Construction method of dynamic calculation model of substance / charge transport rate of copper-nickel alloy barrier layer based on first principle

Abstract

The application discloses a construction method of a dynamic calculation model of a copper-nickel alloy barrier layer substance / charge transmission rate based on a first principle, and belongs to the technical field of electrochemical corrosion, and comprises the following steps: (a) a first principle calculation method based on a density functional theory is used to construct a cuprous oxide and doped nickel cuprous oxide single cell model, the single cell model is cut (100) crystal surface, and a vacancy formation energy model of the doped nickel cuprous oxide is obtained; (b) on the basis of the cuprous oxide single cell, a 2*2*3 super cell model is constructed, and a cation migration rate calculation model is constructed on the basis of the super cell model; (c) on the basis of steps (a) and (b), a substance transmission rate calculation model is constructed; and (d) on the basis of the cuprous oxide and doped nickel cuprous oxide single cell, a charge migration rate calculation model is constructed.

Description

Technical Field

[0001] This invention belongs to the field of electrochemical corrosion technology, specifically relating to a method for constructing a kinetic calculation model of the material / charge transport rate of a copper-nickel alloy barrier layer based on first-principles calculations. Background Technology

[0002] Electrochemical corrosion typically acts on the material surface, causing electron transfer between atoms and leading to redox reactions. The ease with which atoms or ions diffuse through the barrier layer on the alloy surface determines the material's corrosion resistance; generally, the more stable the barrier layer, the higher the corrosion resistance. Therefore, the stability of the barrier layer has always been a key focus for corrosion researchers.

[0003] Bouzoubaa etc. [1] The relationship between nickel oxide film (NiO) and Cl was studied. - OH - Through interactions, it was found that the oxide film thinning mechanism induced by surface particle adsorption and the interfacial vacancy mechanism induced by infiltration can explain the damage behavior of the surface passivation film. Ng et al. [2] The corrosion characteristics of Fe2O3 and Cr2O3 surfaces were calculated. The results showed that Cr2O3, as an effective passivation layer, has a high overpotential. By doping with Mo, the redox reaction can be suppressed. This study provides a new perspective for designing highly corrosion-resistant passivation films.

[0004] However, these studies focus on the thermodynamic stability of the system and the interfacial reaction barrier, occurring at the barrier layer and solution interface. They do not consider another crucial factor affecting the corrosion resistance of the barrier layer: transport rates, including mass transport and charge transport rates. Considering the corrosion resistance of the barrier layer solely from an interfacial perspective is insufficient to explain the long-term performance of the material; therefore, new methods need to be designed to study the kinetic processes of mass and charge transport.

[0005] [1]Bouzoubaa A, Diawara B, Maurice V, et al. Ab initio modeling of localized corrosion: study of the role of surface steps in the interaction of chlorides with passivated nickel surfaces[J]. Corrosion Science, 2009, 51(9): 2174-2182.

[0006] [2]Ng MF,Blackwood DJ,Jin H,et al.DFT study of oxygen reductionreaction on chromia and hematite:insights into corrosion inhibition[J].TheJournal of Physical Chemistry C,2020,124(25):13799-13808. Summary of the Invention

[0007] The purpose of this invention is to provide a method for constructing a kinetic calculation model of the mass / charge transport rate of a copper-nickel alloy barrier layer based on first principles, so as to explain the influence of alloying elements on the corrosion resistance of the material barrier layer.

[0008] The technical solution adopted to achieve the above objectives is as follows:

[0009] A method for constructing a kinetic calculation model of the mass / charge transport rate of a copper-nickel alloy barrier layer based on first-principles calculations includes the following steps:

[0010] (a) Using a copper-nickel alloy barrier layer as the research object, first-principles calculations based on density functional theory were used to construct single-cell models of cuprous oxide and nickel-doped cuprous oxide. Based on this, the single-cell model was cut with a (100) crystal plane to obtain a vacancy formation energy model for nickel-doped cuprous oxide. (b) Based on the cuprous oxide single-cell model, a 2×2×3 supercell model was constructed, and a cation migration rate calculation model was built based on the supercell model. (c) Based on steps (a) and (b), a mass transport rate calculation model was constructed. (d) Based on the single-cell models of cuprous oxide and nickel-doped cuprous oxide, a charge migration rate calculation model was constructed.

