Native defects and erbium impurities in CaWO4

This study employs hybrid density functional calculations to characterize the energetics, optical properties, and migration barriers of native defects and erbium impurities in CaWO$_4$, revealing that oxygen-related defects drive optical transitions while erbium's stability and emission quality depend on its charge state, complex formation, and the removal of interstitials through annealing.

The Gist

Imagine Calcium Tungstate (CaWO₄) as a high-end, ultra-stable hotel designed to host very special guests: Erbium ions. These guests are like tiny, glowing lightbulbs that can hold onto quantum information (like a secret code) for a long time. This makes the hotel a promising candidate for building the future "quantum internet."

However, even in a perfect hotel, things can go wrong. Sometimes the building materials are missing (defects), or the wrong guests show up (impurities). This paper is like a detailed architectural inspection report that uses powerful computer simulations to figure out exactly what's happening inside the walls of this hotel.

Here is what the researchers found, broken down into simple concepts:

1. The "Missing Bricks" and "Extra Bricks" (Native Defects)

In a perfect crystal, every atom sits in its exact spot. But in reality, atoms sometimes go missing (vacancies) or squeeze into spots where they don't belong (interstitials).

• The Oxygen and Calcium "Missing Persons": The study found that the most common problems are missing Oxygen atoms and missing Calcium atoms. It's like having holes in the floor or missing pillars.
  • The Oxygen Hole: When an oxygen atom is missing, the surrounding atoms shift around. If this hole has a positive charge, it acts like a tiny magnet that can spin, creating "noise" that disturbs the quantum guests.
  • The Calcium Hole: When a calcium atom is missing, it leaves a negative charge behind.
  • The "Handshake": Interestingly, the positive oxygen hole and the negative calcium hole are like magnets; they are very likely to find each other and stick together, forming a pair (a complex). This pairing changes how the material behaves.
• The "Tungsten" Mystery: The researchers checked if Tungsten atoms (the heavy metal in the crystal) were missing or extra. They found that Tungsten-related problems are extremely unlikely to happen. The Tungsten atoms are very happy staying put.
• The "Wandering" Atoms: Some of these missing or extra atoms are like restless toddlers. Specifically, extra Calcium, missing Oxygen, and extra Oxygen can move around very easily, even at room temperature. They are so mobile that they might wander out of the crystal entirely or bump into other defects.

2. The "Glow" of the Crystal (Optical Properties)

When you shine light on this crystal, it absorbs some colors and glows in others. Scientists have seen these glows in experiments but didn't know exactly which "flaw" was causing them.

• The Culprit: The computer simulations suggest that most of the strange glows and light absorption seen in experiments are caused by the Oxygen-related defects (the missing or extra oxygen atoms).
• The Explanation: It's like looking at a stained-glass window. The paper argues that the specific colors you see aren't coming from the glass itself, but from the tiny cracks and scratches (the oxygen defects) in the glass.

3. The Special Guest: Erbium (Er)

The main reason people study this crystal is to host Erbium atoms, which are the "quantum lightbulbs."

• The Perfect Seat: Erbium loves to sit in the Calcium seat. It fits in perfectly and stays in a positive charge state. This is the ideal spot because it's stable and doesn't get distracted by electrical noise from other parts of the building.
• The Wrong Seats: Erbium rarely tries to sit in the Tungsten seat or squeeze in as an "interstitial" (squeezing between the walls). If it does, it's unstable.
• The "Buddy System" Problem: Even if Erbium sits in the right seat, it can get "deactivated" if it forms a complex with a missing Calcium atom or an extra Oxygen atom. It's like the Erbium guest getting stuck in a hug with a neighbor, preventing it from doing its job.

4. The "Fix-It" Process (Annealing)

One of the most practical findings in the paper explains why heating the crystal (a process called annealing) makes the Erbium light stable.

