Low-loss tunable metamaterials using superconducting circuits with Josephson junctions

0.42 MB·English

by  Philipp Jung

Low-loss tunable metamaterials using superconducting
circuits with Josephson junctions
P. Jung and S. Butz
Physikalisches Institut, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany
S. V. Shitov
National University of Science and Technology MISIS, Moscow 119049, Russia and
Institute of Radio Engineering and Electronics (IREE RAS), Moscow 125009, Russia
3
1 ∗
A. V. Ustinov
0
Physikalisches Institut, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany and
2
National University of Science and Technology MISIS, Moscow 119049, Russia

n (Dated: December 11, 2013)
a
J We report on experiments with superconducting metamaterials containing Josephson junctions.
In these structures, split-ring resonators used in conventional metamaterials are replaced by su-
3
perconducting loops that are interrupted by Josephson junctions, so called rf-SQUIDs. Like the

] split-ringresonators, theseelementscanbeseenasLC-resonatorsthatcoupletothemagneticfield.
n The advantage of superconducting thin-film metamaterials is that, due to the tunable intrinsic
o inductance of the Josephson junction, the resonance frequency of the rf-SQUID can be changed
c by applying an external dc magnetic field. We present experimental results that demonstrate the
-
tunability of theresonance frequency of these devices.
r
p
u
s One of the main limitations of many metamaterial de- sides ofthe junction arounda constantvalue ϕ′ is trans-
. ′
t signs is the restriction of their usability to a narrow fre- latedintoasmallvariationofthecurrentI =Icsinϕ+δI
a
quency band. By using resonant elements in order to flowing through it. In this way, current and voltage are
m
achieve the desired effects (such as a negative magnetic related to each other by [9, 10]:
-
permeability µ ) the meta-atoms are usually made for a
d r
n fixedoperationfrequencyrangebydesign. Tocircumvent Φ0 d ′ d
V = I ≡L(ϕ) I (1)
o this limitation, several approaches have been developed 2πI cosϕ′dt j dt
c
c [1],manyofwhicharebasedontheideaofintroducinga
[ nonlinear element into the resonantcircuit, thus making This equation takes the form characteristic to an in-
1 ittunable[2,3]. SuperconductorsingeneralandJoseph- ductive element. That is, the voltageoverthe element is
v son junctions in particular are ideal constituents of tun- proportionaltothetimederivativeofthecurrentthrough
0 able meta-atoms due to the strong dependence of their it. Inthis model,thevalueofthe so-calledJosephsonin-
4 inductance on external parameters such as temperature ductance Lj depends on the constant part of the phase
4 and magnetic field. The tunability by varying tempera- difference across the junction and therefore on the bias
0
ture has already been demonstrated with superconduct- currentthroughit. Thus,itworksasatunableinductor.
.
1 ing split-ring resonators (SRRs)[4, 5]. In this work, we AcircuitconsistingofaJosephsonjunctionplacedinto
0
move one step further and experimentally demonstrate
3
tunability via magnetic field by replacing the SRR by a
1
superconducting quantum interference device (SQUID).
:
v This approachhas previously been suggested in the the-
i
X oretical work by Lazarides and Tsironis [6] and further
investigated by Maimistov and Gabitov [7].
r
a
The key element used in this work is the Josephson
junctionwhichconsistsoftwosuperconductorsseparated
FIG.1: (a)Asuperconductingloop interruptedbyaJoseph-
by a thin insulating layer. The current and voltage over
son junction can be decomposed into a (b) network model
such a junction is described by the two Josephson rela-
using the restively and capacitively shunted junction model
tions [8–10]. The Josephson junction with a critical cur-
(RCSJ) [10]. The red cross indicates the Josephson junction
rentIc canbetreatedasanonlinearnetworkelement. A forwhichvoltageandcurrentrelationaregivenbytheJoseph-
simple description is provided by a small signal approx- son relations. (c) Small signal network model of theSQUID.
imation. A small variation δϕ of the phase difference Lgeo presents the geometric inductance of the superconduct-
ϕ between the superconducting wave functions on both ing loop. Lj is the Josephson inductance, see Eq.(1). R and
C arethenormalresistanceandcapacitanceoftheJosephson
junction,respectively. Inoursample, thejunction isshunted
by a large additional capacitance to decrease the resonance
∗Electronicaddress: [email protected] frequency,which is added in parallel toC.
2
a superconducting loop (cf. Fig.1a) is a single junc- The time average of the solution of this equation yields
tion(rf-)SQUID.Inthiscase,magneticfluxquantization the complex ac flux susceptibility of the single SQUID
