Possible Benefits from Phonon/Spin-Wave Induced Gaps below or above E_F for Superconductivity in High-T_C Cuprates

Abstract

A phonon of appropriate momentum $k_F$ will open a band gap at the Fermi energy $E_F$. The gap within the electronic density-of-states (DOS), $N\left(E_F\right)$, leads to a gain in electronic energy and a loss of elastic energy because of the gap-generating phonon. A BCS-like simulation shows that the energy gain is larger than the loss for temperatures below a certain transition temperature, $T_C$. Here, it is shown that the energy count can be almost as favorable for gaps a little below or above $E_F$. Such gaps can be generated by auxiliary phonons (or even spin- and charge-density waves) with k-vectors slightly different from $k_F$. Gaps not too far from $E_F$ will add to the energy gain at the superconducting transition. In addition, a DOS-peak can appear at $E_F$ and thereby increase $N\left(E_F\right)$ and $T_C$. A dip in the DOS below $E_F$ will result for temperatures below $T_C$, which is similar to what often is observed in cuprate superconductors. The roles of spin waves and thermal disorders are discussed.

1. Introduction

Phonons and electron-phonon coupling, $\lambda$, is driving superconductivity in most “low-$T_C$” superconductors, such as some elementary metals, transition metal nitrides and carbides, A15-compounds, etc. The superconducting $T_C$ for these materials has been calculated by the BCS equation [1], or by the strong-coupling McMillan form [2], and the resulting $T_C$’s show a reasonable correlation with measured $T_C$’s among many superconductors with low or intermediate $T_C$ [3,4,5,6,7]. The high $T_C$ in pressurized H${}_3$S was predicted from such calculations even before the measured confirmation [8,9]. A large $T_C$ needs a large $\lambda$ and a large electronic density-of-states (DOS) at the Fermi energy, $E_F$. However, the $T_C$’s in doped cuprates and pnictides are high despite a low DOS at $E_F$ and $N\left(E_F\right)$, and this fact cannot be understood from the above-mentioned methods. Several ab-initio calculations for cuprates have shown that $\lambda$ of the order 0.1 to 0.2 are not sufficiently large to explain the high $T_C$’s of these materials [10]. A similar discrepancy between calculations and observations can be observed for superconducting low-$T_C$, low-DOS near-insulators such as doped diamond [11,12].

Here, we investigate how dynamic waves, in particular phonons, make peaks and dips in the DOS close to $E_F$, and how this can be helpful for superconductivity. Phonons with propagation of atomic distortions along the lattice induce variations of the electron potential ($W\left(r\right)$), which can deform the band structure so that a bandgap appears. The wave length of the potential perturbation determines the energy of the gap center, and the size of the gap depends on the amplitude of the variations of $W\left(r\right)$. The DOS is peaked on both sides of the gap. These features appear in the real band structure of cuprates [13] as well as in the nearly-free-electron model. Static displacements (and periodic distributions of dopants, see later) deform the electron potential in the lattice and can make DOS peaks at $E_F$ too, but gaps and flat bands are not spread isotropically over the whole Fermi surface. A static distortion means a new ground state without a possible gain of energy from a phonon excitation, but the $T_C$ of a material with such distortions will be higher than for an undistorted material if $N\left(E_F\right)$ is enhanced. This result supports the proposal that superconductivity will be favored by Van Hove singularities, with peaks in the DOS emerging as a consequence of structural lattice incommensurate modulations at the nanoscale, associated with frozen charge- or spin-density waves, in complex cuprate superconductors [14,15,16].

2. Theory

The atomic displacements in a phonon are different for different sites depending on the type of atoms, the type of movements (transverse and longitudinal), and on the phonon wavelength [17]. The electron potential within a displaced atom will alter (mainly because of the Madelung part) compared to a not-moving atom at the node position of the phonon, etc., so that the potential has a perturbation ($W\left(r\right)$) along the direction (r) of the phonon wave. The wavelength of the phonon, with wave vector q, determines what part of the electronic band is susceptible to have a gap. If W is smooth enough to be described by a single Fourier component and if the electron band is free-electron like, then gaps appear at $\pm \frac{1}{2}q$ and the gap is 2V, where V is the maximal amplitude of W [18]. This is in one dimension, but complementary phonons with other directions of q will span the gap on a 3-dimensional Fermi surface. The movement of a single atom is almost harmonic with a restoring force proportional to the displacement. Assuming that the force constant is the same for each atom and that the direction of the displacements is irrelevant, then we can estimate that the elastic energy of the distorted lattice is $K{u}^2$ per atom, where K is the (positive) force constant for each atom in the lattice and u is a representative averaged amplitude for all atomic displacements. A negative K for static displacements is not considered here. It depends on the electron-phonon coupling if a large u is able to make V large.

