In this paper we consider nonlinear model to describe manufacturing complementary horizontal field-effect heterotransistor. Based on analytical solution of the considered boundary problems some recommendations have been formulated to optimize technological processes.
ON OPTIMIZATION OF MANUFACTURING PLANAR DOUBLE-BASE HETEROTRANSISTORS TO DECR...ijaceeejournal
In this paper we consider an approach of manufacturing of double-base hetero transistors to decrease their
dimensions. Framework the approach it should be manufactured a heterostructure with specific configuration.
Farther it is necessary to dope certain areas of the hetero structure by diffusion or by ion implantation.
After finishing of the doping process the dopant and/or radiation defects should be annealed. We consider
an approach of optimization of dopant and/or radiation defects for manufacturing more compact double base
heterotransistors.
On optimization ofON OPTIMIZATION OF DOPING OF A HETEROSTRUCTURE DURING MANUF...ijcsitcejournal
We introduce an approach of manufacturing of a p-i-n-heterodiodes. The approach based on using a δ-
doped heterostructure, doping by diffusion or ion implantation of several areas of the heterostructure. After
the doping the dopant and/or radiation defects have been annealed. We introduce an approach to optimize
annealing of the dopant and/or radiation defects. We determine several conditions to manufacture more
compact p-i-n-heterodiodes
An Approach to Analyze Non-linear Dynamics of Mass Transport during Manufactu...BRNSS Publication Hub
In this paper, we introduce an approach to increase integration rate of elements of a hybrid comparator with the first dynamic amplifying stage and the second quasi-dynamic latching stage. Framework the approach, we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized
INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERSZac Darcy
In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a
heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the
considered p-n-junction have been formulated.
OPTIMIZATION OF MANUFACTURE OF FIELDEFFECT HETEROTRANSISTORS WITHOUT P-NJUNCT...ijrap
It has been recently shown, that manufacturing p-n-junctions, field-effect and bipolar transistors, thyristors
in a multilayer structure by diffusion or ion implantation under condition of optimization of dopant and/or
radiation defects leads to increasing of sharpness of p-n-junctions (both single p-n-junctions and p-njunctions,
which include into their system). In this situation one can also obtain increasing of homogeneity
of dopant in doped area. In this paper we consider manufacturing a field-effect heterotransistor without pn-
junction. Optimization of technological process with using inhomogeneity of heterostructure give us
possibility to manufacture the transistors as more compact.
ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE TO INCREASE DENSITY OF ELEMENTS ...ijoejournal
We introduce an approach for increasing density of voltage restore elements. The approach based on
manufacturing of a heterostructure, which consist of a substrate and an epitaxial layer with special configuration.
Several required sections of the layer should be doped by diffusion or ion implantation. After
that dopants and/or radiation defects should be annealed.
On Approach to Increase Integration rate of Elements of a Circuit Driver with...BRNSS Publication Hub
In this paper, we introduce an approach to increase the integration rate of elements of a driver with 2-tap de-emphasis and impendence matching. Framework the approach, we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized.
Optimization of technological process to decrease dimensions of circuits xor ...ijfcstjournal
The paper describes an approach of increasing of integration rate of elements of integrated circuits. The
approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one
should manufacture a heterostructure with specific configuration. After that several special areas of the
heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of dopant
and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation
defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical
approaches of modeling of technological process.
ON OPTIMIZATION OF MANUFACTURING PLANAR DOUBLE-BASE HETEROTRANSISTORS TO DECR...ijaceeejournal
In this paper we consider an approach of manufacturing of double-base hetero transistors to decrease their
dimensions. Framework the approach it should be manufactured a heterostructure with specific configuration.
Farther it is necessary to dope certain areas of the hetero structure by diffusion or by ion implantation.
After finishing of the doping process the dopant and/or radiation defects should be annealed. We consider
an approach of optimization of dopant and/or radiation defects for manufacturing more compact double base
heterotransistors.
On optimization ofON OPTIMIZATION OF DOPING OF A HETEROSTRUCTURE DURING MANUF...ijcsitcejournal
We introduce an approach of manufacturing of a p-i-n-heterodiodes. The approach based on using a δ-
doped heterostructure, doping by diffusion or ion implantation of several areas of the heterostructure. After
the doping the dopant and/or radiation defects have been annealed. We introduce an approach to optimize
annealing of the dopant and/or radiation defects. We determine several conditions to manufacture more
compact p-i-n-heterodiodes
An Approach to Analyze Non-linear Dynamics of Mass Transport during Manufactu...BRNSS Publication Hub
In this paper, we introduce an approach to increase integration rate of elements of a hybrid comparator with the first dynamic amplifying stage and the second quasi-dynamic latching stage. Framework the approach, we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized
INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERSZac Darcy
In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a
heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the
considered p-n-junction have been formulated.
OPTIMIZATION OF MANUFACTURE OF FIELDEFFECT HETEROTRANSISTORS WITHOUT P-NJUNCT...ijrap
It has been recently shown, that manufacturing p-n-junctions, field-effect and bipolar transistors, thyristors
in a multilayer structure by diffusion or ion implantation under condition of optimization of dopant and/or
radiation defects leads to increasing of sharpness of p-n-junctions (both single p-n-junctions and p-njunctions,
which include into their system). In this situation one can also obtain increasing of homogeneity
of dopant in doped area. In this paper we consider manufacturing a field-effect heterotransistor without pn-
junction. Optimization of technological process with using inhomogeneity of heterostructure give us
possibility to manufacture the transistors as more compact.
ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE TO INCREASE DENSITY OF ELEMENTS ...ijoejournal
We introduce an approach for increasing density of voltage restore elements. The approach based on
manufacturing of a heterostructure, which consist of a substrate and an epitaxial layer with special configuration.
Several required sections of the layer should be doped by diffusion or ion implantation. After
that dopants and/or radiation defects should be annealed.
On Approach to Increase Integration rate of Elements of a Circuit Driver with...BRNSS Publication Hub
In this paper, we introduce an approach to increase the integration rate of elements of a driver with 2-tap de-emphasis and impendence matching. Framework the approach, we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized.
Optimization of technological process to decrease dimensions of circuits xor ...ijfcstjournal
The paper describes an approach of increasing of integration rate of elements of integrated circuits. The
approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one
should manufacture a heterostructure with specific configuration. After that several special areas of the
heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of dopant
and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation
defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical
approaches of modeling of technological process.
On Approach to Increase Integration Rate of Elements of a Current Source CircuitBRNSS Publication Hub
In this paper, we introduce an approach to increase integration rate of elements of a current source circuit.
Framework the approach, we consider a heterostructure with special configuration. Several specific
areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/
or radiation defects should be optimized.
ON OPTIMIZATION OF MANUFACTURING OF FIELD-EFFECT HETEROTRANSISTORS FRAMEWORK ...ijoejournal
In this paper we introduce an approach to increase density of field-effect transistors framework a voltage reference. Framework the approach we consider manufacturing the inverter in heterostructure with specific configuration. Several required areas of the heterostructure should be doped by diffusion or ion implantation. After that dopant and radiation defects should by annealed framework optimized scheme. We also consider an approach to decrease value of mismatch-induced stress in the considered heterostructure. We introduce an analytical approach to analyze mass and heat transport in heterostructures during manufacturing of integrated circuits with account mismatch-induced stress.
AN APPROACH TO OPTIMIZE OF MANUFACTURING OF A VOLTAGE REFERENCE BASED ON HETE...JaresJournal
In this paper we introduce an approach to increase density of field-effect transistors framework a voltage
reference. Framework the approach we consider manufacturing the inverter in heterostructure with specific configuration. Several required areas of the heterostructure should be doped by diffusion or ion implantation. After that dopant and radiation defects should by annealed framework optimized scheme. We also
consider an approach to decrease value of mismatch-induced stress in the considered heterostructure. We
introduce an analytical approach to analyze mass and heat transport in heterostructures during manufacturing of integrated circuits with account mismatch-induced stress.
On Approach to Increase Integration Rate of Elements of an Operational Amplif...BRNSS Publication Hub
In this paper, we introduce an approach to optimize manufacturing of an operational amplifier circuit based on field-effect transistors. Main aims of the optimization are (i) decreasing dimensions of elements of the considered operational amplifier and (ii) increasing of performance and reliability of the considered field-effect transistors. Dimensions of considered field-effect transistors will be decreased due to manufacture of these transistors framework heterostructure with specific structure, doping of required areas of the heterostructure by diffusion or ion implantation, and optimization of annealing of dopant and/or radiation defects. Performance and reliability of the above field-effect transistors could be increased by optimization of annealing of dopant and/or radiation defects and using inhomogeneity of properties of heterostructure. Choosing of inhomogeneity of properties of heterostructure leads to increasing of compactness of distribution of concentration of dopant. At the same time, one can obtain increasing of homogeneity of the above concentration. In this paper, we also introduce an analytical approach for prognosis of technological process of manufacturing of the considered operational amplifier. The approach gives a possibility to take into account variation of parameters of processes in space and at the same time in space. At the same time, one can take into account nonlinearity of the considered processes.
On Approach to Increase Integration Rate of Elements of a Switched-capacitor ...BRNSS Publication Hub
In this paper, we introduce an approach to increase integration rate of elements of a switched-
capacitor step-down DC–DC converter. Framework the approach, we consider a heterostructure with
special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion
implantation. Annealing of dopant and/or radiation defects should be optimized.
MODIFICATION OF DOPANT CONCENTRATION PROFILE IN A FIELD-EFFECT HETEROTRANSIST...msejjournal
In this paper we consider an approach of manufacturing more compact field-effect heterotransistors. The
approach based on manufacturing a heterostructure, which consist of a substrate and an epitaxial layer
with specific configuration. After that several areas of the epitaxial layer have been doped by diffusion or
ion implantation with optimized annealing of dopant and /or radiation defects. At the same time we introduce
an approach of modification of energy band diagram by additional doping of channel of the transistors.
We also consider an analytical approach to model and optimize technological process.
Modeling of manufacturing of a field effect transistor to determine condition...ijcsa
In this paper we introduce an approach to model technological process of manufacture of a field-effect
heterotransistor. The modeling gives us possibility to optimize the technological process to decrease length
of channel by using mechanical stress. As accompanying results of the decreasing one can find decreasing
of thickness of the heterotransistors and increasing of their density, which were comprised in integrated
circuits.
AN APPROACH TO OPTIMIZE MANUFACTURE OF AN ACTIVE QUADRATURE SIGNAL GENERATOR ...antjjournal
In this paper we introduce an approach to increase density of field-effect transistors framework an active
quadrature signal generator. Framework the approach we consider manufacturing the generator in heterostructure
with specific configuration. Several required areas of the heterostructure should be doped by diffusion
or ion implantation. After that dopant and radiation defects should by annealed framework optimized
scheme. We also consider an approach to decrease value of mismatch-induced stress in the considered
heterostructure. We introduce an analytical approach to analyze mass and heat transport in heterostructures
during manufacturing of integrated circuits with account mismatch-induced stress.
On Optimization of Manufacturing of a Sense-amplifier Based Flip-flopBRNSS Publication Hub
The paper describes an approach for increasing of density of field-effect heterotransistors in a sense-amplifier based flip-flop. To illustrate the approach, we consider manufacturing of an amplifier of power in a heterostructure with specific configuration. One shall dope some specific areas of the heterostructure by diffusion or ion implantation. After that, it should be done optimized annealing of radiation defects and/or dopant. We introduce an approach for decreasing of stress between layers of heterostructure. Furthermore, it has been considered an analytical approach for prognosis of heat and mass transport in heterostructures, which can be take into account mismatch-induced stress.
ON OPTIMIZATION OF MANUFACTURING OF ELEMENTS OF AN BINARY-ROM CIRCUIT TO INCR...JaresJournal
In this paper we introduce an approach to increase integration rate of elements of an binary-ROM circuit. Framework the approach we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized.
On Decreasing of Dimensions of Field-Effect Heterotransistors in Logical CMOP...BRNSS Publication Hub
In this paper, we introduce an approach to decrease the dimensions of CMOP voltage differencing inverting buffered amplifier based on field-effect heterotransistors by increasing density of elements. Dimensions of the elements will be decreased due to manufacture heterostructure with a specific structure, doping of required areas of the heterostructure by diffusion or ion implantation, and optimization of annealing of dopant and/or radiation defects.
On prognozisys of manufacturing double basemsejjournal
In this paper we introduce a modification of recently introduced analytical approach to model mass- and
heat transport. The approach gives us possibility to model the transport in multilayer structures with account
nonlinearity of the process and time-varing coefficients and without matching the solutions at the
interfaces of the multilayer structures. As an example of using of the approach we consider technological
process to manufacture more compact double base heterobipolar transistor. The technological approach
based on manufacturing a heterostructure with required configuration, doping of required areas of this heterostructure
by diffusion or ion implantation and optimal annealing of dopant and/or radiation defects. The
approach gives us possibility to manufacture p-n- junctions with higher sharpness framework the transistor.
In this situation we have a possibility to obtain smaller switching time of p-n- junctions and higher compactness
of the considered bipolar transistor.
ON INCREASING OF DENSITY OF ELEMENTS IN A MULTIVIBRATOR ON BIPOLAR TRANSISTORSijcsitcejournal
In this paper we consider an approach to increase density of elements of a multivibrator on bipolar transistors.
The considered approach based on manufacturing a heterostructure with necessity configuration,
doping by diffusion or ion implantation of required areas to manufacture the required type of conductivity
(p or n) in the areas and optimization of annealing of dopant and/or radiation defects to manufacture more
compact distributions of concentrations of dopants. We also introduce an analytical approach to prognosis
technological process.
On Analytical Approach to Prognosis of Manufacturing of Voltage Divider Biasi...BRNSS Publication Hub
In this paper, we introduce an approach for prognosis of manufacturing of voltage divider biasing common emitter amplifier based on bipolar transistors with account mismatch-induced stress. Based on this prognosis, we formulate some recommendations for optimization of manufacturing of the amplifier. Main aims of the optimization are as follows: (1) Decreasing dimensions of elements of the considered operational amplifier and (2) increasing of performance and reliability of the considered bipolar transistors. Dimensions of considered bipolar transistors will be decreased due to manufacture of these transistors framework heterostructure with specific structure, doping of required areas of the heterostructure by diffusion or ion implantation, and optimization of annealing of dopant and/or radiation defects. Performance and reliability of the above bipolar transistors could be increased by optimization of annealing of dopant and/or radiation defects and using inhomogeneity of the properties of heterostructure. Choosing of inhomogeneity properties of heterostructure leads to increasing of compactness of distribution of concentration of dopant. At the same time, one can obtain increasing of homogeneity of the above concentration. In this paper, we also introduce an analytical approach for prognosis of technological process of manufacturing of the considered operational amplifier. The approach gives a possibility to take into account variation of parameters of processes in space and at the same time in space. At the same time, one can take into account nonlinearity of the considered processes.
ON OPTIMIZATION OF MANUFACTURING OF FIELD EFFECT HETEROTRANSISTORS FRAMEWORK ...antjjournal
We consider an approach for increasing density of field-effect heterotransistors in a single-stage multi-path
operational amplifier. At the same time one can obtain decreasing of dimensions of the above transistors.
