Influence of Recombination Centers on the Phase Portraits in Nanosized Semiconductor Films ()
1. Introduction
Operation recombination processes allow to control the number of excess charge carriers in semiconductor. Control of the concentration of charge carriers has special importance at the production of semiconductor devices. There are many types of recombination such as linear recombination and quadratic one, recombination through recombination centers and radiative recombination [1] - [3] . When recombination goes through recombination centers, the transition of the charge carriers from a
free state
to a bound one is independent of the presence of excess charge carriers of opposite sign. This means that there is no direct connection of an electron and the hole, that is, to complete the act first capturing one sign carrier by trap takes place and then capturing of the opposite sign does. There are many internal and external factors contributing to the growth and reduction of recombination centers. Changing the number of recombination centers may be undesirable for semiconductor devices and can disable them. In order to prevent interruption in the semiconductor devices work, it is necessary to conduct diagnostic of generation-recombination processes, and if it is necessary, replace them. The process of generation and recombination is similar to the oscillatory process, since the increase and the decrease of concentration of charge carriers occur periodically. For analysis of oscillatory processes, phase portraits (PhP) are used effectively, because they give the most complete picture of what is happening. In [4] using PhP the influence of frequency of variable deformation on the concentration of charge carriers in semiconductor illuminated by its forbidden zone’s light, it is shown that the frequency of the variable deformation has a strong influence on the shape of the PhP. However, in [4] influence of recombination centers in the form of PhP is not investigated. In this paper, we investigate influence of the change of recombination centers in the semiconductor by PhP.
2. The Continuity Equation Taking into Account the Combined Effects of Light and Variable All-Round Deformation
The continuity equation expressing the change of concentration of charge carriers is described by the following expression
(1)
where g―rate of generation of charge carriers, r―rate of recombination of charge carriers, q―charge of electron,
―the current density of electrons. In uniform sample, the continuity equation has the following form
(2).
when illuminated by its forbidden zone’s light semiconductor becomes sensitive to external influences change of the absorption coefficient contributes to that [5] . At radia-
tion generation is expressed by the following expression
. Here, α-absorption
coefficient, I―intensity of the light, h―Planck’s constant,
―the frequency of the light. With the express permission of the transition frequency dependence of the absorption coefficient is 
where
―the frequency of its forbidden zone’s light [1] [3] [6] A-certain coefficient which is defined by the following
expression 
for direct allowed transitions 
[6] , here
―is the reduced effective mass,
―the effective mass of electrons
and holes, respectively
―the index of refraction of the light, c―velocity of the light.
Let’s consider the case when deformation of all-round strain effects on the semiconductor. When the deformation of band gap changes as follows [7] - [9] 
, here
―the constant of the deformation potential, ε―relative deformation. In the case
, then if
, 
the absorption coefficient becomes
. If the deformation changes periodically![]()
, whereas ![]()
here ωd frequency of variable deformation, here
generation at light![]()
. Equation (2) for this case is as follows:
![]()
,![]()
(3).
At direct unpermitted transition frequency dependence of the absorption coefficient ![]()
where![]()
―frequency of its forbidden zone’s light,
B―coefficient that is defined by the following expression![]()
, ![]()
[6] .
Equation (2) for direct unpermitted transition will be as follows:
![]()
,![]()
(4).
Taking into account the permitted and unpermitted direct transition continuity equation takes the form
![]()
(5).
