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Manifestation of superconducting gap symmetry in the optical spectrum
14 November 1994 PHYSICS LETTERS A
El XEV!ER
Physics Letters A 194 (1994) 405-412
Manifestation of superconducting gap symmetry in the optical spectrum Yurii E. Lozovik 1, Andrei V. Poushnov Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region ~42092, Russian Federation
Received 2 August i994; accepted for publication 23 September 1994 Communicatedby V.M. Agranovich
Abstract The manifestation of the superconducting gap symmetry in optical spectra at frequencies w >> A,/h corresponding to interband transitions is investigated. The problem is treated in the framework of the BCS model with mixed sd-wave pairing. The characteristic structure of the imaginary part of the dieleeLric function depending on the symmetry of the superconducting order parameter is obtained. The cases when the imaginary part of the dielectric function has a two-peak structure near the Fermi level are considered, in these cases complete information about the superconducting order parameter may be obtained from the difference optical spectra extracted from femto~oecondlaser spectroscopy data.
1. Introduction The nature of the superconducting state and the superconducting order parameter in high-Tc superconductors is still a matter o f controversy. Theoretical model studies (see Refs. [ 1-3 ] ) have indicated that the superconducting gap of high-To superconductors might have a d-wave symmetry. Different models of d-wave superconductors were considered to describe the properties of cuprate oxide materials in the superconducting state [ 4 - 6 ] . Although many experimental results can be interpreted in the framework of some d-wave superconductor model [ 7 - 9 ], a lot of experiments [ 10,11 ] call d- wave symmetry of the superconducting order parameter into question. Most o f the traditional experiments devoted to the investigation of the superconducting energy gap anisotropy (for example, tunneling spectroscopy) are very sensible to the quality of a sample's surface. Thermomodulational spectroscopy [ 12-14] promises to be an excellent bulk probe of the superconducting order parameter. This technique has been used in femtosecond laser pump-probe experiments [ 15,16] for the investigation of the YBa2CH307_~ film reflectivity in the optical region related to interband transitions. The difference spectra of the imaginary part of the dielectric function o f pumped and unpumped YBazCu3OT-a films both above and below the critical temperature Te have been extracted from the experimental results. The structure o f the difference spectra near the Fermi level (FL) was interpreted in Ref. [ 16] as a possible manifestation of the I E-mail: lozovik@~san.msk.su. 0375-9601/94/$07.00 (~) 1994 Elsevier Science B.V. All righls reserved SSD10375-9601 (94) 00789-6
Yu.E. Lozovik.A. K Poushnov/ PhysicsLettersA 194 (1994)405-412
406
superconducting gap. A theoreticM study of this problem for s-wave superconductors [17] confirmed this assumption. In the present paper we study the possible manifestation of an anisotropy of the superconducting energy gap in the optical spectra for frequencies ~o ~2 Ao/h corresponding to trans:.tions from a certain valence band to the conduction band. We show that the information about the type of symmetry of the superconducting gap may be obtained from spectra of the imaginary part of the dielectric function near the FL obtained by pump-probe femtosecond spectroscopy. In particular, we have found the cases of mixed sd-wave pairing when a two-peak structure appears in the spectra of the imaginary part of the dielectric function near the FL. In these c~ses not only relative s-wave and d-wave contributions in the superconducting order parameter but also their possible relative phase may be obtained. The differences in absorption spectra of pumped and unpurnped samples can be investigated by femtosecond laser spectroscopy; the contribution of transitions from a valence band to the region near the FL in the absorption spectrum may be extracted. Subtracting the difference spectrum of the pumped and unpumped system at T > Tc from that at T < Tc gives the characteristic structure near the FL. In our point of view, it enables one, in particular, to investigate the superconducting gap symmetry by femtosecond laser spectroscopy. For simplicity we analyze the optical properties of the model system in the framework of the BCS theory with the anisotropic order parameter /t(p) = A0(s + d ei/~cos 2q5),
( 1)
where s and d (s > 0 , d > 0, s + d = 1) are the relative contributions of the s-wave and d-wave pairing interactions and/3 is their relative phase; Ao is the energy constant, depending on temperature T. At s = 0, Eq. (1) describes the gap symmetry investiga~.ed in Refs. [4,5]. When s = d a n d / 3 = ,r/2, the superconducting order parameter is the same as that discussed in Refs. [2,6]. The rest of the paper is organized as follows. In Section 2 interband transitions between bands with different effective masses (including different signs of tl2:e effective masses) are considered. It is obtained that the transitions from the heavy effective mass valence band to the light effective mass conduction band are most favorable for observing the symmetry of the superconducting energy gap. In Section 3 the transitions from the heavy effective mass valence band to the light effective mass conduction band for the mixed sd-wave superconductor are considered. In Section 4 the possibility of the experimental determination of the type o f superconducting gap symmetry is discussed.
