WO2011036952A1 - Electronic state calculation system and program - Google Patents

Abstract

Disclosed is a system capable of a primary theoretic high precision calculation of the electronic state of material having electrons with a high localization. A technique is provided for applying a pseudopotential method for the calculation of an interaction parameter Ueff value for localized orbitals, and by calculations applying the pseudopotential method using the Ueff value that has been calculated using the method, the electronic state of the material is calculated. More specifically, for the case in which the effective potential for localized electrons is minute compared to the DFT+U energy, the effective potential and the energy gap are calculated on the basis of the DFT+U calculation.

Description

Electronic state calculation system and program

The present invention relates to a simulation technique for obtaining the structure and properties of a material by numerical calculation. For example, the present invention relates to an electronic state calculation system and a program.

In recent years, first-principles methods that predict the structure and properties of materials at the atomic level have attracted attention, and many material simulation systems based on first-principles methods have been developed. The structure and properties of a material are determined by the quantum states of the nuclei and electrons that make up the material. The first principle method is a method for predicting the structure and properties of a material by approximately calculating the quantum state of electrons constituting the material. Material simulation based on the first-principles method is effective in designing materials without trial and error by experiment.

The first principle method includes a wave function method for obtaining a wave function of the entire electrons constituting the material and a density functional (DFT) method for calculating the electron density.

The wave function method is a method for approximately obtaining the wave function of the entire electrons constituting the material by considering the electron wave function as a variable describing the quantum state of the electrons constituting the material. There are various types of wave function methods, but the most basic one is the Hartley Fock (HF) method. In the HF method, one electron orbit is introduced for each electron constituting the material, and the wave function is represented by an anti-symmetrization product (Slater determinant) of the one electron orbit, and the expected energy value of the Slater determinant is minimum. In this way, one electron orbit is obtained to calculate the electronic state. Since each electron interacts with other electrons, the calculation of one electron orbit for each electron requires self-consistent calculation. The details of the wave function method including the HF method are described in Non-Patent Document 1.

On the other hand, the DFT method is a method of finding the electron density that minimizes the electron energy by considering the electron density as a variable describing the quantum state of the material electrons. It is guaranteed by the Hohenberg-Kohn theorem that the wave function method and the DFT method are equivalent.

The Kohn-Sham method is usually used for calculating the electron density. The Kohn-Sham method assumes that the electron density created by N Coulomb interacting electrons constituting the material is given by the electron density created by N independent quasielectrons (Kohn-Sham electrons). This is a method for obtaining the Kohn-Sham electron orbital that gives the electron density that minimizes. The explanation about the DFT method is described in Chapter 4 and Chapter 5 of Non-Patent Document 2.

Since the Kohn-Sham electron orbit is determined by the electron density, and the electron density is determined by the Kohn-Sham orbit, it is necessary to repeatedly calculate the electron density and the Kohn-Sham orbit. However, since the Kohn-Sham orbit is an independent quasi-electron orbit, self-consistent calculation of the orbit is not required as in the HF method. For this reason, the DFT method is easier to calculate than the HF method. Further, in the case of the DFT method, it is possible to take into account an electron correlation that is ignored in the HF method.

Thus, the DFT method is easier to calculate than the wave function method, and the electronic state of the material can be calculated in consideration of the effect of electron correlation. For this reason, the DFT method is currently employed in many material simulation systems.

However, the DFT method has a problem of self-interaction of Kohn-Sham electrons (Self-interactionSof Kohn-Sham ら れ electrons) that is not found in the wave function method including the HF method. An electron self-interaction is a non-physical interaction in which an electron interacts with itself.

Due to self-interaction, the DFT method tends to underestimate the solid band gap. For example, when calculated by the DFT method of local density approximation of silicon crystal, a value of 0.6 eV is given to 1.12 eV actually measured. In the case of a germanium crystal, the DFT calculation gives a value of 0.01 eV with respect to the actually measured 0.74 eV. Since band gaps determine the electrical and optical properties of solids, the DFT method often cannot correctly predict these properties. This problem is serious because the band gap is an important physical quantity that governs the physical properties of the material.

Also, due to self-interaction, the DFT method often cannot correctly describe chemical reactions. In the analysis of chemical reactions, the orbits of electrons located outside the molecule need to be calculated with high accuracy, whereas the Kohn-Sham electron orbitals tend to spread outside the actual orbitals due to self-interaction. This is because the state of electrons outside the molecule involved in the reaction cannot be described correctly.

Therefore, in recent years, studies have been made to correct such problems of the DFT method by correcting the self-interaction of Kohn-Sham electrons in the DFT method. There are various methods for reducing the self-interaction in the DFT method. Currently, the hybrid DFT method and the DFT + U method combining the DFT method and the HF method are most widely used.

The self-interaction in the DFT method is due to the fact that the exchange / correlation potential describing the quantum mechanical interaction of Kohn-Sham electrons is approximate.

The hybrid DFT method is a method for reducing the self-interaction by partially describing the exchange potential component for the Kohn-Sham electron from the HF exchange potential not including the self-interaction. As an example of the hybrid DFT method, there is a B3LYP method widely used in the field of quantum chemistry. The hybrid DFT is described in Chapter 6 of Non-Patent Document 2.

In the hybrid DFT method, the DFT exchange potential component for Kohn-Sham electrons is partially replaced with the HF exchange potential, but the replacement ratio must be determined based on experimental data. In the B3LYP method, the ratio is determined so as to reproduce the value of the G2 database, which is the molecular heat generation database.

