HAL Id: jpa-00247076
https://hal.archives-ouvertes.fr/jpa-00247076
Submitted on 1 Jan 1995
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Calculated Superconducting Gap Dependence on Doping in Single Layered Copper Oxides
T. Hocquet, J.-P. Jardin, P. Germain, J. Labbé
To cite this version:
T. Hocquet, J.-P. Jardin, P. Germain, J. Labbé. Calculated Superconducting Gap Dependence on Doping in Single Layered Copper Oxides. Journal de Physique I, EDP Sciences, 1995, 5 (4), pp.517- 524. �10.1051/jp1:1995144�. �jpa-00247076�
Classification Physics Abstracts
75.loL 74.70V 74.20
Calculated Superconducting Gap Dependence on Doping in Single
Layered Copper Oxides
T- Hocquet (1), J.-P. Jardin (~), P. Germain (1) and J. Labbé (1)
(~) Laboratoire de Physique de la Matière Condensée de l'Ecole Normale Supérieure, 24 rue
Lhomond, 75231 Paris Cedex 05, France
(~) Laboratoire PMTM (CNRS), Université Paris Nord, 93430 Villetaneuse, France
(Received 21 September 1994, received in final form 13 December 1994, accepted 15 December
1994)
Abstract. We use a total Hamiltonian containing both an electron-phonon induced attrac- tive part of the interaction between electrons, and a Coulomb repulsive part formulated in the
Hubbard model- By diagonalising it in the Bogoliubov and Valatin mean field approximation,
we obtain equations for
a two valued superconducting gap function, with a much more precise
statement about the repulsive cut off energy than in the Morel and Anderson model. By applying
these equations to our bidimensional electronic model for a Cu02 plane, we find that the cal- culated superconducting gap decreases very slowly with increasing doping x in La2-~Sr~Cu04,
and we compare to the behaviour of the antiferromagnetic gap.
1. Introduction
In a previous paper [ii we explained why, in an itinerant electron model, trie antiferromagnetic phase of La2-~Sr~Cu04 was stable only for small values of trie doping ratio i. In trie present
paper, we show that, on the contrary, in trie same model, trie superconducting phase exists for much larger values of i.
From the begmning, trie true physical nature of the electronic structure of this type of com~
pounds is highly controversial, one of the most important questions being to know whether the electrons are localised by the correlations, or not. But very recent experimental photoemission data [2-4] clearly show, not only the existence of a Fermi energy, but also the existence of
a bidimensional saddle point in the energy spectrum, indicating the itinerant nature of the electron gas and its bidimensional character- Of course, various methods have been used to calculate the band structure of these materials [5-7]. However, smce our purpose was to car culate the variation of the superconducting gap as trie Fermi level continuously moves in the very neighbourhood of the Van Hove singularity of the electronic density of states, we preferred
to use a completely analytical expression of the latter, and we used a simplified tight-binding
model.
© Les Editions de Physique 1995
518 JOURNAL DE PHYSIQUE I N°4
To explain the existence of an antiferromagnetic phase and of a superconducting phase in
the saine compound, according to trie value of i, we start from a mortel Harniltonian which
is assumed to contain both trie Hubbard intra-atomic repulsive term Un~in~j, nia being trie number operator of electrons on atomic site1 with spin a and U trie positive repulsive Coulomb parameter, and trie usual B-C-S- attractive terms, with an eoEective coupling constant Vkk, depending on trie wave vector k and k'. By introducing trie creation (annihilation) operator
c[~(ck«) for non-interacting electrons with energies Ek, our total Hamiltonian is
H=Ho+Hi+H2 (1)
with
Ho = £Ekc[~cka, (2)
k,a
~l "
) £
CÎ4l~k2lCÎ3Î~kijôki+k2,k3+k4> (3)
k1k2k3k4
H2 "
£ Vk,k'CÎIC~ kjc-k'jck'l> (4)
k,k,
where N is trie number of unit cells in a Cu02 Plane, and trie symbol of Kronecker ôk,k,
ensures trie wave vector conservation. Expression (3) was obtained by Fourier~transforming
trie Hubbard Hamiltonian. It can be noticed that Vkk, is itself proportional to iIN.
