Operation of kinetic-inductance travelling wave parametric amplifiers at millimetre wavelengths

We constrain the superconducting material parameters for use in our model by fitting the measured 'unpumped' S₂₁ transmission spectrum of our Ka-band device [11]. That device was fabricated based on a transmission line with the periodic structure depicted in figure 1. The central microstrip line is shunted with a series of open stub sections, the length of which is modulated to produce a stopband in the transmission spectrum around 38 GHz, as shown in figure 2. The location of the stopband depends sensitively on the superconducting properties. Both the central microstrip line and the stub sections were formed using a 250 nm wide, 35 nm thick NbTiN conductor layer, which is deposited and patterned on a 100 µm thick high-resistivity silicon substrate ($\epsilon_{\mathit{r}} =$11.9). A 60 nm amorphous silicon (aSi, $\epsilon_{\mathit{r}} =$10.3) dielectric layer deposited on top of the conductor. Finally, a 200 nm NbTiN 'sky-plane' layer is deposited to form an inverted microstrip geometry.

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We modeled the device using a commercially available 3D electromagnetic simulator Ansys^® high-frequency structure simulator (HFSS), with exactly all the dimensions and topology listed in [11] and BCS functions built into the simulations. The latter is performed by applying the Mattis-Bardeen surface impedance as impedance boundary condition on the perfect conductor modelling the transmission line [14, 15]. For completeness, we listed all the dimensions of the inverted microstrip structure used for the simulation in table 1 with reference to figure 1. The NbTiN film is expected to have a critical temperature of $T_c = 12.5$ K, corresponding to a gap parameter $\Delta = 2.05$ meV, given a ratio $2 \Delta / k_b Tc \approx 3.8$ [16], which is slightly high than the weak coupling BCS value. The normal state resistivity $\rho \approx 200\,\mu\Omega$cm [11].

Table 1. The critical dimensions of the inverted microstrip structure used to form the unit cell and the entire Ka-band TWPA.

w	g	$w_{\mathrm{stub}}$	l0	$l_{\mathrm{mod}}$	No. of	Total no.
(nm)	(nm)	(nm)	(µm)	(µm)	stubs, n0	of cells
250	500	250	6	0.1	51	550

However, if we used the expected film parameters, we found that the theoretical model would poorly predict the behaviour of the measured S₂₁ spectrum; the stopband position would be off by several GHz. To match the stopband position, we can either retain the normal state resistivity of the NbTiN film as estimated in Shu et al [11], but vary the gap voltage from the designated 4.1 meV; or retain that estimate but varying ρ (all other parameters are well constrained in our fabrication process). In either case, we are simply adjusting the surface inductance per square value to match that of the actual film. The reason behind the former consideration is that the $T_{\mathrm{c}}$ was measured in an experimental setup where the temperature sensor was anchored at the sample holder, and we expect that the device chip may be at a warmer temperature than the measured 12.5 K; while the latter related to the uncertainties in measuring the actual ρ of the film across the entire wafer. Although not shown in figure 2, if we adjust ρ to 168 $\mu\Omega$cm (with $T_{\mathrm{c}}$ remaining at 12.5 K), the simulated transmission curve would match the position of the stopband extremely well, as well as the width of the stopband too. Similarly, if we adjust the $T_{\mathrm{c}}$ to 15.0 K but retain ρ at 200 $\mu\Omega$cm. In reality, the actual surface impedance could be at any value between the two limits (ρ = 168–200 $\mu\Omega$cm & $T_{\mathrm{c}} = 12.5$–15.0 K).

As our KITWPA device is designed to operate in the DC-biased 3-wave (DC3WM) mixing regime, it is obvious that the stopband position would shift when a DC current is applied. Using the recovered surface impedance value from above (here we use only the case where $\rho = 168\,\mu\Omega$cm and $\Delta = 2.05$ meV), we managed to match the measured transmission curve when DC current is applied accurately by simply multiplying the surface inductance calculated by the BCS (Bardeen–Cooper–Schrieffer) and Mattis–Bardeen equation, by the expected nonlinear inductance i.e. $L_{\mathrm{k}} = L_{0}\times[(1+({{I_{\mathrm{dc}}}}\!/\!{{I^*}})^2+({{I_{\mathrm{dc}}}}\!/\!{{I^{*^{{\prime}}}}})^4]$ where $I^* = 4.3$ mA and $I^{*^{{\prime}}} =$2.8 mA, with the DC current applied at 0.57 mA, as indicated in [11], and shown in figure 2, without including the BCS surface resistive loss from the device. In practice, the quartic term is typically negligible compared to the qudratic term due to the critical current of the device, $I_c \lesssim 0.2 (I_*, I_*^{{\prime}})$. Note that the simulated curves in figure 2 were made by taking into account the 50 dB attenuation (from various stages of attenuators installed in the line in the experiment), and the frequency-dependent attenuation of the RF cables measured experimentally at 0.098 dB GHz⁻¹ m⁻¹ (or 0.49 dB GHz⁻¹) [12].

