You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We theoretically investigate the electronic and optical properties of semiconductor Cu$_{3}$PSe$_{4}$. We also report diffuse reflectance spectroscopy measurements for Cu$_{3}$PSe$_{4}$ and Cu$_3$PS$_4$, which indicate a band gap of 1.40 eV for the former. Hybrid functional calculations agree well with this value, and reveal that the band gap is direct. Calculations yield an optical absorption spectrum similar to GaAs in the visible region, with $\alpha > 5 \times 10^4$ cm$^{-1}$ for $\lambda < 630$ nm. We conclude that the optical properties of Cu$_{3}$PSe$_{4}$ are within the desired range for a photovoltaic solar absorber material.'
author:
- 'D. H. Foster'
- 'V. Jieratum'
- 'R. Kykyneshi'
- 'D. A. Keszler'
- 'G. Schneider'
title: 'Electronic and optical properties of potential solar absorber Cu$_{3}$PSe$_{4}$'
---

The need for low cost solar photovoltaic cells continues to drive a global effort to find, evaluate, and refine materials that offer both cost effective and highly efficient solutions. Within the search among the inorganic semiconductors, one of the most exciting families has been the ternary and quaternary copper chalcogenides consisting of CuCh$_4$ and ACh$_4$ tetrahedra, where Ch is one of (S, Se, Te) and A represents one or more other elements. Examples include the commercially widespread Cu(In,Ga)Se$_2$ (CIGS), as well as Cu$_2$ZnSnS$_4$, which has the benefit of consisting of low-toxicity, readily available elements. One material in this family which has not received much attention in this regard is Cu$_{3}$PSe$_{4}$. The structure of Cu$_{3}$PSe$_{4}$ has been determined[@garin_crystal_1972; @ma_synthesis_2002], and a photoelectrochemical analysis[@marzik_photoelectronic_1983] of isostructural compounds Cu$_3$PS$_4$ and Cu$_3$PS$_3$Se has indicated indirect bandgaps of 2.38 eV and 2.06 eV respectively. However the bandgap of Cu$_{3}$PSe$_{4}$ itself has not been reported previously. In this Letter we examine this material, theoretically in the scope of low temperature and zero defects, and experimentally in the scope of room temperature measurements on powder samples.

In our synthesis of Cu$_{3}$PSe$_{4}$ and Cu$_3$PS$_4$, elemental powders of Cu (Cerac, 99.5%), P (Alpha Aesar 99%), Se (Cerac, 99.6%), and S (Cerac, 99.999%), are ground and sealed in evacuated fused silica tubes. Polycrystalline powders are obtained via solid state reaction at 480 $^\circ$C (Cu$_{3}$PSe$_{4}$) and 600 $^\circ$C (Cu$_3$PS$_4$) for 24 h. The XRD patterns, collected on a Rigaku Ultima IV, of the reacted samples are similar to the ISCD card $\#$095412 for Cu$_{3}$PSe$_{4}$ and $\#$412240 for Cu$_3$PS$_4$. Hole majority carrier type is confirmed by positive thermoelectric voltage measurements. The optical bandgaps are determined from diffuse reflectance measurements (see Figure \[fig\_exp\_and\_struct\]) collected with a spectrometer equipped with an Ocean Optics HR4000 UV-VIS detector and a balanced deuterium/tungsten halogen source (DH-2000-BAL). The Kubelka-Munk model is employed to determine the optical absorption edges of 1.40 eV and 2.38 eV for Cu$_{3}$PSe$_{4}$ and Cu$_3$PS$_4$, respectively. The Cu$_{3}$PSe$_{4}$ bandgap is within the desired range for solar photovoltaic devices (1.0 eV to 1.6 eV), while the Cu$_3$PS$_4$ bandgap agrees precisely with Ref. \[\].

Calculations based on density functional theory (DFT) are performed using the projector augmented wave (PAW) method[@bloechl_projector_1994; @*kresse_ultrasoft_1999] as implemented in the plane wave code `VASP`[@kresse_efficient_1996]. For the exchange-correlation functional we use the generalized gradient approximation (GGA) in the PW91 parametrization[@perdew_atoms_1992] for accurate total energy calculations. To avoid the bandgap underestimation common to standard DFT calculations[@stampfl99den], we use the Heyd-Scuseria-Ernzerhof[@heyd_hybrid_2003; @*heyd_erratum:_2006] (HSE) hybrid functional, which combines Hartree-Fock (HF) exchange with GGA exchange and includes an empirical shielding of the HF exchange. The HSE functional has yielded respectable estimates for semiconductor bandgaps[@peralta_spin-orbit_2006; @*kim_accurate_2009; @*chan_efficient_2010], although it is generally not as reliable as methods based on the $GW$ approximation.[@shishkin2007sel] We will see below that HSE performs particularly well for calculations on Cu$_{3}$PSe$_{4}$. For brevity and the expressed application interest, we present theoretical analysis for Cu$_{3}$PSe$_{4}$ but not Cu$_3$PS$_4$. Both materials crystallize in the wurtzite-based enargite structure with a simple orthorhombic unit cell with 16 atoms admitting the space group $Pmn2_1$ (FIG. \[fig\_exp\_and\_struct\]).

