不完整晶体的电子结构计算
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摘要
不完整晶体含有缺陷、掺杂或者无序。缺陷存在于晶体生长的过程中并对晶体的性质产生很大影响；掺杂是为了拓展晶体用途人为地往晶体掺入某些物质，可被视为某种特殊的缺陷；无序常出现在混晶中，而很多混晶材料属于高科技材料。由于不完整晶体用途日益广泛，对它们的研究自然成为凝聚态物理的一个重要课题。但由于失去周期性边界条件，不完整晶体的理论计算相当困难，我们只得对不同的研究对象采用不同的近似计算方法。
对含有点缺陷的不完整晶体，通常使用原子团簇(cluster)方法或超原胞模拟计算方法，两种方法各有其优缺点。原子团簇方法能够提供缺陷能级的相对位置，但无法确定它们相对于价带顶或导带底的确切位置。超原胞模拟计算则可以确定缺陷能级的具体位置，但为了减小周期性边界条件的影响，超原胞的尺度必须尽量地大，这无形中增加了非常大的计算量。为了取得良好的计算效果，我们在本论文中提出了以下两条准则：
(1)由于缺陷的存在，超原胞中即使同一种原子也会有不同的电荷密度。因此必须根据对称性及其与缺陷中心的距离远近，再对超原胞中同种原子进行必要的分类。这样当然更有利于获取体系更多的有用信息。
(2)远离缺陷中心的原子的净电荷应与基质中同种原子的净电荷相差不大，这样才能确保周期性边界条件对缺陷中心的影响较小，使计算结果较可靠，否则还需进一步扩大超原胞。
另一方面，在混晶计算中，已往常采用刚带近似(RBA)，虚晶近似(VCA)，相干势(CPA)等方法。这些近似方法是通过对某些物理量取平均的办法来恢复周期性边界条件。取平均的物理量可能是原子序数，原子紧束缚参数，原子势参数，甚至是态密度本身。上述近似方法通常适用于金属键混晶系统，但对于共价键或离子键混晶系统计算结果则不理想。在本论文中，我们尝试川一种全新的计算方法——统计超原胞法来计算混晶的
摘 要
电子结构。这种新方法是将计算过程分为两个阶段来进行：先计算混晶中微晶的电子结
构，再利用统计理论将无序效应包括进来。
本论文由五章组成。在全文中我们使用标量相对论性的LMTO－ASA方法来计算能带
结构，交换关联势则选择Hedn－Lundqvist势。LMTO－ASA方法之所以较适用于大型超原
胞模拟计算，是因为使用这种方法时计算量较小而结果又能满足一般的精度要求。在本
论文第一章，我们集中介绍了本文所用的计算原理和方法，其中除了LMTO－ASA方法外，
我们还详细介绍了特殊k点方法和统计超原胞方法。
在第二章，作为研究发光中心的一个例子，我们计算了纤锌矿硫化锌掺铜的电子结
构。计算结果表明，铜发光中心的能级与铜的价态和是否加入共掺杂剂有关。在硫化锌
掺铜掺氯和硫化锌掺铜掺铝时，铜受主能级反常地深，比较靠近导带底。而在硫化锌只
掺人一价铜（这时有硫空位存在），铜发光中心为一缔合中心。至于硫化锌掺杂二价铜，
此时铜的类d 带则位于价带顶上方。上述不同情形的铜发光中心的发光机制分别与
Lambe－Klick模型、WIlliams－Prener模型及Sob6n－Klasens模型相符合。
在第三章，作为研究晶体点缺陷的一个例子，我们计算了砷化嫁中反位砷的电子结
构。砷化镜中的反位砷因与ELZ中心紧密相关而倍受人们的关注。我们的主要计算结果
是，带隙态主要是由反位的砷原于与邻近的砷原于通过反键键合而产生的，且带隙态由
类山和类TZ态组成，其能级分别在巳一巳＋0．刀＊V和N＝EAI＋1刀7＊V处。我们的
计算结果与实验相符，也与其它方法的理论计算结果相一致。
在第四章，我们研究了TNx混晶系统。根据氮一钛合成相图，Tffox往往具有双相混
合结构，即要么是a—Ti与e－TiZN的双相混合结构，要么是6一n 与。一n 的双相混
合结构。我们先用LMTO－ASA方法和超原胞模拟法计算微晶a—Ti，e一卫 和6一Tffo
的电子结构，然后再应用上述微晶的统计分布规律计算所有低化学计量的TINx的电子结
构。为了进一步与实验结果作比较，我们又分析了Tffox电阻率随组分X的变化。我们发
现理论预言的变化总趋势与实验事实基本一致。
在第五章，我们研究了另一混晶系统B＜；－xKxBIO。。当X＜0．3时，BIO。八面体的呼
－2－
＠8
吸模式形变形成了两种不等价的钒原子。为了模拟Bi（l的＋3价和Bitll）的＋5价，在
我们的计算中将Bi（l）的6S态处理成芯态，而将Bi（II的6S态处理作价态。我们首先计
算了某些Ba；．xKxBIO。有序合金的电子结构，然后再借助统计超原胞法涵盖钾替换钡的无
序效应。我们的计算结果在以卜几方面与实验事实一致：一、无掺杂的BaBIO3是带隙为
1石电于伏的半导体（实验得出的带隙是 2刀电于伏）；二、X＜0．3时 BS，xKxBIO。系统显
示出半导体性，且禁带宽度随着组分 X的增加而减小：三、当 0．3＜X＜0．5时，BX；．xKxBIO。
系统具有金属的性质，但Fermi能处的总态密度较小；四、不等价韧原子间的电荷转移
Imperfect crystals contain defects, dopants or disorder. Defects exist natively in the process of the crystal growth and produce a great effect on the crystal properties. Dopants are artificially put into a crystal to widen its use fields. They can be considered as a special kind of defects. Disorder appears in mixed crystals, which constitute a large number of high-technology materials. Owing to the increasingly extensive use of imperfect crystals, the investigations on them become a very important issue in the condensed matter. However, theoretical researches on them are confronted with gigantic difficulty because of the loss of periodic boundary condition. We have to use differently approximate methods to calculate different objects of study.