[0011] The aforementioned vacancy formation energy model is implemented according to the following steps:

[0012] Step 1: Set the lattice parameters, mainly the lattice constants of the cuprous oxide and nickel-doped cuprous oxide unit cell models, with the nickel doping atomic ratio being Cu:Ni = 3:1;

[0013] Step 2: For the set lattice parameters, the PBE of the generalized gradient approximation (GGA) is selected as the exchange-dependent generalized function adapted to the model. The interactions between valence electrons and atomic nuclei are described using OTFG ultrasoft. The plane wave energy cutoff is 571.4 eV. The convergence energy tolerance after geometry optimization is set to 1.0E-5 eV / atom, and the self-consistent field (SCF) used to control the electron minimization algorithm is set to 1.0E-6 eV / atom. The force and stress convergence tolerances are respectively set to... and 0.05 GPa. Structural optimization was performed under the above conditions.

[0014] Step 3: Using the constructed single-cell model, cut the surface according to the (100) crystal plane, set the number of layers to 3.75, and construct a vacuum layer as follows. The bottom two layers of atoms were fixed to obtain a surface model, and the vacancy defect states of copper and nickel were constructed using the surface model.

[0015] Step 4, construct the lattice with point group P1 and parameters as follows: The cubic structure was used to draw copper and nickel atoms at the origin, respectively, to obtain an isolated atom model;

[0016] Step 5: Select the PBE of the generalized gradient approximation (GGA) as the exchange-related generalized function adapted to the model, and calculate it for the isolated atom model, surface model and vacancy defect model to obtain the optimization results;

[0017] Step 6: Calculate the energies of isolated atoms, the surface model, and the vacancy defect model, and substitute them into the formula: E VAC =E slab-VAC +E atom -E slab The vacancy formation energy E is obtained. VAC In the formula, E slab-VAC and E slab E represents the total energy of the system with vacant spaces and the system without vacant spaces, respectively. atom The energy of the isolated atom to be removed;

[0018] The aforementioned cation migration rate calculation model is implemented according to the following steps:

[0019] Step 1: Based on the cuprous oxide single cell, construct a 2×2×3 supercell model and replace the 5 copper atoms with high symmetry with nickel atoms;

[0020] Step 2: For the supercell model, the PBE of the generalized gradient approximation (GGA) is selected as the exchange-dependent generalized function adapted to the model. The interactions between valence electrons and atomic nuclei are described using OTFG ultrasoft. The plane wave energy cutoff is 571.4 eV. The convergence energy tolerance after geometry optimization is set to 1.0E-5 eV / atom, and the self-consistent field (SCF) used to control the electron minimization algorithm is set to 1.0E-6 eV / atom. The force and stress convergence tolerances are respectively set to... and 0.05 GPa. Structural optimization was performed under the above conditions;

[0021] Step 3: Construct copper vacancy defects with high symmetry in the supercell model;

[0022] Step 4: Based on the vacancy defect structure of copper, construct a transition state search model for the migration of copper and nickel with shared oxygen and non-shared oxygen.

[0023] Step 5: Perform density generalized function theory (DFT) calculations, selecting the PBE of the generalized gradient approximation (GGA) as the exchange-correlation generalized function. In this calculation, the interactions between valence electrons and atomic nuclei are described using OTFG ultrasoft. The plane wave energy cutoff is 571.4 eV. The geometrically optimized convergence energy tolerance is set to 1.0E-5 eV / atom, and the self-consistent field (SCF) used to control the electron minimization algorithm is set to 1.0E-6 eV / atom. The force and stress convergence tolerances are respectively set to... and 0.05 GPa. The maximum number of secondary synchronization transfer steps is selected as 5. Perform a transition state search under the above conditions;

[0024] Step 6: Based on the transition state search model, calculate the energy barriers (referred to as the copper and nickel migration barriers) E for copper and nickel migration to copper vacancies. a ;

[0025] The aforementioned mass transport rate calculation model is implemented according to the following steps:

[0026] Step 1: In the transition state search model, measure the distance between the positions before and after ion migration to obtain the migration distance l;

[0027] Step 2, calculate the migration energy barriers E of copper and nickel. a Substitute the atomic masses and migration distance l of Cu and Ni into the formula:

[0028] v=(E a / 2ml 2 ) 1 / 2

[0029] The vibration frequency ν is calculated.

