Cutoff Frequency
Cutoff frequency is the frequency above which a transistor's gain falls too low to be useful, and the most quoted version is the transition frequency, fT, the frequency at which the small-signal current gain of the transistor drops to one. At low frequencies a bipolar transistor might multiply base current by a hundred; as frequency rises, internal capacitances and the finite time carriers need to cross the device make that gain fall steadily, and fT marks the point where the transistor stops amplifying current at all. It is therefore a speed rating, comparable across devices and technologies, and a first clue about what a transistor can do in a radio or a fast switch.
Alpha and beta cutoff describe the same limit from two circuit points of view. The alpha cutoff frequency is where the common-base current gain falls to about 70 percent (3 decibels down) of its low-frequency value; the beta cutoff frequency is the same point for the common-emitter current gain, which is far lower because beta is much larger than alpha. If beta is 100, the beta cutoff comes at about a hundredth of the alpha cutoff. Above the beta cutoff, current gain declines roughly in inverse proportion to frequency, so the product of gain and frequency is nearly constant and equals fT. This is a gain-bandwidth product, in the same spirit as the one quoted for operational amplifiers, and gives a quick rule: at a frequency f above the beta cutoff, the current gain is about fT divided by f.
What sets fT is the total time a carrier needs to get through the device. The transition frequency is approximately one divided by two times pi times the sum of several delays: the time to charge the emitter junction, the time to diffuse across the base, the time to cross the collector depletion region, and the time to charge the collector capacitance. In a good design, base transit time dominates, and it scales with the square of the base width, which is why thinning the base lifts speed and also why punch-through limits how thin it can go. Doping the base with a built-in field, as diffused-base transistors do, helps carriers along. The Kirk effect and high-level injection mean fT peaks at a particular collector current and falls beyond it, so a quoted figure is only valid at a stated bias.
The history of the transistor is partly a history of rising fT. The point-contact transistor of 1947 and the early alloy-junction devices of the 1950s worked at audio and low radio frequencies, with cutoffs ranging from the kilohertz range to a few megahertz, enough for hearing aids and AM broadcast radios such as the Regency TR-1 of 1954 but not for television tuners or computers. Diffusing the base and etching a mesa, techniques of the late 1950s, shortened the base dramatically and pushed germanium and silicon devices into the hundreds of megahertz. The 2N2222 of the early 1960s, a general-purpose workhorse, specified a transition frequency of a few hundred megahertz. In the 1980s and 1990s silicon-germanium heterojunction bipolar transistors, an idea associated with Herbert Kroemer, who shared the 2000 Nobel Prize in Physics, brought values in the hundreds of gigahertz, and MOSFETs have their own transition frequencies, set by transconductance divided by gate capacitance.
fT is not the same as bandwidth, and fmax is often the better guide for radio design. A voltage amplifier with a gain of 100 cannot work at fT; with feedback and capacitances it will usually be limited to roughly a hundredth of it or less. The maximum oscillation frequency, fmax, is where the power gain falls to one, and depends on base resistance and collector-base capacitance as well as on fT, so two transistors with the same fT can differ widely in radio performance. Another pitfall is to compare an fT quoted at an optimum current with a circuit run at a different bias. The number matters in modern technology as a figure of merit for radio-frequency chips in phones, wireless links and radar, where designers choose process options by their fT and fmax.
Read cutoff frequency as a transistor's speed limit expressed as a gain-bandwidth product: the carrier transit time sets fT, the alpha and beta cutoffs are views of it from different configurations, and fmax and bias decide how much of it can be turned into a working amplifier.