Analysis of Semiconductor Band Gap Energy Characteristics

Study Case Analysis of Semiconductor Band Gap Energy Characteristics using the Pseudopotential Method Michael Kurniawan (2018123456) University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH 45221, United States kurniaml@mail.uc.edu Abstract Semiconductors are key components in the development of various modern electronic devices. Understanding of semiconductors, including the calculation of band energy or band gap, continues to evolve. One method used to determine band gap is pseudopotential analysis. This article reviews the concept of the pseudopotential method that has been developed to analyze the band gap of various semiconductor materials. In general, this method has advantages over other methods, especially in terms of calculation efficiency through the computational solution of the one-electron Schrödinger wave equation. In addition, this method only considers the outermost shell electrons, making the calculation more effective without involving the entire atomic nucleus. Keywords: Review, Pseudopotential, Semiconductor, Band gap INTRODUCTION The development of materials science is one of the main drivers of technological advancement, so that materials research continues to be carried out. Various electronic devices require specific materials as components. The type of material used directly affects the performance of the device. To improve the quality of the device, new materials are discovered through various approaches, ranging from laboratory experiments to computational simulations that model the physical properties and properties of materials using advanced algorithms. In materials research, whether through laboratories or computational simulations, the main focus is on understanding the physical properties of materials and their MICHAEL KURNIAWAN 1 Study Case predictions. One of the physical properties of concern is electron configuration, which reflects the interaction between atomic nuclei and electrons. In simple materials, electron configuration can be solved with Schrödinger's equation, taking into account the physical parameters in the quantum system. However, in complex materials— including semiconductors—the calculations become much more complicated because they involve many particles. Semiconductors play an important role in modern technology and electronic devices, ranging from smartphones and household appliances to transportation systems. Semiconductors are materials with electrical conductivity caused by the movement of electrons, falling between conductors and insulators. Band gap energy is the main indicator for distinguishing semiconductors from other materials. Each material has two energy bands: the valence band and the conduction band, which are the pathways for electron transfer. Band gap energy indicates the minimum energy required to excite electrons from the valence band to the conduction band. Knowing the band gap energy allows for more efficient material processing. The pseudopotential method is an approach to modeling electron interactions in a material to study its physical properties. One of the properties that can be analyzed using this method is band gap energy. The advantage of the pseudopotential method is that it focuses on the outer shell electrons, while the deeper core electrons are ignored because they tend to be rigid, not easily polarized, and tightly bound to the atomic nucleus [1][2]. Previous studies have used this method to determine the band gap energy of semiconductor materials such as silicon (Si) and germanium (Ge). In this study, the band gap energy of zinc oxide (ZnO) semiconductors will be analyzed using the pseudopotential method. METHODOLOGY The empirical pseudopotential method is one approach to determining the atomic characteristics of a material. This method is capable of producing calculations that approximate the accuracy of other methods such as Density Functional Theory (DFT) [1]. The main principle of the pseudopotential method is to study the properties of materials by only considering the outermost electrons, which are treated as Nearly Free Electrons (NFE), while the influence of core electrons is frozen [2]. This potential forms the electron wave function in the form of plane waves, which is expressed by the MICHAEL KURNIAWAN 2 Study Case following equation: 𝜓(𝐫) = ∑𝐤 𝑐𝐤 𝑒 𝑖𝐤⋅𝐫 (1) Pseudopotential calculations are performed using the Orthogonalized Plane Wave (OPW) approach, in which the wave function is separated between the atomic nucleus and the outermost electrons: 𝜓 = 𝜓core + 𝜓valence (2) Here, ψ is the total wave function, ψ_“valence” is plane waves, while ψ_“core” is a linear combination of atoms at the center of the nucleus. The development of the pseudopotential equation follows Cohen and Bergstresser, starting from the time-dependent Schrödinger equation: ∂ ̂ 𝜓(𝐫, 𝑡) (3) ℏ ∂𝑡 𝜓(𝐫, 𝑡) = 𝐻 Next, Bloch's Theorem is used to express