2. The interband contribution in the imaginary part of the didectrie function for a d-wave superconductor To obtain the interband contribution to the imaginary part of the dielectric function for frequencies a: 1/'-"--/A/~,o we consider interband transitions from an arbitrary valence band to a certain conduction band. Only transitions from a narrow spectral region of some valence band to the states near the FL will be considered. (The consideration of transitions from a conduction band to an upper empty band is equivalent.) This problem differs from traditional considerations of intraband transitions in superconductors. In our case the initial state o f an electron does not depend on the superconducting state parameters as opposed to the IR spectroscopy of the superconducting gap where intraband transitions with to ~ d 0 / h are essential and both initial and final electron states depend on A. We calculate the electromagnetically induced interband current density j corresponding to the contribution of the transitions fi-om the valence band (v) to the conduction band (c):
J((s) =
4e2TIA[2m2eX22~ ,£t = - - o o
Z p
Gec(fs, + (s/2;p)Gw,((s, - (s/2;p)a(fs) ,
(2)
Yu.E Lozovik, A. I,: Poushnov / Physics Letters A 194 (1994} 405-412
407
where Gee and Gvv are the unperturbed Green functions in the c- and v-bands; A is the matri.~ element of the m o m e n t u m between the corresponding Bloch states (as we are interested in a narrow region near the FL we suppose that A is independent of the momentum p ) ; ,(2 is the elementary cell volume; (s = 2",rsT are Matsubara frequencies; ~'s~ = ( 2 s ~ + 1 - s)~rT; m is the mass of the free electron; h = 1. We take Gcc and Gvv in the Bloch function representation and use the Fourier transformation for the Matsubara variable. A corresponding linear response at real frequencies is obtained by the analytical continuation o f Eq. (2) : j ( t o ) = - Q ( t o ) A ( t o ) , where Q(to) is the general susceptibility: • e21AI2 fat a n h ( , ' o ' / 2 T ) Q(to) = -]~--~-~cO2 + [GvRv(to,)
{ [ G ~ ( t o ' ) - ~ec(to .-A , )]Gvv(to a , - to)
A ' ,~. , d2p Gvv(to )]G,,c(oo + t o ) } dto / (271-)2"
-
(3)
¢7.R(A)
Here v~(vv ) are the retarded and advanced Green functions in the c- and v-bands. To represent the layered structure: common to copper oxides we assume the dispersion laws of valence and conduction bands to be two-dimensional and isotropic. We also suppose that they have extrema in the same point of the Brillouin zone (see, for example, Ref. [ 18] ). The advanced and retarded Green function can be written as
~,~
,,~
,.;~.,~ (to, p)
GRe(A) ( t o ' P ) = to -- (P 4- i0 i- ~ + ep 4- i 0 '
--vv
=
l to -- 17vV4- i 0 '
(4)
where
p2 rleP = ~me - Ixsgn (m¢ ) ,
p2 r/v~ = - 2m---~- Ixsgn (me) - Eg.
Here mc and mv are the effective mass~.s of an electron in the c- and v-bands; ix is the Fermi level; Eg is the c - v band gap; /i(p) is the superconducting gap. Taking into account the nonresonant contribution of the other bands, the dielectric function can be written as 41rc - ~ - 0 ( )to.
~(oJ)=~0
(5)
The imaginary part of Q(to) determines the energy dissipation. Its negative sign corresponds to the positive sign o f the dielectric function. From Eqs. (2) and (4) the imaginary part o f the dielectric function e2(to) is defined as 2~" o o
0
- [tanh
0
(~/2r) -
tanh (r#/2T)] o26(e + a, - r#)} dxd~b,
(6)
where x = lp212md, g = me~my, 1+
e = ~ ¢ / ( x - Ix)2 + zl2 ,
~1 = x l g l s g n ( m O + Ixsgn (m~) + E g ,
A = 2e2i;t121m~llm21"l 2 , ,
l--
/.
Yu.E. Lozovik, A. K Poushnov / Physics Letters A 194 (1994) 405-412
408
Further we consider transitions from a valence band with negative dispersion of electrons. (The consideration of transitions from a valence band with positive dispersion of electrons is equivalent.) Integrating over the variable x, we obtain
e~V(o~,T)=2o~2[l+g )
I +tanh \
2T (i _ g2)
} j ~sgn(mc) -)
+0(,,
] (7)
I~ F_~ (7} we dene:ed ~"2 = Ae(1 _ g 2 ) ,
z~ = ZI~]I - - g e l ,
6' -- "-, - Eg - s g n ( m c ) / x ( 1 + g ) ,
toz.2 = + V ~
+ ~12 -. #igl.
At T = 0 we have e~v (~o, T = 0) = e~,Vo~-~v (co, T = 0), where 4~A ~ ; 0 = ,o211 + g l '
21r
=1
+ sgn(g)O (~ - ,~Ol) 0 (!g[ - 1)I J
(8)
At Ig[ < (/x2 - A~)/(/z2 + A~) the quantity Eev 2,0 is the contribution o f the interband transition in the imaginary part of the dielectric function of a normal metal. We obtain for a d-wave superconductor
Yu.E. Lozovik, A. V. Poushnov/ PhysicsLettersA 194 (1994) 405-412
409
'jl
3 r
t t
i
:,o. oeq
o II I-v
0.0
05
l.O
I 1~
2.0
Frequency (g/Ao) Fig. 1. Contributionof interband transitions between a valence band and a conductionband in the imaginarypart of the dielectric function for a d-wave ( ) and s-wave ( . . . . ) ~apcrconductor.Igl < (/~2 _ 4~)/(/~2 + 4~).