• The Problem with Implantation: When scientists force Erbium into the crystal (using a process called implantation), many of them end up in the wrong spots (interstitials) or get stuck in "hugs" with defects. This causes the light to flicker (blinking) and change color randomly (spectral diffusion).
• The Heat Solution: The paper explains that these misplaced Erbium atoms are like people stuck in a crowded hallway. When you heat the crystal to a modest temperature (around 300°C or 573 K), it gives the atoms enough energy to move.
  • The misplaced Erbium atoms "kick" their way into the correct Calcium seats.
  • The wandering defects (the extra atoms or holes) move away.
• The Result: Once the Erbium is in the right seat and the neighbors have moved away, the light becomes stable and steady. However, if you heat it too much (around 800°C), the Erbium starts moving too much and leaves its seat, causing the light to disappear.

Summary

Think of this paper as a guide for building a perfect quantum hotel. It tells us:

1. Don't worry about Tungsten; it's stable.
2. Watch out for Oxygen and Calcium missing or extra; they move around and cause noise.
3. Erbium wants to sit in the Calcium seat, but it needs to be alone (not stuck in a complex with a defect).
4. Heat is the key: A moderate amount of heat helps the Erbium find its perfect seat and clears out the wandering defects, resulting in a stable, glowing quantum signal.

Technical Summary

Technical Summary: Native Defects and Erbium Impurities in CaWO$_4$

Problem Statement
Calcium tungstate (CaWO$_4$), crystallizing in the scheelite structure, is a promising host material for rare-earth ions, particularly erbium (Er), due to its potential for quantum communication and information technologies. Specifically, Er-doped CaWO$_4$ exhibits long coherence times for spin qubits and emits photons in the telecom band (1.5 $\mu$m). However, the performance of these systems is limited by decoherence and noise arising from native defects and impurities. While previous studies have utilized density functional theory (DFT) with generalized gradient approximation (GGA) or GGA+U to investigate defects in CaWO$_4$, a detailed understanding of the microscopic structure, energetics, and charge-state dependence of these defects remains incomplete. Furthermore, the specific mechanisms governing the stability and optical properties of Er dopants, including their interaction with native defects and the effects of annealing, require a more rigorous theoretical description.

Methodology
The authors performed first-principles calculations using the Heyd-Scuseria-Ernzerhof (HSE) hybrid density functional, implemented within the VASP code using the projector augmented-wave method. The HSE functional was chosen for its demonstrated accuracy in predicting band structures and defect formation energies in wide-band-gap materials.

• Computational Setup: Bulk calculations utilized a 24-atom conventional unit cell, while defect calculations employed 96-atom supercells (2 $\times$ 2 $\times$ 1 multiples). Spin polarization was included, and atomic positions were relaxed until forces were below 0.01–0.02 eV/Å.
• Defect Energetics: Formation energies were calculated as a function of the Fermi level and atomic chemical potentials ($\mu_{Ca}$, $\mu_{W}$, $\mu_{O}$). The chemical potential space was constrained by the stability conditions of CaWO$_4$, CaO, and WO$_3$, with specific points selected to represent O-rich, W-rich, and Ca-rich growth environments (e.g., conditions relevant to pulsed laser deposition and molecular beam epitaxy).
• Properties Analyzed: The study determined defect formation energies, charge-state transition levels, optical transition energies (using the Frank-Condon principle), and migration barriers (calculated via the climbing-image nudged elastic band method).
• Erbium Investigation: The properties of substitutional Er (Er$_{Ca}$), interstitial Er (Er$_i$), and their complexes with native defects (VCa, Oi) were examined.

