Parametric resonance in superconducting micron-scale waveguides
N. V. Fomin,a) O. L. Shalaev, and D. V. Shantsev
A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia
~Received 26 July 1996; accepted for publication 20 February 1997!
A parametric resonance due to temperature oscillations in superconducting micron-scale
waveguides is considered. Oscillations of superconductor temperature are assumed to be induced by
the irradiation of the waveguide with a laser beam. The laser power and parameters of the
waveguide providing a possibility of parametric excitation have been calculated. It is shown that for
a waveguide made of a YBa2Cu3O7 microstrip with resonant frequency of 10 GHz a laser with a
power of about 70 W/cm2 is needed to excite oscillations. The effect can be used for the creation of
high-sensitivity tuneable filters and optoelectric transformers on superconducting microstrips in the
GHz range. © 1997 American Institute of Physics. @S0021-8979~97!02411-0#
I. INTRODUCTION
Since the discovery of high-T c superconductors ~HTSC!,
much attention has been paid to both theoretical and experimental investigations of their application in microwave technology. The basic advantage of superconductors over conventional materials is that they provide ultralow-loss
performance in applications in microwave band. One of the
most important properties of a waveguide with a superconducting wire is the dependence of its characteristics on external parameters, mainly, on temperature and magnetic
field. This dependence appears since a change of external
parameters leads to a change in the concentration of superconducting electrons and, therefore, to a change of the impedance of the superconductor. Superconducting electrons,
moving along the wire without dissipation provide an inductive type of the waveguide impedance that is often referred to
as kinetic inductance. It adds to the ‘‘vacuum’’ waveguide
impedance and influences the wave propagation in the waveguide. Various effects and their possible applications based
on the existence of kinetic inductance have been discussed in
the literature,1 such as signal slowing down and increase of
the wave impedance in superconducting wires and
interconnections,2 a bolometer based on a superconductor in
the nonresistive state,3,4 and impact waves in superconducting waveguides.5
In the present work we report a new effect associated
with the kinetic inductance in superconductors: parametric
resonance in a superconducting micron-scale resonator. It is
known6 that in oscillating systems with periodically changing parameters, standing waves appear in some frequency
ranges. This is the so-called parametric resonance effect. In
our case the resonance system is a superconducting microstrip. Due to a contribution from the superconducting electrons, its resonant frequency depends on the temperature.
Therefore a periodic change of the superconductor temperature can lead to a parametric excitation of current in the
microstrip. In this paper we assume the modulation of temperature to be caused by a laser with an amplitude modulation. The conditions of parametric excitation and the values
a!
Electronic mail: fomin@theory.ioffe.rssi.ru
J. Appl. Phys. 81 (12), 15 June 1997
of the current, and the power dissipated when parametric
resonances appears, have been found.
The paper is organized as follows. In Sec. II the resonant
frequencies and Q factor of the system are determined. In
Sec. III the equations for oscillations in the microstrip with
periodically modulated temperature are deduced. In Sec. IV
the heat equation is solved and the spatial distribution of
temperature in the waveguide is calculated. In Sec. V the
results are analyzed and some estimates are given.
II. SUPERCONDUCTING MICRON-SCALE WAVEGUIDE
AS AN OSCILLATING SYSTEM
Due to extremely low dissipation in superconductors, a
piece of a micron-scale superconducting waveguide can
serve as a resonator with high Q-factor. Let us find out how
the basic characteristics of such an oscillating system, the
resonant frequency and the Q-value, depend on temperature
and geometry. First we introduce some computation for the
case of no temperature oscillations. Let us consider a micronscale waveguide, consisting of a superconducting microstrip,
an isolating buffer layer and a superconducting base ~see Fig.
1!. Propagation of electromagnetic waves in the waveguide
for the principal mode can be described by telegraph
equations.7 They connect the potential difference V between
the strip and the base and the current J in the strip:
C
]V
]J
52 ,
]t
]x
L ]J
]V
52 2
2E,
]x
c ]t
~1!
E5 u Es ~ d ! u 1 u Es ~ d1d buff! u .
Here Es and Eb are electric field strengths in the strip and the
base; C and L are the distributed capacity and inductance of
the waveguide satisfying the relation:
LC5 e ,
~2!
c is the speed of light, e is the dielectric permeability of the
buffer layer. Substituting running wave solutions J,V
}exp@i(v/n)x2ivt# ~x is the waveguide direction! into the
telegraph equations ~1! and introducing the impedance
Z( v ) of the waveguide at frequency v, one can obtain the
following expression for the signal velocity
0021-8979/97/81(12)/8091/6/$10.00
© 1997 American Institute of Physics
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E s ~ z ! 5E s ~ 0 ! cosh
SD
z
,
l
E b ~ z ! 5E b ~ d1d buff! exp
0,z,d,
S
D
d1d buff2z
,
l
~10!
d1d buff,z,`.
