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Structural instability and high temperature superconductivity
PhysicaC153 155 (1988) 194 195 North-Holland, Amsterdam
STRUCTURAL INSTABILITY AND HIGH TEMPERATURESUPERCONDUCTIVITY V.L.AKSENOV and N.M.PLAKIDA Jo i n t I n s t i t u t e f o r Nuclear Research,Dubna, 141980, USSR The influence of structural i n s t a b i l i t y on superconducting pairing in new oxide metals is investigated. I t is shown that the i n t e r a c t i o n of electrons with highly anharmonic oxygen-ion vibrations results in a s u f f i c i e n t l y large coupling constant and high T e. 1. INTRODUCTION A common feature of new oxide metals is the appearance of a structural phase t r a n s i t i o n (SPT) of displacive type in doped L2CuO4 and of order-disorded type in YBa2CusO7. In t h i s paper the influence of structural i n s t a b i l i t y on superconducting pairing is investigated. The discussion is mostly concerned with doped La2CuO4 in which the c o r r e l a t i o n between structural ins t a b i l i t y and superconductivity reveals i t s e l f more c l e a r l y . Application of the model to the higher-To materials is discussed also. 2. MODEL DESCRIPTION In the La2_x(Ba,Sr)xCuO 4 the SPT takes place at T=T d from a high temperature tetragonal phase (space group D~) to an orthorombic phase (space group D ~ ) . The Td decreases nearly l i nearly with the concentration x. On the basis of the symmetry consideration i t has been shown ( I ) that the SPT is induced by the condensation of a soft t i l t i n g mode at the X-point of the B r i l l o u i n mode observed in the neutron experiment (2). This SPT is determined by a two component order parameter (CI,C2) which is connected with the one-dimensional i r r e d u c i b l e representation r 3 of a two-arm star of the wave vector at X-point: ql = (~/a)(~,~, O) and qe = = (:/a)(-g,~,O). The order parameter is physic a l l y r e a l i z e d by the t i l t i n g of the CuO6 octahedra around the [ I I 0 ] axis f o r the arm ql and around the [ i i 0 ] axis fo r the arm q2 of the wave vector at X-point. By taking into account the above mentioned symmetry consideration the dynamics of the oxygen ions in soft r o t a t i o n a l - t y p e modes can be described by a model Hamiltonian (3):
Hs
Z
( l / 2 ) Z h ~ f~;)Rh(~)Rh(f')
+
Z
' " ," v, A A1"'" A t t /' )"R x2 c"~' ") R A , ~ ."~ ' /
+
~;
(1/4) F h h ( [ g ' ) [ R x ( f )
+
+
where R(~) are the local normal coordinates associated with the r o t a t i o n of the Cu06 octahedra around ~=x.y axis, K ~ , V ~ , , F~A are determined by i n t e r a c t i o n parameters in the l a t t i c e . The dynamics of the model given by e l. ( I ) was investigated in reference (3). The addition of Ba and Sr impurities brings a change in the parameters of e l . ( 1 ) . P a r t i c u l a r l y , according to (3) (p is concentration)
vxx,(ql.p) =vAx,(q2, p) = -A(p) 8AX,. Thus doping leads to s t a b i l i z a t i o n of the l a t t i c e . At the same time l i n e a r in R(~) terms do not appear in e l . ( 1 ) as i t follows from the symmetry of the system. In the case of strong anharmonicity one can consider only two lowest energy levels of the anharmonic double-well potential in the Hamiltonian given by e l . ( 1 ) . In t h i s case using pseudo-spin representation we obtain e f f e c t i v e transverse Ising model Hamiltonian (4). The proposed model can be applied to the YBa2Cu307 to describe oxygen ions vibrations perpendicular to a CuO bond in a remote from the SPT-point region where anomalously large and strongly anisotropic thermal factors have been observed. 3. EFFECTIVE COUPLING CONSTANT Bond-bending oxygen ions v i b r a t i o n s on a Cu-O bond lead to a r e l a t i v e l y weak i n t e r a c t i on due to additional hy d r id iz at ion of the Cu-d(x~-y ~) and O-Pz o r b i t a l s . The overlap in tegral t ~ J . x ( ~ , k ) , where the oxygen ions displacements X(~,k) ~I =
(I)
= (2/2.v'2m)
(i, 0, 0),
~2=(~ i, 0).
