646 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 42, NO. 4, APRIL 1995

Theoretical Analysis of Transient Behavior of Optoelectronic Integrated Devices

Yu Zhu, Susumu Noda, Member, IEEE, and Akio Sasaki, Member, IEEE

Abstruct- Transient behavior of an optoelectronic integrated device composed of a heterojunction phototransistor and a light- emitting diode is studied theoretically. The transient behavior of the integrated device is investigated by considering: 1) the fre- quency response of a phototransistor and a light-emitting diode, and 2) the optical feedback inside the devices. The analytical expressions describing the transient response of the integrated device are derived, and the rise times in both the amplification and the switching modes also are calculated. By increasing the optical feedback, the rise time in the amplification mode is increased along with an increasing output, while that in the switching mode can be reduced effectively with a saturated output.

I. INTRODUCTION HERE! has been much interest in optoelectronic integrated T devices (OEIDs) [l], [2], which are important for optical

information processing and optical computing. A band diagram of one such device is shown in Fig. 1, where a heterojunction phototransistor (HPT) and a light-emitting diode (LED) [ 11, [3]-[12] or a laser diode (LD) [2], [13]-[22] are vertically and directly integrated. The HPT converts an input light to the amplified current, and the LED or LD driven by the current emits an intensified output light. In addition to the input light, the HPT also responds to the light emitted from the LED (LD)-this is called an optical feedback inside the OEID. The optical feedback plays an important role in realizing various optical functions such as light amplification and optical switching [8]-[ 101. In the amplification mode, the output light changes linearly with the input light; while in the switching mode, the output light jumps abruptly from the low-current state to the high-current state when the input light exceeds a threshold value.

Thus far, a lot of experimental results on the fabrication and the characteristics of the OEID have been reported [1]-[22], and theoretical analysis on the static behavior of the OEID has also been implemented [l], [lo], [16]. However, little attention has been paid to the transient behavior of the OEID; for example, it is still unclear how to estimate the rise time of the OEID.

This paper is focused on the theoretical analysis of the transient behavior of the OEID consisting of HPT and LED. In Section 11, the frequency responses of the HPT and the

Manuscript received June 6, 1994; revised October 11, 1994. The review

Y. Zhu is with the Research Dept. 2, Central Research Laboratories, SHARP

S. Noda and A. Sasaki are with the Department of Electrical Engineering,

IEEE Log Number 9409052.

of this paper was arranged by Associate Editor P. K. Bhattacharya.

Corporation, Tenri, Nara 632, Japan.

Kyoto University, Kyoto 606, Japan.

HPT LED OR LD

INPUT OUTWT LIGHT LIGHT

ABSORPTION RAD1 ATlON

Fig. 1. Energy band diagram of OEID.

Fig. 2. Block diagram of OEID with optical feedback.

LED are briefly described, and then the frequency response of the integrated device is derived. In Section 111, the transient response of the OEID is presented, from which the rise time of the OEID is calculated. The results obtained are summarized in Section IV.

11. FREQUENCY RESPONSE OF OPTOELECTRONIC INTEGRATED DEVICE

Characteristics of an optoelectronic device depend on both the frequency (wavelength) and the modulation signal fre- quency of an input light. The former is called the spectral response, and the latter is the frequency response. In this paper, v denotes the frequency of the light, and w denotes the angular modulation frequency.

The block diagram of the OEID with optical feedback, which is considered as a linear system [23], is shown in Fig. 2, and the frequency response of the optical gain G ( w ) of the OEID can be expressed as

(1) G(w) = s(w>rl(w> 1 - k ( w ) g ( w h ( w )

where g(w) denotes the conversion gain of the HPT, ~ ( w ) the external quantum efficiency of the LED. q f ( w ) the internal quantum efficiency of the LED for the feedback light, and k ( w ) the ratio of the photons which reach the HPT to those emitted by the LED inside the OEID.

0018-9383/95$04.00 0 1995 IEEE

. I

ZHU et al.: THEORETICAL ANALYSIS OF TRANSIENT BEHAVIOR 641

The frequency response of the conversion gain of the HPT is expressed as [241-[291

(2 ) go

g(w) = 1 + j w / w p

where go = ,&7]ho denotes the conversion gain of the HPT at low-frequency regime, and Po = I c / I p and T)ho = ( Ip /q) / (Pi /hui ) are the current gain and the quantum efficiency of the HPT in the low frequency regime, respectively, Ip and I, are the primary photocurrent and the output current, respectively; Pi and hu; are the power and the photon energy of the input light; and wp is the beta cutoff frequency. Since the time constant for the generation of the primary photocurrent is negligible compared to that for the current amplification [24], [30], the quantum efficiency T)h(W)

of the HPT was approximated to be independent of frequency, i.e., m ( w ) = Vho.

