平均键能物理内涵与肖特基势垒和异质结带阶的研究
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摘要
金属—半导体接触在技术上十分重要，在半导体器件和集成电路中广泛地利用着各种不同性质的金属—半导体接触，因而器件与集成电路的质量和可靠性在很大程度上依赖于电路中金属—半导体接触的性质：半导体异质结的价带带阶ΔE_p和导带带阶ΔE_c决定着量子阱和超晶格的势阱深度及其基本特性，预言、调节或控制带阶的“能带剪裁”新技术是当今“能带工程”的重要组成部分。金属—半导体接触Schottky势垒和异质结带阶的实验和理论研究与“表面与界面”新兴学科的研究进展紧密相关，基于不同实验结果的不断发现，先后提出Schottky势垒和异质结带阶的不同理论模型。关于Schottky势垒或异质结带阶，已提出的理论模型和计算方法多种多样，尚未统一，各种不同理论模型的计算中采用的参考能级也不相同。在本课题研究中，我们研究了自由电子能带和金属能带中平均键能E_(IN)与费米能级E_F的内在联系，发现平均键能相当于费米能级；在半导体能带中，着重研究了平均键能E(IN)与半导体“固有费米能级”所具有的相同物理内涵，揭示了平均键能的物理实质，建立了一种同时适用于Schottky势垒和异质结带阶的理论计算方法，即“平均键能方法”；并结合有关形变势的研究，将“平均键能方法”推广应用于应变层异质结带阶的理论计算，同时根据平均键能方法的特点，提出应变层异质结带阶的简化计算方案。本论文由五章组成，第1章介绍涉及本课题的研究背景和理论基础，第2至5章研究内容与结果的提要如下。
第2章，研究自由电子能带模型中的平均键能与费米能级间的关系。对于Si、Ge、GaP、InP、AlAs、GaAs、InAs、A1Sb、GaSb和InSb等10种面心立方半导体晶体和六角结构的Ti、Zr、Hf与体心四方的Sn等4种金属晶体，将它们的价电子近似为自由电子。一方面根据自由电子填充的费米球半径计算它们的费米能级E_F；另一方面，计算这些晶体在平衡时以及流体静压力应变和单轴应变状态下的自由电子能带结构，由电子在能带中的“最高填充态”计算其费米能级E_F~(ID)。与此同时，根据自由电子能带本征值的计算结果，引用正文中式(1．2．11)计算它们的平均键能E_(IN)，着重研
论文摘 要
究了Ec、E。’“与E＿三者间的关系。数值计算中发现，不但晶体处于平衡时其平均
键能 E＿值非常接近于费米能级 Er（或 E。““）值，而且在晶体发生流体静压力应变
和单轴应变时，E。与EF随应变状态的变化规律也是相同的。研究结果表明，在自由
电子能带模型中平均键能民相当于费米能级E。，我们建立的E。计算公式可以用来
计算自由电子系统的费米能级E。。
第3章，研究金属能带中的平均键能与费米能级间的关系。对于六角结构（h。p）
Ti、Zr和Hf以及体心四方（bio）p七n等4种不同的金属晶体，采用第一原理赝势法
计算它们的能带结构，由价电于“最高填充态”确定其费米能级厂厂，并计算其平均
键能厂。值，探讨Et与尸F的关系。研究结果表明，在上述4种金属能带中，不仅E。
与厂；n在数值上非常接近，E。与厂P两者随能带计算中采用的平面波基函数的数目的
变化趋势也相当接近，而且 E。与肛 k晶体的流体静压力应变和单轴应变的变化规律
也是相同的。所以与自由电子能带中所发现的情况类似，金属能带中的E。也相当于费
米能级 E尸，平均键能 E。的计算公式也可以用来计算这些金属晶体的费米能级。
第4章，研究半导体能带中平均键能的物理内涵及其在肖特基势垒和异质结带阶
理论计算中的应用。在能带理论中，以布里渊区中整体的能带结构表征晶体的价电子
态。对于出现禁带的半导体能带，4个价带填满价电子、所有导带是空态。我们把这种
价电子状态的费米能级称为半导体“固有费米能级”，以区别于半导体物理中己经定
义的“本征半导体费米能级”。研究中，我们先说明通常在能带计算中采用的由“最
高填充态”确定费米能级的方法不适用于半导体，半导体物理中根据简化能带定义的
“本征半导体费米能级”也不同于半导体“固有费米能级”；然后着重研究了平均键
能E。的物理内涵。研究中发现：半导体的平均键能厂。随计算能带的哈密顿矩阵阶数
Np的变化规律与金属的情况相同；E。和E。随单轴应变及流体静压力应变的变化规律
也相当接近，两者的形变势a。与a／户ee）符号相同、数值接近。这些研究结果表
明，半导体E。具有费米能级的一些重要的物理内涵；从计算方法考虑，在计算E。的
公式中，各个物理量都与是否出现带隙没有直接联系。因此若它的计算值在自由电于
能带和金属能带中都相当于费米能级，在半导体中也应该相当于“固有费米能级”。
－Vlll
论文 摘 要
这是一个推论，我们将其应用于金属－半导体接触势垒高度和异质结带阶的实际计算
中，获得了比较合理的计算结果，证实了推论的合理性。
第5章，探讨平均键能方法在应变层异质结带阶理论计算中的应用。如果半导体
中平均键能相当
Metal-semiconductor contacts are very important in technology. They are widely used in semiconductor devices and integrate circuits. The quality and reliability of semiconductor devices and integrate circuits greatly rely on the properties of metal-semiconductor contacts. The valence band offset AE, and conduction band offset A of heterojunction interface determine the depth of potential well and other basic properties of quantum well and superlattice. It is an important ingredient of modern "energy band project" to predict, modify or control the band offset (namely energy band