On the other hand, the DFT + U method is a method of describing a d orbit or f orbit localized on an atom by an approximate HF method. HF calculation is performed using the interaction parameter U _eff between electrons. By describing d electrons and f electrons by the HF method, the self-interaction of these localized electrons can be removed. The DFT + U method is mainly used in the field of solid state physics to calculate strongly correlated materials. The theory of the DFT + U method is described in Non-Patent Document 3. Hereinafter, calculation processing based on the DFT + U method is referred to as DFT + U calculation.

The U _eff value of the DFT + U method is generally determined so as to reproduce the measured band gap, but can also be obtained by calculation. A method for obtaining the U _eff value by calculation is described in Non-Patent Document 4. The method of Non-Patent Document 4 is a method called a restricted DFT method. In the limited DFT method, as will be described later, DFT calculation is performed by changing the number of localized electrons, and a U _eff value is calculated as a secondary differential coefficient related to the number of electrons of energy. In this case, the U _eff value can be determined without using experimental values.

The DFT + U method is superior to the hybrid DFT method in that the calculation is possible without depending on the experimental value.

A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, MacMillan Publishing Co. Inc., New York (1982). Wolfram Koch and Max C. Holthausen, A Chemist ’s Guide to Density Functional Theory, Wiley-VCH, Weinheim (2000). V. I. Anisimov and A. I. Lichtenstein, Strong Coulomb Correlations in Electronic Structure Calculations; Beyond the Local Density Approximation, Gordon and Breach Science Publishers, Amsterdam (2000); Chapter 2. A. K. McMahan, R. M. Martin, and S. Satpathy, “Calculated effective Hamiltonian in La2CuO4 and solution in the impurity Anderson approximation”, Physical Review B, volume 38, No.10, pp.6650 ). C. Persson and A. F. da Siva, “Strong polarization effects on rutile TiO2 electronic band edges”, Applied Physics Letters, volume 39, No.10, pp.1708 (1989). M. Akasaka et al. “The theomoelectric properties of bulk crystalline n and p-type Mg2Si prepared by the vertical Bridgman method”, Journal of Applied Physics, volume 104, pp.013703 (2008). Materials Studio version 4.4, Accelrys, San Diego (2008). PHASE ver 8.0, Innovative Simulation Research Center, University of Tokyo Institute of Industrial Science (2009).

However, the method of Non-Patent Document 4 has a problem that it is not suitable for the pseudopotential method that is currently widely used. In this document, the U _eff value of an orbital localized on an atom is obtained using an all-electron state calculation method called the LMTO method. In the LMTO method, a material is divided into a spherical atomic region and an outer region of the atom, the electron's orbit in the atomic region is represented by an atomic orbit, and the electron's orbit in the outer region of the atom is represented by a plane wave. Is a method of calculating The LMTO method has a clear atomic orbit, and is suitable for the restricted DFT method in which the number of electrons in the local atomic orbital is changed.

On the other hand, in the pseudopotential method, since the electron orbit is expanded using a plane wave, the concept of the atomic orbit is not clear. For this reason, it is difficult to calculate the U _eff value by the limited DFT method in the pseudopotential method, and in the DFT + U calculation by the pseudo potential method, an empirical U _eff value is often used for the local atomic orbitals. As a result, there is a problem that the calculation is arbitrary. However, the pseudopotential method can be calculated in consideration of only the valence electrons of the atoms constituting the material. In other words, the pseudo-potential method has the feature that the potential of valence electrons can be calculated in a first-principle manner, and can perform highly accurate calculation in a shorter time than the LMTO method. For this reason, the pseudopotential method is used in many material design calculations.

In view of the above technical background, the present inventor newly provides a method of applying the pseudopotential method to the calculation of the interaction parameter U _eff value for the localized electron orbit, and the U _eff value calculated using the method. The technology to calculate the electronic state of a material (solid) by the pseudopotential method using More specifically, the present invention provides a method for calculating the effective potential and the energy gap based on the pseudopotential calculation when the effective potential is small compared to the pseudopotential energy.

According to the present invention, the effective potential of an electron system having a localized electron orbit can be calculated with high accuracy without actually measuring the energy gap.

Diagram for explaining the calculation of the interaction parameter U _eff value at a 3d orbital of Ti which constitute the rutile TiO _2. It shows the calculated electron band structure of rutile TiO ₂ using the technique of the present invention. It shows the calculated electron band structure of rutile TiO ₂ using conventional techniques. The figure explaining comparatively the band gap value calculated about various transition metal oxides and rare earth oxides. The figure which shows the system configuration example of an electronic state calculation system. The flowchart explaining the calculation procedure of interaction parameter _Ueff value. The figure which shows the example of a display of input data GUI. The figure which shows the example of a display of calculation result GUI. The figure which shows the hardware structural example of an electronic state calculation system.

Hereinafter, embodiments of the present invention will be described with reference to the drawings. It should be noted that the contents of the device configuration and processing operation described below are examples for explaining the invention, and the present invention relates to an invention in which a known technology is combined with the device configuration and processing operation described later, and the device configuration and processing operation described later. It also includes inventions that partially replace known techniques.

(Conceptual composition)
In the following, s electrons and p electrons with high itinerant (delocalization) among the electrons constituting the material (solid) are handled by the density functional (DFT) method and localized to the atoms constituting the material (solid). In the DFT + U electronic state calculation system (program) that handles the d-electrons and f-electrons to be converted by the Hartree Fock wave function method, the interaction parameter U _eff between the localized electrons required for the DFT + U calculation is first-principles (numerical) A DFT + U electronic state calculation system (program) that can calculate the electronic state of a material (solid) by calculating the following.