As trie ranges of trie interactions contained in Hi and H2 are dioEerent, their respective
constants U and Vkk> do not have the same dependence on the wave vector: the intra-atomic
repulsive term U gives rise to a coupling constant which extends to the entire baud, whereas Vkk> is drastically restricted to a narrow domain close to the Fermi energy.
In Section 2, we include the repulsive terms of Hi when diagonahsing H in the mean.
field approximation, m the superconducting state. This leads to a set of two equations when
calculating the superconducting gap Ak, which has two distinct values Ai and A2 whether the energy is smaller or larger than the cut oOE energy contained in Vkk,.
In Section 3, we numerically solve the previous equations for Ai and A2 as functions of the doping ratio i, at the absolute zero, for a Cu02 Plane, by using our previous tight~binding mortel [8]. Dur important result is that Ai and A2 decrease only slowly as the Fermi energy
moues away from the logarithmic singularity, when z increases. In Section 4, we give an
approximate version of our exact results, and discuss its vahdity.
2. Equations for trie Superconducting Gap
Ta apply the meanfield approximation to the superconducting state, a well known procedure
consists in expressing the B-C-S- part Ho + H2 of H in terms of the Bogoliubov.Valatin (BV)
fermion operators
7~i " ~kCÎi "kC-kj
(5)
'ÎÎj " ~kCÎj + VkC-kl
where the real coefficients uk and vk must be chosen so as to diagonalise Ho + H2 in the self-consistent meanfield approximation, with ~( + v(
= i.
Dur purpose is to diagonalise by the sonne procedure our total Hamiltonian H including
the repulsive part Hi Thus we have to express Hi itself in terms of the same BV operators,
leading to
~ll "
~ ("k4'1'4i ~ ~k4'Î-k41) "k2'1k21
~ ~k2'1~k21)
kik~k~k~ (6)
X ~"k3'1'31 ~3'Î-k3i) ("ki'ÎkiÎ Vki'Î~ki1) ~ki+k2,k3+k4
In trie mean-field approximation, trie BV operators are assumed to describe independent quasi-particles. But trie exact expression (6) of Hi is a sum of terms which are products of four BV operator terms. Then it must be linearised by replacing, in each one of these terms, trie
product of two of trie four BV operators by its averaged value. These two operators are chosen in ail trie possible ways arnong trie four operators, trie anticommutation rules for fermions being
taken into account. Trie operators 1)~ are looked for sa as ta diagonalise trie total mean-field Hamiltonian. Thus trie involved averaged values are:
l'lla'lk'a' " fkôkk'ôaa', 'Î'a'Î'i~> " l'fka'Îk'a')
" Ù,
where fk will appear at trie end as trie Fermi-Dirac occupation factor. Trie resulting simplified expression Hi of Hi is
Hi =
~ ~ uÎ ~'f)a'fka
+ VÎ
2 ~ )a'fka)
k a a
+2ukVk 6'f)i'f~
ki +'f-kj'fki) NU~
+àc~ (UÎ VÎ) 6'f)i'f~
ki +'f-ki'fki) ~~~
k
~2 +2ukvk
1
~i)~ika) N#
a
~
where
Ac =
~ ~
ukvk (1 2fk) (8)
N
~
and where n
= 1- x is trie averaged number of electrons per copper atoms, with nN =
~ (c)~cka
" 2 ~ (u(fk + V((1 fk) (9)
ka k
Adding expression (7) of Éi ta trie well-known similarly linearised expression Éo + É2 of
Ho + H2 (9], one gets trie linearised expression É of H. Then, introducing trie chemical
potential p, and trie total number operator Mi one gets ù prit = wo + £ E~i[~i~a + £ r~ (i[~it
~~ + i-~i,~~) (io)
~a k
with
wo = £ 2 Ek +
~~
p v( (A(°~ Ac) (1+ 2fk) ukvk NU~
j
(II)
2
k
Ek = Ek + ~~ pl(u( v() + 2 A~°~ Ac) ukvk, (12)
2
520 JOURNAL DE PHYSIQUE I N°4
rk = 2ukvk lEk +
~~
p) A~°~ Ac) (u( VI) ,
(13)
2 and
Af~ = £ VkkJukJvkJ (1 2fk,) (14)
~>
Wo is trie ground state energy, Ek is trie energy of an elementary excitation, and Af~ would be trie superconducting gap if trie Coulomb repulsion were net taken into account. We see from the above expressions (10-13) that the elfect of adding the repulsive part Hi ta Ho + H2 is
simply ta replace Af~ by
Ak = A(°~ Ac (15)
and Ek by Ek + U).
