Electronic crystals and quasicrystals in semiconductor quantum wells: an AI-powered discovery

The gist

This work develops an unbiased AI-powered method, specifically attention-based Neural Network Variational Monte Carlo (NN-VMC), to study quantum wells in three dimensions for various electron densities and well thicknesses, revealing a new quantum phase of matter termed the electronic quasicrystal in the emergent bilayer regime

Methodology

The researchers employ an unbiased attention-based NN-VMC method to quantitatively determine the many-body ground states of interacting electrons in a semiconductor quantum well from first principles. This approach involves generating many-electron variational wavefunctions from a trainable, attention-based neural network whose weights and biases are variational parameters. The system is modeled by a Hamiltonian (1) describing spin-polarized electrons confined in a quantum well with thickness d along the out-of-plane direction z.

The implementation details include:

= The confinement of the electrons inside the well extending from −dwell/2 to dwell/2 along the z direction is implemented via the potential Hwell = X N i=1 Vwell 2 1 + tanh zi − dwell/2 w (2) characterized by the well depth Vwell and step width w. For our main text simulations we set w = 0.05 × dwell, while Vdepth is fixed to be the largest energy scale in the system. The electrostatic interaction between electrons is given by He−e = Nϕ0 2aB + 1/2aB X N i=1 X i!=j ϕ(ri − rj) (3). The Coulomb kernel ϕ(r) and Madelung constant ϕ0 in planar periodic boundary conditions with open z boundary condition are written with the Ewald summation formula for a slab geometry [62, 63] as ϕ(r) = X L erfc[α p (ρ + L) 2 + z 2] p (ρ + L) 2 + z 2 − 2 √π A / α e − α 2 z squared + √π zerf(αz) + π A X G!=0 e iG·ρ G e Gz erfc G 2α + αz + e −Gz erfc G 2α - αz, (4). The periodic boundary condition is implemented through the feature layer f(ri) = (sin(b1 · ri), sin(b2 · ri), cos(b1 · ri), cos(b2 · ri), zi/dbg). (6) Using a combination of trainable multilayer perceptron and attention layers, we generate a collection of generalized (many-body correlated) single particle orbitals ϕ m j (ri, r!=i) indexed by 1 ≤ j ≤ N, 1 ≤ m ≤ Ndet. The wave function is then constructed as a sum of Ndet Slater determinants of these orbitals. To ensure that the wave function is localized along z and respects the Coulomb cusp condition, we include a Gaussian envelope and a Jastrow factor, arriving at the general expression Ψθ(r1,..., rN) = exp (− X i,j β α 2 α + d(ri − rj)) exp −σ X N i=1 z squared i! N Xdet m=1 detij [ϕ m j (ri, r!=i)] (7) with additional trainable parameters α, β, σ. For the Jastrow factor, we make use of the periodized norm d(r) = p z squared + Mu squared, u = 1/2π Hρ mod 1, M = [a1 a2], H = [b1 b2] T, and initialize the cusp parameter to its default 2D value β = (3aB) −1. (8) The one-body reduced density matrix elements are expressed in a form that can be evaluated by Monte Carlo sampling a batch of configurations B from the probability distribution Ψ 2, yielding ρ1(nz, k; n′z, k′) ≈ X N i=1 X r i d 3 r i 3 r'i (Y j!=i d 3 r j) Ψ(· · ·, ri, · · ·)Ψ

Φ∗ nz,k(ri)Φn′z,k'(r'i). (10)

Results and Findings

The NN-VMC simulation reveals a rich phase diagram parametrized by the electron density and well thickness. The calculations reveal metallic phases as well as electron crystals in both monolayer and bilayer regimes. Increasing the well thickness at fixed electron density enhances correlation effects, thereby promoting crystal states and opening a promising route toward their experimental realization.

Key observations from the results include:

**= In the two-dimensional limit d → 0, electrons are tightly confined to the z = 0 plane and the resulting 2D Hamiltonian (1) displays only two energy scales, Ek ∼ r−2 2D and Ec ∼ r−1/2D, with r2D = (π n2D) −1/2 and n2D the areal density. As is well known, such 2DEG is then characterized by a single dimensionless parameter rs ≡ r−1/aB; the ground state is a Fermi liquid at high density and a Wigner crystal at low density, with the transition occurring at rs ∼ 30. The well thickness introduces a third energy scale Ew ∼ d −2, corresponding to the energy gap separating the different subbands associated with the quantized motion along z. In narrow wells Ew is the largest energy scale and only the lowest subband is occupied, as indicated by the presence of a single Fermi surface in the metallic state (Fig. 1 (c)). As a result, the charge density profile is monolayer, as shown in Fig. 1 (a) for our NN-VMC simulations. In wider quantum wells, when Ew becomes comparable to the kinetic and Coulomb energies, higher subbands become populated, which allows electrons to spread out along z and stay away from each other. Indeed, the charge density distribution develops into the bilayer profile displayed in Fig. 1 (b) and, at sufficiently large densities, two distinct Fermi surfaces emerge using the one-body reduced density matrix (1RDM) [37] obtained from our NN-VMC simulations. The presence of two Fermi surfaces of different sizes leads to two different frequencies in quantum oscillation as observed experimentally [40]. We note that, while the emergence of a bilayer charge distribution in a single quantum well can be qualitatively described by density functional theory or mean field approaches [6, 10, 41], these methods fail to capture strong correlation effects at small densities, as we will show below. The bilayer spacing ∆ grows monotonically upon increasing the electron density or the well thickness. Such bilayer charge distribution is induced by electron-electron interaction in a wide quantum well; it does not occur in the noninteracting limit even when higher subbands are occupied. Notably, increasing quantum well thickness has multifaceted effects on the competing ground states of interacting electrons. Compared with the strictly two-dimensional limit (d = 0), a small well thickness leads to the smearing of the electron wave-packet in the outof-plane direction, which effectively softens the Coulomb energy to Ec ∼ (r squared 2D + d 2) −1/2/aB [42]. The critical value of r2D/aB for Wigner crystallization therefore increases with thickness ∝ (1 + d 2/r2D) −1/2. On the other hand, in the bilayer regime where the total charge splits equally on two sides of the well, the inplane interparticle distance within each layer is reduced by ∼ 1/√2, leading to enhanced ratio of interaction to kinetic energy. This allows for a bilayer crystal to form already at electron areal densities well above the critical density for Wigner crystallization in strictly two dimensions. The monolayer electron crystal is realized at low densities rs ≳ 30, while the bilayer crystal is present already at larger densities rs ≳ 20 for well thicknesses d/aB ≳ 60. Remarkably, the two crystalline phases are separated by an intervening Fermi Liquid phase (blue dots) that follows the monolayer-bilayer transition line in Fig. 2 (a). Thus, in a sufficiently thick well, the system undergoes a sequence of phase transitions upon reducing the density: bilayer Fermi liquid → bilayer crystal → monolayer Fermi liquid → monolayer Wigner crystal. In particular, the bilayer crystal that intervenes between two metallic phases should be manifested as a resistance peak at intermediate densities. The lattice structures found are a triangular lattice in the monolayer regime, corresponding to the 2D Wigner crystal at d = 0.

1. Neural network variational Monte Carlo (NN-VMC) can be used to quantitatively determine many-body ground states of interacting electrons in a semiconductor quantum well from first principles by solving the Hamiltonian (1). This allows AI systems to predict and map out phase diagrams parameterized by electron density and well thickness, revealing metallic phases, electron crystals in both monolayer and bilayer regimes, and the novel electronic quasicrystal.

2. The attention-based architecture used in the neural network can be adapted for other strongly correlated systems like chiral superconductivity (29) or fractional quantum Hall states (27). This enables AI systems to discover new quantum phases by utilizing machine learning methods that solve the many-body problem in continuous space with a large number of variational parameters.

3. The NN-VMC method can be applied to model experimental observables, such as transport measurements and optical detection of exciton umklapp scattering, which should manifest as resistivity peaks in insulating crystalline phases (Fig. 2). This allows the AI system to bridge the gap between theoretical predictions and measurable physical properties in semiconductor quantum wells and van der Waals heterostructures.

4. The classical analysis comparing honeycomb, square, and quasicrystal energies can guide the design of more efficient neural network architectures for finding ground states. This provides a benchmark for training deep learning models aimed at accurately capturing quantum fluctuation effects that stabilize the electronic quasicrystal.

The homogeneous electron gas is a cornerstone of quantum condensed matter physics, providing the foundation for developing density functional theory and understanding electronic phases in semiconductors. However, theoretical understanding of strongly-correlated electrons in realistic semiconductor systems remains limited. In this work, we develop a neural network based variational approach to study quantum wells in three dimensional geometry for a variety of electron densities and well thicknesses. Starting from first principles, our unbiased AI-powered method reveals metallic and crystalline phases with both monolayer and bilayer charge distributions. In the emergent bilayer, we discover a new quantum phase of matter: the electronic quasicrystal.