clipping). The experimental and theoretical study of Schottky barrier of metal-semiconductor contact and band offset of heterojunction is closely related to the progress of new subject on surface and interface. Based on different experimental results, different theoretical models on Schottky barrier and heterojunction band offset have been proposed using different reference levels. In this thesis, we studied the intrinsic relationship between average-bond-energy Em and Fermi-level Eh- in energy bands of free electrons and metals. We find that the average-bond-energy is equivalent to Fermi-level. For semiconductor, average-bond-energy and "innate Fermi-level" have same physical connotation. A theoretical calculation method was established that can be applied to both Schottky barrier and heterojunction band offset, i.e. average-bond-energy method. Combined with the deformation potential, the average-bond-energy method was extended to the study of strained layer heterojunction band offset. Based on the character of average-bond-energy
method, we brought forward a simplified model to calculate strained layer heterojunction band offset. This dissertation is composed of five chapters. Chapter 1 is concerned with the research background and theoretical basis. The main contents and results of Chapters 2 - 5 are summarized as follows.
In Chapter 2, we investigated the relationship of average-bond-energy and Fermi-level EF (free) in free electronic energy band by studying ten face-centered cubic semiconductor crystals, Si, Ge, GaP, InP, AlAs, GaAs, InAs, AlSb, GaSb and InSb, three hexagonal metal crystals,Ti, Zr, Hf, and body-centred tetragonal metal crystal, P - Sn. The valence electrons were approximated by free electrons. The Fermi-level EF(free) was calculated according to the radius of Fermi sphere filled by free electrons. The free electronic energy bands, under the equilibrium state and the strain states of hydrostatic pressure and the uniaxial strain, respectively, were considered. The Fermi-level EFID(free) was computed based on the highest filled state of electrons in the free electronic energy band. According to the calculated eigenvalues of free electronic energy band, the average-bond-energy Em was obtained using the equation (1.2.11) in this dissertation. The results indicate that not only the average-bond-energy Em and the Fermi-level EF(free) (or E,,'D(free)) are very close to each other under the equilibrium states, but also they obey the same variation rule under the strained states. Therefore, Em is equivalent to Eh-(free) in free electron energy band model. Equation (1.2.11) can be used to calculate the Fermi-level EF(free) of free electron system.