Estimations giveot ~ V p ~ ( X / a ) = 2 eV and a = 2 A leading to J - Vpd/a = I
gg'h
~k'(R(g +a~k)-R(g))]z,
, where Vpd~
(eV/A).
- RA(~ , ) ] 4 ,
0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
In t h i s case the electron-electron coupling constant
V.L. Aksenov and N.M. Plakida / S t r u c t u r a l instability
As = N ( 0 ) ~ X s where
N(0)
~ ~s~s is
the
f r e q u e n c y - s . Then a r e l a t i v e change in T c is given by reference (5) as
,
density
of
electron
states
on the Fermi l e v e l ; j2 the deformation potent i a l averaged over the Fermi surface becomes large due to the high l a t t i c e s u s c e p t i b i l i t y (4) ~s = d2/h~
s .
Here 2d is the distance between minima in the double-well p o t e n t i a l , ~s the mean frequency of anharmonic v i b r a t i o n s . In a harmonic lattice Xpb =
/h~pb ~
1/m~
~
and the r e l a t i o n of the coupling constants As and Aph = N(O) j 2 y p h = np h ~ph has the form ~s/Aph = Ys/Xph = d 26/~s > %0,
(2)
where < u e > = I0 -s ~2, d 2 10-2 X2 are obtained from the Debye-Waller f a c t o r at low temperatures f o r harmonic and anharmonic quasi-local v i b r a t i o n s , respectively. =
4. SUPERCONDUCTINGTEMPERATURE Let us estimate the superconducting temperature in the case of anharmonic v i b r a t i o n s only using the Allen-Dynes formula: kT c = 0.18~s
~/As
195
'
For t h i s one should know the f o l l o w i n g parameters: ~s ns and d . The v a l u e s ~ f ~s and d are no = I eV/A 2, d ( 0 . 2 + 0 . 3 ) A being about the same (4) f o r the compounds La-(Ba,Sr)-Cu-O and Y-Ba-Cu-O. There is some i d e n t i t y in the determination of ~s f o r the Y-Ba-Cu-O. Using some averaged value ~ s = 1 0 ÷ 2 0 meVwe obtain As= = 4 ÷ 9 and To = (80 ÷120) K. 5. ISOTOPE EFFECT An important test f o r the phonon theories is an isotope e f f e c t . Contrary to the conventional theory the e f f e c t i v e coupling constant As in our model depends on mass M through anharmonic
~ T c = AT c / T O :
(3)
: -(AM/M) [ 1 + (M/O~s) (d~s/dM) ] R(A, ~ * ) ,
where R ( A , ~ * ) i s a function of model parameters. On s u b s t i t u t i o n of 160 with 180 eq.(3) gives f o r To= 40 K ~T o = -10-8+ 2.5xi0 -2 ( in the harmonic theory ~To= 6.25xi0 -2) and f or Tc = 90 K is y i e l d s ~Tc = -10 -2 . These results are in q u a l i t a t i v e agreement with experiment a l data on the doped La2CuO4 and not in cont r a d i c t i o n with the data on the YBa2Cu307 which are not s u f f i c i e n t l y r e l i a b l e at present. 6. CONCLUSION Thus the presence of structural i n s t a b i l i ty in new superconductors leads to the appearance of the strongly anharmonic quasi-local v i b r a t i o n s c h a r a c t e r i s t i c s f or the perovskitel i k e structure. In t h i s case the large coupl i n g constant As (and high To) is reached
not due to a strong electron-phonon interaction but rather due to the high l a t t i c e susc e p t i b i l i t y for these vibrations. A more detailed investigation must take into account anisotropy effects and the possibility of the appearance of short-order clusters changing the dynamics of the system (6). REFERENCES (1) N.M.Plakida and V.S.Schahmatov, JINR P17-87-488 (Dubna, 1987). (2) R.J.Birgeneau, C.Y.Chen, D.R.Gable, P.H.Jenssen, M.A.Kastner, C.J.Peters, P.J.Picone, Tineke Thio, T.R.Thurston, H . L . T u l l e r , J.D.Axe, P.Boni and G.Shirane, Phys.Rev.Lett. 59 (1987) 1329. (3) V.L.Aksenov, N.M.Plakida and S.Flach, JINR, P17-87-44 (Dubna, 1987). (4) N.M.Plakida, V.L.Aksenov and S.Drechsler, Europhys.Lett. 4 (1987) 1309. (5) S.Drechsler and N.M.Plakida, phys. stat. sol . (b), to be published. (6) V.L.Aksenov, M.Bobeth, N.M.Plakida and J.Schreiber, J.Phys.C, 20 (1987) 375.
