Nanometric constrictions in superconducting coplanar waveguide resonators
Mark David Jenkins, Uta Naether, Miguel Ciria, Javier Sesé, James Atkinson, Carlos Sánchez-Azqueta, Enrique del Barco, Johannes Majer, David Zueco, Fernando Luis
NNanoscale constrictions in superconducting coplanar waveguide resonators
Mark David Jenkins,
1, 2
Uta Naether,
1, 2
Miguel Ciria,
1, 2
Javier Ses´e,
3, 2
James Atkinson, CarlosS´anchez-Azqueta, Enrique del Barco, Johannes Majer, David Zueco,
1, 2 and Fernando Luis
1, 2, a) Instituto de Ciencia de Materiales de Arag´on, CSIC - Universidad de Zaragoza, 50009 Zaragoza,Spain Departamento de F´ısica de la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza,Spain Instituto de Nanociencia de Arag´on, Universidad de Zaragoza E-50009 Zaragoza,Spain Department of Physics, University of Central Florida, Orlando, FL 32816,USA Dpto. de Ingenier´ıa Electr´onica y Telecomunicaciones, Universidad de Zaragoza, 50009 Zaragoza,Spain Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna,Austria (Dated: 11 September 2018)
We report on the design, fabrication and characterization of superconducting coplanar waveguide resonatorswith nanoscopic constrictions. By reducing the size of the center line down to 50 nm, the radio frequencycurrents are concentrated and the magnetic field in its vicinity is increased. The device characteristics are onlyslightly modified by the constrictions, with changes in resonance frequency lower than 1 % and internal qualityfactors of the same order of magnitude as the original ones. These devices could enable the achievement ofhigher couplings to small magnetic samples or even to single molecular spins and have applications in circuitquantum electrodynamics, quantum computing and electron paramagnetic resonance.The field of cavity quantum electrodynamics (QED)studies the interaction of photons in resonant cavitieswith either natural or ”artificial” atoms, such as quan-tum dots and superconducting qubits, having a nonlinearand discrete energy level spectrum.
For applicationsin spectroscopy and especially quantum information pro-cessing a major goal is to maximize the coupling strength g of the atom to either electric or magnetic cavity fields,making it larger than the decoherence rates of both thecavity and the atom (strong coupling regime).One of the most direct ways of enhancing g is to reducethe cavity size and therefore increase the electromagnetic (a) FIG. 1. (a) Microscope photograph of a coplanar waveg-uide resonator fabricated with niobium deposited on a sap-phire substrate. The length of the resonator segment is44 mm, which corresponds to a resonance frequency f (cid:39) . − . a) Electronic mail: fl[email protected] energy density and field strength at the atom site. This isthe idea behind circuit QED, in which the classical threedimensional resonant cavities are replaced by sections ofmicrowave transmission lines. The prime example of thistype of cavity is a coplanar waveguide (CPW) resonator,i.e., a section of coplanar waveguide capacitively coupledto external feed lines (see Fig. 1). As a first approxima-tion, these systems can be considered as one dimensionalresonators, where the resonant frequency f is controlledby the length of the transmission line segment. The pho-ton energy is concentrated in and around the center line,leading to field strengths up to 100 times larger than intypical three dimensional cavities. Furthermore, if thelines are superconducting the resistive losses can be sup-pressed to achieve quality factors Q of up to 10 . In thesecases, the losses and dephasing of the composite systemare often limited only by the atom properties.Previous studies have shown the performance of thesedevices, and their strong coupling with different typesof quantum two-level systems, such as superconductingqubits or quantum dots, at the single photon level.Strong coupling has also been achieved to collective spinstates of, e.g., nitrogen vacancy centers in diamond andother magnetic systems. Electron spins are attrac-tive due to their usually longer coherence times, as com-pared with electrical degrees of freedom, which couldallow longer storage times for quantum information. Strong coupling to single spins or small ensembles ofthem has not been achieved yet, due to their weaker cou-pling to the electromagnetic radiation. Achieving thislimit would open new possibilities in quantum informa-tion and related fields. Artificial molecular magnets, syn-thesized by chemical methods, are of especial interest a r X i v : . [ c ond - m a t . s up r- c on ] O c t as they provide realizations of well-defined and identi-cal qubits and quantum logic gates. In a previouswork, it was proposed that strong magnetic couplingcould be achieved, even for single molecular magnets, bynarrowing the center line of a resonator down to nanome-ter length scales, as long as the resonator characteristicscan be maintained. A similar procedure has been pro-posed to couple superconducting resonators to the spinsof donor defects in silicon and to flux qubits. In thiswork we design, fabricate and test superconducting CPWresonators with this type of constrictions and show thattheir characteristics are stable within a relatively broadrange of constriction geometries.Our devices are fabricated on 500 µ m thick C-planesapphire wafers and consist of a 150 nm thick niobiumlayer deposited by radio frequency (RF) sputtering andthen patterned by either photolithography and lift-off orreactive ion etching. Nanoscale constrictions were madeat the midpoint of the center line by etching it with a fo-cused beam of Ga + ions, using a commercial dual beamsystem. The ionic current was kept below 20 pA to max-imize the resolution in the fabrication process and tominimize the Nb thickness, of order 10 −
15 nm, thatis implanted with Ga.