In Chapter 3, we studied the relationship of average-bond-energy Em and Fermi-level Ef in metal energy band. The energy band of the metal crystals with hexagonal close-packed structure, Ti, Zr, Hf, and body-centred tetragonal P - Sn were calculated by the first principle pseudopotential method. The Fermi-level E'?was determined by the highest filled state of valence electrons. It is found that the average-bond-energy Em of these four metals is also very close to the Fermi-level Ef . The variation trends of Em and Ef with the number Np of
plane wave base function adopted in the energy band calculation are similar. The regularity of EHI and E'f changing with the strained states of hydrostatic pressure and uniaxial strain are the same. Therefore, as in the case of free
第2章，研究自由电子能带模型中的平均键能与费米能级间的关系。对于Si、Ge、GaP、InP、AlAs、GaAs、InAs、A1Sb、GaSb和InSb等10种面心立方半导体晶体和六角结构的Ti、Zr、Hf与体心四方的Sn等4种金属晶体，将它们的价电子近似为自由电子。一方面根据自由电子填充的费米球半径计算它们的费米能级E_F；另一方面，计算这些晶体在平衡时以及流体静压力应变和单轴应变状态下的自由电子能带结构，由电子在能带中的“最高填充态”计算其费米能级E_F~(ID)。与此同时，根据自由电子能带本征值的计算结果，引用正文中式(1．2．11)计算它们的平均键能E_(IN)，着重研
论文摘 要
究了Ec、E。’“与E＿三者间的关系。数值计算中发现，不但晶体处于平衡时其平均
键能 E＿值非常接近于费米能级 Er（或 E。““）值，而且在晶体发生流体静压力应变
和单轴应变时，E。与EF随应变状态的变化规律也是相同的。研究结果表明，在自由
电子能带模型中平均键能民相当于费米能级E。，我们建立的E。计算公式可以用来
计算自由电子系统的费米能级E。。
第3章，研究金属能带中的平均键能与费米能级间的关系。对于六角结构（h。p）
Ti、Zr和Hf以及体心四方（bio）p七n等4种不同的金属晶体，采用第一原理赝势法
计算它们的能带结构，由价电于“最高填充态”确定其费米能级厂厂，并计算其平均
键能厂。值，探讨Et与尸F的关系。研究结果表明，在上述4种金属能带中，不仅E。
与厂；n在数值上非常接近，E。与厂P两者随能带计算中采用的平面波基函数的数目的
变化趋势也相当接近，而且 E。与肛 k晶体的流体静压力应变和单轴应变的变化规律
也是相同的。所以与自由电子能带中所发现的情况类似，金属能带中的E。也相当于费
米能级 E尸，平均键能 E。的计算公式也可以用来计算这些金属晶体的费米能级。
第4章，研究半导体能带中平均键能的物理内涵及其在肖特基势垒和异质结带阶
理论计算中的应用。在能带理论中，以布里渊区中整体的能带结构表征晶体的价电子
态。对于出现禁带的半导体能带，4个价带填满价电子、所有导带是空态。我们把这种
价电子状态的费米能级称为半导体“固有费米能级”，以区别于半导体物理中己经定
义的“本征半导体费米能级”。研究中，我们先说明通常在能带计算中采用的由“最
高填充态”确定费米能级的方法不适用于半导体，半导体物理中根据简化能带定义的
“本征半导体费米能级”也不同于半导体“固有费米能级”；然后着重研究了平均键
能E。的物理内涵。研究中发现：半导体的平均键能厂。随计算能带的哈密顿矩阵阶数
Np的变化规律与金属的情况相同；E。和E。随单轴应变及流体静压力应变的变化规律
也相当接近，两者的形变势a。与a／户ee）符号相同、数值接近。这些研究结果表
明，半导体E。具有费米能级的一些重要的物理内涵；从计算方法考虑，在计算E。的
公式中，各个物理量都与是否出现带隙没有直接联系。因此若它的计算值在自由电于
能带和金属能带中都相当于费米能级，在半导体中也应该相当于“固有费米能级”。
－Vlll
论文 摘 要
这是一个推论，我们将其应用于金属－半导体接触势垒高度和异质结带阶的实际计算
中，获得了比较合理的计算结果，证实了推论的合理性。
第5章，探讨平均键能方法在应变层异质结带阶理论计算中的应用。如果半导体
中平均键能相当
Metal-semiconductor contacts are very important in technology. They are widely used in semiconductor devices and integrate circuits. The quality and reliability of semiconductor devices and integrate circuits greatly rely on the properties of metal-semiconductor contacts. The valence band offset AE, and conduction band offset A of heterojunction interface determine the depth of potential well and other basic properties of quantum well and superlattice. It is an important ingredient of modern "energy band project" to predict, modify or control the band offset (namely energy band clipping). The experimental and theoretical study of Schottky barrier of metal-semiconductor contact and band offset of heterojunction is closely related to the progress of new subject on surface and interface. Based on different experimental results, different theoretical models on Schottky barrier and heterojunction band offset have been proposed using different reference levels. In this thesis, we studied the intrinsic relationship between average-bond-energy Em and Fermi-level Eh- in energy bands of free electrons and metals. We find that the average-bond-energy is equivalent to Fermi-level. For semiconductor, average-bond-energy and "innate Fermi-level" have same physical connotation. A theoretical calculation method was established that can be applied to both Schottky barrier and heterojunction band offset, i.e. average-bond-energy method. Combined with the deformation potential, the average-bond-energy method was extended to the study of strained layer heterojunction band offset. Based on the character of average-bond-energy
method, we brought forward a simplified model to calculate strained layer heterojunction band offset. This dissertation is composed of five chapters. Chapter 1 is concerned with the research background and theoretical basis. The main contents and results of Chapters 2 - 5 are summarized as follows.
