JP2005268726A - Program for making computer perform exciton quantum level computation, and computer readable recording medium that has recorded program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a program that can make a computer execute exciton quantum level computation with a high precision, and a computer readable recording medium that has recorded the program. <P>SOLUTION: In computing a quantum level of an exciton made up of an electron and a hole confined within a semiconductor microstructure such as a quantum well, respective wave functions ψ<SB>e</SB>and ψ<SB>h</SB>for an electron and hole about which a many-body mutual effect among charged-particles is taken into consideration, and their energy E<SB>e</SB>and E<SB>h</SB>are computed (Step S101). The computed wave functions ψ<SB>e</SB>and ψ<SB>h</SB>for the electron and hole and the envelop functions ϕ<SB>e-h</SB>showing a relative movement of the electron and the hole are used to compute the exciton's wave function Ψ by a variation method, thus finding the exciton's energy (S102). <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

  The present invention relates to a program for causing a computer to execute the calculation of quantum levels of excitons composed of electrons and holes confined in a fine structure, and a computer-readable recording medium on which the program is recorded.

  Electrons and holes excited in a semiconductor microstructure such as a quantum well, a quantum wire, and a quantum dot form a bound state called an exciton by Coulomb interaction that works between electrons and holes. Such behavior of excitons is important as a basic process responsible for light emission / absorption phenomena in semiconductor devices.

The derivation of the exciton quantum level (exciton level) is performed by designing and analyzing the optical element, for example, by analyzing the emission spectrum of the light emitting element, or by optimizing the element by simulating the operation of the light emitting / receiving element. It becomes important in the above. In addition, the derivation of exciton levels is useful in proposing optical elements having new functions and new structures. As a method for deriving such an exciton level, for example, there is a method described in Non-Patent Document 1 below.
DA Kleinman, "Binding energy of biexcitons and bound excitons in quantum wells", Phys. Rev. B28, p.871 (1983).

  In the method described in the above-mentioned document 1, a wave function of electrons and holes in a quantum well that does not include a many-body interaction effect between a plurality of charged particles is used, and using them, an electron-positive The exciton wavefunction including the interaction between holes is calculated by the variational method.

  This derivation method is based on the image that electrons and holes are paired one by one and are combined by Coulomb force to form electrically neutral excitons. For this reason, even when a plurality of electrons / holes exist in the quantum well, the exciton level obtained does not change. Therefore, in such a method, there is a problem that a part depending on the many-body interaction cannot be reproduced in the energy change due to the excitation density of the exciton.

  The present invention has been made to solve the above-described problems, and a program capable of causing a computer to execute the calculation of exciton quantum levels with high accuracy, and a computer-readable program storing the program An object is to provide a recording medium.

  In order to achieve such an object, a program according to the present invention is a program for causing a computer to execute a quantum level calculation of excitons composed of electrons and holes confined in a fine structure. 1) first calculation processing for calculating respective wave functions of electrons and holes in a fine structure in consideration of the many-body interaction effect between a plurality of charged particles; and (2) obtained in the first calculation processing. Using the electron and hole wave functions, the computer calculates the exciton wave function by a variational method, and causes the computer to execute a second calculation process for obtaining the exciton energy.

  The recording medium according to the present invention is a computer-readable recording medium that records the above-described program.

  In the program and the recording medium according to the present invention, a calculation method in which the quantum levels of electrons and holes excited in a fine structure such as a quantum well, a quantum wire, and a quantum dot are considered in consideration of the many-body interaction effect, respectively. Use to calculate. And the calculation about the quantum level of an exciton is performed using the obtained wave function including the many-body interaction effect of an electron and a hole. As a result, even when a plurality of electrons and holes are present in the fine structure, the exciton level can be obtained with high accuracy.

  Here, for the calculation of the quantum level of the exciton in the second calculation process, the wave function of the exciton, the wave function of the electrons and holes obtained in the first calculation process, and the relative motion of the electrons and holes are determined. And an envelope function set to a function including a variation parameter. Thereby, the calculation of the exciton level by the variational method can be suitably executed.

  Moreover, it is preferable to perform the calculation by the variational method of the exciton wave function in the second calculation process using the exciton energy and its derivative under the condition that the exciton energy is minimized. Thereby, the derivation of exciton levels can be executed efficiently.

  As for a specific method for deriving the quantum levels of electrons and holes, the first calculation process for calculating the wave functions of electrons and holes, which are particles, uses a Hamiltonian that does not include many-body interaction effects. Calculates the initial wave function of the particle, considers the process of discretizing the initial wave function into a numerical sequence composed of N (N is a natural number) components, the wave function by N components, and the many-body interaction effect A method including calculating a particle wave function using a variational method so that the energy of the entire system is minimized by using a Hamiltonian including the nonlinear term can be used.

