When a particle is constrained to move in a honeycomb-lattice structure, its properties change dramatically: it behaves as though it has no mass and travels at the speed of light. Such particles, known as massless Dirac fermions, were first observed1,2 in atomic graphene in work that spawned a frenzy of research into its properties, both to understand the fundamental science and for technological applications. In a pair of letters published in this issue, Gomes et al.3 (page 306) and Tarruell et al.4 (page 302) describe how, for the first time, they have created synthetic analogues of graphene in two different systems.
Atomic graphene is a single layer of carbon atoms organized into a honeycomb structure. The new graphene analogues3,4 share this honeycomb topology and offer substantial advantages in manipulation and readout of material properties over what is possible with atomic graphene. Gomes et al.3 assembled 'molecular graphene' from individually placed carbon monoxide (CO) molecules, and studied the bizarre effects that arise from small variations in the material's lattice structure (Fig. 1a,b). Tarruell et al.4 show that a clever arrangement of laser beams (an optical lattice) produces a honeycomb structure to confine an ultracold gas of potassium atoms, and investigated how the lattice structure controls the mass of the atoms (Fig. 1c,d).
The first step in studying a graphene analogue is to show that its particles are massless. In everyday life, an object's mass is measured by weighing it on a scale. That is impossible for an extremely light object such as an electron, particularly when its mass is inextricably tied to the material in which it resides. Gomes and colleagues3 circumvented this difficulty by studying the energy gap between the valence and conduction energy bands in their molecular graphene. This quantity reflects the energy required to create an electron–hole pair out of the vacuum (where a hole is a charge carrier created by the absence of an electron), and thus the electron's mass. The authors measured the gap using scanning tunnelling microscopy and spectroscopy, and observed no separation between the valence and conduction bands — only a sharp dip at the Dirac point, at which the bands touch. The dip indicates a vanishing number of quantum states at zero energy, and the absence of a gap is a clear signature of massless Dirac fermions.
Having established the existence of massless electrons in their molecular graphene, Gomes and colleagues employed their exquisite control of the material to study how slight modifications to the lattice geometry change the properties of its resident electrons. First, they introduced a periodic arrangement known as a Kekulé structure to reimbue their massless Dirac fermions with mass, and detected this mass through the appearance of an energy gap in the material's electron-energy spectrum. They then showed that a distortion of the lattice structure, akin to squeezing it along several axes, makes the electrons act as if they are in a magnetic field5. The appearance of an isolated, zero-energy quantum state in the presence of this apparent field serves as striking confirmation of the existence of massless Dirac fermions in the molecular graphene.
Meanwhile, Tarruell and colleagues4 demonstrated that the potassium atoms in their material behave as massless Dirac fermions by watching them undergo a transition between the conduction and valence bands. Such transitions require substantial energy input for particles that have mass, to overcome the particles' rest mass and bridge the energy gap. But for a massless particle there is no energy gap, so the particle can move freely between valence and conduction bands at the Dirac point.
Holding a massless Dirac fermion near a Dirac point in order to observe transitions between bands would be extremely challenging in the solid state, because motional damping is difficult to control, and scattering off lattice impurities and phonons (quasiparticles associated with lattice vibrations) regularly randomizes the Dirac fermions' momentum. Ultracold atomic gases in optical lattices are ideally suited to overcoming this problem, as the atoms move without dissipation in extremely clean and tightly controlled environments6.
Tarruell and co-workers took advantage of these features in their work. They started with a cloud of fermionic (half-integer spin) potassium atoms in the valence band. These atoms exhibited a spread in momentum due to the Pauli exclusion principle, according to which two or more fermionic particles cannot occupy the same quantum state. The authors then subjected the atomic cloud to a magnetic-field gradient that gently accelerated the atoms, slowly varying their momentum. In this setting, any atom whose momentum trajectory crosses a Dirac point should be transferred to the conduction band (Fig. 1d).
Tarruell et al. elegantly observed the existence of Dirac points by imaging the momentum distribution of their cloud. They achieved this by allowing the cloud to expand and then taking an image of its spatial distribution, noting that atoms in the cloud that have more momentum move more quickly in this expansion. Before the acceleration, the atoms are distributed smoothly around zero momentum, whereas after acceleration the distribution has gaps corresponding to atoms that have been transported through a Dirac point, and hence transferred to the conduction band.
To address the question of how the Dirac fermions arise, Tarruell and colleagues smoothly tuned the energy potential of their structure to transform the underlying honeycomb lattice, which has two Dirac points, into a dimer lattice, which has none. To probe for the existence of Dirac points in each lattice, they accelerated the potassium atoms and measured the fraction of atoms transferred to the conduction band. What they find is an impressive verification of a theoretically predicted topological phase transition7,8,9: the atom fraction transferred drops abruptly to zero at precisely the point at which the dimer lattice forms and the two Dirac points merge and annihilate one another.
These studies3,4 pave the way for a new realm of condensed-matter physics, in which materials with exotic topological properties can be built to order from the ground up. However, some of the most interesting properties arise when interactions between particles cause them to self-organize into intricate patterns determined by the laws of quantum mechanics. To this end, it will be essential to suppress the role of the copper substrate underlying Gomes and colleagues' material, because it screens out the repulsive Coulomb interaction between the electrons and limits the electrons' lifetime.
By contrast, the ultracold atoms described by Tarruell et al.4 are extremely long lived and can exhibit strong interactions, but so far have been studied only in synthetic magnetic fields10 that are not strong enough to investigate the physics of highly correlated Dirac fermions. Much stronger fields should be achievable through temporal modulation of the Dirac points11. Once these limitations are overcome, the techniques developed by Tarruell, Gomes and their colleagues should point the way to a new generation of quantum materials.
About this article
Noncatalytic hydrogenation of decene-1 with hydrogen accumulated in a hybrid carbon nanostructure in nanosized membrane reactors
Russian Journal of Physical Chemistry A (2014)