11.2.4. Condition for Validity of the Equation

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 is higher than V_th; that means, the channel is existing. And then we apply the other potential, so we do have the V_DS which is providing the lateral field. Then, if you see here I think it is let me go with intuitive way that I it is proportional to what? It is proportional to W. In fact, it will be proportional to because if you see here this is the L and this is the orthogonal dimension it is the W. So, if you are having higher length for everything is remaining same it is expected that the resistance here it will increase. So, as a result the corresponding current it will decrease. So, on the other hand if the W is increasing the corresponding resistance it will decrease. So, you may say directly that I_DS is proportional to or you can say that aspect ratio of the channel.

The expression for the current in a MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor) is determined by several parameters, including the width (W) and length (L) of the channel, as well as the gate-source voltage (V_GS) and drain-source voltage (V_DS). We start from the assumption that the voltage V_GS is greater than the threshold voltage V_th, which allows the channel for current flow to exist. If W increases while keeping L constant, the resistance within the channel decreases, which leads to an increase in the drain-source current (I_DS). Conversely, if L increases with constant W, the channel resistance increases, and I_DS decreases. Thus, I_DS is proportional to the width-to-length ratio of the channel, directly affecting the overall current through the device.

Think of water flowing in a pipe. If the pipe is wider (greater W), more water can flow through it simultaneously, which increases the flow rate (I_DS). However, if you extend the length of the pipe (greater L), it becomes harder for the water to flow, reducing the flow rate. This analogy helps to visualize how dimensions of the MOSFET channel can influence current flow.

Now, how about the other parameters? So, this will be proportional to the conductivity in the channel regions which is controlled by this V_GS - V_th which means that whatever the excess voltage you do have beyond the threshold voltage that is effectively contributing to the conductivity or it is helping to increase the conductivity in the channel. So, we can say V_GS is V_GS - V_th directly increasing this current. And also we do have the lateral field which is getting produced by V_DS, so we can also say that this is proportional to V_DS.

In addition to the geometric parameters, the voltage difference between the gate-source voltage (V_GS) and the threshold voltage (V_th) is critical for determining the current I_DS. The excess voltage above the threshold voltage increases the channel's conductivity, thereby allowing more current to flow. The voltage V_DS, which creates a lateral electric field in the device, also affects this current. Consequently, the relationship dictates that I_DS is influenced by both how much higher V_GS is compared to V_th and the applied V_DS.

Imagine a car traveling downhill (representing the current flow). The degree of the slope (V_GS - V_th) determines how fast the car can go. If that slope is steep (high voltage difference), the car accelerates quickly (more current). Similarly, V_DS affects the speed of the car – it must have a proper incline (lateral field) to maintain momentum.

So, if I combine all of them, so what we can say here it is I_DS is say 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.

Combining the influence of the geometric factors (W and L) and the voltage factors (V_GS - V_th and V_DS), we find that the drain-source current can be expressed mathematically. The expression takes the form I_DS = K × (V_GS - V_th) × V_DS. Here, K represents a constant that includes various device parameters, such as the mobility of the charge carriers (electrons), which significantly influences how readily current flows through the device.

Consider an assembly line where the speed of production (I_DS) depends on the efficiency of the machines (K), the raw materials available (V_GS - V_th), and the flow of products through the line (V_DS). If all these factors are ideal, the production rate increases accordingly.

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, but this is valid probably in this portion.

It's crucial to understand the conditions under which the derived expression for current I_DS holds true. The assumptions include that V_DS should be small compared to V_GS - V_th. If V_DS becomes substantial, the current may need to be recalculated, as the assumptions regarding channel conductivity and electric fields would no longer be valid. Thus, for accurate modeling of I_DS, these conditions must be strictly adhered to.

Imagine trying to maintain a consistent water flow through a faucet. If you suddenly open the faucet fully (high V_DS), the initial flow predictions (I_DS) may not apply anymore because of turbulence and overflow. Maintaining a balance ensures the predictions remain accurate.

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.

As the drain-source voltage (V_DS) is increased significantly while keeping V_GS constant, a point may be reached where the conductivity of the channel effectively approaches zero. This occurs because the electric field created by V_DS overwhelms the existing conductive channel. As a result, at this critical condition, the expression for I_DS becomes invalid since the assumptions regarding the channel's conductivity are violated.

Think of a river reaching a dam. When water flow increases to a point where it cannot be contained (high V_DS), the water eventually spills over or can't pass through effectively (conductivity approaching zero). This illustrates how excess voltage changes the effective behavior of MOSFET conductivity.

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