Chapter 4 Description of bulk inversion asymmetry in the effective bond orbital
4.4 Bulk GaSb
The considerations above are illustrated with an example calculation of bulk GaSb.
Figure 4.1 shows the bands of bulk GaSb for a few special directions. The dashed and dotted lines in plot (a) correspond to the EBOM model without the zincblende symmetry corrections. The dotted line is obtained under the original requirement [9]
that the separationXhl = 4.0 eV is used to obtain Esx. This will be calledX model.
The dashed line is obtained taking Esx = 4aP , with P obtained from the value of the effective mass given by Eq. (5.17). This will be calledP model. With the parameters used, looking at the X and L points, in the P model the conduction (CB) and split off (SO) bands are pushed further apart than in the X model. In the P model, the
-14 -12 -10 -8 -6 -4 -2 0 2 4
U,K Σ
∆ Λ
L Γ X Γ
Energy [eV]
-14 -12 -10 -8 -6 -4 -2 0 2 4
U,K Σ
∆ Λ
L Γ X Γ
Energy [eV]
Figure 4.1: Band structure of GaSb calculated with EBOM under different assump- tions for the parameters. The dotted line in plot (a) is obtained under the original requirement [9] that the separation Xhl = 4.0 eV is used to obtain Esx. The dashed line is obtained takingEsx= 4aP . A term describing BIA has been included in plot (b), which otherwise uses the same set of parameters as in the calculations represented by the dashed line.
positions of the CB and SO bands are very sensitive to the value of the CB effective mass. For example, changing m∗c in InAs from 0.025 to 0.024 changes the position of the SO band at the X point from about -10 eV to about -6.5 eV. Going one step further and setting m∗c = 0.023 makes the SO band anticross with the heavy and light holes, and the spurious heavy hole (HH) - light hole (LH) crossing described in Ref. [9] appears. Therefore, it is reasonable to assume that very small changes in the value of them∗c parameter can not only get rid of spurious solutions present in the P model but also tune the position of the CB at the X point.
Plot (b) in Fig. 4.1 is generated under the same conditions as model P, but with BIA effects turned on by letting Es,xy = −B/a2. This will be called P B model. In agreement with predictions from the character tables for theTd group [17], the bands become spin split in the Σ direction because of the breakdown of Kramers degeneracy.
However, the correct description of the zincblende symmetry is made at the cost of
-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.000
0.005 0.010 0.015
CB Splitting Fit to Splitting=γck3
Energy [eV]
k [2π/a]
-0.04 -0.02 0.00 0.02 0.04
-0.8 -0.4 0.0 0.4 0.8 1.2
[100]
[110]
E n e rg y [ e V ]
k [2 π /a]
Figure 4.2: Bands close to the zone center showing the spin splitting, calculated with EBOM. The inset shows the amount of CB splitting and its k3 dependence at low values of k.
the loss of accuracy for the CB and SO bands, specially along the Λ line, where they take values quite far from pseudopotential calculations [19]. The preference of having a correct description of the bands near the Γ point or the ∆ line including spin—with its ability to describe short period 100 superlattices—or a more accurate full zone description will determine the model to be used. The inclusion of second nearest neighbors matrix elements [20] might reconcile the energy values at the L point in
splitting in the bands along the [110] direction. The inset shows the splitting in the conduction band along the Σ line for the CB, and a fit using
Splitting =γck3, (4.18)
where γc is the k3 splitting proportionality constant. The value used for γc is 186 eV·˚A3, in good agreement with the measured value of 187 eV·˚A3 [22]. This shows that the parameter B determines the CB splitting near the zone center in the P B model in the same way as it does in the k·pmethod, as expected from the derivation in Sec. 4.3. A look at the inset reveals that, for GaSb, expression (4.18) is good until about 2% of the zone edge. The only qualitative aspect of the bulk bands that the extension in Sec. 4.3 cannot incorporate is the linear spin splitting in the valence bands close to the zone center [23] (cf. Fig. 5.2). In k·p, this is described by a parameter C coming from second-order mixedk·p and spin-orbit terms in the perturbation expansion [14, 24]. It is this relation of C to the spin orbit interaction that makes it impossible to include its effects in the EBOM method. This is because the starting tight binding formulation includes spin orbit effects only in a limited and ad hoc fashion. Boykin [25] has extended the tight binding method to include the linear k splittings in the valence band, but the fact that four extra parameters are needed in his treatment while a single one does the job in k·p suggests that a tight binding formulation with spin orbit effects included from the beginning should yield the linear splitting naturally and reveal constraints due to symmetry between Boykin’s parameters. In any case, its effects are normally small, and its importance for heterostructures is studied, for a particular case, in the following section.
0.00 0.01 0.02 0.03 0.04 0.05 0
2 4 6 8 10 12 14 16 18
αk.p=22x10-12 eV.m αEBOM=20x10-12 eV.m
k || [100] [Å-1]
Splitting E1 [meV]
-0.04 -0.02 0.00 0.02 0.04
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
HH2 LH1 HH1 E1
k.p EBOM
Energy [eV]
k || [100] [Å-1]
Figure 4.3: Comparison of EBOM and k·p superlattice bands. The structure is an 8/8/8 AlSb/GaSb/AlSb SL. The solid (dotted) lines are the k·p (EBOM) results.
The bands are spin split away from Γ due to the bulk inversion asymmetry. The inset shows, for both methods, the amount of splitting in the E1 band and the values for the splitting coefficients as defined in Eq. (5.69).