VNA measurements of UTC-PDs

Figure 2 shows a broadband, low-cost, and high-responsivity evanescent waveguide UTC-PD fabricated at III-V Lab using all 2-in InP processing including etched facet with on-wafer anti-reflection coating. The input optical waveguide can be coupled with a $3.5\, {\rm \mu } {\rm m}$-mode diameter lensed fiber for inserting light into the UTC-PD [Reference Achouche, Magnin, Harari, Lelarge, Derouin, Jany, Carpentier, Blache and Decoster43–Reference S45]. The generated electrical signals are guided to the output through a conductor-backed coplanar waveguide (CB-CPW) where the ground pads of the CB-CPW are connected to the n-contact of the PD. Reflection coefficient measurements are conducted with the support of a probe station and a vector network analyzer (VNA). The VNA is connected to the pads of this single port device via an internal bias-tee enabling the application of reverse DC bias to the UTC-PDs. Then, the bias-tee is linked to an RF cable, and a ground-signal-ground (GSG) microwave probe [Reference Wu, Shi, Wu, Huang, Chan, Huang and Xuan46]. In order to optimize the experimental processes, VNA calibration is performed allowing the elimination of errors on the measured $S$-parameters that stem from the VNA and the interconnected RF components due to losses and reflections [Reference Rumiantsev and Ridler47–Reference Teppati and Ferrero49]. This procedure can be complex for optoelectronic devices due to the different nature of their ports (optical input, electrical output) [Reference Elamaran, Pollard and Iezekiel50, Reference Zhang, Zhang, Wang, Liu, Peters and Bowers51]. However, the goal of this study is the acquisition of $S_{11}$ parameters and the electrical characterization of UTC-PDs under the dark regime where no optical illumination is applied. Thus, the calibration is conducted on a commercial wafer substrate provided by the GSG probe vendor where the standards used are open, short, and load (OSL) [Reference Chen, Wang, Liu and Zhu52]. Through the OSL correction technique of the VNA, the reference measurement plane is moved to the tip of the microwave probe [Reference Souria, Parra, Montoriol and Landez53]. All the data described are based on the same OSL calibration ensuring a fair comparison between the presented de-embedding methods. An alternative solution to the performed process is based on the replacement of the open and load with two additional offset shorts [Reference Scalzi, Slobodnik and Roberts54, Reference Cho, Kang, Lee and Koo55]. The three-offset shorts (SSS) approach is capable of achieving single port calibration on transmission media where it is difficult to realize loads or opens. It is important to note that this method is highly frequency dependent and each short should have a unique length that is precisely known and it is not proportional to multiples of $\lambda /2$ [Reference Rytting56, Reference Hoffmann, Leuchtmann and Vahldieck57]. Once calibrated, the VNA acquires information on the reflection parameters of a PD ($\Gamma _m$) that is illustrated in Fig. 3(a). $\Gamma _m$ shows a capacitive behavior and this is confirmed by the blue curve on the Smith chart of Fig. 4 for a measured $5\times 25\, {\rm \mu } {\rm m}^2$ UTC-PD at a reverse bias of $-2\;{\rm V}$. If the $S$-parameters of the TML $S_{TML} = [ S_{ij}] ,\; i,\; j = \{ 1,\; 2\}$ are known, then the reflection coefficient of the active region within the UTC-PD ($\Gamma _{UTC\text {-}PD}$) can be extracted analytically based on equation (1). However, the TML properties are not always easy to measure and thus de-embedding techniques are employed in order to eliminate these parasitics and move the reference measurement plane to the active region of the UTC-PDs i.e. isolate $\Gamma _{UTC\text {-}PD}$ [Reference Zhang, Xiong, Wang, Hu and Li58]:

(1)$$\Gamma_{UTC\hbox{-}PD} = {S_{11}-\Gamma_{m}\over S_{11}S_{22}-S_{22}\Gamma_{m}-S_{12}S_{21}}$$

Fig. 2. High-speed UTC-PD fabricated in III-V Lab connected to a VNA through an RF probe with $150\, {\rm \mu } {\rm m}$ pitch while its optical waveguide is coupled to a lensed fiber.

