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Optical conductivity in superconducting layered cuprate oxides
Physica C 170 (1990) 505-512 North-Holland
Optical conductivity in superconducting layered cuprate oxides S.Takahashi
and M. Tachiki
lnstitutefor Materials Research, Tohoku University, Katahira 2-l-l. Aoba-ku. Sendai 980, Japan
Received 3 August 1990
Various properties of the cuprate oxides are explained by a layer model in which normal and superconducting layers are alternately stacked and the bulk superconductivity is maintained by the proximity effect. By using this model, the optical conductivity of the cuprate oxides is calculated. The result for YBa&usO, is that the optical conductivity for the light polarized parallel to the CuO chain has finite values even for frequencies below the gap, although the conductivity for the light perpendicular to the chain vanishes below the gap, as in the conventional superconductors. The character of the optical conductivities is seen in the results of the optical conductivities measured by Schlesinger et al. using untwinned YBaQtsO, single crystals.
1. Introduction The Cu02 layers in the layered cuprate oxides are considered to be important for high-T, superconductivity. Many experimental results indicate that the CuOZ layers and the other normal layers are alternately stacked, and suggest that the bulk superconductivity is maintained by the proximity effect. On the other hand, since the superconductivity in the oxides appears in a metallic state near the MottHubbard metal-insulator transition, it is believed that the correlation effect of electrons is important for the occurrence of the high-T, superconductivity in the oxides. Both the layer structure and the correlation effect influence the electronic and superconducting properties of the oxides. Therefore, when we try to learn the mechanism of superconductivity in the oxides from the experimental results, we need to extract the properties which originate from the layer structure itself. The optical conductivity is a quantity appropriate for investigating the electronic and superconducting state of the oxides [ l-6 1, along with other quantities such as the tunneling conductance [ 7- 121, and the relaxation time and the Knight shift of NMR [ 1318 1. Therefore, we calculate the optical conductivity of the layered cuprate oxides using a model of the layer structure and the proximity effect. In section 2 taking the YBazCu30, system as an 092 l-4534/90/$03.50
example of the oxides, we formulate the optical conductivity of this system. In section 3 we make a numerical calculation of the optical conductivity and discuss the electronic and superconducting state of YBa2Cu307, comparing our numerical results with the experimental results which were obtained by Schlesinger et al. using untwinned YBa$&O, single crystals [ 11.
2. Model for the layered cuprate oxides and formulation for the optical conductivity Photoemission [ 19,201 and EELS [ 211 experiments show that when holes are doped into crystals of the cuprate oxides, the holes mainly occupy the p orbitals of the oxygens. Neutron scattering experiments imply that the d electrons of the Cu ions are almost localized due to the strongly intra-atomic Coulomb interaction, and the Cu ions have only the spin degree of freedom [ 221. In this mode, the single particle excitations of low energy may be described by the freedom of the p holes of the oxygens. The spins exert a renormalization effect on the p holes through the interaction between the spins and the p holes. Let us consider YBazCu30, as an example of the copper oxide superconductors. As shown in fig. 1, the crystal of YBa2Cu30, has a stacking structure
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
506
S. Takahashi, M. Tachiki /Optical conductivity in superconducting oxides
layers are much smaller than the transfer integrals within the CuOZ layer and the CUO~ chain layer. Then, to diagonalize the terms with the large transfer integrals in Ho the following transformation is introduced.
C
t
-t3,31 where (5)
’ a Fig. 1. Definition of the oxygen sites and the transfer integrals between them in the unite cell of YBa2Cu307 crystal.
U,=
\ ’
which has two Cu02 layers and one Cu03 chain layer in the unit cell. The oxygens in the CuOi layer are denoted by 0( 2) and 0( 3), those in the adjacent CuOZ layer are denoted by O(2’) and 0( 3’ ), and those in the CuO, chain layer are denoted by 0( 1 ), O(4) andO(4’) asshowninlig. l.Theaandbaxes are taken perpendicular and parallel to the chain in the CuO, chain layer, respectively, and the c axis is taken perpendicular to the layer. The kinetic energy of the p electrons is described by HO
=
1 i,o
%dfaPia+
C
i,j,o
ti,jPfrJpjg
3
where pjO is the annihilation operator of the p electron with spin cr at the i site, E,, the energy level of the p electron at the i site, and tij the direct transfer integral between nearest-neighbor oxygen sites, as shown in fig. 1. The indirect transfer energy via the Cu ions is not included in eq. ( 1 ), since the magnitude of the indirect transfer is estimated to be much smaller than that of the direct transfer energy [ 23 1. In the following we assume that the energy levels of the p electrons in the CuOZ layer are equal and those in the CuOs chain layer are equal, and that the transfer integrals between the CuOZ layer and the CuO, chain layer and those between the adjacent CuOZ
