Chapter 5 Bulk inversion asymmetry effects on the bands of zincblende
5.3.3 Analytic expressions of the energy values near the zone centercenter
5.3.3 Analytic expressions of the energy values near the zone
[100] ∆ xˆ
[110] Σ yˆ−xˆ
[111] Λ yˆ−xˆ
Table 5.6: Quantization axis for the basis functions of the Hamiltonian.
After the Hamiltonian has been expressed in the basis with the new quantization axis, another change of basis consisting of a rearranging of the vectors is applied in or- der to leave it in its final block diagonal form. The basis states are reordered from the set ©
|Γ6,+12i,|Γ6,−12i,|Γ8,+32i,|Γ8,+12i,|Γ8,−12i,|Γ8,−32i,|Γ7,+12i,|Γ7,−12iª into
©|Γ6,+12i,|Γ8,−32i,|Γ8,+12i,|Γ7,+12i,|Γ6,−12i,|Γ8,+32i,|Γ8,−12i,|Γ7,−12iª
for the
∆ directions, or ©
|Γ6,+12i,|Γ8,+32i,|Γ8,−12i,|Γ7,−12i,|Γ6,−12i,|Γ8,−32i,|Γ8,+12i,
|Γ7,+12iª
for the Σ and Λ directions.
Taking one of the 4 ×4 blocks, the secular equation to obtain the eigenvalues would be a fourth degree polynomial. Instead, the standard procedure of taking the unperturbed energy for the bands that are not under consideration will require the solving of two first-degree equations—for the conduction and the split off bands—
and one second-degree polynomial for the heavy and light hole bands to yield the approximate energies near the zone center.
Bands along the [100] direction
The energy dispersion relation for the conduction band (CB) along the [100] direction up to second order in kis given by
ECB(k) = Eg+ ~2k2x
2mem∗CB, (5.16)
where me is the free electron mass,m∗CB is the conduction band effective mass, with a value of
1
m∗CB = 1 + 1 3
EP
Eg
3Eg + 2∆SO
(Eg + ∆SO), (5.17)
C=0 C≠0
-0.05 0.00 0.05
0.40 0.45 0.50 0.55
0.60 Light Holes
Heavy Holes
E n e rg y [ e V ]
k || [100] [Å
-1]
Figure 5.1: Linear splitting of the HH and LH bands along [100] near the zone center due to bulk induced asymmetry (BIA) effects. The inset compares the bands with BIA included C 6= 0 to the bands with BIA not included C = 0. The value ofC has been artificially augmented 1,000-fold for C 6= 0.
where the symbols have the meaning listed in Table 5.4. This effective mass is isotropic—the same along any direction on kspace.
For the heavy hole (HH) band, the dispersion relation is
EHH(k) =Ckx− (3Egγ1~2/me+ 2P2)k2x 6Eg
, (5.18)
while for the light hole (LH) band
ELH(k) = −Ckx−(3Egγ1~2/me+ 2P2)kx2
6Eg . (5.19)
These results indicate that, very close to the Γ point, the valence bands have a linear behavior. Also, the effective mass seems to be quite different from the usual
the Γ point for GaSb, except that the valence band splitting parameter C has been increased by a factor or 1,000 to show more clearly its effects. It is seen that very close to the zone center the HH and LH bands split linearly instead of quadratically, following the behavior described in Eqs. (5.18)-(5.19); and it is not until farther into the Brillouin zone that the bands recover the usual quadratic behavior. The inset in Fig. 5.1 compares these bands to the case where the splitting has been set to zero.
It is seen that even with the inclusion of the bulk inversion asymmetry effects, the bands recover soon the shape of the C = 0 case. The only difference then is that the HH-LH separation is slightly bigger than predicted in the models not taking C into account.
Setting C = 0 in the analytical Hamiltonian and expanding it to second order yields the more usual expressions for the HH and LH effective masses
1
m∗HH =γ1−2γ2
1
m∗LH =γ1+ 2γ2+ 4 3
EP
Eg
, (5.20)
which have been expressed in terms of the Kane parameters. In terms of the Luttinger parameters (see Eq. (5.11) ), they adopt the form
1
m∗HH =γ1L−2γ2L
1
m∗LH =γ1L+ 2γ2L. (5.21)
Finally, the spin-orbit split off (SO) band has the following dispersion relation
ESO(k) = −∆SO − ~2kx2
2mem∗SO, (5.22)
where m∗SO is the split off band effective mass, with a value of 1
m∗SO =γ1+ 2 3
EP
(Eg+ ∆SO). (5.23)
The effective mass for the SO band is also isotropic.
