## 1. Introduction

The development of new integrated high-speed Si receivers is requested for short distance optical data link and emerging optical storage (OS) systems, notably for the Gb/s Ethernet standard [1] - [8] and Blue DVD (Blu-Ray, HDDVD) [3], [4], [9]. As requirements on bandwidth, gain, power consumption as well as low read-out noise and cost are quite severe, an optimal design strategy of a monolithically integrated solution, i.e. with on-chip photodetector and transimpedance amplifier (TIA), is required.

In optical communication, however, non integrated detectors are usually employed [2] - [8] since the particular indirect energy band properties of Silicon make this semiconductor not very efficient for optical reception at 850nm wavelength. As Si is the most widely used and low cost semiconductor material in electronics and due to the availability of low-cost 850nm transmitters, there is yet a great interest and challenge to integrate such receivers. 1 to 10 Gb/s, high sensitivity and low complexity, low-cost silicon photodetectors for the monolithic integration of optical receivers for short distance applications at 850nm are really an issue as the Si absorption thickness required for high-speed (low transit time and low capacitance) favors thin-film technologies for which the responsivity is low. Some solutions exist but at the price of more costly and complex fabrication processes [10-16]. At the system level, owing to its low dark current (pA range) [17], low capacitor (10fF) for the photodetector [1] and possibility to integrate this detector with high-performance low-capacitance transistors, global thin-film SOI monolithically integrated photoreceivers have potentially higher gain and lower noise performances which in turn, as we will show here, can increase the IC-sensitivity and alleviate this requirement on the photodetector itself. Furthermore only SOI photodiodes have so far achieved bandwidth compatible with the 10Gb/s specification and even higher data rate among the "easy to integrate" Si photodetectors [1], [15], [16] and [18].

In the blue and UV wavelengths, these diodes achieve a high responsivity [17] and then combine all the advantages of high speed, low dark current and finally high sensitivity [1]. This makes SOI receivers the best candidate for blue DVD applications and future optical storage generation. This also suggests that blue wavelength for multi Gb/s short reach optical communication could be used in a near future under the condition that the recent progresses in blue emitting sources make them available [17, 19].

We present here a top-down design methodology, fully validated by Eldo circuit simulations [20] and experimental measurements, which allows to predict and optimize, starting from the speed requirements and the technological parameters, the architecture and performances of the receiver. Our approach generalizes the one proposed in [21] to all inversion regimes. In addition our design strategy is based on the
*m* Partially-Depleted (PD) SOI CMOS implementation in terms of gain, sensitivity, power consumption, area and noise.

In section 2 (Optical Receivers Basics), the simple resistor system is first presented as well as its limitations. The transimpedance amplifier is then introduced and its basic theory and concepts such as transimpedance gain, bandwidth and stability are derived. Important parameters to compare transimpedance amplifiers are also discussed as well as architectures most often used in the high speed communication area. Then in section "Design of Multistage Transimpedance Amplifiers", we present our top-down methodology to design transimpedance amplifiers in the case where the voltage gain of the voltage amplifier used in the TIA is independent of the feedback resistor *R _{f}
*. This is usually the case when the TIA bandwidth is not too close to the transistors frequency limit

*f*of a given technology and leads to a multi-stage approach. Our design procedure is then applied to the design of a 3 stages 1GHz bandwidth transimpedance amplifier in a 0.13

_{t}*m*PD-SOI CMOS technology. Finally, in section "Single stage Transimpedance Amplifier Modeling", we present a top-down methodology to design transimpedance amplifiers when the voltage gain depends on

*R*. This is the case for very high-speed single-stage transimpedance amplifiers. Our design procedure is then applied to the design of a single stage 10GHz bandwidth transimpedance amplifier in a 0.13

_{f}*m*PD-SOI CMOS technology and to the design of a 1GHz bandwidth single stage TIA in a 0.5

*m*FD-SOI CMOS technology.

## 2. Optical Receivers Basics

### 2.1 The simple Resistor Optical Receiver

The optical receiver is a key element in the optical link. It performs the optical to electrical conversion. The receiver consists of a photodetector followed by a preamplifier and eventually one or more post-amplifiers. The performance of an optical receiver is mainly determined by the preamplifier - photodiode combination. In high-speed communication links, the two most important specifications are speed and sensitivity. In many cases, the speed is fixed by the application, while the sensitivity has to be maximized. The ultimate limitation is noise. The main noise sources are the photodiode and the preamplifier. In a good design, the latter contribution is minimal. Little noise is added when no active components are used in the preamplifier. This is the case for the simplest preamplifier possible presented in fig. 1a: a simple resistor *R _{L}
* that performs both the current - voltage conversion and the preamplification. In this figure, the simple receiver is followed by a buffering amplifier with gain A. Its major drawback is the limited maximal achievable bandwidth when low noise is important. For an input current

*i*, the output voltage of the simple optical receiver of fig. 1 a) is given by:

_{ph}in which *C _{ph}
* is the total photodiode capacitance and

*C*is the input capacitance of the voltage amplifier. The bandwidth of this simple receiver is thus given by:

_{inA}The bandwidth is limited by *C _{ph}
* which is often much bigger than

*C*and the transimpedance-gain

_{inA}*R*of the very first stage. As a result, the required bandwidth constrains the maximal transimpedance-gain

_{L}*R*and accordingly the achievable sensitivity of this receiver. Indeed, the equivalent input noise current spectral density of this front-end is given by [21]:

_{L}in which the first term is due to the resistor and the second term to the voltage amplifier. The noise of the latter is concentrated in the equivalent spectral density voltage source at the gate of the input transistor, *Svg _{Amp}
*. As the noise is inversely proportional to

*R*, low-noise operation implies that the pole is located at a relatively low frequency. This receiver is therefore not suitable for high speed communication applications. To achieve high speed in combination with a large resistor, the latter is used as a feedback resistor with an inverting voltage amplifier. The resulting structure is a transimpedance amplifier.