It is instructive to compare the energy required for a phonon distortion, $K{u}^2$ (where K and u are the representative values for the force constant and the distortion amplitude, respectively), to the gain in electronic energy (also calculated per atom) because of the gap (V) generated by the phonon [19]. The gain is usually smaller than the elastic energy and no phonons will be excited. However, the opposite case is interesting when energy can be gained through the activation of phonon vibrations. The distortions u are repeatedly zero for optic phonons and their gaps can therefore fluctuate in time, at least for adiabatic conditions. Acoustic phonons can support stable gaps, since they always have non-zero displacements at some points along the atomic rows. A gap is opened in the DOS if the gain of kinetic electron energy from the activated phonon is larger than the cost from the atomic vibrations. For a free-electron band, this leads to the following equation, as was shown previously [19]:

$$\frac{K{u}^2}{N{V}^2}=(\int fϵdϵ-\int fϵ|ϵ|/\sqrt{({ϵ}^2-V^2)}dϵ)/V^2=J(T,V)$$

(1)

where the energy limits in the integration J are $\pm \hslash \omega$ (the reach of $\lambda$), and the interval −V to +V is excluded. The DOS, N, is assumed to be constant, and $f(E_F,ϵ,T)$ is the Fermi–Dirac occupation. The coupling parameter $\lambda =N\left(E_F\right)I^2/M{\omega }^2$, where M is an atomic mass, $\omega$ is a phonon frequency and I is the matrix element $<dW/du>$; the change in $W\left(r\right)$ is due to the displacement $du$. Furthermore, $M{\omega }^2=K$, and $I=V/u$ (assuming I constant all over the Fermi surface, FS) for harmonic oscillations, and Equation (1) can be written $1/\lambda =J(T,V)$. The analytic solution for $T=0$, when f is a step function and the gap is at $E_F$, is the same as the BCS result for the superconducting gap at $T=0$; $V=2\hslash \omega e^{-1/\lambda }$ [19]. The logarithm of the BCS equation for $T_C$ is:

$$1/\lambda =ln(1.13\hslash \omega /T_C)$$

(2)

The numerical limit of the J integral for $V\to 0$ turns out to be very close to the logarithm in Equation (2). Thus, Equation (1) describes the logarithm of the BCS $T_C$ equation [19]. (The perturbative derivation of Equation (1) has no self-consistent feedback, no strong coupling effects, and is based on pure band structure, but because the final result is like BCS, we can, if we wish, refer to the derivation of BCS for gaps at $E_F$.)

The advantage of a numerical solution is that a transition temperature for a gap below or above the chemical potential (or for a non-constant DOS) can be determined. Such solutions are different from the BCS result and their gaps below or above $E_F$ are not associated with superconductivity, but they might appear anyway if the balance between the gap energy and the vibrational energy is favorable. The particle-hole channel in BCS remains open for gaps slightly aside $E_F$ when $df(E_F,ϵ,T)/dT$ is wide enough for finite T.

It is generally considered that $\lambda$ is coming from coupling to phonons only, but coupling to spin-fluctuations can be considered from complementary corrections in the development [20,21,22,23,24]. Calculations for cuprates demonstrate that spin waves and electron-phonon coupling can enforce each other and produce deep pseudogaps [13]. The same equations as for electron-phonon coupling can be used with appropriate parameters for ${\lambda }_{spin}$ instead of ${\lambda }_{phon}$, but quantitative results based on the local-spin density approximation (LSDA) are not satisfying, see later.

Charge-density waves (CDW) have been demonstrated to reduce the superconducting $T_C$ and the superconducting gap at $E_F$ [25]. The doping dependence of the superconducting gap versus the pseudogap in cuprates appears as a result of the competition between a superconducting gap and a CDW gap [26]. The potential and the charge in CDW can be modulated without distortions of the atomic positions, and they (as well as spin waves) can therefore exist at higher frequency than atomic vibrations. Formation of CDW is an interesting subject, but an estimate of a corresponding ${\lambda }_{cdw}$ has not been made and such waves are not discussed here. Instead, we consider how waves with gaps $\mathit{aside}$ $E_F$ can lead to $T_C$-enhancements for certain temperatures and gap positions.

