Dr Paul Kinsler. [Acknowledgements & Feedback]

This is part of a hyperlinked notes maze -- please read the index-file for important information about the nature and reliability of this document.

XKEYWORD: boundary-condition

# Boundary Conditions

## DC-model

(..., Babiker-Z-1998) has continuity in EM fields (?check) E_x and epsilon E_z, I assume (or polarizations).

## HD-model

stress and/or pressure

## What do we match across boundaries?

Ridley-1993 hybrid-phonon model insists we have to match both EM and mechanical boundary-conditions. This can be relaxed as an approximation if the energy involved in one of these components is negligible compared to the other.

Babiker-1986 (HD-model) insists on continuity in the normal component of the velocity, which for the LO-phonon in terms of u_z is that rho^{-1/2}(d slash dt) U_z is continuous (U_z = rho^{1/2} W_z, W_z is the true ionic displacement). Similarly, Babiker-1986 insists on continuity of the pressure, which is that beta^2 rho^{1/2} (grad . u) is continuous.

Ridley-B-1991, on LO-phonon boundary-conditions, says: there must be continuity in stress in z (continuity in u_z) and pressure pi (from optical strain grad . u) (continuity in bar{rho} v^2 (grad . u) ). These are stated as being equivalent to those in Babiker-1986. Further, they state "a similar argument leads to HD-model boundary-conditions for purely TO-phonon modes".

Ridley-ACB-1994 use define optical strains

```S_{alpha beta} = - (1 / 2) ( du_{alpha} / d{beta} + du_{beta} / d{alpha} )

alpha, beta = x, y, z
```
Ignoring the y variable, these give three equations. S_{xx} reduces to continuity of C_x; S_{zz} swaps modes (+ to -) and weights each with k_L, k_T, or k_z; but both are easily written in matrx form. S_{xz} however gives a weighted sum of x and z components, and cannot be written in the TFM-model matrix form. However, Ridley-ACB-1994 use T_{alpha beta}, the "elastic stress", which they do not define (or reference), but is "related to S_{alpha beta}".

## The hybrid-phonon model

(Ridley-1993) fixes in u_x, u_z, E_x, epsilon E_z at the (boundary-conditions) slab sides.

19971209: Assuming that the various papers with their varied boundary-conditions are all correct, it seems that which boundary-conditions are appropriate depend on the particular (er) boundary-conditions (system) chosen -- eg "free standing slab" (Ridley-ACB-1994) uses "stress free" ones. So - what should be used for a general heterostructure? Perhaps "stress free" on the edges, and others (what?) at the layer- layer interfaces.

## Other

My TFM-model needs to match boundary-conditions from one side of a heterostructure interface to another.

19971127 1209 1210

Email Feedback: Dr.Paul.Kinsler@physics.org

LOGBUG