Dimensions of the elements could be decreased by manufacturing of these elements in a heterostructure
with specific structure. The manufacturing is doing by doping of required areas of the heterostructure by
diffusion or ion implantation with future optimization of annealing of dopant and/or radiation defects.
MODELING OF REDISTRIBUTION OF INFUSED DOPANT IN A MULTILAYER STRUCTURE DOPANT...mathsjournal
In this paper we used an analytical approach to model nonlinear diffusion of dopant in a multilayer structure with account nonstationary annealing of the dopant. The approach do without crosslinking solutions at
the interface between layers of the multilayer structure. In this paper we analyzed influence of pressure of
vapor of infusing dopant during doping of multilayer structure on values of optimal parameters of technological process to manufacture p-n-junctions. It has been shown, that doping of multilayer structures by
diffusion and optimization of annealing of dopant gives us possibility to increase sharpness of p-n-junctions
(single p-n-junctions and p-n-junctions within transistors) and to increase homogeneity of dopant distribution in doped area.
Optimization of Technological Process to Decrease Dimensions of Circuits XOR,...ijfcstjournal
The paper describes an approach of increasing of integration rate of elements of integrated circuits. The
approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one
should manufacture a heterostructure with specific configuration. After that several special areas of the
heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of dopant and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation
defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical
approaches of modeling of technological process.
ON OPTIMIZATION OF MANUFACTURING OF AN AMPLIFIER TO INCREASE DENSITY OF BIPOL...ijoejournal
In this paper we consider a possibility to increase density of bipolar heterotransistor framework an amplifier
due to decreasing of their dimensions. The considered approach based on doping of required areas of
heterostructure with specific configuration by diffusion or ion implantation. The doping finished by optimized
annealing of dopant and/or radiation defects. Analysis of redistribution of dopant with account redistribution
of radiation defects (after implantation of ions of dopant) for optimization of the above annealing
have been done by using recently introduced analytical approach. The approach gives a possibility
to analyze mass and heat transports in a heterostructure without crosslinking of solutions on interfaces
between layers of the heterostructure with account nonlinearity of these transports and variation in time of
their parameters.
On Approach to Increase Integration Rate of Elements of a Current Source CircuitBRNSS Publication Hub
In this paper, we introduce an approach to increase integration rate of elements of a current source circuit.
Framework the approach, we consider a heterostructure with special configuration. Several specific
areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/
or radiation defects should be optimized.
ON OPTIMIZATION OF MANUFACTURING OF FIELD-EFFECT HETEROTRANSISTORS FRAMEWORK ...ijoejournal
In this paper we introduce an approach to increase density of field-effect transistors framework a voltage reference. Framework the approach we consider manufacturing the inverter in heterostructure with specific configuration. Several required areas of the heterostructure should be doped by diffusion or ion implantation. After that dopant and radiation defects should by annealed framework optimized scheme. We also consider an approach to decrease value of mismatch-induced stress in the considered heterostructure. We introduce an analytical approach to analyze mass and heat transport in heterostructures during manufacturing of integrated circuits with account mismatch-induced stress.
AN APPROACH TO OPTIMIZE OF MANUFACTURING OF A VOLTAGE REFERENCE BASED ON HETE...JaresJournal
In this paper we introduce an approach to increase density of field-effect transistors framework a voltage
reference. Framework the approach we consider manufacturing the inverter in heterostructure with specific configuration. Several required areas of the heterostructure should be doped by diffusion or ion implantation. After that dopant and radiation defects should by annealed framework optimized scheme. We also
consider an approach to decrease value of mismatch-induced stress in the considered heterostructure. We
introduce an analytical approach to analyze mass and heat transport in heterostructures during manufacturing of integrated circuits with account mismatch-induced stress.
On Approach to Increase Integration Rate of Elements of an Operational Amplif...BRNSS Publication Hub
In this paper, we introduce an approach to optimize manufacturing of an operational amplifier circuit based on field-effect transistors. Main aims of the optimization are (i) decreasing dimensions of elements of the considered operational amplifier and (ii) increasing of performance and reliability of the considered field-effect transistors. Dimensions of considered field-effect transistors will be decreased due to manufacture of these transistors framework heterostructure with specific structure, doping of required areas of the heterostructure by diffusion or ion implantation, and optimization of annealing of dopant and/or radiation defects. Performance and reliability of the above field-effect transistors could be increased by optimization of annealing of dopant and/or radiation defects and using inhomogeneity of properties of heterostructure. Choosing of inhomogeneity of properties of heterostructure leads to increasing of compactness of distribution of concentration of dopant. At the same time, one can obtain increasing of homogeneity of the above concentration. In this paper, we also introduce an analytical approach for prognosis of technological process of manufacturing of the considered operational amplifier. The approach gives a possibility to take into account variation of parameters of processes in space and at the same time in space. At the same time, one can take into account nonlinearity of the considered processes.
On Approach to Increase Integration Rate of Elements of a Switched-capacitor ...BRNSS Publication Hub
In this paper, we introduce an approach to increase integration rate of elements of a switched-
capacitor step-down DC–DC converter. Framework the approach, we consider a heterostructure with
special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion
implantation. Annealing of dopant and/or radiation defects should be optimized.
MODIFICATION OF DOPANT CONCENTRATION PROFILE IN A FIELD-EFFECT HETEROTRANSIST...msejjournal
In this paper we consider an approach of manufacturing more compact field-effect heterotransistors. The
approach based on manufacturing a heterostructure, which consist of a substrate and an epitaxial layer
with specific configuration. After that several areas of the epitaxial layer have been doped by diffusion or
ion implantation with optimized annealing of dopant and /or radiation defects. At the same time we introduce
an approach of modification of energy band diagram by additional doping of channel of the transistors.
We also consider an analytical approach to model and optimize technological process.
Modeling of manufacturing of a field effect transistor to determine condition...ijcsa
In this paper we introduce an approach to model technological process of manufacture of a field-effect
heterotransistor. The modeling gives us possibility to optimize the technological process to decrease length
of channel by using mechanical stress. As accompanying results of the decreasing one can find decreasing
of thickness of the heterotransistors and increasing of their density, which were comprised in integrated
circuits.
AN APPROACH TO OPTIMIZE MANUFACTURE OF AN ACTIVE QUADRATURE SIGNAL GENERATOR ...antjjournal
In this paper we introduce an approach to increase density of field-effect transistors framework an active
quadrature signal generator. Framework the approach we consider manufacturing the generator in heterostructure
with specific configuration. Several required areas of the heterostructure should be doped by diffusion
or ion implantation. After that dopant and radiation defects should by annealed framework optimized
scheme. We also consider an approach to decrease value of mismatch-induced stress in the considered
heterostructure. We introduce an analytical approach to analyze mass and heat transport in heterostructures
during manufacturing of integrated circuits with account mismatch-induced stress.
On Optimization of Manufacturing of a Sense-amplifier Based Flip-flopBRNSS Publication Hub
The paper describes an approach for increasing of density of field-effect heterotransistors in a sense-amplifier based flip-flop. To illustrate the approach, we consider manufacturing of an amplifier of power in a heterostructure with specific configuration. One shall dope some specific areas of the heterostructure by diffusion or ion implantation. After that, it should be done optimized annealing of radiation defects and/or dopant. We introduce an approach for decreasing of stress between layers of heterostructure. Furthermore, it has been considered an analytical approach for prognosis of heat and mass transport in heterostructures, which can be take into account mismatch-induced stress.
ON OPTIMIZATION OF MANUFACTURING OF ELEMENTS OF AN BINARY-ROM CIRCUIT TO INCR...JaresJournal
In this paper we introduce an approach to increase integration rate of elements of an binary-ROM circuit. Framework the approach we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized.
On Decreasing of Dimensions of Field-Effect Heterotransistors in Logical CMOP...BRNSS Publication Hub
In this paper, we introduce an approach to decrease the dimensions of CMOP voltage differencing inverting buffered amplifier based on field-effect heterotransistors by increasing density of elements. Dimensions of the elements will be decreased due to manufacture heterostructure with a specific structure, doping of required areas of the heterostructure by diffusion or ion implantation, and optimization of annealing of dopant and/or radiation defects.
On prognozisys of manufacturing double basemsejjournal
In this paper we introduce a modification of recently introduced analytical approach to model mass- and
heat transport. The approach gives us possibility to model the transport in multilayer structures with account
nonlinearity of the process and time-varing coefficients and without matching the solutions at the
interfaces of the multilayer structures. As an example of using of the approach we consider technological
process to manufacture more compact double base heterobipolar transistor. The technological approach
based on manufacturing a heterostructure with required configuration, doping of required areas of this heterostructure
by diffusion or ion implantation and optimal annealing of dopant and/or radiation defects. The
approach gives us possibility to manufacture p-n- junctions with higher sharpness framework the transistor.
In this situation we have a possibility to obtain smaller switching time of p-n- junctions and higher compactness
of the considered bipolar transistor.
ON INCREASING OF DENSITY OF ELEMENTS IN A MULTIVIBRATOR ON BIPOLAR TRANSISTORSijcsitcejournal
In this paper we consider an approach to increase density of elements of a multivibrator on bipolar transistors.
The considered approach based on manufacturing a heterostructure with necessity configuration,
doping by diffusion or ion implantation of required areas to manufacture the required type of conductivity
(p or n) in the areas and optimization of annealing of dopant and/or radiation defects to manufacture more
compact distributions of concentrations of dopants. We also introduce an analytical approach to prognosis
technological process.
On Analytical Approach to Prognosis of Manufacturing of Voltage Divider Biasi...BRNSS Publication Hub
In this paper, we introduce an approach for prognosis of manufacturing of voltage divider biasing common emitter amplifier based on bipolar transistors with account mismatch-induced stress. Based on this prognosis, we formulate some recommendations for optimization of manufacturing of the amplifier. Main aims of the optimization are as follows: (1) Decreasing dimensions of elements of the considered operational amplifier and (2) increasing of performance and reliability of the considered bipolar transistors. Dimensions of considered bipolar transistors will be decreased due to manufacture of these transistors framework heterostructure with specific structure, doping of required areas of the heterostructure by diffusion or ion implantation, and optimization of annealing of dopant and/or radiation defects. Performance and reliability of the above bipolar transistors could be increased by optimization of annealing of dopant and/or radiation defects and using inhomogeneity of the properties of heterostructure. Choosing of inhomogeneity properties of heterostructure leads to increasing of compactness of distribution of concentration of dopant. At the same time, one can obtain increasing of homogeneity of the above concentration. In this paper, we also introduce an analytical approach for prognosis of technological process of manufacturing of the considered operational amplifier. The approach gives a possibility to take into account variation of parameters of processes in space and at the same time in space. At the same time, one can take into account nonlinearity of the considered processes.
ON OPTIMIZATION OF MANUFACTURING OF FIELD EFFECT HETEROTRANSISTORS FRAMEWORK ...antjjournal
We consider an approach for increasing density of field-effect heterotransistors in a single-stage multi-path
operational amplifier. At the same time one can obtain decreasing of dimensions of the above transistors.
Dimensions of the elements could be decreased by manufacturing of these elements in a heterostructure
with specific structure. The manufacturing is doing by doping of required areas of the heterostructure by
diffusion or ion implantation with future optimization of annealing of dopant and/or radiation defects.
MODELING OF REDISTRIBUTION OF INFUSED DOPANT IN A MULTILAYER STRUCTURE DOPANT...mathsjournal
In this paper we used an analytical approach to model nonlinear diffusion of dopant in a multilayer structure with account nonstationary annealing of the dopant. The approach do without crosslinking solutions at
the interface between layers of the multilayer structure. In this paper we analyzed influence of pressure of
vapor of infusing dopant during doping of multilayer structure on values of optimal parameters of technological process to manufacture p-n-junctions. It has been shown, that doping of multilayer structures by
diffusion and optimization of annealing of dopant gives us possibility to increase sharpness of p-n-junctions
(single p-n-junctions and p-n-junctions within transistors) and to increase homogeneity of dopant distribution in doped area.
Optimization of Technological Process to Decrease Dimensions of Circuits XOR,...ijfcstjournal
The paper describes an approach of increasing of integration rate of elements of integrated circuits. The
approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one
should manufacture a heterostructure with specific configuration. After that several special areas of the
heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of dopant and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation
defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical
approaches of modeling of technological process.
ON OPTIMIZATION OF MANUFACTURING OF AN AMPLIFIER TO INCREASE DENSITY OF BIPOL...ijoejournal
In this paper we consider a possibility to increase density of bipolar heterotransistor framework an amplifier
due to decreasing of their dimensions. The considered approach based on doping of required areas of
heterostructure with specific configuration by diffusion or ion implantation. The doping finished by optimized
annealing of dopant and/or radiation defects. Analysis of redistribution of dopant with account redistribution
of radiation defects (after implantation of ions of dopant) for optimization of the above annealing
have been done by using recently introduced analytical approach. The approach gives a possibility
to analyze mass and heat transports in a heterostructure without crosslinking of solutions on interfaces
between layers of the heterostructure with account nonlinearity of these transports and variation in time of
their parameters.
MODIFICATION OF DOPANT CONCENTRATION PROFILE IN A FIELD-EFFECT HETEROTRANSIST...msejjournal
In this paper we consider an approach of manufacturing more compact field-effect heterotransistors. The
approach based on manufacturing a heterostructure, which consist of a substrate and an epitaxial layer
with specific configuration. After that several areas of the epitaxial layer have been doped by diffusion or
ion implantation with optimized annealing of dopant and /or radiation defects. At the same time we introduce an approach of modification of energy band diagram by additional doping of channel of the transistors. We also consider an analytical approach to model and optimize technological process.
On prognozisys of manufacturing doublebaseijaceeejournal
In this paper we introduce a modification of recently introduced analytical approach to model mass- and
heat transport. The approach gives us possibility to model the transport in multilayer structures with account
nonlinearity of the process and time-varing coefficients and without matching the solutions at the
interfaces of the multilayer structures. As an example of using of the approach we consider technological
process to manufacture more compact double base heterobipolar transistor. The technological approach
based on manufacturing a heterostructure with required configuration, doping of required areas of this
heterostructure by diffusion or ion implantation and optimal annealing of dopant and/or radiation defects.
The approach gives us possibility to manufacture p-n- junctions with higher sharpness framework the transistor.
In this situation we have a possibility to obtain smaller switching time of p-n- junctions and higher
compactness of the considered bipolar transistor.
On Decreasing of Dimensions of Field-Effect Transistors with Several Sourcesmsejjournal
We analyzed mass and heat transport during manufacturing field-effect heterotransistors with several
sources to decrease their dimensions. Framework the result of manufacturing it is necessary to manufacture
heterostructure with specific configuration. After that it is necessary to dope required areas of the heterostructure by diffusion or ion implantation to manufacture the required type of conductivity (p or n). After
the doping it is necessary to do optimize annealing. We introduce an analytical approach to prognosis mass
and heat transport during technological processes. Using the approach leads to take into account nonlinearity of mass and heat transport and variation in space and time (at one time) physical parameters of these
processes
ON APPROACH OF OPTIMIZATION OF FORMATION OF INHOMOGENOUS DISTRIBUTIONS OF DOP...ijcsa
We introduce an approach of manufacturing of a field-effect heterotransistor with inhomogenous doping of channel. The inhomogenous distribution of concentration of dopant gives a possibility to change speed of transport of charge carriers and to decrease length of channel.
