《电子类论文》word版.doc

Opt Quant ElectronDOI 10.1007/s11082-011-9506-3A exible, plane-wave based multiband k p model一个灵活的，基于平面波的K P模型Oliver Marquardt Stefan SchulzChristoph FreysoldtSixten BoeckTilmann Hickel Eoin P. OReillyJrg NeugebauerAbstractIn this work, we present a highly generalized implementation of multiband k pmodels. We have achieved a high efciency of our approach by incorporating it in a plane-wave framework within the Density Functional Theory package S/PHI/nX. To demonstratethe exibility and applicability of our code, we have chosen two example studies that aredirectly accessible with the standard eight-band k p model. By employing a 14-band k pmodel for the description of pyramidal InAs/GaAs quantum dots (QDs), we show that thismodel is able to accomodate for the correct symmetry of the underlying zincblende lattice,which is not reected in the standard eight-band model. Our second example provides adescription of site-controlled (111)-oriented InGaAs/GaAs QDs. The extremely small aspectratio of these QDs makes a description using conventional k p Hamiltonians computation-ally highly expensive. We have therefore rotated the standard eight-band Hamiltonian, to suitthe description of these systems. The studies of electronic properties of the above mentionedmodel systems demonstrate the efciency and exibility of our approach.摘要：在这项工作中，我们提出了一个多波段的高度概括的可执行得kP模型。我们已经取得了用我们的方法把它整合到一个高效率的平面密度泛函理论包的S /PHI/NX波的框架内。为了演示我们的代码的灵活性和适用性，我们选择了两个可直接理解的例子，直接与标准的8频段KP模型的研究。通过采用一个14波段Kp锥体InAs / GaAs量子点的描述模型，我们发现，这个模型能适应底层的闪锌矿晶格的正确对称性，但是不能体现在标准的八带模型。我们的第二个例子提供了一个说明以现场控制为导向的砷化铟镓/砷化镓量子点。以这些量子点非常小的方面的比例作出说明，使用传统的非常昂贵的kP哈密顿计算。我们因此替换标准的8频段哈密顿量，以适应这些系统的描述。上面提到的电子性质的研究模型系统，证明我们的方法是有效的和灵活的。KeywordsNanostructures Electronic properties Multiband k p formalism关键词：纳米 电子性质 多频带K P形式体系1 IntroductionSemiconductor nanostructures such as quantum wells, -wires and -dots (QDs) of variousmaterial compositions have been successfully grown in the past and their optoelectronic prop-erties are still a matter of high scientic interest. The remarkable number of such systems,ranging from InAs and InGaAs QDs and quantum wires (Stier 2000), wurtzite III-NitrideQDs grown on polar, semipolar and nonpolar surfaces (Daudin 2008), coupled ternary-alloyednanostructures (Marquardt et al. 2011) or Si/Ge-nanostructures that exhibit an indirect bandgap, indicates a strong demand for a simulation tool that allows for theoretical descriptions ofall these systems with high reliability, efciency and exibility. The eight-band kp formalismis nowadays one of the standard tools for the description of electronic and optical propertiesof such systems (Fonoberov and Balandin 2003; Stier et al. 1999; Schliwa et al. 2007; Pryor1998; Zhao and Mei 2011). This approach is, due to its relatively low computational costs,excellently suited to perform studies of not only single nanostructures, but also of series ofpossible modications, such as different sizes, shapes or material compositions, as these fea-tures are commonly not exactly determined from experimental evidence. On the other hand,the k p model provides only a continuum description of the system under investigation andtherefore lacks a proper description of the underlying atomistic structure. This simplicationcan, in certain systems, induce articially degenerate electronic states, that would be foundto be nondegenerate in an atomistic picture (Bester and Zunger 2005), in particular in smallsystems, where the atomistic nature of the interfaces between nanostructure and surroundingmatrix material becomes important. Moreover, existing k p codes are typically optimisedfor a xed Hamiltonian that is implemented in the code, and thus lack the exibility to choosethe level of sophistication with respect to the scope of the study that is to be performed andto describe different crystal structures or even orientations in a given crystal structure. Anexample where this exibility is important are site-controlled InGaAs/GaAs QDs, grownon the (111)-surface. These QDs are highly promising for the generation of entangled pho-tons (Stock et al. 2010), as the structure geometry, the crystal lattice and strain as well aspolarisation potentials exhibit a C3v -symmetry in the (111)-plane (Schulz et al. 2011). Theabove discussed drawbacks of the standard eight-band k p model do not