E.1 Definitions

Quantitative definitions of , e.g., in terms of a surface potential displacement, drain current, or gate capacitance can be found in almost any text book relevant to the subject. The particular problem of threshold voltage definition is that there are many different definitions which are not consistent, even though many of them are physically well-motivated. Worse yet, some of the definitions are ambiguous as they demand the selection of other threshold parameters or definitions of effective geometry parameters. The following list gives the definitions relevant to this work:

1.

The strong-inversion threshold voltage : $\vert\ensuremath{\psi _{\mathit{s}}}\xspace (\ensuremath{V_{\mathit{GS}}}\xspace )\vert = 2\vert\ensuremath{\Phi _{\mathit{B}}}\xspace \vert$

The surface potential displacement is twice the built-in potential , i.e., the band structure is inverted at the surface.

This definition applies only to long-channel devices with constant channel doping. cannot be measured directly.

2.

The threshold current method : $\ensuremath{I_{\mathit{D}}}\xspace (\ensuremath{V_{\mathit{GS}}}\xspace ) = \ensuremath{I_{\mathit{T}}}\xspace \frac{W_\mathit{eff}}{L_\mathit{eff}}$

For a fixed drain-source voltage the drain current equals some threshold current at $\ensuremath{V_{\mathit{GS}}}\xspace = \ensuremath{V_{\mathit{T}}}\xspace$.

This definition is ambiguous because of the definition of and is only restrictly applicable to small devices because of the uncertainty in $L_\mathit{eff}$ and $W_\mathit{eff}$.

3.

The $\sqrt{\ensuremath{I_{\mathit{D}}}\xspace }$ method : $\sqrt{\ensuremath{I_{\mathit{D}}}\xspace (\ensuremath{V_{\mathit{GS}}}\xspace )... ...opto (\ensuremath{V_{\mathit{GS}}}\xspace -\ensuremath{V_{\mathit{T}}}\xspace )$

In saturation ( $\ensuremath{V_{\mathit{DS}}}\xspace >\ensuremath{V_{\mathit{GS}}}\xspace -\ensuremath{V_{\mathit{T}}}\xspace$) the tangent to $\sqrt{\ensuremath{I_{\mathit{D}}}\xspace (\ensuremath{V_{\mathit{GS}}}\xspace )}$ at the highest slope intersects the -axis at .

This definition applies only to long-channel devices.

4.

The fitted threshold voltage : $\left<\vert\ensuremath{I_{\mathit{D}}}\xspace (\ensuremath{V_{\mathit{GS}}}\xsp... ...)-f(\ensuremath{V_{\mathit{GS}}}\xspace ,\ldots)\vert^2\right> \rightarrow\min$

A given model function $f(\ensuremath{V_{\mathit{GS}}}\xspace ,\ldots)$ is fitted to IV data via the set of parameters (including ).

This definition is ambiguous because it depends on the choice of the model function, parameter set, and operating points.

5.

The linear threshold voltage : $\ensuremath{I_{\mathit{D}}}\xspace (\ensuremath{V_{\mathit{GS}}}\xspace ) \prop... ...-\ensuremath{V_{\mathit{T}}}\xspace -Vds/2)\ensuremath{V_{\mathit{DS}}}\xspace$

In linear operation ( $\ensuremath{V_{\mathit{DS}}}\xspace <\ensuremath{V_{\mathit{GS}}}\xspace -\ensuremath{V_{\mathit{T}}}\xspace$, usually 50mV) the tangent to $\ensuremath{I_{\mathit{D}}}\xspace (\ensuremath{V_{\mathit{GS}}}\xspace )$ at the highest slope intersects the -axis at $\ensuremath{V_{\mathit{T}}}\xspace +Vds/2$.

This definition is unique ( $\lim_{\ensuremath{V_{\mathit{DS}}}\xspace \to 0}$) and universal.

6.

The saturation threshold voltage : $\ensuremath{I_{\mathit{D}}}\xspace (\ensuremath{V_{\mathit{GS}}}\xspace ) = \ensuremath{I_{\mathit{T}}}\xspace$, $\ensuremath{I_{\mathit{T}}}\xspace \leftarrow \ensuremath{I_{\mathit{D}}}\xspace (\ensuremath{V_{\mathit{GS}}}\xspace =\ensuremath{V_{\mathit{T,lin}}}\xspace )$

For a fixed drain-source voltage the drain current equals the threshold current at $\ensuremath{V_{\mathit{GS}}}\xspace = \ensuremath{V_{\mathit{T}}}\xspace$. For $\ensuremath{V_{\mathit{GS}}}\xspace =\ensuremath{V_{\mathit{T,lin}}}\xspace$ the tangent to $\ensuremath{I_{\mathit{D}}}\xspace (\ensuremath{V_{\mathit{DS}}}\xspace )$ at the smallest slope intersects the -axis at .

This definition is unique (because of the unique definition of ), universal, and consistent with (for long-channel devices in saturation).