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We used microwaves to apply an electric field to semiconductor samples without using electrical contacts. High-frequency electric fields ionize impurities just as d.c. fields do4, but they do not impose inhomogeneities like electric contacts. We find that in thin n-type gallium arsenide (GaAs) epitaxial layers subjected to a uniform microwave field, circular spots of enhanced free electron density with sharp boundaries are spontaneously formed above a critical microwave threshold power. This new type of self-organized free-carrier density pattern is different from current filaments in that they are currentless; they also differ from electron-hole drops5 as only one type of charge carrier is involved.

The spatial patterns in free electron density were made visible by photoluminescence quenching6. Samples cooled to low temperatures (1.8 K) were illuminated by interband light and photographed in the spectral range of the luminescence of exciton recombination and donor-acceptor transitions. With increasing microwave power P, a decrease in photoluminescence occurs at a threshold value P+ as a result of an almost circular spot of enhanced electron density with a diameter (D+) of about 1 mm (Fig. 1b). The diameter of the spot at the threshold is always finite and independent of the size of the semiconductor sample. This pattern formation was observed in doped samples with impurity densities of about 1015 per cm3, but not in ‘ultrapure’ material with an impurity density of about 5×1012 per cm3.

Figure 1: Self-organized formation of spots of high electron density in an n-doped gallium arsenide epitaxial layer as a result of impact ionization of shallow donors in a microwave electric field.
figure 1

a-d, Images of near-infrared luminescence for microwave power increasing from zero (a) to about 30 mW (d). Shaded stripes show the walls of a waveguide. e, Spot diameter as a function of microwave power. Red and blue triangles indicate increasing and decreasing power, respectively. P+ is of the order of 10 mW.

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When the microwave power is increased above the threshold P+, the diameter of the original spot increases (Fig.1b,c); at certain values of P, additional spots appear at a distance of 1 to 3 mm (Fig.1d). Decreasing the microwave power after the first spot has formed makes it smaller until it vanishes at a power of P-<P+, at a diameter D-<D+(Fig. 1e). The pattern formation process shows a hysteretic behaviour, as it is characteristic for a first-order phase transition.

The quenching of the photoluminescence in the spots indicates that the average energy of electrons is high enough to ionize impurities and excitons in the spots. The observed structures therefore correspond to spots of high free electron density. We verified this using a sample with two parallel stripe contacts at opposite edges. When a voltage was applied across the contacts, a current filament was formed in addition to the microwave-induced spot. The current filament goes through the spot, indicating that the observed structures are regions of high free electron density.

Our findings indicate that the physical background of microwave-induced pattern formation in the electron density is the same as that of current filamentation. Both of these phenomena are based on a bistability of conductivity combined with a constraint of the external driving mechanism. In the case of current filaments, the constraint is provided by the finite current applied to the sample, whereas for microwave-induced pattern formation the external constraint must be the limited power supply from the field.

Bistability and the possibility of spatial modulation of semiconductor conductivity have been attributed to three different physical mechanisms: multilevel generation and recombination kinetics of impurities2, runaway of electron energy due to energy-dependent electron scattering7, and electron density-dependent screening of ionized impurity scattering4. For all three mechanisms, bistability vanishes below a critical density of impurities, which explains the qualitatively different behaviour of doped and ultrapure materials, for which pattern formation has not been observed.