leads to the relation which is χ = hΦ/Φ i − 1. There is a strong fre-
φ ext t
quency dependence of real and imaginary components
Φ
0
ϕ +L I =Φ . (2) of χ around the resonance frequency ω .
geo ext φ 0
2π
For an effective medium consisting of these elements,
The electric circuit equivalent of an rf-SQUID is de- themagneticsusceptibilityisgivenbyχ =F˜χ . The
mag φ
picted in Fig.1b. In addition to the Josephsonrelations, coefficient F˜ depends on the arrangement of the meta-
one has to account for the loop inductance, the capac- atoms in the medium. Provided that the coupling be-
itance between the superconducting electrodes and the tween the SQUIDs is small (i.e. F˜ ≪1) [6], the relative
(resistive) single electron tunneling current. permeability of the medium is
Once an external magnetic flux Φ (t) = Φ +
ext e0
Φeacos(ωt) is applied to the system and the driving ac µr =1+χmag =1+F˜χφ. (5)
amplitude Φ ≪ I /L is small, the small signal ap-
ea c geo In this work, we experimentally verify the microwave
proximationisvalid. ThetotalinductanceL comprises
tot propertiesofaprototypecircuitrepresentingasinglesu-
thegeometricinductanceL andthetunableJosephson
geo perconducting meta-atom of such a medium. We have
inductanceL (Φ ),seeFig.1c. Theresonancefrequency
j e0 designed and fabricated a circuit consisting of a single
of the system is given as a function of the external bias
rf-SQUID placed inside the gap of a coplanar waveguide
flux Φ
e0 (CPW).Inthissetting,theSQUIDloopisperpendicular
−1 to the magnetic field, as can be seen in Fig.2a, and thus
ω (Φ ) = (L (Φ )C) 2 . (3)
0 e0 tot e0
couplestothequasi-TEMmodeoftheCPW.Thearrows
Inthepresentexperiment,theadditionalkineticinduc- denoted by E~, H~ and S~ indicate the orientation of the
tance [9] of the superconducting ring is negligible com- electricandmagneticfieldcomponentsandthePoynting
pared to its geometric inductance. It should be noted, vector of the incoming wave, respectively. An external
though, that close to the critical temperature of the su- magnetic field is applied perpendicular to the substrate
perconductor,the kinetic inductance will increase signif- plane either through an external coil or by driving a dc
icantly and influence the resonance frequency. Due to currentthroughthecentralstripofthetransmissionline.
this fact and the temperature dependence of the criti- In either case, the SQUID loop is biased with a dc mag-
cal current of the junction, the rf-SQUID also shows an netic flux Φ thus making the resonance frequency tun-
e0
intrinsic tunability by temperature which, however, will able in situ according to Eq.(3).
not be detailed here. The SQUID loop used in the presented measurement
TheapproximationoftheJosephsonjunctionbyatun- is made from niobium (which has a superconducting
able inductor workswell within the mentionedsmallsig- transition temperature of T = 9.2K) and includes a
c
nallimit. However,inordertoobtainacompletedescrip- Nb/AlO /Nb circular Josephson tunnel junction with a
x
tion,theJosephsonnonlinearityandlosstermshavetobe radius of 0.5nm and a critical current I ≈4.5µA . The
c
included. By applying Kirchhoff’s laws, the two Joseph- geometric loop inductance L =63.5pH is comparable
geo
sonrelationsandEq.(2)tothecircuitdepictedinFig.1b, to the zero-flux Josephsoninductance L (0)=74pH. By
j
a second order nonlinear differential equation is derived: shuntingtheJosephsonjunctionwithanadditionallarge
capacitor C =1.7pF we designed the resonance fre-
shunt
quency to be tunable between about 21GHz and 5GHz,
2π 1
Φ+L I sin Φ + Φ˙ +CΦ¨ =Φ (ω,t). (4) for aninfinitesimally smallΦ . The measurementswere
geo c ext ea
(cid:20) (cid:18)Φ (cid:19) R (cid:21)
0 performed using a cryogenic amplifier and a vector net-
work analyzer while the sample was immersed in liquid
4He at a temperature of 4.2K.
The experimental results for low-power transmission
presentedinFig.3ademonstratethefrequencytunability
oftheresonancedip. Thedecreasedtransmissionisillus-
tratedbythedarkercolorintheplot. Theresonancefre-
quency changes periodically as a function of the applied
magnetic flux Φ with a period of Φ . The dashed blue
e0 0
line represents a fit of the small signal model of Eq.(3).
As discussed earlier, the model should yield good fit re-
sults only for verysmall values of the driving amplitude.
The powerusedinthis measurementwas -85dBmwhich
FIG. 2: Sketch of a single rf-SQUID embedded into a copla- corresponds to an ac flux amplitude of Φ ≈ Φ /1600
ea 0
nartransmissionline. ThedarkerareasrepresenttheNbelec- and can be safely assumed to be small enough for the
trodes. The inset on the right shows an optical microscope approximationtohold. Thevalues,obtainedfromthefit
image of a single rf-SQUID. forL andC agreewithin5%withthevaluesexpected
