1 Introduction

The 60, 90 and 120 GHz millimeter wave (MMW) bands are of great interest for wireless data transmission as these frequency bands are ‘license free’ (Shi et al. 2001). MMW signal can be easily transported by radio over fiber (ROF) (Yao 2012) from central to the multiple remote antenna unit (RAU) base stations (Hekkala et al. 2010). UTC photodiode-based photonic-wireless transmitter has the potential to be a key component for 75–180 GHz wireless transmission (Shi et al. 2001). The MMW transmitter (Shi et al. 2001) consists of a high-power and large band width photodiodes, built-in amplifier and antenna on the same guiding structure radiates signal directly to the end user. Such approach facilitates microcellular system (Zhu 2011); with several hundred meters diameter communication range provides higher spectrum utilization and capacity. In order to integrate UTC-PD with other photonic and radio frequency (RF) components and to optimize the system performance an electrical circuit model of the device is required. Additionally, the device must have good linearity. In order to gain insight to all the above factors it is important to have a tool to analyze the device performance in particular, to predict harmonics and inter-modulation distortion products \(({\textit{IMD}}_{3})\). Hence techniques for high bandwidth UTC-PD must be investigated without sacrificing its high power.

Efforts have largely been devoted to optimize the device performance through various experiments (Pan et al. 2010; Ishibashi et al. 2000; Ito et al. 2004; Chtioui et al. 2012). However, till date very few comprehensive physics based circuit models of UTC-PD have been reported. Brief accounts of the shortcoming of the existing models (Beling et al. 2008; Kuo et al. 2009) are as follows: Beling et al. (2008) proposed an equivalent circuit of UTC-PD to analyze frequency characteristics of third order inter-modulation distortion product at high photocurrents. Transit-time effects of carriers have not been considered in their work. As a result dependency of bandwidth on absorption layer width could not be shown. Kuo et al. (2009) in their work presented an equivalent circuit model of UTC-PD to study the internal dynamic behaviors particularly bandwidth enhancement effect of GaAs-AlGaAs based UTC-PD under zero bias by combining transit time and RC delay time where \(R\) is the load resistance and \(C\) is the capacitance at the junction. No results have been found on inter modulation distortion and linearity study. To achieve high photocurrent thick absorption layer (1,200 nm) may be employed but this limits the linearity and bandwidth to less than 30 GHz (Chtioui et al. 2012). A solution is urgently required to achieve high photocurrent as well as high bandwidth of the device. On the other hand researchers have found that independently large bandwidth and high photocurrent can be achieved using waveguide photodiode (WGPD) structure. This is reported in Lasaosa et al. (2004). However photocurrent can still be limited by saturation effect of the carriers. Maintaining a constant optical power along the device in an integrated amplifier based WGPD (Piprek et al. 2003) problem can be largely solved but at the cost of high amplified spontaneous emission noise (ASE) during the pre-amplification process. In this method the slow movement of holes does not restrict the carrier speed due to adaptation of very thin absorption layer of the device. In UTC-PD, the problem of slow movement of holes is eliminated by a diffusion block layer near the p-contact.

In this work, a comprehensive UTC-PD equivalent circuit model is developed from the simple integral rate equation (Ishibashi et al. 2001) involving electron and hole movement in the device. Incorporating the effect of self-induced field \((E_{{\textit{ind}}})\) (Ishibashi et al. 2001), the total frequency response of the device is determined. From the total frequency response sub-blocks are formed which are implemented by the circuits containing current controlled, voltage controlled sources and by other passive elements. The circuit implementation of UTC-PD and simulation techniques to measure inter-modulation distortion and frequency response are discussed in detail. Optimum absorption layer width of the device can be obtained from the intersection curves of 3 dB cutoff frequency and output photocurrent. It is shown here that by inserting an inductor in series with load resistance a substantial improvement of 3 dB cutoff frequency can be achieved without sacrificing the output photocurrent and device linearity to a large extent.

The paper is organized as follows: Section 2 is devoted to circuit modeling and simulation methods. Section 3 reports the results on some important characteristics of UTC-PD and the bandwidth enhancement effect due to inductor. Conclusions are drawn in Sect. 4.

2 Circuit model and simulation techniques

2.1 Circuit modeling of UTC-PD

Schematic of UTC-PD is shown in Fig. 1a. Details of the device structure can be found in Ishibashi et al. (2001).