On Optimization of Manufacturing of Field-Effect Heterotransistors Frame-work...antjjournal
We consider an approach for increasing density of field-effect heterotransistors in a single-stage multi-path operational amplifier. At the same time one can obtain decreasing of dimensions of the above transistors. Dimensions of the elements could be decreased by manufacturing of these elements in a heterostructure with specific structure. The manufacturing is doing by doping of required areas of the heterostructure by diffusion or ion implantation with future optimization of annealing of dopant and/or radiation defects.
OPTIMIZATION OF MANUFACTURING OF LOGICAL ELEMENTS "AND" MANUFACTURED BY USING...ijcsitcejournal
In this paper we introduce an approach to decrease dimensions of logical elements "AND" based on fieldeffect
heterotransistors. Framework the approach one shall consider a heterostructure with specific structure.
Several specific areas of the het
ON OPTIMIZATION OF MANUFACTURING OF FIELD-EFFECT HETERO TRANSISTORS A THREE S...jedt_journal
In this paper we introduce an approach to increase density of field-effect hetero transistors framework a three-stage
amplifier circuit. At the same time one can obtain decreasing of dimensions of the above transistors. Dimensions of the elements will be decreased due to manufacture heterostructure with specific structure, doping of required areas of the hetero structure by diffusion or ion implantation and optimization of annealing of dopant and/or radiation defects.
ON APPROACH TO DECREASE DIMENSIONS OF FIELD-EFFECT TRANSISTORS FRAMEWORK ELEM...ijfcstjournal
In this paper we consider manufacturing of elements SRAM with increased density of field-effect transistors
consisting these elements. The approach based on manufacturing of the elements in heterostructure with
specific configuration. We consider doping of several required areas of the heterostructure by diffusion or
by ion implantation. After that dopant and radiation defects have been annealed framework optimized
scheme.
ON DECREASING OF DIMENSIONS OF FIELDEFFECT TRANSISTORS WITH SEVERAL SOURCESmsejjournal
We analyzed mass and heat transport during manufacturing field-effect heterotransistors with several sources to decrease their dimensions. Framework the result of manufacturing it is necessary to manufacture heterostructure with specific configuration. After that it is necessary to dope required areas of the heterostructure by diffusion or ion implantation to manufacture the required type of conductivity (p or n). After
the doping it is necessary to do optimize annealing. We introduce an analytical approach to prognosis mass
and heat transport during technological processes. Using the approach leads to take into account nonlinearity of mass and heat transport and variation in space and time (at one time) physical parameters of these processes
ON DECREASING OF DIMENSIONS OF FIELDEFFECT TRANSISTORS WITH SEVERAL SOURCESmsejjournal
We analyzed mass and heat transport during manufacturing field-effect heterotransistors with several
sources to decrease their dimensions. Framework the result of manufacturing it is necessary to manufacture
heterostructure with specific configuration. After that it is necessary to dope required areas of the heterostructure by diffusion or ion implantation to manufacture the required type of conductivity (p or n). After
the doping it is necessary to do optimize annealing. We introduce an analytical approach to prognosis mass
and heat transport during technological processes. Using the approach leads to take into account nonlinearity of mass and heat transport and variation in space and time (at one time) physical parameters of these
processes
Influence of Overlayers on Depth of Implanted-Heterojunction RectifiersZac Darcy
In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a
heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the
considered p-n-junction have been formulated.
Similar to An Approach to Optimize Regimes of Manufacturing of Complementary Horizontal Field-Effect Transistor (20)
On the Unification of Physic and the Elimination of Unbound Quantitiesijrap
This paper supports Descartes' idea of a constant quantity of motion, modernized by Leibniz. Unlike Leibniz, the paper emphasizes that the idea is not realized by forms of energy, but by energy itself. It remains constant regardless of the form, type, or speed of motion, even that of light. Through force, energy is only transformed. Here it is proved that force is its derivative. It exists even at rest, representing the object's minimal energy state. With speed, we achieve its multiplication up to the maximum energy state, from which a maximum force is derived from the object. From this point, corresponding to Planck's Length, we find the value of the force wherever we want. Achieving this removes the differences between various natural forces. The new idea eliminates infinite magnitudes. The process allows the laws to transition from simple to complex forms and vice versa, through differentiation-integration. For this paper, this means achieving the Unification Theory.
Gravity Also Redshifts Light – the Missing Phenomenon That Could Resolve Most...ijrap
In this paper I discover that gravity also redshifts light like the velocity of its source does. When light travels towards a supermassive object, its waves (or photons) undergo continuous stretching, thereby shifting towards lower frequencies. Gravity redshifts light irrespective of whether its source is in motion or static with respect to its observer. An equation is derived for gravitational redshift, and a formula for combined redshift is presented by considering both the velocity, and gravity redshifts. Also explained is how frequencies of electromagnetic spectrum continuously downgrade as a light beam of mix frequencies passes towards a black hole. Further, a clear methodology is provided to figure out whether expansion of the universe is accelerating or decelerating, or alternatively, the universe is contracting.
In this paper I present a new theory that explains as to when and how dark energy is created as mass is destroyed. The theory extends Einstein’s mass energy equation to a more generic form in order to make it work even in high gravity conditions. It also explains why dark energy is created. Further, it is proved Einstein’s mass energy equation holds good only when the destroyed mass has no supermassive object in its close vicinity. The relationship between dark energy and dark matter is unveiled. An extended mathematical form of Einstein’s mass energy equation is derived, based on which the conditions leading to dark energy creation are explained. Three new physical parameters called dark energy discriminant, dark energy radius and dark energy boundary are introduced to facilitate easy understanding of the theory. It is explained in detail that an extremely superdense object has two dark energy boundaries, outer and inner. Mass destroyed only between these two boundaries creates dark energy. Dark energy space, the space between the two aforementioned boundaries, shrouds visible matter in obscurity from optical and electromagnetic telescopes. This theory identifies Gargantuan as a superdense black hole currently creating fresh dark energy, which could be the subject of interest for the astronomical research community having access to sophisticated telescopes, and working on dark energy. It also upholds dark energy and denies the existence of dark matter. Dark matter is nothing but the well-known visible matter positioned in dark energy space. An important relationship is derived between a photon’s frequency and its distance from a black hole to demonstrate the effect of gravity on light. Another important fact revealed by this theory is gravity stretches out light, thereby causing redshift, which is unaccounted in the computation of velocities of outer galaxies. Whether the universe is undergoing accelerated or decelerated expansion, or accelerated contraction can precisely be determined only after accounting for the redshift caused by gravity
International Journal on Soft Computing, Artificial Intelligence and Applicat...ijrap
International Journal on Soft Computing, Artificial Intelligence and Applications (IJSCAI)
is an open access peer-reviewed journal that provides an excellent international forum for sharing
knowledge and results in theory, methodology and applications of Artificial Intelligence, Soft
Computing. The Journal looks for significant contributions to all major fields of the Artificial
Intelligence, Soft Computing in theoretical and practical aspects. The aim of the Journal is to
provide a platform to the researchers and practitioners from both academia as well as industry to
meet and share cutting-edge development in the field.
Authors are solicited to contribute to the journal by submitting articles that illustrate research
results, projects, surveying works and industrial experiences that describe significant advances in
the areas of Database management systems.
SOME THEORETICAL ASPECTS OF HYDROGEN DIFFUSION IN BCC METALS AT LOW TEMPERATURESijrap
Purpose of the work is to discuss some theoretical aspects of the diffusion of hydrogen atoms in the crystal
lattice of BCC metals at low temperatures using the methods of statistical thermodynamics. The values of
the statistical model calculations of H diffusion coefficients in α-Fe, V, Ta, Nb, K are in good agreement
with the experimental data. The statistical model can also explain deviations from the Arrhenius equation
at temperatures 300-100 K in α-Fe, V, Nb and K. It was suggested that thermally activated fast tunnelling
transition of hydrogen atoms through the potential barrier at a temperature below 300 K provides an
almost free movement of H atoms in the α-Fe and V lattice at these temperatures. The results show that
quantum-statistical effects play a decisive role in the H diffusion in BCC metals at low temperatures. Using
the statistical model allows for the prediction of the diffusion coefficient for H in BCC metals at low
temperatures, where it’s necessary to consider quantum effects.
MASSIVE PHOTON HYPOTHESIS OPENS DOORS TO NEW FIELDS OF RESEARCHijrap
Mass, an inherent property of matter, is calculated directly for the photon particle from the very classical
principles of the kinetic theory of gases. It is not an end result with no perspective nor other outcome.
Quite the opposite, a single ponderable tiny photon frees the mind of old ways of thinking and opens up
new paths to a broad field of investigation where the very large can then be described and explained by the
very small. This reality of a non-zero mass suddenly shows up in the interpretation of many experiments
which become clear and simple to comprehend. Besides, that same key particle has the potential to unlock
and solve some long lasting major observational issues or enigmas. All this converges upon its
acknowledgement and acceptance.
PHENOMENOLOGICAL METHOD REGARDING A THIRD THEORY OF PHYSICS “THE EVENT:THE TH...ijrap
The quest for a third theory uniting macro-cosmos (relativity) and micro-cosmos (quantum mechanics) has coexisted with the denial of feminine/subjective polarity to masculine/objective. The dismissal of electromagnetism as the tension of opposites in quest of inner/outer unity is sourced in the denial of the feminine qualia -- the negative force field attributed to dark energy/dark matter. However, a conversion philosophy sourced in the hieros gamos and signified by the Mobius strip has formulated an integral consciousness methodology producing quantum objects by means of embracing the shadow haunting contemporary physics. This Self-reflecting process integrating subject/object comprises an ontology of kairos as the “quantum leap.” An interdisciplinary quest to create a phenomenological narrative is disclosed via a holistic apparatus of hermeneutics manifesting image/text of a contemporary grail journey. Reflected in this Third space is the sacred reality of autonomous number unifying polarities of feminine/subjective (quality) and objective/masculine (quantity) as new measurement apparatus for the quantum wave collapse.
3rd International Conference on Integrating Technology in Education (ITE 2022)ijrap
3rd International Conference on Integrating Technology in Education (ITE 2022) This forum also aims to provide a platform for exchanging ideas in new emerging trends that needs more focus and exposure and will attempt to publish proposals that strengthen our goals.
A SPECIAL RELATIONSHIP BETWEEN MATTER, ENERGY, INFORMATION, AND CONSCIOUSNESSijrap
This paper discusses the advantages of describing the universe, or nature, in terms of information and consciousness. Some problems encountered by theoretical physicists in the quest for the theory of everything stem from the limitations of trying to understand everything in terms of matter and energy only. However, if everything, including matter, energy, life, and mental processes, is described in terms of information and consciousness, much progress can be made in the search for the ultimate theory of the universe. As brilliant and successful as physics and chemistry have been over the last two centuries, it is important that nature is not viewed solely in terms of matter and energy. Two additional components are needed to unlock her secrets. While extensive writing exists that describes the connection between matter and energy and their physical basis, little work has been done to learn the special relationship between matter, energy, information, and consciousness.
This paper discusses the advantages of describing the universe, or nature, in terms of information and consciousness. Some problems encountered by theoretical physicists in the quest for the theory of everything stem from the limitations of trying to understand everything in terms of matter and energy only. However, if everything, including matter, energy, life, and mental processes, is described in terms of information and consciousness, much progress can be made in the search for the ultimate theory of the universe. As brilliant and successful as physics and chemistry have been over the last two centuries, it is important that nature is not viewed solely in terms of matter and energy. Two additional components are needed to unlock her secrets. While extensive writing exists that describes the connection between matter and energy and their physical basis, little work has been done to learn the special relationship between matter, energy, information, and
consciousness.
THE CONCEPT OF SPACE AND TIME: AN AFRICAN PERSPECTIVEijrap
Understanding the concept of space and time is critical, essential, and fundamental in searching for theall-encompassing theory or the theory of everything (ToE). Some physicists argue that time exists, whileothers posit that time is only a social or mental construct. The author presents an African thought systemon space and time conception, focusing on the African (Bantu) view of space and time. The author arguesthat before the advent of the Western linear view of space and time, Africans had their own visionregarding these two concepts. Their conception of time appears to be holistic, highly philosophical, non-linear, and thought-provoking. The author hopes that exploring these two concepts from an African perspective will provide a new and more in-depth insight into reality's nature. A scientific investigation of space and time from an African-centered perspective is a worthy and necessary endeavor in the quest forthe ToE
Learning to Pronounce as Measuring Cross Lingual Joint Orthography Phonology ...ijrap
Machine learning models allow us to compare languages by showing how hard a task in each language might be to learn and perform well on. Following this line of investigation, we explore what makes a language “hard to pronounce” by modelling the task of grapheme-to-phoneme (g2p) transliteration. By training a character-level transformer model on this task across 22 languages and measuring the model’s proficiency against its grapheme and phoneme inventories, we show that certain characteristics emerge that separate easier and harder languages with respect to learning to pronounce. Namely the complexity of a language's pronunciation from its orthography is due to the expressive or simplicity of its grapheme-to phoneme mapping. Further discussion illustrates how future studies should consider relative data sparsity per language to design fairer cross-lingual comparison tasks.
THE CONCEPT OF SPACE AND TIME: AN AFRICAN PERSPECTIVEijrap
Understanding the concept of space and time is critical, essential, and fundamental in searching for the all-encompassing theory or the theory of everything (ToE). Some physicists argue that time exists, while others posit that time is only a social or mental construct. The author presents an African thought system on space and time conception, focusing on the African (Bantu) view of space and time. The author argues
that before the advent of the Western linear view of space and time, Africans had their own vision
regarding these two concepts. Their conception of time appears to be holistic, highly philosophical, nonlinear, and thought-provoking. The author hopes that exploring these two concepts from an African
perspective will provide a new and more in-depth insight into reality's nature. A scientific investigation of space and time from an African-centered perspective is a worthy and necessary endeavor in the quest for the ToE.
International Journal of Recent advances in Physics (IJRAP)ijrap
International Journal of Recent advances in Physics (IJRAP) is a peer-reviewed, open access journal, addresses the impacts and challenges of Physics. The journal documents practical and theoretical results which make a fundamental contribution for the development of Physics.
The Concept of Space and Time: An African Perspectiveijrap
Understanding the concept of space and time is critical, essential, and fundamental in searching for the all-encompassing theory or the theory of everything (ToE). Some physicists argue that time exists, while others posit that time is only a social or mental construct. The author presents an African thought system on space and time conception, focusing on the African (Bantu) view of space and time. The author argues that before the advent of the Western linear view of space and time, Africans had their own vision regarding these two concepts. Their conception of time appears to be holistic, highly philosophical, nonlinear, and thought-provoking. The author hopes that exploring these two concepts from an African perspective will provide a new and more in-depth insight into reality's nature. A scientific investigation of space and time from an African-centered perspective is a worthy and necessary endeavor in the quest for the ToE.
The majority of physicists take it for granted that the universe is made up of matter. In turn, matter is composed of atoms; atoms are made up of particles such as electrons, protons, neutrons, etc. Also, protons
and neutrons are composed of quarks, etc. Furthermore, that everything in nature is governed by the known laws of physics and chemistry. The author only partially shares this view. He argues that many phenomena in the universe may depend on rules or factors as yet incorporated by the physical sciences.
