Band structure of CuO Cu4O4 unit cell calculated with basic... LIST OF ABBREVIATIONS DFT : Density functional theory GGA : Generalized gradient approximation LDA : Local density approxim
INTRODUCTION
CuO thin films and its potential applications
Metal oxide semiconductors (MOS) have gained significant attention in materials research, particularly cupric oxide (CuO) CuO stands out for its unique structural, chemical, optical, and electrical properties, making it a valuable candidate for various applications.
CuO nanostructures have diverse applications across multiple fields, including gas sensors, bio-sensors, photodetectors, electrochromic devices, and supercapacitors They are also utilized in nanofluids, field emission devices, the removal of inorganic pollutants, photocatalysis, magnetic storage media, and antimicrobial applications.
Copper oxide (CuO) stands out among semiconductors due to its nontoxic nature, cost-effectiveness, and ease of synthesis Various methods have been developed for fabricating CuO thin films, including thermal evaporation, sputtering, electro-deposition, chemical vapor deposition, chemical bath deposition, spin coating, spray pyrolysis, solution processing, thermal oxidation, pulsed laser deposition, and reactive ion sputtering.
The solution processing method for thin film fabrication is favored across various fields due to its low cost, simplicity, and efficiency This technique does not require a vacuum environment and allows for easy molecular level doping While it may lack the precision needed for very small chips, it is sufficiently accurate for applications requiring micrometer-level precision, such as electronic displays and energy conversion systems like power inverters.
CuO is a native p-type semiconductor, which is significant given that most metal oxide semiconductors are n-type The advancement of p-type metal oxide semiconductors like CuO can enhance various photovoltaic technologies, including bifacial and multijunction solar cells, as well as serve as potential back contacts for cadmium telluride and organic solar cells.
Bui Nguyen Quoc Trinh et al (2020) developed a thin film transistor utilizing solution-processed CuO thin films as the channel layer, which were found to be polycrystalline with a monoclinic structure according to XRD analysis The study revealed that the grain size of the thin films increased with higher solution concentrations, achieving a minimum electrical resistivity of 0.0359 Ωcm at 0.30 M When employed as the channel layer in the thin film transistor, the CuO thin films demonstrated p-type operation, characterized by an on/off current ratio of 10² and a saturation mobility of approximately 10⁻⁴ cm² V⁻¹ s⁻¹.
Figure 1.1 Transfer characteristics of the CuO TFTs with various channel lengths The inset is a microscope image of the CuO TFT fabricated (Bui Nguyen Quoc Trinh et al
Copper oxide (CuO) can effectively convert heat into electricity through the thermoelectric Seebeck effect, although most materials exhibit a minimal or negligible effect The ideal thermoelectric materials possess a high Seebeck coefficient and electrical conductivity while maintaining low thermal conductivity However, materials with high electrical conductivity, like metals, also conduct heat efficiently, as electrons are responsible for both This challenge can be mitigated by using materials where phonons predominantly carry heat Increasing the band gap allows electrons to conduct less heat, and techniques such as nanostructuring can suppress phonon flow without significantly reducing electronic conductivity Additionally, appropriate doping can enhance electronic properties while further reducing thermal conductivity Notable thermoelectric materials include tellurides like Bi2Te3 and PbTe, which have been optimized through doping to achieve high zT values suitable for practical applications.
Due to the harmful effects of lead and the limited availability of tellurium, these materials are not viable for large-scale production Consequently, research has shifted towards alternative compound groups, including sulfides like CuS and oxides such as ZnO, Cu2O, CuO, and SnO.
CuO-based semiconductors are effective gas sensors for detecting a range of target molecules, including volatile organic compounds (VOCs), carbon monoxide (CO), carbon dioxide (CO2), hydrogen sulfide (H2S), nitrogen oxides (NOx), and hydrogen (H2) These semiconductors can be enhanced through various doping elements such as nickel (Ni), aluminum (Al), zinc (Zn), platinum (Pt), chromium (Cr), indium (In), and iron (Fe), and they come in diverse structures, including nanosheets, nanowires, thin films, nanocubes, nanoparticles, and hollow nanofibers.
Stephan Steinhauer (2021) has summarized the lists of copper oxide-based gas sensors for those gaseous molecules [1] Table 1.1 shows the example for the volatile organic compounds
Table 1.1 A summary of sensing volatile organic compounds (VOCs) with copper oxide-based devices [1].
DFT simulation approach to design and fabrication
1.2.1 The role of DFT simulation approach in design and fabrication
The diverse range of target molecules, doping elements, and structures in CuO-based semiconductor gas sensors presents a challenge in exploring all possible combinations efficiently Utilizing simulation approaches can significantly streamline this process by identifying promising combinations for practical applications For instance, research by Limei Fan et al (2022) demonstrates how simulations of structures with and without carbon monoxide (CO) allow for the calculation of various properties These properties can then be used to derive macroscopic measurements, enabling the detection of CO concentration through observed changes in these properties.