For imperfect crystals with point defects, the cluster method and the supercell simulation are two methods in common use. Both methods have their strong and weak points. The cluster method can provide relative positions of defect levels, but it is unable to determine their concrete positions with respect to the top of valence band or to the bottom of conduction band. The supercell simulation approach can establish the defect level positions. But, hi order to reduce the affection of periodic boundary condition, the size of supercell must be taken as large as possible. At the same time, the computational load increases dramatically. We pose two following rules for the best results of calculations in this thesis.
(1) Due to the existence of defects, the chemically same atoms in the supercell may have differently effective charges. They should be classified again in the light of their symmetry and their distances from the defect center. More useful information can be obtained if so.
(2) The effective charge of an atom far from the defect center should be almost the same as the same kind atom in the host. Only hi this way can the effect of the periodic boundary condition on the defect center guarantee to be omitted and the calculating results be trustworthy. If it is not so, we have to enlarge the size of the
supercell in use.
For mixed crystal systems, on the other hand, the rigid-band approximation (RBA), the virtual crystal approximation (VGA) and the coherent potential approximation (CPA) have been constantly utilized in previous calculations. In these approximate methods, some average quantities are used to re-validate the periodic boundary conditions, such as the atomic number, atomic tight-binding parameters, potential parameters of atomic spheres, and even density of states. Usually, they are suitable to mixed crystals with metallic bond but not to those with covalent bond or ionic bond. In this thesis, we attempt a new method, the statistical supercell method, to calculate the electronic structure of a mixed crystal. This new method divides the calculations into two steps, first to calculate the electronic structure of micro-crystals appearing in a mixed crystal system and then to include the effect of disorder by means of statistical theory.
