Superconducting hetero-junction made of the p-wave superconductor and the normal s-wave superconductor has been considered. Bogoliubov transformation in the p-wave superconductor has been obtained by diagonalizing the superconducting Hamiltonian. Moreover, the value of the Josephson current that flows in the hetero-junctions of p-wave/s-wave superconductors has been calculated by this Bogoliubov transformation, and proved to be zero in the first order perturbation

Nakayama Akiyoshi

Okabe Yoichi

中山 明芳

岡部 洋一

UTokyo Repository

IEEE TRANSACTIONS ON MAGNETICS, VOL. 27, NO. 2, MARCH 1991 
CALCULATION OF THE JOSEPHSON CURRENT 
IN P-WAVE AND S-WAVE SUPERCONDUCTING HETERO-KJNCTIONS 
BY BOGOLIUBOV TRANSFORMATION 
Akiyoshi NAKAYAMA 
Department of Electrical Engineering, Faculty of Engineering, Kanagawa University 
and 
Yoichi OKABE 
Research Center for Advanced Science and Technology, University of Tokyo 
4-6-1 Komaba, Meguro-ku, Tokyo, 113, Japan 
3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, 221, Japan 
Abstract 
Superconducting hetero-junction made of the p-wave 
supcrconductor and the normal s-wave superconductor has been 
considered. Bogolinbov transformation in the p-wave 
superconductor has been obtained by diagonalizing the 
superconducting Hamiltonian. Moreover, the value of the 
Josephson current that flows in the hetero-junctions of p-wave/s- 
wave superconductors has been calculated by this Bogoliubov 
transformation, and proved to be zero in the first order perturbation. 
Introduction 
Discovery of high-temperature oxide superconductors has 
enhanced the field of superconducting electronics. Though the 
mechanism of the superconductivity of high-temperature oxides is 
not clear yet, devices with high-temperature oxide superconductors 
are very important. Moreover, it is thought that “heavy-fermion’’ 
compounds such as CeCu2Si2, UBel3, UPt3, and UPtzC, are 
likely not to be normal “singlet” superconductors, but to be 
“triplet” superconductors. Superconducting hetero-junctions such 
as metal-superconductor / oxide-superconductor junction and 
“triplet” superconductor/“singlet” superconductor junction are so 
useful for electrical application and also interesting from physical 
point. Maximum dc Josephson current1 of s-wave/s-wave homo- 
junction was first calculated by Ambegaokar and Baratoff,2 and 
was expressed in the frequency domain by Werthamer.3 In this 
paper, as a example that is typical of the hetero-junction of 
“triplet” superconductor / “singlet” superconductor, p-wave- 
supercomdutor/s-wave-superconductor hetero-junction has been 
considered. Josephson current between p-wave and s-wave 
superconducting electrodes has been calculated with a generalized 
Bogoliubov transformation,475 which has been obtained from diag- 
onalizing the superconducting Hamiltonian. As the p-wave state 
we have considered Balian-Werthamer (BW) state,5 which also 
appears in He3 liquid. 
Josephson Current in Tunnel Junctions 
Following Ambegaokar and Baratoff,2 calculation of the 
tunneling current begins with the assumption of the tunneling 
interaction Hamiltonian HT. The junction is described by the total 
Hamiltonian H = HL+HR+HT, as shown in Fig. 1, where HL and 
HR describe the Hamiltonian of each left and right superconducting 
electrode, and HT transfers electrons from one superconductor to 
the other: 
HT =E TbazDbqa+ h.c., (1) 
kqc 
where ako (bqo) denotes the annihilation operator of a electron of 
1341 
wave vector k (4) and spin state a i n  the left (right) electrode, and 
Tk4 is a tunneling matrix element connecting a k-state on the left 
with a q-state on the right. In the tunneling process conservation of 
the direction of spin is assumed, and this assumption will later play 
an important role in the calculation of the Josephson current. We 
calculate the tunneling current I from the expectation value of the 