Fig. 1
figure 1

a Schematic of top illuminated UTC-PD and b different mathematical subsections of UTC-PD

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The normalized frequency response of the electron injection current from the absorption layer to the collection layer can be calculated as (Ishibashi et al. 2001)

$$\begin{aligned} J_{N }(\omega )&= \frac{J_{tot} (\omega )}{J_{DC}}\nonumber \\&= \frac{W_A }{W}\left[ {\frac{1}{1+j\omega \tau _A }} \right] .\left[ {1-\frac{j\omega \tau _R }{1+j\omega \tau _R }\left( {1-\frac{k}{J_{DC} }} \right) } \right] \nonumber \\&+\,\frac{W_C }{W}\left[ {\frac{1}{1+j\omega \tau _A }} \right] Sinc \left( \frac{\omega \tau _C }{2}\right) exp \left( -\frac{j\omega \tau _C }{2}\right) \end{aligned}$$
(1)

where the output dc current \(J_{DC}\) is \(\frac{\eta qP_i }{h\upsilon }\), the constant term \(k\) used to derive Eq. (1) is given by \((q\mu _{n}n_{o}E_{ind})/2\) and the efficiency \(\eta \) is (Leavittt and Bradshaw 1991) \(\left( {exp\left( -\frac{W_A }{L_e }\right) -\exp \left( -\alpha W_A \right) } \right) \) considering only drift motion of photo generated minority electrons in the n-type InP collection layer. The descriptions of other device parameters used in Eq. (1) are given in Table 1. In deriving Eq. (1) we have taken into account the self induced electric field (Ishibashi et al. 2001) which develops due to accumulation of holes by the diffusion block layer. This causes the absorption layer quasi-neutral. Due to this field the UTC-PD can be operated even at zero bias condition (Ito et al. 2004) which is not possible in a conventional photodiode. Normalized frequency response of UTC-PD can be represented by small subblocks in Fig. 1b. The upper branch in Fig. 1b represents the absorption layer response due to travelling time of electrons in the absorption layer and dielectric relaxation time of the majority carrier holes and the lower branch contains terms involving travelling time of electrons in the absorption and collection layers. Careful analysis of Eq. (1) shows that it contains some constant and frequency dependent terms written as sum of the product. The basic subsystem elements which construe Eq. (1) are of the following types:

$$\begin{aligned} \hbox {J}_\mathrm{out}(\omega )&= \hbox {J}_\mathrm{in}.\hbox {K}, \hbox {K}= \hbox {constant}{\ldots }\end{aligned}$$
(2a)
$$\begin{aligned} \hbox {J}_\mathrm{out}(\omega )&= \hbox {J}_\mathrm{in} \frac{1}{1+j\omega \tau } {\ldots }\end{aligned}$$
(2b)
$$\begin{aligned} \hbox {J}_\mathrm{out}(\omega )&= \hbox {J}_\mathrm{in} \frac{j\omega \tau }{1+j\omega \tau } {\ldots }\end{aligned}$$
(2c)
$$\begin{aligned} \hbox {J}_\mathrm{out}(\omega )&= \hbox {J}_\mathrm{in}. {\textit{exp}} (-j\omega \tau ) {\ldots } \end{aligned}$$
(2d)
Table 1 Parameter values used for simulation
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The subsystem blocks given by Eqs. (2a)–(2d) can be realized following the basic rules of circuit theory. The first block in Eq. (2a) can be realized by a current controlled current source (CCCS) with gain K. This is shown in Fig. 2a. The term in Eq. (2b) can be realized by the sub circuit in Fig. 2b. In Fig. 2b we use current controlled voltage source represented by CCVS of unit gain converting input current to an equivalent voltage \(\hbox {J}_\mathrm{in}\); CCVS is followed by an R-L circuit to obtain an output current \(\frac{\hbox {J}_\mathrm{in}}{1+j\omega \tau }\), where \(\tau \) is equal to L Henry. The output of this block is connected to CCCS because next sub-block is driven by the current. The subsystem in Eq. (2c) can also be built in a similar way as in Eq. (2b) except inductor (L) is replaced by capacitor (C) equal to \(\tau \) Farad. Finally, the term in Eq. (2d) can be implemented by a CCCS with a delay \(\tau \) s. Multiplicative and additive sub blocks are connected in series and parallel respectively. The negative sign can be realized by connecting the positive end of one sub-circuit with the negative end of the other sub-circuit block. Using the rules for building basic subcircuit blocks shown in Fig. 2 the complete equivalent circuit model of UTC-PD is implemented from Eq. (1). Complete circuit model is shown in Fig. 3a. Here electron-hole pair generation due to uniform optical excitation is represented by a current source. The absorption region is modeled with series inductance L representing the carrier transit time \((\tau _A)\) and C representing the relaxation time \((\tau _R)\) of photo-generated holes in the absorption layer.

