固体材料中正电子自洽与非自洽场理论计算
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摘要
正电子是电子的反粒子,同时是人类发现的第一个反粒子,正电子与电子相遇会以两γ光子为主要形式发生湮没。这些湮没光子为研究固体材料的基础特性提供了一种途径。在过去半个多世纪里,在无数科研工作者不懈努力下,已发展出一门新型独立学科——正电子湮没谱学(Positron Annihilation Spectroscopy),它是一种无损探测手段,并且已广泛应用于材料科学如原子层次点缺陷、费米面、元素鉴别等,医学如正电子断层扫描等众多领域。
实验上这些诸多成功极大促使了正电子湮没理论研究。为了能更深刻、定量分析这些实验结果,需要全面的、广泛的理论研究。正电子理论模拟计算是正电子材料研究中的一个重要而不可或缺的研究领域,只有通过正电子定量的理论计算,才能从实验上分析出更多物理结果,同时更能可靠地判断材料的微观缺陷结构、化学环境等等。事实上,在正电子各实验探测技术发展的同时,正电子相关理论模拟计算同时也在发展。
这里我们认为没有涉及任何自洽计算过程的方法称为非自洽场方法,如经验赝势方法、中性原子叠加方法等,相反则称为自洽场方法,如第一性原理赝势平面波方法等。
无论什么方法,到目前为止,基本上都是将正电子与电子这“二态”分开处理,这就是所谓的" conventional scheme ",本文将详细讲述非自洽场中的中性原子叠加方法与自洽场中的赝势平面波方法。在正电子理论计算领域,其它众多的方法虽然有所不同,但处理流程却基本相同,掌握这两种方法,其它计算方法就容易掌握。
本文在参阅国内外大量正电子文献及前人工作基础上搭建了正电子相关计算系统,以相关晶体为例进行全面测试,并与实验结果或文献理论计算结果进行比较,对比结果表明这些计算系统的正确性与可靠性,并可用于计算其它复杂材料中正电子湮没行为,为理论研究固体材料打下一定的基础。本博士论文取得的主要成果如下：
(1)通过对单质的大量模拟计算发现,在超热化阶段,正电子超热化时间很短,与其在固体材料中的寿命相比可完全忽略,注入深度满足很好的指数关系,这一点与文献符合,但我们计算发现在某些轻元素单质如单晶Si中,正电子注入轮廓不能很好地满足Makhov形式；对于热化过程,计算发现固体材料在低温情况下(小于100K),超热化时间一般大于5ps,在实验上是可以探测到的,因此在分析低温寿命谱时,必须考虑热化时间的影响,也就是说在通过传统的方法计算得到正电子寿命后,需要考虑热化时间的修正,才能与低温正电子寿命实验结果进行对比。
(2)通过对闪锌矿结构和氯化钠结构的化合物计算发现,对于相同晶体结构的化合物,正电子寿命与晶格常数存在简单的关系,这为避免繁琐的寿命计算提供了一种途径。同时应用中性原子叠加方法计算了复杂体系如AgBiSe2体系中寿命情况,计算得到的Ag单空位理论结果是195ps,与我们的实验结果198ps符合很好,从而确定材料中存在Ag空位,而正是Ag空位的存在,使得体系在升温过程中,Ag和Bi双金属原子位置发生交换,这是AgBiSe2体系发生p-n-p转换的重要原因。
(3)运用Generator Coordinate Hartree Fock方法可以计算出元素周期表中几乎所有的原子波函数。本文首次将此方法用于正电子理论计算领域,从而可以理论计算出元素周期表中几乎所有元素的正电子波函数参量,这一点弥补了芬兰开发的MKA/Doppler程序包严重的不足；在此自由原子波函数基础上,进一步计算得到的Si多普勒展宽谱与实验结果符合较好；通过与第一性原理赝势平面波方法的对比计算发现,由于其理论模型的根本问题,由中性原子叠加方法计算得到的多普勒展宽谱有些高估了低动量湮没成分,而低估了高动量湮没成分。
(4)在考虑带隙修正与虚电荷修正的基础上,计算了半导体Si和AlN中的单空位电荷态缺陷形成能,计算得到的Si单空位形成能与正电子寿命实验获得的缺陷形成能符合很好；而在AlN中,计算得到的N空位形成能很高,因此AlN呈现n型半导体的原因并不是N空位引起的。
初步搭建的正电子理论计算系统可用于计算众多晶体结构材料中正电子热化、捕获、湮没等行为,为以后正电子研究固体材料提供理论平台。
Positron is the antiparticle of electron and is the first kind of antimatter discovered by human, if it collide with an electron, they will annihilate mainly with the production of two gamma photons. These annihilation photons provide a new way for studying the basic physics property in solid materials. In the past more than half a century, a new independent discipline that is positron annihilation spectroscopy has been formed under the unremitting efforts with countless people, it is a non-destructive technique and has been widely used in material science such as point defects in atom level、Fermi surface、elemental specificity et al., medical science, such as positron emission tomography, and many other areas.
