The current research focuses on high-Tc materials, superconducting fullerenes and the very recently discovered superconductivity in MgB2. Surprisingly a single optical phonon mode seems to provide the dominant interaction in the two-band superconductor MgB2. While the traditional concepts of electron-phonon mediated superconductivity seem to apply in the case of MgB2, questions are raised for the intercalated fullerenes, which exhibit particu-larly high transition temperatures up to 117 K in C602 CHBr3. Neutron scattering experi-ments using tiny mm3-size single crystals of Tl2Ba2CuO6δprovided first evidence for the resonance mode in the spin excitation spectrum of a single layer material. A novel high-resolution neutron spectrometer was successfully tested. Further progress in this field relies on the controlled growth of single crystals of complex multilayer cuprate structures as well as on the synthesis of new superconducting compounds.
Superconductivity and specific heat in MgB
2Y. Kong, O.V. Dolgov, O. Jepsen, O.K. Andersen, and J. Kortus;
A.A. Golubov (University of Twente, The Netherlands); B.J. Gibson, K. Ahn, and R.K. Kremer The discovery of superconductivity at 39 K in
MgB2immediately raised the question if such a relatively high critical temperature could result from the conventional electron-phonon cou-pling or whether more exotic mechanisms are involved like in the copper-oxide high-Tc super-conductors.
In order to answer this question we calculated the electronic structure, the phonon spectrum, and the electron-phonon (e-ph) interaction us-ing the linear-response theory within the full-potential LMTO density functional method.
The crystal structure of MgB2is hexagonal with graphite-like boron layers stacked on top with Mg in between.
The electronic structure near and below the Fermi level consists of two B pz π-bands and three quasi-2D B–B bonding σ-bands.
The σ and π bands do not hybridize when kz= 0 and πc. The π-bands lie lower with respect to the σ-bands than in graphite and have more kz-dispersion due to the influ-ence of Mg, the on-top stacking, and the
smaller ca-ratio. This causes the presence of pσl= 0.056 light and pσh= 0.117 heavy holes near the doubly-degenerate top alongΓA of the σ-bands. For the density of states atεF0, we find: N0= Nσl0+ Nσh0+ Nπ0= 0.048 + + 0.102 + 0.205 = 0.355 states / (MgB2eVspin).
AlongΓA there is a doubly-degenerateσ-band of symmetry E which is slightly above the Fermi level and its eigenvectors are given in the two insets at the bottom of Fig. 55. The corresponding Fermi surface sheets are warped cylinders.
The phonon dispersions ωmqand density of states Fω are shown in Fig. 55. The agree-ment between our Fω and those obtained from inelastic neutron scattering is excellent;
our peaks at 260 and 730 cm1(32 and 90 meV) are seen in the experiments at 32 and 88 meV.
For the frequencies of the optical Γ-modes we get: 335 cm1E1u, 401 cm1A2u, 585 cm1E2g, and 692 cm1B1g. The all-important E2gmodes are doubly degenerate op-tical B–B bond-stretching modes (obs).
Figure 55: Left: Calculated phonon dispersion curves in MgB2. The area of a circle is proportional to the contribution to the electron-phonon coupling constant,λ. The insets at the bottom show the twoΓAE eigenvectors (un-normalized), which apply to the holes at the top of theσ-bands (bond-orbital coefficients) as well as to the optical bond-stretching phonons (relative change of bond lengths). Right: F(ω) (black and bottom scale),α2(ω) F(ω) (green) andα2tr(ω) F(ω) (purple).
Close to ΓA, they have exactly the same symmetry and similar dispersions as the light and heavyσ-holes, although with the opposite signs. The E eigenfunctions shown at the bot-tom of Fig. 55 now refer to displacement pat-terns, e.g.,101has one bond shortened, another bond stretched by the same amount, and the third bond unchanged. These E displace-ment patterns will obviously modulate the elec-tronic bond energy, such as to split the light- and heavy-hole bands. It was judged that the cor-responding electron-phonon (e-ph) matrix ele-ment, gσobs, will be the dominating one.
