According to the result in this thesis, we have three new questions to study. The first is the transmission property through the potential that breaks the pseudo-parity. The transverse position operator breaks the pseudo-parity symmetry. So the transmission of quantum point contact or a side gate is different to the case we consider. The potential may can split the spin of the edge bands so we may have another way to modulate the spin and the side of the edge bands.
The second is what is the origin of the surface states at the boundary between nor-mal and topological non-trivial materials. The topological physics predicts we can find surface states at the boundary between the topological trivial and topological non-trivial materials. According to our study in the chapter 3, we find out that the bulk energy gap closing can change decay direction of the wave vector, vertical to the surface. Besides, the gap-closing can also change the topological number.[19] Therefore, the decay wave vectors having the same pseudo-spin direction at different region have opposite decay direction.
The pair of decay wave vectors form a surface state at the surface. Is it the origin of the surface states?
The final is what is the transport of the edge channels in a very dirty quantum bar.
In a quantum channel, DSOI makes the edge channels that have different pseudo-parity can be coupled by normal impurity. The phase difference bewteen the edge channels determines the spin polarization of the edge states. The spin of the edge state may not depend on Lc. On the other hand, the finite size effect makes the edge channels can be back scattered. DSOI lets the all edge channels can be coupled by impurity only. This can occur without any bulk channel at the Fermi energy. The transport of the edge channels in a very dirty system may not be a trivial question.
Detail derivation: From basic band formulation to effective 2D
Hamiltonian
In this appendix, we start our derivation from the 3D 8 × 8 Kane Hamiltonian with DSOI.[6] The energy range we focus upon is far away from the split-off bands. To simplify the problem, we reduce the Hamiltonian into the 3D 6 × 6 Kane Hamiltonian. Some of the perturb terms are smaller than the others so we justify and drop the smaller terms.
Then, we show how to derive an effective 2D Hamiltonian with C terms that are orig-inated from DSOI. For checking the band structure, we derive two effective 2D Hamilto-nians. One is with the basis independent of DSOI terms and the other is with the basis depending on the Ckz terms in H6×63D. Then we discuss the isotropy and the gap closing of the band structure. At the final section, we derive the effective 2D 4 × 4 Hamiltonian that we use in this thesis for describing the band structure of conduction and heavy hole bands.
Our result shows the energy crossing of the conduction and heavy hole bands is at well thickness dc ≈ 6.6nm and it is not equal to ref.[1] This difference doesn’t comes from keeping the correction of effective mass but comes from the difference of band parameter of Kane Hamiltonian. (In the ref[1], they use HgCdTe instead of CdTe.) Only the energy band gap of the effective four band Hamiltonian is much different to the other groups’
EFFECTIVE 2D HAMILTONIAN
result and it doesn’t change the topological physics (See chapter 6.).
Ckz terms perform like constant in the effective 2D Hamiltonian so it must be kept when we keep γ terms (effective Luttinger parameters) in H6×63D. C terms remove the band crossing at Γ point but it doesn’t make the global band gap always opened. It make the Dirac point becomes the Dirac ring with critical kcand the gap is still closed at d ≈ 6.6nm.
Even C and γ terms both make the band structure be anisotropic. The energy dif-ference of conduction and heavy hole bands versus the direction of k is still small. By dropping the neglectable terms in H10×103D , we obtain the effective 4 × 4 Hamiltonian Hef f that has been announced at ref[7].
A.1 3D 8 × 8 Hamiltonian
The lattice structures of HgTe and CdTe are zinc blende structure so they do not have center of inversion. It makes those materials be BIA. This symmetry broken makes a spin orbital interaction well known as DSOI. We introduce DSOI effect by adding C terms in Kane Hamiltonian[6]. Those terms appear at Hh and Hhs listed below.
