The maximum efficiencies of ideal solar cells are calculated for both single and multiple energy gap cells using a standard air mass 1.5 terrestrial solar spectrum. The calculations of efficiency are made by a simple graphical method, which clearly exhibits the contributions of the various intrinsic losses. The maximum efficiency, at a concentration of 1 sun, is 31%. At a concentration of 1000 suns with the cell at 300 K, the maximum efficiencies are 37, 50, 56, and 72% for cells with 1, 2, 3, and 36 energy gaps, respectively. The value of 72% is less than the limit of 93% imposed by thermodynamics for the conversion of direct solar radiation into work. Ideal multiple energy gap solar cells fall below the thermodynamic limit because of emission of light from the forward‐biased p‐n junctions. The light is radiated at all angles and causes an entropy increase as well as an energy loss.

Limiting efficiencies of ideal single and multiple energy gap terrestrial solar cells

A rigorous proof for a reciprocity theorem that relates the spectral and angular dependences of the electroluminescence of solar cells and light emitting diodes to the spectral and angular quantum efficiency of photocarrier collection is given. An additional relation is derived that connects the open circuit voltage of a solar cell and its electroluminescence quantum efficiency.

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Reciprocity relation between photovoltaic quantum efficiency and electroluminescent emission of solar cells

Theory predicts appreciable bandgap narrowing in silicon for impurity concentrations greater than about 1017 cm−3. This effect influences strongly the electrical behaviour of silicon devices, particularly the minority carrier charge storage and the minority carrier current flow in heavily doped regions. The few experimental data known are from optical absorption measurements on uniformly doped silicon samples. New experiments in order to determine the bandgap in silicon are described here. The bipolar transistor itself is used as the vehicle for measuring the bandgap in the base. Results giving the bandgap narrowing (ΔVg0) as a function of the impurity concentration (N) in the base (in the range of 4.1015–2.5 1019 cm−3) are discussed. The experimental values of ΔVg0 as a function of N can be fitted by: δV g0 = V 1 ln N N 0 + ln 2 N N 0 +C where V1, N0 and C are constants. It is also shown how the effective intrinsic carrier concentration (nie) is related with the bandgap narrowing (ΔVg0).

Measurements of bandgap narrowing in Si bipolar transistors

This paper presents simple but accurate closed form equations for Band Gap Narrowing (BGN) for n and p type, Si, Ge, GaAs and GexSi1−x alloys and strained layers The equations are derived by identifying the four components of BGN: exchange energy shift of the majority band edge, correlation energy shift of the minority band edge and impurity interaction shifts of the two band edges In the simple parabolic band approximation, the BGN is determined by the effective masses of the carriers and the relative permittivity of the semiconductor For real semiconductors, known corrections due to anisotropy of the bands, due to multi-valleys in a band and due to interactions between sub-bands are used The values of BGN for n Si, n Ge and n and p GaAs calculated using this simple formulation agree closely with the theoretical values calculated by other authors using advanced but complex many body methods and the Random Phase Approximation for screening effects For p Si and p Ge ours appear to be the first calculations taking all interactions into account Experimental values of BGN for all semiconductors except for p Ge for which no data could be found, are also in very good agreement with our theory The Fermi level for n and p Si and p GaAs is determined using the published luminescence data In heavily doped p type semiconductors, the values are found to be considerably smaller than those calculated using the known values of the effective density of states The values of apparent BGN for n and p Si and p GaAs calculated using experimentally determined Fermi levels are in remarkably good agreement with the experimental values derived from device measurements All results are presented in a form which lends itself to numerical computer simulation studies

A simple expression for band gap narrowing (BGN) in heavily doped Si, Ge, GaAs and GexSi1−x strained layers

An analytical model of the band gap narrowing (BGN) in silicon was derived from a non-self-consistent finite-temperature full random-phase approximation (RPA) formalism. Exchange-correlation self-energy of the free carriers and correlation energy of the carrier-dopant interaction were treated on an equal basis. The dispersive quasi-particle shift (QPS) in RPA quality was numerically calculated for a broad range of densities and temperatures. The dispersion was found to be smooth enough for the relevant energies to justify the rigid shift approximation in accordance with the non-self-consistent scheme. A pronounced temperature effect of the BGN only exists in the intermediate density range. The contribution of the ionic part of the QPS to the total BGN decreases from 1/3 at low densities to about 1/4 at very high densities. Based on the numerical results, Pade approximations in terms of carrier densities, doping, and temperature with an accuracy of 1 meV were constructed using limiting cases. The analytical expression for the ionic part had to be modified for device application to account for depletion zones. The model shows a reasonable agreement with certain photoluminescence data and good agreement with recently revised electrical measurements, in particular for p-type silicon. The change of BGN profiles in a bipolar transistor under increasing carrier injection is demonstrated.