_{L}### 2.2 The Transimpedance Amplifier

The transimpedance amplifier (TIA) (fig. 1b) is the most widely used preamplifier for high-speed optical receivers. It is based on an inverting voltage amplifier with open-loop gain A and a feedback resistor *R _{f}
* to convert and amplify the input current

*i*from a photodiode to an output voltage

_{ph}*v*. The transimpedance amplifier is a circuit with shunt-shunt feedback. As a general rule, this reduces both the amplifier input and output impedances by the loop-gain of the amplifier [23]. In turn, this reduction of input impedance allows one for improving the sensitivity and speed trade-off, we faced in the simple resistor amplifier. In this section, various important aspects of transimpedance amplifiers are analyzed.

_{out}#### 2.2.1 Bandwidth of the Transimpedance Amplifier

To analyze the TIA frequency behavior, the most important capacitors have been included in fig. 1b. The total input capacitance of the circuit *C _{inT}
* consists of the photodiode capacitance

*C*, eventually with its associated parasitics, and the voltage amplifier input capacitance

_{ph}*C*. The output capacitance

_{inA}*C*is the sum of the amplifier output capacitance

_{outT}*C*and the input capacitance of the subsequent stage,

_{outA}*C*. The closed-loop transimpedance-gain is given by:

_{next}For a well designed voltage amplifier with sufficiently large gain A and small output impedance *R*
_{out} this can be simplified to:

This compares well with the simplified transfer function of a system with two well separated real poles *p _{1}
* and

*p*, which is given by [24]:

_{2}The TIA dominant pole *p _{1}
* can theoretically be situated either at the input or at the output node. In between those two extreme cases, both poles approach each other and give rise to a complex conjugated pole pair. This intermediate poles placement is best avoided as it results in a bump in the frequency response with overshoot and long settling times of the transients [24] (see also fig. 2a).

In the case of high speed circuits where the diode capacitance *C _{ph}
* usually dominates the other capacitances and where we try to have the resistance

*R*value as high as possible to increase the transimpedance-gain, the dominant pole is usually located at the input node and the transimpedance bandwidth is then given by:

_{f}Comparing this bandwidth with that given by the simple resistor amplifier one (eq. 2), we see an improvement by a factor A of the transimpedance-gain-bandwidth product of the transimpedance amplifier. This means that for a given bandwidth, a TIA will have a transimpedance gain increased by A compared to the simple resistor amplifier. To be stable the gain A has however to be constrained, as we will see below.

#### 2.2.2 Stability of the Transimpedance Amplifier

As the transimpedance amplifier contains a feedback loop, its stability has to be assured. To analyze this problem, the shunt-shunt feedback amplifier is presented by its equivalent circuit of fig. 2b [23]. From this figure, the open-loop gain and the feedback factor are easily derived:

The loop gain, which is an important factor in the amplifier stability analysis is given by:

To obtain a stable system, the non-dominant pole of the amplifier loop gain has to be sufficiently higher than the 0 dB crossing frequency, which is given (the dominant pole is assumed at the input node) by:

The latter is equal to the TIA closed-loop bandwidth (Eq. 7). Actually, the equivalence of these two equations demonstrates that a TIA behaves as a voltage amplifier in unity-gain feedback configuration. The stability analysis of both structures is therefore identical [21]. To achieve a reasonable phase-margin, non-dominant poles have to be sufficiently higher than the receiver bandwidth. This implies for the second pole in this structure [24]:

where *X _{A}
* is a factor ranging from 2 to 3 depending on the required phase margin [25]. In general, some extra poles may be present on the internal nodes of the voltage amplifier. These must obviously also be considered during the design and be placed at sufficiently high frequencies to guarantee stability. These requirements ultimately limit the maximal achievable transimpedance-gain and bandwidth.

To summarize, the transimpedance amplifier allows one, owing to its feedback mechanism, to increase the frequency of the dominant pole to a value that is A times higher that the one we would have with a simple resistor. However, by increasing the gain A, the dominant pole approaches the second pole. In order to avoid stability problems, we have to limit the gain A such that the dominant pole stays around 3 times smaller that the second pole (the factor 3 corresponds to the classical 60 phase margin condition in open loop configuration).

### 2.3 Comparison of Transimpedance Amplifiers

In analogy with the gain-bandwidth product used for an amplifier, the transimpedance-gain (*R _{f}
*) - bandwidth (

*BW*) product or ZBW has been proposed as the parameter that measures both the speed and the sensitivity performance of the transimpedance amplifier [21]:

_{trans}In a given device, ZBW is a constant. This tells us that, to some extent, transimpedance gain can simply be traded for bandwidth and vice-versa by tuning the feedback resistor. This substitution is only limited by stability considerations. The maximal achievable bandwidth has to be a factor smaller than the first non-dominant pole of the receiver to maintain sufficient phase-margin. This factor of merit also emphasizes the need in critical high speed applications for photodiodes integrated on chip along with the receivers which can reduce the photodiode capacitance drastically (by a factor up to ten times) by removing the bonding capacitor. SOI has here an advantage over other Si integrated technologies for high speed applications (e.g. 10 GBps) because both the photodiode and the amplifier will have smaller capacitances, and also because SOI photodiodes themselves can achieve such high transit time frequencies [1].

### 2.4 Architecture of Transimpedance Amplifiers

Transimpedance amplifier may differ by the kind of voltage amplifiers they use. In the high-speed communications field, two amplifier architectures are mostly used: the single transistor voltage amplifier (STVA) and the CMOS inverter used as an amplifier. A three stages TIA version of both cases is shown in fig. 3a and b. The inverter case has the advantage of auto biasing (the current depends on the gate voltage and the size of the P and NMOS transistor) and then requires no additional bias circuit. The inverter based amplifier has, however, less design flexibility as the gate voltages of the PMOS and NMOS transistors are equal, on the contrary to the case of the SVTA circuit. In very high speed applications, in order to improve crosstalk immunity, differential versions of the transimpedances are also employed and then differential pairs or two single inverters with a dummy branch can be used [26], [27] and [7].