Once we obtain the DC-biased transmission curve, it is straightforward to simulate the gain curve. Here, we use our coupled-mode equation (CME) model as presented in [17] but modified to operate in the DC3WM mode. As shown in figure 2, we successfully recover a gain curve that matches very well with the measurement, not only the level of gain but the bandwidth and the overall gain-bandwidth profile. The RF pump frequency was set at 38.8 GHz with an amplitude of −27.5 dBm as indicated in the experiment. We stress that in our model, we do not alter any parameters other than recovering the actual surface reactance via ρ.

Adopting the values for the normal metal resistivity and the gap parameter adopted above, the surface impedance $Z_s(\omega, T)$ may be found using the Mattis–Bardeen expressions for the complex conductivity $\sigma(\omega, t) = \sigma_1(\omega, t) - i\sigma_2(\omega, t)$ [13]. The integral expressions for $\sigma(\omega, t)$ are valid up to the gap frequency, beyond which the attenuation jumps abruptly, as photons beyond that energy are able to directly break Cooper pairs. For the device designs considered here, the film thickness t is much less than the penetration depth, so the current is nearly uniform through the film and the thin film limit is valid. In this case, the surface impedance is

$$\begin{align} Z_s \equiv R_s + i\omega L_s = \frac{1}{\sigma t} = \frac{\sigma_1}{t\sigma_2^2} + \frac{i} {t\sigma_2}, \end{align} \tag{ 1 }$$

from which we can calculate the loss and propagation velocity of an NbTiN microstrip line as a function of frequency. For the geometries of interest for our KITWPA, the Pearl length $2\lambda^2 / t$, which is the length scale over which the current varies laterally across the film [18], is much longer than the microstrip conductor width, w. In this case, the current is uniform across the microstrip conductor, and the inductance per unit length can be written in terms of a sum of magnetic and kinetic contributions as $\mathcal{L} = \mathcal{L}_{m} + L_s / w + L_{s,gp} / w_{gp}$, where $\mathcal{L}_{m}$ is the inductance associated with the magnetic field, $L_{s,gp}$ is the surface inductance of the ground plane and w_gp is the effective width of the current distribution in the ground plane [12]. Due to the large kinetic inductance of NbTiN and narrow geometry of the microstrip, $\mathcal{L}_{m} \ll L_s / w$, and the ground plane contribution is also small in comparison to the conductor's inductance as the ground plane currents are spread over a much wider area. The propagation velocity $v_{ph} = 1 / (\mathcal{L} \mathcal{C})^{1/2}$ and characteristic impedance $Z_0 = (\mathcal{L} / \mathcal{C})^{1/2}$ can be found using an expression [19] for the capacitance per unit length of a microstrip line. The power attenuation constant is then given by $\alpha = 2 R_s / (w Z_0)$ or $20\pi \log_{10}(e) \,\alpha\,v_{ph} / \omega$ in terms of dB per wavelength. These quantities are plotted as a function of frequency in figure 3. Interestingly, for a fixed number of wavelengths, the loss due to the surface resistance stays roughly constant through the mm-wave region at a very low level before increasing abruptly at the gap frequency. Nevertheless, the propagation velocity shows dispersion due to the frequency dependence of σ₂, varying more rapidly as the gap frequency is approached, hence may be more of a concern for mm-wave operation, which we shall address later. We note that the behavior very near the gap frequency may be somewhat different from that shown in the figure due to disorder induced broadening of the singularity in the quasiparticle density of states near the gap energy [20]. That effect would be expected to smear the abrupt onset of increased dissipation and slightly lower the maximum frequency of the KITWPA. Another source of broadening of the density of states singularity is effect of the current associated with the strong RF pump tone used to drive the KITWPA [21]. Generally for KITWPAs, including for the designs considered later in this report, the nonlinearity at the operating point is relatively weak, resulting in only a percent level effect.

