Analytical expressions for the design of twin junction tuning in SIS mixers

We derive general analytical expressions for matching the complex impedance of the feeding radio frequency (RF) circuit to a twin junction device in Superconductor-Insulator-Superconductor (SIS) mixers. A unique feature of our analysis is that it allows for junctions of different admittance and an RF circuit with complex admittance. This provides important design flexibility and accommodates device fabrication tolerances. We then focus on special cases of identical junctions, in particular those cases that yield simple expressions for matching the twin junction to a real impedance. Finally, we derive design curves that allow the determination of the twin junction parameters for a given complex impedance of the feeding circuit. These curves allow the designer to choose different solutions than the commonly used quarter-wavelength transmission line separating the two SIS junctions, which our analysis shows is not always the optimal solution. Moreover, even when commercial software is used for designing the circuits, our analytical equations will act as an important guide to understanding the physics and guide the designer to the optimal solution.


Introduction
In recent years, SIS (Superconductor-Insulator-Superconductor) detectors have revolutionised the heterodyne receiver technology of millimetre (mm) and sub-mm astronomy. These receivers are used in large interferometer telescopes such as ALMA (The Atacama Large Millimetre Array), which played a major role in the recently reported remarkable imaging of two black holes, one in the M87 galaxy and the other in our own galaxy [1,2]. ALMA, participating as a member of the 'Event Horizon Telescope', employed SIS mixers operating at 230 GHz to provide the data that led to the black holes imaging.
In fact, modern heterodyne mm/sub-mm telescopes are equipped with SIS mixers operating in the frequency range between 200 GHz to 1 THz [3]. This is because SIS mixers provide quantum-limited sensitivity only restricted by the uncertainty principle, which allows for a minimum noise temperature of hν/k B [4,5]. To achieve sensitivities close to the quantum limit, however, requires considerable effort in the device fabrication and the design of the mixer circuit. This is because the SIS device is a tunnel junction consisting of two superconductors sandwiching a ≈10 Å insulator. While modern lithographic and e-beam techniques allow for the repeatable design of devices with sub-micron area, the device remains highly capacitive at mm and sub-mm wavelengths where such devices are needed.
An important step in the design of SIS mixers, therefore, is the cancellation of the junction's capacitance (commonly known as tuning) which would otherwise short high-frequency signals; and to maximise the coupling of the feeding RF circuit to the SIS tunnel junction(s). This is by no means an easy task since the mixer circuits consist of superconducting transmission lines of stringent dimensions (e.g. microstrip lines with a metallisation thickness equal to the dielectric thickness), and quantum treatment of the mixer is needed because of the high photon energy [6,7]. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
A commonly used method to achieve this tuning is by using a ≈λ/4 transmission line terminated by a stub [8,9]. While this method serves the purpose in many working receivers, it implies that the bandwidth of the mixer will not only be limited by the RF transmission line but also by the stub bandwidth [10]. Moreover, the large capacitance of the stub can severely limit the down-converted intermediate frequency (IF) bandwidth, especially in mixers that aim to operate at high IF bandwidths [11].
A more elegant tuning mechanism that has been proposed by Belitsky et al [12] is the twin junction tuning method which has been successfully employed in SIS mixers at several frequency bands [13,14]. It consists of two parallel SIS tunnel junctions connected by a transmission line, typically a microstrip line. The principle of operation is based on the fact that when a transmission line with a carefully selected electrical length and characteristic impedance separating the two SIS junctions is used, the reactances of the two junctions can be made to cancel each other, and the input impedance of the device becomes real, hence can easily be matched to the feeding RF circuit. A strong feature of this design is its fabrication simplicity which enables the mixer to achieve sufficient RF bandwidth for a typical astronomical heterodyne receiver without the addition of complex tuning circuits on the mixer chip.
Previous publications gave only tentative descriptions of the twin junction design, focusing mainly on the performance of the SIS mixer [15][16][17]. It is also evident that in recent years, SIS mixer designers use commercial electromagnetic software packages, such as Ansys High Frequency Structure Simulator (HFSS), to optimise the mixer design. In those packages, the SIS tunnel junction is modelled as a lumped element circuit of a capacitance connected in parallel to a resistance, while the rest of the mixer circuits are rigorously modelled as planar superconducting transmission lines [18]. While this method is effective and can lead to accurate modelling of the mixer performance, it has the disadvantage that it obscures the intuitive understanding of the design, which may, in fact, lead to missing out effective designs. For example, we will later show that a quarter-wavelength transmission line separating the two junctions is not the only solution for good matching and, indeed, may not be the best solution for particular values of the line characteristic impedance dictated by the limitation of fabrication. In fact, we will show that other length choices may provide more flexibility in choosing the impedance of the transmission line separating the two junctions or other output impedance values of the feeding RF circuit.
Another assumption that previous publications made is that the two junctions are identical [14,15]. While this assumption is quite reasonable considering that the two junctions are closely located on the mixer chip, it would be interesting to investigate the effect of possible differences in admittance due to fabrication tolerances of the two junctions on the tuning and, indeed, if there are any advantages of using a twin junction device with different junction admittances. Also, common mixer designs aim at having a real input impedance of the twin junction device by mutual cancellation of the reactance of two identical junctions [19,20]. This, however, may not be sufficient for an optimum mixer design and, in fact, may not even be the best choice for designing an SIS mixer with large RF bandwidth that, in general, has complex input impedance. For example, mixers that use complex RF circuits, such as an integrated local oscillator feed by using the harmonic of an SIS device [21], have complex output impedance. Therefore, an additional circuit may be needed to match it to the twin junction device. A real input impedance of the twin junction device, in this case, may not constitute a significant advantage.