The last few years have led him to reflect on the many unsolved physics problems, such as the quest for the theory of everything (ToE), the arrow of time, the interpretation of quantum mechanics, the fine-tuned
universe, etc. to mention just a few. The author posits that a field carries information, performs various mathematical and computational operations, and behaves as an intelligent entity embedded with consciousness.
Call For Papers - International Journal of Recent advances in Physics (IJRAP)ijrap
International Journal of Recent advances in Physics (IJRAP) is a peer-reviewed, open access journal, addresses the impacts and challenges of Physics. The journal documents practical and theoretical results which make a fundamental contribution for the development of Physics.
Call For Papers - International Journal of Recent advances in Physics (IJRAP)ijrap
International Journal of Recent advances in Physics (IJRAP) is a peer-reviewed, open access journal, addresses the impacts and challenges of Physics. The journal documents practical and theoretical results which make a fundamental contribution for the development of Physics.
Call For Papers - International Journal of Recent advances in Physics (IJRAP)ijrap
International Journal of Recent advances in Physics (IJRAP) is a peer-reviewed, open access journal, addresses the impacts and challenges of Physics. The journal documents practical and theoretical results which make a fundamental contribution for the development of Physics.
Call For Papers - International Journal of Recent advances in Physics (IJRAP)ijrap
International Journal of Recent advances in Physics (IJRAP) is a peer-reviewed, open access journal, addresses the impacts and challenges of Physics. The journal documents practical and theoretical results which make a fundamental contribution for the development of Physics
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
PRESENTATION ABOUT PRINCIPLE OF COSMATIC EVALUATION
An Approach to Optimize Regimes of Manufacturing of Complementary Horizontal Field-Effect Transistor
1. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
DOI : 10.14810/ijrap.2014.3205 55
AN APPROACH TO OPTIMIZE REGIMES OF
MANUFACTURING OF COMPLEMENTARY
HORIZONTAL FIELD-EFFECT TRANSISTOR
E.L. Pankratov1
and E.A. Bulaeva 2
1
Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950,
Russia
2
Nizhny Novgorod State University of Architecture and Civil Engineering, 65 Il'insky
street, Nizhny Novgorod, 603950, Russia
ABSTRACT
In this paper we consider nonlinear model to describe manufacturing complementary horizontal field-effect
heterotransistor. Based on analytical solution of the considered boundary problems some recommendations
have been formulated to optimize technological processes.
KEYWORDS
Horizontal field-effect transistor, modelling of manufacturing of transistor, recommendations for
optimisation of manufacturing of transistor
1. INTRODUCTION
In the present time it is intensively increasing degree of integration of elements of integrated
circuits [1-8]. At the same time it is obtaining decreasing of dimensions of the elements. To
decrease dimensions of elements of integrated circuits it is traditionally using some approaches.
Two of them are laser and microwave types of annealing of dopants and/or radiation defects
during manufacturing p-n-junctions, field-effect and bipolar transistors, thyristors [9-15]. Another
way to increase degree of integration of elements of integrated circuits is using of inhomogeneity
of heterostructures on the basis of which integrated circuits are manufactured [13-19]. However
in this case it is practicably to optimize annealing of dopant and/or radiation defects. It is known,
that distribution of concentrations of dopants in elements of integrated circuits and their discrete
analogs will be changed under influence of radiation processing (for example, during ion
implantation) [20]. Because of this to decrease dimensions of elements of integrated circuits and
their discrete it is attracted an interest radiation processing of materials [21,22].
In this paper we consider manufacturing of complementary field-effect heterotransistor. Structure
of the heterotransistor is presented on the Fig. 1. The heterostructure consist of a substrate and
epitaxial layer. The epitaxial layer has several sections, which have been manufactured by using
another materials. Some dopants have been infused or implanted in the sections to manufacture
required types of conductivity (p or n). Farther we consider annealing of dopant (for doping by
diffusion) and/or radiation defects (during ion doping). Main aim of the present paper we
analyzed dynamics of redistribution of dopant and radiation defects to formulate conditions,
which correspond to manufacture more thin heterotransistor with smaller dimensions into another
dimensions.
2. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
56
Substrate
Drain Source
Source Drain
Gate Gate
p p n n
Fig.1. Heterostructure with a substrate and epitaxial layer with several sections
2. METHOD OF SOLUTION
To solve our aims we determine spatio-temporal distribution of concentration of dopant. We
determine the distributions by solving the second Fick’s law [1,3-5]
( ) ( ) ( ) ( )
+
+
=
z
t
z
y
x
C
D
z
y
t
z
y
x
C
D
y
x
t
z
y
x
C
D
x
t
t
z
y
x
C
C
C
C
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
(1)
with boundary and initial conditions
( ) 0
,
,
,
0
=
∂
∂
=
x
x
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
= x
L
x
x
t
z
y
x
C
,
( ) 0
,
,
,
0
=
∂
∂
=
y
y
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
= y
L
x
y
t
z
y
x
C
, (2)
( ) 0
,
,
,
0
=
∂
∂
=
z
z
t
z
y
x
C
,
( ) 0
,
,
,
=
∂
∂
= z
L
x
z
t
z
y
x
C
, C(x,y,z,0)=f(x,y,z).
Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant; T is the temperature
of annealing; DС is the dopant diffusion coefficient. Value of dopant diffusion coefficient depends
on properties of materials in layers of heterostructure, speed of heating and cooling of hetero-
structure (with account Arrhenius law). Dependences of dopant diffusion coefficient on
parameters could be approximated by the following relation [23-25]
( ) ( )
( )
( ) ( )
( )
+
+
+
= 2
*
2
2
*
1
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
V
t
z
y
x
V
V
t
z
y
x
V
T
z
y
x
P
t
z
y
x
C
T
z
y
x
D
D L
C ς
ς
ξ γ
γ
, (3)
where DL (x,y,z,T) is the spatial (due to inhomogeneity of heterostructure) and temperature (due to
Arrhenius law) dependences of dopant diffusion coefficient; P (x,y,z, T) is the limit of solubility
of dopant; parameter γ depends on properties of materials and could be integer in the following
interval γ ∈[1,3] [23]; V(x,y,z,t) is the spatio-temporal distribution of concentration of vacancies;
V*
is the equilibrium distribution of concentration of vacancies. Concentrational depen-dence of
dopant diffusion coefficient has been discussed in details in the Ref. [23]. It should be noted that
doping of heterostructure by diffusion did not leads to generation of radiation damage and ζ1=ζ2=
3. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
57
0. Spatio-temporal distributions of concentrations of point radiation defects we determine by
solving the following system equations [24,25]
( ) ( ) ( ) ( ) ( ) +
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
y
t
z
y
x
I
T
z
y
x
D
y
x
t
z
y
x
I
T
z
y
x
D
x
t
t
z
y
x
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
(4)
( ) ( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
I
T
z
y
x
k
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
I
T
z
y
x
D
z
I
I
V
I
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
,
, −
−
∂
∂
∂
∂
+
( ) ( ) ( ) ( ) ( ) +
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
y
t
z
y
x
V
T
z
y
x
D
y
x
t
z
y
x
V
T
z
y
x
D
x
t
t
z
y
x
V
V
V
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
V
T
z
y
x
D
z
V
V
V
I
V ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
,
, −
−
∂
∂
∂
∂
+
with initial
ρ(x,y,z,0)=fρ (x,y,z) (5a)
and boundary conditions
( ) 0
,
,
,
0
=
∂
∂
=
x
x
t
z
y
x
ρ
,
( ) 0
,
,
,
=
∂
∂
= x
L
x
x
t
z
y
x
ρ
,
( ) 0
,
,
,
0
=
∂
∂
=
y
y
t
z
y
x
ρ
,
( ) 0
,
,
,
=
∂
∂
= y
L
y
y
t
z
y
x
ρ
,
( ) 0
,
,
,
0
=
∂
∂
=
z
z
t
z
y
x
ρ
,
( ) 0
,
,
,
=
∂
∂
= z
L
z
z
t
z
y
x
ρ
. (5b)
Here ρ =I,V; I (x,y,z,t) are the spatio-temporal distributions of concentrations of radiation
interstitials and radiation vacancies; Dρ(x,y,z,T) are the diffusion coefficients of the interstitials
and vacancies; terms V2
(x,y,z,t) and I2
(x,y,z,t) correspond to generation of divacancies and
diinterstitials, respectively; kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) are parameters of
recombination of point defects and generation of their complexes, respectively.
Spatio-temporal distributions of concentrations of divacansies ΦV (x,y,z,t) and diinterstitials ΦI
(x,y,z,t) we determine by solution the following system of equations [24,25]
( ) ( ) ( ) ( ) ( ) +
Φ
+
Φ
=
Φ
Φ
Φ
y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x I
I
I
I
I
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k
z
t
z
y
x
T
z
y
x
D
z
I
I
I
I
I ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
, −
+
Φ
+ Φ
∂
∂
∂
∂
(6)
( ) ( ) ( ) ( ) ( ) +
Φ
+
Φ
=
Φ
Φ
Φ
y
t
z
y
x
T
z
y
x
D
y
x
t
z
y
x
T
z
y
x
D
x
t
t
z
y
x V
V
V
V
V
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
( ) ( ) ( ) ( ) ( ) ( )
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k
z
t
z
y
x
T
z
y
x
D
z
V
V
V
V
V ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 2
, −
+
Φ
+ Φ
∂
∂
∂
∂
with boundary and initial conditions
4. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
58
( )
0
,
,
,
0
=
∂
Φ
∂
=
x
x
t
z
y
x
ρ
,
( )
0
,
,
,
=
∂
Φ
∂
= x
L
x
x
t
z
y
x
ρ
,
( )
0
,
,
,
0
=
∂
Φ
∂
=
y
y
t
z
y
x
ρ
,
( )
0
,
,
,
=
∂
Φ
∂
= y
L
y
y
t
z
y
x
ρ
,
( )
0
,
,
,
0
=
∂
Φ
∂
=
z
z
t
z
y
x
ρ
,
( )
0
,
,
,
=
∂
Φ
∂
= z
L
z
z
t
z
y
x
ρ
, ΦI(x,y,z,0)=fΦI (x,y,z), ΦV(x,y,z,0)=fΦV (x,y,z). (7)
Here DΦI(x,y,z,T) and DΦV(x,y,z,T) are diffusion coefficients of complexes of point radiation
defects; kI(x,y,z,T) and kV (x,y,z,T) are parameters of decay of complexes of point radiation
defects.
To determine spatio-temporal distributions of concentrations of point radiation defects we used
recently elaborated approach [16,19,22]. Framework the approach we transform approximations
of diffusion coefficients of point radiation defects to the following form:
Dρ(x,y,z,T)=D0ρ[1+ερgρ(x,y,z,T)], where D0ρ are the average values of the diffusion coefficients,
0≤ερ< 1, |gρ(x,y,z,T)|≤1, ρ =I,V. We used the same transformation for approximations of
parameters of recombination of point radiation defects and generation of their complexes:
kI,V(x,y,z,T)=k0I,V[1+εI,V gI,V(x,y,z,T)], kI,I(x,y,z,T)= k0I,I[1+εI,I gI,I(x,y,z,T)] и kV,V(x,y,z,T) = k0V,V [1+
εV,V gV,V(x,y,z,T)], where k0ρ1,ρ2 are the appropriate average values, 0≤εI,V< 1, 0≤εI,I < 1, 0≤εV,V<1, |
gI,V(x,y,z,T)|≤1, | gI,I(x,y,z,T)| ≤1, |gV,V(x,y,z,T)|≤1. Let us introduce the following dimensionless
variables: χ = x/Lx, η = y/Ly, φ = z/Lz, ( ) ( ) *
,
,
,
,
,
,
~
I
t
z
y
x
I
t
z
y
x
I = , ( ) ( ) *
,
,
,
,
,
,
~
V
t
z
y
x
V
t
z
y
x
V = ,
2
0
0 L
t
D
D V
I
=
ϑ , V
I
V
I D
D
k
L 0
0
,
0
2
=
ω , V
I D
D
k
L 0
0
,
0
2
ρ
ρ
ρ =
Ω . The introduction leads to
modification of Eqs.(4) and conditions (5)
( ) ( )
[ ] ( ) ( )
[ ]
×
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
T
g
I
T
g
D
D
D
I
I
I
I
I
V
I
I
,
,
,
1
,
,
,
~
,
,
,
1
,
,
,
~
0
0
0
φ
η
χ
ε
η
χ
ϑ
φ
η
χ
φ
η
χ
ε
χ
ϑ
ϑ
φ
η
χ
( ) ( )
[ ] ( ) ( ) ×
−
∂
∂
+
∂
∂
+
∂
∂
× ϑ
φ
η
χ
η
ϑ
φ
η
χ
φ
η
χ
ε
φ
φ
ϑ
φ
η
χ
,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
0
0
0
0
0
0
I
I
T
g
D
D
D
D
D
D
I
I
I
V
I
I
V
I
I
( )
[ ] ( ) ( ) ( )
[ ]
T
g
I
V
T
g I
I
I
I
I
V
I
V
I ,
,
,
1
,
,
,
~
,
,
,
~
,
,
,
1 ,
,
2
,
, φ
η
χ
ε
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε
ω +
Ω
−
+
× (8)
( )
( )
[ ] ( )
( )
[ ]
×
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
T
g
V
T
g
D
D
D
V
V
V
V
V
V
I
V
,
,
,
1
,
,
,
~
,
,
,
1
,
,
,
~
0
0
0
φ
η
χ
ε
η
χ
ϑ
φ
η
χ
φ
η
χ
ε
χ
ϑ
ϑ
φ
η
χ
( ) ( )
[ ] ( ) ( ) ×
−
∂
∂
+
∂
∂
+
∂
∂
× ϑ
φ
η
χ
η
ϑ
φ
η
χ
φ
η
χ
ε
φ
φ
ϑ
φ
η
χ
,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
0
0
0
0
0
0
I
V
T
g
D
D
D
D
D
D
V
V
V
V
I
V
V
I
V
( )
[ ] ( ) ( ) ( )
[ ]
T
g
V
V
T
g V
V
V
V
V
V
I
V
I ,
,
,
1
,
,
,
~
,
,
,
~
,
,
,
1 ,
,
2
,
, φ
η
χ
ε
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε
ω +
Ω
−
+
×
( ) 0
,
,
,
~
0
=
∂
∂
=
χ
χ
ϑ
φ
η
χ
ρ
,
( ) 0
,
,
,
~
1
=
∂
∂
=
χ
χ
ϑ
φ
η
χ
ρ
,
( ) 0
,
,
,
~
0
=
∂
∂
=
η
η
ϑ
φ
η
χ
ρ
,
( ) 0
,
,
,
~
1
=
∂
∂
=
η
η
ϑ
φ
η
χ
ρ
,
( ) 0
,
,
,
~
0
=
∂
∂
=
φ
φ
ϑ
φ
η
χ
ρ
,
( ) 0
,
,
,
~
1
=
∂
∂
=
φ
φ
ϑ
φ
η
χ
ρ
, ( )
( )
*
,
,
,
,
,
,
~
ρ
ϑ
φ
η
χ
ϑ
φ
η
χ
ρ ρ
f
= . (9)
We determine solution of Eqs.(8) and conditions (9) by approach from Refs. [16,19,22], i.e. as the
following power series
( ) ( )
∑ ∑ ∑Ω
=
∞
=
∞
=
∞
=
0 0 0
,
,
,
~
,
,
,
~
i j k
ijk
k
j
i
ϑ
φ
η
χ
ρ
ω
ε
ϑ
φ
η
χ
ρ ρ
ρ . (10)
5. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
59
Substitution of the series (10) into Eqs.(8) and conditions (9) gives us possibility to obtain
equations for initial-order approximations of concentrations of point radiation defects
( )
ϑ
φ
η
χ ,
,
,
~
000
I and ( )
ϑ
φ
η
χ ,
,
,
~
000
V and corrections for them ( )
ϑ
φ
η
χ ,
,
,
~
ijk
I and ( )
ϑ
φ
η
χ ,
,
,
~
ijk
V , i ≥1,
j ≥1, k ≥1. The equations and conditions for them are presented in the Appendix. Solutions of
them have been obtained by standard approaches (see, for example, [26,27]). The solutions have
been obtained in the Appendix.
