Effects of anharmonicity of current-phase relation in Josephson junctions
The aim of this review is the analysis of dynamical properties of Josephson junctions (JJ) with anharmonic current-phase relation (CPR). Firstly, discussion of theoretical foundation of anharmonic CPR in different Josephson structures and their experimental observation are presented. The influence o...
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irk-123456789-1220562017-06-27T03:03:02Z Effects of anharmonicity of current-phase relation in Josephson junctions Askerzade, I.N. The aim of this review is the analysis of dynamical properties of Josephson junctions (JJ) with anharmonic current-phase relation (CPR). Firstly, discussion of theoretical foundation of anharmonic CPR in different Josephson structures and their experimental observation are presented. The influence of anisotropy and multiband effects on CPR of JJ are analyzed. We present recent theoretical study results of the anharmonic CPR influence on I–V curve, plasma frequency, and dynamics of long JJ. Results of study of Shapiro steps in I–V curve of anharmonic JJ are also presented. Finally, CPR anharmonicity effect on characteristics of JJ-based qubits is discussed. 2015 Article Effects of anharmonicity of current-phase relation in Josephson junctions (Review Article) / I.N. Askerzade // Физика низких температур. — 2015. — Т. 41, № 4. — С. 315-337. — Бібліогр.: 91 назв. — англ. 0132-6414 PACS: 74.50.+r, 03.67.-a, 85.25.Cp http://dspace.nbuv.gov.ua/handle/123456789/122056 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The aim of this review is the analysis of dynamical properties of Josephson junctions (JJ) with anharmonic current-phase relation (CPR). Firstly, discussion of theoretical foundation of anharmonic CPR in different Josephson structures and their experimental observation are presented. The influence of anisotropy and multiband effects on CPR of JJ are analyzed. We present recent theoretical study results of the anharmonic CPR influence on I–V curve, plasma frequency, and dynamics of long JJ. Results of study of Shapiro steps in I–V curve of anharmonic JJ are also presented. Finally, CPR anharmonicity effect on characteristics of JJ-based qubits is discussed. |
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Askerzade, I.N. Effects of anharmonicity of current-phase relation in Josephson junctions Физика низких температур |
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Askerzade, I.N. |
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Askerzade, I.N. |
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Effects of anharmonicity of current-phase relation in Josephson junctions |
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Effects of anharmonicity of current-phase relation in Josephson junctions |
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Effects of anharmonicity of current-phase relation in Josephson junctions |
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Effects of anharmonicity of current-phase relation in Josephson junctions |
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Effects of anharmonicity of current-phase relation in Josephson junctions |
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effects of anharmonicity of current-phase relation in josephson junctions |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Effects of anharmonicity of current-phase relation in Josephson junctions (Review Article) / I.N. Askerzade // Физика низких температур. — 2015. — Т. 41, № 4. — С. 315-337. — Бібліогр.: 91 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT askerzadein effectsofanharmonicityofcurrentphaserelationinjosephsonjunctions |
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2025-07-08T21:03:42Z |
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1837114194333270016 |
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© I.N. Askerzade, 2015
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4, pp. 315–337
Effects of anharmonicity of current-phase relation in
Josephson junctions
(Review Article)
I.N. Askerzade
Computer Engineering Department and Center of Excellence of Superconductivity Research of Turkey,
Ankara University, Ankara 06100, Turkey
Institute of Physics of Azerbaijan National Academy of Sciences, Baku Az-1143, Azerbaijan
E-mail: Iman.Askerzade@science.ankara.edu.tr
Received September 18, 2014, published online February 23, 2015
The aim of this review is the analysis of dynamical properties of Josephson junctions (JJ) with anharmonic
current-phase relation (CPR). Firstly, discussion of theoretical foundation of anharmonic CPR in different Jo-
sephson structures and their experimental observation are presented. The influence of anisotropy and multiband
effects on CPR of JJ are analyzed. We present recent theoretical study results of the anharmonic CPR influence
on I–V curve, plasma frequency, and dynamics of long JJ. Results of study of Shapiro steps in I–V curve of an-
harmonic JJ are also presented. Finally, CPR anharmonicity effect on characteristics of JJ-based qubits is dis-
cussed.
PACS: 74.50.+r Tunneling phenomena; Josephson effects;
03.67.-a Quantum information;
85.25.Cp Josephson devices;
Keywords: Josephson junction, two-band superconductors, qubits, current-phase relation, anharmonicity.
Contents
1. Introduction .......................................................................................................................................... 315
2. Influence of anisotropy and multiband effects of superconducting state on the CPR of JJ .................. 316
2.1. JJ based on d-wave superconductors ............................................................................................ 316
2.2. JJ between two-band superconductors ......................................................................................... 318
2.3. CPR relation for JJ structures with FM and AFM layers .............................................................. 319
2.4. Experimental results of CPR investigations in different Josephson structures ............................. 320
3. Influence of anharmonic effects of CPR on JJ dynamics .................................................................... 321
3.1. Anharmonic effects in I–V curve .................................................................................................. 321
3.2. Plasma frequency of JJ with anharmonic CPR ............................................................................. 325
3.3. Shapiro steps in I–V curve of JJ with anharmonic CPR ............................................................... 326
3.4. Inluence of anharmonic CPR on long JJ dynamics ...................................................................... 328
4. Qubits based on JJ with anharmonic CPR ............................................................................................ 329
4.1. Qubits ........................................................................................................................................... 329
4.2. Influence of anharmonic CPR on qubit characteristics ................................................................. 331
5. Conclusions .......................................................................................................................................... 334
References ................................................................................................................................................ 335
1. Introduction
The Josephson effect was discovered by Brian Josephson
[1]. The stationary Josephson effect was first observed exper-
imentally by Rowell [2], and the nonstationary Josephson
effect was observed by Yanson et al. [3]. Since that time,
there has been a continuously growing interest in the funda-
mental physics and applications of this effect. The achieve-
ments in Josephson-junction (JJ) technology have made it
possible to develop a variety of sensors for detecting ultralow
magnetic fields and weak electromagnetic radiation; they
have also enabled the fabrication, testing, and application of
ultrafast digital rapid single flux quantum (RSFQ) circuits as
well as the design of large-scale integrated circuits for signal
processing and general purpose computing [4,5].
I.N. Askerzade
316 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4
It is clear that the Josephson effect, the 50th anniver-
sary of which was celebrated in 2012, remains one of the
most spectacular manifestations of quantum mechanics in
all of experimental science. At its most fundamental level
the Josephson effect is nothing more than the electronic
analogue of interference phenomena in optical physics.
But from this humble premise springs a huge range of
physical phenomena and electronics applications which
placed Josephson devices at the heart of physics research
during the second half of the century of superconductivity
and beyond.
The Josephson effect may be observed in a variety of
structures. To realize such structures it is enough to fabri-
cate a “weak” place interrupting the supercurrent flow in a
superconductor or suppress the ability of a superconductor
to carry a current, e.g., by deposition of a normal metal on
its top, by implantation of impurities within a restricted
volume, or by changing the sample geometry. One main
characteristic of a JJ is the current-phase relation (CPR).
Only in few cases CPR reduces to classical sinusoidal form
with critical current cI [6,7]:
( ) = sin .S cI I (1)
Modern aspects of the supercurrent SI dependence on
the phase difference and the forms this dependence takes
in Josephson junctions of different types ( superconductor–
normal–superconductor, superconductor–insulator–super-
conductor, double barrier, superconductor–ferromagnet–
superconductor, superconductor–two-dimensional electron
gas–superconductor junctions, and superconductor–con-
striction–superconductor point contacts) were discussed in
[8,9]. CPR manifestations related to unconventional sym-
metry in the order parameters of a high-Tc superconductors
were also widely investigated during last years [10–13]. As
it follows from reviews [8–12] supercurrent SI dependence
on the phase difference can be presented in general as
1
( ) = ( sin cos ).S c n
n
I I n J n (2)
The shape of supercurrent ( )SI does not only depend on
temperature and the distance between electrodes, but also
on the critical temperature and transport parameters of both
superconductors and the interface layer in JJ structures.
Detailed analysis of CPR in different JJ structures was car-
ried out in [9]. The pairing symmetry in superconducting
state also strongly influences on CPR [11].
Simple sinusoidal form of CPR (1) was widely used to
study the dynamics and ultimate performance of analogous
and digital devices based on JJ up to recent time [4–14].
Above mentioned reviews [8–12] have been devoted to the-
oretical basis for the study of CPR in different Josephson
structures. Results of these studies reveal fundamental phys-
ical mechanisms for control and experimental investigation
of CPR. It is clear that modification of CPR in different JJ
structures leads to changing of dynamical properties of Jo-
sephson circuits. Several recent research papers (see below)
have been devoted to study dynamical effects in JJ with an-
harmonic CPR. Recent progress in the theoretical study and
experimental investigation of dynamical properties of such
junctions justifies an overview of the fundamentals of JJ
dynamics with anharmonic CPR.
