The novelty and surface to volume ratio dependent electron band structure in semiconductor nanowire

THE NOVELTY AND SURFACE-TO-VOLUME-RATIO DEPENDENT ELECTRON BAND STRUCTURE IN SEMICONDUCTOR NANOWIRE YAO DONGLAI (Master of Science) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2011 THE NOVELTY AND SURFACE-TO-VOLUME-RATIO DEPENDENT ELECTRON BAND STRUCTURE IN SEMICONDUCTOR NANOWIRE YAO DONGLAI 2011 Acknowledgement i Acknowledgement This thesis summarizes my research work that has been done since I came to Professor Li Baowen’s group in 2006. During my PhD study, I have worked with quite a lot of people whose contribution in assorted ways to the research and the making of this thesis deserved special mention. It is my pleasure to show my gratitude to them all in my humble acknowledgment. In the first place I would like to record my gratitude to Li Baowen for his supervision, advice, and guidance from the very early stage of this research as well as giving me extraordinary experiences through out the work. Above all and the most needed, he provided me unflinching encouragement and support in various ways. Anytime I was in confusion or lost direction in my study, he will rectify my mistake and guide me to the right way. His truly scientist intuition has made him as a constant oasis of ideas and passions in science, which exceptionally inspire and enrich my growth as a student and a researcher. I also thank him for giving me continuous support and help on applying NUS research scholarship and President Graduates Fellowship, China Overseas Excellent Graduates Awards, Research Assistant position in National University of Singapore and IME Astar Singapore, which support me Acknowledgement ii from the very base during my whole candidature of my PhD. Without him, I can never reach here. I thank him from my deep heart. I gratefully acknowledge Professor Zhang Gang, my co-supervisor, for his advice, supervision, and crucial contribution, which made him a backbone of this research and so to this thesis. His involvement with his originality has triggered and nourished my intellectual maturity that I will benefit from, for a long time to come. Professor Zhang, I am grateful in every possible way and hope to keep up our collaboration in the future. Furthermore, I thank him for using his precious times to read this thesis and gave his critical comments about it. I am indebted to him more than he knows. Many thanks go in particular to Professor Wang Jian-sheng, Professor Gong Jiangbin. I am much appreciated for their valuable advice in science discussion, supervision in courses of computational physics, advanced quantum dynamics. I have also benefited by advice and guidance from Porfessor Wang, who also kindly grants me his time even for answering some of my unintelligent questions. I would like to thank Department of Physics, Centre for Computation Physics Acknowledgement iii in National University of Singapore (NUS). It is them who provide me such a good research environment and financial support. A lot of thanks go to Dr. Zhang Xinhuai and Shin Gen, who gives me a lot of help on the high performance computing in SVU and CCSE in NUS. I also benefited a lot from Professor Li Zhenya, Professor Gao Lei, Professor Shen Mingrong, Professor Jiang Qin, Professor Wu Yinzhong, Professor Zhu Shiqun, Professor Gu Jihua, Professor Gan Zhaoqiang, Professor Nin Zhaoyuan, Professor Fang Liang, Professor Mu Xiaoyong and President Zhu Xiulin during my bachelor and master study in Suzhou University. I specially thank Professor Li Zhenya for his mentorship during my master degree, and Professor Gao Lei for his recommendation to NUS. Many thanks go to Professor Guo Guangyu in National Taiwan University for his patient explanation and detailed instructions on my very first step in the field of ab-init computational physics. To all the group member: Wang Lei, Wu Gang, Lan Jinhua, Li Nianbei, Yang Nuo, Wu Xiang, Chen Jie, Zhang Lifa, Ren Jie, Shi Lihong, Ni Xiaoxi, Zhang Kaiwen, Xie Rongguo, Xu Xiangfang, Zhu Guimei, Zhang Xun, Ma Jing, Feng Lin, I thank you so much for your useful discussion, sincere comments, Acknowledgement iv and instructive suggestions not only in the weekly group meeting but also in our personal conversation. I am proud to record that I had several years to work with you all. Where would I be without my family? My parents deserve special mention for their inseparable support. My mother, Yu Huiyu, in the first place is the person who put the fundament my learning character, showing me the joy of intellectual pursuit ever since I was a child. My father, Yao Huaxiang, is the one who sincerely raised me with his caring and gently love. Words fail me to