## Abstract

A back-illuminated mesa-structure InGaAs/InP modified uni-traveling-carrier photodiode (MUTC-PD) is fabricated and its frequency response is investigated. A bandwidth of 40 GHz and a saturation photocurrent up to 33 mA are demonstrated. A photocurrent-dependent equivalent circuit model is proposed to analyze the frequency response of the high power MUTC-PDs. The influences of the space-charge screening, self-induced electric field and over-shoot effects are discussed in detail based on the model. Fitted curves obtained from the simple equivalent circuit model are found to be in good agreement with the data measured under different bias voltages and photocurrents.

© 2015 Optical Society of America

## 1. Introduction

In the analog domain, wide bandwidth photodiodes (PDs) with high power handling capability and high responsivity are favored by many microwave applications, such as phased array antennas, cable televisions and wireless-over-fiber systems [1]. Various device structures have been proposed to improve the device bandwidth and saturation photocurrent, such as uni-traveling-carrier (UTC) [2], dual-depletion region [3], and partially depleted-absorber [4]. In a UTC-PD, a highly doped p-layer is adopted for light absorption and only electrons drift in the depletion region, thus greatly enhancing the bandwidth and output radio frequency (RF) power due to the higher mobility of electrons over that of holes [2]. A modified UTC-PD is developed by introducing an undoped InGaAs absorption layer in the depletion region to increase its responsivity and bandwidth [5], and it has become one of the most promising PD structures to achieve wide bandwidth and high saturation performance simultaneously. Previous studies have shown that the bandwidth of MUTC-PDs increases along with increasing photocurrent and decreases under excessively high bias voltage. Furthermore, the velocity overshoot effect and self-induced electric field in the absorption region are assumed to help improve the frequency response [5, 6 ]. However, to our best knowledge, there has been no detailed analysis on the bandwidth variation of MUTC-PDs.

In this work, a photocurrent dependent equivalent circuit model is developed to study the high frequency response of MUTC-PDs. Compared with the equivalent circuits used to analyze p-i-n [7] and near-ballistic UTC-PDs [8], circuit elements depicting the influences of photo-generated carriers under high power illumination are included in the proposed model. The model takes both carrier transit time and resistance-capacitance (RC) time constant effects into consideration. Circuit elements are extracted by fitting the microwave scattering parameters (*S*
_{22} and *S*
_{21}) with the measured data under different operating conditions. The frequency response associated with space-charge screening effect, self-induced electric field in the absorption layer, and velocity overshoot effect are thoroughly discussed based on the model. The analysis reveals that the wide bandwidth of the device comes from the self-induced electric field and high electron drift velocity under moderate bias voltage and photocurrent.

## 2. Device structure and results

The epitaxial layer structure of our MUTC-PD is shown in Fig. 1 , which consists of 300 nm graded highly-doped InGaAs absorption region, 200 nm InGaAs absorbing depletion region and 635 nm nonabsorbing depletion region. Similar to the fabrication process described in [9], back-illuminated mesa-structure PDs with 14-μm diameter are fabricated. The responsivity of our MUTC-PD is about 0.5 A/W at 1.55 μm. Figure 2(a) plots the output RF power of the device versus frequency, measured with a two-laser heterodyne beating system (modulation depth ≈100%) [6]. The ripples in the frequency response curve originate from RF signal reflection at the probe, the bias-tee, and the RF sensor head. Since such ripples are not intrinsic to the frequency response of the device, in case of strong ripple, some smoothing is carried out so as to obtain a reasonable bandwidth. The bandwidth of the device reaches 40 GHz for photocurrent ranging from 15 to 25 mA under a reverse bias of 3 V. At a lower reverse bias of 2 V, the bandwidth decreases significantly with increased photocurrent due to the space-charge screening effect, as shown in Fig. 2(b). When the bias is increased to 4 V, there is a slight decrease in the bandwidth compared with that obtained at 3 V. Similar variation of bandwidth has been observed previously [5, 6 ]. The reduced in bandwidth at high bias voltage is attributed to the decreased drift velocity under high electric field in the depletion region, and this will be discussed in the following discussion.