fi
kb
eikdl
kb,i&l
’
lkbl
1kb 1
-i
0
--e Jj:
kb
i&l
i’bl i
-’ fi
kb
,-ik,cl
lkbl 1
(6) cl being half the distance between the O(4) and 0(4’) oxygens. In eqs. (2), (3) and (4), the subscripts -, 0 and + indicate the antibonding, nonbonding and bonding bands, respectively. In the oxides the Fermi level lies in the antibonding bands. In the following we will consider the excitations near the Fermi level, so that we retain only the operators, A_,(k), B_,(k), and A’_,(k) in Ho and we will omit the subscript - hereafter:
Ho= ;ieA(k)
[d(k)&(k) +@(k)&(K) 1
+~B(k)B~(k)B,(k)+t(k) [‘a~)&(~) +B:(k)A&(k)]+t’(k)A&t(k)A,(k)+h.c.},
(7)
where e,(k)=e,,+41t,,sin(k,a/2)sin(kba/2)I
,
507
S. Takahashi, M. Tachiki /Optical conductivity in superconducting oxides
quency in the normal state. If the damping of the p electrons is caused by the scattering with these fluctions, the damping constants or the imaginary parts of the self-energies are proportional to w in the normal state [ 271 as shown in Appendix A: 2’ (k) = - f2,2zeik3 ,
Irn~~(o)=-(Y]c0],
a being the lattice constant in the CuO, layer, c, the distance between the O(4) oxygen and the CuO,? layer, and c3 the distance between the adjacent CuO, layers. We assume that the superconducting pairing interaction acts between the p electrons in the CuOZ layer, and add the interaction term to the Hamiltonian (7). Introducing the Nambu representation
Imf,(o)=-P]o]
_
A+(k)
A(k)=[ AI
1 ’
we write the total Hamiltonian Ha = z{eA(k)
+t(k)
[/i+(k)r,zi(k)+ii’+(k)~,/i’(k)]
[a+(k)r,&k)+l?+(k)r,ii’(k)]
+A[~+(k)r,~(k)+/i'+(k)q,i'(k)]}
+h.c.
,
(9)
where r, and r3 are the Pauli matrices and A is the superconducting order parameter in the Cu02 layer. The derivation of Green’s function for the p electrons from the Hamiltonian (9 ) is given in Appendix A. These Green’s functions are used for the calculation of the conductivity. As seen in eqs. (A. 1 ), (A.2) and (A.3), the equations for Green’s functions include the self-energiessA (0) and fB( w ). The self-energies are caused by the many-body effect due to the spin fluctuations of the Cu ions and the charge fluctuations accompanying the Cu-0 charge transfer. The imaginary part of the self-energy corresponds to the damping constant of the p electron. The experimental results of the optical conductivity [ l51 and the Raman scattering [24-261 suggest that the charge and spin fluctuations are nearly independent of frequency in a considerable range of fre-
.
(11)
The coefficients (Yand p are estimated to be 0.3-0.4 from the reflectivity measurements [ 11. On the other hand, the real part of the self-energies causes the mass renormalization for the p electron. The mass renormalization is understood to be included in eA(k) and eB(k). In the superconducting state, opening of the energy gap in the Cu02 layers suppresses the low frequency part of Im fA (0). For Zm zA (w) we use the expression
- adln
as
(10)
All01 l-J_
>
sign(w)r,
,
(12)
for IWI > A, and zero otherwise in the CuOz layer. For the derivation of eq. ( 12), see Appendix A. In Appendix A, we assume that if the scattering of the p electrons is caused by the spin fluctuations, the scattering is strongly suppressed as the superconducting order parameter develops. the self-energy of the Cu03 chain layers remains unchanged by the onset of the superconductivity. The optical conductivity in the long wave limit is expressed by [ 28 ] cr~v(o)=
-1m
npY(w)
(
0
> ’
(13)
where 17,,(w) is given by the analytical continuation (iw,+o+iS) of the the current-current correlation function I/T 17,,(iw,)
= -
s
dr
(TTLjp(71.iy(0)l)eiOnr. (14)
0
The current operator of the system is expressed the long wave limit as
in
S. Takahashi, M. Tachiki /Optical conductivity in superconductrng oxides
508
j= -e ${UA (k)
[A+(k)A(k)+A’+(k)A’ (k)]
+vB(k)B+(k)B(k) +
dt(k) -+(k)&k)+h.c.
dkA
+ dt(k) dk + dt’(k) dkA
-+ B (k)A^’ (k) + h.c. -,+
(k)&k)+h.c.},
(15)
wherey,(k)~d~A(k)/dkandv,(k)~d~B(k)/dkare the velocities of the p electrons in the CuOZ layer and the Cu03 chain layer, respectively. For the electric field of the light polarized along the a axis, the first term in the curly brackets of eq. ( 15 ) contributes to the current, while for the electric field along the b axis, both the first and second terms contribute to the current. The last three terms have little contribution to the current so long as the electric field lies in the a-b plane, and thus these terms are neglected for the calculation of the conductivity in the u-b plane. Inserting eq. ( 15) into eq. ( 14) and calculating the current-current correlation function 17,” (iw ) to one-loop approximation, we have the expression for the optical conductivities for the CuOZ layer and the Cu03 chain layer
x (tanh(v/2T)-tanh(
(v-o)/2T)),
space, ,Q,,(k, o) is a matrix of 2 x 2. The optical conductivities along the a- and b-axes are approximately given by a,(o)=2a$(w) and r~~(w)=rz~,(o)+ a&(o), respectively, if the transfer integrals between the Cu02 layer and the Cu03 chains are considerably small. When the electric field is polarized along the c axis, the last three terms in the curly brackets of eq. ( 15) contribute to the current. Thus the optical conductivity along the c-axis is given by 7 dv{21%1 -cc
+2
I I2 I- I’ dt(k)
TrkMk ~MA( k v-w) 1
x
‘
+2
dt’ (k)
W&(k
dkc
v)b (4 v--o)
x (tanh(v/2T)-tanh(
(y--w)/2T)).
