The correct boundary conditions fo r grow th along the [001] direction w ere derived by Forem an [For93]. W hen working with the ordered form o f the Ldw din interaction (3.7), one finds that it is convenient to replace the previously used L uttinger param eters, which w ere chosen to relate directly to the effective m asses involved, by a new set o f three param eters w hich explicitly identify the contribution corresponding to remote bands o f a specific sym m etry Fy, F /j o r Fy2 (neglecting the sm all contribution o f F2j) [For93]. Using the tight-binding picture as done in subsection 2.2.2 where the atom ic orbitals upto d corresponding to the various sym m etry groups w ere presented, it is straightforw ard to show that the only non-zero contributions to the band param eters L M J ^ in the param etrised H am iltonian (2.47) are o f the form
a = o / 3 " < o ) i i ( % H V ( ^ v - £ )
It = ( l / 3 m o ) S |( ^ |p , l v ) |7 ( £ v - E ) (3.8) V
5 = ( l / 6 m „ ) i | ( x | p » | 7 ( £ v - £ )
Each of these interactions will generate terms in the Hamiltonian with a specific ordering of the differential parameters with respect to the band parameters. Identifying the contribution to the Luttinger parameters 7/, 7 2, 7j, one finds
71 = - l + 2 a + 4ti +4Ô
7 2 = 0 - 7 1 + 2 5 (3.9)
7 3 = 0 +71 - 5
One can now proceed as done in (2.47-2.52) and derive the ordered Hamiltonian elements for a specific growth direction using the appropriate angular momentum basis for growth on [001]. Integrating this ordered Hamiltonian across a boundary, one then obtains the boundary condition of continuity of F and D F where D is given by [For93]
(1) (3)
0 j i k + { a ' (yi+2y2) ^ V3'^_(o -6]n - V 3 * + [ ^ ( o - 2i ^Y2
0 0 (y I "^Yz)^ \ 0
- J ^ i k + ( a - 5 ) -2V2Y2 V3^*+j^j(a- 6 ) + j j i J 0
V
Yl ^ i - J i - ô ) |0 >/3Â:_[^|(a- 6 ) + -|nj -2V2Y2 ; -/*+ ((J-n -5 ); ""
(3.10) The above (correct) boundary condition matrix differs considerably from what one would have obtained had one used the conventional symmetrisation procedure. This boundary condition matrix exhibits
1) a considerable asymmetry with respect to the diagonal, something one would not have found using a symmetrisation procedure.
2) reduced coupling between the heavy hole band and the light/split-off hole bands as these interactions do not feature the dominant contribution from the remote states with s-type symmetry (o)
The boundary condition problem
As a result, significant changes in the dispersion relations or related quantities like subband effective masses are expected when adopting the new boundary conditions. F or example, it was shown in [For93] that, using the conventional symmetrised boundary conditions, the calculated zone-centre effective mass for the second subband in an Ino^GaogAs/GaAs QW can have triple the value of that obtained using the new boundary conditions (fig.3.4). Furthermore, it was shown in [Men94] that 4- and 6-bands calculations which do not implicitly include the conduction band improve significantly when adopting the new boundary conditions when compared to an 8-band model (where the higher 5-orbitals are explicitly included) (fig.3.5). The latter results clearly show that the new boundary conditions are more physical than the old ones, as one does not expect :
• the effective mass to be so sensitive with respect to well width
• the valence band dispersion to display large variations when comparing the various (4-, 6-, 8- band) models. Obviously, one expects the accuracy to increase when including more bands in the expansion but with a gentle convergence.
-30 iS -60 -90 0 T £ -60 -90 0 8 12 0.8 N 0.6 HH2 HH1 0.4 HHI 0.2 0.0 0 20 40 60 80 100
Quantum well width (Â)
▲ F ig .3 .4 : Z o n e c e n tr e h e a v y h o le e ffe c tiv e m a s s f o r an Ino^G ao s A s /G a A s Q W . S h o w n a r e th e c a lc u la te d r e s u lts u sin g th r e e d iffe r e n t b o u n d a r y c o n d itio n s , B u r t-F o r e m a n (s o lid ) , s y m m e tr is e d (d a s h e d ) a n d u n c o u p le d ( d o tte d ) (a fte r [ F o r 9 3 ] ) .
4 8 12 K [1 0 'cm ']
F ig .3 .5 : V a le n c e h a n d d is p e r s io n f o r a IOOÂ Ino.](,Gao.84A so.j4Sho.86lA lo 4G ao 6As Q W (s o lid :B u r t-F o r e m a n b o u n d a r y c o n d itio n s , d a s h e d : s y m m e tr is e d ) . T h e f o l lo w i n g b a n d s w e r e e x p lic itly in c lu d e d in th e e x p a n s io n : (a ) C B .H H f M ,S O ; ( b ) C B ,H H J M ;
A t this point, the need fo r implementing the new boundary conditions has been established as it was show n that valence band effective m odels that do no t explicitly include the interaction w ith the conduction band states (such as the 4- and 6-band models th at are used throughout this thesis) are very sensitive to the choice o f boundary conditions. D espite the num erical evidence and new theoretical insight in the origin o f the boundary conditions, it w as found th at the sym m etric boundary conditions w ere still being used in the literature when looking at non-[001] grow th as the exact envelope function theory had so fa r only been applied to [001]. T his w as addressed in two papers [Sta97,D al98], in which the first one provides the boundary conditions fo r [110] grow th and the second one extends the formalism to arbitrary grow th directions. T hese theoretical results will be discussed in detail in the next chapter. The relevance o f using the correct o p erato r ordering will be highlighted again in chapter 6 where dispersion relations are presented fo r non-[001] grow th directions fo r the various boundary conditions, along with its effects on theoretical hole m obility calculations.