Fig. 3. (a) A block diagram of the active region of an on-wafer UTC-PD connected to the waveguide pads that need to be de-embedded and (b) the short and open circuit structures.

Fig. 4. Reflection coefficients mapped on a Smith chart for the open ($\Gamma _{OC}$) and short ($\Gamma _{SC}$) structures as well as for a $5\times 25\, {\rm \mu } {\rm m}^2$ UTC-PD at a reverse bias of $-2\;{\rm V}$.

Retrieving data from InP on-wafer structures with the VNA

The mathematical extraction of the $\Gamma _{UTC\text {-}PD}$ is based on the de-embedding principle that includes the measurements of the TML while it is terminated by an open (open test structure, $\Gamma _{OC}$) and by a short circuit (short test structure, $\Gamma _{SC}$) [Reference Velayudhan, Pistono and Arnould16]. The block diagrams of these components are depicted in Fig. 3(b). Based on the Smith chart of Fig. 4, $\Gamma _{OC}$ (in red) has a capacitive behavior and $\Gamma _{SC}$ (in yellow) has an inductive behavior. Both $\Gamma _{SC}$ and $\Gamma _{OC}$ contribute to the curvature of $\Gamma _m$.

Ideally, the retrieved reflection coefficient curves of both the short and open should coincide with the outer circle of the Smith chart where the load impedances $Z_L = 0$ and $Z_L = \infty$ are located respectively. However, since the on-wafer components are not ideal, the measured parameters have a deviation that is also visible in the magnitude and phase of $\Gamma _{OC}$ and $\Gamma _{SC}$ in Fig. 5(a). Optimally, it is expected that both magnitudes to be equal to $1$ for all frequencies. Nonetheless, losses in the conductors arise due to the finite conductivity of the metals leading to the decrease of $\vert \Gamma \vert$. In addition, the $\Gamma _{OC}$ curve shows a reduction in magnitude for lower frequencies while it reverts back to higher values over 30 GHz. This may be an indication of radiation losses and parasitic coupling with neighboring on-wafer structures that are closely spaced or direct coupling between the structures and the GSG probe [Reference Phung, Schmückle and Heinrich59, Reference Schmückle, Doerner, Phung, Heinrich, Williams and Arz60]. Moreover, multiple reflections may occur before the signal reaching the end of the structure as a result of impedance mismatch between the probe tip and the on-wafer design [Reference Spirito, Gentile and Akhnoukh61–Reference Torres-Torres, Hernandez-Sosa, Romo and Sanchez63]. Another cause can be due to material characteristics as well as fabrication imperfections on the InP substrate. Thus, the representation of the TML with lumped components becomes more and more complex making the de-embedding techniques a compelling solution for extracting the active region properties.

Fig. 5. (a) Magnitude and phase of $\Gamma _{OC}$ and $\Gamma _{SC}$ and (b) the magnitude and phase of $\Gamma _{m}$ for measurements of diodes with different sizes at a reverse bias of $-2\;{\rm V}$.

Figure 5(b) calculates the magnitude and phase of $\Gamma _{m}$ for UTC-PDs with different sizes at an applied bias voltage of $-2\;{\rm V}$. It is observed that the negative slope increases for diodes with wider surfaces. This resides on the fact that the values for the real part of the reflection coefficients ($\hbox{Re} [ \Gamma _{m}]$) are higher for smaller devices due to their larger series resistance [Reference Yeganeh, Rahmatallahpur, Nozad and Mamedov64–Reference Tang, Liu, Wang and Wang66]. Moreover, the imaginary part of the $\Gamma _{m}$ values ($\hbox{Im}[ \Gamma _{m}]$) also show the same trend where the capacitive behavior of the active region is more dominant for wider structures comparing to the parasitics of the TML [Reference Wang, Pan, Tang and Liu67–Reference Doussin, Bajon, Wane, Magnan and Parra69]. The opposite occurs for small diodes (e.g. $4\times 10\, {\rm \mu } {\rm m}^2$) where the TML principally influences the magnitude and phase of $\Gamma _m$ that is equal to $\sqrt {( \hbox{Re}[ \Gamma _{m}] ) ^2 + ( \hbox{Im}[ \Gamma _{m}] ) ^2}\angle ( \hbox{Re}[ \Gamma _{m}] ,\; \hbox{Im}[ \Gamma _{m}] )$.

