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Quantum chemistry and high-Tc superconductivity
Physica C 153-155 (1988) 1165-1166 North-Holland, Amsterdam
QUANTUM CHEMISTRY AND HIGH-Tc SUPERCONDUCTIVITY
K.H. JOHNSON, M.E. McHENRY, C. COUNTERMAN, R.C. O'HANDLEY, and G. KALONJI
Massachusetts
Institute
of Technology,
A. COLLINS,
Cambridge,
M.M. DONOVAN,
MA 02139,
U.S.A.
A "real-space" quantum-chemlcal basis f o r Cooper p a i r i n g i s r e v i e w e d and applied to high-Tc oxide superconductors. Calculated Tc's, isotope effects, c o h e r e n c e l e n g t h s , and c r i t i c a l f i e l d s a r e i n good a g r e e m e n t w i t h e x p e r i m e n t .
I. QUANTUM CHEMISTRY AND SUPERCONDUCTIVITY Rapidly expanding research on high-To superconductivity in La2Cu04 and YBa2CumO7-x oxides, for which BCS theory seems to be inadequate, has prompted us to investigate these systems with a novel quantum-chemical basis for Cooper pairing proposed in 1983 (i). This basis is the presence (in all known types
electronic pairing mechanism; a static J-T distortion (lattice instability) can compete with Cooper pairing for 0 < ~ s I/4. The s o l u t i o n of SchrSdinger's equation for the pair potential (1) leads to simple formulae for calculating Tc f r o m B and d ( 1 ) : ksTc = 1 . 1 4 h ~ c e x p [ - 1 / ( A
of superconductors) of degenerate (or nearly degenerate) "real-space" molecular orbitals at the Fermi energy (EF) which are bonding within spatially extended "tubes" (or "layers") of alternating phase, such as those obtained from linear
combinations
of
overlapping p~or ( i ) . Figs. I and 2 show how, under the influence of strong Cu(d)-O(p) ~-antibonding, the O(p~) orbltals in St-doped ia2Cu04 and in YBa2Cu307 overlap at EF to form O(p)-O(p) ~-bonding "tubes" parallel to the Cu-O planes. Denoting the extended tubular molecular orbitals of opposite phase as ~÷ and ~-, one can construct a Cooper-pair singlet wavefunction that describes the dynamical "antiferromagnetic" correlation of an electron pair of opposite spins over the ~+ and Ctubes, respectively, induced by dynamic or pseudo Jahn-Teller (J-T) displacements ~ of the lattice ions (e.g. oxygen ions in Figs. I and 2). The resulting pair potential is (1,2)
dS-bonding atomic opbitals
V(r) ~ -e26/d 2 ~ -e2(m/M)~/d m -e2/1
(I)
over the distance d < r < 1 = (M/m)~d. Here d is the "correlation distance" between neighboring tubes (related to the lattice constant; see Figs. I and 2), beyond which the electron-electron repulsion at EF is la/'gely screened out by the intervening ion cores; M and m are the lattice ion (e.g. oxygen in Figs. I and 2) and electron masses, respectively; is a measure of the J-T coupling, which is a function of the bond overlap at EF along the tubes ~+ and ¢-. Dynamic J-T coupling occurs for I/4 < ~ ~ I/2 (small bond overlap at EF) and yields BCS-like ("phonon"-induced) pairing; pseudo J-T coupling can occur for 0 < ~ ~ I/4 (large bond overlap at EF) and mimics an
0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
- ~)],
(2)
h~)c = h2(m/M)~14nmd 2, A = 2me2d/h2;
(3)
~ = A(m/M) ~.