We start our study from Kane Hamiltonian that contains Γ6(conduction band), Γ8(hole band) and Γ7(split-off band) subbands. In the order of basis vector (|Γ6, 1/2i, |Γ6, −1/2i,
|Γ8, 3/2i, |Γ8, 1/2i, |Γ8, −1/2i, |Γ8, −3/2i, |Γ7, 1/2i, |Γ7, −1/2i), 3D 8 × 8 Kane Hamil-tonian H8×83D is of the form:
H8×83D =
Hc Hch Hcs Hch† Hh Hhs† Hcs† Hhs Hs
. (A.1)
Hc = Ec+ T k2. Hs = Es− γ1k2.
Hcs =
−√13P kz −√13P k−
−√1
3P k+ √1
3P kz
.
Hch= depend on the material and their values are listed at Table A.1.
Table A.1: The band structure parameter of HgTe and CdTe
Ec Ev Es P F
HgTe −0.303eV 0eV −1.08eV 8.46eV ·˚A 0 CdTe 1.036eV −0.57eV −1.48eV 8.46eV ·˚A −0.09
γ1 γ2 γ3 C[13]
HgTe 4.1 0.5 1.3 −74.6meV ·˚A
CdTe 1.47 −0.28 0.03 −23.4meV ·˚A
a
All parameters except C are listed at Ref.[12]
b
We set E
vHg= 0 and E
vHg− E
vCd= 0.57eV [12]
EFFECTIVE 2D HAMILTONIAN
A.2 3D 6 × 6 Hamiltonian
The split-off band is far away the energy range we want to analyze. Therefore we drop split-off band but keep 2nd order terms by Lo ¨wdin perturbation theory. We call the new Hamiltonian as 3D 6 × 6 Hamiltonian H6×63D.
In the next chapter, we want to derive the 2D effective Hamiltonian by expanding H6×63D with the eigenfunction of H6×63D (0, 0, −i∂z). The kz term is like constant and the Kane Hamiltonian describes the band structure near Γ point. Therefore we can drop some terms in 2nd order perturb terms because they are smaller than the others for small k. We drop k3 and k4 terms but keep kxk2z and kyk2z terms. Furthermore, the P terms are much larger than the others so the effect of P0 terms in 2nd order perturb term are small enough to be neglected.
The 3D 6 × 6 Hamiltonian is
H6×63D = H(0)+ H(1)+ H(2). (A.2)
Where H(i) is ith order term. Their forms are
(H(0))mn= Hmn0 , (A.3a)
(H(1))mn= Hmn0 , (A.3b)
(H(2))mn= 1 2
X
l
H0mlH0ln
1
Em− El + 1 En− El
. (A.3c)
H0 is a Hamiltonian with eigenenergies Ei of Γ6, Γ8, and Γ7 subbands. Hmn0 is (H8×83D)mn
except for Ei. The variables m and n indicate Γ6 and Γ8 subbands and the variable l indicates Γ7 subbands. H6×63D contains the eigenenergy(H(0)) of Γ6 and Γ8 subbands and all interaction between those subbands (H(1)).
H(0)+ H(1) =
Hc Hch Hch† Hh
. (A.4)
To simplify H(2), we let R and Q as those forms:
Q†=
Hcs† Hhs
, (A.5a)
R† =
Hcs†
Ec−Es
Hhs
Eh−Es
. (A.5b)
We can rewrite H(2) as this form:
H(2) = 1
2 R†Q + Q†R =
Hcs†Hcs
Ec−Es Hcs†Hhs∆chs Hhs† Hcs∆chs H
† hsHhs
Ev−Es
. (A.6)
Where ∆chs = 12
1
Ec−Es +E 1
v−Es
. Hhs† Hhs contributes P0 terms so we drop it. Therefore the 3D 6 × 6 Hamiltonian is
H6×63D =
Hc+ Hc(2) Hch+ Hch(2) Hch† + Hch(2)† Hh
. (A.7)
The Hc(2) and Hch(2) come from 2nd order perturbation term. The Hc(2) term is the effective mass correction term of the conduction band and is of the form:
Hc(2) = P2
3 (Ec− Es)k2. (A.8)
We define Tc= T + 3(EP2
c−Es) so Hc+ Hc(2) becomes this form:
Hc+ Hc(2) = Ec+ Tck2. (A.9)
After dropping the terms we neglected, Hch(2) is of the following form:
EFFECTIVE 2D HAMILTONIAN
We consider a CdTe/HgTe/CdTe quantum well structure showed in Fig. A.1. In z-direction, there is a HgTe layer thickness of d nm between two CdTe layers. We set z = 0 at the middle of HgTe layer. For deriving the 2D effective Hamiltonian, we separate the 3D 6 × 6 Hamiltonian H6×63D into two parts, H0 and H0. Here we treat all kx and ky dependent terms in H6×63D as perturbation(H0). In the set of basis vector (|Γ6, 1/2i,
|Γ6, −1/2i, |Γ8, 3/2i, |Γ8, 1/2i, |Γ8, −1/2i, |Γ8, −3/2i), H0 is of the form:
Figure A.1: It shows the CdTe/HgTe/CdTe quantum well structure. We set z = 0 at the middle of HgTe region in the following calculation.