rate of change of the number operator 
for electrons on the left side. 
method, we obtain6.7 
By the first order perturbation 
r t  
where e = le1 is the absolute value of electron charge, Tr( ) is the 
trace of the operator inside the brackets, [A,B] = AB - BA, and 
symbol < > denotes an expectation value referred to the 
unperturbed Hamiltonian HO = HL+HR: 
+ v -  
(4) 
Figure 1. The tunnel junction structure. 
superconductors are coupled by the tunneling Hamiltonian HI. 
The left and right 
Manuscript received September 24, 1990. 
001 8-9464/91/03~-1341$01 .OO 0 199 1 IEEE 
1342 
and u(t), b(t) and H(t) are the 
representation, such as 
operators in the interaction and 
(6) "+ U ku(r) = eievt/b+kd& 
;ii d a m  =; W O ,  4dt)l> ( 5 )  where V is the voltage difference between the left and right 
electrodes. After the integration, a positive small value tends to 
zero. 
In particular, dc Josephson current is obtained as: 
In this section we discuss the Diagonalization of each 
assume that the left electrode has a s-wave superconducting state 
and that the right electrode has a p-wave superconducting state. 
Hamiltonian H L  and H R  for the left and right electrodes. We ak = 
In this section, the expression of Josephson current of eq. (7) 
has been obtained. b k l  2= ~ k ' d k  = d121 , ( 1 2 )  
uk 0 0 -vkeie 
0 uk vkeie 0 
0 -vke-'e uk 
Diaponalizinp Suuerconducting Hamiltonians Hence, 
Bogoliubov transformation: 
the Hamiltonian H L  can be diagonalized by the usual 
where a and cl+ are a quasi-particles annihilation and creation 
operator, respectively, and quantities with superscript k are 
matrices in either 2- or 4-dimensional spin space, following Balian 
and Werthamer.5 In the Hartree-Fock approximation, the 
Hamiltonian of the left superconducting electrode is as follows: 
(9)  
where I is a unit matrix, ek is the kinetic energy of electron 
measured from the Fermi level, and Ak is the 2 x 2 gap matrix. We 
can diagonalize this su erconducting Hamiltonian by the 
Bogoliubov transformation45 in the form: 
ak = ukak, (10) 
where Uk is a 4 x 4 unitary matrix. 
state, then the gap matrix is exhibited: 
The left superconductor is assumed to have normal s-wave 
Ak= Aleie( -' ) , 
1 0  
in which Ek is the energy of the quasi-particle of the wave vector k 
in the s-wave superconductor. 
P-wave Superconductors 
We assume that the right p-wave superconductor has a Balian- 
Werthamer (BW) state.5 In this right electrode, we denote the wave 
vector by q and the annihilation operators of an electron and a 
quasi-particle by b and j?, respectively. The gap manix Aq can 
be written as 
where ei@ is a phase factor, Aq2 is a isotropic positive value that 
means the magnitude of the gap, x and I+Y are the usual latitude 
and longitude angles defining the direction of the wave vector q as 
show? in Fig. 2,and the functions Y1,m are spherical harmonics, 
and q denotes 4 = 41191. 
The Hamiltonian HR of the BW-state of the right electrode 
is 
where e i o  is a phase factor and A1 is a positive value which 
means the magnitude of the gap: 
1343 
and can be diagonalized by the Bogoliubov transformation as: 
where 
E,+% 4 2  (19) e w  = n { (u,+v&(o*-@sin x b-+qtb;T+vqcos Xeie'b-+qlblt) 
i2E,(E,+e,s v q = m J q = J  4 
(Uq+#@ '+@sin x b.+qLb,+l+~q~~~ Xeio'b-+,tb,+) 1 (22) 
and 
E,=- , (20) 
in which Eq is the energy of the quasi-particle of the wave vector 
in the BW-state superconductor. Ambiguity factors in coefficients 
of unitary matrix of the Bogoliubov transformation are removed so 
that &, can satisfy the relation Pqq + b,, when the absolute 
value of the wave vector q tends to infinite. 
In this section Bogoliubov transformation in the s-wave and P- 
wave superconductors has been obtained by the diagonalization of 
the superconducting Hamiltonian. The ground state of the BW- 
state superconductor at 0 K has also been obtained in the second- 