The open-short de-embedding

The open-short method introduced by Koolen [Reference Koolen34] is based on the conversion of all measured $S$-parameters into admittances ($Y$) and calculates the impedance of a DUT by implementing equation (2) [Reference Tiemeijer, Pijper and van der Heijden75]:

(2)$$Z_{Koolen} = ( Y_{m}-Y_{OC}) ^{-1}-( Y_{SC}-Y_{OC}) ^{-1}$$

This technique is applied to a $5\times 25\, {\rm \mu } {\rm m}^2$ UTC-PD and the Smith chart of Fig. 7(a) is obtained. It is evident that the de-embedded (red) curve of $\Gamma _{Koolen}$ has also a capacitive behavior while the parasitics introduced by the waveguide are removed. The differences between the two reflection coefficients are also calculated in Fig. 7(b) based on magnitude and phase. As previously stated, the added parasitics from the TML increase the negative slope of the $\Gamma _{m}$ comparing to the de-embedded $\Gamma _{Koolen}$.

Fig. 7. (a) Impact of the open-short method on a $5\times 25\, {\rm \mu } {\rm m}^2$ UTC-PD leading to the removal of the waveguide parasitics at $-2\;{\rm V}$ reverse bias plotted on a Smith chart and (b) the magnitude and phase of $\Gamma _{m}$ and $\Gamma _{Koolen}$.

With regard to the lumped equivalent circuit of the active region, it is expected that $Z_{Koolen} = Z_{UTC\text {-}PD}$. However, in order to acquire a perfect optimization in the circuit simulations software and minimize $\epsilon$, an inductor ($L_p$) needs to be added in series with the $Z_{UTC\text {-}PD}$ of Fig. 6(a) after the TML. This additional element contradicts with the predicted circuit model of the active area within a PD [Reference Lucovsky, Lasser and Emmons76]. The origin of $L_p$ is thoroughly analyzed in the following section.

Analysis of the inaccuracies introduced by the open-short method

In order to verify the existence and magnitude of the additional inductance, the open-short de-embedding is investigated analytically. As depicted in Fig. 6(b), the TML connected to the DUT can be based on the distributed model. The transfer matrix for this type of ideal waveguide with length $l$ is given by equation (3), with a propagation constant $\gamma = \sqrt {zy}$ and $z = R + j\omega L\approx j\omega L$, y = jωC [Reference Cortés-Hernández, Sánchez-Mesa, Sejas-García and Torres-Torres77]. Finally, $Z_c = \sqrt {z/y}$ is the characteristic impedance of the line. Each variable is measured in units per transmission length:

(3)$$\eqalign{T & = \left[\matrix{A & B\cr C & D}\right] = \left[\matrix{ \cosh( \gamma l) & Z_c\sinh( \gamma l) \cr \displaystyle{\sinh( \gamma l) \over Z_c} & \cosh( \gamma l)}\right]\cr Z_{m} & = {AZ_{DUT} + B\over CZ_{DUT} + D},\; \quad Z_{OC} = {A\over C},\; \quad Z_{SC} = {B\over D} }$$

By converting the impedances of this symmetric model ($S_{11} = S_{22}$) in equation (3) to admittances through inversion and replacing them into the equation (2) result in:

(4)$$Z_{Koolen} = Z_{DUT}\cosh^2( \gamma l) $$

The term $\cosh ^2( \gamma l)$ can be approximated with a Taylor series and thus,