  Alternatively, the first calculation processing for calculating the wave functions of each of the electrons and holes that are particles includes a processing for calculating the initial wave function of the particles using a Hamiltonian that does not include a many-body interaction effect, and an initial wave function Using the obtained particle distribution to calculate the potential by solving the Poisson equation and calculating the new wave function of the particle using the Hamiltonian with the obtained potential added to the interaction term Can do.

  According to the program and the recording medium of the present invention, the quantum levels of electrons and holes in the fine structure are calculated in consideration of the many-body interaction effect, and the obtained wave functions of the electrons and holes are obtained. , A program capable of causing a computer to execute the calculation of the quantum level of the exciton with high accuracy by performing an operation on the quantum level of the exciton, and a computer-readable recording medium storing the program Is realized.

  Hereinafter, preferred embodiments of a program and a recording medium according to the present invention will be described in detail with reference to the drawings. In the description of the drawings, the same elements are denoted by the same reference numerals, and redundant description is omitted.

  FIG. 1 is a diagram schematically illustrating the generation of excitons in a fine structure. When electrons (e: electron) and holes (h: hole) are excited by light absorption in a semiconductor microstructure such as a quantum well, a quantum wire, or a quantum dot (FIG. 1 (a)), the electrons and Holes form a bound state by Coulomb interaction acting between electrons and holes (FIG. 1B), and then emit light (emit light) again (FIG. 1C). The composite particles generated from the electrons and holes are called excitons. The program and recording medium according to the present invention are for calculating the quantum level of such excitons.

FIG. 2 is a flowchart showing calculation processing in a program for causing a computer to execute calculation of exciton quantum levels according to the present invention. In this calculation process, first, wave functions indicating respective quantum levels of electrons and holes confined in the fine structure are calculated by referring to calculation conditions such as conditions given for the fine structure. In particular, in the present embodiment, the calculation of the electron and hole wave functions is performed in consideration of the many-body interaction effect between a plurality of charged particles (step S101, first calculation process). By this calculation, the electron wave function ψ _e , the hole wave function ψ _h , the electron energy E _e , and the hole energy E _{h in} consideration of the many-body interaction effect are obtained. The calculation method of the wave functions of electrons and holes in consideration of the many-body interaction effect will be specifically described later.

Next, the quantum levels of excitons are calculated using the electron and hole wave functions ψ _e and ψ _h obtained in step S101 (S102, second calculation process). Here, the wave function of the exciton is expressed by a trial function, the wave function is calculated by applying a variational method, and the energy of the exciton is obtained.

ここで、ｚは量子井戸等の微細構造の成長方向（半導体微細構造における半導体層の積層方向）の座標、ｘ、ｙはｚ軸に垂直な面内での座標である。また、ｚ_ｅ、ｚ_ｈは、それぞれ電子、正孔のｚ方向における位置を表す。 When the electron and hole wave functions ψ _e and ψ _h are given, the wave function ψ of the entire system of excitons can be expressed by the following equation (1) (S103).

Here, z is a coordinate in the growth direction of a fine structure such as a quantum well (semiconductor layer stacking direction in the semiconductor fine structure), and x and y are coordinates in a plane perpendicular to the z-axis. Z _e and z _h represent positions of electrons and holes in the z direction, respectively.

In the approximation of the equation (1), φ _e−h is a wave function representing the relative motion of electrons and holes that form a bound state serving as an exciton, and is called an envelope. Also, here, the envelope function is set to a function including a variation parameter, and the exciton wave function Ψ is calculated by the variation method.

ここで、ｒ^２＝ｘ^２＋ｙ^２である。また、λ、ζは変分パラメータである。 Specifically, when the calculation is performed for the 1s state which is the ground state of the exciton, the envelope function φ _e−h expressed by the equation (2) can be used.

Here, r ² = x ² + y ² . Λ and ζ are variational parameters.

この式（３）において、μはｘｙ面内での電子＋正孔系の換算質量であり、電子及び正孔の有効質量ｍ_ｅ ^⊥、ｍ_ｈ ^⊥に対し、

によって定義される。また、ｅは電気素量、εは媒質の誘電率、Ｎは規格化因子である。 Furthermore, using this equation (2), the Hamiltonian that is a function that gives the exciton energy E _EXC is expressed by the following equation (3) as a function E _EXC (λ, ζ) of the variation parameters λ and ζ. (S104).

