Design of a resonant-cavity-enhanced photodetector for high-speed applications

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Design of a Resonant-Cavity-Enhanced

Photodetector for High-Speed Applications

Hsin-Han Tung and Chien-Ping Lee,

Senior Member, IEEE

Abstract—We present a theoretical study of the effects of light field distribution on the frequency response of a resonant-cavity-enhanced p-i-n photodetector. Taking advantage of the flexibility of cavity design, one can tailor the light field distribution in the absorption region. Because of the difference in velocities of the carriers, the speed performances of the detector depend on the field distribution and the cavity design. The results of our work indicate that when the maximum of light field intensity happens near the p+ edge of the depletion layer, the device shows the best speed performance. The frequency response, the impulse response, and the step response have been calculated for different structures to demonstrate the importance of the field distribution. Index Terms—Frequency response, impulse response, quantum efficiency, resonant-cavity-enhanced photodector, step response.

I. INTRODUCTION

H

IGH-SPEED and high-sensitivity photodetectors are im-portant for high-bit-rate optical communication systems [1], [2]. The detectors are required to have a wide band-width and a high quantum efficiency. For conventional p-i-n photodiodes, the quap-i-ntum efficiep-i-ncy is limited by the absorption coefficient and the absorbing layer thickness . The highest obtainable for detectors with a perfect

antireflecting coating is . To achieve

high a thick absorbing layer is required. However, a thick absorbing layer means a long carrier transit time which reduces the device’s speed. Resonant-cavity-enhanced (RCE) photodetectors provide a solution to this problem [3], [4]. For such detectors, a thin absorbing layer is put inside a Fabry–Perot cavity. The feedback mirrors of the Fabry–Perot cavity are usually comprised of quarter wavelength stacks (QWS’s) [5] with a periodic modulation of the refractive index. At resonance, construction interference is built up within the cavity to enhance the internal optical field intensity. Since the generation rate, of the electron-hole pairs is proportional to the light intensity i.e.,

can be greatly enhanced. So only a very thin absorbing layer is needed to achieve high quantum efficiency. RCE photodetectors with nearly 100% quantum efficiency [6], [7] have been demonstrated.

Manuscript received July 8, 1996; revised January 22, 1997. This work was supported by the National Science Council of the Republic of China under Contract NSC84-2215-E009-039.

H.-H. Tung is with the Department of Electrical Engineering, National Lien-Ho Junior College of Technology and Commerce, Miao Li, Taiwan, R.O.C.

C.-P. Lee is with the Department of Electronics Engineering, National Chiao Tung University, Hsin Chu, Taiwan, R.O.C.

Publisher Item Identifier S 0018-9197(97)03065-0.

The speed of a photodetector is primarily determined by the combined effect of the capacitance of the detector and the transit time for the photocarriers to move through the absorbing layer. For a high-speed detector, the transit time has to be very short. When the area of the detectors is made sufficiently small, the influence from the capacitance is reduced and the effect of the transit time dominates. Wey

et al. have demonstrated a 2 m 2 m GaInAs–InP p-i-n photodiode with a 3-dB bandwidth as high as 110 GHz [8], [9]. Bowers and Burrus have indicated that bandwidths in excess of 200 GHz should be possible [10].

For high-speed operation, the electric field throughout the intrinsic layer should be very high so that the carriers travel at the saturation velocity during most of the transit. Usually the electron saturation velocity is greater than the hole saturation velocity; taking advantage of this property, one can improve speed performance for proper device designs. On the other hand, the distributions of photogenerated carriers can also have significant influence on the speed response because the average distance that carriers travel depends on the distribution. In this paper, the performances of RCE photodetectors with different carrier profiles as a result of different cavity design are compared. We found that by proper design, we can tailor the carrier profile in the absorbing region to maximize the speed of the detector.

