11.2.3. Combination of Variables

So, what will be the expression of this I? So, we do have, so this is the big question. First of all let me quickly put the biases. For vertical field we do have V here, so that creates vertical field. And let me assume that this V it is higher than V. So, the first assumption is that this is higher than V; that means, the channel is existing. And then we apply the other potential, so we do have the V which is providing the lateral field.

In this chunk, we're focusing on determining the expression for the drain-source current (I_DS) in a MOSFET. Firstly, we establish two voltage conditions: V_GS (gate-source voltage) must be higher than the threshold voltage (V_th) for the channel to exist. When the gate voltage exceeds the threshold, a conductive path forms—this is essential for the current to flow. We also introduce V_DS (drain-source voltage), which creates a lateral electric field helping drive the current from the drain to the source.

Think of a MOSFET like a water pipe. V_GS is like turning on a water valve, allowing water to flow through. If you don't turn the valve (i.e., V_GS is less than V_th), no water (current) can flow through the pipe (the MOSFET) no matter how much pressure (V_DS) you apply.

So, on the other hand, if the W is increasing the corresponding resistance it will decrease. So, you may say directly that I is proportional to or you can say that aspect ratio of the channel. Now, how about the other parameters? So, this will be proportional to the conductivity in the channel regions which is controlled by this V – V_th.

This chunk explains the influence of the width (W) and length (L) of the MOSFET channel on the current. Specifically, if the channel width increases while keeping everything else constant, the resistance decreases, leading to an increase in the drain current (I_DS). Moreover, we discuss the concept of conductivity, which is proportional to the excess voltage (V_GS - V_th). Greater excess voltage results in higher conductivity, thus enhancing the current flow.

Consider W as the width of a highway and L as the length. If you increase the width of the highway (W), more cars (current) can travel simultaneously, reducing congestion (resistance). If you add more lanes (increase W), traffic flows more efficiently, just like a higher current in a MOSFET.

So, if I combine all of them, so what we can say here it is I_DS is proportionality constant say × (V_GS - V_th) × V_DS, ok. And this K, this K it encapsulates whatever the device parameter is there. In fact, this K if you see, if the mobility of the electrons is flowing in this way. So, if the mobility of the electron is higher of course, the current it will be better.

Here, we combine the effects of the previously mentioned parameters into a single expression for I_DS: it is proportional to the excess gate voltage (V_GS - V_th), the drain-source voltage (V_DS), and a constant K that includes the characteristics of the device. This constant K encapsulates parameters like the mobility of electrons—higher mobility leads to a higher current as the charge carriers can flow more freely.

Imagine a sports team where K is the coach's effectiveness. The quality of the team (current) depends on both the coach's strategy (K) and how well the team plays under pressure (V_GS - V_th and V_DS). A great coach helps the team to perform better, just like a better K helps the MOSFET conduct more current.

But one important thing we are missing here it is that, whenever we say that V_DS is higher than V_th and whatever the excess amount we have it is contributing for towards the conductivity of the channel.

In this section, we address a limitation of the previously derived current equation. It is valid only when the drain-source voltage (V_DS) remains relatively small compared to the effective gate voltage (V_GS - V_th). If V_DS becomes too large, it significantly affects the channel conductivity and requires modifications to the current expression. Essentially, high V_DS leads to a change in the voltage distribution along the channel, thereby affecting the current flow.

Think of this as adding too much water pressure to a hose. If the pressure is low, the water flows smoothly. But as you increase the pressure (akin to a higher V_DS), the flow changes, and at a certain point, it might even create a back pressure or burst. The MOSFET behaves similarly: if the V_DS is too high, the behavior of the current flow must be reassessed.

Now, what happens in a critical situation when we are just making this voltage higher and higher keeping this V may be constant and such that the conductivity here it is approaching towards 0, which means that if it is V_GD is exactly equal to V_th.

In this chunk, we discuss the 'pinch-off' condition, which occurs when the drain voltage (V_D) approaches the gate-source voltage minus the threshold voltage (V_GS - V_th). Under these conditions, the channel conductivity can fall to zero, significantly modulating current flow. Even as the channel weakens, there may still be a current due to the strong electric field at the drain. However, as V_D exceeds a certain threshold, the effects of pinch-off must be considered, modifying the current expression.

Imagine a camera shutter that closes to control the amount of light hitting the sensor. As you increase the pinching mechanism (V_D), less light (electric current) gets through until it almost stops. However, even when closed, there may still be a trickle of light (electrons) due to the intense light beam (high electric field) directed at the sensor.

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