7.5 Photovoltaic effect

The photovoltaic effect converts light into electricity within semiconductors and forms the physical basis of solar cell technology. Understanding this effect requires connecting several condensed matter concepts: photon absorption, carrier generation, charge separation at junctions, and transport through the device. This topic ties together much of what you've seen in earlier units on band structure, doping, and p-n junctions.

Fundamentals of photovoltaic effect

The photovoltaic effect occurs when photons absorbed by a semiconductor generate electron-hole pairs that are then separated by an internal electric field, producing a voltage and driving a current. Unlike a battery, the energy source is external (light), and the semiconductor structure itself provides the mechanism for converting that energy into useful electrical work.

Photoelectric vs photovoltaic effect

These two effects are related but distinct. The photoelectric effect ejects electrons entirely out of a material's surface when photons with sufficient energy strike it. Einstein's 1905 explanation of this effect (photon energy $E = h\nu$ must exceed the work function) earned him the Nobel Prize and established the quantum nature of light.

The photovoltaic effect, by contrast, generates voltage and current within a material. Photons create electron-hole pairs inside the semiconductor, and the built-in field of a junction separates them. No electrons leave the material. Key differences:

• The photoelectric effect typically occurs in metals; the photovoltaic effect occurs in semiconductors
• The photoelectric effect requires photon energy above the work function; the photovoltaic effect requires photon energy above the bandgap
• The photovoltaic effect produces sustained current through an external circuit, which is what makes solar cells possible

Light-matter interaction basics

When electromagnetic radiation hits a semiconductor, three things can happen: absorption, reflection, or transmission. Only absorption contributes to the photovoltaic effect.

A photon's energy is set by its wavelength:

$E = \frac{hc}{\lambda}$

where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength. Shorter wavelengths carry more energy. For a photon to be absorbed and promote an electron across the bandgap, its energy must satisfy $E \geq E_g$. Photons with energy below $E_g$ pass through the material without being absorbed. Photons with energy well above $E_g$ are absorbed, but the excess energy is lost as heat (thermalization).

Light intensity determines how many photons arrive per unit time, which directly affects how many electron-hole pairs can be generated.

Semiconductor band structure

The band structure determines which photons a semiconductor can absorb and how efficiently it converts them.

• The valence band is the highest energy band that is fully occupied by electrons at $T = 0$
• The conduction band is the next band up, empty (or nearly empty) at equilibrium
• The bandgap $E_g$ is the energy gap between them. For silicon, $E_g \approx 1.12 \text{ eV}$; for GaAs, $E_g \approx 1.42 \text{ eV}$

Direct vs. indirect bandgap matters enormously for photovoltaics. In a direct bandgap material like GaAs, the conduction band minimum and valence band maximum occur at the same crystal momentum ($k$-value), so a photon can excite an electron directly. In an indirect bandgap material like Si, these extrema occur at different $k$-values, meaning a phonon must also participate to conserve momentum. This makes absorption weaker in Si, requiring thicker wafers to capture the same amount of light.

Doping introduces energy levels within the bandgap (donor levels near the conduction band for n-type, acceptor levels near the valence band for p-type), which is essential for forming the p-n junction that drives the photovoltaic effect.

Photon absorption mechanisms

Absorption is the first step in the photovoltaic process. The absorption coefficient $\alpha$ describes how quickly light intensity drops as it penetrates the material. High $\alpha$ means the light is absorbed in a thin layer; low $\alpha$ means you need more material.

Direct vs indirect transitions

In a direct transition, an electron absorbs a photon and jumps from the valence band to the conduction band without changing its crystal momentum. This is a two-particle process (photon + electron), and it happens readily. Materials like GaAs and CdTe have direct bandgaps and correspondingly high absorption coefficients (on the order of $10^4 \text{ cm}^{-1}$ near the band edge). Only a few micrometers of material are needed to absorb most of the sunlight.

