11.2. Expression of the Current

So, welcome back here again the second part of today’s module. What we are looking for is the expression of the current as a function of the W’s and L’s and V and V. V and V of course, they are applied here. And also, just to get an idea that how this current depends on the device parameter.

In this introduction, we are setting the stage for understanding how the current in a MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor) is expressed. The terms W (width) and L (length) refer to the dimensions of the transistor, while V represents the gate-source voltage (V_GS) and the drain-source voltage (V_DS). This section emphasizes that the expression we are about to derive is based on these parameters, and we aim to explore how variations in these factors influence the current that flows through the device.

Consider a water hose: The width of the hose (analogous to W) and its length (L) determine how water flows through it. The pressure from the tap (representing V_GS) and any restriction at the end (analogous to V_DS) also influence the water flow rate. Just as adjustments to these factors can change water flow, so too can changes in W, L, and V affect the electrical current in a MOSFET.

So, what will be the expression of this I_DS? This is a big question. First of all, let me quickly put the biases. For vertical field we do have V_GS here, so that creates vertical field. And let me assume that this V_GS is higher than V_th. This means the channel is existing. Then we apply V_DS, which is providing the lateral field.

This segment discusses the basic assumptions involved in deriving the expression for the drain-source current (I_DS). First, we establish that the gate-source voltage (V_GS) must be higher than the threshold voltage (V_th) in order for a conductive channel to exist. This ensures that the MOSFET is 'on' or conducting. Additionally, the drain-source voltage (V_DS) creates a lateral electric field that influences the flow of current through the channel. Understanding these conditions is crucial as we outline the expression for I_DS.

Imagine turning on a faucet—once you pull the handle (akin to applying V_GS), water starts flowing through a pipe that must be clear (just like the channel in the MOSFET). If the faucet is off (V_GS < V_th), no water flows, regardless of the pressure within the pipe (V_DS).

Intuitively, I_DS is proportional to what? It is proportional to W. If you are having higher length for everything remaining the same, the resistance will increase, and the current will decrease. Conversely, if W is increasing, the current will increase.

This portion establishes a direct proportional relationship between the width (W) of the MOSFET channel and the drain-source current (I_DS). An increase in W leads to lower resistance, thereby allowing more current to flow. On the other hand, if the length (L) of the channel increases (while keeping everything else constant), the resistance to current flow increases, which reduces I_DS. This highlights the significance of the aspect ratio (W/L) in determining the behavior of current in a MOSFET.

Think of W as the width of a river; wider rivers allow more water to flow with less resistance. If a river gets narrower (analogous to a smaller W), there is more restriction, and water flow decreases. Similarly, adjusting W in a MOSFET directly affects how much current can flow.

The current will be proportional to the conductivity in the channel regions controlled by V_GS - V_th. The excess voltage contributes to the conductivity, increasing I_DS as V_GS increases.

Here, the discussion focuses on the role of voltage beyond the threshold voltage (V_GS - V_th) in increasing conductivity within the MOSFET's channel. Essentially, a larger difference between the gate-source voltage and the threshold voltage allows more charge carriers to be available, thereby increasing the current flow. This aspect indicates how crucial the voltage applied at the gate is in controlling the electrical characteristics of the MOSFET.

Picture adding more and more water pressure to a water balloon. Initially, the water does not fill it much when there isn't enough pressure (V_GS < V_th). Once you apply sufficient pressure (V_GS > V_th), the balloon fills and expands, allowing more water (current) to flow in.

So, if I combine all of them, we can say that I_DS is proportional to a constant K times (V_GS - V_th) times V_DS.

In this chunk, we summarize the relationships established previously into a coherent formula for I_DS. The formula states that the drain-source current is directly proportional to the product of the excess gate voltage (V_GS - V_th), the drain-source voltage (V_DS), and a proportionality constant K that encapsulates various device parameters including electron mobility and capacitance per unit area. This puts all our earlier observations together into a single expression that predicts current flow based on specified voltages and dimensions.

This is like calculating how much water will flow through a pipe based on its diameter (W), the pressure difference at the ends (V_GS - V_th), and the length of the pipe (L), taking into account any restrictions or features (represented by K) that affect flow. All these variables combine in a formula to give an accurate prediction of the flow rate.

This equation assumes that V_GS is higher than V_th and that V_DS is much smaller than V_GS - V_th. This assumption is critical for the validity of the derived expression for I_DS.

In this concluding section, we discuss the important conditions under which our expression for I_DS holds true. Specifically, it is crucial that the gate-source voltage remains above the threshold voltage, indicating that the MOSFET is on. Additionally, V_DS must be small compared to the gate drive conditions for the equation to accurately predict current behavior. Without these conditions, one may receive incorrect predictions about current flow.

Returning to our water analogy, think of it like this: if you have a full faucet (V_GS > V_th), you're likely to get a good flow. But if the faucet is partially closed (akin to a large V_DS), then flow might not be as expected. The conditions must be right for a reliable prediction of how much water flows.

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