•r c e ~ 1 - \ a- - ~ J
O(ot) ,
(9)
where K ( x ) is the complete elliptic integral, ~ = &/A0 (see Fig. 1 ). This formula is correct near the FL (for & < /z(1 - igl))- In contrast to an s-wave superconductor, the interband absorption o f a d-wave superconductor begins when ~o = Eg + s g n ( m c ) ~ ( 1 + g). In the framework of the BCS model the imaginary part o f the dielectric function has a logariffav.ic peak at ~ = A3. The function ~ v ( w , T = 0)[a increases linearly near & = 0. Note that at Igl < ( ~ 2 _ 4 ] ) / ( ~ 2 + 4]) the structure o f the imaginary part o f the dielectric function near the FL shows the density of states of superconducting electrons in the conduction band (see, for example, Ref. [4] ). At g = / z / ~ the logarithmic peak in the imaginary part of the dielectric function at 6~ = A0 vanishes and when Igl > 1 and 6~ > ~ + A2 _ ~lgl
~v(o',r=O) ld=~+~. 2./-gr-~
~
sgn(g).
(lo)
In this case there is no qualitative difference in the imaginary part of the dielectric function of s-wave and dwave superconductors. Therefore, the interband transitions with ]gl < (t z2 - ~ ) / ( t ~2 + 4]) are mo,,;t favorable for observing the superconducting gap anisotropy.
3. The ~tse of mixed sd-wave p a i r i n g Let us consider a superconductor with the order parameter A(p) --. Zlo(s + deiBcos2~b). In the case ]gl < (/z 2 - z ~ ) / ( / z 2 + 4 ] ) the imaginary part of the dielectric function near the tm~ is
Yu,E.
Lozovik, A. V.. Poushnov I Physics Let ters A 194 (1994,) 405-4 ] 2
410
Ii 1I II I I
2.0
> O 20
•q
IJ ~
1.5
0 II I--
II II II
II
/I
l-
I I
I./
g
~
/
1.0
0.5
0.0
#
0~ II 15
7
0000
I
i 0.5
10
1.5
2,0
Frequency(~/Ao)
Frequency (~/~o)
Fig. 2. Contribution of interbend transitions between a valence band and a conduction band in the imaginary part of the dielectric function for a sd-wave superconductor: s = 0.3, d = 0.7, ,8 = 0 ( ~ );s=O.3, d = O . 7 , ~ = z r / 4 ( . . . . ). Fig. 3. Contribution of interband transitions between a valence band and a conduction band in the imaginary pan-; "~f the dielectric function for a sd-wave superconductor: s = 0.7, d = 0 . 3 , / / = 0 ( - ); s = 0,7, d = 0.3, ~8 =-rr/3 ( . . . . ).
~g(o,,r=0) lsd=~.2
O(scos•'-a)
g x/
+
where
x=74dk/~'-s2sin217', B' =fl =¢r-fl = fl - ¢r = 2~- - 13
y-=72d,lo~'-s'sin2~'+o~'-s'+d2,
if O
L e t u s c o n s i d e r this f o r m u l a w i t h d i f f e r e n t v a l u e s o f s, d a n d 0 < 9 < ¢ r / 2 in d e t a i l ( w e r e c a l l t h a t s + d = 1 ) . W h e n d = 0 a n d d = 1 it c o i n c i d e s w i t h e x p r e s s i o n a o b t a i n e d f o r t h e i m a g i n a r y p a r t o f t h e d i e l e c t r i c f u n c t i o n of s-wave (see Ref. [ 17] 3 and d-waxe superconductors respectieely (see Section 2).
Yu.E. Lozovik, A. I'~ Po'~shnov / Physics Letlers A 194 (1994) 405-412
411
When scos/3 < d the absorption begins at ~ = "J0ssin/3 in accordance with the mil~imum absolute value of the superconductin~ energy gap. Besides, a two-peak structure appears in the spectrum of the imaginary part of the dielectric function. Ap~,'t from the logarithmic peak at bT~= za0( 1 - 4sdsin2/3/2) I/2 anodaer logarithmic peak appears at ~ = ~o[ (s - d) 2 4- 4sdsin 2/3/2] 1/2 (see Fig. 2). At/3 = 7r/2 these peaks coincide, generating a single logarithmic peak at ~ = zi0(s 2 + d 2) 1/2 In the case s cos/3 .'~ d the absorption begins at ~ = ~o[ ( s - d ) 2 + 4sdsin 2/3/2] 1/2 The imaginary part of the dielectric function has a single logarithmic peak at ~ = ,~o(1 - 4 s d s i n 2 / 3 / 2 ) J/2. If d < scos/3 < [ 7d 2 _ s 2 + (s 4 _ 47d 4 + 82s2d 2 ) I/2 ] / 12d a peak of finite value appears at & = ~o [ ( s - d) 2 + 4sd sin 2/3/2] l/2 (see Fig. 3). The case o f s cos/3 = d is peculiar. At ~o = 71o[ (s - d) 2 + 4 s d sin 2/3/2] 1/2 the imaginary part of the dielectric function diverges as (~r2 - s 2 sin 2/3) -i/4. When s = d and/3 = 0 the absorption begins at ~ = 0. In this case the imaginary part of the dielectric function increases as v ~ . Therefore, at s cos ,8 ~< d and at d < s cos,8 < [ 7d 2 - s 2 + (s 4 - 47d 4 + 82s2d 2) 1/2 ] / 12d the two-peak structure appears in the spectrum o f the imaginary part of the dielectric function.