Images of these constrictions,as those shown in Fig. 1, were obtained in situ by scan-ning electron microscopy. The microwave transmissionmeasurements were done using a programmable networkanalyzer at 4 . µ m center line and200 µ m gaps that narrow down to around 14 µ m and7 µ m, respectively after going through gap capacitors.Several types of gap capacitors with a finger design (seeFig. 1) were fabricated to allow for differently coupledsystems. The gaps between the fingers are of 4 µ m whilethe finger lengths are of 100 µ m. The length of the cavityis chosen to be 44 mm by making the waveguide meanderacross the surface. Although sapphire has an anisotropicdielectric constant, taking an average value of (cid:15) r = 10is sufficient for our circuit calculations. These parame-ters give a waveguide characteristic impedance Z (cid:39)
50 Ωand an unloaded f (cid:39) . . .
405 GHz) andhave higher Q (from 100 to 30000) the less coupled theyare to the feed lines. The gap capacitances C gap havebeen estimated, as illustrated in Fig. 2, by fitting thetransmission S vs frequency data with a simple circuitmodel, which consists of a segment of transmission linewith two identical capacitors connected to the input andoutput ports. They range from 70 to 5 fF, of the order ofmagnitude of values reported in the literature. After the devices were characterized, the center linewas narrowed down from about 10 µ m to minimumwidths w of 50 nm along distances δ of up to 50 µ m (seeFig. 1). Once the constrictions are made, special caremust be taken both with the device manipulation andwith the scanning electron microscope imaging since elec-trostatic buildup and discharge can easily destroy the (b)(c) × -25-20-15-10-5 -5-10-15-20-25 1.374 1.376 1.378 1.380 Frequency (GHz) S ( d B ) Original (a) (d)
FIG. 2. (a) Comparison of transmission resonances measuredon a superconducting resonator before and after the fabrica-tion of a nanoloop constriction at its center line. ∆ f , ∆ Q and ∆ T are the variations found in the resonance frequency,quality factor and maximum transmission (in linear scale),respectively. In this particular case, the original parameterswere f = 1 .
378 GHz and Q = 2700 while ∆ f /f = − . Q/Q = − .
7% and ∆ S /S = − L A and R A that ac-count, respectively, for the reduced f and the enhancementof electromagnetic losses. (c) Variation of resonance param-eters for 100 nm wide constrictions of varying lengths (withoriginal values of f (cid:39) .
35 GHz and Q (cid:39) µ m long constrictions of varyingwidths (with original values of f (cid:39) .
31 GHz and Q (cid:39) nanowires. The present study uses two series of threeidentical resonators. In each series, we make constrictionsvarying either δ or w and keeping the other constant. Thefirst series has 100 nm wide constrictions and lengths of1 µ m, 5 µ m and 15 µ m. Similarly, the second series keepsa constant δ = 1 µ m while w is varied through 300 nm,100 nm and 50 nm. Other geometries, such as loop con-strictions, can also be obtained as we show in Fig. 1(c).The performances of these devices are analyzed by com-paring values of f , Q and maximum S measured beforeand after the constrictions were made. The internal qual-ity factor Q int , which parameterizes the intrinsic losses ofthe resonator, has been estimated from the insertion loss,determined from Q and S as described in [5]. The val-ues found are shown in Figs. 2 (c) and (d). Althoughall the results we present are from transmission measure-ments ( S , S ), the same behavior can be seen in thereflection signals ( S , S ).The decrease of f , of order 1%, likely results from theenhancement of the inductance at the constriction. Asimple approach is to model the constriction by a regionof length δ and effective inductance per unit length, l (cid:48) ,larger than its value l outside this region. As it is de-scribed in detail in [24], this effect makes the center lineeffectively longer for the propagation of RF currents andleads to a close to linear decrease of f with increasing δ ∆ f f (cid:39) − δL (cid:18) l (cid:48) l − (cid:19) (1)The experimental data shown in Fig. 2(c) are compatiblewith l (cid:48) /l = 10 .
