Theory of quantum transport in Josephson junctions with a ferromagnetic insulator
We investigate the Josephson transport through ferromagnetic insulators (FIs) by taking into account the band structure of FIs explicitly. Using the recursive Green's function method, we found the formation of a π-junction in such systems. Moreover the atomic-scale 0–π oscillation is induced by...
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| Cite this: | Theory of quantum transport in Josephson junctions with a ferromagnetic insulator / S. Kawabata, Y. Asano // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1143–1148. — Бібліогр.: 62 назв. — англ. |
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| citation_txt | Theory of quantum transport in Josephson junctions with a ferromagnetic insulator / S. Kawabata, Y. Asano // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1143–1148. — Бібліогр.: 62 назв. — англ. |
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| description | We investigate the Josephson transport through ferromagnetic insulators (FIs) by taking into account the band structure of FIs explicitly. Using the recursive Green's function method, we found the formation of a π-junction in such systems. Moreover the atomic-scale 0–π oscillation is induced by increasing the thickness of FI and its oscillation period is universal, i.e., just single atomic layer. Based on these results, we show that stable π-state can be realized in junctions based on high-Tc superconductors with La₂BaCuO₅ barrier. Such FI-based Josephson junctions may become an element in the architecture of future quantum computers.
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© Shiro Kawabata and Yasuhiro Asano, 2010
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11, p. 1143–1148
Theory of quantum transport in Josephson junctions
with a ferromagnetic insulator
Shiro Kawabata
Nanosystem Research Institute (NRI), National Institute of Advanced Industrial Science and Technology (AIST),
Tsukuba, Ibaraki 305-8568, Japan
E-mail: s-kawabata@aist.go.jp
CREST, Japan Science and Technology Corporation (JST), Kawaguchi, Saitama 332-0012, Japan
Yasuhiro Asano
Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
Received April 5, 2010
We investigate the Josephson transport through ferromagnetic insulators (FIs) by taking into account the band
structure of FIs explicitly. Using the recursive Green's function method, we found the formation of a π-junction
in such systems. Moreover the atomic-scale 0–π oscillation is induced by increasing the thickness of FI and its
oscillation period is universal, i.e., just single atomic layer. Based on these results, we show that stable π-state
can be realized in junctions based on high-Tc superconductors with La2BaCuO5 barrier. Such FI-based Joseph-
son junctions may become an element in the architecture of future quantum computers.
PACS: 74.50.+r Tunneling phenomena; point contacts, weak links, Josephson effects;
72.25.–b Spin polarized transport;
85.75.–d Magnetoelectronics; spintronics: devices exploiting spin polarized transport or integrated
magnetic fields;
03.67.Lx Quantum computation architectures and implementations.
Keywords: ferromagnetic insulator, Josephson junction, superconductor.
1. Introduction
There is an increasing interest in the novel properties of
interfaces and junctions of superconductors and ferromag-
netic materials [1,2]. One of the most interesting effects is
the formation of a Josephson π-junction in superconduc-
tor/ferromagnetic-metal/superconductor (S/FM/S) hetero-
structures [3]. In the ground-state phase difference between
two coupled superconductors is π instead of 0 as in the
ordinary 0-junctions. In terms of the Josephson relation-
ship
= sin ,J cI I φ (1)
where φ is the phase difference between the two super-
conductor layers, a transition from the 0 to π states implies
a change in sign of cI from positive to negative. Such a
negative cI was originally found in the Josephson effect
with a spin-flip process [4–6]. In S/FM/S junctions, such a
sign change of cI is a consequence of a phase change in
the pairing wave-function induced in the FM layer due to
the proximity effect. The existence of the π-junction in
S/FM/S systems has been confirmed in experiment by
Ryanzanov et al. [7] and Kontos et al. [8].
Recently, a quiet qubit consisting of a superconducting
loop with a S/FM/S π-junction has been proposed [9–11].