4. The possibility of experimental determination of ttte type of the superconducting gap symmetry in the spectra of the imaginary part of the dielectric function The comparison of Figs. ! - 3 shows that not only the existence of the superconducting gap anisotropy but also s-wave and d-wave contributions in the superconducting order parameter may be obtained from the characteristic structure of the imaginary part of the dielectric function corresponding to the transitions to the states near the FL. Moreover, if s c o s f l < d the relative phase difference of s-wave and d-wave contributions may be determined (see Fig. 3 ). The spectral width of the characteristic structure containing the information about the superconducting energy gap symmetry is ~d0v/l - g2 (Igl < 1 ). Therefore, the transitions between bands with Ig] << 1 are most favorable for observing the superconducting order parameter's symmetry. The results presented above were obtained for a clean superconductor. Impurities smoolh peculiarities of the imaginary part of the dielectric function corresponding to the superconducting gap. Moreover, if the impurity concentration is su~ciently high, the shape of peculiarities may be distorted. Nonmagnetic impurities and surface interfaces distort the midgap density of states and the imaginary part of the dielectric function only in the case of d-wave pairing. (Bound states appear within the superconducting gap, see, e.g., Ref. [ 19].) In the case of s-wave pairing only paramagnetic impurities may destroy the superconducting energy gap [20]. In this connection it is interesting to investigate the dependence of the imaginary part o f the dielectric function on the impurity concentration in the case of mixed sd-wave pairing. The realistic band structure of high-Te cuprate superconductors is substantially more complex than the twoband model considered above [ 18 ]. Nevertheless, femtoseecnd thermomodulational spectroscoFy (the pumping o f only the electro.-, subsystem) makes it feasible to select transitions from some valence band to the states near the FL in a narrow spectral region [ 15,16,21,22]. In fact, the electron temperature increases because of the action o f a pump pulse and the population o f the states below the FL decreases because of the pumping. Therefore, e~v(~o) increases at frequencies corresponding to the states above the FL and decreases at frequencies corresponding to transitions to the states above the FL. Thus the PL is obtained by the equation Ae2(~o) = 0, where AE2(OJ) is the difference in the ima#nary p-art of the dielectric function between the pumped and unpumped system. The peculiarities of the spectrum of the imaginary part of the dielectric function at T < Tc corresponding to such selected transitions are promising to defining experimentally the type o f the superconducting order parameter's symmetry. Measuring the transmission and reflection spectra o f the superconductor, one may obtain the imaginary part of the dielectric function at T < Tc and T > Tc. Studying the difference of zle2 for frequencies
412
Yu.E. Lozovik, A. V. Poushnov/ PhysicsLetters A 194 (1994) 405-412
corresponding to interband transitions to the states near the FL one may obtain a spectrum s i m i l a r to the ones shown in Figs. 1-3.
5. Conclusion We have shown that the interband optical spectra o f a sd-wave superconductor for co ->> Ao/h have a characteristic structure near the FL containing information about the s y m m e t r y o f the superconducting order parameter. I f s c o s f l <~ d and at d < s c o s , B < ( 7 d 2 - s 2 + v/s 4 - 4 7 d 4 + 82s2d2)/12d where s and d are the relative contributions of the s- and d-wave i_n_teractions a n d / 3 is their relative phase a t w o - p e a k structure appears in the spectrum o f the imaginary part of the d2,~lectric function near the FL. Not o n l y the relative s-wave and d-wave contributions in the superconducting order parameter but also '._heir relative phase may be extracted from the spectrum o f the i m a g i n a r y pair o f the dielectric function obtained from laser femtosecond spectroscopy data.
Acknowledgement We a c k n o w l e d g e V.M. Farztdinov for useful discussions. The work is partially supported by Grant No. 93233 on High-~~ superconductivity and by the International Science Foundation (Grant No. M U 6 0 0 0 ) .
References [1] [2] [3] [4] [5] [6] [71 [81 [91
P. Monthoux, A. Balatsky and D. Pines, Phys. Rev. ten. 67 (1991) 3448. G. Kotliex, Phys. Rev. B 37 (1988) 3664. N. Balut and D.J. Scalapino, Phys. Rev. Lett. 67 (1991) 2898. H. Won and K. Maki, Phys. Rev. B 49 (1994) 1397. K. Maki and H. Won, Phys. Rev. Lett. 72 (1994) 1758. Q.E Li, B.E. Koltenbah and R. JoynL Phys. Rev. B 48 (1993) 437. W.Hardy, D.A. Bonn, D.C. Morgan, R. L,_'angand K. Zhang, Phys. Rev. tetl. 70 (1993) 3999. J. Liu, Y. Li and Ch. Lieber, Phys. Rev. B 49 (1994) 6234. Z.-X. Shen, D.S. Dessau, B.O. Wells, D.M. King, W.E. Spicer, A.J. Arko, D. Marshal, L.W. Lombardo, A. Kapitulnik, P. Dickinson, S. Doniach, J. DiCarlo, A.G. Loeser and C.H. Park, Phys. Rev. Lett. 70 (1993) 1553. [ 10] T.E. Mason, G. Aeppli, S.M. Hayden, A.P. Ramirez and H.A. Mook, Phys. Rev. tea. 71 (1993) 919. [11] D. Chandhari and Shaw-Yu Lin, Phys. Rev. Len. 72 (1994) 1084. [12] R.W. Schoenlein, W.Z. Lin and J.G. Fujimototo, Phys. Rev. I..etL 58 (1987) 1680. [13] G.L. Eesley, Phys. Rev. B 33 (1986) 2144. [ 14] S.D. Brorson, A. Kazeroonian, J.S. Moodera, D.W. Face, T.K. Ch~',g, E.E Ippen, ~'.LS. Dcesselhau.¢.and G. Dresselhaug. Phys. Rev. Lett. 64 (1990) 2172. [ 15] M.E. Gershenson, V.V. Golovlev, I.B. Kedich, V.S. Letokhov, Yu.E. Lozovik, Yu.A. Matveets, E.G. Sil'kis, A.G. Stepanov, V.D. Titov, M.I. Faley, V.M. Farztdinov, S.V. Chekalin and A.P. Yartsev, Piz'ma Zh. Eksp. Teor. F!z 52 (1990) I 189. [ 16] S.V. Chekalin, V.M. Farztdinov, V.V. Golovlev, V.S. Letokhov, Yu.E. Lozovik, Yu.A. Matveets and A.G. Stepanov, Phys. Rev. Len. 67 (1991) 3860. [17] A.U Dob~-akov, V.M. Farztdinov, Yu.E. Lozovik and V.S. Letokhov, Opt. Commun. 105 (1994) 309. [18] W.E. PieketL Rev. Mod. Phys. 61 (1989) 433. [19] C.-R. Hu, Phys. Rev. Lea. 72 (1994) 1526. [ 20] A.A. Abrikosov and L.P. Gorkov, Zh. Eksp. Teor. Fiz. 39 ( 1961 ) 1781; K. Maki, in: Superconductivity, ed. R.D. Parks (Dekker, New York, 1969) p. i035. [21 ] S.G. Han, Z.V. Vardeny, K.S. Wong, O.G. Syrnko and G. Koren, Phys. Rev. Lett. 65 (1990) 2708. [22] A. Kazeroonian, T.K. Cheng, S.D. Brorson, Q. Li, E.P. lppen, X.D. Wn, T Venkatesan, S, Etemad and G. Dresselhans, Solid State Commun. 78 (1991) 95.
