4.2.4 Channel Region

As the charge transport in the channel is governed by the same laws exactly as in a non split-drain MOSFET, it is desirable to observe how quantities change when a magnetic field is applied. Plots of the electron mobility, electron concentration, potential and electric field are shown. Because the variations are very small in the whole MAGFET structure, these quantities are presented from cuts along the MAGFET width and at 1 nm from the silicon oxide-silicon interface, that is, along the length of the MAGFET (see Fig. 4.1), from the end of the source towards the drains, cuts at 20 $\mu$m, 50 $\mu$m, 80 $\mu$m, and 110 $\mu$m from the source side are made. Then, the quantities are shown at 1 nm from the silicon oxide-silicon interface where deviations can be observed.

Figures 4.10 through 4.15 show the potential inside the channel at 20 $\mu$m from the source and at 1 nm from the silicon oxide-silicon interface at 300 K for different gate voltages and magnetic fields. Because the cut is close to the source which is biased to zero volts, the potential is almost constant when no magnetic field is applied. As soon as a magnetic field is applied, the potential is bent upwards if the magnetic field points inside the MAGFET sensor ($y$ component of the magnetic field is -50 mT) or downwards on the contrary case ($y$ component of the magnetic field is +50 mT). For a gate voltage of zero volts this bent is not noticeable, because the magnitude of the moving charge is very low, so the magnetic field has no visible effect on the potential.

Figure 4.10: Potential in the channel
@(20$\mu$m,300K,Vg=0.0V).

Figure 4.11: Potential in the channel
@(20$\mu$m,300K,Vg=1.0V).

Figure 4.12: Potential in the channel
@(20$\mu$m,300K,Vg=2.0V).

Figure 4.13: Potential in the channel
@(20$\mu$m,300K,Vg=3.0V).

Figure 4.14: Potential in the channel
@(20$\mu$m,300K,Vg=4.0V).

Figure 4.15: Potential in the channel
@(20$\mu$m,300K,Vg=5.0V).

From the previous plots, it is expected that the potential should be bent around a common point when a magnetic field is applied. Because the cut has been made at 20 $\mu$m from the source side of the MAGFET, the gate has more control over the channel than the other contacts. This explains the asymmetry in these figures. Figures 4.16 through 4.21 show the potential inside the channel at 50 $\mu$m from the source and at 1 nm from the silicon oxide-silicon interface at 300 K for different gate voltages and magnetic fields. The potential with zero magnetic field shows a small curvature coming from the drains, because the cut is approaching the drains. As can be seen, the potentials are symmetric with respect to the direction of the magnetic field.

Figure 4.16: Potential in the channel
@(50$\mu$m,300K,Vg=0.0V).

Figure 4.17: Potential in the channel
@(50$\mu$m,300K,Vg=1.0V).

Figure 4.18: Potential in the channel
@(50$\mu$m,300K,Vg=2.0V).

Figure 4.19: Potential in the channel
@(50$\mu$m,300K,Vg=3.0V).

Figure 4.20: Potential in the channel
@(50$\mu$m,300K,Vg=4.0V).

Figure 4.21: Potential in the channel
@(50$\mu$m,300K,Vg=5.0V).

It is expected that a cut closer to the drains will show a strong influence of the drains on the potential.Figures 4.22 through 4.27 show the potential inside the channel at 80 $\mu$m from the source and at 1 nm from the silicon oxide-silicon interface at 300 K for different gate voltages and magnetic fields. A symmetry is roughly noticeable. Because the cut has been made in a zone between the middle of the MAGFET structure and the drains, in this zone both the gate and the drains fight for controlling this zone of the channel.

Figure 4.22: Potential in the channel
@(80$\mu$m,300K,Vg=0.0V).

Figure 4.23: Potential in the channel
@(80$\mu$m,300K,Vg=1.0V).

Figure 4.24: Potential in the channel
@(80$\mu$m,300K,Vg=2.0V).

Figure 4.25: Potential in the channel
@(80$\mu$m,300K,Vg=3.0V).

Figure 4.26: Potential in the channel
@(80$\mu$m,300K,Vg=4.0V).

Figure 4.27: Potential in the channel
@(80$\mu$m,300K,Vg=5.0V).

Figures 4.28 through 4.33 show the potential inside the channel at 110 $\mu$m from the source and at 1 nm from the silicon oxide-silicon interface at 300 K for different gate voltages and magnetic fields. The cut has been made only at 15 $\mu$m from the drains and their influence on the potential is quite remarkable. A quasi perfect symmetry can be seen in these figures.

Figure 4.28: Potential in the channel
@(110$\mu$m,300K,Vg=0.0V).

Figure 4.29: Potential in the channel
@(110$\mu$m,300K,Vg=1.0V).

Figure 4.30: Potential in the channel
@(110$\mu$m,300K,Vg=2.0V).

Figure 4.31: Potential in the channel
@(110$\mu$m,300K,Vg=3.0V).