(4)
For ~ s I/2 (very small bond overlap at EF), h ~ ~ h~D (2) (the Debye energy), ~ << A, and Eq. (2) reduces to the BCS formula. For ~ O. 15 (large bond overlap at EF), h ~ ~ 0.1EF, and formula (2) resembles one for a purely electronic ("excitonic") pairing theory. Calculated Te's for superconductors havfng a wide range of d's and ~'s are compared with experiment in Table I. Plots of Te vs. ~ (M = 16) for increasing d are shown in Fig. 3 and reveal that the upper limit to Tc is ~ 240°K, that high Tc's are attainable only for relatively large d and small ~ (large bond overlap at EF), and that the isotope effect is zero at the Tc-vs-B peak for any d-value. Formulae for calculating crltlcal magnetic fields from d and ~ have also been applied with good results to many superconductors (3). TABLE I. Tc values from formulae (2)-(4). Superconductor
~
d(A)
T~heO(°K)
O.SO 1 . 7 6 0.43 1.$3 Molybdenum 0.36 1.36 Vanadium 0.26 1.51 Niobium 0 . 2 1 s 1.65 Nb3Sn 0.19 1.87 BaPbo.7Bio.303 0.43 3.03 (Sr)La2Cu04 0.28 2.73
1.16 0.90 0.92 6.34 9.22 18.2 12.6 41.2
Aluminum Zinc
YBa2Ou307
f3.86(d2) 139.2l
0.17 [3.0
(dr)
90.0 J
T~xpt(°K)
1.18 0.90 0.92 5.43 9.25 18.0 ~13 ~40 90+
K.H. Johnson et al. / Quantum chemistry and high-T¢ superconductivity
1166
2. APPLICATIONS TO HIGH-Tc OXIDES SCF-Xm-SW cluster molecular orbitals (4) have been calculated for Sr-doped La2Cu04 and YBa2Cu307. Orbital topologies at EF are shown in Figs. I and 2. The key feature of both are the oxygen p~ orbitals that couple in phase as O(p)-O(p) m-bonding tubes parallel to the Cu-O planes along the and directions, respectively. Cu(d)-O(p) ~-antibonding at EF "compresses" the O(p~) orbitals to promote 0-0 p~-bond overlap. Also, some delocalized La and Ba d~-bonding character at EF couples with the 0-0 p~-bonding tubes to "phase-lock" the tubular p~-bend overlap. Thus the detailed composite real-space coordination chemistry at EF i n these oxides ( l a r g e l y i g n o r e d in k-space band-structure and model-Hamiltonian analyses) is absolutely crucial to the understanding of their high-To superconducting properties.
250 :
225
z
• Cu
FIGURE I. Orbital Topology of La2CuO¢ at EF.
Z
0
Ba
M =
16 f: d e: d d:d= c: d b: d
f
200
175 ~
~x~ 150 ~3 i ~ 125
'/~ fl ff
\
/ /
11
I00
\\ \(: \ \ \\\\
b \\\\
' ~\\\at
75
= 12A = 8A 6A = 4A = 5A
Zero isotope effect
peaks
?¢u¢o J - .~..~'b~.nomic J - T
50
25i .0
•La
e
0.1
0.2
/3
0.3
0.4
0.5
FIGURE 3. Graph of Tc vs. ~ for increasing d.
J-T ~-values for La2Cu04 and YBa2Cu307, derived from the 0-O p~-bond overlap at EF, are listed in Table I, along with the correlation distances d obtained from Figs. 1 and 2. Note, in Fig. 2 and Table I, that two values of d in YBa2Cu307 explain observed 90+ Tc's and suggest anisotropic pairing, ia2CuO¢ is barely within the dynamic J-T ("phonon") region of Fig. 3, while YBa2Cu307 is well inside the pseudo J-T ("electronic") region. To's, calculated from formulae (2)-(4), agree well with experiment. The ~-value for YBa2Cu307 is near the peak of its Tc-vs-~ curve in Fig. 3, where the isotope effect: Tc ~ M vanlshes; from ~Eqns. (2)~4), = -OlnTc/OlnM =~{1-10.46A(m/M)W/d[l-(m/M)~W]} ~ for = 0.06, for 100Z O isotope substitution. La2CuO¢ lies off the peak of its Tc-vs-~ curve in Fig. 3, giving ~ = 0.22, within the range ~f experiment. The "coherence length" 1 = (M/m)'~d is 17-22A for YBa2CusOv and 49A for ta2Cu04, the orders of magnitude observed. Critical magnetic fields have also been calculated (3). 3. CONCLUSIONS
R e a l - s p a c e c o m p o s i t e O(p~)-O(p~) b o n d i n g / Cu(d~)-O(p~) a n t i b o n d i n g a t EF, t o g e t h e r w i t h dynamic and pseudo J-T c o u p l i n g , e x p l a i n s t h e properties
of high-Tc oxide superconductors.
ACKNOWLEDGEMENTS This work was sponsored by the Office Naval Research and the IBM Corporation.