In this formula (3), μ is the converted mass of the electron + hole system in the xy plane, and for the effective masses m _e ^電子 , m _h ^{の of} electrons and holes,

Defined by Further, e is the elementary electric quantity, ε is the dielectric constant of the medium, and N is a normalization factor.

Although the dependence of the energy E _{EXC on} the variation parameters λ and ζ is not clearly shown on the right side of Equation (3), in reality, G (a), J (a), and K (a) are envelopes. It is a function of parameters λ and ζ through the function φ _e−h .

規格化因子Ｎも、同様に以下の式

によって定義される。また、Ｆ（ａ）は

で与えられる。以上の式（５）〜式（１０）をＥ_ＥＸＣの式（３）に代入することにより、励起子のエネルギーが変分パラメータλ、ζの関数として与えられることとなる。 In the following, the definitions of these expressions and the expressions that are evaluated in actual operations by modifying the expressions are summarized.

Similarly, the normalization factor N

Defined by F (a) is

Given in. By substituting the above formulas (5) to (10) into E _EXC formula (3), the exciton energy is given as a function of the variation parameters λ and ζ.

ここで、∇はλ、ζに関する偏微分をまとめて表したもので、実際には∂／∂λ、または∂／∂ζを意味している。この式（１１）において必要となる各式を以下に示す。

以上により、励起子のエネルギーＥ_ＥＸＣ、及びその偏導関数∇Ｅ_ＥＸＣが得られる。 In step S102 in which the exciton quantum level is calculated, in order to obtain the minimum value of the exciton energy, derivatives of the parameters λ and ζ of the exciton energy E _EXC are also required. The derivative ∇E _EXC is obtained as shown in the following equation (11) by differentiating the above equations with respect to λ and ζ.

Here, ∇ collectively represents partial differentials related to λ and ζ, and actually means ∂ / ∂λ or ∂ / ∂ζ. Each formula required in this formula (11) is shown below.

Thus, the exciton energy E _EXC and its partial derivative ∇ E _EXC are obtained.

Further, parameters λ ₀ and ζ ₀ that minimize the exciton energy E _EXC are obtained by using the exciton energy E _EXC and the partial derivative ＣE _EXC (S105). The parameter values λ ₀ and ζ ₀ can be calculated by a multivariable nonlinear optimization method such as a conjugate gradient method. Then, by substituting the obtained parameter values λ ₀ and ζ ₀ into the equation (3), the final exciton energy is obtained (S106).

  FIG. 3 is a block diagram showing an example of a hardware configuration of an arithmetic system (computer) used for executing a program for calculating the quantum level of excitons including the arithmetic processing shown in the flowchart of FIG. Each arithmetic processing of the program necessary for obtaining the exciton quantum level is performed by the CPU 10 of the arithmetic system 1.

  Connected to the CPU 10 are a ROM 11 storing a program for calculating the quantum level of excitons executed in the calculation system 1 and a RAM 12 storing data temporarily during execution of the program. Has been. In addition, an external storage device 13 such as a hard disk is connected and used to hold necessary data such as parameters and calculation results.

  Then, an input device 14 used for inputting necessary parameters and the like and a display device 15 used for displaying calculation results are connected to the CPU 10 and the like, thereby constituting the calculation system 1. . As the input device 14, for example, a pointing device such as a mouse or a keyboard is used. Moreover, as the display device 15, for example, a CRT display, a liquid crystal display, or the like is used.

  A program for calculating the quantum level of excitons for realizing each calculation process performed by the CPU 10 can be recorded on a computer-readable recording medium and distributed. In such a recording medium, for example, a magnetic medium such as a hard disk and a flexible disk, an optical medium such as a CD-ROM and a DVD-ROM, a magneto-optical medium such as a floppy disk, or a program instruction is executed or stored. Specially arranged hardware devices such as RAM, ROM, and semiconductor nonvolatile memory are included. Further, in response to reading or execution of a program from such a recording medium, as shown in FIG. 3, a recording medium reading drive 16 (for example, a flexible disk drive or the like) reads the program from the recording medium as necessary. May be connected to the CPU 10.

  The effects of the program and the recording medium according to the above embodiment will be described.

In the program for executing the arithmetic processing shown in FIG. 2 and the recording medium on which the program is recorded, in step S101, the quantum level of electrons and holes excited in a fine structure such as a quantum well, a quantum wire, or a quantum dot is obtained. The position is calculated using a calculation method that takes into account the many-body interaction effect. In step S102, using the wave functions ψ _e and ψ _h including the many-body interaction effect of electrons and holes obtained in S101, the exciton quantum level, specifically, the exciton wave function Calculations for Ψ and energy E _EXC are performed.

