Cutoff frequency is often measured and presented in academic papers, for example, for high electron mobility transistors. How is this defined exactly? Is it defined as a frequency at which the current gain is unity, or in terms of voltage gain, or perhaps I_out/V_in?
Also, how is this measured in lab? In a common source configuration?
And lastly, what does higher cutoff frequency imply? Does it simply mean that it can operate at higher frequency?
The definition of cutoff frequency depends on the parameter being measured. It is commonly defined as the frequency the value is 3 dB lower than the value at DC (or the lower frequency limit).
-3dB is used because the amplitude of the signal is 0.707 the DC value, and therefore the power (which depends on V^2) is half the DC value.
Note that cutoff frequency defined this way (in fact in nearly any way) is usually only useful for systems where the next frequency effect is much farther away.
Refer to this unity gain frequency discussion. If the device cannot provide current gain at a certain frequency, it can be replaced with passive components; hence it is not useful as an amplifier. fT is also called 'transition frequency' or 'current gain cutoff frequency' in literature.
OTOH fmax is the unity power gain frequency also called 'maximum oscillation frequency' or 'power gain cutoff frequency'
how is this measured in lab
These parameters are measured at the fab and are part of process characterization. Schroder's book "Semiconductor Material and Device Characterization" will be of help if you want to explore the topic further.
fT / fmax both vary with current.
1) If you want increase the fT by increasing the bias current, the device size increases, parasitics increase and reduce the frequency that the device can work at.
2) If you have less bias current, speed suffers as device must overcome its own parasitics before it can drive anything useful.
Does it simply mean that it can operate at higher frequency?
fT is determined by process. In RF circuits, fT/30 performance would be be achievable for commercial circuits over process corners and for reasonably simple circuits only. These numbers show the efficacy of the process. Actual circuits have more going on. For instance, interconnects need to be charged; meaning you have to have larger devices on layout and that means more parasitics.
I think there is no general definition. But it is common practice to use some typical properties of a frequency-dependent device or circuit for defining the start (highpass behaviour) resp. end (lowpass behaviour) of the "passband region".
In some cases such a typical property is the frequency where the magnitude of a sinusoidal signal is 3dB less than the maximum value. This is, for example, the case for lowpass and highpass filters with a "maximum flat" characteristics (Butterworth response). For other filter circuits having a peak (or ripple) within the passband (Chebyshev responses) we have other definitions (based on the ripple).
In some other cases we even can define the cut-off frequency in the time domain (Bessel filters, equalizer,...). In these cases, the cutoff is defined at a frequency where the group delay has a certain value.
Conclusion: There is no fixed definition of a "cut-off frequency" because such a parameter is application-oriented.
Comment 1: There are circuits with an upper as well as lower cut-off frequency (bandpass response). More than that, many amplifiers with capacitive coupling do not work for very low frequencies (low cut-off) and very high frequencies (high cut-off).
Comment 2: The frequency where the gain (current gain for transistors or voltage gain for amplifiers) reaches the value of "1" (0 dB) is called "transit frequency" (not cut-off).