(19)
3. Numerical results and comparison with experiments We normalize all parameter values by the value of t2,3. Considering the layer nature of the oxides, we assume the following set of parameter values: t,+= 1, $2 = Ep3 = 0, t2.2.= t3.3.= 0.15, t2,4=t3,‘$=0.3, t,, = t,,=0.4, and the Fermi level tF= 1.7. As seen in fig. 2, we have two kinds of Fermi surfaces in the Brillouin zone; the almost two-dimensional Fermi surfaces with the character of 0 ( 2 ) and 0 ( 3 ) around
(16)
where the suffix /i takes A for the Cu02 layer and B for the Cu03 chain layer, and rj,, (k, o) is the spectral function defined by
S
Y
i,,, &,(k, w) = - i Im eBB(k, w) . Since Green’s
function
is expressed
(18) in the Nambu
r Fig. 2. Calculated
X Fermi surfaces.
S. Takahashi, M. Tachiki / Optical conductivity in superconducting oxides
the S point and the almost one-dimensional Fermi surfaces with the character of 0 ( 1) and 0 (4) along the T-X direction. These Fermi surfaces are in good agreement with those obtained by the experiments of angle-resolved photoemission [ 201 and positron annihilation [29] of YBa2Cu@_a. This agreement may justify the neglect of the indirect transfer terms in eq. ( 1). If the indirect transfer terms were dominant, the Fermi surfaces would be quite different from those in fig. 2. Using eqs. ( 16) and ( 19 ), we calculate the optical conductivities for the electric fields parallel to the a, b and c axes. Figures 3(a) and 3 (b) show the optical conductivities a,(o) and ~~b(o), when loI ~0.4, I/31 ~0.3, and A/t,,,=0.08, and the same values for other parameters as those in fig. 2 are taken. The dashed curves indicate the optical conductivities in the normal state. The peaks around o N 0 correspond to the Drude conductivity, and the broak peaks around w/2A- 1.5 correspond to the conductivity which arises from the interband tran-
509
sition between the two CuOz bands split by the weak interlayer transfer interaction. The optical conductivities in the superconducting state are shown by the solid curves. A superconducting energy gap of 24 appears in the spectrum of a, (0). The optical conductivity G~(w) also has a gap structure around 24, but does not vanish even below 24. The finite conductivity below 24 is the contribution from the conductivity in the chain layer. The behavior of the optical conductivities a,(w) and ran shown in fig. 3 is in qualitative agreement with that of the optical conductivity measured by Schlesinger et al. using untwinned crystals [ 11. To understand the origin of the frequency dependence of the conductivities obtained above, we calculate the local density of states of the p electrons in the Cu02 layer and the CUO~ chain layer. The local densities of states in the CuOz layer and the Cu03 chain layer are respectively given by
(20) (21)
I \
(4
\ ‘1 \
\
\ \ \ \
0
Using eqs. (20) and (2 1), and the same parameter values as in fig. 3, we calculate the local densities of states and show the result in fig. 4. As seen in the figure, a large energy gap opens in the CuOz layer. Roughly speaking, the gap appearing in the spectrum of CJ,(w ) corresponds to the excitation process of an
E/j a-axis
\
(b) E//b-axis
2
1
3
W/ZA Fig. 3. Optical conductivities as a function of frequency: (a) shows the optical conductivity a,(w) for the electrical field E parallel to the a-axis, and (b) shows the optical conductivity u~( W) for E parallel to the b-axis. The solid curves indicate the optical conductivity in the superconducting state and the dashed curves indicate the optical conductivity in the normal state.
I -5
I
I
I
I
I
0
III
II 5
E/A Fig. 4. Densities of states. Figures (a) and (b) show the densities of states at the CuOz layer and the CuO, chain layer, respectively.