express my appreciation to my wife, Hu Wei, whose dedication, love and persistent confidence in me, has taken the load off my shoulder. I owe her for being unselfishly let her intelligence, passions, and ambitions collide with mine. Therefore, I would also thank my parents in-law for letting me take her hand in marriage, and accepting me as a member of the family, warmly. Finally, I would like to thank everybody who was important to the successful realization of thesis, as well as expressing my apology that I could not mention personally one by one. Table of Content Acknowledgement ····································································i Abstract (Summery) ·································································v Publications ··········································································vii List of Tables ·······································································viii List of Figures ·······································································ix Chapter Introduction ·······················································1 1.1 General background from nanotechnology to silicon nanowires. ··············································································1 1.2 Literature review ····················································3 1.3 Introduction to our work ···········································6 References ·······························································10 Chapter Modelling and Methodology ··································13 2.1 Density Functional Theory········································13 2.2 Tight Binding Method ·············································19 2.3 Density Functional Tight Binding ·······························21 2.4 DFT applied to Silicon nanowires ·······························22 2.5 Discussion ···························································24 References ·······························································26 Chapter A Universal Expression of Band Gap for Silicon Nanowires of Different Cross-Section Geometries ·····································31 3.1 Introduction ·························································32 3.2 SVR(Surface-to-Volume Ratio)··································32 3.3 Density Functional Tight Binding (Methodology) ············33 3.4. Results and discussion ············································36 3.5 Conclusions ·························································42 References ·······························································44 Chapter Impacts of size and cross-sectional shape on surface lattice constant and electron effective mass of silicon nanowires ···············53 4.1 Introduction ·························································54 4.2 Methodology ························································56 4.3 Results and discussion ·············································59 4.4 Conclusions ·························································64 References ·······························································66 Chapter Direct to Indirect Band Gap Transition in [110] Silicon Nanowires ·······································································73 5.1 Introduction ·························································74 5.2 Density Functional Theory and DMol3 ·························76 5.3 Results and discussion ·············································77 5.4 Conclusions ·························································81 References ·······························································83 Chapter Conclusion and Future Research ·····························89 6.1 Conclusion ··························································89 6.2 Future Research ····················································92 References ·······························································96 Abstract v ABSTRACT THE NOVELTY AND SURFACE-TO-VOLUME-RATIO DEPENDENT ELECTRON BAND STRUCTURE IN SEMICONDUCTOR NANOWIRE By Donglai Yao In the field of nanotechnology, we focus this thesis on the novelty and surface-to-volume ratio dependent electronic band structure in semiconductor nanowires by means of first principle calculation. Silicon nanowires (SiNWs) in [110] growth direction is main research object, whose cross-sectional geometrics and surface-to-volume ratio dependence on the electronic band gap, effective mass are covered in this thesis. We have found that there is a universal band gap expression which is only related to surface-to-volume ratio for nanowires with dimension up to nm. Most interestingly, this expression is a linear dependence of band gap on surface-to-volume ratio, which is independent