FRACTAL PROPERTIES OF NANOSTRUCTURED SEMICONDUCTOR FILMS
Z.Zh. Zhanabaev, T,Yu, Grevtseva
Al-Farabi Kazakh National University, Al-Farabi Avenue, 71, Almaty 050040, Kazakhstan
Zeinulla Zh. Zhanabaev is a doctor of physical and mathematical
sciences, professor. He is the head of the Laboratory of Nonlinear
Physics at Research Institute of Experimental and Theoretical
Physics (at al-Farabi Kazakh National University). He received
the degree of doctor of physical and mathematical sciences from
al-Farabi Kazakh National University in 1995. Research interests
fall into the main themes: problems of chaos and theory of
information. He established theoretically criteria of selforganization in complex systems and suggested the new equation
for evolution of fractal measure. He is an author of research works
on turbulence, dynamical chaos in radio electronics, electrical and
optical properties of nanocluster semiconductors, dynamics in
neural networks. He is a scientific supervisor of bachelor students,
master students and Ph.D. students.
Tatyana Yu. Grevtseva is a candidate of physical and
mathematical sciences, senior research scientist. She works at
physical and technical faculty of al-Farabi Kazakh National
University. She received the degree of candidate of physical and
mathematical sciences from al-Farabi Kazakh National University
in 2009. Her research efforts are focused on application of ideas of
nonlinear physics for theoretical description of physical,
particularly surface, electrical and optical properties of
nanostructured semiconductor films. She is an author of research
works devoted to this subject area. She is a senior lecturer and
scientific supervisor of bachelor students and master students and
Ph.D. students.
Our work is devoted to the description of basic statements of our own theory and
algorithms for modeling of physical phenomena in nanostructured semiconductors. We suggest
new equations for the description of fractal evolution of concentration of charge carriers and
energy of excitonic formations. Optical processes are considered on the base of fluctuationdissipative relation. Fluctuations defined via power spectra of charge carriers’ concentration.
Dissipation of energy is expressed via equilibrium distribution of photons. Criteria of selfsimilarity and self-affinity of chaotic systems are theoretically defined for fixed points of
information and entropy. These criteria define difference between fractal and topological
dimensions used for characterization of nanostructures. These criteria are the key parameters of
the theory. Also we suggest a new approach for the description of morphology of semiconductor
films with nanostructures and electrical conductivity of quantum nanowires. [1-5].
Nanostructured semiconductors have been attracting considerable attention because they
may have various applications in new electronic devices. Modern methods of microscopy
demonstrate the nanocluster structure of semiconductor films. Such structures are irregular, selfaffine and self-similar. So, nanoclusters can be considered as fractal and multi-fractal objects.
Self-similarity means that similarity factors are equal each other for all variables. Self-affinity
corresponds to different values of similarity factors for different variables. We can change
properties of an electronic system containing nanoclusters by variation of geometrical size and
configuration of the clusters. So, we have an opportunity to control such characteristics of
structures as energy spectra of current carriers and phonons. Therefore, we can control optical
properties of nanostructures.
Development of new technologies requires knowledge about physical properties of
nanomaterials which have quantum and fractal properties. Values of de Broglie’s wavelength of
charge carriers in nanocluster are relatively small (about several nanometers), so, quantum
effects are important in such structures. But nanoclusters differ from other structures not only by
size but also by existence of dependence of geometrical (length, area and volume) parameters
and physical characteristics on scale of measurements. Such objects have irregular shape.
Therefore, geometrical and physical measures which are characteristics of the objects depend on
time according to nonlinear laws. This effect is noticeable for different scales of measurement,
and can be taken it into account at uniquely determined characteristics of scale invariance. So,
computer simulation alongside with experimental study of nanostructured semiconductors is
very important due to develop nanotechnologies. Nanostructures have ambiguous properties. So,
it is impossible to develop effective and reliable technologies by use of physical experiment
only. Characteristics of nanostructures cannot be measured precisely because they depend on
scale of measurement and can be defined as fractal measures. Expensive experimental
investigations without of corresponding descriptions via theoretical and computer models cannot
lead to great results. Thereby, theoretical analysis and computer simulation of electrophysical,
optical, radiation properties of nanostructured semiconductors are very important.
Nanostructures can be classified into the following types: quantum dots, quantum wires
and superlattices (quantum wells). At the present time these structures are widely used in
electronics for creation of lasers, sensitive detectors, quick-operating computer techniques, solar
cells and so on. Recently, problems of nanoelectronics often connect with problems of physics of
exciton formations. Excitons and biexcitons can be described as two quantum bits which can
interact with each other. So, exciton and biexciton can be used as basic units for quantum gates.
