Superconductivity in an extended Hubbard model with attractive interaction

1.3 MB·English

by  E. J. Calegari

Superconductivity in an extended Hubbard model with attractive
interaction
E. J. Calegari,1 S. G. Magalha˜es,2 C. M. Chaves,3,∗ and A. Troper3
1
Laborat´orio de Teoria da Mat´eria Condensada,
1
Departamento de F´ısica - UFSM, 97105-900
1
0
Santa Maria, RS, Brazil
2

n 2Instituto de F´ısica, Universidade Federal Fluminense
a
J
Av. Litorˆanea s/n, 24210, 346, Niter´oi, Rio de Janeiro, Brazil

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1 3Centro Brasileiro de Pesquisas F´ısicas,

]
Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brazil
l
e
-
r (Dated: January 12, 2011)
t
s
. Abstract
t
a
m
In this work, a two-dimensional one-band Hubbard model is investigated within a two-pole
-
d
approximation. Themodelpresentsanon-localattractive potentialU(U < 0)thatallowsthestudy
n
o
c of d-wave superconductivity and also includes hopping up to second-nearest-neighbors. The two-
[

pole scheme has been proposed to improve the Hubbard-I approximation. The analytical results
1
v
show a more complex form for the gap ∆(T), when compared to the one obtained in the latter
9
2
1 approximation. Indeed,newanomalouscorrelationfunctionsassociatedwiththesuperconductivity
2
. areinvolved inthecalculation of∆(T). Numericalresultsinarangeoftemperaturesarepresented.
1
0
Moreover, thestructureofthequasiparticlebandsandthetopologyoftheFermisurfacearestudied
1
1
: in detail in the normal state. Connections with some experimental results are also included.
v
i
X
PACS numbers: 71.27.+a,71.10.-w,74.25.-q
r
a
∗
Electronic address: [email protected]
1
I. INTRODUCTION
Superconductivity in strongly correlated systems is a field with plenty of challenging
problems. Several non-usual properties of high temperature superconductors (HTSC) [1]
still needed to be properly clarified. In particular, some experimental systems exhibit im-
portant deviations from the standard BCS theory. For instance, the superconductor gap
behaviour as a function of temperature in some borocarbides display a non-monotonic fea-
ture at lower temperatures. In fact, such gap decreasing in RNi B C(R = Dy,Ho,Er,Tm)
2 2
and ErNi B C [2] when the temperature decreases towards to T = 0. This effect ascribed
2 2
due to a competition between the superconductivity and antiferromagnetic correlations,
which are absent in a weak coupling BCS-like superconductivity, appear naturally from the
formulation which is shown in this work. Another quite interesting feature of HTSC sys-
tems is the behaviour found in some cuprates [3] in the low doping region, the so called
pseudo-gap region, which gives rise to an anomalous Fermi surface leading to a pseudo-gap.
In this very complex problem, a number of theories has been proposed in order to explain
the presence of a pseudo-gap region [1]. In the present formulation, we claim that the
appearance of a pseudo-gap can be ascribed to a more detailed many-body treatment in
which superconductor and AF correlations compete.
Although BCS-like approach [4] has been widely used to describe these physical systems,
it is well recognized that superconductivity is a two-dimensional problem in which strong
correlations play a fundamental role [5]. Thereby, we apply a two-pole approximation [6–8]
to deal with the strong interaction coupling. Here we consider a d-wave symmetry gap and
