Analysis of the Efficiency of a Solar Cell with Nano-Dimensional Hetero Transitions

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Introduction The paper analyzes the efficiency of a solar cell with nano-dimensional hetero junctions (SC with NHJ), based on its light-voltage characteristic (LVC). A special feature of the research is that nano-dimensional hetero junctions are created on the surface of a substrate made of technical silicon. Usually, this modification of silicon is practically not used to convert solar radiation into electricity. However, in our previous works [1-7]: the choice of technical silicon (after additional nano-technological impact) as a substrate of a solar cell is justified in detail; the technology of creating hetero contact structures on its surface is considered; the requirements for contacting materials are defined; the conditions under which the efficiency of a solar cell with nano-dimensional hetero junctions will be commensurate with the efficiency of traditional solar cells based on a crystalline modification of silicon are estimated. A solar cell with nano-dimensional hetero junctions is able to significantly improve the process of converting solar radiation into electricity, since the transformation is carried out by nanoscale hetero contact structures that are formed in accordance with the natural phenomenon of self-organization [8-10]. The creation of self-organized nano-sized semiconductor hetero contacts on the surface of a solar cell is realized most successfully if the degree of crystallinity of the contacting materials is close to each other [11]. It was shown in [12] that lead chalcogenides are close to silicon single crystals in their crystal structure and are capable of creating nano-dimensional heterocontact systems on its surface based on the phenomenon of self-organization. The study of the structure of real homogeneous technical silicon [13] showed the presence in it in small proportions (within one percent) of uniformly distributed nano-sized crystalline areas on the surface. By the method of molecular beam epitaxy, it is these areas of the silicon crystal due to the manifestation of elastic stresses at the contact boundary of the materials that are the centers of the formation of stable nano-dimensional heterocontact structures (the so-called “islands” up to several hundred million in one cm2 [8-11]). In this paper, the transformative properties of a solar cell with nano-dimensional hetero junctions are investigated based on the analysis of its light-voltage characteristic. Light voltage characteristic When studying the optical properties of a solar cell with nano-dimensional hetero junctions, each of them is considered as an ideal p-n-junction [14-19]. They are located on the surface in a strict staggered order, the equivalent scheme of functioning of which is easy to imagine in the form of a chain of homogeneous photodiodes connected to each other in parallel. Each of them has an independent “generator” G of the light current IL. Such effective converters with many p-n-junctions are studied in detail in [20-21]. From the equivalent circuit diagram of the electrical circuit (Fig. 1), it can be seen that the light current IL of a separate photodiode branches into the Io current that ensures its operation, and into the Is current flowing through the resistance p = RsRsh/(Rs + Rsh) external to the contact structure (with subsequent flow to the resistance of the external load R), that is, IL = Is + Io. Here Rs is the resistance of the single-crystal section of the nano-dimensional p-n-junction, and Rsh is the resistance of the structureless part of the substrate and the contact of the Schottky barrier. Fig. 1. An equivalent scheme of a solar cell with a nanoscale hetero junction In turn, Iо = Id + Uр/Rр, where Up/Rp is the current of the diode through the internal parallel resistance Rp, and Id = Ioo (eαUp - 1) is the diode current. Thus, the light voltage characteristic of each single contact structure on the surface of a solar cell has the form: Is = IL - Id - Uр/Rр. (1) Directly on the surface of a thin substrate with a thickness of 100 to 250 microns, n nano-sized contact structures with embedded electrostatic fields are formed at distances of about 1-2 microns from each other (Fig. 2). The current I through the external load R is formed from these n independent and identical microcurrents Is from each single contact structure, functioning due to an independent source of electromotive force ε (the function of the current generator G). Therefore, it can be represented as an algebraic sum of n independent microcurrents Is, that is: I = nIs. (2) Hence, the current-voltage characteristic of a real solar cell with parallel connected n nano sized p-p-transitions will be of the form: I = nIs = n(IL - Id - Uр/Rр). (2*) To obtain relevant and reliable data circuit consisting of conductive regions, should be closed. It can be seen from (2*) that I = nIs - the current, as a function of the voltage drop on the diode Up, decreases from the value of the short - circuit current (Isc = nIL) to zero when Up changes from zero to U* - the no-load voltage. Moreover, both values of Isc and U* are easily measured parameters maximum volt-ampere characteristics: when Up = 0, Isc = nIL - short-circuit current is equal to the light current; when Up = U*, I = 0, the current through the resistance of the external load is zero, and Isc/n = IL = Id* + U*/Rр. Fig. 2 Knowing the easily measured values U* and Isc = nIL, it is possible to estimate from this equality the Rp value of the parallel resistance of the diode part of the equivalent circuit. It is equal to Rр = U*/(Isc/n - Id*) = U*/(IL - Id*), where Id* = Ioo(eαU* - 1) - is the diode current at the no-load voltage U*, and Ioo is its amplitude value. So, U* and Isc determine the boundary of the change in the parameters of the current-voltage characteristic: The current varies from Isc = nIL to zero, and the voltage from zero to