## Rise and fall times, 3 dB frequency, and bandwidth in linear circuit theory

A simple RC circuit is shown in Fig. 24.1 (a). When subjected to a step-function input pulse, the output voltage increases according to

Vout(t) = V0[1 - exp(-1/T1)] (24.1)

where t1 = RC is the time constant of the RC circuit. When the input voltage returns to zero, the output voltage decreases according to

Vout(t) = V0 exp(-1 / T2) . (24 2)

For an RC circuit, it is т1 = t2. The rise and fall times are defined as the time difference between the 10% and 90% points of the voltage, as shown in Fig. 24.1 (b). The rise and fall times of the signal are related to the time constants t1 and t2 by

Tr = (ln9) T1 « 2.2 T1 and Tf = (ln9) T2 « 2.2 T2 . (24.3)

The voltage transfer function H(©) is given by

H (и) = (1 + i ©t)-1 . (24.4)

The bandwidth of the system, Af corresponds to the frequency at which the power transmitted through the system is reduced to half of its low-frequency value. This condition can be written as | H(2 n f ) |2 = 1/2. Thus the bandwidth of the RC circuit is given by

uss 1 ln9 ln9 0.70

A f = f3dB = -— = --------- = ——+—T ~ 7—+—T. (24 5)

2пт 2птг n (Tr +Tf) (Tr + Tf)

The bandwidth is also called the 3 dB frequency, since the power transmitted at this frequency is reduced by 3 dB compared with its low-frequency value.

Next, consider an LED with a rise time Tr as illustrated in Fig. 24.1 (c). As a step-function input current is applied to the LED, the optical output power increases according to

Pout(t) = P0[1 - exp(-t/Tr)] . (24.6)

In analogy to Eqs. (24.1) and (24.4), the power transfer function is given by

HLED(ra) = (1 + i ®T) 1 . (24.7)

The absolute value of the power transfer function is reduced to half at the 3 dB frequency of the LED. Thus the 3 dB frequency of an LED is given by

Af = f3dB = JL = = ^ln9 . _12_ . (24.8)

2пт 2птг п (тг + Tf) (тг + Tf)

Exercise: Derivation of equations. Derive Eqs. (24.3), (24.4), (24.5) and (24.8).

Exercise: Rise and fall time and 3 dB frequency. Consider an LED with a rise time of 1.75 ns. Assume that the fall time of the LED is identical to the rise time. What is the 3 dB frequency of the device? Give the physical reasons as to why Eq. (24.8) gives only an approximate value of the 3 dB frequency.

Solution: A 3 dB frequency of 343 MHz is expected on the basis of Eq. (24.8). In practice the 3 dB frequency can be lower or higher than the calculated value since the rise and fall are frequently not exponential. As a practical rule, the numerical factor 1.2 in the numerator of Eq. (24.8) can vary between

1.0 and 1.5.

24.2 Rise and fall time in the limit of large diode capacitance In diodes used for solid-state lamp applications, the current-injected p-n junction area is large, sometimes as large as the entire LED die. Such diodes have a large capacitance. Denoting the diode capacitance as C and the overall series resistance of the drive circuit and the diode as R, the rise and the fall time of the diode are equal and these times are given by the RC time constant.

In communication LEDs, the current-injected active region is much smaller, so that the spontaneous lifetime rather than the diode capacitance limits the maximum modulation frequency. As a result, communication LEDs can be modulated. Since LEDs do not exhibit strictly exponential changes in power, as predicted by Eq. (24.6), Eq. (24.8) is only an approximation.

Consider an LED in which the p-n junction region extends over the entire area of the die. The LED has a small contact area that determines the size of the light-emitting spot. Such LEDs are
used for communication applications. At zero bias, the capacitance of the LED is given by the depletion capacitance (space-charge capacitance) of the diode. Since the area of the diode is large (e. g. 250 x 250 ^m), the capacitance is large and can amount to 200-300 pF.

As the diode is turned on, the current crowds in the area below the contact. When the p-n junction is forward biased, the capacitance of the LED is given by the diffusion capacitance. The relevant area is, however, not the entire LED die but just the region injected with the diode current.

The reduction of the diode capacitance increases the LED modulation bandwidth. The depletion capacitance can be reduced by mesa etching. However, the mesa should be larger than the contact size in order to avoid surface recombination effects.

There is no viable way to reduce the diffusion capacitance. The diffusion capacitance can be reduced by purposely introducing defects that act as luminescence killers. Such defects reduce the minority carrier lifetime and thereby the diffusion capacitance. Such LEDs can be modulated at several GHz. However, the light output intensity decreases as well so that the overall benefit of such lifetime killers is questionable.

24.3 Rise and fall time in the limit of small diode capacitance Next, we discuss the rise and fall time in the limit of small diode capacitance. This consideration applies, for example, to surface-emitting communication LEDs that have a small-area active region. Consider an LED driven by a constant current that is switched on at t = 0. Electrons are injected into the active region and the carrier concentration builds up. At the same time, the optical output intensity of the LED increases. In the case of the monomolecular recombination model, the light output intensity is directly proportional to the injected minority carrier concentration.

The monomolecular rate equation is given by

(249)

 і

where na is the carrier concentration in the active region, A is the current-injected area of the active region, and d is the thickness of the active region. The steady-state current flow of magnitude I causes a steady-state minority carrier concentration na = It / (e A d). The mean lifetime of the carriers is the spontaneous recombination lifetime t.

Next, consider that the diode is initially in the “off” state and, starting at t = 0, the diode is
injected with a constant current I. When the diode is in the “off’ state, the minority carrier concentration in the active region is very low and we approximate the concentration with na « 0.

Solving the differential Eq. (24.9) for the initial condition na « 0 at t = 0 for a constant injection current yields that the carrier concentration in the active region increases according to

"a(t) = "a (1 - e-'1T) = ejj (' - Є-''T) . (24.10)

The equation reveals that it takes the spontaneous recombination time t to fill the active region with the steady-state carrier concentration. The light output intensity follows the minority carrier concentration directly. Thus, the rise time is given by the spontaneous recombination time.

A similar consideration applies to the fall time of the diode. Once the diode has been switched off, the decay constant for emission is, of course, the spontaneous recombination lifetime. Thus, the fall time of an LED is given by the spontaneous recombination lifetime.

In the case of an undoped active region, the monomolecular recombination equation no longer applies and the bimolecular recombination equation must be used to describe the carrier dynamics. Also, the carrier lifetime is no longer a constant. In this case, the shortest lifetime, i. e. the lifetime that applies when the carrier concentration is at the highest level, can be used.

It should be noted that there are methods to reduce both the rise and the fall times below the limit of the spontaneous recombination lifetime. The rise time can be reduced by current shaping. The fall time can be reduced by carrier sweep-out. Both methods will be discussed below.

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