therefore, a non-local attractive interaction is used [9, 10]. The net attractive interaction
(U < 0) may result, for example, from the elimination of the electron-phonon like coupling
through a canonical transformation [11] or, alternatively, from an electronic mechanism
proposed by Hirsch [12] which may produce, for a certain range of parameters, an effective
attractive interaction.
In this work, we focused on the many-body renormalized normal state of these systems.
Our obtained Fermi surface is consistent with recent claims in the literature [3, 13] about the
presence of hole-pockets due to antiferromagnetic correlations. Moreover, we discuss some
thermodynamical properties of the superconducting regime, namely the critical temperature
T , the zero temperature superconducting gap ∆ and temperature dependence of the gap
c 0
2
∆(T) for various dopings δ ≡ 1 − n (with n = n + n ) and interaction U. The n
T T σ −σ σ
represents the average occupation per site of electrons with spin σ =↑,↓.
This paper is organized as follows. In the section II, we present a general formulation
describing the model as well as the ingredients of the normal state, e.g. the quasi-particle
and the special characteristics of the Fermi surface (FS). In section III, we present the
superconducting state which appears from the application of the two-pole approximation.
InsectionIV,weexhibitself-consistentnumericalresultsandconclusionsforboththenormal
and the superconducting states . The Appendix A briefly describes the main points involved
in the two-pole approach whereas in the Appendix B the correlation functions involved in
the Green’s functions governing the superconducting and the normal states are displayed.
II. GENERAL FORMULATION
The Hamiltonian studied here is, in a standard notation
H = t d† d +U nd nd −µ d† d (1)
ij iσ jσ i,σ j,−σ iσ iσ
hhXijiiσ hXijiσ Xiσ
where hh...ii indicates the sum over the first and the second-nearest-neighbors of i and µ is
the chemical potential. The two-dimensional dispersion relation is given by:
ε = 2t(cos(k a)+cos(k a))+4t cos(k a)cos(k a). (2)
~k x y 2 x y
In the present work, we adopted the two-pole approximation [6, 7] which consists in
choosing a set of operators describing the most important excitations of the system. The
details of the method are given in Appendix A. In the present case, the set of operators
considered is d ,nd d ,d† ,nd d† . The first two are associated with the normal
i,σ i,−σ i,σ i,−σ i,σ i−σ
n o
state whereas the last two are associated with the superconductivity [7, 8]. Following the
method exhibited in the Appendix A, the one-particle Green’s function for the normal state
is:
Z Z
Gdd (~k,ω) = 1~kσ + 2~kσ (3)
Nσ ω −ω ω −ω
1~kσ 2~kσ
with
1 U −2U −ε +W
Z = + 1 ~k ~kσ, (4)
1~kσ 2 2X
~kσ
Z = 1−Z (5)
2~kσ 1~kσ
3
and
U +n (U −2U )
2 −σ 1 2
U = . (6)
n (1−n )
−σ −σ
The quasiparticle bands are:
U +ε −2µ+W X
ω = ~k ~kσ − ~kσ, (7)
1~kσ 2 2
ω = ω +X (8)
2~kσ 1~kσ ~kσ
where
X = (U −ε +W )2 +4U (ε −W )+U (9)
~kσ ~k ~kσ 1 ~k ~kσ
q
e
and
4U (U −U )
2 2 1
U = . (10)
n (1−n )
−σ −σ
e
The effective interactions U , U and the band shift W are defined in the Appendix A
1 2 ~kσ
(see Eqs. (A5)-(A7)). Here, as we are assuming a paramagnetic state of a translationally
invariant system, hn i = hn i = hn i. It should be noticed that, due to many body
i,σ i,−σ −σ