In an indirect transition, the conduction band minimum sits at a different $k$-value from the valence band maximum. The electron needs both a photon (for energy) and a phonon (for momentum) to make the jump. This three-particle process is less probable, giving indirect materials like Si and Ge lower absorption coefficients near the band edge. Silicon solar cells therefore need absorber layers on the order of 100–300 μm, much thicker than direct-bandgap cells.

Band structure engineering through alloying (e.g., varying the In/Ga ratio in InGaAs) can tune the bandgap and absorption properties for specific spectral ranges.

Exciton formation

When a photon creates an electron-hole pair, the two carriers can remain bound to each other by their Coulomb attraction, forming an exciton. Whether this matters for device physics depends on the binding energy relative to thermal energy $k_BT$ (about 26 meV at room temperature).

• Wannier-Mott excitons occur in inorganic semiconductors. They have large radii (spanning many lattice constants) and low binding energies (typically a few meV in Si). At room temperature, thermal energy easily dissociates them into free carriers, so you can often ignore exciton effects in conventional inorganic solar cells.
• Frenkel excitons occur in organic semiconductors. They are tightly bound (binding energies of 0.1–1 eV) and localized to one or a few molecules. In organic photovoltaics, exciton dissociation is a major design challenge, requiring donor-acceptor interfaces to split the exciton into free charges.

Carrier generation and recombination

Generation is straightforward: a photon with $E \geq E_g$ is absorbed, creating a free electron in the conduction band and a free hole in the valence band.

Recombination is the reverse, and it's the main enemy of solar cell efficiency. The three dominant mechanisms are:

1. Radiative recombination: An electron falls back into the valence band and emits a photon. This is the unavoidable, thermodynamically required process. It sets the fundamental efficiency limit.
2. Auger recombination: The energy released when an electron and hole recombine is transferred to a third carrier (another electron or hole) instead of being emitted as a photon. This process becomes significant at high carrier concentrations (heavily doped regions, concentrated sunlight).
3. Shockley-Read-Hall (SRH) recombination: Defects and impurities create trap states within the bandgap that act as stepping stones for recombination. This is the dominant non-radiative loss in most real devices and is why material quality matters so much.

Surface recombination is particularly damaging in thin-film cells where the surface-to-volume ratio is high. Passivation layers (e.g., $\text{SiO}_2$ or $\text{Al}_2\text{O}_3$ on Si) reduce dangling bonds at surfaces and interfaces, suppressing this loss channel.

The carrier lifetime $\tau$ characterizes how long a carrier survives before recombining. The diffusion length $L = \sqrt{D\tau}$ (where $D$ is the diffusion coefficient) tells you how far a carrier can travel before it's lost. For efficient collection, the diffusion length must be comparable to or larger than the device thickness.

Charge separation and collection

Generating electron-hole pairs is only useful if you can separate them before they recombine. The p-n junction provides the built-in electric field that accomplishes this.

Built-in electric fields

When p-type and n-type semiconductors are joined, electrons diffuse from the n-side into the p-side and holes diffuse the other way. This leaves behind fixed ionized dopants, creating a depletion region with a built-in electric field pointing from n to p.

The built-in potential $V_{bi}$ is determined by the difference in Fermi levels before contact:

$V_{bi} = \frac{k_BT}{q} \ln\left(\frac{N_A N_D}{n_i^2}\right)$

where $N_A$ and $N_D$ are the acceptor and donor concentrations and $n_i$ is the intrinsic carrier concentration. This field is what sweeps photogenerated carriers apart: electrons toward the n-side, holes toward the p-side.

Field strength can be enhanced through heterojunctions (junctions between different semiconductor materials) or graded doping profiles.

p-n junction dynamics

The depletion region is the active zone for charge separation. Band bending across this region creates the potential energy slope that drives carriers apart.

Carriers generated within the depletion region are immediately swept out by the field. Carriers generated outside the depletion region must diffuse to it before recombining. This is why minority carrier diffusion length is so critical: in the p-type region, photogenerated electrons (minority carriers) need to reach the depletion region to be collected.

Under illumination, the photocurrent flows in the reverse-bias direction (opposing the diode's forward current). The net current under illumination is described by the ideal diode equation modified with a photocurrent term:

$I = I_0\left(\exp\frac{qV}{k_BT} - 1\right) - I_{ph}$

where $I_0$ is the dark saturation current and $I_{ph}$ is the photogenerated current.