In the quiet qubit, a quantum two-level system (qubit) is
spontaneously generated and therefore it is expected to be
robust to the decoherence by the fluctuation of the external
magnetic field. From the viewpoint of the quantum dissipa-
tion, however, the structure of S/FM/S junctions is inhe-
rently identical with S/N/S junctions (N is a normal non-
magnetic metal). Thus a gapless quasiparticle excitation in
the FM layer is inevitable. This feature gives a strong dis-
sipative effect [12–14] and the coherence time of S/FM/S
quiet qubits is bound to be very short. Therefore Josephson
π junctions with a nonmetallic interlayers are highly de-
sired for qubit application.
On the other hand, a possibility of the π-junction for-
mation in Josephson junctions through ferromagnetic insu-
Shiro Kawabata and Yasuhiro Asano
1144 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11
lators (FIs) have been theoretically predicted [15] and in-
tensively analyzed by use of the quasiclassical Green's
function techniques [16,17]. Recently, by extending these
results, we have proposed superconducting phase [18] and
flux qubits [19–21] based on S/FI/S π-junctions. Moreover
we have also showed that the effect of the dissipation due
to a quasi-particle excitation on macroscopic quantum
tunneling is negligibly small [20]. These results clearly
indicate the advantage of the FI based π-junction for qubit
applications with longer coherence time.
However, up to now, a simple δ -function potential [15]
has been used in order to model the FI barrier. In this phe-
nomenological model, the up (down) spin electrons tunnel
through a positive (negative) delta-function barrier. There-
fore, strictly speaking, this model describes not ferromag-
netic insulators but half metals with infinitesimal thick-
ness. Moreover the possibility of the π-junction formation
in the finite barrier thickness case is also an unresolved
problem. In order to resolve above issues, we formulate a
numerical calculation method for the Josephson current
through FIs by taking into account the band structure and
the finite thickness of FIs explicitly. In this paper we
present our recent numerical results [21–23] on the forma-
tion of the π-coupling for the Josephson junction through a
FIs, e.g., La2BaCuO5 and K2CuF4 and show that the me-
chanism of the π-junction in such systems is in striking
contrast to the conventional S/FM/S junctions.
2. Magnetic and electronic properties of ferromagnetic
insulators
In this section, we briefly describe the magnetic proper-
ties and the electronic density of states (DOS) of FIs. The
typical DOS of FI for each spin direction is shown sche-
matically in Fig. 1. One of the representative material of FI
is half-filled La2BaCuO5 (LBCO) [24–26]. The crystal
structure of LBCO has tetragonal symmetry with space
group 4 /P mbm . In 1990, Mizuno et al., found that LBCO
undergoes a ferromagnetic transition at 5.2 K [24]. The
exchange splitting exV is estimated to be 0.34 eV by a
first-principle band calculation using the spin-polarized
local density approximation [27]. Since the exchange split-
ting is large and the bands are originally half-filled, the
system becomes FI.
An another example of FPFI is K2CuF4 compounds in
which the two-dimensional Heisenberg ferromagnet is rea-
lized [28,29]. The ferromagnetic behavior of this materials
has been experimentally confirmed by the magnetic sus-
ceptibility [30] and neutron diffraction measurements [31].
Moreover a result of the first-principle band calculation
[32] indicated that K2CuF4 compounds with Jahn-Teller
distortion have the electronic structure similar to Fig. 1. In
the followings, we calculate the Josephson current through
such FIs numerically.
3. Numerical method
In this section, we develop a numerical calculation me-
thod for the Josephson current of S/FI/S junctions based on
the recursive Green's function technique [33–35]. Let us
consider a two-dimensional tight-binding model for the
S/FI/S junction as shown in Fig. 2. The vector
= j m+r x y (2)
points to a lattice site, where x and y are unit vectors in
the x and y directions, respectively. In the y direction,
we apply the periodic boundary condition for the number
of lattice sites being W .
Fig. 1. The density of states for each spin direction for a ferro-
magnetic insulator, e.g., LBCO. exV is the exchange splitting and
8t is the band width.
