263

8Coherent Quantum Phase Slips

8.1Introduction

In the last few years, experiments aimed at demonstrating the existence of coherentquantum phase slips (cQPS) have flourished. In particular, in 2012, three experi-ments published within a few months of each other provided convincing evidencethat the effects of cQPS are readily observable and the duality proposed by Mooijand Nazarov [1], and later by Khlebnikov [2, 3], has experimentally observable con-sequences. Those experiments are:

A single-charge transistor based on the charge-phase duality of a superconduct-ing nanowire circuit

Quantum phase slip (QPS) phenomenon in ultranarrow superconductingnanorings

Coherent quantum phase slip.

8.2A Single-Charge Transistor Based on the Charge-Phase Duality of a SuperconductingNanowire Circuit

In 2006, Mooij and Nazarov suggested that “if coherent quantum phase slips exist,they are the exact dual to Josephson tunneling” [1]. In their paper, they also sug-gested a device to be used as a standard of current – dual to the voltage standardbased on the Josephson junction (JJ). From this suggestion and the later proposalby Hriscu and Nazarov [4], Hongisto and Zorin [5] demonstrated a QPS transistor,dual to a DC SQUID.

From the circuit presented in Figure 8.1a, we can see that the transistor iscomposed of two QPS elements embedded in a high-dissipative environment(R1 � R2 � 0.4 MΩ): the modulation is provided by a voltage gate symmetric withrespect to the two QPS elements. Looking at Figure 8.2, we can see that the twohalves of the circuit (left and right of the central capacitor) are each equivalent to acurrent biased JJ (Josephson junction). Remembering that the dual of a capacitor

One-Dimensional Superconductivity in Nanowires, First Edition. F. Altomare and A.M. Chang© 2013 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2013 by WILEY-VCH Verlag GmbH & Co. KGaA.

264 8 Coherent Quantum Phase Slips

Figure 8.1 (a) The layout with three fragmentsof the SEM image and (b) the simplified elec-tric circuit diagram of the QPS transistorembedded, in fact, in the four-terminal net-work of on-chip resistors. The device includes

two QPS elements denoted by diamond sym-bols, kinetic inductances of the nanowire seg-ments, and a capacitive gate. Thicknessesof Nbx Si1�x and Cr films are 10 and 30 nm,respectively. From [5].

Figure 8.2 (a) Current-biased Josephson junction; (b) voltage biased Josephson junction; (c)current-biased QPS junction; (d) voltage-biased QPS junction. Circuit (a) is the exact dual ofcircuit (d). Circuit (b) is the exact dual of circuit (c). From [1].

is an inductance and the dual of the gate voltage will be a current flowing in theinductance – and hence a flux – the equivalence of this circuit with a DC SQUIDis clear.

As the DC SQUID is sensitive to the phase across the junctions, it depends onthe critical current and capacitance of the JJs, it depends on the shunting resistoracross the JJs and it operates in a low impedance environment; the QPS transistoris sensitive to the charge through the QPS elements, it depends on the voltagesdrop and inductances of the QPS, it has series resistances and it operates in a high-impedance environment.

To make sure that the nanowires were in the 1D regime, the authors fabricated an18 nm wide wire, smaller than the superconducting coherence length (� 20 nm).If the circuit is asymmetric, as the fabrication of an 18 nm wide wire is nontrivial,that is, one of the two QPS junctions has an energy higher than the other (EQPS1 �

8.2 A Single-Charge Transistor 265

EQPS2), the equation of motion describing this circuit becomes ((4.16) with Q D�2eq)

Lk RQ C R PQ C Vm(Qg) sin

πQe

�D Vb , (8.1)

with L k D L k1 C L k2, R D R1 C R2, Vm the modulation of the blockade voltage,Qg D q1 � q2, Q D q1,2 ˙ Qg/2, L k i the kinetic inductance of each of the wiresegments, Ri the resistors, qi the charge of each QPS element, and Vb the drivingvoltage. Equation (8.1) describes the dynamics of a nonlinear oscillator with finitedamping and is fully analogous to the equation describing the DC SQUID (see [7],for example).

Two devices (differing from each other for the length of the island and nanowire)were reported by Hongisto and Zorin [5]. Device fabrication consisted of the initialdeposition of chromium resistors together with gold micropads that were usedto make contact to the superconductor Nbx Si1�x . This is deposited in a subse-quent step in a sputtering chamber in which the relative composition of the twotargets had been calibrated. Finally, the nanowire elements and the island are de-fined with e-beam lithography using negative tone resist hydrogen silsesquioxane(HSQ). Once exposed with the electron beam during the lithography step, HSQwould harden and the unexposed region would be washed away by the developer.Dry etching is then used to transfer the pattern in the film, etching away the excesssuperconductor. The main experimental results are shown in Figure 8.3.