The main emphasis of this review is the investigation of
CPR influence on dynamical properties of Josephson junc-
tions. Firstly, we will briefly discuss influence of anisotro-
py and multiband effects of the order parameter in super-
conducting electrodes on the shape of CPR of JJ. The
experimental investigations results concerning Josephson
structures with anharmonic CPR are also reviewed. In se-
cond section we present detailed results of study of anhar-
monic CPR influence on I–V curve and on the plasma fre-
quency of JJ. This section contains study of Shapiro steps
in anharmonic JJ. Properties of long JJ with anharmonic
CPR are also described in this section. Third section is
devoted to detailed investigation of anharmonicity effects
on characteristics of JJ qubits. Finally, conclusions are
presented.
2. Influence of anisotropy and multiband effects of
superconducting state on the CPR of JJ
2.1. JJ based on d-wave superconductors
For the calculation of Josephson current in such struc-
tures it is easy to use a Ginzburg–Landau theory of d-wave
superconducting state based on its symmetry properties.
Figure 1 presents schematic diagram of a JJ between pure
2 2x y
d superconductors. The gap states are assumed to
align with the crystalline axes, which are rotated on angles
L and R with respect to the junction normals Ln and
Rn on the left and right-hand sides, respectively. It is well
known that d-wave order parameter symmetry was ob-
served in cuprate superconductors [10–12]. In this sym-
metry, we use a complex order parameter, which behaves
the same way as the pair wave function [10]
( ) = cos cosx yk kk . (3)
The Ginzburg–Landau free-energy functional F for d-
wave superconductors has a form [15]
2 4 2 2
1= [ ( ) (| | | | )x yF dV A T K D D
2
2
1
| | ( 2 )] .
8
zK D B B H (4)
Fig. 1. Schematic diagram of a JJ between d-wave superconductors.
Effects of anharmonicity of current-phase relation in Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4 317
The real coefficients and iK are phenomenological
parameters, and ( ) = ( )cA T T T changes its sign at the
superconducting transition temperature Tc. The symbols
D denote the components of the gauge-invariant gradient
D 0[ (2 / )]iA , where A is the vector potential
( = ).B A For the calculation of the Josephson cur-
rent, it is useful to introduce the coupling between the
order parameters of two linked superconductors (Fig.1).
Coupling between superconductors can be expressed
through addition of term
coup 1 1 2 2 1 2 2 1= ( ) ( ){ }F t dS n n , (5)
where t is a real parameter denoting the coupling strength.
The functions ( )j jn are symmetry functions of the inter-
face normal vector jn in the crystal basis of the side j.
For the current density perpendicular to the interface we
can get
1 1 2 2 1 2
0
4
( ) = ( ) ( ) sin .
ct
I n n (6)
For the d-wave superconductors usage of symmetry func-
tion as
2 2
1 1( ) = x yn nn [16] leads to final Sigrist–Rice
result for clean JJ
( ) = cos(2 )cos(2 )sins L RI A , (7)
where sA is a constant characteristic of the junction [12].
For the dirty limit of JJ the relation is:
( ) = cos2( )sin .s L RI A (8)
Basing a Green's function method the Josephson current
in a d-wave superconductor/insulator/d-wave superconduc-
tor ( / / )d I d junction is calculated taking into account the
anisotropy of the pair potentials explicitly [17,18]
____________________________________________________
/2
1 1
, ,/2
( , , ) ( , , )
( ) = | ( ) | | ( ) | cos ,N n n
N L L
L L
n
R kT a i a i
R I d
e
(9)
where
2 2
, , = ( ) .n L L n The quantity NR denotes the normal resistance and NR is expressed as
/2 2
1 0
2 22
0 0/2
4
= cos ; = ,
(1 ) ( ) 4 cosh( )sinh
N N N
i i
Z
R d
Z d Z d
(10)
_______________________________________________
2 2
0 0
2 2
cos
= 1 ; Z =cos
1 cos
. (11)
Here, N denotes the tunneling conductance for the in-
jected quasiparticle when the junction is in the normal
state. The quantity = 2 ( 1/2)n kT n denotes the Mat-
subara frequency. The Andreev reflection [19] coefficient
1( , , )na i is obtained by solving the Bogoliubov equa-
tion [21], and 1( , , )na i is obtained substituting ,
– ,L and – R for , ,L and R into 1( , , ),na i
respectively. If we take only the = 0 component, the
magnitude of the Josephson current is proportional to
cos(2 )cos(2 ), and the phenomenological theory by
Sigrist and Rice [16] is reproduced.
In general, supercurrent in JJ with d-wave superconduc-
tors ( )I can be decomposed into the series of sin ( )n
and cos( )n using above presented Eq. (2). This equation
includes the Josephson current component carried by the
multiple Andreev reflection processes at the interface. In
the above equation, the current components with index n
correspond to the amplitudes of the nths reflection pro-
cesses of quasiparticles. For ~ 0,N supercurrent ( )I is
proportional to sin ( ) and the classical results of
Ambreokar–Baratoff theory [22] are reproduced, while for
= 1,N above Eqs. (5)–(11) reproduce the previous re-
sults of Kulik and Omel’yanchuk theory [23,24]. On the
other hand, for a fixed phase difference between two su-
perconductors, the component of the Josephson critical
current becomes either positive or negative depending on
the injection angle of the quasiparticle (as it follows from
Tanaka's analysis [17,19]). In some situations, the phase
difference 0 , which gives the free energy minima, is lo-
cated at neither zero nor . When the crystal axis is tilted
from the interface normal, zero-energy states i.e. midgap
states, are formed near the interface depending on the an-
gle of the crystal axis and the injection angle of the
quasiparticle. This effect leads to enhancement of the Jo-
sephson current at low temperatures.
Negative Josephson coupling was first noted by Kulik
many years ago [3]. He discussed the spin-flip tunneling
through an insulator with magnetic impurities. Late
Bulaevskii et al. [20] proved that under some conditions
such spin-flip tunneling prevails over the direct tunneling
and leads to a -junction. The junction energy achieves
minimum at the phase difference , and a spontaneous
supercurrent may appear in a circuit containing the junc-
tion. Two possible directions of the supercurrent reflect the
doubly degenerated ground state. In contrast to the usual
junction such a state is achieved without application of an
external field.
I.N. Askerzade
318 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4
2.2. JJ between two-band superconductors
Multiband superconductivity became a hot topic in
condensed matter physics in 2001, when the two-band su-
perconductivity in MgB2 with anomalous high = 39KcT
was discovered [25]. It is striking that the pairing mecha-
nism had electron-phonon origin in magnesium diboride
and that order parameters, which are attributed to super-
conducting energy gaps, have s-wave symmetry. Iron-
based superconductors, which have been discovered not
long ago, and nonmagnetic borocarbides [15,26] can be
classified as multiband systems. In this section the station-
ary Josephson effect in SCS (superconductor–constriction–
superconductor) junction is presented. The behavior of such
junctions even in the case of one-band superconductors, as
revealed in [23,24], has the qualitative differences compar-
ing to SIS (superconductor–insulator–superconductor) tun-
nel junctions. The microscopic theory of the “dirty” SCS
junction for two-band superconductors is built, which gener-
alizes the Kulik–Omel’yanchouk theory in this case [27].
The case of dirty two-band superconductor with strong intra-
band scattering rates by impurities (dirty limit) and weak
interband scattering is investigated [27]. In the dirty limit
superconductor is described by the Usadel equations for
normal and anomalous Green's functions g and f, which for
two-band superconductor take the form presented in [27]
(Fig. 2). Calculation of the Josephson current between two-
band superconductors in the absence of inter-band scatter-
ing leads to:
____________________________________________________
1 1
2 2 2 22 21 >0 1 1
2 2
2 2 2 22 22 >0 2 2
cos ( /2) sin ( /2)4
( ) = arctan
( /2) ( /2)cos cos
cos ( /2) sin ( /2)4
arctan .
( /2) ( /2)cos cos
N
N
T
I
eR
T
eR
(12)
_______________________________________________
As it follows from Eq. (12) current flows independently
from the first band to the first one and from the second
band to the second one. This equation is a straightforward
generalization of Ambreokar-Baratoff results for one-band
superconductor [22]. Introducing the total resistance
1 2 1 2= /( )N N N N NR R R R R and normalizing the current
on the value 0 = (2 / )N cI eR T the current-phase relations
for different values of 1 2= /N Nr R R and temperature T
are plotted in Fig. 3(a, b). Results of calculation of critical
current temperature dependence for two-band based JJ are
presented in Fig. 4. For the calculation of ( )I and ( )cI T
the parameters for two-band superconductor MgB2 without
inter-band interaction [15,27] were used. The deformation
of ( )I curve depending on different parameters of JJ
based on two-band superconductors is clear.
