Quantum-confined Stark effect (QCSE)

The quantum-confined Stark effect (QCSE) is a very strong and clear quantum mechanical phenomenon that is seen in quantum well semiconductor structures with applied electric fields. It leads to large changes in optical absorption with applied voltage. It is easily seen at room temperature in quantum wells made from a wide range of semiconductors, such as the III-V semiconductors GaAs and InGaAs, or, more recently germanium. In practice, the quantum wells are contained in diode structures, and reverse biasing the diodes with voltages ~ 1 – 10 V allows strong optical modulation. The mechanism is fast, having been tested to picosecond time scales, and it is arguably the strongest high-speed optical modulation mechanism, allowing compact, low-energy devices. QCSE modulators (discussed in more detail here) are used extensively in optical fiber telecommunications.

QCSE mechanism

The QCSE was discovered, explained, and named by David Miller and his colleagues. The mechanism is seen primarily in the shift of the lowest energy valence-to-conduction band optical transitions, with the majority of the shift coming from the “particle-in-a-box” quantum mechanics as the “boxes” or quantum well potentials for electrons and holes are skewed by the electric field applied perpendicular to the quantum well layers, leading to changes in the energies of the confined states.

The detailed mechanism exploits also the excitonic effects in the material. The actual optical absorption is strongly enhanced by the fact that the absorbed photons create electron-hole pairs that are attracted to one another by Coulomb electrostatic attraction (excitons). That attraction increases the probability of finding the electron and hole in the same place, which, in the excitonic model of optical absorption, greatly increases the strength of the transitions that create these excitons, giving excitonic enhancement of optical absorption. Notably, that absorption leads to particularly strong absorption peaks just below the nominal separation energy of electron and hole states. The main peak is usually associated with creating the so-called 1S exciton, a bound electron-hole pair state analogous to a 1S hydrogen atom, but with a much lower binding energy (on the scale of a few meV to a few 10’s of meV in many semiconductors) and a larger size scale (e.g., ~ 10 nm).

Excitonic effects are present in normal “bulk” semiconductors, and enhance the optical absorption in them even at room temperature, but for most bulk semiconductors those peaks are only clear at low temperature. In quantum wells, however, because the excitons are “squeezed” even smaller by the quantum well layers, the excitonic peaks are visible clearly even at room temperature. In bulk semiconductors, applying electric fields leads to very rapid field-ionization of excitons, so in practice electric fields cause those peaks to broaden (a kind of “lifetime broadening”) and disappear. However, in quantum wells, the walls of the well hold the exciton together even as the field is applied. As a result, the peaks do not broaden much with field, and they continue to enhance the absorption even as they are shifted.

The correct physical description of the shifts of the absorption with electric field is a shift of the excitonic state energy with field, which, by analogy with the hydrogen atom, is a Stark shift. Because the exciton continues to exist even to very high fields (because the walls of the wells hold it together), extremely large such Stark shifts are possible, up to even many times the original binding energy of the exciton. Hence the mechanism is named the quantum-confined Stark effect.

As mentioned above, a relatively good approximation to the shifts can be obtained by calculating the shifts of the individual electron and hole levels in the wells; the exciton peaks remain “attached” to steps in the optical absorption that we would calculate in a non-excitonic model. There is also a slight shift in the exciton binding energy – it is reduced because the electron and hole are pulled apart from one another towards the opposite walls of the well – but this is generally a small correction (a few meV) and is often neglected. However, the excitonic effects in the quantum well case are particularly important and special in greatly enhancing the optical absorption, so the underlying excitonic physics is important.

In addition to the shift of the lowest energy transition, the “overlap integral” between the electron and hole confined wavefunctions reduces as the wavefunctions are pulled to opposite sides of the well. This causes a reduction of the “step height” in the optical absorption. There is, however, and overall sum rule on these step heights, and the step height lost on the lowest transition is picked up on “forbidden” transitions – so, for example, on the transition from the second confined hole level to the first confined electron level. These nominally forbidden transitions, for which the overlap integral is small or zero at low field, become partly allowed with increasing field.

The QCSE is usually observed in simple “rectangular” quantum wells, but other shapes of potentials show related effects – for example, coupled quantum wells.

The same effect will also exist in quantum “wires” (structures confined in two dimensions) and quantum “dots” or “boxes” (structures confined in all three dimensions) for fields applied along any of the “confined” directions. (Quantum dots or boxes are also known as microcrystallites.)

Electro-refractive (“electro-optic”) effects

The QCSE does also show strong associated refractive effects that are also used for interferometric devices. Changes in refractive index are essentially inevitable whenever there are changes in the absorption spectrum. Such changes can be calculated using the Kramers-Kronig relations based on measured absorption spectra. In this early theoretical paper by David Miller and his colleagues [a], Kramers-Kronig calculations, supported by electroabsorption sum rules [b], are used to predict the size and spectral behavior of such effects. Calculations and modeling in this work showed the feasibility of such an electro-refractive effect (or what is more commonly called an electro-optic effect) in quantum wells for compact high-speed phase shifters for use in optical modulators, such as Mach-Zehnder interferometers (MZIs).

Now, generally, such changes in refractive index can be accompanied by some amount of change in absorption, and absorption, and changes in it, are generally undesirable in phase shifting devices. Importantly, though, this work clarified that, by moving somewhat to photon energies further below the bandgap energy (i.e., to longer wavelengths), the absorption and changes in it decrease faster than the resulting changes in refractive index. So, phase modulation with little or no absorption or absorption change can be accomplished in quantum wells by the choice of photon energy and/or bandgap energy. Moving the photon energy further below the bandgap energy does reduce the amount of refractive index change, requiring longer devices, but the mechanism is already so strong that lengths ~ 500 microns, for example, are quite sufficient for large optical phase shifts with only small absorption effects.