The frequency response of an LED can be expressed as [31l, 1321

(3)

where vsp0 a r / T , denotes the quantum efficiency for the spontaneous emission in the low-frequency regime; r and 7,. are the lifetimes for the whole and radiative recombination processes, respectively; w1 is the cutoff frequency of the LED; and w;' is the same order of the magnitude as 7.

We now examine the frequency response of the optical feedback. Since the distance between the integrated HPT and LED is on the order of pm, the delay time caused by the light transmission from the LED to the HPT is in the order of a femtosecond and can be neglected compared to the delays caused by the HPT and the LED. Within the regime of the frequency response of the HPT, the optical feedback is assumed independent of the frequency as

k ( w ) = Lo. (4)

The value of ko is 0 5 ko 5 1, which is determined by the geometrical arrangement of the HPT and LED, and by the overlap of the spectrum responses of the HPT and the LED.

The frequency response of the OEID can thus be expressed as

TABLE I DEPENDENCE OF A1 AND ON THE OFTICAL FEEDBACK

f=O O<f<l f=l f>l

6o04 .- f=0.8 a t- .

fz0.5

f = O I I I I I

0 1 0 0 200

TIME (ns)

Fig. 3. Transient response of OED in amplification mode.

where y = uo/ui and U , and ui are the frequencies of the output and input lights, respectively, and

(7) - ( u p + W I ) + d ( w p + ~ 1 ) ~ - 4 ( 1 - f ) w p w ~

2 A1 =

The dependance of AlandX:! on the optical feedback are listed in Table I. From the characteristics of HPT and LED reported [31]-[38], wp = lo8 Hz, w1 = lo1' Hz, and goq0 = 100 are used in the calculation below.

The transient response of the OEID for f # 1 can be obtained from the inverse Laplace transforms of (6) as

D O

- ( l + j w / w p ) ( l f + ~ w / w , )

Since the cutoff frequency of the LED w1 is higher than that of the HPT WO. the relations X 1 > A2 and [ A l l < 1x21 were used to derive the (9).

written as X1 = -(1 - f ) w p and A2 = -w1. Thus, (9) becomes

( l+ jw/wg, )P l+ jw/wd G(w) = k,g,v ( 5 )

where 77, denotes the quantum efficiency for the output, and q f o the feedback light at low-frequency regime.

When f < 1, (7) and @) can be

111. " W E N T BEHAVIOR OF O E D

The transient response of the OEID for 0 < f < 1 is shown in Fig. 3. It can be seen that the output light of the OEID approaches a definite value Po = y g o q o P ; / ( l - f ) proportional to the input light. Thus, the OEID is stable and in the amplification mode. m e optical gain in the amplification mode can be obtained as

A. Transient Response of OEID FOf simplicity of notation, we set f = kogorlf0 in the

following discussion. When the input light is assumed as a step function in time, the Laplace transform of the output light can be obtained from (5) as

I -

648 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 42, NO. 4, APRIL 1995

TIME (ns)

Fig. 4. Transient response of OEID in switching mode.

The transient response of the OEID with f 2 1 is shown in Fig. 4. When f > 1, we have A 1 > 0. As shown in (9), the output of the OEID increases exponentially with time, which corresponds to the jump in the switching mode. In the case of 1 < f < 3, we have A1 N (f - l)wp, and thus (9) can be simplified as

l )wpt] - 1).

When f = 1, the transient response of the OEID can be obtained from the inverse Laplace transforms of (8) as

1 + exp (-wit)]. (13)

For t > l / w ~ , (13) can be simplified as

Po(t) YgorloPiwpt- (14)

As shown in Fig. 4, the output of the OEID increases with time linearly, which also corresponds to the jump in the switching mode. Since w1 >> wp, the period of l/w1, which is neglected in (14), is much shorter than the rise time of the OEID.

Since the maximum current in the OEID and thus the maximum output light power of the OEID are limited by the external conditions, the output light power in the switching mode increases with time, at first, and saturates when it reaches a maximum. In the switching mode, the voltage across the load resistance-the resistance connected in series with the OEID-is usually much larger than that across the OEID; when the OEID is in the high-current state, the maximum current and the maximum output light power of the OEID can be written as I,, = E / R L and Po,,, = hvoVoIm,/q, respectively, where E and RL are the bias voltage and load resistance, respectively, and h the Planck constant. Thus, the optical gain in the switching mode can be expressed as

when Pi is larger than the threshold value.

