Quantum phase slip interference device based on superconducting nanowire
aa r X i v : . [ c ond - m a t . s up r- c on ] O c t Quantum phase slip interference device based on superconducting nanowire
T. T. Hongisto and A. B. Zorin
Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany (Dated: October 31, 2011)We propose a transistor-like circuit including two serially connected segments of a narrow super-conducting nanowire joint by a wider segment with a capacitively coupled gate in between. Thiscircuit is made of amorphous NbSi film and embedded in a network of on-chip Cr microresistors en-suring a sufficiently high external electromagnetic impedance. Assuming a virtual regime of quantumphase slips (QPS) in two narrow segments of the wire, leading to quantum interference of voltageson these segments, this circuit is dual to the dc SQUID. Our samples demonstrated appreciableCoulomb blockade voltage (analog of critical current of the SQUIDs) and periodic modulation ofthis blockade by an electrostatic gate (analog of flux modulation in the SQUIDs). The model of thisQPS transistor is discussed.PACS numbers: 73.63.Nm, 73.23.Hk, 74.78.Na, 85.25.Am, 74.81.Fa
The discovery of the Josephson effect [1] triggered thedevelopment of superconductive electronic devices oper-ating on the principle of the classical behavior of thecollective quantum variable, i.e. the superconductingphase difference. These include rf and dc SQUIDs [2],single-flux-quantum logic circuits [3], Josephson voltagestandard arrays [4], microwave receivers [5], etc. The op-eration of these devices is based on the nonlinear (2 π periodic) dependence of the supercurrent I = I c sin ϕ on the phase difference ϕ on the Josephson junction(JJ), where I c is the critical current. The electromag-netic impedance of the JJ for a small harmonic sig-nal is therefore expressed in terms of Josephson induc-tance L J which varies with ϕ also in a periodic fashion, L − J ( ϕ ) = (2 πI c / Φ ) cos ϕ , where Φ = h/ e is the fluxquantum.Recently, a new type of nonlinear superconducting ele-ment, dual to the JJ, - the quantum phase slip (QPS)nanowire (see, for example, the review on the phys-ical properties of superconducting nanowires Ref. [6],and references therein) - was proposed in Refs. [7, 8].Being included in a superconducting ring with flux-bias near degeneracy points, Φ b = ( m + 0 . , m is an integer number, the QPS element enables cou-pling of quantum states | m i and | m + 1 i . This leadsto anticrossing of corresponding levels with the gap∆ E = E QPS . For a uniform nanowire of length l ,the characteristic coupling energy E QPS = lǫ S with ǫ S ≈ ( k B T c /ξ )( R Q /R ′ ξ ) / / exp(0 . αR Q /R ′ ξ ), where T c is the critical temperature, ξ the coherence length, R ′ thenormal-state resistance per unit length, R Q = h/ e ≈ . α ∼ E J = (Φ / π ) I c [9]).Coupling of the nanowire to a high-ohmic environ-ment through the appropriate boundary conditions im-poses dramatic changes in its dynamics [10]. Basedon the quantum duality of the Josephson and QPS ef- fects stemming from conjugation of the phase and chargevariables, Mooij and Nazarov had further predicted [8]that in the charge-bias regime, the QPS element shoulddemonstrate the Coulomb blockade behavior with pe-riodic dependence of voltage on the injected charge q ,viz. V QPS ( q ) = V c sin( πq/e ) with V c = 2 πE QPS / e .Thus, this element should behave as a nonlinear capac-itor, C − ( q ) = dV QPS /dq = ( πV c /e ) cos( πq/e ) with a2 e -periodic dependence on q .It is clear that such a QPS element could be exploitedin the engineering of electronic devices controlled by thecharge in the classical manner. These devices are dualto the Josephson ones controlled by the classically be-haved phase. For example, applying microwave irradia-tion of frequency f should lead to the formation of thecurrent steps in the I - V curve at I = 2 nef ( n is in-teger), enabling a fundamental standard of current [8].These steps are dual to Shapiro voltage steps in a JJ I - V (b)(a) NbSi nanowireCr resistors Cr resistorsNbSi gate w=18nm= 6 m l m w=18nm= 6 m l m w=30nm= 12 m l m w=110nm= 100 m l m w=110nm= 100 m l m w=60nm QPSQPS L k2 L k1 R R V /2 b V g C g C -V /2 b ++ + AuPd pad AuPd pad
