Abstract
We consider electrons in the presence of interfaces with different effective electron mass, and electromagnetic fields in the presence of a high-permittivity interface in bulk material. The equations of motion for these dimensionally hybrid systems yield analytic expressions for Green’s functions and electromagnetic potentials that interpolate between the two-dimensional logarithmic potential at short distance, and the three-dimensional r−1 potential at large distance. This also yields results for electron densities of states which interpolate between the well-known two-dimensional and three-dimensional formulas. The transition length scales for interfaces of thickness L are found to be of order Lm/2m* for an interface in which electrons move with effective mass m*, and for a dielectric thin film with permittivity in a bulk of permittivity . We can easily test the merits of the formalism by comparing the calculated electromagnetic potential with the infinite series solutions from image charges. This confirms that the dimensionally hybrid models are excellent approximations for distances r ≳ L/2.
Highlights
When we suppress motion of particles in certain directions through confining potentials, e.g. in quantum wells or quantum wires, we often model the residual low energy excitations in the system through low-dimensional quantum mechanical systems. Prominent examples of this concern layered heterostructures, and one instance where the number d of spatial dimensions enters in a manner which is of direct relevance to technology is in the density of states
I.e. if the states only probe length scales smaller than the transition length scale ‘, we find the two-dimensional density of states properly rescaled by a dimensional factor to reflect that it is a density of states per three-dimensional volume, 8mE‘2 ) h2 : m1 .ðE; z0Þ ! HðEÞ4ph2‘ 1⁄4 4‘.ðd1⁄42ÞðEÞ: ð12Þ
I.e. if the states probe length scales larger than ‘, we find the three-dimensional density of states
Summary
When we suppress motion of particles in certain directions through confining potentials, e.g. in quantum wells or quantum wires, we often model the residual low energy excitations in the system through low-dimensional quantum mechanical systems. This limiting behavior for interpolation between two and three dimensions is consistent with what is observed for the zero-energy Green’s function in the interface, see equations (21–22) below. We cannot infer from the previous section that the zero energy limit of the inter-dimensional Green’s function calculated there yields a dimensionally hybrid potential, because we were dealing with solutions of Schrodinger’s equation instead of the Gauss law. The zero-energy Green’s function in the interface is given in terms of a Struve function and a Neumann function1, This yields logarithmic behavior of interaction potentials at small distances r ( ‘ and 1/r behavior for large separation r ) ‘ of charges in high-permittivity thin films,
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