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Figure 2.10 Effect of electron density, N=l, 2, 5, 10 (x 1010) cm- 2 , on the distribution function of the lowest subband of the infinite quantum well of Fig. 2.7
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Figure 2.11 Effect of electron density on the quasi Fermi energy describing the distribution of the ground state E\
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whose only physical meaning is to describe the population within a subband it is not the true Fermi energy of the complete system. Fig. 2.10 displays the distribution functions for a range of carrier densities, for this same ground state and at a lattice temperature of 77 K. Although not obvious
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from the mathematics, f FD (E) at any particular energy E appears to scale with N. The corresponding Fermi energy is illustrated in Fig. 2.11. Clearly, the Fermi energy starts below the subband minima at this mid-range temperature, as discussed above, and as expected increasing numbers of carriers in the subband increases the Fermi energy, i.e. the energy of the state whose probability of occupation is 1/2. 2.5 FINITE WELL WITH CONSTANT MASS
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Figure 2.12 Solutions to the finite well potential
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While the infinitely deep confining potential has served well as a platform for developing the physics of two-dimensional systems, more relevant to alternating layers of dissimilar semiconductors is the finite quantum well model, which under both the effective mass and envelope function approximations looks like Fig. 2.12. In particular, a layer of GaAs 'sandwiched' between two thick layers of Gai-xA1xAs would form a type-I finite quantum well, where the conduction band has the appearance of Fig. 2.12, with the potential energy V representing the discontinuity in the conduction band edge between the materials. Again taking the simplest starting case of a constant electron mass m* throughout the dissimilar layers, and neglecting movement within the plane of the layers, then the standard Schrodinger equation can be written for each of the semiconductor layers as follows:
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Considering solutions to the Schrodinger equation for the central well region, then as in the infinite well case, the general solution will be a sum of sine and cosine terms.
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As the potential is symmetric, then the eigenstates will also have a definite symmetry, i.e. they will be either symmetric or antisymmetric. With the origin placed at the centre of the well, the symmetric (even parity) eigenstates will then be in cosine terms, while the antisymmetric (odd parity) states will be as sine waves. For states confined to the well, the energy E must be less than the barrier height V, thus rearranging the Schrodinger equation for the right hand barrier:
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Therefore, a function / is sought which when differentiated twice gives +/. The exponential function fits this description, therefore consider a sum of growing exp (+KZ) and decaying exp ( K Z ) exponentials. In the right-hand barrier, z is positive, and hence as z increases the growing exponential will increase too and without limit. The probability interpretation of the wave function requires that:
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These boundary conditions for states confined in wells will be used again and again and will be referred to as the standard boundary conditions. Using this result, the growing exponential components must be rejected and the solutions are for the even parity states, which would follow as:
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Note for later that these wave functions are real, and that the eigenfunctions of this confined system carry no current and hence are referred to as stationary states. Using these trial forms of the wave function in their corresponding Schrodinger equations, gives the, as yet unknown constants:
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In order to proceed it is necessary to impose boundary conditions. Recalling the constant mass kinetic energy operator employed in equations (2.56)-(2.58), then in order to avoid infinite kinetic energies: both must be continuous
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