Note that the expressions of the effective masses of the CB and SO bands are not affected by the inclusion of BIA effects.
Bands along the [110] direction
Along the [110] direction, the energy of the electrons in the CB as a function of the wavevector kis, up to third order,
ECB(k) =Eg + ~2k2
2mem∗CB ± 1
2γck3, (5.24)
withm∗CB given by Eq. (5.17), and thek3 splitting coefficient (see Eq. (5.4) ) in terms of the model parameters given by
γc = P 3
2BEg∆SO−√
3CP (Eg+ ∆SO)
Eg2(Eg+ ∆SO) . (5.25)
It is easy to show that the contribution toγc of the part containingCis only about 4% for InSb and InAs. That contribution goes down to about 0.3% for GaAs and AlAs, and it drops to a mere 0.03% for GaSb. Therefore, it is a good approximation to consider that all the splitting in the conduction band is due to the nonvanishing bulk inversion parameter B, which has its source in the momentum coupling of the conduction and valence bands via remote Γ5 states [15].
Note that, in order to turn off BIA effects, both parameters B and C need to be set to zero.
As in the [100] case, the inclusion of C 6= 0 changes the characteristics of the bands very close to the Γ point respect to the more common assumption of C = 0.
In particular, it provides them with a small linear component. But, in opposition to the [100] case, here the HH and LH bands are not doubly degenerate. The LH1–
-0.010 -0.005 0.000 0.005 0.010 0.540
0.545 0.550 0.555 0.560 0.565
∆LH ∆HH
LH2 HH2 LH1 HH1
E n e rg y [ e V ]
k
x=k
y[Å
-1]
Figure 5.2: Linear splitting of the HH and LH bands along [110] near the zone center.
The value of the parameterC has been artificially augmented 1,000-fold.
LH2 and HH1–HH2—the number indicating subbands—linear splittings (∆LH[110]
and ∆HH[110] respectively) turn out to be the same for both HH and LH bands (see Fig. 5.2), and are given by
∆HH[110] = ∆LH[110] =√
3Ck. (5.26)
This result is slightly different from the one indicated in Ref. [9] in their Eq. (7.5).
A numerical diagonalization of the Hamiltonian has been performed to check the validity of Eq. (5.26). The discrepancy arises because the splittings in Ref. [9] are valid in the region where the quadratic (effective mass) splitting predominates, while the result obtained here is valid in the region where the linear splitting dominates.
In the materials studied, one needs not go far from Γ to enter a regime where the bands basically behave according to the standard behavior, described by the effective
masses
1
m∗HH =γ1L− q
γ2L2 + 3γ3L2 (5.27)
1
m∗LH =γ1L+ q
γ2L2 + 3γ3L2 , (5.28)
which agree with the expressions in Ref. [30]. In this regime, there is some LH and HH splitting proportional to k3 due to the effect of B:
δHH[110] = BP 3Eg
γ2L−3γ3L+p
γ2L2 + 3γ3L2
pγ2L2 + 3γ3L2 k3 (5.29) δLH[110] = BP
3Eg
3γ3L−γ2L+p
γ2L2 + 3γ3L2
pγ2L2 + 3γ3L2 k3. (5.30) Note that the k3 splitting in the heavy hole band is a good indicator of the anisotropy of the hole effective masses, because it should vanish for a material with isotropic hole effective masses (γ2L =γ3L).
The SO band also presents k3 splitting ∆SO[110], proportional to the B parameter only:
∆SO[110] = 2BP
3 (Eg+ ∆SO)k3. (5.31) Bands along the [111] direction
Along this direction, in the region where the linear splitting dominates, the heavy hole (HH) band has the dispersion relation
EHH(k) = ±√
2Ckx− ~2k2 2me
(γ1L−2γ3L) +O(k4), (5.32) while for the light holes
ELH(k) = −~2k2 2me
(γ1L+ 2γ3L) +O(k4), (5.33)
-0.04 -0.02 0.00 0.02 0.04 0.0
0.5 1.0
HH LH
SO
Energy [eV]
k || [100] [Å-1]
Figure 5.3: Band structure for GaSb near Γ along [100].
from which
1
m∗HH =γ1L−2γ3L (5.34)
1
m∗LH =γ1L+ 2γ3L. (5.35)
The light hole, conduction and split-off bands are degenerate along the [111] di- rection, as can also be deduced by group theory arguments [10]. The heavy hole band shows a linear splitting near the zone center.