From ~6! and ~8! one can obtain the relation between the
current and the field and its derivative on the surface:
J5b
E
z2
z1
FIG. 1. Schematic of micron-scale superconducting waveguide.
v5
c
Ae 1 @ iCZ ~ v ! c 2 / v #
~3!
.
For a strip-like waveguide with the strip width b much
greater than the buffer thickness d buff ,
C5
be
.
4 p d buff
Z5Z s 1Z b ,
E s 5Z s J,
E b 5Z b J.
~5!
Let the z axis be directed along the normal to the base as
shown in Fig. 1 and let the strip occupy the space
0,z,d, the buffer layer d,z,d1d buff , and the superconducting base z.d1d buff . To obtain the formula for the signal propagation, one can use Maxwell equations
~ “3E! 52
1 ]H
,
c ]t
u “3Hu 5
4p
j,
c
~6!
j5 s E,
l 2 5const
~7!
~where l is the complex penetration depth!. For the harmonic dependance on time and neglecting terms of the order
of l v /c, we obtain an equation for the x-projection of the
field in the strip and the base:
] 2E~ z ! E~ z !
5 2 .
]z2
l
~8!
]E~ 0 !
50,
]z
u E ~ ` ! u ,`,
solutions of equation ~8! are given by:
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J. Appl. Phys., Vol. 81, No. 12, 15 June 1997
~9!
c2 ]E~ z !
4piv ]z
l 4piv
1
E s~ 0 !
52
,
J
b c 2 tanh ~ d/l !
Z b5
l 4piv
E b ~ d1d buff!
52
,
J
b c2
Z5Z s 1Z b 52
U
z2
,
z1
S
D
1
l 4li v
.
2
b c
tanh ~ d/l !
As an example of a resonator we consider a piece of superconducting microstrip with a length of L 0 . In this case the
boundary conditions for the current are
J u x50 5J u x5L 0 50.
Then, using relations ~3! and ~4!, the resonant frequencies of
the waveguide are given by
v 0 5n
p
p c
v 5n
L0
L 0 Ae
F
11
l
d buff
S
1
1
11
tanh ~ d/l !
D
1/2 ,
~11!
where n is an arbitrary integer. In case the base and the strip
have different complex penetration depths l s and l b , respectively, formula ~11! should be replaced by
v 0 5n
p
p c
v 5n
L0
L 0 Ae
F
11
l
d buff
S
1
1
1l b
tanh ~ d/l s !
D
1/2 .
Making use of this expression in the framework of twoliquid model one can estimate the Q-factor of the oscillating
system. Here we restrict our consideration to the case of a
zero applied magnetic field. In a nonzero field a contribution
exists to the impedance arising from the vortex lattice.8 The
summation of the conductivities of normal and superconducting electrons gives
1
1
1
5 2 ,
l 2 d 2s d 2n
Under the boundary conditions
z1
E ~ z ! dz52b
Z s5
and the London equation
c2
s 52
,
4pivl2
E
z2
where z 1 and z 2 set the limits of the area in question ~for the
strip z 1 50, z 2 5d, and for the base z 1 5d1d buff , z z 5`!.
From the formulas ~5! the following relations for the impedance can be obtained:
~4!
The distributed impedance of the waveguide Z is given by
the sum of the impedances of both superconductors composing the waveguide: the strip impedance Z s and the base impedance Z b . Both of them are coefficients of proportionality
between the current and the field on the surface.
j ~ z ! dz5b s
~12!
where d s is the London penetration depth in the superconductor, d n is the normal skin-layer which is determined by
the expression similar to that for s ~7!:
Fomin, Shalaev, and Shantsev
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s n5
c2
~13!
2,
4 pv d n
where s n is the conductivity of normal electrons. To estimate d n one may use the value of electric conductivity just
above the critical temperature. Then s n 5431015 s21 @the
value for YBa2Cu3O7 ~Ref.9!# and assuming the strip length
to be L 0 51 cm, we obtain v 51010 s21 and, from ~13!,
d n5
c
A4 pvs n
~14!
