Unidirectional distributed acoustic reflection transducers for quantum applications

Surface acoustic waves (SAWs) are typically launched and received using interdigital transducers (IDTs), comprising parallel metal fingers evenly spaced on a piezoelectric substrate. The center frequency f₀ of the transducer corresponds to an acoustic wavelength λ₀ twice the lateral spacing p of the fingers, where the frequency and wavelength are related by $f0=ve/λ0$⁠, where v_e is the effective surface velocity in the transducer structure. The transducer bandwidth scales inversely with the number N of transducer fingers.¹ These structures have been used for many years, providing a convenient, inexpensive, and highly flexible approach to integrating acoustic wave resonators, delay lines, and pulse shaping with semiconductor electronics. Recently, in a pioneering experiment,² a superconducting qubit was coupled to surface acoustic waves (SAWs) using an IDT transducer, the circuit fabricated on a GaAs substrate with a center frequency of about 5 GHz. This experiment allowed the low-temperature observation of quantum effects mediated by SAW phonons, the quanta of mechanical vibrations, coupled to the qubit through the IDT structure. More recently, other advances in the quantum control of surface acoustics have been demonstrated,^3–5 showing promise for the expansion of this area of quantum physics.

A general challenge associated with uniform IDTs is that a voltage tone applied to the IDT will result in symmetric, oppositely directed SAWs being emitted from the two ends of the IDT, effectively acting as a three-port device, with one electrical and two acoustic channels. This is appropriate for resonant structures where the IDT is placed between a pair of Fabry-Perot-like acoustic mirrors,^1,3–5 but leading to the loss of half the acoustic power when used, for example, in a delay line geometry. For this reason, unidirectional transducers (UDTs) are typically used to preserve acoustic power, using a more complex finger geometry or multiple phase-controlled electrical signals to emit and receive waves in a single direction. In the limit of very high directivity, a UDT can be viewed as a two-port device with one electrical channel and one acoustic channel, transducing signals between the two modes without loss. Future quantum applications would greatly benefit from high directivity UDTs working in the gigahertz frequency band.

SAW directionality in UDTs is usually achieved using variants of IDT finger geometries designed to give constructive interference for waves emitted in one direction and destructive interference for those in the opposite direction. Multiphase UDTs offer very high directivity¹ but have complex finger geometries and require two or more phased voltage sources, making both fabrication and operation challenging. By contrast, single-phase UDTs (SPUDTs) achieve reasonable directivity using only a single voltage source, by using an asymmetric finger geometry. Since their introduction in 1976,⁶ these transducers have been implemented in a number of different formats.^7–15 

One interesting approach to SPUDTs, termed a distributed acoustic reflection transducer (DART),^1,9 is a design that achieves high directivity by shifting the center of transduction (TC) from the center of reflection (RC) by a distance 3λ₀/8, yielding the desired oppositely directed constructive and destructive interference. Traditionally, these have only been used at lower frequencies, below 100 MHz, with isolated examples up to 500 MHz.^12,14 The great simplicity of fabrication and operation makes these devices appealing for quantum operations if they can be made to work at higher (gigahertz) frequencies, compatible with superconducting qubits. Reaching such frequencies requires a careful iterative determination of the transducer parameters and solving some challenges in the fabrication of long and narrow fingers on somewhat uncooperative substrates. Here, we demonstrate a DART design that works well at frequencies approaching 5 GHz, an order of magnitude increase from previous work.

We designed and measured various DART designs, measuring single transducers as well as paired transducers in a delay line configuration, to fully characterize the transducer properties. The standard design is shown in Fig. 1, comprising driven transducer fingers of width λ₀/8 spaced by λ₀, interspersed with a 3λ₀/8- and a λ₀/8-wide pair of grounded fingers with λ₀/8 interfinger spacing. It has been shown that smaller thickness fingers not only lead to less surface-to-bulk mode conversion¹⁶ but also decrease the reflectivity,¹ and so a compromise is made on the metal transducer thickness. We pattern the UDTs as a single layer of aluminum of thickness t ≈ 28 nm on top of a single-side polished LiNbO³–128YX wafer of thickness h = 500 μm and room-temperature permittivity ϵ_r = 56. The aluminum film is lift-off patterned by electron-beam lithography, using a polymethylmethacrylate (PMMA) bilayer composed of a 100 nm thick 495 kD weight bottom layer and a 100 nm thick 950 kD weight top layer. A 10 nm thick layer of Au is thermally evaporated on the top PMMA layer to reduce charging effects. The DART geometry is kept constant along the entire transducer length, with center wavelength λ₀ = 800 nm, corresponding to a center frequency f₀ just below 5 GHz.