STRUCTURAL INSTABILITY AND HIGH TEMPERATURESUPERCONDUCTIVITY V.L.AKSENOV and N.M.PLAKIDA Jo i n t I n s t i t u t e f o r Nuclear Research,Dubna, 141980, USSR The influence of structural i n s t a b i l i t y on superconducting pairing in new oxide metals is investigated. I t is shown that the i n t e r a c t i o n of electrons with highly anharmonic oxygen-ion vibrations results in a s u f f i c i e n t l y large coupling constant and high T e. 1. INTRODUCTION A common feature of new oxide metals is the appearance of a structural phase t r a n s i t i o n (SPT) of displacive type in doped L2CuO4 and of order-disorded type in YBa2CusO7. In t h i s paper the influence of structural i n s t a b i l i t y on superconducting pairing is investigated. The discussion is mostly concerned with doped La2CuO4 in which the c o r r e l a t i o n between structural ins t a b i l i t y and superconductivity reveals i t s e l f more c l e a r l y . Application of the model to the higher-To materials is discussed also. 2. MODEL DESCRIPTION In the La2_x(Ba,Sr)xCuO 4 the SPT takes place at T=T d from a high temperature tetragonal phase (space group D~) to an orthorombic phase (space group D ~ ) . The Td decreases nearly l i nearly with the concentration x. On the basis of the symmetry consideration i t has been shown ( I ) that the SPT is induced by the condensation of a soft t i l t i n g mode at the X-point of the B r i l l o u i n mode observed in the neutron experiment (2). This SPT is determined by a two component order parameter (CI,C2) which is connected with the one-dimensional i r r e d u c i b l e representation r 3 of a two-arm star of the wave vector at X-point: ql = (~/a)(~,~, O) and qe = = (:/a)(-g,~,O). The order parameter is physic a l l y r e a l i z e d by the t i l t i n g of the CuO6 octahedra around the [ I I 0 ] axis f o r the arm ql and around the [ i i 0 ] axis fo r the arm q2 of the wave vector at X-point. By taking into account the above mentioned symmetry consideration the dynamics of the oxygen ions in soft r o t a t i o n a l - t y p e modes can be described by a model Hamiltonian (3):
Hs
Z
( l / 2 ) Z h ~ f~;)Rh(~)Rh(f')
+
Z
' " ," v, A A1"'" A t t /' )"R x2 c"~' ") R A , ~ ."~ ' /
+
~;
(1/4) F h h ( [ g ' ) [ R x ( f )
+
+
where R(~) are the local normal coordinates associated with the r o t a t i o n of the Cu06 octahedra around ~=x.y axis, K ~ , V ~ , , F~A are determined by i n t e r a c t i o n parameters in the l a t t i c e . The dynamics of the model given by e l. ( I ) was investigated in reference (3). The addition of Ba and Sr impurities brings a change in the parameters of e l . ( 1 ) . P a r t i c u l a r l y , according to (3) (p is concentration)
vxx,(ql.p) =vAx,(q2, p) = -A(p) 8AX,. Thus doping leads to s t a b i l i z a t i o n of the l a t t i c e . At the same time l i n e a r in R(~) terms do not appear in e l . ( 1 ) as i t follows from the symmetry of the system. In the case of strong anharmonicity one can consider only two lowest energy levels of the anharmonic double-well potential in the Hamiltonian given by e l . ( 1 ) . In t h i s case using pseudo-spin representation we obtain e f f e c t i v e transverse Ising model Hamiltonian (4). The proposed model can be applied to the YBa2Cu307 to describe oxygen ions vibrations perpendicular to a CuO bond in a remote from the SPT-point region where anomalously large and strongly anisotropic thermal factors have been observed. 3. EFFECTIVE COUPLING CONSTANT Bond-bending oxygen ions v i b r a t i o n s on a Cu-O bond lead to a r e l a t i v e l y weak i n t e r a c t i on due to additional hy d r id iz at ion of the Cu-d(x~-y ~) and O-Pz o r b i t a l s . The overlap in tegral t ~ J . x ( ~ , k ) , where the oxygen ions displacements X(~,k) ~I =
(I)
= (2/2.v'2m)
(i, 0, 0),
~2=(~ i, 0).
Estimations giveot ~ V p ~ ( X / a ) = 2 eV and a = 2 A leading to J - Vpd/a = I
gg'h
~k'(R(g +a~k)-R(g))]z,
, where Vpd~
(eV/A).