Carrier diffusion and drift

Two forces drive carrier transport in a solar cell:

• Drift: Carrier motion driven by electric fields. The drift velocity is $v_d = \mu \mathcal{E}$, where $\mu$ is the carrier mobility and $\mathcal{E}$ is the electric field. Drift dominates inside the depletion region.
• Diffusion: Carrier motion driven by concentration gradients, described by Fick's law. Diffusion dominates in the quasi-neutral regions outside the depletion zone.

Ambipolar diffusion describes the coupled motion of electrons and holes when both are present in significant concentrations. The faster carrier is slowed down and the slower carrier is sped up by the internal electric field they create between them.

The diffusion length $L = \sqrt{D\tau}$ is the key figure of merit. In high-quality monocrystalline silicon, electron diffusion lengths can exceed 1 mm, far larger than the wafer thickness, enabling efficient collection.

Solar cell operation principles

A solar cell is a p-n junction diode operated under illumination. Its performance is fully described by the current-voltage (I-V) characteristic, from which all key figures of merit are extracted.

I-V characteristics

The I-V curve of an illuminated solar cell is the dark diode curve shifted downward by the photocurrent $I_{ph}$. The curve passes through two important points:

• At $V = 0$: the current equals the short-circuit current $I_{sc}$
• At $I = 0$: the voltage equals the open-circuit voltage $V_{oc}$

Between these points lies the maximum power point (MPP), where the product $P = IV$ is maximized. Real solar cells are operated at or near the MPP using maximum power point tracking (MPPT) electronics.

Non-ideal effects distort the I-V curve:

• Series resistance $R_s$ (from contacts, bulk resistance, interconnects) reduces current at high voltages, rounding off the "knee" of the curve
• Shunt resistance $R_{sh}$ (from manufacturing defects, leakage paths) allows current to bypass the junction, reducing voltage at low currents

Open-circuit voltage

$V_{oc}$ is the maximum voltage the cell can produce (at zero current). From the diode equation:

$V_{oc} = \frac{k_BT}{q} \ln\left(\frac{I_{ph}}{I_0} + 1\right)$

Several things follow from this expression:

• $V_{oc}$ increases logarithmically with light intensity (through $I_{ph}$)
• Lower dark saturation current $I_0$ gives higher $V_{oc}$, which is why material quality and surface passivation matter
• $V_{oc}$ is always less than $E_g/q$. The difference is called the "voltage deficit" and minimizing it is a major goal in cell design
• Temperature increases $I_0$ exponentially, so $V_{oc}$ drops with increasing temperature (typically 2–2.5 mV/°C for Si)

In multi-junction cells, each subcell contributes its own $V_{oc}$, and the total voltage is the sum.

Short-circuit current

$I_{sc}$ is the maximum current the cell can deliver (at zero voltage). It depends on:

• Light intensity: $I_{sc}$ is directly proportional to the number of incident photons with $E \geq E_g$
• Active area: Larger cells produce more current
• Absorption: The absorption coefficient and cell thickness determine what fraction of above-bandgap photons are absorbed
• Collection efficiency: What fraction of photogenerated carriers actually reach the contacts (determined by diffusion length relative to cell thickness)

$I_{sc}$ can be enhanced through anti-reflection coatings (reducing front-surface reflection from ~30% to <5% in Si), surface texturing for light trapping, and back-surface reflectors.

Fill factor and efficiency

The fill factor quantifies how close the I-V curve is to an ideal rectangle:

$FF = \frac{P_{max}}{V_{oc} \times I_{sc}} = \frac{V_{mpp} \times I_{mpp}}{V_{oc} \times I_{sc}}$

Typical values range from 0.7 to 0.85 for good cells. Higher $FF$ means less power is lost to resistive and recombination effects. Series resistance reduces $FF$ most noticeably.