Using perturbation theory in the first approximation for
Green’s functions in each band for the case of nonzero
interband scattering, the corrections to the current (12)
1 2=I I I were obtained [27]:
____________________________________________________
2
12 2 1 1
1
2 2 3 2 2 2 22 21
1 2 1
2
2 1
2 2 2 2 2 22
1 1 2
2 ( e )cos ( /2) sin ( /2)
= arctan
( ( /2)) ( /2)cos cos
( e )sin1
,
2 ( )( ( /2))cos
i
N
i
T
I
eR
(13)
2
12 1 2 2
2
2 2 3 2 2 2 22 22
1 2 2
2
1 2
2 2 2 2 2 22
2 2 1
2 ( e )cos ( /2) sin ( /2)
= arctan
( ( /2)) ( /2)cos cos
( e )sin1
.
2 ( )( ( /2))cos
i
N
i
T
I
eR
(14)
Fig. 2. Schematic diagram of a Josephson junction between two-
band superconductors.
Effects of anharmonicity of current-phase relation in Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4 319
where 12 is parameter of the interband scattering, and is
a phase shift. When the interband scattering is taken into
account and the phase shift 0, the phases of Green’s
functions d0 not coincide with phases of order parameters
.i From the above-mentioned discussion it follows that
CPR of JJ based on two-band superconductors also devi-
ates from simple sinusoidal form (1) (see Fig. 3).
2.3. CPR relation of JJ structures with FM and AFM
layers
In this part we pay attention to the approach based on
the Usadel equation and consider the S/F/S junction with F
layer of thickness 2df (Fig. 5). The following formula for
the supercurrent was used in [8,28]
( ) = (0) f
dF dF
I ieN D TS F F
dx dx
, (15)
where anomalous Green’s function ( )F x depends only on
one coordinate x, and this function meets following condi-
tions:
/2
*
2 2
e
( , ) ( , – ) : ( ) = ,
i
s fF x h F x h F d
/2
2 2
e
( ) =s fF d ,
S is the junction cross-section area, and (0)N is the elec-
tron density of state for one-spin projection. The last ex-
pression gives the usual sinusoidal current-phase depend-
ence with the critical current [28]:
2
2 2 2
2 / cosh (2 )
= (0) ,
tanh (2 ) (1 ) 2
f
c f
f
k kd
I eN D TS
kd k k
(16)
where = / .B N sG Generalization of the presented
theory on the case of the different interface transparencies
is presented in [9,29]. It gives:
0 cos2 sinh(2 ) sin 2 cosh(2 )
= 4
cosh(4 ) cos4
c
N
V y y y y
I y
R y y
, (17)
where = ( / ) /(2 ),F F cy d H T H is the exchange energy
in F layer. The S/F/S junctions reveal the nonmonotonic
behavior of the critical current as a function of the F layer
thickness. Vanishing of the critical current signals the tran-
sition from the state 0 to the state . It occurs at
2 = 2.36cy which is exactly the critical value of the F
layer thickness in the S/F/S multilayer system correspond-
ing to the 0– -state transition (Fig. 6) [8,9].
CPR in different structures such as SFcFS and double-
barrier SIFIS are presented in [30], where the non-
monotonic temperature dependence of the critical current
is analyzed. Deformation of CPR of double-barrier SIFIS
junctions for different exchange integrals in F layer is pre-
sented in Fig. 7. Similar change in CPR was experimental-
ly observed in [31]. One of the interesting properties of
SFS systems is the rotation of the magnetization vector of
F layer under action of an external magnetic field [32–34].
Details of CPR of different Josephson structures with F
Fig. 3. CPR of MgB2/MgB2 junction for different temperatures T:
0(1); 0.5Tc(2); 0.9Tc(3) and ratios of resistances r = RN1/RN2.
Fig. 4. Temperature dependencies of critical current Ic for differ-
ent values of r = RN1/RN2: 0.1(1); 1(2); 10(3).
Fig. 5. Schematic description of S/F/S junction.
I.N. Askerzade
320 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4
layer were presented in excellent reviews [8,9]. The study
of the CPR is also important for understanding the funda-
mental properties of superconducting materials, such as
symmetry of the superconducting correlation and peculiari-
ties of the spin transport in multilayer systems based on
superconducting and ferromagnetic materials.
Despite wide discussion about JJ with FM layers, there
are few papers concerning investigations of structures with
AFM layers. Firstly such structure was studied by Gor’kov
and Kresin [35]. They found that the critical current
strongly depends on external magnetic field. The analytical
expression can be written as
0
2
( ) = cos
4
c s c s
s
I M I M
M
, (18)
where 1 is related to characteristic of AFM layer,
0 < < 1sM is parameter of AFM ordering, 0cI is critical
current in the absence of external magnetic field coinciding
with corresponding critical current in SNS junctions. It is
useful to note the importance of study of JJ with magneti-
cally ordered layers. As mentioned in [5], such Josephson
structures may allow substantial savings in the Josephson
circuit area.
2.4. Experimental results of CPR investigations
in different Josephson structures
As it follows from above presented theoretical review,
the general case CPR in Josephson structures is determined
by the types of JJ. At high temperatures ( << )c cT T T de-
viation of CPR ( )I from sin law is negligible for any
type of JJ. At low temperatures ( << )cT T the relation
( ) sinI takes place for SIS junctions [36]. In early in-
vestigations [36], the high accuracy realization of sinusoidal
character of CPR was shown using plasma resonance tech-
nique. Recently, Gronbech-Jensen et al. [37] studied the
dynamics of the tunnel JJ simultaneously carrying dc and ac
currents by measuring the statistics of switching of low-
temperature Nb–NbAlOx–Nb type tunnel junctions to the
resistive state. The critical current statistics in this system,
which is controlled by thermal fluctuations at the bottom of
the potential well ( ) = ( 1 cos ),JU E i was deter-
mined for 10000 events. By changing the amplitude of ac
current, it was possible to control the dc current correspond-
ing to a peak in the switching events distribution.
A new method for CPR measurement and some of its
practical applications were presented in [38,39]. Most
commonly for the experimental investigation of the CPR,
the weak link of interest is incorporated in a superconduct-
ing ring with a sufficiently small inductance L. This circuit
is usually called a single-junction interferometer [4]. Under
limitations / << 1,NL R and
2 << 1LC the supercon-
ducting part ( ) = ( )s cI I f of the current exceeds essen-
tially all other components, so the following equation is
valid for single junction interferometer [38,39]:
= ( ),e lf (19)
where l is the normalized inductance, 02 / .cl LI There
is a more precise method to determine the CPR using radio
frequency (rf) technique. It was proposed many years ago
[40,41]. Further development of this method was presented
Fig. 6. Critical current of S/F/S junctions versus =y
/ /(2 ),F F cd H T where H is the exchange energy.
Fig. 7. Deformation of CPR of SIFIS junction for different ex-
change energies.
Fig. 8. Experimental CPR of symmetric /4 grain-boundary
junction.
Effects of anharmonicity of current-phase relation in Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4 321
by Ii’ichev et al. in [42]. They have shown that the CPR
and the phase-dependent conductance can be extracted
from experimental data. Results of measurements of Jo-
sephson current through a junction as a function of the
phase difference in symmetric 45 grain-boundary high-
Tc junction are presented in Fig. 8 [43]. Recent achieve-
ments in fabrication of the JJ based on high-Tc superconduc-
tors were described in [44]. Measurements reveal that
YBCO-based grain boundary tunnel junctions fabricated in
[44] are highly hysteretic and Fig. 9 shows the ratio of coef-
ficients 1I and 2I determined by a Fourier analysis of the
CPR at various temperatures. With decreasing T, value of
2I grows monotonically down to = 4.2K,T while the 1I
component exhibits only a weak temperature dependence
[44]. Very recent review of physical properties of JJ based
on high-Tc superconductor was presented in [45]. Further-
more anharmonic CPR in graphene JJ was reported recently
in experimental research [46] and corresponding theoretical
calculations were proposed in [47].
Very recently, topological insulators attached to super-
conductors have attracted great interest of researchers. The
topological insulator offers a new state of matter, which is
topologically different from the conventional band insula-
tor [48–50]. When SF junctions are deposited on a topo-
logical insulator, surface Dirac fermions gain a domain
wall structure of the mass. The CPR shows 4 periodicity,
i.e., the shape of supercurrent has a form of sin ( /2)
[51,52]. JJ in hybrid superconductor-topological insulator
devices revealing two peculiarities was reported in [53].
c NI R products for this structures is inversely proportional
to the width of the junction. Another property is related to
a low characteristic magnetic field needed for suppression
of supercurrent, i.e., Fraunhofer capture is different from
traditional dependence ( )cI H [4]. The shape of CPR for
such a junction is presented in Fig. 10. Detailed analysis of
the superconductor-topological insulator junctions is the
subject of future investigations.