El XEV!ER
Physics Letters A 194 (1994) 405-412
Manifestation of superconducting gap symmetry in the optical spectrum Yurii E. Lozovik 1, Andrei V. Poushnov Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region ~42092, Russian Federation
Received 2 August i994; accepted for publication 23 September 1994 Communicatedby V.M. Agranovich
Abstract The manifestation of the superconducting gap symmetry in optical spectra at frequencies w >> A,/h corresponding to interband transitions is investigated. The problem is treated in the framework of the BCS model with mixed sd-wave pairing. The characteristic structure of the imaginary part of the dieleeLric function depending on the symmetry of the superconducting order parameter is obtained. The cases when the imaginary part of the dielectric function has a two-peak structure near the Fermi level are considered, in these cases complete information about the superconducting order parameter may be obtained from the difference optical spectra extracted from femto~oecondlaser spectroscopy data.
1. Introduction The nature of the superconducting state and the superconducting order parameter in high-Tc superconductors is still a matter o f controversy. Theoretical model studies (see Refs. [ 1-3 ] ) have indicated that the superconducting gap of high-To superconductors might have a d-wave symmetry. Different models of d-wave superconductors were considered to describe the properties of cuprate oxide materials in the superconducting state [ 4 - 6 ] . Although many experimental results can be interpreted in the framework of some d-wave superconductor model [ 7 - 9 ], a lot of experiments [ 10,11 ] call d- wave symmetry of the superconducting order parameter into question. Most o f the traditional experiments devoted to the investigation of the superconducting energy gap anisotropy (for example, tunneling spectroscopy) are very sensible to the quality of a sample's surface. Thermomodulational spectroscopy [ 12-14] promises to be an excellent bulk probe of the superconducting order parameter. This technique has been used in femtosecond laser pump-probe experiments [ 15,16] for the investigation of the YBa2CH307_~ film reflectivity in the optical region related to interband transitions. The difference spectra of the imaginary part of the dielectric function o f pumped and unpumped YBazCu3OT-a films both above and below the critical temperature Te have been extracted from the experimental results. The structure o f the difference spectra near the Fermi level (FL) was interpreted in Ref. [ 16] as a possible manifestation of the I E-mail: lozovik@~san.msk.su. 0375-9601/94/$07.00 (~) 1994 Elsevier Science B.V. All righls reserved SSD10375-9601 (94) 00789-6
Yu.E. Lozovik.A. K Poushnov/ PhysicsLettersA 194 (1994)405-412
406
superconducting gap. A theoreticM study of this problem for s-wave superconductors [17] confirmed this assumption. In the present paper we study the possible manifestation of an anisotropy of the superconducting energy gap in the optical spectra for frequencies ~o ~2 Ao/h corresponding to trans:.tions from a certain valence band to the conduction band. We show that the information about the type of symmetry of the superconducting gap may be obtained from spectra of the imaginary part of the dielectric function near the FL obtained by pump-probe femtosecond spectroscopy. In particular, we have found the cases of mixed sd-wave pairing when a two-peak structure appears in the spectra of the imaginary part of the dielectric function near the FL. In these c~ses not only relative s-wave and d-wave contributions in the superconducting order parameter but also their possible relative phase may be obtained. The differences in absorption spectra of pumped and unpurnped samples can be investigated by femtosecond laser spectroscopy; the contribution of transitions from a valence band to the region near the FL in the absorption spectrum may be extracted. Subtracting the difference spectrum of the pumped and unpumped system at T > Tc from that at T < Tc gives the characteristic structure near the FL. In our point of view, it enables one, in particular, to investigate the superconducting gap symmetry by femtosecond laser spectroscopy. For simplicity we analyze the optical properties of the model system in the framework of the BCS theory with the anisotropic order parameter /t(p) = A0(s + d ei/~cos 2q5),
( 1)
where s and d (s > 0 , d > 0, s + d = 1) are the relative contributions of the s-wave and d-wave pairing interactions and/3 is their relative phase; Ao is the energy constant, depending on temperature T. At s = 0, Eq. (1) describes the gap symmetry investiga~.ed in Refs. [4,5]. When s = d a n d / 3 = ,r/2, the superconducting order parameter is the same as that discussed in Refs. [2,6]. The rest of the paper is organized as follows. In Section 2 interband transitions between bands with different effective masses (including different signs of tl2:e effective masses) are considered. It is obtained that the transitions from the heavy effective mass valence band to the light effective mass conduction band are most favorable for observing the symmetry of the superconducting energy gap. In Section 3 the transitions from the heavy effective mass valence band to the light effective mass conduction band for the mixed sd-wave superconductor are considered. In Section 4 the possibility of the experimental determination of the type o f superconducting gap symmetry is discussed.