The FS’s of the cuprates are quite 2-dimensional (2D) because of the weak electron interaction between different CuO-planes, and the FS takes an almost cylindrical shape at certain dopings [27]. This is in contrast to the complicated 3-dimensional (3D) multi-structured FS’s in transition metals and most conventional superconductors. The FS for the 2D cuprate can be approximated by a free-electron band to give a circular FS with radius $k_F$, which is well adapted for the model calculation below or for wires of 1D superconductors [28]. Equation (1) describes a gap on a single (one dimensional) band from a phonon perturbation with a suitable k-vector, k. This requires (like in the application of BCS to real 3-dimensional bands) that phonons with other directions and amplitudes of k-vectors have to be included in order to follow a gap over the entire FS. All band structures have inversion symmetry, and a gap at k will also appear at $-k$. This fulfills one condition of nesting, even if other types of nesting are possible between different bands in multi-band materials.

3. Results

First, we consider the normal case when $k=k_F$ (the free electron band opens a gap at a zone boundary $k_F$) and the energy of the gap center coincides with $E_F$ in order to determine the superconducting of $T_C$. The bold (green) line is in the equation. Figure 1 shows the result for $J\left(T\right)$ for a small value of V (1/2 meV) and $\hslash \omega =100$ meV. For instance, a $T_C$ of 5 meV requires a $\lambda$ of about 1/3.

Now, consider (‘auxiliary’) phonons with gaps below or above $E_F$. Their J-solutions, shown in Figure 1, are very close to the green curve for intermediate $\lambda$, so that an energy gain is possible for temperatures just slightly lower than the superconducting $T_C$. (Here, $T_C$ denotes the transition temperature for a gap at $E_F$, i.e. a superconducting gap, as well as the transition temperatures for non-superconducting gaps below or above $E_F$). This is because the Fermi–Dirac function decreases slowly over a wide energy range for sufficiently large T, and the exact position of a gap relative to $E_F$ is not so important for the energy balance between electronic and vibrational energies. At low T, when f is almost a step function, there is very little energy gain from gaps below or above $E_F$, and that is why the two curves turn down at low $T_C$ in Figure 1. An auxiliary phonon can adjust its k-vector so that the gap will optimize its position relative to $E_F$ for the best energy gain at a given T.

The separation between the blue and red curves in Figure 1 indicates that a gap above $E_F$ has a larger energy gain than a gap below $E_F$. This is because $E_F$ itself is pushed down together with the DOS peak induced by the gap above $E_F$ (see Figure 2), and this increases the gain in kinetic energy. The situation is less favorable for a gap below $E_F$ when the latter is pushed upwards by the gap. These effects due to the shift of the Fermi–Dirac edge are largest at low T and/or large gaps. A difference of the electric potential between a gapped and a normal region is possible if the two regions could be put together, similar to the thermo-electric effect.

Gaps aside $E_F$ can enforce the superconducting gap at $E_F$. First, this is because of the positive energy count mentioned above. Secondly, these gaps appear at almost the same temperature as the superconducting transition and they generate a DOS peak at $E_F$ with a higher $N\left(E_F\right)$. The energy spent on additional phonons will be paid back by a larger gain in electronic energy at a higher $T_C$. This second mechanism can widen a superconducting gap even as $T\to 0$, despite a negative energy count from the gaps. Phonons with gaps very far from $E_F$ are not helpful, since no gain of kinetic energy is possible and since the DOS peaks are too far from $E_F$. A third mechanism to increase $T_C$ is to generate a DOS-peak at $E_F$ through a periodic distribution of dopants [16,29,30], but such a peak is fixed (no T-dependence) and the mechanism of energy gain is absent. The same can be achieved for a completely softened phonon (K negative) so that the distortions u become static. This can produce a DOS peak at $E_F$ if the periodicity of the distortions in the lattice is optimal.

Figure 2 shows the un-broadened DOS function for a potential perturbation with $k>k_F$ and the resulting gap above $E_F$. Likewise, the gap appears below $E_F$ if $k<k_F$. Figure 3 illustrates how two non-superconducting gaps of equal strengths below and above $E_F$ are superimposed in a common DOS (without mutual interactions between the two gaps). The DOS has a sharp double peak near $E_F$, since the two gaps work against each other in the energy range near $E_F$. A stronger upper gap could push the DOS peak near $E_F$ to lower energy, as was discussed above, and therefore lead to an additional energy gain. However, all phonon gaps near $E_F$ probably have similar values of $V/u$ and a symmetric DOS structure around $E_F$.

