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wave p-Ge THz laser was realized [193] under weak electric field pumping, accompanied by a complete theoretical interpretation [194].

The theory of population inversion based on the formation of resonant states has been given in detail very recently [195]. Under uniaxial stress the heavy hole band (HHB) in p-Ge lies lower than the light hole band (LHB) by an amount Edef. Dictated by the symmetry properties, one set of impurity levels is attached to the edge of the HHB, and another set is attached to the edge of the LHB. If Edef is greater than the hole binding energy of the lowest impurity level attached to the HHB edge, this impurity level overlaps the Bloch states in the LHB. Resonant states are then formed. Under an external electric field, the impact-ionized holes are accelerated toward the resonant level. With the proper combination of impurity concentration, temperature, electric field, stress, and scattering strength of phonons and impurities, holes have a large probability of occupying resonant states [192, 194, 195]. A resonant level population inversion with respect to those impurity levels which attach to the LHB edge is then formed, and THz lasing occurs [191, 193].

The externally applied uniaxial stress can be replaced by the strain in a QW with lattice mismatch, and a Si/GexSi1-X/Si QW with x < 0.2 and a boron-doped well was proposed [196, 197]. In such a structure the well is stretched along the growth direction, so the HHB lies above the LHB. When impurities are added to the SiGe/Si QW system, all relevant energy levels are affected by the electric field produced by the charge redistribution in the system. It is plausible that a strong electric field in the QW may favor the formation of resonant states for THz lasing. Such a field can be achieved by employing two aspects of the sample structure. First, the Si buffer layer and the Si cap layer on each side of the well will be 8 doped with boron acceptors. Second, the existence of a thin SiO2 layer on top of the cap layer creates interface states between them. In such a structure THz lasing was indeed detected [198].

Because the carriers in a SiGe/Si QW are electrically pumped, electric transport in the well has been investigated in detail in order to clarify the relevant physical processes [197, 199]. Under a weak dc bias, the temperature behavior of the conductivity a, plotted as ln(a) as a function of the inverse temperature 1/T, exhibits two different linear regimes, below and above T & 20 K, respectively, as shown by Figure 1 in [197]. Further experiments on magnetocon-ductivity and Hall mobility [197, 199] have indicated that the low temperature conductivity may be due to hopping. In the high temperature region, the slope of the ln(a) vs 1/T curves suggests an activation energy of about 12 meV for Ge content x = 0.1 and 18 meV for x = 0.15, if indeed the activation process exists. It was suggested in [197, 199] that the possible hole activation is between the two impurity levels Ehs and E[s attached to each respective band. The present study has disproved this suggestion. The mechanism of the high temperature conductivity and its relation to the THz lasing are the questions to be answered.

In [189] an extensive numerical study of a Si/Ge015Si0 85/Si QW was performed. The system structure consisted of, in sequence, an n-Si substrate, an i-Si buffer layer of 130 nm thickness, a Ge015Si0 85 well of 20 nm thickness, and an i-Si cap layer of 60 nm thickness. Over the cap layer a thin layer of SiO2 typically appears, on which ohmic contacts were installed. The middle of the well was 8 doped with a boron concentration of 6 x 1011 cm-2. In both the buffer layer and the cap layer, at a distance of 30 nm from the respective QW interface, a 8 layer of boron was doped with a concentration of 3 x 1011 cm-2. In this system Edef = 31 meV [200]. Using a one band variational approach given in [201], the binding energies E[s and Ehs of the impurity levels attached to the LHB edge and the HHB edge, respectively, were calculated and both were found to have a value of about 27 meV.

One very important feature of the QW system is the interface states between the SiO2 layer and the Si cap layer, which can accumulate almost all holes from the 8 doping in the cap later. This resulted in pinning of the chemical potential at the interface state energy level, which lay in the bandgap at A & 0.4 eV measured from the valence band top [202]. This charge redistribution inside the sample built up a strong electric field in the QW, which was detected experimentally [197, 199].

The energy level structure in the QW calculated is shown in Figure 69. The zero energy was set at the chemical potential ¡¡. The solid curve marks the HHB edge and the dotted curve the LHB edge. The lowest quantization energy levels Ehhb in the HHB and Ejhb in the LHB were indicated, along with the lowest impurity states attached to each band. In the case depicted (T = 4 K), the electric field strength in the QW was 19 kV/cm, and Ejhhb - E[s = 5 meV. The overlap of the impurity level E[s with the Ejhhb 2D subband made it possible to form the resonant state required for THz lasing. Once they proved that a resonant impurity state could be formed in the structure, the theory of population inversion developed for strained bulk p-Ge [195] could be applied directly to explain the origin of the observed lasing in the SiGe/Si QW.

In order to understand the transport properties of the SiGe/Si QW system, which is relevant for the electric pumping of carriers into the resonant states, they also calculated the concentration of free holes, pv(T), as a function of the temperature, and their results are plotted as the solid curve in the upper panel of Figure 70. The temperature

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