Extending the theory of phonon-mediated superconductivity in quasi-2D
arXiv:0704.0365v1  [cond-mat.supr-con]  3 Apr 2007
7 Extending the theory of phonon-mediated
superconductivity in quasi-2D
J.P.Hague
Department of Physics, Loughborough University, Loughborough, LE11 3TU
Abstract. I present results from an extended Migdal–Eliashberg theory of electron-phonon inter-
actions and superconductivity. The history of the electron-phonon problem is introduced, and then
study of the intermediate parameter regime is justified from the energy scales in the cuprate su-
perconductors. The Holstein model is detailed, and limiting cases are examined to demonstrate the
need for an extended theory of superconductivity. Results of the extended approximation are shown,
including spectral functions and phase diagrams. These are discussed with reference to Hohenberg’s
theorem, the Bardeen–Cooper–Schrieffer theory and Coulomb repulsion. [Published in: Lectures
on the physics of highly correlated electron systems X, p255-264, AIP Conference Proceedings
vol. 846 (2006)]
INTRODUCTION
Over the past half-century, the study of the role of electron-phonon interactions in
condensed matter physics has been an active and controversial field. Initially of interest
from the point of view of thermal properties, early models of the interactions between
lattice vibrations and electrons included the continuum Fröhlich model [1]. Interest in
electron-phonon interactions increased dramatically when in 1957, Bardeen, Cooper and
Schrieffer (BCS) published their famous theory of superconductivity [2], which directly
implicated phonons as the microscopic mechanism for the low temperature absence of
resistivity in a variety of metals. Until the discovery of the cuprate superconductors by
Bednorz and Müller in 1986 [3], the BCS picture was found to account well for all
superconducting materials - a remarkable success for a simple mean-field theory which
is only applicable at weak coupling!
Soon after the realisation that phonons were responsible for superconductivity, Eliash-
berg extended the theoretical description beyond the absolute weak coupling theory with
the famous Eliashberg equations [4]. In doing this, he built on the earlier work of Migdal,
who argued that a simple resummation of a certain class of Feynmann diagrams should
be sufficient to describe the limit of low phonon frequency [5]. Eliashberg’s theory can
be argued to be one of the first applications of the dynamical mean-field theory (DMFT)
[6], since (in its original sense) it ignores spatial fluctuations (momentum dependence)
in the self-energy, while keeping frequency dependent (dynamical) effects.
The purpose of this paper is to describe an extension to the theory of superconductivity
from electron-phonon interactions. The approach goes beyond the Eliashberg theory by
introducing the effects of spatial fluctuations and higher order terms in the perturbation
theory. The aim is to develop a theory which can be used for systems with stronger
coupling, larger phonon frequencies and reduced dimensionality. I begin by motivating
http://arxiv.org/abs/0704.0365v1
the need for a more sophisticated theory from the experimental viewpoint. I also discuss
limiting cases of the Holstein model, and how the large phonon frequency limit of that
model implies that the conventional theories of superconductivity are incomplete. I then
introduce the approximations needed to develop a more sophisticated theory. Finally
I present some results from the new approximation, and discuss them in relation to
Cuprate superconductors, and also with regard to conventional theories and the exact
Hohenberg theorem [7].
MOTIVATION
When the high-temperature cuprate superconductors were discovered in 1986 [3], the
possibility that phonons could be attributed to the microscopic mechanism was quickly
discounted by many people. In part, this was due to the absence of an isotope effect at
optimal doping, and also an assumption that phonon-mediated superconductivity could
not occur above 30K. The mechanism for high-TC superconductors remains highly
controversial, and many different hypotheses are suggested (some examples are spin
fluctuations [8] and exotic phonon mechanisms such as bipolarons [9]). An increasing
body of evidence shows that phonons as well as Coulomb repulsion have an effect on the
physics of the cuprate materials. I shall give a brief review of the current experimental
situation in this section, and argue that (1) Electron-phonon interactions need to be
treated on an equal footing to Coulomb repulsion if the Cuprates are to be understood,
and (2) In order to treat the phonons in the Cuprates, extensions to the current theories
of electron-phonon interactions and phonon-mediated superconductivity are required.