This thesis is composed of five chapters. Throughout this thesis, the band structure calculations are performed by using the scalar-relativistic linear muffin-tin orbital (LMTO) method in the atomic-spheres approximation (ASA) with Hedin-Lundqvist exchange-correlation potential. LMTO-ASA method is suitable to making self-consistent calculation of a big supercell since its computational load is relatively small but its calculating precision is able to meet usual demands. In Chapter 1, the computational principles and methods used in this thesis are presented. Besides the LMTO-ASA method, the special k points and the statistical supercell method are also introduced in detail.
In Chapter 2, as an instance of studying luminescence centers, we investigated the electronic structure of wurtzite ZnS doped with Cu. The calculating results show that, the levels of Cu luminescence centers depend on the oxide-state of copper and the existence of codopants. In the cases of ZnS:Cu,Cl and ZnS:Cu,Al, Cu acceptor states are anom
对含有点缺陷的不完整晶体，通常使用原子团簇(cluster)方法或超原胞模拟计算方法，两种方法各有其优缺点。原子团簇方法能够提供缺陷能级的相对位置，但无法确定它们相对于价带顶或导带底的确切位置。超原胞模拟计算则可以确定缺陷能级的具体位置，但为了减小周期性边界条件的影响，超原胞的尺度必须尽量地大，这无形中增加了非常大的计算量。为了取得良好的计算效果，我们在本论文中提出了以下两条准则：
(1)由于缺陷的存在，超原胞中即使同一种原子也会有不同的电荷密度。因此必须根据对称性及其与缺陷中心的距离远近，再对超原胞中同种原子进行必要的分类。这样当然更有利于获取体系更多的有用信息。
(2)远离缺陷中心的原子的净电荷应与基质中同种原子的净电荷相差不大，这样才能确保周期性边界条件对缺陷中心的影响较小，使计算结果较可靠，否则还需进一步扩大超原胞。
另一方面，在混晶计算中，已往常采用刚带近似(RBA)，虚晶近似(VCA)，相干势(CPA)等方法。这些近似方法是通过对某些物理量取平均的办法来恢复周期性边界条件。取平均的物理量可能是原子序数，原子紧束缚参数，原子势参数，甚至是态密度本身。上述近似方法通常适用于金属键混晶系统，但对于共价键或离子键混晶系统计算结果则不理想。在本论文中，我们尝试川一种全新的计算方法——统计超原胞法来计算混晶的
摘 要
电子结构。这种新方法是将计算过程分为两个阶段来进行：先计算混晶中微晶的电子结
构，再利用统计理论将无序效应包括进来。
本论文由五章组成。在全文中我们使用标量相对论性的LMTO－ASA方法来计算能带
结构，交换关联势则选择Hedn－Lundqvist势。LMTO－ASA方法之所以较适用于大型超原
胞模拟计算，是因为使用这种方法时计算量较小而结果又能满足一般的精度要求。在本
论文第一章，我们集中介绍了本文所用的计算原理和方法，其中除了LMTO－ASA方法外，
我们还详细介绍了特殊k点方法和统计超原胞方法。
在第二章，作为研究发光中心的一个例子，我们计算了纤锌矿硫化锌掺铜的电子结
构。计算结果表明，铜发光中心的能级与铜的价态和是否加入共掺杂剂有关。在硫化锌
掺铜掺氯和硫化锌掺铜掺铝时，铜受主能级反常地深，比较靠近导带底。而在硫化锌只
掺人一价铜（这时有硫空位存在），铜发光中心为一缔合中心。至于硫化锌掺杂二价铜，
此时铜的类d 带则位于价带顶上方。上述不同情形的铜发光中心的发光机制分别与
Lambe－Klick模型、WIlliams－Prener模型及Sob6n－Klasens模型相符合。
在第三章，作为研究晶体点缺陷的一个例子，我们计算了砷化嫁中反位砷的电子结
构。砷化镜中的反位砷因与ELZ中心紧密相关而倍受人们的关注。我们的主要计算结果
是，带隙态主要是由反位的砷原于与邻近的砷原于通过反键键合而产生的，且带隙态由
类山和类TZ态组成，其能级分别在巳一巳＋0．刀＊V和N＝EAI＋1刀7＊V处。我们的
计算结果与实验相符，也与其它方法的理论计算结果相一致。
在第四章，我们研究了TNx混晶系统。根据氮一钛合成相图，Tffox往往具有双相混
合结构，即要么是a—Ti与e－TiZN的双相混合结构，要么是6一n 与。一n 的双相混
合结构。我们先用LMTO－ASA方法和超原胞模拟法计算微晶a—Ti，e一卫 和6一Tffo
的电子结构，然后再应用上述微晶的统计分布规律计算所有低化学计量的TINx的电子结
构。为了进一步与实验结果作比较，我们又分析了Tffox电阻率随组分X的变化。我们发
现理论预言的变化总趋势与实验事实基本一致。
在第五章，我们研究了另一混晶系统B＜；－xKxBIO。。当X＜0．3时，BIO。八面体的呼
－2－
＠8
吸模式形变形成了两种不等价的钒原子。为了模拟Bi（l的＋3价和Bitll）的＋5价，在
我们的计算中将Bi（l）的6S态处理成芯态，而将Bi（II的6S态处理作价态。我们首先计
算了某些Ba；．xKxBIO。有序合金的电子结构，然后再借助统计超原胞法涵盖钾替换钡的无
序效应。我们的计算结果在以卜几方面与实验事实一致：一、无掺杂的BaBIO3是带隙为
1石电于伏的半导体（实验得出的带隙是 2刀电于伏）；二、X＜0．3时 BS，xKxBIO。系统显
示出半导体性，且禁带宽度随着组分 X的增加而减小：三、当 0．3＜X＜0．5时，BX；．xKxBIO。
系统具有金属的性质，但Fermi能处的总态密度较小；四、不等价韧原子间的电荷转移