Figure 4.32: Potential in the channel
@(110$\mu$m,300K,Vg=4.0V).

Figure 4.33: Potential in the channel
@(110$\mu$m,300K,Vg=5.0V).

The following quantity to be shown is the electron concentration inside the channel. As it is known from the Hall effect, the carriers pile up in one of the planes parallel to the applied magnetic field depending on their direction and the direction of the magnetic field. In the two-drain MAGFET, because the magnetic field is applied perpendicular to the channel, the electrons pile up in one side of the channel within the MAGFET structure. Figures 4.34 through 4.39 show the electron concentration inside the channel at 20 $\mu$m from the source and at 1 nm from the silicon oxide-silicon interface at 300 K for different gate voltages and magnetic fields. For the gate voltages from 0 V, 1 V, and 2 V, a symmetry in the electron concentration due to the magnetic fields can be seen. For gate voltages larger than 3 V, the electron concentration with zero magnetic field is lower than the corresponding electron concentration with a magnetic field. This effect can be explained in terms of the driving force that exerts the gate. Because the electrons are attracted to the silicon oxide-silicon interface and at the same time swept by the Hall electric field, their concentration is higher in one side than in the other. Then, a symmetry point cannot be seen.

Figure 4.34: Electron concentration in the channel
@(20$\mu$m,300K,Vg=0.0V).

Figure 4.35: Electron concentration in the channel
@(20$\mu$m,300K,Vg=1.0V).

Figure 4.36: Electron concentration in the channel
@(20$\mu$m,300K,Vg=2.0V).

Figure 4.37: Electron concentration in the channel
@(20$\mu$m,300K,Vg=3.0V).

Figure 4.38: Electron concentration in the channel
@(20$\mu$m,300K,Vg=4.0V).

Figure 4.39: Electron concentration in the channel
@(20$\mu$m,300K,Vg=5.0V).

Figure 4.40: Electron concentration in the channel
@(50$\mu$m,300K,Vg=0.0V).

Figure 4.41: Electron concentration in the channel
@(50$\mu$m,300K,Vg=1.0V).

Figure 4.42: Electron concentration in the channel
@(50$\mu$m,300K,Vg=2.0V).

Figure 4.43: Electron concentration in the channel
@(50$\mu$m,300K,Vg=3.0V).

Figure 4.44: Electron concentration in the channel
@(50$\mu$m,300K,Vg=4.0V).

Figure 4.45: Electron concentration in the channel
@(50$\mu$m,300K,Vg=5.0V).

Figures 4.40 through 4.45 show the electron concentration inside the channel at 50 $\mu$m from the source and at 1 nm from the silicon oxide-silicon interface at 300 K for different gate voltages and magnetic fields. As stated for the potential, the cuts are approaching the drains, so a stronger influence from the drains on the electron concentration can be seen. A maximum is observed in the middle of the channel (with respect to the width of the MAGFET) for zero magnetic field, and it is caused by the combination of the lateral electric field which concentrates on the drains and the stronger transversal electric field. The scale at which the electron concentrations are plotted should be noticed. Because the applied magnetic field is quite low (50 mT), the changes with respect to the zero magnetic field case are very small. Stronger variations in the electric field and carrier concentration can be seen for stronger magnetic inductions or for devices with higher carrier mobilities [5,24].

Figure 4.46: Electron concentration in the channel
@(80$\mu$m,300K,Vg=0.0V).

Figure 4.47: Electron concentration in the channel
@(80$\mu$m,300K,Vg=1.0V).

Figure 4.48: Electron concentration in the channel
@(80$\mu$m,300K,Vg=2.0V).

Figure 4.49: Electron concentration in the channel
@(80$\mu$m,300K,Vg=3.0V).

Figure 4.50: Electron concentration in the channel
@(80$\mu$m,300K,Vg=4.0V).

Figure 4.51: Electron concentration in the channel
@(80$\mu$m,300K,Vg=5.0V).

Figure 4.52: Electron concentration in the channel
@(110$\mu$m,300K,Vg=0.0V).

Figure 4.53: Electron concentration in the channel
@(110$\mu$m,300K,Vg=1.0V).

Figure 4.54: Electron concentration in the channel
@(110$\mu$m,300K,Vg=2.0V).

Figure 4.55: Electron concentration in the channel
@(110$\mu$m,300K,Vg=3.0V).

Figure 4.56: Electron concentration in the channel
@(110$\mu$m,300K,Vg=4.0V).

Figure 4.57: Electron concentration in the channel
@(110$\mu$m,300K,Vg=5.0V).

Figures 4.46 through 4.51 show the electron concentration inside the channel at 80 $\mu$m from the source and at 1 nm from the silicon oxide-silicon interface at 300 K for different gate voltages and magnetic fields. Comparing these figures with Figures 4.34 through 4.39, the electron concentration for zero magnetic field shows a maximum. Actually there is indeed a maximum but it is hardly noticeable due to the scale used for the plots and because the cut is localized far away from the drains. In Figures 4.52 through 4.57 the electron concentration inside the channel at 110 $\mu$m from the source and at 1 nm from the silicon oxide-silicon interface at 300 K for different gate voltages and magnetic fields is shown. The strong influence from the drains and the gate can be seen on the electron concentration. In addition, a perfect symmetry can be observed.