(a) (b)

Figure 8.3 (a) The charge-modulated I–Vcurves of sample A recorded in a current-bias regime for two gate voltages shifted bya half-period. In the region of small currents(enlarged in the inset), one can see the mod-ulation with period Δ I D 13.5 pA, which isdue to the asymmetry of off-chip biasing cir-cuitry, resulting in the current dependence ofthe electric potential of the transistor island,δV D (Rbias1 � Rbias2)I and, therefore, of theeffective gate charge, δQg D (Cg C C0)δV .The green dashed line shows the shape of thebare I–V curve given by the RSJ model (Equa-

tion 10 in [5]) with fixed Qg. (b) The I–Vcurves of sample B measured in the voltagebias regime at different values of gate voltageVg. The bottom right inset shows details of theCoulomb blockade corner. Upper left inset:the gate voltage dependence of the transistorcurrent measured at different bias voltages Vb,providing a steady increase of hV i from 0.321up to 0.481 mV in 20 μV steps (from bottomto top). Adapted from [5]. For a color versionof this figure, please see the color plates at thebeginning of the book.

266 8 Coherent Quantum Phase Slips

As can be seen from Figure 8.3a,b, both devices qualitatively exhibit the same be-havior. The I–V curves of both devices show the Coulomb blockade (hI i D 0) until acertain critical voltage is reached. For larger voltage, the nanowires become normal.This is precisely the reverse of what takes place in a JJ, for which the voltage is zerountil a critical current is reached. As seen in many superconducting nanowires, theintercept between the linear extrapolation of the I–V curve at high voltage with thecurrent axis is positive, that is, a fraction of the electrons are still superconducting;this is a clear indication that the gating effect has its origin in the superconductingnanowire portion of the circuit (e.g., [8]). The insets in Figure 8.3b clearly showthe effect due to the gating voltage. However, while the period of modulation forsample A is about 2 mV, somewhat consistent with the estimate of the capacitance,for sample B, it is 150 mV with a capacitance two orders of magnitude smallerthan expected. While the authors cannot explain this discrepancy, this experimentdefinitely demonstrates the behavior expected from a QPS-based transistor and itssensitivity to the gate voltage.

8.3Quantum Phase-Slip Phenomenon in Ultranarrow Superconducting Nanorings

As it is well-known, a superconducting ring can sustain a persistent current.Changing the flux through the ring will change the value of the persistent currentuntil the critical current of the ring is reached. If one were to further increasethe flux above this value, the total flux through the ring will increase by one ormore flux quanta and its persistent current will drop. This simple picture is heavilymodified by the presence of quantum phase slips as suggested by Matveev et al. [9]:“The superconductivity in very thin rings is suppressed by quantum phase slips.As a result, the amplitude of the persistent current oscillations with flux becomesexponentially small, and their shape changes from sawtooth to a sinusoidal one.”

As shown in Figure 8.4, the effect of quantum phase slips is to open a gap wherethe energy levels cross so that the persistent current, for a system evolving in fluxwhile following the ground state, becomes sinusoidal. This effect was derived fornanorings composed of a chain of weak links: however, for high transparency be-tween the grains, this is equivalent to a uniform nanowire in a ring geometry [10].Starting from this observation, Arutyunov et al. [11] fabricated nanorings of bothaluminum and titanium. The nanoring is embedded between two highly resistivecontacts in order to perform tunnel spectroscopy of the system as a function of themagnetic field. Starting from a nanoring composed of a large diameter nanowire(not to be confused with the diameter of the nanoring) and using the ion millingtechnique [12], the tunneling current through the nanoring is measured as a func-tion of the magnetic field. The results for aluminum and titanium nanorings areshown in Figures 8.5 and 8.6.

In the case of an aluminum nanoring for a wide nanowire (Figure 8.5), the tun-neling current exhibits a behavior similar to the one in Figure 8.4: the currentincreases until a current close to the critical current is reached at which point

8.4 Coherent Quantum Phase Slip 267

Figure 8.4 (a) The energy of a nanoring as afunction of the flux Φ through it. In the limitof the relatively large nanowire cross sectionA, the energy shows the classical behaviorshown by a solid line. Quantum phase slipsresult in level splitting, shown by a dashedline, and then lead to sinusoidal dependenceof the ground state energy on the flux (dotted

line). (b) The persistent current in the nanor-ing, I(Φ ) D cdE/dΦ (in cgs units). In theclassical limit, the current shows sawtoothbehavior, shown by a solid line. As the wirebecomes thinner, the sawtooth is rounded andeventually transforms to a sinusoidal oscilla-tion. From [9].

