NANOMETROLOGY: OPTICAL PROPERTIES OF SI-NANOCLUSTERS

Проведені дослідження торкаються квантоворозмірних систем, таких, як напівпровідникові нанокластери, зокрема, кремнію (Si-НК). Останні представляють собою нанофрагменти, що складаються з декількох, або десятків атомів. Симетрія таких систем може співпадати з геометрією об’єму. Але, як правило, завдяки великому відсотку поверхневих атомів і, відповідно, станів, НК демонструють залежні від геометричних розмірів специфічні оптичні, або електричні властивості. Істотні зміни цих властивостей корелюють зі зменшенням розміру системи. Варіація ширини забороненої зони, що викликана зміною розмірів напівпровідникового НК для даної температури, викликає відповідний перехід у металевий стан. Такий перехід відбувається при відносно великих розмірах напівпровідникового НК у порівнянні з металами, діелектриками або молекулярними кристалами. Показано, що кінетична стабільність тетрасилатетрагедрану (Si4H4), гексасилаприсману (Si6H6) і октасилакубану (Si8H8) залежить від просторової об’ємності замісників, які пасивують розірвані хімічні зв’язки Si-матриці. Силілзаміщений НК типу SinYm (де Y – замісник пасиватор t Bu) є стабільним в інертній атмосфері, але окислюється у повітрі шляхом виникнення безбарвних твердих речовин. Крім цього, триметил-пропіл-заміщений НК типу SinYm (Y C Me2CH Me2) є надзвичайно стабільним, навіть у повітрі і існує протягом двох тижнів Призматичні НК, що утворені атомами Si, або Ge виявляють колір від жовтого до помаранчевого. Такі НК структури здійснюють оптичне поглинання у видимому діапазоні. НК-Si6H6 має смугу поглинання з максимумом від 241 нм до 500 нм. Смуга поглинання Ge6Y6 (Y 2,6 i Pr2C6H3) має максимум при 261 нм, що відповідає зсуву у червону область. Виходячи з цього, слід підкреслити, що експериментальні дослідження виявленої люмінесценції por-Si, дозволяють пов’язати можливості використання Si-НК у оптоелектронній техніці. Слід врахувати те, що формування нанокристалітів у вигляді Si-НК в об’ємному матеріалі напівпровідника, сприяє частковому порушенню правил оптичного відбору і, таким чином, обмежень щодо прояву цим матеріалом ефекту люмінесценції. Найбільш вражаючою властивістю напівпровідникових НК є суттєва зміна оптичних властивостей пористого кремнію (por-Si), як функції розміру Si-НК. При зменшенні розмірів НК, електронні переходи зміщуються до більш високих енергій. Генерація сконцентрована лише в декількох переходах. Описані фізичні явища квантової локалізації виникають внаслідок змін густини електронних станів і можуть бути зрозумілими з позиції зв’язку між станами і моментами вільних і обмежених частинок. Для вільної частинки, або частинки в періодичному потенціалі твердого тіла, імпульс може бути точно визначеним, тоді як локалізація у просторі є невизначеною. І навпаки: для локалізованої у просторі частинки невизначеність імпульсу збільшується. Наведені теоретичні розрахунки ширини забороненої зони НК-Si і порівняння їх з даними експерименту дозволяють зробити висновки про те, що існує можливість різних радіаційних каналів для рекомбінації носіїв у пористий кремній. УДК 530.145+678.9

Наведені теоретичні розрахунки ширини забороненої зони НК-Si і порівняння їх з даними експерименту дозволяють зробити висновки про те, що існує можливість різних радіаційних каналів для рекомбінації носіїв у пористий кремній. R esearch into semiconductor clusters is focused on the properties of quantum dots (QD) -fragments of semiconductor (for example, Si) consisting of some to hundreds of atoms -with the bulk bonding geometry and with surface states eliminated by enclosure. QD exhibit strongly size-dependent optical and electrical properties [1][2][3]. Two peculiar character istics of semiconductors influence the ways, in which we think of an ideal semiconductor cluster, which is often called a QD. First, it is important to realize, that in any material, substantial variation of fundamen tal electrical and optical properties with reduced size will be observed, when the elec tronic energy level spacing exceeds the temperature. In semiconductors, this transi tion occurs for a given temperature at a relatively large size compared to metals, insulators, or molecular crystals.
The luminescence observed for por-Si raises an interesting problem related to the possibility of using Si in optoelectronics [2]. One likely explanation is quantum confinement, induced by the formation of nanocrystallites, whose effect is to break partially the optical selection rules and allow the material to luminesce.
The most striking property of semiconduc tor nanocrysials is the massive change in optical properties as a function of size. As size is reduced, the electronic ex citations shift to higher energy, and the oscillator strength is concentrated into just a few transitions. These basic physical phenomena of quantum confinement arise as a result of changes in the density of electronic states and can be understood by considering the relation between position and momen tum in free and confined particles. For a free particle, or a particle in the periodic potential of an extended solid, the energy and the crystal momentum can both be precisely defined, whereas the position can not. For a localized particle, the energy may still be well defined, but the uncertainty in position decreases, so that momentum is no longer well defined.
For example, the kinetic stability of tetrasilatetrahedrane (Si 4 H 4 ), hexasilaprismane (Si 6 H 6 ) and octasilacubane (Si 8 H 8 ) depends strongly on the steric bulkiness of the substituents (matrix). The silylsubstituted Si n Y m ( ) is stable in an inert atmosphere, but is oxidized in air to give colourless solids. The 1,1,2-trimethylpropyl-substituted Si n Y m ( ) is very stable even in air and survives for two weeks in the solid state. The prismanes with Si and Ge skeletons are yellow to orange. These prismanes have absorptions tailing into the visible region. So, Si 6 H 6 has an absorption band with a maximum at 241 nm tailing to ca 500 nm. The absorption band of Ge 6 Y 6 ( ) has a maximum at 261 nm, which is red-shifted compared to that of Ge 6 Y 6 because of the higher-lying orbitals of the Ge-Ge bonds [4].
The discrete energy eigenfunctions of the particle may then be viewed as superpositions of bulk momentum states. Given the relation between energy and momentum in the bulk solid, one can see how a series of nearby transitions occur ring at slightly different energies in the bulk be compressed by quantum confine ment into a single, intense transition in a QD.
The experimental data reveal a more complex situation probably characteristic of several radiative channels. Our main aim of this paper is to review the relevant theoretical information in order to identify radiative channels.

