The ballistic acceleration of a supercurrent in a superconductor
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Published in Phys. Rev. Lett. , 077001 (2009).
The ballistic acceleration of a supercurrent in a superconductor
Gabriel F. Saracila and Milind N. Kunchur ∗ Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208 (Dated: Submitted on 26 October 2008; accepted on 21 January 2009)One of the most primitive but elusive current-voltage (I-V) responses of a superconductor is whenits supercurrent grows steadily after a voltage is first applied. The present work employed a mea-surement system that could simultaneously track and correlate I ( t ) and V ( t ) with sub-nanosecondtiming accuracy, resulting in the first clear time-domain measurement of this transient phase wherethe quantum system displays a Newtonian like response. The technique opens doors for the con-trolled investigation of other time dependent transport phenomena in condensed-matter systems. PACS numbers: 74.25.Op,74.25.Qt,74.25.Fy,72Keywords: ballistic,acceleration,kinetic,inductance,supercurrent,superfluid,superconductor,superconductivity
A particle under the action of a single applied forceaccelerates ballistically in accordance with Newton’s sec-ond law. In the presence of a frictional force, an appliedforce will ultimately maintain a constant velocity ratherthan produce acceleration. Analogously, an externallyapplied voltage V maintains a constant current I in thecase of a resistive conductor, whereas it can ballisticallyaccelerate the superfluid in a superconductor, leading toa supercurrent that grows with time. This accelerationphase of the supercurrent lasts for a very brief period—until flux motion, phase slip centers, or other dissipativephenomena set in—making it extremely difficult to ob-serve in the time domain in a correlated current-voltagemeasurement. In the present work, an electrical measure-ment system was developed that could resolve and corre-late the time evolutions of I ( t ) and V ( t ) on a subnanosec-ond time scale. For the sample pattern, extremely longmeander geometries (with length-to-width aspect ratiosin the thousands) were employed to prolong the acceler-ation time while maintaining V at a manageable level.The combination of these two measures facilitated thesuccessful observation of the acceleration phase.The acceleration of the supercurrent density j s is givenby (from the London equations [1, 2]) djdt ≃ dj s dt = Ee n s m ∗ = Eµ λ L , (1)where E is the local internal electric field, e is the elec-tronic charge, m ∗ is the effective mass and n s is the super-fluid density (related to the number of electrons per vol-ume participating in the condensate); the far right handside of the equation relates n s to the London magnetic-field penetration depth λ L ; we can take j = j s + j n ≃ j s because the normal current density j n is a negligible com-ponent of the total current density j . This supercurrentacceleration phase lasts for the duration ∆ t ≈ j c µ λ L /E ,where j c is the critical current density that marks theonset of resistance. The inductance-like proportionalitybetween dj/dt and E in Eq. 1, arising from the inertia ofthe superfluid, is referred to as the kinetic inductance L k .In terms of the geometrical length l and cross sectional area A , it is given by L k = µ λ lA , (2)where λ is a more general penetration depth, which in-cludes effects such as impurity scattering ( λ ≥ λ L ).Kinetic inductive effects are small except close to thetransition temperature T c , where their signatures havebeen seen in the high-frequency ac response or as non-equilibrium inductive voltage spikes during abrupt cur-rent steps [3–11]. In the present work, timescales werechosen to be short enough to have a sufficient magni-tude of V while long enough (compared to characteristictimescales such the gap-relaxation and electron-phononscattering times) to avoid non-equilibrium effects. Vari-ations in fields occured at length scales that were longcompared with both λ and the coherence length ξ , soas to avoid non-local effects. Thus the conditions wereoptimum for observing the simplest limiting behavior ofan accelerating condensate as predicted by the Londonequations, i.e., Eq. 1.The samples used in this work were niobium films de-posited on silicon substrates with DC magnetron sputter-ing. The films were patterned into long narrow meandersby electron-beam lithography using the lift-off technique.Sample A had a thickness of t = 70 ± w = 12 . ± . µ m, and a length (between voltage probes)of l = 4 . ± .