Farther we determine spatio-temporal distributions of concentrations of complexes of point
radiation defects. To obtain the concentrations we transform approximations of diffusion
coefficients to the following form: DΦρ(x,y,z,T)=D0Φρ[1+εΦρgΦρ(x,y,z,T)], where D0Φρ are the
average values of diffusion coefficients. After this transformation the Eqs.(6) takes the form
( ) ( )
[ ] ( )
( )
[ ] ( ) ( )
[ ]
( ) ( ) ( ) ( ) ( )
( ) ( )
[ ] ( )
( )
[ ] ( ) ( )
[ ]
( ) ( ) ( ) ( ) ( )
−
+
Φ
×
×
+
+
Φ
+
×
×
+
Φ
+
=
Φ
−
+
Φ
×
×
+
+
Φ
+
×
×
+
Φ
+
=
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
Φ
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k
D
z
t
z
y
x
T
z
y
x
g
z
y
t
z
y
x
T
z
y
x
g
y
D
x
t
z
y
x
T
z
y
x
g
x
D
t
t
z
y
x
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k
D
z
t
z
y
x
T
z
y
x
g
z
y
t
z
y
x
T
z
y
x
g
y
D
x
t
z
y
x
T
z
y
x
g
x
D
t
t
z
y
x
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
V
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
,
,
,
1
,
,
,
2
,
0
0
0
2
,
0
0
0
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂
∂
∂
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂
∂
∂
ε
∂
∂
∂
∂
We determine solutions of the above equations as the following power series
( ) ( )
∑ Φ
=
Φ
∞
=
Φ
0
,
,
,
,
,
,
i
i
i
t
z
y
x
t
z
y
x ρ
ρ
ρ ε . (11)
Substitution of the series (11) into Eqs.(6) and appropriate boundary and initial conditions gives
us possibility to obtain equations for initial-order approximations of concentrations of complexes
of point of radiation defects Φρ0(x,y,z,t), corrections for them Φρi(x,y,z,t), i ≥1, boundary and
initial conditions for all functions Φρi(x,y, z,t), i≥0. The equations and conditions are presented in
the Appendix. Solutions of the equations have been solved by standard approaches [26,27] and
presented in the Appendix.
Spatio-temporal distribution of dopant concentration we determine framework the same approach
as for determination of concentrations of radiation defects. Framework the approach we transform
approximation of dopant diffusion coefficient in the following form: DL(x,y,z,T)=
D0L[1+εLgL(x,y,z,T)], D0L is the average value of dopant diffusion coefficient, 0≤εL< 1, |gL(x,y,z,
T)|≤1. We determine solution of Eq.(1) as the following power series
6. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
60
( ) ( )
∑ ∑
=
∞
=
∞
=
0 1
,
,
,
,
,
,
i j
ij
j
i
L
t
z
y
x
C
t
z
y
x
C ξ
ε .
Substitution of the series into Eq.(1) and conditions (2) gives us possibility to obtain equations for
initial-order approximation of concentration of dopant C00(x,y,z,t), corrections for the
approximation Cij(x,y,z,t) (i≥1, j≥1), boundary and initial conditions for all functions Cij(x,y,z,t) (i
≥0, j ≥0). All these equations and conditions for them are presented in the Appendix. The above
equations have been solved by standard approaches (see, for example, [26,27]). The solutions are
presented in the Appendix.
Analysis of spatio-temporal distributions of concentrations of dopant and radiation defects have
been done analytically by using the second-order approximation by all parameters, which have
been used in considered power series. The second-order approximation is usually enough good
approximation to make qualitative analysis and to obtain some quantitative results. All analytical
results have been checked by numerical simulation.
3. DISCUSSION
In this section based on relations, which have been calculated in previous section, we analyzed
dynamics of redistribution of dopant and radiation defects during the annealing. Figs. 2 and 3
shows distributions of concentrations of dopants (for diffusive and ion types of doping) in
neighborhood of interface between layers of heterostructure under condition, when dopant
diffusion coefficient in the epitaxial layer is larger, than in the substrate. The figure shows, that
the interface gives us possibility to manufacture more thin field- effect transistors. Similar
choosing of properties of sections in the epitaxial layer gives us possibility to manufacture more
compact transistors in other directions.
Fig.2. Distributions of concentration of infused dopant in heterostructure from Figs. 1 and 2 in
direction, which is perpendicular to interface between epitaxial layer substrate. Increasing of
number of curve corresponds to increasing of difference between values of dopant diffusion
coefficient in layers of heterostructure under condition, when value of dopant diffusion
coefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate
7. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
61
x
0.0
0.5
1.0
1.5
2.0
C(x,
Θ
)
2
3
4
1
0 L/4 L/2 3L/4 L
Epitaxial layer Substrate
Fig.3. Distributions of concentration of implanted dopant in heterostructure from Figs. 1 and 2 in
direction, which is perpendicular to interface between epitaxial layer substrate. Curves 1 and 3
corresponds to annealing time Θ = 0.0048(Lx
2
+Ly
2
+Lz
2
)/D0. Curves 2 and 4 corresponds to
annealing time Θ= 0.0057(Lx
2
+Ly
2
+Lz
2
)/D0. Curves 1 and 2 corresponds to homogenous sample.
Curves 3 and 4 corresponds to heterostructure under condition, when value of dopant diffusion
coefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate
It should be noted, that interface between layers of heterostructure leads to influence on
distribution of concentration of dopant at appropriate value of annealing time: annealing time of
dopant should be neither much, no small. In this situation it should be done optimization of
annealing. The optimization of annealing has been done framework recently introduce criterion
[13-19,21,22]. By using the optimization we obtain values of optimal annealing time.
Dependences of the values are presented on the Figs. 4 and 5. It should be noted, that after ion
implantation one shall make annealing of radiation defects. One can find spreading of distribution
of concentration of dopant during the annealing. In the ideal case parameters of technological
process should be chosen so, that after finishing the annealing dopant should achieve interface
between layers of heterostructure. If after finishing of the annealing dopant did not achieved the
interface, it is attracted an interest additional annealing of dopant. In this situation additional
optimal annealing time of dopant decreases in the case of ion doping.
0.0 0.1 0.2 0.3 0.4 0.5
a/L, ξ, ε, γ
0.0
0.1
0.2
0.3
0.4
0.5
Θ
D
0
L
-2
3
2
4
1
8. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
62
Fig.4. Dependences of dimensionless optimal annealing time for doping by diffusion, which have
been obtained by minimization of mean-squared error, on several parameters. Curve 1 is the
dependence of dimensionless optimal annealing time on the relation a/L and ξ=γ =0 for equal to
each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is the
dependence of dimensionless optimal annealing time on value of parameter ε for a/L=1/2 and ξ =
γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of parameter ξ
for a/L=1/2 and ε =γ =0. Curve 4 is the dependence of dimensionless optimal annealing time on
value of parameter γ for a/L=1/2 and ε=ξ=0
0.0 0.1 0.2 0.3 0.4 0.5
a/L, ξ, ε, γ
0.00
0.04
0.08
0.12
Θ
D
0
L
-2
3
2
4
1
Fig.5. Dependences of dimensionless optimal annealing time for doping by ion implantation,
which have been obtained by minimization of mean-squared error, on several parameters. Curve 1
is the dependence of dimensionless optimal annealing time on the relation a/L and ξ = γ = 0 for
equal to each other values of dopant diffusion coefficient in all parts of heterostructure. Curve 2 is
the dependence of dimensionless optimal annealing time on value of parameter ε for a/L=1/2 and
ξ = γ = 0. Curve 3 is the dependence of dimensionless optimal annealing time on value of
parameter ξ for a/L=1/2 and ε = γ = 0. Curve 4 is the dependence of dimensionless optimal
annealing time on value of parameter γ for a/L=1/2 and ε=ξ=0
4. CONCLUSION
In this paper we introduce an approach to manufacture thinner field-effect heterotransistor with
decreasing of their dimensions into another directions.
ACKNOWLEDGEMENTS
This work is supported by the contract 11.G34.31.0066 of the Russian Federation Government,
Scientific School of Russia SSR-339.2014.2 and educational fellowship for scientific research.
REFERENCES
[1] I.P. Stepanenko. Basis of Microelectronics (Soviet Radio, Moscow, 1980).
[2] A.G. Alexenko, I.I. Shagurin. Microcircuitry (Radio and communication, Moscow, 1990).
[3] V.G. Gusev, Yu.M. Gusev. Electronics (Moscow: Vysshaya shkola, 1991, in Russian).
[4] N.A. Avaev, Yu.E. Naumov, V.T. Frolkin. Basis of microelectronics (Radio and communication,
Moscow, 1991).
9. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
63
[5] V.I. Lachin, N.S. Savelov. Electronics (Phoenix, Rostov-na-Donu, 2001).
[6] A. Kerentsev, V. Lanin. "Constructive-technological features of MOSFET-transistors" Power
Electronics. Issue 1. P. 34-38 (2008).
[7] A.N. Andronov, N.T. Bagraev, L.E. Klyachkin, S.V. Robozerov. "Ultrashallow p+−n junctions in
silicon (100): electron-beam diagnostics of sub-surface region" Semiconductors. Vol.32 (2). P. 137-
144 (1998).
[8] S.T. Shishiyanu, T.S. Shishiyanu, S.K. Railyan. "Shallow p−n junctions in Si prepared by pulse
photon annealing" Semiconductors. Vol.36 (5). P. 611-617 (2002).
[9] V.I. Mazhukin, V.V. Nosov, U. Semmler. "Study of heat and thermoelastic fields in semiconductors
at pulsed processing" Mathematical modelling. Vol. 12 (2), 75 (2000).
[10] K.K. Ong, K.L. Pey, P.S. Lee, A.T.S. Wee, X.C. Wang, Y.F. Chong. "Dopant distribution in the
recrystallization transient at the maximum melt depth induced by laser annealing" Appl. Phys. Lett.
Vol. 89 (17), 172111 (2006).
[11] J. A. Sharp, N. E. B. Cowern, R. P. Webb, K. J. Kirkby, D. Giubertoni, S. Genarro, M. Bersani, M. A.
Foad, F. Cristiano, P. F. Fazzini. "Deactivation of ultrashallow boron implants in preamorphized
silicon after nonmelt laser annealing with multiple scans" Appl.Phys. Lett. Vol. 89, 192105 (2006).
[12] Yu.V. Bykov, A.G. Yeremeev, N.A. Zharova, I.V. Plotnikov, K.I. Rybakov, M.N. Drozdov, Yu.N.
Drozdov, V.D. Skupov. "Diffusion processes in semiconductor structures during microwave
annealing" Radiophysics and Quantum Electronics. Vol. 43 (3). P. 836-843 (2003).
[13] E.L. Pankratov. "Redistribution of dopant during microwave annealing of a multilayer structure for
production p-n-junction" J. Appl. Phys. Vol. 103 (6). P. 064320-064330 (2008).
[14] E.L. Pankratov. "Optimization of near-surficial annealing for decreasing of depth of p-n-junction in
semiconductor heterostructure" Proc. of SPIE. Vol. 7521, 75211D (2010).
[15] E.L. Pankratov. "Decreasing of depth of implanted-junction rectifier in semiconductor heterostructure
by optimized laser annealing" J. Comp. Theor. Nanoscience. Vol. 7 (1). P. 289-295 (2010).
[16] E.L. Pankratov. "Influence of mechanical stress in semiconductor heterostructure on density of p-n-
junctions" Applied Nanoscience. Vol. 2 (1). P. 71-89 (2012).
[17] E.L. Pankratov, E.A. Bulaeva. "Application of native inhomogeneities to increase compactness of
vertical field -effect transistors" J. Comp. Theor. Nanoscience. Vol. 10 (4). P. 888-893 (2013).
[18] E.L. Pankratov, E.A. Bulaeva. "An approach to decrease dimensions of field-effect transistors"
Universal Journal of Materials Science. Vol. 1 (1). P.6-11 (2013).
[19] E.L. Pankratov, E.A. Bulaeva. "Doping of materials during manufacture p-n-junctions and bipolar
transistors. Analytical approaches to model technological approaches and ways of optimization of
distributions of dopants" Reviews in Theoretical Science. Vol. 1 (1). P. 58-82 (2013).
[20] V.V. Kozlivsky. Modification of semiconductors by proton beams (Nauka, Sant-Peterburg, 2003, in
Russian).
[21] E.L. Pankratov. "Decreasing of depth of p-n-junction in a semiconductor heterostructure by serial
radiation processing and microwave annealing" J. Comp. Theor. Nanoscience. Vol. 9 (1). P. 41-49
(2012).
[22] E.L. Pankratov, E.A. Bulaeva. "Increasing of sharpness of diffusion-junction heterorectifier by using
radiation processing" Int. J. Nanoscience. Vol. 11 (5). P. 1250028-1--1250028-8 (2012).
[23] Z.Yu. Gotra. Technology of microelectronic devices (Radio and communication, Moscow, 1991).
[24] P.M. Fahey, P.B. Griffin, J.D. Plummer. "Point defects and dopant diffusion in silicon" Rev. Mod.
Phys. 1989. V. 61. № 2. P. 289-388.
[25] V.L. Vinetskiy, G.A. Kholodar', Radiative physics of semiconductors. ("Naukova Dumka", Kiev,
1979, in Russian).
[26] A.N. Tikhonov, A.A. Samarskii. The mathematical physics equations (Moscow, Nauka 1972) (in
Russian).
[27] H.S. Carslaw, J.C. Jaeger. Conduction of heat in solids (Oxford University Press, 1964).