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In the previous Ka-band KITWPA analysis, we did not include the effect of loss tangent of the dielectric, which could potentially be important at high frequencies. To investigate the high-frequency dielectric loss, we conducted a separate experiment where we measured the resonance curves of W-band half-wavelength microstrip resonators fabricated using the same processes and film stack-up as the Ka-band KITWPA. The chip containing the resonators was designed to fit in a split-block waveguide machined from copper (see figure 4). E-plane waveguide probes on either end of the chip were utilized to couple power between the WR-10 waveguides and an on-chip 1 µm wide microstrip line. The resonators are formed with 'U' shaped sections of the same microstrip line and are gap-coupled to the readout line. The resonators have total lengths between 33 and 57 µ, corresponding to resonance frequencies in the W-band, given a propagation velocity close to that shown in figure 3.

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The transmission S₂₁ parameters of the resonators were measured using a vector network analyzer with W-band waveguide extenders. The transmission curves for one resonator, measured at 0.9 K, are shown in figure 4(c) for a range of excitation powers. For the higher excitation powers, the resonances are distorted due to the nonlinear kinetic inductance and evolve into the characteristic shape of a Duffing resonator. The internal loss versus the energy stored in the resonator is extracted by fitting to a nonlinear resonator model [22] and is shown in figure 4(d) as a function of U_int, the energy stored in the resonator, which is determined from the excitation power P_in through

$$\begin{align} \frac{U_{\textrm{int}}}{\hbar\omega_r} = \frac{2 Q_r^2 P_{\textrm{in}}}{\hbar \omega_r^2 Q_c}, \end{align} \tag{ 2 }$$

where ω_r and Q_r are the frequency and the quality factor of the resonator respectively, and Q_c is the coupling factor. The decrease in Q at low power is the expected behavior of loss due to two-level-systems (TLS) [23] in the aSi dielectric and is described by the standard TLS model. For a KITWPA, the limiting low-power value of the loss is of interest, which we estimate by fitting the data to the equation

$$\begin{align} Q = Q_{\textrm{TLS}} \left(1 + U_{\textrm{int}} / U_c\right)^\phi \end{align} \tag{ 3 }$$

where U_c is the critical energy above which there is a power dependence to the quality factor. The parameter φ used in the standard TLS theory is 0.5, but lower values have been found for low-loss amorphous dielectrics [24], so we include it as a fitting parameter. The fit, shown in the figure 4(d), suggests that $Q_{\textrm{TLS}} = 23\,000$ and φ = 0.31. The power attenuation due to dielectric loss in a transmission line of length l is $P(l) = P(0) \exp( -\tan(\delta) k l)$, where $k = 2\pi/ \lambda$ is the propagation constant, and $\tan(\delta) = 1 / Q_{\textrm{TLS}}$ is the loss tangent. Expressed in dB, the loss is $20 \pi \log_{10}e / Q_{\textrm{TLS}} = 0.0012$ dB per wavelength, which is relatively small given that the length of a KITWPA is on the order of a few hundreds of wavelengths, resulting in minimal excess noise or heating from the applied pump. Nevertheless, for subsequent analysis in the mm-wave range, we include this dielectric loss in all our models.

For operation in mm/sub-mm wavelength, apart from the dielectric loss, there are several design considerations that need to be taken into account e.g. it is most likely that the device will have to be coupled using a waveguide feed [25] instead of the commonly used coaxial cables at microwave regime such as SMA or SMK cables. Therefore, the characteristic impedance of the TWPA need not be restricted to firmly 50 Ω, but should be optimised to match the output impedance of the feeding antenna. Furthermore, the choice of substrate used to support the TWPA becomes a crucial issue. For most waveguide antennas, such as the probe antenna [25] (the most commonly used antenna in this frequency range), a thinner substrate with a lower dielectric constant is preferable such that it does not load the waveguide and supports unwanted substrate modes.

One potential solution to this is to use a membrane technology such as silicon-on-insulator or silicon nitride where only the silicon handle layer underneath the antenna is etched away to form the membrane but retain the handle to support the remaining TWPA circuit. Nevertheless, to avoid the need for sophisticated fabrication technology, we opt for a simpler solution by fabricating the device on top of a thin 40 µm quartz substrate with $\epsilon_{\mathit{r}} = 3.78$. Figure 5 shows the optimised design of the feeding antenna embedded inside a $0.5\times1.0$ mm rectangular waveguide with a cutoff frequency near 150 GHz. The overall device makes use of two of these antennas, one at either end of TWPA, as was done for the W-band resonator measurement device depicted in figure 4. The dimensions of the probe antenna with an optimised output impedance of 67.7 Ω, and the supporting substrate, are tabulated in table 2. As can be seen in figure 5, our probe antenna can achieve close to 0 dB coupling from 170–300 GHz, broad enough to cover the bandwidth of our TWPA.

