Optical Nonlinearities in Semiconductors Enhanced by Carrier Transport

Elsa Garmire    Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire

I Introduction

This chapter is based on the photoexcitation of free carriers, in which some fraction of the light incident on a semiconductor is absorbed, creating electron–hole pairs. This photogeneration of carriers causes a change in the optical properties of the semiconductor so that subsequent light sees a change in absorption and/or refractive index as a result of the existence of the photocarriers. A brief review of such semiconductor nonlinearities was published in Physics Today (Garmire, 1994).

This chapter includes only resonant processes and excludes virtual excitations, in which no physical electronic transitions take place and which may be nonresonant. The requirement of real physical excitation of photocarriers means that the nonlinearities must have the speed of electronic excitation times, typically picoseconds. The nonlinearities will last as long as the optically excited carriers remain. Depending on the device structure, this can be anything from 10 ps to milliseconds. The ability to control the response time by device design is one advantage of semiconductor nonlinearities that use carrier transport, the focus of this chapter. The applications that motivate the study of these nonlinearities are those that require a high degree of parallelism, with low operating intensities and moderate speeds. That is, this study will exclude the femtosecond and picosecond virtual processes that require very high optical field strengths, and consequently face issues of multiphoton absorption. The high degree of parallelism means that two-dimensional geometries are required, ruling out waveguides. Therefore, the geometry under consideration in this chapter is thin semiconductor films, grown epitaxially and illuminated at normal incidence, such as shown in Fig 1. By convention, the direction of crystal growth is labeled z, and the thickness of quantum wells is designated Lz.

Direct-band semiconductors are usually chosen, because their band edge is sharper than indirect-band semiconductors and the optical absorption between direct bands is strongly influenced by electric field.

1 LOCAL NONLINEARITIES ENHANCED BY CARRIER TRANSPORT

In semiconductors, optical nonlinearities can be local or nonlocal (Garmire et al., 1989). In local nonlinearities, optical excitation changes the absorption and refractive index of the semiconductor locally; that is, photogenerated carriers stay where they are and fill available states, reducing the absorption, as discussed in the preceding chapter. The amount of state filling depends on the photocarrier density. Thus, if photocarriers can live longer, the optical intensity required to maintain the same level of optical nonlinearity can be lower. That is, the material has a higher sensitivity. Carrier recombination rates can be reduced if photogenerated holes can be separated from electrons, which reduces the probability of recombination.

The localized filling of available states in bulk semiconductors (Lee et al., 1986) and in quantum wells (Chemla et al., 1984) causes a change in the absorption coefficient α that can be described heuristically by a simple saturation as a function of photocarrier density N (Kawase et al., 1994, Park et al., 1988):

N=αu+αs1+N/Ns

consisting of an unsaturable part αu and a saturable part αs, where Ns is the saturation value of the photocarrier density. Writing δα(N) ≡ α(0) − α(N) = αs(N/Ns)/(1 + N/Ns), experimental data can be most easily fit to the saturation form if 1/δα is plotted vs 1/N. Then the intercept gives 1/αs, and the saturation carrier density can be determined from the slope.

The refractive index has a similiar form:

N=nu+ns1+N/Ns

The photocarrier density N is related to the incident intensity I by the absorption and the recombination time of the carriers, τ:

=αIτ/hv

where hv is the photon energy. The time τ is determined by the joint probability of electrons and holes being available for recombination. Therefore, if carrier transport can be used to increase τ, then the required intensity I decreases, since the scaling is proportional to Iτ. If τ can be controlled through carrier transport, then the nonlinearity can be optimized to achieve as much sensitivity as possible for a given required response time.

When experimental data are given in terms of the intensity dependence of the absorption change, the saturated absorption change, αs, can be most easily found by plotting log(δα) vs. log(I) and extrapolating to large log(I). The saturation intensity occurs when the absorption change reaches half its saturation value, and can be found from the value at which log(δα) = log(αs/2) = log(αs) − 0.3.

The use of carrier transport to control recombination time by separating photogenerated electrons and holes in semiconductors can be understood with reference to Fig. 2(a). An undoped quantum well (QW) sits in an n region, surrounded by the tilted bands from two back-to-back p-i-n junctions, as shown. The tilt of the conduction and valence bands (C.B. and V.B., respectively) is created by the equalization of the Fermi levels. The undoped QW has a smaller band gap than the rest of the material, and its resonant absorption creates photocarriers in the QW. The electrons remain trapped in the well, but the holes tunnel out and float up to the p-doped region. The resulting local concentration of electrons in the QW causes the available states to fill, as discussed in Chap. 1, whereas removal of the holes means that recombination times become very long. This results in very sensitive state-filling nonlinearities, although at the sacrifice of much slower response times, which can be lengthened from nanoseconds to milliseconds, decreasing saturation intensities from kilowatts to milliwatts.

Since state filling depends on the electron density, and each photon creates a single electron that lasts until it recombines, there is a direct tradeoff between sensitivity (defined by the required intensity to fill available states) and reponse time. In fact, for times less than the recombination time, the fluence F at the excitonic resonance wavelength required to cause a given change in absorption or refractive index would be the same as in undoped material, since Iτ in Eq. 3 becomes F.

2 NONLOCAL NONLINEARITIES

Carrier transport in the presence of static fields causes electrons and holes to move out of their local region, with the resultant charge separation screening the static field. This nonlocal self-modulation of the electric field results in a large nonlocal change in refractive index and/or absorption near the...