APPENDIX
Equations for the functions ( )
ϑ
φ
η
χ ,
,
,
~
ijk
I and ( )
ϑ
φ
η
χ ,
,
,
~
ijk
V , i ≥0, j ≥0, k ≥0 and conditions for
them are
10. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
64
( ) ( ) ( ) ( )
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0
000 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
( ) ( ) ( ) ( )
2
000
2
0
0
2
000
2
0
0
2
000
2
0
0
000 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V
;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
00
2
0
0
2
00
2
0
0
2
00
2
0
0
00 ,
,
,
~
,
,
,
~
,
,
,
~
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
χ i
V
I
i
V
I
i
V
I
i I
D
D
I
D
D
I
D
D
I
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+ −
−
η
ϑ
φ
η
χ
φ
η
χ
η
χ
ϑ
φ
η
χ
φ
η
χ
χ
,
,
,
~
,
,
,
,
,
,
~
,
,
, 100
0
0
100
0
0 i
I
V
I
i
I
V
I I
T
g
D
D
I
T
g
D
D
( )
( )
∂
∂
∂
∂
+ −
φ
ϑ
φ
η
χ
φ
η
χ
φ
,
,
,
~
,
,
, 100
0
0 i
I
V
I I
T
g
D
D
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
00
2
0
0
2
00
2
0
0
2
00
2
0
0
00 ,
,
,
~
,
,
,
~
,
,
,
~
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
χ i
I
V
i
I
V
i
I
V
i V
D
D
V
D
D
V
D
D
V
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+ −
−
η
ϑ
φ
η
χ
φ
η
χ
η
χ
ϑ
φ
η
χ
φ
η
χ
χ
,
,
,
~
,
,
,
,
,
,
~
,
,
, 100
0
0
100
0
0 i
V
I
V
i
V
I
V V
T
g
D
D
V
T
g
D
D
( )
( )
∂
∂
∂
∂
+ −
φ
ϑ
φ
η
χ
φ
η
χ
φ
,
,
,
~
,
,
, 100
0
0 i
V
I
V V
T
g
D
D , i≥1;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
0
010 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
I
I
D
D
I
V
I
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 000
000
,
, V
I
T
g V
I
V
I
+
−
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
010
2
2
010
2
2
010
2
0
0
010 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
V
V
D
D
V
I
V
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 000
000
,
, V
I
T
g V
I
V
I
+
− ;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
020
2
2
020
2
2
020
2
0
0
020 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
I
I
D
D
I
V
I
( )
[ ] ( ) ( ) ( ) ( )
[ ]
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
1 010
000
000
010
,
, V
I
V
I
T
g V
I
V
I +
+
−
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
020
2
2
020
2
2
020
2
0
0
020 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
V
V
D
D
V
V
I
( )
[ ] ( ) ( ) ( ) ( )
[ ]
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
1 010
000
000
010
,
, V
I
V
I
T
g V
I
V
I +
+
− ;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
0
001 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
I
I
D
D
I
V
I
( )
[ ] ( )
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
1 2
000
,
, I
T
g I
I
I
I
+
−
11. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
65
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
001
2
2
001
2
2
001
2
0
0
001 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
V
V
D
D
V
I
V
( )
[ ] ( )
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
1 2
000
,
, V
T
g I
I
I
I
+
− ;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
110
2
0
0
2
110
2
0
0
2
110
2
0
0
110 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
η
ϑ
φ
η
χ
φ
η
χ
η
χ
ϑ
φ
η
χ
φ
η
χ
χ
,
,
,
~
,
,
,
,
,
,
~
,
,
, 010
0
0
010
0
0 I
T
g
D
D
I
T
g
D
D
I
V
I
I
V
I
( )
( )
( )
[ ]×
+
−
∂
∂
∂
∂
+ T
g
I
T
g
D
D
I
I
I
I
I
V
I
,
,
,
1
,
,
,
~
,
,
, ,
,
010
0
0
φ
η
χ
ε
φ
ϑ
φ
η
χ
φ
η
χ
φ
( ) ( ) ( ) ( )
[ ]
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
100
000
000
100 V
I
V
I +
×
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
110
2
0
0
2
110
2
0
0
2
110
2
0
0
110 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
η
ϑ
φ
η
χ
φ
η
χ
η
χ
ϑ
φ
η
χ
φ
η
χ
χ
,
,
,
~
,
,
,
,
,
,
~
,
,
, 010
0
0
010
0
0 V
T
g
D
D
V
T
g
D
D
V
I
V
V
I
V
( )
( )
( )
[ ]×
+
−
∂
∂
∂
∂
+ T
g
V
T
g
D
D
V
V
V
V
V
I
V
,
,
,
1
,
,
,
~
,
,
, ,
,
010
0
0
φ
η
χ
ε
φ
ϑ
φ
η
χ
φ
η
χ
φ
( ) ( ) ( ) ( )
[ ]
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ
ϑ
φ
η
χ ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
100
000
000
100 I
V
I
V +
× ;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
0
0
2
002
2
0
0
2
002
2
0
0
002 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
, I
I
T
g I
I
I
I
+
−
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
002
2
0
0
2
002
2
0
0
2
002
2
0
0
002 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
, V
V
Е
g V
V
V
V
+
− ;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
101
2
0
0
2
101
2
0
0
2
101
2
0
0
101 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
η
ϑ
φ
η
χ
φ
η
χ
η
χ
ϑ
φ
η
χ
φ
η
χ
χ
,
,
,
~
,
,
,
,
,
,
~
,
,
, 001
0
0
001
0
0 I
T
g
D
D
I
T
g
D
D
I
V
I
I
V
I
( )
( )
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε
φ
ϑ
φ
η
χ
φ
η
χ
φ
,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
, 000
100
001
0
0
V
I
T
g
I
T
g
D
D
I
I
I
V
I
+
−
∂
∂
∂
∂
+
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
101
2
0
0
2
101
2
0
0
2
101
2
0
0
101 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V
12. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
66
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
η
ϑ
φ
η
χ
φ
η
χ
η
χ
ϑ
φ
η
χ
φ
η
χ
χ
,
,
,
~
,
,
,
,
,
,
~
,
,
, 001
0
0
001
0
0 V
T
g
D
D
V
T
g
D
D
V
I
V
V
I
V
( )
( )
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε
φ
ϑ
φ
η
χ
φ
η
χ
φ
,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
, 100
000
001
0
0
V
I
T
g
V
T
g
D
D
V
V
V
I
V
+
−
∂
∂
∂
∂
+ ;
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
011
2
0
0
2
011
2
0
0
2
011
2
0
0
011 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ I
D
D
I
D
D
I
D
D
I
V
I
V
I
V
I
( )
[ ] ( ) ( )−
+
− ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 010
000
,
, I
I
T
g I
I
I
I
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
, V
I
T
g V
I
V
I
+
−
( ) ( ) ( ) ( )
−
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
011
2
0
0
2
011
2
0
0
2
011
2
0
0
011 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
φ
ϑ
φ
η
χ
η
ϑ
φ
η
χ
χ
ϑ
φ
η
χ
ϑ
ϑ
φ
η
χ V
D
D
V
D
D
V
D
D
V
I
V
I
V
I
V
( )
[ ] ( ) ( )−
+
− ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 010
000
,
, V
V
T
g V
V
V
V
( )
[ ] ( ) ( )
ϑ
φ
η
χ
ϑ
φ
η
χ
φ
η
χ
ε ,
,
,
~
,
,
,
~
,
,
,
1 001
000
,
, V
I
t
g V
I
V
I
+
− ;
( )
0
,
,
,
~
0
=
∂
∂
=
x
ijk
χ
ϑ
φ
η
χ
ρ
,
( )
0
,
,
,
~
1
=
∂
∂
=
x
ijk
χ
ϑ
φ
η
χ
ρ
,
( )
0
,
,
,
~
0
=
∂
∂
=
η
η
ϑ
φ
η
χ
ρijk
,
( )
0
,
,
,
~
1
=
∂
∂
=
η
η
ϑ
φ
η
χ
ρijk
,
( )
0
,
,
,
~
0
=
∂
∂
=
φ
φ
ϑ
φ
η
χ
ρijk
,
( )
0
,
,
,
~
1
=
∂
∂
=
φ
φ
ϑ
φ
η
χ
ρijk
(i≥0, j≥0, k≥0);
( ) ( ) *
000 ,
,
0
,
,
,
~ ρ
φ
η
χ
φ
η
χ
ρ ρ
f
= , ( ) 0
0
,
,
,
~ =
φ
η
χ
ρijk (i≥1, j≥1, k≥1).
Solutions of these equations with account boundary and initial conditions could be written as
( ) ( ) ( ) ( ) ( )
∑
+
=
∞
=1
000
2
1
,
,
,
~
n
n
n e
c
c
c
F
L
L
ϑ
φ
η
χ
ϑ
φ
η
χ
ρ ρ
ρ ,
where ( ) ( ) ( ) ( )
∫ ∫ ∫
=
1
0
1
0
1
0
*
,
,
cos
cos
cos
1
u
d
v
d
w
d
w
v
u
f
w
n
v
n
u
n
F n
n ρ
ρ π
π
π
ρ
, cn(χ) = cos (π n χ), ( )=
ϑ
nI
e
( )
I
V D
D
n 0
0
2
2
exp ϑ
π
−
= , ( ) ( )
V
I
nV D
D
n
e 0
0
2
2
exp ϑ
π
ϑ −
= ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
∂
∂
−
−
=
∞
=
−
1 0
1
0
1
0
1
0
100
0
0
00
,
,
,
~
2
,
,
,
~
n
i
n
n
nI
nI
n
V
I
i
u
w
v
u
I
v
c
u
s
e
e
c
c
c
n
D
D
I
ϑ τ
τ
ϑ
φ
η
χ
π
ϑ
φ
η
χ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫ ∫
−
−
×
∞
=1 0
1
0
1
0
0
0
2
,
,
,
n
n
n
nI
nI
n
V
I
I
n v
s
u
c
e
e
c
c
c
D
D
d
u
d
v
d
w
d
T
w
v
u
g
w
c
ϑ
τ
ϑ
φ
η
χ
π
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑
−
∫
∂
∂
×
∞
=
−
1
0
0
1
0
100
2
,
,
,
~
,
,
,
n
nI
n
V
I
i
I
n e
c
c
c
n
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
w
c
n ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( )
∫ ∫ ∫ ∫
∂
∂
−
× −
ϑ
τ
τ
τ
0
1
0
1
0
1
0
100 ,
,
,
~
,
,
, d
u
d
v
d
w
d
w
w
v
u
I
T
w
v
u
g
w
s
v
c
u
c
e i
I
n
n
n
nI
13. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
67
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∑ ×
∫ ∫ ∫ ∫
∂
∂
−
−
=
∞
=
−
1 0
1
0
1
0
1
0
100
0
0
00
,
~
2
,
,
,
~
n
i
n
n
n
nI
nV
n
I
V
i
u
u
V
w
c
v
c
u
s
e
e
c
c
c
n
D
D
V
ϑ τ
τ
ϑ
φ
η
χ
π
ϑ
φ
η
χ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∑ ×
∫ ∫ ∫ ∫
−
−
×
∞
=1 0
1
0
1
0
1
0
0
0
2
,
,
,
n
n
n
n
nI
nV
n
I
V
V w
c
v
s
u
c
e
e
c
c
c
D
D
d
u
d
v
d
w
d
T
w
v
u
g
ϑ
τ
ϑ
φ
η
χ
π
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫
−
−
∂
∂
×
∞
=
−
1 0
1
0
0
0
100
2
,
~
,
,
,
n
n
nI
nV
n
I
V
i
V u
c
e
e
c
c
c
D
D
d
u
d
v
d
w
d
v
u
V
T
w
v
u
g
n
ϑ
τ
ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( )
( )
∫ ∫
∂
∂
× −
1
0
1
0
100 ,
~
,
,
, τ
τ
d
u
d
v
d
w
d
w
u
V
T
w
v
u
g
w
s
v
c
n i
V
n
n , i ≥1,
where sn(χ)=sin(πnχ);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∑ ∫ ∫ ∫ ∫ ×
−
−
=
∞
=1 0
1
0
1
0
1
0
010 2
,
,
,
~
n
n
n
n
n
n
n
n
n w
c
v
c
u
c
e
e
c
c
c
ϑ
ρ
ρ τ
ϑ
φ
η
χ
ϑ
φ
η
χ
ρ
( )
[ ] ( ) ( ) τ
τ
τ
ε d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
T
w
v
u
g V
I
V
I ,
,
,
~
,
,
,
~
,
,
,
1 000
000
,
,
+
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]×
∑ ∫ ∫ ∫ ∫ +
−
−
=
∞
=1 0
1
0
1
0
1
0
,
,
0
0
020 ,
,
,
1
2
,
,
,
~
n
V
I
V
I
n
n
n
n
n
V
I
T
w
v
u
g
e
e
c
c
c
D
D ϑ
ρ
ρ ε
τ
ϑ
φ
η
χ
ϑ
φ
η
χ
ρ
( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ] τ
τ
τ
τ
τ d
u
d
v