articially increasethe systems symmetry. However, experimental evidence indicates an extremely small aspectratio of these QDs with base lengths of up to 80 nm and typical heights of only 1.5 nm. Aconventional k p Hamiltonian for the description of QDs grown on (001)-surfaces, thusrequires a huge unit cell with an extremely narrow discretisation in all three dimensions,massively raising the computational costs. It is therefore required to modify the existingeight-band Hamiltonian.In this paper, we present a highly generalised and efcient approach to perform multibandk p calculations for a range of semiconductor nanostructures that reects the wide varietiesof experimentally observed systems. We will demonstrate the high exibility of our approachby employing different k p models to address the above described drawbacks of the standardeight-band model. The paper is organised as follows: We will provide a brief description of aplane wave (PW)-based implementation of multiband k p formalisms in Sect. 2. Results ofour calculations on multiband k p models providing the correct symmetry of crystal latticeand nanostructure geometry, as well as calculations on (111)-oriented InGaAs/GaAs QDsusing a rotated eight-band k p model, will be presented in Sect. 3. We summarise our workin Sect. 4.1 介绍 半导体纳米结构如量子井，线和点（量子点）的各种物质成分已在过去成功地成长了，光电特性仍然是一个具有很高的科学兴趣的问题。许多这种卓越的系统，从砷化铟和InGaAs量子点和量子线，纤锌矿III族氮化合物量子点生长的极性，半极性和非极性表面，加上三元合金化纳米结构或硅/锗纳米结构，表现出一种间接的差距，表明一个模拟工具强烈要求允许理论的描述所有这些具有高可靠性，有效率的和灵活性的系统。这个8频段的Kp形式体系是现在的一个标准的工具之一的电子和光学特性描述这样的系统（Fonoberov和Balandin2003年，公牛等人，1999年。Schliwa等2007年，普赖尔1998年，赵梅2011）。这种方法是，由于其相对较低的计算成本，很好地适合于执行不仅单个纳米结构的研究，也适合于一系列可能的修改，如不同尺寸，形状或材料的成分，因为这些特点通常是不完全确定的实验证据。另一方面，这个 KP模型提供了只有在调查中一个连续系统的描述和因此缺乏一个基本的原子结构的正确描述。这种简化可以在某些系统中，人为的诱发退化的电子态，会发现非退化的原子论图片（贝斯特和Zunger2005年），特别是在小系统，纳米结构和周围之间的接口，原子的性质和基质材料变得很重要。此外，现有的KP码通常是了的优化实现代码，从而缺乏灵活的选择一个固定的哈密顿与复杂性水平方面的研究范围，要执行一个给定的晶体结构来描述不同的晶体结构，甚至方向。这个例子的灵活性是很重要的，其中站点控制砷化铟镓/砷化镓量子点，在表面上增长。这些量子点是非常有前景的对于杂乱光子的产生，像几何结构，晶格应变以及极化潜力表现出一个C3v对称平面（舒尔茨等2011）。上面讨论的标准的8频段KP模型的弊端不是人为地增加了该系统的对称性。然而，实验证据表明，一个非常小的方面这些基数长度可达80纳米和典型的高度只有1.5纳米的量子点的比例。一传统的Kp哈密顿表面上增长量子点的描述，从而需要在所有三个方面极为狭窄的半离散化的一个巨大的晶胞，大规模提高计算成本。因此，需要修改现有的8频段的哈密顿。在本文中，我们提出了一个高度概括的和有效的方法进行多波段kP计算的半导体纳米结构的范围，反映了广泛的不同种类的实验观察到的系统。我们将证明我们的方法的高度灵活性是采用不同的KP模型，以解决上述标准的8频带模型弊端。该文的安排如下：1.我们将提供一个简要的说明以平面波为基础实施多波段kP形式体系。2.我们关于多波段kP模型计算的结果将提供正确的晶格对称和纳米结构，以及以几何计算为导向的砷化铟镓/砷化镓量子点使用一个旋转的8频段KP模型，将呈现在第三部分。我们把总结我们的工作放在第4节。2 A generalised implementation of multiband k p models in a plane-wave frameworkWe have implemented our generalised scheme of continuum elasticity theory and multibandk p formalism in the PW framework of the Density Functional Theory (DFT) softwarelibrary S/PHI/nX (www.sphinxlib.de, Boeck et al. 2011). This enables us to benet from thehighly optimised minimisation routines commonly available in such a DFT software package.123A exible, plane-wave based multiband k p modelMoreover, a calculation of gradients can be performed in reciprocal space in a PW picturewith significantly less numerical effort than in real space using nite differences. We havepreviously demonstrated the applicability and efciency of our implementation of continuumelasticity theory and eight-band k p theory (Marquardt et al. 2010). Based on this, we havegeneralised our model to arbitrary N -band k p Hamiltonians, that are constructed from aset of physically meaningful basic elements:1. Spatial derivative operators / k, 2/ k kacting on the wave function.2. External r-dependent elds and potentials.3. Material-dependent parameters, such as effective masses and band offsets.4. Constant complex numbers.These physically meaningful elements can be combined by multiplication, division, additionand subtraction. The key feature of this generalised implementation is that the Hamiltonianis dened in an input le in a human-readable meta-language (more details and exampleswill be published elsewhere Marquardt et al.). To set up a new k p model, therefore, doesnot require additional coding. Similarly, we allow for a variety of possible nanostructures.A general nanostructure may consist ofNcdifferent compounds (compatible with theemployed k p model) and their alloys. The nanostructure is set up in the form of a com-position map that denes the local composition ci(r), where i=1 . . . Ncandci=1.Material parameters are interpolated