We designed arrays of transducers in delay line configurations, each delay line containing a pair of DARTs, typically with a center-to-center distance L = 400 μm = 500 λ₀. In a typical test set of devices, we would pattern an array of samples with different transducer apertures W = {18, 21, 24} μm and different numbers of repeat cells, N = {125, 130, 135}. Each set of transducers was fabricated in two delay line configurations, one where the DARTs are designed to emit toward one another and the other in the opposite configuration, where they emit away from one another [Fig. 1(d)].

Measurements were made at room-temperature with a vector network analyzer, using a calibrated microwave probe station. We measured both reflection $S11≅S22$ and transmission $S21≅S12$ between pairs of DARTs in a delay line configuration, with typical room-temperature results shown in Fig. 2.

For the transmission measurements, we show both the “towards” and “away” configurations, allowing us to extract the transducer directivity, a measure of the directionality of the acoustic power emission.

Over a broad frequency range, the reflection S₁₁ and transmission S₂₁ show the expected behavior, which away from the design frequency f₀ is dominated in reflection by the interdigital capacitance of the transducers and in transmission by the stray electrical coupling between the two transducers. For our design, the stray electrical coupling is less than −60 dB for most of the frequency range of the measurement from 10 MHz to 7 GHz. Near the design center frequency f₀ = 4.8 GHz, we see the expected detailed frequency response for both reflection and transmission. There is a pronounced difference in the transmission for the towards and away transducer configurations, as expected for these directional transducers.

The detailed response near resonance is accurately captured by a model that takes into account the properties of the LiNbO₃ wafer, the aluminum electrodes, and the microwave measurement circuitry. A schematic of the equivalent circuit model is shown in Fig. 1(c), including the transducers and the delay line equivalent impedances $Z1−3$⁠, and an approximate embedding circuit comprising a series resistance R_x ≈ 15 Ω associated with the electrodes and a stray capacitive coupling C_s ≈ 0.9 pF between the two DARTs. These two parameters provide the broad frequency response for both S₁₁ and S₂₁, away from the design frequency f₀.

To model the details of the DART response, we use the quasistatic approximation¹ to calculate the surface wave power as well as the electrical charge density and the DART capacitance. The wide 3λ₀/8 grounded electrode of the DART is modeled as two electrodes, spaced by λ₀/8, an approximation that has been shown to be valid at lower frequencies.¹⁷ The effective wave velocity v_e is estimated using a first-order expansion developed for single finger pair IDTs and used here as an approximation,¹ where v_e = v_f + Δv_e + Δv_m, with v_f being the free-surface SAW velocity, Δv_e the velocity change due to electrical loading, and Δv_m the velocity change due to mechanical loading. The reflection, transduction, conductance, and susceptance of the DARTs are modeled through the coupling-of-modes (COM) theory,¹ which models the DART cells as equally spaced point contacts that couple the SAW modes traveling in one direction to the modes traveling in the opposite direction, through the electrical current flowing in the microwave circuit. The COM theory has two critical parameters, the transduction and reflection of each DART cell. Following Ref. 1, we use the reflective array method (RAM) to relate the transduction to the surface wave power and electrostatic charge density; while the RAM theory was not developed for DART-style transducers, elsewhere it has been shown that this is a reliable modeling approach.¹⁸ The reflection is kept as a fitting parameter, which we restrict to be imaginary-valued, equivalent to assuming that transmission loss in the substrate dominates over transducer-associated losses.

In the reflection signal near resonance, the response consists of two local minima near 4.77 GHz and 4.84 GHz, separated by a broader local maximum. This is a characteristic of UDT-style transducers.¹ The reflection minima are at the frequencies where transmission is maximal, and the small ∼6 MHz ripples between the minima are due to interference between single and triple transit signals between the two transducers.