of the specific cross sectional shape. We also explore the electron effective mass of [110] silicon nanowires with different cross sectional shapes. We found that the electron effective mass decreases with the SiNW transverse dimension (cross sectional area) increases. With the same cross sectional area, Abstract vi the triangular cross section SiNW has larger electron effective mass than that of rectangular cross section SiNW. We also trying to find the direct to indirect band gap transition in [110] SiNWs. We successfully estimated the critical dimension where this direct-indirect band gap transition takes place by using the gauge of SVR and the DFT calculation results. It is found that tri-SiNW has the largest transition dimension up to 14 nm in diameter. [Chapter 5. Direct to Indirect Band Gap…]                                                81  undergoes a change from direct to indirect band gap as wire diameter increases, and the transition takes place at a considerably small diameter of nm.[14] In sharp contrast, our finding suggests that large-diameter, up to 14 nm [110] SiNW is still a direct gap material. This finding is remarkable and shows the important application of [110] SiNWs in optoelectronic and solar cell areas. 5.4 Conclusion In this Chapter, the transition of band structure from direct to indirect behavior of small diameter hydrogen-terminated [110] SiNWs has been studied. We successfully estimated the critical dimension where this direct-indirect band gap transition takes place by using the gauge of SVR and the DFT calculation results. It is found that tri-SiNW has the largest transition dimension up to 14 nm in diameter. Therefore tri-SiNW has a wider range of high efficient optoelectronic applications. In this chapter, because we focus on the transition point of the second minimum in the band curve, our calculation are based on the DFT, which is known as more accurate method, while in the chapter 2, we use the DFTB. [Chapter 5. Direct to Indirect Band Gap…]                                                82  Acknowledgment. The work is supported in part by grant R-144-000-222-646 from National University of Singapore, and by a SERC Grant, A*STAR, Singapore. [Chapter 5. Direct to Indirect Band Gap…]                                                83  References [1] A. M. Morales, C. M. Lieber, Science 279, 208 (1998). [2] Y. Cui, X. F. Duan, J. T. Hu, C. M. Lieber, J. Phys. Chem. B 104, 5213 (2000). [3] Y. Cui, Q. Q. Wei, H. K. Park, C. M. Lieber, Science 293, 1289 (2001). [4] G.-J. Zhang, G. Zhang, J. H. Chua, R.-E. Chee, E. H. Wong, A. Agarwal, K. D. Buddharaju, N. Singh, Z. Gao, N. Balasubramanian, Nano Lett. 8, 1066 (2008). [5] A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, P. Yang, Nature 451, 163 (2008). [6] A. I. Boukai, Y. Bunimovich, J. T. Kheli, J.–K. Yu, W. A. Goddard III, J. R. Heath, Nature 451, 168 (2008). [7] N. Yang, G. Zhang, B. W. Li, Nano Lett. 8, 276 (2008). [8] G. Zhang, Q. X. Zhang, C. T. Bui, G. Q. Lo, B. W. Li, Appl. Phys. Lett. 94, 213108 (2009). [9] B. M. Kayes, H. A. Atwater, N. S. Lewis, J. Appl. Phys. 97, 114302, (2005). [10] L. Hu, G. Chen, Nano Lett. 7, 3249 (2007). [Chapter 5. Direct to Indirect Band Gap…]                                                84  [11] J. S. Li, H. Y. Yu, S. M. Wong, G. Zhang, X. W. Sun, G. Q. Lo, D. L. Kwong, Appl. Phys. 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[Chapter 5. Direct to Indirect Band Gap…]                                                85  Tri-SiNW Rect-SiNW Hex-SiNW Figure 5.1: Cross-section view of the types of SiNWs: Tri-SiNW (Triangular cross-section SiNW), Rect-SiNW (Rectangular cross-section SiNW) and Hex-SiNW (Hexagonal cross-section SiNW). a, b and c are the lateral facets, which are (100), (110) and (111), respectively. In Tri-SiNW, the angle α is 70.6º and β is 54.7º. The blue dotted lines represent the virtual cages used to construct the SiNWs. Si and H atoms are represented in yellow and white, respectively. [Chapter 5. Direct to Indirect Band Gap…]                                                86  Figure 5.2: The dependence of conduction band edges on the SVR for hex-NWs. Inset is the band structure of hex-NW with cross sections area of 7.29 nm2. [Chapter 5. Direct to Indirect Band Gap…]                                                87  Figure 5.3: (Color online) ΔE versus surface-to-volume ratio (SVR) for tri-, rect-, and hex-SiNWs. [Chapter 5. Direct to Indirect Band Gap…]                                                88  Table 5.1: ΔE versus SVR relationship, critical SVR and diameters for tri-, rect-, and hex-SiNWs studied in this work. Tri-SiNW ΔE (eV) -0.059+0.108×SVR -1 SVR|ΔE=0(nm ) 0.546±0.037 D|ΔE=0(nm) 