Physics of Heavy Fermions

The following sections are included:

• Contents
• Preface

Conduction electrons are described by the one-electron band theory in which the electron-electron interactions (correlations) are simply neglected at the start. The Schrödinger wave function is solved for an electron moving in a potential which varies periodically with distance. The corresponding Bloch state with a band width of a few eV develops in the whole crystal. The conduction electrons are thus well described in momentum (k) space. The conduction electrons have energies up to a Fermi energy ε_F, which is simply expressed by ${\epsilon }_{\mathrm{F}}=\frac{1}{2}m{v}_{\text{F}}^2={\left(\hslash k_{\text{F}}\right)}^2/2m$. The Fermi surface is thus defined as a constant energy surface in k space of which the energy is equal to ε_F. Even though the electron-electron interactions may change the topology of the Fermi surface, the volume of the Fermi surface is kept invariant. It is the boundary of the electronic momentum distribution in the ground state, dividing k space sharply into two regions which electrons occupy or do not occupy. Note that the mass of the correlated conduction electron is changed from the bare mass m or the band mass m_b to an effective mass m^*.

The f electrons in rare earth and actinide compounds are localized at high temperatures. The so-called RKKY interaction leads to long-range magnetic ordering of the f electrons at low temperatures, where the indirect f-f interaction is mediated by the spin polarization of the conduction electrons. The corresponding ordered moment is based on the crystalline electric field (CEF) scheme of the f electrons. In some CEF cases, the quadrupole moment becomes dominant instead of the magnetically ordered moment, which leads to quadrupole ordering.

High-quality single crystals are grown by various methods such as the Czochralski method, floating zone method, flux method, Bridgman method, and chemical transport method. They are explained in this chapter. For heavy fermion compounds, low-temperature measurements are essential to clarify the electronic properties of the sample because the typical heavy fermion state is formed below 1–10K. In the experiments, we mainly measure the electrical resistivity, magnetic susceptibility magnetization, specific heat, thermoelectric power, thermal expansion coefficient, thermal conductivity, NQR-NMR, neutron scattering, μSR, magnetoresistance, Hall coefficient, dHvA, and photoelectron spectroscopy, together with the electrical resistivity under pressure. We also describe the methods of measurement in this chapter.

The 3d electrons in transition metal compounds are itinerant and become conduction electrons. The Fermi surfaces are thus mainly composed of the 3d electrons. Correlations between these conduction electrons are reflected in a relatively large electronic specific heat coefficient and cyclotron effective mass, together with the magnetic moment and magnetic ordering. This is derived from a large partial density of states in the 3d electrons. The ordered moment is thus produced from a difference between the spin-up(majority) and spin-down( minority) densities of states containing the 3d electrons.

Localized-f electrons at high temperatures in some Ce and Yb compounds or actinide compounds change into f-itinerant electrons at low temperatures due to the Kondo effect, forming heavy fermions. Correspondingly, the energy scale is changed from the Fermi energy of ε_F = 10⁴ K to the Kondo temperature of T_K = 0−100 K. Since the energy scale is small in magnitude, the electronic state can be appreciably changed by temperature, magnetic field, and pressure.

The microscopic BCS theory of superconductivity, which was proposed by Bardeen, Cooper and Schrieffer in 1957, is based on the following idea [177]. When an attractive interaction between fermions is present, the stable ground state is no longer the degenerate Fermi gas but a coherent state in which the electrons are combined into pairs of spin-singlets with zero total momentum (k ↑, −k ↓: Cooper pairs). A conduction electron distorts the lattice by moving in the lattice and attracting the positive ion. This distortion attracts another conduction electron. Namely, the interaction between two electrons mediated by the phonon forms the Cooper pair of the two electrons. A BCS-type superconductor has an isotropic superconducting gap which is opened over the entire Fermi surface. Superconductivity was regarded as one of the more well-understood many-body problems. The BCS-superconductivity has been, however, altered in heavy fermion superconductors. The pairing mechanism is not due to phonons. These alternative mechanisms are as follows: pairing mechanism correlated with antiferromagnetism (or antiferromagnetic fluctuations), ferromagnetism vs superconductivity, quadrupole interaction vs superconductivity, and a novel pairing state in the non-centrosymmetric crystal structure.

The following sections are included:

• References
• Appendix: A.1 230 space groups
• Appendix: A.2 Brillouin zones
• Index