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Exchange-mediated pairing: gap anisotropy and a narrow-band limit for hybridized electrons
J. Spalek, P. Gopalan
To cite this version:
J. Spalek, P. Gopalan. Exchange-mediated pairing: gap anisotropy and a narrow- band limit for hybridized electrons. Journal de Physique, 1989, 50 (18), pp.2869-2893.
�10.1051/jphys:0198900500180286900�. �jpa-00211109�
Exchange-mediated pairing: gap anisotropy and a narrow-
band limit for hybridized electrons
J. Spa~ek (*) and P. Gopalan (**)
Department of Physics, Purdue University, West Lafayette, Indiana 47907, U.S.A.
(Reçu le 22 mai 1989, révisé le 19 juin 1989, accepté le 20 juin 1989)
Résumé. 2014 Nous dérivons analytiquement la forme de la bande interdite supraconductrice 0394k dans l’approximation de couplage faible (BCS) pour un mécanisme d’appariement par
échange. Pour des appariements de type d-d ou p-p, la bande interdite a la même forme que pour
une onde s étendue. Dans le cas d’appariements hybrides : 3d-2p (systèmes à haut Tc) ou 4f-5d (systèmes de fermions lourds), l’anisotropie de 0394k reflète la symétrie de l’élément de matrice
d’hybridation Vk. La valeur de la température de transition supraconductrice est estimée dans la limite de bande étroite pour des électrons hybridés. La forme générale de l’hamiltonien
d’appariement pour des électrons corrélés et hybridés est aussi discutée.
Abstract. 2014 We derive analytically the form of the superconducting gap 0394k within the exchange-
mediated pairing mechanism in the weak-coupling (BCS) approximation. For the d-d or p-p types of pairing, the gap is in the form of extended s-wave. In the case of hybrid pairings : 3d-2p (for high- Tc systems) or 4f(5d (for heavy-fermion systems), the anisotropy of 0394k reflects the symmetry of the hybridization matrix element Vk. The value of the superconducting transition temperature is estimated in the narrow-band limit for the hybridized electrons. A general form of the pairing
Hamiltonian for correlated and hybridized electrons is also discussed.
Classification
Physics Abstracts
75.10L
1. Introduction.
Soon after the discovery [1] of superconductivity in the La2-,,Ba,,CU04 system Anderson [2]
proposed a real-space spin-singlet type of pairing among electrons in a narrow band induced
by an antiferromagnetic kinetic exchange (superexchange) interaction [3]. The principal point
of this mechanism of pairing is the treatment on the same footing and within a single
theoretical framework of antiferromagnetism (AF), metal-insulator transition of the Mott type and superconductivity (SC). In this approach the presence of antiferromagnetic ordering
in a parent compound (e.g., in La2Cu04) is as crucial as is the isotope effect in classic
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180286900
superconductors. In the actual situation AF ordering may not be completely compatible whith
SC. This is because the former state requires a formation of stable spin sublattices in real space and hence is possible only when electrons constitute well-defined stationary magnetic
moments, whereas the latter state requires metallicity of the electron system. Thus, the relatively small number 6 - 0.05 of holes in the Mott insulator destroys long-range AF order [4]. In effect, AF and SC may coexist only in a rather narrow range of hole concentration [5] ;
hence, the question concerning the stability of exchange-mediated SC state in the absence of AF ordering has a well defined meaning.
In this paper we consider the exchange mediated pairing taking into account the full form of the kinetic exchange interaction, i.e., with the three-site processes included [6, 7]. This will provide us with an explicit expression for the superconducting gap anisotropy (i.e., its wave
vector k dependence). Furthermore, we also address the problem of deriving the effective d-d (in high-Tc oxides) and f-f (in heavy-fermion compounds) types of pairing from the Anderson lattice model in the mixed-valence or heavy-fermion limits. We believe that our approach supplies a proper scheme of discussing the spin-fluctuation mediated pairing in the limit of
strongly correlated and hybridized electrons.