QCSE in visible light-emitting diodes

The same QCSE physical mechanism is also seen in semiconductor structures that have built-in electric fields resulting from the underlying symmetry properties of the materials. In semiconductors with wurtzite crystal structures, such as the InGaN family of materials used in visible light emitting diodes, such fields exist and the resulting separation of electron and hole wavefunctions can reduce the strength of the optical emission transitions.

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood and C. A. Burrus, “Bandedge Electro-absorption in Quantum Well Structures: The Quantum Confined Stark Effect,” Phys. Rev. Lett. 53, 2173‑2177 (1984).

D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood and C. A. Burrus, “Electric Field Dependence of Optical Absorption near the Bandgap of Quantum Well Structures,” Phys. Rev. B32, 1043‑1060 (1985).

J. S. Weiner, D. A. B. Miller, D. S. Chemla, T. C. Damen, C. A. Burrus, T. H. Wood, A. C. Gossard and W. Wiegmann, “Strong Polarization‑Sensitive Electroabsorption in GaAs/AlGaAs Quantum Well Waveguides,” Appl. Phys. Lett. 47, 1148‑1150 (1985).

W. H. Knox, D. A. B. Miller, T. C. Damen, D. S. Chemla, C. V. Shank and A. C. Gossard, “Subpicosecond Excitonic Electroabsorption in Room-Temperature Quantum Wells,” Appl. Phys. Lett. 48, 864‑866 (1986).

D. A. B. Miller, D. S. Chemla and S. Schmitt-Rink, “Relation Between Electroabsorption in Bulk Semiconductors and in Quantum Wells: The Quantum-Confined Franz-Keldysh Effect,” Phys. Rev. B33, 6976‑6982 (1986).

[b] D. A. B. Miller, J. S. Weiner and D. S. Chemla, “Electric Field Dependence of Linear Optical Properties in Quantum Well Structures: Waveguide Electroabsorption and Sum Rules,” IEEE J. Quantum Electron. QE‑22, 1816‑1830 (1986).

[a] J. S. Weiner, D. A. B. Miller, and D. S. Chemla, “Quadratic Electro-Optic Effect due to the Quantum-Confined Stark Effect in Quantum Wells,” Appl. Phys. Lett. 50, 842‑844 (1987).

M. N. Islam, R. L. Hillman, D. A. B. Miller, D. S. Chemla, A. C. Gossard, and J. H. English, “Electroabsorption in GaAs/AlGaAs Coupled Quantum Well Waveguides,” Appl. Phys. Lett. 50, 1098‑1100 (1987).

I. Bar-Joseph, C. Klingshirn, D. A. B. Miller, D. S. Chemla, U. Koren, and B. I. Miller, “Quantum-Confined Stark Effect in InGaAs/InP Quantum Wells Grown by Organometallic Vapor Phase Epitaxy,” Appl. Phys. Lett. 50, 1010‑1012 (1987).

D. S. Chemla, D. A. B. Miller, and S. Schmitt-Rink, “Generation of Ultrashort Electrical Pulses through Screening by Virtual Populations in Biased Quantum Wells,” Phys. Rev. Lett. 59, 1018‑1021 (1987)

D. A. B. Miller, D. S. Chemla, and S. Schmitt-Rink, “Electroabsorption of highly confined systems: Theory of the quantum-confined Franz-Keldysh effect in semiconductor quantum wires and dots,” Appl. Phys. Lett. 52, 2154‑2156 (1988).

S. Schmitt-Rink, D. S. Chemla, and D. A. B. Miller, “Linear and nonlinear optical properties of semiconductor quantum wells,” Adv. Phys. 38, 89‑188 (1989)

I. Bar-Joseph, K. W. Goossen, J. M. Kuo, R. F. Kopf, D. A. B. Miller, and D. S. Chemla, “Room-temperature electroabsorption and switching in a GaAs/AlGaAs superlattice,” Appl. Phys. Lett. 55, 340‑342 (1989)

S. Schmitt-Rink, D. S. Chemla, W. H. Knox, and D. A. B. Miller, “How fast is excitonic electroabsorption?” Optics Lett. 15, 60‑62 (1990)

S. L. Chuang, S. Schmitt-Rink, D. A. B. Miller, and D. S. Chemla, “Exciton Green’s-function approach to optical absorption in a quantum well with an applied electric field,” Phys. Rev. B, 43, 1500‑1509, (1991)

A. Partovi, A. M. Glass, D. H. Olson, R. D. Feldman, R. F. Austin, D. Lee, A. M. Johnson, and D. A. B. Miller, “Electroabsorption in II‑VI Multiple Quantum Wells,” Appl. Phys. Lett., 58, 334‑336, (1991)

Y.-H. Kuo, Y.-K. Lee, Y. Ge, S. Ren, J. E. Roth, T. I. Kamins, D. A. B. Miller and J. S. Harris, “Strong quantum-confined Stark effect in germanium quantum-well structures on silicon,” Nature 437, 1334-1336 (2005) https://doi.org/10.1038/nature04204

R. K. Schaevitz, E. H. Edwards, J. E. Roth, E. T. Fei, Y. Rong, P. Wahl, T. I. Kamins, J. S. Harris, and D. A. B. Miller, “Simple Electroabsorption Calculator for Designing 1310nm and 1550nm Modulators Using Germanium Quantum Wells,” IEEE J. Quantum Electron. 48, 187 – 197 (2012)  https://doi.org/10.1109/JQE.2011.2170961