14.69±1.00 Rect-SiNW -0.108+0.136×SVR 0.794±0.064 7.12±0.58 Hex-SiNW -0.134+0.178×SVR 0.753±0.019 6.84±0.17 [Chapter 6. Conclusion and Future Research]                                              89  Chapter Conclusion and Future Research 6.1 Conclusion In this thesis, we have been concerned with calculation from first principles, in the framework of density functional theory and density functional based tight binding method, the electronic structure of [110] oriented SiNWs. Detailed studies of the impacts of size, cross sectional shape and surface-to-volume ratio on the electronic band gap, effective mass are carried out. In Chapter 3, we firstly systematically studied the electronic structure of [110] oriented SiNWs. [110] SiNWs remain direct band gap with transverse dimension up to nm. The band gap of SiNW increases as the transverse dimension is decreased. With the same transverse dimension, the tri-SiNW has the largest band gap among those of rect- and hex-SiNWs. Most interestingly, a linear dependence of band gap on SVR is found for the first time, which is independent of the specific cross sectional shape. In other words, a universal band gap expression: [Chapter 6. Conclusion and Future Research]                                              90  EG = E0 + aS where E0 corresponds to the band gap of bulk silicon, a is an adjustable parameter and S is the value of SVR in unit of nm-1. This expression is demonstrated for [110] SiNWs with any cross sectional shapes. The using of SVR is of practical importance as it allows us to avoid the ambiguous definition of the nanowire’s diameter. Intrinsically, the linear SVR dependence is related to the inverse relation between the band gap and the transverse dimension in small SiNWs. Our results show that in addition to the important factor for the reactivity in chemical reactions, SVR is also a key role on electronic band structure of nano materials. It is the first formula which relates quantitatively the band gap to SVR of SiNWs. The band gap of SiNWs are usually difficult to measure, but their transverse cross sectional shape and dimension are easy to know, so it is of significance to predict the band gap values of SiNWs by using the above expression. It is noted that the linear SVR relation with band gap does not depend on the specific calculation methods. In Chapter 4, we investigated the lattice constant and the electron effective mass of [110] oriented hydrogen-passivated silicon nanowires (SiNWs) of different cross sectional shapes by using the first-principles tight binding method. We explore the properties of SiNWs with transverse cross sectional [Chapter 6. Conclusion and Future Research]                                              91  area up to 18 nm2, which is larger than the structures studied with common used DFT methods. It is found that the surface lattice constants of SiNWs increases with the transverse dimension of the wire increases. The electron effective mass decreases with the SiNW transverse dimension (cross sectional area) increases. With the same cross sectional area, the tri-SiNW has larger electron effective mass than that of rect-SiNW. And in tri-SiNW, the dependence of electron effective mass on cross section area is more obvious than that in rect-SiNW. The quantum confinement effects on surface lattice constant and effective mass, and the impacts of transverse cross sectional shape are well explained by the concept of surface-to-volume ratio. SVR has the impacts of reducing the surface lattice constant and increasing the electron effective mass. At the same transverse dimension, tri-SiNW has larger SVR than that of the rect-SiNW. As a result, it has larger electron effective mass than that of rect-SiNW. Our results demonstrate that rect-SiNW has obvious advantage over tri-SiNW in application for higher ON-current in SiNW. In Chapter 5, we are trying to find the direct to indirect band gap transition in [110] SiNWs. As we know, [110] SiNWs have a direct band gap in the ultrathin diameter regime, whereas the energy difference between the indirect and direct fundamental band gaps progressively decreases as the wire size [Chapter 6. Conclusion and Future Research]                                              92  increases, indicating that larger [110] SiNWs could have an indirect band gap. We successfully estimated the critical dimension where this direct-indirect band gap transition takes place by using the gauge of SVR and the DFT calculation results. It is found that tri-SiNW has the largest transition dimension up to 14 nm in diameter. Therefore tri-SiNW has a wider range of high efficient optoelectronic applications. The SVR again proved to be an important and useful gauge parameter in the research of material in nano-scale regime. In a summery, we obtained a universal expression of band gap for silicon nanowires [110] of different cross-section geometries, based on the quantity of SVR (surface-to-volume ratio); we studied the impacts of size and cross-sectional shape on surface lattice constant and electron effective mass of silicon nanowire; we also try to find out the direct to indirect band gap transition dimension in [110] silicon nanowires. 