S. Takahashi, M. Tachiki /Optical conductivity in superconducting oxides
510
1.5
I
I
-
E/l
3 2 ”
c-axis
l.O-
* 2
0.5-
b”
a$ b
0’
0
I
I
I
1
2
3
w/2A
0/2A Fig. 5. Optical conductivity of the CuOr chain layers. The solid curve indicates the optical conductivity in the superconducting state and the dashed curve indicates the optical conductivity in the normal state.
electron crossing a gap of 24 in the density of states. We mentioned above that the finite value of u~( o) below 24 is the contribution from the chain layer conductivity &,b<0). The calculated conductivity is shown in fig. 5. As seen from the figure, the conductivity in the superconducting state denoted by the solid curve has no gap structure and thus contributes to the finite value of Q(W) below 2 A. The gaplessness of a&(w) is understood in the following way. As seen in fig. 4(b), the density of states at w= 0 is finite, although it has a gap structure which arises from the superconductivity induced by the proximity effect ” . The gapless character of a&(w) comes from the finiteness of the density of states around w= 0. The gap structure in the density of states causes a decrease of a&(w) in the superconducting state from o&(w) in the normal state, shown by the dashed curve in fig. 5. The optical conductivity along the c axis is shown in fig. 6. The parameter values are the same as those in fig. 3. The optical conductivity a,(w) gradually decreases as the frequency decreases below 24. The first term in the brackets of eq. ( 19) contains both the spectral functions of the CuOz layer and the CUO~
Fig. 6. Normalized optical conductivity for the electric field E parallel to the c-axis as a function of frequency. Q.,(W) is the optical conductivity in the normal state.
chain layer, and thus Us shows an intermediate behavior between the conductivities of both layers. The magnitude of oc( o) is about one tenth of a,( w ) at frequencies larger than the gap energy 24. The magnitude and the frequency dependence of a,(w) are consistent with the experimental results obtained by Collins et al. using single crystals of YBazCu307
[21.
Acknowledgements The authors would like to express their sincere thanks to Prof. F. Steglich, Prof. H. Matsumoto, Dr. T. Koyama and Mr. M. Machida for their valuable discussions. This work was supported by a Grant-inAid from the Ministry of Education, Science and Culture, Japan.
Appendix A Green’s functions, GM(k, iw,), 6,(k, iw,), and im,,), satisfy the equations of motion
GA,,(k,
io, - CA(k)?3 -AT, -&(w)
l~~(k, iw) = l+t(k)e,(k,
PI
The flat Fermi surface in fig. 2 is that of the CUO~ chain band. The magnitude of the superconducting gap is large in the region of the Fermi surface near the X point, since the mixing with the Cu02 bands is large. The superconducting gap in the region near the rpoint almost vanishes, since the mixing with the CuOZ bands is negligibly small.
+f*(k)&A(k,
[iOn-dkh
iw,)r3
(A.11
io,)r3,
--J%(w) l&A(k bJ =t*(k)eM(k,
iw,)r,
S. Takahashi. M. Tachiki /Optical conductivity in superconducting oxides
+f(k)eAsA(kim,)T3, [iw,-c,(k)r,
-dr,
(A-2)
-fA(cO)]GA,A(k,
ion)
~t(k)G~~(k, +t’(k)GM(kr
io,)73,
io,)r3 (A.3)
where fA (0) and fB( o) are given in eqs. ( 12 ) and (ll),respectively.Fromeqs. (A.l)-(A.3),wehave eM(k,
icG=
X
(A.4)
ZIY-,~T
where g is the coupling constant between the p electrons and the fluctuations, F(V) is the spectral function of the fluctuations, and & (k, co) is the spectral function of the p electron which is given by
&A(4 w)= -iIm
1 o+i&-eA(k)r3-Arl
-.$A(w)
1*
If the F(Y) is independent of v, integrating with respect to k we have
(B.2) \ ,
eq. (B. 1)
0
where ImfA(o)=-aRe X=io,
511
-CITY
--AT, -fA(o)
Ill(k -73ico,-cEA(k)T3-Ar, Y=io,-c,(k)r,
0
x -fA(w)
“’
w-v-~~,,(~-v)+(d-~~,2(w-v))7, ,/(o- v-&,(wv))*- (A-x&,z(w-
-fB(o)
t*(k)t’*(k) +T3 . m,--Ez(k)r3 -A7l
v))* ’ (J3.3)
where cr= rrg2NA(0)F, N,(O) being the density of states of the CuOZ layer at the Fermi level in the normal state. If we put _??A(o- v) =O in the right-hand side of eq. (B.3), we have for all o
It(k Z=t(k)T,
dv
s
-s,(w)
” ’
ImZ,(o)=-culwlJ .%t*(k)r3+r3.
t(k)t’ lo,-tA(k)T3
(k) -AT, -&(w)
“.
- cvdln
A/lWl l-J_
>
Similarly, &,(k,
&,z,
(A-5)
x_iy_,z.
(A.61
iw,) = y
eA,,,(k, ion)=
It is easily proved that eM( k, ion) = GA.,, (k, ion).
ImS,(w)=-cll]wl
by 0 Im~,(W)=--g2~IduF(Y)tea(k,o-ll)r3, k
0 03.1)
,
(B-4)
for I o I > A, and zero otherwise. Even when the self-energy is caused by the magnetic fluctuations, the imaginary part of the self-energy is approximated by the form of eq. (B.4), if the interaction with the magnetic fluctuations is strongly suppressed below the energy gap A. In the normal state, eq. (B.4) is reduced to
Appendix B In the following derivation, we assume that the fluctuations which interact with the p electrons are nonmagnetic. For T= 0 and o > 0, the imaginary part of the p-electron self-energy which is caused by the emission and absorption of the fluctuations is given
sign(w)T,
.