- RA(~ , ) ] 4 ,
0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
In t h i s case the electron-electron coupling constant
V.L. Aksenov and N.M. Plakida / S t r u c t u r a l instability
As = N ( 0 ) ~ X s where
N(0)
~ ~s~s is
the
f r e q u e n c y - s . Then a r e l a t i v e change in T c is given by reference (5) as
,
density
of
electron
states
on the Fermi l e v e l ; j2 the deformation potent i a l averaged over the Fermi surface becomes large due to the high l a t t i c e s u s c e p t i b i l i t y (4) ~s = d2/h~
s .
Here 2d is the distance between minima in the double-well p o t e n t i a l , ~s the mean frequency of anharmonic v i b r a t i o n s . In a harmonic lattice Xpb =
1/m~
~
and the r e l a t i o n of the coupling constants As and Aph = N(O) j 2 y p h = np h ~ph has the form ~s/Aph = Ys/Xph = d 26/
(2)
where < u e > = I0 -s ~2, d 2 10-2 X2 are obtained from the Debye-Waller f a c t o r at low temperatures f o r harmonic and anharmonic quasi-local v i b r a t i o n s , respectively. =
4. SUPERCONDUCTINGTEMPERATURE Let us estimate the superconducting temperature in the case of anharmonic v i b r a t i o n s only using the Allen-Dynes formula: kT c = 0.18~s
~/As
195
'
For t h i s one should know the f o l l o w i n g parameters: ~s ns and d . The v a l u e s ~ f ~s and d are no = I eV/A 2, d ( 0 . 2 + 0 . 3 ) A being about the same (4) f o r the compounds La-(Ba,Sr)-Cu-O and Y-Ba-Cu-O. There is some i d e n t i t y in the determination of ~s f o r the Y-Ba-Cu-O. Using some averaged value ~ s = 1 0 ÷ 2 0 meVwe obtain As= = 4 ÷ 9 and To = (80 ÷120) K. 5. ISOTOPE EFFECT An important test f o r the phonon theories is an isotope e f f e c t . Contrary to the conventional theory the e f f e c t i v e coupling constant As in our model depends on mass M through anharmonic
~ T c = AT c / T O :
(3)
: -(AM/M) [ 1 + (M/O~s) (d~s/dM) ] R(A, ~ * ) ,
where R ( A , ~ * ) i s a function of model parameters. On s u b s t i t u t i o n of 160 with 180 eq.(3) gives f o r To= 40 K ~T o = -10-8+ 2.5xi0 -2 ( in the harmonic theory ~To= 6.25xi0 -2) and f or Tc = 90 K is y i e l d s ~Tc = -10 -2 . These results are in q u a l i t a t i v e agreement with experiment a l data on the doped La2CuO4 and not in cont r a d i c t i o n with the data on the YBa2Cu307 which are not s u f f i c i e n t l y r e l i a b l e at present. 6. CONCLUSION Thus the presence of structural i n s t a b i l i ty in new superconductors leads to the appearance of the strongly anharmonic quasi-local v i b r a t i o n s c h a r a c t e r i s t i c s f or the perovskitel i k e structure. In t h i s case the large coupl i n g constant As (and high To) is reached
not due to a strong electron-phonon interaction but rather due to the high l a t t i c e susc e p t i b i l i t y for these vibrations. A more detailed investigation must take into account anisotropy effects and the possibility of the appearance of short-order clusters changing the dynamics of the system (6). REFERENCES (1) N.M.Plakida and V.S.Schahmatov, JINR P17-87-488 (Dubna, 1987). (2) R.J.Birgeneau, C.Y.Chen, D.R.Gable, P.H.Jenssen, M.A.Kastner, C.J.Peters, P.J.Picone, Tineke Thio, T.R.Thurston, H . L . T u l l e r , J.D.Axe, P.Boni and G.Shirane, Phys.Rev.Lett. 59 (1987) 1329. (3) V.L.Aksenov, N.M.Plakida and S.Flach, JINR, P17-87-44 (Dubna, 1987). (4) N.M.Plakida, V.L.Aksenov and S.Drechsler, Europhys.Lett. 4 (1987) 1309. (5) S.Drechsler and N.M.Plakida, phys. stat. sol . (b), to be published. (6) V.L.Aksenov, M.Bobeth, N.M.Plakida and J.Schreiber, J.Phys.C, 20 (1987) 375.