The success in experiment greatly encourages the theoretical study of positron annihilation. Experimental methods need a comprehensive theory for a deep, quantitative understanding of the experimental results. Only through the theoretical calculations, can we obtain more physical information from the experimental results. Positron calculation is very important and necessary in the research of materials with positron techniques. In fact, with the development of positron experimental techniques, positron theory is also developing.
Here, we define that the calculation method which does not involve the consistent calculation process is the non-consistent field method, such as empirical pseudo-potential method, superposed-neutral-atom method and so on, otherwise, it is the consistent field method, such as the first principle pseudo-potential plane wave method.
No matter what kind of method, most of them take the self-consistent calculation for electron, but not for positron. Positron and electron, the "two components" are treated separately, this is the so called "conventional scheme", in this thesis, we will introduce the superposed-neutral-atom method and the pseudo-potential plane wave method in detail for calculating the positron quantum state, although many other methods have something different, they have similar calculating process, if we know the above two methods, it is easy to master the other approaches.
In this thesis, some positron related calculation systems based on many references and previous works are finished to study the positron thermal and epithermal process, positron lifetime, positron Doppler broadening and so on in solids. Comprehensive test in done with some related crystalloids, these calculation results agree well with experimental ones, indicating the rationality and correctness of the positron calculation systems. In addition, they can also be used to study the positron annihilation behavior in complex solid materials, laying the foundation for better researches of solids. The main achievements of this thesis are as follows:
(1) We use the Monte Carlo simulation to study the positron epithermal process, from much simulation of many simple substances, we find that in epithermal process, the positron epithermal time is very small, it can be ignored compared to positron lifetime; the positron stopping profile obey well the form originally suggested by Makhov in1961for electrons, the mean penetration depth depends on the incident positron energy rather accurately as the power which first suggested by Mills and Wilson in1982, this coincidence with the references; however, in some light elements, such as in Silicon, the positron stopping profile does not agree well with the Makhov form. For thermal process, from many calculations, we find that in low material temperature (usually lower than100K), the positron thermal time usually larger than5ps, this could be detected in positron lifetime experiment, that is to say the thermal time should be considered when we analyze the low temperature lifetime spectrum,
(2) From our calculations of positron lifetime, we find that in the compounds which have the same crystal structure, the positron bulk lifetime may have rather simple relationship with the lattice constant. This provides a rather easy way to calculate the positron lifetime. Besides, we use the superposed-neutral-atom method (ATSUP) to calculate the positron lifetime of some complex compounds such as AgBiSe2, the lifetime of Ag-mono vacancy is195ps, agree well with our experimental one which is about198ps. This confirms that there are Ag-vacancies in this material, exists of these Ag-vacancies provide a way for the exchange of the positions of Ag and Bi, which is an important mechanism for the p-n-p transition of AgBiSe2.
(3) The Generator Coordinate Hartree Fock method could be used to calculate the free atom wave-function of almost all the elements in periodic table. We adopt this method for positron theoretical research, and it can be used to calculate the positron wavefunction parameters of almost all the elements in periodic table, this makes up the serious weakness of MIKA/Doppler software which developed by Finland group; Base on the free atom wave-function, the broadening spectrum could be obtained, we take the semiconductor silicon as an example, the calculation result of Doppler broadening agrees very well with the experimental one; besides, by compared the first principle pseudo-potential calculation result, we find that because of the defect of the theoretical model, the ATSUP calculation results are a little larger than the experimental ones in the low momentum region, but a little smaller in the high momentum range.
(4) Based on the band gap correction and the image charge correction, we calculate the mono-vacancy defect formation energy in different charge states of semiconductor Si and AlN. The calculation result of the mono-vacancy defect formation energy in Si agree well with the formation energy obtained form positron lifetime experiment; In AlN, we find that the defect formation energy of N mono-vacancy is very high, and so, the n-type in AlN semiconductor is not caused by N mono-vacancy.