Figure 1.2 Configurations of Al-doped CuO nanoparticle interacting with CO [50]
Table 1.2 Calculated properties of different Configurations of Al-doped CuO nanoparticle with and without the CO molecule [50]
The DFT simulation method can uncover the thermoelectric properties of p-type semiconductors, which play a crucial role in converting heat into electricity through the thermoelectric Seebeck effect, as highlighted by Jarno Linnera et al.
In 2018, research utilizing hybrid density functional theory calculations revealed that Cu2O, CuO, and NiO exhibit comparable theoretical maximum thermoelectric (TE) performance under the constant relaxation approximation Notably, the power factor of NiO at 650 K was enhanced by three orders of magnitude with just a 2.4% addition of lithium, indicating potential for further improvement Additionally, transport coefficients for CuO were calculated based on carrier concentration, as illustrated in Figure 1.3.
Figure 1.3 Transport coefficients for CuO as a function of carrier concentration ρ
Top: Electrical conductivity calculated with respect to the electronic relaxation time Middle: Seebeck coefficient Bottom: Power factor S 2 σ with respect to the electronic relaxation time [49]
Density-functional theory (DFT) is a quantum mechanical computational modeling method widely used in physics, chemistry, and materials science It primarily focuses on investigating the ground state of simulated systems and is applicable to the electronic and nuclear structures of many-body systems, including atoms, molecules, and condensed phases.
Density Functional Theory (DFT) calculations enable the prediction and analysis of material behavior without the need for higher-order parameters, relying instead on fundamental material properties The electronic structure is assessed by evaluating a potential that acts on the system's electrons, which is formulated as the sum of the system's elemental composition and an effective potential (V_eff) that accounts for inter-electronic interactions Additionally, external potentials (V_ext), determined solely by the material's structure, are included in the analysis This approach allows for the examination of a supercell representing a material with n electrons as a set of n individual problems.
9 electron Schrửdinger-like equations Those equations are also known as Kohn–Sham equations
The DFT method is based on two Hohenberg–Kohn theorems:
Theorem I: The external potential V ext is uniquely determined (apart from an additive constant) by the electron density in the ground state of the system
In details, theorem I means: o V determines the electron density ρ(r) o The electron density ρ(r) determines V
If the density ρ(r) corresponds to two different potentials V 1 and V 2, corresponding Hamiltonians H1 and H2 the ground-state wavefunctions : Ψ1 and Ψ2, and the energy is described by the equation (1.1)
We have the relation between 𝐻 1 and 𝐻 2 described by equation (1.2)
Then we can consider the form of 𝐸 1 and 𝐸 2 to get the comparison in inequations (1.3) and (1.4)
From those inequations, we can deduce the inequation (1.5)
This is a contradiction Therefore, we must have 𝑉 1 = 𝑉 2
Theorem II states that the ground state energy, viewed as a functional of density, reaches its minimum when the density is correctly normalized This indicates that the energy is optimized when the density reflects its true value.
The Kohn–Sham equation is derived from key theorems and represents a non-interacting Schrödinger-like equation for a system of non-interacting particles, typically electrons This equation is designed to produce the same density distribution in space as a system of interacting particles, leading to N coupled equations.
2𝑚∇ 2 +𝑉 ion (𝐫) + 𝑉 H (𝐫) + 𝑉 xc (𝐫)] 𝜙 𝑖 (𝐫) = 𝜀 𝑖 𝜙 𝑖 (𝐫) (1.6) with i goes from 1 to N, and 𝑉 𝑒𝑓𝑓 (𝒓) is defined by equation (1.7)
And the total energy can be described as a functional of the charge density, as shown by equation (1.8)
The term 𝑉 xc , with 𝑉 xc (𝐫) ≡ 𝛿𝐸 𝑥𝑐 [𝜌]
The exchange-correlation potential, denoted as 𝛿𝜌(𝐫), represents a key unknown in the Kohn-Sham approach to density functional theory To address this term, approximations like the Local-Density Approximation (LDA) and the Generalized Gradient Approximation (GGA) are utilized These approximations, known as "functionals," can yield varying results in calculations.
Albeit not compulsory to DFT, there are also some other approximations to make the calculation feasible:
Pseudo-potential simplifies calculations by treating electrons in inner shells and the nucleus as a single source of potential, significantly reducing computational costs This approach allows for an approximation of the real potential in relevant regions, excluding areas too close to the nucleus—where outer shell electrons are unlikely to be found—and regions too far away, where potential approaches zero.