One can distinguish different type of TIAs by their number of stages. Typical designs use either a single-stage or multi-stage approach. It can be shown [21] and will be further developed that applications requesting a high bandwidth compared to the technology's *f _{t}
* are limited to single-stage amplifiers as there is no room for placing the extra poles that are inevitably introduced by the multiple-stage designs. However, for frequencies at least 20 to 30 times lower than

*f*, a multi-stage design is the way to go.

_{t}## 3. Design of Multistage Transimpedance Amplifiers

### 3.1 Optimizing the number of stages of STVA transimpedance amplifiers

We will now describe our methodology to optimize the stages number N that maximizes the voltage gain in transimpedance amplifiers based on N identical STVA amplifiers. It is based on the

where *V _{ea}
* is the Early voltage by

*m*of length and

*L*is an equivalent length in

_{eq}*m*. In the case of the amplifier of figure 3a, we have:

The whole gain of the amplifier is then:

For the single stage amplifier, with the dominant pole at the input and only one non-dominant pole at the output, the condition of stability was given by eq. 11. In the case of a multi-stages amplifier, we now have a multiple pole roll-off. To achieve same phase margin with a N pole roll-off than with a single pole, we must now place each of these N poles at a frequency *X _{N}.X_{A}
* higher than the dominant pole frequency.

*X*is given by [21]:

_{N}As *X _{A}
* has to be larger than 2 to 3,

*X*is approximately equal to N while:

_{N}As a result, the i^{
th
} stage bandwidth of the N-stages voltage amplifier used in a TIA has to be at least (the drain capacitance of transistor M2, *C _{d2
}
* has been neglected here. This will be addressed latter):

By multiplying both sides of this equation by the gain of 1 stage, *A _{Nsi
}
*, and isolating the gain on the right side, we obtain that to be stable the gain of 1 stage has to be lower than:

*GBW _{Nsi
}
* is the gain-bandwidth product of an individual stage i:

By introducing:

where the parameter *f _{t}
* in function of the normalized drain current,

*i*, or of the parameter gm/Id, is a technological curve for a given length which tends towards a maximum in strong inversion.

_{dn}*X*is also a technological curve which depends on the transistor inversion degree and is minimum in strong inversion. We can then rewrite

_{C}*GBW*and the stability condition on the maximum value of the gain as a function of a technological curve

_{Nsi }*vs*. the inversion degree of the transistor:

This last equation combined with the equation of the gain *A _{Nsi
}
* (eq. 13) allows us to write the following condition on

*BW*, a given N, and a given

_{trans}*X*:

_{A}This implicit equation of
*f _{Nstab}
* for this

*f*, we then increase

_{Nstab}*f*and compare and iterate while

_{Nstab}*f*or while

_{Nstab}The method is illustrated graphically in fig. 4. As the *f _{Nstab}
* vs.

*N*exists. In fig. 4a, we can see that for a bandwidth of 1GHz in a typical 0.13

_{opt}*m*PD-SOI technology, eq. 25 features a solution in terms of

*f*over

_{t}*f*(This ratio is not constant over the

_{ttrans }*f*/

_{tmax }*f*)). Therefore, the curve of

_{ttrans }*A*, vs. N and the optimal number of stages,

_{Ns}*N*, can be deduced from there (fig. 5 (b)). In the case of fig. 5 (b),

_{opt}*N*is equal to 9. In the case of fig. 4b, if we increase the transimpedance bandwidth specification to 10 GHz and then work at

_{opt}*f*/

_{tmax }*f*ratio of about 10, eq. 25 has no solution for N greater than 1 and

_{ttrans }*N*is then equal to 1 (To have stable solutions for N greater than 1 would require in fact to have

_{opt}### 3.2 Current mirror impact on optimal stages number and gain

We have neglected so far the current mirror drain capacitance, *C _{d2
}
*, in the calculation of the output pole as it gave us useful indications on the maximal gain vs. the number of stages for a given transimpedance bandwidth without having to choose neither W1 nor W2. However,

*C*can lead to significant and unacceptable loss in phase margin. As the drain currents of M1 and M2 are the same, we can relate W2 to W1 by:

_{d2 }and then introduce a factor *X _{C2}
* independent of W1 and W2 to add to Xc:

As L1 and L2 are usually kept to their minimum value in order to minimize capacitances and area and to maximize speed performances, the maximum value of
*X _{C2}
*:

The selection of
*X _{C2}
* on

*R*, we see that we need

_{f}*i*to be large compared to

_{dn2}*i*. This implies to have

_{dn1}*Vds*. In strong inversion, the latter can be approximated by:

_{sat2 }Once
*X _{C2}
* increases with N due to the decrease of the ratio

*i*/

_{dn1}*i*with

_{dn2}*vs*. N on fig. 6c) and fig. 5b). These values are nearly identical until N=3. This indicates that, as long as its size is kept small because of a good

### 3.3 Noise

To analyze the TIA noise performance, we have to consider its major noise sources which are the feedback resistor, *R _{f}
*, thermal noise, modeled by the voltage noise source,

For the amplifier power spectral density referred to its input, in a first approximation, we can only take into account the noise of the first stage, as, from stage to stage, the relative importance of the noise of each subsequent stage is approximatively divided by the gain of each preceding stage.

where

where the parameter has the well known 2/3 value in strong inversion and 1/2 in weak inversion [28]. We then derive:

The total TIA output noise power spectral density is finally given by:

To compare with the noise of the simple resistor system (eq. 3), we can calculate the equivalent input referred current noise power spectral density of the transimpedance amplifier:

As *R _{f}
* is A (the voltage gain) times higher than

*R*for same bandwidth and same input capacitance, the TIA noise power is reduced by about a factor A. The shape of

_{L}*f*is plotted in fig. 7a. At low frequencies, the noise is usually dominated by the resistor noise (

However, at medium frequency in the bandwidth of the transimpedance, the amplifier contribution to the noise

exhibits a zero at the frequency

To simplify the calculations, as was done in [21], the complete integral is approximated by the sum of three components. They are, respectively:

the integral of the noise of the feedback resistor, which we can approximate by:

Since:

we have:

the integral of the noise of the amplifier up to the transimpedance amplifier bandwidth, which can be approximated by:

This can be rewritten as:

the integral of the amplifier noise from the transimpedance amplifier bandwidth up to infinity, which we can approximated by:

Since [29]:

by defining:

we have:

Finally, the TIA RMS value of the output noise is given by:

(48) |

The
*f _{amp}
*) to the output noise, we see that we need a value of

*gm*(and then of

_{eq}*gm*) as well as a value of

_{1}*R*as high as possible. The RMS value of the equivalent input noise current reduces with

_{f}*R*and is given by:

_{f}It can be easily verify that for same bandwidth and same input capacitance, the input noise current reduces by about a factor
*A _{Ns}
* compared to the simple resistor system. Additional noise sources arise from the photodiode dark current

*I*, the dark current shot noise,

_{dark}*in*and the photocurrent shot noise itself

_{in,dark}*in*. The latter is inherent to the mechanism of photodetection and set the ultimate detection threshold of the ideal detector. We have:

_{in,ph}
*I _{dark}
* can be calculated from [17] and is, for a typical SOI diode with a total length and width of 50m, smaller than 10

^{-12}A. From eq. 50), this yields a dark current shot noise of around 10

^{-11}A for a transition frequency of 1GHz. Both are usually negligible when compared to the other sources of noise. The total input noise is given by:

As illustrated in fig. 7b, the transimpedance readout noise usually dominates at low input current, while at high input current it is the photocurrent shot noise that dominates.

### 3.4 BER, sensitivity and dynamic range

In order to achieve a given BER (Bit Error Rate), *i _{ph}
* must be at least a factor Q times larger than the RMS value of the input noise. For a typical BER of 10

^{-12}, Q must be equal to 7.

By definition, the sensitivity, *S*, is equal to the minimum input power to achieve a given BER. Since *i _{ph}
*=

*R*.

*P*, the sensitivity is given by:

_{in}where *R* is the responsivity of the photodiode. It can be calculated with the model developed in [17]. To ensure that the receiver can achieve the required BER, we must have:

where *P _{inmax
}
* is the maximum input power before saturation of the receiver (due to its dynamic range limitation). In the dark, the current flowing through

*R*is negligible and the output voltage is about equal to the gate voltage of M1. When a photocurrent

_{f}*i*crosses

_{ph}*R*, the maximum reachable output voltage is Vdd, the supply voltage, minus the saturation voltage of the transistor M2,

_{f}*Vd*. We can derive the maximum photocurrent and from there the maximum input power before saturation of the receiver:

_{sat2 }### 3.5. Power consumption

The power consumption of the TIA is equal to the power consumption of the N stages plus the power consumption of the biasing branch:

For a given W1, the power consumption increases with N due to the increased number of stages but also due to the increase of power consumption per stage related to the decrease of
*i _{dn1}
* with N for for a given transition frequency. Once N and the related

### 3.6 Diode capacitance Optimization

The diode capacitance *C _{ph}
* must be accounted for when optimizing the detector. As discussed in [1],

*C*is proportional to the diode area,

_{ph}*A*, which is fixed by the diameter D of the optical fiber that carries the light signal, and inversely proportional to the length

_{d}*L*of the intrinsic zone. To decrease this capacitance, we could increase

_{i}*L*. However this increases the transit time of the carriers in the diode and thus decreases its cut-off frequency

_{i}*f*

_{ctr }[1]. We will then fix the value of

*L*such that

_{i}*f*>

_{ctr }*ft*where

_{trans}*ft*is the desired cut-off frequency of the receiver. Note that as

_{trans}*f*depends on the bias of the diode [1], which is equal to the gate voltage of the input transistor, M1, of the amplifier,

_{ctr }*L*and

_{i}*C*depend on

_{ph}*L*fixes the capacitance

_{i}*C*and the number of finger, m, of the diode, and influences

_{ph}*C*. As the amplifier gain is chosen to fulfill the stability requirements, the value of

_{in}*R*is then adapted in order to maintain the RC frequency of the dominant pole.

_{f}### 3.7 Design of a 1GHz, 3 stages SVTA Transimpedance Amplifier in 0.13 *m* PD-SOI CMOS technology

We will now apply the design methodology presented above to the design of a 1GHz, 3-stages SVTA transimpedance amplifier in a typical 0.13 *m* PD-SOI CMOS technology. The model has been implemented using Matlab. We assume that the photodiode has a total length and width of 50m, which is typical. We first have to choose a value for
*X _{C2
}
* increases compared to

*X*. As a result, the maximum value of

_{C}*V*. However, the ratio of

_{th}Once we have chosen the value of
*i _{dn1}
* are constant. As the bias current increases and

*i*are constant, W2 also increases with W1. Since W1 and W2 are increased, the input capacitance is increased and then

_{dn2}*R*must decrease accordingly, in order to keep the transition frequency at the required value of 1GHz. Note that while the output capacitance is also increased by the increase of W1 and W2, the output pole stays at the same frequency as the output impedance is automatically reduced accordingly by the increase of the bias current. The stability requirement (eq. 18) is thus satisfied. Larger transistors are intrinsically less noisy as gm is increased (see eq. 32). As a result, the TIA noise is reduced when W1 and W2 are increased. This leads to a better receptor sensitivity. A W1 of 4

