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The metal-oxide superconducting state symmetry and I–V characteristics of the tunnel junctions
Solid State Communications, Printed in Great Britain.
Vol. 78, No. 3, pp. 227-231, 1991.
0038-1098/91 $3.00 + .OO Pergamon Press plc
THE METAL-OXIDE SUPERCONDUCTING STATE SYMMETRY AND Z-l’ CHARACTERISTICS OF THE TUNNEL JUNCTIONS V.N. Kostur and S.E. Shafranjuk Institute for Metal Physics Academy of Sciences, Ukrainian S.S.R., USSR (Received 12 October 1989 by A. Pinczuk)
It is shown that while interpreting the shape of the I-l’-characteristics of metal-oxide superconductor (MOS) - isolator - superconductor tunnel junction, it is necessary to take into account the possibility of pairing with a nonzero orbital moment (or spin) in the MOS. The density of states for p- and d-pairing is calculated on the basis of the order parameter symmetry classification. ALONG with the searching for the microscopic pairing mechanisms in anisotropic superconductors, such as high temperature superconductors or heavy fermion superconductors, there is the number of questions concerning general properties of these superconductors which could be determined only by the internal symmetry of the superconductivity state and do not depend upon pairing mechanism. To answer this question one can consider the symmetry properties of the anisotropic superconducting order parameter [l] Act/I(p) = (c,,c_,~), where cpol= the annihilation operator of a superconducting pair. Then we shall consider the Fermi statistic case of these particles which (having a half-integer spin) can form singlet (s = 0) or triplet (s = 1) pairs. It should be mentioned that possibility of triplet pairing also was discussed for the heavy fermion superconductors (see [2, 31). Despite the experiments on Josephson effects the triplet pairing possibility for metal oxide cannot be excluded entirely because the phase coherence between singlet and triplet superconductors can be established for example by strong spin-orbital interaction or due to magnetically active interface [4-61. The wave function transformation properties of the particle pairs in the superconducting condensate, coinciding in meaning with order parameter A,,(p), determine special features of superconducting phase under the influence of different symmetry transformations. The symmetry analysis of possible states is made by the method given in [l]. Also the similar speculations were presented at calculation of the impurity effects on the p-wave superconductivity in [3]. The corresponding classification of the order parameter for high temperature superconductors was made in [7]. Particularly, in the mentioned work the type of the basis wave function of condensate is found at the finite values of the orbital moment of Cooper 227
pair 1 = 0,l and 2 for the point groups of the different representation. Further, we shall consider the case of group D2,,. For DZh there are four one-dimensional representations corresponding to every type of pairing (singlet or triplet). In generally used notations it is A,, B,, , B,, B,, for s = 0; 1 = 0, 2 . . and A,, B,, , Bau, B,, for s = 1; 1 = 1, 3 . . . where 1 is the pair orbital quantum number. Usually for the case of Cooper pairing in the presence of the spin-orbital interaction or inhomogeneousness a pair spin s turns to be a “bad” quantum number and that is why it is more correct to speak about parity. So, further we shall call the state with even 1 singlet and with odd 1 triplet. In our case the full group of symmetry consist of the point group of crystal symmetry DIh, operation of inversion time R and the gauge transformation group U(1). It is a well-known fact [l] that the superconductor order parameter (energy gap function) can be represented in general case as a matrix (2 * 2) on spin indexes, for singlet pairing (s = 0) : Ap = rl/(p)iay and for triplet pairing (s = 1) : Ar, = (d(p), b)iGy, where Ii/(p)-even scalar function, d(p)-odd vector function. In present work we shall consider the tunnel quasiparticle characteristics of high T, superconductors implying the different kinds of symmetry for superconducting condensate wave function. Similar calculations were performed in [8] for the isotropic BW-type p-wave superconductor and some results for the special cases of anisotropic pairing in oxides were presented in brief report [9] (and then for the d-wave pairing in [IO]). To calculate the single particle density of states of the anisotropic superconductor N,(o) below T, in more general suppositions and then Z-V characteristics of tunnel junction S-Z-S,, where S(S,) = the isotropic (anisotropic) superconductor it is necessary to take into account the order parameter symmetry. It
228
THE METAL-OXIDE
SUPERCONDUCTING
is convenient to take into account the anomaly average (singlet or triplet) while computing the full Green function G(o) with the help of the matrix representation. The necessary calculations can be made at quite general suppositions as to pairing mechanism. To make it more obvious in further calculations we shall neglect the renormalization effects of the single electron spectra and electrochemical potential shift for the frequencies changing on the energy gap scale AaS(p). In our task the metering of the electrochemical potential delay and shift can be done in parallel with taking into account the anisotropy effects. Thus the full electron Green function can be represented in our model in the following way: G,(o)
=
(a + &V’(A)
(1)
in which P(8)
=
(w’ -
<,z -
x (w2 + ?$ -
@(A+ 8) + A+ A)/[o’
@(A+ A)) + det (A’ A)]
-
t;)
=
(N(O)/271) Im SJJ j d&
(3)
(N(0) = the density of electronic states at Fermi level, G,‘(o) = the retarding Green function and (. . .)p = the averaging at angle of electron impulse). Using formulas (l)-(3), we shall get the following expression for density of states N(o)
=
x (co -
A&))/(&
‘y(p)
-
A;(p))“’
=
a&&z
‘P(p)
u,,k,k,.