Let’s consider the case where the recombination through recombination centers takes place. According to statistics of the Shockley-Read, the rate of recombination is
described by the expression: ![]()
Here, Nt―the concentra-
tion of recombination centers, ![]()
and ![]()
coefficients capture electrons and holes, respectively [1] [3] . In this case, the continuity equation is
![]()
(6)
3. Analysis of Phase Portraits
Let’s consider the effect of deformation on a illuminated semiconductor. Let’s assume the following values: the temperature T = 300 K, the band gap Eg = 1.1 eV, the relative deformation ε = 10−6, the deformation potential’s constant is Ξ = 11.4 eV, lightl’s intensity I = 1018 cm−2∙sek−1, the effective mass off electron ![]()
= 0.4∙m0, were m0―mass of free electron, the effective mass of holes ![]()
= 0.541∙m0, the coefficient of direct allowed transitions A = 2 × 104 [6] , the coefficient of unpermitted direct transition B = 1.3 × 104 [6] , the concentration of recombination centers Nt = 2 × 1013 cm−3, the electron capture coefficient cn = 4.4 × 10−10 cm3/sek, hole capture coefficient cp = 6.2 × 10−9 cm3/sek, the intrinsic concentration ni = 2.2 × 1013 cm−3 [6] [10] [11] . Let the frequency of the variable all-round compression is ωd = 250 Hz. Let’s consider one period of variable strain. Let’s consider the phase portrait in 0.3 seconds after the start of a periodic deformation. The beginning of counting out is t = 0.3 sek, because by this time the carrier concentration becomes the stable value, ![]()
in which will returns periodically at the process of variable deformation of the sample (see Figure 1 and Figure 2, point 1). As the deformation increases of the carrier concentration will increase and reach its peak ![]()
at point 2 as shown in Figure 1 and Figure 2. After this, decrease of the strain begins and the concentration at the point 3 has again significance![]()
. Further, when the strain is completely takes off the concentration takes minimum value ![]()
at point 4. Upon completion of the period of hydrostatic compression at point 5 the concentration again returns to its primary value![]()
. This process will continue in this way until all parameters of the oscillatory system remain constant, and the phase portrait will take the form of a closed loop. ![]()
At prolonged effect of the strain, especially if the deformation is variable there is the probability of increase of recombination centers. The appearance of new defects and structural changes in the semiconductor caused by fatigue and wear material promote it. Let’s consider the effect of changes in the concentration of recombination centers in the form of phase portraits. Let the concentration of recombination centers grows increases from 2 × 1013 cm−3 to 8 × 1013 cm− 3 , in this case the phase portrait will not be in the form of a closed loop, but it will curl into a spiral form (see Figure 3). The maximum value of the carrier concentration will be ![]()
when![]()
, and at ![]()
the maximum concentration of charge carriers will be
![]()
Figure 1. Dynamics of changes of the concentration of charge the carriers in the period of variable strain.
![]()
Figure 2. Phase portrait of the concentration (n) of charge carriers versus the rate of change of the charge carrier concentration (dn/dt).
![]()
Figure 3. Phase portrait n versus dn/dt, for the case when the concentration of recombination centers variers from ![]()
to![]()
.
![]()
. Such decrease of the concentration of charge carriers can be explained by the fact that the generation changes in a constant range, and the range of the recombination increases with time.
Figure 4 shows the phase portrait for the case when the concentration of recombination centers decreases from 2 × 1013 cm−3 to 2 × 1012 cm− 3 . In this case, when ![]()
the maximum concentration is![]()
, and when ![]()
the maximum concentration is![]()
. Increasing concentration of charge carriers is caused by that the range of the recombination term R in the Equation (2) decreases but the range of generation term g remains constant.
Consideration of the phase picture’s transformation at the change of the system parameters is very important for understanding the physical processes in the system. Looking at the “phase portrait” under certain given values of the parameters it is possible to imagine all the possible movements in the system for any initial values. While you observe the modification in the picture at the change of the parameters, you represent all advances that the given physical system can have for all possible values of the parameters. For example, the location and nature of the singular points on the phase plane make possible to do a number of conclusions about the processes in the system.
Research of generational process by phase portraits method has practical importance because the phase portraits give the most complete picture of what is happening in the semiconductor. This allows make diagnostics of semiconductor devices and to replace them timely for preventing interruption of their work.
![]()
Figure 4. Phase portrait n versus dn/dt, for the case when the concentration of recombination centers variers from ![]()
to![]()
.
4. Conclusions
Thus, the study of generation-recombination processes on the basis of the phase portraits allows us to make the following conclusions:
・ Increasing recombination centers in the semiconductor causes decrease of carrier concentration, and the phase trajectory rolls spirally toward the lowest values, both on the axis n and on the axis dn/dt.
・ Decreasing of recombination centers in the semiconductor causes with increasing carrier concentration, and the phase trajectory is set in a spiral towards the largest values as along the axis n as along the axis dn/dt.
・ The phase portraits permitted to conduct diagnostics of semiconductor devices
Acknowledgements
This work was supported by government grants of Uzbekistan F2-OT-O-15494 “Improving the efficiency of the radiators, solar cells and other optoelectronic devices based on quantum dot research and hetero nanostructure exciton and bipolaron transport phenomena”, F2-FA-0-97004 “Dynamics of solitons in ultracold quantum gases, and optical systems and synergistic processes in multicomponent semiconductors” and F2-21 “Mathematical modelling of the determination of the density of surface states at the semiconductor-insulator”.