11 anyway) This lets us use pseudo-potentials with much simpler expressions compared to the real potential
Figure 1.4 Illustration of pseudo-potential
Periodic boundary conditions are essential for effectively applying the DFT method to systems with thousands of atoms, as directly calculating electron configurations for such large systems is impractical By implementing these conditions, we can analyze a smaller unit cell while allowing interactions between atoms and electrons at the boundaries with those on the opposite side This approach simulates an infinite space of repeating unit cells, enabling the study of larger patterns and mechanisms that may not be accurately represented in smaller unit cells.
The choice of considering or ignoring various other aspects, such as: spin, vacancy, deformation, etc
Accurate simulation of specific systems often requires tailored approximations that depend on the system's nature or mechanisms In some instances, researchers may explore various approximations until they identify one that aligns closely with experimental data While such results may not always have a solid theoretical basis, the chosen approximation could potentially be applicable to similar systems, leading to insights that may yet be uncovered through experimentation.
1.2.3 Obstacle in applying DFT simulation approach to design and fabrication CuO based semiconductors
The DFT method would give different results compared to experimental data of CuO based semiconductors because of the aspects:
DFT, or Density Functional Theory, is utilized to calculate the ground state and determine the electron band structure However, when electrons are excited, the band structure may change unpredictably This is particularly relevant for CuO-based semiconductors, where factors such as room temperature and electron transitions between energy levels must be considered.
According to John P Perdew (1986), the fundamental gap in the exact Kohn-Sham band structure of an insulator does not equal the true gap I-A due to a discontinuous change in the exchange-correlation potential as the conduction band fills Various numerical calculations indicate that the error in the gap from the local density approximation is likely to persist in the exact Kohn-Sham band structure.
The choice of functional for each system remains unclear due to the lack of concrete evidence The strong exchange-correlation potential of electrons in the 3d orbitals of copper (Cu) atoms further complicates this issue.
When optimizing configurations to find the ground state, we often encounter local minima instead of the desired global minimum This challenge arises because we can only assess the energy gradient relative to configuration changes, making it impossible to determine if the resulting configuration is indeed the global minimum This issue is particularly pronounced in materials with spin considerations, as numerous local minima can emerge from various high-energy spin configurations To address this problem, one approach is to initiate calculations from multiple starting configurations; however, this significantly increases computational costs.
Research motivation
Choosing functionals and pseudo-potentials that yield results close to experimental data often involves arbitrary selections Researchers typically try various combinations and select those that best match experimental outcomes However, this approach, which relies on parameters fitted to experimental results, lacks predictive power Without a thorough understanding of which approximations to use, there is no assurance that the same set will produce accurate results in similar cases.
Therefore, in this thesis, I have attempted to:
By intuitively selecting a set of approximations and considering factors like magnetic spin and vacancy, I can gain a clearer understanding of the key aspects of CuO that significantly influence its properties, particularly the band gap.
do the “method of using parameters fitted to experimental results” for reference
This study applies various approximation sets to Sn-doped CuO, comparing the calculated band gap values with experimental data It evaluates the applicability of these approximations to similar systems and their potential for yielding useful predictions The aim is to determine whether a more intuitive selection of approximation sets can outperform traditional, arbitrary functionals in predicting outcomes for similar materials.
fabricate the Sn-doped CuO and measure it experimentally for comparison
Experimental results by P.J.M Isherwood et al indicate that Sn-doped CuO behaves as a p-type semiconductor, exhibiting similar lattice parameters to CuO This similarity suggests that the existing models for CuO could potentially be applied to Sn-doped CuO, making it a promising candidate for further research.
METHODOLOGY
Experimental procedures
2.1.1 Steps of thin-film fabrication
The main steps of thin-film fabrication are:
Figure 2.1 Main steps of thin-film fabrication
2.1.2 Raw material for synthesis of precursors
The host semiconductor in this thesis was cupric oxide (CuO), while the transition metal-doped oxide semiconductors were Tin (Sn) Copper (II) acetate monohydrate and
Tin chloride dihydrate were combined in ethanol as starting materials and MEA as a stabilizer for the reaction process
The main chemical compounds used in this process are:
Figure 2.2 Main chemical compounds used to prepare the precursor solution
To prepare precursor solutions for doping, 0.004 mol of Cu(CH3COO)2.H2O and SnCl2.2H2O are combined in a cup containing 19.5 ml of ethanol The ratio of Cu(CH3COO)2.H2O to SnCl2.2H2O is adjusted across different samples to achieve varying doping percentages It is essential to add the ethanol first to prevent the powders from adhering to the bottom of the cup, which could result in inaccurate doping percentages.
