Black light test of Dawn's triple-junction gallium arsenide solar cells^[1]

Multi-junction (MJ) solar cells are solar cells with multiple p–n junctions made of different semiconductor materials. Each material's p-n junction will produce electric current in response to different wavelengths of light. The use of multiple semiconducting materials allows the absorbance of a broader range of wavelengths, improving the cell's sunlight to electrical energy conversion efficiency.

Traditional single-junction cells have a maximum theoretical efficiency of 33.16%.^[2] Theoretically, an infinite number of junctions would have a limiting efficiency of 86.8% under highly concentrated sunlight.^[3]

As of 2008 the best lab examples of traditional crystalline silicon (c-Si) solar cells had efficiencies between 20% and 25%,^{[4][obsolete source]} while lab examples of multi-junction cells have demonstrated performance over 46% under concentrated sunlight.^[5][6][7] Commercial examples of tandem cells are widely available at 30% under one-sun illumination,^[8][9] and improve to around 40% under concentrated sunlight. However, this efficiency is gained at the cost of increased complexity and manufacturing price. To date, their higher price and higher price-to-performance ratio have limited their use to special roles, notably in aerospace where their high power-to-weight ratio is desirable. In terrestrial applications, these solar cells are emerging in concentrator photovoltaics (CPV), but can not compete with single junction solar panels unless a higher power density is required.^[10]

Tandem fabrication techniques have been used to improve the performance of existing designs. In particular, the technique can be applied to lower cost thin-film solar cells using amorphous silicon, as opposed to conventional crystalline silicon, to produce a cell with about 10% efficiency that is lightweight and flexible. This approach has been used by several commercial vendors,^[11] but these products are currently limited to certain niche roles, like roofing materials.

Description

Basics of solar cells

Figure A. Band diagram illustration of the photovoltaic effect. Photons give their energy to electrons in the depletion or quasi-neutral regions. These move from the valence band to the conduction band. Depending on the location, electrons and holes are accelerated by E_drift, which gives generation photocurrent, or by E_scatt, which gives scattering photocurrent.^[12]

Traditional photovoltaic cells are commonly composed of doped silicon with metallic contacts deposited on the top and bottom. The doping is normally applied to a thin layer on the top of the cell, producing a p-n junction with a particular bandgap energy, E_g.

Photons that hit the top of the solar cell are either reflected or transmitted into the cell. Transmitted photons have the potential to give their energy, hν, to an electron if hν ≥ E_g, generating an electron-hole pair.^[13] In the depletion region, the drift electric field E_drift accelerates both electrons and holes towards their respective n-doped and p-doped regions (up and down, respectively). The resulting current I_g is called the generated photocurrent. In the quasi-neutral region, the scattering electric field E_scatt accelerates holes (electrons) towards the p-doped (n-doped) region, which gives a scattering photocurrent I_pscatt (I_nscatt). Consequently, due to the accumulation of charges, a potential V and a photocurrent I_ph appear. The expression for this photocurrent is obtained by adding generation and scattering photocurrents: I_ph = I_g + I_nscatt + I_pscatt.

The J-V characteristics (J is current density, i.e. current per unit area) of a solar cell under illumination are obtained by shifting the J-V characteristics of a diode in the dark downward by I_ph. Since solar cells are designed to supply power and not absorb it, the power P = VI_ph must be negative. Hence, the operating point (V_m, J_m) is located in the region where V > 0 and I_ph < 0, and chosen to maximize the absolute value of the power |P|.^[14]

Loss mechanisms

The Shockley-Queisser limit for the efficiency of a single-junction solar cell. It is essentially impossible for a single-junction solar cell, under unconcentrated sunlight, to have more than ~34% efficiency. A multi-junction cell, however, can exceed that limit.

The theoretical performance of a solar cell was first studied in depth in the 1960s, and is today known as the Shockley–Queisser limit. The limit describes several loss mechanisms that are inherent to any solar cell design.

The first are the losses due to blackbody radiation, a loss mechanism that affects any material object above absolute zero. In the case of solar cells at standard temperature and pressure, this loss accounts for about 7% of the power. The second is an effect known as "recombination", where the electrons created by the photoelectric effect meet the electron holes left behind by previous excitations. In silicon, this accounts for another 10% of the power.