![\[fig\_exp\_and\_struct\](Color online) Measured diffuse reflectance of Cu$_{3}$PSe$_{4}$. Inset: The wurtzite-based enargite structure. For Cu$_{3}$PSe$_{4}$, $a=7.685$ Å, $b=6.656$ Å, $c=6.377$ Å.](dataandcell.eps)

For electronic and optical calculations, we use experimentally determined lattice and atomic parameters.[@ma_synthesis_2002] Ionic and lattice relaxations are discussed below. In all calculations, we carefully consider the convergence of the result with respect to plane wave basis cut-off energy, $k$-point density, and time saving approximations.

The HSE determined band structure is shown in FIG. \[fig\_bands\].

![\[fig\_bands\]Band structure of Cu$_{3}$PSe$_{4}$ determined by HSE calculation. The direct bandgap of 1.38 eV lies at $\Gamma$.](bandplot.eps)

The HSE calculations used a $\Gamma$-centered $8 \times 8 \times 8$ $k$-point grid with $2\times$ down-sampling for one of the $k$-point summations in the evaluation of the Hartree-Fock exchange potential.[@paier_screened_2006] The band structure plot itself (FIG. \[fig\_bands\]) is formed by interpolation of the $8^3$ $k$-point grid using a Fourier method.[@pickett88smo] The band structure most importantly shows a direct bandgap of 1.38 eV at the $\Gamma$ point, in excellent agreement with the experimental value of $1.40$ eV. The band structure also shows low lying valleys ($\sim0.1$ eV) in both the valence band (at Y) and the conduction band (along $\Gamma$-Z). We note that an HSE calculation for Cu$_3$PS$_4$ at experimental parameters[@pfitzner_refinement_2002] gives these two valleys as being the valence band maximum (VBM) and conduction band minimum (CBM), with theoretical and experimental[@marzik_photoelectronic_1983] indirect bandgaps of 2.55 eV and 2.38 eV respectively. We calculate electron and hole effective masses of 0.17 and 0.60 times the electron mass, respectively. Spin-orbit coupling was verified to be small in a single GGA calculation which showed VBM splitting and a bandgap decrease of 0.030 eV relative to the spinless GGA bandgap of 0.29 eV.

An analysis of the partial density of states indicates that the valence bands from the VBM to about $-5.5$ eV below (see FIG. \[fig\_bands\]) have similar orbital composition to the corresponding bands of Cu$_2$ZnSnS$_4$.[@paier09cu2] In particular, the valence band edge has anti-bonding $t_{2}$ Cu-d/Se-p$^*$ hybrid character, with the corresponding bonding states lying in the second valence band. Between these regions, at the bottom of the first valence band, lies the very dense bundle of non-bonding $e$ Cu-d orbitals. The two states in the lowest conduction band show hybridization among Se-p, P-s, and Cu-d orbitals. The charge density distributions of these states indicate significant anti-bonding character between P and Se ions. This will have important effects on relaxation calculations, as discussed below.

To obtain the optical properties of Cu$_{3}$PSe$_{4}$, the complex dielectric tensor $\epsilon$ has been calculated[@gajdos_linear_2006] in the random phase approximation from HSE wavefunctions without including local effects, which are generally small.[@footnote1] The direction-averaged dielectric function $\text{Tr}(\epsilon) / 3$ and optical absorption coefficient $\alpha$ are shown in Figure \[fig\_abseps\]. Optical absorption is compared with experimental data[@palik_handbook_1991; @paulson_optical_2003] for GaAs and polycrystalline CuIn$_{1-x}$Ga$_{x}$Se$_2$ for $x=0.31$. (In photovoltaic applications, CIGS stoichiometries near $x = 0.3$ yield the most efficient power conversion.[@kemell05thi]) For $520 < \lambda < 660$ nm, the absorption in Cu$_{3}$PSe$_{4}$ is calculated to be higher than in GaAs. The greater absorption of CIGS is primarily due to its absorption curve being shifted to lower energies by its smaller bandgap (1.17 eV). While allowing absorption of a greater fraction of the solar spectrum, the CIGS bandgap limits its solar cell voltage—and its maximum theoretical efficiency—relative to larger bandgap materials. Absorption is above $5\times 10^4$ cm$^{-1}$ for wavelengths $\lambda < 630$ nm ($\hbar \omega > 2.0$ eV) and above $1 \times 10^5$ cm$^{-1}$ for $\lambda < 480$ nm ($\hbar \omega > 2.6$ eV).

