In Constant field model, the scaling factor of switching energy per gate would be
1/α3
In constant-field (full) scaling of MOSFETs, all dimensions (L, W, tox, junction depth) and the supply voltage are reduced by the same factor α (α > 1), so the electric field inside the device stays constant.
Under this scaling, the gate capacitance \(C = \varepsilon WL/t_{ox}\) scales as \((1/\alpha)(1/\alpha)/(1/\alpha) = 1/\alpha\), and the supply voltage V scales as 1/α.
The switching energy per gate is the energy needed to charge the gate capacitance: \(E = \frac{1}{2}CV^2\).
So \(E \propto \frac{1}{\alpha}\cdot\frac{1}{\alpha^2} = \frac{1}{\alpha^3}\). (For comparison, gate delay scales as 1/α and power per gate as 1/α², while power density stays constant.)
Thus the switching energy per gate falls by the factor 1/α³.
Hence, the answer is 1/α3.
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