This solution is stirred on a magnetic stirrer for 10 minutes
Then 0.5 ml of MEA (monoethanolamine) is added to the cup This leads to a total of 0.2M concentration of Cu and Sn cations, and 0.4M concentration of MEA It is expected that the complexes will be formed as shown in Figure 2.3 [55]
The precursor solution is stirred on a magnetic stirrer for 15 minutes and then stabilized at 75°C for 60 minutes, allowing Cu2+ ions to react with MEA and form a 4-species complex that links copper ions together As a result, the precursor develops a dark blue color, indicating it is ready for thin film deposition.
The substrate used in this thesis is glass
To ensure optimal cleanliness, the glass must undergo a thorough cleaning process Initially, it is immersed in acetone and subjected to ultrasonic stirring for 5 minutes This is followed by a similar treatment in ethanol for another 5 minutes Finally, the glass is rinsed in pure water with ultrasonic stirring for an additional 5 minutes, completing the cleaning procedure.
To prepare the glass for reuse on various substrates, it is first dried to prevent excess water from contaminating the dilute Hydrofluoric acid (HF) solution The glass is then immersed in the dilute HF solution for one minute, after which it is thoroughly rinsed with water and dried again.
2.1.5 Deposition of CuO and Sn-doped CuO thin films
Sn-CuO thin films were coated on the cleaned glass utilizing a spin-coating system
Step 1: The precursor solution is dripped onto a spinning substrate with a rotation speed of 1500 rpm for 50 seconds There are more steps before and after that duration with rotation speed of 1500 rpm to avoid sudden acceleration, as shown on the screen of the spin-coating system in Figure 2.5 A surplus amount of fluid is expelled on the substrate's surface
Step 2: The sample was heated to 100°C for 3 minutes by using a hot plate To achieve the desired thickness of the CuO thin film, the coating and drying process was done three times
Step 3: After the final coating, the films were annealed in the air at 550°C for 30 minutes to form a crystalline film
20 a) b) c) Figure 2.5 Spin-coating system (a), hot plate (b), annealing furnace (c)
The film characterization employs X-ray diffraction (XRD) to analyze crystal structures and unit cell sizes, while UV-Vis spectroscopy is utilized to assess optical transmittance and absorbance, ultimately determining the optical band gap.
The X-ray diffraction (XRD) method is a strong instrument for determining crystal structures and unit cell sizes in great detail X-rays are electromagnetic radiation having wavelengths 1Å (1Å = 10 -10 meters) X-rays are useful for exploring inside crystals because their wavelengths are close to the size of atoms
X-rays have higher energy than visible light because their wavelength is shorter As a result of their higher intensity, X-rays can penetrate matter more effectively than visible light The diffraction of x-rays can determine the lattice parameters, crystallite size, strain, and dislocation density of polycrystalline materials of powders, thin films samples
The technique of XRD is ideal for thin film analysis for two reasons:
- X-ray scattering techniques are non-destructive to the thin film
- The wavelengths of X-rays are close to the atomic distances in the matter, so this device is used as structural probes
Figure 2.6 X-ray diffractometer (XRD, Bruker, D2)
Figure 2.7 Diffraction of X-rays by a crystal
The diffracted wave, illustrated in Figure 2.7, occurs when X-rays interact with atomic planes, leading to interference and the formation of a diffracted beam Additionally, a portion of the beam is reflected by the electrons in the atoms Measurements of the diffractometer reflections were conducted in the 30 to 70° range at room temperature This phenomenon is described by Bragg's Law, represented by the equation nλ = 2d sinθ, which defines the relationship between the angle of the incident X-ray and the resulting diffraction.
λ is the wavelength of the x-ray,
d is the inter-planar spacing,
n is an integer that represents the order of the peak (n = 1, 2, 3,…),
The crystallites size was calculated from Debye-Scherer’s equation formula, as shown in equation (2.1), based on the width of the diffraction peak and the diffraction angle of θ
𝛽 cos 𝜃 (2.1) β in equation (2.1) is the FWHM (full width at half maximum) of diffraction peaks, θ is the incidence angle, λ is the X-ray wavelength, and D is the crystallite size
Figure 2.8 UV-Vis Spectroscopy (Shimadzu 2450)
UV-Vis spectroscopy is a quantitative method used to measure the optical transmittance and absorbance of various materials By comparing the light that passes through a sample to that of a control sample, this technique effectively analyzes a diverse range of samples, including liquids, solids, and thin films, across a photon wavelength spectrum of 190 nm to 1100 nm.
The experimental transmittance (T%) and absorbance (Abs) values were utilized to calculate the optical band gap energy Additionally, the optical band gap and absorption coefficient were determined through a series of specific calculations.