The power conversion efficiency is:

$\eta = \frac{P_{max}}{P_{in}} = \frac{FF \times V_{oc} \times I_{sc}}{P_{in}}$

where $P_{in}$ is the incident light power. Under standard test conditions (STC) (AM1.5G spectrum, 1000 W/m², 25°C), record efficiencies for single-junction Si cells are around 26.8%, while multi-junction concentrator cells have exceeded 47%.

Materials for photovoltaics

Material choice determines the bandgap, absorption properties, carrier lifetimes, and ultimately the cost-per-watt of a solar cell. The ideal single-junction bandgap for the AM1.5G solar spectrum is around 1.1–1.4 eV.

Silicon-based solar cells

Crystalline silicon (c-Si) dominates the market with over 95% share, thanks to decades of manufacturing optimization inherited from the microelectronics industry.

• Monocrystalline Si: Single-crystal wafers grown by the Czochralski process. Highest efficiencies (~26%) due to low defect density and long carrier lifetimes. More expensive to produce.
• Polycrystalline Si: Cast into blocks and sliced into wafers. Grain boundaries introduce recombination centers, reducing efficiency (~22%), but lower cost.
• Amorphous Si (a-Si): Disordered structure with a direct-like bandgap of ~1.7 eV. Used in thin-film applications (calculators, building-integrated PV). Lower efficiency (~13%) but can be deposited on flexible substrates.
• Silicon heterojunction (SHJ) cells: Combine c-Si wafers with thin a-Si passivation layers. The a-Si provides excellent surface passivation, enabling very high $V_{oc}$ values. Record efficiencies above 26%.

Thin-film technologies

Thin-film cells use only 1–5 μm of absorber material (compared to ~180 μm for c-Si wafers), reducing material costs.

• CdTe: Direct bandgap of 1.45 eV, near-optimal for single-junction cells. Lab efficiencies ~22%. Concerns about cadmium toxicity and tellurium scarcity.
• CIGS ($\text{CuIn}_{1-x}\text{Ga}_x\text{Se}_2$): Tunable bandgap (1.0–1.7 eV depending on Ga content). Flexible substrates possible. Lab efficiencies ~23%.
• GaAs: Direct bandgap of 1.42 eV. Highest single-junction efficiencies (~29%) but very expensive due to epitaxial growth. Used primarily in space applications where cost-per-watt is less important than power-per-kilogram.

Emerging photovoltaic materials

• Perovskites (e.g., $\text{CH}_3\text{NH}_3\text{PbI}_3$): Tunable bandgap, high absorption coefficients, and solution-processable. Efficiencies have risen from ~3.8% in 2009 to over 26% in single-junction cells. Stability and lead toxicity remain challenges.
• Organic photovoltaics (OPVs): Use conjugated polymers or small molecules. Flexible, lightweight, and potentially very cheap. Efficiencies around 19%. Frenkel exciton dissociation at donor-acceptor interfaces is the key physics.
• Quantum dot solar cells: Semiconductor nanocrystals whose bandgap is tunable via quantum confinement (smaller dots = larger bandgap). Potential for multiple exciton generation, where one high-energy photon creates more than one electron-hole pair.
• Dye-sensitized solar cells (DSSCs): Use a dye molecule adsorbed on a nanostructured $\text{TiO}_2$ electrode. The dye absorbs light and injects electrons into the oxide. Efficiencies around 13%.

Advanced photovoltaic concepts

These approaches aim to surpass the single-junction Shockley-Queisser limit by capturing more of the solar spectrum or reducing thermalization losses.

Multi-junction cells

Multi-junction cells stack two or more p-n junctions with decreasing bandgaps from top to bottom. The top cell absorbs high-energy photons, and lower-energy photons pass through to be absorbed by subsequent cells.

The subcells are connected in series through tunnel junctions (heavily doped, thin p-n junctions that allow carriers to tunnel through). Because the subcells are in series, the total voltage is the sum of individual $V_{oc}$ values, but the current is limited by the lowest-current subcell. Current matching between subcells is therefore critical and is achieved by adjusting layer thicknesses and bandgaps.