In Chapter 2, we investigated the relationship of average-bond-energy and Fermi-level EF (free) in free electronic energy band by studying ten face-centered cubic semiconductor crystals, Si, Ge, GaP, InP, AlAs, GaAs, InAs, AlSb, GaSb and InSb, three hexagonal metal crystals,Ti, Zr, Hf, and body-centred tetragonal metal crystal, P - Sn. The valence electrons were approximated by free electrons. The Fermi-level EF(free) was calculated according to the radius of Fermi sphere filled by free electrons. The free electronic energy bands, under the equilibrium state and the strain states of hydrostatic pressure and the uniaxial strain, respectively, were considered. The Fermi-level EFID(free) was computed based on the highest filled state of electrons in the free electronic energy band. According to the calculated eigenvalues of free electronic energy band, the average-bond-energy Em was obtained using the equation (1.2.11) in this dissertation. The results indicate that not only the average-bond-energy Em and the Fermi-level EF(free) (or E,,'D(free)) are very close to each other under the equilibrium states, but also they obey the same variation rule under the strained states. Therefore, Em is equivalent to Eh-(free) in free electron energy band model. Equation (1.2.11) can be used to calculate the Fermi-level EF(free) of free electron system.
In Chapter 3, we studied the relationship of average-bond-energy Em and Fermi-level Ef in metal energy band. The energy band of the metal crystals with hexagonal close-packed structure, Ti, Zr, Hf, and body-centred tetragonal P - Sn were calculated by the first principle pseudopotential method. The Fermi-level E'?was determined by the highest filled state of valence electrons. It is found that the average-bond-energy Em of these four metals is also very close to the Fermi-level Ef . The variation trends of Em and Ef with the number Np of
plane wave base function adopted in the energy band calculation are similar. The regularity of EHI and E'f changing with the strained states of hydrostatic pressure and uniaxial strain are the same. Therefore, as in the case of free
引文
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[2]夏建白 朱邦芬《半导体超晶格物理》上海科学技术出版社，1995年出版
[3]J. M. Andrew, J. Vac. Sci Technol. 11 972 (1974)
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[6]N.F. Mott, Proc. Roy. Soc.(London) A177, 27 (1939)
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[2]夏建白 朱邦芬《半导体超晶格物理》上海科学技术出版社，1995年出版
[3]J. M. Andrew, J. Vac. Sci Technol. 11 972 (1974)
[4]F. Braun, Pogg. Ann. 153 556 (1874)
[5]W. Schottky, Z. Phys. 113, 367 (1939), 118, 539 (1942)
[6]N.F. Mott, Proc. Roy. Soc.(London) A177, 27 (1939)
[7]J. Bardeen, Phys. Rev. 71 717 (1947)
[8]A.M. Cowley,S.M. Sze, J. Appl. Phys. 36,3212 (1965)
[9]F.G. Allen and G.W. Gobeli, Phys. Rev. 127, 150 (1962)
[10]F.G. Allen and G.W. Gobeli, J. Appl. Phys. 35, 594 (1964)
[11]L.F. Wggner and W.E. Spicer, Pyhs. Rev. Lett. 28, 1381 (1972)
[12]D.E. Eastman and W. D. Grobman, Pyhs. Rev. Lett. 28, 1378 (1972)
[13]J. Van Laar and J.J. Scheer, Surface Sci. 8, 342 (1967)
[14]P.E. Gregoor and W.E. Spicer, Phys. Rev. B 13, 725 (1976)
[15]V. Heine, Phys. Rev., 138A, 1689 (1965)
[16]C. Tejedor and F. Flores, J. Phys., C 11, L79 (1978)
[17]J. Tersoff, Phys. Rev. Lett. 52, 465 (1984); Phys. Rev. Lett. 56, 2755 (1986)
[18]J. Tersoff, Phys. Rev. B 32, 6968 (1985)
[19]N.E. Christensen, Phys. Rev. B 37, 4528 (1988)
[20]C.G. Van de Walle, R.M. Martin, Phys. Rev. B 35, 8154 (1987)
[21]S.H. Wei and A. Zunger, Appl. Phys. Lett. 72,2011 (1998)
[22] W.A. Harrson, J. Tersoff, J.Vac.Sci.Technol. B 4,1068 (1986)
[23] M. Cardona, N.E. Christensen , Phys. Rev. B 35, 6182 (1988)
[24] J. Tersoff, Phys. Rev. B 30, 4874 (1984) , Vac.Sci.Technol B 3, 1157 (1985)
[25] 王仁智 黄美纯，中国科学，A缉10期1073(1992) ，物理学报，40，1683(1991)
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