However, the dominant loss mechanism is the inability of a solar cell to extract all of the power in the light, and the associated problem that it cannot extract any power at all from certain photons. This is due to the fact that the photons must have enough energy to overcome the bandgap of the material.

If the photon has less energy than the bandgap, it is not collected at all. This is a major consideration for conventional solar cells, which are not sensitive to most of the infrared spectrum, although that represents almost half of the power coming from the sun. Conversely, photons with more energy than the bandgap, say blue light, initially eject an electron to a state high above the bandgap, but this extra energy is lost through collisions in a process known as "relaxation". This lost energy turns into heat in the cell, which has the side-effect of further increasing blackbody losses.^[15]

Combining all of these factors, the maximum efficiency for a single-bandgap material, like conventional silicon cells, is about 34%. That is, 66% of the energy in the sunlight hitting the cell will be lost. Practical concerns further reduce this, notably reflection off the front surface or the metal terminals, with modern high-quality cells at about 22%.

Lower, also called narrower, bandgap materials will convert longer wavelength, lower energy photons. Higher, or wider bandgap materials will convert shorter wavelength, higher energy light. An analysis of the AM1.5 spectrum, shows the best balance is reached at about 1.1 eV (about 1100 nm, in the near infrared), which happens to be very close to the natural bandgap in silicon and a number of other useful semiconductors.

Multi-junction cells

Cells made from multiple materials layers can have multiple bandgaps and will therefore respond to multiple light wavelengths, capturing and converting some of the energy that would otherwise be lost to relaxation as described above.

For instance, if one had a cell with two bandgaps in it, one tuned to red light and the other to green, then the extra energy in green, cyan and blue light would be lost only to the bandgap of the green-sensitive material, while the energy of the red, yellow and orange would be lost only to the bandgap of the red-sensitive material. Following analysis similar to those performed for single-bandgap devices, it can be demonstrated that the perfect bandgaps for a two-gap device are at 0.77 eV and 1.70 eV.^[16]

Conveniently, light of a particular wavelength does not interact strongly with materials that are of bigger bandgap. This means that you can make a multi-junction cell by layering the different materials on top of each other, shortest wavelengths (biggest bandgap) on the "top" and increasing through the body of the cell. As the photons have to pass through the cell to reach the proper layer to be absorbed, transparent conductors need to be used to collect the electrons being generated at each layer.

Figure C. (a) The structure of an MJ solar cell. There are six important types of layers: pn junctions, back surface field (BSF) layers, window layers, tunnel junctions, anti-reflective coating and metallic contacts. (b) Graph of spectral irradiance E vs. wavelength λ over the AM1.5 solar spectrum, together with the maximum electricity conversion efficiency for every junction as a function of the wavelength.^[17]

Producing a tandem cell is not an easy task, largely due to the thinness of the materials and the difficulties extracting the current between the layers. The easy solution is to use two mechanically separate thin film solar cells and then wire them together separately outside the cell. This technique is widely used by amorphous silicon solar cells, Uni-Solar's products use three such layers to reach efficiencies around 9%. Lab examples using more exotic thin-film materials have demonstrated efficiencies over 30%.^[17]

The more difficult solution is the "monolithically integrated" cell, where the cell consists of a number of layers that are mechanically and electrically connected. These cells are much more difficult to produce because the electrical characteristics of each layer have to be carefully matched. In particular, the photocurrent generated in each layer needs to be matched, otherwise electrons will be absorbed between layers. This limits their construction to certain materials, best met by the III-V semiconductors.^[17]

Material choice

The choice of materials for each sub-cell is determined by the requirements for lattice-matching, current-matching, and high performance opto-electronic properties.

For optimal growth and resulting crystal quality, the crystal lattice constant a of each material must be closely matched, resulting in lattice-matched devices. This constraint has been relaxed somewhat in recently developed metamorphic solar cells which contain a small degree of lattice mismatch. However, a greater degree of mismatch or other growth imperfections can lead to crystal defects causing a degradation in electronic properties.

Since each sub-cell is connected electrically in series, the same current flows through each junction. The materials are ordered with decreasing bandgaps, E_g, allowing sub-bandgap light (hc/λ < eE_g) to transmit to the lower sub-cells. Therefore, suitable bandgaps must be chosen such that the design spectrum will balance the current generation in each of the sub-cells, achieving current matching. Figure C(b) plots spectral irradiance E(λ), which is the source power density at a given wavelength λ. It is plotted together with the maximum conversion efficiency for every junction as a function of the wavelength, which is directly related to the number of photons available for conversion into photocurrent.

