Observation of negative absolute resistance in a Josephson junction

0.44 MB·English

by  J. Nagel

Observation of negative absolute resistance in a Josephson junction
J. Nagel,1 D. Speer,2 T. Gaber,1 A. Sterck,1 R. Eichhorn,2
P. Reimann,2 K. Ilin,3 M. Siegel,3 D. Koelle,1 and R. Kleiner1,∗
1Physikalisches Institut – Experimentalphysik II and Center for Collective Quantum Phenomena and their Applications,
Universit¨at Tu¨bingen, Auf der Morgenstelle 14, D-72076 Tu¨bingen, Germany
2Fakult¨at fu¨r Physik, Universit¨at Bielefeld, 33615 Bielefeld, Germany
3Institut fu¨r Mikro- und Nanoelektronische Systeme, Universit¨at Karlsruhe (TH),
Hertzstraße 16, D-76187 Karlsruhe, Germany
(Dated: February 4, 2008)
Weexperimentallydemonstratetheoccurrenceofnegativeabsoluteresistance(NAR)uptoabout
−1Ω in response to an externally applied dc current for a shunted Nb-Al/AlOx-Nb Josephson
8 junction,exposed toamicrowave currentat frequenciesin theGHzrange. Therealization (ornot)
0 of NAR depends crucially on the amplitude of the applied microwave current. Theoretically, the
0 system is described by means of the resistively and capacitively shunted junction model in terms
2 of a moderately damped, classical Brownian particle dynamics in a one-dimensional potential. We
n findexcellent agreement of theexperimental results with numerical simulations of themodel.
a
J PACSnumbers: 05.45.-a,05.40.-a,05.60.Cd,74.50.+r

8
2
When a static force is applied to a system consisting jected to multiplicative white noise [12]. While each of

] of mobile particles, these particles usually move in the the different ingredients of the model is quite realistic
D direction of the force, i. e., they show positive mobility, in itself, their combined realization in an experimental
C whichleadsto,e.g.,apositiveconductanceorresistance system seems difficult. In particular, the main physical
. in electrical systems. Also well known is the fact that mechanism is based on collective effects of at least three
n
such a system can exhibit regions of negative differential interacting particles [13]. An entirely different mecha-
i
l mobility/resistance [1, 2, 3, 4, 5, 6]. However, the ab- nism was later on suggested theoretically for a realistic,
n
solute mobility/resistance usually remains positive. The classicalmodeldynamicsofasingleBrownianparticlein
[
opposite response, i. e., a motion againstthe static force a suitably tailored, two-dimensional potential landscape

1 istermednegativeabsolutemobilityornegativeabsolute in Ref. [14] and subsequently realized experimentally in
v
resistance(NAR).Thisisclearlyaquitecounterintuitive Refs. [11, 15]. As a firstapplicationofNAR, the separa-
0
effectwhich,atfirstglance,mightseemeventobeincon- tionof different particle species has been realizedin Ref.
7
flict with Newton’s laws and thermodynamic principles [16]. Whiletheunderlyingbasicphysicalmechanismstill
3
4 [7]. Yet, nonlinear systems being driven far from equi- requires at least two spatial dimensions, very recently,
. librium can indeed exhibit not only a negative differen- NAR has been analyzed and predicted theoretically to
1
tial resistance but also a NAR effect. Unambiguous and occur also in the simplest possible case of a single Brow-
0
8 convincing experimental observations of NAR are still nian particle dynamics in one dimension [17, 18]. More
0 quite scarce, involving systems consisting of electrons in precisely,twobasicallydifferentphysicalmechanismsca-
: a sample of bulk GaAs [8], electrons in semiconductor pable of generating NAR in such systems have been un-
v
heterostructures[9],electronsinlowdimensionalconduc- raveled,namely apurelynoiseinduced effectinRef. [17]
i
X tors [10], and charged Brownian particles in structured andatransientchaosinducedeffectinRefs. [18]. Inboth
r microfluidicdevices[11]. Apartfromthelowdimensional cases,anexperimental realizationby means of a Joseph-
a
conductors,the system was always drivenout of equilib- sonjunctionsubjectedtosuitabledcandaccurrentshas
rium by means of an ac driving force and then its re- beenproposed. InthisLetterweshowthatamoderately
sponse to an externally applied static perturbation was damped Josephson junction being driven by microwaves
studied. On the theoretical side, a considerably larger indeed shows NAR of the type predicted in Refs. [18].