_{f}*m*seems to be a good trade-off. Note that the output dynamic range is constant with W1 as it is only a function of

*R*is decreased. However, in any case, S is well below

_{f}*P*and the required sensitivity is then achieved. A very good sensitivity of approximately 1W is achieved at a wavelength of 405 nm. This demonstrates the potential of SOI thin-film photodiodes for blue DVD and short-distance optical applications. The bias current is about 200A which leads to a power consumption of:

_{inmax }Modeled performances are in table 1. Numerical simulations of the designed circuit have then be carried out using Eldo and a BSIMSOI model fitted for ST transistors. Both values of transimpedance gain given by Matlab and by Eldo are very close from each other and around 330 k (fig. 10a). However, the BSIMSOI model used in Eldo is slightly different from that we have used in the Matlab design. In the latter, we have indeed used technological curves (*f _{t},
* transistor capacitances, transconductance, output conductances...) vs.

*V*which results in an increase in

_{ea},*A*and in the TIA bandwidth up to 2.5GHz. This reduces the phase margin of the TIA compared to Matlab and leads to a peaking of around 2.5dB in the transfer function given by Eldo. In order to increase the phase margin and reduce the peaking, we can increase the value of

_{Ns}*R*in the Eldo simulation. By increasing

_{f}*R*up to 700k, the peaking is reduced to a value below 0.1dB, the TIA bandwidth reduces to 1.2 GHz, while the TIA transimpedance gain is increased by 2.1.

_{f}We then performed transient simulation of the transimpedance circuit (fig. 10b) for both *R _{f}
* values. The variation of the output voltage,

*v*to the 1ns width photocurrent pulses confirms that in a simulation point of view, both circuits (identical amplifier, but different

_{out}*R*values) are suitable for 1 GBps applications. Compared to the circuit with

_{f}*R*=700k , that with

_{f}*R*=332.8k has steeper response (as its bandwidth is higher) but higher overshooting that requires a longer time to vanish (as its phase margin is lower). The multistage amplifier voltage gain

_{f}*A*that fixes the TIA bandwidth and phase margin, hence its stability, depends on

_{Ns}*V*by about a factor 2. The possibility to tune the feedback resistor in the design is another very interesting and efficient solution. This can be achieved by using a transistor to implement

_{ea}*R*and by tuning its gate voltage.

_{f}Noise simulations were also performed with Eldo. Eldo spectral power density of output noise voltage are close to Matlab results (fig. 10c). Below 200kHz, both Matlab and Eldo total power densities are identical and are dominated by the feedback resistance contribution. Above, the high frequency peak values are related to the low voltage amplifier power densities multiplied by the voltage gain of the amplifier. The higher peak in Eldo is then explained by the higher voltage gain compared to Matlab. The equivalent RMS value of the input integrated noise given by Eldo is 41.5nA for *R _{f}
*=332.8k . This is a little higher than the 27.5nA predicted with Matlab due to the peak. For the Eldo simulations with

*R*=700k , this noise reduces to 19.1nA. As can be expected, by increasing

_{f}*R*, the input noise and sensitivity performance of the receiver are improved.

_{f}### 3.8 Design methodology Summary

To summarize, in our multistage top-down design methodology, we first determine the optimal number of stages (which is greater than 1 when the TIA bandwidth is not too close to the transistors frequency limit *f _{t}
* of a given technology) to optimize the voltage gain of the amplifier and hence

*R*and the transimpedance gain. This maximum voltage gain was shown to be limited by stability considerations and to impose a value on

_{f}## 4. Single-stage Transimpedance Amplifier Modeling

In this section we will focus on the design of a high-speed transimpedance receiver. The term "high-speed" is in fact here related to technology and means that we have to design at a ratio (*f _{tmax
}
*/

*ft*)) close to ten or less. In a 0.13

_{trans}*m*SOI CMOS technology as used here, it corresponds to 10GBps application. As just discussed in the section 3.1, for this kind of applications, the optimal number of stages is 1. This is simply due to the fact that there is no room to place extra poles in the feedback loop to achieve stability. In fig. 4b, we can also see that the

*R*, will be low and comparable with the value of the output impedance of the amplifier. In this case the voltage gain will become dependent on

_{f}*R*. A typical solution to this problem is to use an output buffer. In our case, however, this has to be rejected as it would add an extra pole in the feedback loop. We thus have to accordingly adapt our design scheme to take into account the voltage gain dependency on

_{f}*R*. In the text, we will call this new case of figure the degenerated case compared to the previous one that we will refer as the non-degenerated case.

_{f}### 4.1 Design Methodology of single-stage Transimpedance Amplifier

The schematic view of the single stage transimpedance is presented in fig. 11a. As previously, to simplify the analysis of this shunt-shunt feedback circuit, we can rewrite its small signal schematic in its canonical form [25] (fig. 11b). The amplifier is presented as a controlled current source with an output impedance to take account of *R _{f}
* in the expression of the voltage gain Ao (fig. 11b). The input capacitance,

*C*, consists of the amplifier input capacitance,

_{in}*C*, and the photodiode capacitance,

_{g1}*C*. The transimpedance gain in closed loop,

_{ph}*Z*is then:

_{cl}As previously, assuming that the dominant pole is that of the input, the bandwidth of the transimpedance is given by:

Dynamic range (eq. 54) and power consumption expressions (eq. 55) are not changed compared to the non-degenerated case, while voltage gain Ao and non dominant output pole, which is also the bandwidth of the amplifier, *BW _{1s}
* now depend on

*R*or its inverse

_{f}*g*:

_{f}By making these substitutions, however, noise analysis stays essentially the same. The total output noise power spectral density of the transimpedance (eq. 34), is now given by:

where

Even though the gain and the non-dominant pole now both depend on the value of *R _{f}
*, the dependence of one compensating the other, the condition of stability can be rewritten as a condition on a maximum gain

*Ao*exactly as in the non-degenerated case:

_{maxstab }Again, we see that, for a given bandwidth and a given phase margin, i.e. for a given *X _{A}
*, the maximal achievable amplifier gain dictated by stability conditions is imposed by technology and inversion degree but is independent of

*R*. It is this gain, which is maximum near strong inversion (

_{f}*f*is decreasing,

_{t}*X*is increasing and

_{C}*X*is decreasing with

_{C2}*R*, we cannot directly specify a value of

_{f}where *g _{f}
* is unknown and linked to the gain by eq. (58). As we will now see, for a fixed gain

*Ao*, one can derive the input transistor width, W1, from its length,

_{spec}*L1*, its

*Ao*, we find by eq. (58) a condition on the

_{spec}*R*product to satisfy the temporal specification:

_{f}C_{in}By injecting this equation in the expression of the gain eq. (63), we find:

We can write:

where *C _{g1o}
* is equal to the input capacitor of the amplifier,

*C*, divided by W1 and L1 and is related to

_{g1}Finally, we choose to satisfy the constraints and to optimize the performances of the circuit:

where kc is a proportionality factor strictly smaller than 1 according to eq. (67). fig. 11c shows the evolution of the gain in function of the

### 4.2 Design of a 10GHz single stage TIA in a 0.13*m* PD-SOI CMOS technology

We now apply the design methodology presented in subsection 4.1 to the design of a 10GHz, single stage SVTA transimpedance amplifier in a typical 0.13*m* PD-SOI CMOS technology. The model has been implemented using Matlab. Again we used a typical photodiode with a total length and width of 50m. We first have to choose a value for
*X _{C2
}
*, compared to

*X*(the

_{C}*Ao*that we can choose to ensure stability also decreases. As a result,

_{maxstab }*Ao*decreases. As we are working at the technology frequency limit, the stability limitations prevail over the feasibility (which is by the way independent on

_{spec}*Ao*induces a reduction of W1 (see eq. 66) and then of Id1 and

_{spec}*C*. The reduction of

_{in}*C*is however not sufficient to compensate the decrease of the value of

_{in}*R*due to the reduction of

_{f}*Ao*(eq. 64). Noise and then sensitivity performances are also degraded when increasing

_{spec}^{-1}is required. Increasing

^{-1}seems to be a good trade-off in between all these considerations for a

^{-1}.

However, we also have to determine the value of
*Ao _{maxstab
}
*. The curve is a non-monotonic one as

*f*and

_{t}*X*(as the ratio of W2/W1) mainly decrease, while

_{C2}*X*decreases with

_{C}*V*, which corresponds to a minimum in W1 and a maximum in

^{-1}*R*(eq. 58). The current Id1 strongly decreases with

_{f}*i*and W1 decrease. When further increasing

_{dn1}*i*still decreases but W1 now increases. The input noise also presents a minimum value of around 1.5A that is mainly related to the maximum value of

_{dn1}*R*and, then, a minimum in sensitivity in the range of

_{f}^{-1}.

Note that the small decrease in *R _{f}
* around

^{-1}, while

*Ao*still increases is due to the fact that

_{spec}*C*is increased (and then also

_{ph}*C*) by a decrease in

_{in}*L*in order to maintain the transit time in the diode despite the reduction of

_{i}*Vg1*due to the increase of

*Vds*decreases with

_{sat1}*V*seems a good trade off. Finally, in this range, results shows that it is better in terms of performances (

^{-1}*R*, noise, ...) to take a higher value of

_{f}*V*and a

^{-1}^{-1}. Modeled performances are summarized in table 2.

Numerical simulations of the designed circuit have then be carried out using Eldo and a BSIMSOI model fitted for the ST transistors. The simulated transimpedance gain is around 600 , while the transimpedance bandwidth is 10 GHz (fig. 14a). We then performed transient simulation of the transimpedance circuit (fig. 14b). The variation of the output voltage, *v _{out}
* related to the 0.1ns width photocurrent pulses confirms that in a simulation point of view, the designed circuit is suitable for 10 Gb/s applications. Noise simulations were also performed with Eldo. Eldo spectral power density of output noise voltage is in very good agreement with Matlab results (fig. 14c). Below a few GHz, the total power densities are dominated by the feedback resistance contribution. Above, the high frequency peak values are related to the low voltage amplifier power densities multiplied by the voltage gain of the amplifier. The equivalent RMS value of the input noise integrated up to 10

^{13}Hz given by Eldo is 1.76A which is in perfect agreement with the 1.75A predicted with Matlab.

### 4.3. Design of a 1GHz single stage TIA in a 0.5*m* SOI CMOS technology and scaling perspectives

In order to compare technologies and draw some conclusions on scaling, we have applied the Matlab model to the design of a 1GHz single stage transimpedance in a fully depleted 0.5 *m* SOI CMOS technology [30]. Again we are working at the technology frequency limit here such that the stability limitation prevails compared to the feasibility. The evolution of the performances with
*m* technologies for the design of 1GHz TIA (see table 1 and table 3), we first notice that the use of a more advanced technology has allowed to use 3 stages instead of one owing to a better *f _{t}
*. This allows for a real increase in performance of the transimpedance (transimpedance gain

*Z*increased by more than 16 times, sensitivity improved by 7.5 times). The total transimpedance area should not increase since the use of more stages should be compensated by a reduction of the length of the transistors while the width is about the same. The power consumption is however increased in the 0.13

_{cl}*m*technology by a factor of 4. The higher bias current is mainly due to a much higher

*m*design by reducing W1 by 4 and then be equal to the 0.5

*m*design. In this case

*R*but also the input noise will increase leading to a slightly reduced sensitivity of 1.56W (see fig. 9) but still more than 5 times better than the sensitivity of the 0.5

_{f}*m*design. The use of 0.13

*m*technology also allows for an increase of the photodiode performance (smaller capacitance and higher photosensitive to total area ratio owing to the reduction of the size of the P+ and N+ areas).