CALCULATIONS OF QUANTUM CONFINE MENT EFFECT
A number of calculations have been performed over the last few years, as for quantum dots as for silicon clusters, since both possibilities have been in voked for porous silicon. They essentially belong to four classes: effective mass approximation (EMA), empirical tight binding (ETB), empirical pseudopotential (EPS) and finally ab initio local density functional theory (LDFT).
In Fig. 1 we give the predicted band gaps versus size as obtained from LDFT calculations compiled in Ref. [5] for hydro gen-terminated Si-clusters, wires and slabs.
They are compared to the results obtained in our group using DFT approach with parameters (PDFT [3]) providing an extremely good fit to the bulk band structure. One can notice a good agreement between the LDFT and PDFT predic tions, which gives some con- НАНОМЕТРОЛОГІЯ fidence into the reliabil ity of these theoretical values. At this stage it is important to notice, that the LDFT gap values in clude a rigid shift of 0,6 eV, since it is known that LDFT underestimates the bulk band gap by this amount. Note, that the theoretical calculation grossly overestimate the blueshift. They must then be discarded, since EMA can be considered as an approximation to the best ETB or EPS descrip tions, which match the effective masses. One can, however, wonder why parameterized techniques should provide quantitative estimates of the one-electron gap. The basic point is that they are based on the postulate of transferability of the parameters from the known bulk band structure (to which they are fitted) to the unknown crystallite case. If this is accepted, then an essential criterion, by which a particular semiempirical model can be judged is how well it describes the bulk band struc ture. So from Fig. 1 we could conclude, that PDFT as well as corrected LDA techniques are likely to give reliable predictions for crystallites. Fig. 2 presents a compilation of data showing, that observed luminescence energies on porous silicon or silicon nanocrystals in an oxide matrix are consistently lower, that the predicted optical gaps of Fig. 1. On the other hand, they qualitatively agree with optical absorption data. Recent results also show, that the luminescence of fresh porous silicon samples is subject to a large red shift, when it is exposed to air and when the average size of the nanocrystals is smaller than 3 nm (Fig. 3). On the other hand, recent luminescence measure ments on silicon crystals obtained by silane decomposition are in good agreement with theory, but the luminescence is only observed for the largest crystallites. The situation is thus complex, even if it seems, that the degree of oxidation of the samples plays an impor tant role in the recombination mechanisms. All these results suggest, that other channels for the radiative recombination are possible. Large Stokes shifts might be consistent with the eventual exist ence of deep luminescent centers. The prob lem is that nothing is presently known regar ding the nature and origin of these states. Both from PDFT and LDA calculations, that such states indeed exist under the form of self-trapped exactions, most prob ably at the surface. A possible situation is the trap ping of an exciton on a Si-Si bond of a surface dimer, whose dangling bonds are saturated by hy drogen atoms. We have found ano ther interesting situation with very small crystals, conta ining less than about 50 silicon atoms, where we systematically obtain a large atomic relaxation in the excited state, which induces an important reorgan ization of the bonds in the cluster. The consequence is a large Stokes shift between the absorption and the emission energies. Therefore, small nanocrystals could play a role in the luminescence of porous silicon.

CHANNELS FOR THE RADIATIVE RECOMBINATION
We are presently investigating the possible existence of defect states in the band gap, induced by the oxidation of the surface. Among different systems, that we have studied, preliminary results show, that an oxy gen atom doubly bonded to a silicon atom (Si=O) at a nanocrystal surface is a good candidate to be involved in the luminescence of porous silicon. It gives rise to a deep level below the conduc tion band minimum, which could explain the evolu tion with size of the luminescence peak in Fig. 3.