AMPLIFICATION SWITCHING MODE MODE

-----.---)

OPTICAL FEEDBACK f

Fig. 5. the switching mode used in the calculation are G, = lo3, lo4, and lo5.

Dependence of optical gain on optical feedback. The optical gains in

For a given input light power, the dependence of the optical gain on the optical feedback for both the amplification and the switching modes is shown in Fig. 5. The optical gain increases with optical feedback when f < 1, and is independent of the optical feedback when f 2 1.

B. Derivative of Output of OEID When input light is incident on the OEID, its output light

power changes continuously from the initial state to the final state. The rise time of the OEID-the time needed for the OEID to reach 90% of the light power at the final state-can be expressed as

O . 9 P O f d p T=s,. *- (16)

Here Poi and Pof are the output light power at the initial and the final states, respectively. As described above, the difference between the output light power at the initial state and that at the final state, namely, the integral interval in (16), increases with increasing the optical feedback in the amplification mode, and is independent of the optical feedback in the switching mode.

The derivative of the output light power of the OEID with respect to time denoted by w in (16) is expressed by

dPo (t) v( t ) = - dt which describes how fast the output of the OEID changes with time, and can be obtained from (9) and (14) as

for f # 1 and

v ( t ) = Ygo%PiWp (19)

for f = 1, respectively, where we considered the derivative of the output for t > l/w1.

ZHU et al.: THEORETICAL ANALYSIS OF TRANSIENT BEHAVIOR 649

1 N / L f = l 4

I . . . . I . . . . J 0 5

TIME (ns) 10

0

Fig. 6. Derivative of output of OED. vo = goqoPiwp is the derivative of output of OED for f = 1.

The time dependence of the derivative of the output is shown in Fig. 6. In the amplification mode with f < 1, the output of the OEID changes with a decreasing derivative and thus will approach a definite value ygovoPi/(l - f ) whether there is an external limitation or not. In the switching mode with f 2 1, the output of the OETD changes with a constant or increasing derivative, and thus will not approach a definite value unless there is an external limitation. It can also be seen that at a given time, the derivative increases with increasing optical feedback.

C. Rise Time

f # 1 can be given by By using (9), (16), and (19), the rise time of the OEID for

where G = P,,/yPi is the optical gain of the OEID. The dependence of the rise time on the optical feedback is shown in Fig. 7.

When f < 1, the rise time in the amplification mode TA can be obtained from (11) and (20) as

The rise time of the OEID in the amplification mode is the same as that of the HPT itself when f = 0 because w1 >> wp. By increasing the optical feedback, the increase in the rise time in the amplification mode is due to the increase of the difference between the output light powers at the initial and the final state. Since the linearity between the input and the output power of the OEID becomes poor with increasing optical feedback, the optical feedback is usually weakened in the amplification mode by inserting an absorption layer between the HPT and LED, and thus the rise time in the amplification

AMPLIFICATION SWITCHING MODE MODE

10-1 100 1 0 1 102 1 0 - 1

OPTICAL FEEDBACK f

Dependence of rise time on optical feedback. The optical gains in Fig. 7. the switching mode used in the calculation are G , = lo3 , lo4, and lo5.

When f > 1, the rise time in the switching mode T, can be obtained by setting G = G, in (20), which can further be simplified as

in the case of 1 < f < 3. The rise time in the switching mode for f = 1 can be obtained from (14) and (16) as

Different from the amplification mode, the rise time in the switching mode decreases with increasing the optical feedback. By increasing the optical feedback, the decrease in the rise time in the switching mode with saturated output power is due to the increase of the derivative of the output.

As shown in (22) and (23), the rise time and the optical gain in the switching mode are mutually related. As shown in Fig. 7, for a given optical feedback, the rise time can be reduced at the expense of losing the optical gain, or the optical gain can be enhanced at the expense of increasing the rise time. Therefore, in order to characterize the OEID in the switching mode, it is better to measure and to present both the rise time and the optical gain at the same time.

In the derivation of (20), go, qo, and f are assumed to be constant during the transient response of the OEID. Because of the current dependence of go and q,, the variation in qo, go, and f are sometimes not negligible. In this case, go and v0, and f in the integration shown in (16), should first be expressed as the functions of the current according the characteristics of HFT and LED; then the rise time of OEID can be obtained by performing the integration analytically or numerically.