The diagonalisation of É pA'is achieved by imposing the condition rk " 0, leading ta
Ek =
fi~, (16)
~~ ~~ ~~~
~~
~~~~
~k
i + ~Ek/kBT' ~~~~
where we have introduced the self-consistent Hartree-Fock one-partiale energy ck
# Ek + U ~
2 p, referred ta the chemical potential p, which actually depends on the doping ratio x
= 1 n.
Of course, the above results (16-18) have exactly the same form as if the Coulomb repulsion
were net taken into account. But, from equations (8), (14) and (15) of Ac, Af~ and Ak
respectively, the implicit equation for the gap Ak is modified and becomes
~~ ~ ~~~' ~ Î) ~J ~~
~2ÎÎ'T) ~~~~
The following step is ta assume, as usually, that the attractive coupling constant V~~> has a
non-vanishing value -VIN, with V > 0, only if bath (ck( and (ckJ are smaller than some cut off energy &wo. Then it results from equation (19) that the energy gap Ak has two distinct
constant values Ai and A2 according to the sign of (ck( &wo:
Ai = VC(T) UD(T) if (ck( 1 &wo
Ak = (20)
A2 = -UD(T) if (ck( > &wo,
with ~~ ~~
~~~~~N ~
2Ek~~ 2kBT
lekl<&Wo (~i)
~~~~ ÎÎÎ~É ~~ ~211T)
From their respective definitions (8) and (14), Ac has a constant value on the entire electronic
bond, but Af~ has a non-vanishing constant value only for (ck( < &wo.
It clearly appears from the previous equations that the repulsive part of the interaction,
as it results from the Hubbard model, extends ta the entire electronic band, whatever the Fermi energy is. This statement is more precise thon the Morel and Anderson prescription ta
introduce a phenomenological large cut off energy for this repulsive part [loi.
3. Exact Solution in a Cu02 Plane at T
= 0 K
In this section, we apply the equations of Section 2 ta a single Cu02 plane, using the same model for the electronic density of states, per copper atom and per spin, as in Dur previous paper iii
~~~~ 112t
~~ ~Î' ~~~~
where E is the bare single partiale energyj as it appears in equation (2), with (E( < 4t. Thus,
at T
= 0 K, equation (21) becomes
~~~~ /Ît/~~~
~~~ ~] ~~ (c Î~
+ /L( ~~~~
~~~~ ~~~~ ~~~~ 4~~~Î
~M
~ ~~
Î~ ~~ ~
Î~Î (~~)
~ÎÎtÎÎ~~ ~ c/)
/~j ~~ je Î~
+ p['
where c
= E + U~
p. Trie chemical potential p is related ta trie doping ratio z = 1 n by
~ 2
equation (9), which, with equation (17), leads ta:
~~~~~ ~~ ~ ~~
Î~ ~~ ~
Î~Î
~~
Î~ jj/jp ~ ~~
Î~
~
~ LÎ ~~ ~~~~
~~~"o
~ ~ ~ ~~
~ ~~ ~
Î~Î
~~
The parameter à = U] p, which appears in trie above equations, is trie opposite of trie Fermi level shift from the logarithmic singularity at trie middle U] of trie band. Dur purpose is to
calculate the variations of Ai and A2 as functions of the doping ratio x
= 1- n. From equation (20) with T
= 0, we have
C(°) =
~~
~
~~,
~
(26) D(°) ~ ~@'
For any given values of à, we first calculate Ai and A2 from the implicit equations (23, 24)
and (26), and then x from equation (25).