[0030] Step 3: The migration distance l, vibrational frequency ν, and vacancy formation energies E of copper and nickel in the nickel-doped cuprous oxide model are used. VAC The migration energy barrier E of copper and nickel a Substitute into the formula:

[0031]

[0032] k B T and K represent Boltzmann's constant and absolute temperature, respectively. B =1.38E-23J / K, T is chosen as 298.15K, and the diffusion coefficient D of copper and nickel migrating to copper vacancies is calculated;

[0033] The charge transfer rate calculation model is implemented according to the following steps:

[0034] Step 1: Based on the optimized cuprous oxide and nickel-doped cuprous oxide single cells, calculate the band structure and select consistent Fermi surface symmetry points and paths.

[0035] Step 2: Based on the band structure, calculate the band gap for cuprous oxide and nickel-doped cuprous oxide, respectively. and

[0036] Step 3: Based on the band structure, select the G-symmetric point in the reciprocal lattice space and calculate the effective mass of holes near the top of the valence band. and

[0037] Step 4: Substitute the calculated band gap and effective mass into the following formula:

[0038]

[0039] The derivation of the formula is as follows:

[0040] For semiconductors, the formula for calculating their ohmic current density is:

[0041] J σ =en i (μ n +μ p E

[0042] The intrinsic carrier concentration is:

[0043]

[0044] The carrier mobility is:

[0045]

[0046] Therefore, the Ohmic current density can be written as:

[0047]

[0048] Since the two crystal structures in the computational model of this invention are similar, both based on cuprous oxide, the differences in effective density of states N and scattering time τ can be ignored, and C is introduced. J =e 2 (N C N V ) 1 / 2 τ simplifies the ohmic current density to:

[0049]

[0050] The ratio of the limiting ohmic current density in cuprous oxide and nickel-doped cuprous oxide is obtained, thereby enabling a comparison of charge transport rates.

[0051] The beneficial effects of this invention are:

[0052] (a) This invention simulates the formation of vacancies and the migration of cations in the barrier layer at the atomic level, and calculates the vacancy formation energy and migration energy barrier to quantitatively compare the ease of formation and migration of nickel and copper in the barrier layer.

[0053] (b) This invention utilizes the effect of nickel doping on the band structure of cuprous oxide to calculate the band gap and effective mass, thereby achieving a quantitative comparison of charge migration rates in the two substances.

[0054] (c) All data in this invention are derived from first-principles calculations, eliminating the need for experiments, thus reducing operational difficulty and saving experimental costs. Attached Figure Description

[0055] Figure 1 This is the cuprous oxide single-cell model in the embodiments of the present invention;

[0056] Figure 2 This is the nickel-doped cuprous oxide single-cell model in the embodiments of the present invention;

[0057] Figure 3 This is a vacancy defect model obtained by cutting the (100) surface of a nickel-doped cuprous oxide single crystal cell in an embodiment of the present invention.

[0058] Figure 4 This is the single-atom copper single-cell model in the embodiments of the present invention;

[0059] Figure 5 This is the single-atom nickel single-cell model in the embodiments of the present invention;

[0060] Figure 6 This is a 2×2×3 supercell model obtained by expanding cuprous oxide cells and doping it with nickel in this embodiment of the invention, named M1;

[0061] Figure 7 In this embodiment of the invention, the copper migration barrier for shared oxygen is established in a 2×2×3 supercell model and named M1_Cu_Oxygen-sharing.

[0062] Figure 8 In this embodiment of the invention, the copper migration barrier for non-shared oxygen is established in a 2×2×3 supercell model and named M1_Cu_Non-oxygen-sharing.

[0063] Figure 9In this embodiment of the invention, the nickel migration barrier for shared oxygen is established in a 2×2×3 supercell model and named M1_Ni_Oxygen-sharing.

[0064] Figure 10 In this embodiment of the invention, the nickel migration barrier with non-shared oxygen is established in a 2×2×3 supercell model and named M1_Ni_Non-oxygen-sharing.

