Physics of Thin-Film Photovoltaics. Victor G. Karpov

is transparent to light (shown in waving lines). The presented built-in electric field E is caused by the dark positive and negative charges shown as + and −. R is a load resistor. For specificity, the diagram presents a two semiconductor layer design, such as p- and n- materials with the field E in their junction proximity. However, sufficient electric fields can exist as well in the proximities of semiconductor-metal junctions.

      Another reading of the latter statement is that photovoltaics do not necessarily have to be related to or understood in terms of p-n junctions. In reality, any (not only p-n) junctions of different materials produce built-in electric fields. Some of them, but not all, create photovoltaic effect.

      For example, a junction of two metals produces the built-in field underlying the phenomenon of thermoelectricity, but not suitable for PV because the light does not penetrate in a metal deep enough and because that field is screened (by metal electrons) beyond a nanometer thin region, insufficient for light absorption. However the built-in fields of metal/semiconductor (rather than metal/metal) junctions can make good diodes and PV devices.

      As another example, we point at a charge acquired by a solid particle immersed in a liquid. Curiously, that example explains how wines and many other liquid products consist of charged micro particles suspended in a somewhat ionized water. We will return to that example later in the book describing the so called “red wine effect” in PV.

Note that junction fields require electronic exchange between two materials while their direct physical contact is not necessary. As an illustration, chemically different metal electrodes of a capacitor will spontaneously acquire opposite charges proportional to the difference in their work functions. That process often called the ‘dielectric absorption’ takes time sufficient for the inter-plate electron transfer. Along the same lines, two different metals separated by a dielectric or semiconductor layer will exchange electrons forming the built-in electric field throughout that layer as described more in detail next (Sec. II).

      B. Fundamental Material Requirements

      A well known feature of all PV devices is that at least some of their constituting materials must have a not-too-wide forbidden gap, say G ~ 1–2 eV. The origin of that criterion is that larger band gaps exceeding the energies of most of sun spectra photons would not allow efficient light absorption necessary to produce enough electrons and holes. However, there is also the opposite requirement of those gaps being not too small, so that efficiently absorbing narrow band semiconductors are not good for PV either. As a result, semiconductors with substantial but not-too-wide forbidden gaps G ~ 1–2 eV, such as Si, Ge, CdTe, CIGS and some others are most suitable for PV.

      The not-too-narrow gap limitation is less intuitive and may be worth explaining here. It is dictated by PV functionality as a power source. The power is a product of electric current and voltage, P = IV where both I and V components must be not too small. I is proportional to the light induced charge generation rate favoring significant absorption coefficients in the sun spectral region of ħω ~ 0.5–2 eV. Forbidden gaps G ~ 1–2 eV of many semiconductors fall in that region. To the contrary, the typical dielectric gaps G ≳ 4 eV are too wide for sufficient absorption (for example, window glass that is practically transparent and does not absorb light).

The requirement that undermines the suitability of narrow gap semiconductors (despite their strong absorption coefficients) is related to the voltage component: V ≤ G/q, where q is the electron charge. To explain the latter inequality we note that V can be related to the difference in Fermi energies in two metal electrodes as illustrated in Fig. 2. We recall that the Fermi (quasi-Fermi) energy describes the energy change due to removing one particle from the system. Therefore, ΔE_F ≡ E_F1 − E_F2 gives the energy change due to transferring one electron between the two metals, and so does qV. On the other hand, the metal Fermi energies must not overlap with the semiconductor conduction or valence bands because such an overlap would mean that the semiconductor has in reality metal conductivity (no gap between the Fermi level and forbidden gap edges). As illustrated in Fig. 2, G presents the upper limit of the difference between the electron and hole Fermi levels allowing their steady state spatial separation. Increasing V beyond G/q would shift the Fermi levels beyond the forbidden gap turning the semiconductor into metal thus shorting the built in field and the circuit.

Fig. 2 Band diagram of a metal/semiconductor/metal structure where E_F1 and E_F2 represent quasi-Fermi levels determined by voltage drop V across the semiconductor.

      Another wording of the same is that electrons and holes recombine very efficiently when the two Fermi levels are close thus suppressing their spatial separation and the built-in field. Assuming point defect mediated recombination, it can be shown more quantitatively [12] that the electron-hole recombination rate strongly accelerates with decrease of the gap G. As a result, there exists a range of gaps optimizing PV performance, G ≲ 2 eV. It should be remembered however, that the latter optimum gap prediction was derived under certain assumptions about the nature of (defect facilitated) recombination, lack of traps, insignificant leakage due to shunts, and others clearly outlined in the original work [12].

      C. Charge Transport. Definition of Thin Film PV

We now consider charge carriers transport in the junction region where the built in field is significant; that region is often called the space charge region, simply because the field and charge density are ∇E = 4πρ (where E and ρ are correspondingly the field and charge density, and we use the Gaussian system of units). The built-in field extends through a limited thickness l_E determined by the condition that the change of electron energy through l_E equals the difference in Fermi levels ΔE_F of the materials involved. In the order of magnitude, the latter condition yields where n is the charge concentration (cm⁻³) and ε is the dielectric permittivity (for simplicity, we neglect possible differences in n and ε between the junction components). Using the ballpark values, ΔE_F ~ 1 eV, ε ~ 10, n ~ 10¹⁶ cm⁻³ yields l_E ~ 1 μm. This estimate is consistent with the published data for various junctions.

      In the built-in field region, the charge carrier transport is dominated by drift with velocity v = μF where μ is the mobility and F = qE = qV/l_E is the electric force. The diffusion with coefficient D = μkT (according to the Einstein formula where k is the Boltzmann constant and T is the temperature) is insignificant contributing to the transport time a relatively small fraction of kT/qV ≪ 1 assuming the typical V ~ 1 V and kT ~ 0.025 eV (room temperature).

      The drift transport approximation is sufficient if the light absorption is limited to the space charge region, which applies to many thin-film structures. That approximation is well justified when the light absorption length is short enough, l_abs ≲ l_E, limiting charge carrier generation to the built-in field region, which takes place in CdTe, CIGS, and a variety of other thin-film PV materials. However, some crystalline structures, such as Si and Ge, possess the non-direct band structures with absorption lengths in the hundred of microns range. In such materials, charge carriers generation extends way beyond the built-in region, and the transport is dominated by diffusion rather than drift. We recall that the characteristic diffusion distance and time are related as