Reflection coefficient between the RF circuit and twin junction circuit
In figure 1, we show the Norton equivalent circuit model of an SIS mixer using twin junction tuning. As usual, the SIS tunnel junction is modelled as a lumped element which has been shown by both experimental and simulation results to be an excellent approximation, assuming the dimensions of the device are small relative to the length of transmission lines [22]. The reflection coefficient between the twin junction circuit of admittance Y in and the feeding RF circuit of admittance Y RF is given by [ In these equations, Y 1 and Y 2 are the complex admittances of the junctions, and l, β, and Y 0 are respectively the length, propagation constant and characteristic admittance of the transmission line connecting the two junctions, assumed lossless. To ensure maximum power coupling from the RF feeding circuit to the twin junction circuit, we request that Expressing the admittances as Y = G + iB, where G is the conductance and B is the susceptance, then in the matched case Γ = 0, we obtain the solution The above expressions give the admittance that the designer may use for adjusting the output admittance of the feeding RF circuit in order to maximise the coupling to the twin junction device. In what follows, we shall consider the special case of two identical junctions with Y 1 = Y 2 = Y = G + iB. In this case, the admittance for the matched case can be expressed as We shall now discuss the following two options:

Matching to a real input impedance
This is the commonly used case in which the reactance of the tunnel junctions is cancelled, and the input impedance of the twin junction device viewed from the RF side becomes real. This case may also be applied when the RF circuit has complex output impedance, but with an intermediate circuit between the RF circuit and the junction circuit, such that the impedance seen by the twin junction device is real. In either case, it is interesting to consider the following cases that yield simple analytical expressions that can easily be analysed: (i) The half-wavelength multiples case where T = 0: It corresponds to a transmission line satisfying βl = nπ where n is an integer. The twin junction admittance is then given by This is simply a parallel circuit of shunt admittances which is not influenced by the characteristic admittance of the transmission line Y 0 . The special case βl = 0 can be interpreted as doubling the junction area. However, it is clear from the equation above that the imaginary part of Y in is B in ≠ 0, hence it is impossible to match the twin junction device to a real impedance of the RF feeding circuit.   The behaviour of the input impedance of a twin junction as a function of the characteristic impedance Z 0 = 1/Y 0 at 230 GHz is shown in figure 2 for the special cases described above. We have chosen an SIS junction with junction capacitance of 75 fF and normal resistance of R N = 20 Ω, which are typical values for a 1.0 μm 2 Nb/AlO x /Nb tunnel junction used in SIS mixers at mm/sub-mm wavelengths. Such a junction will have an when the junction is modelled as a lumped element. A twin junction with the two SIS junctions separated by half a wavelength (T = 0) halves the required RF circuit admittance independent of Z 0 , which is disadvantageous since the RF circuit impedance needs to have low resistance in a highly capacitive circuit. In fact, it can be seen from figure 2 that when the length of the transmission line is half a wavelength, the input impedance of the twin junction cannot possibly be made real and hence cannot be matched to a real value. In contrast, a twin junction circuit with a transmission line separation satisfying the conditions given in cases 2 (T = ∞ ) or 3 (T = ± 1) can be made to transform the large capacitance of the tunnel junctions into a pure resistance by carefully choosing βl and Z 0 .