Here the dependence v~l! is determined by Eq. ~11!. In the
limits of a very thin d! d s and very thick d@ d s strip this
expression can be rewritten as
S
1
d
dd buff
12
1 2
2
2ds
ds
S
d buff
2ds
DS D
DS D
dn 2
,
ds
dn
,
ds
d! d s ,
It can be seen from ~10! that the resonant frequencies
depend on the value of the complex penetration depth l,
which in turn depends on the external conditions: temperature and magnetic field. This fact gives the key idea of the
parametric excitation of oscillations in the system. As shown
in the previous section, a resonance mode described by the
spatial dependence J(x,t)5J(t)sin(npx/L0) satisfies the
equation
~15!
Now let us deduce the equation similar to ~15! for the case
when l changes with time because of a change in external
conditions ~e.g. under modulated laser irradiation!. We will
restrict ourselves to the simplest case when l does not depend on the coordinate x along the waveguide and has only
two different values: l s on the inner surface of the strip and
l b on the surface of the base. The applicability of this assumption in the case of the periodical heating of the strip
surface is discussed in the following section ~see ~24!!. The
independence of l of x allows us to continue using the same
expression for the resonant frequency ~11!, while the independence l of z permits an explicit analytical solution of the
problem. It will be shown below that one can not simply
rewrite ~15! in the form
J. Appl. Phys., Vol. 81, No. 12, 15 June 1997
~16!
In order to calculate the electric field E we use speculations
similar to those used in the previous section for the calculation of impedances. Making use of the Maxwell and London
equations we have
] 2E 4 p ] j
5
]z2 c2 ]t
and
E5
S
D
] 4pl2 j
.
]t
c2
E s ~ z,t ! 5A s ~ t ! cosh
U
z
,
l~ t !
~17!
] J~ t ! c 2b ] E d
c 2b
d
5
5A s ~ t !
sinh
,
]t
4p ]z 0
4pl~ t !
l~ t !
] J~ t ! c 2b ] E
5
]t
4p ]z
III. PARAMETRIC EXCITATION OF OSCILLATIONS IN
RESONATOR
] 2J~ t !
1 $ v @ l ~ t !# % 2 J ~ t ! 50.
]t2
c2
Cc 2 ] E
50.
J1
e
e ]t
E b ~ z,t ! 5A b ~ t ! exp
d@ d s .
] 2J~ t !
1 @ v ~ l !# 2 J ~ t ! 50.
]t2
2
The solution of these equations is given by
1 Re~ v 0 !
1
Q52
.
52
2 Im~ v 0 !
] ln v 0 / ] ln l
Q thick5 11
S D
] 2J
np
1
]t2
L0
>1022 cm.
Since d s is of the order of 1025 – 1024 cm, it can be seen that
( d s / d n ) 2 !1, and the dominant contribution to the conductivity arises from the superconducting component. Keeping
only the terms of the order of ( d s / d n ) 2 , one can easily find
from Eq. ~12! the expression for the Q-factor of the system:
Q thin5
Excluding the voltage V from the telegraph equations ~1!,
one obtains
U
d1d buff2z
,
l~ t !
`
5A b ~ t !
d1d buff
~18!
c 2b
.
4pl~ t !
From formulas ~17! and ~18! we find the electric field E:
E ~ t ! 5E s ~ d,t ! 2E b ~ d1d buff ,t !
5
S
D
1
]J~ t !
4pl~ t !
.
11
2
c b
tanh @ d/l ~ t !#
]t
Substitution E in ~16! gives the equation for oscillations in
the system:
J1
S
D
]
1
]J
50,
2
] t $ v @ l ~ t !# % ] t
~19!
where v~l! is defined according to expression ~11!.
Let us consider the small harmonic modulation of parameters at the frequency close to twice the resonant frequency of the microstrip. One can write:
v 2 ~ t ! 5 v 20 @ 11h cos~ 2 v 0 1 d v ! t # ,
dv!v0 .
0,h!1,
Substituting the solution in the form of J'e st cos$@v0
1(dv/2) # t2 f % in ~19!, one obtains
cos 2f 522
dv
,
hv0
s56 21 A~ h v 0 /2! 2 d v 2 .
It is well known that if dissipation is taken into account, the
range of instability narrows. In our case the resonance will
take place in the range
2
AF S D
h
2
2
2
1
Q2
G
1/2
,
dv0
,
v0
AF S D
h
2
2
2
1
Q2
G
1/2
,
~20!