There are several experiments demonstrating strong electron-phonon coupling in the
cuprates. The most compelling is the existence of a strong isotope effect on exchanging
O16 for O18 [10]. There are also some more recent experiments which demonstrate
the effects of electron-phonon interactions in a transparent manner. Figure 1 shows
schematic representations of electron and phonon dispersions in the cuprates. Panel
(a) details the main features of the electronic dispersion measured by Angle-Resolved
Photo-Emmission Spectroscopy (ARPES) in the [11] direction [11]. At energies close to
the Fermi-surface, there are coherent excitations with a long lifetime. As εk = |ω0 − εF |
is approached, the gradient of the dispersion changes at a sharp kink. The phonon is
of the transverse optic variety, and its frequency (ω0) is of the order of 100meV. It
suffices here to mention that this is very large. The ratio of the gradients above and
below the kink is related to the dimensionless coupling constant (λ = g2/tω0), and it is
found that λ can take values of up to 2 [11]. Panel (b) shows a schematic representation
of some neutron scattering results measuring the phonon dispersion [12, 13]. Above
the transition temperature, this looks like the solid line, but as the system moves from
normal to superconducting state, the spectral weight in the circled area vanishes. This
indicates that the superconductivity (bound pairs of electrons) affects the phonons, and
is additional evidence for a strong electron-phonon coupling.
A frequent misconception about the cuprates is that electron-phonon terms in the
Hamiltonian can be neglected on the basis that they are small. To demonstrate that this
is not the case, figure 2 shows approximate energy scales in the cuprates. The largest
energy by far is the Coulomb repulsion (or Hubbard U ) which weighs in at some 10eV.
(a) (b)
INCOHERENT
COHERENT
WEIGHT LOST ON
TRANSITION
kk F 0
PHONON DISPERSIONELECTRON DISPERSION
FIGURE 1. Schematics showing the effect of electron-phonon interactions on the electron and phonon
dispersions in the cuprates. Both panels describe measurements along the [11] direction. Panel (a) shows
a schematic representation of the electronic dispersion measured by Angle-Resolved Photo-Emmission
Spectroscopy (ARPES) [11]. At energies close to the Fermi-surface, there are coherent excitations with
a long lifetime. As εk = |ω0 − εF | is approached, the gradient of the dispersion changes and a kink is
introduced. The phonon is of the transverse optic variety, and its frequency (ω0) is ∼ 75meV. The ratio
of the gradients above and below the kink is related to the coupling constant [11]. Panel (b) shows a
schematic representation of some neutron scattering results measuring the phonon dispersion [12, 13].
Above the transition temperature, this looks like the solid line, but as the system moves from the normal
to the superconducting state, the spectral weight in the shaded area vanishes. This indicates that the
superconducting state affects the phonons, and is further evidence for strong electron-phonon coupling.
Next is the intersite hopping integral t, which is of the order of 1eV. Using a simple
2nd order perturbation theory at strong coupling, an effective exchange interaction is
generated [14], with J = t2/U of the order of 100meV. This J often used to argue for a
spin-fluctuation theory of high-TC superconductivity that neglects phonons. The problem
with this viewpoint is immediately clear if one reviews the experimental data. First, the
energies of the phonons are also approximately 100meV, so they cannot be treated as
a small energy scale. Second, a dimensionless coupling constant of order unity implies
dimensionfull coupling g with similar magnitude. Thus with three very close energy
scales, it is important that the contributions from both phonon and Coulomb mechanisms
are treated on equal footing in a theory for the cuprates. Unfortunately, as I discuss in the
next section, current theories of electron-phonon interactions are not capable of handling
the large phonon energies and coupling constants in the cuprates. The remainder of this
paper focuses on how the theory can be extended to describe this regime.