The structure of this paper is as follows. In section 2 we review briefly the narrow-band d-d
pairing theory [2, 6]. The results obtained there will be used for a later comparison with those
for interorbital (hybrid) pairing in the narrow-band limit ; the latter pairing is discussed in sections 3 and 4. Finally, in section 5 we provide a critical overview of various types of exchange-mediated pairings (d-d, d-p, and p-p) and relate the microscopic formulation with the Ginzburg-Landau theory for the case of space homogeneous superconducting gap. Our paper can be regarded as providing a mean-field discussion of the exchange mediated hybrid pairing, and of its relation to d-d or f-f pairings.
2. Anisotropic gap for the d-d pairing : a critical assessment.
In this section we discuss the role of the 3-site term in the kinetic-exchange interactions among correlated itinerant electrons [6]. We start from the effective Hubbard Hamiltonian projected
onto the subspace containing only singly occupied site configurations. It has the following
form [3, 6]
In this expression tij is the hopping (Bloch) integral, ai and ai, are creation and annihilation operators of electrons in the Wannier state 1 i > , Ni u == ai£ ai,, and Si the spin operator in the second-quantization representation. The first term describes the motion of electrons as a
series of electron hops between the sites j and i, the second expresses exchange interaction between the electrons located on those sites, and the last represents a hopping from site i to
site j via doubly occupied state at site k. Those three terms are schematically shown in figure 1, where the corresponding real hopping processes (corresponding to the first and the third terms) are depicted by the diagrams b) and c), respectively, while the virtual hoppings depicted by the diagram a) lead to the so-called kinetic exchange interaction represented by
Fig. 1. - Hopping processes in a narrow-band system in partial band filling case. Processes (a) represent virtual hopping between the occupied sites, whereas (b) and (c) illustrate real hoppings between occupied and empty sites.
the second term in (1). This model contains a single dimensionless parameter U/W, where U
is the magnitude of the intraatomic part of the Coulomb repulsion between electrons, and W= 2 tij is the band-width of bare (uncorrelated) electrons in the tight-binding scheme.
7(0
The Hamiltonian is valid only for (WIU) « 1 ; this condition defines the regime of strong corrélations.
By introducing projected one- and two-particle operators
and
one can recast (1) into a more compact and equivalent form
One sees that in (4) the dynamics of the correlated electron system is decomposed into that of
single itinerant holes and that of pairs. However, this decomposition is not complete since the
two parts do not commute. Also, the operators {biu} and {b/.} do not anticommute to a number.
To demonstrate the importance of the three-site contribution to (4) we use a simple scheme
based on the Gutzwiller approximation [8]. This amounts to replacing the operators (2) by the
fermion operators a:’ and ai,, respectively with simultaneous renormalization of tij in the first term of (4) in such a way as to reflect the restriction on the hopping processes imposed by the
removal of the doubly occupied site configurations from the Fock space. This is achieved by introducing [9] fermion quasiparticles with energies Ek = CPEk, where
4S = (1 - n)l (1 - n/2), n -= (Ni T + Ni j ) is the average number of particles per atomic
site, and E is the band energy for bare (noninteracting, U = 0) particles. In effect, the
Hamiltonian (4) is approximated by
where now
Taking the space Fourier transform to reexpress (4’) in terms of reciprocal (k) space variables we obtain
with
We now employ the commonly used approximation of neglecting all terms with q :A 0 (they
do not change the results appreciably since the density of pair states with k + k’ = 0 is sharply peaked). In effect, we then obtain the Hamiltonian (5) which now contains a separable pairing potential Vkk, of the type
The corresponding self-consistent equation for the superconducting gap àk in the mean-field
(BCS) approximation has the form
where 8 = (kB T )-1 is the inverse temperature (in energy units), and Ék is the quasiparticle
energy in the superconducting phase. From (8) it follows that the gap dk is of the form
..:1k (T) = à (T) E k, i.e., the gap has the same k-dependence as the band energy Ek. In the case
of d-d pairing in high-Tc superconductors modeled by a single Cu02 plane, Ek =
zt (cos (kx a ) + cos (ky a ) ) ; thus, the gap is of extended s-form [10]. In the present approach
this is the only admissible solution. Also, the equation for the gap amplitude 2l has the form
where
and 1£ is the chemical potential. This equation must be supplemented with equation determining li in the superconducting phase, i.e.,
The mean-field solution for à(T) obtained from equations (9)-(11) has one unphysical feature, namely a finite gap in the limit of the Mott insulator, i.e., for n --* 1 [11]. This point is
demonstrated in figure 2, where .d (0) is plotted as a function of n for flat density of states in
the bare band. One can also, estimate the value of the transition temperature Tc by solving equations (9) and (11) for L1 (Tc) = 0. For n - 1 the solution for T, approaches the value
Fig. 2. - The superconducting gap ,¿1 ( T = 0) as a function of the number of holes in a narrow band.