[0065] Figure 11 This is a band structure diagram obtained by analyzing a single crystal cell of cuprous oxide in an embodiment of the present invention;

[0066] Figure 12 This is the band structure diagram obtained by analyzing a nickel-doped cuprous oxide single cell in an embodiment of the present invention. Detailed Implementation

[0067] A method for constructing a kinetic calculation model of the mass / charge transport rate of a copper-nickel alloy barrier layer based on first-principles calculations includes the following steps:

[0068] (a) Using a copper-nickel alloy barrier layer as the research object, first-principles calculations based on density functional theory were used to construct single-cell models of cuprous oxide and nickel-doped cuprous oxide. Based on this, the single-cell model was cut with a (100) crystal plane to obtain a vacancy formation energy model for nickel-doped cuprous oxide. (b) Based on the cuprous oxide single-cell model, a 2×2×3 supercell model was constructed, and a cation migration rate calculation model was built based on the supercell model. (c) Based on steps (a) and (b), a mass transport rate calculation model was constructed. (d) Based on the single-cell models of cuprous oxide and nickel-doped cuprous oxide, a charge migration rate calculation model was constructed.

[0069] For the cation migration rate calculation model, the specific steps are as follows:

[0070] Step 1, as follows Figure 1 and Figure 2 Set the lattice parameters, mainly the lattice constants of the cuprous oxide and nickel-doped cuprous oxide unit cell models, with the nickel doping atomic ratio being Cu:Ni = 3:1;

[0071] Step 2: For the set lattice parameters, the PBE of the generalized gradient approximation (GGA) is selected as the exchange-dependent generalized function adapted to the model. The interactions between valence electrons and atomic nuclei are described using OTFG ultrasoft. The plane wave energy cutoff is 571.4 eV. The convergence energy tolerance after geometry optimization is set to 1.0E-5 eV / atom, and the self-consistent field (SCF) used to control the electron minimization algorithm is set to 1.0E-6 eV / atom. The force and stress convergence tolerances are respectively set to... and 0.05 GPa. Structural optimization was performed under the above conditions;

[0072] Step 3, as follows Figure 3 Using the constructed single-cell model Figure 1 According to the (100) crystal plane, the number of layers is set to 3.75, and a vacuum layer is constructed as follows: The bottom two layers of atoms were fixed to obtain a surface model, and the vacancy defect states of copper and nickel were constructed using the surface model.

[0073] Step 4, as follows Figure 4 and Figure 5 The lattice parameters for constructing a point group of P1 are: The cubic structure was used to draw copper and nickel atoms at the origin, respectively, to obtain an isolated atom model;

[0074] Step 5: Select the PBE of the generalized gradient approximation (GGA) as the exchange-related generalized function adapted to the model, and calculate it for the isolated atom model, surface model and vacancy defect model to obtain the optimization results;

[0075] Step 6: Calculate the energies of isolated atoms, the surface model, and the vacancy defect model, and substitute them into the formula: E VAC =E slab-VAC +E atom -E slab The vacancy formation energy E is obtained. VAC In the formula, E slab-VAC and E slab E represents the total energy of the system with vacant spaces and the system without vacant spaces, respectively. atom The energy of the isolated atom removed, E VCu = 3.74727 eV, E VNi = 5.30506 eV; (This part calculates the energy of defect-free structures, structures with copper atom defects, and structures with nickel atom defects, as well as isolated copper and nickel atom systems, using the formulas above.) VCu and E VNi );

[0076] Step 7, as follows Figure 6 Based on the single unit cell of cuprous oxide, a 2×2×3 supercell model was constructed, and copper atoms with high symmetry were selected to replace nickel atoms.

[0077] Step 8: For the supercell model, the PBE of the generalized gradient approximation (GGA) is selected as the exchange-dependent generalized function adapted to the model. The interactions between valence electrons and atomic nuclei are described using OTFG ultrasoft. The plane wave energy cutoff is 571.4 eV. The convergence energy tolerance after geometry optimization is set to 1.0E-5 eV / atom, and the self-consistent field (SCF) used to control the electron minimization algorithm is set to 1.0E-6 eV / atom. The force and stress convergence tolerances are respectively set to... and 0.05 GPa. Structural optimization was performed under the above conditions.

[0078] Step 9: Construct copper vacancy defects at different locations in the supercell model;

[0079] Step 10: Based on the vacancy defect structure of copper, construct a transition state search model for the migration of copper and nickel with shared oxygen and non-shared oxygen.

