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Building quantum materials, one layer at a time

From the Stanford Physics Department Annual Newsletter 2024/25

In primary school, we are told that there are three phases of matter: solid, liquid and gas. We later learn that solids can act as metals, insulators, or semiconductors, and that in most cases, their electronic character can be understood by considering one electron at a time. However, both simplified pictures are upended when interactions between electrons become important. Then materials that might have been expected to be metals can turn insulating, and more exotic phases from magnetism to superconductivity may arise. A major aim of condensed matter physics, and a motivating goal in the Feldman Lab, is to better understand such interactions and their consequences.

Most strongly correlated materials are chemically complex, making them difficult to grow and study. The discovery that atomically thin two-dimensional (2D) crystals can be isolated using Scotch tape has opened new avenues to engineer similar behavior in a simpler setting. Individual 2D materials can be mixed and matched like Lego blocks to generate emergent properties which are not present in either system on its own. And it’s even better than that: because there is weak bonding between layers, they can also be twisted relative to one another. This produces a beating pattern in the spatial arrangement of atoms in each layer known as a moiré superlattice, which can quench the kinetic energy of electrons so that interactions dominate. Furthermore, moiré systems are tremendously flexible because they can be tuned by twist angle, electrostatic gates, or applied electromagnetic fields.

In our lab, we use micromanipulators and rotation stages to pick up 2D flakes and stack them with controlled interlayer twist. In this way, we build up synthetic quantum materials, one layer at a time. To study them, we use a scanning probe microscope capable of imaging electronic properties at the nanoscale and at cryogenic temperatures. As a sensor, we fabricate a single-electron transistor (SET) at the tip of a tapered quartz rod (Fig. 1a). The SET is an exquisite probe of local electrostatic potential and charge because its conductance depends sensitively on the electrostatic environment. By bringing it close to a sample of interest (Fig. 1b), we can then extract thermodynamic quantities such as electronic compressibility, providing complementary insight to other techniques with a spatial resolution on the order of 100 nm.

In one recent work led by graduate students Carlos Kometter and Jiachen Yu, we studied a low-twist heterobilayer composed of two different monolayer semiconductors, WSe2 and MoSe2. This system combines an especially strong moiré potential with a long moiré wavelength. As a result, it can be understood in the moiré atomic limit: individual carriers reside in far-separated potential wells and populate electronic states that are analogous to atomic orbitals, but on the moiré scale (Fig. 1c, left). Studying the behavior of this system as a function of applied magnetic field and carrier density, we found distinct regimes in which the electrons themselves form solid or fluid ground states (Fig. 1d).

This rich phase diagram can be explained by the interplay of flat moiré bands (with heavy carriers) and dispersive moiré bands (with light carriers) that respectively arise from the hybridization of localized and extended orbitals of the moiré atoms (Fig. 1c). Their relative energies and therefore occupations are tuned by the magnetic field. Occupation of the flat bands favors periodic arrangements of charge, i.e. electron solids. In contrast, Hofstadter states, i.e. topological fluids, are preferred as the dispersive bands are being filled. Most surprisingly, in an intervening region where both types of bands are changing occupation, charge-ordered states reappear, implying a cooperative crystallization of light and heavy carriers. Our work provides new insight into how microscopic characteristics dictate the nature of correlated ground states, and it demonstrates intriguing interplay between localized moments and itinerant carriers.

Going forward, we seek to push the bounds of scanning probe microscopy. This includes designing new approaches to apply an electric displacement field – a critical tuning knob to control quantum phases in many materials systems – while maintaining high spatial resolution of the scanning SET. Likewise, we have recently demonstrated SET operation at milliKelvin temperatures in a dry dilution refrigerator, and we plan to conduct thermodynamic measurements with spatial resolution down to the atomic scale. We anticipate that pushing toward these new frontiers will reveal fundamentally new quantum states and elucidate their provenance.

-- Ben Feldman

 

Figure 1

Figure 1. a, Scanning electron micrograph of a single-electron transistor (SET) tip. b, Schematic of the SET measurement setup showing a WSe2/MoSe2 heterobilayer encapsulated in hexagonal boron nitride (hBN). c, Schematic of electronic states in moiré potential wells that are similar to s and p atomic orbitals. Hybridization of these states leads to flat and dispersive bands, respectively, whose energy offsets depend on on-site Coulomb repulsion U and single-particle offset D. d, Electronic compressibility dm/dn as a function of moiré filling factor n and perpendicular magnetic field B. Incompressible charge-ordered states (vertical) and Hofstadter states (diagonal) appear in different regimes.

Benjamin Ezekiel Feldman

Benjamin Ezekiel Feldman, Assistant Professor of Physics

The Feldman lab is interested in exploring correlated electronic states and topological phenomena that emerge in nanoscale quantum materials and devices. We use a variety of tools ranging from transport measurements to scanning probe microscopy to study these effects from both a local and global perspective. Please see the research section for more detail about our interests and tools.