(5)$$\eqalign{Z_{Koolen} = {1\over 2}Z_{DUT}\left[1 + \sum_{n = 0}^{\infty}{( 2\gamma l) ^{2n}\over ( 2n) !}\right]\cr Z_{Koolen} = Z_{DUT}[ 1 + ( \gamma l) ^2] + R_2}$$

is extracted. The residual error ($R_2$) for this Taylor approximation is presented in equation (6). It is a constant that multiplied by an impedance results in negligible values and therefore can be omitted:

(6)$$R_2 = \sum_{n = 2}^{\infty}( \! - \!1) ^n( 10) ^{-9n}\left[{2^{2n}\over ( 2n) !}\right]$$

If the DUT is a UTC-PD i.e. $Z_{UTC\text {-}PD} = R_s-j( {1}/{\omega C_j})$, and by replacing it in equation (5), then,

(7)$$Z_{Koolen} = Z_{UTC\hbox{-}PD}-R_s( \gamma l) ^2 + j\omega {LC\over C_j}l^2$$

is obtained. Consequently, equation (7) agrees with the previous statement that $Z_{Koolen}\neq Z_{UTC\text {-}PD}$ for both the real and imaginary parts of the UTC-PD. For low frequencies, the real parasitic term $\vert R_s( \gamma l) ^2\vert = \vert R_s\omega ^2LCl^2\vert$ that depends on the characteristics of the TML ($L,\; C,\; l$) can be omitted ($< 1\!{\rm m\Omega }$). However, in higher frequencies its quadratic impact increases reducing the series resistance. The additional inductance ($L_p = ( {LC}/{C_j}) l^2$) that is calculated it is inversely proportional to the junction capacitance of the device. The exact same result for $L_p$ is measured if the equation of the open-short method is implemented to the Pi and T equivalent models of a waveguide [Reference Pozar78]. Based on this process it is proven that a systematic error is added to $Z_{DUT}$ while using the Koolen de-embedding technique. In principle, since the device model of Fig. 6(a) is related to different types of diodes such as PN, PIN, and Schottky, it is expected that the same error would occur once the open-short technique is applied [Reference Alamoud, Al-Mashary and Ragaie79–Reference Semple, Georgiadou, Wyatt-Moon, Gelinck and Anthopoulos81].

These analytical calculations concerning the additional inductance are confirmed through circuit simulations based on experimental data for diodes with different reverse bias applied, sizes, and frequency of operation. In theory, the value of $C_{j_{th}} = \epsilon A_j/l_j$ is proportional to the junction area ($A_j$) and inversely proportional to the junction length that is directly linked to the reverse bias. In the case where the bias is not enough to fully deplete the active region of a UTC-PD leads to a decrease of $l_j$ increasing $C_j$. In addition, the deteriorating junction area of a PD reduces the value of $C_j$ while $L_p$ increases. In Fig. 8, the junction capacitance and inductance of a $5\times 25\, {\rm \mu } {\rm m}^2$ UTC-PD is extracted for different values of the reverse bias. As expected, the form of the curve of $C_j$ (blue) is approximately a reflection of $L_p$ (red) as a reference to the $x$-axis, verifying the previous observations on the inverse proportional relationship between these two lumped components.

Fig. 8. Descending behavior of the additional inductance ($L_p$) extracted from the open-short method as a function of the applied reverse bias that is inverse to the curve for the junction capacitance.

Furthermore, in the ideal case, the values of the junction capacitance are expected to be constant as a function of frequency. However, since $C_j$ is extracted based on $S$-parameter measurements it is expected to show a small deviation for different frequencies with $C_j$ being higher at low frequencies due to parasitic effects [Reference Natrella, Liu, Graham, van Dijk, Liu, Renaud and Seeds82]. This deviation (in percentages) is depicted in Fig. 9(a) where for the same diodes, the junction capacitance calculated over the range 0–$35\, {\rm GHz}$ shows higher values comparing to the data measured between 0 and 50 GHz. The maximum percentage of difference at $1.65\%$ is obtained for a diode with size of $4\times 15\, {\rm \mu } {\rm m}^2$. This directly affects the calculation of $L_p$ that is plotted in Fig. 9(b) as a function of increasing $A_j$ for different measured UTC-PDs at a bias of $-2\;{\rm V}$. Then, a hyperbolic curve fitting is used in order to show the inverse proportionality of $L_p$ to the sizes of the diodes while $LCl^2$ is constant. As previously discussed, this curve is shifted upward for higher frequencies.

