Basics of solar cells 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
Edrift, which gives generation
photocurrent, or by
Escatt, which gives scattering photocurrent. 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,
Eg.
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 ≥
Eg, generating an electron-
hole pair. In the depletion region, the drift electric field
Edrift accelerates both electrons and holes towards their respective n-doped and p-doped regions (up and down, respectively). The resulting
current Ig is called the generated
photocurrent. In the quasi-neutral region, the scattering electric field
Escatt accelerates holes (electrons) towards the p-doped (n-doped) region, which gives a scattering photocurrent
Ipscatt (
Inscatt). Consequently, due to the accumulation of
charges, a potential
V and a photocurrent
Iph appear. The expression for this photocurrent is obtained by adding generation and scattering photocurrents:
Iph =
Ig +
Inscatt +
Ipscatt. 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
Iph. Since solar cells are designed to supply power and not absorb it, the power
P =
VIph must be negative. Hence, the operating point (
Vm,
Jm) is located in the region where and , and chosen to maximize the
absolute value of the power .
Loss mechanisms 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. 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 0.77eV and 1.70eV. 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. 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. 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%. 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 (
Eg = 1.8–1.9eV),
InGaAs for the middle sub-cell (
Eg = 1.4eV), and
Germanium for the bottom sub-cell (
Eg = 0.67eV). 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. 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
JSC by matching currents of the three subcells. 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 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. 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. It is simply a wide band gap, highly doped diode. The high doping reduces the length of the depletion region because : 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 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. 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: : 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.
EgTunnel >
EgMiddleCell.
Window layer and back-surface field 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 Ed, 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.,
EgWindow >
EgEmitter and
EgBSF >
EgEmitter. Furthermore, the lattice constant must be close to the one of InGaP and the layer must be highly doped (
n ≥ 1018cm−3).
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 : 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 J_\text{SC,1} and J_\text{SC,2} are the short circuit currents of the individual cells in the stack, \Delta J_\text{SC} is the difference between these short circuit currents, and 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 it results in the same relationship for the short-circuit current of the MJ solar cell:
JSC = min(
JSC1,
JSC2,
JSC3) where
JSC
i(
λ) is the short-circuit current density at a given wavelength
λ for the subcell
i. Because of the impossibility to obtain
JSC1,
JSC2,
JSC3 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
φi(
λ) be the photon flux of corresponding incident light in subcell
i and
QEi(
λ) be the quantum efficiency of the subcell
i. By definition, this equates to: : 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 QE_i(\lambda) is obtained by linking it with the absorption coefficient \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: 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 [W/m2·eV] to [number of photons/m2·s·eV]). For this intermediate unit conversion, the following points have to be considered: A photon has a distinct energy which is defined as follows. : (1)
Eph =
hf =
h(
c/
λ) where
Eph is photon energy,
h is the Planck constant (),
c is speed of light (),
f is frequency, and
λ is wavelength. Then the photon flux per photon energy, d
nph/d
hν, with respect to certain irradiance
E [W/m2·eV] can be calculated as follows. : (2) \frac{dn_\text{ph}}{dhv} = \frac{E}{E_\text{ph}} = \frac{E}{\frac{hc}{\lambda}} \, =
