JPH0644210A - Method for finding band gap of crystal - Google Patents

Abstract

PURPOSE:To precisely find out the band gap of crystal of various materials including an unknown material in a short time without actually manufacturing the crystal. CONSTITUTION:Cluster calculation is carried out by using a molecule track method by a DV-Xalpha method (discrete calculus of variation for local density approximating method) for a cluster selected for the crystal whose band gap is to be found, thereby finding the band gap of the crystal. For this cluster calculation, it is considered that all atoms of the same kind in the cluster are equivalent in charge density distribution and when an atom track function as the base function of the molecule track method by the DV-Xalpha method is found, the value of the coefficient alpha of a term of electron-electron exchanging operation is adjusted.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for determining the band gap of a crystal, and is suitable for use in, for example, predicting the band gap of a new substance.

[0002]

2. Description of the Related Art Conventionally, when a band gap of a new substance is obtained, the substance is actually prepared and the band gap is directly measured by an optical absorption method, or the band gap is calculated by the first principle band calculation or cluster calculation. Was calculated.

[0003]

The method for actually measuring the bandgap by the above-mentioned optical absorption method can accurately determine the bandgap, but prior to the measurement, the substance is converted into a highly pure and clean crystal. Since it needs to be manufactured, it is time-consuming and expensive. In particular, when searching for a substance having a certain property, many substances may be trial-produced, which requires a huge amount of time and cost.

In addition, in the method of calculating the band gap by the above-mentioned first principle band calculation, the band structure itself can be obtained by using the periodic boundary condition, but the amount of calculation becomes enormous, so that the current computer With the ability, the time required for the calculation becomes remarkably long. Moreover, an error of about 50% occurs in the band gap value itself.

Further, in the method of calculating the band gap by the above cluster calculation, a part of the crystal for which the band gap is to be obtained is cut out and regarded as a virtual molecule (cluster), and the molecular orbital calculation is performed to calculate the band gap. As expected, if this calculation is done properly, in many cases, the required band gap will be larger than the actual band gap unless the cluster size is made extremely large. For example, silicon (S
In the case of i), a band gap close to the actual band gap cannot be obtained unless the cluster size is set to about 1000 atoms. However, it is difficult to accurately determine the band gap by this method because the current computer capacity is limited to the calculation of cluster sizes up to about 100 to 200 atoms.

Therefore, an object of the present invention is to obtain a crystal band gap of a crystal of various substances including an unknown substance, which can be accurately obtained in a short time without actually producing the crystal. It is to provide a method of seeking.

[0007]

In order to achieve the above object, the present invention provides a cluster calculation using a molecular orbital method by a discrete variation local density approximation method for a cluster selected for a crystal whose band gap is to be obtained. This is a method for determining the band gap of a crystal by performing the above, and when performing the cluster calculation, it is considered that the charge density distributions of all the atoms of the same type in the cluster are equivalent. , The value of the coefficient of the term of the electron-electron exchange interaction is adjusted when the atomic orbital function which is the basis function in the molecular orbital method by the discrete variational local density approximation method is obtained.

[0008]

According to the method of obtaining the band gap of the crystal according to the present invention having the above-described structure, the band gap is obtained by calculation, so that it is not necessary to actually produce the crystal. In addition, since the cluster calculation is performed by using the molecular orbital method based on the discrete variational local density approximation method, it is necessary to perform the calculation for each of many wave number spaces like the case of calculating the band gap by the band calculation. Since there is no such calculation, the time required for calculation is short, and the band gap can be obtained with high accuracy.

[0009]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of the present invention will be described below with reference to the drawings.

In this embodiment, a band gap of a semiconductor material is obtained by performing cluster calculation using a discrete variational local density approximation method (hereinafter referred to as "DV-Xα method") as a molecular orbital method. .

