Holographic Measurement of Strain and Macroscopic Potentials in
GaN
Heterostrucutres
Authors: Narodovitch Michael , Niermann Tore , Lehmann Michael 
Published: December 2016

The phase of an electron wave transmitted through a sample can be measured with the transmission electron microscope (TEM) using electron holography. This phase information is used to determine electrostatic potential distributions within the specimen, e.g. a semiconductor device [1]. Increasing the resolution towards anatomic scale makes it further possible to measure the lattice constants and the strain state of the crystalline specimen.

In this contribution we present a simulation study on the obtainable phase information using the crystal structure, the macroscopic electrostatic potential, and the strain‐state as input parameters. This input is used to compute the potential distribution within the specimen. Then, we use the multislice‐algorithm to propagate the electron wave through the sample. The output of the simulation, i.e. the actual amplitude and phase of the exit‐wave, is analyzed by Fourier filtering of selected reflections. Figure 1 illustrates this approach.

We apply this approach to simulate the measurement of the polarization induced electrostatic fields in a strained Al
0.2
Ga
0.8
N thin‐layer, embedded in a GaN matrix. Figure 2a shows the scheme of the specific simulation setup described in this contribution. This material system is of particular interest, because of its applications to optoelectronics. The actual field‐strengths in this material system are still a matter of scientific discussion.

For a holographic measurement in a TEM, a sample thickness of 200‐300 nm is normally used. This has the advantage, that a large phase signal can be expected. But the thick samples lead to strong dynamic diffraction conditions and thus a complicated material contrast. Hence, sample‐tilt and local thickness variations have a significant impact on the measurement result.

Figure 2b shows a typical simulation result in the case of a 10 nm Al
0.2
Ga
0.8
N thin‐layer, grown on a c‐plane of the Wurzite GaN matrix. The strain of the layer is set to 1% and the polarization induced field to 1.5 MV/cm, according to the order of magnitude expected from literature [2]. The polarization induced field corresponds to an interface charge density of ±7.7 *10
12
cm
‐
2
.

The simulation results were filtered around the (0000)‐beam and the ±(0002)‐reflections (figure 3a). Typical results are shown in figure 3b. The phase profiles of the (0000)‐beam and diffracted ±(0002)‐beams show a slope within the Al
0.2
Ga
0.8
N ‐ layer. This slopes contain information about the actual polarization field and strain of the layer. We show that, despite the dynamic diffraction conditions, the slope can be evaluated with the standard approach from kinematic diffraction theory yielding precisely the strain and field strength, as given as input to the model.

Nevertheless, one has to consider the dynamical diffraction conditions because they define the material contrast at the hetero interface and the actual signal amplitude within the Al
0.2
Ga
0.8
N ‐ layer. By adjusting the sample thickness and sample tilt the conditions can be optimized with respect to the signal amplitude. We suggest to use wedge shaped samples in order to obtain more experimental flexibility.