The output RF power of the device at 40 GHz under various reverse biases is plotted in Fig. 3 . The saturation photocurrent, defined as the photocurrent of the device at 1-dB compression point, is 22, 31 and 33 mA under 2, 3 and 4 V reverse bias, with corresponding output RF power of 7.9, 10.9 and 11.4 dBm, respectively. The frequency response of the device will degrade when the photocurrent is higher than the saturation value, and it will be further analyzed in the following discussion.

## 3. Photocurrent-dependent equivalent circuit model

In order to study the significant bandwidth variation along with the photocurrent and bias voltage, an equivalent circuit model is established, which takes both the RC-delay (Region 2) and the carrier transit time (Region 1) into consideration, as shown in Fig. 4
. Both port 1 and port 2 are matched to 50 Ω. *R*
_{1} represents the bulk material resistance, while *R*
_{2} represents the p- and n-contact resistance. *R _{u}* and

*C*are the photo-carriers related resistance and capacitance of the absorbing depletion region, respectively.

_{u}*R*and

_{j}*C*are the resistance and capacitance of the nonabsorbing depletion region (shown in Fig. 1), respectively, which are closely related to the photo-carrier accumulation in the nonabsorbing depletion region. The parallel resistance and capacitance of the two regions are similar to those in a p-i-n PD [7]. The RF electrode geometry is similar with that in [9].

_{j}*C*is the parasitic capacitance of the p-electrode, and the parasitic inductance and capacitance of the coplanar waveguide electrodes are given by

_{p}*L*and

_{c}*C*, respectively.

_{c}The alternating photocurrent flowing to the external circuit (Region 2) is controlled by the time delay *R _{t}C_{t}* circuit (Region 1), which models the transit time response of the photodetector. The current source of Region 2 is related to the voltage

*V*(

*ω*) across the capacitance

*C*as

_{t}*I*=

*g*⋅

_{m}*V*(

*ω*), where

*g*is a constant adjusted to the conversion efficiency [8].

_{m}In our analysis, the values of circuit elements in Region 2 of the equivalent circuit are determined by fitting the measured reflection parameter *S*
_{22}. Thus the RC time constant limited bandwidth *f _{RC}* can be determined. Then the conversion parameter

*S*

_{21}of the entire circuit is adjusted by tuning

*R*and

_{t}*C*to fit the measured frequency response curve. The carrier transit time limited bandwidth

_{t}*f*= 1/(2

_{t}*πR*) can be obtained accordingly. The 3-dB bandwidth of the PD

_{t}C_{t}*f*

_{3dB}can be approximated as: $\frac{1}{{f}_{3dB}^{2}}=\frac{1}{{f}_{t}^{2}}+\frac{1}{{f}_{RC}^{2}}$ [7].

## 4. Fitting results and discussion

The measured *S*
_{22} parameters with photocurrents increasing from 5 to 35 mA under different bias voltages are shown in Fig. 5
. Obvious variation in *S*
_{22} occurs when the photocurrents are greater than the saturation photocurrent. On the other hand, the measured *S*
_{22} under different biases remains almost unvaried for photocurrent lower than the saturation value.

In order to investigate the influence of space-charge screening effect on the parallel resistances and capacitances, the measured and fitted *S*
_{22} parameters under 2 V are plotted on a Smith chart, as shown in Fig. 6(a)
, while the corresponding RF responses and fitted *S*
_{21} curves are shown in Fig. 6(b). The values of the carrier transit time related *R _{t}* and

*C*in Region 1 are so chosen as to obtain the best fitting between the measured and the fitted

_{t}*S*

_{21}curves. The bandwidth is reduced from 35 to 13 GHz as the photocurrent increases from 5 to 30 mA. The significant degradation in bandwidth is due to the fact that the depletion region becomes partially depleted when the photocurrent is larger than the saturation value, which is confirmed by using commercial tool Nextnano [9] to simulate the electric field distribution within the depletion region.

