1. Introduction

Opto-electronic mixers (OEMs) based on InP/GaInAs HBT are attractive components for optical sub carrier multiplexed systems^[1]. InP/GaInAs HBTs exhibit a large inherent nonlinearity^[2] that's a key property for mixing operation efficiency. Both single and cascode configuration of HBTs have been examined for better performances in frequency response. In this paper, first of all, in Section 2, we develop a software simulator, to simulate behaviors of a cascode opto-electronic mixer which has already been fabricated and empirically tested by Betser et al.^[3]. In Sections 3 to 5, we use the experimental results to verify our software with suitable fitting parameters. Then in Section 6, we take a further look at the physical structure of a typical HBT to modify its physical dimensions in a way to obtain better performances. Physical dimensions are converted into $Y$-parameters (systematic model), and then this model is used in our developed software to prove better performances of the new proposed HBT. We also have a conclusion section.

2. Setting up a simulation work space

The model we used for simulation was based on the P-SPICE charge control model. The P-SPICE large-signal model^[4] is based on the T model shown in Fig. 1.

The following relationships are used in this model^[4]:

$i_{\rm b}=\frac{I_{\rm s}}{\beta_{\rm F}}\left[ \exp\left( \frac{V_{\rm BE}}{nV_{\rm T}}\right)-1 \right] + \frac{I_{\rm s}} {\beta_{\rm R}} \left[ \exp \left(\frac{V_{\rm BC}}{nV_{\rm T}}\right)-1 \right],$

(1)

$i_{\rm T}=I_{\rm s} \left[ \exp\left(\frac{V_{\rm BE}}{nV_{\rm T}}\right)-1 \right] -I_{\rm s} \left[ \exp \left(\frac{V_{\rm BC}}{nV_{\rm T}}\right)-1 \right],$

(2)

$i_{\rm T}$ is a controlled current source that is controlled with base–emitter and base–collector voltages. The complete model that is used in the large-signal P-SPICE model is shown in Fig. 2.

$i_{\rm B}$ and $i_{\rm C}$ are obtained from the T model expressed before in Fig. 1:

$i_{\rm C}=i_{\rm T}- \frac{I_{\rm s}}{\beta_{\rm R}} \left( \exp \frac{V_{\rm BC}}{nV_{\rm T}}-1 \right),$

(3)

$i_{\rm E}=i_{\rm T}+\left( \frac{I_{\rm s}}{\beta_{\rm F}} \right) \left( \exp \frac{V_{\rm BE}}{nV_{\rm T}}-1 \right).$

(4)

Base–emitter and base–collector capacitances are variable capacitances which include both diffusion and depletion capacitances.

The diffusion part of base–emitter capacitance is expressed as:

$C_{\rm de}=\tau_{\rm F}g_{\rm m},$

(5)

in which

$\tau_{\rm F}=\frac{W^2}{2D_{\rm n}},$

(6)

and the depletion part can be expressed as^[4]:

$C_{\rm je}=\frac{C_{\rm je0}}{\left( 1-\dfrac{V_{\rm BE}}{V_{\rm 0e}} \right)^{ m}}\cong 2 C_{\rm je0}.$

(7)

$W$ is the base width and $D_{\rm n}$ is the diffusivity of electrons in base.

Our model is the mentioned model in Fig. 2 that is to some extent simplified. The above model works in both active and saturation regimes, but our research is confined to the active mode, so the model for the transistor can be simplified. In Ref. [3] the base emitter capacitor is expressed as:

$C_{\rm BE}=C_{\rm BE0}+\frac{\tau I_{\rm s}\exp \dfrac{V_{\rm BE}}{V_{\rm T}}} {V_{\rm T}} \cong 2 C_{\rm je0}.$

(8)

$\tau$ is the collector-to-emitter transit time, and $C_{\rm BE0}$ is the base–emitter capacitance for $V_{\rm BE}$ $=$ 0.

The model we used for the transistor is shown in Fig. 3.

$R_{\rm Bi}$ is the intrinsic base resistance that is expressed as^[5]:

$R_{\rm Bi}=\frac{1}{12}\rho_{\rm b} \frac{W_{\rm e}}{L_{\rm e}X_{\rm b}},$

(9)

in which $\rho_{\rm b}$ is the resistivity of the base material^[5]:

$\rho_{\rm b}= \frac{1}{q \mu_{\rm b}N_{\rm b}}.$

(10)

$W_{\rm e}$, $L_{\rm e}$ and $X_{\rm b}$ are emitter width, emitter length and base thickness.