literature is available, most notably on different types of A first hint along these lines can be found in Fig. 13 of
semiconductors and semiconductor heterostructures [7]. [19], although without further explanation or discussion
In all those cases (except [11]) NAR is based on purely andnodirectreferencetotheresistivelyandcapacitively
quantum mechanical effects which cannot be transferred shunted junction model.
into the realm of classical physics. For classical systems,
TomodeltheJosephsonjunctionweusetheresistively
a first theoretical demonstration of the effect was pro-
and capacitively shunted junction model [20, 21]. It de-
vided in the context of a spatially periodic and symmet-
scribes the equation of motion for the difference δ of the
ric model system of interacting Brownian particles, sub-
phasesofthesuperconductingorderparameterinthetwo
electrodes
∗Electronicaddress: [email protected] I = Φ0Cδ¨+ Φ0 δ˙+I sinδ+I . (1)
0 N
2π 2πR
2
Here, C, R, and I denote the junction capacitance, re- with an area of 200µm2, cf. upper left inset in Fig.
0
sistance and maximum Josephson current, respectively, 1(a). The junctions were shunted by a AuPd strip with
dots indicate time-derivatives, Φ is the magnetic flux resistanceR=1.27Ωandintegratedinacoplanarwave-
0
quantum, and I = I +I sin(ωt) is the total current guide. WedenotethecriticalcurrentI asthemaximum
dc ac c
applied to the junction, consisting of a dc and a high dc current for which V = 0. In general, I is a function
c
frequency ac component. The first term on the right of I and fluctuations. By measuring I (I = 0) and
ac c ac
hand side of Eq. (1) describes the displacement current matchingitwithsimulationswedeterminedI =197µA,
0
CU˙, where U is the voltage across the junction, and has yielding I0R = 250µV, fc = 121GHz and Γ = 9 10−4.
·
been rewritten in terms of δ˙ using the Josephson rela- The Josephson length is about 40µm, i. e., well above
tion δ˙ = 2πU/Φ . The second term describes the cur- the 16µm diameter of our junctions assuring the short
0
rent through the resistor R, the third term the Joseph- junction limit. The design value of the capacitance was
son current, and the last term the noise current arising 8.24pF, yielding fpl = 43GHz, and βc = 7.9. The ac-
from Nyquist noise in the resistor. Its spectral power tual value used in the simulations shown below is some-
density is assumed to be white with SI(f) = 4kBT/R, what smaller, namely βc =7.7, reproducing particularly
where T is the temperature and k Boltzmann’s con- well the hysteretic IVC in the absence of microwaves.
B
stant. The model (1) implicitly assumes that magnetic The transport measurements have been performed with
fields created by circulating supercurrents can be ne- a standard four terminal method, using filtered leads.
glected (short junction limit). This holds when the lat- Microwavesbetween 8 and 35GHz, with variable output
eral junction dimensions are below about 4λJ, where power Pm, were applied through a semirigid cable that
λ =(Φ /4πµ j λ )1/2 istheJosephsonlengthinterms wascapacitivelycoupledtothe50Ωcoplanarwaveguide.
J 0 0 0 L
ofthecriticalcurrentdensityj ,theLondonpenetration The samples were electromagnetically shielded and sur-
0
depth λ , and the magnetic permeability µ . rounded by a cryopermshield, to reduce static magnetic
L 0
ByintegratingEq.(1)oneobtainsδ and,bytimeaver- fields.