=
=
NO) j (d2p/v,) x (w’ -
where
the
C o&w *
A;(p))“’
following
-
d(p)
=
(S = 1) symbols
(5) are
used:
=
a, + a,e’“kz + a,e’“(kL: -
J$)
(6)
and contains five real parameters, depending on Ip] : ai > 0; 0 < a, CC’< 271. Then we shall consider
+ b2elrkJe,. + h,k,e,
d(p)
=
k,[b,(e,. + ie;)/2”*
d(p)
=
k,[b,(ie,
d(p)
=
k,[&(e,
(8)
+ b,e’“e,)
+ e,)/2”* + b,e18e,,) + ie,)/2”2
+ b,e@e,).
(9)
These phases are similar to polar phase 3He. Thus, the form of the single particle density of states in S for each representation A,, B, p3g, A,, B, _3ucan be found with the account of obvious form of basic functions (6)-(9). In a general case the pairing with a nonzero orbital moment, as a rule, results in sufficient anisotropy of superconducting order parameter [l, 71. We shall further consider the idealized situation, when order parameter in area S, making the main contribution to the tunnel current, can be considered as homogeneous. Standard speculations lead to the following expression for the quasi-particle tunnel current:
=
s
(~*cAJj (d2dyJ4p, 4) (1/2~2eRn) X
j
dw [f(o) - f(w + eV1 Im j d5,
x Cd (co + eV) Im jd$SpG,‘(w)
(10)
where R,, = junction resistance in normal state; q and p = the electron moments in S and S, accordingly; f(o) = the distribution Fermi function; A(p, q) = the non-negative function describing the tunnelling matrix element anisotropy; (1(p, q))p,4 = 1. As far as in our case superconductivity in S can be considered to be isotropic, we can use the function A(p) = (I@, q))4. Then expression (10) can be written as:
A,,(P) = IVP)L A,(P) = (IMP)? f [d*(pM(p)lI)“*. Parameters A,,(p) and A*(p) can be calculated for each of the representations of group D2,,. In case of the singlet pairing the basic function for one-dimensional representation A, when the angular moment 1 = 0, 2 1, can be written as the following (k = p/lpi): ‘Y(p)
b,e’Bk,e,
also contains five real parameters. The basic functions Blu, B2, and B,, are the following:
(s = 0)
A+(P))/
(7)
In the case of triplet pairing with spin orbital interaction the basic function of one-dimensional representation A, when 1 = I:
(4) N,(o)
Vol. 78, No. 3
a&k;
‘Y(P) =
I,
NO) j (~‘P/~p)~~
SYMMETRY
special cases when one or several parameters are equal to zero. Basic function for the representations B,,, BZK, B3Kare accordingly written down in the following way:
(2)
where SJJ means the operation of the taking trace and det means determinant. We consider the order parameter anisotropy more essential than other kinds of anisotropy (for example, the Fermi surface anisotropy). Then single-particle density of states of the anisotropic superconductors is calculated by formula J%(o)
STATE
=
(l/e&)
j do[f(o)
- f(o
+ eU1
x (Iw + eVI - A) x ((lo
+ eVl/((o
+ eV)’ -
Az)“2)N,(o) (11)
Vol. 78, No. 3
THE METAL-OXIDE
SUPERCONDUCTING
where A = the energy gap in S film, and the tunnel density of electron states N,(o) normalized on the density of states above T,, has the following form N,(w)
=
j sinq dq d4 F-‘(8, x oq(o
N,(O) = j
-
A&
4)&e) ~))/[o’
sin 8 de db I--‘(&
x we(w -
-
4)n(e)
A;(&
4)1”* (12)
1 -*A*+
A+(e, +))/]o*
(e, Cj)]“2 (13)
where F(0, 4) = the Gauss curvature of the Fermi surface at the point at the angle coordinates q, jI A(0) determined by the angular dependence of the barrier transparency. The dependence can be investigated on different model mechanisms for electron transfer through tunnel barrier. A0(8, 4) and A+(8, 4) are the order parameter values on Fermi surface. Then we shall assume that order parameter angular dependence is more sufficient than angular dependence of the tunnel matrix elements. Besides, we shall assume that near Fermi surface the order parameter dependence on IpI is weak. Then special features of the density of states N,(w) will develop in tunnel density N,(o) sufficiently noticeably and, consequently, superconducting order parameter symmetry type will play an important role in the calculation of the tunnel junctions characteristics on the basis of polycrystal or single-crystal anisotropic superconductor. While studying the behavior and dependence of single-particle density of states on frequency it is convenient to introduce anisotropy gap function m,(z) (see for example [12, 131) for singlet pairing Q(z)
=
j sin 0 de d4 r-‘(0,
c$)~(z
-
and for triplet pairing
r-‘(8, c#I)zC+~(~- A:
Then the single-particle density of states for anisotropic superconductor No(w), normalized on the density of states above the critical temperature can be written as: N(o)
=
j: dz(o/[o*
-
z*]“‘)@(z).
