Franz-Keldysh Electro-Absorption Energy (ħΘ_FK) Modulator Formula

How a 100 kV/cm electric field turns a transparent semiconductor opaque below its bandgap!

Independently predicted in 1958 by Walter Franz and Leonid V. Keldysh, a strong electric field F tilts the conduction and valence bands of a bulk semiconductor so Airy-function electron and hole wavefunctions leak into the forbidden gap—enabling sub-bandgap photon absorption and 100 Gbaud telecom Electro-Absorption Modulators (EAMs)! The characteristic electro-optic tunneling energy scale is ħΘ_FK = ((e^2 * ħ^2 * F^2) / (2 * m_r * m_0))^(1/3)!

In practical optoelectronics units—electric field F in kV/cm (= 10^5 V/m) and reduced electron-hole effective mass ratio m_r = (m_e * m_hh) / (m_e + m_hh) in units of free electron mass m_0—the exact SI prefactor equals 0.72495 meV·(cm/kV)^(2/3): ħΘ_FK = 0.72495 * (F^2 / m_r)^(1/3) meV! In direct-gap GaAs (m_e = 0.067, m_hh = 0.45 -> m_r = 0.0583), biasing a PIN junction to F = 100 kV/cm yields ħΘ_FK = 40.28 meV (~1.56 k_B*T at 300 K), while ramping reverse bias to F = 250 kV/cm expands the absorption tail to 74.20 meV!

Franz-Keldysh Characteristic Electro-Optic Energy ħΘ_FK (meV)

Evaluates 0.72495 * ((F_kV_cm ^ 2) / m_r) ^ (1 / 3) in millielectronvolts from electric field B2 (kV/cm) and reduced electron-hole mass ratio B3 (m_r / m_0).

=IF(OR(B2 <= 0, B3 <= 0), "Invalid Electro-Optic Inputs", ROUND(0.72495 * (((B2 ^ 2) / B3) ^ (1 / 3)), 2))

Reduced Electron-Hole Effective Mass Ratio m_r = (m_e * m_h) / (m_e + m_h)

Computes the reduced interband mass ratio from conduction electron mass B4 (0.067) and heavy-hole mass B5 (0.45).

=IF(OR(B4 <= 0, B5 <= 0), "Invalid Mass Inputs", ROUND((B4 * B5) / (B4 + B5), 4))

Further reading

• Microsoft Support — ROUND function