The absorption coefficient (α) was deduced from the Beer-Lambert law in the spectral region of the light’s absorption using the equation (2.2)
In in equation (2.2), A is the absorbance, d is the thickness of the film, α is the absorption coefficient, and T is the transmittance
The optical band gap energy E g was calculated using the absorption coefficient’s (α) dependence on incident photon energy (hν), as shown by equation (2.3)
In the equation (2.3), A represents an energy-independent constant, while α denotes the absorption coefficient of the material The variable m takes on values of 1/2, 1/3, 2, and 3, which correspond to different types of transitions: direct allowed, direct forbidden, indirect allowed, and indirect forbidden, respectively.
The value of m is 1/2, indicating that CuO exhibits a direct permitted transition To determine the bandgap energy (E g), a graph is plotted with (𝛼ℎ𝜈) on the y-axis against hν on the x-axis, and E g is found by extrapolating the linear section of the spectrum to where 𝛼ℎ𝜈 equals 0.
Simulation method
In this thesis, there are 2 different program used for the DFT calculation
PHASE-SYSTEM is a comprehensive suite of software packages designed for simulating nanoscale materials It includes PHASE for first-principles electronic structure calculations, UVSOR for dielectric function analysis, ASCOT for quantum transport properties, and CIAO for all-electron atom calculations and pseudopotential generation Additionally, PHASE Viewer serves as a graphical user interface (GUI) for the PHASE package, enhancing user experience and accessibility.
PHASE is an advanced electronic structure calculation program that utilizes density functional theory (DFT) and the pseudopotential approach, enabling it to predict the physical properties of materials without relying on experimental parameters This program achieves high accuracy in forecasting properties not yet observed in experiments and can compute various physical quantities from calculated wave functions In addition to determining electronic states, PHASE also calculates total energy and atomic forces, allowing users to minimize forces for stable structures and conduct molecular dynamics simulations to analyze the time evolution of systems.
PHASE is run on a Unix-like cluster, access through Secure Shell Protocol (SSH), and it is controlled by command lines, as well as input and output files
Figure 2.9 Command line interface of PHASE
The other program used is Material Studio, and more specifically, the CASTEP package in it
BIOVIA Materials Studio is an advanced modeling and simulation platform designed for research in materials science and chemistry It enables researchers to predict and analyze the connections between a material's atomic and molecular structure and its resulting properties and behavior.
Materials Studio is a client–server model software package, including:
Microsoft Windows-based PC clients
Windows and Linux-based servers, which run on PCs, Linux IA-64 workstations
(including Silicon Graphics (SGI) Altix) and HP XC clusters
CASTEP is a versatile shared-source software package utilized in both academic and commercial settings, grounded in density functional theory with a plane wave basis set It enables first-principles calculations of electronic properties for various structures, including crystalline solids, molecules, surfaces, liquids, and amorphous materials The software supports geometry optimization and finite temperature molecular dynamics, incorporating implicit symmetry and geometry constraints while providing a comprehensive range of derived electronic configuration properties.
Figure 2.10 Microsoft Windows-based PC client user interface of Materials Studio
2.2.2 Main steps of DFT calculation
For a specific system, there are many aspects to choose what set-up we want to use, which would dramatically change the calculated results as well as the computational cost
The CuO film examined in this thesis is not sufficiently "thin" to qualify as a "2D" material, as it requires periodic boundary conditions in all directions for simulating infinitely repeating CuO cells The primitive unit cell of CuO is defined by two copper (Cu) atoms and two oxygen (O) atoms within a triclinic crystal system However, this classification overlooks the potential differences in properties, such as spin, among the Cu atoms, which may necessitate a larger unit cell to accurately represent the repeating spin pattern Further exploration of the spin aspect will be provided in section 2.2.2.1.5.
Figure 2.11 Primitive unit cell of CuO, Cu in blue, O in red
Figure 2.12 unit cell of Cu15SnO16 to simulate the Sn-doped CuO, Cu in blue, O in red, Sn in grey
To create Sn-doped CuO, a supercell is constructed by substituting one or more copper (Cu) atoms with tin (Sn) As illustrated in Figure 2.12, the unit cell of Cu15SnO16 represents a 10% weight doping of Sn in CuO, assuming that the Sn atoms are organized in a perfectly repeating pattern due to periodic boundary conditions.
Larger unit cells can unveil properties and mechanisms that smaller unit cells may not capture, as these characteristics can manifest in larger or even non-repeating patterns However, utilizing significantly larger unit cells demands considerably more computational resources For instance, while experimental data suggests that spin configurations form domains approximately 10 μm in size, it is impractical to use Density Functional Theory (DFT) to compute all atoms within a single domain, much less to analyze multiple domains simultaneously.
Selecting the appropriate unit cell formula involves balancing accuracy with computational cost Insights into the expected properties and mechanisms can guide us in determining the minimum necessary size Additionally, adjusting the size—either increasing or decreasing it—can help achieve optimal results.