[0080] Step 11: Perform density generalized function theory (DFT) calculations, selecting the PBE of the generalized gradient approximation (GGA) as the exchange-correlation generalized function. In this calculation, the interactions between valence electrons and atomic nuclei are described using OTFG ultrasoft. The plane wave energy cutoff is 571.4 eV. The geometrically optimized convergence energy tolerance is set to 1.0E-5 eV / atom, and the self-consistent field (SCF) used to control the electron minimization algorithm is set to 1.0E-6 eV / atom. The force and stress convergence tolerances are respectively set to... And 0.05GPa. The maximum number of secondary synchronization transfer steps is selected as 5. Perform transition state search under the above conditions.

[0081] Step 12: Based on the transition state search model, calculate the energy barrier E for the migration of copper and nickel to copper vacancies. a The result is as follows Figures 7 to 10 The summary is shown in the table below:

[0082]

[0083] Step 13: In the transition state search model, measure the distance between the positions before and after ion migration to obtain...

[0084]

[0085] Step 14, lower the migration energy barrier E of copper and nickel in M1. a Substitute the atomic masses and migration distance l of Cu and Ni into the formula:

[0086] v=(E a / 2ml 2 ) 1 / 2

[0087] The calculated vibration frequency ν is shown in the table below:

[0088]

[0089] Step 15: The migration distance l, vibrational frequency ν, and vacancy formation energies E of copper and nickel in the nickel-doped cuprous oxide model are used. VAC The migration energy barrier E of copper and nickel a Substitute into the formula:

[0090]

[0091] k B T and K represent Boltzmann's constant and absolute temperature, respectively. B =1.38E-23J / K, T is chosen as 298.15K, the diffusion coefficient D of copper and nickel migrating to copper vacancies is calculated and shown in the table below:

[0092]

[0093] The results show that nickel migrates at a much lower rate in the barrier layer than copper.

[0094] The charge transfer rate calculation model is implemented according to the following steps:

[0095] Step 1, as follows Figure 1 and Figure 2 Based on the optimized cuprous oxide and nickel-doped cuprous oxide single-cell structures, the band structure was calculated, such as... Figure 11 and Figure 12 Choose consistent Fermi face symmetry points and paths;

[0096] Step 2: Based on the band structure, calculate the band gap for cuprous oxide and nickel-doped cuprous oxide, respectively. and

[0097] Step 3: Based on the band structure, select the G-symmetric point in the reciprocal lattice space and calculate the effective mass of holes near the top of the valence band. and

[0098] Step 4: Substitute the calculated band gap and effective mass into the following formula:

[0099]

[0100] The results show that nickel doping reduces the current density in the barrier layer.

Claims

1. A method for constructing a kinetic calculation model of the mass / charge transport rate of a copper-nickel alloy barrier layer based on first-principles calculations, characterized in that, Includes the following steps: (a) Using the first-principles calculation method based on density functional theory, a single-cell model of cuprous oxide and nickel-doped cuprous oxide was constructed. Based on this, the single-cell model was cut into crystal planes to obtain a vacancy formation energy model of nickel-doped cuprous oxide. Follow these steps: Step 1: Set the lattice constants of the cuprous oxide and nickel-doped cuprous oxide unit cell models, with the nickel doping atomic ratio being Cu:Ni = 3:1; Step 2: For the set lattice parameters, the PBE of the generalized gradient approximation is selected as the exchange-related generalized function adapted to the model. The interaction between valence electrons and atomic nuclei is described using OTFG ultrasoft. The plane wave energy cutoff value is 571.4 eV. The convergence energy tolerance after geometric optimization is set to 1.0E-5 eV / atom, and the self-consistent field used to control the electron minimization algorithm is set to 1.0E-6 eV / atom. The force and stress convergence tolerances are set to 0.03 eV / Å and 0.05 GPa, respectively. Structural optimization is performed under the above conditions. Step 3: Using the constructed single-cell model, cut the surface according to the crystal plane, set the number of layers to 3.75, construct a vacuum layer of 15 Å, and fix the atoms of the bottom two layers to obtain the surface model. In addition, use the surface model to construct the vacancy defect states of copper and nickel. Step 4: Construct a cubic structure with point group P1 and lattice parameters of , and draw copper and nickel atoms at the origin to obtain isolated atom models. Step 5: Select the PBE of the generalized gradient approximation as the exchange correlation generalized function adapted to the model, and calculate it for the isolated atom model, surface model and vacancy defect model to obtain the optimization results; Step 6: Calculate the energies of isolated atoms, the surface model, and the vacancy defect model, and substitute them into the formula: E VAC =E slab-VAC +E atom -E slab The vacancy formation energy E is obtained. VAC In the formula, E slab-VAC and E slab E represents the total energy of the system with vacant spaces and the system without vacant spaces, respectively. atom The energy of the isolated atom to be removed; (b) Based on the cuprous oxide single cell, a 2×2×3 supercell model was constructed, and a cation migration rate calculation model was constructed based on the supercell model. (c) Based on steps (a) and (b), construct a material transport rate calculation model; (d) Based on the single cell of cuprous oxide and nickel-doped cuprous oxide, a charge transfer rate calculation model is constructed.

