Tunable negative permeability in a three-dimensional superconducting metamaterial

0.52 MB·English

by  C. Kurter

Tunable negative permeability in a three-dimensional superconducting metamaterial
C. Kurter,1,2 T. Lan,1,3 L. Sarytchev,1 and Steven M. Anlage1,4
1Center for Nanophysics and Advanced Materials, Department of Physics,
University of Maryland, College Park, MD 20742-4111
2Department of Physics, Missouri University of Science and Technology, Rolla, MO 65409
3Department of Materials Science and Engineering,
Northwestern University, Evanston, IL 60208, USA
4Department of Electrical and Computer Engineering,
University of Maryland, College Park, MD 20742-3285
5 (Dated: March 31, 2015)
1
0 We report on highly tunable radio frequency (rf) characteristics of a low-loss and compact three
2 dimensional (3D) metamaterial made of superconducting thin film spiral resonators. The rf trans-
mission spectrum of a single element of the metamaterial shows a fundamental resonance peak at
r
∼24.95MHzthatshiftstoa25%smallerfrequencyandbecomesdegeneratewhena3Darrayofsuch
a
M elementsiscreated. Themetamaterialshowsanin-situtunablenarrowfrequencybandinwhichthe
realpartoftheeffectivepermeabilityisnegativeoverawiderangeoftemperature,whichrevertsto
gradually near-zero and positive values as the superconducting critical temperature is approached.
8
Thismetamaterialcanbeusedforincreasingpowertransferefficiencyandtunabilityofelectrically
2
small rf-antennas.

]
n
o INTRODUCTION tennas. Antennas integrated with compact rf metamate-
c rials were speculated to have enhanced radiation power,
-
r highdirectivityandreducedreturnlossduetotheunique
p A three dimensional (3D) array of sub-wavelength
electromagnetic properties of metamaterials [23, 24]. Zi-
u elements composing a metamaterial can demonstrate
olkowski[25,26]hasproposedusingametamaterialwith
s unique electromagnetic properties which are not directly
. negativerefractiveindextoboosttherealradiatedpower
t accessible in nature, such as negative permeability [1, 2]
a of an electrically short dipole antenna (size λ /1000) by
0
m and zero refractive index [3, 4]. Most metamaterials
almost two orders of magnitude compared to the dipole-
achieve these unusual effects by means of tailored geo-
- in-free-space case. A large imaginary part (reactance) of
d metricalstructureswithoutrelyingontheintrinsicprop-
radiation impedance Z =R +iX of an electrically
n rad rad rad
ertiesofmaterialsusedinmetamaterialfabrication. Con-
o small antenna limits the radiation efficiency because free
ducting non-magnetic split-ring resonators (SRRs), and
c space has a purely real impedance [27]. When the an-
[ their derivatives, have been employed to demonstrate tenna is placed in near-field proximity to a negative re-

negative permeability by strongly coupling to the mag-
2 fractive index medium, the Xrad of the antenna can be
netic field component of incident electromagnetic waves
v cancelled out. Therefore the metamaterial acts as a nat-
1 at microwave [5] or higher frequency bands [6, 7]. How- ural impedance matching network for the dipole, which
2 ever, when lower operational frequencies are needed,
maximizesthetransferofpowerfromtheantennatofree
7 these resonator-based metamaterials require impracti-
space in the far-field. The bandwidth and the operat-
3
cally large dimensions to be able to compensate for
0 ing frequency of the antennas can also be reconfigured
ohmic losses [8]. One way to address this miniaturiza-
. with tunable metamaterials, which is crucial for modern
1 tion problem is to use superconductors since they show
wireless communication systems relying on multi-band
0
verylowsurfaceresistancebelowtheirtransitiontemper-
5 operation [24].
ature [9, 10]. Moreover, tunability of the resonant fea-
1 In this work, we demonstrate a tunable 3D rf meta-
: turescanbepreciselyachievedwithoutlumpedelements
v bymeansoftemperatureandmagneticfield[11–13]. For material composed of electrically small (∼ λ0/2300) su-
i perconducting thin film resonators. The single resonant
X instance,theplasmafrequencyofsuperconductingspiral
element is a planar spiral constructed by w= 3 µm wide
r meta-atomshavebeentunedtotheMHzrangebymeans
semi-circular strips with increasing radii. The center to
a
of temperature variation, and the plasmonic properties
center separation between two adjacent turns is s= 6
have been studied at essentially fixed frequency [14].
