Probing the electrostatic potential of charged dislocations in $n\text{−}\mathrm{Ga}\mathrm{N}$ and $n\text{−}\mathrm{Zn}\mathrm{O}$ epilayers by transmission electron holography

Epitaxial layers frequently contain a high density of threading dislocations, which may exceed values of ${10}^9{\mathrm{cm}}^{-2}$ for ZnO and GaN. To study the electrical activity of single dislocations off-axis electron holography in a transmission electron microscope was applied. The electrostatic potential in the vicinity of charged dislocations was determined from the reconstructed phase of the electron wave, which also provides access to the charge density at the dislocation. On the basis of the electrostatic potential for a screened line charge given by Read [Philos. Mag. 45, 775 (1954)] a refined model is proposed in which a charge density with a finite spatial distribution at the dislocation core is assumed. Comparing the measured and theoretically expected potential, charge densities between $5\times {10}^{19}{\mathrm{cm}}^{-3}\mathrm{and}5\times {10}^{20}{\mathrm{cm}}^{-3}$ in cylinders with radii up to $5\mathrm{nm}$ around the dislocation lines were found. An important aspect of our work is the analysis of dislocations in cross-section transmission electron microscopy samples where the dislocation lines are oriented perpendicular to the electron beam. The advantages and drawbacks of the cross-section geometry are discussed. Electrostatic potentials due to piezoelectric charges in the dislocation strain field are considered and found to be insignificant with respect to dislocation charges.

Figure 1

Schematic model of the charging of dislocations in a crystal. The additional electronic states introduced by the dislocation bend locally the band structure of the crystal generating a potential barrier $e{V}_B$.

Figure 2

Schematic drawing illustrating the charge configuration for the refined Read potential and the interaction of an incident plane electron wave with a thin TEM sample containing a charged dislocation.

Figure 3

Bright-field TEM images of a GaN cross-section sample taken with imaging vectors (a) $\vec{g}=\left[0002\right]$ and (b) $\vec{g}=\left[11\overline{2}0\right]$.

Figure 4

Analysis of a dislocation with $\vec{b}=1∕3⟨11\overline{2}3⟩$ in $n\text{−}\mathrm{Ga}\mathrm{N}$: (a) gray-scale coded reconstructed phase of the electron wave with the dislocation line direction indicated by the dashed line, (b) phase plotted as a function of the $x$ direction as defined in (a) perpendicular to the dislocation line. To reduce the noise, the phase was averaged along the $y$ direction within the frame marked in (a). (c) Phase (upper curve) and amplitude profiles parallel to the dislocation line. The variation of the phase is proportional to the thickness variation while the amplitude is opposite to it.

Figure 5

Analysis of an edge dislocation in $n\text{−}\mathrm{Zn}\mathrm{O}$: (a) gray-scale coded reconstructed phase of the electron wave with the dislocation line direction indicated by the dashed line, (b) phase plotted as a function of the $x$ direction as defined in (a) perpendicular to the dislocation line. To reduce the noise the phase was averaged along the $y$ direction within the frame marked in (a).

Figure 6

Phase shift (dashed line) and projected potential (continuous line) due to piezoelectric charges on a surface parallel to a screw dislocation. The dislocation core is situated at a distance of $10\mathrm{nm}$ from the surface.

Figure 7

Maximum phase shift due to a charged dislocation oriented perpendicular to the electron beam represented as a surface depending on the screening charge density $N_s$ and the line charge density $q$. Due to the limited field of view and the phase detection limit of $0.08\mathrm{rad}$ the surface is truncated. The projection of the truncated phase surface provides the detectable range of line charges $q$ in dependence of the available density of screening charges $N_s$. The gray-scale coding of the projected surface represents the maximum phase shift.