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Figure 2.7 The density of states as a function of energy for a 200 A GaAs quantum well surrounded by infinite barriers
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carriers are fermions, then clearly the probability of occupation of a state is given by Fermi-Dirac statistics; hence:
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where the integral is over all of the energies of a given subband and, of course:
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Note that Ep is not the Fermi energy in the traditional sense [1]; it is a 'quasi' Fermi energy which describes the carrier population within a subband. For systems left to reach equilibrium, the temperature T can be assumed to be the lattice temperature; however this is not always the case. In many quantum well devices which are subject to excitation by electrical or optical means, the 'electron temperature' can be quite different from the lattice temperature, and furthermore the subband population could be non-equilibrium and not able to be described by Fermi-Dirac statistics. For now, however it is sufficient to discuss equilibrium electron populations and assume that the above equations are an adequate description. Given a particular number of carriers within a quantum well, which can usually be deduced directly from the surrounding doping density, it is often desirable to be able to describe that distribution in terms of the quasi-Fermi energy EF. With this aim substitute the two-dimensional density of states appropriate to a single subband from equation (2.46) into equation (2.48), then the carrier density, i.e. the number
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Figure 2.8 Effect of temperature on the distribution functions of the subband populations (all equal to 1 x 1010cm~2) of the infinite quantum well of Fig. 2.7
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Figure 2.9 Effect of temperature on the quasi-Fermi energy describing the electron distribution of the ground state E1
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per unit area, is given by:
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By putting:
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equation (2.50) becomes:
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which is a standard form (see for example, Gradshteyn and Ryzhik [23] equation 2.313.2, p. 112)
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Hence:
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Evaluation then gives:
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The minimum of integration E^n is taken as the subband minima and the maximum JE^max can either be taken as the top of the well, or even Ep + lOKT, say, with the latter being much more stable at lower temperatures. Given a total carrier density N, the quasi-Fermi energy EF is the only unknown in equation (2.55) and can be found with standard techniques. For an example of such a method, see Section 2.5 Fig. 2.8 gives an example of the distribution functions fFD(E1) for the first three confined levels within a 200 A GaAs infinite quantum well. As the density of carriers, in this case electrons, have been taken as being equal and of value 1 x 1010cm-2, then the distribution functions are all identical, but offset along the energy axis by the confinement energies. As mentioned above, at low temperatures the carriers tend to occupy the lowest available states, and hence the transition from states that are all occupied to those that are unoccupied is rapid as illustrated by the 2 K data for all three subbands. As the temperature increases the distributions broaden and a range of energies exist in which the states are partially filled, as can be seen by the 77 and 300 K data. Physically this broadening occurs due mainly to the increase in electronphonon scattering as the phonon population increases with temperature (more of this in 9). Fig. 2.9 displays the Fermi energy EF as a function of temperature T for the ground state of energy E\ = 14.031 meV. At low temperatures, EF is just above the confinement energy, since the electron density is fairly low (1 x 1010 cm- 2 ). As the temperature increases, EF falls quite markedly and below the subband minima. If this seems counterintuitive, it must be remembered that EF is a quasi-Fermi energy
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