V. S. Egorov

Recent results on Condon domains in metals

Abstract Recent progress in spectroscopy of the Condon domain phase in metals, in particular in beryllium and white tin, is reviewed. The observed variation of domain magnetizations with temperature and applied magnetic field is consistent with the theory, though understanding of the phase diagram in the case of Be requires a generalized form of the Lifshitz–Kosevich formula. For tin, two domain-generating de Haas–van Alphen modes were seen in overlapping field regions. Besides in Be, Sn and Ag, Condon domains are expected to appear in pure single crystals of all sp metals and quasi-2D conductors. To complement spectroscopy, techniques suited for studying the spatial structure of the Condon state have to be developed.

DIRECT EVIDENCE FOR DIA- AND PARAMAGNETIC DOMAINS IN BERYLLIUM

The first spectroscopic evidence for dia‐ and paramagnetic domains (Condon domains) in beryllium metal is presented. The domains, detected by the splitting of the μSR line, arise and disappear periodically in each de Haas–van Alphen cycle as the field \bf H, normal to the single crystal Be plate and parallel to its [0001] axis, is tuned near \bf H_0\approx 2.7\ T. The intensity of the lines in the doublet reflect the ratio of dia‐ to paramagnetic regions. For the difference in induction within the domains we obtain \Delta B\approx 30\mbox--40\ G in the investigated field range at T=0.8 K.

Condon domains in aluminum and lead

Formation of Condon domains was observed in highly perfect single crystals of Al and Pb by the periodic, sharp broadening of the spectral linewidth. Unlike in Be, domains in these metals can arise only via strong anharmonicity of the quantum oscillations, at very low temperatures.

Observation of dia- and paramagnetic domains in beryllium and white tin by muon spin rotation spectroscopy

Dia- and paramagnetic (Condon) domains in simple metals, produced at low temperatures and in high magnetic fields via the cooperative effect of electrons in Landau orbitals, are observable by the muon spin rotation (μSR) technique. For beryllium, domains were seen for T<3.5 K in fields 1<H<3 T through a well resolved splitting of the muon precession frequency, the split lines reappearing in each de Haas–van Alphen (dHvA) period as the applied field H varies. The beat in the dHvA oscillations allowed varying of the amplitude χ0 of the differential susceptibility, the key parameter determining domain properties. The data points in the (B,T) plane show that the standard formula for χ0(B,T) fails to describe the phase diagram, a consequence of the nearly cylindrical Fermi surface of Be. For white tin, preliminary data indicate the presence of domains by the oscillatory behavior of the μSR linewidth at T=0.1 K in fields of H≈1 T, arising from electrons in the pocket of the Fermi surface in the sixth zone.

Effect of the boundary conditions on the properties of the two-dimensional electron gas and the quantum Hall effect

Abstract The properties of the two-dimensional electron gas (2DEG) with a fixed number of electrons, in magnetic fields at low temperatures, turn out to be drastically different from those of the 2DEG with fixed chemical potential. If the electron number is constant, the position of the Fermi level oscillates in the magnetic field and additional charges near the sample edge arise due to the oscillations of the contact potential. These charges move along the edge in a certain direction. Whereas the charges themselves form a Shottky-type potential barrier, their motion gives rise to an edge current which is noticeably larger than the magnetization current in de Haas–van Alphen effect (dHvA), the former being half period shifted with respect to the latter. The Lorentz force acting on this current creates an additional oscillating pressure. As a result, the 2DEG state equation for small Landau level number i becomes similar to the van der Waals equation. Within the instability regions, the 2DEG undergoes a transition into a mixed state with inhomogeneous charge density. In this state, the 2DEG appears to be identical with the hypothetical sample in Laughlins “gedanken” experiment. In the mixed state the interface electron density, pressure, longitudinal and transversal conductances remain unchanged within some interval of the magnetic field (plateau). This model accounts for the integer quantum Hall effect (QHE) within the framework of conventional metal theory. For the boundary condition of fixed chemical potential, the electron number oscillates in the magnetic field but the 2DEG remains homogeneous everywhere and the above effects do not occur.

Superconducting intermediate state in white tin near Hc: new results by μSR

A depression of the critical field Hc, predicted for the transition from the normal (N) to the intermediate (I) state in thin plates of type 1 superconductors, was spectroscopically confirmed for the first time. In an ultra-high purity single-crystal plate of white tin oriented normal to the applied field H, the N–I transition occurs at H=HcI≈0.94Hc. No hysteresis was seen as the field varied (in steps of 10 G) upwards and downwards. The average field B in the N domains is found to be HcI at the transition, increasing towards Hc with further lowering of H. The huge asymmetrical peak in the damping rate of the precession signal, observed in the range of the initial growth of the S domains, can be a sign of a topological transition from a tubular to a laminar structure.

Observation of diamagnetic domains in beryllium

We have observed diamagnetic domains (Condon domains) in a beryllium single crystal in magnetic fields H⩽3 T (H∥[0001]) at liquid-helium temperatures. The formation of the domain structure was determined according to magnetic-breakdown quantum oscillations of the resistance thermoelectric power as well as according to the splitting of the resonance peak of the free spin precession frequency of muons (µSR). The alternation of a uniform state (with one µSR peak) and a state with domain structure (with two peaks) is consistent as regards the periodicity with the de Haas-van Alphen effect, the period is ΔH≅78 Oe, and the range of existence of domains and the difference in their magnetizations are ΔB=4πΔM=B2−B1≅30 Oe.