The primary challenge of Density Functional Theory (DFT) lies in the fact that, apart from the free-electron gas scenario, the exact functionals for correlation and exchange are not known However, by employing approximations of these functionals, we can achieve accurate calculations of various physical quantities.
We have been advancing our understanding of these functionals through both theoretical and experimental methods; however, the complexity of the subject remains high, and we lack definitive evidence to determine which functional is best suited for each specific case.
The usually used functionals include:
Local-density approximations (LDA) are a method used to approximate the exchange-correlation (XC) energy functional in quantum mechanics These approximations rely exclusively on the electron density at the specific location where the functional is assessed.
In Local Density Approximation (LDA), the exchange–correlation energy is divided into two components: the exchange part, known as Dirac or Slater exchange, represented as 𝜀 X ∝ 𝑛 1/3, and the correlation part, which can take various mathematical forms Recent quantum Monte Carlo simulations of jellium have led to the development of precise formulae for the correlation energy density.
The constructed 𝜀 C (𝑛 ↑ , 𝑛 ↓ ) highlights the limitations of the Local Density Approximation (LDA), which assumes uniform density throughout This assumption often leads to an overestimation of correlation energy and an underestimation of exchange energy However, the errors from these two components tend to partially offset each other.
The Generalized Gradient Approximations (GGA) method addresses the shortcomings of Local Density Approximations (LDA) by incorporating the gradient of electron density, which helps correct the overestimation of correlation energy and the underestimation of exchange energy This approach accounts for the non-homogeneity of the true electron density by basing corrections on density variations away from a given coordinate, resulting in a more accurate representation of exchange-correlation energy as described by equation (2.5).
The Hubbard model, also referred to as DFT+U, describes how each electron is influenced by opposing forces: one repels it from neighboring electrons, while the other encourages tunneling to adjacent atoms The Hubbard correction, typically represented as U eff, is utilized in first-principles simulations based on Density Functional Theory (DFT) To determine the value of U eff, researchers often adjust it to align with experimental data Incorporating the Hubbard correction in DFT simulations enhances the accuracy of electron localization predictions, thereby preventing the erroneous prediction of metallic conduction in insulating materials.
RESULTS AND DISCUSSION
Experiment results
By using XRD analysis, as shown in Figure 3.1, we can calculate the lattice parameters.
Figure 3.1 XRD analysis of CuO and Sn-doped CuO thin-films
The lattice parameters are roughly the same, and they are: o a = 4.7 Å; o b = 3.4 Å; o c = 5.1 Å; o β = 99.35°
This result seems to match the published experimental data by Jing Wu et al., as shown in Figure 3.2 [59]
Figure 3.2 XRD analysis of CuO and Sn-doped CuO thin-films by Jing Wu et al [59]
The UV-Vis measurements indicate that even a small percentage of Sn can significantly impact the results, although no clear trend is observed across different Sn concentrations As shown in Table 3.1 and Figure 3.3, this variability may be attributed to random fluctuations due to the limited sample size The band gap value of CuO is influenced by several preparation factors, including precursor concentration, thickness, and annealing temperature Any variations in these factors can lead to inconsistencies in the results Therefore, further experiments are necessary to enhance accuracy and confirm the relationship between Sn concentration and its effects.
Table 3.1 Band gap of CuO and Sn-doped CuO thin-films
39 Figure 3.3 Band gap of CuO and Sn-doped CuO thin-films
Figure 3.4 Obtaining band gap from UV-Vis measurement
Our findings on the band gap values align well with the published research of Tariq Jan et al (2015) and Jing Wu et al., as illustrated in Figures 3.5 and 3.6, respectively, indicating the validity of our results.
Figure 3.5 Band gap of Sn-doped CuO by Tariq Jan et al (2015), the doping percentage is shown by mol of Sn/(Sn+Cu) [64]
Figure 3.6 Band gap of Sn-doped CuO by Jing Wu et al (2016) [59]
Computational results
In this study, I calculated the density of states (DoS) and band structure of copper oxide (CuO) using a standard density functional theory (DFT) setup The approach employed the GGA PBE functional, an ultra-soft pseudopotential, a conventional unit cell, and did not incorporate spin or Hubbard corrections.
When I first looked up the lattice parameter of CuO, I found some publications showing that:
In comparing my experimental data with the materialsproject database, it is surprising to find that, despite some agreement in the experimental results, the database presents significantly different values This discrepancy raises questions about the reliability of computational calculations in this context.
In this study, we performed unit cell optimization to achieve a structure with a minimal energy state We calculated the unit cell optimization for both configurations to explore potential differences in local minima The results indicate variations in energy and lattice parameters throughout the optimization process.
