## 3 Results and Discussions  
### Wall Erosion When the Value of \(\alpha\) Is 0  
When the electric field \(E\) is perpendicular to the magnetic field \(B\), the values of the tilt angle \(\alpha\) and the axial magnetic field component \(B_{z}\) are 0. In this orthogonal-field configuration, the electrons form a stable \(E\times B\) drift in the azimuthal direction. The drift velocity can be expressed as: [27]:  
\[v_{E\times B}=\frac{E\times B}{B^{2}} \tag{9}\]  
In HETs, these drift velocities of electrons are as high as \(10^{6}\) m/s [20]. However, the ions in the channel can be considered unmagnetized due to their large mass relative to the electrons, and their drift velocity along the azimuth direction is almost zero. The large velocity difference between the electrons and the ions produces a strong instability that displays ion acoustic-like characteristics along the azimuthal direction, which is also known as electron cyclotron drift instability (ECDI). The ECDI is a short wavelength, high-frequency kinetic instability that manifests as large potential and density oscillations [27].  
As shown in Figure 4, obvious periodic modulations of electric potential and electron density driven by ECDI are observed within the quasi-neutral region of the near-sheath layer. The wavelength and the frequency of the instability are 1.4 mm and 57 MHz, respectively. Moreover, the sheath with an electric potential of \(-37\) V is formed on the wall in Figure 4 because the electrons move faster than the ions.  
Due to the potential gradient between the azimuthal periodic modulations, a local electric field formed in the \(Ox\) direction. As shown in Figure 5a, the value of the electric field along the azimuth direction alternates between positive and negative. Therefore, this local electric field is defined as \(\widetilde{\mathbf{E}}_{x}\), where \"\(\sim\)\" indicates that the electric field is oscillating. Figure 5b shows the distributions of the electric field along the azimuthal direction at the distances of \(20\lambda_{De}\) and \(40\lambda_{De}\) from the wall. It can be seen that there is a phase difference between the two electric fields due to the inclination angle of the wave structure of ECDI along the traveling direction. It should be noted that the average value of \(\widetilde{\mathbf{E}}_{x}\) is not zero, and the average value of the electric field on the two lines is very close and both are approximately 1.2 \(\times\) 10\({}^{3}\) V/m.  
For HETs, the presence of the near-wall sheath increases the velocity of ions to the wall and prevents the approach of electrons, which induces the radial divergence of the ion beam and intensifies the wall erosion. By collecting the sputtering parameters of energetic ions that hit the wall at different moments, the erosion rate can be obtained. Figure 6 shows  
Figure 4: The 2D spatial distribution of (**a**) electric potential and (**b**) electron density with \(E\bot B\).  
Figure 5: Distribution of electric field in (**a**) two-dimensional and (**b**) one-dimensional space along azimuth at y = 0.84 mm and y = 1.68 mm with \(E\bot B\).  
the sputtering erosion rate along the azimuthal direction in different time periods with the orthogonal electric fields \(\mathbf{E}\) and magnetic fields \(\mathbf{B}\). Here, the erosion rate was averaged in \(\Delta T=3.2\)\(\upmu\)s for accuracy. It can be found from Figure 6 that the sputtering erosion rate of the wall was basically stable after the simulation had converged, and the average value of the sputtering erosion rate is approximately 2.9 \(\times\) 10\({}^{-10}\) m/s.