains both elastically and inelastically scattered electrons but does not contain the image phase; the ±1 side bands are of our interest as they contain the Fourier spectrum of the complete image wave and convoluted around ⃗q=±⃗qc, respectively.

The two side bands contain only elastically scattered electrons and the amplitude and phase are linearly related to the object properties [64].

From the quantum mechanical description, the incident electron wave experiences a phase shift upon traveling though an electromagnetic potential that can be expressed (in 1D) as where the incident electron beam direction z is perpendicular to x, C_E is an inter-action constant with a value of 6.53×10⁶radV⁻¹m⁻¹ at an accelerating voltage of 300 kV,V is the electrostatic potential andA_z is the component of magnetic vector potential along z. The magnetic vector potential A⃗ is related to the magnetic flux density byB⃗ =∇ ×A⃗. In the absence of long range charge redistribution and elec-trostatic fringing fields around the specimen,V mainly comprises of the mean inner potential (MIP) of the material V₀, which is dependent on its composition, density and ionicity. Taking MIP constant in the electron beam direction in the specimen with thickness t, the electrostatic contribution to the phase can be simplified to ϕ_E =C_EV₀t. the magnetic contribution to the phase can be written as

4.5 TEM modes

ϕ_M =−2πe h

I

Adl (4.22)

where the integral is performed around a rectangular loop that is formed by two parallel electron trajectories crossing the sample and joined at infinity by segments perpendicular to their trajectories. different approaches can be employed to separate the magnetic and MIP contributions to the phase. In this work, the approach involves reversing the magnetization direction in the sample in-situ in the electron microscope and subsequently selecting the pairs of holograms that differ only in the opposite directions of the magnetization in the specimen. Thus, the magnetic contribution to the phase can be obtained by taking half of the difference between the two phase images. The in-situ magnetization reversal can be achieved by exciting the microscope objective lens and tilting the specimen to apply a precalibrated magnetic field [65].

5 Experimental techniques

5.1 Thin Film Growth

This section describes the different growth techniques employed to deposit La0.7Sr0.3MnO3

(LSMO) and BaTiO3 thin films on Pb(Mg1/3Nb2/3)0.7Ti0.3O3(PMN-PT-(001)) and Nb doped SrTiO3(Nb:STO(001)) substrates, respectively. Different techniques used to characterize the thin film heterostructures structurally and magnetically are dis-cussed here.