In Fig. 7
, the extracted values of the circuit elements are plotted as a function of the output photocurrent, which are derived by fitting the *S*
_{22} parameters. The measured reflection parameter *S*
_{22} remains almost unvaried for photocurrent lower than saturation values. As a result, the parallel resistances and capacities remain almost constant, as shown in Fig. 7. At small photocurrent, the extracted values of resistances and capacitances are *R _{u}* = 20 kΩ,

*R*= 180 Ω,

_{j}*C*= 161 fF and

_{u}*C*= 35 fF, respectively. During the fitting process, the parasitic capacitance

_{j}*C*is kept at 32 fF, the bulk material resistance

_{p}*R*

_{1}is assumed to be 5 Ω and the contact resistance

*R*

_{2}is 3 Ω, meanwhile

*L*= 78 pH and

_{c}*C*= 1.4 fF. By dividing the fitting process into several stages and carefully limiting the value range of the circuit elements, the values of the circuit elements in our model can be uniquely determined.

_{c}The variation of resistances (*R _{u}* and

*R*) can be explained by the following equations:

_{j}*I*

_{0}is the saturation photocurrent value (

*I*

_{0}= 22, 31 and 33 mA under 2, 3 and 4 V reverse biases, respectively).

*d*and

*s*are the thickness and area of depletion region, respectively.

*q*and

*μ*are the charge and drift mobility of electron, respectively. Under depletion condition,

*n*

_{0}is considered as the residual charge density, which is on the order of 10

^{12}cm

^{−3}for the absorbing depletion region, while 10

^{14}cm

^{−3}for the nonabsorbing depletion region, and it’s far lower than the doping density.

*n*is the accumulated electron density in the undepleted region. For photocurrent lower than

*I*

_{0}, the fitted resistances remain almost constant when there is no charge accumulation. When the photocurrent exceeds

*I*

_{0}, part of the depletion region (

*d*

_{1}) remains depleted, while part of it (

*d*

_{2}) becomes undepleted due to electric field collapse, so the increase of

*n*plays a dominant role in the reduction of

*R*and

_{j}*R*. The values of

_{u}*d*

_{1}and

*d*

_{2}are related to the photocurrent and bias voltage.

*R*is much smaller than

_{j}*R*, since the residual charge density in the InP depletion region are much higher than that in the InGaAs depletion region. Since the charge screening effect is more serious in the absorbing depletion region,

_{u}*R*decreases more rapidly than

_{u}*R*. As shown in Fig. 7(a), when the photocurrent increases to 35 mA, the values of

_{j}*R*and

_{u}*R*decrease to 700 and 110 Ω under 2 V, respectively. At a photocurrent

_{j}*I*= 30 mA under 2V,

*n*is calculated to be ~10

^{16}cm

^{−3}for

*R*(assuming

_{u}*d*

_{2,}

*= 6*

_{u}*d*

_{1,}

*, as confirmed in the following electric field analysis) and ~10*

_{u}^{15}cm

^{−3}for

*R*(assuming

_{j}*d*

_{2,}

*= 1/2*

_{j}*d*

_{1,}

*). Due to the suppressed space charge effect, the reduction of resistance is less obvious in the above photocurrent range of observation under 3 and 4 V [10].*

_{j}The variation of capacitances (*C _{u}* and

*C*) can also be explained by the following equations:

_{j}*ε*is the dielectric constant, $\Delta U$is the voltage variation across the depletion region induced by the alternating photocurrent delivered to the series resistance and load resistance, and $\Delta n$is the variation of the accumulated electron density. $\raisebox{1ex}{$\epsilon s$}\!\left/ \!\raisebox{-1ex}{$d$}\right.$ represents the junction capacitance, and remains almost constant for photocurrent lower than saturation values (

*I ≤ I*

_{0}). $qs{d}_{2}\cdot \Delta n$ represents the variation of accumulated charge. When the photocurrent exceeds