effects, in the pole structure of the Green’s functions, in the normal paramagnetic phase,
there is a spin-spin correlation function which exhibit only antiferromagnetic (AF) short
range correlations. That is not in contradiction in our previous paramagnetic assumption.
Moreover, in order to simplify the notation, we write hn i = n .
−σ −σ
From the Green’s function in equation (3), we find the spectral function
1
A (~k,ω) = − Im[Gdd (~k,ω)]. (11)
σ π Nσ
~
The Fermi surface is obtained from A (k,ω = 0).
σ
III. THE SUPERCONDUCTING STATE
In the superconducting state, the Green’s function Gdd is written as:
A′(ω)−(ω +E )(1+n2 U)2∆2
Gdd(~k,ω) = 11 −σ θ ~k (12)
Sσ P(ω)
where
A′(ω) = α +α ω +α ω2+α ω3 (13)
0 1 2 3
4
with
α = (E2 −E E )(E −2n E +n2 E ) (14)
0 12 11 22 22 −σ 12 −σ 11
α = 2n nE E −(n+2n )n E2 −n3 E2
1 −σ 11 12 −σ −σ 12 −σ 11
e−E [E +e2n (n E −2E )] (15)
22 22 −σ −σ 11 12
α = n2 (1−n )2E (16)
2 −σ −σ 11
α = n2 (1−n )2 (17)
3 −σ −σ
and n = (1+n ). The E are elements of the energy matrix (A2) and the quantity P(ω),
−σ nm
is defined as:
e
P(ω) = [(ω −E )(n ω −E )−(n ω −E )2]
11 −σ 22 −σ 12
×[(ω +E )(n ω +E )−(n ω +E )2]
11 −σ 22 −σ 12
+∆2(A −A ω) (18)
~k 1 2
with
2
U U U
A = a +a + a +∆2(1+n2 )2 (19)
1 0 1 θ (cid:20) 2 ~k −σ θ (cid:21)(cid:18)θ (cid:19)
and
2
U U
A = (1+n2 )2 +n2 (1−n )2 . (20)
2 −σ θ −σ −σ (cid:18)θ (cid:19)
The quantities a , a , a and θ, are given by:
0 1 2
a = E2 (21)
0 11
a = 2E (2n E −E ) (22)
1 12 −σ 11 12
a = E2 −4n E E +2n2 (E2 +E E ) (23)
2 22 −σ 12 22 −σ 12 11 22
and
θ = tn −U(D +2hSzSzi) (24)
01σ 01σ 1 0
with the correlation functions n , D and hSzSzi defined in the Appendix B.
01σ 01σ 1 0
The main reason that we are adopting the d-wave symmetry is that we are following
reference [7] where it is claimed that for a large number of HTSC material d-wave gap
symmetry is the most relevant.
Moreover, the d-wave symmetry follows also from the fact that in our Hamiltonian we
consider an attractive delocalized interaction term. Actually the extended s-wave symmetry
is more favoured for an attractive local interaction as discussed in [9].
5
For the particular case of pairing with d-wave symmetry, the gap function is
∆ = 2∆[cos(k )−cos(k )], (25)
~k x y
where ∆ is the gap function amplitude. Following the procedure described in references
[7, 8], the self-consistent gap function has been obtained from the Green’s function:
∆ (β +β ω2)
Gdd†(~k,ω) = − ~k 0 1 (26)
σ P(ω)
in which,
1
∆ = −2θ∆ [cos(q )−cos(q )]2F (27)
x y 1~qσ
L
X
q~
and
1 β +β ω2
0 1
F = f(ω)[ ]dω. (28)
1~qσ
2πi I P(ω)
The β and β are
0 1
U
β = n2 (1−n )2 (29)
0 −σ −σ θ
U U
β = E2 −[E2 −2E E +∆2(1+n2 )2] (30)
1 1 2 1 2 ~q −σ θ θ
with E = E −n E and E = E −n E .
1 12 −σ 11 2 22 −σ 12
IV. SELF-CONSISTENT RESULTS AND CONCLUSIONS
Firstly, we discuss the numerical results for the normal state (T > T ). Figure 1 shows
c
the FS for four different doping values δ. In (a), δ = 0.30, a well defined electron-like FS
is found. However, in (b) when δ is decreased to δ = 0.20, the topology of the FS changes,
with the emergence of a hole-pocket enclosing the nodal point (π, π). As a consequence, due
2 2
to low spectral intensity, a pseudogap appears near the antinodal points (π,0) and (0,π).
Further decrease in δ intensifies the presence of the pseudo-gap as shown in (c)-(d). This
unusual behaviour is due to the strong antiferromagnetic correlations coming from the band
shift Wd [7, 14] defined in equation (A7).