Ec, Ev, Tc, TH and TL are function of z and all they are in this form
χ (z) = χHg
θ
z +d
2
− θ
z − d
2
+ χCd
θ
−z + d 2
+ θ
z − d
2
. (A.12)
Where θ (z) is step function. χHg is band paraneter of HgTe and χCd is band parameters of CdTe.
The |Γ8, ±3/2i subbands are decoupled to the other subbands and form the heavy hole subbands |Hi, ±i. Because P terms are much larger than Tc and TL terms, we must keep them for calculating the eigenenergy. The |Γ6, ±1/2i and |Γ8, ±1/2i subbands are coupled by q
2
3P kz terms and form the conduction subband |E1, ±i and light hole sub-band |L1, ±i. kz is an odd parity operator. The symmetry of |Γ6, ±1/2i and |Γ8, ±1/2i subbands is different in z-direction. The |Γ6, ±1/2i part of E1 subband is even. The
|Γ6, ±1/2i part of L1 subband is odd. The detial of the basis is in appendix B.
The eigenenergy of H1, H2, H3, E1 and L1 is showed in Fig. A.2. The energys of conduction and light hole subband are different from ref[1]. The energy crossing of the conduction and heavy hole bands is shifted to dc≈ 6.58nm. It is not because we have keep the effective mass corretion terms but because the band parameter of Kane Hamiltonian they use are different.
The effective mass corretion not only effects the eigenenergy but determines the sign of the M1E in H10×102D we will define later. If we are to drop this terms, the M1E will always be negative. According to it, we can see that effective mass corretion is already kept
EFFECTIVE 2D HAMILTONIAN
Figure A.2: This picture shows the eigenenergies of the subbands versus the well thickness d. The blue line is E1. The red line is H1. The red dash line is H2. The red dot line is H3. The green line is L1.
in ref[1]. In their system, they doesn’t use CdTe to be the barrier material. They use Hg0.3/Cd0.7Te to be the barrier material so the band parameter of Kane Hamiltonian is different. It mainly effects the eigenenergy of conduction and light hole subbands.
We define the set of basis vectors (|E1, +i, |H1, +i, |E1, −i, |H1, +i, |H3, +i, |H3, −i,
|H2, +i, |H2, −i, |L1, +i, |L1, −i). The effective 2D 10×10 Hamiltonain H10×102D is defined as
H10×102D (kx, ky)
ij = hi| H6×63D (kx, ky, kz) |ji . (A.13) Where |ii is ith element of basis set. In our calculation we have treat the kz dependent terms as
χkz = 1
2{χ, kz} . (A.14)
χkz2 = kzχkz. (A.15)
Where χ is some band structure parameters like γ. In general, the band structure param-eters are not the same in two materials so kz operating on χ will precdures a δ-function on the boundary. This δ-function contributes a term propertional to χHg − χCd. For
example, the matrix element [H10×102D ]12 is of the form:
Because γ depends on materials and is in the form of equation(A.12), third term of equation (B.5) doesn’t vanish.