In the transmission signal near resonance, the overall signal level increases by more than 40 dB over the background stray coupling, with fine ripples due to interference effects within each transducer. The most striking feature is the strong difference in transmission between the toward and away DART configurations, due to the strong directivity of the transducer design. This directivity-dominated feature is more than 30 dB over a span of about 15 MHz centered at 4.785 GHz. The asymmetry of the response about the operating frequency is thought to be due to SAW velocity dispersion and the frequency-dependent reflectivity.¹ There is a slight frequency difference in the maximum transmission for the toward configuration compared to the minimum in the away configuration, attributed to the impedance mismatching of the transducers to the system 50 Ω impedance.

Fitting the detailed model to the data, we can extract the free velocity v_f = 3865 m s⁻¹, 3% slower than in the literature,¹ and the piezoelectric coupling Δv/v = 1.9%, where for a similar two-finger SPUDT design, but at a lower center frequency, a comparison value¹⁴ is Δv/v = 1.5%. The shape of the overall response gives the imaginary reflectivity r_s = i 0.04, which is roughly 30% higher than the RAM estimate for this geometry. We note that the positive sign for the reflectivity is expected for this type of substrate and is responsible for inverting the preferred emission direction for this DART design compared to other substrate choices.¹ The fit geometric DART capacitance of 1.43 pF is 22% smaller than the RAM estimate. While this represents a sizeable discrepancy, this is consistent with other experiments we have completed with significantly different designs but at similar frequencies on similar substrates.⁵ 

The rapid oscillations in the transmission signal near the operating frequency are due to interference of signals in the delay line and give us a very precise way to measure the effective distance between the two transducers, which is 400.08 μm, 0.02% longer than geometric center-to-center design distance. The position of the minimum dip in the away configuration is determined by the distance between the transduction (TC) and reflection (RC) centers, 3λ₀/8 by design, and is used as a fit parameter in the RAM theory. The numerical comparison yields a RC-TC distance 4% larger than the design value.

The DART directivity is defined¹ as the ratio of the relevant P-matrix couplings, $D=|P13/P23|$⁠. As these are not experimentally accessible, we instead use the approximate expression

the square root of the ratio of the transmission scattering parameters in the towards (t) and away (a) configurations. The directivity is shown in Fig. 3, where in (a) we show the directivity from the full frequency-dependent signal, and in (b), we show the directivity when the delay line response is filtered in the time domain to isolate the response from the first transit signal through the delay line, as shown in (c). We note that without filtering, the apparent peak directivity is as high as 37 dB, but by selecting only the signal associated with the first transit, we find a more accurate measure of the peak directivity of 23 dB. This device has a bandwidth of 15 MHz, defined as the frequency range over which the directivity exceeds 15 dB.

We have measured a number of different DART configurations and tabulated representative results in Table I, using the first transit filtered data for each design. The highest directivity was measured to be D ≈ 27 dB for an aperture W = 18 μm, with N = 125 cells. The largest bandwidth was ≅17 MHz. The directivity shows a weak dependence on DART aperture, with larger apertures giving slightly lower directivity and smaller bandwidth, with the change mostly due to an increase in the away transmission.

In conclusion, we have fabricated DARTs with a high center frequency of 4.8 GHz, demonstrating a directivity greater than 15 dB over a bandwidth of 17 MHz. While highly promising, more studies should be devoted to understanding the performance as a function of the transducer electrode dimensions. Future work should study the dependence of the directivity and bandwidth on parameters such as the metal electrode thickness, as well as the electrode and interelectrode width ratio, which here were kept close to unity. The bandwidth of the DART could be increased by using slanted fingers.¹⁹ A weighted or resonant design^1,20 could also sharpen and flatten the DART transduction peak response. The performance demonstrated here is promising for experiments in the quantum limit.

See the supplementary material for the derivation of the approximate directivity formula [Eq. (1)] and a discussion of its validity.

Devices and experiments were supported by the Air Force Office of Scientific Research and the Army Research Laboratory, and material for this work was supported by the Department of Energy (DOE). É.D. was supported by LDRD funds from Argonne National Laboratory, K.J.S. was supported by NSF GRFP (No. NSF DGE-1144085), and A.N.C. was supported by the DOE, Office of Basic Energy Sciences. This work was partially supported by the UChicago MRSEC (No. NSF DMR-1420709) and made use of the Pritzker Nanofabrication Facility, which receives support from SHyNE, a node of the National Science Foundation's National Nanotechnology Coordinated Infrastructure (No. NSF NNCI-1542205).