However, the Hartley Fock method here means an approximate Hartley Fock method that considers only the interaction between d electrons and f electrons belonging to the same atom. The DFT + U electronic state calculation system also includes a graphical user interface (GUI) used for inputting data necessary for calculation and displaying the calculation result, an arithmetic unit for executing the calculation, and data and calculation results necessary for the calculation. It is assumed that a storage device that stores data is included.

The DFT + U electronic state calculation system (program) here expresses the effect of the inner core electrons of the atoms constituting the material (solid) on the valence electrons as a pseudo-potential, and expresses the wave function of the valence electrons as a plane wave. It is assumed that it has a function of calculating an electronic state of a material (solid) by expanding into a linear combination. That is, the DFT + U electronic state calculation system (program) ignores the inner core electrons of atoms and handles only valence electrons.

The electronic state calculation system (program) has pseudo-potential data of atoms required for calculation, and has a function of automatically setting the pseudo-potential data as input data at the time of calculation execution. It is preferable.

The electronic state calculation system (program) also provides information on the crystal structure of a solid (lattice constant, space group, atomic species and position) required for the calculation and information on atoms having localized d electrons or f electrons. It is preferable that a computer has a function of inputting through a mouse, keyboard, or other input device, and a function of displaying a solid electronic band structure and / or band gap on the operation screen after completion of the calculation.

The electronic state calculation system (program), by performing DFT + U calculated stepwise changing the interaction parameter u _eff value of the localized electrons near zero, the U _eff value for localized electrons, the DFT + U calculations It is preferable to calculate the second derivative of the DFT component of the electron energy defined as the difference between the total electron energy component obtained and the Hartley Fock electron energy component represented by the number of localized electrons. Here, u _eff is an interaction parameter between localized electrons as a variable. Further, the u _eff value for the localized electrons is preferably changed stepwise by 0.1 eV in a range of −0.5 eV to 0.5 eV, 0 eV to 0.5 eV, or −0.5 eV to 0 eV.

Further, the electronic state calculation system (program) uses the DFT component E _DFT of the electron energy, the number of localized electrons as n, and the least square parameters A, B and C as E _DFT (n) = An (n−1 ) When expressed as a function of + B + C, it is preferable to calculate the secondary differential coefficient regarding the number of localized electrons of the DFT component E _DFT as 2A. However, electronic state calculation system (program), the second derivative about the localized electrons the number of DFT components E _DFT, may be calculated as the difference of the localized electron number n of the electron energy of the DFT components E _DFT.

The localized electrons in the electronic state calculation system (program) described above are preferably d electrons of transition metal elements or f electrons of rare earth elements. In other words, the material (solid) in the electronic state calculation system (program) described above is preferably a transition metal oxide or a rare earth oxide. Further, the electronic state calculation system (program) described above is suitable for application to a material design system for a high-k gate insulating film of a CMOS transistor, a thermoelectric conversion material, and a metal oxide catalyst.

(Expected effect)
As described above, in the electronic state calculation system (program) proposed in this specification, the interaction parameter U _eff for the localized electron orbit is calculated in the first principle for the pseudopotential method, and the calculated interaction parameter is calculated. By executing DFT + U calculation by the pseudopotential method using U _eff , the electronic state of a solid having a localized electron orbit can be calculated in a first principle. As a result, the electronic state in the transition metal oxide or rare earth oxide can be calculated more accurately than in the conventional method.

On the other hand, in the calculation of the electronic state by the conventionally used DFT method, there is a problem that the band gap of the material is underestimated compared to the actual measurement value due to the self-interaction of Kohn-Sham electrons. Further, in the conventional technique, the DFT + U method based on pseudo-potential method, an interaction parameter U _eff values used for calculation can not be determined in the first principle, it is necessary to use empirical U _eff value. For this reason, the latter conventional method has a problem in which it is inevitable that the calculation result is arbitrary.

Thus, the first-principles calculation based on the pseudopotential method that can be used conventionally has a problem that the electronic state of the material cannot be accurately described.

On the other hand, in the case of the electronic state calculation system proposed herein (program), and enabling the determination of the U _eff value first principle for orbital localized on the atom by pseudopotential method, precisely by DFT + U method It is possible to calculate the electronic state.

(Example)
The electronic state calculation system (program) proposed in this specification has a DFT + U electronic state calculation program based on a pseudopotential method, a GUI interface necessary for calculation, an arithmetic device, and a storage device as a basic configuration. A known technique can be used for these basic components. The characteristic structure of the electronic state calculation system (program) proposed in this specification is the loading of a function for first-principles calculation of interaction parameter U _eff values for localized electrons required for DFT + U calculation. .

The DFT + U electronic state calculation program described above expresses the effect of the core electrons of atoms constituting the solid on the valence electrons as a pseudo potential for the valence electrons, and expands the orbit of the valence electrons into a linear combination of plane waves. It is a program of electronic states based on the pseudopotential method for calculating electronic states. The pseudo-potential method can calculate the electronic state of a material with high accuracy at a relatively low calculation cost.

The calculation of electronic states by the pseudopotential method requires atomic pseudopotential data. Since the pseudopotential data of an atom differs for each atomic species, the system (program) proposed in this specification automatically uses the potential data necessary for the calculation as input data for the electronic state calculation program from the pseudopotential data library. Equipped with the function to set to.

In the case of the system (program) proposed in this specification, information on the crystal structure of a solid (lattice constant, space group, atomic species and position) and localized d-electrons or f-electrons necessary for calculating the electronic state A function to set the position information of atoms through the screen displayed on the computer display, and a function to display the solid electronic band structure and band gap calculated after the calculation on the computer display (GUI function) is doing. This GUI function allows a user to input material data necessary for DFT + U electronic state calculation on a computer display. Note that an operation input device such as a mouse, a keyboard or the like is used for the operation input required for the setting.