(B.5)
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Optical conductivity in superconducting layered cuprate oxides S.Takahashi
and M. Tachiki
lnstitutefor Materials Research, Tohoku University, Katahira 2-l-l. Aoba-ku. Sendai 980, Japan
Received 3 August 1990
Various properties of the cuprate oxides are explained by a layer model in which normal and superconducting layers are alternately stacked and the bulk superconductivity is maintained by the proximity effect. By using this model, the optical conductivity of the cuprate oxides is calculated. The result for YBa&usO, is that the optical conductivity for the light polarized parallel to the CuO chain has finite values even for frequencies below the gap, although the conductivity for the light perpendicular to the chain vanishes below the gap, as in the conventional superconductors. The character of the optical conductivities is seen in the results of the optical conductivities measured by Schlesinger et al. using untwinned YBaQtsO, single crystals.
1. Introduction The Cu02 layers in the layered cuprate oxides are considered to be important for high-T, superconductivity. Many experimental results indicate that the CuOZ layers and the other normal layers are alternately stacked, and suggest that the bulk superconductivity is maintained by the proximity effect. On the other hand, since the superconductivity in the oxides appears in a metallic state near the MottHubbard metal-insulator transition, it is believed that the correlation effect of electrons is important for the occurrence of the high-T, superconductivity in the oxides. Both the layer structure and the correlation effect influence the electronic and superconducting properties of the oxides. Therefore, when we try to learn the mechanism of superconductivity in the oxides from the experimental results, we need to extract the properties which originate from the layer structure itself. The optical conductivity is a quantity appropriate for investigating the electronic and superconducting state of the oxides [ l-6 1, along with other quantities such as the tunneling conductance [ 7- 121, and the relaxation time and the Knight shift of NMR [ 1318 1. Therefore, we calculate the optical conductivity of the layered cuprate oxides using a model of the layer structure and the proximity effect. In section 2 taking the YBazCu30, system as an 092 l-4534/90/$03.50
example of the oxides, we formulate the optical conductivity of this system. In section 3 we make a numerical calculation of the optical conductivity and discuss the electronic and superconducting state of YBa2Cu307, comparing our numerical results with the experimental results which were obtained by Schlesinger et al. using untwinned YBa$&O, single crystals [ 11.
2. Model for the layered cuprate oxides and formulation for the optical conductivity Photoemission [ 19,201 and EELS [ 211 experiments show that when holes are doped into crystals of the cuprate oxides, the holes mainly occupy the p orbitals of the oxygens. Neutron scattering experiments imply that the d electrons of the Cu ions are almost localized due to the strongly intra-atomic Coulomb interaction, and the Cu ions have only the spin degree of freedom [ 221. In this mode, the single particle excitations of low energy may be described by the freedom of the p holes of the oxygens. The spins exert a renormalization effect on the p holes through the interaction between the spins and the p holes. Let us consider YBazCu30, as an example of the copper oxide superconductors. As shown in fig. 1, the crystal of YBa2Cu30, has a stacking structure
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
506
S. Takahashi, M. Tachiki /Optical conductivity in superconducting oxides
layers are much smaller than the transfer integrals within the CuOZ layer and the CUO~ chain layer. Then, to diagonalize the terms with the large transfer integrals in Ho the following transformation is introduced.
C
t
-t3,31 where (5)
’ a Fig. 1. Definition of the oxygen sites and the transfer integrals between them in the unite cell of YBa2Cu307 crystal.
U,=
\ ’
which has two Cu02 layers and one Cu03 chain layer in the unit cell. The oxygens in the CuOi layer are denoted by 0( 2) and 0( 3), those in the adjacent CuOZ layer are denoted by O(2’) and 0( 3’ ), and those in the CuO, chain layer are denoted by 0( 1 ), O(4) andO(4’) asshowninlig. l.Theaandbaxes are taken perpendicular and parallel to the chain in the CuO, chain layer, respectively, and the c axis is taken perpendicular to the layer. The kinetic energy of the p electrons is described by HO
=
1 i,o
%dfaPia+
C
i,j,o
ti,jPfrJpjg
3
where pjO is the annihilation operator of the p electron with spin cr at the i site, E,, the energy level of the p electron at the i site, and tij the direct transfer integral between nearest-neighbor oxygen sites, as shown in fig. 1. The indirect transfer energy via the Cu ions is not included in eq. ( 1 ), since the magnitude of the indirect transfer is estimated to be much smaller than that of the direct transfer energy [ 23 1. In the following we assume that the energy levels of the p electrons in the CuOZ layer are equal and those in the CuOs chain layer are equal, and that the transfer integrals between the CuOZ layer and the CuO, chain layer and those between the adjacent CuOZ
