Box 1.

Box 1.

Semiconductor Elementary Excitations: Length and Energy Scales

The fundamental light-induced elementary excitations in semiconductors are Wannier excitons (X). These are objects with an internal structure. In an absorption process a photon promotes an electron from a valence state (mostly p-like) to a conduction state (mostly s-like), leaving a hole behind. However, because this process occurs in a crystal and not in the real vacuum, the particles have very small effective masses and feel the very large background dielectric constant. Thus, relative to the hole, the electron spreads over many sites following a distribution described by an hydrogenic envelope wavefunction. The exciton Bohr radius, a_o, and Rydberg, R_y, determine, respectively the volume it occupies and the distribution of electron and hole Bloch states it is made of. The situation sketched here for the exciton is similar to that encountered in the case of the other light-induced elementary excitations shown in Fig. 1. Much of the physics of semiconductors and their heterostructures is determined by these envelope wavefunctions whose energy, time, and length scales are very different from those of an atom. Order of magnitude considerations show that excitons and related quasi-particles are indeed unusual objects. For example in the model material GaAs, R_y = 4.2 meV and a_o = 14 nm, meaning that the exciton is distributed over several hundred thousand atomic sites. The energy of the photon necessary to create an exciton bound state is less than the energy gap between the top of the valence and the bottom of the conduction band. Thus these bound states appear in the absorption spectrum as strong resonances whose energy follow a Rydberg series E_n = E_g − R_y/(n²), followed by a continuum of unbound, but yet interacting, scattering states with a strongly enhanced absorption strength (see Fig. 1). It is worth noting that although the correlation energy scale is small, ∝ R_y ≈(10⁻³ − 10⁻²) × E_g, the Coulomb interaction produces major effects.

The fact that the natural energy and length scales are determined by the envelope wavefunctions offers the very interesting opportunities to perform on the elementary excitations operations that would require extreme conditions if performed on atomic systems. Again in the example of GaAs, the X ionization field, ℰ_I∝R_y/ea_o ≈1 V/1 μm, is less than the field existing in the active region of a diode, the ionization temperature is T = R_y/k_B = 50 K is such that X ionize at room temperature, the magnetic field at which the cyclotron radius equals a_o, B_c ≈3.4 T, is easily obtained with a modest commercial magnet, whereas for the hydrogen atom this magnetic field, B_c ≈10⁴ T, only exists at the surface of neutron stars. Another type of geometrical confinement impossible to achieve on atomic systems is now easily performed in artificial quantum structures, quantum wells, wires, and boxes, when one or more of the spatial dimensions, L, becomes comparable or smaller than a_o. Confined X then look like pancakes or cigars with modified properties (Box B2). In these quantum structures a number of other parameters become accessible to experiments. As discussed in the text, this is the case for the period of the Bloch oscillations and the amplitude of electronic wavepackets.

A further practical, but technologically crucial, interest of semiconductors for all modern opto-electronic applications stems from the fact that because of Coulomb enhancement, the absorption/gain coefficients are of the order of 10⁴ − 10⁵ cm⁻¹, i.e., very significant over 1-μm length scale combined with the latitude to manipulate and affect the envelope wavefunctions and therefore control the interaction of light with the material.

Semiconductor Elementary Excitations: Length and Energy Scales

The fundamental light-induced elementary excitations in semiconductors are Wannier excitons (X). These are objects with an internal structure. In an absorption process a photon promotes an electron from a valence state (mostly p-like) to a conduction state (mostly s-like), leaving a hole behind. However, because this process occurs in a crystal and not in the real vacuum, the particles have very small effective masses and feel the very large background dielectric constant. Thus, relative to the hole, the electron spreads over many sites following a distribution described by an hydrogenic envelope wavefunction. The exciton Bohr radius, a_o, and Rydberg, R_y, determine, respectively the volume it occupies and the distribution of electron and hole Bloch states it is made of. The situation sketched here for the exciton is similar to that encountered in the case of the other light-induced elementary excitations shown in Fig. 1. Much of the physics of semiconductors and their heterostructures is determined by these envelope wavefunctions whose energy, time, and length scales are very different from those of an atom. Order of magnitude considerations show that excitons and related quasi-particles are indeed unusual objects. For example in the model material GaAs, R_y = 4.2 meV and a_o = 14 nm, meaning that the exciton is distributed over several hundred thousand atomic sites. The energy of the photon necessary to create an exciton bound state is less than the energy gap between the top of the valence and the bottom of the conduction band. Thus these bound states appear in the absorption spectrum as strong resonances whose energy follow a Rydberg series E_n = E_g − R_y/(n²), followed by a continuum of unbound, but yet interacting, scattering states with a strongly enhanced absorption strength (see Fig. 1). It is worth noting that although the correlation energy scale is small, ∝ R_y ≈(10⁻³ − 10⁻²) × E_g, the Coulomb interaction produces major effects.

The fact that the natural energy and length scales are determined by the envelope wavefunctions offers the very interesting opportunities to perform on the elementary excitations operations that would require extreme conditions if performed on atomic systems. Again in the example of GaAs, the X ionization field, ℰ_I∝R_y/ea_o ≈1 V/1 μm, is less than the field existing in the active region of a diode, the ionization temperature is T = R_y/k_B = 50 K is such that X ionize at room temperature, the magnetic field at which the cyclotron radius equals a_o, B_c ≈3.4 T, is easily obtained with a modest commercial magnet, whereas for the hydrogen atom this magnetic field, B_c ≈10⁴ T, only exists at the surface of neutron stars. Another type of geometrical confinement impossible to achieve on atomic systems is now easily performed in artificial quantum structures, quantum wells, wires, and boxes, when one or more of the spatial dimensions, L, becomes comparable or smaller than a_o. Confined X then look like pancakes or cigars with modified properties (Box B2). In these quantum structures a number of other parameters become accessible to experiments. As discussed in the text, this is the case for the period of the Bloch oscillations and the amplitude of electronic wavepackets.

A further practical, but technologically crucial, interest of semiconductors for all modern opto-electronic applications stems from the fact that because of Coulomb enhancement, the absorption/gain coefficients are of the order of 10⁴ − 10⁵ cm⁻¹, i.e., very significant over 1-μm length scale combined with the latitude to manipulate and affect the envelope wavefunctions and therefore control the interaction of light with the material.