U*. Accordingly, the P-power of a solar cell with n independent single nano-dimensional hetero contact structures is determined by the product of the current current I = nIs by the current voltage Up, that is, Р = IUр = nIsUр. To analyze the obtained relations, it is convenient to present them in a normalized form by dividing the current Uр, I and Р by the corresponding boundary values U*, Isc and Р. By performing this operation, we obtain a dimensionless normalized voltage characteristic: (3) For ease of analysis, we introduce some dimensionless coefficients: τ = αU* еαU* = еτ еαUр = еτу With these dimensionless parameters, the dimensionless normalized current-voltage characteristic of a solar cell will now be: f = [Ω(1 + аy) - еτу]·[1/(Ω - 1)], (4) and the power: Р = IUр = Uр{IL(1 - Uр/Uсо) + Id*(Uр/Uсо - Id/Id*)} becomes non-dimensional form looks like: Ψ = Р/U* · Isc = уf = у[1/(Ω - 1)][Ω(1 + аy) - еуτ]. (5) Dividing the current Up, Is and P by the corresponding boundary values U*, Isc and Pm, we obtain a dimensionless view of the current-voltage characteristic and power of the solar cell: The power, as a function of voltage Ψ = Ψ(y), increases with increasing y (voltage) from zero to a certain maximum Ψm at a certain current fm and voltage Um. Then the power Ψ = Ψ(y) is rapidly reduced to zero at y = 1 (or Up = U*). The main indicators of the efficiency of a solar cell are the parameters of the maximum power: fm, Ψm and уm (or Im, Pm and Um). If we find уm, then by formulas (4) and (5) we find fm and Ψm. Therefore, by equating to zero dP/dUp - the first derivative of the solar cell power by the voltage Up (or in the dimensionless form dΨ/dy), we obtain the transcendental equation еуmτ = Ω(1 + 2yma)/(1 + τym). (6) The solution of the transcendental equation will allow us to determine ym = Um/U*, as well as fm, and Ψm. The values of the maximum power Pm, voltage Um and current Im at the same time, in an explicit and dimensionless form, are equal: Pm = ImUm Um Ψm = fmуm ym The power of a solar cell divided by IscU* determines its efficiency - the efficiency coefficient η = P/IscU*, which in dimensionless form coincides by definition with the normalized power and has the form: At the point of maximum power, the efficiency coefficient in the dimensionless form is equal to: (8) Analysis of the results obtained The analysis of the obtained results shows that the value of the maximum power efficiency is significantly determined by the parameters of a solar cell with nano-dimensional hetero transitions: Ω, τ and the ratio between them. It follows from the transcendental equation (6) that the left growing exponential part of еуmτ is equal to the right, asymptotically decreasing part with respect to ym: But it follows from (4) that in the entire range of voltage changes from 0 to U*(y ≤ 1), the value of the normalized current f must be positive. And this is possible under the condition еτу > Ω, which is guaranteed to be fulfilled near the boundary y ≤ 1. Note that еτу and Ω are large numbers in themselves, but they are quite close to each other. Since Ω >1 and а = еτ/Ω - 1, the decrease of the right part of (6) is possible for Thus, the combination of the conditions: еτу > Ω, Ω > 1, (1 + 2yma)/(1 + τym) < 1 and 2а < τ, allows us to obtain a relation connecting Ω, τ and y: еτу > Ω > 2еτ/(2 + τ) It follows from this inequality that the greatest efficiency of a solar cell with nano-dimensional hetero transitions is manifested when (9) and when the value of the maximum voltage Um is close to the boundary voltage U*, that is, Um ≤ U*. It is precisely with the simultaneous fulfillment of inequalities (9) and Um ≤ U* that the maximum value of the ηm - the efficiency indicator of a solar cell with nano-dimensional heterojunctions can be achieved. We list several combinations of the numbers Ω = 1 + IL/Ioo and eτ = ееU*/(AkT), whose substitution in ηm, уm, fm and Ψm will give a high efficiency of a single element with nano-dimensional heterojunctions. It is only necessary to observe the conditions eτ > Ω each time. Ω = 8 and eτ = 10; Ω = 21 and eτ = 24; Ω = 51 and eτ = 56; Ω = 101 and eτ =111; Ω = 201 and eτ = 203. Temperature variations or changes in the non-ideality coefficient of the diode A affect the value τ = еU*/(AkT). For example, with an increase in AkT or due to temperature, or a change in the A - coefficient of imperfection, the type of current-voltage characteristic changes dramatically. Conclusions The optimal ratios of the parameters of a solar cell with nano-dimensional hetero transitions that provide maximum power efficiency are determined. It follows from the obtained relations that a solar cell with nano-dimensional hetero transitions can be constructed in such a way that its efficiency will always have the required high level. It is shown that such a controlled situation is possible in relation to a solar cell with nano-dimensional p-p-transitions created due to the phenomenon of self-organization on a substrate made of technical silicon. Such solar cells can contribute to improving the efficiency of solar energy, the formation of highly efficient, cheap photovoltaic converters.

About the authors

Erkin Z. Imamov

Tashkent University of Information Technologies named after Muhammad al-Khorazmiy

Email: erkinimamov@mail.ru
Dr. Sci. (Phys.-Math.), Professor Tashkent, Republic of Uzbekistan

Ramizulla A. Muminov

Institute of Physics and Technology of the Scientific and Production Association “Physics-Sun” of the Academy of Sciences of the Republic of Uzbekistan

Email: detector@uzsci.net
Dr. Sci. (Phys.-Math.), Academician of the Academy Sciences of the Republic Uzbekistan Tashkent, Republic of Uzbekistan

Rustam Kh. Rakhimov

Institute of Materials Science of the Scientific and Production Association “Physics-Sun” of the Academy of Sciences of the Republic of Uzbekistan

Email: rustam-shsul@yandex.com
Dr. Sci. (Eng.); Head of laboratory No. 1 Tashkent, Republic of Uzbekistan

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