3. Influence of anharmonic effects of CPR
on JJ dynamics
3.1. Anharmonic effects in I–V curve
For a long period, the real shape of the CPR was not
considered as an important factor affecting dynamical
properties of JJ. Tunnel Josephson junction of SIS struc-
ture reveals ( ) = sin ,SI which was observed experimen-
tally with high precision for such junctions (see above).
The shape of the CPR, or more explicitly the energy and
phase dependencies of spectral current density have be-
come important parameters in the analysis of the dynamic
properties of Josephson junction circuits. Small deviation
from harmonic case does not essentially affect the response
of the junctions on a steady magnetic field and may be
taken into account in the circuit design as an additional
intrinsic inductance (see [54]), which must be added to the
geometrical one in the circuit simulation. In this section,
we study I–V curve of JJ with considerably anharmonic
CPR using relation = (sin sin 2 ).J cI I The resistive-
ly, capacitively, inductively shunted Josephson junction
(RCLSJ) circuit shown in Fig. 11 is shunted by a small
external resistor isatisting s nR R [4,55,56]. Here nR
and sR denote the normal state and shunt resistances, re-
spectively. As shown in the Fig. 11 the Josephson tunnel
junction is replaced by three parallel current channels. The
total current through JJ is represented as a sum of the
supercurrent ( ),I the displacement current =DI
( / ),C dV dt and the normal current due to quasiparticles
Fig. 9. YBCO based grain boundary junction: I2/I1 versus tem-
perature.
.
Fig. 10. CPR of superconductor-topological insulator junction.
Fig. 11. Circuit model of RCLSJ.
I.N. Askerzade
322 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4
= / ( ).NI V R V The voltage-dependent junction resistance
[4,55] is assumed to be:
sg
if | | > ,
( ) =
if | | .
n g
g
R V V
R V
R V V
(20)
where = 2 /gV e is the gap voltage that depends on the
energy gap (i.e., ) of superconductor, nR is the normal
state resistance and sgR is the sub-gap resistance of the JJ
in the superconducting state. The applied bias current dcI
is carried by the sum of the listed components ( , , ).J N SI I I
= (sin sin 2 )J cI I , = / ( )NI V R V , = ( / ).DI C dV dt
= D N J sI I I I I where sI denotes the current in the
shunt branch. For simplicity, we ignore the effect of ther-
mal noise throughout this study fluc( = 0).I
Equations that correspond to the circuit in Fig. 11 in
dimensionless form are
=
d
d
v ,
( ) ( ) = ,C s
d
g f i i
d
v
v v (21)
= ,s
L s
di
i
d
v
where 2= (2 / )C c se I CR is the McCumber capacitance
parameter; ( ) = / ( )sg R Rv v is the normalized tunnel
junction conductance; = /s s ci I I is dimensionless shunt
current; = / ci I I is dimensionless external dc bias current;
= (2 / )L c se I L is the dimensionless inductance; = ct
is the normalized time, = (2 / )c ce V is the characteristic
frequency, and =c c sV I R is the characteristic voltage.
The relationship between C and c can be written as
2= ( / (0))C c p where plasma frequency (0) =p
2 / .ceI C
The solutions of Eqs. (21) are numerically obtained using
MATLAB routine based on adaptive Runge–Kutta method
[57]. The time-averaged voltage for the determination of I–V
curve can be evaluated using the expression:
0
1
= / = ( )
rng
rng
d d dv v , (22)
where rng is the sampling range. Note that rng in
Eq. (22) is taken much longer than the period of Josephson
oscillations as well as relaxation oscillations. For that rea-
son, the time-averaged voltage in Eq. (22) is sometimes
called in literature as long-time averaged voltage.
In order to study the influence of second harmonic on
the dynamics of the JJ, we firstly evaluated the critical cur-
rent related to the amplitude of both harmonics. In such
way, the normalized critical current can be found as an
extremum of the function ( ) (sin sin 2 )f
0
/ = max( ( ))c cI I f , (23)
where
0c
I is the critical current at = 0. The normalized
critical current with respect to anharmonicity parameter is
plotted in Fig. 12. As shown in the figure, cI is nonlinear
for small , whereas it is linear for large values. Moreo-
ver, the linear dependence of critical current cI was exper-
imentally observed at large in YBCO-based JJ [43]. In
addition, a similar plot in Fig. 12 was obtained using analyt-
ical expression for critical current given in [58].
Two types of dynamics of RCLSJ circuit presented in
Fig. 13 can be explained using load-line analysis associat-
ed with I–V curve of the JJ. The first case is shown in
Fig. 13(a) and corresponds to relaxation oscillations in the
circuit with parameters =1.1i ,
0
= 1.11C ,
0
= 21.7L at
= 0. Relaxation oscillations in Josephson circuits have
been studied by many authors [55] and [59]. A similar re-
laxation generator was used to study the dynamical proper-
ties of tunnel JJ comparator in [60]. The second regime
corresponds to the regular ac Josephson oscillations [55]
and it is shown in Fig. 13(b) with parameters =1.1i ,
0
= 2.22C ,
0
= 43.4L , at = 0 . For nonzero anhar-
monicity parameter such as = 0.4 (see Fig. 13(c)), the
amplitude of the Josephson oscillations becomes smaller,
which is related to the effective capacitive properties of JJ.
Such situations can be explained by the increase of the
critical current of JJ with anharmonic CPR (the detailed
discussion is given below).
Fig. 12. (Color online) Normalized critical current versus
anharmonicity parameter .
Effects of anharmonicity of current-phase relation in Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4 323
In paper [57] numerical analysis of I–V characteristics
for different values was performed as shown in Fig. 14.
In the figure, we have plotted I–V characteristics of the
system with different
0C and
0L values using the same
,sR sg ,R and nR from Table 1, similar [55]. Note that
0C and
0L refer to the harmonic case of the CPR (i.e.,
= 0). Similar results were obtained in [61] for = 0.2
case only due to the limited nature of the analytical calcu-
lations. Furthermore, it is difficult to study directly the
details of the dynamics both experimentally and analytical-
ly, therefore, we can rely on numerical solutions of
Eqs. (21) to study the influence of anharmonicity parame-
ter . As can be seen from Fig.14, the width of the hystere-
sis in the I–V curve becomes larger with an increase of
anharmonicity parameter . Consequently, the presence of
anharmonic CPR impacts the inertial properties of the JJ as
an undesirable effect. In addition, we repeat few simula-
tions using opposite sign of the anharmonicity parameter
in relation to our calculations presented here. We observed
that the hysteresis of the I–V characteristics decreases
compared to its counter part in the presented plots. In gen-
eral case, the sign of is determined by the physical prop-
erties of barrier layer in Josephson structure [10,20,58].
The size of hysteresis in I–V curve is characterized by
the return current, at which the JJ switches from R-branch
to S-branch in the I–V curve. The relationship between the
return current and high values of McCumber parameter
C can be obtained using simple resistive model [4]:
4 1
= .R
c C
I
I
(24)
If we consider Eq. (20), the return current vs
McCumber parameter qualitatively reveals a similar be-
havior [4]. The deviation from the expression in Eq. (24)
becomes larger when the ratio of sg / nR R increases. On the
other hand, the return current RI is not only a function of
Table 1. Fabrication parameters of Josephson junctions (see [55])
T, K Ic0, mA Vg, mV Rsg, Rn, R,
4.22 0.550 2.91 50 3 1.1
7.60 0.275 2.09 15 3 1.1
Fig. 13. (Color online) Time dependence of dynamical variables:
voltage d /d and current I through shunt branch. Computational
parameters are: i = 1.1, C0 = 1.11, L0 = 21.7, = 0 (a); i = 1.1,
C0 = 2.22, L0 = 43.4, = 0 (b); i = 1.336, C0 = 2.696, L0 =
= 52.701, = 0.4 (c).
Fig. 14. (Color online) I–V curves for various values at C0 =
10; L0 = 1 (a); L0 = 10(b); L0 = 30 (c).
I.N. Askerzade
324 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4
C but also a function of and L . However, it is diffi-
cult to obtain an explicit analytical expression for it. For
this reason, numerical simulations are performed to ana-
lyze the influence of anharmonicity parameter and di-
mensionless inductance L on the normalized return cur-
rent /R cI I at two different values of McCumber parameter
(e.g.,
0
= 5C and
0
=10).C
First of all, we will discuss the relationship between
/R cI I and which is shown in Fig. 15 for various
0L
values. In general, the influence of the capacitance and
inductance of the JJ on the junction impedance is of oppo-
site character. That is why one reactive element will damp
the influence of the other. This implies that corresponding
I–V curve and associated hysteresis will be determined by
the resulting impedance. At fixed
0L , the value of the
Josephson inductance = ( /2 )/c cL e I decreases with
increasing. As a result, the influence of the junction capaci-
tance at nonzero on the I–V curve becomes dominant
compared to harmonic case. If we compare the plots here
with the results in [61], the normalized return current
/ ,R cI I is defined here accurately in contrast to the result
therein. The reason is that the author in introduced [61] an
approximate solution using analytical approach.