At last, we consider the rise time of the OEID composed of HPT and LD. The output of the LD driven by a pulse current exhibits damped relaxation oscillation as [39]-[41]

mode is of the same order as that of the HPT itself. Po(t) = Po[I - cos (wTt) exp (-wmt)] (24)

650 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 42. NO. 4, APRIL 1995

where Po is the output light power at steady state, and w, and w, are the relaxation oscillation frequency and the damping constant, respectively. w,, which is larger than w,, is on the order of lolo Hz [34]. While the transient response of the LD should be described by both w, and w,, the delay time between the current pulse and a steady-state stimulated emission is determined only by the damping constant U,. The frequency response similar to that of LED can be obtained from (24) by neglecting the oscillation and using w, instead of wl. Therefore, (20) and (23), derived for the OEID composed of H F and LED, can also be used to calculate the rise time of the OEID composed of HPT and LD.

IV. CONCLUSION The transient behavior of the OEID has been analyzed by

considering the frequency responses of the HPT, the LED, and the optical feedback inside the OEID. The analytical expressions describing the transient response of the OEID have been derived. The OEID is in the amplification (stable) mode when the optical feedback f < 1, and in the switching (unstable) mode when f 2 1. The rise time in both the amplification and the switching modes have been calculated. In the amplification mode, both the optical gain and the rise time increase with increasing the optical feedback. The rise time in the amplification mode is the same as or longer than

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ifomia, Berkeley. He is Engineering, Kyoto Ur

Akio Sasaki (S’63-M’66) received the B.S. and M.S. degrees in electrical engineering from Kyoto University, Kyoto, Japan, in 1955 and 1957, re- spectively; the Ph.D. degree from the University of Califomia, Berkeley, in 1966; and the Dr.Eng. degree from Kyoto University, in 1976.

From 1957 to 1962 he was engaged in investiga- tions of microwave devices at the Kobe Industries Corporation, Kobe, Japan (now Fujitsu Ltd.). From 1963 to 1966 he was a Research Assistant with the Electronics Research Laboratory, University of Cal-

currently a Professor with the Department of Electrical iiversity, and is engaged in research on heteroepitaxy,

crystalline microstructure, characterization, and quantum optoelectronic prop- erties and devices of HI-V semiconductors. He was a Senior Visiting Fellow in 1985 at the University of Sheffield, Sheffield, U.K., and a representative coordinator of the Spec~al Project Research “Alloy Semiconductor Physics and Electronics” supported hy the Ministry of Education, Science and Culture of Japan, from 1985 to 1987.

Dr. Sasaki is a member of SID, the IEICE of Japan, the Institute of Electncal Engineers of Japan, the Japan Society of Applied Physics, and the Japanese Association of Crystal Growth. He received the Society for Information Display (SID) President’s Citation Awards in 1985 and 1990, and the Achievement Award of the Institute of Electronics, Information, and Communication Engineers (IEICE) of Japan in 1987

Yu Zhu received the B.S. and M.S. degrees In ap- plied physics from Shanghai Jiao Tong University, Shanghai, China, and the Ph.D. degree in electrical engineering from Kyoto University, Kyoto, Japan. His doctoral research covered epitaxy growth and characterization of 111-V semconductors, modeling and fabrication of optoelectronic integrated devices.

In 1992 he joined the Central Research Labo- ratory, Sharp Corporation, where he is currently working on the development of high-speed semicon- ductor devices for microwave and millimeter-wave applications.

Susumu Noda (M’92) was born in Kyoto, Japan, on March 28, 1960. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from Kyoto University, Kyoto, Japan, in 1982, 1984, and 1991, respectively.

From 1984 to 1988 he was with the Central Re- search Laboratory, Mitsubishi Electric Corporation, Hyogo, Japan, and was engaged in the research and development of AlGaA-GaAs distributed feedback (DFB) laser, multiple-quantum-well (MQW) DFB laser, and surface-emitting MQW DFB lasers. He is

now an Associate Professor with the Department bf Electrical Engineering, Kyoto University, and is engaged in research on quantum nanostructures using lattice-mismatched semiconductor systems, nonlinear optical devices using multiple energy levels in QW structure, and optoelectronic integrated functional devices.

Dr. Noda is a member of the Institute of Electronics and Communication Engineers of Japan and the Japan Society of Applied Physics. He received the Ando Incentive Prize for the study of “Light-controlled optical devices” from the Foundation of Ando Laboratory in 1991.