# Body effect

## What is the body effect?

You might have forgotten, but a transistor is a 4-terminal device. Gate, drain and source are the 3 terminals that are used to control the transistor, but the bulk or body, if not properly biased, may put the transistor inoperable. PN junctions are diodes

The pn junctions defined by source-bulk and drain-bulk, which are basically two diodes, must be reverse-biased to stop them from leaking current from the source/drain to the substrate. That means that the source potential must always be equal or greater than the bulk potential. Since drain voltage is always greater or equal than source voltage, we don't even consider the drain-bulk junction. The body effect

When $V_S > V_B$, the depletion width of the pn junction increases (remember the pn junction depletion region?). That makes it more difficult to create a channel with the same $V_{GS}$, effectively reducing the channel depth. In order to return to the same channel depth, $V_{GS}$ needs to increase accordingly.
The body effect can be seen as a change in threshold voltage and it is modeled as just that: $$V_{TH} = V_{T0_n} + \gamma \left( \sqrt{2\phi_f+V_{SB}} - \sqrt{2\phi_f} \right)$$ where $V_{T0_n}$ is the threshold voltage when $V_{SB} = 0$, $\phi_f$ is a physical parameter ($2\phi_f \approx 0.6V$ for NMOS and $0.75V$ for PMOS) and $\gamma$ is a process parameter called body-effect parameter ($\gamma \approx 0.4 V^{1/2}$ for NMOS and $-0.5 V^{1/2}$ for PMOS).
For PMOS, the bulk voltage should always be higher than the source because the pn junction is in the opposite direction (source $p^+$ and bulk $n^-$). Then, the threshold voltage should be rewritten as: $$V_{TH} = V_{T0_p} + \gamma \left( \sqrt{2\phi_f+V_{BS}} - \sqrt{2\phi_f} \right)$$

If you want to see what is the influence of the body effect in the small signal model, you should check the small signal model topic.

## Difference between NMOS and PMOS

NMOS transistors share the same the substrate. For that reason, the bulk of all NMOS transistors need to be connected to the same voltage, usually the lowest voltage in the circuit. NMOS share the same bulk

However, PMOS transistors are created inside wells that can be separated from each other. Each well, which is the bulk of the PMOS transistors inside it, can be biased to a different voltage. That way, if we want to avoid the body effect for a particular PMOS by connecting the bulk and the source together, all we need to do is isolate it in a n-well, and connect that n-well to the source of the PMOS. PMOS can be created in different n-wells with independent voltages for the bulks

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