MODEL AND LIMITS
A generic model of electron-phonon interactions includes the motion of the electrons
Hel, the motion of the ions (or phonons) Hph and the interaction between the electrons
and the phonons (which may be absorbed or emitted) which is denoted Hel−ph. In this
Energy: 10meV 100meV 1eV 10eV
T J t U
FIGURE 2. Schematic showing the energy scales in the cuprates. The largest energy by far is the
Coulomb repulsion (or Hubbard U) of order 10eV. The intersite hopping integral t, is ∼1eV. Using a
simple 2nd order perturbation theory, an effective exchange interaction is generated, with J = t2/U of
the order of 100meV. This J is then used to argue for the spin-fluctuation theory of high TC. However,
the energies of the phonons are also approximately 100meV and the dimensionful coupling g has around
the same value. Thus with 3 similar energy scales, it is important that the contributions from both spin-
fluctuations and phonon mechanisms are treated on equal footing.
way, H = Hel +Hel−ph +Hph is the total Hamiltonian.
Hel = ∑
kck ≈− ∑
<i j>σ
tc†iσ c jσ (1)
Hel−ph =−∑
k−qck(b
q +b−q)≈−∑
niσ gri (2)
Hph = ∑
b†kbk +
Mω20 r
The first term in the Hamiltonian is the general form for free electrons, i.e. the total
energy is the sum of the kinetic energies of all occupied states. In a special case,
which is known as the Holstein Hamiltonian, the electrons in a tight binding model
may hop between nearest-neighbour sites only, and εk = −2t ∑Di=1 cos(ki), where t is
the overlap integral. In the generic form of the electron-phonon interaction, an electron
may be scattered by absorbing a phonon with momentum −q or emitting a phonon with
momentum q. An additional approximation uses a momentum independent electron-
phonon coupling, g, and in that case the Fourier transform shows that the second
term connects the local ion displacement, ri to the local electron density. Finally, the
free phonon term may be simplified by using the Einstein approximation ωk ≈ ω0
and Fourier transforming, the bare phonon Hamiltonian is shown to be a series of
independent simple harmonic oscillators at each site index. The creation of electrons
and phonons is represented by c† and b† respectively, pi is the ion momentum and M the
ion mass. By choosing t = 0.25, a bandwidth of W = 2 is chosen. A small interplanar
hopping of t⊥ = 0.01 is included to remove the logarithmic singularity in the 2D density
of states at ε = 0.
Figure 3 shows the parameter space of the Holstein model. For very large phonon
frequency, the effective interaction is instantaneous, and a Lang–Firsov transformation
[15] results in an attractive Hubbard model (which is one of the standard models for
correlated electron systems) [16]. Alternatively, taking the limit of very small phonon
frequency, a fast moving electron cannot ‘see’ the nuclei move in the time it takes to
EXTENDED THEORY
CUPRATE?
FIGURE 3. Parameter space of the Holstein model. For very large phonon frequency, the effective
interaction is instantaneous, and a Lang–Firsov transformation results in an attractive Hubbard model.
Alternatively, taking the limit of very small phonon frequency, a fast moving electron cannot ‘see’ the
phonons move, and the problem maps to a static disorder problem (similar to the Falikov–Kimball model
[19]). This makes the phonon problem extremely hard, and little is known about the middle of the
parameter space. The range of the Eliashberg theory is shown in the bottom left corner. The expected
position of the cuprates is shown as the single diamond. The expected validity of an extended theory
including all 2nd order Feynman diagrams is also shown.
traverse many sites, so the problem maps to a static disorder problem (which is essen-
tially uncorrelated). One may therefore think of the phonon frequency as possessing the
ability to “tune” the effect of correlations, and one therefore obtains a second motivation
for the study of electron-phonon systems of trying to understand electronic correlations
[17]. The correlation tuning makes the phonon problem extremely hard, and little is
known about the intermediate regime of the parameter space. The range of the Eliash-
berg theory is shown in the bottom left corner. Contrary to Migdal’s assumption, the
theory cannot extend beyond intermediate coupling since renormalisation of the effec-
tive mass reduces εF invalidating the condition (Migdal’s theorem) ω0 ≪ εF [18, 9]. The
approximate position of the phonon parameters in the cuprates is shown as the single di-
amond. It is essential to correct the theory for weak to intermediate coupling at larger
phonon frequencies. The extension is clear by looking at the large phonon frequency
limit. The Hubbard limit requires that all 2nd order processes in U are included in the
self-energy, or the incorrect weak coupling limit is found. An extended theory including
all 2nd order Feynman diagrams is required to understand the weak coupling limit, from
small to large phonon frequency.