quantization formalism. 
Calculation of the JoseDhson Current 
We define F (k,r-r) and F '(q,t-t) as follows: 
X 
Figure 2. Representation of the direction of the wave vector q in  
the spherical coordinates. x and ty are the usual latitude and 
longitude angles defining the direction of the wave vector q. and 
With the following relations: 
PF '(q,t-t') 
TF '(q,t-t') = <b-,~(t)b,t(t)> = - F '(q,t-r'),(31a,b,c) 
<b-,T(r)b,L(r')> = - F '(q,r-t), 
PTF '(q,r-t) E <b,L(r)b,t(f)> = F '(q,r-r'), 
Especially at 0 K, the ground state of the BW-state where 
respectively, such as 
P and T are the parity and time reversal opertor,8 
superconductor satisfies the relations: 
('q,O) Pqo. @EW = 0 , (21) Pk?= -k?, Tk?= -k&, PTk?= k&. (32a,b,c) 
The dc Josephson current Js  of eq. (7) is calculated as follows: because no quasi-particle exists in the ground state at 0 K. From 
these relations we obtain the BW ground state at 0 K as follows: 
1344 
dr'e7t'Tk,T..q( - F (k,r-t) F'(q,r-r) - Ey; (k,r-t) PF'(q,t-t) - 7F (k,t-r) TF'(q,r-r) - PTF (k,t-t) PTF'(q,t-t) 
+ F(k, l - t )  F'(q,t'-t) + IF (k,t'-t) P$(q,t'-t) + TF(k,t'-t) TF'(q,t'-t) + PTF (k,t'-t) PTF'(q,r'-t)} 
With the relations (30) and (31), the magnitude of the Josephson 
current is zero in the fiist order perturbation. 
The BW state is, in general, degenerated; the state constructed 
by replacing Yl,m(G) in eq. (16) with Yl,,(RG) has the same energy, 
where R is a rotation in the momentum space only. The gap matrix 
of this degenerated state is as follows: 
where x 'and y ' are the latitude and longitude angles defining the 
direction of the wave vector q'= Rq. In this case, the Hamiltonian 
HR can be diagonalized by the following Bogoliubov 
transformation: 
References 
1. B.D.Josephson, ''Supercurrents Through Barriers", Adv. 
2. V.Ambegaokar and A.Baratoff ''Tunneling Between Super- 
conductors'', Phys. Rev. Lett., 10, 486-489, 1963, Errata, 11, 104, 
1963. 
3. N.R.Werthamer "Nonlinear Self-Couplig of Josephson Radi- 
ation in Superconducting Tunnel Junctions", Phys. Rev., 147, 255- 
263, 1966. 
4. N.N.Bogoliubov, "A New Method in the Theory of Super- 
conductivity", Sov. Phys. JETP, 7,41-46, 1958. 
5. R.Balian and N.R.Werthamer, "Superconductivity with Pairs 
in a Relative p Wave", Phys. Rev., 131, 1553-1564, 1963. 
6. S.Nakajima: Chodendo Nyumon ( Introduction to 
Superconductivity ) [in Japanese], Baifukan, Tokyo, 197 1. 
7. A.Barone and G. Patemo, Physics and Applicatiom of the 
Josephson Effect, Wiley-Interscience, New York, 1982. 
8. P.W.Anderson, ''Structure of "mplet" superconducting ener- 
gy gaps", Phys. Rev., B 30,4000-4002, 1984. 
Phys., 14,419-451, 1965. 
then the similar relations with eqs. (30)-(31) are obtained. Hence, 
also in this case Js  is zero in the first order perturbation. 
The degenerated ground state of the BW-state superconductor 
at 0 K can be obtained by replacing x and y in eq. (22) with the 
latitude angle x 'and the longitude angle y 'of the direction of the 
wave vector q' = Rq, respectively. 
Conclusions 
In conclusions, Josephson currents have been calculated with 
Bogoliubov transformations, which have been obtained by the 
diagonalization of the Hamiltonians of the p-wave and the s-wave 
superconducting electrodes, and proved to be zero in the first order 
perturbation in the p-wave/s-wave superconducting hetero- 
junctions at 0 K and at the finite temperature. Moreover, the 
ground state of the BW-state superconductor at 0 K has been 
obtained in the second-quantization formalism. 


CALCULATION OF THE JOSEPHSON CURRENT IN P-WAVE AND S-WAVE SUPERCONDUCTING HETERO-JUNCTIONS BY BOGOLIUBOV TRANSFORMATION

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CALCULATION OF THE JOSEPHSON CURRENT IN P-WAVE AND S-WAVE SUPERCONDUCTING HETERO-JUNCTIONS BY BOGOLIUBOV TRANSFORMATION

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