d
w
d
w
v
u
V
w
v
u
I
w
v
u
V
w
v
u
I
w
c
v
c
u
c n
n
n ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
010
000
000
010 +
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0
1
0
1
0
1
0
001 2
,
,
,
~
n
n
n
n
n
n
n
n
n w
c
v
c
u
c
e
e
c
c
c
ϑ
ρ
ρ τ
ϑ
φ
η
χ
ϑ
φ
η
χ
ρ
( )
[ ] ( ) τ
τ
ρ
ε ρ
ρ
ρ
ρ d
u
d
v
d
w
d
w
v
u
T
w
v
u
g ,
,
,
~
,
,
,
1 2
000
,
,
+
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∑ ∫ ∫ ×
∫ ∫
−
−
=
∞
=1 0
1
0
1
0
1
0
002 2
,
,
,
~
n
n
n
n
n
n
n
n
n w
c
v
c
u
c
e
e
c
c
c
ϑ
ρ
ρ τ
ϑ
φ
η
χ
ϑ
φ
η
χ
ρ
( )
[ ] ( ) ( ) τ
τ
ρ
τ
ρ
ε ρ
ρ
ρ
ρ d
u
d
v
d
w
d
w
v
u
w
v
u
T
w
v
u
g ,
,
,
~
,
,
,
~
,
,
,
1 000
001
,
,
+
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0
1
0
1
0
1
0
0
0
110 ,
,
,
2
,
,
,
~
n
I
n
n
n
nI
nI
n
n
n
V
I
T
w
v
u
g
u
c
v
c
u
s
e
e
c
c
c
D
D
I
ϑ
τ
ϑ
φ
η
χ
π
ϑ
φ
η
χ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫ ∫
−
−
∂
∂
×
∞
=
−
1 0
1
0
1
0
0
0
100
2
,
,
,
~
n
n
n
nI
nI
n
n
n
V
I
i
v
s
u
c
e
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
I
n
ϑ
τ
ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
×
−
∫
∂
∂
×
∞
=
−
1
0
0
1
0
100
2
,
,
,
~
,
,
,
n
nI
n
n
n
V
I
i
I
n e
c
c
c
n
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
u
c ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
−
∫ ∫ ∫ ∫
∂
∂
−
×
∞
=
−
1
0
1
0
1
0
1
0
100
2
,
,
,
~
,
,
,
n
n
n
n
i
I
n
n
n
nI c
c
c
d
u
d
v
d
w
d
w
w
v
u
I
T
w
v
u
g
u
s
v
c
u
c
e φ
η
χ
τ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]
∫ ∫ ∫ ∫ ×
+
−
×
ϑ
τ
τ
τ
τ
τ
ϑ
0
1
0
1
0
1
0
100
000
000
100 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
w
v
u
V
w
v
u
I
w
v
u
V
w
v
u
I
v
c
u
c
e
e n
n
nI
nI
( )
[ ] ( ) τ
ε d
u
d
v
d
w
d
w
c
T
w
v
u
g n
V
I
V
I ,
,
,
1 ,
,
+
×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0
1
0
1
0
1
0
0
0
110 ,
,
,
2
,
,
,
~
n
V
n
n
nV
nV
n
n
n
I
V
T
w
v
u
g
v
c
u
s
e
e
c
c
c
n
D
D
V
ϑ
τ
ϑ
φ
η
χ
π
ϑ
φ
η
χ
14. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
68
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
∂
∂
×
∞
=
−
1 0
1
0
0
0
100
2
,
,
,
~
n
n
nV
nV
n
n
n
I
V
i
n u
c
e
e
c
c
c
D
D
d
u
d
v
d
w
d
u
w
v
u
V
u
c
ϑ
τ
ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ −
−
∫ ∫
∂
∂
×
∞
=
−
1 0
0
0
1
0
1
0
100
2
,
,
,
~
,
,
,
n
nV
nV
I
V
i
V
n
n e
e
D
D
d
u
d
v
d
w
d
v
w
v
u
V
T
w
v
u
g
u
c
v
s
n
ϑ
τ
ϑ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
∑ ×
−
∫ ∫ ∫
∂
∂
×
∞
=
−
1
1
0
1
0
1
0
100
2
,
,
,
~
,
,
,
n
n
i
V
n
n
n
n
n
n c
d
u
d
v
d
w
d
w
w
v
u
V
T
w
v
u
g
u
s
v
c
u
c
c
c
c
n χ
τ
τ
φ
η
χ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]
∫ ∫ ∫ ∫ ×
+
−
×
ϑ
τ
τ
τ
τ
τ
ϑ
φ
η
0
1
0
1
0
1
0
100
000
000
100 ,
,
,
~
,
,
,
~
,
,
,
~
,
,
,
~
w
v
u
V
w
v
u
I
w
v
u
V
w
v
u
I
u
c
e
e
c
c n
nV
nI
n
n
( ) ( ) τ
d
u
d
v
d
v
c
w
d
w
c n
n
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0
1
0
1
0
1
0
0
0
101 ,
,
,
2
,
,
,
~
n
I
n
n
n
nI
nI
n
n
n
V
I
T
w
v
u
g
w
c
v
c
u
s
e
e
c
c
c
n
D
D
I
ϑ
τ
ϑ
φ
η
χ
π
ϑ
φ
η
χ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫
−
−
∂
∂
×
∞
=1 0
1
0
1
0
0
0
001
2
,
,
,
~
n
n
n
nI
nI
n
n
n
V
I
v
s
u
c
e
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
I ϑ
τ
ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
−
∫
∂
∂
×
∞
=1
0
0
1
0
001
2
,
,
,
~
,
,
,
n
nI
n
n
n
V
I
I
n e
c
c
c
n
D
D
d
u
d
v
d
w
d
v
w
v
u
I
T
w
v
u
g
w
c ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
−
∫ ∫ ∫ ∫
∂
∂
−
×
∞
=1
0
1
0
1
0
1
0
001
2
,
,
,
~
,
,
,
n
n
n
I
n
n
n
nI c
c
d
u
d
v
d
w
d
w
w
v
u
I
T
w
v
u
g
w
s
v
c
u
c
e η
χ
τ
τ
τ
ϑ
( ) ( ) ( ) ( ) ( ) ( )
[ ] ( ) ( ) ×
∫ ∫ ∫ ∫ +
−
×
ϑ
τ
τ
ε
τ
ϑ
φ
0
1
0
1
0
1
0
000
100
,
, ,
,
,
~
,
,
,
~
,
,
,
1 w
v
u
V
w
v
u
I
T
w
v
u
g
v
c
u
c
e
e
c V
I
V
I
n
n
nI
nI
n
( ) τ
d
u
d
v
d
w
d
w
cn
×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0
1
0
1
0
1
0
0
0
101 ,
,
,
2
,
,
,
~
n
V
n
n
nV
nV
n
n
n
I
V
T
w
v
u
g
v
c
u
s
e
e
c
c
c
n
D
D
V
ϑ
τ
ϑ
φ
η
χ
π
ϑ
φ
η
χ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
∂
∂
×
∞
=1 0
1
0
0
0
001
2
,
,
,
~
n
n
nV
nV
n
n
n
I
V
u
c
e
e
c
c
c
n
D
D
d
u
d
v
d
w
d
u
w
v
u
V
w
c
ϑ
τ
ϑ
φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
−
∫ ∫
∂
∂
×
∞
=1
0
0
1
0
1
0
001
2
,
,
,
~
,
,
,
n
n
n
n
I
V
V
n
n c
c
c
n
D
D
d
u
d
v
d
w
d
v
w
v
u
V
T
w
v
u
g
w
c
v
s φ
η
χ
π
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑
−
∫ ∫ ∫ ∫
∂
∂
−
×
∞
=1
0
1
0
1
0
1
0
001
2
,
,
,
~
,
,
,
n
n
V
n
n
n
nV
nV c
d
u
d
v
d
w
d
w
w
v
u
V
T
w
v
u
g
w
s
v
c
u
c
e
e χ
τ
τ
τ
ϑ
ϑ
( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ] ( ) ( ) ×
∫ ∫ ∫ ∫ +
−
×
ϑ
τ
τ
ε
τ
ϑ
φ
η
0
1
0
1
0
1
0
100
000
,
, ,
,
,
~
,
,
,
~
,
,
,
1 w
v
u
V
w
v
u
I
T
w
v
u
g
v
c
u
c
e
e
c
c V
I
V
I
n
n
nV
nV
n
n
( ) τ
d
u
d
v
d
w
d
w
cn
× ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]
{
∑ ∫ ∫ ∫ ∫ ×
+
−
−
=
∞
=1 0
1
0
1
0
1
0
,
,
011 ,
,
,
1
2
,
,
,
~
n
I
I
I
I
n
n
nI
nI
n
n
n T
w
v
u
g
v
c
u
c
e
e
c
c
c
I
ϑ
ε
τ
ϑ
φ
η
χ
ϑ
φ
η
χ
( ) ( ) ( )
[ ] ( ) ( )}×
+
+
× τ
τ
ε
τ
τ ,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
~
000
001
,
,
010
000 w
v
u
V
w
v
u
I
T
w
v
u
g
w
v
u
I
w
v
u
I V
I
V
I
( ) τ
d
u
d
v
d
w
d
w
cn
×
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]
{
∑ ∫ ∫ ∫ ∫ ×
+
−
−
=
∞
=1 0
1
0
1
0
1
0
,
,
011 ,
,
,
1
2
,
,
,
~
n
V
V
V
V
n
n
nV
nV
n
n
n T
w
v
u
g
v
c
u
c
e
e
c
c
c
V
ϑ
ε
τ
ϑ
φ
η
χ
ϑ
φ
η
χ
15. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
69
( ) ( ) ( )
[ ] ( ) ( )}×
+
+
× τ
τ
ε
τ
τ ,
,
,
~
,
,
,
~
,
,
,
1
,
,
,
~
,
,
,
~
001
000
,
,
010
000 w
v
u
V
w
v
u
I
T
w
v
u
g
w
v
u
V
w
v
u
V V
I
V
I
( ) τ
d
u
d
v
d
w
d
w
cn
× .
Inutial-order approximations of distributions of concentrations of complexes of radiation defects
Φρ0(x,y,z,t), corrections for the approximations Φρi(x,y,z,t) i≥1, boundary and initial conditions for
them
( ) ( ) ( ) ( )+
Φ
+
Φ
+
Φ
=
Φ
Φ
Φ
Φ 2
0
2
0
2
0
2
0
2
0
2
0
0 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
D
y
t
z
y
x
D
x
t
z
y
x
D
t
t
z
y
x I
I
I
I
I
I
I
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( )
t
z
y
x
I
T
z
y
x
k
t
z
y
x
I
T
z
y
x
k I
I
I ,
,
,
,
,
,
,
,
,
,
,
, 2
, −
+
( ) ( ) ( ) ( )+
Φ
+
Φ
+
Φ
=
Φ
Φ
Φ
Φ 2
0
2
0
2
0
2
0
2
0
2
0
0 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
D
y
t
z
y
x
D
x
t
z
y
x
D
t
t
z
y
x V
V
V
V
V
V
V
∂
∂
∂
∂
∂
∂
∂
∂
( ) ( ) ( ) ( )
t
z
y
x
V
T
z
y
x
k
t
z
y
x
V
T
z
y
x
k V
V
V ,
,
,
,
,
,
,
,
,
,
,
, 2
, −
+ ;
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ
Φ
Φ 2
2
0
2
2
0
2
2
0
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
D
y
t
z
y
x
D
x
t
z
y
x
D
t
t
z
y
x i
I
I
i
I
I
i
I
I
i
I
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )
+
Φ
+
Φ
+
−
Φ
Φ
−
Φ
Φ
y
t
z
y
x
T
z
y
x
g
y
D
x
t
z
y
x
T
z
y
x
g
x
D
i
I
I
I
i
I
I
I
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
1
0
1
0
( )
( )
Φ
+
−
Φ
Φ
z
t
z
y
x
T
z
y
x
g
z
D
i
I
I
I
∂
∂
∂
∂ ,
,
,
,
,
,
1
0
( ) ( ) ( ) ( )
+
Φ
+
Φ
+
Φ
=
Φ
Φ
Φ
Φ 2
2
0
2
2
0
2
2
0
,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
D
y
t
z
y
x
D
x
t
z
y
x
D
t
t
z
y
x i
V
V
i
V
V
i
V
V
i
V
∂
∂
∂
∂
∂
∂
∂
∂
( )
( )
( )
( )
+
Φ
+
Φ
+
−
Φ
Φ
−
Φ
Φ
y
t
z
y
x
T
z
y
x
g
y
D
x
t
z
y
x
T
z
y
x
g
x
D
i
V
V
V
i
V
V
V
∂
∂
∂
∂
∂
∂
∂
∂ ,
,
,
,
,
,
,
,
,
,
,
,
1
0
1
0
( )
( )
Φ
+
−
Φ
Φ
z
t
z
y
x
T
z
y
x
g
z
D
i
V
V
V
∂
∂
∂
∂ ,
,
,
,
,
,
1
0 , i≥1;
( )
0
,
,
,
0
=
∂
Φ
∂
=
x
i
x
t
z
y
x
ρ
,
( )
0
,
,
,
=
∂
Φ
∂
= x
L
x
i
x
t
z
y
x
ρ
,
( )
0
,
,
,
0
=
∂
Φ
∂
=
y
i
y
t
z
y
x
ρ
,
( )
0
,
,
,
=
∂
Φ
∂
= y
L
y
i
y
t
z
y
x
ρ
,
( )
0
,
,
,
0
=
∂
Φ
∂
=
z
i
z
t
z
y
x
ρ
,
( )
0
,
,
,
=
∂
Φ
∂
= z
L
z
i
z
t
z
y
x
ρ
, i≥0; Φρ0(x,y,z,0)=fΦρ (x,y,z), Φρi(x,y,z,0)=0, i≥1.
Solutions of the above equations could be written as
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑
+
∑
+
=
Φ
∞
=
∞
=
Φ
Φ
1
1
0
2
2
1
,
,
,
n
n
n
n
n
n
n
n
n
n
z
y
x
z
y
x
z
c
y
c
x
c
n
L
t
e
z
c
y
c
x
c
F
L
L
L
L
L
L
t
z
y
x ρ
ρ
ρ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]
∫ ∫ ∫ ∫ ×
−
−
× Φ
Φ
t L L L
I
I
I
n
n
n
n
x y z
w
v
u
I
T
w
v
u
k
w
v
u
I
T
w
v
u
k
v
c
u
c
e
t
e
0 0 0 0
2
, ,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
ρ
ρ
( ) τ
d
u
d
v
d
w
d
w
cn
× ,
16. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
70
where ( ) ( ) ( ) ( )
∫ ∫ ∫
= Φ
Φ
x y z
L L L
n
n
n
n u
d
v
d
w
d
w
v
u
f
w
c
v
c
u
c
F
0 0 0
,
,
ρ
ρ
, ( )
+
+
−
= Φ
Φ 2
2
2
0
2
2 1
1
1
exp
z
y
x
n
L
L
L
t
D
n
t
e ρ
ρ
π ,
cn(x)=cos(πnx/Lx);
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )×
∑ ∫ ∫ ∫ ∫
−
−
=
Φ
∞
=
Φ
Φ
Φ
1 0 0 0 0
2
,
,
,
2
,
,
,
n
t L L L
n
n
n
n
n
n
n
z
y
x
i
x y z
T
w
v
u
g
v
c
u
s
e
t
e
z
c
y
c
x
c
n
L
L
L
t
z
y
x ρ
ρ
ρ
τ
π
ρ
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
Φ
×
∞
=
Φ
Φ
−
1 0 0
2
1 2
,
,
,
n
t L
n
n
n
n
n
n
z
y
x
i
I
n
x
u
c
e
t
e
z
c
y
c
x
c
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
w
c τ
π
τ
∂
τ
∂
ρ
ρ
ρ
( ) ( ) ( )
( )
( ) ( ) ( ) ×
∑
−
∫ ∫
Φ
×
∞
=
−
Φ
1
2
0 0
1 2
,
,
,
,
,
,
n
n
n
n
z
y
x
L L
i
I
n
n z
c
y
c
x
c
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
T
w
v
u
g
w
c
v
s
y z π
τ
∂
τ
∂ ρ
ρ
( ) ( ) ( ) ( ) ( )
( )
( )
∫ ∫ ∫ ∫
Φ
−
× Φ
−
Φ
Φ
t L L L
i
I
n
n
n
n
n
x y z
d
u
d
v
d
w
d
T
w
v
u
g
w
w
v
u
w
s
v
c
u
c
e
t
e
0 0 0 0
1
,
,
,
,
,
,
τ
∂
τ
∂
τ ρ
ρ
ρ
ρ
, i≥1,
where sn(x)=sin(πnx/Lx).