E [W/m2⋅eV] ×
λ [nm]/(1.998 × 10−25 [J⋅s⋅m/s]) =
Eλ × 5.03 × 1015 [(no. of photons)/m2⋅s⋅eV] As a result of this intermediate unit conversion, the AM1.5 spectral irradiance is given in unit of the photon flux per photon energy, [no. of photons/m2·s·eV], as shown in Figure 1. Fig._1_Photon_flux_per_photon_energy_vs._photon_energy.tif|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) n_\text{ph}(E_g) = \int_{E_\text{g}}^{\infty} \frac{dn_\text{ph}}{dhv} \, dhv = \sum_{i=E_\text{g}}^{\infty}(hv_{i+1} - hv_i) \frac{1}{2} \left[\frac{dn_\text{ph}}{dhv} (hv_{i+1}) + \frac{dn_\text{ph}}{dhv} (hv_i)\right] \, As a result of this numerical integration, the AM1.5 spectral irradiance is given in unit of the photon flux, [number of photons/m2/s], as shown in Figure 2. Fig. 2 Photon flux vs. photon energy.tif|Figure 2. Photon flux from standard solar energy spectrum (AM of 1.5). There are no photon flux data in the small photon energy ranges 0–0.3096eV because the standard (AM1.5) solar energy spectrum for
hν rad, first. According to Shockley and Queisser method,
Jrad can be approximated as follows. : (4) J_\text{rad} = A \exp\left(\frac{eV - E_\text{g}}{kT}\right) \, : (5) A = \frac{2\pi\,\exp\left(n^2 + 1\right)E_\text{g}^2 kT}{h^3 c^2} \, where
Eg is in electron volts and
n is evaluated to be 3.6, the value for GaAs. The incident absorbed thermal radiation
Jth is given by
Jrad with
V = 0. : (6) J_{th} = A \exp\left(\frac{-E_\text{g}}{kT}\right) \, The current density delivered to the load is the difference of the current densities due to absorbed solar and thermal radiation and the current density of radiation emitted from the top surface or absorbed in the substrate. Defining
Jph =
enph, we have : (7)
J =
Jph +
Jth −
Jrad The second term,
Jth, is negligible compared to
Jph for all semiconductors with
Eg ≥ 0.3eV, as can be shown by evaluation of the above
Jth equation. Thus, we will neglect this term to simplify the following discussion. Then we can express J as follows. : (8) J = en_\text{ph} - A \exp\left(\frac{eV - E_\text{g}}{kT}\right) The open-circuit voltage is found by setting
J = 0. : (9) eV_\text{OC} = E_\text{g} - kT\ln\left(\frac{A}{en_\text{ph}}\right) The maximum power point (
Jm,
Vm) is found by setting the derivative \frac{dJV}{dV} \, = 0. The familiar result of this calculation is : (10) eV_\text{m} = eV_\text{OC} - kT \ln\left(1 + \frac{eV_\text{m}}{kT}\right) : (11) J_\text{m} = \frac{en_\text{ph}}{1 + kT/eV_\text{m}} Finally, the maximum work (
Wm) done per absorbed photon is given by : (12) W_\text{m} = \frac{J_\text{m} V_\text{m}}{n_\text{ph}} = \frac{eV_\text{m}}{1 + kT/eV_\text{m}} = eV_\text{m} - kT Combining the last three equations, we have : (13) W_\text{m} = E_\text{g} - kT\left[\ln\left(\frac{A}{en_\text{ph}}\right) + \ln\left(1 + \frac{eV_\text{m}}{kT}\right) + 1\right] \, Using the above equation,
Wm (red line) is plotted in Figure 3 for different values of
Eg (or
nph). Fig. 3 Maximum Work by Multi-Junction Solar Cells.tif|Figure 3. Maximum work by ideal infinite multi-junction solar cells under standard AM1.5 spectral irradiance. Now, we can fully use Henry's graphical QE analysis, taking into account the two major intrinsic losses in the efficiency of solar cells. The two main intrinsic losses are radiative recombination, and the inability of single junction solar cells to properly match the broad solar energy spectrum. The shaded area under the red line represents the maximum work done by ideal infinite multi-junction solar cells. Hence, the limiting efficiency of ideal infinite multi-junction solar cells is evaluated to be 68.8% by comparing the shaded area defined by the red line with the total photon-flux area determined by the black line. (This is why this method is called "graphical" QE analysis.) Although this limiting efficiency value is consistent with the values published by Parrott and Vos in 1979: 64% and 68.2% respectively, there is a small gap between the estimated value in this report and literature values. This minor difference is most likely due to the different ways how to approximate the photon flux over 0–0.3096eV. Here, we approximated the photon flux as 0–0.3096eV as the same as the photon flux at 0.31eV. == Materials ==