4.4 Extracted Device Quantities: Invariants

The well known definition for the current gain cut-off frequency ${\it f}_\mathrm{T}$ is given in (4.10).

$$\vert h_\mathrm {21}\vert _{{\it f}={\it f}_\mathrm{T}}= 1$$ (4.10)

This definition of ${\it f}_\mathrm{T}$ has various approximations in terms of intrinsic small-signal equivalent elements. The most important approximations for a HEMT are (assuming the parasitic inductances and capacitances to be stripped off) according to Baeyens in [23]:

$${\it f}_\mathrm{T}= \frac{{\mit g}_{\mathrm{mi}}}{2\pi \cdot [{\i... ...it R}_{\mathrm{S}} +{\it R}_{\mathrm{D}})}= \frac{1}{2\pi\cdot \tau_\mathrm{T}}$$ (4.11)

and even simpler in (4.12) neglecting the parasitic resistances and the output conductance:

$${\it f}_\mathrm{T}= \frac{{\mit g}_{\mathrm{mi}}}{2\pi \cdot ({\it C}_{\mathrm{gs}} +{\it C}_{\mathrm{gd}})}$$ (4.12)

On the contrary, the inclusion of the pad capacities ${\it C}_{\mathrm{pad}}$ and the fringe part ${\it C}_{\mathrm{fringe}}$ from (4.4) in (4.11) yield an extrinsic charging time and extrinsic ${\it f}_\mathrm{T}$, respectively.

$$\tau_{ext} = \frac{1}{2\pi \cdot {\it f}_{\mathrm{T},ext}}= \tau_... ... g}_{\mathrm{m,ext}}} +\frac{{\it C}_{\mathrm{fringe}}}{{\mit g}_{\mathrm{mi}}}$$ (4.13)

The intrinsic transconductance ${\mit g}_{\mathrm{mi}}$ and the extrinsic transconductance ${\it g}_{\mathrm{m,ext}}$ are then related by:

$${\mit g}_{\mathrm{mi}}= \frac{{\it g}_{\mathrm{m,ext}}}{1-{\it g}... ...m{S}}+{\it R}_{\mathrm{D}})-{\it g}_{\mathrm{m,ext}}\cdot {\it R}_{\mathrm{S}}}$$ (4.14)

where ${\it g}_{\mathrm{ds,ext}}$ is the extrinsic output conductance. Using (4.11)-(4.13) ${\it f}_\mathrm{T}$ is precisely defined for different levels of deembedding.

The quantity ${\it f}_\mathrm{max}$ is defined in several manners depending on the invariants used for its definition [111]. Defining the Unilateral Power Gain ${\it U}$ allows for the highest values of ${\it f}_\mathrm{max}$ in a device representing the maximum gain in a lossless reciprocal deembedding [111].

$${\it U}({\it f})= \frac{\displaystyle \bigg\vert \frac{S_\mathrm ... ...g\vert- 2\cdot {\text {Re}} \bigg(\frac{S_\mathrm {21}}{S_\mathrm {12}} \bigg)}$$ (4.15)

Second, the Maximum Available Gain (MAG) and the Maximum Stable Gain (MSG) can be used for defining ${\it f}_\mathrm{max}$:

$$MAG({\it f})= \bigg\vert\frac{S_\mathrm {21}}{S_\mathrm {12}}\bigg\vert\cdot ({\it k}\pm \sqrt{{\it k}^2-1})$$ (4.16)

for the stability factor ${\it k}$$\geq$ 1. ${\it f}_\mathrm{max}$ is then determined as:

$$\vert MAG\vert _{{\it f}= {\it f}_\mathrm{max}} = 1$$ (4.17)

The MAG drops with a slope of -20 dB/dec as a function of frequency near $\vert MAG\vert$ =1. Kurokawa's stability factor ${\it k}$ is defined from the S-parameters as [111]:

$${\it k}({\it f})= \frac{1- \vert S_\mathrm {11}\vert^2 - \vert S_... ...mathrm {21}\vert^2}{2 \cdot \vert S_\mathrm {12}\vert\vert S_\mathrm {21}\vert}$$ (4.18)

The Maximum Stable Gain (MSG) is used for ${\it k}$$\leq$ 1 is defined as:

$$MSG= \bigg\vert\frac{S_\mathrm {21}}{S_\mathrm {12}}\bigg\vert$$ (4.19)

The MSG drops with -10 dB/dec as a function of frequency. The transition between the two for ${\it k}$= 1 defines the frequency ${\it f}_\mathrm{c}$, from which ${\it f}_\mathrm{max}$ based on MAG/MSG eventually can be extrapolated with a slope of -20 dB/dec for a given gate width ${\it W}_{\mathrm{g}}$. The frequency:

$${\it f}_{c}= {\it f}({\it k}=1)= \frac{\displaystyle 1}{\displays... ...}{{\it C}_{\mathrm{gs}}}+{\it g}_{\mathrm{ds}}{\it R}_{\mathrm{S}}\big)}\bigg)}$$ (4.20)

defines the stability point and is expressed as a function of small-signal elements [162]. ${\it f}_\mathrm{c}$ depends the single finger gate width and is a critical quantity for amplifier design, especially for mm-wave applications. It will be analyzed in Chapter 7 with respect to statistical changes of mm-wave devices.