In a word, the positron theoretical calculation systems are built. They can be applied to calculate the positron thermal, trapping, annihilation process and so on in many solid materials of different crystal structures. They also provide the theoretical basis for further positron researches.
实验上这些诸多成功极大促使了正电子湮没理论研究。为了能更深刻、定量分析这些实验结果,需要全面的、广泛的理论研究。正电子理论模拟计算是正电子材料研究中的一个重要而不可或缺的研究领域,只有通过正电子定量的理论计算,才能从实验上分析出更多物理结果,同时更能可靠地判断材料的微观缺陷结构、化学环境等等。事实上,在正电子各实验探测技术发展的同时,正电子相关理论模拟计算同时也在发展。
这里我们认为没有涉及任何自洽计算过程的方法称为非自洽场方法,如经验赝势方法、中性原子叠加方法等,相反则称为自洽场方法,如第一性原理赝势平面波方法等。
无论什么方法,到目前为止,基本上都是将正电子与电子这“二态”分开处理,这就是所谓的" conventional scheme ",本文将详细讲述非自洽场中的中性原子叠加方法与自洽场中的赝势平面波方法。在正电子理论计算领域,其它众多的方法虽然有所不同,但处理流程却基本相同,掌握这两种方法,其它计算方法就容易掌握。
本文在参阅国内外大量正电子文献及前人工作基础上搭建了正电子相关计算系统,以相关晶体为例进行全面测试,并与实验结果或文献理论计算结果进行比较,对比结果表明这些计算系统的正确性与可靠性,并可用于计算其它复杂材料中正电子湮没行为,为理论研究固体材料打下一定的基础。本博士论文取得的主要成果如下：
(1)通过对单质的大量模拟计算发现,在超热化阶段,正电子超热化时间很短,与其在固体材料中的寿命相比可完全忽略,注入深度满足很好的指数关系,这一点与文献符合,但我们计算发现在某些轻元素单质如单晶Si中,正电子注入轮廓不能很好地满足Makhov形式；对于热化过程,计算发现固体材料在低温情况下(小于100K),超热化时间一般大于5ps,在实验上是可以探测到的,因此在分析低温寿命谱时,必须考虑热化时间的影响,也就是说在通过传统的方法计算得到正电子寿命后,需要考虑热化时间的修正,才能与低温正电子寿命实验结果进行对比。
(2)通过对闪锌矿结构和氯化钠结构的化合物计算发现,对于相同晶体结构的化合物,正电子寿命与晶格常数存在简单的关系,这为避免繁琐的寿命计算提供了一种途径。同时应用中性原子叠加方法计算了复杂体系如AgBiSe2体系中寿命情况,计算得到的Ag单空位理论结果是195ps,与我们的实验结果198ps符合很好,从而确定材料中存在Ag空位,而正是Ag空位的存在,使得体系在升温过程中,Ag和Bi双金属原子位置发生交换,这是AgBiSe2体系发生p-n-p转换的重要原因。
(3)运用Generator Coordinate Hartree Fock方法可以计算出元素周期表中几乎所有的原子波函数。本文首次将此方法用于正电子理论计算领域,从而可以理论计算出元素周期表中几乎所有元素的正电子波函数参量,这一点弥补了芬兰开发的MKA/Doppler程序包严重的不足；在此自由原子波函数基础上,进一步计算得到的Si多普勒展宽谱与实验结果符合较好；通过与第一性原理赝势平面波方法的对比计算发现,由于其理论模型的根本问题,由中性原子叠加方法计算得到的多普勒展宽谱有些高估了低动量湮没成分,而低估了高动量湮没成分。
(4)在考虑带隙修正与虚电荷修正的基础上,计算了半导体Si和AlN中的单空位电荷态缺陷形成能,计算得到的Si单空位形成能与正电子寿命实验获得的缺陷形成能符合很好；而在AlN中,计算得到的N空位形成能很高,因此AlN呈现n型半导体的原因并不是N空位引起的。
初步搭建的正电子理论计算系统可用于计算众多晶体结构材料中正电子热化、捕获、湮没等行为,为以后正电子研究固体材料提供理论平台。
Positron is the antiparticle of electron and is the first kind of antimatter discovered by human, if it collide with an electron, they will annihilate mainly with the production of two gamma photons. These annihilation photons provide a new way for studying the basic physics property in solid materials. In the past more than half a century, a new independent discipline that is positron annihilation spectroscopy has been formed under the unremitting efforts with countless people, it is a non-destructive technique and has been widely used in material science such as point defects in atom level、Fermi surface、elemental specificity et al., medical science, such as positron emission tomography, and many other areas.
