Condon Domains Research Articles

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.

Recent results on Condon domains in metals

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.

Diamagnetic domains and magnetostriction in beryllium

Magnetostriction was for the first time studied under the conditions of formation of diamagnetic domains (Condon domains). Transverse magnetostriction oscillations on a beryllium single crystalline plate oriented normally to magnetic field were measured in magnetic fields up to 7 T at temperatures down to 1.5 K. The relative amplitude of oscillations increased almost as the square of magnetic field and reached 10−5. The signal had a sawtoothed shape corresponding to alternation of homogeneous and inhomogeneous (domain) states in the region of the existence of magnetic domains. The arising of domains was accompanied by singularities in the observed signals which is explained by an anomalous increase in the compressibility coefficient of the domain state: coefficient oscillations were more than 100 times larger than the value predicted by the standard theory. The observed relation between magnetization current and deformation led us to conclude that the compressibility of the metal was fully determined by conduction electrons. Magnetostriction then exactly compensated Fermi level oscillations. The position of the Fermi level therefore remained constant under magnetic field variations. In addition, the domain wall thickness had to increase as the plate grew thicker.

Evidence for Condon domains in white tin with two de Haas–van Alphen periods

Evidence for Condon domains in white tin with two de Haas–van Alphen periods

Anomalous compressibility and magnetostriction of beryllium under the conditions of diamagnetic domain formation

Magnetostriction oscillations are measured for a single-crystal beryllium sample shaped like a plate perpendicular to the direction of the magnetic field. In the range of 2–5 T at a temperature of 1.5 K, i.e., in the region of diamagnetic domain formation (Condon domains), the striction signal has the saw-tooth shape corresponding to the alternation of homogeneous and nonhomogeneous (domain) states. The formation of the domain structure is accompanied by an anomalous increase in compressibility; the oscillations in this coefficient are more than one hundred times greater than the value given by the standard theory. An analysis of the results indicates that the domain wall width should increase with increasing plate thickness.

Diamagnetic domains in beryllium observed by muon-spin-rotation spectroscopy

Dia- and paramagnetic (Condon) domains were observed in single crystal beryllium for applied fields $\mathit{H}||[0001]$ of 1--3 T, at temperatures $T=0.1--3$ K, by muon-spin-rotation $(\ensuremath{\mu}\mathrm{SR})$ spectroscopy at PSI. On varying H, the domains with magnetization parallel and opposite to $\mathit{H}$ reappear in each de Haas--van Alphen period $\ensuremath{\Delta}H,$ detected by the doublet splitting in the precession frequency spectrum. As H varies within the ``domain section'' $\ensuremath{\delta}H$ of a dHvA cycle, the domain inductions ${B}_{1}{,B}_{2}$ stay constant, while the volume fractions of the para- and diamagnetic regions vary linearly. The beat in the susceptibility amplitude $a(B)=4\ensuremath{\pi}{(dM/dB)}_{\mathrm{max}},$ with a period of $\ensuremath{\approx}33\ensuremath{\cdot}\ensuremath{\Delta}H,$ allowed us to determine ${B}_{i}$ and $\ensuremath{\delta}H$ for different regimes. As a increases by a factor of $\ensuremath{\approx}3,$ the sections $\ensuremath{\delta}H$ grow, $|{B}_{2}\ensuremath{-}{B}_{1}\stackrel{\ensuremath{\rightarrow}}{|}\ensuremath{\Delta}H,$ and the uniform state between two domain sections shows an increasingly strong diamagnetism, $\ensuremath{-}4\ensuremath{\pi}\ensuremath{\chi}\ensuremath{\gg}1.$ The domains persist up to $T\ensuremath{\approx}3$ K and at fields down to $H\ensuremath{\approx}1$ T. On varying H, a slight ``field overheating'' was observed. The domain area in the phase diagram is much larger than expected, while other predictions of the theory proved to be qualitatively correct.

Order Parameter for a Diamagnetic Phase Transition

A theory of the magnetic induction bifurcation at a diamagnetic phase transition in normal metals is developed. Our results are in good agreement with the recent observation of the muon spin rotation resonance splitting in beryllium due to Condon domains, Solt et al. [Phys. Rev. Lett. 76, 2575 (1996); Hyperfine Interact. 104, 257 (1997)], being the first measurement of the onset of a diamagnetic phase transition. Temperature dependence of the order parameter, the magnetization in each Condon domain, is explained and reproduced quantitatively. A new kind of phase diagram for diamagnetic phase transitions is constructed.

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.

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.

De Haas–van Alphen effect as a first-order electronic topological transition

The de Haas–van Alphen effect and the behaviour of a superlattice in a quantising magnetic field can be described in terms of an electronic topological transition. Near the transition, the thermodynamic stability condition is shown to break down, thus eliminating the 11/2-order transition and giving rise to a first-order phase transition. The latter leads to the formation of diamagnetic Condon domains.

Nuclear magnetic resonance in the Condon domain state

Calculations of splittings of nuclear magnetic resonance (NMR) lines at diamagnetic phase transitions are carried out for a three-dimensional electron gas. There is a good agreement between the theory and the experimental results in silver. The size of the Condon domains and the width of the domain walls are estimated. Experimental results on the de Haas-van Alphen effect in silver are analyzed with the help of the temperature-magnetic-field diagram. The results of the analysis are in agreement with the NMR data. The NMR splitting in a two-dimensional electron gas due to Condon domains is calculated.

Perfect differential paramagnetism (condon domains) in a graphite intercalation compound

The ground state of an electron gas in a strong magnetic field can be inhomogeneous. In a nearly filled Landau level, magnetic interaction can reduce the energy of the next higher Landau level to such an extent that domains of this higher level can nucleate. In a two-dimensional electron (or hole) gas, these Condon domains are predicted to be closely analogous to the quantum Hall effect. This paper discusses these theoretical predictions and connects them to the experimental observations of Condon domains in Br 2-intercalation compounds. The extreme sharpness of the observed transitions allows detailed studies of domain-wall (soliton) dynamics in the two-phase regime.

Giant magnetic interaction (Condon domains) in two dimensions.

Observations tres fines d'oscillations de Haas-van Alphen et Shubinkov-de Haas, mettant en evidence la formation d'un etat a deux phases, induit par le champ magnetique, dans le compose d'insertion graphite-Br 2 . Observation de manifestations possibles de la dynamique des parois de domaines dans le regime a deux phases