Operacionalización de variables

In the tight binding approxim ation the valence band states are o f the form \ x ) = R { r ) x , \ y ) = R { r ) y , \ z ) = R { r ) z. Rew riting these states as spherical harm onics Yun, it is easy to prove using the operators Lz,L^,L. [Gas74] that operating the an g u lar m om entum operators

Symmetry considerations

X = 0 X = - f Z X = zT

Y = zZ Y = 0 Y = -z X (2.23)

Z = - i ¥ Z = fX Z = 0

S im ilaily, it can be show n that

Î T 5 /

i

i

2

2

U sing (2.23) and (2.24) and orientating the angular mom entum J along the z-dinection, the following set |/,my) o f band edge eigenfunctions is obtained fo r the H am iltonian including spin- orbit interaction.

1 -4 )

i ' + i ) = - ; | ^ [ ( X + < T ) i - 2 z T ]

14)

(2.25)

As m entioned earlier, the |3/2,±3/2) states correspond to the heavy, |3 /2 ,± l/2 ) to the light and |l/2 ,± l/2 ) to the split o ff holes. It is im portant to note that the above set is no t uniquely defined as each elem ent can be m ultiplied w ith a phase factor a + / p o f unit length w ithout changing any o f the physics. One therefore finds different sets in the literature; com pare the above set to, for exam ple, [Lut55]. This issue will be discussed in a subsequent section.

A t this point, the choice o f orientating the angular momentum J along the z-direction is rather arbitrary as a different ‘preferred’ direction could ju st as well have been chosen. It will prove advantageous to choose J along the grow th direction when setting up a bandstructure model for com positionally varying structures as will be done in the following section. T he above set will therefore be o f im portance when looking at [100]-grow th only. A dopting a C artesian reference fram e w ith unit vectors a = ( a j,0 2,0 3), b = (b i,b2,b3), £=(C],C2,C3), the general form o f the set \J,m) in w hich the angular m om entum is orientated along the vector c w as presented in [Dal98] and has the form

^ » + ^ ) = ; ^ | («1 + i b ^ ) x + (û2 + ib i ) y + (û3 + ) z ) T

| , + | ) = ; ^ k , X + C 2y + C 3 Z ) î - ; ~ | ( û . + / ô J X + ( û 2 + / ^ ) î ^ + ( û 3 + i ^ ) 2 ) j ' (2.26)

^ = - ^ | c i X + C27 + C3z ) t - ^ | ( f l , + ( û2 + (ûf3 + « ^3) ^ ) ' ^

The corresponding \I -m j) states can be constructed using the K ram ers o perator [K an56] on their positive counterparts. Using the general form defined in (2.21), one obtains th at the negative counterparts are given b y '

I / . - « , ) = \ x \ ^ + x 7 + x \ z ) ' t - \ x \ x - ^ x \ y + x l 2 ) i (2.27) F o r the standard reference fram e, i.e. inserting the vectors a= (1 0 0 ), 6= (010) and £= (001), these equations reduce to the set (2.25). E xplicit form s fo r the case w here the quantisation axis is orientated along (110) o r (111) can for exam ple be found in [L au71,Sta97].

A t this point, its w orthw hile to sum m arise w hat has been achieved in this chapter. So fa r :

• a one-electron Schrôdinger equation operating on cell-periodic functions has been obtained that will be the starting point o f the derivation o f the bandstructure m odel in the next section.

• knowledge o f the degeneracies and sym m etries o f the various band edge levels h as been acquired w hen excluding spin-orbit coupling.

• inclusion o f spin-orbit coupling has been shown to only consist o f fonriing linear com binations o f the eigenfunctions o f each level.

• an explicit representation o f the valence band states at the band edge { k - 0 ) has b e a i derived

including spin-orbit splitting.

• the freedom to choose a preferred direction for the angular m om entum in this representation has been show n to be a pow erful m eans to optim ise the set w hen looking at finite m om entum .

U sing this inform ation, the next section deals with the derivation and underlying m otivations o f setting up a bandstructure m odel within the so-called effective m ass (or k-p) fram ew ork.

Symmetry considerations

2 .3

> The k'p approacb

,

The Schrodinger equation and its one-electron equivalent that are the starting point o f any bandstructure model were presented in the first chapter. As outlined in the previous section, bandstructure calculations are numerically very demanding even in the simplest case when dealing with bulk material. As this thesis is concerned with compositionally varying structures, which is expected to make the physical problem even more difficult, a method that partially relies on material constants rather than a first principles method was chosen. Effective mass theory, also often referred to as k-p (‘k-dot-p’) theory, provides such a method. The latter approach significantly simplifies the physical problem and provides an easy interpretation, of course at the cost of requiring some input in the form o f physical parameters.

The method was pioneered by Kane who used it to calculate the valence bands in bulk germanium and silicon [Kan56] as well as the conduction and valence bands in Indium Antimonide [Kan57]. Although this model is only applicable to bulk materials, it proves to be a useful tool to understand the physics behind the processes that shape the bandstructure. As it is much simpler than its more elaborate counterpart the Luttinger-Kohn [Lut55] model (which is also applicable to compositionally varying structures) and since the latter model reduces to the Kane model in the bulk case, this review of k-p theory will start with the Kane model.

Starting with the simplest bulk model, various improvements are introduced and the transition to compositionally varying structures is discussed. Special attention is paid to the use of this model to non-conventional growth directions such as [110] or [111]. Finally, section 2.4 deals with the inclusion of strain effects which is essential when modelling non-lattice matched compounds like InGaAs on GaAs.

Although this section covers k-p theory in extensive detail, the intrinsic problem of how to connect wavefunctions across an interface in a compositionally varying structure will not be discussed here. Because o f its complexity, it was chosen to address the latter subject separately in chapter 3.