  According to such a calculation method, even when a plurality of electrons / holes exist in the fine structure, the exciton level is increased in consideration of the influence of the interaction between them. It can be obtained with accuracy. In particular, this many-body interaction effect is greatly affected in a type II quantum well / superlattice structure in which electrons and holes occupy spatially different positions. Therefore, a program and a recording medium for performing the above-described calculation are very important for analyzing device characteristics of the light emitting element and the light receiving element. Specific examples of the type II material include an InAs-GaSb system, a GaAs-AlAs system under certain conditions, and a doping superlattice (well).

Here, in the above-described calculation method, the envelope function φ _e is set to a function that represents the relative motion of electrons and holes and the variation parameters λ and ζ as well as the electron and hole wave functions ψ _e and ψ _{h. -H} is used to represent the exciton wave function Ψ by the equation (1), and the exciton is calculated using this wave function Ψ. Thereby, the calculation of the exciton level by the variational method can be suitably executed.

で示す関数をエンベロープ関数としても良い。 In such a calculation method, the envelope function φ _e−h is not limited to the function shown in the equation (2), and other functions may be used. For example, in the calculation of the exciton ground state, the following formula (16) is used instead of formula (2).

The function indicated by may be an envelope function.

ここでは、α、λ、ζが変分パラメータである。なお、これらの式（１６）、（１７）のようにエンベロープ関数を変更する場合には、エンベロープ関数を用いて表される他の式についても同様に変更する必要がある。 Moreover, when calculating about states other than the ground state of an exciton, it is preferable to use the envelope function according to the state. For example, when calculating about the 1st excitation state (2s state) of an exciton, envelope function (phi) _eh represented by Formula (17) can be used.

Here, α, λ, and ζ are variational parameters. In addition, when changing an envelope function like these Formulas (16) and (17), it is necessary to change similarly about the other formulas expressed using an envelope function.

In the above-described calculation method, the exciton wave function Ψ is calculated by a variational method using the exciton energy E _EXC and its derivative ∇E _EXC, and using the multivariable nonlinear optimization technique. This is done under conditions that minimize the energy. Thereby, the derivation of exciton levels can be executed efficiently. As the multivariable nonlinear optimization technique, various methods such as the Newton method, the steepest descent method, and the annealing method can be used in addition to the above-described conjugate gradient method.

  A calculation method of electron and hole quantum levels in consideration of the many-body interaction effect between a plurality of charged particles in step S101 in FIG. 2 will be described together with a specific example thereof.

  FIG. 4 is a flowchart showing an example of an electron and hole quantum level calculation method used for calculation of exciton quantum levels. FIG. 5 is a graph showing an example of a fine quantum well in which electrons and holes as particles are confined, and a wave function of particles (electrons and holes) in the quantum well. In this graph, the horizontal axis indicates the position, and the vertical axis indicates the energy. The quantum well 50 includes a quantum well layer 51 and quantum barrier layers 52 and 53 located on both sides of the well layer 51. In such a quantum well 50, the particles confined in the well layer 51 form an energy level 55.

  In the calculation method shown in FIG. 4, the Schrodinger equation is solved using the Hamiltonian that does not include the many-body interaction effect between the plurality of charged particles, and the wave function ψ of the particle in the quantum well 50 is obtained. This wave function becomes the initial wave function in the quantum level calculation (step S201). Then, starting from the initial wave function ψ, a wave function that minimizes the energy of the entire system is obtained based on the principle of the variational method. In this case, the energy of the whole system is defined as the calculated value of the Hamiltonian including a nonlinear term considering the many-body interaction effect for a given wave function.

Specifically, as shown in FIG. 5, the end of one barrier layer 52 of the quantum well 50 to the end of the other barrier layer 53 is divided into a plurality of points x _{1 to} x _N (N is a natural number), Corresponding to each of the divided points x _i (i = 1,..., N), the wave function ψ is discretized into _N wave function components ψ ₁ ,..., Ψ _i ,. The points x _{1 to} x _N are preferably set so that the distances Δx between adjacent points are all equal.

In this calculation method, a wave function discretized into a numerical sequence composed of the above-described wave function components ψ _{1 to} ψ _N and a Hamiltonian including a nonlinear term considering the many-body interaction effect are used. Each of the components ψ ₁ to ψ _N is calculated so as to be minimized. According to such a method, the wave function of particles (electrons and holes) in the quantum well 50 in which the many-body interaction effect is considered can be suitably calculated. Hereinafter, this calculation method will be specifically described. Here, a case where the Coulomb interaction of particles is considered as a nonlinear term introduced into the Hamiltonian will be described.