FIGURE 4. Series of Feynman diagrams used in the current approximation. Σ is the electron and Π the
phonon self-energy. Series (a) is the Migdal-Eliashberg approximation and (b) the vertex corrected series.
EXTENDING THE ELIASHBERG THEORY
Extending the Eliashberg theory involves inserting the lowest order vertex corrections
into the electron and phonon self energies. In the Eliashberg theory, emitted phonons
are reabsorbed in a last-out-first-in order. Vertex corrections essentially allow this order
to be changed once. Such contributions are shown diagrammatically in figure 4. All
the diagrams must be included in the calculation, or electron number would not be
conserved. Momentum dependence is included in the approximation, which is essential
in low-dimensions. The inclusion of vertex corrections leads to double 2-fold integration
over the Brillouin zone in combination with a double sum over matsubara frequencies,
which is time consuming for the numerics. In order to reduce the number of points in k-
space while maintaining the thermodynamic limit, the dynamical cluster approximation
is applied [20]. Additionally, superconducting states can be considered by using the
Nambu formalism. The full details of the implementation of the extended approximation
can be found in references [21] and [22].
Using a maximum entropy technique, it is possible to compute the spectral function
from the Matsubara axis Green function. Figure 5 shows the spectral function of the
Holstein model calculated using the extended Migdal–Eliashberg theory. The results are
qualitatively similar to ARPES measurements of the cuprates. In particular the change
between incoherent and coherent particles occurs at the phonon frequency (shown as the
dashed line), associated with a kink in the [11] direction. It is noted here that the effect
of the phonon self-energy is a softening of the phonon mode. In the standard ME theory
in 2D, the mode at the (π ,π) point is completely softened, leading to a fatal instability
of the theory. However, the vertex corrections act against this softening, and relieve the
instability. In such a way, it is clear that a vertex corrected Eliashberg theory is essential
for the study of quasi-2D materials [21].
One can also compute properties in the superconducting state. One such property is
the momentum-dependent pairing density, ns(k) = T ∑n F(iωn,k), where F(iωn,k) is
the anomalous Green function associated with the pairing of electrons with momentum
k and −k. It is possible to transform the momentum dependent order parameter to
determine the magnitude of individual spherical harmonics. Figure 6 shows such a
decomposition. A cluster size of NC = 64 is used, with U = 0.6 and ω0 = 0.4. Note
ω0=0.2, U=0.3, DCA(VC)
-2 -1
A(k,ω)
FIGURE 5. Spectral function of the Holstein model in the extended Migdal–Eliashberg theory. The
results are qualitatively similar to ARPES measurements of the cuprates. In particular the change between
incoherent and coherent particles occurs at the phonon frequency, associated with a kink in the [11]
direction. ©Institute of physics publishing 2003 [21].
how higher order harmonics develop as the filling is increased. In particular, it can be
seen that no single harmonic (such as the s-wave symmetry) is sufficient to describe the
order parameter. Some of the higher order terms come about due to increased pairing at
momentum k = (π/2,π/2), in particular, pairs with angular momentum.
Finally, by varying the temperature and chemical potential, the phase diagram can
be computed. Figure 7 shows phase diagrams of the Holstein model for the different
approximations. U = 0.6 and ω0 = 0.4.The top diagram shows the result from the
Eliashberg approximation (dynamical mean-field theory NC = 1). On the bottom the
results from the current approximation with NC = 4 are shown. The superconducting
order is suppressed close to half filling. Assuming a form for the density of states in 2D
(with small interplane hopping) of D(ε) = (1− t log((ε2+ t2⊥)/16t
2))/tπ2 (for |ε|< 4t)
[23], which matches the full density of states with reasonable accuracy. From this the
BCS result may be calculated using the expression
TC(n) = 2ω0 exp(−1/|U |D(µ(n)))/π , (4)
with the chemical potential taken from the self-consistent solution for a given n. This
result also drops off monotonically. Results in the dilute limit are in good agreement with
the BCS result (line with points). Close to half-filling, the DMFT result is significantly
smaller than the BCS result (which predicts TC(n = 1)> 0.07). The difference in results
-0.025
-0.02
-0.015
-0.01
-0.005
 0.005
 0.01
 1  1.1  1.2  1.3  1.4  1.5  1.6  1.7
s, m=0
d, m=0
g, m=0
g, m=4,-4
FIGURE 6. Decomposition of the order parameter into spherical harmonics. A cluster size, NC = 64 is
used, with U = 0.6 and ω0 = 0.4. Note how higher order harmonics develop as the filling is increased. In
particular, the g harmonics can be almost as strong as the s harmonics at n = 1.45. ©Institute of physics
publishing 2005 [22].