II. RCE p-i-n PHOTODIODE

Fig. 1 shows the schematic diagram of a RCE p-i-n pho-todiode used in this study. The structure is similar to that designed by Tang et al. [7]. The thickness of the p , i, n layers are denoted as and , respectively. The front mirror is composed of a half- -thick SiO plus one Si-SiO quarter-wave pair, and the back mirror consists of a -thick GaAs layer and a 27-period GaAs-AlAS QWS, where is the designed wavelength, 1.3 m divided by the refractive index of the material. The absorption coefficient of the intrinsic InGaAs layer is assumed to be 1.16 10 cm at 1.3 m. The quantum efficiency of the resonant detector can be calculated numerically by using the transfer-matrix method (TMM) [11] and is equal to where is the power reflectivity and is the power transmission. Fig. 2 shows the calculated quantum efficiency versus wavelength for 3/4 m 1/4 m and 5/4 m. The dashed curve is for the reference photodiode without a resonant cavity. It is noticed that the quantum efficiency is significantly increased by a factor of 10 (from 10% to nearly 100%) at 1.3 m. 0018–9197/97$10.00  1997 IEEE

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754 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 5, MAY 1997

Fig. 1. Schematic representation of the studied RCE p-i-n photodetector. Here is the design wavelength, 1.3 m; divided by the refractive index of each layer.

Fig. 2. Calculated quantum efficiency for the studied RCE photodiode with

Lp = 3=4; Li = 1=4; and Ln = 5=4 (solid line). The diode

is designed at 1.3-m wavelength. Quantum efficiency for conventional photodiode without resonant cavity is also shown for comparison (dashed line).

The quantum efficiency of the resonant detector can be also obtained approximately as [12] (1), shown at the bottom of the page, where and are reflectivities of the top and bottom mirrors, respectively, is the round-trip phase delay of the resonant mode within the cavity, and is the resonant mode. The effective absorption coefficient, and the factor is between 0 and 2 depending on the standing wave effect. From [12], we know that when the condition

(2) is satisfied, the quantum efficiency at a certain value is maximized. The parameters we use in the present calculation

at 1.3 m yield

and Under such conditions, (2)

is approximately satisfied. Substituting these values into (1), we get 100%. This means the studied RCE photodiode

structure has been optimized and the resonant wavelength is 1.3 m.

To obtain high quantum efficiency, RCE photodetectors have to be operated at the resonant condition. But even at resonance, one still has the flexibility to change the thickness of individual layers as long as the round trip phase delay is . As will be described later, one can vary and therefore optimize the device’s speed performance while keeping the devices at resonance with high quantum efficiencies by ad-justing the thickness of certain layers. If we vary the ratio

but keep and all other structure

parameters the same, we can obtain the quantum efficiencies for devices with various combinations of and at the resonant condition. Fig. 3 shows the quantum efficiency for the RCE photodiode versus the ratio calculated by the TMM method. It can be seen that is nearly 100% at and . The quantum efficiency changes periodically

whenever (or changes by . Because of the

nature of this periodic property, there are only two distinct

ratios (namely, and

which have (nearly) 100% quantum efficiency but different field distributions in the cavity. Fig. 4 shows the light field intensity as a function of position of these two combinations. For , we find the maximum of light intensity occurs near the p edge of the i region, and the result is shown

in Fig. 4(a). On the contrary, for the

maximum of the field intensity occurs near the n edge of the i region [see Fig. 4(b)]. The two different and nonuniform light intensity distributions have different influence on the speed response of this diode and will be discussed in next section. Also shown in Fig. 4 is the light intensity of a conventional p-i-n photodiode for comparison (dashed line); a significant increase in the light intensity can be clearly seen.

III. FREQUENCY RESPONSE OF RCE PHOTODIODE From Figs. 3 and 4, we know that at the resonant con-dition when there are two different light field distributions for two different ratios. Since the carrier generation rate is proportional to the light intensity, we can thereby obtain different distributions of photogenerated carriers within the i layer. For convenience, we call the light field profile in Fig. 4(a) “Profile 1” and the light field profile in Fig. 4(b) “Profile 2.” Fig. 5 shows the photocarrier distributions for these two different light field profiles. In Fig. 5(a), most of the carriers are generated near the p edge of the i layer, while in Fig. 5(b) most of the carriers are generated near the n edge of the i layer. Taking into account the difference in the drift velocities of the carriers and the nonuniform distribution of the generated carriers in the i region, we calculate the frequency response of the diode below.