*I*

_{0}, the additional capacitance induced by $\Delta n$must be considered with such abundant accumulation of photon-generated electron, and the changing trend of capacitance is mainly determined by the increase of $\raisebox{1ex}{$qs{d}_{2}\cdot \Delta n$}\!\left/ \!\raisebox{-1ex}{$\Delta U$}\right.$. For photocurrent around 30 mA under 2V, is calculated to be ~10

^{16}cm

^{−3}for

*C*and ~10

_{u}^{15}cm

^{−3}for

*C*, assuming$\Delta U$ = 0.5V. As shown in Fig. 7(b),

_{j}*C*and

_{u}*C*increase to 690 and 53 fF at 35 mA under 2 V, respectively. This variation of capacitance is similar to the case of p-i-n PDs [7].

_{j}In the above discussion, we can conclude that the saturation phenomenon is closely related to the photocurrent dependent parallel resistances and capacitances. For photocurrent higher than the saturation value, similar changing trends of parallel resistances and capacitances can also be obtained when the bias voltage is elevated to 3 or 4 V. The differences of resistances and capacitances between different voltages for photocurrent lower than the saturation value are hard to be reflected in our model, because of the almost overlapped *S*
_{22} parameters.

The measured *S*
_{22} parameters at 15 mA photocurrent under different reverse biases are shown in Fig. 8(a)
. It is seen that the *S*
_{22} parameters under different reverse biases almost overlap with each other. As a result, the resistances and capacitances under different bias voltages remain the same. Since the parallel resistances and capacitances under different biases remain constant for photocurrent lower than the saturation value, it is concluded that the bandwidth variation is mainly determined by the transit time, which can be extracted by adjusting *R _{t}* and

*C*to fit the RF response curve. A maximum bandwidth of 40 GHz is recorded for 15 mA photocurrent under 3 V reverse bias, as shown in Fig. 8(b), where the fitted

_{t}*S*

_{21}curve is obtained with

*R*= 21 Ω and

_{t}*C*= 35 fF, corresponding to a carrier transit time of about 4.6 ps.

_{t}To analyze the bandwidth variation due to the transit time of electron, the extracted transit time is plotted as a function of the photocurrent in Fig. 9(a)
. Under the reverse bias of 3 V, the transit time decreases from 11.4 to 4.5 ps as the photocurrent increases from 5 to 15 mA, remains almost constant for photocurrent ranging between 15 to 25 mA, and then increases again to 8.3 ps for a photocurrent of 30 mA. The decrease of transit time is attributed to self-induced field in the p-type absorption layer and velocity overshoot effect in the depletion region. The self-induced electric field in the p-type absorption region is${E}_{\text{ind}}=\frac{{I}_{photon}}{q{\mu}_{h}{p}_{0}s}$ [11], where *I _{photon}* is the output photocurrent,

*q*is the electron charge, ${\mu}_{h}$is the hole mobility,

*p*is the doping density of the absorption region, and

_{0}*s*is the area of the MUTC-PD. The value of the electric field is proportional to the output photocurrent. When the photocurrent is low, the electric field in the p-type absorption region can be ignored, and the electron will diffuse through the absorption layer. The diffusion time is given by $\tau =\frac{{W}_{A}}{3{D}_{e}}+\frac{{W}_{A}}{{\upsilon}_{th}}$ [12], where

*W*is the thickness of the p-type absorption region,

_{A}*D*= 200 cm

_{e}^{2}/s is the diffusion coefficient of electrons, and${\upsilon}_{th}$ = 5.5 × 10

^{7}cm/s is the thermionic velocity. The calculated diffusion time is about 3.4 ps for photocurrents of 5 and 10 mA. When the photocurrent increases to more than 15 mA, the electric field in the p-type absorption region can reach to several kV/cm, which means that electrons will drift instead of diffusing through this region. Assuming a drift velocity of 3.0 × 10

^{7}cm/s [11, 13 ], the transit time of the p-type absorption region will decrease to 1 ps, which is much smaller than the diffusion time of 3.4 ps at 5 and 10 mA.