We then calculated the e-ph interaction which is represented by the Eliashberg spectral func-tionα2ωFωwhich for superconductivity as well as for transport is shown in the right panel of Fig.55. The dominance of theσ–σcoupling
via the optical bond-stretching mode is clearly seen in the left panel of the figure, where the area of a red circle is proportional to the tribution to the electron-phonon coupling con-stant from the particular mode. This gives rise to the huge peak in the two Eliashberg func-tions around ωobs= 590 cm1= 73 meV. The total λ=λσ+λπ= 0.62 + 0.25 = 0.87 is moder-ately large.
We first assumed isotropic pairing and obtained a Tc of 40 K by solving the Eliashberg equa-tions for µ= 0.10 which is at the low end of what is found for simple sp metals. The temper-ature dependence of the specific dc-resistivity calculated with the standard Bloch-Gr¨uneisen expression is nearly isotropic and in accord with recent measurements on dense wires over the entire temperature range from 40 to 300 K.
We finally solved the Eliashberg equations for a number of temperatures from 0 K to above Tc for a fixed µ in the superconducting as well as in the normal state. From this we calcu-late the difference in specific heat in the two states which is compared to our original mea-surements and to some more recent measure-ments on high quality samples in Fig. 56. It may be seen that in the calculations the specific heat jump is considerably larger at Tcthan in the ex-periments and also that the exex-periments show larger heat capacity at low temperatures.
Figure 56: Experimental data of the heat capac-ity difference∆Cp=Cp(0 Tesla) –Cp(9 Tesla) mea-sured by us (Æ) and by Bouquet et al. [Physical Review Letters 87, 047001 (2001)] (). The dash-dotted line is the theoretical result of the one-band model and the thick solid line corresponds to the two-band model from the solution of the Eliashberg equations. The two-band model repro-duces much better the specific heat jump as well as the low temperature behavior.
We therefore relaxed the assumption of an isotropic one-band model and instead consid-ered a two-band model with the possibility of different gaps on the σ and π Fermi sur-face sheets. The reason for discarding this model originally was the experimental fact that a two orders of magnitude change in scatter-ing rate changed Tc by merely2 K. With in-terband scattering and a two-band model Tc would change much more. However, if the in-terband scattering is small and only intraband
scattering is responsible for the change in scat-tering rate then a small change of Tc would be possible.
We therefore recalculated the Eliashberg func-tions allowing for two different order parame-ters for theσandπ-electrons. The correspond-ing Eliashberg functions are shown in Fig. 57.
As expected the most significant contribution comes from the coupling of the bond-stretching phonon modes to theσ-band.
Figure 57: The four superconducting Eliashberg functionsα2F(ω) obtained from first-principles cal-culations for the effective two-band model are com-pared with the isotropic Eliashberg function for the one-band model. The coupling constant of the isotropic one-band model has a value ofλiso= 0.87.
Besides the spectral functions we need the in-formation of the Coulomb matrix element µij. With the help of the wavefunctions from our first-principles calculations we can approxi-mately calculate the ratio of the µ-matrix. The ratio betweenσσ,ππandσπwere 2.23/2.48/1.
This allows one to express µij by these ratios and one single free parameter which is fixed to get the experimental Tc of 39.4 K from the so-lution of the Eliashberg equations.
Using our calculated Eliashberg functions on the imaginary (Matsubara) axis to-gether with the above matrix µij we obtain the gap values ∆σ= lim
T0∆σiπT7.1 meV, and ∆π2.7 meV, which corresponds to 2∆σkBTc= 4.18 and 2∆πkBTc= 1.59 and
which are in good agreement with several ex-periments. We solved the two-band Eliashberg equations as function of temperature and calcu-lated the difference between the specific heat in the superconducting state and the normal state
which is shown in Fig. 56. It may be seen that not only the specific heat jump at Tc but also the low temperature behavior is in much better agreement with experiments than the results of the one-band model.