2. The interband contribution in the imaginary part of the didectrie function for a d-wave superconductor To obtain the interband contribution to the imaginary part of the dielectric function for frequencies a: 1/'-"--/A/~,o we consider interband transitions from an arbitrary valence band to a certain conduction band. Only transitions from a narrow spectral region of some valence band to the states near the FL will be considered. (The consideration of transitions from a conduction band to an upper empty band is equivalent.) This problem differs from traditional considerations of intraband transitions in superconductors. In our case the initial state o f an electron does not depend on the superconducting state parameters as opposed to the IR spectroscopy of the superconducting gap where intraband transitions with to ~ d 0 / h are essential and both initial and final electron states depend on A. We calculate the electromagnetically induced interband current density j corresponding to the contribution of the transitions fi-om the valence band (v) to the conduction band (c):
J((s) =
4e2TIA[2m2eX22~ ,£t = - - o o
Z p
Gec(fs, + (s/2;p)Gw,((s, - (s/2;p)a(fs) ,
(2)
Yu.E Lozovik, A. I,: Poushnov / Physics Letters A 194 (1994} 405-412
407
where Gee and Gvv are the unperturbed Green functions in the c- and v-bands; A is the matri.~ element of the m o m e n t u m between the corresponding Bloch states (as we are interested in a narrow region near the FL we suppose that A is independent of the momentum p ) ; ,(2 is the elementary cell volume; (s = 2",rsT are Matsubara frequencies; ~'s~ = ( 2 s ~ + 1 - s)~rT; m is the mass of the free electron; h = 1. We take Gcc and Gvv in the Bloch function representation and use the Fourier transformation for the Matsubara variable. A corresponding linear response at real frequencies is obtained by the analytical continuation o f Eq. (2) : j ( t o ) = - Q ( t o ) A ( t o ) , where Q(to) is the general susceptibility: • e21AI2 fat a n h ( , ' o ' / 2 T ) Q(to) = -]~--~-~cO2 + [GvRv(to,)
{ [ G ~ ( t o ' ) - ~ec(to .-A , )]Gvv(to a , - to)
A ' ,~. , d2p Gvv(to )]G,,c(oo + t o ) } dto / (271-)2"
-
(3)
¢7.R(A)
Here v~(vv ) are the retarded and advanced Green functions in the c- and v-bands. To represent the layered structure: common to copper oxides we assume the dispersion laws of valence and conduction bands to be two-dimensional and isotropic. We also suppose that they have extrema in the same point of the Brillouin zone (see, for example, Ref. [ 18] ). The advanced and retarded Green function can be written as
~,~
,,~
,.;~.,~ (to, p)
GRe(A) ( t o ' P ) = to -- (P 4- i0 i- ~ + ep 4- i 0 '
--vv
=
l to -- 17vV4- i 0 '
(4)
where
p2 rleP = ~me - Ixsgn (m¢ ) ,
p2 r/v~ = - 2m---~- Ixsgn (me) - Eg.
Here mc and mv are the effective mass~.s of an electron in the c- and v-bands; ix is the Fermi level; Eg is the c - v band gap; /i(p) is the superconducting gap. Taking into account the nonresonant contribution of the other bands, the dielectric function can be written as 41rc - ~ - 0 ( )to.
~(oJ)=~0
(5)
The imaginary part of Q(to) determines the energy dissipation. Its negative sign corresponds to the positive sign o f the dielectric function. From Eqs. (2) and (4) the imaginary part o f the dielectric function e2(to) is defined as 2~" o o
0
- [tanh
0
(~/2r) -
tanh (r#/2T)] o26(e + a, - r#)} dxd~b,
(6)
where x = lp212md, g = me~my, 1+
e = ~ ¢ / ( x - Ix)2 + zl2 ,
~1 = x l g l s g n ( m O + Ixsgn (m~) + E g ,
A = 2e2i;t121m~llm21"l 2 , ,
l--
/.