The exact value of return current is sensitive to the
characteristics of the junction in the subgap region. Usually
a switching from R-state to S-state leads to an exponential
decay of the voltage transient waveform. As mentioned in
[4], the voltage transient from R-state to S-state is accom-
panied by a slowly damped plasma oscillation. For this
reason, we observed inaccuracies at some points on the
curves presented in Fig. 15. The accuracy of our calcula-
tions for the return current in Fig. 15 was roughly estimat-
ed as 5%.
The relationship between return current /R cI I and di-
mensionless inductance L is illustrated in Fig. 16 for var-
ious . The junction circuit shown in Fig. 11 is shunted by
a serially connected inductance sL and shunt resistor sR .
This means that the junction is shunted by impedance
2
= 1 .s s
s s
Z L
R R
(25)
At vanishing shunt inductance (i.e., sg ),s sL R R
we come to standard resistively shunted junction (RSJ)
Fig 15. (Color online) Return current dependence on anhar-
monicity parameter , at C0= 5 (a); C0 = 10 (b).
Fig. 16. (Color online) Return current dependence on dimension-
less inductance L0 at C0= 5 (a); C0 = 10 (b).
Effects of anharmonicity of current-phase relation in Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4 325
model with hysteresis on I–V curve controlled by
McCumber parameter 2= (2 / ) .C c se I CR With increasing
inductance sL the impedance sZ also increases. On the
other hand, if the shunting effect is in the high shunting
inductance limit (i.e., )s sL R the impedance sZ ap-
proaches .sL In this case, the impedance sZ becomes
much greater than sgR and nR (i.e., sg ).s nZ R R As
a result, McCumber parameter C can be determined by
sub-gap resistance sg :R 2
sg sg( ) = (2 / ) .C cR e I CR Due to
that reason, sg( )C R will become greater than .C The
return current /R cI I approximately reaches the constant
values lower than corresponding value for .s sL R It is
shown in Fig. 16, that the crossover from one regime to
another reveals a peak character. Furthermore, as it follows
from Fig. 16, peaks width increases for large McCumber
parameter .C The inaccurate behavior similar to that in
Fig. 16 is also observed on the curves in Fig. 15 due to the
transient plasma oscillation in R → S switching.
3.2. Plasma frequency of JJ with anharmonic CPR
It is useful to discuss the influence of CPR anhar-
monicity on small perturbations of the S state of JJ, i.e., the
possible phase motion in the vicinity of equilibrium state
0. It is known, that JJ dynamics has much in common
with the motion of a particle in potential of the washboard
type [4]
( ) = (cos 1)jU E i , (26)
where i is the dc current expressed in the cI (critical cur-
rent) units, is the Josephson phase, and = /2j cE I e is
the Josephson energy. If the capacitance of the junction is
sufficiently large, the junction may exhibit slowly decay-
ing oscillations of the plasma phase at the bottom of the
potential well (26). The frequency of these oscillations
(plasma frequency) depends on the dc current and is given
by formula (see, e.g., [4]),
1/2
2 1/42
= (1 )c
p
eI
i
C
. (27)
Relation (27) is usually satisfied for the JJ connected to a
dc voltage source. If = sin ,cI I the theory exhibits per-
fect agreement with experiment [36]. However, there are
deviations from the behavior predicted by Eq. (27) in the
JJ characterized by anharmonic CPR. The anharmonicity
may be caused by the simultaneous passage of both dc
and ac currents of large amplitudes via the junction. Re-
cently, Gronbech-Jensen et al. [37] studied the statistics
of S → R switching of low-temperature tunnel JJ of the
Nb–NbAlOx–Nb type. The critical current in this system
was determined for 10000 events.
The obtained results confirms the validity of relation
(27) for the plasma frequency for small amplitudes of the
ac current component. However, as the ac current ampli-
tude grows, the agreement of formula (27) with the exper-
imental values measured deteriorates, which can be related
to the anharmonic character of the potential
( ) = (cos cos2 1)jU E i at large ac current
amplitudes. The theoretical investigation of the alternating
current effect on the plasma frequency of the tunnel JJ
simultaneously carrying dc and ac currents is presented
below. The dynamics of a JJ can be described using the
following equation (in this equation we use time units
= (0) ):p t
1
sin sin 2 = sind di t . (28)
For the calculation we will use 0 1= , where 1
obeys equation
1 = sind di i t . (29)
Using mathematical expressions described in [62]
2cos ( sin ) = exp( ); = ( )n n ka t A in t A J a , (30)
2 1sin ( sin ) = exp( ); = ( )n n ka t B in t B J a (31)
we obtain the following expression for plasma frequency
of JJ with anharmonic CPR
2
pl
0 0 02
pl
( )
= cos 2 ( )cos 2
(0)
a
J a . (32)
In the last expression 0 ( )J a is the Bessel function of
zero order, (0) = 2 / ,p ceI C 2 2= /( ).d da i Equi-
librium value 0 is determined from relation
0
0 0
0
= sin sin 2
( )
i
J a
. (33)
According to Eq. (32), an increase in the ac current compo-
nent di results in decrease in the plasma frequency .p
Thus, the presence of the ac component leads to renormaliza-
tion of the plasma frequency (27) of the tunnel JJ.
The results of calculations according to Eqs. (32) and
(33) are presented in the Fig. 17 by the solid and dashed
curves, respectively, and compared with the experimental
data (black circles) taken from [37]. Result of calculations of
2 2
pl pl( )/ (0) as function of anharmonicity parameter is
presented in Fig. 17. Nonsymmetric character of
2
pl ( ) is
clear from calculations for different values of a. There is a
minimum of
2 2
pl pl( )/ (0) 0.794 at negative = 0.3
[63]. At positive values of anharmonicity parameter plas-
ma frequency
2
pl ( ) decreases with increasing amplitude
of oscillating part .di At negative values of anharmonicity
parameter influence of a on the plasma frequency
2
pl ( )
is very small.
Results of calculations of
2
pl ( )a at different
anharmonicity parameter are presented in Fig. 18. At
small , the change in plasma frequency
2
pl ( )a is negli-
I.N. Askerzade
326 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4
gible, while at high the influence of anharmonicity is
important. At negative influence of anharmonicity of
CPR on plasma frequency is decreased.
3.3. Shapiro steps in I–V curve of JJ with anharmonic CPR
When radio-frequency signal is applied to Josephson
junction, its I–V curve shows a set of Shapiro steps result-
ing from phaselocking of Josephson oscillations [64]. Ana-
lytical description of the Shapiro step dependence on the
signal amplitude was obtained only for a high-frequency
limit in the frame of RSJ model describing an overdamped
junction with McCumber parameter << 1C [4]. In par-
ticular, a nonsinusoidal CPR results in the generation of
subharmonic Shapiro steps [65], which may lead to insta-
bilities in modes of operation of Josephson voltage stand-
ards. Results of analytical and computational investigations
of high-frequency dynamics of JJ characterized by nonzero
capacitance ( > 1)C and the second harmonic in the CPR
are presented in [66]. Above presented Eq. (28) gives the
result for step amplitude in harmonic case ( = 0)
2
= 2
( ) 1
n n
C
a
i J , (34)
where ( )nJ x is the Bessel function, a and are the am-
plitude and frequency of applied rf signal. The case of
= 0C coincides with the well known RSJ model [4]. In
contrast to harmonic case ( = 0),C there are subharmonic
steps in I–V curve with amplitude (according to [66])
1
2 2
(2 1)/2
2
( ) 1 ( ) 1
= 2
( ) /4 1
n n
C C
n
C
a a
J J
i .
(35)
For JJ with anharmonic CPR ( 0) the following ex-
pression for harmonic Shapiro step amplitudes is obtained
as [66]:
____________________________________________________
2
2 2
= 2max sin sin 2
( ) 1 ( ) 1
n n n
C C
a a
i J J , (36)
and for subharmonic steps as
1 0
2 2
1/2 1
2 2
0 2
2 2
2
2
2
( ) 1 ( ) 12
= 2max sin
( ) 1 ( ) /4 1
2 2
( ) 1 ( ) 1
4 cos .
( ) 1
C C
C
C C
C C
C
C
a a
J J
a
i J
a a
J J
(37)
Fig. 17. Plasma frequency of JJ as function of anharmonicity
parameter.
Fig. 18. Plasma frequency of JJ as function of amplitude of ex-
ternal ac current for different anharmonicity parameter.
Effects of anharmonicity of current-phase relation in Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4 327
Figures 19 and 20 present the analytical results, as well as
experimental data for both c-oriented and c-tilted
Nb/Au/YBCO junctions formed on NdGaO substrates (junc-
tion areas ranged from 10·10 m
2
to 30·30 m
2
) [67,68].
Similar results for subharmonic Shapiro steps were obtained
in [69] for c axis YBa2Cu3O7–x/Pb tunnel junctions.