2. The method for constructing a kinetic calculation model of the mass / charge transport rate of a copper-nickel alloy barrier layer based on first-principles calculations, as described in claim 1, is characterized in that... The aforementioned cation migration rate calculation model is implemented according to the following steps: Step 1: Based on the cuprous oxide single cell, construct a 2×2×3 supercell model and replace the 5 copper atoms with high symmetry with nickel atoms; Step 2: For the supercell model, the PBE (Generalized Gradient Approximation) is selected as the exchange-dependent generalized function adapted to the model. The interactions between valence electrons and atomic nuclei are described using OTFG ultrasoft. The plane wave energy cutoff is 571.4 eV. The convergence energy tolerance after geometric optimization is set to 1.0E-5 eV / atom, and the self-consistent field used to control the electron minimization algorithm is set to 1.0E-6 eV / atom. The force and stress convergence tolerances are set to 0.03 eV / Å and 0.05 GPa, respectively. Structural optimization is performed under the above conditions. Step 3: Construct copper vacancy defects with high symmetry in the supercell model; Step 4: Based on the vacancy defect structure of copper, construct a transition state search model for the migration of copper and nickel with shared oxygen and non-shared oxygen. Step 5: Perform density generalized function theoretical calculations, selecting the PBE approximation of the generalized gradient as the exchange-correlation generalized function; in this calculation, the interaction between valence electrons and atomic nuclei is described using OTFG ultrasoft; the plane wave energy cutoff is 571.4 eV; the geometrically optimized convergence energy tolerance is set to 1.0E-5 eV / atom, and the self-consistent field used to control the electron minimization algorithm is set to 1.0E-6 eV / atom; the force and stress convergence tolerances are set to 0.03 eV / Å and 0.05 GPa, respectively; the maximum number of second-order synchronous transfer steps is selected as 5; under the above conditions, a transition state search is performed; Step 6: Calculate the migration energy barriers E for copper and nickel based on the transition state search model. a .

3. The method for constructing a kinetic calculation model of the mass / charge transport rate of a copper-nickel alloy barrier layer based on first-principles calculations, as described in claim 2, is characterized in that... The aforementioned mass transport rate calculation model is implemented according to the following steps: Step 1: In the transition state search model, measure the distance between the positions before and after ion migration to obtain the migration distance l; Step 2, calculate the migration energy barriers E of copper and nickel. a Substitute the atomic masses and migration distance l of Cu and Ni into the formula: The vibration frequency ν is calculated. Step 3: The migration distance l, vibrational frequency ν, and vacancy formation energies E of copper and nickel in the nickel-doped cuprous oxide model are used. VAC The migration energy barrier E of copper and nickel a Substitute into the formula: k B T and K represent Boltzmann's constant and absolute temperature, respectively. B =1.38E-23J / K, T is chosen as 298.15K, and the diffusion coefficient D of copper and nickel migrating to copper vacancies is calculated.

4. The method for constructing a kinetic calculation model of the mass / charge transport rate of a copper-nickel alloy barrier layer based on first-principles calculations, as described in claim 1, is characterized in that... The charge transfer rate calculation model is implemented according to the following steps: Step 1: Based on the optimized cuprous oxide and nickel-doped cuprous oxide single cells, calculate the band structure and select consistent Fermi surface symmetry points and paths. Step 2: Based on the band structure, calculate the band gap and band structure of cuprous oxide and nickel-doped cuprous oxide, respectively. and ; Step 3: Based on the band structure, select the G-symmetric point in the reciprocal lattice space and calculate the effective mass of holes near the top of the valence band. and ; Step 4: Substitute the calculated band gap and effective mass into the following formula: The ratio of the limiting ohmic current density in cuprous oxide and nickel-doped cuprous oxide is obtained, thereby enabling a comparison of charge transport rates.

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