~kσ
The effects described above, are corroborated by the quasiparticle band calculation ex-
hibited in figure 2 which displays the quasiparticle band for distinct doping values δ. In
fact, while in the higher doping regime the quasiparticle bands cross the Fermi level near
6
(a) nT=0.70 (π,π) 40 (b) nT=0.80 (π,π) 35
35 30
30 25
25
20
ky 20 15
15
10 10
5 5
0 0
(0,0) (0,0)
(c) nT=0.85 (π,π) 35 (d) nT=0.90 (π,π) 35
30 30
25 25
20 20
ky 15 15
10 10
5 5
0 0
(0,0) (0,0)
kx kx
FIG. 1: (Color)The spectral function A(~k,ω = 0) representing the Fermi surface for different
dopings δ = 1−n . The model parameters considered here are U = 8t, t = −1.0 eV, t = 0.3|t|
T 2
and k T = 0.1|t| (k is the Boltzmann constant).
B B
1
0
-1
/|t|k -2
ω1 -3
-4 δ=0.30
δ=0.20
-5 δ=0.15
δ=0.10
-6
(0,0) (π,π) (0,π) (0,0)
k
FIG. 2: (Color) The quasiparticle bands intercepted by the chemical potential (µ = 0). The model
parameters and the temperature are the same as in figure 1.
(π, π) and near the antinodal point (0,π), in the lower doping regime the quasiparticle
2 2
band crosses the Fermi level only near the nodal point (π, π). Such a behavior gives rise
2 2
to a pocket around (π, π). On the other hand, as the quasiparticle band does not touch
2 2
the Fermi level near (0,π), a pseudogap emerges in that region. As far as we know, the
emergence of the pseudogap due to an attractive U in a strong correlation regime, is for the
first time presented here. The kink observed near the (π,π) point of the quasiparticle band
is caused by the strong antiferromagnetic correlations associated with hS~ · S~ i, which are
j i
maximum in Q= (π,π). The Q is the antiferromagnetic wave-vector.
Now we discuss some thermodynamical properties associated to the superconducting
7
U=8t t=-1.0 eV t2=0.3|t|
0.9
0.669
0.8 nnTT==00..9800 nT=0.90
0.7 ∆/|t| 0.666
0.6
0.663
0.5
∆/|t| 0.4 0.528 nT=0.80
0.3
0.2 ∆/|t| 0.525
0.1
0 0.522
0 1 2 3 4 5 0 0.2 0.4 0.6 0.8
kBT/|t| kBT/|t|
FIG. 3: (Color)The main figure shows the gap function amplitude versus the temperature for
U = 8t, t = −1.0 eV, t = 0.3|t| and two different occupations n . The small figures show the
2 T
regions of low temperatures where the gap presents an unusual behavior.
state. In figure 3 we describe the gap function amplitude ∆(T) for U = 8t and two different
occupations n , in the lower doping regime. One sees that for a given U (in a character-
T
istic strong coupling regime |U| >> 1), the zero temperature gap for n = 0.90 is higher
t T
than the corresponding one for n = 0.80. Furthermore, the temperature where a non-
T
superconducting phase arises is higher for n = 0.90, i.e., T (n = 0.90) > T (n = 0.80).
T c T c T
It should be noted that in both cases for n , in the region of low T, there is a increase in
T
the value of the gap amplitude as compared to the zero gap amplitude value. This unusual
behavior is due to the effect of the strong correlations, since that in the BCS weak correlated
regime, ∆(0) is always greater than ∆(T).
We stress that in our case this non-monotonic behavior at low temperatures is mainly
due to correlation functions in the pole structure of the superconduction Green’s function
(see equations (12) and (26)). To be more precise, we have found in our self-consistent
calculation a complex interplay between the SC gap behaviour and the AF type short range
correlations.