この式（１８）において、右辺の第２項は多体相互作用を含んだクーロン相互作用項である。また、ε（ｘ）は位置ｘ（図５参照）における誘電定数であり、−ｅ及びｍはそれぞれ電子の電荷及び質量である。また、ｄ（ｙ）は位置ｙにおける固定ドナーの体積密度であり、固体ドナーは＋ｅの電荷を有するものと仮定している。 First, as described above, the initial wave function ψ of a particle (for example, an electron) is calculated using a Hamiltonian that does not include a many-body interaction effect, that is, a Hamiltonian that includes a kinetic energy term and a potential term due to an external electric field. (S201). The calculation of the initial wave function is performed using a method such as a transfer matrix method, an S matrix method, or a shooting method. The total system energy to be minimized is given by the following equation (18).

In this equation (18), the second term on the right side is a Coulomb interaction term including a many-body interaction. Further, ε (x) is a dielectric constant at the position x (see FIG. 5), and −e and m are the charge and mass of electrons, respectively. Also, d (y) is the volume density of the fixed donor at position y, and it is assumed that the solid donor has a charge of + e.

この式（１９）においてＮｅは電子数であり、Ｖ_ｉ’はクーロンポテンシャルである。また、Ｎｔは規格化因子であり、

で与えられる。 The above equation (18) represents the energy of the entire system in the quantum well 50, but since it is difficult to perform an operation using this equation, the space is discretized and exists in the entire system. Standardize by the number of particles to be used. That is, the obtained wave function ψ is spatially discretized into the _N wave function components ψ _{1 to} ψ N described above, and is normalized by the number of electrons in the entire system. The following formula (19) representing the per unit energy is obtained.

In this formula (19), Ne is the number of electrons, and V _i ′ is a Coulomb potential. Nt is a normalization factor,

Given in.

ただし、

である。 This equation (19) represents the energy per one electron existing in the system, and is hereinafter referred to as “cost function” in accordance with the term of the optimization problem. Then, the equation (19) is partially differentiated by each of the discretized _N wave function components ψ _{1 to} ψ _N to calculate N derivatives. Thereby, Formula (21) is obtained.

However,

It is.

In the formula (21), E _int means nonlinear mutual energy per particle. Then, obtaining the equations (19) and (21) from the equation (18) corresponds to calculating the cost function and the derivative of the cost function (S202).

Here, the expression (21) is not a partial differentiation of both sides of the expression (19) by each of the wave function components ψ _{1 to} ψ _N , but a partial differentiation of the Hamiltonian H ′ in consideration of the Coulomb interaction of the expression (19). It has become. This is because the wave function is calculated so that the energy of the entire system is minimized in consideration of the Coulomb interaction. Therefore, the calculation is performed by reflecting the influence of the Coulomb interaction to the maximum by partially differentiating the Hamiltonian H ′. It is what I do.

As described above, in step S202, the energy S [{ψ _i }] of the entire system is calculated by substituting the discretized wave functions ψ _{1 to} ψ _N into the cost function (equation (19)). Further, the derivative of the cost function is calculated by partial differentiation of the Hamiltonian H ′ considering the many-body interaction effect with each of the wave function components ψ _{1 to} ψ _N (Equation (21)).

この式（２４）において、ηは最小エネルギーの演算が収束するようにするためのスケーリングファクターである。また、式（２４）の右辺第２項は、クーロン相互作用を考慮したハミルトニアンＨ’を波動関数ψの各成分によって偏微分して式（２１）を演算し、その演算した式（２１）に波動関数ψ^ｏｌｄを代入して演算される。 Subsequently, a new wave function of the particle is calculated using the calculated N derivatives and the following equation (24) (S203).

In this equation (24), η is a scaling factor for causing the minimum energy calculation to converge. Further, the second term on the right side of the equation (24) calculates the equation (21) by partially differentiating the Hamiltonian H ′ considering the Coulomb interaction with each component of the wave function ψ, and the calculated equation (21) It is calculated by substituting the wave function ψ ^old .

The new wave function ψ _i ^new according to the equation (24) is obtained by adding the change due to the Coulomb interaction (the second term on the right side of the equation (24)) to the already calculated wave function ψ ^old . Therefore, in this method, the new wave function ψ _i ^new is calculated by reflecting the change in Coulomb interaction. The new wave function ψ _i ^new is calculated for each of the N discrete wave functions ψ _{1 to} ψ _N.