When the photocurrent is greater than 15 mA under 3 V, the velocity overshoot effect in the depletion region begin to play a leading role in reducing the transit time. As the photocurrent increases, the space-charge effect and the load ac voltage swing effect reduce the net electric field in the depletion layer to around the critical value (5 ~30 kV/cm) for overshoot transport [14, 15
]. This is confirmed by the electric field distributions within the MUTC-PD, which are calculated by solving the photocurrent-Poisson equation under different reverse bias voltages and photocurrents [9]. The velocity overshoot effect under 3 V can be clearly seen in Fig. 9(b), where the average drift velocity of electron in the 835-nm thick depletion region is plotted as a function of photocurrent. For photocurrent ranging from 15 to 25 mA under 3 V, the drift time in the p-type absorption region is about 1 ps, and the total transit time is 4.6 ps, thus leaving 3.6 ps for the electron to drift though the depletion region, and the calculated average drift velocity through the 835-nm-thick depletion region is about 2.3 × 10^{7} cm/s. When the photocurrent increases to 30 mA, the charge screening effect results in a very low electric field in the undoped InGaAs region. Consequently, the average drift velocity of the electron reduces to 1.0 × 10^{7} cm/s. However, because of the self-induced field in the p-type absorption region, the transit time at 30 mA is smaller than that at 5 mA.

When the reverse voltage is increased to 4 V, only at a photocurrent of 25 mA is the electric field in depletion region adequate for velocity overshoot, as confirmed by Fig. 10(c)
. For photocurrent in the range of 5 ~15 mA, the electric field in the absorbing depletion region is larger than the critical value, so the electron drifts through this region at the saturation velocity of about 1.0 × 10^{7} cm/s. On the other hand, the electric field at a photocurrent of 10 ~20 mA under 3 V allows the electron to drift at the overshoot velocity through the absorbing depletion region, thus leading to a shorter transit time. Consequently, the small signal response of the device under 4 V is slightly degraded compared with that obtained under 3 V. It bears pointing out that the reduced drift velocity in the depletion region under 4 V also has something to do with the scattering and thermal effect under high voltage. The above discussions indicate that further increase in bias voltage is adverse for the MUTC-PD to achieve wide bandwidth, although higher saturation photocurrent can be demonstrated under high voltage. When the photocurrent increases to 30 mA, the electric field is too low, so the drift velocity is reduced. The electric field at different photocurrents under 2 V are also plotted, which shows the electric field around 15mA is adequate for velocity overshoot.

In the above discussion, we can get a conclusion that the device can obtain a higher bandwidth performance only in moderate photocurrent and voltage conditions. On the other hand, in order to achieve higher saturation characteristics, the higher bias voltage is better, but the bandwidth will also degrade under elevated voltage. So there is a balance between wide bandwidth and high saturation performance. According our discussion, the key point is to keep the high overshoot velocity under high voltage and high photocurrent condition, and this problem can only be solved by putting forward devices with new structures.

## 5. Conclusion

MUTC-PDs with a bandwidth of 40 GHz and corresponding saturation photocurrent up to 33 mA are demonstrated. A novel equivalent circuit model is proposed for high power MUTC-PD to investigate its dynamic frequency response. Circuit parameters are extracted by fitting the measured frequency response of the device. The space-charge screening effect, self-induced electric field in the absorption region and overshoot effect are analyzed based on the model. The self-induced electric field and high electron drift velocity are rewarding to achieve wide bandwidth performance. The model has been proved valuable for the design and simulation of high performance MUTC-PDs.

## Acknowledgments

This work is supported in part by the National Basic Research Program of China (Grant Nos. 2012CB315605, and 2014CB340002), the National Natural Science Foundation of China (Grant Nos. 61176015, 61176059, 61210014, 61321004 and 61307024), and the Open Fund of State Key Laboratory on Integrated Optoelectronics (Grant Nos. IOSKL2012KF08 and IOSKL2014KF09).

The authors would like to thank T. Shi and Z. F. Liu for helpful discussion on the circuit model. The authors would also like to thank L. S. Ye for support in device fabrication.

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