For a quarter-wavelength multiples transmission line βl = (1/2 + n)π, which is the most commonly used choice, R in is almost three times that of a single SIS junction at Z 0 ≈ 8.4 Ω, where the reactance crosses through 0 Ω. At large values of Z 0 , the input impedance approaches the impedance of the first junction, as can easily be seen from equation (2).
As we have indicated previously, the quarter-wavelength multiples transmission line defined by βl = (1/ 2 + n)π (or T = ∞ ) is not always the best choice for matching the twin junction device. From figure 2 we can see that the slope of the T = ∞ reactance curve at the characteristic impedance where X in = 0 Ω is much sharper than that for the curve corresponding to T = 1. This makes it much less tolerant to the choice of Z 0 and hence to tolerances in the device fabrication. Moreover, for Z 0 > 10 Ω, the reactance for the T = ∞ case becomes highly capacitive, while it remains close to zero for T = 1 for all impedance values in the range from 0 Ω to 22 Ω. It is, however, evident from figure 2 that the input resistance values for the T = 1 line are too low for Z 0 < 15 Ω. Hence, the final choice will depend on the preferred Z 0 value.

Matching to a complex impedance
This is the general case where we would match the complex impedance of the twin junction directly to the impedance of the RF circuit, which is, in general, complex. Given tunnel junction parameters and known admittance of the input RF feeding circuit G RF and B RF , equation (1) and (2) can be solved for Y 0 and βl of the transmission line connecting the two tunnel junctions. For the case G RF ≠ G 1 + G 2 and Γ = 0, we obtain The expression above becomes singular for G = G 1 + G 2 , however, expressions for this case have already been derived and analysed in the previous section. The analytical solution above is given graphically as design curves in figure 3. The curves are plotted for junction admittances used in the previous section and typical characteristic impedance values employed in SIS mixer circuits. These plots allow the designer to match the twin junction device to both real and complex impedances. For example, assume that the output impedance of the RF circuit is given by Z RF = (11 − 4i)Ω. The designer will therefore notice from figure 3 that the matched twin junction would be given by points on the blue curves corresponding to βl = 3π/8 and, therefore, to a transmission line of length l = 3/16λ and characteristic impedance Z 0 = 15 Ω. Computed curves showing the input impedance of the twin junction as a function of characteristic impedance Z 0 for the discussed lengths T of the transmission lines connecting the junctions, where R in is the input resistance and X in is the input reactance of the twin junction. The y-axis is normalised to the normal resistance of the SIS junction R N , and R in and X in are shown as solid and dashed lines, respectively. The curves were computed from equations (3) and (4) assuming identical junctions.

Conclusion
We have derived analytical expressions for the design of the twin junction mixer, for both identical and nonidentical tunnel junctions. We have shown that the commonly employed quarter-wavelength transmission line length that connects the junctions is not always the ideal choice, even if matching to a real RF circuit impedance is sought. Other solutions may be more suitable depending on the parameters of the junction. In fact, tuning out the junction's capacitance may not be the ideal design procedure for mixers that employ complex RF circuit impedance. The analysis given in this paper, therefore, contributes to additional miniaturising of the mixer circuits and hence allows the integration of important receiver circuits such as balanced and double sideband receivers. Notice that our mathematical derivations are based on modelling the mixer using lossless transmission lines with the SIS devices as a lumped element. This method has shown to work very well in SIS mixers operating below the superconducting gap (700 GHz for niobium) and using small SIS devices (≈1 μm 2 ), which are necessary for large RF bandwidth.