Fig. 9. (a) Percentage of difference between the values of $C_j$ extracted up to $35$ and $50\;{\rm GHz}$ affecting the calculation of $L_p$ and (b) the additional inductance ($L_p$) introduced by the open-short technique as a function of area at a reverse bias voltage of $-2\;{\rm V}$ for different UTC-PDs up to $35$ and $50\, {\rm GHz}$.

A correction to the open-short method

To eliminate the additional inductive behavior of equation (7), a correction to the open-short technique needs to be introduced, isolating $Z_{UTC\text {-}PD}$. Therefore, the term $\cosh ^2( \gamma l)$ in (4) is expressed as a function of the two known variables imported from the open ($Y_{OC}$) and short ($Y_{SC}$) structures:

(8)$$Y_{OC} + Y_{SC} = {\cosh^2( \gamma l) + \sinh^2( \gamma l) \over Z_c\sinh( \gamma l) \cosh( \gamma l) }$$

Calculating $Y_{OC} + Y_{SC}$ in equation (8), as well as taking into account $\cosh ^2( \gamma l) + \sinh ^2( \gamma l) = 2\cosh ^2( \gamma l) -1$ of the hyperbolic trigonometric identities and equation (4), results in equation (9) which introduces a correction forming an impedance-based de-embedding capable of removing the systematic error imported in equation (2) for any single-port-based DUT:

(9)$$Z_{Corrected} = Z_{OC}{Z_{m}-Z_{SC}\over Z_{OC}-Z_{m}}$$

The de-embedded reflection coefficients based on the open-short technique and its correction are calculated and plotted in the Smith chart of Fig. 10(a) for the same $5\times 25\, {\rm \mu } {\rm m}^2$. The curves for $\Gamma _{Koolen}$ and $\Gamma _{Corrected}$ overlap while the one for the open-short is longer due to $L_p$ added by the method itself. A similar observation can be extracted from Fig. 10(b) where the magnitude and phase of the two obtained reflection coefficients are plotted. Consequently, equation (9) is capable of isolating the active region of a UTC-PD leading to the valid design of its circuit model matching the impedance $Z_{UTC\text {-}PD}$ and can be expanded to all devices using open and short structures for de-embedding.

Fig. 10. (a) Impact of the corrected method to a $5\times 25\, {\rm \mu } {\rm m}^2$ UTC-PD leading to the removal of the waveguide parasitics at $-2\;{\rm V}$ reverse bias depicted on a Smith chart; the corrected curve is compared to $\Gamma _{Koolen}$ showing the additional inductance $L_p$ and (b) the magnitude and phase of $\Gamma _{Koolen}$ and $\Gamma _{Corrected}$.

The $S$-parameter-based de-embedding method

An alternative process that can be implemented in order to de-embed the parasitics of the UTC-PDs is the $S$-parameter-based (SPb) method. This mathematical approach resides on primarily reducing the unknown $S$-parameter variables of the TML ($S_{TML}$) in equation (1) from four to two. This is initially accomplished by implementing the property of passive behavior of the TML connected to the UTC-PD where $S_{12} = S_{21}$. Moreover, by assuming the symmetry of the TML, the equality between $S_{11}$ and $S_{22}$ is acquired. The validity of this hypothesis is substantial and it will be investigated even further through simulation processes in the following section. The de-embedding equation is finally derived in equation (10) as a function of $\Gamma _{m},\; \Gamma _{OC}$ and $\Gamma _{SC}$ by replacing and rearranging the variables in equation (1) from the data obtained for the open structure, $\Gamma _{DUT} = 1,\; \Gamma _{m} = \Gamma _{OC}$ and for the short $\Gamma _{DUT} = -1,\; \Gamma _{m} = \Gamma _{SC}$. The Smith chart of Figs 11(a) and (b) presents the impact of the $S$-parameter-based method on a $5\times 25\, {\rm \mu } {\rm m}^2$ UTC-PD where the TML parasitics are removed in the curve of $\Gamma _{SPb}$:

(10)$$\Gamma_{SPb} = {\Gamma_{OC} + \Gamma_{SC}-2\Gamma_{m}-\Gamma_{m}( \Gamma_{OC}-\Gamma_{SC}) \over 2\Gamma_{OC}\Gamma_{SC} + \Gamma_{SC}-\Gamma_{OC}-\Gamma_{m}( \Gamma_{OC} + \Gamma_{SC}) }$$

Fig. 11. (a) Impact of the $S$-parameter-based method to a $5\times 25\, {\rm \mu } {\rm m}^2$ UTC-PD leading to the removal of the waveguide parasitics at $-2\;{\rm V}$ reverse bias mapped on a Smith chart and (b) the magnitude and phase of $\Gamma _{SPb}$ and $\Gamma _{m}$.

Asymmetry dependence in $S$-parameter-based method

Figure 13 examines the percentage of difference between the junction capacitance and series resistance extracted in both equation (1) that provides the exact values of $C_j$ and $R_s$ and the resulting lumped elements calculated by equation (10) as a function of the increasing asymmetry. The curves are linear and maximum difference reaches $7.1\%$ for $C_j$ and $8.2\%$ for $R_s$ once $a = 1$. Similar behavior is demonstrated in the percentage differences in the mean values of magnitude, phase, real and imaginary parts of the diode's reflection coefficients ($\Gamma _{UTC\text {-}PD}$).

Fig. 13. Percentage difference of $C_j$, $R_s$, and of the characteristic properties of $\Gamma _{UTC\text {-}PD}$ between the $S$-parameter-based and direct methods as a function of the increasing TML asymmetry.

Thus, the extracted data confirm that the TML symmetry influences the $S$-parameters ($\Gamma _{SPb}$) of the de-embedded circuits as well as the values of the lumped components $C_j$ and $R_s$. However, the percentage of difference does not exceed the $8.2\%$ for all the tested variables while the performance of the technique improves while the additional branch is added on the TML. Identical values are observed once the equation for the corrected method ($Z_{Corrected}$) is used. Since there is not an effective divergence between the de-embedded elements and the original data, the $S$-parameter-based method can be considered as an alternative de-embedding leading to the removal of the TML parasitics from a DUT.

Asymmetry dependence in open-short method

Similar procedure is applied while implementing the open-short de-embedding in order to analyze the impact of asymmetry. The results of this procedure are shown in Fig. 14 for the series resistance, junction capacitance, and additional inductance.

Fig. 14. Percentage difference of $C_j$, $R_s$, and $L_p$ between the open-short and direct methods as a function of the increasing TML asymmetry.

While the percentage of difference on the extraction of $C_j$ is low varying between $-0.33$ and $0.21\%$, the real part of the intrinsic region ($Re[ Z_{UTC\text {-}PD}]$) is directly affected by the type of the TML linked to the active area of the UTC-PD. The value for $Rs$ is $19.1\%$ lower than the actual once the $C_{TML_2}$ rises. This percentage is linearly reducing for increasing $a$ due to the decrease of the factor $LC$ in the real part of equation (7). The same trend is presented for the value of $L_p$.

These results show that the open-short technique is capable of successfully removing the TML parasitics. However, once a second parallel branch with increasing admittance is shunt in the line model, the process starts demonstrating errors that become significant for low $a$. It is important to underline that the increasing $L_p$ for high values of $C_{TML_2}$ can be a good indication on the type of TML that is experimentally measured as well as on the effectiveness of the open-short especially for the calculation of $R_s$. Therefore, the additional inductance is not only dependent on the diode size, reverse bias voltage applied, and the frequency of operation (Figs 8 and 9(b)) but also on the type of the TML. Finally, this simulation validates the mathematical analysis of equation (7) on this series component i.e. $L_p$ added by the process.