µm. The spiral is densely packed with a large number of
Ofparticularinterestareradiofrequency(rf)metama- turns, N= 200, where the inner and outer diameters are
terials where the free space wavelength λ can be tens of D =4.606mmandD =7mmrespectively. Theelectro-
0 i o
meters. Despite the challenge of scaling the dimensions magnetic response of the resonant spirals can be studied
down, rf metamaterials offer a wide spectrum of applica- withasimplecircuitmodel[28];i.e. anindividualplanar
tionsincludinghighresolutionmagneticresonanceimag- spiralcanbetreatedasamagneticresonatorwithanan-
√
ing [15–17], magneto-inductive lenses [18] and wireless gular resonant frequency of ω =1/ LC where L and C
0
communication [19–22] employing electrically small an- are the total inductance and capacitance, respectively.
2
Possessing large geometrical inductance and capaci- RESULTS AND DISCUSSION
tance (due to large number of turns) over a compact ge-
ometryputstheindividualspiralsintoasignificantlysub-
Figure1showstherf-transmission|S |throughasin-
wavelength self-resonance regime. Therefore the meta- 12
gle Nb spiral along with the 2D and 3D arrays of the
material formed by the 3D array of these spirals can be
coupledspirals. Themeasurementshavebeenperformed
considered as a continuous effective medium character-
by means of two loop antennas sandwiching the samples
izedwitheffectiveconstitutiveparameters;i.e. frequency
(see the sketch in the inset of Fig. 1 illustrating the 3D
dependent permeability µ (ω) and permittivity (cid:15) (ω).
eff eff array of spirals between the antennas; the quartz sub-
Previous theoretical [29] and experimental [30, 31] work
strates are not shown). The antennas are made of semi-
on metamaterials with periodic circular conducting el-
rigid coax cables by shaping the inner conductors into
ements have shown that the fundamental resonant fre-
1.2 cm-diameter loops and attaching them to the outer
quencyofthemetamaterialcanbedrasticallylowerthan
conductors. Theotherendsofthecoaxialcablesarecon-
that of the single constitutive element mainly due to the
nected to a 2-port vector network analyzer to measure
mutual inductance (magnetic coupling) between the res-
the scattering matrix. The magnetic excitation of the
onantelements. However,electrical(capacitive)coupling
sample is achieved via inductive coupling into the top
can also play a role in the collective electromagnetic re-
(drive) loop and the transmitted signal is picked up by
sponseofthemetamaterial,thusinteractionbetweenthe
the bottom (pickup) loop.
individual elements can be complex depending on struc-
The loop antennas are 27.2 mm apart to minimize the
tural details of the metamaterial such as separation be-
direct magnetic coupling between them [34] that can ob-
tween the resonant elements along the substrate, inter-
scure the response of the 3D metamaterial. The single
layer spacing, array size and alignment/arrangement of
spiraldemonstratesasharpfundamentalresonancepeak,
the elements along the stacking direction [32, 33].
f =ω /2π at ∼ 24.95 MHz and a dip at ∼ 25.04 MHz at
0 0
Here we study a 3D superconducting metamaterial
4.3Kwithaloadedqualityfactorof1273. Thepeakand
with sub-wavelength spiral elements operating in the rf
the dip in the transmission spectrum correspond to con-
regime. Wefindthatthemutualinteractionsbetweenthe
structiveanddestructiveinterferencesbetweenthedirect
individual spirals contributes to the effective response of
coupling of the loops and magnetic coupling of the loops
themacroscopic3Dmetamaterial. Moreover,weobserve
to the metamaterial respectively [35]. We have observed
thatboththecenterfrequencyoftheresonantbandwidth
higher order modes corresponding to shorter-wavelength
in the rf transmission and the negative µ (ω) region is
eff rf-current standing waves in the spiral, which have been
highly temperature tunable.