Imperfect crystals contain defects, dopants or disorder. Defects exist natively in the process of the crystal growth and produce a great effect on the crystal properties. Dopants are artificially put into a crystal to widen its use fields. They can be considered as a special kind of defects. Disorder appears in mixed crystals, which constitute a large number of high-technology materials. Owing to the increasingly extensive use of imperfect crystals, the investigations on them become a very important issue in the condensed matter. However, theoretical researches on them are confronted with gigantic difficulty because of the loss of periodic boundary condition. We have to use differently approximate methods to calculate different objects of study.
For imperfect crystals with point defects, the cluster method and the supercell simulation are two methods in common use. Both methods have their strong and weak points. The cluster method can provide relative positions of defect levels, but it is unable to determine their concrete positions with respect to the top of valence band or to the bottom of conduction band. The supercell simulation approach can establish the defect level positions. But, hi order to reduce the affection of periodic boundary condition, the size of supercell must be taken as large as possible. At the same time, the computational load increases dramatically. We pose two following rules for the best results of calculations in this thesis.
(1) Due to the existence of defects, the chemically same atoms in the supercell may have differently effective charges. They should be classified again in the light of their symmetry and their distances from the defect center. More useful information can be obtained if so.
(2) The effective charge of an atom far from the defect center should be almost the same as the same kind atom in the host. Only hi this way can the effect of the periodic boundary condition on the defect center guarantee to be omitted and the calculating results be trustworthy. If it is not so, we have to enlarge the size of the
supercell in use.
For mixed crystal systems, on the other hand, the rigid-band approximation (RBA), the virtual crystal approximation (VGA) and the coherent potential approximation (CPA) have been constantly utilized in previous calculations. In these approximate methods, some average quantities are used to re-validate the periodic boundary conditions, such as the atomic number, atomic tight-binding parameters, potential parameters of atomic spheres, and even density of states. Usually, they are suitable to mixed crystals with metallic bond but not to those with covalent bond or ionic bond. In this thesis, we attempt a new method, the statistical supercell method, to calculate the electronic structure of a mixed crystal. This new method divides the calculations into two steps, first to calculate the electronic structure of micro-crystals appearing in a mixed crystal system and then to include the effect of disorder by means of statistical theory.