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Fig. 3. Thickness dependence of the quantum efficiency for various com-binations of Lp and Ln. To operate the RCE photodiode at the resonant condition,Lp+ Lnis kept as a constant of2.

We start with the linearized continuity equations for the electrons and holes [13]

(3) where p and n are the densities of the holes and electrons, is the volume generation rate, and and are the hole and electron velocities, respectively. We assume a saturated hole velocity of 4.8 10 cm/s and a saturated electron velocity of 6.5 10 cm/s for the intrinsic layer [10]. At the resonant condition, the light intensity distribution within the i region can be expressed as a function of

(see Fig. 4). In the present calculations, and for Profile 1 and for Profile 2). Consider a sinusoidal modulation of optical power illuminate upon the RCE photodiode, the time variation of the generation rate can be expressed in the following form:

(4) where is a modulation index and is a normalization constant. Substituting (4) into (3) together with the boundary

conditions and we get

(5a) (5b)

(a)

(b)

Fig. 4. Light field distribution within the i region of the studied RCE photodiode as a function of position for (a) Lp = ; Ln = and (b)

Lp = 3=4; Ln = 5=4. The dashed curve is for the conventional

photodiode without resonant cavity.

where

(6a)

(6b)

(6c)

(4)

(a)

(b)

Fig. 5. Photocarrier distribution profiles for different combinations of Lp

andLn. (a) WhenLp= ; Ln= ; the maximum occurs at the p+ edge of the i region. (b) WhenLp= 3=4; Ln= 5=4; the maximum occurs at

the n+ edge of the i region.

and are the electron and the hole transit times, respectively, and are given in Appendix A. The current density in the diode is

(7)

where is the electronic charge, is the DC component, is the signal current density, and

(8) where

(8a)

(8b)

The frequency response is given by

(9a)

If the RC effect is included, then

dB

(9b) where is the sum of the load resistance and the diode series resistance and is the diode capacitance plus other parasitic capacitances. In this calculations, is assumed to be 50 and the diode area is assumed to be 2 m 2 m. A reasonable value of is taken to be 10 fF. Fig. 6 plots the frequency response calculated from (9b). If the light field distribution is Profile 2 [Fig. 4(a)], the calculated 3-dB frequency is 204 GHz. For Profile 1 [Fig. 4(b)], the calculated 3-dB frequency is about 234 GHz, and an increase of 30 GHz is observed. In Profile 1, most of the electron-hole pairs are generated near the p edge of the depletion layer. The holes are collected immediately by the p layer while the electrons drift through the depletion layer for a time which depends on the electron saturation drift velocity . If most of the carriers were generated near the n edge of the depletion layer (Profile 2), the transit time and the current would be determined by the hole saturation drift velocity . Since ( 6.5 10 cm/s) is greater than ( 4.8 10 cm/s), a bandwidth increase for Profile 1 is expected. If the ratio of is larger, the difference will be even more evident. The frequency response of the photodiode without a resonant cavity is also shown in Fig. 6 (the dashed–dotted curve) for comparison. The calculated 3-dB frequency is about 216 GHz (Appendix B). For such devices, there is no enhancement for the optical field due to resonance and the light is absorbed almost uniformly in the i region because the i layer is very thin (less than 0.1 m ; photocarriers are generated everywhere in the depletion region. Fig. 7 shows the 3-dB frequency (or bandwidth) as a

function of the ratio. was kept at to

ensure the resonant condition. From Figs. 3 and 7, we obtain the gain–bandwidth product which is shown in Fig. 8. We can

see that (or thicknesses differ by gives

the best gain-bandwidth product and is the optimum design for the present structure. To emphasize further the influence of optical field distribution on the speed of a RCE photodiode, the

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Fig. 6. Calculated frequency response of the studied RCE photodiode. The dashed line shows the response with Profile 1 and the solid line shows the response with Profile 2. Also shown is the calculated response for the conventional photodiode without resonant cavity (dashed–dotted line).

Fig. 7. Thickness dependence of the 3-dB frequency (or bandwidth) for various combinations ofLpandLn. To operate the RCE photodiode at the resonant condition,Lp+ Lnis kept as a constant of2.

response of the photodiode to an impulse of light is computed in the next section.