Fermi enerFermi energygy
VVeexx
8t8t
EE
Fig. 2. A schematic figure of a Josephson junction through the
ferromagnetic insulators on the two-dimensional tight-binding
lattice.
SuperconductorSuperconductor SuperconductorSuperconductorFerromagnetic insulatorFerromagnetic insulator
1 ··· L1 ··· LFF ······
yy
zz
m = 1m = 1
m =m = WW
xx
······ j = 0j = 0
Theory of quantum transport in Josephson junctions with a ferromagnetic insulator
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1145
Electronic states in a superconductor are described by
the mean-field Hamiltonian
††
, ,
, S
1 ˆ ˆ=
2BCSH c h c c h cr r r r r r r r
r r
∗
′ ′ ′ ′
′∈
⎡ ⎤− +⎢ ⎥⎣ ⎦∑ % % % %
† †
S
1 ˆ ˆ .
2
c c c c∗
∈
⎡ ⎤+ Δ − Δ⎢ ⎥⎣ ⎦∑ r r r r
r
% % % % (3)
Here
, | |,1 , 0
ˆ ˆ= ( 4 ) ,s s sh t t′ ′ ′−⎡ ⎤− δ + −μ + δ σ⎣ ⎦r r r r r r (4)
with
( ), ,= , ,c c c↑ ↓r r r% (5)
where †
,c σr ( ,c σr ) is the creation (annihilation) operator of
an electron at r with spin =σ (↑ or ↓ ), c% means the
transpose of c% , and 0σ̂ is 2 2× unit matrix. The chemical
potential sμ is set to be 2 st for superconductors. In super-
conductors, the hopping integral st is considered among
nearest neighbor sites and we choose
2ˆ ˆ= ,iΔ Δσ (6)
where Δ is the amplitude of the pair potential in the s-
wave or d-wave symmetry channel, and 2σ̂ is a Pauli ma-
trix.
We consider FIs as a barrier of the Josephson junction.
The Hamiltonian of the FI barrier is given by a single-band
tight-binding model as
††
, , ,,
, ,
= (4 )FIH t c c t c cr r rr
r r r
′σ σ ↑↑
′ σ
− − − μ +∑ ∑
†
ex ,,(4 ) ,t V c c ↓↓
+ −μ +∑ rr
r
(7)
where exV is the exchange splitting (see Fig. 1). If
ex > 8V t ( ex < 8V t ), this Hamiltonian describes FI (FM).
The chemical potential μ is set to be
ex= 4 .
2
V
tμ + (8)
The Hamiltonian is diagonalized by the Bogoliubov
transformation and the Bogoliubov–de Gennes equation is
numerically solved by the recursive Green function method
[33–35]. We calculate the Matsubara Green function in a
FI region,
ˆˆ ( , ) ( , )
( , ) = ,
ˆ ˆ( , ) ( , )
n n
n
n n
g f
G
f g
ω ω
ω ∗ ∗
ω ω
⎛ ⎞′ ′
⎜ ⎟′
⎜ ⎟′ ′− −⎝ ⎠
r r r r
r r
r r r r
(
(9)
where
= (2 1)n n Tω + π (10)
is the Matsubara frequency, n is an integer number, and T
is a temperature. The Josephson current is given by
=1
( ) = Tr ( , ) ( , ) ,
W
J n n
mn
I ietT G Gω ω
ω
⎡ ⎤′ ′φ − −⎣ ⎦∑∑ r r r r
( (
(11)
with =′ +r r x . The Matsubara Green function in Eq. (9) is
a 4 4× matrix representing Nambu and spin spaces.
Throughout this paper we fix = 0.01 cT T , where cT is the
superconductor transition temperature.