9. The decrease of Q and S indicate thatthe constriction introduces some extra losses into the sys-tem. This is confirmed by the stronger relative variationseen in Q int . The constriction constitutes a defect forthe propagation of electromagnetic radiation and there-fore might enhance reflection to the source. As shownin Fig. 2, the experimental results can be accounted forby a circuit model with two additional lumped elements,an inductance L A that accounts for changes of f , and aresistance R A that accounts for ∆ Q . From fits such asthose shown in Fig. 2(a), we find L A to lie in the rangeof a few tens pH, increasing with δ as expected from theabove considerations, whereas the effective resistance R A is of the order of a few mΩ’s and fairly independent of δ .Despite these additional losses, Q remains mainly limitedby the coupling capacitors C g for all devices studied inthis work (see Fig. 1 in [24]). These results show there-fore that the fabrication of narrow constrictions does notpreclude the attainment of Q int values well above 10 .Another difference observed in resonators with con-strictions is the power dependence of the resonances. Fig-ure 3 shows the transmission through a 100 nm wide and1 µ m long constriction at different excitation powers andfor the first three cavity modes. The fundamental mode(Fig. 3(b)) and the second harmonic (Fig. 3(d)) breakdown when power is increased. This effect can be ex-plained by noting that, since both the fundamental modeand second harmonic have a standing wave with a currentmaximum at the center (as shown by Fig. 3(a)) wherethe cross section has been drastically reduced, the con-striction can eventually become resistive. By contrast, wefind no such effect in the resonance associated with thefirst harmonic since there is almost zero current flowingthrough the constriction for this mode. These qualita-tive arguments can be made quantitative by comparingthe current density at which the resonances break downto the critical current of superconducting niobium. Thecurrent flowing through the nanowire can be estimated byapplying the definition of Q and formulas for the equiv-alent RLC circuit for the resonator. The following ex-pression gives the current amplitude in the resonator I = (cid:114) πQP loss Z , (2)where P loss is the power loss from the resonator (dis-sipated or lost to the feed lines). From the max-imum RF power for which the resonance shown inFig. 3 remains stable we get a critical current den-sity j c (cid:39) . × A cm − (corresponding to about Frequency (GHz) Frequency (GHz) Frequency (GHz) (a)(b) (c) (d) -45 -30-20 -10 0 dBm -45 -30-20-100 dBm
FIG. 3. Diagram (a) schematically shows the center line of aresonator and the standing current waves for the first threeresonant modes. Graphs (b), (c) and (d) show the transmis-sion spectra in a constricted resonator for these three modesand for increasing excitation power. Modes with a currentmaximum at the constriction (fundamental and second har-monic modes) show a loss of resonance when power is in-creased while modes with no current at the constriction (firstharmonic) show very small changes. × photons stored in the cavity), which is of the or-der of magnitude of that found for niobium thin films. Experiments performed on resonators with 50 nm wideconstrictions show a similar power dependence and give j c (cid:39) . × A cm − . These results suggest thatthe the implantation of Ga at the constriction edges,which is consubstantial to focussed ion beam lithogra-phy, does not dramatically affect the superconductivityof Nb. They also evidence that the current is forced toflow through the constrictions and that its density, thusalso the RF magnetic field, is being locally enhanced.In order to explore such enhancement in a more di-rect manner, we locally measured, by means of magneticforce microscopy (MFM), the magnetic field generated bya current flowing via one constriction. A 1 µ m long by100 nm wide constriction was fabricated out of a 100 nmthick gold layer deposited on sapphire. The circuit wasthen mounted on the MFM stage and connected to a di-rect current source at room temperature. MFM images ofthe constriction area were measured while a 2 . φ = 0 . ◦ from maximum to minimum)in the constriction area and a negligible signal (indistin-guishable from the background) in the wider areas of thecircuit. Also, the current was reversed three times dur-ing the image acquisition. Each time, a sharp contrastchange was observed in the phase image, thus showingthat the signal must be magnetic in origin and due to thecirculating current. We observe a constant backgroundsignal probably due to an electrostatic potential differ- Profile 3Profile 2Profile 1Profile 2
FIG. 4. (a) Atomic force microscopy image of a 1 µ m by100 nm constriction made from a 100 nm thick gold layer de-posited on a sapphire substrate. A 2 . ence between the metallized and non-metallized areas. Ifgreater precision were needed, this background could befiltered out using schemes similar to those detailed in where electric potential nulling was used.We conclude that the fabrication of nanoscopic con-strictions in the center line of coplanar superconduct-ing resonators provides a simple and efficient method tolocally concentrate the RF magnetic field, with a mi-nor cost in terms of quality factor. Although the res-onator frequencies explored in this work are relativelylow (1 . . . These devices could potentially be used to de-velop high sensitivity spectrometers for the characteriza-tion of nanoscopic magnetic samples and, provided thatsufficiently long spin decoherence times are attained, toachieve strong coupling to individual magnetic qubits.