01 cm. Sample B had the dimensions t = 85 ± w = 8 . ± . µ m, and l = 4 . ± . T c =6.74 K and T c =7.23 K.The measurements were carried out in a pulsed-tubeclosed-cycle refrigerator in zero applied magnetic field.The electrical measurements were conducted with apulsed signal source and detection electronics, in com-bination with a digital storage oscilloscope. Parts ofthe signal-source and preamplifier circuitry in this setupwere developed and built in-house. An active (buffered)ground arrangement was developed for improving theshielding between the fast changing high-voltage signalin the current leads from the low-voltage sample-signalsensing leads. The entire signal chain up until the digi-tal oscilloscope is analog. Using pulsed signals instead ofcontinuous ac or dc excitations permits a wider range ofcurrents without Joule heating of the sample and a flex-ible control over the waveform shape. The system per-forms simultaneous independent differential four-probemeasurements of I ( t ) and V ( t ) with a relative timing ac-curacy of ∼
100 ps. The stray mutual inductive couplingbetween current and voltage leads has a (temperatureindependent) value of . ± µ V. Thetime interval between digitized samples is 10 − second(the single-shot digitizing sampling rate is 10 GS/s). Thespeed and accuracy with which both I ( t ) and V ( t ) weretracked and correlated in a superconductor in the presentexperiment are, to our knowledge, unprecedented.Some tests and verifications of the measurement sys-tem are shown in Fig. 1. Panel (a) shows the voltage andcurrent (scaled by a constant) for a purely resistive testsample and panel (b) shows the current derivative andvoltage (scaled by a constant) for a purely inductive testsample. The time scales used in the actual experimentwere longer than these test conditions of Fig. 1 so thatthe temporal tracking between the current and voltagesensing circuits was essentially perfect. Some additionalinformation on the apparatus can be found in our previ-ous review articles [12, 13]. (a) V ( m V ) t (ns) V I x R (b) d I/ d t ( M A / s ) t (ns) dI/dt V/L FIG. 1:
Measurement-apparatus temporal accuracy checks.(a) V ( t ) and I ( t ) (multiplied by a constant R =62 Ω ) for atest resistor in place of a superconducting sample. (b) dI ( t ) /dt and V ( t ) (divided by a constant L =15 nH) for a test induc-tor in place of a superconducting sample. The voltages acrossthe purely resistive and inductive loads are seen to track theirrespective current and current-derivative functions with sub-nanosecond accuracy. Figs. 2(a) and (b) show V ( t ) (solid lines) across twoniobium-meander samples in the superconducting stateat one temperature. Panels (c) and (d) show the cor-responding I ( t ) functions, which are seen to acceleratesteadily during the plateaus in V ( t ). The dashed linesin panels (a) and (b) show dI/dt scaled by a constant( L =16.7 and 16.9 nH for samples A and B respectively)and are seen to track V ( t ) in instantaneous detail. Thusthe response is purely inductive, with an inductance thatis independent of I and dI/dt . (a) Sample A
Voltage dI/dt x L V ( m V ) (c) Sample A
Current I ( m A ) Sample B
Voltage dI/dt x L (b) V ( m V ) Sample B
Current t (ns) (d) I ( m A ) t (ns) FIG. 2:
Time dependencies of voltage and current in niobiummeanders at T=3.84 K. Panels (a) and (b) represent V ( t ) and dI/dt (scaled by a constant L ) for samples A and B re-spectively. Panels (c) and (d) show the corresponding I ( t ) functions, which rise steadily during the voltage plateaus ofthe panels above; the time axes for panels (a) and (c) and forpanels (b) and (d) are the same. The ratio between the dI ( t ) /dt and V ( t ) curves in thetop panels of Fig. 2 gives the time and current depen-dent inductance: L ( t ) = V ( t ) / [ dI ( t ) /dt ]. Figs. 3(a) and(b) plot this L ( t ) versus time for various values of T , foreach of the samples. The plateau value of L is seen to in-crease steadily with temperature as is expected becauseof the declining superfluid density and consequent rising λ L . Another interesting trend is that the curves at high-est temperatures show L ( t ) functions that rise with time(i.e., current). This happens because the current sup-presses the superfluid density through its pair-breakingaction, a regime not seen before in any other kind ofmeasurement. Note that continuous-ac probes of n s can-not endure high enough excitation levels to explore thisregime because of Joule heating; and tunneling measure-ments reveal the spectral gap Ω g rather than n s . A sys-tematic study of the suppression of n s by j will be thesubject of a future investigation, since the optimum sam-ple geometry for studying this effect is almost opposite tothe sample geometry required for the present experiment.Figs. 3(c) and (d) plot L (as measured above) versus
11 12 13 14 15 (c) L ( n H ) T (K) (a) L ( n H ) t (ns) (d) T (K) (b) t (ns)