(16)
0
first, let us consider the case with singlet order parameter. The function F(z) differs from zero only in the case, when z changes in the limits from A, to AM,
At
SYMMETRY
229
where A, = minimum, AM = maximum value of the anisotropic energy gap A(0, 4). It is not difficult to show that for all cases of singlet pairing, when the order parameter is determined by the basic function of one from representations B,,, B2K, B,, (i.e. it is described by expressions (7) anisotropy function Q(z) and consequently, density of states N,(o) has the same form and it is a particular case of anisotropy function and density of states accordingly for A, representation. At the same time A,(e, 4) has two lines of zeros on the Fermi surface and apparently, the minimum gap value is equal to zero. Then for the density of states we find: N,(o)
=
(2/n) j dl
ve(v - /l)/(v’ _ ;12)‘/2
0
x (l/(1: + Iz)“‘)K([(l
-
A)/(1 + J)]“‘) (17)
where v = m/AM, and K(k) = is the full elliptic integral of the first kind. Curve 1 on the Fig. 1 is the density of states frequency dependence. For this curve A, = AM. There is the following asymptotic behavior near typical points: v -+ 0, then N, + v In v, when v + 1 + 0, N, + c; - (v - 1)“‘. If o tends to A,,,, on the left, N,(w) behaves as N, + c; x c;(v - I), where CAand c; are the constants. From the above expressions it is clear, that function dNJo)/do has some special features, namely divergences near o = 0 (logarithmic) and to the right of w = A,,,, (rootness, N (w - AM))‘/*). Now let us take the case when superconducting order parameter changes by representation A, from D2,,group. We shall limit ourselves to consideration of axial-symmetric order parameter of this representation Ao(e> =
A,,(& 4)) (14)
Q(Z) = jsin eded
STATE
Ia, + ale’”
cos 281
a = q/a0
(18)
at the beginning we shall put phase a equal to zero. The density of states frequency dependence for this case is shown on Fig. 1 (curve 2) when Al = A,,,, A2 = A,,,. The case, when c1 = rc and the value a < 1 corresponds to curve 3 with the same values Al = A,,,, A, = AM. The divergence near AM has a logarithmic character. The function dN, (o)/dw to the right of the point o = A,,, has the rootness divergence. When parameter c1 > l(a = rr), the minimum value of gap is zero and density of states differs from zero when o case when) 0. Curve 4 is draw for the value a = 1, A, = A,,, . For comparison the density of states for an isotropic superconductor is shown as dotted line on Fig. 1. The behavior N,(o) described in expression (19) is shown on Fig. 2 at values 1 < a < 2 = curve 5 and a > 2 curve 6 (AI is the value of A”(O) and A, = Ao(7c/2) for curve 5, at the time when for curve
230
THE METAL-OXIDE
SUPERCONDUCTING
STATE
SYMMETRY
Vol. 78, No. 3
Fig. 3.
Fig. 1.
6 A, = A,(rc/2), AZ = A&O)). The triplet order parameter leads to the following dependencies (Fig. 3): curve 7 for order parameter changing by representations B,.-B,, of Dzhgroup, curve 8 for A, representation (axial symmetric case) when A, = maximum of A-(0, 4), and A, = max A+(t), 4). For convenience A+ = max A+(0, 4). The divergence at o = A+ for curve 8 is proportional to In (w/A+) - 1 I. Besides, on the same figure there is a plot (curve 9) for “inert” phase of superconducting order parameter. The most simple case is depicted on curve 10 when A_ = 0. The qualitative difference of this case consists in the fact that at frequency tending to zero the density of states N,(o) differs from zero. It is easy to be sure and convince oneself, that for a singlet case such a situation, in principle, is impossible if A,,(p) is finite on Fermi surface. An asymptotic behavior N,(w) in the area of points o = 0, o = A,, o = A, for curves in Table 1, where we used the l-10 is systematized = v, - 1, i = 1, 2; following symbols: v, = o/A,;&,
Fig. 2.