A typical triple-junction cell might use InGaP (1.9 eV) / GaAs (1.4 eV) / Ge (0.67 eV). Record efficiencies exceed 47% under concentration.

Tandem solar cells

Tandem cells are conceptually similar to multi-junction cells but the term often refers to configurations where the subcells are made from different material systems, particularly combinations that don't require lattice matching.

• Two-terminal (monolithic): Subcells grown or deposited on top of each other, connected in series. Requires current matching.
• Four-terminal (mechanically stacked): Each subcell has its own contacts and operates independently. No current matching needed, but more complex wiring.

Perovskite-on-silicon tandems are the most actively researched combination. The perovskite top cell ($E_g \approx 1.7$ eV) absorbs blue/green light while the Si bottom cell ($E_g \approx 1.1$ eV) absorbs red/infrared. Lab efficiencies have exceeded 33%.

Concentration photovoltaics

Concentration photovoltaic (CPV) systems use lenses or mirrors to focus sunlight onto small, high-efficiency cells (typically multi-junction). Concentration ratios range from low (2–10x) to ultra-high (>1000x).

The advantage: you replace expensive semiconductor area with cheap optics. Under concentration, $V_{oc}$ increases logarithmically (more photocurrent), boosting efficiency. The disadvantages: CPV requires precise two-axis sun tracking, effective thermal management (concentrated light generates significant heat), and only works well with direct normal irradiance (not diffuse light), limiting it to high-insolation regions like deserts.

Limitations and loss mechanisms

Every solar cell loses a significant fraction of the incident solar energy. Understanding where these losses occur is essential for improving designs.

Shockley-Queisser limit

The Shockley-Queisser (SQ) limit is the theoretical maximum efficiency for a single-junction solar cell under the assumptions of:

1. One electron-hole pair per absorbed photon
2. Complete thermalization of carriers to the band edges
3. Only radiative recombination (no defect or Auger losses)
4. Perfect absorption of all photons with $E \geq E_g$

Under the AM1.5G spectrum, the SQ limit peaks at about 33.7% for a bandgap of ~1.34 eV. The two biggest loss channels are:

• Thermalization: Photons with $E > E_g$ lose the excess energy as heat (~30% of incident energy)
• Sub-bandgap transparency: Photons with $E < E_g$ aren't absorbed at all (~20% of incident energy)

Strategies to exceed the SQ limit include multi-junction cells, hot carrier extraction (collecting carriers before they thermalize), multiple exciton generation, and intermediate band solar cells.

Recombination losses

In real devices, non-radiative recombination further reduces efficiency below the SQ limit.

• Radiative recombination is unavoidable and sets the thermodynamic voltage limit. In high-quality GaAs cells, radiative recombination dominates, and these cells approach the SQ limit closely.
• Auger recombination scales with carrier density cubed ($R_{Auger} \propto n^2 p$ or $np^2$). It becomes the dominant intrinsic loss mechanism in silicon at high injection levels and sets a practical efficiency ceiling around 29.4% for Si.
• SRH recombination through defect states is the primary extrinsic loss. Reducing impurities and crystallographic defects directly improves cell performance.

Passivation of surfaces and interfaces is one of the most impactful techniques for reducing recombination. Modern high-efficiency Si cells use passivated emitter and rear cell (PERC) or tunnel oxide passivated contact (TOPCon) architectures specifically to minimize surface recombination.

Optical and electrical losses

Optical losses reduce the number of photons that reach the absorber:

• Reflection at the front surface (mitigated by anti-reflection coatings and surface texturing)
• Parasitic absorption in non-active layers like transparent conductive oxides, encapsulants, or metal grids
• Incomplete absorption in thin absorber layers (addressed by light trapping structures that increase the optical path length)

Electrical losses reduce the power extracted from generated carriers:

• Series resistance $R_s$ from metal contacts, bulk semiconductor, and interconnects. Reduces fill factor.
• Shunt resistance $R_{sh}$ from pinholes, edge leakage, or other defect paths that allow current to bypass the junction. Also reduces fill factor.

Characterization techniques

Characterizing solar cells requires a combination of optical and electrical measurements to identify where losses occur and how to fix them.