Key Results

1. Native Defect Energetics and Prevalence:

   • Oxygen and Calcium Vacancies: Under intrinsic conditions, oxygen vacancies (V$_O$) and calcium vacancies (V$_{Ca}$) possess the lowest formation energies. V$_O$ acts as a deep donor with transition levels at 2.71 eV and 1.99 eV below the conduction band minimum (CBM). V$_{Ca}$ acts as a deep acceptor with a (0/2–) transition level at 0.46 eV above the valence band maximum (VBM).
   • Tungsten Defects: Tungsten-related defects (V$_W$, W$_i$, and antisites) are unlikely to form due to high formation energies, even under W-rich conditions.
   • Interstitials: Oxygen interstitials (O$_i$) and calcium interstitials (Ca$_i$) have modest formation energies, particularly under O-rich and Ca-rich conditions, respectively.
   • Complexes: The positively charged V$_O$ and negatively charged V$_{Ca}$ are likely to form stable complexes (V$_{Ca}$–V$_O$) with binding energies of approximately 1.25–1.38 eV.

2. Migration Barriers and Mobility:

   • Calculations reveal that Ca$_i^{2+}$, V$_O^{2+}$, and O$_i^{2-}$ are highly mobile, with migration barriers of 0.51 eV, 0.43 eV, and 0.80 eV, respectively. These low barriers suggest these defects can diffuse even below room temperature.
   • In contrast, V$_{Ca}^{2-}$ has a high migration barrier (1.89 eV), rendering it immobile at room temperature but mobile during annealing.

3. Optical Transitions:

   • The calculated optical transition levels for oxygen-related defects (V$_O$ and O$_i$) align with experimentally observed absorption peaks at 340 nm (3.65 eV) and 520 nm (2.38 eV), as well as emission peaks in the visible range.
   • Specifically, transitions involving V$_O$ and the V$_{Ca}$–V$_O$ complex are suggested to explain absorption features, while Ca interstitials and oxygen vacancies are implicated in higher-energy emission peaks.

4. Erbium Impurities:

   • Substitutional Er: Er easily substitutes on the Ca site (Er$_{Ca}$) in a positive charge state (+1 or +3 depending on the reference), consistent with Er preferring the 3+ oxidation state. The local atomic structure retains high symmetry (S$_4$), which is beneficial for reducing spectral diffusion.
   • Interstitial Er: If introduced via implantation, Er interstitials (Er$_i$) are likely to form. These defects exhibit lower symmetry and can exist in multiple charge states, leading to spectral diffusion and blinking.
   • Complexes and Deactivation: Er$_{Ca}$ can form complexes with V$_{Ca}$ and O$_i$. These complexes are thermodynamically stable in negative charge states and would deactivate the Er emission.
   • Diffusion Mechanisms: Er$_i$ can exchange places with a host Ca atom via an interstitialcy mechanism with a barrier of 0.94 eV, allowing it to move into a substitutional site at modest temperatures.

Significance and Claims
The paper claims to provide a comprehensive theoretical framework for understanding the defect physics of CaWO$_4$, addressing gaps left by previous GGA-based studies. The primary contributions are:

• Defect Identification: Establishing that oxygen and calcium vacancies are the dominant native defects, while tungsten-related defects are negligible.
• Optical Assignment: Attributing experimentally observed absorption and emission peaks primarily to oxygen-related defects and their complexes, rather than intrinsic lattice properties alone.
• Mechanism for Stability: Explaining the role of annealing in Er-doped CaWO$_4$. The authors propose that annealing at modest temperatures (e.g., ~300°C or 573 K) facilitates the migration of Er interstitials into substitutional sites and allows mobile native defects to diffuse away or recombine. This process eliminates the unstable interstitial configurations and defect complexes responsible for blinking and spectral diffusion, resulting in stable, high-quality Er emission suitable for quantum applications.
• Mobility Insights: Highlighting the high mobility of specific native defects (Ca$_i$, V$_O$, O$_i$) even at low temperatures, which influences defect evolution during growth and processing.

The study concludes that controlling the chemical potential during growth and utilizing appropriate annealing protocols are critical for minimizing detrimental defect complexes and maximizing the performance of CaWO$_4$ as a host for quantum emitters.