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where Q is determined by the formula ~14!. When the modulation of resonant frequency is induced by the modulation of
temperature, one can write:
U
] ln v 20
h5
]ls
U
ls5ds
U
] d ~Ls !
] ln v 20
d T s1
]T
]lb
U
lb5db
] d ~Lb !
d T b.
]T
~21!
IV. HEAT TRANSFER IN THE WAVEGUIDE
Let us suppose temperature to depend on the direction
z ~normal to the strip surface!. In the 1D case heat transfer
equation can be written down as
D5
k
,
cr
~22!
where k is a coefficient of thermal conductivity, c is a thermal capacity and r is density ~they can be different for the
strip, the buffer layer and the base!. This equation has been
solved with the boundary condition
]T
5q,
2k
]z
q5q 0 exp~ 2i g t ! ,
g 52 v 0 1 d v ,
~23!
which implies that the heat flow q is absorbed on the upper
surface of the strip. The above-mentioned assumption about
the spatially uniform distribution of temperature in the current carrying regions of the waveguide would be satisfied if
the following inequality holds:
ulu@ds .
~24!
Here, l is the diffusion length defined here as l5 AD/2i v 0
~the value of the square root should be chosen so that
Re@Ai # >0!. For simplicity, we consider the case when the
thermal conductivity of the buffer layer is too high and its
thickness is too small, so that one can treat buffer layer as a
thermoresistance. In other words, we consider the case when
d buff d
! ,
l buff l s
d T ~ z ! 5 @ a 1 exp~ z/l s ! 1b exp~ 2z/l s !# exp~ 2i g t ! ,
0,z,d,
z.d1d buff .
The boundary condition ~23! together with the requirement
U
5k
z5d
]T
]z
U
52
z5d1d buff
1
@ T ~ d ! 2T ~ d1d buff#
RT
~25!
on the continuity of the heat flow on the boundary of two
regions give
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J. Appl. Phys., Vol. 81, No. 12, 15 June 1997
SD
S DG
52
c bD b
b ,
lb 2
~26!
c sD s
2
~ a 1 2b 1 ! 5q 0 ,
ls
where the quantities with indices s and b correspond, respectively, to the strip and the base and R T is a thermoresistance
of the buffer layer. Solving the set of equations ~26! with
respect to a 1 , b 1 , and b 2 the spatial distribution of temperature can be found. To complete our calculations of the parametric resonance effect, the amplitudes of the temperature
oscillations at the inner surface of the strip and the base
surface are needed. They can be written as follows:
d T s5 d T~ d !
5qR s
S
D
R b 1R T
,
@ R s cosh ~ d/l s ! 1 ~ R b 1R T ! sinh ~ d/l b !#
d T b 5 d T ~ d1d buff!
5qR s
S
D
Rb
,
@ R cosh ~ d/l s ! 1 ~ R b 1R T ! sinh ~ d/l s !#
where parameters R s 5l s /c s D s , R b 5l b /c b D b were introduced. To estimate the amplitude of the temperature oscillations we use the value of laser power q520 W/cm2, and the
following values of parameters for high-T c superconductor
YBa2Cu3O7: k520 W/m K, c s 51.531024 J/~g K! ~taken
from Refs. 10 and 11!, v 0 51010 s21, and obtain l s 51.1
mm, d T(d)51.531022 K.
V. RESULTS AND DISCUSSION
Using the results of the previous sections we now determine the frequency band where the excitement of the system
will take place. To calculate h according to ~21!, one must
first determine the derivative ] l/ ] T. This can be done by
using the well-known approximation for d s near T c :
d s ~ T50 !
A12 ~ T/T c !
~27!
.
Substituting this approximation into equation ~21! gives
h5
US
DU
S DU
1 ] ln v 20
2 ] ln l s
1
d T ~ z ! 5 @ b 2 exp~ d1d buff2z ! /l b # exp~ 2i g t ! ,
]T
]z
F
d
c sD s
d
a 1 exp
2b 1 exp 2
ls
ls
ls
d s~ T ! 5
where l buff and l s are diffusion lengths in the strip and buffer
layer, respectively. Then, solutions of ~22! can be written in
the form
k
a 1 exp~ d/l s ! 1b 1 exp2 ~ d/l s ! 2b 2
RT
5
In the following section d T s and d T b are obtained for the
case of the periodical heating of the strip surface.
] T ~ z,t !
] 2 T ~ z,t !
1D
50,
]t
]z2
2
1 ] ln v 20
2 ] ln l b
dTs
ls5ds
T c 2T
dTb
lb5db
T c 2T
U
.