Finite-temperature full random-phase approximation model of band gap narrowing for silicon device simulation

For quasi-neutral regions of semiconductor devices with position-dependent composition, we have derived expressions for the position dependence of the excess minority-carrier density and for relevant recombination currents To make the development concrete, we study nonuniformly and heavily doped emitter regions of silicon p-n junction devices The expressions developed differ from those previously advanced in that they are in the form of a multiple integral series, yielding, by truncation, many different orders of approximation Correspondences exist between some of the different orders of approximation and various solutions previously obtained All of the mechanisms relating to hole and electron transport in position-dependent heavily doped semiconductors are accounted for in the new expressions These mechanisms include bandgap narrowing, majority-carrier degeneracy, Auger recombination lifetime, etc To assess the accuracy of the various orders of approximation, we compare their predictions with a numerical solution We determine that the simplest approximation, which contains only one term of the integral series, is accurate to within about 5 percent of the numerical solution for thin emitters (∼02 µm) provided the surface recombination velocity is less than 105cm/s It thus applies directly to bipolar transistors with polysilicon contacts, and to surface-passivated solar cells This new solution, which we call the zeroth-order or quasi-neutral quasi-equilibrium approximation, is simpler than solutions previously put forward If the emitter junction is deep or the contact is ohmic, the higher-order approximations provide whatever accuracy is needed We separate the emitter recombination current into charge and a characteristic time, enabling the calculation of the contribution of emitter to capacitance and to delay in the time domain or phase shift in the frequency domain

Systematic analytical solutions for minority-carrier transport in semiconductors with position-dependent composition, with application to heavily doped silicon

Utilizing an approach for describing the density of quantum states in a semiconductor as a function not only of energy but also of impurity concentration, we calculate the density of states and the Fermi level in n -type silicon. The calculated densities of states are shown to agree with pertinent experimental results.

Impurity concentration dependent density of states and resulting fermi level for silicon

A methodology for experimentally-based determination of gap shrinkage and effective lifetimes in the emitter and base of p-n-junction solar cells

An experimentally based methodology is described that determines the effective gap shrinkage and lifetime in the emitter of a p-n junction solar cell. It provides the first experimental means available for assessing the importance of gap shrinkage relative to that of large recombination rates in the highly doped emitter. As an additional result of the procedures employed, the base lifetime is also determined. The methodology pertains to a solar cell after the junction is formed. Hence each material parameter determined includes the effects of the processing used in junction fabrication. The methodology consists of strategy and procedures for designing experiments and interpreting data consistently with the physical mechanisms governing device behavior. This careful linking to the device physics uncovers the material parameters concealed in the data. To illustrate the procedures, they are applied to an n+-p solar cell having substrate resistivity of about 0.1 Ω cm.

A methodology for experimentally based determination of gap shrinkage and effective lifetimes in the emitter and base of p-n junction solar cells and other p-n junction devices

Analytical and equivalent‐circuit models based on numerical solutions for a‐Si:H solar cells are presented. The dependencies among such variables as recombination rate and electrical field on terminal and optical excitation are discussed. Based on our physical interpretation of the numerical solutions, we propose an equivalent‐circuit model which separates the currents into photocollected and back injected components. This model clarifies the concept of the limiting carrier and points out that the limiting carrier is the carrier with the least photocollected current. Analytical expressions for uniformly and strongly absorbed light are derived.

F.A. Lindholm

Papers

Systematic analytical solutions for minority-carrier transport in semiconductors with position-dependent composition, with application to heavily doped silicon

Impurity concentration dependent density of states and resulting fermi level for silicon

A methodology for experimentally-based determination of gap shrinkage and effective lifetimes in the emitter and base of p-n-junction solar cells

A methodology for experimentally based determination of gap shrinkage and effective lifetimes in the emitter and base of p-n junction solar cells and other p-n junction devices

Analytical and numerical modeling of amorphous silicon p‐i‐n solar cells