IV. IMPULSE RESPONSE ANDSTEP RESPONSE

A. Impulse Response

Consider a situation in which an impulse of optical power generates electron-hole pairs at according to the light field distribution. As described earlier, the carrier density

Fig. 8. Thickness dependence of the gain-bandwidth product for various combinations of Lp and Ln. The maximum is about 231 GHz at

Lp = ; Ln = (Profile 1).

distribution function can be expressed as

(10) where is a normalization factor and

from which we get

(11)

After a period of time part of the carriers drift out of the depletion region, and the average electron density left within the i layer becomes

(6)

where is the unit step function. The electron current density is

(12a) Similarly, the number of holes inside the i layer at time is

and the hole current density is

(12b) The total current density is then

(13) Fig. 9 shows the calculated current impulse response. An important figure of merit for digital circuit applications is the fall time, which is defined as the time when the current density falls from 90% of the initial value to 10% of the initial value. The calculate is 1.570 ps for Profile 2 and 1.086 ps for Profile 1. A 50% improvement in is obtained. Since most of the holes in Profile 2 have to take a longer time to drift out of the i region, a clear “tail” is observed in Fig. 9, and that explains why Profile 2 has a much longer . So, for the purpose of high-bit-rate communication, it is a better choice to use Profile 1 in the design of a RCE photodetector. Also shown in Fig. 9 is the impulse response for the photodiode without a resonant cavity; the calculated is 1.289 ps (see Appendix C).

B. Step Response

Since the continuity equations in (3) are linear, the response to an arbitrary input function is then found by convolving impulse response, with [14]:

(14)

Fig. 9. Impulse response for the studied RCE photodiode. The dashed line shows the response with Profile 1 and the solid line shows the response with Profile 2. Also shown is the calculated response for the conventional photodiode without resonant cavity (dashed–dotted line).

For unit step response,

(15)

Substituting (13) and (15) into (14), we have

(16) where

(17a)

(17b) The calculated step response of (16) is plotted in Fig. 10. If we define the rise time as the time when changes from 10% of the final value to 90% of the final value, the calculated is 0.945 ps for Profile 1, 1.234 ps for Profile 2, and 1.078 ps for the conventional photodiode (see Appendix D), respectively. The influence of the light field distribution on the performance of step response is again very clear.

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Fig. 10. Step response for the studied RCE photodiode. The dashed line shows the response with Profile 1 and the solid line shows the response with Profile 2. Also shown is the calculated response for the conventional photodiode without resonant cavity (dashed–dotted line).

V. CONCLUSION

We have studied the high-speed performance of a double-heterostructure RCE p-i-n photodiode. At the resonant condi-tion, by varying the length of while keeping

constant, we can obtain different light field distributions. By properly choosing we can obtain an optimum structure where the maximum light intensity occurs near the p edge of the i layer. In such a structure, most of the generated holes are collected immediately by the p layer while the electrons drift through the depletion layer with a larger drift velocity. By theoretically solving the linearized continuity equations, the calculated 3-dB frequency, gain-bandwidth product, impulse response, and step response all demonstrate the superiority of Profile 1 over Profile 2. So in a RCE structure, one can take advantage of the flexibility of cavity design to tailor the light field distribution in the absorption region to improve the device speed. This concept, demonstrated in a photodetector in this work, should also be useful for other optoelectronic devices.

APPENDIX A The parameters used in (6) are

APPENDIX B

The frequency response of a conventional p-i-n photodetec-tor with light incident from the p side is (B1) [15], shown at the bottom of the page. For n-side illumination, (B1) is valid if the subscripts p and n are switched.

APPENDIX C

For a p-side illuminated p-i-n photodetector without a resonant cavity, we have

(C1) From (C1), we get

The average electron density within the i layer is

and the electron current density is

Similarly,

(8)

and the hole current density is

The total current density is

(C2) APPENDIX D

The step response for a conventional p-i-n photodetector with p-side illumination can be calculated directly by convolv-ing the impulse response (C2) with unit step function . Substituting (C2), into (14), we obtain

where

and

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Hsin-Han Tung, photograph and biography not available at the time of

publication.

Chien-Ping Lee (M’80–SM’94), photograph and biography not available at