~28!
The parametric excitement of the current in the system will
take place in the frequency band g P(2 v 2 d v ,2v 1 d v ).
The width of this band is determined by formula ~20!. The
dependences of dv/v on d buff at various d and T are shown
in Figs. 2 and 3. It is seen that for easy excitement the buffer
layer thickness should be minimal. Then the volume occupied by the magnetic field would be maximal and the contribution from kinetic inductance would be dominant over that
Fomin, Shalaev, and Shantsev
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FIG. 2. The dependence of the width of frequency band where parametric
excitation can be observed on the thickness of the buffer layer. All calculations were carried out for a waveguide made of YBa2Cu3O7, for T577 K
and for different strip thickness: ~1! d 5 0.1 mm; ~2! d50.2 mm; ~3! d
50.3 mm.
from geometrical inductance. As a result, the depth of modulation increases. The effect also becomes stronger at T
'T c . This is related to the fact that as the transition approaches, the concentration of superconducting electrons decreases and, therefore, the influence of its modulation becomes more substantial. It should be noted that our
speculations are not applicable too far from the transition
region because the thermal conductivity of the superconductor is determined by normal electrons only. On the other
hand, they are not applicable in the vicinity of the superconducting transition, either, since in this case the amplitude of
the resonant frequency modulation h will not be small ~see
~28!!.
For simplicity, when making the calculations, we ignored variations in the direction of the strip width ~along y
axis!. In practice, the current density distribution is inhomogeneous in this direction: the current crowding at the edges
of the strip should occur. However, for the set of parameters
used in this paper, the London penetration depth is of the
order of the strip thickness. Therefore, the effect of inhomogeneous current distribution would lead to a change of our
estimates by a small numerical factor. In particular, the laser
power required to observe parametric oscillations would
slightly increase.
Let us now estimate the voltage, current density and
electric power in the excited system. It is assumed that the
oscillation growth stops when the dissipated Joule power becomes comparable to the laser beam power. Estimations will
be carried out in the effective cross section approach.12 This
means that the total current in the strip is J5 j(d)S * , where
j(d) is current density and S * is the effective cross section
where the current flows. The latter can be approximated as
J. Appl. Phys., Vol. 81, No. 12, 15 June 1997
FIG. 3. The dependence of the width of frequency band where parametric
excitation can be observed on the thickness of the buffer layer for a strip
thickness d50.3 mm and for different temperatures: ~1! T577 K; ~2! T
580 K; ~3! T583 K.
S * 'b d s (12e 2d/ d s ) which for thin strip (d! d s ) gives S *
'bd. Then the power dissipated in the unit area is Q' 21
s n u E u 2 d. Setting it equal to the heat flow q from the laser
beam one obtains u E u ' Aq/ s n d. Hence, u Iu ' u s s u • u Eu •S* .
For q565 W/cm2, s n 5531015 s21, d s (T577 K)53.7
31025 cm, e 510, d50.3 mm, d buff51 mm, b51022 cm,
we have E530 V/cm, j523108 A/cm2. Thus, one can see
that the obtained value of the current density in the excited
waveguide exceeds substantially its critical value. This
means that the excitation grows until the current density in
the superconductor reaches j c . The growth of the excitement
will proceed with the characteristic time 1.731027 s.
In conclusion, a new effect, the parametric resonance in
superconducting micron-scale waveguides due to the temperature oscillations, is considered. Its possible application is
a high-sensitivity tunable filter of modulated laser beam in
the GHz range. The adjustment of the resonant frequency can
be achieved by changing the temperature of the resonance
system resulting in a change of the penetration depth d s . The
high sensitivity of the filter is provided by a high Q-factor of
superconductor. Besides this, the considered effect can be
used for the creation of an optoelectronic transformer on
HTSC films. It should also be noted that instead of the laser
beam a source of magnetic field modulated at the same frequency can serve for excitation.
ACKNOWLEDGMENTS
This work is supported by the Russian National Program
on HTSC, Project No. 94048 and Grant No. 03229 ~Interconnection! of the Russian Ministry of Science. The authors
wish to thank R. A. Suris, A. P. Cawthorn, M. A. Zukerman,
Fomin, Shalaev, and Shantsev
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N. B. Mironkov, B. Y. Averboukh, N. M. Zukerman, A. S.
Polkovnikov, and K. R. Sofer for helpful discussions and
technical assistance.
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J. Appl. Phys., Vol. 81, No. 12, 15 June 1997
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Fomin, Shalaev, and Shantsev
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