Yu.E. Lozovik, A. K Poushnov / Physics Letters A 194 (1994) 405-412
408
Further we consider transitions from a valence band with negative dispersion of electrons. (The consideration of transitions from a valence band with positive dispersion of electrons is equivalent.) Integrating over the variable x, we obtain
e~V(o~,T)=2o~2[l+g )
I +tanh \
2T (i _ g2)
} j ~sgn(mc) -)
+0(,,
] (7)
I~ F_~ (7} we dene:ed ~"2 = Ae(1 _ g 2 ) ,
z~ = ZI~]I - - g e l ,
6' -- "-, - Eg - s g n ( m c ) / x ( 1 + g ) ,
toz.2 = + V ~
+ ~12 -. #igl.
At T = 0 we have e~v (~o, T = 0) = e~,Vo~-~v (co, T = 0), where 4~A ~ ; 0 = ,o211 + g l '
21r
=1
+ sgn(g)O (~ - ,~Ol) 0 (!g[ - 1)I J
(8)
At Ig[ < (/x2 - A~)/(/z2 + A~) the quantity Eev 2,0 is the contribution o f the interband transition in the imaginary part of the dielectric function of a normal metal. We obtain for a d-wave superconductor
Yu.E. Lozovik, A. V. Poushnov/ PhysicsLettersA 194 (1994) 405-412
409
'jl
3 r
t t
i
:,o. oeq
o II I-v
0.0
05
l.O
I 1~
2.0
Frequency (g/Ao) Fig. 1. Contributionof interband transitions between a valence band and a conductionband in the imaginarypart of the dielectric function for a d-wave ( ) and s-wave ( . . . . ) ~apcrconductor.Igl < (/~2 _ 4~)/(/~2 + 4~).
•r c e ~ 1 - \ a- - ~ J
O(ot) ,
(9)
where K ( x ) is the complete elliptic integral, ~ = &/A0 (see Fig. 1 ). This formula is correct near the FL (for & < /z(1 - igl))- In contrast to an s-wave superconductor, the interband absorption o f a d-wave superconductor begins when ~o = Eg + s g n ( m c ) ~ ( 1 + g). In the framework of the BCS model the imaginary part o f the dielectric function has a logariffav.ic peak at ~ = A3. The function ~ v ( w , T = 0)[a increases linearly near & = 0. Note that at Igl < ( ~ 2 _ 4 ] ) / ( ~ 2 + 4]) the structure o f the imaginary part o f the dielectric function near the FL shows the density of states of superconducting electrons in the conduction band (see, for example, Ref. [4] ). At g = / z / ~ the logarithmic peak in the imaginary part of the dielectric function at 6~ = A0 vanishes and when Igl > 1 and 6~ > ~ + A2 _ ~lgl
~v(o',r=O) ld=~+~. 2./-gr-~
~
sgn(g).
(lo)
In this case there is no qualitative difference in the imaginary part of the dielectric function of s-wave and dwave superconductors. Therefore, the interband transitions with ]gl < (t z2 - ~ ) / ( t ~2 + 4]) are mo,,;t favorable for observing the superconducting gap anisotropy.
3. The ~tse of mixed sd-wave p a i r i n g Let us consider a superconductor with the order parameter A(p) --. Zlo(s + deiBcos2~b). In the case ]gl < (/z 2 - z ~ ) / ( / z 2 + 4 ] ) the imaginary part of the dielectric function near the tm~ is
Yu,E.
Lozovik, A. V.. Poushnov I Physics Let ters A 194 (1994,) 405-4 ] 2
410
Ii 1I II I I
2.0
> O 20
•q
IJ ~
1.5
0 II I--
II II II
II
/I
l-
I I
I./
g
~
/
1.0
0.5
0.0
#
0~ II 15
7
0000
I
i 0.5
10
1.5
2,0
Frequency(~/Ao)
Frequency (~/~o)
Fig. 2. Contribution of interbend transitions between a valence band and a conduction band in the imaginary part of the dielectric function for a sd-wave superconductor: s = 0.3, d = 0.7, ,8 = 0 ( ~ );s=O.3, d = O . 7 , ~ = z r / 4 ( . . . . ). Fig. 3. Contribution of interband transitions between a valence band and a conduction band in the imaginary pan-; "~f the dielectric function for a sd-wave superconductor: s = 0.7, d = 0 . 3 , / / = 0 ( - ); s = 0,7, d = 0.3, ~8 =-rr/3 ( . . . . ).
~g(o,,r=0) lsd=~.2
O(scos•'-a)
g x/
+
where
x=74dk/~'-s2sin217', B' =fl =¢r-fl = fl - ¢r = 2~- - 13
y-=72d,lo~'-s'sin2~'+o~'-s'+d2,
if O
L e t u s c o n s i d e r this f o r m u l a w i t h d i f f e r e n t v a l u e s o f s, d a n d 0 < 9 < ¢ r / 2 in d e t a i l ( w e r e c a l l t h a t s + d = 1 ) . W h e n d = 0 a n d d = 1 it c o i n c i d e s w i t h e x p r e s s i o n a o b t a i n e d f o r t h e i m a g i n a r y p a r t o f t h e d i e l e c t r i c f u n c t i o n of s-wave (see Ref. [ 17] 3 and d-waxe superconductors respectieely (see Section 2).