The bare width W = 1, and values of U are specified in eV.
This value is comparable to the mean-field value of the Néel temperature TN for the Heisenberg antiferromagnet corresponding to n = 1 limit which is kB TN = W2/ (4 zU), where
z is the number of nearest neighbors. From the fact that both â (0) and T, are nonzero in the limit n - 1 one draws the conclusion that the mean-field (BCS) approximation does not lead
to correct results when combined with the Gutzwiller ansatz [9]. This is one of the reasons why the holon-spinon decomposition [12] of the projected operators (2) has been introduced and studied in detail [13].
The approximation introduced on replacing the effective Hamiltonian (4) by (4’) has one
additional defect. Namely, we have renormalized the first term in (4) but the part representing the hopping of electron pairs contained in the second term and with i =,A k has been left unchanged. The importance of the latter renormalization may be seen from the following qualitative argument. The anticommutation relations for bi, and b’, operators are
where sr == ai£ ai _ == bi bi - (T. Taking the expectation value of the projected operators on both sides of (13), we obtain
By defining the renormalized fermion operators a ’ == (1 - n)- 1/2 b:’ , and aiQ = (1 - n)- 112 biu we obtain from (14) fermion anticommutation rules as well as a renormali- zation factor 4S ’ = 1 - n in equation (4’). However, in the present scheme also the hopping
term contained in the pairing part is renormalized by the factor (1 - n ) [6].
The ambiguities present in the renormalization of the model parameters [14], together with
the possibility of introducing holon-spinon formalism in different ways [15] forced us to
reconsider the narrow-band limit starting from a more general Anderson lattice Hamiltonian.
This consideration discussed below removes not only the above mentioned ambiguities but
also provides a proper superconducting solution already in the mean-field approximation.
3. Anisotropic hybrid (interorbital) pairing.
We now consider a model involving the hybridized 2p-3d appropriate to the high- Tc systems and of 5d-4f states for heavy-fermion systems. For this purpose we discuss first the effective Hamiltonian derived earlier [7] from the Anderson lattice Hamiltonian which in the
tight-binding approximation has the form
where the (i, j) label atomic a - 3d or 4f states, the (m, n ) label delocalized c - 2p or 5d
states, and cj and Ciu are the creation and annihilation operators for the itinerant states. The first term describes the band energy of the delocalized c states, the second and third the
single-particle and Coulomb (atomic) energies for a states, whereas the last represents the hybridization energy involving both intra-atomic (i = m ) and inter-atomic (i :o m) parts.
The matrix elements tmn and Vim represent hopping and hybridization integrals, respectively.
In deriving the effective Hamiltonian with real-space pairing we assume that the
intraatomic Coulomb interaction U is by far the largest parameter in the system. However, unlike some other authors [16] we assume that in general, Et and V may become comparable.
In those circumstances the interorbital charge transferts (1 -+ c and f -+ (1 can be divided into low- and high-energy processes, as illustrated in figure 3. Explicitly, we introduce a decomposition
Fig. 3. - Mixing (hybridization) induced processes in a hybridized system composed of an atomic (a ) level placed at E f and a conduction (c ) band of width W.
into both parts in the last term of (15). The first term of (16) represents the mixing of the
a and c states via processes which do not involve the energy U in any order, whereas the second represent a part of the a - c mixing involving U. Explicitly, treating the whole hybridization term in (15) as a perturbation it is easy to see that the first part of (16) leads to higher-order terms -- V k/ (Ef - Ek) while the second part of (16) leads to terms -- Vk/(Ef + U - Ek)’ as was shown before [18]. Note that Vk and Ek are space Fourier transforms of Vim and tmn, respectively. Now, if the bare level is placed in the band of
c states, then the denominator (,Ef - El,) diverges which means that the corresponding low-
energy cannot processes be treated as a perturbation. Rather, those low-energy mixing
processes lead to a new type of ground state called the fluctuating-valence (or heavy-fermion)
state. This is characterized by a nonzero expectation value of the charge-transfer amplitude
(ai£ cmu>, thus eliminating the distinction between the bare a and c states.
