built-in potential

**Built-in Potential (V_bi)** is the **equilibrium electrostatic potential difference that develops across a p-n junction without any applied bias** — arising from the diffusion of carriers across the junction and the resulting charge separation of ionized dopants, it determines the depletion width, the diode turn-on voltage, and the maximum open-circuit voltage achievable by a solar cell. **What Is Built-in Potential?** - **Definition**: The potential difference V_bi = (kT/q) * ln(N_A * N_D / ni^2) established across a p-n junction at equilibrium, equal to the separation of the quasi-Fermi levels on both sides divided by the electron charge. - **Formation Mechanism**: Holes from the p-side diffuse to the n-side (and electrons from n to p) down their concentration gradients. As they leave, they expose fixed ionized dopant charges — negative acceptors on the p-side and positive donors on the n-side — that create an electric field opposing further diffusion until equilibrium is reached. - **Typical Values**: In silicon p-n junctions, V_bi ranges from approximately 0.55V for low doping (10^15 cm-3 on both sides) to 0.95V for high doping (10^20 cm-3), increasing logarithmically with doping product N_A*N_D. - **Unmeasurable by Voltmeter**: Metal contacts in equilibrium develop compensating contact potentials that exactly cancel V_bi — the total terminal voltage of an unbiased junction is zero, making V_bi immeasurable by any external technique and accessible only indirectly through C-V measurements. **Why Built-in Potential Matters** - **Depletion Width**: The depletion width W = sqrt(2*epsilon*V_bi/q * (1/N_A + 1/N_D)) is set by V_bi — larger built-in potential produces a wider depletion region, stronger built-in field, and larger junction capacitance for a given total applied voltage. - **Diode Turn-On Voltage**: Under forward bias, the applied voltage reduces the effective barrier from V_bi to (V_bi - V_applied). Significant current flows when the barrier is reduced to a few kT/q, which occurs near 0.6V for typical silicon junctions — the familiar "0.6V diode drop" reflects V_bi. - **Solar Cell Open-Circuit Voltage**: The theoretical maximum open-circuit voltage of a p-n junction solar cell cannot exceed V_bi — it is limited further by recombination but bounded by the built-in potential, motivating high-doping junction designs and wide-bandgap materials to maximize V_bi. - **Heterojunction Band Alignment**: In heterojunction devices (HBT, HEMT, III-V solar cells), V_bi depends on both the doping profile and the band offset between the two semiconductor materials, requiring careful alignment engineering to achieve the desired band structure. - **Depletion Approximation Foundation**: The standard depletion approximation for diode analysis assumes abrupt boundaries of the depletion region and uses V_bi as the total barrier height — virtually all analytical diode and transistor models are built on this foundation. **How Built-in Potential Is Used in Device Design** - **C-V Profiling**: Applying an AC voltage to a reverse-biased junction and measuring capacitance versus bias allows extraction of V_bi from the Mott-Schottky plot, which is the standard technique for doping profile measurement and V_bi characterization. - **Band Diagram Construction**: V_bi appears as the total band bending at a p-n junction in the equilibrium band diagram — the foundation for visualizing carrier transport and designing band structures for desired device characteristics. - **Solar Cell V_oc Optimization**: Maximizing V_bi through heavier doping and high-quality junction formation is one design lever for improving open-circuit voltage in photovoltaic cells. Built-in Potential is **the self-organizing electrostatic foundation of all p-n junction devices** — the automatic band bending that forms without applied voltage determines depletion physics, diode turn-on, and solar cell voltage limits, making V_bi the starting point for understanding and designing every semiconductor junction from a simple diode to a multi-junction concentrator solar cell.