the current drops. Further increasing the flux through the nanoring, produces asimilar trace. The increase in the baseline is attributed to a gap reduction due tothe applied magnetic field. Reducing the cross-sectional area from σ1/2

fit D 91 nmto σ1/2

fit D 43 nm produces a reduction of the overall amplitude of the tunnelingcurrent. A further reduction in area completely changes the characteristics of thetunneling current and the shape of the persistent current. Even more impressiveare the results for a titanium nanoring in Figure 8.6: even the σ1/2

fit D 67 nm, com-parable to the superconducting coherence length in Ti [6] (see Table 6.1), showa behavior close to the QPS regime. A simple reduction of 10% in cross-sectionalarea, a device that can still easily be fabricated with conventional lithography, showsa changes in the amplitude of the oscillations by a factor of � 6. The appearanceof cQPS at such relatively large cross sectional area, is probably to be expectedgiven the results on titanium nanowires reported by the same group (see [6] andSection 7.4.2).

8.4Coherent Quantum Phase Slip

In 2005, Mooij and Harmans [13] suggested a (relatively) simple circuit, a supercon-ducting loop interrupted by a nanowire, in which the effect of coherent quantumphase slips would change the number of quantized flux in a loop. Recently, Astafievet al. [14] not only fabricated this circuit, but showed that its operation is consistent

268 8 Coherent Quantum Phase Slips

Figure 8.5 Oscillations of the normalizedtunnel current jItun(W ) � I min

tun j/I mintun in the

external magnetic flux Φ/Φ0 (Φ0 D „/ein SI units) Al-AlOx -Al-AlOx -Al structureswith the same area of the aluminum loopS D 19.6 μm2. Experimental data are shownby circles (ı), calculations – by lines. (a)Large period (Φ/Φ0 D 8) and magnitude(Δ Itun/I min

tun � 0.8) oscillations in the struc-ture with loops formed by 110 nm � 75 nmwire, Vbias D 780 μV, Tbath D 65 ˙ 5 mK,Te D 70 mK, σ1/2

fit D 90.8 nm. Themonotonous increase of the base line is dueto the gap reduction by the magnetic fieldnoticeable at biases close to the gap edgeeVbias � 2(Δ1/Δ2). (b) Oscillations with thevariable period in the narrower (ion-milled)sample, Tbath D 52 ˙ 5 mK, Vbias D 608 μV,σ1/2

fit D 42 ˙ 30 nm. The solid line representscalculations at the intermediate limit withTe D 70 mK and σ1/2

fit D 12.49 nm, resultingin the ΔΦ/Φ0 D 3 period, dashed line – cal-culated ΔΦ/Φ0 D 1 oscillations in a slightly

narrower loop σ1/2fit D 12.37 nm. (c) The same

sample as in (b), but further gently ion-milledat Tbath D 54 ˙ 5 mK and Vbias D 666 μV.The solid line corresponds to calculationsin the QPS limit with σ1/2

fit D 12.15 nm andTe D 70 mK with the same parameters usedto fit R(T ) dependencies of Al nanowires.Left insets – flux dependencies of the char-acteristic energies: superconducting pairingpotential Δ (�), spectral gap G (˘), depairingenergy Γ (4), rate of quantum fluctuationsEQPS (�) and the corresponding thermal en-ergy kB T (�). Right insets – calculated fluxdependence of the persistent current Is(Φ )normalized by the critical current Ic(0). Notethe change of the shape, reduction of themagnitude and the period of both the tun-nel and the persistent current oscillations. Inthe classic (a) and intermediate (b) regimes,the periodicity of oscillations is defined byEquation 1 in [11], while in the essentially QPSregime (c) – by Equation 4 in [11]. For furtherdetails, see the text. Adapted from [11].

with the theoretical model. The circuit together with the theoretical dependence ofthe energy levels on the flux is shown in Figure 8.7.

8.4 Coherent Quantum Phase Slip 269

Figure 8.6 Tunnel current Itun oscillations inexternal magnetic flux Φ/Φ0 of Al-AlOx -Ti-AlOx -Al structure with the area of the titaniumloop S D 18.9 μm2. Top data – for non ion-milled sample σ1/2

fit D 66 ˙ 8 nm, bottom data

– for sputtered sample σ1/2fit D 62 ˙ 8 nm,

Tbath D 65 ˙ 5 mK at the same biasVbias D 105 μV. Solid lines represent QPSlimit calculations, assuming Te D 70 mK,σ1/2

fit D 63 and 56 nm, respectively, and the

same parameters used to fit R(T ) depen-dencies of Ti nanowires [6]. Note the dropof the oscillation amplitude by a factor of� 6 when the diameter σ1/2

fit is reduced byjust � 10%. The inset shows simulation: forthe same fitting parameters of the ion-milledsample, which would be the tunnel currentoscillations in the intermediate limit (similarto Figure 8.5b). For further details see the textand [11]. Adapted from [11].