As described above, the greatest feature of the system (program) proposed in this specification is that the U _eff value for localized electrons can be calculated in a first principle by the pseudopotential method. This point will be described in detail below.

The DFT + U method is a method in which s electrons and p electrons with high itinerary are described by the DFT method, and d electrons and f electrons with high localization are described by the HF method. The U _eff value is a parameter that describes the interaction between these localized electrons. The U _eff value can be calculated using a method called a restricted DFT method as described in Non-Patent Document 4. In Non-Patent Document 4, the U _eff value is calculated using Equation 1 shown below.

Here, E is the total electron energy of the material calculated by the restricted DFT method, const is a constant, ε ₀ is the energy of localized electrons calculated by the DFT method, n is the number of localized electrons, and O (n ³ ) Is a third-order negligible term for n.

The limited DFT method is a method of calculating the electronic state of a material by the DFT method by changing the number of localized electrons n. In the normal DFT calculation, n is determined so that the electron energy is minimized, but the limited DFT method is characterized in that the DFT electronic state calculation is performed using n as an external parameter. In this case, the interaction parameter U _eff value is calculated as a second derivative with respect to the number of localized electrons n of the electron energy E.

In Non-Patent Document 4, a limited DFT calculation is performed using the LMTO method, and a U _eff value is obtained using Equation 1. In the LMTO method, since the electron orbit of an atom is represented by a localized atomic orbital, it is easy to calculate an electronic state in which n is changed. In Non-Patent Document 4, the limited DFT calculation is performed so that the localized electron orbit in the atomic region does not hybridize with other delocalized orbits.

However, in the pseudopotential method, a delocalized plane wave is used to represent an electron trajectory, so that it is not easy to perform a limited DFT method that calculates an electronic state by changing the number of localized electrons. For this reason, when the DFT + U calculation is performed by _{obtaining the} U _eff value by the pseudo-potential method, it is necessary to perform the calculation using an empirical U _eff value. As a result, the calculation is arbitrary and it is difficult to perform the DFT + U calculation on the first principle.

Therefore, in the system (program) proposed in this specification, this problem is solved by executing approximate limited DFT calculation by the suspicion potential method.

In general, it is known in the DFT + U calculation that n changes by changing the U _eff value for the local orbit. Therefore, if E (n) can be obtained in some form from the DFT + U calculation when U _eff is changed, the U _eff value can be calculated by Equation 1.

It is known that the electron energy E _{DFT + U} (n) obtained by the DFT + U calculation is given by the following equation.

Here, E _DFT (n) is the DFT component of E _{DFT + U} (n), and the second term on the right side is the Hartley Fock component of E _{DFT + U} (n). Rewriting Equation 2 with U _eff as a variable yields Equation 3.

Here, u _eff is a U _eff value as a variable, and n ′ is n as a variable. When u _eff = 0, the DFT + U calculation coincides with the DFT calculation, and E _DFT (n ₀ ) = E (n ₀ ). n ₀ is the number of localized electrons obtained by DFT calculation, and E is the energy obtained by DFT calculation. When u _eff ≠ 0, generally E _{DFT + U} (n ′) ≠ E (n ′), but when u _eff is close to 0, approximate by E (n ′) ≈E _DFT (n ′) it can.

In the system (program) proposed in this specification, the U _eff value is obtained according to Equation 4 from n ′ and E _DFT (n ′) obtained by performing DFT + U calculation by slightly changing u _eff near 0. I did it.

Here, n ′ is the number of electrons in a localized orbit obtained by DFT + U calculation using a small u _eff value, E _DFT (n ′) is a DFT component of electron energy obtained by calculation, and U _eff is localized. U _eff value for electrons. The U _eff value is calculated as a second derivative with respect to n ′ of E _DFT (n ′). Here, the u _eff value and U _eff are different.

The second-order differential calculation is performed by calculating E _DFT (n ′) by a quadratic function of n ′ using the least square method and calculating a second-order differential coefficient with respect to n ′ of the function, or E _DFT (n ′ ) To obtain a numerical difference by n ′.

Hereinafter, application examples of the system (program) proposed in this specification will be shown. Here, rutile TiO ₂ will be described. Rutile TiO ₂ is composed of titanium (Ti) atoms and oxygen (O) atoms, and the Ti atoms have localized 3d orbitals. The u _eff value for the 3d orbital of Ti atoms that make up the rutile TiO ₂ from -1.0 eV 1.0 eV to the electron number n 'of the 3d orbital of Ti in the case where by changing the 0.1eV every performed DFT + U calculations of the electron energy The relationship of the DFT component E _DFT (n ′) is shown in Table 1. The exchange correlation function used for the calculation is LDA. It can be confirmed that n ′ and E _DFT (n ′) change when the u _eff value is changed. The change width is not limited to 0.1 eV, and may be 0.05 eV or 0.001 eV.

FIG. 1 plots the relationship between the number of electrons n ′ and the DFT component E _DFT (n ′) in Table 1. The Y axis in FIG. 1 is E _DFT (n ′) − E _DFT (n′0). n′0 is the value of n ′ when the value of u _eff is 0. Each data point in the plot is based on a DFT + U calculation with a different u _eff . Here, a quadratic function (E _DFT (n ′) = An ′) in which the DFT component E _DFT (n ′) of the electron energy is expressed using the number of localized electrons n ′ and the least square parameters A, B, and C. When (n′−1) + Bn + C) is fitted to the least squares, the following function is obtained.