constants c!‘) are determined by huge expressions and we shall not present them. It is necessary to mention that for curves 1 and 4 A, = A,,,(A,,, = 0); for curves 2 and 3 A, = A,,,, A2 = AM; for 5 A, = A,,(O), A, = A0(7r/2), for 6 A, = A,(rc/2), A, = A,,(O); for curves 7, 8 and 9 A, = A_ ; AZ = A+ and for 10 A, = A+ (A = 0). From the above calculations it is clear that the tunnel density of states of anisotropic superconductor can contain a number of special features, which qualitatively differ from special features in the isotropic superconductor density of states. These special features develop only at the frequency values which are equal to the parameters determined by gap anisotropy. At these points tunnel density of states or its derivative can contain divergences of two types: rootness and logarithmic. At A, = 0 (zero points or lines are on Fermi surface), tunnel density of states in singlet case differs from zero at frequencies o > 0 and always tends to zero at w -+ 0. On the other hand, in a triplet case the situation is possible (for example, at A. = 0) when density of states has a finite value even at w = 0. Before we have considered the ideal situation with the homogeneous order parameter inside oxide superconductor. The shape of characteristics calculated above would be true only for case when the tunnelling current is dominated by bulk electronic states. To show the role of the order parameter inhomogeneity near the tunnel junction wall let us consider the siniplest case of the specularly reflecting surface for the p-wave pairing when the perpendicular component of the order parameter tends to zero as it approaches the wall. In considering the situation to obtain the dependence d:(k, z) (we assume z-direction is perpendicular to wall and take into account the quasiclassical condition a/& < 1; z = the “slow” coordinate) it is necessary to solve the self-consistency equation together with quasiclassical equations for Green function G,(w). The solution roughly speaking can be rep-
Vol. 78, No. 3
THE METAL-OXIDE
SUPERCONDUCTING
resented as d=(k, z) = (d,,;(k)/4&,)z at z d 4& and d,(k, z) = d,,) = d,,,(k) at z > 45,(&(k) - the bulk value of d,(k)). Such solution leads to the some increasing of the N,(w) at w < A, in comparison with the homogeneous case that means the coordinate dependence of the triplet pairing order parameter leads to possibility of the additional excitation below “energy gap”. Unfortunately the above approach is not applicable in case when I, % &, N a and when the quasiclassical equations are false. We should note that last case claims the special consideration and accepts the alternative in interpretation of the tunnelling data. The effect of the nonmagnetic impurities on the tunnel density of states in isotropic triplet pairing case have been considered in [8]. The analogous calculation here for anisotropic triplet pairing showed the same especialities: the smearing of the sharp peaks and the arising of impurity bands. One can hope, that the comparison of the special features of the tunnel density of states found as a result of the symmetry classification of superconducting order parameter with the special features of the experimentally found tunnel characteristics of clean single crystal samples, will make it possible to get the information of pairing type and symmetry of anisotropic superconductor order parameter.
- The authors are thankful for his attention and discussions.
to
SYMMETRY
231
REFERENCES 1. 2. 3.
4.
5. 6.
;: 9.
:;: 12. 13. 14. 15.
Acknowledgement V.G. Baryakhtar
STATE
16.
G.E. Volovik & L.P. Gor’kov, Zh. Eksp. Teor. Fiz. 88, 1412 (1985). C.M. Varma, Bull. Am. Phys. Sot. 29, 857 (1984). K.Ueda & T.M. Rice, in Theory of Heavy Fermions and Valence Fluctuations (Edited by T. Kasuya & T. Sago), Springer, Berlin (1985). E.W. Fenton, Solid State Commun. 54, 709 (1985); E.W. Fenton, Solid State Commun. 60, 347 (1986). V.B. Geshkenbein & A.I. Larkin, JETP Lett. 43, 395 (1986). A. Millis, D. Rainer & J.A. Sauls, Phys. Rev. B38, 4504 (1988). M. Siegrist & T.M. Rice, Z. Phys. B 68,9 (1987). L.J. Buchholtz & G. Zwicknagl, Phys. Rev. B23, 5788 (1981). d V.N. Kostur & S.E. Shafranjuk, Ukrainian Physics Journal 34, 224 (229), (in Russian). G.D. Mahan, Phys. Rev. B&l, 11317 (1989). M. Tinkham, Group Theory and Quantum Mechanics, McGraw-Hill (1964). A.J. Bennet, Phys. Rev. 140, 1902 (1965). A.E. Gorbonosov & 1.0. Kulik, Zh. Eksp. Teor. Fiz. 55, 876 (1968). V. Ambegaokar, P.G. de Gennes & D. Rainer, Phys. Rev. A9, 2676 (1974). L.J. Buchholtz & D. Rainer, Physik B35, 151 (1979). B. Ashauer, G. Kieselmann & D. Rainer, J. Low Temp. Phys. 63, 349 (1986).