The distance between the energy bands is designed to be in the GHz range; bytuning the external flux, it is possible to map the difference between the ground andthe first excited state. Notice that the minimum difference in energy is equal to theamplitude of the quantum phase-slip process EQPS; this provides an important con-sistency check with the value of the QPS provided by the theory. The Hamiltonianof the circuit is

H D � 12

EQPS (jN C 1ihN j C jNihN C 1j) C En jNihN j , (8.2)

where jNi is the flux state with Φ D N Φ0 in the loop, En D (Φext � N Φ0)2/2Lk,and Lk is the kinetic inductance of the loop. By exchanging EQPS with Ej (theJosephson energy), considering N the number of Cooper pairs on a superconduct-ing island En D (Vext � N Vc)/eCisland, this equation becomes the Hamiltonian ofa Cooper pair box which is the dual of this circuit. This naturally suggests that thesame technique used to measure a Cooper box can (and was) used to measure theQPS qubit. One important observation is that coupling of the jNi and jN C 1istates is only possible if EQPS ¤ 0; this is therefore a direct proof of the existenceand a direct measurement of the amplitude of the quantum phase-slips process.The main results of the paper are reported in Figure 8.8.

A small superconducting loop (of area A D 32 μm2), in which a nanowire ofdimension w D 40 nm, h D 35 nm and length L D 400 nm long is galvanicallyconnected to a transmission line with characteristic impedance Z1 D 1600 Ω. Thisis galvanically coupled to a transmission line of impedance Z0 D 50 Ω. Given

270 8 Coherent Quantum Phase Slips

Figure 8.7 The device. (a) Energies of theloop versus external flux, Φext . Blue and redlines: ground and excited energy levels, re-spectively. The degeneracy between stateswith N and N C 1 flux quanta (Φ0), seenat Φext D (N C 1/2)Φ0, is lifted by thephase-slip energy, ES. (b) InOx loop with anarrow wire segment on the right side is at-tached to the resonator (horizontal line) at

the bottom. (c) False-color scanning-electronmicrograph of the narrow InOx segment. (d)Step-impedance resonator comprising a 3-mm-wide InOx strip with wave impedanceZ1 � 1600 Ω galvanically coupled to a goldcoplanar line with impedance Z0 D 50 Ω. Theboundaries of the resonator are defined bythe strong impedance mismatch (Z1 Z0).From [14].

the large difference in impedance, a standing wave, with maximum current at theboundaries, is formed. As the loop is placed in the center of the step-impedanceresonator, only even modes of the standing wave with frequency f m D m�2.4 GHzwill couple with the loop. The mode with m D 4 and f4 D 9.08 GHz is used tomeasure the transmission coefficient (jtj) as a function of the applied magneticfield in Figure 8.8b; the dips in jtj correspond to absorption of the microwave bythe QPS qubit. The observed flux period is consistent, within the error, with thearea of the loop with a 40 nm nanowire. In fact, in each resonator, several loops,with varying loop areas and nanowire widths are embedded in the resonator, butno signal has been observed from a loop with a wider nanowire.

Using two-tone spectroscopy, with a tone at f D f4 to monitor the state of thestep-impedance resonator and a second tone to monitor the qubit state, Astafievet al. were able to map the energy spectrum as a function of the applied field inFigure 8.8c. Here, they also report the theoretical results assuming an energy split-

ting ΔE/„ D 4.9 GHz. The agreement between theory (δE Dq

(2IpδΦ )2 C E 2QPS

with Ip D Φ0/2Lk) and experiment is quite good. The single trace at Φ /Φ0 D 0.52is reported in Figure 8.8d, together with a Gaussian fit, suggesting low-frequencyGaussian noise as the mechanism of decoherence.