the wave function in a plane wave basis due to the periodic crystal structure: 𝜓𝐤 (𝐫) = 𝑒 𝑖𝐤⋅𝐫 𝑢𝐤 (𝐫) (4) By utilizing the periodicity of the wave function and potential, the plane wave expansion can be written as an infinite series, summed over the material's reciprocal lattice vector: 𝑉(𝐫) = ∑𝐆 𝑉(𝐆)𝑒 𝑖𝐆⋅𝐫 (5) 𝜓(𝐫) = ∑𝐆 𝑐𝐆 𝑒 𝑖(𝐤+𝐆)⋅𝐫 (6) Here, the parameters V(G) and c_G are Fourier coefficients that are entered into the lattice vector. By substituting equations (5) and (6) into the Schrödinger equation, we obtain: ∑𝐆 ′ 𝐻𝐆,𝐆′ 𝑐𝐆′ = 𝐸𝑐𝐆 (7) Equation (7) is then multiplied by the orthogonal function and integrated over the crystal volume, yielding the Kronecker delta function, which was previously exponential: ∑𝐆 ′ (𝐻𝐆,𝐆′ − 𝐸𝛿𝐆,𝐆′ )𝑐𝐆′ = 0 (8) In this equation, the summation over identical elements can be eliminated, yielding the final form: 𝐇𝐜 = 𝐸𝐜 (9) Equation (9) is the eigenvalue equation, where H is the Hamiltonian matrix. If the Fourier series is not limited, the matrix becomes infinite; however, the final calculation usually uses a 124 × 124 matrix. The Fourier expansion must be symmetric about the MICHAEL KURNIAWAN 3 Study Case origin for the obtained eigenvalues to be valid. To solve equation (10), the Hamiltonian matrix is diagonized using standard numerical methods. The eigenvalues of this matrix are the energies E(k), which must be calculated for each vector k in the band diagram. The following equation allows the calculation of each matrix element and the selection of the potential value V: ℏ2 𝐻𝐆,𝐆′ = 2𝑚 ∣ 𝐤 + 𝐆 ∣2 𝛿𝐆,𝐆′ + 𝑉(𝐆 − 𝐆′ ) (11) Since there are two atoms per cell in the FCC structure for Zincblende or Diamond materials, each atom contributes through the Cohen and Bergstresser convention, with the midpoint between the atoms as the starting point in FCC. These midpoints can be decomposed into symmetric and antisymmetric components: 𝐑𝑗 = 𝐑 0 ± 𝐝 (12) Here, R_0 is the absolute offset of each atom. The symbols V_s and V_a are empirically obtained form factors: V_s,V_a (calculated empirically) (13) Theoretical studies of semiconductor band gap energy using the empirical pseudopotential method were conducted by reviewing the characteristics of the calculation results, including for materials such as Si and Ge. The steps for calculating band gap energy using the pseudopotential method are: 1. Determine the form factor of the semiconductor material. 2. Enter the form factor into the calculation algorithm and process it using supporting software. The material form factor can be seen in the following table. TABLE 1. Pseudopotential Form Factors (in Rydberg), obtained from band split energy experiments MICHAEL KURNIAWAN 4 Study Case The calculation results will be visualized in the form of an energy band graph. RESULTS AND DISCUSSION Silicon (Si) is a semiconductor with an indirect band gap. The primary gap, which is the minimum gap, is calculated from the maximum valence band at the Γ point to the minimum conduction band along the 85% direction from the Γ to X distance. The calculation results show that the Si band gap is E_g^Si=1.08 eV, which is consistent with experimental findings. Germanium (Ge) is also an indirect band gap semiconductor, with the gap determined from the peak of the valence band at Γ to the minimum conduction band at point L. The calculation results show that the band gap of Ge is E_g^Ge=0.73 eV. For the direct band gap, which is measured from the maximum valence band at Γ to the minimum conduction band at Γ, the values obtained are 3.27 eV for Si and 0.82 eV for Ge, respectively. It should be noted that the curvature of the valence band of Ge is greater than that of Si, which is consistent with the fact that the effective hole mass in Si is greater than that in Ge. In addition, the inclusion of spin-orbit interactions will lift the triplet degeneracy in MICHAEL KURNIAWAN 5 Study Case the band at the Γ point, resulting in heavy hole and light hole bands with double degeneracy, while the split-off band moves to a lower energy by several tens of meV, depending on the material being analyzed. FIGURE 1. Silicon Band Gap FIGURE 2. Germanium Band Gap CONCLUSION Overall, the local empirical pseudopotential method described is quite effective for determining the band gap energy of semiconductors with good accuracy. However, as noted by Chelikowsky and Cohen [2], when this local method is used to calculate the electron density in the valence band, the results obtained are less satisfactory. This discrepancy arises due to the neglect of core electrons in the previous pseudopotential derivation, which allowed the use of a simple plane wave basis. To overcome this limitation, a non-local energy-dependent correction term was added to the local atomic potential. This approach increases the number of parameters used but results in better convergence and a more accurate description of the band structure. MICHAEL KURNIAWAN 6 Study Case REFERENCE MICHAEL KURNIAWAN 7