Semiconductor surfaces containing quantum wires with defined geometrical and topological
characteristics have attracted considerable attention in recent years. Using of such
semiconductors can lead to increasing of efficiency of optoelectronic devices. These
semiconductor films can be used as covering of surfaces of optoelectronic devices (for example,
solar cells) for reduction of scattering and reflection of photons. So, coefficient of light
absorption increases because of reduction of scattering and reflection of light. So, it is possible to
increase efficiency of solar cells by use of semiconductors films containing quantum wires with
specific parameters as cover. At the present time we haven’t a completed simple theory for the
description of optical processes in nanostructured matter. It makes some difficulty at
experimental researches. As usual, optical processes are described by use of the Kramers-Kronig
relations, Uhrbach formula, Tauc formula and so on. These methods have a great importance for
the description of optical phenomena in semiconductors. But we cannot take into account
nanocluster structure of matter and described above nonlinear effects. So, by use of such
methods we cannot describe resent experimental results completely. For example, at the present
time theory for the description the new effect of light localization in crystalline medium is
insufficient developed. Also we have no completed theory for explanation of interaction of
excitons and biexcitons in quantumsize structures, etc.
We have suggested a system for fractal evolution of concentration of current carriers
(electrons and holes) and admixtures in nanostructured semiconductors. Fractal dimension of a
real chaotic object cannot be defined correctly in experiments. So, it is effective to use criteria of
self-similarity and self-affinity. We have theoretically defined them in our previous works. These
criteria related with fractional parts of fractal dimension of an object and can be considered as
numerical values of Kolmogorov-Sinay entropy. We have obtained some of our results by taking
into account influence of stochastic perturbation on physical processes in nanostructures. Also
we have suggested a nonlinear map for chaotic oscillations which can be described as
“accumulation-bursting.” By use of our approach based on the theory of dynamical chaos we get
models of surfaces of nanostructured films. We can model quantum dots, quantum wires and
wells by use of corresponding values of fractal dimensions. We describe optical processes of
absorption, reflection and transmission of photons in nanostructured semiconductor films on the
base of quantum form of fluctuation-dissipative relation. Correlations of charge carriers
correspond to fluctuations of physical values. Dissipation can be defined via relation for
equilibrium photon distribution.
Our results of computer simulation based on the mentioned original equations describe
various results of recent physical experiments. Our calculations we had chosen the corresponding
values of such parameters as wavelength, bandgap, concentration of charge carriers, and so on.
The original equations used for modeling of physical properties of nanostructures are
based on two statements. At first, all measures (measure is an additive, measurable physical
value) are considered as fractal values. At second, the derivative with respect to an argument is
considered as limited quantity due to the Lipshitz-Hölder condition defining by fractal dimension
of the set of argument values. These equations correctly describe main regularities of
morphology of surfaces of nanostructures, kinetics and temperature dependence of concentration
of charge carriers.
Via the new two-dimensional map for the description of chaotic bursting we can model
point, lined, spatial and volumetric nanostructures. For this aim we use quantitative criteria of
self-similarity and self-affinity of sets established in our works.
The new form of fluctuation-dissipative relation for quasi-stationary fluctuations for the
description of optical processes in nanostructured semiconductors has been suggested. We can
take into account phonon and excitonic mechanism for photon absorption. The new formula
describing measure of a nonlinear fractal for the description of hierarchical structure of excitons
(biexcitons and trions) has been suggested. Nonlinearity of a fractal means the dependence of its
measure on itself at external influence.
Reflection of photons from nanostructures is described precisely if mathematical
expression for spectral density of correlations is chosen as a nonlinear fractal measure.
By use of the described approaches we can explain new experimental data on light
localization (photon delay) in nanostructured semiconductors, complex regularities of interaction
of light with surfaces of porous (with quantum wires) silicon.
We have proved that singularities of current-voltage characteristic of vertical silicon
quantum nanowires can be explained by dependence of potential of fractal clusters on external
voltage. Fractality of geometry of wire-like formations leads to multi-barrier tunneling effect in
nanowires containing in homogeneous sample (silicon). For this reason current-voltage
characteristic of silicon nanowires contains regions with negative differential resistance and
hysteresis loops.
Our theoretical results are in a good agreement with corresponding experimental data.
So, we hope that results of the present work can be used for further theoretical and
experimental studies in optoelectronics, photonics, nanoelectronics and other perspective
branches of technologies.
References
1. Zhanabayev Z.Zh., Grevtseva T.Yu. Fractal Properties of Nanostructured Semiconductors // Physica
B: Condensed Matter. – 2007. - Vol. 391, No 1. - P. 12-17.
2. Zhanabayev Z.Zh., Grevtseva T.Yu. Fractality of Nanostructured Semiconductor Films // e-Journal of
Surface Science and Nanotechnology. – 2007. - Vol. 5. - P. 132-135.
3. Zhanabaev Z.Zh., Grevtseva T.Yu., Danegulova T.B., Assanov G.S. Optical Processes in
Nanostructured Semiconductors // Journal of Computational and Theoretical Nanoscience. –2013. Vol. 10, No 3. – P.673-678.
4. Zhanabaev Z.Zh., Grevtseva T.Yu. Physical Fractal Phenomena in Nanostructured Semiconductors //
Reviews in Theoretical Science. – 2014. – Vol. 2, No 3. – P. 211-259.
5. Zhanabaev Z.Zh., Grevtseva T.Yu., Ibraimov M.K. Morphology and electrical properties of silicon
films with vertical nanowires // Journal of Computational and Theoretical Nanoscience. – 2015 (in
press).