Quantum efficiency measurements

External quantum efficiency (EQE) is the ratio of collected carriers to incident photons at each wavelength:

$EQE(\lambda) = \frac{\text{electrons collected}}{\text{photons incident}}$

Internal quantum efficiency (IQE) corrects for reflection losses:

$IQE(\lambda) = \frac{EQE(\lambda)}{1 - R(\lambda)}$

where $R(\lambda)$ is the reflectance. IQE isolates the electrical collection efficiency from optical losses.

Interpreting QE curves:

• Low EQE at short wavelengths (blue) suggests poor collection near the front surface (surface recombination or dead layer)
• Low EQE at long wavelengths (red/infrared) suggests poor collection deep in the cell (short diffusion length or insufficient absorption)
• Bias-dependent QE measurements reveal voltage-dependent collection problems, such as field-dependent carrier extraction

Spectral response analysis

Spectral response is closely related to QE but expressed in units of A/W rather than electrons/photon. It measures current output per unit of incident optical power at each wavelength.

This measurement is particularly useful for:

• Optimizing anti-reflection coatings by identifying which wavelengths suffer the most reflection loss
• Current matching in multi-junction cells by comparing the spectral response of each subcell
• Identifying absorption edges that reveal the effective bandgap

Measurements under bias light (additional steady-state illumination) simulate realistic operating conditions, which is important because carrier concentrations and recombination rates change under illumination.

Photoluminescence spectroscopy

Photoluminescence (PL) is a non-destructive, contactless technique. You shine a laser on the sample and measure the emitted light.

• Steady-state PL intensity correlates with material quality: brighter PL means less non-radiative recombination (more of the recombination is radiative). A high-quality solar cell material should luminesce strongly.
• PL emission spectrum reveals the bandgap energy and can identify sub-bandgap defect emission.
• Time-resolved PL (TRPL) measures how quickly the PL signal decays after a pulsed excitation, directly giving the carrier lifetime $\tau$.
• PL imaging maps spatial variations in carrier lifetime across a wafer, identifying regions with high defect density or poor passivation.

Applications and future prospects

Photovoltaic technology has moved well beyond rooftop panels. The physics of the photovoltaic effect now intersects with grid engineering, architecture, and even space exploration.

Grid-connected vs off-grid systems

Most installed PV capacity is grid-connected. These systems feed power into the utility grid, and net metering policies allow owners to offset their electricity costs. Grid-connected systems don't need batteries, which significantly reduces cost.

Off-grid systems require energy storage (typically lithium-ion batteries) to provide power when the sun isn't shining. These are essential in remote locations without grid access.

Hybrid systems combine grid connection with local battery storage, providing backup during outages. Microgrids integrate local PV generation, storage, and loads into a semi-autonomous system that can operate independently from the main grid when needed.

Building-integrated photovoltaics

Building-integrated photovoltaics (BIPV) replace conventional building materials with PV-active elements: solar roof tiles, semi-transparent PV windows, and PV facade cladding. The dual function (building envelope + power generation) can offset the higher cost of the PV material.

Challenges include non-optimal orientation and tilt angles, partial shading from surrounding structures, and aesthetic requirements that may limit cell efficiency (e.g., colored or semi-transparent cells absorb less light). The goal is energy-positive buildings that generate more electricity than they consume over a year.

Next-generation solar technologies

• Bifacial cells capture reflected light from the ground (albedo) on their rear side, boosting energy yield by 5–20% depending on ground reflectivity and mounting height
• Floating solar (floatovoltaics) on reservoirs and lakes avoids land-use conflicts and benefits from water cooling, which improves $V_{oc}$
• Solar water splitting uses photovoltaic or photoelectrochemical cells to drive the hydrogen evolution reaction, producing green hydrogen as a storable fuel
• Photovoltaic-thermal (PVT) systems capture waste heat from the cell for water or space heating, improving total energy utilization
• Space-based solar power would collect solar energy in orbit (no atmosphere, no night on geostationary orbit) and beam it to Earth via microwave or laser. Still largely conceptual, but the physics is sound