Application of transfer matrices to surface states in finite periodic systems of quantum wells
For a system of N coupled quantum wells an exact energy-dispersion relation in terms of composite functions was derived by Liboff and Seidman 1 (1990). The essential qualitative feature of this equation was the emergence of a band structure for only a few quantum wells. Therefore one can consider this an embryonic band structure of any lattice with short range peridocity. Using transfer matrices one can find a parallel result for the wave function, that is, an exact closed form solution with N as an explicit variable. Based on a result of Abelès2 (1950), the transfer matrix for N arbitrary barriers can be expressed in terms of the transfer matrix of a single barrier multiplied by Chebyshev polynomials of degree N - 1 and N - 2. This expression for the wavefunction allows one to write an exact energy-dispersion relation for a finite-periodic system of N-coupled quantum wells for a general barrier shape. Once the transfer matrix for this barrier is determined, an exact algebraic expression can be written, the roots of which are the eigenenergies. This result represents a generalization of the expression in Liboff & Seidman (1990) since there the expression was specifically for rectangular quantum wells whereas this expression will work for any particular quantum well so long as the transfer matrix across the barrier separating the wells is known. From the eigenergies, the wavefunctions can then be written in terms of the recursive expression for the transfer matrix to determine if there are surface states (or surface resonances). A comparison of this methodology is made to that of Tamm's direct matching procedure for a semi-infinite lattice.
1R. L. Liboff & S. R. Seidman, "Exact energy-dispersion relations for N-well superlattice configurations," Phys Rev B, vol. 42, 1990, p. 9552. 2F. Abelès, "Reserches sur la propagation des onde électromagnétiques sinusoidales dans les milieux stratifiés", Ann. Phys. Paris, vol. 5, 1950 p. 706.