STRUCTURAL DEPENDENCE OF THE BAND GAP
G. Allan with coworker's shows that the radiative recombination rate in spherical silicon nanocrystals (calculated as in Ref. [4]). It is low and it decreases for smaller band gap because of the indirect bulk band gap. In this regard, it would be of interest to use a direct gap phase of silicon such as Si-III (BC8) or to use materials  like SiGe alloys or amorphous silicon because the disorder breaks the selection rules. But an essential question arises about the existence of quantum confinement effects in disordered materials. Here we describe recent results, that we have obtained on these problems. The Si-III (BC-8) crystal phase is obtained for t bulk samples by releasing the pressure on the high-pressure betta-tin phase (Si-II) [5]. Existing theoretical calculations show, that the valence band maximum and the conduction band minimum occur at the same H point in the Brillouin zone. BC-8 silicon is thus a direct gap material but the calculations conclude, that it is close to a zero gap semiconductor. To calculate the electronic structure of BC-8 crystallites with size in the 1-3 nm range, we have chosen the same non-orthogonal EТВ technique as used for silicon crystallites with the diamond structure, but we have developed a specific parameterization for that struc ture.
Our results show, that the confinement effect is quite similar for BC-8 and diamond clusters. The only difference, when one goes from the BC-8 clus ter to the diamond one with the same size comes from the bulk gap value, which simply shifts the cluster gap energy. We have also performed PDFT calculations. One can see, that the values cal culated with PDFT for two small clusters and shifted by 0.6 eV to take into account the underestimation of the bulk gap, are in very good agreement with our ETB calculation. This confirms the transfer-ability of the ETB parameters from the bulk mater ial to clusters.
Experimentally, it was shown [2,6,7] than the BC-8 structure is obtained upon release of a high pressure on porous silicon. But the luminescence band remains practically unchanged except perhaps for a small shift (of order 0.1 -0.2 eV) after release of the pressure. This finding completely disagrees with our predictions, where this redshift should amount to ~ 1 eV for crystallites of the same size. This would rule out quantum confinement as the origin of the observed luminescence band and favor other possibilities.
We compare the variation of the recombina tion rate as a function of the cluster gap for the BC-8 [5] and the diamond structures [8]. Because it has a direct bulk band gap, the recombination rate in the BC-8 phase remains constant and pretty high, when the cluster size increases and the blue shift decreases. It is of the order of a few ms -1 (i.e. more then 10 3 times larger than in the diamond phase below 2 eV), but remains however lower than the result for GaAs (~ ns -1 ), which is also a direct gap semiconductor. However, the luminescence yield must be strongly improved for the BC-8 structure compared to the dia-mond structure.
With improved optical properties compared to silicon, Si x Ge 1-x alloys are also interesting materials. We have studied the strong confinement effects in SiGe clusters performing ETB calcu lations with the parameters. We con sider spherical clusters passivated by hydrogen atom, where the atomic sites are occupied randomly by Si or Ge atoms following the composition x. Our results shows, that the band gaps of Si 0,8 Ge 0,2 and Si clusters are quite close, with comparable blue-shift. This is due to the fact, that the electronic states in bulk SiGe alloys are still delocalized, so they experience the full confinement effect as for crystal line Si (c-Si).
We now analyze the case of stronger disorder as obtained in amorphous silicon (a-Si). It raises ex tremely interesting problems related to the confine ment induced blue shift of the energy gap: (a) does it exist in clusters of a-Si and is it comparable to what is obtained for c-Si; (b) what is the behavior of disorder-induced localized states in this regard. It has been often assumed, that quantum confine ments effects are small in a-Si nanostructures due to the short coherence length of free carriers in these materials. We will see that it is not true.
We calculate the electronic structure of a-Si and a-Si:H spherical clusters using the ETB and PDFT model. The interaction parameters are limited to first-nearest neighbors and the usual d -2 Harrison law can be used to calculate their variation with in teratomic distance d. The starting structure for the a-Si or a-Si: H clusters is obtained by selecting the atoms belonging to the respective atoms unit cell. Due to the new boundary conditions the struc ture is no more in equilibrium and we have thus relaxed the atomic positions, using a Keating poten tial.
A generally accepted picture of the electronic structure of a-Si is that it is still composed of valence and conduction bands separated by an energy gap but with bandtails of defect or dis order-induced localized states extending into the gap. For what follows we find it useful to classify the electronic states into three categories: delocalized states, experiencing the full confine ment effect as for c-Si; strongly localized states with extension in space much smaller than the cluster diameter and energies deep in the gap, insensitive to the confine ment effect and showing no blue shift; weakly localized states with extension in space of the order of the cluster diameter and energies near the gap limits, subject to an intermediate blue shift.
To characterize the luminescence of our a-Si clusters with 1-2.5 nm size we have first computed their fundamental gap, i.e. the distance in energy be tween the HOMO (highest occupied molecular orbital) and