4. Josephson current for low- cT superconductors
In this section we show numerical results of the Joseph-
son current for low-Tc superconductor/FI/low-Tc super-
conductor junctions and discuss the physical origin of the
π-junction formation in such systems [21–23]. In the cal-
culation, we assume = st t and set = 25W , and
= = 0.01s tΔ Δ . The phase diagram depending on the
strength of exV ( ex0 / 8V t≤ ≤ for FM and ex / > 8V t for
FI) and FL is shown in Fig. 3. The black (white) regime
corresponds to the π(0)-junction, i.e.,
= ( ) sin .J cI I− + φ (12)
In the case of FI, the π-junction can be formed. Moreover,
the 0–π transition is induced by increasing the thickness of
the FI barrier FL and the period of the transition is univer-
sal and just single atomic layer [23]. We also found that
the atomic-scale 0–π transition is also thermally stable
[36]. On the her hand, in the case of FM, the oscillation
period strongly depends on exV and the temperature [1,2].
A physical origin of the appearance of the π-junction
and the atomic scale 0–π transition can be explained as
follows [23]. In the high barrier limit ( exV t ), Joseph-
son critical current is perturbatively given by [20,21]
* .cI T T↓ ↑∝ (13)
Here ( )T↑ ↓ is a transmission coefficient of the FI barrier
for up (down) spin electron. In the case of the single-cite
FI (i.e., =1FL ), the transmission coefficients are analyti-
cally by use of the transfer matrix method [37–39] as
Fig. 3. The phase diagram depending on the strength of exV and
FL for FM ( ex0 / 8V t≤ ≤ ) and FI ex( / > 8)V t . The black and
white regime correspond to the π - and 0-junction, respectively.
1212
88
44
00
11 22 33 44 55 66
FIFI
FMFM
LLFF
VV
/
t
/
t
eexx
Shiro Kawabata and Yasuhiro Asano
1146 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11
1
ex
= ,tT
V↑ α (14)
1
ex
= ,tT
V↓ −α (15)
where 1α is a spin-independent complex number. There-
fore the sigh of the critical current
2
2
1
ex
| |c
tI
V
⎛ ⎞
∝ − α ⎜ ⎟
⎝ ⎠
(16)
becomes negative , so the π-junction is formed in the case
of single-cite FI barrier.
On the other hand, the transmission coefficients for an
arbitrary value of 1FL ≥ can be expressed by
ex
= ,
LF
LF
tT
V↑
⎛ ⎞
α ⎜ ⎟
⎝ ⎠
(17)
ex
= ,
LF
LF
tT
V↓
⎛ ⎞
α −⎜ ⎟
⎝ ⎠
(18)
where LFα is a complex number. So the sign of the criti-
cal current
2
2
ex
( 1) | |
LF
LFc LF
tI
V
⎛ ⎞
∝ − α ⎜ ⎟
⎝ ⎠
(19)
becomes negative for the odd number of FL and positive
for the even number of FL . Therefore we can realize the
atomic-scale 0–π transition with increasing the thickness of
the FI barrier FL as demonstrated in Fig. 3.
5. Josephson current for high- cT superconductors
We would like to show an experimental set-up for ob-
serving the π-junction using LBCO in Fig. 4. From the
perspectives of the FI/superconductor interface matching
and the high-temperature device-operation, the usage of
high-Tc cuprate superconductors (HTSC), e.g., YBa2Cu3O7–δ
and La2–xSrxCuO4 (LSCO) is desirable. Recent develop-
ment of the pulsed laser deposition technique enable us to
layer-by-layer epitaxial-growth of such oxide materials
[40,41]. Therefore, the experimental observation of the
0–π transition by increasing the layer number of LBCO
could be possible.
In order to show the possibility of π-coupling in such
realistic HTSC junctions, we have numerically calculated
the c -axis Josephson critical current cI based on a three-
dimensional tight binding model with aL and bL being
the numbers of lattice sites in a and b directions [Fig. 4,a]
[23,42]. In the calculation we have used a hard wall boun-
dary condition for the a and b direction and taken into
account the d-wave order-parameter symmetry in HTSC,
i.e.,
= (cos cos ).