The diode relationship is expressed as:

$i_{\rm b}=I_{\rm s}\exp \frac{V_{\rm BE}}{nV_{\rm T}}.$

(11)

$\beta$ is the collector current gain. Base collector capacitance is reported to be 43 fF^[3]. Values of the equivalent circuit components^[3] are shown in Table 1.

3. Experimental arrangement

A schematic diagram of the epitaxial layer structure and mesa structure is shown in Fig. 4. The epitaxial layers were grown on a semi-insulating substrate by a compact molecular beam epitaxy (MBE). The emitter and base dimensions were 4 $\times$ 11 $\mu$m$^2$ and 9 $\times$ 23 $\mu$m$^2$ respectively. Obtained $F_{\rm T}$ and $F_{\rm max}$ of the single device were 70 and 50 GHz respectively.

Small signal $S$-parameters of the single device and the cascode pair were measured up to 40 GHz^[3]. The schematic diagram of the experimental arrangement is shown in Fig. 5. The 5 $\times$ 6 $\mu$m$^2$ optical window was located on the base mesa of HBT Q1 in Fig. 5.

The light detection, mixing and amplifying were performed by HBT Q1, while HBT Q2 served as a low input resistance unity. In Fig. 5 the base of Q1 is the input port and collector of HBT Q2 is the output. A 50 $\Omega$ local oscillator and a DC voltage source were connected via bias T to the base port. The output power was measured using a spectrum analyzer. The base of HBT Q2 was connected to a dc voltage source using a DC probe with a 120 pF capacitor to provide a radio frequency (RF) ground. A distributed feedback laser emitting at 1.55 $\mu$m was externally modulated by a Mach Zender modulator. The modulated light was amplified using an erbium doped fiber amplifier (EDFA). The light was focused onto the optical window using a microscope. The modulation index was reported to be 27% ($-6.7$ dB). In the experiment an average optical power typically of about –15.3 dBm is incident on HBT's optical window. Both intrinsic and extrinsic conversion gains are useful figures of merit.

The intrinsic conversion gain $G_{\rm int}$ is defined as the ratio of the output power to $P_{\rm prime}$, the primary photo-detected RF power. $P_{\rm prime}$ is the photo induced RF electrical power detected by the base–collector junction that was measured by shorting the base–emitter junction. The extrinsic conversion gain is defined as the ratio of the output power of up or down converted signal to the equivalent electrical RF power $P_{\rm in}$ that would have been detected by an ideal photo diode with equal load resistance. The relation between the incident peak modulated component of the optical power, $P_{\rm mod}$, and the equivalent electrical input power is $p_{\rm in}=(qP_{\rm mod} hv)^2 \left(\frac{R_{\rm LOAD}}{2}\right)$, where $q$ is the electric charge, $hv$ is the photon energy and $R_{\rm LOAD}$ $=$ 50. $P_{\rm prime}$ and $P_{\rm in}$ are related by the external quantum efficiency of the base–collector photo diode ($\eta$), $G_{\rm ext}=G_{\rm int} \eta^2$. The external quantum efficiency^[6] was reported to be $\eta =$ 29% or $\eta^2 =$ –10.8 dB^[3].

4. Simulation of the experiment of single HBT

The simulation is based on the previous transistor schematic. Here we used an inductance and capacitance with large magnitude for simulating bias T of the base. The output bias T is omitted for simplification of calculations. The schematic of the simulation is shown in Fig. 6.

For single HBT mixer $R_{\rm Bi}$ was calculated from Eq. (9) to be 31 $\Omega$. The current source $I_{\rm OPT}$ is the representation of photo-induced current that is produced in the base. The current source $I_{\rm OPT}$ is a sinusoidal current source with 3 GHz frequency^[7]. For every $V_{\rm BE}$, the base–emitter capacitance $C_{\rm BE}$ is calculated from Eq. (8). $V_{\rm LO}$ is a sinusoidal voltage source with 3.5 GHz frequency. The diode relationship is that mentioned in Eq. (11). $C_{\rm BC}$ was 43 fF in simulation and $L_{\rm T}$ and $C_{\rm T}$ are bias T components. The current controlled current source, is the diode current multiplied by $\beta$. The simulation was repeated for various $V_{\rm be}$'s. After this section the simulation results were compared to the experimental results. A simulation fitting coefficient was exerted on the simulation results. Figure 7 shows the comparison of simulation results and experimental results in one diagram.

