P-N junction in solar cell

To understand the p-n junction, we first look at its energy band alignment:
In an isolated p-type semiconductor, the Fermi level ($E_F$) lies close to the valence band. In an isolated n-type semiconductor, the Fermi level lies close to the conduction band.
When p-type and n-type materials are brought into contact, a concentration gradient drives a spontaneous flow of charge carriers: free electrons diffuse from the n-region into the p-region, while holes diffuse from the p-region into the n-region. This initial movement is called the diffusion current.
As electrons and holes cross the boundary, they recombine near the interface. This leaves behind uncompensated, fixed donor ions ($+$) on the n-side and acceptor ions ($-$) on the p-side, forming a thin space charge region (depletion region).
This localized charge distribution creates an internal electric field directed from the n-side to the p-side, resulting in a built-in potential barrier ($V_bi$) that eventually opposes further net diffusion and aligns the Fermi levels across the junction at equilibrium.

Equalization of Fermi levels upon p-n contact

Diffusion current and formation of the internal electric field

The defining property of a p-n junction is its ability to rectify current—allowing charge to flow easily in only one direction.
Forward Bias: When the positive terminal of an external voltage source is connected to the p-type material and the negative terminal to the n-type material, the external potential opposes the built-in barrier. The potential barrier is lowered, allowing majority carriers to flow across the junction easily, causing the current to increase exponentially with applied voltage.
Reverse Bias: When the p-side is connected to the negative terminal and the n-side to the positive terminal, the external potential reinforces the internal electric field. The potential barrier widens, preventing the flow of majority carriers.
Under reverse bias, only a tiny current composed of thermally generated minority carriers flows through the junction. This is known as the reverse saturation current ($I_0$), typically around $10^{-9}\text{ A}$ (or $\text{nA}$ range) for silicon at room temperature.

Charge carriers and internal electric field in the space charge region

Current-Voltage (I-V) characteristic curve of a p-n junction

The net current $I_j$ flowing through a p-n junction is determined by the movement of both majority and minority charge carriers.
Charge carriers are divided into two categories:
1. Minority carriers: Electrons on the p-side ($n_p$) and holes on the n-side ($p_n$). They easily slide down the potential barrier across the junction.
2. Majority carriers: Holes on the p-side ($p_p$) and electrons on the n-side ($n_n$). To cross the junction, majority carriers must possess sufficient thermal energy to overcome the net potential barrier:

The fraction of majority carriers having enough thermal energy to surpass this barrier follows the Maxwell-Boltzmann distribution factor:

To define the current components crossing the junction area:

• $n_1$: Minority electron density on the p-side
• $n_2$: Majority hole density on the p-side
• $n_3$: Majority electron density on the n-side
• $n_4$: Minority hole density on the n-side
• $I_1$: Drift current of minority electrons (p $\rightarrow$ n)
• $I_2$: Diffusion current of majority holes overcoming the barrier (p $\rightarrow$ n)
• $I_3$: Diffusion current of majority electrons overcoming the barrier (n $\rightarrow$ p)
• $I_4$: Drift current of minority holes (n $\rightarrow$ p)

Assuming proportional constants $k_1, k_2, k_3, k_4$:

General Junction Current Equation:

Case 1: Equilibrium ($U = 0$, unilluminated junction)
In thermal equilibrium without external voltage, net current is zero ($I_j = 0$). Thus, the drift and diffusion currents balance out:

Case 2: Reverse Bias ($U \ll 0$)
For large negative voltage values, the Boltzmann term $e^{\frac{eU}{kT}} \to 0$. The majority carrier diffusion ceases completely, leaving only the reverse saturation drift current:

Combining these conditions yields the standard Shockley Diode Equation for an unilluminated p-n junction:

Understanding the physics of the p-n junction in the dark provides the baseline for solar cell modeling. When light is introduced, photogenerated current is added to this equation, forming the complete PV operational model.

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