linearly or quadratically from the pure bulk compounds.Quadratic interpolation of course requires bowing parameters, and is thus effectively limitedto ternary alloyed systems. Our implementation does not pose any restrictions on the numberof bands in the model, the number of compounds, or the number and names of materialparameters used in the model Hamiltonian. The exibility of this generalised model, on topof a well-tested underlying PW framework provides an excellent basis for comparing andapplying multiband k p models to a huge variety of different nanostructures.2 一个广义的多波段KP模型在平面波框架里的实施 我们已经实施了我们的连续弹性理论和多波段Kp形式体系的广义计划在密度泛函理论的平面波框架软件库中的S /菲律宾/NX。这使我们能够从高度优化的尽量减少的程序中受益通常在这样的DFT软件包中。 此外，一个的梯度计算可以在倒易的空间在平面波图片比在现实空间大大减少数值用有限差分的努力中执行。我们先前已经表明了我们连续实施的弹性理论和8频段kP理论的适用性和效率性。在此基础上，我们已经把任意n-波段Kp哈密顿模型一般化，从设置物理意义的基本要素中构造：1在波函数中的空间导数运算符/K，2 /KK。2外部R -依赖的领域和潜力。3材料相关的参数，如有效的群众和能带偏移。4. 恒定复杂的数。这些物理意义的元素可结合乘法，除法，加法和减法。这个广义的执行的主要特点是，哈密顿量是指在一个人类可读的元语言的输入文件中（更多细节和例子将刊登在别处夸特等）。因此要成立一个新的KP模型，不需要额外的编码。相似的，我们允许各种各样可能的纳米结构。一个一般的纳米结构可能由NC不同的化合物（和受雇的KP模型兼容）和他们的合金构成。纳米结构是在构成地图的形式中成立的，定义的局部的组成为CI（r），其中i = 1Ncand,CI =1。材料参数从纯粹的大部分的化合物以线性或二次方得内插值。二次方插值当然要求屈服于参数，从而有效地限制三元合金系统。我们的实施不构成任何限制在这个模型的数字频率中，以及化合物的数量，或数量和材料参数的名称使用在哈密顿模型中。这个广义模型的灵活性，是在一个行之有效的基本平面波框架之上的提供一个良好的基础作比较和在一个巨大的各种不同的纳米结构应用到多波段KP模型中。3 Model applications3.1 Bulk inversion anisotropy in multiband k p modelsWe have previously compared atomistic empirical tight-binding method (ETBM) andcontinuum k p approaches for the description of electronic properties in cubic GaN/AlNQDs (Marquardt et al. 2008). While we have observed an excellent quantitative agreement ofthe electron and hole eigenenergies, we could furthermore observe a small splitting betweenthe rst two excited, p-like electron states in the ETBM, that none of the employed k p mod-els was able to reproduce, similar to previous simulations on InAs QDs (Bester and Zunger2005). This is a direct consequence of the bulk inversion asymmetry (BIA) of the underlyingzincblende (ZB) lattice, which is not contained in the standard eight-band k p model. Forstructures such as (truncated) pyramidal or lens-shaped QDs, this increases the symmetryarticially to C4v. However, atomistic models, e.g. the ETBM or the empirical pseudopoten-tial method (EPM), are able to reproduce the correct C2v-symmetry of the underlying ZBlattice (Bester and Zunger 2005; Marquardt et al. 2008). This shortcoming of the standardeight-band model can be overcome by taking a higher number of explicitly treated bands intoaccount. As explained inJancu et al. (2005) and Hermann and Weisbch (1977), a 14-bandk p model is able to reproduce the correct BIA in the ZB lattice, even in a continuum picture.We were able to employ our generalised and exible multiband k p code to compare theeffective mass approximation (EMA), and six- and eight-band k p models with a more123O. Marquardt et al.accurate 14-band model with respect to their capability of describing the BIA in ZB nano-structures. Subsequently, we have studied the energy splitting of the p-like electron states ina pyramidal InAs/GaAs QD with these models. We have employed the k p HamiltoniansfromBahder (1990) (eight-band model) and Pfeffer and Zawadzki (1996) (14-band model)with the parameters fromSchliwa et al.(2007) and Jancu et al.(2005). Effective mass(two-band) approximation and six-band simplications were achieved by setting the Kanematrix parameter EPin the eight-band model to zero. The band structure of InAs reproducedfrom the different models is shown in Fig. 1 (left). It can be seen that the EMA (two-bandmodel) provides a good description of the lowest conduction band (CB) only in the closestvicinity of the-point. Similarly, the six-band description (EMA and six-band descriptionsare shown as black, short-dashed lines in Fig. 1.) of the three highest valence bands (VB)is significantly modied already by taking the CB-VB-coupling via the Kane parameter EPinto account in an eight-band model (blue, dashed line). Taking the next six higher CBs intoaccount (14-band model) not only provides a more accurate description of the band structuretowards the outer regions of the Brillouin zone (solid and thin dashed red lines), but alsoleads to a splitting of the lowest conduction band and the valence bands in the vicinity of the-point which increases further towards the