When a new wave function ψ _i ^new is calculated by the equation (24), it is substituted for the wave function ψ _i of the equation (19), and the cost function and its derivative by the new wave function are calculated. (S204). Then, it is determined whether or not the energy of the entire system converges by determining whether or not the cost function increases or by determining whether or not all the N derivatives calculated in step S204 are zero. (S205).

  In step S205, when the cost function does not increase or when all the N derivatives are not zero, it is determined that the energy of the entire system has not converged, and steps S203 to S205 are repeatedly executed. This is because when the cost function does not increase, the increase in the cost function is zero or the cost function is decreasing, and the energy of the entire system may further decrease. Further, when the derivative of the cost function is not zero, it indicates that the cost function is changing, and in this case, the energy of the entire system may be further reduced.

  On the other hand, in step S205, when the cost function increases or when all the N derivatives are zero, it is determined that the energy of the entire system has converged. Then, the wave function one step before is output (S206). This is because the cost function is minimized by using the wave function calculated one step before.

とを用いて全系のエネルギーを導出し（Ｓ２０７）、全体の演算処理が終了する。 When N wave function components that minimize the energy of the entire system are determined in step S206, the wave function (final wave function) including the determined N components and the Coulomb interaction are considered. Formula (25) that is a Hamiltonian

Are used to derive the energy of the entire system (S207), and the entire calculation process is completed.

  Of the arithmetic processing described above, steps S202 to S206 represent processing for one particle. Therefore, in this calculation method, when N components are determined so as to minimize the energy of the entire system, attention is paid to one particle, and the influence of Coulomb interaction is minimized in the wave function for the one particle. N components are determined as follows. When the wave function for one particle is determined, the determined wave function is applied to the particles of the entire system to calculate the energy of the entire system. As a result, even if the number of electrons confined in the well layer 51 increases due to an increase in the doping amount in the barrier layers 52 and 53 of the quantum well 50 or the well layer 51, the influence of the interaction of electrons increases. The calculation can be performed so that the energy of the entire system converges.

  A specific embodiment of the above-described calculation process of exciton quantum levels will be described.

(Example 1)
FIG. 6 is a graph showing the structure of a quantum well obtained by calculating the exciton quantum level in Example 1. In this graph, the horizontal axis indicates the position in the quantum well growth direction in terms of the number of atomic layers (ML), and the vertical axis indicates energy (meV). In FIG. 6, the upper graph shows the conduction band, and the lower graph shows the valence band.

ここで、１ＭＬは２．８３Å、演算時の空間分割数は２０点／ＭＬ（Δｚ＝０．１４１５Å）である。また、ｍ_０は真空中での電子の質量ｍ_０＝９．１０９３９×１０^−３１ｋｇ、ε_０は真空中の誘電率ε_０＝８．８５４１９×１０^−１２Ｆ／ｍである。また、外部電界は０ｋＶ／ｃｍ、温度は室温とした。 In this embodiment, as shown in FIG. 6, a combination of GaAs · 8ML (ML: atomic layer) and AlAs · 30ML is sandwiched between Al _0.44 Ga _0.56 As barrier layers having a thickness of 40 ML. Type II quantum well
Al _0.44 Ga _0.56 As (40ML) -GaAs (8ML) -AlAs (30ML) -Al _0.44 Ga _0.56 As (40ML)
The operation was performed. Table 1 shows the parameters used in the calculation.

Here, 1ML is 2.83Å, and the number of space divisions at the time of calculation is 20 points / ML (Δz = 0.1415Å). Further, m ₀ is the mass of electrons in vacuum m ₀ = 9.103939 × 10 ⁻³¹ kg, and ε ₀ is the dielectric constant ε ₀ = 8.85419 × 10 ⁻¹² F / m in vacuum. The external electric field was 0 kV / cm and the temperature was room temperature.

  In such a configuration, holes are heavy holes confined in the GaAs layer, whereas electrons are X electrons confined in the AlAs layer. The gap between the conduction band and the valence band, that is, the energy difference between the X electron level and the heavy hole level is 1.642 eV.

Based on the above calculation parameters, the electron and hole quantum levels are calculated while changing the electron-hole density (S101 in FIG. 2), and further, calculations are performed on the exciton quantum levels (S102). ). FIG. 7 is a graph showing an example of calculation results of electron and hole quantum levels. Here, wave functions ψ _e and ψ _h of electrons and holes when the electron-hole density is 1.132 × 10 ¹¹ / cm ² are shown. Further, the energy of electrons and holes at this time is E _e = 11.061 meV and E _h = 77.371 meV.