Vol. 78, No. 3, pp. 227-231, 1991.
0038-1098/91 $3.00 + .OO Pergamon Press plc
THE METAL-OXIDE SUPERCONDUCTING STATE SYMMETRY AND Z-l’ CHARACTERISTICS OF THE TUNNEL JUNCTIONS V.N. Kostur and S.E. Shafranjuk Institute for Metal Physics Academy of Sciences, Ukrainian S.S.R., USSR (Received 12 October 1989 by A. Pinczuk)
It is shown that while interpreting the shape of the I-l’-characteristics of metal-oxide superconductor (MOS) - isolator - superconductor tunnel junction, it is necessary to take into account the possibility of pairing with a nonzero orbital moment (or spin) in the MOS. The density of states for p- and d-pairing is calculated on the basis of the order parameter symmetry classification. ALONG with the searching for the microscopic pairing mechanisms in anisotropic superconductors, such as high temperature superconductors or heavy fermion superconductors, there is the number of questions concerning general properties of these superconductors which could be determined only by the internal symmetry of the superconductivity state and do not depend upon pairing mechanism. To answer this question one can consider the symmetry properties of the anisotropic superconducting order parameter [l] Act/I(p) = (c,,c_,~), where cpol= the annihilation operator of a superconducting pair. Then we shall consider the Fermi statistic case of these particles which (having a half-integer spin) can form singlet (s = 0) or triplet (s = 1) pairs. It should be mentioned that possibility of triplet pairing also was discussed for the heavy fermion superconductors (see [2, 31). Despite the experiments on Josephson effects the triplet pairing possibility for metal oxide cannot be excluded entirely because the phase coherence between singlet and triplet superconductors can be established for example by strong spin-orbital interaction or due to magnetically active interface [4-61. The wave function transformation properties of the particle pairs in the superconducting condensate, coinciding in meaning with order parameter A,,(p), determine special features of superconducting phase under the influence of different symmetry transformations. The symmetry analysis of possible states is made by the method given in [l]. Also the similar speculations were presented at calculation of the impurity effects on the p-wave superconductivity in [3]. The corresponding classification of the order parameter for high temperature superconductors was made in [7]. Particularly, in the mentioned work the type of the basis wave function of condensate is found at the finite values of the orbital moment of Cooper 227
pair 1 = 0,l and 2 for the point groups of the different representation. Further, we shall consider the case of group D2,,. For DZh there are four one-dimensional representations corresponding to every type of pairing (singlet or triplet). In generally used notations it is A,, B,, , B,, B,, for s = 0; 1 = 0, 2 . . and A,, B,, , Bau, B,, for s = 1; 1 = 1, 3 . . . where 1 is the pair orbital quantum number. Usually for the case of Cooper pairing in the presence of the spin-orbital interaction or inhomogeneousness a pair spin s turns to be a “bad” quantum number and that is why it is more correct to speak about parity. So, further we shall call the state with even 1 singlet and with odd 1 triplet. In our case the full group of symmetry consist of the point group of crystal symmetry DIh, operation of inversion time R and the gauge transformation group U(1). It is a well-known fact [l] that the superconductor order parameter (energy gap function) can be represented in general case as a matrix (2 * 2) on spin indexes, for singlet pairing (s = 0) : Ap = rl/(p)iay and for triplet pairing (s = 1) : Ar, = (d(p), b)iGy, where Ii/(p)-even scalar function, d(p)-odd vector function. In present work we shall consider the tunnel quasiparticle characteristics of high T, superconductors implying the different kinds of symmetry for superconducting condensate wave function. Similar calculations were performed in [8] for the isotropic BW-type p-wave superconductor and some results for the special cases of anisotropic pairing in oxides were presented in brief report [9] (and then for the d-wave pairing in [IO]). To calculate the single particle density of states of the anisotropic superconductor N,(o) below T, in more general suppositions and then Z-V characteristics of tunnel junction S-Z-S,, where S(S,) = the isotropic (anisotropic) superconductor it is necessary to take into account the order parameter symmetry. It
228
THE METAL-OXIDE
SUPERCONDUCTING
is convenient to take into account the anomaly average (singlet or triplet) while computing the full Green function G(o) with the help of the matrix representation. The necessary calculations can be made at quite general suppositions as to pairing mechanism. To make it more obvious in further calculations we shall neglect the renormalization effects of the single electron spectra and electrochemical potential shift for the frequencies changing on the energy gap scale AaS(p). In our task the metering of the electrochemical potential delay and shift can be done in parallel with taking into account the anisotropy effects. Thus the full electron Green function can be represented in our model in the following way: G,(o)
=
(a + &V’(A)
(1)
in which P(8)
=
(w’ -
<,z -
x (w2 + ?$ -
@(A+ 8) + A+ A)/[o’
@(A+ A)) + det (A’ A)]
-
t;)
=
(N(O)/271) Im SJJ j d&
(3)
(N(0) = the density of electronic states at Fermi level, G,‘(o) = the retarding Green function and (. . .)p = the averaging at angle of electron impulse). Using formulas (l)-(3), we shall get the following expression for density of states N(o)
=
x (co -
A&))/(&
‘y(p)
-
A;(p))“’
=
a&&z
‘P(p)
u,,k,k,.