The success in experiment greatly encourages the theoretical study of positron annihilation. Experimental methods need a comprehensive theory for a deep, quantitative understanding of the experimental results. Only through the theoretical calculations, can we obtain more physical information from the experimental results. Positron calculation is very important and necessary in the research of materials with positron techniques. In fact, with the development of positron experimental techniques, positron theory is also developing.
Here, we define that the calculation method which does not involve the consistent calculation process is the non-consistent field method, such as empirical pseudo-potential method, superposed-neutral-atom method and so on, otherwise, it is the consistent field method, such as the first principle pseudo-potential plane wave method.
No matter what kind of method, most of them take the self-consistent calculation for electron, but not for positron. Positron and electron, the "two components" are treated separately, this is the so called "conventional scheme", in this thesis, we will introduce the superposed-neutral-atom method and the pseudo-potential plane wave method in detail for calculating the positron quantum state, although many other methods have something different, they have similar calculating process, if we know the above two methods, it is easy to master the other approaches.
In this thesis, some positron related calculation systems based on many references and previous works are finished to study the positron thermal and epithermal process, positron lifetime, positron Doppler broadening and so on in solids. Comprehensive test in done with some related crystalloids, these calculation results agree well with experimental ones, indicating the rationality and correctness of the positron calculation systems. In addition, they can also be used to study the positron annihilation behavior in complex solid materials, laying the foundation for better researches of solids. The main achievements of this thesis are as follows:
(1) We use the Monte Carlo simulation to study the positron epithermal process, from much simulation of many simple substances, we find that in epithermal process, the positron epithermal time is very small, it can be ignored compared to positron lifetime; the positron stopping profile obey well the form originally suggested by Makhov in1961for electrons, the mean penetration depth depends on the incident positron energy rather accurately as the power which first suggested by Mills and Wilson in1982, this coincidence with the references; however, in some light elements, such as in Silicon, the positron stopping profile does not agree well with the Makhov form. For thermal process, from many calculations, we find that in low material temperature (usually lower than100K), the positron thermal time usually larger than5ps, this could be detected in positron lifetime experiment, that is to say the thermal time should be considered when we analyze the low temperature lifetime spectrum,
(2) From our calculations of positron lifetime, we find that in the compounds which have the same crystal structure, the positron bulk lifetime may have rather simple relationship with the lattice constant. This provides a rather easy way to calculate the positron lifetime. Besides, we use the superposed-neutral-atom method (ATSUP) to calculate the positron lifetime of some complex compounds such as AgBiSe2, the lifetime of Ag-mono vacancy is195ps, agree well with our experimental one which is about198ps. This confirms that there are Ag-vacancies in this material, exists of these Ag-vacancies provide a way for the exchange of the positions of Ag and Bi, which is an important mechanism for the p-n-p transition of AgBiSe2.
(3) The Generator Coordinate Hartree Fock method could be used to calculate the free atom wave-function of almost all the elements in periodic table. We adopt this method for positron theoretical research, and it can be used to calculate the positron wavefunction parameters of almost all the elements in periodic table, this makes up the serious weakness of MIKA/Doppler software which developed by Finland group; Base on the free atom wave-function, the broadening spectrum could be obtained, we take the semiconductor silicon as an example, the calculation result of Doppler broadening agrees very well with the experimental one; besides, by compared the first principle pseudo-potential calculation result, we find that because of the defect of the theoretical model, the ATSUP calculation results are a little larger than the experimental ones in the low momentum region, but a little smaller in the high momentum range.
(4) Based on the band gap correction and the image charge correction, we calculate the mono-vacancy defect formation energy in different charge states of semiconductor Si and AlN. The calculation result of the mono-vacancy defect formation energy in Si agree well with the formation energy obtained form positron lifetime experiment; In AlN, we find that the defect formation energy of N mono-vacancy is very high, and so, the n-type in AlN semiconductor is not caused by N mono-vacancy.
In a word, the positron theoretical calculation systems are built. They can be applied to calculate the positron thermal, trapping, annihilation process and so on in many solid materials of different crystal structures. They also provide the theoretical basis for further positron researches.
引文
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