Fig. 19. Dependencies of the 1/2- and 3/2-step amplitudes on the applied signal amplitude a at frequencies = 0.611, = 35 and = 0.
Solid line corresponds to Eq. (35), filled dots correspond to numerical simulation, and empty dots correspond to experimental results for
the c-oriented Nb/Au/YBCO junctions (a). Dependencies of the critical current amplitude i/2 (0-step) and the 1-step amplitude i (in
inset) on the applied signal amplitude a at frequency = 1.62 and = 4. Dashed and solid lines correspond to formula (36) at = 0 and
= 1, the filled dots correspond to experimental results for the c-tilted Nb/Au/YBCO junctions (b).
Fig. 20. Dependence of the 1/2-step amplitude i on the applied signal of amplitude a at = 4 for frequencies = 1.62 (a) and
= 2.2 (b). Dashed, solid and dotted lines correspond to the step behavior according to formula (37) with = 0, = 0.14, and = 0.3,
respectively. The filled dots are experimental data for the c-tilted Nb/Au/YBCO junction.
I.N. Askerzade
328 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4
3.4. Inluence of anharmonic CPR on long JJ dynamics
Physical properties of magnetic flux dynamics in long
JJ play an important role in the modern superconductivity
related electronics. We consider a long JJ, where the word
“long” means that we take into account the variation of the
phase along one of the spatial coordinates, i.e., along x.
Further, it will become clear that opposite to the case of the
usual JJ with only first harmonic in CPR, there is no uni-
versal length scale at which the phase changes or at which
the weak magnetic field is screened. Moreover, depending
on the state, the characteristic scale of x variation affecting
magnetic field screening can be different, so the junction
can be small if it is in one state and it can be long in anoth-
er state. At the same time, in order to neglect the spatial
variation of the phase along the junction width (y direc-
tion), we assume that the junction is short in the y direction
in all states. The calculation of static magnetic flux distri-
butions in the long JJ with consideration of the anharmonic
CPR was carried out in [58,70–72]. This model is de-
scribed by the double sine-Gordon equation (2SG) for
magnetic flux distribution in the static regime
sin sin 2 = ; ( , )x l l (38)
with the boundary conditions in the following form
( ) = el h . (39)
The magnitude is the external current, l is the
semilength of the junction. cI and are parameters cor-
responding to contributions of 1st and 2nd harmonic, re-
spectively. he is external magnetic field. Stability analysis
of ( , )x p is based on numerical solution of the corre-
sponding Sturm–Liouville problem
( ) = ; ( ) = 0q x l , (40)
with potential ( ) = cos 2 cos 2 .q x x x The minimal the
eigenvalue 0 ( ) > 0p corresponds to the stable solution.
In case when 0 ( ) < 0p the solution ( , )x p is unstable.
The case when 0 ( ) = 0p indicates the bifurcation with
respect to one of parameters = ( , , , ).ep l h Results of
investigations carried out in [70–72] shows that considera-
tion of the second harmonic significantly changes the
shape and stability properties of trivial and fluxon static
distribution in long JJ.
In the “traditional” case = 0 two trivial solutions
= 0 and = (denoted by 0M and ,M respectively)
are known at = 0 and 0.eh Consideration of the se-
cond harmonic sin 2 leads to appearance of two addi-
tional solutions = arccos( 1/2 ) (denoted as M ).ac
The corresponding 0 as functions of 2SG-equation coeffi-
cients have the form 0 0[ ] =1 2 ,M 0[ ] = 1 2 ,M
and 2
0[ ] [1 (1/2 ) ].M ac The exponential stability
of these constant solutions is determined by the signs of the
parameters and by the [70–72] (Fig. 21). The full energy
associated with the distribution of ( )x is calculated using
the expression:
2
( ) = 1 ( )
2
l
e
l
F p q x dx h , (41)
Fluxon solution of Eq. (38) in the case of = 0eh and
= 0 at l has a form [73]:
= ( ) = 4arctan(exp( )) 2x x n (42)
where “+” sign corresponds to fluxon, “–” sign corre-
sponds to antifluxon. At small external fields eh such dis-
tributions are fluxon 1, antifluxon 1 and their bound
states 1 1 and 1 1. As external magnetic field he
grows, more complicated stable fluxon and bound states
appear: n and nn ( =1,2,3,...).n
The energy of one-fluxon distribution 1 converges to
unity ( 0) 1F which corresponds to an energy of a
single fluxon 1 in a traditional “infinite” junction model
at = 1. With change of the number of fluxons
1
( ) = ( )
2
l
l
N p x dx
l
(43)
corresponding to the distribution 1 is conserved, i.e.,
/ 0.N Here we have 1[ ] = 1N . Results of influence
of second harmonic CPR in 0( )eh were calculated in
[70–72] and presented in Figs. 22, 23. Simulation results
show that consideration of the second harmonic in CPR
Fig. 21. Stability region as a function of anharmonicity parameter.
Fig. 22. Dependence of 0(he) for
1
with increasing of
anharmonicity parameter . 2l = 10, = 0.
Effects of anharmonicity of current-phase relation in Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4 329
significantly changes the shape and stability properties of
fluxon static distribution in long JJ.
4. Qubits based on JJ with anharmonic CPR
4.1. Qubits
The great majority of Josephson and SQUID research
since the beginning of the XXI century has focused on
possible applications in the field of quantum computation
[74,75]. In classical digital computation, the processor
takes as its input ones and zeros (coded, in the case of sili-
con integrated circuits, as two distinct voltage levels) and
derives an output by performing some kind of classical
Boolean logical operation on this input. In contrast to it, in
quantum computation the processor takes as its input a
quantum coherent superposition of ones and zeros [74,75].
The quantum processor then performs a quantum mechani-
cal operation on this input state in order to derive an output
which is also a quantum coherent superposition. The basic
element of a quantum computer is known as a qubit. The
state of the qubit, | is a linear superposition of the two
quantum basis states | 0 and |1 [74,75]. Realization of
qubits based on JJ and their application requires the
millikelvin temperature region. As follows from above
presented discussion, the anharmonic character of CPR
becomes important at this temperature and therefore
anharmonicity must be taken into account in discussion of
JJ qubits. This conclusion is also supported by investiga-
tions [44,76].
In order to analyze the qubits with JJ, one has to solve
the corresponding stationary Schrödinger equation with an
appropriate boundary condition
=H E , (44)
where H is the Hamiltonian operator, is the
wavefunction, and E is the eigenenergy. Quantum dynam-
ics of an isolated JJ is described with the Mathieu–Bloch
picture for a particle moving in a periodic potential, similar
to the electronic solid state theory [4]. In this section, we
shall describe the quantum dynamics of two types of
qubits: phase and charge qubits. Such qubits have distinct
limiting regimes: the phase regime ,j cE E is analogous
to the tight-binding approximation, and the charge regime,
,j cE E is analogous to the near-free particle approxi-
mation. At the end of this section, we also shall discuss a
flux qubits using low inductance interferometer with an-
harmonic JJ [77,78,80,90].
As mentioned in previous studies (i.e., [4, 81,82]), the
wavefunction should satisfy the periodic boundary condi-
tion ( ) = ( 2 ). Therefore, the required boundary
condition for solving Eq. (44) can be expressed as
( ) = ( ), ( ) = ( )a b a b ,
where min=a and min= 2b are the lower and up-
per bounds such that a and b depend on the variation of bi
as well as . Note that the value of a and b are different for
phase and charge qubits. Additional details about a and b
are given below.
4.1.1. Phase qubit with anharmonic CPR. A circuit
model of a phase qubit system using single JJ is shown in
Fig. 24. Corresponding Hamiltonian of the system [81,83]
associated with anharmonic CPR can be written as [82]
2
2
= [ cos cos 2 ]
2
c j bH E E i , (45)
where = /b b ci I I is the ratio of the bias currents applied to
the system, denotes the phase difference, cE is the elec-
trostatic energy, and jE is the Josephson coupling energy.
In some models suggested in [78] and [80], CPR of Joseph-
son junctions includes second and third harmonics.
The presence of second harmonic in CPR leads to hump
like shape of potential energy which seems to be very im-
portant for the manipulation with phase qubit. Figure 25
illustrates the influence of second harmonic on the potential
Fig. 23. Dependence of 0(he) for one fluxon states
1
and
1*
with increasing of anharmonicity parameter . 2 l = 10, = 0.
Fig. 24. Circuit model of a phase qubit. The crossed lines indicate
the junction. Ib is the bias current source.
I.N. Askerzade
330 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4
( ) = 1 cos( ) cos(2 )
2 2
U , (46)
for various values of . For instance, the potential has a
single minimum at = 0 for 0.5. and double minima
for > 0.5 in the ranges < < 0 and 0 < < , re-
spectively. The authors in [58] and [84] also discussed how
( )U changes from single potential well to double poten-
tial well in the case of junctions (i.e., the junctions with
negative critical current).