Figure 4.58: Electric field in the channel
@(20$\mu$m,300K,Vg=0.0V).

Figure 4.59: Electric field in the channel
@(20$\mu$m,300K,Vg=1.0V).

Figure 4.60: Electric field in the channel
@(20$\mu$m,300K,Vg=2.0V).

Figure 4.61: Electric field in the channel
@(20$\mu$m,300K,Vg=3.0V).

Figure 4.62: Electric field in the channel
@(20$\mu$m,300K,Vg=4.0V).

Figure 4.63: Electric field in the channel
@(20$\mu$m,300K,Vg=5.0V).

The electric field is also a quantity of interest. Actually, its shape is correlated to the carrier concentration and the potential. However, the electric field is a vectorial quantity and because this subsection is presenting two-dimensional cuts from full three-dimensional simulations of a two-drain MAGFET, one component is missing. The cuts are made perpendicular to the $y$-$z$ plane (see Figure 4.1) so the lateral component of the electric field due to the drains to source bias is missing ($x$ component). The following plots only show the absolute value of the $y$ and $z$ components at 1 nm of the silicon oxide-silicon interface.

Figure 4.64: Electric field in the channel
@(50$\mu$m,300K,Vg=0.0V).

Figure 4.65: Electric field in the channel
@(50$\mu$m,300K,Vg=1.0V).

Figure 4.66: Electric field in the channel
@(50$\mu$m,300K,Vg=2.0V).

Figure 4.67: Electric field in the channel
@(50$\mu$m,300K,Vg=3.0V).

Figure 4.68: Electric field in the channel
@(50$\mu$m,300K,Vg=4.0V).

Figure 4.69: Electric field in the channel
@(50$\mu$m,300K,Vg=5.0V).

Figures 4.58 through 4.63 show the electric field inside the channel at 20 $\mu$m from the source and at 1 nm from the silicon oxide-silicon interface at 300 K for different gate voltages and magnetic fields. As explained for the electron concentration plots at the same cut distance, a symmetry in the electric field line can be observed for low gate voltages but this symmetry is lost as soon as the gate voltage is larger than 3 V. Because the cut is close to the source which is set to zero volts, the dominant component of the electric field is due to the gate. Although the $x$ component of the electric field is missing in these figures, its influence can be noticed for the following figures as the cut approaches the drains.

Figure 4.70: Electric field in the channel
@(80$\mu$m,300K,Vg=0.0V).

Figure 4.71: Electric field in the channel
@(80$\mu$m,300K,Vg=1.0V).

Figure 4.72: Electric field in the channel
@(80$\mu$m,300K,Vg=2.0V).

Figure 4.73: Electric field in the channel
@(80$\mu$m,300K,Vg=3.0V).

Figure 4.74: Electric field in the channel
@(80$\mu$m,300K,Vg=4.0V).

Figure 4.75: Electric field in the channel
@(80$\mu$m,300K,Vg=5.0V).

Figures 4.64 through 4.69 show the electric field inside the channel at 50 $\mu$m from the source and at 1 nm from the silicon oxide-silicon interface at 300 K for different gate voltages and magnetic fields whereas Figures 4.70 through 4.75 show the electric field inside the channel at 80 $\mu$m from the source and at 1 nm from the silicon oxide-silicon interface at 300 K for different gate voltages and magnetic fields. Because a magnetic field deflects the electron current lines inside the channel, a Hall electric field across the current path must counteract the deflection.

Figure 4.76: Electric field in the channel
@(110$\mu$m,300K,Vg=0.0V).

Figure 4.77: Electric field in the channel
@(110$\mu$m,300K,Vg=1.0V).

Figure 4.78: Electric field in the channel
@(110$\mu$m,300K,Vg=2.0V).

Figure 4.79: Electric field in the channel
@(110$\mu$m,300K,Vg=3.0V).

Figure 4.80: Electric field in the channel
@(110$\mu$m,300K,Vg=4.0V).

Figure 4.81: Electric field in the channel
@(110$\mu$m,300K,Vg=5.0V).

As soon as the electrons leave the source, the magnetic field acts on them but not instantaneously. After the electrons have run through some of their paths, they are turned, so it is expected that the Hall electric field is not uniform. The electric field line in Figures 4.58 through 4.75 show this Hall electric field (when a magnetic field is applied). If the device were a MOS Hall plate, this line should be a straight line. That is the reason why the Hall contacts in a MOS Hall plate should be placed close but not next to the drain.

Figures 4.76 through 4.81 show the electric field inside the channel at 110 $\mu$m from the source and at 1 nm from the silicon oxide-silicon interface at 300 K for different gate voltages and magnetic fields. The influence of the drains is evident in this zone of the channel.