Comparison $S$-parameter-based – corrected

In order to compare the de-embedding methods for the corrected and $S$-parameter-based technique the de-embedded $S$-parameters for $\Gamma _{Corrected}$ and $\Gamma _{SPb}$ of Figs 7 and 11 respectively are superimposed on a Smith chart leading to an overlap between the two red curves. Moreover, the error difference between the magnitudes of the two reflection coefficients is negligible. That is due to the fact that both the techniques in equations (9) and (10) are extracted based on the symmetry of the TMLs. With further mathematical analysis the conversion equation from $Z$ to $S$ parameters in equation (11) [Reference Frickey84] is implemented:

(11)$$Z = \sqrt{Z_0}( 1-\Gamma) ^{-1}( 1 + \Gamma) \sqrt{Z_0}$$

By substituting equations (11) to (9) then,

(12)$$Z_0{1 + \Gamma_{Corrected}\over 1-\Gamma_{Corrected}} = Z_0{1 + \Gamma_{OC}\over 1-\Gamma_{OC}}{\frac{1 + \Gamma_{m}}{1-\Gamma_{OC}}-\frac{1 + \Gamma_{SC}}{1-\Gamma_{SC}}\over \frac{1 + \Gamma_{OC}}{1-\Gamma_{OC}}-\frac{1 + \Gamma_{m}}{1-\Gamma_{m}}}$$

is obtained.

If equation (12) is rearranged and $\Gamma _{Corrected}$ is extracted, the corrected method becomes identical to the $S$-parameter-based ($\Gamma _{Corrected} = \Gamma _{SPb}$). Therefore, the two de-embedding processes converge and both $Z$ and $S$ parameters can be used to calculate a valid equivalent for the active area of UTC-PDs.

Comparison Koolen – corrected

A similar process is conducted for the comparison between the open-short and the corrected method. As a first step, equation (2) is analyzed based on $Z$-parameters in equation (13):

(13)$$Z_{Koolen} = Z_{OC}^2{Z_m-Z_{SC}\over ( Z_{OC}-Z_{SC}) ( Z_{OC}-Z_{m}) }$$

Then, the error ratio of difference between $Z_{Koolen}$ and $Z_{Corrected}$ is expressed in equation (14) as a function of $Z_{OC}$ and $Z_{SC}$:

(14)$${Z_{Koolen}-Z_{Corrected}\over Z_{Corrected}} = {Z_{SC}\over Z_{OC}-Z_{SC}} = \alpha + j\beta$$

In the ideal case where the ratio is a real number, i.e. $\beta = 0$ the relationship between the lumped elements for $R_s$ and $C_j$ extracted by the two methods is given in equation (15):

(15)$$\eqalign{ C_{\,j, Koolen} = ( 1 + \gamma^2) C_{\,j, Corrected}\cr R_{s, Koolen} = {1\over 1 + \gamma^2}R_{s, Corrected} }$$

For the two techniques, both $R_s$ and $C_j$ are dependent on the propagation constant of the waveguide ($\gamma$) with $C_{j, Koolen}> C_{j, Corrected}$ and $R_{s, Koolen}< R_{s, Corrected}$. However, the measurement results contradict with the ideal scenario since it is calculated that $R_{s, Koolen}> R_{s, Corrected}$ clarifying that it is important to take into account the impact of the imaginary part of equation (14), i.e. $\beta \neq 0$. This is because, as shown in Figs 4 and 5(a), the open and short circuit curves are not ideal. Therefore, by taking into account that the product $\beta \omega R_{s, Corrected} C_{j, Corrected}\approx 10^{-6}$ the results of equation (16) are obtained:

(16)$$\eqalign{C_{\,j, Koolen}& = {C_{\,j, Corrected}\over \alpha( \omega) -\beta\omega R_{s, Corrected} C_{\,j, Corrected}}\cr & \approx{C_{\,j, Corrected}\over \alpha( \omega) }\cr R_{s, Koolen}& = \alpha( \omega) R_{s, Corrected} + {\beta\over \omega C_{\,j, Corrected}}\cr & = \alpha( \omega) R_{s, Corrected} + \rho( \omega,\; C_j) }$$