Thermal disorder and zero-point motion (ZPM) of the lattice will smear the normal state DOS. This has been shown for several materials, where it can be detected [31,32,33] and be important for the physical properties in some materials [34,35,36,37]. The effect from disorder will also smear the gaps in normal-state DOS, and distort the DOS that surrounds the superconducting gap. The bold red line in Figure 3 shows the smeared two-gap DOS. The smearing (∼50 meV) is close to what is typical in other materials at low T from thermal disorder of the lattice, but effects of thermal spin disorder will add to the smearing. The smeared gaps remain as dips in the DOS, such as what is found from tunneling measurements in many cuprates [38], while the DOS is peaked at $E_F$.

4. Possible Spin-Waves

Band gaps at and being close to $E_F$ require waves with almost the same wave length, and it is therefore tempting to believe that such waves are of different origin in order to minimize any destructing interaction between them. High-frequency spin waves and CDW are decoupled from slow phonon modes, but spin fluctuations can develop and follow the atomic distortions of a phonon. In fact, anti-ferro magnetic (AFM) fluctuations on Cu-sites and phonons are mutually enforced in cuprates if the waves have equal wavelengths, while different wavelengths generate gaps in the DOS at different energies (The Hamiltonian for calculating band gaps from phonons is obtained by having the atomic positions distorted (u) within a supercell. Instead, for band gaps from spin waves, the perturbations consist of having magnetic fields applied within the atomic spheres along the cell [13]). An AFM gap appearing below or above $E_F$ will be stabilized by electronic energy gains as described above for phonons, but with an additional energy from the local (Ferro-magnetic, FM) spin splitting. The loss of elastic energy for phonons is replaced by a loss of kinetic energy when the electron gas is spin split. The Stoner condition for FM is that the gain of exchange energy ($N^2{I}_s{\xi }^2$) has to be larger than the loss of kinetic energy ($N{\xi }^2$), where $\xi$ is the spin splitting and $I_s$ is an exchange integral [39,40,41].

FM spin-fluctuations are of interest if the two energies are approximately equal, but calculated values of the DOS and $I_s$ in cuprates are too small for that. Observations of very weak FM moments in over-doped cuprates might be related to intrinsic disorder [42,43].

However, an AFM wave with alternating up and down spins along the lattice can, via the spin part of the potential, open a gap in the band structure of the two spins, similar to the effect of a potential modulation along a phonon distortion. The opening of gaps in the (two spin) DOS constitutes a gain in kinetic energy in addition to what might result from the local exchange-energy term, but as for phonons, the efficiency of these gains becomes weaker for gaps far from $E_F$. AFM and fluctuating AFM moments exist at moderate doping, but band calculations for cuprates based on the LSDA do not find AFM [10]. Simplified additions of higher spin-density gradients as well as kinetic energy corrections to the LSDA in band structure calculations show a stronger tendency towards AFM on Cu sites and make an opening of a weak gap in the band structure [44]. State-of-the-art inclusions of such corrections into what is called a meta-GGA potential [45], produce a stabilization of the magnetic moments on Cu in cuprates [46].

Calculations of the key parameters $I_s$ and $\xi$ by use of the meta-GGA potential for doped cuprates are needed to know whether AFM waves can be stabilized at a certain T and produce gaps in the DOS. Therefore, in the absence of such calculations, we cannot say if the superconducting gap and other gaps below and above $E_F$ can be caused by pure phonons or spin-waves (or CDW), or whether the waves are mixtures between lattice distortions and magnetic excitations. Strong central peaks with weaker side spots have been detected by neutron scattering [47], which is an indication that complex magnetic interactions with different wavelengths might be present.

5. Conclusions

The message of this paper is that selected phonons, spin waves, or CDW, can generate partial gaps near $E_F$ and thereby improve the conditions for a superconducting gap within a boosted DOS peak at $E_F$. Model calculations with reasonable values of the coupling demonstrate that a gap below or above $E_F$ can appear at a temperature very close to the superconducting $T_C$. The superconducting gap and $T_C$ can be boosted by these gaps through two effects: the increased gain of total energy from the side gaps and the higher $N\left(E_F\right)$. Auxiliary gaps appear and disappear together with the superconducting gap, and the energy position of the these gaps depend on the wavelengths of the generating waves. The wavelength is not fixed but can be modified so the gaps are able to find an optimized position as a function of T. This work has not demonstrated whether AFM spin-fluctuations (or CDW) are more likely than phonons to be responsible for gaps close to $E_F$. Gaps produced by phonons have been discussed here because their coupling parameters are better known than for the other types of waves. Further works are needed to adress the questions about different types of waves as well as the energetics of the simultaneous opening of gaps at and aside $E_F$.