Particles in a substance randomly jiggle about more vigorously at higher temperatures, causing solids to melt into liquids above a critical temperature. In thermodynamics, higher temperatures favour the formation of states that have larger amounts of entropy, a measure of disorder. The liquid state of a substance typically has a larger entropy than has the solid state, because the movement of atoms is more disordered. However, an exception occurs for helium-3, which freezes into a solid as the temperature rises1. This behaviour is known as the Pomeranchuk effect, and occurs because solid 3He has a larger entropy than does the liquid form — a phenomenon associated with the fluctuation of the spin (angular momentum) of 3He atoms. Writing in Nature, Saito et al.2 and Rozen et al.3 now describe a similar effect in a graphene system, in which electrons are found to ‘freeze’ as the temperature increases.
The system in question consists of two stacked sheets of graphene — single layers of carbon atoms, in which the atoms form a hexagonal lattice. The top sheet is twisted out of alignment with the sheet below, yielding a periodic arrangement of atoms called a moiré pattern (Fig. 1a). At a twist angle of about 1° (the ‘magic’ angle), the energy bands of electrons in the twisted bilayer graphene become almost flat4; in other words, the velocity of the electrons becomes considerably lower than normal.
As a result, the behaviour of the electrons is dominated by the repulsive (Coulomb) interaction between them, which leads to the emergence of phases that do not exist in single layers of graphene5–8. At low temperatures (below 5–10 kelvin), when the electron number is tuned to fill one or more quarters of the flat bands, the system typically forms an electrically insulating phase owing to the interactions between electrons. By contrast, when the electron number deviates from quarter fillings, the system becomes either a metal (low electrical resistance) or a superconductor (zero resistance).
A metal can be broadly regarded as a liquid state of electrons, which physicists often call a Fermi liquid. By contrast, one can view an insulator as a solid state of electrons, in which electrons are frozen in position and aligned into ordered arrays. In most cases, insulator states have lower entropy than do metal states, because the electrons are more ordered. Insulators are therefore usually expected to become metals as the temperature increases.
Saito et al. and Rozen et al. observed exactly the opposite phenomenon in magic-angle twisted bilayer graphene. By measuring electrical transport in this system, both groups find that, with increasing temperature, magic-angle twisted bilayer graphene transitions from a metal to a high-resistance phase that is close to being an electrical insulator, when the electron number is tuned to nearly one-quarter filling of the flat bands. This transition happens at a temperature of about 10 K, and the near-insulating phase persists up to about 70–100 K.
The two experiments thus reveal a Pomeranchuk effect for electrons, analogous to the phenomenon observed for 3He atoms1. To understand the origins of the effect, Saito et al. and Rozen et al. measured the entropy of one-quarter-filled twisted bilayer graphene, and find that the entropy per electron of the high-temperature near-insulating phase is greater than that of the low-temperature metal phase by an amount that is a fraction of the Boltzmann constant (kB, which is 1.38 × 10–23 joules per kelvin) — about 0.2kB in Saito and colleagues’ case, up to 0.8kB in Rozen and co-workers’ experiments. This is roughly equal to the entropy contribution of a free electron’s spin.
The electrons in twisted bilayer graphene carry both spin and a valley degree of freedom (a local minimum in the electronic energy-band structure of single-layer graphene), which together can be viewed as an isospin — a generalization of spin that involves more than three dimensions. Therefore, the two research teams suggest that the high-temperature phase is close to being a ferromagnetic insulator with an extremely low isospin stiffness — that is, the electron isospins are broadly aligned in the same direction, but the alignment is weakly constrained (Fig. 1b). By contrast, the electrons in the low-temperature metal are thought to be strongly constrained to have equal numbers of isospins in opposite directions, so that the overall sum of isospins is zero. Thus, the extra entropy from electron isospins in the near-insulating phase favours the formation of that phase at high temperatures.
This picture is supported by experiments in which a magnetic field applied parallel to the graphene sheets was found to polarize the spin part of the electron isospin in the insulator, without perturbing electron motion. In their experiments, Saito et al. observed that a large magnetic moment arose in the insulator, whereas Rozen et al. found that the entropy of the near-insulating phase dropped by roughly the amount expected to be contributed by free-electron spins. Both observations agree well with the idea that the isospin stiffness is low — that is, that the alignment of the isospins in the near-insulating state is easily perturbed by a magnetic field.
Moreover, Saito et al. observed a reduction in the number of electrons that can simultaneously occupy an energy level when a perpendicular magnetic field was applied to the system as it transitioned from the metal to the near-insulating phase. Rozen et al. observed a sharp peak in the electron compressibility (a measure of how difficult it is to increase the density of electrons) under the same conditions. These phenomena indicate that the way in which electrons occupy the graphene system resets, passing from a metal phase in which the electrons lack overall isospin polarization to an isospin-polarized ferromagnetic phase. Neither the resetting nor the ferromagnetic phase would be possible in the absence of interactions between electrons.
The discovery of the electronic Pomeranchuk effect by the two teams sheds light on the phases that occur in magic-angle twisted bilayer graphene. Further careful measurements of the isospin stiffness are now needed to determine the energy required to switch electron isospins from being unpolarized to being polarized, and to work out whether the isospin fluctuations of the near-insulating phase enhance or harm the superconductor phase of this graphene system — thereby increasing our understanding of the mechanism and tunability of superconductivity in this system.
The new findings also leave many open questions. For example, is the low-temperature metal separated from the high-temperature near-insulating phase by a first-order phase transition (characterized by an abrupt change in thermodynamic properties), or is there a smoother transition (a crossover)? Another question is why the electronic Pomeranchuk effect is absent at other quarter fillings of the band structure of magic-angle twisted bilayer graphene (that is, when the bands are half and three-quarters full), given that similar behaviour could occur at these fillings. The answers to these questions might help physicists to uncover and design further exciting phases of matter in this system, and in the many other moiré systems currently being studied.