Loss Measurement

Chapter V: Other Experiments

5.1 Loss Measurement

All of our parametric amplifiers over the past several years have been designed with a uniform basis for the microstrip line: a 250 nm wide, 35 nm thick NbTiN layer deposited on a Silicon substrate separated from the ground plane by a layer of amorphous Silicon. Because the central conductor and adjacent materials remain identical across our device, we can accurately estimate the loss across all of our devices through a careful measurement of a single similar test structure.

We accomplish this by fabricating an on-chip Fabry-Pérot interferometer centered at 8.3 GHz consisting of a 93 mm transmission line in between two Bragg reflec- tors.[64] The device, shown in Fig. 5.1, acts as an etalon for frequencies within the stop band. This creates transmission peaks in the𝑆₂₁at frequencies

𝜔_𝑛 = 𝑛𝜋 𝑣_ph

𝐿

(5.1) where 𝐿 is the distance between the reflectors.[64] From the measured frequency spacing in Figure 5.1 (c), we find that𝑣_ph =0.0077𝑐, in perfect agreement with the value measured in our low-frequency parametric amplifiers. The transmission of the etalon may be expressed as

𝑆₂₁ = 𝑡²𝑒^{−𝛾 𝐿} 1−𝑟²𝑒^{−2𝛾 𝐿}

, (5.2)

where𝑡and𝑟are the frequency dependent Bragg reflector transmission and reflection amplitudes and𝛾is the propagation constant on the internal transmission line section as given in 3.29. Near the resonance frequencies, 𝜔_𝑛, the transmission peaks are nearly Lorentzian and follow

𝑆₂₁(𝜔_𝑛+𝛿𝜔) ≈ 𝑄_𝑟 𝑄_𝑐

𝑡² 1−𝑟²

𝑒^{−𝛾 𝐿} 1−2𝑖𝑄_𝑟𝛿𝜔/𝜔_𝑛

, (5.3)

for some small 𝛿𝜔 where 𝑄_𝑟 is quality factor measured by the full width half maximum of the resonance and can be decomposed into the quality factor from the internal losses,𝑄_𝑖, and from the coupling,𝑄_𝑐, via

𝑄⁻¹

𝑟 =𝑄⁻¹

𝑐 +𝑄⁻¹

𝑖 . (5.4)

+ +

+ +

+ +

transmission line x666 Bragg reflector

taper x8 taper Bragg reflector

taper x8 taper

(a)

(b)

(c)

Figure 5.1: (a) The schematic structure of the on-chip Fabry-Pérot interferometer.

The capacitive fingers have an average length of 26𝜇m spaced 2 𝜇m apart and are sinusoidally modulated with an amplitude of 12 𝜇m and periodicity 140 𝜇m for a total of 8 periods to form the Bragg reflectors, and this modulation is then tapered across 2 periods to decrease the impedance mismatch with the transmission line.

(b) The calculated𝑆₂₁according to the method in section 3.1. (c) The measured𝑆₂₁ at 1 K with baseline loss removed from fitting the higher/lower frequencies.[64]

We can further express these two individual quality factors as 𝑄_𝑐 = 𝜔_𝑛𝐿

𝑣_ph 𝑟² 1−𝑟²

(5.5) and

𝑄_𝑖 = 𝛽 2𝛼

(5.6) where𝛼 and 𝛽 are the real and imaginary components of𝛾 = 𝛼+𝑖 𝛽. Neglecting loss in the Bragg reflectors by taking

𝑟²+𝑡² =1 (5.7)

and normalizing the transmission to that of a single pass through the internal trans- mission line section (dividing by 𝑒^{−𝛾 𝐿}), the above expression reduces to simply

𝑆₂₁ ≈ 𝑄_𝑟 𝑄_𝑐

1

1−2𝑖𝑄_𝑟𝛿𝜔/𝜔_𝑛

. (5.8)

The 𝑆₂₁ transmission of the device was measured in a setup similar to Figure 4.6 at 1 K without the unnecessary pump channel and directional coupler. The result, shown in Figure 5.1, has great agreement with the theoretical prediction.

We then extracted 𝑄_𝑖 using two different methods. In the first “circuit model method,” we apply the lossless circuit model used to obtain the predicted 𝑆₂₁ in Figure 5.1 (b) to extract𝑄_𝑐 according to equation 5.8 (nothing that in the lossless calculation,𝑄_𝑟 =𝑄_𝑐). With this calculated value of𝑄_𝑐at hand, we fit our measured 𝑆₂₁ data to extract 𝑄_𝑖. In the second “resonance height method,” we instead note that the maximum resonance height

max(|𝑆₂₁|) = 𝑄_𝑟 𝑄_𝑐

(5.9) which when combined with equation 5.4 can be solved for

𝑄_𝑖 = 𝑄_𝑟

1−max(|𝑆₂₁|) (5.10)

using the measured 𝑆₂₁ and fitting the full width half maximum of each resonator for𝑄_𝑟. We only perform this analysis for resonances near the center of the stop band because the higher𝑄_𝑐 makes them more sensitive to changes in𝑄_𝑖. The results of both of these calculations are shown in Figure 5.2 (a).

The large variance and slope of the circuit model calculation suggests that the coupling quality factors,𝑄_𝑐, deviate from the idealized values in our model. At 8.4

(a)

(b)

(c)

Figure 5.2: (a) The frequency dependent attenuation factor when 𝐼_𝐷𝐶 = 0. (b) Quality factor and corresponding attenuation as a function of current. The device loss for a one-way parametric amplifier with the same 93 mm length. (c) The frequency shift of a single resonance by increasing DC current and corresponding extracted nonlinearity scale factors. [64]

GHz where the two methods give consistent results, we obtain an𝛼=3.7 dB/m and 𝑄_𝑖 = 2.8∗10⁴. This result is lower than the𝑄_𝑖 > 10⁵ than has been observed in resonators using an identical amorphous Silicon dielectric,[79] indicating that some of the losses may result from the NbTiN film itself.

Repeating the measurements across a range of applied DC currents reveals a degra- dation in the quality factor at high currents (Figure 5.2 (b)) and measured frequency shift of the resonators due to the change in𝑣_ph(Figure 5.2 (c)). While the latter result provides an excellent measure of𝐼_∗and𝐼_∗^′for our gain calculations, the source of of the increased attenuation at higher currents is unclear. The temperature dependent loss (previously shown in Figure 4.15) for NbTiN is too small near 1 K to account for heating effects to account for the three-fold increase in attenuation. Furthermore, a calculation of 𝑄_𝑖 using the Usadel and Nam’s equations remains above 10⁸ for the highest current we applied, so it is not explained by the changing density of states.[105, 106, 107, 108] One potential explanation is the presence of magnetic fields perpendicular to the microstrip, where a 3 mT field has been reported to degrate𝑄_𝑖in NbTiN resonators from 10⁵to 10³.[109] The magnetic field generated from our DC current is 0.84 mT, and the geometry of the meandering microstrip and induced ground plane currents may yield a similar, smaller effect that we see in our measurements. Further experiments are needed to confirm or refute this hypothesis.