fi
kb
eikdl
kb,i&l
’
lkbl
1kb 1
-i
0
--e Jj:
kb
i&l
i’bl i
-’ fi
kb
,-ik,cl
lkbl 1
(6) cl being half the distance between the O(4) and 0(4’) oxygens. In eqs. (2), (3) and (4), the subscripts -, 0 and + indicate the antibonding, nonbonding and bonding bands, respectively. In the oxides the Fermi level lies in the antibonding bands. In the following we will consider the excitations near the Fermi level, so that we retain only the operators, A_,(k), B_,(k), and A’_,(k) in Ho and we will omit the subscript - hereafter:
Ho= ;ieA(k)
[d(k)&(k) +@(k)&(K) 1
+~B(k)B~(k)B,(k)+t(k) [‘a~)&(~) +B:(k)A&(k)]+t’(k)A&t(k)A,(k)+h.c.},
(7)
where e,(k)=e,,+41t,,sin(k,a/2)sin(kba/2)I
,
507
S. Takahashi, M. Tachiki /Optical conductivity in superconducting oxides
quency in the normal state. If the damping of the p electrons is caused by the scattering with these fluctions, the damping constants or the imaginary parts of the self-energies are proportional to w in the normal state [ 271 as shown in Appendix A: 2’ (k) = - f2,2zeik3 ,
Irn~~(o)=-(Y]c0],
a being the lattice constant in the CuO, layer, c, the distance between the O(4) oxygen and the CuO,? layer, and c3 the distance between the adjacent CuO, layers. We assume that the superconducting pairing interaction acts between the p electrons in the CuOZ layer, and add the interaction term to the Hamiltonian (7). Introducing the Nambu representation
Imf,(o)=-P]o]
_
A+(k)
A(k)=[ AI
1 ’
we write the total Hamiltonian Ha = z{eA(k)
+t(k)
[/i+(k)r,zi(k)+ii’+(k)~,/i’(k)]
[a+(k)r,&k)+l?+(k)r,ii’(k)]
+A[~+(k)r,~(k)+/i'+(k)q,i'(k)]}
+h.c.
,
(9)
where r, and r3 are the Pauli matrices and A is the superconducting order parameter in the Cu02 layer. The derivation of Green’s function for the p electrons from the Hamiltonian (9 ) is given in Appendix A. These Green’s functions are used for the calculation of the conductivity. As seen in eqs. (A. 1 ), (A.2) and (A.3), the equations for Green’s functions include the self-energiessA (0) and fB( w ). The self-energies are caused by the many-body effect due to the spin fluctuations of the Cu ions and the charge fluctuations accompanying the Cu-0 charge transfer. The imaginary part of the self-energy corresponds to the damping constant of the p electron. The experimental results of the optical conductivity [ l51 and the Raman scattering [24-261 suggest that the charge and spin fluctuations are nearly independent of frequency in a considerable range of fre-
.
(11)
The coefficients (Yand p are estimated to be 0.3-0.4 from the reflectivity measurements [ 11. On the other hand, the real part of the self-energies causes the mass renormalization for the p electron. The mass renormalization is understood to be included in eA(k) and eB(k). In the superconducting state, opening of the energy gap in the Cu02 layers suppresses the low frequency part of Im fA (0). For Zm zA (w) we use the expression
- adln
as
(10)
All01 l-J_
>
sign(w)r,
,
(12)
for IWI > A, and zero otherwise in the CuOz layer. For the derivation of eq. ( 12), see Appendix A. In Appendix A, we assume that if the scattering of the p electrons is caused by the spin fluctuations, the scattering is strongly suppressed as the superconducting order parameter develops. the self-energy of the Cu03 chain layers remains unchanged by the onset of the superconductivity. The optical conductivity in the long wave limit is expressed by [ 28 ] cr~v(o)=
-1m
npY(w)
(
0
> ’
(13)
where 17,,(w) is given by the analytical continuation (iw,+o+iS) of the the current-current correlation function I/T 17,,(iw,)
= -
s
dr
(TTLjp(71.iy(0)l)eiOnr. (14)
0
The current operator of the system is expressed the long wave limit as
in
S. Takahashi, M. Tachiki /Optical conductivity in superconductrng oxides
508
j= -e ${UA (k)
[A+(k)A(k)+A’+(k)A’ (k)]
+vB(k)B+(k)B(k) +
dt(k) -+(k)&k)+h.c.
dkA
+ dt(k) dk + dt’(k) dkA
-+ B (k)A^’ (k) + h.c. -,+
(k)&k)+h.c.},
(15)
wherey,(k)~d~A(k)/dkandv,(k)~d~B(k)/dkare the velocities of the p electrons in the CuOZ layer and the Cu03 chain layer, respectively. For the electric field of the light polarized along the a axis, the first term in the curly brackets of eq. ( 15 ) contributes to the current, while for the electric field along the b axis, both the first and second terms contribute to the current. The last three terms have little contribution to the current so long as the electric field lies in the a-b plane, and thus these terms are neglected for the calculation of the conductivity in the u-b plane. Inserting eq. ( 15) into eq. ( 14) and calculating the current-current correlation function 17,” (iw ) to one-loop approximation, we have the expression for the optical conductivities for the CuOZ layer and the Cu03 chain layer
x (tanh(v/2T)-tanh(
(v-o)/2T)),
space, ,Q,,(k, o) is a matrix of 2 x 2. The optical conductivities along the a- and b-axes are approximately given by a,(o)=2a$(w) and r~~(w)=rz~,(o)+ a&(o), respectively, if the transfer integrals between the Cu02 layer and the Cu03 chains are considerably small. When the electric field is polarized along the c axis, the last three terms in the curly brackets of eq. ( 15) contribute to the current. Thus the optical conductivity along the c-axis is given by 7 dv{21%1 -cc
+2
I I2 I- I’ dt(k)
TrkMk ~MA( k v-w) 1
x
‘
+2
dt’ (k)
W&(k
dkc
v)b (4 v--o)
x (tanh(v/2T)-tanh(
(y--w)/2T)).