One aspect on which we have not focused our attention is the particular choiceof superconductor. While the experiment reported in [14] only discussed resultsobtained in devices fabricated in InOx (� D 20�30 nm), the NEC group has fab-ricated and measured other devices with similar qubit geometry and different ma-terials (O.V. Astafiev, personal communication). In particular, while devices fabri-cated in NbTi did manifest a qubit behavior, a device with titanium nanowire of size

8.4 Coherent Quantum Phase Slip 271

Figure 8.8 Experimental data. (a) Powertransmission through the resonator mea-sured within the bandwidth of our experi-mental setup. Peaks in transmission powercoefficient, jtj2, correspond to resonatormodes, with mode number m indicated foreach peak (a.u., namely, arbitrary units).(b) Transmission through the resonator asfunction a of external magnetic field Bextat m D 4 ( f4 D 9.08 GHz). The peri-odic structure in amplitude (jtj) and phase(arg(t)) corresponds to the points where thelowest-level energy gap ΔE/ h matches f4.The period ΔB D 0.061 mT (D Φ0/S)indicates that the response comes from theloop (shown in Figure 1b) with the effective

loop area S D 32 μm2. (c) The two-levelspectroscopy line obtained in two-tone mea-surements. The phase of transmission, arg(t),through the resonator at f4 is monitored,while another tone with frequency fprobe froman additional microwave generator, and Bext ,are independently swept. The plot is filteredto eliminate the contribution of other reso-nances (2 < m < 6), visible as horizontalred features. The dashed line is the fit to theenergy splitting, with ΔE/ h D 4.9 GHz,Ip D 24 nA. (d) The resonant dip is mea-sured at Φ/Φ0 D 0.52. The red curve is theGaussian fit. From [14]. For a color version ofthis figure, please see the color plates at thebeginning of the book.

t D 20�30 nm, w D 30�60 nm, l D 2�5 μm (thickness t, width w and length l)and TC varying between 0.4 and 0.7 K depending on the evaporation conditions,did not exhibit qubit behavior. A similar fate was suffered by devices fabricated inArutyunov’s group (coauthor of the study) in aluminum with the ion-beam millingtechnique and size t D 10 nm, w . 10�20 nm, l D 2�5 μm, although, in thiscase, wire reliability might have been an issue. The superconductor used in thestudy is a crucial issue as emphasized by Astafiev et al. [14] and Bezriadyn [15].InOx is a very disordered superconductor and is not properly described by the BCStheory. In this material, which is very close to the superconductor-insulator transi-tion, electrons form localized pairs even in the absence of superconductivity. Oncethe superconducting transition takes places, “lakes” of small condensate form, instark contrast to the BCS superconductor in which we have only one “lake” en-compassing the entire superconductor. This influences the QPS rate which for a

272 8 Coherent Quantum Phase Slips

moderately disordered (kF lmfp > 1) BCS conductor is ((3.37) and (3.46)) [16, 17]

γQPS � Δ„

RQ

R�

L�

exp

�αRQ

R�

�, (8.3)

with RQ D h/e2 and R� , the resistance of a wire with length equal to the coherencelength [18]. While moderate agreement can be reached between this expression andthe experiment, a better agreement is obtained using the expression for disorderedsuperconductors in the supplementary information of [14]. However, the exponen-tial factor in (8.3) possibly provides a clue as to why neither aluminum or titaniumin such a short nanowire manifested a qubit behavior. For these materials, this ra-tio is extremely large, thus suppressing the rate below what is measurable. In somesense, we are not properly designing the device and we are letting the material usedin the device dictate the energy scale for QPS.

8.5Conclusion

The final question that can be asked [15] is whether it is possible to create qubitsout of more common superconductors such as Al, Ti, Mo1�x Gex or Nb, insteadof a somewhat exotic superconductor such as InOx or Nbx Si1�x , both close to thesuperconductor–insulator transition. This would be important for practical applica-tions. While it can be argued that the presence of cQPS is necessary for the resultsof Arutyunov et al. in nanorings, the fact that QPS flux-qubits fabricated in Ti andAl have not, so far, shown a qubit behavior (O.V. Astafiev, personal communication)remains a mystery. As we have mentioned, the gap between the ground state andthe first excited state is a direct measurement of Es: however, if Es in Ti were toosmall, the nanoring experiment would not have succeeded.

We would like to argue that this is a definite possibility, or at least we hope so.The key, in our opinion, lies in a more profound exploitation of the duality betweenJosephson junction devices and QPS devices. Several researchers have publisheddiagrams of equivalent circuits between Josephson devices and QPS devices (seee.g., [4, 19]), however, in the works presented above, the scale of ES is determinedby the material properties. However, what we have presented here is just the be-ginning, akin to the initial development of superconducting qubits in early 2000when the initial proof of the concepts were demonstrated. An entire class of de-vices dual to the more common Josephson-based devices is waiting to be exploitedand the papers briefly presented in this chapter, together with the seminal work ofMooij and Nazarov, and Khlebnikov, will be regarded in a few years from now asthe foundation for a new type of electronics based on quantum phase-slip devices.

References 273

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