Finally, the layers must be electrically optimal for high performance. This necessitates usage of materials with strong absorption coefficients α(λ), high minority carrier lifetimes τ_minority, and high mobilities µ.^[18]

The favorable values in the table below justify the choice of materials typically used for multi-junction solar cells: InGaP for the top sub-cell (E_g = 1.8–1.9 eV), InGaAs for the middle sub-cell (E_g = 1.4 eV), and Germanium for the bottom sub-cell (E_g = 0.67 eV). The use of Ge is mainly due to its lattice constant, robustness, low cost, abundance, and ease of production.

Because the different layers are closely lattice-matched, the fabrication of the device typically employs metal-organic chemical vapor deposition (MOCVD). This technique is preferable to the molecular beam epitaxy (MBE) because it ensures high crystal quality and large scale production.^[14]

Structural elements

Metallic contacts

The metallic contacts are low-resistivity electrodes that make contact with the semiconductor layers. They are often aluminum. This provides an electrical connection to a load or other parts of a solar cell array. They are usually on two sides of the cell. And are important to be on the back face so that shadowing on the lighting surface is reduced.

Anti-reflective coating

Anti-reflective (AR) coating is generally composed of several layers in the case of MJ solar cells. The top AR layer has usually a NaOH surface texturation with several pyramids in order to increase the transmission coefficient T, the trapping of the light in the material (because photons cannot easily get out the MJ structure due to pyramids) and therefore, the path length of photons in the material.^[12] On the one hand, the thickness of each AR layer is chosen to get destructive interferences. Therefore, the reflection coefficient R decreases to 1%. In the case of two AR layers L₁ (the top layer, usually SiO
₂) and L₂ (usually TiO
₂), there must be ${\displaystyle n_{\text{L2}}=n_{\text{AlInP}}^{\frac {1}{2}}n_{\text{L1}}}$ to have the same amplitudes for reflected fields and n_L1d_L1 = 4λ_min, n_L2d_L2 = λ_min/4 to have opposite phase for reflected fields.^[19] On the other hand, the thickness of each AR layer is also chosen to minimize the reflectance at wavelengths for which the photocurrent is the lowest. Consequently, this maximizes J_SC by matching currents of the three subcells.^[20] As example, because the current generated by the bottom cell is greater than the currents generated by the other cells, the thickness of AR layers is adjusted so that the infrared (IR) transmission (which corresponds to the bottom cell) is degraded while the ultraviolet transmission (which corresponds to the top cell) is upgraded. Particularly, an AR coating is very important at low wavelengths because, without it, T would be strongly reduced to 70%.

Tunnel junctions

Figure D: Layers and band diagram of the tunnel junction. Because the length of the depletion region is narrow and the band gap is high, electrons can tunnel.

The main goal of tunnel junctions is to provide a low electrical resistance and optically low-loss connection between two subcells.^[21] Without it, the p-doped region of the top cell would be directly connected with the n-doped region of the middle cell. Hence, a pn junction with opposite direction to the others would appear between the top cell and the middle cell. Consequently, the photovoltage would be lower than if there would be no parasitic diode. In order to decrease this effect, a tunnel junction is used.^[22] It is simply a wide band gap, highly doped diode. The high doping reduces the length of the depletion region because

${\displaystyle l_{\text{depl}}={\sqrt {{\frac {2\epsilon (\phi _{0}-V)}{q}}{\frac {N_{\text{A}}+N_{\text{D}}}{N_{\text{A}}N_{\text{D}}}}}}}$

Hence, electrons can easily tunnel through the depletion region. The J-V characteristic of the tunnel junction is very important because it explains why tunnel junctions can be used to have a low electrical resistance connection between two pn junctions. Figure D shows three different regions: the tunneling region, the negative differential resistance region and the thermal diffusion region. The region where electrons can tunnel through the barrier is called the tunneling region. There, the voltage must be low enough so that energy of some electrons who are tunneling is equal to energy states available on the other side of the barrier. Consequently, current density through the tunnel junction is high (with maximum value of ${\displaystyle J_{P}}$, the peak current density) and the slope near the origin is therefore steep. Then, the resistance is extremely low and consequently, the voltage too.^[23] This is why tunnel junctions are ideal for connecting two pn junctions without having a voltage drop. When voltage is higher, electrons cannot cross the barrier because energy states are no longer available for electrons. Therefore, the current density decreases and the differential resistance is negative. The last region, called thermal diffusion region, corresponds to the J-V characteristic of the usual diode:

${\displaystyle J=J_{S}\left(\exp \left({\frac {qV}{kT}}\right)-1\right)}$

In order to avoid the reduction of the MJ solar cell performances, tunnel junctions must be transparent to wavelengths absorbed by the next photovoltaic cell, the middle cell, i.e. E_gTunnel > E_gMiddleCell.

Window layer and back-surface field

Figure E: (a) Layers and band diagram of a window layer. The surface recombination is reduced. (b) Layers and band diagram of a BSF layer. The scattering of carriers is reduced.

A window layer is used in order to reduce the surface recombination velocity S. Similarly, a back-surface field (BSF) layer reduces the scattering of carriers towards the tunnel junction. The structure of these two layers is the same: it is a heterojunction which catches electrons (holes). Indeed, despite the electric field E_d, these cannot jump above the barrier formed by the heterojunction because they don't have enough energy, as illustrated in figure E. Hence, electrons (holes) cannot recombine with holes (electrons) and cannot diffuse through the barrier. By the way, window and BSF layers must be transparent to wavelengths absorbed by the next pn junction; i.e., E_gWindow > E_gEmitter and E_gBSF > E_gEmitter. Furthermore, the lattice constant must be close to the one of InGaP and the layer must be highly doped (n ≥ 10¹⁸ cm⁻³).^[24]

J-V characteristic

In a stack of two cells, where radiative coupling does not occur, and where each of the cells has a JV-characteristic given by the diode equation, the JV-characteristic of the stack is given by^[25]

${\displaystyle J={\frac {1}{2}}\left(J_{\text{SC,1}}+J_{\text{SC,2}}\right)-{\sqrt {{\frac {1}{4}}{\Delta J_{\text{SC}}}^{2}+J_{0}^{2}\mathrm {e} ^{\frac {qV}{kT}}}},}$

where ${\displaystyle J_{\text{SC,1}}}$ and ${\displaystyle J_{\text{SC,2}}}$ are the short circuit currents of the individual cells in the stack, ${\displaystyle \Delta J_{\text{SC}}}$ is the difference between these short circuit currents, and ${\displaystyle J_{0}^{2}=J_{\mathrm {0,1} }J_{\mathrm {0,2} }}$ is the product of the thermal recombination currents of the two cells. Note that the values inserted for both short circuit currents and thermal recombination currents are those measured or calculated for the cells when they are placed in a multijunction stack (not the values measured for single junction cells of the respective cell types.) The JV-characteristic for two ideal (operating at the radiative limit) cells that are allowed to exchange luminesence, and thus are radiatively coupled, is given by^[25]

${\displaystyle J={\frac {1}{2}}\left(J_{\text{SC,1}}+J_{\text{SC,2}}\right)+{\frac {1}{2}}T^{-}\Delta J_{\text{SC}}-\left(1-T^{+}\right){\sqrt {{\frac {1}{4}}{\Delta J_{\text{SC}}}^{2}+{\tilde {J}}_{0}^{2}\mathrm {e} ^{\frac {qV}{kT}}}}.}$

Here, the parameters ${\displaystyle T^{-}}$ and ${\displaystyle T^{+}}$ are transfer coefficients that describes the exchange of photons between the cells. The transfer coefficients depend on the refractive index of the cells. ${\displaystyle {\tilde {J}}_{0}^{2}}$ also depend on the refractive index of the cells. If the cells have the same refractive index ${\displaystyle n_{\text{r}}}$, then ${\displaystyle {\tilde {J}}_{0}^{2}=\left(1+2n_{\text{r}}^{2}\right)\left(J_{0,2}+2n_{\text{r}}^{2}J_{0,1}\right)J_{0,1}}$.

For maximum efficiency, each subcell should be operated at its optimal J-V parameters, which are not necessarily equal for each subcell. If they are different, the total current through the solar cell is the lowest of the three. By approximation,^[26] it results in the same relationship for the short-circuit current of the MJ solar cell: J_SC = min(J_SC1, J_SC2, J_SC3) where J_SCi(λ) is the short-circuit current density at a given wavelength λ for the subcell i.