FIG. 1: (Color online) (a) The layout and (b) the simpli-fied electric circuit diagram of the QPS transistor embeddedin the 4-terminal network of on-chip resistors. The deviceincludes two QPS elements denoted by diamond symbols, ki-netic inductances of the nanowire segments and a capacitivegate. Thicknesses of NbSi and Cr films are 10 nm and 30 nm,respectively. curve at V = n Φ f [11]. Developing such an analogy, inthis paper we proposed and realized a QPS-based singlecharge transistor which is dual to dc SQUID and can beoperated as an electrometer.Our device (see schematic diagram in Fig. 1a) has asymmetric in-line configuration and includes two nar-row pieces of superconducting nanowire joined by an is-land (a wider middle part of the same nanowire) witha capacitively coupled gate. This transistor is embed-ded in the network of compact high-ohmic resistors, R ∼ . ≫ R Q , which ensure a sufficiently high elec-tromagnetic impedance seen by the transistor. Due tothis improvement, a classical charge regime of operationwith significant damping of dynamics is realized. (Notethat the recent proposal by Hriscu and Nazarov [12] dealswith a complementary QPS transistor, without resistors,operating in a quantum regime, which, in contrast to ourcircuit is phase biased and, therefore, not strictly dual tothat of the dc SQUID.)The Kirchhoff equation for the equivalent electric cir-cuit presented in Fig. 1b has the form V + V = V b , V i = L ki ¨ q i + R i ˙ q i + V QPS i ( q i ) , i = 1 , . (1)Here, V QPS i ( q i ) = V ci sin( πq i /e ) are the 2 e -periodic volt-ages on the QPS elements on either side of the island.The charge conservation relation takes the form q − q = − C g V g − ( C g + C )( V − V ) . (2)This equation reflects the balance of the charges injectedin the island and the polarization charges induced by thegate and floating potential of the island having a self-capacitance C with respect to ground. In the case ofsufficiently small gate- and self-capacitances, C g , C ≪ e/V c ,c , (3)and small injected currents (viz. ± ˙ q , ), the differencecharge q − q = − C g V g ≡ Q g is totally controlled by thegate voltage V g .One can easily find that equations Eqs. (1-3) are dualto those of the dc SQUID, see, for example, Ref. [13]. Thecharges q and q are dual to the phases across individ-ual JJs, whereas amplitudes V c ,c play the roles of thecritical currents. To continue this analogy we can put incorrespondence the pairs of parameters L k ,k and JJ ca-pacitances, R , and shunting resistors, etc., for the QPStransistor and dc SQUID, respectively. The condition ofsmall gate- and self-capacitances Eq. (3) is dual to therequirement of a small inductance of the SQUID loop( L ≪ L J ). Similar to external flux control of quantuminterference in the SQUID [13], the voltage on the QPStransistor is resulted from quantum interference of volt-ages on individual QPS elements and controlled by thegate charge Q g , V = V QPS1 + V QPS2 = V m ( Q g ) sin( πQ/e ) , (4) where Q = q + eη/π = q + eη/π − Q g is the ”average”charge, tan η = 2 V c tan( πQ g / e ) V c + + V c − tan ( πQ g / e ) , (5) V c ± = V c ± V c , and the Coulomb blockade voltage V m = V c + V c + 2 V c V c cos( πQ g /e ) . (6)In the case of a symmetrical circuit, V c = V c = V c , Q = q , ± Q g /
2, Eq. (6) yields the maximum modulation ofthe blockade voltage, V m = 2 V c | cos( πQ g / e ) | . In the caseof a highly asymmetrical circuit, say V c /V c = a ≪ Q ≈ q and the blockade voltage is only slightlymodulated by the gate charge Q g , V m = V c [1 + a cos( πQ g /e )] . (7)Thus the equation of motion Eq. (1) takes a form of theresistively shunted junction (RSJ) model for dc SQUID, L k ¨ Q + R ˙ Q + V m ( Q g ) sin( πQ/e ) = V b (8)with L k = L k + L k and R = R + R .Equation (8) describes the dynamics of a nonlinearoscillator with finite damping. It yields a dc IV -curvewith both the static- and running-charge (the oscillatingregime, ω = π h ˙ Q i /e = π h I i /e ) branches, dependent on Q g in a periodic fashion. The dimensionless parameter,crucial for the dynamics, introduced earlier in Ref. [8], β QPS = ω c L k /R, (9)where ω c = πV m ( Q g ) /eR is a characteristic circular fre-quency. Eventually, β QPS is the analog of the Stewart-McCumber parameter in the Josephson dynamics. Inthe most realistic case of large damping, β QPS <