2
d
x yk a k a
Δ
Δ − (20)
The tight binding parameters t and g have been deter-
mined by fitting to the first-principle band structure calcu-
lations [27]. Figure 5 shows the FI thickness FL depen-
dence of cI at = 0.01 cT T for a LSCO/LBCO/LSCO
junction with ex / = 28V t , / = 0.6d tΔ , and = =a bL L
100= . As expected, the atomic scale 0–π transitions can
be realized in such oxide-based c-axis stack junctions.
The formation of the π-junction can be experimentally
detected by using a HTSC ring [see Fig. 4,b]. The phase
quantization condition for the HTSC ring is given by
ext
1 2
0
2 = 2 ,n
Φ −Φ
π + φ + φ π
Φ
(21)
where 1φ and 2φ are the phase difference across the junc-
tion 1 and 2, Φ is the magnetic flux penetrating through
Fig. 4. Schematic picture of c-axis stack high-Tc superconduc-
tor/LBCO/high-Tc superconductor Josephson junction (a) and
high-Tc ring which can be used in experimental observations of
the π-junction (b).
aa
bb
High-High- layerlayerTTcc
High-High- ringringTTcc
High-High- layerlayerTTcc
Fig. 5. The Josephson critical current cI as a function of the
FI thickness FL at = 0.01 cT T for a c-axis stack
LSCO/LBCO/LSCO junction with ex / = 28V t , / = 0.6d tΔ , and
= = 100a bL L . The large (small) circles indicate the π(0)-
junction.
11
1010
–2–2
1010
–4–4
1010
–6–6
1010
–8–8
11 22 33 44
LLFF
|I|I
(L(L
)/
I
)/
I
(1
)|
(1
)|
cc
FF
cc
Theory of quantum transport in Josephson junctions with a ferromagnetic insulator
Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1147
the ring, 0Φ is the flux quantum, and n is an integer. The
current passed through the ring divides between the junc-
tion 1 and 2, i.e.,
1 1 2 2= sin sin .c cI I Iφ + φ (22)
Applied external magnetic flux extΦ depletes phases
1φ and 2φ causing interference between currents through
the junctions 1 and 2. For a symmetric ring with
1 2 =c c cI I I≈ and negligible geometric inductance
( = 0)L , the total critical current as a function of extΦ is
given by
00 ext
0
= = 2 cos ,c c cI I Iππ ⎛ ⎞Φ
π⎜ ⎟Φ⎝ ⎠
(23)
for the case that FL of the both junctions are same. If FL
of the junction 1(2) is even and FL of the junction 2(1) is
odd, we get
0 0 ext
0
= = 2 sin .c c cI I Iπ π ⎛ ⎞Φ
π⎜ ⎟Φ⎝ ⎠
(24)
Therefore the critical current of a 0–π (0–0) ring has a
minimum (maximum) in zero applied magnetic field [43].
Experimentally, the half-periodic shifts in the interference
patterns of the HTSC ring can be used as a strong evidence
of the π-junction. Such a half flux quantum shifts have
been observed in a s-wave ring made with a S/FM/S [44]
and a S/quantum dot/S junction [45].
It is important to note that in the case of c-axis stack
HTSC Josephson junctions [46,47], no zero-energy An-
dreev bound-states [48] which give a strong Ohmic dissi-
pation [49–51] are formed. Moreover, the harmful influ-
ence of nodal-quasiparticles due to the d -wave order-
parameter symmetry on the macroscopic quantum dynam-
ics in such c-axis junctions is found to be weak both theo-
retically [52–56] and experimentally [57–60]. Therefore
HTSC/LBCO/HTSC π-junctions would be a good candi-
date for quiet qubits.
6. Summary
To summarize, we have studied the Josephson effect in
S/FI/S junction by use of the recursive Green's function
method. We found that the π-junction and the atomic scale
0–π transition is realized in such systems. By use of the
transfer matrix calculation, the origin of the π-junction
formation can be attributed to the π phase difference of the
spin-dependent transmission coefficient for the FI barrier.