(19)
3. Numerical results and comparison with experiments We normalize all parameter values by the value of t2,3. Considering the layer nature of the oxides, we assume the following set of parameter values: t,+= 1, $2 = Ep3 = 0, t2.2.= t3.3.= 0.15, t2,4=t3,‘$=0.3, t,, = t,,=0.4, and the Fermi level tF= 1.7. As seen in fig. 2, we have two kinds of Fermi surfaces in the Brillouin zone; the almost two-dimensional Fermi surfaces with the character of 0 ( 2 ) and 0 ( 3 ) around
(16)
where the suffix /i takes A for the Cu02 layer and B for the Cu03 chain layer, and rj,, (k, o) is the spectral function defined by
S
Y
i,,, &,(k, w) = - i Im eBB(k, w) . Since Green’s
function
is expressed
(18) in the Nambu
r Fig. 2. Calculated
X Fermi surfaces.
S. Takahashi, M. Tachiki / Optical conductivity in superconducting oxides
the S point and the almost one-dimensional Fermi surfaces with the character of 0 ( 1) and 0 (4) along the T-X direction. These Fermi surfaces are in good agreement with those obtained by the experiments of angle-resolved photoemission [ 201 and positron annihilation [29] of YBa2Cu@_a. This agreement may justify the neglect of the indirect transfer terms in eq. ( 1). If the indirect transfer terms were dominant, the Fermi surfaces would be quite different from those in fig. 2. Using eqs. ( 16) and ( 19 ), we calculate the optical conductivities for the electric fields parallel to the a, b and c axes. Figures 3(a) and 3 (b) show the optical conductivities a,(o) and ~~b(o), when loI ~0.4, I/31 ~0.3, and A/t,,,=0.08, and the same values for other parameters as those in fig. 2 are taken. The dashed curves indicate the optical conductivities in the normal state. The peaks around o N 0 correspond to the Drude conductivity, and the broak peaks around w/2A- 1.5 correspond to the conductivity which arises from the interband tran-
509
sition between the two CuOz bands split by the weak interlayer transfer interaction. The optical conductivities in the superconducting state are shown by the solid curves. A superconducting energy gap of 24 appears in the spectrum of a, (0). The optical conductivity G~(w) also has a gap structure around 24, but does not vanish even below 24. The finite conductivity below 24 is the contribution from the conductivity in the chain layer. The behavior of the optical conductivities a,(w) and ran shown in fig. 3 is in qualitative agreement with that of the optical conductivity measured by Schlesinger et al. using untwinned crystals [ 11. To understand the origin of the frequency dependence of the conductivities obtained above, we calculate the local density of states of the p electrons in the Cu02 layer and the CUO~ chain layer. The local densities of states in the CuOz layer and the Cu03 chain layer are respectively given by
(20) (21)
I \
(4
\ ‘1 \
\
\ \ \ \
0
Using eqs. (20) and (2 1), and the same parameter values as in fig. 3, we calculate the local densities of states and show the result in fig. 4. As seen in the figure, a large energy gap opens in the CuOz layer. Roughly speaking, the gap appearing in the spectrum of CJ,(w ) corresponds to the excitation process of an
E/j a-axis
\
(b) E//b-axis
2
1
3
W/ZA Fig. 3. Optical conductivities as a function of frequency: (a) shows the optical conductivity a,(w) for the electrical field E parallel to the a-axis, and (b) shows the optical conductivity u~( W) for E parallel to the b-axis. The solid curves indicate the optical conductivity in the superconducting state and the dashed curves indicate the optical conductivity in the normal state.
I -5
I
I
I
I
I
0
III
II 5
E/A Fig. 4. Densities of states. Figures (a) and (b) show the densities of states at the CuOz layer and the CuO, chain layer, respectively.