FIGURE 2. Orbital Topology of YBa2Cu307 at EF.
of
REFERENCES (I) K.H. Johnson and R.P. Messmer, Synth. Metals 5 (1983) ISI. (2) V.F. W e i s s k o p f , Contemp. Phys. 22 (1981) 376. (3) D.P. C l o u g h e r t y and K.H. J o h n s o n , t h i s volume (4) J . C . S l a t e r and K.H. J o h n s o n , Phys. Rev BS (1972) 844.
QUANTUM CHEMISTRY AND HIGH-Tc SUPERCONDUCTIVITY
K.H. JOHNSON, M.E. McHENRY, C. COUNTERMAN, R.C. O'HANDLEY, and G. KALONJI
Massachusetts
Institute
of Technology,
A. COLLINS,
Cambridge,
M.M. DONOVAN,
MA 02139,
U.S.A.
A "real-space" quantum-chemlcal basis f o r Cooper p a i r i n g i s r e v i e w e d and applied to high-Tc oxide superconductors. Calculated Tc's, isotope effects, c o h e r e n c e l e n g t h s , and c r i t i c a l f i e l d s a r e i n good a g r e e m e n t w i t h e x p e r i m e n t .
I. QUANTUM CHEMISTRY AND SUPERCONDUCTIVITY Rapidly expanding research on high-To superconductivity in La2Cu04 and YBa2CumO7-x oxides, for which BCS theory seems to be inadequate, has prompted us to investigate these systems with a novel quantum-chemical basis for Cooper pairing proposed in 1983 (i). This basis is the presence (in all known types
electronic pairing mechanism; a static J-T distortion (lattice instability) can compete with Cooper pairing for 0 < ~ s I/4. The s o l u t i o n of SchrSdinger's equation for the pair potential (1) leads to simple formulae for calculating Tc f r o m B and d ( 1 ) : ksTc = 1 . 1 4 h ~ c e x p [ - 1 / ( A
of superconductors) of degenerate (or nearly degenerate) "real-space" molecular orbitals at the Fermi energy (EF) which are bonding within spatially extended "tubes" (or "layers") of alternating phase, such as those obtained from linear
combinations
of
overlapping p~or ( i ) . Figs. I and 2 show how, under the influence of strong Cu(d)-O(p) ~-antibonding, the O(p~) orbltals in St-doped ia2Cu04 and in YBa2Cu307 overlap at EF to form O(p)-O(p) ~-bonding "tubes" parallel to the Cu-O planes. Denoting the extended tubular molecular orbitals of opposite phase as ~÷ and ~-, one can construct a Cooper-pair singlet wavefunction that describes the dynamical "antiferromagnetic" correlation of an electron pair of opposite spins over the ~+ and Ctubes, respectively, induced by dynamic or pseudo Jahn-Teller (J-T) displacements ~ of the lattice ions (e.g. oxygen ions in Figs. I and 2). The resulting pair potential is (1,2)
dS-bonding atomic opbitals
V(r) ~ -e26/d 2 ~ -e2(m/M)~/d m -e2/1
(I)
over the distance d < r < 1 = (M/m)~d. Here d is the "correlation distance" between neighboring tubes (related to the lattice constant; see Figs. I and 2), beyond which the electron-electron repulsion at EF is la/'gely screened out by the intervening ion cores; M and m are the lattice ion (e.g. oxygen in Figs. I and 2) and electron masses, respectively; is a measure of the J-T coupling, which is a function of the bond overlap at EF along the tubes ~+ and ¢-. Dynamic J-T coupling occurs for I/4 < ~ ~ I/2 (small bond overlap at EF) and yields BCS-like ("phonon"-induced) pairing; pseudo J-T coupling can occur for 0 < ~ ~ I/4 (large bond overlap at EF) and mimics an
0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
- ~)],
(2)
h~)c = h2(m/M)~14nmd 2, A = 2me2d/h2;
(3)
~ = A(m/M) ~.