In Fig. 7, down conversion gain is plotted versus base–emitter voltage. The figure shows that the maximum power gain can be obtained in $V_{\rm BE}$ $=$ 0.78 V. In large base–emitter voltages, simulation can't show the drop of experimental results, because our simulator doesn't take into account the saturation effects. The figure shows that for various $V_{\rm BE}$s down conversion gain varies widely with base emitter bias, because of the nonlinearity effect of input impedance that varies widely with base–emitter bias (the mixing efficiency is dependant on the nonlinearity of the circuit).

5. Simulation of the experiment of the cascode HBT mixer

As in the previous section, the bias T components are shown with an inductance and a capacitance and for simplification the output base T is omitted. Schematic of the simulation for the cascode pair and comparison with experiment is shown in Fig. 8.

The base of Q2 is connected to a 2 V DC voltage source. The simulation was repeated for various $V_{\rm BE}$. Then the simulation results were compared with the experimental results. A fitting coefficient was exerted on the simulation results. Figure 9 shows the comparison of simulation results and experimental results in one diagram.

In Fig. 9 down conversion gain is plotted versus base-emitter voltage. The figure shows that the maximum power gain can be obtained in $V_{\rm BE}$ $=$ 0.8 V. The figure shows that for various $V_{\rm BE}$s, down conversion gain varies widely with base emitter bias; this is because of the nonlinearity effect of input impedance that varies widely with base–emitter bias (the mixing efficiency is dependant on the nonlinearity of the circuit).

6. Improving the transistor function

For hetero junction bipolar transistors, one of the important parameters for verifying small signal parameters is the base, collector and emitter materials. Function of this kind of transistors is based on the difference between base and emitter energy band gaps. Double hetero junction transistors use different kinds of materials for both the emitter and the collector from the base, whereas in common HBTs, base and collector materials are the same and differ from the emitter.

The common emitter $y$-parameters of a typical single HBT may be described using physical parameters, then we are able to convert physical parameters to small signal parameters. Here we only bring the results^[5]:

$\text{$\scriptsize [y]_{\rm e}= \begin{bmatrix} g_{\rm e}(1-\alpha_{\rm T0})+{\rm j}\omega (C_{\rm je}+C_{\rm jc})+{\rm j}g_{\rm e} \frac{\omega T_{\rm m}}{2} +{\rm j}g_{\rm e} \frac{\omega}{\omega_0}&-{\rm j}\omega C_{\rm jc} \\[4mm] \alpha_{\rm T0}g_{\rm e}\left[1-{\rm j}(1-m)\dfrac{\omega}{\omega_0}\right] \exp\frac{\omega \tau_{\rm m}}{2} -{\rm j}\omega C_{\rm jc}&{\rm j}\omega C_{\rm jc} \end{bmatrix}. $}$

(12)

Now we define the parameters that are used in the relationship of Eq. (12). $g_{\rm e}$ is the emitter transconductance that is defined as:

$g_{\rm e}=\frac{qI_{\rm E}}{nKT},$

(13)

where $q$ is the electron charge, $I_{\rm E}$ the emitter current, $n$ the ideality factor, $k$ Boltzmann constant and $T$ the absolute temperature. $\alpha_{\rm T0}$ is the DC base transport factor (zero means at zero frequency) and is given by:

$\alpha_{\rm T0}=1-\frac{X_{\rm B}^2}{2L_{\rm n}^2},$

(14)

where $X_{\rm B}$ is the base thickness and $L_{\rm n}$ is the diffusion length of the minority electron in the base and is given by:

$L_{\rm n}=\sqrt{D_{\rm nB}\tau_{\rm n}},$

(15)

where $D_{\rm nB}$ is the diffusion coefficient of the minority electron in the base and $\tau_{\rm n}$ is the minority electron recombination life time. $\omega$ is the frequency in which the transistor is working multiplied by 2$\pi$.

$C_{\rm je}$ is the base–emitter capacitance:

$C_{\rm je}=A_{\rm E}\frac{\varepsilon_{\rm s}}{X_{\rm depE}},$

(16)

where $A_{\rm E}$ is the emitter area, $\varepsilon_{\rm s}$ is the dielectric constant of the emitter and $X_{\rm depE}$ is the depletion thickness of base–emitter junction:

$X_{\rm depE}=\sqrt{\frac{2\varepsilon_{\rm s}}{qN_{\rm E}} \left(\phi_{\rm BE}-V_{\rm BE}\right) },$

(17)

where $N_{\rm E}$ is the emitter doping. $\phi_{\rm BE}$ is the built-in potential of the base–emitter junction:

$\phi_{\rm BE}=E_{\rm g}\left|_{\rm emitter}- \Delta E_{\rm V} \right|_{\rm emitter|base}.$

(18)

$E_{\rm g}\left|_{\rm emitter}\right.$ is the band gap of the emitter and $\left. \Delta E_{\rm V} \right|_{\rm emitter|base}$ is the difference between emitter and base valence bands' energy. $C_{\rm jc}$ is the base collector capacitance:

$C_{\rm jc}=A_{\rm c} \frac{\varepsilon_{\rm s}}{X_{\rm dep}}.$

(19)

$A_{\rm C}$ is the collector junction area; $X_{\rm dep}$ is the depletion thickness of the base collector junction:

$X_{\rm dep}=\sqrt{\frac{2\varepsilon_{\rm s}}{qN_{\rm C}} \left(\phi_{\rm CB}+V_{\rm CB}\right) }.$

(20)

$N_{\rm C}$ is the collector doping. $\phi_{\rm CB}$ is the built-in potential of the base–collector junction:

$\phi_{\rm CB}=\frac{E_{\rm g}({\rm base})}{2}+\frac{kT}{q} \ln \frac{N_{\rm C}}{n_{\rm i}}.$

(21)

$n_{\rm i}$ is the collector intrinsic carrier concentration. $\tau_{\rm m}$ is the collector transit time.

$\tau_{\rm m}=\frac{X_{\rm dep}}{V_{\rm sat}}.$

(22)

$V_{\rm sat}$ is the saturation velocity; $\omega_0$ is the base transit frequency:

$\omega_{0}=\frac{2D_{\rm n}}{X_{\rm b}^2}.$

(23)

This model can be converted to hybrid-$\pi$ model. The hybrid-$\pi$ model is shown in Fig. 10.

In Fig. 10 $R_{\rm Bi}$ is the base parasitic resistance:

$R_{\rm Bi}=\frac{1}{12}\rho_{\rm b}\frac{W_{\rm e}}{L_{\rm e}X_{\rm b}},$

(24)

where $\rho_{\rm b}$ is the base resistivity:

$\rho_{\rm b}=\frac{1}{q\mu_{\rm p}N_{\rm b}}.$

(25)

$N_{\rm b}$ is the base doping. $W_{\rm e}$ and $L_{\rm e}$ are emitter width and length respectively. $\beta_{\rm dc}$ is the dc current gain:

$\beta_{\rm dc}=\frac{D_{\rm nB}X_{\rm e}N_{\rm e}}{D_{\rm pE}X_{\rm b}N_{\rm b}} \exp \frac{\Delta E_{\rm V}}{kT/q}.$

(26)

Usually $\beta_{\rm dc}$ is a large number. $m$ is a constant between 0 and 1, typically equal to 0.66. Using these relationships we wrote a program that converts physical parameters to $y$-parameters and h-parameters. This program enabled us to verify the effects of changing several physical parameters. We first verified the transistor that was used in the experiment. By the hybrid-$\pi$ model and our simulation schematic we were able to draw the function of mixer versus RF frequency. Figures 11 and 12 show the down conversion gain versus frequency for single and cascade HBT respectively.

This calculation was for $V_{\rm BE}$ $=$ 0.7 V and not for the maximum attainable gain base–emitter voltage. This point was selected because at this point we had a coincidence between experimental and simulation results exactly for single HBT. At this point we tested several physical parameters for better hybrid-$\pi$ parameters. We found a better set of parameters for a wider bandwidth operation of single and cascode mixer. The set of parameters which are changed are shown in Table 2.

New proposed transistor schematic is shown in Fig. 13.

Table 3 shows a comparison between some equivalent circuit components estimated theoretically in this work, with previous work^[3]. It demonstrates considerably better performance with this new work. By the means of the new proposed transistor structure the simulation was repeated and a better bandwidth response was obtained.

In Figs. 14 and 15, new diagrams for the single and cascode HBT down conversion gains are plotted. These figures demonstrate better bandwidth operation of new proposed transistor.

RF frequency and comparison with previous transistor.

7. Conclusions

The simulation of single and cascode opto-electronic mixer HBT was done. The down conversion gain was simulated and compared to the experimental results for an RF optical intensity modulation frequency of 3 GHz and local oscillator frequency of, 3.5 GHz. A better transistor was proposed using the simulation results and the program that converts physical parameters to hybrid parameters. The new transistor was simulated to work as an opto-electronic mixer. The new HBT operation was plotted and compared to the existing transistor operation.