K-point. Such a splitting does not occur in any ofthe other models, which indicates a correct description of the BIA in the 14-band case (thindashed red line). We have calculated the rst four electron states using eight and 14-bandk p models for a pyramidal QD for base lengths of 615 nm and heights of 37.5 nm, whichis in the range of previous studies (Stier 2000; Bester and Zunger 2005) (See Fig. 1). As wewere solely interested in the correct symmetry description of the bulk electronic structureHamiltonian, strain and piezoelectric potentials have been neglected, keeping in mind thatthe latter contribution would independently reduce the symmetry of the system to a C2v -symmetry also in an eight-band or EMA picture. The absolute eigenenergies of the rst fourexcited electron states agree in a range of less than 10 meV between the eight- and 14-bandmodel for the systems under consideration. The 14-band model, however, is in addition ableto produce a splitting between the p-like rst two excited electron states that is not observedusing the eight-band model. The splitting between these states E(2e)E(1e)is shown as afunction of the QD base length in Fig. 1 (right). It agrees qualitatively with previous stud-ies using atomistic models (Bester and Zunger 2005) and is found to decrease with largerQD dimensions, where the energies generally converge towards the bulk CB of InAs. It canthus be concluded that a 14-band k p model is able to reproduce the C2v -symmetry of ZBnanostructures, even without an atomistic description of the system.Fig. 1Band structure of InAs reproduced from k p models of different level of approximation (left). Right:splitting between the rst two excited, p-like electron states, E(2e)E (1e), calculated using a 14-band k pmodel as a function of the QD base length in a square-based, pyramidal InAs QD in GaAs (colour online)123A exible, plane-wave based multiband k p modelFig. 2The four hole states closest to the band gap and binding energies in a (111)-oriented In0.35Ga0.65AsQD with base length of 80 nm and a height of 1.5 nm in top view on the (111)-surface. Dark purple (lightblue) isosurfaces represent 50% (20%) of the maximum charge density (colour online)3.2 Site-controlled InGaAs/GaAs QDs grown on (111)-surfacesAs discussed in the introduction, a description of the electronic properties of realistic, site-controlled InxGa1x As QDs in Pelucchi et al. (2007) and Zhu et al. (2007) is computationallyhighly expensive using a standard eight-band k p model (Healy et al. 2009). This problemcan be overcome by employing a Hamiltonian that is suited for the description of QDs grownalong the 111-axis, as this procedure allows to discretise the simulation cell along the growthaxis more accurately than along the in-plane directions. As the symmetry of the combinedsystem of crystal structure and triangular QD shape is C3v-symmetric (Baer et al. 2007),an eight-band model will provide a sufciently good description here and the C3v-symme-try allows for a generation of entangled photons. We have therefore analytically rotated theeight-band k p Hamiltonian to suit a description of such systems. A detailed descriptionof such a formalism will be published elsewhere (Marquardt et al.), the correspondinglyrotated continuum elasticity model can be found in Schulz et al. 2011. Employing the rotatedHamiltonian in our multiband k p model, we have calculated the electronic properties ofrealistic InxGa1xAs QDs grown along the 111-direction. Our model QD has a base lengthof 80 nm and a height of 1.5 nm, with an In content of x= 0.35, following experimentalevidence (Mereni et al. 2009). Our simulations yield only a single localised electron statewith the eigenenergy E(0e) = 813.2 meV (GaAs conduction band: Ecb (GaAs) = 892 meV).The stronger localised hole ground and rst three excited states are shown in Fig. 2, togetherwith their corresponding eigenenergies (GaAs valence band: Evb(GaAs) = 627 meV). Wend the rst two excited hole states to be degenerate, due to the C3v-symmetry of the systemunder consideration. In the above analysis, strain and piezoelectric potentials were neglected.A more detailed discussion of symmetry properties of (111)-oriented Inx Ga1xAs QDs willbe published elsewhere (Marquardt et al.).3 模型的应用 3.1 大多数倒置的各项异性多波段KP模型 我们以前已经比较了原子论的经验以紧密结合的方法（ETBM）和