Here, the least square parameters A, B and C are 5.3309, −24.142 and 40.738, respectively. At this time, if both sides of Expression 4 are second-order differentiated with respect to n ′, the following expression is obtained.

From Equation 6, approximately 10.66 is obtained as the U _eff ′ value of the 3d orbit of Ti. In this example, the u _eff value is changed from −1.0 eV to 1.0 eV to perform DFT + U calculation, and the U _eff value for the 3d orbit of Ti is calculated from the obtained n ′ and E _DFT (n ′). However, it may be subjected to the same calculations while changing the U _eff value u _eff value from -0.5eV to 0.5 eV. Also in this case, about 10.66 eV is obtained as the U _eff value.

Further, when the DFT + U calculation is performed by changing the u _eff value from 0.0 eV to 0.5 eV or from −0.5 eV to 0.0 eV, 10.42 eV or 10.56 eV is obtained as the U _eff value, respectively.

The U _eff value obtained in this way is very close to the empirical U _eff value (10 eV) for the 3d orbit of Ti constituting rutile TiO ₂ described in Non-Patent Document 5.

As described above, the system (program) proposed in this specification performs the DFT + U calculation by changing the u _eff value discontinuously near 0. For this reason, the number of changes in the u _eff value can be reduced, and the amount of DFT + U calculation can be reduced. That is, the required calculation amount can be reduced.

On the other hand, if the number of changes is decreased, the data points used for calculating the second derivative are also reduced, which increases the possibility of an error in the derivative calculation.

Therefore, in the system (program) proposed in this specification, in order to reduce the calculation amount and simultaneously reduce the error, the u _eff value is changed in steps of 0.1 eV from −0.5 eV to 0.5 eV by DFT + U. Calculation was performed to obtain the U _eff 'value for the localized orbitals. Also, so as to obtain a U _eff 'values for localized trajectory by changing the u _eff value from 0eV or 0.0eV from -0.5eV to 0.5 eV. In this way, by using the u _eff value near 0 (zero), it is possible to reduce the calculation error while suppressing the calculation amount small.

In these examples, the least-square fitting function shown in Equation 5 is analytically calculated to obtain the U _eff value, but Equation 5 is calculated by the difference method using the data in Table 1 to obtain the U _eff value. It is also possible. When the u _eff values shown in Table 1 are −0.1 eV, 0.0 eV, and 0.1 eV, the differential method of Equation 6 is evaluated by Equation 7 using n ′ and E _DFT (n ′). About 10.67 eV is obtained as the U _eff 'value of the 3d orbit of Ti. This value is approximately equal to the value 10.67 eV obtained by analytically calculating Equation 5 using the least square fitting function of Equation 5.

　　　…（式７）

... (Formula 7)

Here, E _DFT (n ′) _{Ueff = 0.1 eV} and n ′ _{Ueff = 0.1 eV} are an E _DFT (n ′) value and an n ′ value when the u _eff value is 0.1 eV, respectively. The same applies to other variables corresponding to the cases where the u _eff values are −0.1 eV and 0.0 eV.

When U _eff value for the 3d orbital of the obtained Ti by calculation electronic state of rutile TiO ₂ using (10.66eV) calculated by LDA + U pseudopotential method, the band structure shown in FIG. 2 is obtained. The band gap Eg in FIG. 2 is 2.98 eV. This value is very close to the measured band gap (3.0 eV).

On the other hand, FIG. 3 shows the band structure when the existing LDA pseudopotential method is used to calculate the band gap Eg. In the case of FIG. 3, the hand gap Eg is 1.66 eV, which is about half that of the actually measured gap.

Thus, when using the electronic state calculation system proposed herein (program), numerically determine the U _eff value against localized electrons by pseudopotential method, the band gap of the material with the result It can be seen that Eg can be accurately predicted. Such a calculation method is unprecedented. Therefore, the electronic state calculation system (program) proposed in this specification is based on the DFT method that underestimates the band gap Eg of the material, the DFT + U method that uses empirically required U _eff, and other conventional techniques. It turns out that it is excellent as a state calculation method.

In the above description, the method of calculating the U _eff value of the 3d orbital of Ti has been described using rutile TiO ₂ as an example. However, when the solid to be calculated is a rare earth oxide, The U _eff value for the 4f orbit of rare earth elements can be calculated using the proposed electronic state calculation system (program). For example, when the U _eff value for 4f electrons of La in hexagonal La ₂ O ₃ is similarly calculated, 9.3 eV is obtained. When the band gap Eg of 6 HoAkiragata La ₂ O ₃ by using the U _eff value calculated by DFT + U method, a value of 4.4eV is obtained. This value is very close to the actually measured value of 4.3 eV.

As described above, by using the electronic state calculation system (program) proposed in this specification, U _eff values for localized electrons of transition metal oxides and rare earth oxides can be calculated in principle. It can be seen that the electronic state can be accurately calculated by the DFT + U method.

In FIG. 4, the electronic states of various transition metal oxides and lanthanum oxides (La ₂ O ₃ ), which are rare earth oxides, are calculated by the electronic state calculation system (program) proposed in this specification and obtained by calculation. The result of having compared the band gap Eg with the measured value is shown. Note that the correspondence between the calculation result obtained by the electronic state calculation system (program) proposed in this specification and the actually measured value is indicated by ■ (squares shown in solid) in FIG. For the calculation of TiO ₂ , LDA was used for the DFT exchange correlation potential, and for the calculation of other substances, GGA was used for the potential. The calculation of the localized electron of the transition metal oxide was performed for the d orbital, and the calculation of the localized electron of the lanthanum oxide was performed for the f orbital.