Fig. 2
figure 2

Circuit realization of different functional block of UTC-PD

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2.2 Linearity and inter-modulation distortion \(({\textit{IMD}}_{3})\)

Linearity and high optical-power handling are the key requirements for high-performance UTC-PD based photonic wireless transmitter applications. Thus, from the design standpoint, it is important to have tools able to evaluate the performance of UTC-PD and in particular, predict harmonics and \({\textit{IMD}}_{3}\) generation. The two modulated signals are combined at the UTC-PD and the generated inter-modulation products give the measure of linearity. In order to investigate \({\textit{IMD}}_{3}\) in simulation, two tone input signals \(I_{1}\) and \(I_{2}\) with frequency \(f_{1}\) and \(f_{2}\) (typically assumed here as 60 GHz \((f_{1})\) and 60.1 GHz \((f_{2})\) respectively with 100 MHz separation) are fed to the UTC-PD. A transient a.c. analysis is carried out for a signal of duration 0–100 ns. For accurate estimation of the frequency, at least ten cycles of 100 MHz (difference between two input MMW frequencies) equivalent to 10 ns time span must be taken as shown in Fig. 4a. Otherwise, error in frequency estimation will occur. Then Fourier Transform of these time cycles is performed to obtain spectrum of inter-modulation products. This is shown in Fig. 4b. Employing this method accurate simulation of inter-modulation distortion can be found out.

Fig. 3
figure 3

a Equivalent circuit model of UTC-PD \((R, \hbox {R}_{1}=1\Omega )\) and b inductance \(\hbox {L}_\mathrm{S}\) in series with load

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Fig. 4
figure 4

a Time cycles and b spectrum of inter-modulation product

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2.3 Bandwidth enhancement technique

Increased thickness of the absorption layer provides high photocurrent but it reduces the bandwidth of the device. To enhance the bandwidth we propose to connect a peaking inductance \(\hbox {L}_\mathrm{S}\) in series with the load as shown in Fig. 3b. This may be easily implemented in integrated circuit fabrication of the device. This is a well known technique of increasing bandwidth in electronic video circuit. In integrated circuit solution, this additional inductance may be incorporated by means of short-end coplanar waveguide (CPW) stub (Simons 2001), which can also act as impedance transformer (Ito et al. 2003) to the antenna input.

3 Results and discussions

The electrical circuit model of UTC-PD shown in Fig. 3a is implemented in Capture CIS OrCAD_10.5 simulation software. The device parameter values used for circuit simulation are listed in Table 1.

3.1 Output photocurrent and bandwidth enhancement by inductor peaking

To obtain 3 dB cut off frequency from the circuit model of the device, an a.c. sweep from 10 to 1,000 GHz is applied to the input of the circuit shown in Fig. 3a. Input power is kept fixed at 0 dBm. The 3-dB cutoff frequency is noted down for each absorption layer width of the device. The solid black circles and solid black diamond points in Fig. 5 show the 3-dB cutoff frequencies \((f_\mathrm{3dB})\) and output photocurrent respectively at different absorption width. The analytical plot of \(f_{\mathrm{3dB}}\) (Ito et al. 2004) is shown by green squares connecting line in Fig. 5. The output photocurrent is the responsivity times the input power. The analytical output current versus absorption layer width is shown by blue connecting circles in Fig. 5. There is a tradeoff between output photocurrent and 3 dB bandwidth of the device. As \(f_{\mathrm{3dB}}\) increases with increase in absorption layer width the output photo current decreases. The circuit model of UTC-PD provides a tool to optimize bandwidth and output photocurrent with respect to absorption layer thickness and other device parameters. Experimentally (Ishibashi et al. 2001) obtained \(f_\mathrm{3dB}\) at 220 and 90 nm absorption layer width \((W_{A})\) shown by red stars is plotted in Fig. 5 and a close match between experimental and simulation results is found. When peaking inductance \(\hbox {L}_\mathrm{S} (= 5\hbox { pH})\) is incorporated in series with the load (refer to circuit model in Fig. 3b), the \(f_\mathrm{3dB}\) point increases (shown by open red circles) but photocurrents remain same (values are superimposed on black diamond points) as shown in Fig. 5. The simulation results are given in Table 2. We see from Table 2 that with \(\hbox {L}_\mathrm{S} = 5\hbox { pH}\), at 300 nm absorption width, the bandwidth enhancement is 186 GHz when photocurrent is nearly 0.5 mA.