The analysis above confirms that the relationship of $R_s$ and $C_j$ between the two methods is more complex and it depends on both $a( \omega )$ and $\rho ( \omega ,\; C_j)$ that includes the parameter $\beta$ and the $C_{j, Corrected}$ that is directly connected to the UTC-PD's area ($A_j$). In Fig. 15(a), the factors $\alpha ( \omega )$ and $\rho ( \omega ,\; C_j)$ are plotted as a function of frequency showing that $a( \omega )$ decreases at higher frequencies while the opposite occurs to $\rho ( \omega ,\; C_j)$. In order to get a better perspective on the impact of the different PD sizes to the magnitude of these factors, averaging over frequency is implemented removing $\omega$ from the variables of freedom in $\alpha$ and $\rho$. Thus, $a( \omega )$ as well as each curve of $\rho ( \omega ,\; C_j)$ is represented by a single value. Figure 15(b) shows that ${\alpha }$ remains constant as a function of increasing $C_j$ while ${\rho }( C_j)$ decreases exponentially. Therefore, based on the calculated data it is expected that $C_{j, Koolen}\approx C_{j, Corrected}$ while $R_{s, Koolen}> R_{s, Corrected}$ since the impact of the of ${\rho }$ is high especially for smaller junctions.

Fig. 15. (a) $\alpha ( \omega )$ and $\rho ( \omega )$ as a function of increasing frequency for different diode capacitances and (b) ${\alpha }$ and ${\rho }$ as a function of $C_j$.

These characteristics are verified in Fig. 16 where the percentage of difference between the lumped components of the active region ($C_j,\; R_s$) for the open-short and the corrected method are plotted using real measurements for diodes with different sizes. The percentage of error difference between $C_{j, Koolen}$ and $C_{j, Corrected}$ increases as a function of size not exceeding the absolute value of $1.8\%$. The negative percentage implies that in the corresponding junction area, the lumped element has higher magnitude once it is obtained by the corrected method. Concerning the series resistance, the difference between the two methods starts with a significant difference of $15.2\%$ and decreases to values of approximately $1.6\%$ for high size devices such as a $6\times 50\, {\rm \mu } {\rm m}^2$ UTC-PD.

Fig. 16. Percentage of difference of the lumped element components between the open-short and corrected methods.

The extracted data of these elements are also shown in Fig. 17 for the different de-embedding techniques. Furthermore, the theoretical curve of the junction capacitance ($C_{j_{th}}$) is also calculated. It can be observed that for all the different junction areas, the theoretical curve is shifted downward by a constant equal to $5\, {\rm fF}$. The difference between the dotted line for $C_{j_{th}}$ and the actual measurement data can be caused due to a parasitic capacitance generated by leakage of electric field at the edges of the diodes’ intrinsic region toward the neighboring layers of the collector layer forming a fringe capacitor in parallel to $C_j$ [Reference Chuang, Wang, Chu, Chang and Hu85–Reference Yu, Sun, Yu and Beling87]. In summary, the classical method of open-short provides both the $R_s$ and $C_j$ at the price of adding a virtual inductance ($L_p$) with reduced values for $R_s$ in high frequencies leading to an imprecise circuit model of the intrinsic area within the photodetectors. This error is fixed by using an updated method (corrected) and its equivalent $S$-parameter-based technique. The difference between these three methods in terms of the accuracy of the equivalent circuit and its component magnitudes are mainly based on the assumptions made for the equation calculations, the TML characteristics, the frequency of operation as well as the quality and precision of the on-wafer measurements conducted with the VNA [Reference Khelifi, Reveyrand, Lintignat, Jarry, Quéré, Lapierre, Armengaud and Langrez88]. Overall, the use of these de-embedding methods provides with a good estimation for the values of the lumped elements within the active region of the diodes that are a valuable tool for the characterization and analysis of their physical properties.

Fig. 17. Impact of the corrected method to a $5\times 25\, {\rm \mu } {\rm m}^2$ UTC-PD leading to the removal of the waveguide parasitics at $-2\;{\rm V}$ reverse bias.