According to the symmetry of basis, the effective 2D 10 × 10 Hamiltonain H10×102D is of the form:
EFFECTIVE 2D HAMILTONIAN
A.4 2D Hamiltonian with Dresselhaus spin-orbital interaction (DSOI)
Considering the 3D 6 × 6 Hamiltonian H6×63D including DSOI, the Hamiltonian at Γ point is of the form: The C and Pcterms are much smaller than the others showed in equation (A.18) so we can treat them as perturbation or keep them in calculating the basis vector of effective 2D Hamiltonian. If we treat them as perturbation, the basis vector is as the same as what we derived in the last section. Because the symmetry of DSOI term is different to the others in H6×63D, C and Pc contribute nothing to the terms already existing in H10×102D (equation
is of the form:
EFFECTIVE 2D HAMILTONIAN
Figure A.3: This figure shows the eigenenergy of H10×103D at Γ point with DSOI. E1, H1 and H3 are coupled by Ckz and blue lines show the energy at Γ point for them. L1 and H2 are coupled by Ckz and denote red lines.
H22D =
Becuase δi terms are constants and they couple E1 and H1 subbands, the energy crossing at Γ point will be anti-crossing (See Fig. A.3).
this new set of basis, we can derive a new effective 2D 10 × 10 Hamiltonain ˜H10×102D . For deriving the new basis vector, we define the new H0 as equation (A.19) with Pc = 0.
The Pc terms are smaller than C terms so we treat them as perturbation to simplify the equation.
P kzcouples |Γ6, ±1/2i and |Γ8, ±1/2i subbands. Ckzcouples |Γ8, ±1/2i and |Γ8, ∓3/2i subbands. So |Γ6, ±1/2i and |Γ8, ∓3/2i component have the same symmetry and are real.
|Γ8, ±1/2i component have the opposite symmetry of the others and is pure imaginary.
|Γ6, ±1/2i , |Γ8, ±1/2i and |Γ8, ∓3/2i are coupled together to form
subbands’ |Γ6, ±1/2i component is even function and
Ai, ±˜ E subbands’ |Γ6, ±1/2i component is odd function. The detial of the basis is in appendix C.
The eigenenergy of ˜S and ˜A subbands are showed at Fig. A.4. The P kz and Ckz terms make the curve of S subband not cross to each other. The energy at Γ point of two methods are similar(See Fig. A.5). It shows we can treat C and Pc terms as perturbation and the band structure of two method are comparable.
When d is larger than dc.
S2, ±˜ E
is similiar to |E1, ±i and
S1, ∓˜ E
is similiar to
|H1, ±i. For comparing the result, we define the set of basis vector(
˜i is ith element of basis set. According to the symmetry property of ˜S and ˜A subbands, ˜H10×102D is of the form:
EFFECTIVE 2D HAMILTONIAN
Figure A.4: It shows the eigenenergy of ˜S and ˜A subbands. The blue line is the eigenenergy of ˜S subband versus d. The red line is the eigenenergy of ˜A subband versus d.
Figure A.5: The blue solid line is the eigenenergy of H10×103D at Γ point versus d. The red dash line is the eigenenergy of ˜H10×103D at Γ point versus d.
H˜ef f2D =
EFFECTIVE 2D HAMILTONIAN
A.5 2D band structure: Isotropy and gap-closing
In the section 3.2, we have showed that the DSOI terms make the E1 and H1 subbands anti-crossing. But It doesn’t mean that the global energy band gap between conduction (E1) and heavy hole (H1) bands is always opened. To study whether the gap is cloesd or not, we start our study from the effective Hamiltonian with E1 and H1 subbands only.
For the basis without case, the effective Hamiltonian with E1 and H1 subbands only is of the form:
The eigenenergy Eρµ is of the form (See Appendix E):
E0 = 12(EH1+ EE1). The energy dispersion is isotropy becuase the anisotropy is mainly induced by k+2 and k2− at H12D. Only when we including the effect from H2 and L1 subbands, the bands structure will be anitropy.
The value of aE and a1 are much smaller than A1. The zero-gap exists when the two bands with the same µ touch at the same kc. The value of energy gap is the gap-closing. It makes the Dirac point become the Dirac ring and the global band gap is still closed at a special well width dc.