S. Takahashi, M. Tachiki /Optical conductivity in superconducting oxides
510
1.5
I
I
-
E/l
3 2 ”
c-axis
l.O-
* 2
0.5-
b”
a$ b
0’
0
I
I
I
1
2
3
w/2A
0/2A Fig. 5. Optical conductivity of the CuOr chain layers. The solid curve indicates the optical conductivity in the superconducting state and the dashed curve indicates the optical conductivity in the normal state.
electron crossing a gap of 24 in the density of states. We mentioned above that the finite value of u~( o) below 24 is the contribution from the chain layer conductivity &,b<0). The calculated conductivity is shown in fig. 5. As seen from the figure, the conductivity in the superconducting state denoted by the solid curve has no gap structure and thus contributes to the finite value of Q(W) below 2 A. The gaplessness of a&(w) is understood in the following way. As seen in fig. 4(b), the density of states at w= 0 is finite, although it has a gap structure which arises from the superconductivity induced by the proximity effect ” . The gapless character of a&(w) comes from the finiteness of the density of states around w= 0. The gap structure in the density of states causes a decrease of a&(w) in the superconducting state from o&(w) in the normal state, shown by the dashed curve in fig. 5. The optical conductivity along the c axis is shown in fig. 6. The parameter values are the same as those in fig. 3. The optical conductivity a,(w) gradually decreases as the frequency decreases below 24. The first term in the brackets of eq. ( 19) contains both the spectral functions of the CuOz layer and the CUO~
Fig. 6. Normalized optical conductivity for the electric field E parallel to the c-axis as a function of frequency. Q.,(W) is the optical conductivity in the normal state.
chain layer, and thus Us shows an intermediate behavior between the conductivities of both layers. The magnitude of oc( o) is about one tenth of a,( w ) at frequencies larger than the gap energy 24. The magnitude and the frequency dependence of a,(w) are consistent with the experimental results obtained by Collins et al. using single crystals of YBazCu307
[21.
Acknowledgements The authors would like to express their sincere thanks to Prof. F. Steglich, Prof. H. Matsumoto, Dr. T. Koyama and Mr. M. Machida for their valuable discussions. This work was supported by a Grant-inAid from the Ministry of Education, Science and Culture, Japan.
Appendix A Green’s functions, GM(k, iw,), 6,(k, iw,), and im,,), satisfy the equations of motion
GA,,(k,
io, - CA(k)?3 -AT, -&(w)
l~~(k, iw) = l+t(k)e,(k,
PI
The flat Fermi surface in fig. 2 is that of the CUO~ chain band. The magnitude of the superconducting gap is large in the region of the Fermi surface near the X point, since the mixing with the Cu02 bands is large. The superconducting gap in the region near the rpoint almost vanishes, since the mixing with the CuOZ bands is negligibly small.
+f*(k)&A(k,
[iOn-dkh
iw,)r3
(A.11
io,)r3,
--J%(w) l&A(k bJ =t*(k)eM(k,
iw,)r,
S. Takahashi. M. Tachiki /Optical conductivity in superconducting oxides
+f(k)eAsA(kim,)T3, [iw,-c,(k)r,
-dr,
(A-2)
-fA(cO)]GA,A(k,
ion)
~t(k)G~~(k, +t’(k)GM(kr
io,)73,
io,)r3 (A.3)
where fA (0) and fB( o) are given in eqs. ( 12 ) and (ll),respectively.Fromeqs. (A.l)-(A.3),wehave eM(k,
icG=
X
(A.4)
ZIY-,~T
where g is the coupling constant between the p electrons and the fluctuations, F(V) is the spectral function of the fluctuations, and & (k, co) is the spectral function of the p electron which is given by
&A(4 w)= -iIm
1 o+i&-eA(k)r3-Arl
-.$A(w)
1*
If the F(Y) is independent of v, integrating with respect to k we have
(B.2) \ ,
eq. (B. 1)
0
where ImfA(o)=-aRe X=io,
511
-CITY
--AT, -fA(o)
Ill(k -73ico,-cEA(k)T3-Ar, Y=io,-c,(k)r,
0
x -fA(w)
“’
w-v-~~,,(~-v)+(d-~~,2(w-v))7, ,/(o- v-&,(wv))*- (A-x&,z(w-
-fB(o)
t*(k)t’*(k) +T3 . m,--Ez(k)r3 -A7l
v))* ’ (J3.3)
where cr= rrg2NA(0)F, N,(O) being the density of states of the CuOZ layer at the Fermi level in the normal state. If we put _??A(o- v) =O in the right-hand side of eq. (B.3), we have for all o
It(k Z=t(k)T,
dv
s
-s,(w)
” ’
ImZ,(o)=-culwlJ .%t*(k)r3+r3.
t(k)t’ lo,-tA(k)T3
(k) -AT, -&(w)
“.
- cvdln
A/lWl l-J_
>
Similarly, &,(k,
&,z,
(A-5)
x_iy_,z.
(A.61
iw,) = y
eA,,,(k, ion)=
It is easily proved that eM( k, ion) = GA.,, (k, ion).
ImS,(w)=-cll]wl
by 0 Im~,(W)=--g2~IduF(Y)tea(k,o-ll)r3, k
0 03.1)
,
(B-4)
for I o I > A, and zero otherwise. Even when the self-energy is caused by the magnetic fluctuations, the imaginary part of the self-energy is approximated by the form of eq. (B.4), if the interaction with the magnetic fluctuations is strongly suppressed below the energy gap A. In the normal state, eq. (B.4) is reduced to
Appendix B In the following derivation, we assume that the fluctuations which interact with the p electrons are nonmagnetic. For T= 0 and o > 0, the imaginary part of the p-electron self-energy which is caused by the emission and absorption of the fluctuations is given
sign(w)T,
.
(B.5)
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