Yu.E. Lozovik, A. I'~ Po'~shnov / Physics Letlers A 194 (1994) 405-412
411
When scos/3 < d the absorption begins at ~ = "J0ssin/3 in accordance with the mil~imum absolute value of the superconductin~ energy gap. Besides, a two-peak structure appears in the spectrum of the imaginary part of the dielectric function. Ap~,'t from the logarithmic peak at bT~= za0( 1 - 4sdsin2/3/2) I/2 anodaer logarithmic peak appears at ~ = ~o[ (s - d) 2 4- 4sdsin 2/3/2] 1/2 (see Fig. 2). At/3 = 7r/2 these peaks coincide, generating a single logarithmic peak at ~ = zi0(s 2 + d 2) 1/2 In the case s cos/3 .'~ d the absorption begins at ~ = ~o[ ( s - d ) 2 + 4sdsin 2/3/2] 1/2 The imaginary part of the dielectric function has a single logarithmic peak at ~ = ,~o(1 - 4 s d s i n 2 / 3 / 2 ) J/2. If d < scos/3 < [ 7d 2 _ s 2 + (s 4 _ 47d 4 + 82s2d 2 ) I/2 ] / 12d a peak of finite value appears at & = ~o [ ( s - d) 2 + 4sd sin 2/3/2] l/2 (see Fig. 3). The case o f s cos/3 = d is peculiar. At ~o = 71o[ (s - d) 2 + 4 s d sin 2/3/2] 1/2 the imaginary part of the dielectric function diverges as (~r2 - s 2 sin 2/3) -i/4. When s = d and/3 = 0 the absorption begins at ~ = 0. In this case the imaginary part of the dielectric function increases as v ~ . Therefore, at s cos ,8 ~< d and at d < s cos,8 < [ 7d 2 - s 2 + (s 4 - 47d 4 + 82s2d 2) 1/2 ] / 12d the two-peak structure appears in the spectrum o f the imaginary part of the dielectric function.
4. The possibility of experimental determination of ttte type of the superconducting gap symmetry in the spectra of the imaginary part of the dielectric function The comparison of Figs. ! - 3 shows that not only the existence of the superconducting gap anisotropy but also s-wave and d-wave contributions in the superconducting order parameter may be obtained from the characteristic structure of the imaginary part of the dielectric function corresponding to the transitions to the states near the FL. Moreover, if s c o s f l < d the relative phase difference of s-wave and d-wave contributions may be determined (see Fig. 3 ). The spectral width of the characteristic structure containing the information about the superconducting energy gap symmetry is ~d0v/l - g2 (Igl < 1 ). Therefore, the transitions between bands with Ig] << 1 are most favorable for observing the superconducting order parameter's symmetry. The results presented above were obtained for a clean superconductor. Impurities smoolh peculiarities of the imaginary part of the dielectric function corresponding to the superconducting gap. Moreover, if the impurity concentration is su~ciently high, the shape of peculiarities may be distorted. Nonmagnetic impurities and surface interfaces distort the midgap density of states and the imaginary part of the dielectric function only in the case of d-wave pairing. (Bound states appear within the superconducting gap, see, e.g., Ref. [ 19].) In the case of s-wave pairing only paramagnetic impurities may destroy the superconducting energy gap [20]. In this connection it is interesting to investigate the dependence of the imaginary part o f the dielectric function on the impurity concentration in the case of mixed sd-wave pairing. The realistic band structure of high-Te cuprate superconductors is substantially more complex than the twoband model considered above [ 18 ]. Nevertheless, femtoseecnd thermomodulational spectroscoFy (the pumping o f only the electro.-, subsystem) makes it feasible to select transitions from some valence band to the states near the FL in a narrow spectral region [ 15,16,21,22]. In fact, the electron temperature increases because of the action o f a pump pulse and the population o f the states below the FL decreases because of the pumping. Therefore, e~v(~o) increases at frequencies corresponding to the states above the FL and decreases at frequencies corresponding to transitions to the states above the FL. Thus the PL is obtained by the equation Ae2(~o) = 0, where AE2(OJ) is the difference in the ima#nary p-art of the dielectric function between the pumped and unpumped system. The peculiarities of the spectrum of the imaginary part of the dielectric function at T < Tc corresponding to such selected transitions are promising to defining experimentally the type o f the superconducting order parameter's symmetry. Measuring the transmission and reflection spectra o f the superconductor, one may obtain the imaginary part of the dielectric function at T < Tc and T > Tc. Studying the difference of zle2 for frequencies
412
Yu.E. Lozovik, A. V. Poushnov/ PhysicsLetters A 194 (1994) 405-412
corresponding to interband transitions to the states near the FL one may obtain a spectrum s i m i l a r to the ones shown in Figs. 1-3.
5. Conclusion We have shown that the interband optical spectra o f a sd-wave superconductor for co ->> Ao/h have a characteristic structure near the FL containing information about the s y m m e t r y o f the superconducting order parameter. I f s c o s f l <~ d and at d < s c o s , B < ( 7 d 2 - s 2 + v/s 4 - 4 7 d 4 + 82s2d2)/12d where s and d are the relative contributions of the s- and d-wave i_n_teractions a n d / 3 is their relative phase a t w o - p e a k structure appears in the spectrum o f the imaginary part of the d2,~lectric function near the FL. Not o n l y the relative s-wave and d-wave contributions in the superconducting order parameter but also '._heir relative phase may be extracted from the spectrum o f the i m a g i n a r y pair o f the dielectric function obtained from laser femtosecond spectroscopy data.
Acknowledgement We a c k n o w l e d g e V.M. Farztdinov for useful discussions. The work is partially supported by Grant No. 93233 on High-~~ superconductivity and by the International Science Foundation (Grant No. M U 6 0 0 0 ) .
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