Unlike the general first-principles molecular orbital method, the DV-Xα method is a molecular orbital method in which the calculation amount is reduced by using the local density approximation in the part of the electron-electron exchange interaction, The electronic states of compounds containing semiconductors, transition metals, rare earth elements, etc. can be calculated. Regarding this DV-Xα method, Hirohiko Adachi “Introduction to Quantum Material Chemistry-Approach from DV-Xα Method” (Sankyo Publishing, 1991) and J. Chem. Phys., Vol. 65 (1976) 3
As explained in 629, etc., its outline is shown in Fig. 4. In FIG. 4, H _ij is the resonance integral, S _ij is the overlap integral, ω (r _k ) is the weight of the sample point r _k , and φ _i (r _k ) is the basis function (atomic orbital function) in the step of calculating the matrix elements. , H (r _k ) is a Hamiltonian, and H in the calculation of the secular equation is a matrix whose elements are H _ij , S
Is a matrix having S _ij as an element, and C is a vector having a coefficient of each basis function when the molecular orbital is represented by a linear combination of basis functions.

In this embodiment, the band gap is calculated based on the cluster model, but at this time, the problem is how to process the end portion of the cluster. This terminal portion is generally terminated by a hydrogen (H) atom, and this is the case in this example (see FIG. 1).

When performing the calculation based on this cluster model, the following assumptions (1) and (2) are made.

(1) The charge density distribution of all atoms of the same kind in a cluster or the population (occupation number) of electrons is treated equivalently. Specifically, for example, in the case of Si, the number of electrons in each of 1s, 2s, 2p, 3s, and 3p orbits is changed in association with all Si atoms in the cluster. Also, regarding the number of electrons in the 1s orbit of the H atom that terminates the terminal part of the cluster,
Change all H atoms in tandem.

(2) When the molecular orbital calculation is performed by the DV-Xα method, the coefficient α of the term of the electron-electron exchange interaction is usually set to 0.7, but this calculation is performed. In the cluster model in the example, a value slightly larger than 0.7 is used as the value of α when obtaining the atomic orbital function that is the basis function. Specifically, for each atom in the cluster, the value of α determined so that the value of total energy calculated by the Xα method is equal to the value of total energy calculated by the Hartree-Fock method is To use. Such a value of α has been obtained for most of the atoms on the periodic table, and is 0.9780 for H atoms.
4, helium (He) atoms are 0.77298, lithium (Li) atoms are 0.78147, and then gradually decrease as the atomic number increases, becoming 0.72751 for Si atoms and 2 for heavier atoms. It approaches / 3.

Specific example

71 Si atoms and 6 as shown in FIG.
For a Si ₇₁ H ₆₀ cluster consisting of 0 H atoms,
When the population distribution of electrons of Si atoms and H atoms after the optimization was calculated, the following results were obtained.

─────────────────────── Atomic atomic orbital Electron population ────────────────── ────── Si 1s 2.000 2s 2.000 2p 6.001 3s 1.376 3p 2.530 H 1s 1.110 ─────────────────── ─────

The value of the electron population in the above table is S
All Si and H atoms in the i ₇₁ H ₆₀ cluster were changed in a coordinated manner. Then, under this condition, the value of the coefficient α of the term of the electron-electron exchange interaction was further changed in the following two ways (a) and (b), and the electronic state was investigated.

(A) All values of α are set to 0.7. That is, α = 0.7 for Si atoms and α = for H atoms.
0.7, and α = 0.7 with respect to the molecular orbital (MO).

(B) When obtaining the atomic orbital, a value of α is used so that the total energy value becomes equal to the total energy value obtained when the calculation is performed by the Hartree-Fock method. That is, α = 0.7275 for Si atom
1, α = 0.97804 for H atoms and α = 0.7 for MO.

FIG. 2 shows the energy levels obtained by calculation in the cases (a) and (b) above.

In the case of FIG. 2 (a), the energy difference E _g between the HOMO (highest occupied orbital) and the LUMO (lowest unoccupied orbital) E _g
(Corresponding to the band gap of the crystal) is 3.74 eV, but the measured value of E _{g for} the actual Si crystal is 1.1.
eV, so E obtained by this cluster calculation
_It can be seen that _g is more than three times larger than the measured value of E _g for Si crystal. On the other hand, in the case of FIG. 2B, the value of E _g is 1.12 eV, which is very close to the measured value of E _g of 1.1 eV for Si crystal.