=
=
NO) j (d2p/v,) x (w’ -
where
the
C o&w *
A;(p))“’
following
-
d(p)
=
(S = 1) symbols
(5) are
used:
=
a, + a,e’“kz + a,e’“(kL: -
J$)
(6)
and contains five real parameters, depending on Ip] : ai > 0; 0 < a, CC’< 271. Then we shall consider
+ b2elrkJe,. + h,k,e,
d(p)
=
k,[b,(e,. + ie;)/2”*
d(p)
=
k,[b,(ie,
d(p)
=
k,[&(e,
(8)
+ b,e’“e,)
+ e,)/2”* + b,e18e,,) + ie,)/2”2
+ b,e@e,).
(9)
These phases are similar to polar phase 3He. Thus, the form of the single particle density of states in S for each representation A,, B, p3g, A,, B, _3ucan be found with the account of obvious form of basic functions (6)-(9). In a general case the pairing with a nonzero orbital moment, as a rule, results in sufficient anisotropy of superconducting order parameter [l, 71. We shall further consider the idealized situation, when order parameter in area S, making the main contribution to the tunnel current, can be considered as homogeneous. Standard speculations lead to the following expression for the quasi-particle tunnel current:
=
s
(~*cAJj (d2dyJ4p, 4) (1/2~2eRn) X
j
dw [f(o) - f(w + eV1 Im j d5,
x Cd (co + eV) Im jd$SpG,‘(w)
(10)
where R,, = junction resistance in normal state; q and p = the electron moments in S and S, accordingly; f(o) = the distribution Fermi function; A(p, q) = the non-negative function describing the tunnelling matrix element anisotropy; (1(p, q))p,4 = 1. As far as in our case superconductivity in S can be considered to be isotropic, we can use the function A(p) = (I@, q))4. Then expression (10) can be written as:
A,,(P) = IVP)L A,(P) = (IMP)? f [d*(pM(p)lI)“*. Parameters A,,(p) and A*(p) can be calculated for each of the representations of group D2,,. In case of the singlet pairing the basic function for one-dimensional representation A, when the angular moment 1 = 0, 2 1, can be written as the following (k = p/lpi): ‘Y(p)
b,e’Bk,e,
also contains five real parameters. The basic functions Blu, B2, and B,, are the following:
(s = 0)
A+(P))/
(7)
In the case of triplet pairing with spin orbital interaction the basic function of one-dimensional representation A, when 1 = I:
(4) N,(o)
Vol. 78, No. 3
a&k;
‘Y(P) =
I,
NO) j (~‘P/~p)~~
SYMMETRY
special cases when one or several parameters are equal to zero. Basic function for the representations B,,, BZK, B3Kare accordingly written down in the following way:
(2)
where SJJ means the operation of the taking trace and det means determinant. We consider the order parameter anisotropy more essential than other kinds of anisotropy (for example, the Fermi surface anisotropy). Then single-particle density of states of the anisotropic superconductors is calculated by formula J%(o)
STATE
=
(l/e&)
j do[f(o)
- f(o
+ eU1
x (Iw + eVI - A) x ((lo
+ eVl/((o
+ eV)’ -
Az)“2)N,(o) (11)
Vol. 78, No. 3
THE METAL-OXIDE
SUPERCONDUCTING
where A = the energy gap in S film, and the tunnel density of electron states N,(o) normalized on the density of states above T,, has the following form N,(w)
=
j sinq dq d4 F-‘(8, x oq(o
N,(O) = j
-
A&
4)&e) ~))/[o’
sin 8 de db I--‘(&
x we(w -
-
4)n(e)
A;(&
4)1”* (12)
1 -*A*+
A+(e, +))/]o*
(e, Cj)]“2 (13)
where F(0, 4) = the Gauss curvature of the Fermi surface at the point at the angle coordinates q, jI A(0) determined by the angular dependence of the barrier transparency. The dependence can be investigated on different model mechanisms for electron transfer through tunnel barrier. A0(8, 4) and A+(8, 4) are the order parameter values on Fermi surface. Then we shall assume that order parameter angular dependence is more sufficient than angular dependence of the tunnel matrix elements. Besides, we shall assume that near Fermi surface the order parameter dependence on IpI is weak. Then special features of the density of states N,(w) will develop in tunnel density N,(o) sufficiently noticeably and, consequently, superconducting order parameter symmetry type will play an important role in the calculation of the tunnel junctions characteristics on the basis of polycrystal or single-crystal anisotropic superconductor. While studying the behavior and dependence of single-particle density of states on frequency it is convenient to introduce anisotropy gap function m,(z) (see for example [12, 131) for singlet pairing Q(z)
=
j sin 0 de d4 r-‘(0,
c$)~(z
-
and for triplet pairing
r-‘(8, c#I)zC+~(~- A:
Then the single-particle density of states for anisotropic superconductor No(w), normalized on the density of states above the critical temperature can be written as: N(o)
=
j: dz(o/[o*
-
z*]“‘)@(z).