Then we do the similar procedure to discuss the gap-closing picture in the effective Hamiltonian with the basis incuding C. For d < 8nm, the E1 and H1 couple together to form the ˜S1 and ˜S2 subbands so we study the effective Hamiltonian with S1 and S2 subbands only.
EFFECTIVE 2D HAMILTONIAN
The eigenenergy ˜Eρµ is of the form (See Appendix D):
E˜ρµ= ˜E0+ ˜Dk2+ µ1
2. The band structure is still isotropy when we drop the effect from ˜A1 and ˜A2 subbands.
Because
, the zero-gap exists when the two bands with the same µ touch at the same ˜kc. The value of energy gap is similar to calculation, we find out that the energy gap is closed at d = dc≈ 6.56nm. The difference between this case and the case with the basis without DSOI may come from the high oder terms of Ckz. The high oder terms of Ckz kept in ˜H10×103D is dropped in H10×103D . It
Now we study the gap-closing in the effective 10 × 10 Hamiltonian. We shows the min( ˜E2 − ˜E3) at the 2D k-space versus d in the Fig. A.9. Where ˜E2 and ˜E3 are the energies of middle two bands of conduction and heavy hole bands.(See Fig. A.8.) The energy gap is closed at d ≈ 6.583nm for basis without C and closed at d ≈ 6.558nm for basis with C. We also obtain kc ≈ 4.358 × 10−3nm−1 and ˜kc ≈ 4.781 × 10−3nm−1. The band structure is anisotropy so the value of dc depends on the direction of k. For basis without C the difference of dc is about 10−7nm and the difference of kc is about 10−8nm−1. The difference for basis with C is in the same order. The differences of dc and kc are so small that the dc and kc are isotropy.
We has showed that the dc and kc are isotropy. Is the band structure is isotropy?
In Fig.A.10-A.13, we show the energy difference versus φ, the angle of k vector, with fixed amplitude of k vector. We find out the energy difference is about 10−4meV for k = 0.01nm−1. The energy difference between k = 0 and k = 0.01nm−1 is 2meV. The
structure is nearly isotropy. The isotropy property is because of the anisotropy depending on k2. The band structure is mainly determined by the k linear term so the band structure is isotropy for small k.
EFFECTIVE 2D HAMILTONIAN
Figure A.6: This figure shows the energy dispersion of Hef f2D with d = 6.58nm. The black vertical indicates the kc.
Figure A.7: This figure shows the energy dispersion of ˜Hef f2D with d = 6.56nm. The black vertical indicates the kc.
Figure A.8: This figure shows energy structure of conduction and heavy hole ( ˜S1 and ˜S2) bands. The ˜E1 is the energy of the 1st highest band. The ˜E2 is the energy of the 2nd highest band. The ˜E3 is the energy of the 3rd highest band. The ˜E4 is the energy of the 4th highest band.
Figure A.9: This figure shows min( ˜E2 − ˜E3). The blue solid line is for the Hef f2D. The red dash line is for the ˜Hef f2D.
EFFECTIVE 2D HAMILTONIAN
Figure A.10: This figure shows the energy difference of H10×102D versus φ(angle of k) at d = 6.58nm. Here we show the value of ˜Ei(φ) − ˜Ei(φ = 0) at k = 0.01nm−1.
Figure A.11: This figure shows the energy difference of H10×102D versus φ(angle of k) at d = 7nm. Here we show the value of ˜Ei(φ) − ˜Ei(φ = 0) at k = 0.01nm−1.
Figure A.12: This figure shows the energy difference of ˜H10×102D versus φ(angle of k) at d = 6.56nm. Here we show the value of ˜Ei(φ) − ˜Ei(φ = 0) at k = 0.01nm−1.
Figure A.13: This figure shows the energy difference of ˜H10×102D versus φ(angle of k) at d = 7nm. Here we show the value of ˜Ei(φ) − ˜Ei(φ = 0) at k = 0.01nm−1.
EFFECTIVE 2D HAMILTONIAN