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Table 2. The critical dimensions of the probe antenna with an output resistance of 67.7 Ω.

$r_{\mathrm{p}}$	$w_{\mathrm{p}}$	$w_{\mathrm{sub}}$	$l_{\mathrm{bs}}$	$\theta_{\mathrm{p}}$
(µm)	(µm)	(µm)	(µm)	(°)
235	135	155	326.75	50.7

In this section, we present the design of the mm-wave TWPA that matches the output impedance of the probe antenna using the same substrate and thin film technologies. Here, we retain most of the dimensions used to construct the Ka-band TWPA presented above, except the unit cell length that is adjusted to operate near 220 GHz, the stub's length, and the gap between the stubs modified to match 67.7 Ω, and the stub length modulation calibrated to achieve good phase-matching in the designated frequency band range. The optimized dimensions of the TWPA are tabulated in table 3, and the predicted gain profile of the device is presented in figure 6, while figure 7 illustrates the layout of the entire mm-wave TWPA chip together with the input/output probe antennas, connected to two rectangular waveguides for signal coupling at high frequency.

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Table 3. The critical dimensions of the inverted microstrip structure used to form the unit cell and the entire 220 GHz TWPA utilizing the sinusoidal periodic loading structure for dispersion engineering.

w	g	$w_{\mathrm{stub}}$	l0	$l_{\mathrm{mod}}$	No. of	Total no.
(nm)	(nm)	(nm)	(µm)	(µm)	stubs, n0	of cells
250	250	250	5.9	0.05	16	600

As can be seen from figure 6, the bandwidth of the design presented here is relatively narrow at about 37 GHz. This is due to the resonance frequency of the stubs which creates a large stopband at the very high-frequency end, which in this case, is close to 800 GHz, just only about 4× higher than the central frequency of the device. This high-frequency cutoff inevitably induces a much stronger nonlinear dispersion (β_nl, the divergence of the wave vector away from the otherwise linear relation) and causes a larger total phase mismatch between the propagating tones near the edges of the bands compared to the center of the band. On the other hand, this large nonlinear dispersion helps to suppress the pump's third harmonic generation to ensure optimal gain growth as well as limits the impact of noise from mixing modes to frequencies outside the idler band [26].

To widen the bandwidth, it is possible to shorten the length of the stub while still retaining strong enough nonlinear dispersion at a higher frequency to suppress the pump's harmonics, but this implies that the stub length modulation needs to be proportionally smaller as well, and hence would be very sensitive to fabrication error. Another consideration here is that the transmission line's characteristic impedance no longer would match the impedance of the antenna, and the stubs need to be packed closer to return the $Z0$ to 67.7 Ω. However, from table 3, we see that the gap is already very small at 250 nm, hence any closer packing would undoubtedly cause fabrication issues such as shorts between the stubs.

The simpler solutions to broadening the bandwidth would be to either remove the stubs but widen the width of the microstrip line to match 67.7 Ω, or increase the output impedance of the antenna. Here, we explore the possibility of the prior solution to retain the optimal performance of the probe antenna. In this example, we make use of discrete periodic loading [6, 7] with standard non-inverted microstrip line structure fabricated using the same topology as above, except without the stubs. The unit cell layout is depicted in figure 8(a) where the width of the loading sections is increased and further increased every third loading to excite the sub-stopbands at lower frequencies. By setting the unit cell length at 3$f_{\mathrm{p}}$, we can then pump the TWPA near 220 GHz to achieve optimal gain as shown in figures 8(b) and (c). Table 4 lists the dimensions of the various sections forming the unit cell.

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As shown in figure 8(b), with this method, we successfully widen the operation bandwidth from ~37 GHz to approximately 130 GHz, triple the gain-bandwidth product of the device. The simulation was performed, including the effect of the waveguide antennas in the CME model (similar to the example shown above), hence the slightly narrower bandwidth limited by the cutoff frequency of the rectangular waveguides. It is trivial to amend the waveguide dimension to recover the optimized bandwidth, shown in the dashed curve of figure 8(b) if the waveguide dimension is increased slightly. Nevertheless, as can be seen, with this design, the operational bandwidth is broad enough to cover almost two atmospheric windows of the Atacama Large Millimetre/sub-millimetre Array (ALMA Band 5 & 6) simultaneously. Therefore, it could already potentially be deployed as a pre-amplifier for the SIS mixer front-end detector of the ALMA receivers to improve the receiver sensitivity further [27].