When an exciton is calculated using the electron and hole quantum levels thus obtained, E _EXC = 78.646 meV is obtained as the exciton energy. Further, FIG. 8 shows the result of the calculation of the exciton energy performed in the electron-hole density range of 5.0 × 10 ^{8 to} 1.0 × 10 ¹² / cm ² . In the graph of FIG. 8, the horizontal axis represents the exciton density (/ cm ² ), and the vertical axis represents the obtained exciton energy (meV).

  In FIG. 8, a graph A1 shows the minimum exciton energy obtained by this example, and a graph A2 shows the exciton energy obtained by the conventional calculation method. In the graph A2, since the many-body interaction effect is not taken into consideration, the exciton energy is a constant value without depending on the exciton density.

  On the other hand, in graph A1, it can be seen that the energy increases as the exciton density increases. This is because the many-body interaction effect is taken into consideration in the calculation of the quantum level. In FIG. 8, the exciton energy is shown as a value based on the bottom of the quantum well. For this reason, when converting to the wavelength of light emission emitted from excitons, it is necessary to add 1.642 eV, which is energy between the X conduction band and the valence band.

ここで、外部電界は５５ｋＶ／ｃｍ、温度は２Ｋとした。また、伝導帯−価電子帯間のギャップ、すなわちΓ電子準位−重い正孔準位間のエネルギー差は、１．５１９ｅＶとなっている。また、計算パラメータについては、文献 "Landolt-Bornstein Numerical Data and Functional Relationships in Science and Technology", edited by O. Madelung (Springer, Berlin, 1982) 等を参照している。 (Example 2)
FIG. 9 is a graph showing the structure of a quantum well obtained by calculating the quantum level of excitons in Example 2. In this embodiment, as shown in FIG. 9, a combination of GaAs · 20ML, Al _0.3 Ga _0.7 As · 14ML, and GaAs · 20ML is sandwiched between AlAs barrier layers having a thickness of 15ML, and a quantum having the following configuration. well
AlAs (15ML) -GaAs (20ML) -Al _0.3 Ga _0.7 As (14ML) -GaAs (20ML) -AlAs (15ML)
The operation was performed. Table 2 shows the parameters used in the calculation.

Here, the external electric field was 55 kV / cm, and the temperature was 2K. The gap between the conduction band and the valence band, that is, the energy difference between the Γ electron level and the heavy hole level is 1.519 eV. For calculation parameters, reference is made to the document “Landolt-Bornstein Numerical Data and Functional Relationships in Science and Technology”, edited by O. Madelung (Springer, Berlin, 1982).

Based on the above calculation parameters, the quantum levels of electrons and holes are calculated while changing the electron-hole density, and further, calculations are performed on the quantum levels of excitons. FIG. 10 is a graph showing an example of calculation results of electron and hole quantum levels. The energies of electrons and holes in the wave functions ψ _e and ψ _h are E _e = 116.458 meV and E _h = −63.548 meV. Further, FIG. 11 shows the result of calculating the exciton energy by changing the electron-hole density. In the graph of FIG. 11, the horizontal axis indicates the exciton density (/ cm ² ), and the vertical axis indicates the emission energy (eV) obtained from the exciton energy.

  In FIG. 11, a graph B1 shows the emission energy using the minimum exciton energy obtained by this example, and a graph B2 shows the emission energy obtained by the conventional calculation method. Also, in FIG. 11, the experimental results in the document V. Negoita, DW Snoke, and K. Eberl, Phys. Rev. B61, p.2779 (2000) are shown as data point B3 after slightly adjusting the parameters. ing.

As shown in this graph, at an exciton density of 10 ¹¹ / cm ² or less, it can be seen that the energy value of the graph B1 obtained by calculation in this example and the energy value of the experimental result are in good agreement. . However, at higher densities, the calculated value and the experimental value are shifted. This is considered that another effect is acting in the high density state.

  The program for causing a computer to perform the calculation of the exciton quantum level according to the present invention and the recording medium on which the program is recorded are not limited to the above-described embodiments and examples, and various modifications are possible. . For example, a quantum well is an example of a fine structure in which electrons and holes constituting excitons are confined. In addition, the above-described calculation method for various fine structures such as the quantum wires and quantum dots described above. It is possible to apply.

  Further, in step S101 of the flowchart shown in FIG. 2, as the calculation method for calculating the respective quantum levels of electrons and holes in consideration of the many-body interaction effect, specifically, the method shown in FIG. Besides, various methods may be used.

  FIG. 12 is a flowchart showing another example of the calculation method of the electron and hole quantum levels used in the calculation of exciton quantum levels. Here, the Schrodinger-Poisson method (SP method) that alternately solves the Schrodinger equation and the Poisson equation is used as a method for calculating the quantum levels of electrons and holes that are particles.

