For the first question, to calculate the barrier height of a transistor, you can use the following steps:

Determine the work function of the source and drain electrodes.

Calculate the difference between the work function and the electron affinity of the semiconductor material in the transistor.

Multiply the result by the electronic charge to get the Schottky barrier height.

Here is the formula you can use:

SBH = q(Φs - Φb - χ)

where SBH is the Schottky barrier height, q is the electronic charge, Φs is the work function of the source electrode, Φb is the electron affinity of the semiconductor material in the transistor, and χ is the transistor, and χ is the electronegativity.

For the second question, to calculate the carrier mobility from source to drain in a transistor, you can use the following steps:

Measure the drain current (Id) and the gate-source voltage (Vgs) for different values of drain-source voltage (Vds).

Calculate the transconductance (Gm) using the formula Gm = d(Id)/d(Vgs) at a particular value of Vds.

Calculate the drain conductance (Gd) using the formula Gd = d(Id)/d(Vds) at a particular value of Vgs.

Calculate the carrier mobility (μ) using the formula μ = Gm/(CoxW/LGd), where Cox is the gate oxide capacitance, W is the channel width, and L is the channel length.

Note: hese calculations may require certain assumptions and simplifications depending on the specific device and experimental setup.