1, the IV -curve has a shape with characteristic back bending, h V i = [ V m ( Q g ) + R h I i ] / − R h I i , (10)where h V i is the average voltage on the nanowire.The fabrication method (some modification of themethod developed in Ref. [14]) included the followingsteps. The samples were fabricated on a silicon substrate,having 300 nm thick thermal oxide layer, combining twoprocesses: shadow evaporation and the sputter deposi-tion/etching step. The Cr resistors were fabricated in asingle vacuum cycle with the AuPd contact wires andmicropads utilizing the shadow evaporation techniquethrough a bi-layer PMMA/Copolymer stencil mask. The30 nm thick Cr resistors were evaporated first at a lowresidual pressure of oxygen ( ∼ − mbar) followed by a50 nm thick layer of AuPd from an angle for which thenarrow stencil openings for Cr resistors were overshad-owed by the mask. Using this trick, the formation ofAuPd shadows parallel to the Cr resistors was avoided. -0.10-0.050.000.050.10 -0.1 0.0 0.1 -2 -1 0 1 2 3 4-4-20246 Theory1 mVV g = 0 T = 15 mK C u rr en t I ( n A ) Voltage V (mV)Sample A I off FIG. 2: (Color online) The charge-modulated IV -curves ofSample A recorded in a current bias regime for two gate volt-ages shifted by a half-period. In the region of small currents(blown up in inset) one can see the modulation with period∆ I = 13 . δV = ( R bias1 − R bias2 ) I and,therefore, of the effective gate charge, δQ g = ( C g + C ) δV (cfEq. (2)). The green dashed line shows the shape of the bare IV -curve given by the RSJ model Eq. (10). The Cr resistors and other parts of the circuit were pro-tected by a PMMA mask while 10 nm thick amorphousNb x Si − x film ( x = 0 . , defined by calibrating the sput-ter rate for each element at a given sputter power andperiodically confirmed by EDX measurement using sepa-rately deposited films on Ge substrates) was co-sputteredon the substrate making contact only with the AuPd mi-cropads. These micropads allowed making an electricalcontact of the NbSi film with the Cr resistor circuitry.To remove organic residue and water for the purpose offorming reliable contacts, we cleaned the micropads inRIE with oxygen plasma and baked the sample overnightin N atmosphere at a temperature of 120 C ◦ before NbSideposition. The subtrate was rotated 20 times during aone-minute-long deposition. After the lift-off, the waferwas coated with inorganic negative tone HSQ resist (XR-1541, Dow Corning) patterned with e-beam. After expo-sure, the HSQ resist had characteristics of thermal siliconoxide and could be used as an etch mask to define theNbSi nanowires along with the island and the gate struc-ture made out of the same NbSi film. In a final step, theICP etch process with SF gas was used for etching.The samples were measured in a dilution refrigera-tor with electrical lines equipped with microwave fre-quency filters made of pieces of Thermocoax TM cable.We used battery power sources and home-made electron-ics for either a voltage or a current bias of our circuits.Most of the measurements were performed at the low- -3 -2 -1 0 1 2 3-10-50510 -200 -100 0 100 200 T = 16 mK V g =1.95 3.15mV C u rr en t I ( n A ) Voltage V (mV)
Sample B
Gate voltage V g (mV) V = 0.321 0.481 mV
FIG. 3: (Color online) The IV -curves of Sample B measuredin the voltage bias regime at different values of gate voltage V g . The bottom right inset shows details of the Coulombblockade corner. Upper left inset: the gate voltage depen-dence of the transistor current measured at different bias volt-ages V b providing a steady increase of h V i from 0.321 mV upto 0.481 mV in 4 mV steps (from bottom to top). est temperature of the fridge, T = 15 mK. The layoutallowed independent characterization of both pairs ofon-chip resistors. Their IV -curves at millikelvin tem-peratures were practically linear with resistance about20% above the values measured at room temperature.The NbSi films had the superconducting transition tem-perature T c ≈ x Si − x with sto-ichiometry x = 0 .