In a similar way, the uses of very advanced technology nodes with improved *f _{T}
*, such as the 22nm CMOS one with typical

*f*of about 400 to 500GHz (improved by about a factor 4 compared to that of a 0.13

_{T}*m*technology), could allow for improving the sensitivity of the receiver at 10GHz by allowing for a multi-stage approach and a further reduction of

*C*. Also a single stage 40Gb/s could be attempted. A SOI integrated PIN photodiode with an intrinsic length reduced to below 1

_{ph}*m*could fulfill such requirements in terms of transit time cutoff frequency, but with an increased capacitance [1]. This, coupled with the reduced intrinic voltage gain of the single stage amplifier related to the reduced early voltage of ultra short transistor, might result in too low

*R*and sensitivity values. To compensate a reduced photodiode total surface but with no reduction of input power (i.e. relying on progress at the emitter side: better focused optical signal, higher power density emission at the source,...) might be necessary.

_{f}### 4.4 Design methodology Summary

To summarize our single-stage top-down design methodology, we have shown that, as the voltage gain is now dependent on *R _{f}
*, the maximum voltage gain is limited, on one hand, by stability considerations (which usually prevail in weak or moderate inversion regime), and on a second hand, by feasibility considerations (which usually prevail in strong inversion) and that this results on a constraint on W1. The choice of

*R*, noise and then sensitivity performance. The choice of

_{f}## 5. TIA experimental characterization

A single-stage 10GHz SVTA Amplifier was realized in the ST 0.13*m* SOI CMOS technology. It is co-integrated with a PIN photodiode and a 50 output buffer for the RF measurements (fig. 15a). The latter has a 3mA bias current, a 0.79 voltage gain and a 1A equivalent input noise. Modeled and simulated receiver bandwidths were 10GHz, both before and after insertion of the output buffer. TIA modeled performances are presented in table 4. These expected performances are not as good as those reported in table 2. This is mainly because, in the actual design, we imposed a transit time frequency of the photodiode 1.5 times higher (i.e. 15 GHz) than the bandwidth of the TIA itself in order to make sure to measure the TIA cutoff frequency. This in turn imposed a smaller intrinsic length to the photodiode and a higher diode capacitance. The
*R _{f}
* and

*Z*are decreased which in turn also reduces the noise performances of the TIA.

_{cl}A DC transimpedance gain of 212 was measured which is in very good agreement with the 214 gain predicted by the model. For the AC measurements, we used a modulated 1mW 830 nm laser. The remaining available optical power at the fiber output after modulation and fiber coupling losses was less than 0.01mW which is far below the sensitivity of our receiver and made these measurements impossible. One solution could be to use a femtosecond pulsed laser either at 850nm or 425nm by frequency doubling for example. Because we expected the problem of sensitivity with opto-electrical measurements, we had also designed a "photodiode emulator". The latter was a single transistor voltage amplifier with its output connected at the input of the transimpedance (fig. 15b). The output capacitance of this amplifier was designed to be very similar to that of the PIN diode while its output impedance was kept sufficiently larger than *R _{f}
* in order to have an equivalent input TIA pole as close as possible to the real case. By varying the gate voltage of the input transistor of this diode emulator, an equivalent photocurrent is created at the TIA input. This allows for the electrical characterization of the TIA bandwidth. A photograph of the chip is shown in Fig 16a. S parameters measurements were done in the 40MHz-40GHz band. These measurements were obtained after a calibration to remove the impedance effect of the RF probes and cables used to connect the device to the spectrum analyzer.

The bandwidth of the TIA is characterized by the *S _{21}
* parameter which is equal to

*gm*. It is dominated in the frequency band of interest by the transimpedance poles.

_{a}.Z_{cl}.A_{vout}*A*is the gain of the output buffer which is a little less than one (-1.8dB) in the bandwidth of interest. Measured curve of

_{vout}*S*

_{21}*vs*. frequency is compared to the Eldo simulated curve in fig. 16b. Both are very similar and have a bandwidth slightly exceeding 10GHz. The few dB difference between the low frequency plateau of

*S*is related to a small difference in bias currents. The measured current consumption (equal to 2x

_{21}*Id*+2x

_{B}*Id*+

_{2}*Id*) is equal to 5mA while the simulated one is equal to 6mA. This confirms experimentally the ability of 0.13 m CMOS SOI technology for 10Gb/s transimpedance circuits.

_{out}## 6. Conclusions

In this chapter the design of the transimpedance amplifier, which is the most used receiver front-end for high speed optical applications, has been investigated.

In the Optical Receivers Basics section, the simple resistor system was first presented as well as its limitations. The transimpedance amplifier was then introduced and its basic theory and concepts such as transimpedance gain, bandwidth and stability were derived. Important figures of merit to compare transimpedance amplifiers were also discussed as well as some architectures most often used in the high speed communication area.

Then in the Design of Multistage Transimpedance Amplifiers section, we have presented our top-down methodology to design transimpedance amplifiers in the case where the voltage gain of the voltage amplifier used in the transimpedance amplifier is independent of the feedback resistor *R _{f}
*. We have shown that this happens when the bandwidth of the transimpedance amplifier is not too close to the transistors frequency limit

*f*of a given technology and leads to multi-stages approach. Our systematic design procedure was then applied to the design of a 3-stages 1GHz bandwidth transimpedance amplifier in a 0.13

_{t}*m*PD-SOI CMOS technology. It has allowed us to determine the optimal number of stages (3) to optimize the voltage gain of the amplifier and hence

*R*and the transimpedance gain. This maximum voltage gain was shown to be limited by stability considerations. Model Results were validated by Eldo numerical simulations.