Actually, ail the integrals in equations (23, 24) and (25) are easily performed by introducing
new variables ( or i~ such that c = (Ai sh ( or (A2( sh i~, leading ta:
C(0) = $ 2(1
In ) () il + ~ L12
(e~~(~i~~°~) + L12 (e~~(~i+~°)) ), (27)
ir t 1
~~~~ ~~~~ ~ ÎÎt ~~~~
~ ~~ ~~~~ ~~ (ÎÎ( ~~~~~ ~ ~~~ ~~~
ii II J~ _( l_
L12 (e~ ~ ~" + L12 (e~ ~ +" + L12 (e~ ~ " + L12 (e ~ " (28)
+~ ~j~ ~-2(qi-flo) +~i~ e~2(qi+flo)
2~ ~ ,
522 JOURNAL DE PHYSIQUE I N°4
o.ois
- ",
° 0.01 ".
V ".,
~ 0.005 "",
*Ô "..,_
'clà _:"'
£ 0.005 ;"
A
,"
~ 0 01 ;"
1
,"
*Ô
0.015
0.02 0.04 0.06 0.08 0
~ X
Fig. l. Reduced superconducting energy gaps 61/t and 62/t versus doping z, exactly calculated
(fuit fine)j by expanding D(0) -C(0) (dashed fine) and by trie approximate formula (34) (dotted fine),
with U = V
= 2t and &~o = 0.2t.
i~i~~~i~~i~i~~ i~i~~ ~~i~i~~i~~i~~~~~ ~~~
~ ~
~~~~
~ ~
~+ S( "~~~
)° + Args~ su J~'+ Sh
~l0 where trie new parameters are defined by à
= U] p = (Ai sh (o = (A2 sh ~o, 4t à = (A2(
sh $, 4t + à
= (A2( sh $', and &wo = (Ai sh fi ~ (A21 sh ~i, and where we have introduced the dilogarithm function ilIi:
L12(z) = ~~~~ ~~d9
= ~j
Î~ ~~
9 ~i ~
with the properties L12(z)+L12(-z)
= )L12(z~) and L12(1) = Ç.
Of course, the results depend on the choice of the parameters. For instance, the calculated variation of Ai and A2 ~er8us x, which are shown in Figure 1, bave been obtained from
&wo " 0.2t and U
= V
= 2t. We see that Ai and (A21 monotonously and only slowly decrease
as trie doping ratio x increases. Their maximum values are obtained for x
= 0. With trie
chosen values of trie parameters, they are Aim Ù 0, 016t and (A2~n( t 0, 018t. For instance,
with t
= 1 eV, one finds Ai~n/kB Ù 160 K and (A2m(/kB Ù 180 K.
Obviously, these results do net take into account trie fact that trie antiferromagnetic phase is
more stable for small values of x, forbidding the superconducting phase to exist, as we pointed
eut before iii. But it is precisely because the calculated gap decreases so slowly with x that
one can understand, in our model, that a high Tc superconducting phase can exist once the
antiferromagnetisrn has been destabilised by doping.
4. Approximate Version of our Results
The calculated gaps Ai and (A2 are much smaller than 4t and &wo. The shift -à of the Fermi level from the singularity is much smaller than 4t, and aise than &wo as long as the doping
ratio is net toc large. Furthermore, it is clear from equation (23) that C(0) is very sensitive to the value of the parameter à = U] p, because the logarithmic singularity is contained inside the integration range. On the contrary, the two integrals appearing in equation (24)
extend to the entire bond except the narrow energy range from -&wo to &wo which contains the logarithmic singularity. Thus, D(0) C(0) has Orly a small dependence on à, at least as
long as à is reasonably smaller than &wo.
Then expanding D(0) C(0), one gets as a leading term:
/~~ 4t 64t
D(°) C(°) ~ W ~~~ ~~ (30)
By using the approximate expression (30) instead of the exact expression (28), but by keeping
the exact expression (27) for C(o), our numerical results are essentially net modified. Only
a very small shift in the data appears for the largest values of the doping ratio, as shown in Figure 1.