6.2 Future research In our research group, the main branch is the thermal / heat management, and recent hot topic on the thermal electric materials, so what we did in this thesis [Chapter 6. Conclusion and Future Research]                                              93  is mainly focused on the electronic part, which is also an important factor to the ZT (figure of merit). The future work will be finding the high ZT materials in 2D or 1D system in the size of nano-scale regime. We are looking into the Si/Ge superlattice/multilayer 2D structures thermal cooler in experiments. Effective cooling is essential for many high power or low noise electronic and optoelectronic devices. Thermoelectric (TE) refrigeration is a solid-state active cooling method with high reliability. Unlike conventional air-cooling, it can spot cool discrete or localized devices and reduce the temperature of the device below ambient. For a material to be a good thermoelectric cooler, it must have a high value of the dimensionless figure of merit ZT [1] which is given by ZT=S2σT/κ, where S is the Seebeck coefficient, σ is the electrical conductivity, T is the temperature, and κ is the thermal conductivity. The use of quantum-well structures to increase ZT was proposed by Hicks and Dresselhaus [2]. Since then much work has been done in the study of superlattice thermoelectric properties, mostly for the in-plane direction [3-6]. The physical origin of the increase in ZT comes mainly from the enhanced density of electron states due to the reduced dimensionality. Recent study shows that superlattice thermal conductivity of cross-plane direction is even lower than that of in-plane direction [7], which can further increase ZT. In [Chapter 6. Conclusion and Future Research]                                              94  addition, Shakouri and Bowers proposed that heterostructure could be used for thermionic emission to enhance the cooling [8]. Large ZT improvement is possible for the cross-plane transport [9, 10]. SiGe is a good thermoelectric material especially for high temperature applications [11]. Superlattice structures can enhance the cooler performance by reducing the thermal conductivity between the hot and the cold junctions [7, 12], and by selective emission of hot carriers above the barrier layers in the thermionic emission process [8, 9]. In our preliminary work, single element p-type SiGe/Si and Si/Ge superlattice 2D nano structures (Inversed nanowire structure) with electrical transport in the cross-plane direction is demonstrated in figure 6.1 and figure 6.2 below. This paves the road to make n-type and p-type superlattice coolers electrically in series and thermally in parallel, similar to conventional TE coolers, and thus achieve large cooling capacities with relatively small currents. [Chapter 6. Conclusion and Future Research]                                              95  300nm Figure 6.2: Inversed Si/Ge nanowires array (connected). 300nm Figure 6.2: Inversed Si/Ge nanowires array (separated). [Chapter 6. Conclusion and Future Research]                                              96  Reference: [1]. H. J. Goldsmid, Thermoelectric Refrigeration (Plenum, New York, 1964). [2]. L. K. Hicks and M.S. Dresselhaus, Phys. Rev. B, 47, 12727 (1993). [3]. P. J. Lin_Chung and T. L. Reinecke, Phys. Rev. B, 51, 13244 (1995). [4]. R. Venkatasubramanian, E. Siivola, and T. S. Colpitts, Proceedings of the 17th International Conference on Thermoelectrics, 191 (1998). [5]. T. Koga, T. C. Harman, S. B. Cornin and M. S. Dresselhaus, Phys. Rev. B, 60, 14286 (1999). [6]. T. Koga, X. Sun, S. B. Cronin and M. S. Dresselhaus, Appl. Phys. Lett., 75, 2438 (1999). [7]. G. Chen, S. Q. Zhou, D.-Y. Yao, C. J. Kim, X. Y. Zheng, Z. L. Liu and K. L. Wang, Proceedings of the 17th International Conference on Thermoelectrics, 202 (1998). [8]. A. Shakouri and J. E. Bowers, Appl. Phys. Lett., 71, 1234 (1997). [9]. A. Shakouri, C. Labounty, P. Abraham, J. Piprek, and J. E. Bowers, Material Research Society Symposium Proceedings, 545, 449 (1999). [10]. L. W. Whitlow and T. Hirano, J. Appl. Phys., 78, 5460 (1995). [11]. C. B. Vining, J. Appl. Phys., 69, 331 (1991). [12]. S.