15 [15]), yielding for the narrow seg-ments the specific resistance R ′ ≈
31 kΩ /µ m. The NbSifilm parameters were measured at T = 3 . µ m) and some-what shorter QPS sections (4.5 µ m each). The layout ofthe gate was identical in both samples. Both samplesshowed an appreciable Coulomb blockade, h I i = 0, anda gate effect at small bias. An increase in current h I i & h V i indi-cating a gradual turning of the wire into the normal state.On a large scale, the IV -curves of both samples exhibiteda positive excess current, i.e. had a shifted linear asymp-tote, I off = h I i − h V i ( d h I i /d h V i ) h V i→∞ ≈ . − . IV -curve argues that the observed gate ef-fect has a superconductive origin and allows us to ruleout the strikingly different effect of the Coulomb block-ade in a normal single electron transistor, characterizedby a pronounced voltage offset [17] or, equivalently, anegative ”excess” current.On a small scale, one can see the clear gate mod-ulation of IV -curves, although with very different pe-riods, ∆ V g ≈ e -periodic charge de-pendence in Sample A (see inset to Fig. 2) capacitance C g = 2 e/ ∆ V g takes the value of about 160 aF. Thisvalue is not far from the modelled value of 200 aF. Thedepth of the modulation of about 40% suggests - accord-ing to our model Eq. (7) - the ratio of QPS energies, a = V c /V c = E QPS2 /E QPS1 ≈ . µ eV. Inserting this value in the formulafor the QPS energy and assuming the length of the QPSjunction of the order of the coherence length ξ , we obtainthe ratio R Q /R ′ ξ ≈ ξ ≈
20 nm. This value is close to the values ξ = 10 −
15 nmfound in Ref. [14].The shape of a bare (not modulated) IV -curve is onlyqualitatively similar to that given by Eq. (10) and shownby the dashed line. A back bending weaker than in theorycan be attributed to the effect of stray capacitance of theresistors, resulting in a roll-off of effective impedance R in Eq. (10) with the rise of current ( ∝ ω ). Rounding ofthe Coulomb blockade corner is attributed to the effectof noise, omitted in our model.Sample B demonstrated an even stronger gate effect(see upper left inset of Fig. 3), but also the peculiarproperties. A huge period ∆ V g ≈
150 mV of the gatevoltage dependence corresponds to coupling capacitance C g ≈ E QPS2 /E QPS1 ≈ . E QPS1 ≈ µ eV. A smallhysteresis and almost vanishing back bent part in the IV -curve, seen in the lower bottom inset of Fig. 3, mayindicate that the value of the damping parameter β QPS
Eq. (9) is between 1 and 2 [13], whereas our estimationyields a value of β QPS smaller by an order of magnitude,ensuring a heavily overdamped regime.Interestingly, the observed periodic pattern in Sam-ple B was superimposed on another one having a sub-stantially larger period and a much weaker modulation,discernible in the traces at smaller bias in the upperleft inset of Fig. 3. This behavior can be due to theemergence of an additional (very small) island neighbor-ing in line with the actual island. Thus, in spite of a rather high homogeneity in thickness and width, the en-tire nanowire can be considered as a circuit including sev-eral weaker sections with a local increase in QPS energiesper unit length, ǫ S ( x ). In the spirit of the single chargeinterferometer operation, Eqs. (4-6), the contributions ofthese weak sections to the total QPS energy should besummed up, taking into account phases proportional tothe charges induced by the gate on all intermediate is-lands. The resulting quasiperiodic dependencies can bealso interpreted