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For an $n$-channel silicon $\text{MOSFET}$ with $10\:nm$ gate oxide thickness, the substrate sensitivity $\left ( \partial V_{T}/\partial \left | V_{BS} \right | \right )$ is found to be $50\:mV/V$ at a substrate voltage $\left | V_{BS} \right |=2V$, where $V_{T}$ is the threshold voltage of the $\text{MOSFET}$. Assume that, $\left | V_{BS} \right | \gg 2\Phi _{B}$, where $q\Phi _{B}$ is the separation between the Fermi energy level $E_{F}$ and the intrinsic level $E_{i}$ in the bulk. Parameters given are

  • Electron charge $(q)$ = $1.6 \times 101^{-19}\:C$
  • Vacuum permittivity ($\varepsilon _{0}$) = $8.85 \times 10^{-12}\: F/m$
  • Relative permittivity of silicon ($\varepsilon _{Si}$) = $12$
  • Relative permittivity of oxide ($\varepsilon _{ox}$) = $4$

The doping concentration of the substrate is

  1. $7.37\times 10^{15}\:cm^{-3}$
  2. $4.37\times 10^{15}\:cm^{-3}$
  3. $2.37\times 10^{15}\:cm^{-3}$
  4. $9.37\times 10^{15}\:cm^{-3}$
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