(16)
0
first, let us consider the case with singlet order parameter. The function F(z) differs from zero only in the case, when z changes in the limits from A, to AM,
At
SYMMETRY
229
where A, = minimum, AM = maximum value of the anisotropic energy gap A(0, 4). It is not difficult to show that for all cases of singlet pairing, when the order parameter is determined by the basic function of one from representations B,,, B2K, B,, (i.e. it is described by expressions (7) anisotropy function Q(z) and consequently, density of states N,(o) has the same form and it is a particular case of anisotropy function and density of states accordingly for A, representation. At the same time A,(e, 4) has two lines of zeros on the Fermi surface and apparently, the minimum gap value is equal to zero. Then for the density of states we find: N,(o)
=
(2/n) j dl
ve(v - /l)/(v’ _ ;12)‘/2
0
x (l/(1: + Iz)“‘)K([(l
-
A)/(1 + J)]“‘) (17)
where v = m/AM, and K(k) = is the full elliptic integral of the first kind. Curve 1 on the Fig. 1 is the density of states frequency dependence. For this curve A, = AM. There is the following asymptotic behavior near typical points: v -+ 0, then N, + v In v, when v + 1 + 0, N, + c; - (v - 1)“‘. If o tends to A,,,, on the left, N,(w) behaves as N, + c; x c;(v - I), where CAand c; are the constants. From the above expressions it is clear, that function dNJo)/do has some special features, namely divergences near o = 0 (logarithmic) and to the right of w = A,,,, (rootness, N (w - AM))‘/*). Now let us take the case when superconducting order parameter changes by representation A, from D2,,group. We shall limit ourselves to consideration of axial-symmetric order parameter of this representation Ao(e> =
A,,(& 4)) (14)
Q(Z) = jsin eded
STATE
Ia, + ale’”
cos 281
a = q/a0
(18)
at the beginning we shall put phase a equal to zero. The density of states frequency dependence for this case is shown on Fig. 1 (curve 2) when Al = A,,,, A2 = A,,,. The case, when c1 = rc and the value a < 1 corresponds to curve 3 with the same values Al = A,,,, A, = AM. The divergence near AM has a logarithmic character. The function dN, (o)/dw to the right of the point o = A,,, has the rootness divergence. When parameter c1 > l(a = rr), the minimum value of gap is zero and density of states differs from zero when o case when) 0. Curve 4 is draw for the value a = 1, A, = A,,, . For comparison the density of states for an isotropic superconductor is shown as dotted line on Fig. 1. The behavior N,(o) described in expression (19) is shown on Fig. 2 at values 1 < a < 2 = curve 5 and a > 2 curve 6 (AI is the value of A”(O) and A, = Ao(7c/2) for curve 5, at the time when for curve
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Vol. 78, No. 3
Fig. 3.
Fig. 1.
6 A, = A,(rc/2), AZ = A&O)). The triplet order parameter leads to the following dependencies (Fig. 3): curve 7 for order parameter changing by representations B,.-B,, of Dzhgroup, curve 8 for A, representation (axial symmetric case) when A, = maximum of A-(0, 4), and A, = max A+(t), 4). For convenience A+ = max A+(0, 4). The divergence at o = A+ for curve 8 is proportional to In (w/A+) - 1 I. Besides, on the same figure there is a plot (curve 9) for “inert” phase of superconducting order parameter. The most simple case is depicted on curve 10 when A_ = 0. The qualitative difference of this case consists in the fact that at frequency tending to zero the density of states N,(o) differs from zero. It is easy to be sure and convince oneself, that for a singlet case such a situation, in principle, is impossible if A,,(p) is finite on Fermi surface. An asymptotic behavior N,(w) in the area of points o = 0, o = A,, o = A, for curves in Table 1, where we used the l-10 is systematized = v, - 1, i = 1, 2; following symbols: v, = o/A,;&,
Fig. 2.