Figure 3.7 Change of energy during optimization
43 Figure 3.8 Change of parameter a during optimization
Figure 3.9 Change of parameter b during optimization
44 Figure 3.10 Change of parameter c during optimization
Figure 3.11 Change of parameter y during optimization
Figure 3.12 Change of parameter β during optimization
The parameters obtained from materialsproject.org are significantly better optimized, suggesting that while they may not be experimentally accurate, they are refined for Density Functional Theory (DFT) calculations.
The basic DFT calculation indicates that the unit cell is not in equilibrium at the structure predicted by experimental data.
It is essential to acknowledge the energy-efficient structure concerning the interactions outlined by the basic DFT calculation before proceeding The results of the band structure obtained from this setup are illustrated in Figure 3.13.
Figure 3.13 Band structure of CuO (Cu4O4 unit cell) calculated with basic DFT
A significant issue identified is the absence of a bandgap, with 68 electrons present in the unit cells Upon analysis, it was found that 32 energy bands are entirely below the Fermi level.
Four bands intersect the Fermi level, indicating the presence of four electrons within these bands This configuration allows electrons to occupy a continuous range of energy levels around the Fermi level, resulting in the absence of a bandgap.
I tried to use the Hubbard correction to recreate the band gap of CuO with this Cu4O4 unit cell Figure 3.14 shows the result of DoS calculated with this setup
Figure 3.14 DoS of CuO (Cu4O4 unit cell) calculated with basic DFT and DFT+U
The issue of the Fermi level intersecting a band persists, as DFT+U reveals a band opening near the Fermi level; however, this does not indicate a true bandgap, since the density of states (DoS) at the Fermi level remains non-zero.
The concept of partially filled bands crossing the Fermi level can be understood as acting like "holes." This implies that, in most positions, electrons occupy these bands, leading to unique electronic properties.
CuO is identified as a p-type semiconductor, characterized by a higher concentration of holes than free electrons, aligning with findings in various studies The band structure indicates an optical band gap, suggesting a specific photon energy for absorption, yet the calculated hole-to-atom ratio of 2 holes per 4 Cu atoms leads to a carrier concentration estimate of approximately 10²² carriers/cm³ This conflicts with experimental data, which report a carrier concentration below 10¹⁹ carriers/cm³, indicating a lower ratio of carriers per atom and suggesting insulator behavior in smaller unit cells This behavior may arise from defects, such as Cu atom vacancies, as proposed by Zhiliang Wang (2019) To validate the insulator behavior, it is essential to ensure the density of states is zero at the Fermi level.
3.2.3 Including spin with small unit cell
As mentioned in section 2.2.2.1.5, B.X Yang et al (1988) suggested that CuO has a specific antiferromagnetic spin arrangement [58]
Antiferromagnetic properties can be replicated in smaller unit cells, provided that half of the spins are oriented in one direction and the other half in the opposite direction This approach is beneficial as smaller unit cells allow for faster calculations By examining this structure, we can determine whether the precise antiferromagnetic configuration is essential or if a simplified version with a reduced unit cell—where half of the copper atoms have spins aligned one way and the other half in the opposite direction—will suffice, ultimately conserving computational resources.
With the following computational details, we get the result in Figure 3.1 5:
The study adopted experimental data from S Åsbrink and L.-J Norrby (1970) to analyze the magnetic properties of cupric, which contains four formula units in its unit cell The antiferromagnetic (AFM) configuration is illustrated in Figure 2.14, and the local density approximation (LDA) functional was employed to calculate the exchange correlation energy term.
The density of states (DoS) of CuO, analyzed using linear density approximation (LDA) and the antiferromagnetic structure of Cu4O4, reveals a persistent issue with the Fermi level intersecting the bands To address this, further exploration into a larger unit cell that accurately represents the antiferromagnetic configuration proposed by B.X is necessary.
Yang et al (1988) [58], we may try using hybrid functionals on the small antiferromagnetic unit cell
Using the hybrid functionals (PBE0, HSE06, and B3LYP) on the same small antiferromagnetic unit cell (shown in Figure 2.14), we get the results shown in Figures 3.16, 3.17, and 3.18
Hybrid functionals produced results with band gaps of varying values The B3LYP and HSE06 seem to give reasonable results compared to experimental data
Hybrid functionals, while effective, often rely on arbitrary fitted parameters To enhance the model, I aim to incorporate more intuitive elements, such as spin and vacancies, grounded in fundamental Density Functional Theory (DFT), which has been validated for free electron gas.
