3.3 Exact envelope function theory

The effective m ass equations in chapter 2 were essentially derived fo r bulk structures only. As a result, no specific arrangem ents w ere m ade to allow fo r changes in effective m ass param eters. It is therefore no surprise that only general argum ents (such as the requirem ent th at the Hamiltonian m ust be H erm itian) have so fa r been employed to obtain the correct o perator ordering, as the original derivation o f the effective m ass equations by itself can no t supply such information. H ow ever, as it w as show n that these argum ents cannot provide unam biguous boundary conditions, not even in the relatively sim ple conduction band case, it is clear th at a m ore fundam ental approach is needed.

O bviously, a derivation o f the effective m ass equations which specifically allow s fo r changes in the com position and m ass param eters is required. Such an approach w as first provided by the developm ent o f an exact envelope function theory by B urt [B ur87,B ur92]. T he thorough treatm ent o f the problem in both papers is rather m athem atical and too extensive to be repeated, but as suggested by him self [B ur96,unpublished] there is an alternative and less general approach to show the principles involved.

Starting again from the expansion (2.39)

= (l) (3-2)

j

one w ould now like to explicitly allow fo r changes in the m ass param eters as w ell as link these to the m icroscopic structure o f the sem iconductor m aterials. One now m akes an assum ption involving

The boundary condition problem

the origin of the effective mass parameters. From the bulk Lriwdin-perturbation term (2.37) it follows that any change in mass parameters , either being m* or yj, yz, yj, when crossing an interface that separates material 1 from material 2 can either be caused by

• differences in the form o f the band edge orbitals Ujo(r) in both m aterials, i.e. differences in the momentum matrix elements.

• differences in the energies Ej at which the various bands are located.

Following B urt’s suggestion, the restriction is made that only the energies of the band edge states change when going from material 1 to material 2 and not the orbitals itself. A compositionally varying structure will from now on be described by position dependent band energies Ej but invariant cell-periodic functions up. The important thing to note is that, in contrast to the approach taken in the previous chapter, one does not now implicitly assume all bands to be equally influenced by a change in composition. Obviously, the final equations will still feature a compositionally varying term such as V(r) that describes the energy as a function of position of the band under consideration (either being the conduction or valence bands). However, changes in the mass parameters will now explicitly be allowed through variations in energies o f the remote bands (fig.3.3).

¥

U sing the above approxim ation, the derivation o f the effective m ass equations (2.33-2.38) is now repeated w here particular attention is paid to the operator ordering. Inserting the expansion (3.2) in the general H am iltonian and using that the cell-periodic w avefunctions ujo correspond to the bulk eigenfunctions at energies Ej, one obtains

J - »o ^ 0

w ere Ej(r) is position dependent as one now specifically allows fo r com positional changes. Again the assum ption is m ade th at the expansion can be divided into tw o classes, a dom inant set |/) consisting o f F ; o r Fyj orbitals, /= l..r , which requires explicit treatm ent and a rem ote set |p), p = l .. j , which can be included using perturbation theory. T aking the inner product with respectively a dom inant and a rem ote state gives.

E , ( r ) - E - 2m,0 J 2t72 ^ V f,. V fr = 0 (3.4) mr_{*0 n' ^ 0 f}

where, as a result o f the approxim ation that the orbitals are identical throughout the structure (but not their energies), the mom entum elements P/ji (= (« ji o|p|w/o))» Ppp' and p^J■ do no t have a position dependence. N ote that the inner product pir is zero because o f sym m etry considerations as the explicit expansion in \l) (the dom inant states) is restricted to ju st F ; o r Fyj in this w ork (1-band conduction band, 4- o r 6-band valence band model). N ow , using th at the rem ote states are relatively fa r separated in energy from the energy region o f interest, it follows that E^^-E is the dom inant term in the second equation o f (3.4). N eglecting the kinetic energy term as w ell as the term proportional to as F^^«F| results in the approxim ation

m , ^ E ^ ( r ) - E

(3.5)

Inserting this in the expression fo r the dom inant states, the following system o f coupled differential equations is obtained

E , ( r ) - ^ - E

2m.

Y ' y PryPyr y

^ "EJr)-E ^

Fp =0

The second p art o f the above equation can be seen as the Ldw din interaction term and is very sim ilar to (2.37) ap art from the fact that it is now carrying operators rather than vectors. To be able to com pare the above equation to the original effective m ass, it is rew ritten as

The boundary condition problem D f_jj P V i 9%c + 2 , ( r ) 6 j , ' F ^ ,(r) = £ F . . ( r )

(3.7)

The im portant difference to (2.42) is that an exact^ and unam biguous o perator ordering has been o b ta in e d Q e a rly , one has to keep track o f the operator ordering during the param etrisation o f the effective m ass m odel to be able to derive the correct boundary conditions. Sim ply lum ping the contributions o f all remote bands on the m ass param eters together (to obtain m o r yy.j) and neglecting the operator ordering as done in (2.44,2.47) cannot provide the correct boundary conditions as there are ju st not sufficient basic quantum m echanical constraints to subsequently provide unam biguous operator ordering.

F o r an expansion in a single band, i.e. fo r the conduction band case, using the ordered form (3.7) results in the sam e ordering as one w ould have obtained by using the conventional sym m etrisation rule (3.2). The derivation o f the boundary conditions fo r the valence band are how ever more difficult to obtain, and will be discussed in the next section.

This chapter then ends with som e concluding rem arks about the effects o f the employed approxim ations in the derivation (3.2-3.7) and com ments about m ore recent approaches tow ards the boundary condition problem .