In the electronic state calculation system (program) proposed in this specification, U _eff values for these localized electrons are calculated on a first principle basis, and the electronic state of the material (by the DFT + U method) using the calculated U _eff value ( The band gap Eg) was calculated. In the actual calculation, as described above, u _eff was changed by 0.1 eV in the range of −0.5 eV to 0.5 eV.

FIG. 4 also shows the calculation result of the band gap Eg by the DFT method which is the prior art. In FIG. 4, the correspondence between the calculation result obtained by the conventional method and the actual measurement value is indicated by ○ (white circle). As shown in the figure, it can be seen that the band gap Eg by the DFT method always tends to underestimate the actual measurement value except in the case of hexagonal La ₂ O ₃ .

On the other hand, the calculation results of the DFT + U electronic state by the electronic state calculation system (program) proposed in this specification correctly reproduce the measured values of the band gaps of various transition metal oxides and rare earth oxides. It can also be confirmed from FIG.

Thus, the electronic state calculation system (program) proposed in the present specification can calculate the electronic states of transition metal oxides and rare earth oxides more accurately than in the prior art.

The electronic state calculation system (program) proposed in this specification is more accurate in comparing the electronic states of high-k gate insulating films, oxide-based thermoelectric conversion materials, and metal oxide catalysts for Si transistors compared to the prior art. Can be calculated and applied to the design of these materials.

For example, as seen in FIG. 4, La ₂ O ₃ is a high-k material by DFT method is prior art, monoclinic HfO _2, when calculating the band gap Eg of the cubic HfO _2, calculated values are actually measured values In addition, the magnitude relationship of the measured gap between materials is not accurately reproduced. On the other hand, in the case of the electronic state calculation system (program) proposed in this specification, the magnitude relationship of the measured gap between materials can be accurately reproduced. Therefore, it is effective in predicting the band gap Eg, which is a measure of the insulating property of the high-k material.

The Seebeck coefficient of the thermoelectric conversion material also depends sensitively on the band gap Eg. For this reason, the electronic state calculation system (program) proposed in this specification is effective even when used for the design of thermoelectric conversion materials. The Seebeck coefficient can be calculated from the electron velocity as shown in Non-Patent Document 6.

If this known technique is applied, the electronic state of the material can be calculated by the electronic state calculation system (program) proposed in this specification, and the Seebeck coefficient of the material can be obtained based on the calculated electronic state.

Moreover, in the electronic state calculation system (program) proposed in this specification, the state of d electrons localized in atoms can be described more accurately than in the conventional method. For this reason, it is possible to accurately describe the d-electron state of the transition metal oxide catalyst and more accurately predict the catalytic activity due to the d-electron.

(Example)
Below, the Example of the electronic state calculation system (program) mentioned above is described. FIG. 5 is a system configuration of the electronic state calculation system according to the embodiment. The electronic state calculation system according to the embodiment is executed on a computer including an arithmetic device that executes calculation, a storage device that stores data and calculation results necessary for calculation, and a user interface and other input devices. Realized by a program. FIG. 5 functionally represents the system configuration of the electronic state calculation system.

Therefore, the electronic state calculation system shown in FIG. 5 includes a main system 1, a GUI 2, a material database 3, a pseudopotential (PP) database 4, an input file 5, a U _eff calculation subsystem 6, a DFT + U electronic state calculation program 7, and an output file. 8 is configured. In the figure, the solid line shows the functional configuration of the system, and the broken line shows the flow of calculation data.

The main system 1 corresponds to the main program of the electronic state calculation system according to the embodiment. The main system 1 manages the entire electronic state calculation process. The GUI 2 is a program that supports creation of the input file 5 required by the DFT + U electronic state calculation program 7. As will be described later, the GUI 2 includes structural data (lattice constant, space group, atomic coordinates, and atomic species) required by the DFT + U electronic state calculation program 7, input data related to DFT + U electronic state calculation (type of exchange correlation potential, Provided is a function for displaying on the computer display an operation screen for selectively or descriptively inputting the type of localized orbit and input related to the calculation of U _eff value for the localized orbit. An input file 5 is generated based on data input or set through the operation screen.

The material database 3 is a database in which a crystal structure of a known material is stored in a storage device. The crystal structure data stored in the material database 3 is extracted through the operation of the GUI 2 by the user, and is written in the input file 5 as an input value of the material structure to be calculated.

The pseudo-potential (PP) database 4 is a database that stores the pseudo-potentials of atoms required by the DFT + U electronic state calculation program 7 when calculating electronic states. In the case of this embodiment, the above-described GUI 2 has a function of automatically selecting pseudo-potential data of atoms included in the material input by the user from the pseudo-potential (PP) database 4. Of course, the user can also select individually, but it is desirable to mount an automatic selection function from the viewpoint of operability. In any case, the file name of the selected pseudopotential data is written in the input file 5. Note that the function of the GUI 2 can be configured using a known technique. There is Non-Patent Document 7 as a known technique for realizing a function related to GUI2.