_{f}Finally, in the Single-stage Transimpedance Amplifier Modeling section, we have presented a top-down methodology to design transimpedance amplifiers when the voltage gain is dependent on *R _{f}
*. This is the case for very high speed single stage transimpedance amplifiers. In this case we showed that the maximum voltage gain was limited, on one hand, by stability considerations (which usually prevail in weak or moderate inversion regime), and on a second hand, by feasibility considerations (which usually prevail in strong inversion). Our design procedure was applied to the design of a single-stage 10GHz bandwidth transimpedance amplifier in a 0.13

*m*PD-SOI CMOS technology and to the design of a 1GHz bandwidth single-stage transimpedance amplifier in a 0.5

*m*FD-SOI CMOS technology. Model Results were validated by Eldo numerical simulations and measurements.

Model results for the 3-stage 1GHz bandwidth transimpedance amplifier in a 0.13 *m* PD-SOI CMOS technology showed that *C _{in}
* was very low owing to 1) very good AC performances of thin-film SOI photodiodes (the monolithic integration which avoid bonding capacitor is possible due to the availability of thin-film SOI diodes with 1GHz bandwidth and the very low capacitance of the SOI PIN photodiodes themselves) and 2) the low capacitances of MOS transistors in SOI technology. This fact combined with the high technology

^{-12}) is projected. This shows the adequacy of 0.13

*m*PD-SOI CMOS technology for Blue-DVD applications where the todays specifications are to be able to read input optical power of a few W [9] but at a 4 times lower bandwidth (250MHz). Coupled with the low responsivity of the thin-film SOI photodiodes at 850nm (0.005A/W), a sensitivity of 50W or -13dBm is still achieved which is considered as an adequate sensitivity receiver for optical communications and comparable to CMOS realization reported in the literature using non integrated diodes with a 90 time better responsivity [31]. This shows the potential interest of this technology for monolithically integrated 1Gb/s optical communication receivers at 850nm or even to realize a blue DVD player which is compatible with the previous red standard.

The design of a 1GHz bandwidth single-stage transimpedance amplifier in a 0.5 *m* FD-SOI CMOS technology has allowed us to draw conclusions on the evolution of the performances vs. the CMOS technology scaling down, when compared to the previous design. We first notice that the use of a more advanced technology has allowed us to use 3 stages instead of one owing to its better *f _{t}
* for same TIA bandwidth. This has naturally allowed a real increase of the performances of the transimpedance: transimpedance gain

*Z*by more than 16 times, sensitivity improved by 7.5 times, but at a 4 times higher power consumption in the 0.13

_{cl}*m*technology. At same power consumption the receiver sensitivity in the 0.13

*m*technology, if slightly reduced, still remains more than 5 times better than the sensitivity of the 0.5

*m*design.

Concerning the design of a single stage 10GHz bandwidth transimpedance amplifier in a 0.13 *m* PD-SOI CMOS technology, the low *C _{in}
*, due as previously to the monolithic SOI integration, has enabled us to achieve the highest feedback resistor and one of the highest gain and then the highest transimpedance gain-bandwidth product ZBW (see table 5), as we are achieving one of the highest bandwidth reported for 10Gb/s (theoretically a bandwidth of about 3/4 the data rate is sufficient so that 10 GHz could be related to 13Gb/s operation)). This is achieved without using inductive (LP) or capacitive (CP) peaking techniques which allows to increase ZBW (or the bandwidth for same

*R*) but at the expense of chip area (inductors or capacitors to integrate), design complexity, need for tuning... Noise performances are also very good, especially compared to other CMOS designs. The power consumption of the TIA preamplifier (not the complete receiver) is one of the best achieved so far. However, coupled with the low responsivity of thin-film SOI photodiodes at 850nm, a sensitivity of 1.35

_{f}*mW*or +1.3dBm is achieved for this monolithically integrated receiver for 10Gb/s optical communications at 850nm. At 405nm on the other hand, the same receiver achieves a high sensitivity of -13dBm. In order to increase the sensitivity at 850nm wavelength however, we can try to increase the diode responsivity. Promising solutions such as the use of SOI CMOS compatible GeOI photodiodes have been discussed in [1]. We can also try to further decrease the transimpedance noise by increasing

*R*. With a single-stage approach at a bandwidth of 7.5GHz, our model predicts a 850nm sensitivity of -1.25dBm. Combining this with an avalanche gain of 4 that has been demonstrated with thin-film SOI PIN diodes at 10 Gb/s [18], adequate sensitivity of at least -7dbm [8] should be achieved at 10Gb/s. To further improve the sensitivity, we think that the best solution is to attempt a multistage approach so that the voltage gain (and hence ZBW) could be improved. By reducing the bandwidth to about 7.5GHz and using inductive or capacitive peaking if necessary this could be achieved. If not, the uses of very advanced technologies with improved

_{f}*f*, such as 22nm CMOS ones with

_{t}*f*of about 400 to 500GHz (improved by about a factor 4 compared to that of a 0.13

_{t}*m*technology), could allow for improving the sensitivity of the receiver at 10GHz by allowing for a multi-stage approach and a further reduction of

*C*. Finally, in such technologies higher data rate could be attempted, e.g. a single stage 40Gb/s. A SOI integrated PIN photodiode with a shorter intrinsic length could fulfill such requirements in terms of transit time cutoff frequency [1], but with an increased capacitance value. This, coupled with the reduced intrinic voltage gain of the single stage amplifier related to the reduced early voltage of ultra short transistor, might result in too low

_{ph}*R*and sensitivity values. To compensate a reduced photodiode total surface but with no reduction of input power (i.e. relying on progress at the emitter side) might be necessary.

_{f}