aging, the dc voltage V =Φ δ˙ /2π across the junction. Given I , R, C, T and I , all relevant model param-
0 0 dc
h i
This is the main observable of our present work, which eters are fixed, with the exception of the (frequency de-
is measured when recording V(I ), the current voltage pendent) coupling factor between the microwave ampli-
dc
characteristics(IVC). Fornumericalsimulations,(1) can tude √P applied from the source and the amplitude
m
be rewritten in dimensionless units by normalizing cur- I of the ac current induced across the junction. We
ac
rents to I , voltages to I R, times to t = Φ /(2πI R), havefixedthis factorbycomparingthemeasureddepen-
0 0 c 0 0
and hence frequencies to f =I R/Φ , yielding dence of I (√P ) with the calculated curve I (I ), as
c 0 0 c m c ac
shown in the right inset of Fig. 1, for a microwave fre-
i=β δ¨+δ˙+sinδ+i , (2)
c N quency of 19GHz (f/f 0.16). The experimental and
c
≈
where i = i +i sin(τf/f ) is the normalized applied theoretical curves are in good agreement. In particular,
dc ac c
current, τ = t/t the normalized time, β = (f /f )2 = the main side maxima can be found, both in experiment
c c c pl
2πI R2C/Φ the Stewart-McCumber parameter, f = and simulation. By adjusting the position of these max-
0 0 pl
(I /(2πΦ C))1/2 the Josephson plasma frequency, and ima, we obtain a coupling factor Iac/√Pm(19GHz) =
0 0
i the normalized noise current with spectral density 1.0mA/√mW.
N
S (f/f )=4Γ and noise parameter Γ=2πk T/I Φ . Figure 1(a) shows IVCs under f = 19GHz microwave
i c B 0 0
Inanutshell,thebasicingredientsofNARaspredicted irradiation at three values of I . In the absence of
ac
in[18]areasfollows. Theunperturbeddeterministic dy- microwaves (black line) the IVC is hysteretic, exhibit-
namics (Eq. (2) with i = 0 and i = 0) exhibits two ing a critical current of 195µA and a return current of
dc N
symmetricattractors,carryingcurrentsofoppositesigns 100µA (black arrows). When the microwave field is ap-
(zero crossing Shapiro steps). When an external pertur- plied, the hysteresis decreases with increasing I , and
ac
bationin the formof a static bias i is applied, a subtle step-like features appear on the IVC. At P = 194µW
dc m
interplay of this bias force and the dissipation leads to (I =435µA;magentaline),weobserveNARwithare-
ac
a destabilization of that attractor, whose current points sistanceof 1.07Ω,occurringinaninterval I 20µA
dc
− | |≤
into the same direction as the applied bias. Its remnant (i. e., approximately 10% of I ). When I is increased
0 ac
is a strange repeller, exhibiting transient chaos, hence toP =253µW(I =497µA;greenline)theNARhas
m ac
the name “transient chaos induced NAR” coined in [18]. disappeared. However, centered on a voltage which cor-
The actual realization of NAR along these lines requires responds to the first Shapiro step (V = Φ f 39µV),
1 0
≈
a careful choice of model parameters in (2) within the regions of negative differential resistance appear. In
general regime of frequencies f comparable to f and Fig. 1(b) measured and simulated IVCs for the two mi-
pl
valuesofβ roughlybetween1and100. Toobtainprecise crowave amplitudes 435µA and 497µA are compared.
c
quantitative results, we have solved Eq. (2) numerically For the former case, which is recorded at the microwave
for various such parameter values by integrating and av- amplitude where the maximum NAR has been observed,
eraging over typically 5 103 periods of the ac current. the agreement between the experimental and the theo-
·
For experiments, which were performed at T = 4.2K, retical curve is nearly perfect. For the latter case some
we used circular Nb-Al/AlO -Nb Josephson junctions small differences can be seen, although the agreement is
x
3
FIG. 2: Contour plot of the normalized dc voltage V/Φ0f
across the junction as a function of dc current Idc and mi-
crowave current amplitude Iac. (a) f = 8GHz, experiment;
(b) f = 8GHz, simulation; (c) f = 19GHz, experiment; (d)
f = 19GHz, simulation. For symmetry reasons, Idc 7→ −Idc
implies V 7→ −V, hence negative Idc values are not shown.
FIG.1: Currentvoltagecharacteristics (IVC)oftheJoseph- Blue areas indicate NAR.
son junction at 4.2K in a 19GHz microwave field. (a):
at 3 levels of applied microwave power (0µW, 194µW and
253µW) showing the effect of negative absolute resistance
I 10µA, for all values of I for which NAR shows
dc ac
at 194µW (Iac = 435µA) and of negative differential resis- u|p.|≈In contrast, for f = 19GHz, the I interval for
tance at 253µW (Iac = 497µA). Left inset: image of the dc
NAR decreases with increasing I . When we increased
Josephson junction. Right inset: measured (thick green) and ac
thefrequencyfurtherto35GHz,hystereticShapirosteps
calculated (thin black) dependence of the critical current Ic
appearedontheIVC,crossingthe voltageaxis(I =0).
on the microwave current amplitude Iac. (b): enlargement dc
ofthemeasured IVCsfor194µWand253µW,togetherwith As a consequence, NAR ceases to exist both in the ex-
thesimulated IVCs, cf. legend. periment and the simulations.