![\[fig\_abseps\](Color online) Real (solid) and imaginary (long-dashed) parts of dielectric function $\epsilon$ averaged over principal axes. The absorption coefficient $\alpha$ is shown (scale on right) for Cu$_{3}$PSe$_{4}$ (red solid), GaAs (brown short-dashed), and CIGS (green dashed).](abseps.eps)

As discussed previously, the HSE bandgap using the experimental atomic and lattice parameters agrees well with the measured bandgap. However, using GGA or local density approximation (LDA) relaxed structural parameters, we calculate HSE bandgaps approximately 30% smaller (TAB. \[tab:relax\]). While the bandgap is expected to depend on the lattice parameters, such a large change is unusual, but can readily be understood as follows: An HSE ion relaxation leaves the experimental structure and bandgap nearly unchanged. The LDA and GGA relaxations, however, result in a lengthening of the P-Se bond, accompanied by large reductions of the HSE bandgap. The observed relation between the increase of the P-Se bond length and the underestimate of the calculated bandgaps is a direct consequence of the significant P-s/Se-p$^*$ anti-bonding character of the states at the conduction band edge: an underestimate of the bandgap (as in LDA/GGA) moves the P-s/Se-p$^*$ anti-bonding orbitals to lower energies, resulting in a smaller P-Se bond energy and a correspondingly longer bond. A similar bond length dependence of the bandgap of CuInSe$_2$ is discussed in Ref. \[\].

[llllll]{} relaxation & $E_{\text{g}}$ (HSE) & $E_{\text{g}}$ ($X$) & $V$ & $\langle$P-Se$\rangle$ & $\langle$Cu-Se$\rangle$\
& (eV) & (eV) & (Å$^3$) & (Å) & (Å)\
\
LDA ion & 0.94 & 0.08& 326.19 & 2.30 & 2.39\
LDA full & 1.10& 0.06 & 310.59 & 2.27 & 2.35\
GGA ion & 1.02 & 0.03 & 326.19 & 2.29 & 2.40\
GGA full & 0.90 & 0.08& 338.25 & 2.31 & 2.43\
HSE ion & 1.42 & 1.42 & 326.19 & 2.23 & 2.42\
exp.[@ma_synthesis_2002] & 1.38 & 1.40 & 326.19 & 2.24 & 2.41\

A further structural analysis examines possible alternatives to the enargite structure used in the preceding calculations. While only the enargite form of Cu$_{3}$PSe$_{4}$ has been observed experimentally, it is possible that competing structural phases could form under synthesis techniques appropriate for device construction. To examine structural stability, we have performed a set of GGA calculations (Table \[tab:hof\]) comparing the heat of formation $\Delta H$ for Cu$_{3}$PSe$_{4}$ placed in several structures which are manifested by other compounds having the form A$_3^{+}$B$^{5+}$C$_4^{2-}$. The $\Delta H$ values of the famatinite[@pfitzner_refinement_2002] and enargite structures differ negligibly. This is not surprising, as the essential difference of the structures lies only in the arrangement of the (Cu,P)Se$_4$ tetrahedra: zincblende for famatinite and wurtzite for enargite. Using the lattice and atomic parameters obtained from fully relaxed GGA calculations, we have calculated HSE bandgaps of 0.90 eV (enargite) and 0.84 eV (famatinite) for these potentially competing structures. Though artificially low due to the GGA relaxations, the similarity of these bandgaps indicates that the presence of a famatinite phase would not greatly alter the optical properties of Cu$_{3}$PSe$_{4}$.

[llD[.]{}[.]{}[-1]{}]{} structure & space group &\
enargite (Cu$_3$PS$_4$) & $Pmn2_1$ (no. 31) & -1.324\
famatinite (Cu$_3$SbS$_4$) & $I\bar{4}2m$ (no. 121) & -1.309\
lazarevicite (Cu$_3$AsS$_4$) & $P\bar{4}3m$ (no. 215) & -1.106\
Rb$_3$PS$_4$ & $Pnma$ (no. 62) & -0.500\
Na$_3$PS$_4$ & $P\bar{4}2_1c$ (no. 114) & 0.107

In conclusion, the measurements and calculations reported here indicate that Cu$_{3}$PSe$_{4}$ has optical properties which make it viable for photovoltaic applications. Further measurement and calculation of its properties, thermodynamic stability, and primary defects will be subsequently reported, and will further analyze the potential of Cu$_{3}$PSe$_{4}$ as a photovoltaic or photoelectronic material. This work has been supported by National Science Foundation grant SOLAR DMS-1035513.

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