(4)
For ~ s I/2 (very small bond overlap at EF), h ~ ~ h~D (2) (the Debye energy), ~ << A, and Eq. (2) reduces to the BCS formula. For ~ O. 15 (large bond overlap at EF), h ~ ~ 0.1EF, and formula (2) resembles one for a purely electronic ("excitonic") pairing theory. Calculated Te's for superconductors havfng a wide range of d's and ~'s are compared with experiment in Table I. Plots of Te vs. ~ (M = 16) for increasing d are shown in Fig. 3 and reveal that the upper limit to Tc is ~ 240°K, that high Tc's are attainable only for relatively large d and small ~ (large bond overlap at EF), and that the isotope effect is zero at the Tc-vs-B peak for any d-value. Formulae for calculating crltlcal magnetic fields from d and ~ have also been applied with good results to many superconductors (3). TABLE I. Tc values from formulae (2)-(4). Superconductor
~
d(A)
T~heO(°K)
O.SO 1 . 7 6 0.43 1.$3 Molybdenum 0.36 1.36 Vanadium 0.26 1.51 Niobium 0 . 2 1 s 1.65 Nb3Sn 0.19 1.87 BaPbo.7Bio.303 0.43 3.03 (Sr)La2Cu04 0.28 2.73
1.16 0.90 0.92 6.34 9.22 18.2 12.6 41.2
Aluminum Zinc
YBa2Ou307
f3.86(d2) 139.2l
0.17 [3.0
(dr)
90.0 J
T~xpt(°K)
1.18 0.90 0.92 5.43 9.25 18.0 ~13 ~40 90+
K.H. Johnson et al. / Quantum chemistry and high-T¢ superconductivity
1166
2. APPLICATIONS TO HIGH-Tc OXIDES SCF-Xm-SW cluster molecular orbitals (4) have been calculated for Sr-doped La2Cu04 and YBa2Cu307. Orbital topologies at EF are shown in Figs. I and 2. The key feature of both are the oxygen p~ orbitals that couple in phase as O(p)-O(p) m-bonding tubes parallel to the Cu-O planes along the
250 :
225
z
• Cu
FIGURE I. Orbital Topology of La2CuO¢ at EF.
Z
0
Ba
M =
16 f: d e: d d:d= c: d b: d
f
200
175 ~
~x~ 150 ~3 i ~ 125
'/~ fl ff
\
/ /
11
I00
\\ \(: \ \ \\\\
b \\\\
' ~\\\at
75
= 12A = 8A 6A = 4A = 5A
Zero isotope effect
peaks
?¢u¢o J - .~..~'b~.nomic J - T
50
25i .0
•La
e
0.1
0.2
/3
0.3
0.4
0.5
FIGURE 3. Graph of Tc vs. ~ for increasing d.
J-T ~-values for La2Cu04 and YBa2Cu307, derived from the 0-O p~-bond overlap at EF, are listed in Table I, along with the correlation distances d obtained from Figs. 1 and 2. Note, in Fig. 2 and Table I, that two values of d in YBa2Cu307 explain observed 90+ Tc's and suggest anisotropic pairing, ia2CuO¢ is barely within the dynamic J-T ("phonon") region of Fig. 3, while YBa2Cu307 is well inside the pseudo J-T ("electronic") region. To's, calculated from formulae (2)-(4), agree well with experiment. The ~-value for YBa2Cu307 is near the peak of its Tc-vs-~ curve in Fig. 3, where the isotope effect: Tc ~ M vanlshes; from ~Eqns. (2)~4), = -OlnTc/OlnM =~{1-10.46A(m/M)W/d[l-(m/M)~W]} ~ for = 0.06, for 100Z O isotope substitution. La2CuO¢ lies off the peak of its Tc-vs-~ curve in Fig. 3, giving ~ = 0.22, within the range ~f experiment. The "coherence length" 1 = (M/m)'~d is 17-22A for YBa2CusOv and 49A for ta2Cu04, the orders of magnitude observed. Critical magnetic fields have also been calculated (3). 3. CONCLUSIONS
R e a l - s p a c e c o m p o s i t e O(p~)-O(p~) b o n d i n g / Cu(d~)-O(p~) a n t i b o n d i n g a t EF, t o g e t h e r w i t h dynamic and pseudo J-T c o u p l i n g , e x p l a i n s t h e properties
of high-Tc oxide superconductors.
ACKNOWLEDGEMENTS This work was sponsored by the Office Naval Research and the IBM Corporation.
FIGURE 2. Orbital Topology of YBa2Cu307 at EF.
of
REFERENCES (I) K.H. Johnson and R.P. Messmer, Synth. Metals 5 (1983) ISI. (2) V.F. W e i s s k o p f , Contemp. Phys. 22 (1981) 376. (3) D.P. C l o u g h e r t y and K.H. J o h n s o n , t h i s volume (4) J . C . S l a t e r and K.H. J o h n s o n , Phys. Rev BS (1972) 844.