Figure 4 displays the value of the gap amplitude ∆(T) as a function of temperature for
several values of U, in the strong correlation regime (|U| >> 1). In all cases, for n = 0.90,
t T
we note that when U increases, T increases also, and the same unusual behavior for ∆(T)
c
appears for very low T. We have calculated the gap function amplitude for several values of
U, for different n (n = 0.80 and 0.70), and the same behavior is observed.
T T
A very interesting result is shown in figure 5. Here, we plot ∆ - ∆ for several values
max 0
of |U|. One noted that when U increases, ∆ - ∆ increases also, until a special value (in
t max 0
8
nT=0.90 t=-1.0 eV t2=0.3|t|
0.9
0.669
0.8 UU==87tt U=8t
U=6t
0.7 U=5t ∆/|t| 0.666
0.6
0.663
0.5
0.513
∆/|t| 0.4
0.3 U=6t
0.2 ∆/|t| 0.51
0.1
0 0.507
0 1 2 3 4 5 0 0.2 0.4 0.6 0.8
kBT/|t| kBT/|t|
FIG. 4: (Color)The main figure shows the gap function amplitude versus the temperature for
n = 0.90 and different values of U. The remaining parameters are identical to figure 3. The small
T
figures show the regions of low temperatures for two values of U. For low temperatures, the gap
presents an unusual behavior.
0.0025 nT=0.90 t=-1.0 eV t∆2=0.3|t|
max
0.002 ∆0=∆(T=0.0)/|t| ∆/|t|
0.0015
∆0
-max kBT/|t|
∆
0.001
0.0005
0
4 6 8 10 12 14
|U|/|t|
FIG. 5: (Color)The differencebetween ∆ and∆ as a function of |U|. Theinset shows as ∆
max 0 t max
and ∆ have been obtained.
0
our calculation |U| ≃ 8) where ∆ - ∆ is a maximum and then, ∆ - ∆ tends toward
t max 0 max 0
zero in a very high value of U. When such behavior appears, one has attained the |U| → ∞
limit. Moreover, it should be noted that this unusual behavior occurs for a critical low value
of |U|, (in our calculation |U| ≃ 4), which is a signature of the onset of a characteristic
t
|U| >> 1, signaling the appearance of a strong correlation regime.
t
Our results are in qualitatively agreement concerning the Fermi surface in underdoping
region with other approaches using the t-J model [17]. The reason of such agreement is that
in both approaches the spin spin correlation functions renormalize the band structure giving
9
rise to the appearance of hole-pockets.
Finally, in order to complement our present calculations, we need to discuss the higher
doping regime n . 0.80. Moreover, a detailed discussion of the effect of external pressure,
T
which affects mainly the ratio t2 [15], is needed. These further calculations, are now in
t
progress.
Appendix A: Two-pole approximation
In the present two-pole approximation [6, 7], the Green’s functions are defined as:
G(ω) = N(ωN−E)−1N (A1)
where, E and N are the energy and the normalization matrices given by
E = [A ,H] ,A† (A2)
nm n − m (+)
D E
(cid:2) (cid:3)
and
N = h[A ,A† ] i. (A3)
nm n m (+)
In equations (A2)-(A3), [...,...] denote the (anti)commutator, and h...i, the thermal
(+)−
average. The set of operators {A } must satisfy, within some approximation, the relation
n
[A ,H] = K A .
n − m nm m
P
For the normal state of the model (1), the energy matrix is given by,
ε −µ+U ε −µ+U
~k 1 ~k 2
 
E = (A4)
 
ε −µ+U ε n2 −µ+U +nW 
 ~k 2 ~k σ 2 ~kσ
 
with
U = 2U hn i, (A5)
1 l,−σ
Xl
U = 2U hn n i (A6)
2 l,−σ i,−σ
Xl
and n = n (1−n ). The correlation function D = hn n i is defined in equation
−σ −σ il−σ i−σ l−σ
(B3). The band shift W , can be written as:
~kσ
nWd = − t (n −2m )+ t ei~k·R~jh (A7)
~kσ 0j 0jσ 0jσ 0j jσ
X X
hhj6=0ii hhj6=0ii
with n , m and h defined below.
ijσ ijσ jσ
10