The great advantage of the simplified version (30) is that, when associated with the exact equations (26) and (27), it leads to a very simple relation between Ai and A2, as can been
seen by eliminating C(0) and D(0) between these equations, leading to:
A2 =
~
~~~ Ai, (31)
with
~~
~ U
~4t
64t ~~~~
à ~&wo ~&wo
and where Ai is the solution of the implicit equation
~~~~
~j j~ ~~~
j2 j2 ~ ~~ (~j (~-2(fi-fo)) ~ ~~~ (~-2(fi+io))), (33)
V U* ~ (Ai ~ ° 6 2 ~
Frôm equation (33), the superconducting energy gap Ai is determined by a coupling constant V U*, where V is the attractive part of the interaction, and U* the effective Coulomb
repulsive part. Equation (32) shows how to calculate U* in our model, from the bare Coulomb repulsive parameter U, the bond width 8t and the cut off energy &wo for the attractive part, in a more explicit way than usual previous formulations [la,12]. It leads to large reduction of
the repulsive part, independent of the doping ratio. For instance, with our previous numerical values, one gets U* m U/2.
Another consequence of equation (32) is that, when calculating the isotopic effect, which
in our case appears as the relation between Ai and wo, one must take into account net Orly
the existence of a logarithmic singularity in the electronic density of states [8,13], but aise the influence of wo on U*, which can be large aise.
524 JOURNAL DE PHYSIQUE I N°4
In principle, the exact expression (27) of C(o) can be expanded as we did for D(o) C(o).
But, as we pointed out before, in the case of C(o) this expansion is valid only in the limit of a very small doping ratio. Explicitly, the expansion of C(o) contains a term in (à/Ai)~,
which does not exist in D(o) C(0), and which is small only for very small values of x. As a consequence, the following approximate expression of the superconducting gap
l16t 2 ~~2t ~2 à 2
~~ ~ ~~~ ~~~ ~~&wo ~
V U* 6 ~
Aim
' ~~~~
obtained by expanding C(o), is itself valid only for (ô( much smaller than Aim, and does not
reproduce our exact results for langer à, as shown in Figure 1. Nevertheless, for
x = o and thus
à
= 0, trie analytical expression (34) of Ai is correct.
5. Conclusion
From our results contained in this paper and our previous one iii, we conclude that in our bidimensional itinerant electronic mortel for superconducting copper oxides, trie calculated
superconducting gap decreases with increasing doping much more slowly than trie antiferro- magnetic Slater gap ares. This explains why a high Tc superconducting phase can exist for a
doping large enough to destabilise trie antiferromagnetism.
References
[ii Hocquet T., Jardin J.-P., Germain P. and Labbé J., J. Phys. I France 4 (1994) 423.
[2] Abrikosov A.A., Campuzano J-C- and Gofron K., Physica G 214 (1993) 73.
[3] Gofron K.,.Campuzano J-C-, Ding H., Gu C., Dabrowski B., Veal B-W-, Cramer W. and Jenmngs
G., J. Phys. Ghem. Sohds 54 (1993) l193.
[4] Shen Z.-X., Dessau D.S., Wells B-O- and King D.M., J. Phys. Ghem. Sohds 54 (l193) l169.
[5] Weber W., Phys. Reu. LeU. 58 (1987) 1371.
[6] Freeman A.J. and Yu J., Physica B150 (1988) 50.
[7] Mattheiss L.F., Phys. Reu. LeU. 58 (1987) 1028.
[8] Labbé J. and Bok J., J. Europhys. LeU. 3 (1987) 1225.
[9] Tinkham M., Introduction to superconductivity (Robert E. Krieger Publishing Company, Malabar, Floridaj 1975).
[loi More1P. and Anderson P-W-, Phys. Reu. 125 (1962) 1263.
[Il] Magnus W-j Oberhettinger F. and Sont R-P-j Formulas and Theorems for the Special Functions of Mathematical Physics (Springer Verlag, 1966).
[12] Bok J., Physica G 209 (1993) 107.
[13] Labbé J., Phys. Scr. T29 (1989) 82.
