Wannier based analysis of the direct-indirect bandgap transition by stacking MoS$_2$ layers

This study employs a combined first-principles and Wannier-based model to reveal that the direct-to-indirect bandgap transition in layered MoS$_2$ is driven not only by interlayer $p_z$-$p_z$ coupling but also critically by previously overlooked $p_z$-$p_x$ and $p_z$-$p_y$ orbital interactions between neighboring sulfur atoms.

The Gist

Imagine Molybdenum Disulfide (MoS₂) as a stack of ultra-thin, magical pancakes. Each "pancake" is a single layer of atoms. Scientists have been fascinated by these pancakes because they behave very differently depending on how many you stack on top of each other.

Here is the simple story of what this paper discovered, using some everyday analogies.

1. The Magic Trick: The "Light Switch" Effect

The most famous trick these pancakes can do is change their personality based on their height.

• One pancake (Monolayer): It acts like a direct light switch. When you give it energy, it instantly turns it into light (like a super-efficient LED). This is called a "direct bandgap."
• A tall stack (Multilayer/Bulk): As you stack more pancakes, the magic changes. It becomes a dimmer switch that needs to wiggle around before it lets light out. It becomes less efficient at making light. This is called an "indirect bandgap."

Scientists have known that this happens for a long time, but they didn't fully understand why the stack changes the rules. This paper is like a detective story figuring out exactly which atoms are pulling the strings.

2. The Old Theory: The "Vertical Handshake"

Previously, scientists thought the change happened because of a specific "handshake" between the pancakes.
Imagine the atoms in the pancakes are people holding hands. The sulfur atoms (S) have a special arm sticking straight up and down (called the pz orbital).

• The Old Idea: When you stack the pancakes, these "up-and-down arms" reach out and grab the arms of the atoms in the layer above or below. Scientists thought this "vertical handshake" was the only thing that mattered. It was like saying, "The stack changes because everyone is shaking hands vertically."

3. The New Discovery: The "Side-Step" Dance

The authors of this paper used a super-powerful computer model (like a high-tech crystal ball) to look closer. They found that the old theory was only half-right.

While the vertical handshake is important, it's not enough to explain the whole magic trick. They discovered that the atoms also need to do a side-step dance.

• The New Insight: The sulfur atoms don't just reach up and down; they also reach out to the side (using px and py orbitals).
• The Analogy: Imagine a group of dancers in a line.
  • The Old Theory said the dance changed because everyone held hands with the person directly in front of them.
  • This Paper says, "No, the dance changed because they also had to reach out and tap the shoulder of the person next to them!"

If you only let them hold hands vertically, the dance (the electronic structure) doesn't look right. You must include those side-taps (the in-plane couplings) to get the choreography perfect.

4. Why Does This Matter?

Think of the "bandgap" as the price of admission for electrons to do their job.

• In the single layer, the price is low and easy to pay at the front door (Direct Gap).
• In the stack, the price changes, and the electrons have to take a detour through the back alley (Indirect Gap).

By understanding that both the vertical reach and the side-reach of the atoms are responsible for this price change, engineers can now design better devices.

• For Future Tech: If we want to build super-fast computer chips or super-bright screens using these materials, we need to know exactly how to control these "handshakes" and "side-steps." This paper gives us the blueprint to engineer the material so we can keep it acting like a "direct light switch" even when we stack it up, or switch it off when we want to save energy.

The Bottom Line

This paper solved a mystery about why stacking MoS₂ layers changes how they handle electricity and light. They found that it's not just about atoms reaching up and down between layers; it's also about them reaching side-to-side. It's a reminder that in the microscopic world, you can't just look at one direction; you have to see the whole dance to understand the music.

Technical Summary

1. Problem Statement

Molybdenum disulfide (MoS$_2$) is a prominent two-dimensional van der Waals material. A defining characteristic of MoS$_2$ is its layer-dependent electronic structure:

• Monolayer: Exhibits a direct bandgap (transition occurs at the $K$ point).
• Multilayer/Bulk: Exhibits an indirect bandgap (transition occurs between the valence band maximum at $\Gamma$ and the conduction band minimum at $Q$).

While this phenomenon is well-established, the microscopic mechanism driving the transition, particularly regarding the specific atomic orbital contributions and interlayer interactions, requires deeper quantitative clarification. Existing Tight-Binding (TB) models often fail to accurately reproduce the conduction band structure near the $Q$ point in multilayer systems, limiting their utility for device engineering and bandgap engineering strategies.