The U _eff calculation subsystem 6 is realized as a program that automatically calculates the U _eff value necessary for the DFT + U calculation or the LDA + U calculation using the above-described method. This U _eff calculation subsystem 6 corresponds to the second arithmetic unit in the claims. In this embodiment, U _eff computing subsystem 6, obtained through GUI2 data necessary for calculation of U _eff value. Also, U _eff computing subsystem 6 writes the u _eff values necessary for the calculation of U _eff values in the input file 5. Here, the u _eff calculation subsystem 6 is given u _eff values for localized electrons in the range of −0.5 eV to 0.5 eV, or in the range of 0 eV to 0.5 eV, or in the range of −0.5 eV to 0 eV. In this embodiment, −0.5 eV to 0.5 eV is set as the default value of the setting range. The U _eff calculation subsystem 6 sequentially executes a DFT + U calculation or an LDA + U calculation in which the u _eff value for the local orbit is set, and the DFT component (pseudopotential component) and localization of the electron energy at each u _eff value. Based on the relationship between the number of electrons in the orbit, that is, based on Equation 6 or Equation 7, the U _eff value for the localized electrons is calculated. After calculating the U _eff value, the U _eff calculation subsystem 6 writes the U _eff value for the determined localized electron in the input file 5. Further, the U _eff calculation subsystem 6 monitors the input file 5 and activates the DFT + U electronic state calculation program 7 when it is confirmed that all data required by the DFT + U electronic state calculation program 7 is described.

The DFT + U electronic state calculation program 7 is an electronic state calculation program based on a pseudopotential method having a DFT + U calculation function. The DFT + U electronic state calculation program 7 corresponds to the first arithmetic unit in the claims. The DFT + U electronic state calculation program 7 reads the structural data related to the material structure, the input data related to the electronic state, the u _eff value as a variable, and the pseudo potential data file name from the input file 5, and then refers to the pseudo potential data file name. The corresponding pseudopotential data is read from the pseudopotential (PP) database 4. Thereafter, the DFT + U electronic state calculation program 7 calculates the DFT + U electronic state based on these data. For the DFT + U electronic state calculation program 7, a known technique can be used. For example, the program of Non-Patent Document 78 can be used.

The DFT + U electronic state calculation program 7 outputs the calculation result of the DFT + U electronic state calculation to the output file 8. The output file 8 is read through the GUI 2 and presented to the user who operates the computer display as a calculation result.

FIG. 6 shows an outline of a processing procedure until the U _eff value of the localized electron is calculated by the U _eff calculation subsystem 6. First, the structure of the material whose electronic state is to be obtained is input through GUI 2 (step S1). Next, localized electrons are designated through GUI 2 (step S2). In the case of this example, transition metal oxides and rare earth oxides are targeted. Accordingly, the transition electron d-electron and the rare-earth element f-electron orbit are designated through the GUI 2 as localized electron orbitals (step S3).

Subsequently, a range of u _eff values used for calculation of the U _eff value for the local orbit is input (step S4). As described above, the recommended range is -0.5 eV to 0.5 eV, or 0 eV to 0.5 eV, or -0.5 eV to 0 eV. The default range is -0.5eV to 0.5eV.

When these values are input to the input file 5 through the GUI 2, the U _eff calculation subsystem 6 performs a DFT + U calculation using the u _eff values in the specified range, and performs a DFT component E _DFT (n ') Is least squares fitted with a quadratic function of n (step S5).

When a quadratic function that fits the least squares (for example, Equation 5) is obtained, the U _eff calculation subsystem 6 obtains the second derivative A with respect to n ′, and based on this value, calculates the U _eff value (= 2A). Calculate (step S6). FIG. 6 shows a flowchart for _obtaining the U _eff value as the second derivative of the fitting function. However, as described above, U _eff is calculated using the difference method (Equation 7) without fitting. May be.

FIG. 7 shows an example of an input screen of GUI2. The input screen shown in FIG. 7 includes input GUI windows 9, 10, and 11. Among these, the input GUI window 9 is a window for inputting and displaying a material structure. In this embodiment, the material structure is displayed three-dimensionally. In FIG. 7, the structure of rutile TiO ₂ is displayed.

The input GUI window 10 is a window for inputting material structure data. In this embodiment, a lattice constant (Lattice), a space group (Space Group), and an atom position parameter (Positional Parameter) can be input. In the input GUI window 10, material structure data is input with a keyboard. FIG. 7 is an input example of the structure data of rutile TiO ₂ . Incidentally, in the case of FIG. 7, the positional parameters of titanium (Ti) and oxygen (O) are directly input by the keyboard.

The input GUI window 11 is a window for DFT + U calculation input. In this embodiment, selection of an exchange correlation functional (DFT), specification of a localized orbital (Localized Orbital), specification of a U _eff value, specification of a calculation method, variable range and variable step of a u _eff value may be input. it can. In the case of FIG. 7, radio buttons are used for selection, and direct input using a keyboard is used for specifying values. FIG. 7 shows a state in which LDA is selected as the exchange correlation functional (DFT), and 3d orbitals of Ti atoms constituting rutile TiO ₂ are selected as localized orbitals. Also, selectable options for the U _eff value (U _eff ) include direct input by the user (Input) and calculation by the U _eff calculation subsystem 6 (Calc.). FIG. 7 shows a state in which the calculation option is selected from the two. As described above, the calculation method can be selected from the method using the least square method (Least-Square) and the difference method (numerical). In FIG. 7, the method using the least square method is selected. In FIG. 7, the range of the u _eff value is −0.5 eV to 0.5 eV, and the step width of the u _eff value is set to 0.1 eV. These settings are default values and are recommended settings for u _eff . Of course, the user can directly input these values.

FIG. 8 is an example of a GUI output screen after the calculation is completed. The output screen shown in FIG. 8 includes output GUI windows 12 and 13. The output GUI window 12 is a band structure display window. On the other hand, the output GUI window 13 is a state density display window. In the case of FIG. 8, the calculation result (10.67 eV) of the U _eff value for the 3d orbit of Ti constituting rutile TiO ₂ and the DFT + U band structure using the value are shown. The DFT + U band structure is shown with the position and value of the band gap Eg. The band gap Eg is a direct transition type at point A, and the gap value is 2.98 eV. The output GUI window 13 shows the total state density (total), the local state density (Ti) of the titanium (Ti) site, and the local state density (O) of the oxygen (O) site.