-M Lee, D. G. Cahill and R. Venkatasubramanian, Appl. Phys. Lett., 70, 2957 (1997). [...]... (Color online) Schematic diagrams of the SiNWs used in our calculations From left to right, they are the tri-, rect- and hex-SiNWs In tri-SiNW, the angle α is 70.6º and β is 54.7º where this structure is in , accordance with the nanowires studied in the experimental work in Ref 23 The blue dotted lines represent the virtual cages used to construct the SiNWs Si and H atoms are represented in yellow and white,... where (2.2) is the kinetic energy operator for electrons, is the potential due to the nuclei, and is the internal electron- electron interaction EII is the constant energy of the nucleus-nucleus interactions The Hamiltonian (2.2) is uniquely determined by the external potential, , (which also determines EII ) since and are the same for any N electron problem The properties of the interacting system [Chapter... SiNWs is the smallest among those of the [100], [112] and [111] wires of the same diameter [39] Moreover, so far there is no systematic report on the indirect-direct band gap transition, and its dependence on the geometry of SiNWs As the indirect band gap and consequential weak light absorption remain the bottleneck for their application in optoelectronics/solar PV, a detailed understanding of the indirect -to- direct... electrical and magnetic properties to their bulk 3-D crystalline counterparts Increased surface area, very high density of electronic stats and joint density of states near the energies of their van Hove singularities, enhanced exiton binding energy, diameter -dependent band gap, and increased surface scattering for electrons and phonons are just some of the ways in which nanowires different from their corresponding... sizes of O(1000) atoms can be studied within standard DFT 2.1.1 The many body problem The starting point in the description of a system containing electrons and [Chapter 2 Modeling and Methodology] 14 nuclei is the Hamiltonian (2.1) where lower case subscripts denote electrons, and upper case subscripts denotes nuclei ZI and MI are the charge and mass of the nuclei The inverse of the nuclei masses,... conducting properties Getting these results, we have the ideas that the contributions of the surface energy play an important role in the stable nanowire structure In Ref [9], Rurali et al gives us a detailed report on the size effects in surface- reconstructed and silicon nanowires They performed ab-init calculation on the electronic structure to study the surface reconstructions of and. .. view of the 3 types of SiNWs: Tri-SiNW (Triangular cross-section SiNW), Rect-SiNW (Rectangular cross-section SiNW) and Hex-SiNW (Hexagonal cross-section SiNW) a, b and c are the lateral facets, which are (100), (110) and (111), respectively In Tri-SiNW, the angle α is 70.6º and β is 54.7º The blue dotted lines represent the virtual cages used to construct the SiNWs Si and H atoms are represented in yellow... Energy band structure for tri-SiNWs with transverse dimension of (a) D=1.79 nm and (b) D=4.09 nm The valence band maximum has been shifted to zero The blue dotted lines are drawn to guide the eyes ······49 ······ Figure 3.3: Energy band structure for rect-SiNWs with transverse dimension of (a) D=1.66 nm and (b) D=3.85 nm The valence band maximum has been shifted to zero The blue dotted lines are drawn to. .. nanowires with different diameters The diameter of the small size nanowire is another important factor to influence the band- structure and electronic structure of the nanowire [Chapter 1. Introduction]                                                                              6  1.3 Introduction to our work Extensive investigations have been carried out on the synthesis, properties and applications... silicon nanowires In their papers, they studied quite a few different Silicon Nanowire structures with diameters ranging from 1 to 6nm using the GTBMD scheme [8] The different growth directions ([111], [110], and [100]) of the nanowire are also investigated They found that the tetrahedral type nanowires oriented in the direction are the most stable They also found that the cage-like nanowires . THE NOVELTY AND SURFACE-TO-VOLUME-RATIO DEPENDENT ELECTRON BAND STRUCTURE IN SEMICONDUCTOR NANOWIRE By Donglai Yao In the field of nanotechnology, we focus this thesis on the novelty and. THE NOVELTY AND SURFACE-TO-VOLUME-RATIO DEPENDENT ELECTRON BAND STRUCTURE IN SEMICONDUCTOR NANOWIRE YAO DONGLAI (Master of Science) A THESIS SUBMITTED FOR THE DEGREE OF. section area A (in nm 2 ) and the number of atoms N in the supercell in our calculations. The dimension D is defined as the largest distance between the terminating hydrogen atoms in the cross section