constants c!‘) are determined by huge expressions and we shall not present them. It is necessary to mention that for curves 1 and 4 A, = A,,,(A,,, = 0); for curves 2 and 3 A, = A,,,, A2 = AM; for 5 A, = A,,(O), A, = A0(7r/2), for 6 A, = A,(rc/2), A, = A,,(O); for curves 7, 8 and 9 A, = A_ ; AZ = A+ and for 10 A, = A+ (A = 0). From the above calculations it is clear that the tunnel density of states of anisotropic superconductor can contain a number of special features, which qualitatively differ from special features in the isotropic superconductor density of states. These special features develop only at the frequency values which are equal to the parameters determined by gap anisotropy. At these points tunnel density of states or its derivative can contain divergences of two types: rootness and logarithmic. At A, = 0 (zero points or lines are on Fermi surface), tunnel density of states in singlet case differs from zero at frequencies o > 0 and always tends to zero at w -+ 0. On the other hand, in a triplet case the situation is possible (for example, at A. = 0) when density of states has a finite value even at w = 0. Before we have considered the ideal situation with the homogeneous order parameter inside oxide superconductor. The shape of characteristics calculated above would be true only for case when the tunnelling current is dominated by bulk electronic states. To show the role of the order parameter inhomogeneity near the tunnel junction wall let us consider the siniplest case of the specularly reflecting surface for the p-wave pairing when the perpendicular component of the order parameter tends to zero as it approaches the wall. In considering the situation to obtain the dependence d:(k, z) (we assume z-direction is perpendicular to wall and take into account the quasiclassical condition a/& < 1; z = the “slow” coordinate) it is necessary to solve the self-consistency equation together with quasiclassical equations for Green function G,(w). The solution roughly speaking can be rep-
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resented as d=(k, z) = (d,,;(k)/4&,)z at z d 4& and d,(k, z) = d,,) = d,,,(k) at z > 45,(&(k) - the bulk value of d,(k)). Such solution leads to the some increasing of the N,(w) at w < A, in comparison with the homogeneous case that means the coordinate dependence of the triplet pairing order parameter leads to possibility of the additional excitation below “energy gap”. Unfortunately the above approach is not applicable in case when I, % &, N a and when the quasiclassical equations are false. We should note that last case claims the special consideration and accepts the alternative in interpretation of the tunnelling data. The effect of the nonmagnetic impurities on the tunnel density of states in isotropic triplet pairing case have been considered in [8]. The analogous calculation here for anisotropic triplet pairing showed the same especialities: the smearing of the sharp peaks and the arising of impurity bands. One can hope, that the comparison of the special features of the tunnel density of states found as a result of the symmetry classification of superconducting order parameter with the special features of the experimentally found tunnel characteristics of clean single crystal samples, will make it possible to get the information of pairing type and symmetry of anisotropic superconductor order parameter.
- The authors are thankful for his attention and discussions.
to
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231
REFERENCES 1. 2. 3.
4.
5. 6.
;: 9.
:;: 12. 13. 14. 15.
Acknowledgement V.G. Baryakhtar
STATE
16.
G.E. Volovik & L.P. Gor’kov, Zh. Eksp. Teor. Fiz. 88, 1412 (1985). C.M. Varma, Bull. Am. Phys. Sot. 29, 857 (1984). K.Ueda & T.M. Rice, in Theory of Heavy Fermions and Valence Fluctuations (Edited by T. Kasuya & T. Sago), Springer, Berlin (1985). E.W. Fenton, Solid State Commun. 54, 709 (1985); E.W. Fenton, Solid State Commun. 60, 347 (1986). V.B. Geshkenbein & A.I. Larkin, JETP Lett. 43, 395 (1986). A. Millis, D. Rainer & J.A. Sauls, Phys. Rev. B38, 4504 (1988). M. Siegrist & T.M. Rice, Z. Phys. B 68,9 (1987). L.J. Buchholtz & G. Zwicknagl, Phys. Rev. B23, 5788 (1981). d V.N. Kostur & S.E. Shafranjuk, Ukrainian Physics Journal 34, 224 (229), (in Russian). G.D. Mahan, Phys. Rev. B&l, 11317 (1989). M. Tinkham, Group Theory and Quantum Mechanics, McGraw-Hill (1964). A.J. Bennet, Phys. Rev. 140, 1902 (1965). A.E. Gorbonosov & 1.0. Kulik, Zh. Eksp. Teor. Fiz. 55, 876 (1968). V. Ambegaokar, P.G. de Gennes & D. Rainer, Phys. Rev. A9, 2676 (1974). L.J. Buchholtz & D. Rainer, Physik B35, 151 (1979). B. Ashauer, G. Kieselmann & D. Rainer, J. Low Temp. Phys. 63, 349 (1986).