After substituting Eq.(45) into the Schrödinger Equa-
tion (44), we can obtain Mathieu eigenvalue equation for
zero bias current case:
2
2
cos cos 2 = ,
2
j
c
Ed
Ed
(47)
where = / .cE E Equation (47) is called the Mathieu ei-
genvalue equation, it describes the properties of phase
qubit under the periodic boundary condition with lower
bound = 0a and upper bound = 2 .b
4.1.2. Charge qubit with anharmonic CPR. A circuit
model of a charge qubit system using single JJ is shown in
Fig. 26. The Hamiltonian of the charge qubit system
[81,83] with anharmonic CPR can be written as
2ˆ= ( ) cos cos 2 ,
2
c g j bH E n n E i (48)
where
2= /(2 )c gE Q C is the electrostatic energy (Cooper
pair charge energy) that depends on gate voltage gV and the
capacitor = ,g jC C C and = /(2 )j cE I e is the Joseph-
son coupling energy in terms of critical current of Josephson
junction .cI Introduced ˆ = ( / )n i is the dimensionless
momentum operator that refers to the number of Cooper-pair
on the island and has a physical meaning of charge Q ac-
cumulated on the junction capacitor jC in the units of dou-
ble electronic charges (i.e., ˆ ˆ= 2 ).Q en Furthermore,
= / (2 )g g gn C V e is the dimensionless charge number used
to externally control the system [85]. Figure 26 illustrates a
single Cooper pair box for a charge qubit including a gate
voltage gV and a gate capacitance .gC
Using washboard potential from Eq. (46) for nonzero
bias current as well as , we can determine the upper and
lower bounds of the periodicity interval finding roots of
equation
( ) = = sin sin 2 = 0b
dU
f i
d
that sets condition for ( )U stable minimum min . In this
case, the interval for periodic boundary will be min=a
and min= 2b . Periodical solutions in the case of non-
zero bias current were discussed in detail in [4] and [82]
using wavepacket approach.
After substituting Eq. (48) into the Schrödinger equa-
tion as in Eq. (44), we can obtain a Mathieu-type eigenval-
ue equation:
2
2
= ,
d d
p q
dd
(49)
where = / ,cE E the terms = 2 gp in and
2| |
( ) = cos cos 2
2 4
j
b
c
E p
q i
E
.
Here, we will present the evaluation of expectation values
of the supercurrent operator given in equation ˆ / =s cI I
sin sin 2 and the number operator ˆ = ( / )n i
within the interval [a,b] in the lowest band for the charge
qubit.
First of all, the expectation value of supercurrent [86]
can be obtained from
Fig. 25. Potential energy U( ) of anharmonic JJ. Fig. 26. Circuit model of a charge qubit. The crossed box indicates
the combination of the tunnel element and the junction capacitor Cj
connected parallel. The gate charge is Qg = 2eng = CgVg.
Effects of anharmonicity of current-phase relation in Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4 331
1ˆ= | / | = {sin sin 2 } .
b
s s c
a
i I I d
b a
(50)
Similarly, the expectation value of number operator n̂ can
be determined as
1
ˆ ˆ= | | =
b
a
n n d
b a
. (51)
The Mathieu-type eigenvalue problems defined in
Eqs. (47), (49) can be discretized using the finite difference
approach discussed in [81] on a discrete lattice:
1 1 = , = 0,1,2,..., 1.j j j j je f e j N
The problem is that we want to get rid of the upper and
lower end wavefunctions 1 1( = N and 0= )N
in the form of boundary conditions given in Eq. (44). After
that, through mathematical discretization of these equa-
tions we can approximate the continuum behavior of the
system to obtain the eigenvalue problem of the form:
0 = , = 0,1,2,n n n nA , (52)
where n is an eigenfunction associated with eigenvalue
n for an arbitrary eigenstate n; and 0A is a complex
N N periodic tridiagonal coefficient matrix,
0
1
2
3
0 4
2
1
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
= 0 0 0 0 0
0 0 0 0 0
0 0 0 0
N
N
f e e
e f e
e f e
e f e
e f
f e
e e f
A (53)
The coefficients of the periodic tridiagonal matrix in Eq.
(53) are:
2 2= 1 /2, = ( 2), = , = ,
1
j j
b a
e hp f h q h h
N
(54)
where e is the complex conjugate and h is the step size
in the discretization scheme.
Throughout this study, the numerical solution of Eq. (52)
is calculated using the conventional LAPACK eigenvalue
solver. Note that the coefficient matrix in Eq. (53) requires
an N + 2 data storage space in its present form. In the litera-
ture ([87] and [88]) the linear system solution with a real
symmetric periodic tridiagonal coefficient matrix is widely
studied, but the eigenvalue problem [81] formulated in the
present study requires an efficient design and stable algo-
rithm for finding the eigenvalues of such a matrix. The
number of subintervals is preferred to be N = 3000 with suf-
ficient accuracy, the reason and preference of it will be ex-
plained.
4.2. Influence of anharmonic CPR on qubit characteristics
4.2.1. Phase qubit. The analysis in [78] was limited to
asymptotical solutions of Mathieu equations. Here, we per-
formed a full numerical analysis of Mathieu equation (47)
with inclusion of second harmonic in phase qubit regime
(i.e., / 1).j cE E As follows from the results, all energy
levels split into two sub-levels = .i i i Energy spec-
trum of Mathieu equations for = 0, 1i evaluated using nu-
merical calculations presented in Fig. 27 coincides with the
results in [78]. The ground ( = 0)i and first ( =1)i states of
the energy spectrum were obtained and it is shown that split-
ting in the ground and first excited state depends on the
anharmonicity parameter . For high values of energy scale
/2 ,j cE E the splitting between = 0 and 0 cases be-
comes large. On the other hand, it can be seen from the cal-
culations that the change in splitting of ground state is
Fig. 27. (Color online) Splitting parameters versus anhar-
monicity of the CPR. Splitting of the ground state (a), first
excited state (b).
I.N. Askerzade
332 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4
smaller than the change of the first level. This means that the
first state is more sensitive to anharmonicity parameter .
Furthermore, the numerical modeling is carried out to an-
alyze the influence of the control parameters on the splitting
of energy states = .i i i Figure 27 presents the behav-
ior of the splitting of energy states i for various energy
scales / .j cE E The results for 0 are presented in Fig. 27(a)
within the range 0 2. As was mentioned before, the
authors in [78,89] found similar results for dc-SQUID from
an oscillatory model analytically. Unlike our results, their
findings are limited to the range 0.5 1.5 for 0. In
contrast to [78,89], we observed fine structure in dependence
of 0 ( ) for different /j cE E values. For small anhar-
monicity parameter (i.e., < 0.65), the splitting parameter
i decreases linearly with increasing . The results for i
are in good agreement with findings of solid state theory.
Fixing the amplitude of first harmonic by negative sign of
second harmonic leads to an approximate linear decreasing
of 0 ( ) (1 )jE [90]. Similar behavior of linear de-
creasing in 0 is obtained in our numerical results presented
in Fig. 27(a). However, compared to approximate result, the
vanishing point of 0 is located in the range 0.6 < < 0.9
for various energy scales. This is because numerical results
are more precise than the results obtained from preceding
approximate expression. In addition, ( , )U in Fig. 25 has
a single minimum for 0.5 while it has two minima for
> 0.5.
As discussed before for Fig. 25, the shape of the potential
was switched from a single well to a double well structure
for > 0.5. For higher values of energy scale such as
/ 9,j cE E the behavior of 0 ( ) illustrates different
tendencies. For instance, 0 ( ) keeps decreasing from
= 0.5 to cr= until it vanishes. The values of cr are
determined from our calculation (see Fig. 27(a)) as 0.6231,
0.6007, 0.5784, and 0.5634 for energy scales 9, 15, 30, and
50, respectively. The “hump” of the double well potential is
not so high in this region and the energy levels are strongly
coupled. Such behavior corresponds to two-level crossing.
On the other hand, for cr max< where max is giv-
en in Table 2, 0 ( ) has an increasing tendency as shown
in Fig. 27(a). For max> the “hump” of the double well
increases so the energy levels become weakly coupled. Con-
sequently, the second harmonic in Eq. (46) becomes domi-
nant and leads to a two-level crossing.
For large values of anharmonicity parameter (0.65 < ),
we obtain results similar to those in [78]. The maximum
values of 0( ) peaks at different energy scales are given in
Table 2.
Table 2. Changing of the maximum of 0( ) peaks
EJ/EC max 0 max
9 1.350 0.0045
15 1.125 0.0054
30 0.950 0.0072
50 0.875 0.0090
As also shown in Fig. 27, the peak position of 0 de-
pends on the energy scale / .j cE E Besides, the width of the
peak grows with decreasing energy scale /j cE E in phase
qubit regime. With decreasing / ,j cE E the peak value of
0 ( ) is also suppressed. As shown in the inset of Fig.
