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The above discussion has been a simple introduction to the modelling of I-V curves for barrier structures, but nonetheless it shows some of the features of real devices. For a much more complete and in depth study see, for example, Mizuta and Tanoue [49]. 2.16 EXTENSION TO INCLUDE ELECTRIC FIELD
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An obvious improvement to the above model would be to account for the changes in the transmission coefficient as a function of the applied electric field. By using the substitution as before (equation (2.150)), i.e.
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the solution in each region can then be written as a linear combination of Airy functions, just as for the general electric field case of equation (2.152), i.e.
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while Airy functions can be difficult to work with numerically, an immediate advantage over the zero-field case is that this solution is valid for both E < V and E > V, and hence generalisation to this form produces two benefits. The method of solution is analogous to the zero-field case, in that application of the BenDaniel-Duke boundary conditions yields two equations for each interface, which in this case gives eight equations. The unknown coefficients, A and B, are linked to K and L as before by forming the transfer matrix, and are solved by imposition of a boundary condition, which is again a travelling wave in the direction of +z to the right of the barrier structure. It is left to the interested reader to follow through such a derivation. A very general implementation for multiple barrier structures has been reported in the literature by Vatannia and Gildenblat [39]. 2.17 MAGNETIC FIELDS AND LANDAU QUANTISATION
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If a magnetic field is applied externally to a non-magnetic semiconductor heterostructure then the constant effective mass Hamiltonian (familiar from equation (2.5)):
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which can be written: becomes [50-53]:
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where the kinetic energy operator becomes modified by the magnetic field vector potential A, and the second term produces a splitting, known as the 'gyromagnetic spin splitting' between the spin-up ( sign) and spin-down (+- sign) electrons, g* is known as the 'Landau factor' and is really a function of z as it depends on the material, however it is generally assumed to be constant and approximately 2 for conduction band electrons, HB is the Bohr magneton and B is the magnitude of the magnetic flux density which is assumed aligned along the growth (z-) axis. Although the heterostructure potential V(z) remains one-dimensional, the vector potential means that the wave functions are not necessarily one-dimensional so the Schrodinger equation must be written:
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The magnetic field produces a parabolic potential along one of the in-plane axes, the x-axis say, leaving the particle free to move (with a wave vector ky, say) along the other axis. The standard approach is to employ the Landau gauge A = Bxey, then following the notation of Savic et al. [54] the wave function can be written in the separable form
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where Ly is a normalisation constant (the length of the structure along the y-axis, where is the Landau length, j is an index over the harmonic oscillator solutions of the parabolic potential and w(z) is the usual one-dimensional envelope function of the heterostructure potential without the magnetic field. The harmonic oscillator solutions will be generated numerically in Section 3.5, however they can also be expressed analytically as:
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where Hj is the jth Hermite polynomial, see Liboff [55] page 200 for a detailed description of the analytical solutions of the harmonic oscillator. Taking g* as a constant equal to 2 and the Bohr magneton as 9.274 x 10~24 JT- 1 then it can be seen that even in a relatively high magnetic field of 10 T, the difference in energy between the spin-up and spin-down electrons is:
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