Equation for initial-order approximation of dopant concentration C00(x,t), corrections for the
approximation Cij(x,y,z,t) (i ≥1, j ≥1), boundary and initial conditions of the above functions are
( ) ( ) ( ) ( )
2
00
2
0
2
00
2
0
2
00
2
0
00 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
D
y
t
z
y
x
C
D
x
t
z
y
x
C
D
t
t
z
y
x
C
L
L
L
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
;
( ) ( ) ( ) ( )
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
2
0
2
2
0
2
2
0
2
0
0 ,
,
,
,
,
,
,
,
,
,
,
,
z
t
z
y
x
C
y
t
z
y
x
C
x
t
z
y
x
C
D
t
t
z
y
x
C i
i
i
L
i
( ) ( ) ( ) ( ) +
∂
∂
∂
∂
+
∂
∂
∂
∂
+ −
−
y
t
z
y
x
C
T
z
y
x
g
y
D
x
t
z
y
x
C
T
z
y
x
g
x
D i
L
L
i
L
L
,
,
,
,
,
,
,
,
,
,
,
, 10
0
10
0
( ) ( )
∂
∂
∂
∂
+ −
z
t
z
y
x
C
T
z
y
x
g
z
D i
L
L
,
,
,
,
,
, 10
0 , i≥1;
( ) ( ) ( ) ( ) ( )
( )
×
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
T
z
y
x
P
t
z
y
x
C
x
z
t
z
y
x
C
y
t
z
y
x
C
x
t
z
y
x
C
D
t
t
z
y
x
C
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 00
2
01
2
2
01
2
2
01
2
0
01
γ
γ
( ) ( )
( )
( ) ( )
( )
( )
L
D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
z
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
y
x
t
z
y
x
C
0
00
00
00
00
00 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
× γ
γ
γ
γ
;
( ) ( ) ( ) ( ) ( )
( )
×
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ −
T
z
y
x
P
t
z
y
x
C
z
t
z
y
x
C
y
t
z
y
x
C
x
t
z
y
x
C
D
t
t
z
y
x
C
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 1
00
2
02
2
2
02
2
2
02
2
0
02
γ
γ
( ) ( ) ( ) ( )
( )
( ) +
∂
∂
∂
∂
+
∂
∂
×
−
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
y
x
t
z
y
x
C
t
z
y
x
C
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 00
1
00
01
00
01 γ
γ
( ) ( )
( )
( ) ( )
( )
( )
×
∂
∂
∂
∂
+
∂
∂
∂
∂
+
−
x
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
x
D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
z
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 01
00
0
00
1
00
01 γ
γ
γ
γ
17. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
71
( )
( )
( ) ( )
( )
( )
L
D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
z
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
y
0
01
00
01
00 ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
∂
∂
∂
∂
+
∂
∂
∂
∂
+ γ
γ
γ
γ
;
( ) ( ) ( ) ( ) ( )
( )
×
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂ −
T
z
y
x
P
t
z
y
x
C
x
z
t
z
y
x
C
y
t
z
y
x
C
x
t
z
y
x
C
D
t
t
z
y
x
C
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 1
00
2
11
2
2
11
2
2
11
2
0
11
γ
γ
( )
( )
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
×
−
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
y
x
t
z
y
x
C
t
z
y
x
C
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 00
1
00
10
00
10 γ
γ
( )
( )
( )
( ) ( )
×
∂
∂
∂
∂
+
∂
∂
∂
∂
+
−
x
t
z
y
x
C
x
D
D
z
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
t
z
y
x
C
z
L
L
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 10
0
0
00
1
00
10 γ
γ
( )
( )
( )
( )
( ) ( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
×
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
z
y
t
z
y
x
C
T
z
y
x
P
t
z
y
x
C
y
T
z
y
x
P
t
z
y
x
C ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, 10
00
10
00
00
γ
γ
γ
γ
γ
γ
( )
( )
( )
( )
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
y
t
z
y
x
C
T
z
y
x
g
y
x
t
z
y
x
C
T
z
y
x
g
x
D L
L
L
,
,
,
,
,
,
,
,
,
,
,
, 01
01
0
( )
( )
∂
∂
∂
∂
+
z
t
z
y
x
C
T
z
y
x
g
z
L
,
,
,
,
,
, 01
;
( )
0
,
,
,
0
=
=
x
ij
x
t
z
y
x
C
∂
∂
,
( )
0
,
,
,
=
= x
L
x
ij
x
t
z
y
x
C
∂
∂
,
( )
0
,
,
,
0
=
=
y
ij
y
t
z
y
x
C
∂
∂
,
( )
0
,
,
,
=
= y
L
y
ij
y
t
z
y
x
C
∂
∂
,
( )
0
,
,
,
0
=
=
z
ij
z
t
z
y
x
C
∂
∂
,
( )
0
,
,
,
=
= z
L
z
ij
z
t
z
y
x
C
∂
∂
, i≥0, j≥0;
C00(x,y,z,0)=fC (x,y,z), Cij(x,y,z,0)=0, i≥1, j≥1.
Solutions of the above equations with account boundary and initial conditions could be written as
( ) ( ) ( ) ( ) ( )
∑
+
=
∞
=1
00
2
1
,
,
,
n
nC
n
n
n
nC
z
y
x
z
y
x
t
e
z
c
y
c
x
c
F
L
L
L
L
L
L
t
z
y
x
C ,
where ( )
+
+
−
= 2
2
2
0
2
2 1
1
1
exp
z
y
x
C
nC
L
L
L
t
D
n
t
e π , ( ) ( ) ( ) ( )
∫ ∫ ∫
=
x y z
L L L
C
n
n
n
nC u
d
v
d
w
d
w
v
u
f
w
c
v
c
u
c
F
0 0 0
,
, ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0 0 0 0
2
0 ,
,
,
2
,
,
,
n
t L L L
L
n
n
nC
nC
n
n
n
nC
z
y
x
i
x y z
T
w
v
u
g
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C τ
π
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
∂
∂
×
∞
=
−
1 0 0
2
10 2
,
,
,
n
t L
n
nC
nC
n
n
n
nC
z
y
x
i
n
x
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
w
c τ
π
τ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ×
∑
−
∫ ∫
∂
∂
×
∞
=
−
1
2
0 0
10 2
,
,
,
,
,
,
n
n
n
n
nC
z
y
x
L L
i
L
n
n z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
g
v
c
v
s
y z π
τ
τ
( ) ( ) ( ) ( ) ( ) ( )
( )
∫ ∫ ∫ ∫
∂
∂
−
× −
t L L L
i
L
n
n
n
nC
nC
x y z
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
g
v
s
v
c
u
c
e
t
e
0 0 0 0
10 ,
,
,
,
,
, τ
τ
τ , i≥1;
18. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
72
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0 0 0 0
00
2
01
,
,
,
,
,
,
2
,
,
,
n
t L L L
n
n
nC
nC
n
n
n
nC
z
y
x
x y z
T
w
v
u
P
w
v
u
C
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C γ
γ
τ
τ
π
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
∂
∂
×
∞
=1 0 0
2
00 2
,
,
,
n
t L
n
nC
nC
n
n
n
nC
z
y
x
n
x
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
w
c τ
π
τ
τ
( ) ( )
( )
( )
( )
( ) ( ) ( ) ×
∑
−
∫ ∫
∂
∂
×
∞
=1
2
0 0
00
00 2
,
,
,
,
,
,
,
,
,
n
n
n
n
nC
z
y
x
L L
n
n z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
v
s
y z π
τ
τ
τ
γ
γ
( ) ( ) ( ) ( ) ( )
( )
( )
( )
∫ ∫ ∫ ∫
∂
∂
−
×
t L L L
n
n
n
nC
nC
x y z
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
s
v
c
u
c
e
t
e
0 0 0 0
00
00 ,
,
,
,
,
,
,
,
,
τ
τ
τ
τ γ
γ
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0 0 0 0
01
2
02 ,
,
,
2
,
,
,
n
t L L L
n
n
nC
nC
n
n
n
nC
z
y
x
x y z
w
v
u
C
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C τ
τ
π
( )
( )
( )
( )
( ) ( ) ( ) ( ) ×
∑
−
∂
∂
×
∞
=
−
1
2
00
1
00 2
,
,
,
,
,
,
,
,
,
n
nC
n
n
n
nC
z
y
x
n t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
π
τ
τ
τ
γ
γ
( ) ( ) ( ) ( ) ( )
( )
( )
( )
−
∫ ∫ ∫ ∫
∂
∂
−
×
−
t L L L
n
n
n
nC
x y z
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
w
c
v
s
u
c
e
0 0 0 0
00
1
00
01
,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
τ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
×
∑ ∫ ∫ ∫ ∫
∂
∂
−
−
∞
=1 0 0 0 0
00
2
,
,
,
2
n
t L L L
n
n
n
nC
nC
n
n
n
nC
z
y
x
x y z
w
w
v
u
C
w
s
v
c
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
τ
τ
π
( )
( )
( )
( ) ( ) ( ) ( ) ( )×
∑ ∫ −
−
×
∞
=
−
1 0
2
1
00
01
2
,
,
,
,
,
,
,
,
,
n
t
nC
nC
n
n
n
nC
z
y
x
e
t
e
z
c
y
c
x
c
F
L
L
L
d
u
d
v
d
w
d
T
w
v
u
P
w
v
u
C
w
v
u
C τ
π
τ
τ
τ γ
γ
( ) ( ) ( )
( )
( )
( )
×
∑
−
∫ ∫ ∫
∂
∂
×
∞
=
−
1
2
0 0 0
00
1
00
01
2
,
,
,
,
,
,
,
,
,
,
,
,
n
nC
z
y
x
L L L
n
n F
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
v
c
u
s
n
x y z π
τ
τ
τ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
×
∫ ∫ ∫ ∫
∂
∂
−
×
−
t L L L
n
n
nC
nC
n
n
n
x y z
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
v
c
u
c
e
t
e
z
c
y
c
x
c
n
0 0 0 0
00
1
00
01
,
,
,
,
,
,
,
,
,
,
,
,
τ
τ
τ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
×
∞
=1 0 0 0 0
2
2
n
t L L L
n
n
n
nC
nC
n
n
n
nC
z
y
x
n
x y z
w
c
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
w
s τ
π
τ
( )
( )
( )
( ) ( ) ( ) ( ) ( ) ×
∑ ∫ −
−
∂
∂
×
∞
=1 0
2
01
00 2
,
,
,
,
,
,
,
,
,
n
t
nC
nC
n
n
n
nC
z
y
x
e
t
e
z
c
y
c
x
c
F
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
τ
π
τ
τ
τ
γ
γ
( ) ( ) ( )
( )
( )
( )
( ) ×
∑
−
∫ ∫ ∫
∂
∂
×
∞
=1
2
0 0 0
01
00 2
,
,
,
,
,
,
,
,
,
n
n
nC
z
y
x
L L L
n
n
n x
c
F
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
v
s
u
c
n
x y z π
τ
τ
τ
γ
γ
| ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
∫ ∫ ∫ ∫
∂
∂
−
×
t L L L
n
n
n
nC
nC
n
n
x y z
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
s
v
c
u
c
e
t
e
z
c
y
c
0 0 0 0
01
00 ,
,
,
,
,
,
,
,
,
τ
τ
τ
τ γ
γ
;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫ ∫
−
−
=
∞
=1 0 0 0 0
2
11 ,
,
,
2
,
,
,
n
t L L L
L
n
n
nC
nC
n
n
n
nC
z
y
x
x y z
T
w
v
u
g
v
c
u
s
e
t
e
z
c
y
c
x
c
F
n
L
L
L
t
z
y
x
C τ
π
( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
∂
∂
×
∞
=1 0 0
2
01 2
,
,
,
n
t L
n
nC
nC
n
n
n
nC
z
y
x
n
x
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
u
w
v
u
C
w
c τ
π
τ
τ
( ) ( ) ( )
( )
( ) ( ) ( ) ×
∑
−
∫ ∫
∂
∂
×
∞
=1
2
0 0
01 2
,
,
,
,
,
,
n
n
n
n
nC
z
y
x
L L
L
n
n z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
T
w
v
u
g
w
c
v
s
y z π
τ
τ
19. International Journal of Recent advances in Physics (IJRAP) Vol.3, No.2, May 2014
73
( ) ( ) ( ) ( ) ( ) ( )
( )
×
−
∫ ∫ ∫ ∫
∂
∂
−
×
z
y
x
t L L L
L
n
n
n
nC
nC
L
L
L
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
g
w
s
v
c
u
c
e
t
e
x y z
2
0 0 0 0
01 2
,
,
,
,
,
,
π
τ
τ
τ
( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
×
∑ ∫ ∫ ∫ ∫
∂
∂
−
×
∞
=1 0 0 0 0
10
00 ,
,
,
,
,
,
,
,
,
n
t L L L
n
n
n
nC
nC
n
nC
x y z
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
v
c
u
s
e
t
e
x
c
F
n τ
τ
τ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
×
∑ ∫ ∫ ∫ ∫
−
−
×
∞
=1 0 0 0 0
00
2
,
,
,
,
,
,
2
n
t L L L
n
n
n
nC
nC
n
n
n
z
y
x
n
n
x y z
T
w
v
u
P
w
v
u
C
w
c
v
s
u
c
e
t
e
z
c
y
c
x
c
n
L
L
L
z
c
y
c γ
γ
τ
τ
π
( )
( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫
−
−
∂
∂
×
∞
=1 0 0
2
10 2
,
,
,
n
t L
n
nC
nC
n
n
n
nC
z
y
x
nC
x
u
c
e
t
e
z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
v
w
v
u
C
F τ
π
τ
τ
( ) ( )
( )
( )
( )
( ) ( ) ( ) ×
∑
−
∫ ∫
∂
∂
×
∞
=1
2
0 0
10
00 2
,
,
,
,
,
,
,
,
,
n
n
n
n
nC
z
y
x
L L
n
n z
c
y
c
x
c
F
n
L
L
L
d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
s
v
c
y z π
τ
τ
τ
γ
γ
( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
−
∫ ∫ ∫ ∫
∂
∂
−
×
−
t L L L
n
n
n
nC
nC
x y z
d
u
d
v
d
w
d
u
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
w
c
v
c
u
s
e
t
e
0 0 0 0
00
1
00
10
,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
τ
τ γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( )
×
∑ ∫ ∫ ∫ ∫
∂
∂
−
−
∞
=
−
1 0 0 0 0
00
1
00
2
,
,
,
,
,
,
,
,
,
2
n
t L L L
n
n
n
nC
nC
n
n
n
z
y
x
x y z
v
w
v
u
C
T
w
v
u
P
w
v
u
C
w
c
v
s
u
c
e
t
e
z
c
y
c
x
c
L
L
L
τ
τ
τ
π
γ
γ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ×
∑ ∫ ∫ ∫
−
−
×
∞
=1 0 0 0
2
10
2
,
,
,
n
t L L
n
n
nC
nC
n
n
n
z
y
x
nC
x y
v
c
u
c
e
t
e
z
c
y
c
x
c
n
L
L
L
F
d
u
d
v
d
w
d
w
v
u
C
n τ
π
τ
τ
( ) ( )
( )
( )
( )
∫
∂
∂
×
−
z
L
n
nC d
u
d
v
d
w
d
w
w
v
u
C
T
w
v
u
P
w
v
u
C
w
v
u
C
w
s
F
0
00
1
00
10
,
,
,
,
,
,
,
,
,
,
,
, τ
τ
τ
τ γ
γ
.
Short Biographies:
Pankratov Evgeny Leonidovich was born at 1977. From 1985 to 1995 he was educated in a secondary
school in Nizhny Novgorod. From 1995 to 2004 he was educated in Nizhny Novgorod State University:
from 1995 to 1999 it was bachelor course in Radiophysics, from 1999 to 2001 it was master course in
Radiophysics with specialization in Statistical Radiophysics, from 2001 to 2004 it was PhD course in
Radiophysics. From 2004 to 2008 E.L. Pankratov was a leading technologist in Institute for Physics of
Microstructures. From 2008 to 2012 E.L. Pankratov was a senior lecture/Associate Professor of Nizhny
Novgorod State University of Architecture and Civil Engineering. Now E.L. Pankratov is in his Full Doctor
course in Radiophysical Department of Nizhny Novgorod State University. He has 96 published papers in
area of his researches.
Bulaeva Elena Alexeevna was born at 1991. From 1997 to 2007 she was educated in secondary school
of village Kochunovo of Nizhny Novgorod region. From 2007 to 2009 she was educated in boarding school
“Center for gifted children”. From 2009 she is a student of Nizhny Novgorod State University of
Architecture and Civil Engineering (spatiality “Assessment and management of real estate”). At the same
time she is a student of courses “Translator in the field of professional communication” and “Design
(interior art)” in the University. E.A. Bulaeva was a contributor of grant of President of Russia (grant №
MK-548.2010.2). She has 29 published papers in area of her researches.