extensivelystudiedelsewhere[8,36–39]. AbovetheT of
c
Nb, superconductivity is lost and the resonance is wiped
out at 9.25 K. This demonstrates that such a compact
device is not functional if made with thin films of nor-
EXPERIMENT mal metals due to the large ohmic losses. An analytical
model adapted to a similar experimental configuration
showedthatthetransmissionspectrumintheabsenceof
The metamaterial is patterned out of a 200 nm thick
the spiral resonator between the loops captures the ex-
Nb thin film sputtered on to a 350 µm thick quartz sub-
perimental transmission through the spiral taken above
strate. We define circular spiral-shaped resonators [8]
the T of the superconductor [35].
intotheNbfilmbyusingstandardphotolithographyand c
reactive ion etching using a mixture of CF /O (10 % The low temperature resonance peak shifts to a
4 2
O ). After fabrication, the quartz substrates are diced significantly smaller frequency, 19.15 MHz in the rf-
2
into 0.9 x 0.9 cm2 and 2.6 x 5.2 cm2 pieces to have in- transmission spectrum of the 2D array of 3 x 6 spirals
dividual spirals and 2D arrays of 3 x 6 Nb spirals. To due to a significant change in effective parameters; espe-
be able to extend the structure into three dimensions, ciallyµeff(ω)[32]. Thesmallseparationdistancebetween
four substrates containing 2D arrays are stacked along thespirals∼λ0/1875strengthensthemagneticcoupling
the c-axis of the substrate by using thin sheets of Roha- between the individual spirals that leads to splitting of
cellfoamasspacers. Theplanesofthespiralsarealigned the resonant frequencies [40].
in parallel and each substrate is separated from the next For 3D metamaterials with strong magnetic coupling
by 0.5 mm along the z-direction to produce strong cou- inthestackingdirection,theenhancementincapacitance
plingbetweentheNbspiralsinthethirddimension. The due to interlayer interaction [41] further reduces the res-
experiments have been conducted in an evacuated probe onant frequency at 4.3 K down to 18.83 MHz as seen
inside a cryogenic dewar and the temperature has been in the transmission spectrum of the 3D metamaterial.
precisely adjusted by a temperature controller between The resonant features observed in the low temperature
4.2 K up to well above the superconducting transition spectra of both 2D and 3D arrays of spirals disappear at
temperature, T (cid:39) 9.2 K of Nb. temperatures above 9.25 K as expected.
c
3
by the normal state data taken at 9.25 K) imply the

0.0030 presence of negative permeability. To be able to char-
ar) single acterize the negative magnetic response of our metama-
ne 0.0025 4.3 K terials, we have retrieved µeff(ω) from the experimental
S| (li12 0.0020 34D.3 aKrray stocaattenreitnwgomrkaatrniaxly[4z1er] c(soelleecttheedinbsyettwoof Floiogp.s1)c.onTnheectiend-
n | duced electromotive force in the pickup loop as a func-
o z
ssi 0.0015 y x tion of frequency due to the magnetic coupling of top
mi loop can be written as emf(ω) = -iωAµ H when there
s ref 0 z
Tran 0.0010 24D.3 aKrray s9i.n2g5l eK i-siωnAoµmµeta(mωa)tHeriawlhbeentwtheeernetiwsomelotaompsaatenrdiaelmsafn(ωdw)micehtae=d
0 eff z
between them. Here, A is the area of the pickup coil, µ
0.0005 0
is the free space permeability and H is perpendicular
z
component of the magnetic field. By combining these
0.0000
18 19 20 21 22 23 24 25 twoequationsonecanapproximatethecomplexeffective
Frequency (MHz) permeability as a ratio, µ (ω)=emf(ω) /emf(ω) ∼
eff meta ref
S (ω) /S (ω) . Here it is assumed that H is the
FIG. 1: (Color online) Comparison of transmission |S | 21 meta 21 ref z
12
throughasinglespiral(greencurve),2Darrayof3x6spirals same in both measurements [41].