ACKNOWLEDGMENTS
We acknowledge fruitful discussions with J. J. Garc´ıa-Ripoll and S. Celma. The present work has been partlyfunded by grants MAT2012-38318-C03, MAT2012-31309,and FIS2011-25167 from the Spanish MINECO, DGAgrants E98-MOLCHIP, E19-GEFENOL, and E81, EU project PROMISCE, and Fondo Social Europeo. MarkD. Jenkins work was funded by a JAE-predoc CSICgrant. H. Mabuchi and A. C. Doherty, Science (New York, N.Y.) ,1372 (2002). R. J. Schoelkopf and S. M. Girvin, Nature , 664 (2008). A. Blais, R.-S. Huang, A. Wallraff, S. Girvin, and R. Schoelkopf,Physical Review A , 062320 (2004). L. Frunzio, A. Wallraff, D. Schuster, J. Majer, and R. Schoelkopf,Applied Superconductivity, IEEE Transactions on , 860(2005). M. G¨oppl, A. Fragner, M. Baur, R. Bianchetti, S. Filipp, J. M.Fink, P. J. Leek, G. Puebla, L. Steffen, and A. Wallraff, Journalof Applied Physics , 113904 (2008). A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang,J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature , 162 (2004). J. Majer, J. M. Chow, et al. , Nature , 443 (2007). A. Abdumalikov, O. Astafiev, Y. Nakamura, Y. Pashkin, andJ. Tsai, Physical Review B , 180502 (2008). L. Dicarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow,J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, andR. J. Schoelkopf, Nature , 574 (2010). T. Niemczyk, F. Deppe, et al. , Nature Physics , 772 (2010). K. D. Petersson, L. W. McFaul, M. D. Schroer, M. Jung, J. M.Taylor, A. A. Houck, and J. R. Petta, Nature , 380 (2012). D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio,J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D.Awschalom, and R. J. Schoelkopf, Physical Review Letters ,140501 (2010). Y. Kubo, F. R. Ong, et al. , Physical Review Letters , 140502(2010). I. Chiorescu, N. Groll, S. Bertaina, T. Mori, and S. Miyashita,Physical Review B , 024413 (2010). R. Ams¨uss, C. Koller, et al. , Physical Review Letters , 060502(2011). F. Troiani, A. Ghirri, M. Affronte, S. Carretta, P. Santini,G. Amoretti, S. Piligkos, G. Timco, and R. Winpenny, Phys-ical Review Letters , 207208 (2005). C. J. Wedge, G. A. Timco, E. T. Spielberg, R. E. George,F. Tuna, S. Rigby, E. J. L. McInnes, R. E. P. Winpenny, S. J.Blundell, and A. Ardavan, Physical Review Letters , 107204(2012). M. J. Mart´ınez-P´erez, S. Cardona-Serra, et al. , Physical ReviewLetters , 247213 (2012). F. Luis, A. Repoll´es, et al. , Physical Review Letters , 117203(2011). M. Jenkins, T. H¨ummer, M. J. Mart´ınez-P´erez, J. Garc´ıa-Ripoll,D. Zueco, and F. Luis, New Journal of Physics , 095007 (2013). G. Tosi, F. A. Mohiyaddin, H. Huebl, and A. Morello, AIPAdvances , 087122 (2014). L. Hao, D. C. Cox, and J. C. Gallop, Superconductor Scienceand Technology , 064011 (2009). C. Cast´an-Guerrero, J. Herrero-Albillos, et al. , Physical ReviewB , 144405 (2014). “See supplemental material at http://dx.doi.org/10.1063/1.4899141 for details on the loaded Q-factor, a simple modelto estimate the dependence of the resonant frequency onthe constriction properties, and the power dependence of a50 nm resonator ftp://ftp.aip.org/epaps/appl_phys_lett/E-APPLAB-105-034443 ,”. R. Huebener, R. Kampwirth, R. Martin, T. Barbee, andR. Zubeck, IEEE Transactions on Magnetics , 344 (1975). R. Yongsunthon, J. McCoy, and E. D. Williams, Journal of Vac-uum Science & Technology A: Vacuum, Surfaces, and Films19