Because of the impossibility to obtain J_SC1, J_SC2, J_SC3 directly from the total J-V characteristic, the quantum efficiency QE(λ) is utilized. It measures the ratio between the amount of electron-hole pairs created and the incident photons at a given wavelength λ. Let φ_i(λ) be the photon flux of corresponding incident light in subcell i and QE_i(λ) be the quantum efficiency of the subcell i. By definition, this equates to:^[27]

${\displaystyle QE_{i}(\lambda )={\frac {J_{{\text{SC}}i}(\lambda )}{q\phi _{i}(\lambda )}}\Rightarrow J_{{\text{SC}}i}=\int _{0}^{\lambda 2}q\phi _{i}(\lambda )QE_{i}(\lambda )\,d\lambda }$

The value of ${\displaystyle QE_{i}(\lambda )}$ is obtained by linking it with the absorption coefficient ${\displaystyle \alpha (\lambda )}$, i.e. the number of photons absorbed per unit of length by a material. If it is assumed that each photon absorbed by a subcell creates an electron/hole pair (which is a good approximation), this leads to:^[24]

${\displaystyle QE_{i}(\lambda )=1-e^{-\alpha (\lambda )d_{i}}}$ where d_i is the thickness of the subcell i and ${\displaystyle e^{-\alpha (\lambda )d_{i}}}$ is the percentage of incident light which is not absorbed by the subcell i.

Similarly, because

${\displaystyle V=\sum _{i=1}^{3}V_{i}}$, the following approximation can be used: ${\displaystyle V_{\text{OC}}=\sum _{i=1}^{3}V_{{\text{OC}}i}}$.

The values of ${\displaystyle V_{{\text{OC}}i}}$ are then given by the J-V diode equation:

${\displaystyle J_{i}=J_{0i}\left(e^{\frac {qV_{i}}{kT}}-1\right)-J_{{\text{SC}}i}\Rightarrow V_{{\text{OC}}i}\approx {\frac {kT}{q}}\ln \left({\frac {J_{{\text{SC}}i}}{J_{0i}}}\right)}$

Theoretical limiting efficiency

We can estimate the limiting efficiency of ideal infinite multi-junction solar cells using the graphical quantum-efficiency (QE) analysis invented by C. H. Henry.^[28] To fully take advantage of Henry's method, the unit of the AM1.5 spectral irradiance should be converted to that of photon flux (i.e., number of photons/m²·s). To do that, it is necessary to carry out an intermediate unit conversion from the power of electromagnetic radiation incident per unit area per photon energy to the photon flux per photon energy (i.e., from to ). For this intermediate unit conversion, the following points have to be considered: A photon has a distinct energy which is defined as follows.

(1): E_ph = hf = h(c/λ)

where E_ph is photon energy, h is Planck's constant (h = 6.626×10⁻³⁴ ), c is speed of light (c = 2.998×10⁸ ), f is frequency , and λ is wavelength .

Then the photon flux per photon energy, dn_ph/dhν, with respect to certain irradiance E can be calculated as follows.

(2): ${\displaystyle {\frac {dn_{\text{ph}}}{dhv}}={\frac {E}{E_{\text{ph}}}}={\frac {E}{\frac {hc}{\lambda }}}\,}$ = E × λ /(1.998×10⁻²⁵ ) = Eλ × 5.03×10¹⁵

As a result of this intermediate unit conversion, the AM1.5 spectral irradiance is given in unit of the photon flux per photon energy, , as shown in Figure 1.

• 

  Figure 1. Photon flux per photon energy from standard solar energy spectrum (AM of 1.5).

Based on the above result from the intermediate unit conversion, we can derive the photon flux by numerically integrating the photon flux per photon energy with respect to photon energy. The numerically integrated photon flux is calculated using the Trapezoidal rule, as follows.

(3): ${\displaystyle n_{\text{ph}}(E_g)={\int <!--\; \int \; -->}_{E_{\text{g}}}^{\mathrm{\infty <!--\; \infty \; -->}}\frac{d{n}_{\text{ph}}}{d}}$Zdroj:https://en.wikipedia.org?pojem=Multi-junction_solar_cell
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