Our main task is to describe properly the fluctuating-valence situation by including the dynamic processes associated with the large but finite U to first nontrivial orders. For that
reason we need a perturbation method which allows us to differentiate between the two parts in (16) containing the same coupling constant. Such a method called the canonical
perturbation expansion has been proposed by Spalek et al. for both the Hubbard [3, 6] and
Anderson [7] models ; and for present purposes, the procedure is summarized in appendix A.
It amounts to transforming out canonically only the second term in (16) and to replacing it by
an effective interaction incorporating higher-order virtual processes. In this manner we avoid
singularities that are present in the original Schrieffer-Wolff transformation [18] in the
situation when the bare a level is placed within the band limits of c states. Applying this procedure one obtains the following effective Hamiltonian in the second order
where
The spin operators Si and sm are
defined
in the second quantization scheme, e.g.,Si = atr T ai Sî i i = at! ait T and Si
= -
(Ni - Ni The effective Hamiltonian contains a2 T
renormalization of the band part (first term), of the intraatomic Coulomb interaction (third term), and of the hybridization part (fourth term). More important are the two last terms representing the antiferromagnetic a - c exchange coupling (of the Kondo type) and the spin- flip accompanied hopping process in the conduction band. The antiferromagnetic exchange
interaction originates from virtual hopping a i--± c processes while the spin-flip assisted hopping involves three-site processes. To demonstrate the rotational invariance in the spin
space of the second-order contribution we introduce the following singlet pairing operators [7]
Then, the effective Hamiltonian has a closed form
where the irrelevant renormalization of U has been dropped. One sees that the first four terms represent the Anderson lattice Hamiltonian in the limit U - 00 (i.e., with the projected hybridization onto the subspace of singly occupied localized states) while the last term contains both the exchange energy of binding into a - c singlets (for m = n) and the pair hopping (for m :0 n). The present form of pairing is similar to that in the case of narrow band
[cf. Eq. (4)] ; the pairing takes place between the electrons from different orbitals which
hybridize ; hence, we term this type of pairing a hybrid (interorbital) pairing. The hybrid pairing represents a natural generalization of the Anderson real-space d-d pairing to the
mixed-valent or heavy-fermion situations. In the next section we show that in the heavy-
fermion limit the hybrid pairing reduces to the same type (a - !g) pairing.
The derived effective Hamiltonian (19) must now be diagonalized. The principal difficulty
encountered at this point is a proper treatment of the many-body nature of the residual
(projected) hybridization term. Such treatment should lead to quasiparticle states describing
the normal state properties. Only then can one treat the pairing part in a mean-field (BCS)
type of approximation. In other words, we first introduce quasiparticle hybridized states ;
these are subsequently paired into spin singlets induced by antiferromagnetic interaction of the Kondo type.
To introduce the hybridized quasiparticle states we utilize the Gutzwiller-type ansatz in
U ---> oo limit introduced by Rice and Ueda and others [19], i.e., we renormalize the residual
hybridization term in (19) according to V k --+ Vk in such a manner that the parameter Vk expresses the restriction that only the singly-occupied a states are mixed with the bare band states. Taking additionally the space Fourier transform of (19), one obtains
A brief justification of the effective Hamiltonian with renormalized hybridization
Vk is in order. The factor (1 - nf) in q represents the restriction on single-particle hybridization (a - c mixing) processes : the mixing takes place only if the a state is not occupied. Since the two representations (19) and (20a) of the residual hybridization must yield the same contribution to (jé) we find that
The expectation values can be expressed in the single-site approximation as the following
conditional probabilities that the corresponding final state is empty and that the initial state is
occupied, i.e.,
and
Hence q = (1 - nf)/(1 - n f/2 ). Obviously, this reasoning assumes that (nm) + n f = 1 ; this
condition will be explicitly imposed when adjusting the position of the Fermi level for the
fluctuating-valence configuration.