between the two mean-field theories at half-filling is due to the self-consistency in the
DMFT. When vertex corrections and spatial fluctuations are included, the dilute limit
is relatively unchanged. However at half-filling, there is a huge drop in the transition
temperature. The suppression at half-filling is a manifestation of Hohenberg’s theorem,
which implies that there may be no superconducting order in 2D. Here I have computed
for quasi-2D, so it is interesting that in real materials with low dimensional character the
maximum in superconductivity is shifted away from half-filling.
CONCLUDING REMARKS
I end the paper with a warning for constructing theories of high-temperature supercon-
ductivity using electron-phonon interactions alone, while neglecting the Coulomb re-
pulsion. If one takes the phase diagrams from the previous section, and assigns similar
energy scales to those in the cuprates, it is possible to obtain a temperature in Kelvins
for the maximum in the phase diagram at n = 1.2. This comes out as around 172K - one
could say approximately the TC in the cuprates.
So why isn’t this the solution for the cuprates? Cuprates are very tightly bound ma-
terials, which is why the “Fermi energy” is low, and the ratio ω0/εF is large enough
to justify extending Eliashberg theory. The problem is that a small Fermi energy also
means the the Hubbard U is a comparatively large quantity. On a simple mean-field
level, one can include the Coulomb repulsion in the theory of superconductivity. For ex-
ample, the Eliashberg equations can be extended to include an effective electron-electron
interaction (otherwise known as the Coulomb pseudopotential µC). The effect of this is
to modify λ → λ − µC. Substitution into equation 4 means that the transition temper-
 0.02
 0.04
 0.06
 0.08
U=0.6, ω0=0.4, Nc=4, VC
 0.01
 0.02
 0.03
 0.04
 1  1.2  1.4  1.6  1.8
 0.02
 0.04
 0.06
 0.08
 0.02
 0.04
 0.06
 0.08
 0.01
 0.02
 0.03
 0.04
 1  1.2  1.4  1.6  1.8
 0.02
 0.04
 0.06
 0.08
U=0.6, ω0=0.4, Nc=1
FIGURE 7. Phase diagrams of the Holstein model. U = 0.6 and ω0 = 0.4. The top diagram shows the
result from the Eliashberg approximation (dynamical mean-field theory NC = 1). Also shown is the BCS
result (line with points). On the bottom the results from the current approximation with NC = 4 are shown.
The superconducting order is suppressed close to half filling in the vertex corrected theory. ©Institute of
physics publishing 2005 [22].
ature is considerably reduced, or that superconductivity of the BCS type is completely
destroyed. Any phonon-based mechanism for the cuprates must address this point and
be compatible with the electron-electron interaction. Alternatively (and this is a warn-
ing against the other extreme) on the basis of the similarity of energy scales, any spin-
fluctuation mechanism (which is essentially Coulombic) must also treat the phonons (or
at least be compatible with them) to be plausible.
ACKNOWLEDGMENTS
I sincerely thank the organising committee of the course for their generous financial
support. Aspects of this research were carried out under the MPIPKS guest scien-
tist program, and as a visitor at the University of Leicester. I thank A.S.Alexandrov,
J.L.Beeby, E.M.L.Chung, N.d’Ambrumenil, J.K.Freericks, M.Jarrell, P.E.Kornilovitch,
J.H.Samson and M.Yethiraj for stimulating discussions, both about this work and the
problems of electron-phonon interactions and superconductivity in general. I acknowl-
edge support at Loughborough University under EPSRC grant no. EP/C518365/1.
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