2. Methodology

The authors employed a hybrid approach combining Density Functional Theory (DFT) and Wannier-based Tight-Binding (TB) modeling:

• First-Principles Calculations (DFT):
  • Performed using Quantum ESPRESSO with the PBEsol functional (GGA).
  • Calculated electronic structures for bulk, monolayer, bilayer, and 6-layer MoS$_2$.
  • Used a plane-wave cutoff of 150 Ry and appropriate $k$-point meshes (e.g., $8\times8\times8$ for bulk SCF).
• Wannier-Based TB Model Construction:
  • Utilized Wannier90 to construct a bulk TB model projecting onto Mo-4d and S-3p orbitals, which dominate states near the Fermi level.
  • Multilayer Hamiltonian: Constructed a 2D $k$-space Hamiltonian for $2N$-layer systems by applying open boundary conditions along the $c$-axis and periodic conditions along $a$ and $b$ axes. This allowed the simulation of arbitrary layer numbers by stacking the bulk TB unit cells.
• Orbital Decomposition & Charge Analysis:
  • Analyzed orbital weights at high-symmetry points ($\Gamma$, $K$, $Q$).
  • Calculated matrix elements of interlayer hopping integrals ($t_{ij}$) to quantify coupling strengths.
  • Computed $k$-resolved charge density ($\rho_{nk}$) to visualize the spatial distribution of electrons and interlayer coupling strength.

3. Key Contributions

The paper makes three primary technical contributions:

1. Quantitative Identification of Orbital Roles: It moves beyond the qualitative understanding that "interlayer coupling causes the transition" to a specific orbital-level breakdown.
2. Discovery of In-Plane Coupling Necessity: The authors demonstrate that while the out-of-plane $p_z$-$p_z$ coupling is dominant, a complete quantitative description of the indirect bandgap (specifically the conduction band minimum at $Q$) requires the inclusion of interlayer $p_z$-$p_x$ and $p_z$-$p_y$ couplings.
3. Validation of a Robust TB Model: They successfully constructed a TB model that accurately reproduces the band structures of multilayer MoS$_2$ (including the $Q$ point), overcoming limitations of previous models that relied solely on nearest-neighbor $p_z$ interactions.

4. Results

A. Band Structure Evolution

• Monolayer: Direct gap at $K$ ($E_g \approx 1.72$ eV).
• Bulk: Indirect gap between $\Gamma$ (Valence Max) and $Q$ (Conduction Min) ($E_g \approx 0.958$ eV).
• Transition Mechanism: As layer number increases, the valence band top shifts from $K$ to $\Gamma$, and the conduction band bottom shifts from $K$ to $Q$. This shift is driven by band splitting caused by interlayer interactions.

B. Orbital Contributions & Splitting

• Valence Band ($\Gamma_v$ vs. $K_v$):
  • At $\Gamma_v$ (large splitting), the states are dominated by Mo-$d_{z^2}$ and S-$p_z$ orbitals.
  • At $K_v$ (small splitting), states are dominated by Mo-$d_{xy}/d_{x^2-y^2}$ and S-$p_x/p_y$.
  • Conclusion: The large splitting at $\Gamma$ is driven by strong out-of-plane ($p_z$-$p_z$) interlayer coupling between adjacent Sulfur atoms.
• Conduction Band ($K_c$ vs. $Q_c$):
  • At $K_c$, the band is dominated by Mo-$d_{z^2}$ and S-$p_{x,y}$, with weak interlayer coupling.
  • At $Q_c$, the band exhibits large splitting. While S-$p_z$-$p_z$ coupling is significant, the authors found that S-$p_z$ to S-$p_x/p_y$ couplings are non-negligible.

C. The Critical Role of Mixed Orbital Coupling

The authors tested simplified TB models:

• Model 1 (Only $p_z$-$p_z$): Failed to reproduce the conduction band minimum at the $Q$ point; the minimum remained at $K$.
• Model 2 (Including $p_z$-$p_x$ and $p_z$-$p_y$): Successfully shifted the conduction band minimum from $K$ to $Q$, accurately reproducing the indirect gap.
• Matrix Elements: Table II in the paper shows that at the $Q$ point, the coupling between S2-$p_z$ and S3-$p_y$ (0.250 eV) and S2-$p_z$ and S3-$p_x$ (0.159 eV) are significant, comparable to or larger than other interlayer terms.

D. Visual Confirmation via Charge Density

• $k$-resolved charge density analysis confirmed that at $\Gamma_v$ and $Q_c$ (where splitting is large), electron density spreads significantly between layers, indicating strong interlayer coupling.
• At $K_v$ and $K_c$, electrons are localized within layers, confirming weak interlayer coupling.

5. Significance

• Fundamental Physics: The study clarifies that the direct-to-indirect transition in MoS$_2$ is not solely a result of vertical ($p_z$) stacking but is critically dependent on hybridization between out-of-plane and in-plane orbitals ($p_z$-$p_x/p_y$) across layers.
• Modeling Accuracy: The findings provide a blueprint for constructing accurate Tight-Binding models for MoS$_2$ and other Transition Metal Dichalcogenides (TMDs). Previous models omitting these cross-orbital couplings are insufficient for predicting conduction band properties in multilayer systems.
• Device Engineering: Understanding the specific orbital interactions governing the bandgap allows for more precise bandgap engineering. This is crucial for designing next-generation field-effect transistors (FETs), optoelectronic devices, and flexible electronics where controlling the direct/indirect nature of the gap is essential for efficiency and performance.