This thesis is composed of five chapters. Throughout this thesis, the band structure calculations are performed by using the scalar-relativistic linear muffin-tin orbital (LMTO) method in the atomic-spheres approximation (ASA) with Hedin-Lundqvist exchange-correlation potential. LMTO-ASA method is suitable to making self-consistent calculation of a big supercell since its computational load is relatively small but its calculating precision is able to meet usual demands. In Chapter 1, the computational principles and methods used in this thesis are presented. Besides the LMTO-ASA method, the special k points and the statistical supercell method are also introduced in detail.
In Chapter 2, as an instance of studying luminescence centers, we investigated the electronic structure of wurtzite ZnS doped with Cu. The calculating results show that, the levels of Cu luminescence centers depend on the oxide-state of copper and the existence of codopants. In the cases of ZnS:Cu,Cl and ZnS:Cu,Al, Cu acceptor states are anom
引文
[1] M. Born and J.R. Oppenheimer, Arm. Phys. 87 (1927) 457．
[2] D.R. Hartree, Proc. Cam. Phil. Soc. 24 (1928) 89．
[3] V. Fock, Z. Phys. 61 (1930) 209．
[4] J.C. Slater: Quantum Theory of Atomic Structure (Me Graw-Hill, New York, 1960) .
[5] P. Hohenberg and W. Khon, Phys. Rev. 136 (1964) B864．
[6] W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) A1 133．
[7] F. Bloch, Z. Phys. 52 (1928) 555．
[8] J.C. Slater, Phys. Rev. 51 (1937) 846．
[9] H. L. Skriver: The LMTO Method (Springer-Verlag, 1984) .
[10] O.K. Andersen, Phys. Rev. B 12 (1975) 3060．
[11] O.K. Andersen and R.V. Kasowski, Phys. Rev. B4 (1971) 1064．
[12] E.U. Condon and G.M. Shortley: The Theory of Atomic Spectra (University Press, Cambridge, 1951) .
[13] L. Hedin and B.I. Lundqvist, J. Phys. C4 (1971) 2064．
[14] D.J. Chadi and M.L. Cohen, Phys. Rev. B8 (1973) 5747．
[15] H.J. Monkhorst and J D. Pack, Phys. Rev. B13 (1976) 5188．
[16] C.H. Chen, Solid State Commun. 48 (1983) 235．
[17] 沈耀文、黄美纯，厦门大学学报(自然科学版)，第31卷第4期(1992) ，p360．
[2] D.R. Hartree, Proc. Cam. Phil. Soc. 24 (1928) 89．
[3] V. Fock, Z. Phys. 61 (1930) 209．
[4] J.C. Slater: Quantum Theory of Atomic Structure (Me Graw-Hill, New York, 1960) .
[5] P. Hohenberg and W. Khon, Phys. Rev. 136 (1964) B864．
[6] W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) A1 133．
[7] F. Bloch, Z. Phys. 52 (1928) 555．
[8] J.C. Slater, Phys. Rev. 51 (1937) 846．
[9] H. L. Skriver: The LMTO Method (Springer-Verlag, 1984) .
[10] O.K. Andersen, Phys. Rev. B 12 (1975) 3060．
[11] O.K. Andersen and R.V. Kasowski, Phys. Rev. B4 (1971) 1064．
[12] E.U. Condon and G.M. Shortley: The Theory of Atomic Spectra (University Press, Cambridge, 1951) .
[13] L. Hedin and B.I. Lundqvist, J. Phys. C4 (1971) 2064．
[14] D.J. Chadi and M.L. Cohen, Phys. Rev. B8 (1973) 5747．
[15] H.J. Monkhorst and J D. Pack, Phys. Rev. B13 (1976) 5188．
[16] C.H. Chen, Solid State Commun. 48 (1983) 235．
[17] 沈耀文、黄美纯，厦门大学学报(自然科学版)，第31卷第4期(1992) ，p360．