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Electron interferometry and field mapping

Electron holography is a powerful technique for measuring the phase of the electron wave and can be used to study the local electric and magnetic fields within and around materials at the nanoscale in a fully quantitative way.

When the fast electrons are interacting in a TEM with any electrostatic or magnetic fields within a thin specimen and possibly with stray fields around it, the associated electron wave is phase shifted. This phase shift can be quantitatively measured thanks to the Aharonov - Bohm equation (Eq. 1) :

Where V is the electrostatic potential, Az the components of the magnetic vector potential along the beam direction z, CE a constant dependent of the electron beam energy.

This equation can be rewritten (Eq. 2) :

 

where B is the component of the magnetic induction perpendicular to both x and y.

Principle of an holographic experiment
Principle of an holographic experiment

The holography experiment in a TEM consists in interfering a reference electron beam (ref.) that has passed through the vaccuum with the electron beam (2) that has been phase shifted after interacting with an object and/or the electromagnetic fields surrounding it. This is achieved thanks to a biprism located in the column. The overlap between these two beams results in an interference pattern (the hologram) from which the phase shift of the electron beam (2) can be measured by Fourier analysis.

Projection of the electrostatic and /or magnetic fields within and around the materials under study can then be achieved following Eq.2 allowing their quantitative measurements at the nanometer scale.

Electron microscopes have not in general been optimized for electron holography and only few of them are able to perform EH experiments with sufficient spatial resolution and good sensitivity in phase shift measurements. The I2TEM instrument used within the MIMETIS project has been designed for performing electron holography experiments and provide significantly improved performances. The position of biprisms can be optimized and the capability of using various multiple biprisms configurations provide flexibility concerning field of view and spatial resolution whilst eliminating artifacts from Fresnel fringes. This drastically improves EH and particularly the dark-field electron holography (see below) which in addition would benefit from specific positions for objective apertures in Lorentz mode. 

In addition to studies at remanence, such electron holography can be performed under the application of various external stimuli (magnetic field, electrostatic potential, temperature, stress, electrical currents) thanks to the use of dedicated holders for in-situ experiments that are available within the MIMETIS project. This requires particular sample preparation processes that is also achievable in the MIMETIS project using the dedicated FIB Helios.

Example of charges measurements from Electric field mapping

A electrically charged nanomaterial or device is radiating around it an electric field that can easely be calculated using the Gauss law.

As this electric field can be quantitatively measured by EH we recently showed that EH can therefore be used to determine and and map the density of charges in any materials with a sensitivity of about 1 electron.

This method has been successfully applied for measuring charges in insulating MgO nanocube (see figures) but also to investigated the variation of carriers in carbon nanotubes as a function of the electrical potential they have been brought in-situ up to the value where a cold field emmision occurs. 

Electric field generated by an electrically charged MgO nano cube (a) Holographic setup, (b) Amplified cosinus contour map (13×) of the reconstructed phase of a charged MgO particle (c) False color representation of the reconstructed phase with contours (every 0.3 rad)
Electric field generated by an electrically charged MgO nano cube (a) Holographic setup, (b) Amplified cosinus contour map (13×) of the reconstructed phase of a charged MgO particle (c) False color representation of the reconstructed phase with contours (every 0.3 rad)

Reference:Counting elementary charges on nanoparticles by electron holography,
C. Gatel, A. Lubk, G. Pozzi, E. Snoeck, and M.J. Hÿtch, 
Phys. Rev. Lett. 111, 025501 (2013), http://dx.doi.org/10.1103/PhysRevLett.111.025501

Equation 2 shows that EH gives quantitative access to the components of the magnetic induction projected in the plane parrallel to the electron beam direction. The analysis of the local magnetic configurations in various magnetic nanostructures allows to determine the structure of domain walls (see figure as exemple) and also to quantify the local induction and magnetization. These results are generally comforted by micromagnetic simulations. 

Transverse DW and hybrid magnetic state in 55 and 85 nm Ni nanocylinders (a) and (d) Amplitude image of 55 and 85 nm nanocylinders, respectively. (b) and (e) Experimental magnetic phase shift and corresponding induction field lines. (c) and (f) Magnetic phase shift and corresponding induction field lines calculated from micromagnetic simulation
Transverse DW and hybrid magnetic state in 55 and 85 nm Ni nanocylinders (a) and (d) Amplitude image of 55 and 85 nm nanocylinders, respectively. (b) and (e) Experimental magnetic phase shift and corresponding induction field lines. (c) and (f) Magnetic phase shift and corresponding induction field lines calculated from micromagnetic simulation

The difference of isophase contour is the same for the experiment and simulation, showing the quantitative agreement, and is set to 0.6 and 0.3 rad for 55 and 85 nm, respectively. 

The color bars on the right give the amplitude of the phase shift in radian. 
The dashed lines are a guide for the eye to position the wires. The arrows on the scheme represent a simplified view of the magnetization within the wire.

Reference :
Magnetic Mapping Using Electron Holography,E. Snoeck and C. Gatel,in Transmission Electron Microscopy in Micro-nanoelectronics ed. A. Claverie,Editeur : ISTE Ltd and John Wiley & Sons Inc (18 décembre 2012)

Dark-field electron holography

In 2009, the CEMES researchers has shown that strain fields can also be measured by electron holography and develop a new method for strain mapping : i.e. the dark-field electron holography. Compared to "classical" electron holography, the dark-field electron holography method consists on realizing the interferences between two diffracted beams that are coming from two regions of different strain states one being stated as reference (fully relaxed) crystal, the other being the strained regions that wanted to be analyzed. The phase shift between the two diffracted beams carries the so-called "geometric phase" that allows recovering the deformation of the strained area compared to the reference one. Combining the geometric phases of two non-collinear diffracted beams permit the quantitative mapping of the the strain within the whole region on interest over field of view as large as few microns, with nanometer resolution and strain sensitivity better that 0.1%.

Dark-field electron holography for strain mapping is a major breakthrough, particularly for applications in the microelectronics industry where strained silicon has become an integral part of device technology.

Reference : 
Nanoscale holographic interferometry for strain measurements in electronic devices,M. Hÿtch, F. Houdellier, F. Hüe and E. Snoeck,Nature, 453, 1086-1089(June 2008) DOI : 10.1038/nature07049

Example of strain field mapping by dark field electron holography in a strained semiconducting devices
Example of strain field mapping by dark field electron holography in a strained semiconducting devices