in terms of the Aharonov-Casher effectdemonstrated in the experiments with phase-biased ar-rays of small JJs (i.e. in a lumped-element analog ofsuperconducting nanowire) [18, 19]. Recent calculationsby Vanevi´c and Nazarov [20] support the hypothesis thatapparently homogeneous nanowires may naturally have astrongly inhomogeneous distribution of specific QPS en-ergy ǫ S ( x ) because of the exponential dependence on thelocal parameters. Large spatial fluctuations of the localenergy gap detected by scanning tunneling methods werealso reported for thin disordered films of TiN [21], NbN[22] and InO [23].In conclusion, we have demonstrated the single-chargeeffect in superconducting nanowires having a transistorconfiguration with a capacitively coupled gate, embeddedin a high-impedance environment. A deeper understand-ing and better control of the nanowire parameters deter-mining the characteristics of these transistors and otherpossible circuits is urgently needed and motivates us toconduct further research. Generally, the demonstratedduality of the QPS transistor and the dc SQUID mayopen the way towards interesting applications of QPSnanocircuits in electronics and metrology.The authors acknowledge assistance from ThomasWeimann and Peter Hinze with the fabrication of thesamples, Thomas Scheller for performing NbSi deposi-tions and EDX measurements, and Thorsten Dziombafor AFM measurements. This work was partially sup-ported by the EU through the REUNIAM and SCOPEprojects. [1] B. D. Josephson, Phys. Lett. , 251 (1962).[2] J. Clarke and A. I. Braginski, The SQUID Handbook.Volume I: Fundamentals and Technology of SQUIDs andSQUID Systems (Wiley-VCH, Berlin, 2004).[3] K. K. Likharev and V. K. Semenov, IEEE Trans. Appl.Supercond. , 3 (1991).[4] C. A. Hamilton, Rev. Sci. Instrum. , 3611 (2000).[5] K. K. Likharev and V. V. Migulin, Radio Eng. ElectronPhys. , 1 (1980).[6] K. Yu. Arutyunov, D. S. Golubev, and A. D. Zaikin,Phys. Rep. , 1 (2008).[7] J. E. Mooij and C. J. P. M. Harmans, New J. Phys. ,219 (2005).[8] J. E. Mooij and Yu. V. Nazarov, Nat. Phys. , 169 (2006).[9] Y. Nakamura, Yu. A. Pashkin and J. S. Tsai, Nature , , 067007 (2004).[11] S. Shapiro, Phys. Rev. Lett. , 80 (1963).[12] A. M. Hriscu and Yu. V. Nazarov, Phys. Rev. B ,174511 (2011).[13] K. K. Likharev, Dynamics of Josephson Junctions andCircuits (Gordon and Breach, New York, 1986).[14] T. van der Sar, Quantum Phase Slip. Delft University ofTechnology, Master thesis (2007).[15] C. A. Marrache-Kikuchi, H. Aubin, A. Pourret, K.Behnia, J. Lesueur, L. Berg´e and L. Dumoulin, Phys.Rev. B , 144520 (2008).[16] Compare with behavior of some Josephson weak-linksreviewed in K. K. Likharev, Rev. Mod. Phys. , 101(1979).[17] T. A. Fulton and G. J. Dolan, Phys. Rev. Lett. , 109(1987). [18] V. E. Manucharyan, N. A. Masluk, A. Kamal, J. Koch,L. I. Glazman, M. H. Devoret, arXiv:1012.1928v1 [cond-mat.supr-con].[19] I. M. Pop, B. Dou¸cot, L. Ioffe, I. Protopopov, F.Lecocq, I. Matei, O. Buisson, and W. Guichard,arXiv:1104.3999v1 [cond-mat.mes-hall].[20] M. Vanevi´c and Yu. V. Nazarov, arXiv:1108.3553 [cond-mat.supr-con].[21] B. Sac´ep´e, C. Chapelier, T. I. Baturina, V. M. Vinokur,M. R. Baklanov, and M. Sanquer, Phys. Rev. Lett. ,157006 (2008).[22] M. Mondal, A. Kamlapure1, M. Chand, G. Saraswat, S.Kumar, J. Jesudasan, L. Benfatto, V. Tripathi1, and P.Raychaudhuri, Phys. Rev. Lett. , 047001 (2011).[23] B. Sac´ep´e, T. Dubouchet, C. Chapelier, M. Sanquer, M.Ovadia, D. Shahar, M. Feigel’man, L. Ioffe, Nat. Phys.7