  In this calculation method, first, the Schrodinger equation is solved using a Hamiltonian that does not include the many-body interaction effect, and the initial wave function of the particle is calculated (step S301). In solving the Schrödinger equation, for example, a numerical analysis method such as a shooting method can be used. When the wave function is calculated, the electric field / potential is calculated by solving the Poisson equation using the particle distribution (for example, electron distribution) obtained from the wave function (S302). In this case, the particle distribution is obtained using the fact that it is proportional to the square of the absolute value of the wave function.

  Next, the Schrodinger equation is solved by using the Hamiltonian obtained by adding the potential obtained in step S302 to the interaction term, and a new wave function is calculated (S303). Then, it is determined whether or not to converge depending on whether or not the value of the particle energy obtained using the obtained wave function approaches a constant value or whether or not the change in shape of the wave function becomes small ( S304). Here, if it is determined not to converge, steps S302 and S303 are repeatedly executed using a new wave function. On the other hand, if it is determined to converge, the final energy is derived (S305), and the calculation by the SP method is terminated. Also by such a calculation method, it is possible to suitably calculate the electron and hole quantum levels in consideration of the many-body interaction effect.

  INDUSTRIAL APPLICABILITY The present invention can be used as a program capable of causing a computer to execute the calculation of exciton quantum levels with high accuracy, and a computer-readable recording medium storing the program.

It is a figure which shows typically about the production | generation of the exciton in a fine structure. It is a flowchart which shows the calculation process of the quantum level of the exciton performed in a program. It is a block diagram which shows an example of the hardware constitutions of the arithmetic system used for execution of a program. It is a flowchart which shows an example of the calculation method of the quantum level of particle | grains. It is a graph which shows the example of the wave function of the particle in a quantum well and a quantum well. 4 is a graph showing the structure of a quantum well obtained by calculating the quantum level of excitons in Example 1. It is a graph which shows an example of the calculation result of the quantum level of an electron and a hole. It is a graph which shows the dependence with respect to the exciton density of exciton energy. 6 is a graph showing the structure of a quantum well in which an exciton quantum level is calculated in Example 2. It is a graph which shows an example of the calculation result of the quantum level of an electron and a hole. It is a graph which shows the dependence with respect to the exciton density of luminescence energy. It is a flowchart which shows the other example of the calculation method of the quantum level of particle | grains.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 ... Arithmetic system, 10 ... CPU, 11 ... ROM, 12 ... RAM, 13 ... Storage device, 14 ... Input device, 15 ... Display device, 16 ... Drive,
50 ... quantum well, 51 ... well layer, 52, 53 ... barrier layer, 55 ... energy level.

Claims (6)

A program for causing a computer to perform quantum level calculation of excitons consisting of electrons and holes confined in a fine structure,
A first calculation process for calculating respective wave functions of electrons and holes in the microstructure taking into account the many-body interaction effect between a plurality of charged particles;
Using the electron and hole wave functions obtained in the first calculation process to calculate the exciton wave function by a variational method and causing the computer to execute a second calculation process for obtaining the exciton energy. Program.   In the second calculation process, the wave function of the exciton is changed to a function that includes the variational parameters as well as the electron and hole wave functions obtained in the first calculation process and the relative motion of the electrons and holes. 2. The program according to claim 1, wherein calculation is performed using a set envelope function.   The calculation by the variational method of the wave function of the exciton in the second calculation process is performed by using the exciton energy and its derivative under a condition that minimizes the exciton energy. The program according to claim 1 or 2. The first calculation processing for calculating the wave functions of each of electrons and holes, which are particles,
A process of calculating an initial wave function of a particle using a Hamiltonian not including the many-body interaction effect, and discretizing the initial wave function into a numerical sequence composed of N (N is a natural number) components;
Using the wave function of the N components and a Hamiltonian including a nonlinear term considering the many-body interaction effect, the particle wave function is calculated by a variational method so that the energy of the entire system is minimized, The program according to any one of claims 1 to 3, further comprising a process for obtaining energy of particles. The first calculation processing for calculating the wave functions of each of electrons and holes, which are particles,
Processing to compute the initial wave function of the particles using a Hamiltonian that does not include the many-body interaction effect;
The potential distribution is calculated by solving the Poisson equation using the particle distribution obtained from the initial wave function, and a new wave function of the particle is calculated using a Hamiltonian with the obtained potential added to the interaction term. The program according to any one of claims 1 to 3, wherein the program is included.   The computer-readable recording medium which recorded the program as described in any one of Claims 1-5.

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