FIG. 9 shows a hardware configuration example of the electronic state calculation system according to the embodiment. As described above, the electronic state calculation system (program) according to the embodiment can be realized as the computer 17 having the arithmetic device and the storage device and its peripheral devices (computer display 14, keyboard 15, mouse 16). A computer program for realizing the system shown in FIG. 5 is stored in a hard disk (storage device) in the computer 17. Further, calculation processing and file input / output processing necessary for realizing the system shown in FIG. 5 are performed by a memory (storage device) and an arithmetic unit constituting the computer 17. The GUI window as shown in FIGS. 7 and 8 is displayed on the screen of the computer display 14. In the case of the hardware configuration shown in FIG. 9, a keyboard 15 and a mouse 16 are used for data input operations (including not only alphanumeric input but also selection input) on the GUI screen shown in FIG. .

DESCRIPTION _{OF SYMBOLS} 1 ... Main system, 2 ... Graphical user interface (GUI), 3 ... Material database, 4 ... Pseudopotential database, 5 ... Input file, 6 ... _Ueff calculation subsystem, 7 ... DFT + U electronic state calculation program, 8 ... Output file , 9 ... Input GUI window, 10 ... Input GUI window, 11 ... Input GUI window, 12 ... Output GUI window, 13 ... Output GUI window, 14 ... Computer display, 15 ... Keyboard, 16 ... Mouse, 17 ... Computer.

Claims (13)

Based on the DFT + U method, which treats highly itinerant (non-localized) electrons with density functional (DFT) and treats highly localized electrons with Hartley Fock wavefunction, A first computing device for calculating;
An input device for inputting data necessary for calculation by the DFT + U method;
An output device for outputting a calculation result by the first arithmetic unit;
A storage device for storing the data and the calculation result;
Prior to the calculation by the first arithmetic unit, the interaction parameter U _eff between electrons having high localization of the specified input material is calculated in a first principle by the pseudopotential method, and the calculation result is calculated by the first calculation. An electronic state calculation system, comprising: a second arithmetic device to be given to one arithmetic device. The second arithmetic unit is:
The DFT + U method calculation is performed for each interaction parameter u _eff by changing the interaction parameter u _eff value of highly localized electrons stepwise near zero, and the DFT + U electron energy and heart obtained by the calculation are executed. The interaction parameter U _eff for a highly localized electron is calculated as the second derivative of the DFT component of the DFT component of the electron energy defined as the difference between the Reefock electron energy, and the calculated interaction parameter electronic state calculation system according to claim 1, characterized in providing a U _eff value to the first computing device. The electronic state calculation system according to claim 2, wherein the interaction parameter u _eff is an interaction parameter between localized electrons as a variable. The interaction parameters u _eff value for a high localized electrons, characterized by the range of 0.5eV from -0.5, or 0 to 0.5eV range, or -0.5 be gradually changed within the range of 0eV from The electronic state calculation system according to claim 2. The DFT component E _DFT of the electron energy, localized electron number n and least squares parameter A, with B and _{C, E DFT (n) =} An (n-1) + Bn + C, when expressed as, the secondary The electronic state calculation system according to claim 2, wherein the differential coefficient is calculated as 2A. When the DFT component E _DFT of the electron energy is expressed as E _DFT (n) = An (n−1) + Bn + C using the number of localized electrons n and the least square parameters A, B and C, the second derivative The electronic state calculation system according to claim 2, wherein the coefficient is calculated by calculating a difference between _EDFT and n. The electronic state calculation system according to claim 1, wherein the highly localized electrons are d electrons of a transition metal element or f electrons of a rare earth element. The electronic state calculation system according to claim 1, wherein the designated input material is a transition metal oxide or a rare earth oxide. The high-k gate insulating film, the oxide-based thermoelectric conversion material, or the metal oxide catalyst material is designed based on the electronic state calculated for the specified input material. Electronic state calculation system. The first arithmetic unit includes:
The effect of the core electrons of the atoms constituting the material on the valence electron is expressed as a pseudo potential for the valence electron, and the electronic state of the material is calculated by expanding the orbit of the valence electron into a linear combination of plane waves. The electronic state calculation system according to claim 1, wherein: The storage device has pseudo-potential data of atoms necessary for the DFT + U calculation,
2. The electronic state calculation system according to claim 1, wherein the first arithmetic device automatically acquires the pseudo-potential data related to a designated input material from the storage device. 3. The input device is used to input information on the crystal structure (lattice constant, space group, atomic species and position) of a material specified and input as data necessary for calculation and information on atoms having highly localized electrons. ,
The electronic state calculation system according to claim 1, wherein the output device outputs an electronic band structure and a state density of a material designated and input as a calculation result. On the computer,
The electronic state of a specified input material is calculated based on the DFT + U method in which high itinerant (delocalization) electrons are handled by density functionals and high localization electrons are handled by Hartley Fock wave functions. 1 arithmetic processing,
A process of accepting input of data necessary for calculation by the DFT + U method;
A process of outputting a calculation result by the first calculation process;
Processing to store the data and the calculation result in a storage device;
Prior to the calculation by the first arithmetic processing, the interaction parameter U _eff between the electrons having high localization of the specified input material is calculated in a first principle by the pseudo-potential method, and the calculation result is calculated by the first calculation. A computer program for executing a second arithmetic processing given to one arithmetic device.

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