In the cluster calculation described above, the electron population distributions of all Si atoms and H atoms in the Si ₇₁ H ₆₀ cluster were changed in conjunction with each other. Next, the shape of the cluster was considered. The electronic state calculation was performed again by changing the electron population distribution in conjunction only with the atoms at the equivalent positions in the symmetry (Cs (mirror) symmetry) (the number of atoms in this case is 79). Here, the value of the coefficient α of the term of the electron-electron exchange interaction was set in the same manner as in the above (a) and (b), and the energy level was calculated for each case. The result is shown in FIG.

In the case of FIG. 3A, E _g = 4.09
eV, which is higher than the result (3.74 eV (see (a) in FIG. 2)) when the electron population distribution is changed in conjunction with all Si atoms and H atoms in the Si ₇₁ H ₆₀ cluster. It is a relatively large value. In the case of (b) of FIG. 3, that is, the value of α is
When set as above, E _g = 3.92 eV, which means that there is almost no change compared with the case of FIG.

Summarizing the above results, the above (1),
Only if the condition (2) is satisfied at the same time, the actual Si
It can be seen that it is possible to obtain a bandgap that matches the measured value of the bandgap for the crystal very well.

As described above, according to this embodiment, the charge density distribution of all atoms of the same kind in the cluster selected for the crystal of the semiconductor material whose band gap is to be determined or the electron population is equivalent. In addition, the molecular orbital method by the DV-Xα method was used by adjusting the coefficient α of the term of the electron-electron exchange interaction in obtaining the atomic orbital function which is the basis function in the molecular orbital method. Since the band gap is obtained by performing the cluster calculation, the band gap of the crystal can be obtained accurately without actually producing the crystal. In addition, since it is not necessary to calculate each of many wave number spaces unlike the case of obtaining the band gap by band calculation, the time required for the calculation can be short.

The method according to this embodiment is particularly suitable for application when it is desired to know the band gap of a crystal of a new substance.

Although one embodiment of the present invention has been specifically described above, the present invention is not limited to the above embodiment, and various modifications can be made based on the technical idea of the present invention. .

For example, the Si ₇₁ H ₆₀ cluster used in calculating the band gap of the Si crystal in the above-mentioned embodiment is merely an example, and other clusters can be used.

[0031]

As described above, according to the present invention, the band gaps of crystals of various substances including unknown substances can be accurately obtained in a short time without actually producing the crystals. it can.

[Brief description of drawings]

FIG. 1 is a schematic diagram showing a Si ₇₁ H ₆₀ cluster used as a cluster in one embodiment of the present invention.

FIG. 2 shows that the charge density distributions of all Si atoms and H atoms in a Si ₇₁ H ₆₀ cluster used as a cluster in one embodiment of the present invention are regarded as equivalent DV-
It is a figure which shows the energy level of MO calculated | required by performing the cluster calculation using the molecular orbital method by X (alpha) method.

FIG. 3 shows the energy level of MO obtained by performing cluster calculation using the molecular orbital method by the DV-Xα method, assuming that the charge density distribution of equivalent atoms determined by Cs symmetry is equivalent. It is a figure.

FIG. 4 is a flowchart showing an outline of a molecular orbital method based on the DV-Xα method.

Claims (1)

[Claims] 1. A crystal band gap of a crystal for which a band gap of the crystal is to be obtained by performing cluster calculation on a cluster selected for a crystal for which a band gap is to be obtained by using a molecular orbital method by a discrete variational local density approximation method. A method of obtaining a band gap, which is considered to be equivalent to the charge density distributions of all the atoms of the same kind in the cluster in performing the cluster calculation, and the molecular orbital method based on the discrete variational local density approximation method. A method for obtaining a band gap of a crystal, characterized in that a value of a coefficient of an electron-electron exchange interaction term is adjusted when obtaining an atomic orbital function which is a basis function in.