51 Figure 3.16 DoS and band structure of Cu4O4 calculated with B3LYP hybrid functional
52 Figure 3.17 DoS and band structure of Cu4O4 calculated with HSE06 hybrid functional
53 Figure 3.18 DoS and band structure of Cu4O4 calculated with PBE0 hybrid functional
3.2.4 Including spin with bigger unit cell
We can do the calculation with the big unit cell as B.X Yang et al (1988) suggested
[58] With the unit cell in this set up having 16 Cu atoms and 16 O atoms (as shown in Figure 2.13), we get the results shown in Figure 3.19
Figure 3.19 DoS of CuO using GGA and Cu16O16 antiferromagnetic structure
The band gap is reproduced, albeit much smaller than the value which experimental data suggest
Compared to the result we got with the smaller antiferromagnetic unit cell (without hybrid functional) (as discussed in section 3.2.3), we can check their total energy per atom:
Cu4O4 antiferromagnetic structure, without hybrid functional:
Cu16O16 antiferromagnetic structure, without hybrid functional:
Future study
In future work, I intend to explore further on this subject, including:
Using hybrid functionals on the actual antiferromagnetic structure, as suggested by B.X Yang et al (1988), reveals that while GGA and LDA struggle to accurately reproduce the band gap in smaller antiferromagnetic unit cells, hybrid functionals successfully do so This discrepancy may indicate that hybrid functionals account for factors not captured by the smaller unit cell However, applying these functionals to larger antiferromagnetic unit cells could lead to overcompensation To determine the most effective functional, it is essential to explore various options on the larger unit cell, with a key evaluation criterion being the convergence trend of the chosen functionals as the unit cell size increases.
Vacancy: There are publications suggesting the importance of the vacancies of
Removing a single Cu atom from the unit cell allows for the calculation of the density of states (DoS) and band structure, providing insights into how these properties change despite not perfectly mimicking an actual Cu vacancy However, the computational cost associated with this process is significant, as the unit cell must be sufficiently large to ensure that the vacancy represents a reasonable ratio Additionally, the loss of symmetry from creating a vacancy prolongs the optimization of the unit cell, making the computational effort even more demanding.
To align theoretical predictions with experimental data, we can apply the Hubbard correction to each case, focusing on scenarios that utilize hybrid functionals and exhibit antiferromagnetic arrangements.
To evaluate the effectiveness of hybrid functionals on Sn-doped CuO, it is essential to apply the appropriate Hubbard correction As previously discussed, these hybrid functionals can be somewhat arbitrary, often yielding accurate bandgap results only through parameter tuning to align with experimental data Therefore, we should systematically test different functional sets that correspond to the known experimental outcomes for CuO to determine the most effective option for Sn-doped CuO While the functional that suggests a larger unit cell with an antiferromagnetic spin arrangement may seem more intuitive, it is not guaranteed to be superior Similarly, the B3LYP hybrid functional, which provides an acceptable bandgap value prior to the application of the Hubbard correction, may also not be the best choice.
CuO and Sn-doped CuO thin films were successfully synthesized, with their structural and optical properties analyzed using X-ray diffraction (XRD) and UV-visible spectroscopy The results indicated that all films exhibited a monoclinic crystal structure, prominently featuring the (111) plane.
(002) preferred orientations The CuO thin-film had a bandgap energy of around 2.44 eV, and the Sn-doped (1% to 5% by weight) CuO thin-films had bandgap energies ranged from 2.71 to 2.78 eV
A basic DFT calculation using GGA-PBE-US on the conventional unit cell of CuO demonstrated metallic behavior, with the Fermi level intersecting the bands The introduction of the Hubbard correction increased the energy gap between the band at the Fermi level and the next higher energy band; however, this did not result in a true band gap Additionally, implementing an antiferromagnetic arrangement within the conventional unit cell did not lead to the emergence of a band gap.
The bandgap energies for the antiferromagnetic Cu4O4 and Cu16O16 systems were calculated using various hybrid functionals, yielding different results The B3LYP hybrid functional produced a bandgap of E_g = 2.578 eV, while the PBE0 functional yielded a higher bandgap of E_g = 3.406 eV The HSE06 functional also provided a bandgap of E_g = 2.602 eV for Cu4O4 For the Cu16O16 system, the GGA-PBE-US functional resulted in a bandgap of E_g = 0.49 eV, and with an effective U value of 5 eV, the bandgap increased to E_g = 1.066 eV.
This must be an interesting achievement of correlation between experimental data and DFT calculation at the current stage
Future research will focus on several key areas, including the application of hybrid functionals to the actual antiferromagnetic structure, as suggested by B.X Yang et al (1988) Additionally, the studies will incorporate the effects of copper and oxygen vacancies, as well as implement Hubbard corrections for each configuration of the functional and unit cell of CuO, followed by applying these findings to Sn-doped CuO.
By this, it’s convinced that the formation mechanism of these p-type semiconducting thin films will be well explained and developed further
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