The question now arises whether one can use the present expression for the renormalization
factor q derived in U - ao limit when the pairing part is included ; this part is nonvanishing only for the finite U. One should realize that the only physical condition we impose is the
exclusion of doubly occupied a state configurations. This means that the physically accessible many-particle configurations are those for which (Ni 1 7V ) = 0. It is generally believed that this condition is fulfilled if only U + E f is much larger than either the width W of the bare band states or the hybridization amplitude 1 Vi,, 1 In other words, if U + E f > W and 1 Vin 1 then
the screening of U in the mixed-valence state is regarded as weak. This is certainly true in the
limit nf --+ 1, i.e., for the heavy-fermion state considered in detail later.
For the purpose of subsequent analysis we discuss first the electron states near localization threshold (i.e., for nf - 1), neglecting the last term in (20a). This analysis provides us with
proper quasiparticle states which will form a paired state.
3.1 NORMAL PHASE: QUASIPARTICLE STATES. - In what follows we assume that
Vk is real [20]. Then, the part Ho of the effective Hamiltonian containing first three terms can
be diagonalized with the help of the transformation
with
The Ho part is then of the form
with
One should notice that the quasiparticle energy Ek - depends on the occupancy nf of the a level. Thus, one must calculate self-consistently the occupancy n f = (liN) L (a:q aku) of
ku
the level under the condition that the total number of particles Ne = nN is conserved. In
figures 4 and 5 we have plotted n f as a function of 1 V ] and 1 Efl, respectively (the hybridization was assumed to be k independent). Also, the number n was set as equal to unity
and the density of states in the bare band was taken as uniform, p (E ) = 1/W. Two common
features of those curves should be mentioned. First, for Ef - W/2 and for a small hybridization the atomic state with n f = 1 is stable, i.e., the system is a Mott insulator of the
charge-transfer type [21]. Second, in the limit nf = 1 - 5, with 6 « 1 the density of hybridized
states at E = p, becomes very large, as shown in figure 6. This is the limit of almost-localized
or heavy electrons which will be considered in detail in the next section.
Fig. 4. Fig. 5.
Fig. 4. - The level occupancy nf as a function of the hybridization V ; the number of electrons is
n = 1 per site. Note that nf = 1 insulating configuration is achieved for E 2013 W/2 and small V. All
quantities are in units of W.
Fig. 5. - Same as in figure 4 as a function of the atomic level position ; the values of V are specified.
3.2 MEAN-FIELD SOLUTION: SUPERCONDUCTING GAP ANISOTROPY. - Here, we transform
to the hybridized basis (21a) the pairing part. Taking into account only the (k T , - k 1 ) pairs
we obtain the total Hamiltonian for the electrons in the lower hybridization band
This is a BCS-type Hamiltonian with a pairing potential
where
Fig. 6. - Density of hybridized states (in logarithmic scale) at the Fermi level IL and in the lower band,
as a function of E f. Note a strong enhancement of the density of states when nf --> 1.
The pairing potential vanishes in the limit of the Mott-Hubbard (or charge-transfer) insulator, i.e., when nf --> 1. Otherwise, the Hamiltonian (22) has the same formal structure as its
narrow-band counterpart (7). However, in the present case the pairing takes place between hybridized quasiparticles. Additionally, the pairing also vanishes in the insulating limit, as it
should. The pairing potential Vkk, separates into k and k’ dependent parts ; this property is satisfied only if we take into account three-site interactions, as in the last term of (19).
The Hamiltonian (22) can be solved in the mean-field approximation which yields a self-
consistent equation for the superconducting gap of the form
where Ek is the quasiparticle energy in the superconducting phase, i.e.,
Equation (24) has a solution
where à =A 0 is a solution of the equation
From equation (26) one draws an important conclusion : the gap anisotropy is the same as that
of gk. In particular, the gap vanishes at points for which the hybridization Vk vanishes. Also, if the hybridization is almost k independent (as in the case of dominant intraatomic