Fig. 5
figure 5

Photocurrent and cutoff bandwidth at different absorption layer width \((W_{A})\) without and with \(\hbox {L}_\mathrm{S}\) (5 pH). (Color figure online)

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Table 2 Simulation results obtained from Fig. 5
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Frequency response of UTC-PD obtained from the circuit model for different values of \(\hbox {L}_\mathrm{S}\) is shown in Fig. 6a. Placement of inductor in series with load causes a series resonance in the circuit. This is called inductor peaking. The resonant frequency extends the intrinsic (that without an inductor) 3 dB cutoff frequency of the device as shown in Fig. 6a. However, it is found that the limited range of the inductor between 1 and 5 pH is effective in doing so. Inductor value more than 5 pH causes a higher resonance peak and effectively does not increase the bandwidth further. Bandwidth versus inductor values are plotted in Fig. 6b. It is earlier shown that insertion of \(\hbox {L}_\mathrm{S}\) does not increase the photocurrent. Since 3 dB bandwidth versus \(W_{A}\) plot in Fig. 5 shifts towards right when \(\hbox {L}_\mathrm{S}\) is present we obtain a new optimum \(W_{A}\) corresponding to the intersection point ‘B’ between photocurrent and new 3 dB bandwidth. If absorption layer width \(W_{A}\) is designed at this new optimum point ‘B’ then larger bandwidth and higher photocurrent can be achieved as shown in Fig. 5.

Fig. 6
figure 6

a Frequency response and b bandwidth of UTC-PD with different \(\hbox {L}_\mathrm{S}\)

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3.2 Inter-modulation distortion \(({\textit{IMD}}_{3})\) from the circuit model

Simulation technique to measure inter-modulation products is discussed with detail in Sect. 2. As shown in Fig. 4b, for two fundamental frequencies at 60 GHz \((f_1)\) and 60.1 GHz \((f_2)\) the inter-modulation products are found at 59.9 GHz \((=2f_{1} -f_2 )\) and 60.2 GHz \((= 2{f_2}-{f_1})\) at the output. The amplitude levels of the input current sources at \(f_1\) and \(f_2\) are same and they are varied from 10 to 150 mA. For each input the amplitude of \({\textit{IMD}}_{3}\)’s are noted. Fundamental \((P_0)\) and \({\textit{IMD}}_{3} (P_{03})\) powers at the output are plotted in Fig. 7 and imaginary point of intersection between \(P_0\) and \(P_{03}\) known as third order intercept point \((IP_{3})\) is found out. Then inserting \(\hbox {L}_\mathrm{S} (= 5\hbox { pH})\) at the output load, \({\textit{IMD}}_3\) is found out in a similar way for an absorption layer width 400 nm. \(IP_{3}\) without \(\hbox {L}_\mathrm{S}\) is 60 dBm and with \(\hbox {L}_\mathrm{S}= 5\hbox { pH}\) is 56 dBm which is within the acceptable limit.

Fig. 7
figure 7

\({\textit{IP}}_{3}\) of UTC-PD with and without \(\hbox {L}_\mathrm{S}\)

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4 Conclusion

A physics based comprehensive circuit model of InGaAs/InP UTC-PD is developed. The result is validated through a comparison with the reported experimental data. Simulation techniques to measure linearity and inter-modulation products are given in detail. We have shown a method by inserting a small inductance in the range 1–5 pH in series with the load gives higher photocurrent and three fold improvement in the bandwidth provided the absorption layer width is chosen optimally. Moreover, insertion of inductor does not sacrifice the linearity of the device to a large extent. The circuit model developed here can be useful to optimize different UTC-PD parameters and provides a common platform for simulation of UTC-PD with other RF and optoelectronic devices by electronic design automation (EDA) tools.