In a second series of experiments we applied a mag-
netic field B parallel to the junction plane in order to
still very good. To demonstrate the origin of the NAR, tune (decrease) its Josephson current I , making it a B
0
the grey curve in Fig. 1(b) shows a simulated IVC for dependent function I (B) [22]. Thus all I -dependent
0 0
iac = 2.242µA and Γ = 0, i. e., for the noise-free case. parameters entering the normalized equation (2) aquire
The curve shows n = 1 Shapiro steps to be the cause a B-dependence, in particular i, β , f/f , and Γ. Figure
− c c
of NAR, clearly revealing its nature to be of the type 3showsacomparisonofthemeasuredandcalculatedde-
discussed in [18]. pendenceoftheresistanceuponI andI (B). Again,we
ac 0
Figure 2 compares in more detail the measured find excellent agreement between measurement and the-
and calculated dependence of V on I and on I , ory. Blue regions indicate NAR. Their most remarkable
dc ac
for two frequencies (8GHz and 19GHz). For f = feature is thatthe values ofI ,for whichNAR appears,
ac
8GHz, the comparison between measured I (√P ) practically do not depend on I . Furthermore, we find
c m 0
and simulated I (I ) curves yields a coupling factor that the NAR value can be tuned by I via an applied
c ac 0
I /√P (8GHz) = 0.33mA/√mW. In the graphs, V magnetic field. For our junction parameters we find a
ac m
is normalized to Φ0f, yielding an integer value n for the maximum NAR at I0 ≈ (0.4...0.6)I0(B = 0), which is
n-thShapirostep. Again,the agreementbetweentheory increasing with Iac.
and experiment is very good. There are at least five I In conclusion, we have observed negative absolute
ac
intervals where NAR appears at f = 8GHz, and three resistance (NAR) of up to about 1Ω in a shunted
−
such intervals at f =19GHz. Within those regimes, the Nb-Al/AlO -Nb Josephson junction device subjected to
x
resistanceatI =0reachesvaluesuptoabout 1Ω. In microwaves. To clearly see the effect, a careful choice
dc
−
the case of f =8GHz, the NAR persists up to values of of parameters is required, but still the range of suitable
4
modeldynamics revealsthat the relevantphysicalmech-
anism is of the transient chaos induced NAR type from
[18]. The similarity between our Fig. 1 and Fig. 2 in
[9] suggests that with respect to NAR, purely quantum
mechanical band structure and energy quantization
effects may be imitated by inertia effects in a purely
classical, one dimensional noisy dynamics. Moreover,
ourFig. 3exhibitsmanyfeatureswhicharequitesimilar
to the corresponding plots in [18], while the intuitive
explanation of the almost vertical stripe-pattern in Fig.
3 remains as an open problem. As an application, our
present work opens the intriguing perspective of a new
FIG.3: (a)Contourplotoftheexperimentallymeasured re-
resistor-typeelectronicelementwhichistunablebetween
sistance R:=V(I =5µA)/5µA as a function of theJoseph-
positive and negative resistance via an easily accessible
son current I0(B) and of the microwave amplitude Iac. I0
hasbeen varied byapplyingamagnetic field tothejunction. external control parameter, e. g., the amplitude of an
Graph (b) shows thecorresponding simulated plot. For both ac driving or an externally applied magnetic field in the
graphs parameters at B=0 are thesame as in Fig.2(a),(b). mT range.
parameters is quite large. In all cases, we obtain very
good agreement with theoretical simulations of the ThisworkwassupportedbytheDeutscheForschungs-
resistively and capacitively shunted junction model. gemeinschaft (Grants No. KO 1303/7-1, RE 1344/5-1,
Furthermore a closer inspection of the corresponding and SFB 613).
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