(red curve) and 3D array of the spirals formed by stacking Figure 2 shows the real µ(cid:48) (ω) (red curves) and imag-
eff
four substrates including 2D arrays of spirals (blue curve). inary µ(cid:48)(cid:48) (ω)(blue curves) parts of the retrieved perme-
eff
The lowest resonant frequency band decreases and develops
abilityfor(a)thesinglespiraland(b)the3Dmetamate-
fine splitting by adding more elements to the metamaterial.
rial composed of similar spirals. The overlapping dashed
ThemarooncurveshowsthedatatakenattheT ofNb,9.25
c curves are the Lorentzian fits according to Eq. (1) which
K,forasinglespiral,wheretheresonantpeakdisappearsdue
to enhanced ohmic losses. 2D and 3D arrays also show the reasonably capture the retrieved data at 4.3 K. The fit
same feature when the sample is heated above 9.25 K. The parameters are ω0/2π= 24.955 MHz, Γ/2π= 0.03 MHz
inset shows a schematic of the 3D metamaterial sandwiched and F= 0.0068 for the single spiral and ω /2π= 18.828
r
between two rf-loop antennas which are 27.2 mm apart from MHz, Γ/2π= 0.064 MHz and F= 0.012 for the 3D array.
each other along the z-axis.
The enhanced loss in the 3D array compared to an in-
dividual spiral can be explained in part by the stronger
inductive coupling to the excitation loops [43] and the
By considering the effective medium approximation,
excitation of magneto-inductive waves in the metamate-
the frequency dependent complex effective permeability
rial [44]. We observe that the smooth well-defined reso-
of an array of non-magnetic conducting SRRs has been
nance of the single spiral splits into multiple resonances
characterized with a Lorentzian function as [1]
withmagneticcouplingintootherspiralsinthe3Darray
(see the insets of the plots showing the resonant features
µ (ω)=µ(cid:48) (ω)+iµ(cid:48)(cid:48) (ω) indetail). Justabovetheresonantfrequency, theµ(cid:48) (ω)
eff eff eff eff
Fω2 becomesnegativeinanarrowfrequencyband. Thesefor-
=1− (1) bidden bands of frequencies range from 24.958 to 25.038
ω2−ω2+iΓω
r MHzforthesinglespiralandfrom18.838to18.924MHz
whereµ(cid:48) (ω)andµ(cid:48)(cid:48) (ω)aretherealandimaginaryparts for the 3D metamaterial at 4.3 K.
eff eff
oftheeffectivepermeability,F isthefillingfraction,ω = Having discussed the low temperature magnetic re-
r
2πf istheangularcenterfrequencyoftheresonance,and sponseofthemetamaterial,wenowturntothetunability
r
Γisthedampingtermcharacterizingohmicandradiation of µ (ω) as the metamaterial is heated up to a temper-
eff
losses. The center frequency for the resonant bandwidth ature above the T of Nb. Figure 3(a) and (b) show
c
f isdeterminedbytheresonantparametersofindividual the temperature dependence of the real and imaginary
r
elements f , as well as the mutual interactions between parts of the retrieved µ (ω) for the 3D metamaterial.
