Franz-Keldysh effect

The Franz-Keldysh effect is the modification of interband absorption in a bulk semiconductor by a uniform electric field, predicted independently by W. Franz and L. V. Keldysh in 1958. In the field the electron and hole wavefunctions are no longer plane waves; they become Airy functions whose tails leak into the band gap. A photon with energy slightly below the gap can then create an electron-hole pair with the help of the field, a process equivalent to absorption combined with tunneling. The absorption edge acquires an exponential tail below the band gap, and above the gap the absorption spectrum develops weak oscillations. At fields of order 100 kV/cm, 1 V across 100 nm, the tail extends several tens of meV below a direct gap.

Characteristic energy and the absorption tail

The field enters through the electro-optic energy

$$\hbar\theta = \left(\frac{e^2\hbar^2F^2}{2\mu}\right)^{1/3},$$

where $F$ is the field and $\mu$ the reduced effective mass of the electron-hole pair. Below the gap the absorption coefficient falls approximately as

$$\alpha \propto \exp\!\left[-\frac{4}{3}\left(\frac{E_g-\hbar\omega}{\hbar\theta}\right)^{3/2}\right].$$

For the direct gap of germanium, with an electron mass of 0.038 $m_0$ and a light-hole mass of 0.043 $m_0$, $\mu \approx 0.020\,m_0$. At $F$ = 100 kV/cm this gives $\hbar\theta \approx 57$ meV; for heavy holes, $\mu \approx 0.034\,m_0$, it is 48 meV. The energy scales as $F^{2/3}$, so halving the field reduces it by about 37 %. At 20 meV below the gap the exponential factor is 0.76 at 100 kV/cm and 0.58 at 50 kV/cm, so the tail is strongly field dependent over a span comparable to $\hbar\theta$. Light-mass materials respond most, because a small $\mu$ increases $\hbar\theta$.

Observation in the lab

The effect is measured as a change in transmission or photocurrent with applied bias. A reverse-biased p-i-n diode, with the absorbing layer in the intrinsic region, is the usual structure: reverse bias sets the field, and the photocurrent spectrum at several biases shows the edge broadening toward longer wavelengths. Electroreflectance spectroscopy uses the oscillations above the gap, the Franz-Keldysh oscillations, whose period depends on $\hbar\theta$; fitting them gives the built-in field at a surface or interface and the reduced mass. Excitonic effects modify the simple picture: at low fields a sharp exciton peak broadens and is ionized as the field rises, which the one-electron theory above does not include.

Electro-absorption modulators in germanium and SiGe

In III-V devices the quantum-confined Stark effect is preferred, because confinement keeps the excitonic edge sharp while it shifts, giving a larger absorption change per volt. On silicon, where growing quantum wells is harder, bulk germanium or germanium with a small fraction of silicon uses the Franz-Keldysh effect instead. The direct gap of germanium, 0.80 eV, corresponds to 1550 nm, and tensile strain from growth on silicon pulls it slightly lower; adding silicon moves the edge back so that the steep part of the tail sits at the operating wavelength. Such electro-absorption modulators are compact, have low capacitance and switch on the time scale of the RC time constant, since the absorption change itself follows the field almost instantly. Their drawbacks are a narrow optical bandwidth, set by how far the operating wavelength sits from the edge, insertion loss from residual absorption at zero bias, and strong temperature dependence: the gap shifts with temperature, so a fixed-wavelength modulator drifts relative to the edge.

In silicon itself the Franz-Keldysh effect is weak at 1550 nm, far below the band edge, and plasma dispersion dominates. Germanium photodetectors on silicon also show the effect: under high reverse bias their responsivity at long wavelengths near the edge rises.

Pitfalls

• Field nonuniformity: the formula assumes a uniform field, and doping gradients or depletion edges spread the response.
• Heating: photocurrent in a modulator dissipates power and shifts the gap, which moves the operating point.
• Confusing the two effects: in thin layers the Franz-Keldysh and quantum-confined Stark effects merge, and the label depends on whether confinement energies are small or large compared with $\hbar\theta$.

Common questions

What is the difference between the Franz-Keldysh effect and the quantum-confined Stark effect?

The Franz-Keldysh effect occurs in bulk material and broadens the edge into a tail; the quantum-confined Stark effect occurs in quantum wells, where the barriers keep the exciton bound so that the edge shifts to lower energy while staying sharp.

Does the Franz-Keldysh effect change the refractive index?

Yes. Through the Kramers-Kronig relations the absorption change is accompanied by an index change, largest just below the gap, which produces chirp in electro-absorption modulators.

How does the Franz-Keldysh effect relate to electroabsorption?

The term covers any field-induced change of absorption; the Franz-Keldysh effect is its bulk form.

References: W. Franz, Z. Naturforsch. A 13, 484 (1958); L. V. Keldysh, Sov. Phys. JETP 7, 788 (1958); S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009); J. Liu et al., "Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators," Nature Photonics 2, 433 (2008).