geo
3
20 0 ple, analytical description, we approximated the system
a) bytreatingit asahomogeneous,cylindricallysymmetric
19 -0.1 problem, similar to a coaxial cable. The unit cell of the
waveguide, of length l, contains one SQUID which de-
-0.2
18 fines the interaction with the magnetic field component.
-0.3 Under these assumptions
17
-0.4 µ A2
16 -0.5 F˜ = Lgeoπl0(r22sq−r12). (6)
15 -0.6 Here, A is the area of the SQUID, r and r are the
sq 2 1
outer and inner radius of the gap, as shown in Fig.2a.
14 -0.7
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Using Eqs.(5) and (6), we obtain the effective perme-
ability µ and use it to modify the inductance and thus
r
theimpedance ofthe CPWsection. This,inturn, allows
20 0 us tocalculate the scatteringmatrixcoefficientS21. The
b)
-0.05 result of this calculation, depicted in Fig.3b, is in excel-
19 lent agreement with the measured data. As expected,
-0.1
at so small applied power level, the numerical treatment
18 -0.15 adds very little additional information to the analytical
-0.2 solution of Eq.(3) shown in Fig.3a by the dashed blue
17
-0.25 line. However, at higher power levels, the small signal
approximation of Eq.(1) is no longer valid and the nu-
16 -0.3
-0.35
15
-0.4
14 -0.45
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
FIG. 3: (a) Measured microwave transmission at a power
of P ≈ −85dBm at the sample, that corresponds to Φea ≈
Φ0/1600. Thedarkercolorscorrespondtoareducedtransmis-
sion through theSQUID-loadedwaveguide. Thebluedashed
linerepresentstheanalyticalsolutionextractedfromthesmall
signalmodel(Fig.1c). Itisshownfornegativefluxvaluesonly
inordernottoobscurethedata. (b)Transmission througha
single SQUIDsimulated numerically by solving Eq.(4).
for the fabricationprocedurewhile the value of the criti-
calcurrentI is about6%lessthanthe onemeasuredon
c
test junctions on the same chip.
In order to estimate the magnetic susceptibility and
thereby the relative permeability, Eq.(4) was solved nu-
merically. The parameters L and C were taken from
geo
the fit mentioned in the paragraphabove. The value for
the shunting resistance R was taken from the current-
voltage characteristics measured for a test junction on
the same chip. Additionally, the loss tangent of the
Nb O dielectric of the shunt capacitor was assumed to
2 5
betanδ =1.0·10−3[11]. Tocalculatethescatteringchar-
acteristics versus dc magnetic flux Φ , we employed the
e0
effective medium approach by calculating the effective FIG. 4: Relative (a) magnitude and (b) phase of the trans-
µr of a SQUID-loaded section of the CPW. Analogous mission(S21)throughtheSQUID-loadedcoplanarwaveguide
to the procedure in Ref. [6], our task was to find a suit- at a bias of Φe0 = 0.77Φ0. The measured data (power
able value for F˜. Since the actual field distribution in a P ≈ −110dBm) are shown in black, the simulated curve
superconductingcoplanarwaveguidedoesnothaveasim- (Φea =Φ0/215) in blue.
4
merical treatment will be relevant. metalstructuresemployingsplit-ringresonators. Further
Thefrequency-dependentpropertiesofametamaterial studies ofSQUID propertiesathigher powers,i.e. inves-
can be characterizedby relative magnitude and phase of tigating its nonlinear properties, as well as experiments
the complex transmission coefficient S . Measurement on1Dand2D-arraysofsuchelementsarepresentlybeing
21
andnumericalcalculationofthe transmissionmagnitude conducted.
and phase through the SQUID-loaded coplanar waveg-
uide (cf. Fig.4a and 4b) are in good qualitative agree- Theauthorswouldliketoacknowledgestimulatingdis-
ment. Detailed quantitative comparison in our case is cussions with S. M. Anlage, I. Gabitov, and G. Tsiro-
hindered by difficulty of making precise calibrationmea- nis. This work was supported in part by the Ministry
surements at cryogenic temperatures. ofEducationandScience ofthe RussianFederation,and
In conclusion, we have experimentally and theoreti- alsobytheDeutscheForschungsgemeinschaft(DFG)and
cally investigateda prototype of a magnetically coupled, theState ofBaden-Wu¨rttembergthroughthe DFGCen-
tunable meta-atom made of a thin-film superconducting ter for Functional Nanostructures (CFN). Philipp Jung
circuit. The resonance frequency of a superconducting would like to acknowledge the financial support by the
loop containing a Josephson junction is tunable over a Helmholtz International Research School for Teratron-
wide frequency range both by magnetic field and tem- ics (HIRST), Susanne Butz would like to acknowledge
perature. Therefore, the SQUID metamaterial appears the financial support by the Landesgraduiertenfrderung
as an appealing alternative to the widespread normal- Baden-Wu¨rttemberg.
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