0 eff
them [29, 42]. Thenarrowfrequencybandwheretheµ(cid:48) (ω)isnegative
eff
The wavelength of the electromagnetic excitation we shifts to lower frequency as the temperature increases as
have used is much larger than the dimensions (D , w, s) seen in Fig. 3(a). The same effect is also seen in the
o
and separation of the spirals forming our metamaterial. peaks of the µ(cid:48)(cid:48) (ω) that are often associated with the
eff
Therefore,wecanapplytheeffectivemediumapproxima- resonant frequencies [14, 45] (see Fig. 3(b)). The fre-
tion and characterize the metamaterial with µ (ω) and quencyshiftinthespectraisaresultofadecreaseinthe
eff
(cid:15) (ω). Sinceinductivecouplingisquiteprominenthere, densityofsuperconductingelectrons(superfluiddensity)
eff
the former is the dominant effective parameter. Sharp with temperature n (T) which is inversely proportional
s
dips of the transmission spectra just above the resonant to the square of magnetic penetration depth λ (T). As
L
frequencies in Fig. 1 (below the background signal set thefieldsoftheelectromagneticwavespenetrateintothe
4
FIG. 3: (Color online) Temperature evolution of (a) real
µ(cid:48) (ω) and (b) imaginary µ(cid:48)(cid:48) (ω) parts of the retrieved per-
FIG. 2: (Color online) Real µ(cid:48)eff(ω) and imaginary µ(cid:48)e(cid:48)ff(ω) meeffability for the 3D metamaeffterial. Both the frequency band
partsoftheeffectivecomplexpermeabilityretrievedfromthe wheretheµ(cid:48) (ω)isnegativeandf shifttosmallervaluesof
eff r
experimentalscatteringmatrixat4.3Kfor(a)asinglespiral, frequency with increasing temperature due to the enhanced
(b) 3D array of such spirals. The dashed maroon and cyan inductance and losses.
curves are Lorentzian fits to the retrieved data according to
Eq.(1)demonstratingasignificantagreement. Theinsetsare
shorter frequency range plots of the data showing the details f and Γ are quite constant since there is no significant
r
in the vicinity of the resonant frequency. A single smooth
change in n (T) [48]. As temperature increases, n (T)
resonance peak of the individual spiral splits into multiple s s
starts to decrease leading to a weaker response in oppos-
resonances for the 3D metamaterial.
ing the external magnetic field. Beyond 8.3 K, we see a
slight decrease (increase) in f (Γ) until the temperature
r
is very near T . In the vicinity of T , we observe a steep
c c
metametarialdeeperwithincreasingtemperature,theto-
drop(increase)inf (Γ)mimickingthebehaviorofn (T)
r s
tal inductance of the individual spirals increases due to
(n (T),normalfluiddensity)inthistemperatureregime.
N
the enhanced geometrical and kinetic inductance, L ∝
k Here, λ (T) and thus the total inductance dramatically
L
λ (T) coth[t/λ (T)] where t is the thickness of the Nb
L L increases (i.e. L diverges) and accordingly n (T) ap-
k s
film used to make the metamaterial [46]. The enhanced
proaches zero. At T , ohmic losses are enhanced due to
c
total inductance leads to a reduction in the resonant fre-
thedominanceofn (T)overn (T)andresonantcharac-
N s
quency [47]. Moreover, transmission decreases in mag-
teristics are completely wiped out. To demonstrate this
nitude and is smeared as T is approached due to the
c moreclearlyandtotestthevalidityofourextraction,we
enhancedohmiclossesarisingfromnormalcarriers,char- show 1-[f (T)/f (T )]2 vs. Γ(T)/Γ(T ) in the inset
r r min min
acterized by Γ in Eq. (1).
of Fig. 4. Both quantities are roughly proportional to
The narrow frequency band in the real part of µ (ω) thenormalfluiddensityn (T)sotheplotshouldbeap-
eff N
possesses negative values up to ∼ 8.3 K beyond which it proximatelyastraightline. Thoughweseeanonlinearity
starts to show near zero and then positive values. This in the vicinity of the base temperature (due to the near
crossover temperature corresponds to a similar value at absence of a frequency shift), the trend is quite linear at
which we start to see a discernible downward trend in the higher temperatures, as expected.
the temperature evolution of f (T) and an upturn in
r
the damping term Γ(T). To study this behavior, we ex-
tracted ω (thus f ) and Γ as a function of temperature SUMMARY
r r
through fitting the retrieved µ (ω) of the 3D metama-
eff
terial according to Eq. (1). As seen in the main panel of Wehavedevelopedalow-lossandultrasub-wavelength
Fig. 4, for a long range of temperatures below T , both 3D superconducting metamaterial. The engineered com-
c
5
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