FIG. 3: (a) Measured total inductance, L ( t ) = V ( t ) .dt/dI ( t ) ,versus time for sample A. The curves correspond to the tem-peratures (from top to bottom): T = 6.10, 5.91, 5.54, 5.17,4.73, 4.10, and 3.78 K. (Each plot symbol on these curves cor-responds to separately measured digital voltage sample. Theperiod between samples is 100 ps.) (b) A similar L ( t ) plotfor sample B. For this panel, the temperatures (from top tobottom) are: T = 6.51, 6.4, 6.22, 6.02, 5.84, 5.24, 4.64, and3.84 K. Panels (c) and (d) show the temperature dependen-cies of the above measured L values (taken on the plateausaround t ∼ ns) for samples A and B respectively. Thesymbols show the experimental data and the solid line rep-resents the least-squares fit to the two-parameter function L ( T ) = L g + L k (0) / [1 − ( T /T c ) ] as discussed in the text. T for each sample. This total inductance L = L k + L g has components corresponding to the kinetic inductance L k as well as a geometrical inductance L g (from mag-netic flux change). The temperature dependence ariseschiefly from L k ; the changes in L g with temperature—arising from changes in the current-density profile acrossthe cross section—are relatively small ( ∼ L k ( T ) ≈ L k (0)[1 − ( T /T c ) ] , (3)where L k (0) = µ λ (0) l/A . The solid line curves inFig. 3(c) and (d) correspond to a least-squares fit to thefunction L ( T ) = L g + L k (0) / [1 − ( T /T c ) ]. The valuesof L k (0) and L g obtained from this fitting ( T c is not afitting parameter) are listed in columns 2 and 4 of Ta-ble I. The coefficients of determination of the fits are R =0.9989 and R =0.9994 for samples A and B respec-tively. The standard errors of the fit combined with theerror in the inductance measurement gives the error barsfor L k (0) that are indicated in the table.The third and fifth columns of Table I show, for com-parison, theoretical estimates of L k (0) and L g . For find-ing L k (0) we note that the effective penetration depth ( λ ) L k (0) in nH L g in nHSample Expt. Theor. Expt. Theor.A 2.8 ± ± ± ±
3B 2.8 ± ± . ± ± Experimentally observed values and theoreticallyestimated values of the kinetic and geometrical inductances. becomes lengthened with respect to its clean-limit value( λ L ) in the presence of scattering by static disorder. Thiseffect of impurity scattering can be incorporated throughthe residual resistivity ρ o and expressed in terms of theorder parameter ∆ as [14] λ (0) = s ~ ρ o πµ o ∆(0) . (4)Taking the measured values of ρ o =0.347 µ Ω.m and 0.369 µ Ω.m, and obtaining ∆(0) ≈ k B T c from our measuredvalues of T c , we get λ =223 nm and 222 nm, and L k =3.5nH and 3.7 nH for samples A and B respectively. Thereis an uncertainity in the values ρ o because of the un-certainity in sample dimensions and an uncertainity in∆(0) because of an uncertainity in the absolute value of T c (this is roughly estimated to be around 200 mK). Thisgives rise to the error bars in the theoretical L k valuesthat are stated in the table. The calculated values of L k are somewhat larger than the measured ones of but ofcomparable magnitude.The theoretical geometrical inductances for the mean-ders, tabulated in the last column of Table I, were com-puted numerically by integrating the magnetic flux thatlinks to the path between the voltage probes. The errorbars in the theoretical L g arise from the uncertainities inthe sample dimensions and the approximations inherentin the calculation. It can be seen that the theoreticallyestimated L g values are also in agreement with their mea-sured counterparts.In summary, this work has explored the initial accel-eration phase during which a supercurrent builds up inresponse to an applied voltage. The voltage and currentcurves of Fig. 2 represent the first clear and direct time-domain demonstration of this primitive regime, where thequantum system shows a Newtonian like response. It isalso the first time to observe the non-linear regime wherethe current suppresses the superfluid density, thereby in-creasing the kinetic inductance. The instrumentationdeveloped for this experiment is unique and representsthe first measurement of its kind where both V ( t ) and I ( t ) are tracked in a superconductor with sub-nanosecondtiming accuracy. This technique can reveal more detailedinformation than just an impulse-response measurement,and it can be used to explore time-dependent and non-equilibrium phenomena in condensed-matter systems ina controlled way (some examples of such regimes in su-perconductors would be those related to phase slippage,glassy dynamics, and the nascent stage of a vortex rightafter its nucleation). The present work and its methodshould be distinguished from past experiments in whichan abrupt supercritical current pulse was applied [5, 11]and only the subsequent V ( t ) response was measuredwithout monitoring I ( t ). In those experiments the su-perconductor recoils in a highly non-equilibrium man-ner to the supercritical stimulus. In the present study,the superconducting system is always maintained close toequilibrium by keeping the experimental timescales wellin excess of the gap-relaxation and electron-phonon scat-tering times, while keeping the timescales short enoughto observe the inertia of the superfluid.The authors acknowledge useful discussions with J. M.Knight, B. I. Ivlev, R. A. Webb, R. J. Creswick, T. R.Lemberger, G. Simin, T. M. Crawford, and F. T. Avi-gnone III. This research was supported by the U. S.Department of Energy through grant number DE-FG02-99ER45763. ∗ Corresponding author email: [email protected];URL: [1] F. London and H. London, Proc. Roy. Soc. (London)
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