27(a), 0 ( ) only reveals tendency for growth when
/ < 3j cE E for charge qubit regime. Figure 27(b) illus-
trates similar results for 1. As can be seen from this fig-
ure, 1( ) reveals monotonic decreasing behavior with an
increase in the anharmonicity parameter for all / .j cE E
The influence of the energy scale /j cE E on splitting of
energy state i for fixed value of anharmonicity parame-
ter is presented in Fig. 28. This plot clearly illustrates an
upward trend of i for / > 50.j cE E The reason for re-
stricting /j cE E up to 50 is related to technological
achievement in the realization of JJ with a very small ca-
pacitances (at a level of femto Farad (fF)) [85]. The energy
splitting in ground state 0 increases for energy scale
/ > 3.j cE E The partial derivative of 0 with respect to
energy scale for various is plotted in Fig. 29. As follows
from Fig. 28 and Fig. 29, high slope corresponds to the
Fig. 28. (Color online) Splitting versus energy scale. Splitting
of the ground state (inset shows behavior of 0 for the range
0 Ej/Ec 4) (a). Splitting of the first excited state (inset
shows behavior of 1 for the range 0 Ej/Ec 4) (b).
Effects of anharmonicity of current-phase relation in Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4 333
case of harmonic CPR ( = 0). Increasing up to 2 re-
sults in fall of the slope down to zero.
As follows from the inset of Fig. 28(a), at small
/ < 3,j cE E 0 sharply decreases with an increase in
/ .j cE E Another peculiarity of this region is related to non-
sensitivity of results to changing of amplitude of second
harmonic . The influence of the energy scale parameter
/j cE E on splitting of energy state 1 for fixed value is
presented in Fig. 28(b). The splitting of first state
1( / )j cE E reveals behavior similar to the case 0 ( / ).j cE E
The inset in Fig. 28(b) shows that at small / < 3,j cE E 1
decreases sharply with an increase in /j cE E similar to 0.
Notice that differential plots for the first level also resemble
those shown in Fig. 29.
4.2.2. Charge qubits. As mentioned before, the energy
scale / < 1j cE E corresponds to the single Cooper box
(SCB) charge qubit limit. In this limit, energy spectrum
can be described at quasi-charge approach [4,81] similar to
quasi-momentum representation in solid-state theory [90]
(see also inset Fig. 20(a). Energy gap 0 dependence on
anharmonicity parameter presented in Fig. 30(a) resem-
bles the phase qubit case in Fig. 27. Similarly to 0 which
is the difference between 1 and 0 at = 0.5gn , the “sec-
ondary energy gap” 1 is the difference between 2 and
1 at = 1gn . The detailed description can be found in
[81]. Notice that i refers to energy gap in charge qubit
whereas it refers to splitting of energy states in phase qubit.
In Fig. 30(b), the dependence of gap parameters i on
energy scale /j cE E is illustrated. This result qualitatively
is also in good agreement with Fig. 28. However, in case
of SCB the growth of i with /j cE E has revealed a non-
linear behavior.
The expectation value of number operator n̂ given in
Eq. (51) is plotted with respect to gate number gn in
Fig. 31 at small bias current = 0.1bi . The n̂ vs gn is
experimentally observed for SCB in [91] for junction pa-
rameters = 0.215 meVcE and / = 0.16.j cE E As follows
from Fig. 31 dependence of n̂ vs gn is not sensitive
to . The expectation value of supercurrent ˆ= /s ci I I
versus gn is illustrated in Fig. 32 for different an-
harmonicity parameters . The positions of the peaks in si
vs gn relation is the same as the peaks for = 0 in [81].
Note that the supercurrent is equal to zero when the bias
current bi is set to zero. The peaks at half-integer gn val-
ues correspond to the tunneling of Cooper pairs from one
Fig. 29. (Color online) Differential plot for the splitting parame-
ters in the ground state.
Fig. 30. (Color online) Results for the charge qubit obtained at
ng = 0.5.
Fig. 31. (Color online) n versus ng for ib = 0.1.
I.N. Askerzade
334 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4
electrode to another. When we compare Fig. 32(a) and
32(b), the increasing of bi leads to an increase in the mag-
nitude of si while the increasing of anharmonicity parame-
ter leads to a decrease in the magnitude of .si
It was shown in [54] that the anharmonicity of the CPR
is equivalent to the introduction of an effective inductance
connected in series to the JJ. The value of effective induct-
ance is proportional to the magnitude of the anharmonicity
parameter. This implies that such additional inductance
leads to an increase in the impedance of the circuit for
charge qubit. For that reason, the amplitude of the expecta-
tion value of supercurrent si in Fig. 32 is suppressed with
increasing .
4.2.3. Silent qubit. The silent qubit is just a low-
inductive two-junction interferometer based on JJ with
anharmonic CPR and does not require application of half-
flux quantum. Such a qubit is called “silent” because of
both its high protection against external magnetic field
impact and the absence of any state-dependent sponta-
neous circular currents. The potential energy of such
quantum-mechanical system was described in the papers
[66,78,79,80]. As shown in this work and mentionened in
section 2, presence of second harmonic in CPR of JJ
leads to a two-hump potential
____________________________________________________
0 1 0 21 2( , ) = cos( ) cos( 2 ) cos( ) cos( 2 )
2 2 2 2 2 2
c cI I
U , (55)
_______________________________________________
where 1 2 is difference of the JJ phases,
1 2( )/2, 0= 2 ( / )e are normalized external
fluxes. In the absence of external magnetic field one can
easily obtain the conditions leading to the double-well en-
ergy potential formation (see above). If both junctions
(with the same CPR) are of the same size, the energy po-
tential remains always symmetric and any state-
dependent current is impossible even if an external mag-
netic field is applied. However, at different sizes of the
junction (different critical currents) the external magnetic
field always breaks the potential symmetry and produces
a state-dependent current in the loop. In papers
[66,77,79,80] influence of external magnetic field on the
splitting of energy levels was investigated. It was shown
that the ratio /J QE E also has influence on splitting of
energy levels. The value of external magnetic field ,e
in which splitting parameter remains unchangeable was
found. In these papers examples of some logic operations
using silent qubit are also considered.
5. Conclusions
Results of theoretical and experimental investigations
of different JJ with anisotropic and multiband supercon-
Fig. 32. (Color online) Expectation value of supercurrent I/Ic .
ib =0.05 (a), ib = 0.1 (b). Fig. 33. Energy spectrum of silent qubit.
Effects of anharmonicity of current-phase relation in Josephson junctions
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 4 335
ductors show that in the general case CPR has anharmonic
character. However, up to present time simple sinusoidal
CPR was used to study the dynamics and ultimate perfor-
mance of superconducting devices based on JJ. The main
subject of this review is the investigation of dynamical
properties of JJ with anharmonic CPR. Firstly, numerically
calculated I–V characteristics of externally shunted JJ with
anharmonic CPR were discussed. We conclude that the
second harmonic in the CPR has a strong influence on the
I–V curve of the JJ. Inclusion of anharmonicity parameters
leads to an increase in the critical current and an enhance-
ment of hysteresis in the I–V curves. We confirm that the
shunt inductance in the range of 15L also affects the
dynamics considerably.
Result of calculations of the plasma frequency of JJ
with anharmonic CPR as function of anharmonicity pa-
rameter was presented. Comparison of calculated plas-
ma frequency with experimental data was conducted. Gen-
eralizing formulas for both harmonic and sub-harmonic
Shapiro steps in the presence of nonzero junction capaci-
tance and second harmonic in current-phase relation are
discussed. Experimental results related to Shapiro steps in
YBCO-based JJ are in good agreement with the theory.
Simulations show that consideration of the second harmon-
ic in CPR significantly changes the shape and stability
properties of fluxon static distribution in long JJ.
We conclude that second harmonic in CPR has strong
influence on the characteristics of phase qubits based on
Josephson junctions, which operate at very low tempera-
tures. In contrast to phase qubits, in the limit of charge
qubits no considerable effects of anharmonicity are ob-
served in the characteristic number of Cooper pair n̂
versus gate number .gn It was observed that the influences
of anharmonicity parameter and the bias current bi on
the expectation value of supercurrent si in charge qubit are
opposite. It was demonstrated that splitting of energy lev-
els in phase qubit as well as energy gap in charge qubit
reveals similar behavior with energy scale. Characteristics
and operations of silent qubit using JJ with anharmonic
CPR were also briefly discussed. Finally, we confirm that
anharmonic current-phase relation must be taken into ac-
count in the experimental realizations of Josephson junc-
tion circuits and superconducting qubits.
Acknowledgements
I thank Professors F.M. Hashimzade, R.R. Guseinov,
A. Gencer for the useful discussions, Dr. M. Canturk, Dr.
A. Bozbey for collaboration and Dr. Erkan Bostanci for
technical help. I am also grateful to the Abdus Salam ICTP
for the hospitality during my stay as an associate member.
This work was partly supported by the TUBITAK research
grant No 111E191.
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