Such FI based π-junctions may become an element in the
architecture of quiet qubits.
We would like to point out that the π-junction can be
also realized in the Josephson junction through an another
type of FI, i.e., a spin-filter material, in the case of the
strong hybridization between localized and conduction
electrons [61,62]. It should be also note that FI materials
treated in this paper can be categorized in strongly corre-
lated systems. Moreover, in actual junctions, the influence
of the interface roughness could be important. Therefore
investigation of the atomic-scale 0–π transition in the pre-
sence of the many-body and disorder effect will be also the
subject of future studies.
Acknowledgements
This paper is based on the collaboration works with S.
Kashiwaya, Y. Tanaka, and A.A. Golubov. We would like
to thank J. Arts, A. Brinkman, M. Fogelström, H. Ito,
T. Kato, P.J. Kelly, T. Löfwander, T. Nagahama, F. Nori,
J. Pfeiffer, A.S. Vaenko, and M. Weides for useful discus-
sions. This work was supported by CREST-JST, and a
Grant-in-Aid for Scientific Research from the Ministry of
Education, Science, Sports and Culture of Japan (Grant
No. 22710096).
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|
| id | nasplib_isofts_kiev_ua-123456789-117516 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T15:29:26Z |
| publishDate | 2010 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Kawabata, S. Asano, Y. 2017-05-24T05:14:20Z 2017-05-24T05:14:20Z 2010 Theory of quantum transport in Josephson junctions with a ferromagnetic insulator / S. Kawabata, Y. Asano // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1143–1148. — Бібліогр.: 62 назв. — англ. 0132-6414 PACS: 74.50.+r, 72.25.–b, 85.75.–d, 03.67.Lx https://nasplib.isofts.kiev.ua/handle/123456789/117516 We investigate the Josephson transport through ferromagnetic insulators (FIs) by taking into account the band structure of FIs explicitly. Using the recursive Green's function method, we found the formation of a π-junction in such systems. Moreover the atomic-scale 0–π oscillation is induced by increasing the thickness of FI and its oscillation period is universal, i.e., just single atomic layer. Based on these results, we show that stable π-state can be realized in junctions based on high-Tc superconductors with La₂BaCuO₅ barrier. Such FI-based Josephson junctions may become an element in the architecture of future quantum computers. This paper is based on the collaboration works with S. Kashiwaya, Y. Tanaka, and A.A. Golubov. We would like to thank J. Arts, A. Brinkman, M. Fogelström, H. Ito, T. Kato, P.J. Kelly, T. Löfwander, T. Nagahama, F. Nori, J. Pfeiffer, A.S. Vaenko, and M. Weides for useful discussions. This work was supported by CREST-JST, and a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan (Grant No. 22710096). en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Quantum coherent effects in superconductors and normal metals Theory of quantum transport in Josephson junctions with a ferromagnetic insulator Article published earlier |
| spellingShingle | Theory of quantum transport in Josephson junctions with a ferromagnetic insulator Kawabata, S. Asano, Y. Quantum coherent effects in superconductors and normal metals |
| title | Theory of quantum transport in Josephson junctions with a ferromagnetic insulator |
| title_full | Theory of quantum transport in Josephson junctions with a ferromagnetic insulator |
| title_fullStr | Theory of quantum transport in Josephson junctions with a ferromagnetic insulator |
| title_full_unstemmed | Theory of quantum transport in Josephson junctions with a ferromagnetic insulator |
| title_short | Theory of quantum transport in Josephson junctions with a ferromagnetic insulator |
| title_sort | theory of quantum transport in josephson junctions with a ferromagnetic insulator |
| topic | Quantum coherent effects in superconductors and normal metals |
| topic_facet | Quantum coherent effects in superconductors and normal metals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/117516 |
| work_keys_str_mv | AT kawabatas theoryofquantumtransportinjosephsonjunctionswithaferromagneticinsulator AT asanoy theoryofquantumtransportinjosephsonjunctionswithaferromagneticinsulator |