The impacts of the quantum-dot confining potential on the spin-orbit effect
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y The impacts of the quantum-dot confining potential on the spin-orbit effect
Rui Li,
1, 2, ∗ Zhi-Hai Liu, Yidong Wu, and C. S. Liu Key Laboratory for Microstructural Material Physics of Hebei Province,School of Science, Yanshan University, Qinhuangdao 066004, China Quantum Physics and Quantum Information Division,Beijing Computational Science Research Center, Beijing 100193, China (Dated: May 11, 2018)For a nanowire quantum dot with the confining potential modeled by both the infinite and thefinite square wells, we obtain exactly the energy spectrum and the wave functions in the strongspin-orbit coupling regime. We find that regardless of how small the well height is, there are atleast two bound states in the finite square well: one has the σ x P = − σ x P = 1 symmetry. When the well height is slowly tuned from large to small, the position ofthe maximal probability density of the first excited state moves from the center to x = 0, while theposition of the maximal probability density of the ground state is always at the center. A strongenhancement of the spin-orbit effect is demonstrated by tuning the well height. In particular, thereexists a critical height V c , at which the spin-orbit effect is enhanced to maximal. I. INTRODUCTION
The spin-orbit coupling (SOC), originating from thelacking of space-inversion symmetry in semiconductormaterials [1], has played an important role in the studiesof topological insulators [2, 3], topological superconduc-tors [4–6], cold atom physics [7–9], spin quantum com-putings [10–15], etc. In the presence of SOC, the orbitaldegree of freedom of the electron is no longer separablefrom its spin degree of freedom, such that it is usually dif-ficult to clarify the strong SOC effect in quantum system.It is also of fundamental interest to explore the physicalproperties of the quantum system beyond the weak SOCregime.A semiconductor quantum dot [16], where a conduc-tion electron of the material is localized by the nearbystatic electric gates, can be considered as an artificialatom. Unlike natural atoms, the artificial atom is moreflexible because many system parameters are externallymanipulable. The electronic [17], magnetic [18], and op-tical [19] properties of the semiconductor quantum dothave attracted extensive research interest.For quantum dot confined in quasi-2D with strongSOC, many theoretical works have devoted to solving thesingle electron energy spectrum. If the confining poten-tial is of the cylindrical type, with the help of the Besselfunction, one can get the exact energy spectrum [20–22].If the confining potential is of the harmonic type [23–26], there is no exact solution. For quantum dot con-fined in quasi-1D with strong SOC [13, 14, 27], the sit-uation would be a little different. Note that quantumdot with quasi-1D confinement, e.g., nanowire quantumdot [28, 29], can already be fabricated experimentally. Ifthe confining potential is of the harmonic type, the 1Dquantum dot model can be mapped to the quantum Rabimodel [30], the energy spectrum can be solved using it- ∗ [email protected] x-a a a x-aV ( x ) V ( x ) V (a) (b) FIG. 1. Schematically shown the confining potential of ananowire quantum dot. (a) ISW with width a . (b) FSWwith width a and height V . eration method [30, 31].In this paper, we study the strong spin-orbit effect in aquasi-1D quantum dot with the confining potential mod-eled by both the infinite square well (ISW) and the finitesquare well (FSW). With respect to both the Z sym-metry of the model and the energy region, we obtain aserious of transcendental equations, their solutions giverise to the exact energy spectrum of the quantum dot.The probability density distribution of the eigenstate inthe FSW can be very different from that in the ISW. In-terestingly, when we slowly lower the well height of theFSW, the position of the maximal probability density ofthe first excited state changes from the center to x = 0;while the position of the maximal probability density ofthe ground state is always at the center. Finally, we studythe electric-dipole transition rate between the lowest Zee-man sublevels. A strong enhancement of the transitionrate by lowering the well height is demonstrated. In par-ticular, we find that there exists a critical well height V c ,at which the spin-orbit effect is enhanced to maximal. II. THE MODEL
We consider a model of nanowire quantum dot, wherea conduction electron is confined in a 1D potential welland subject to both the Rashba spin-orbit field [32] andthe external Zeeman field. The Hamiltonian under con-sideration reads [13, 33, 34] (we set ~ = 1) H = p m + ασ z p + ∆ σ x + V ( x ) , (1)where m is the effective electron mass, α is the SOCstrength, ∆ = g e µ B B/ B , and V ( x ) isthe confining potential. In this paper, we only focus onthe strong SOC regime ( mα > ∆), and the quantum-dot confining potential is modeled by both the ISW [seeFig. 1(a)] and the FSW [see Fig. 1(b)], i.e., V I ( x ) = (cid:26) , | x | < a, ∞ , | x | > a, V F ( x ) = (cid:26) , | x | < a,V , | x | > a, (2)where a and V are the width and the height of the well,respectively.Similar to the quantum Rabi model [30], our modelis also invariant under the following Z transformation:( σ x P ) H ( σ x P ) = H , where P is the parity operator.It follows that σ x P and H have common eigenfunctionΨ( x ), i.e., the eigenstates of the quantum dot can bespecified with respect to the Z symmetry. The σ x P = 1symmetry gives Ψ ( x ) = Ψ ( − x ) , (3)and the σ x P = − ( x ) = − Ψ ( − x ) , (4)where Ψ , ( x ) are the two components of the eigenfunc-tion Ψ( x ) = [Ψ ( x ) , Ψ ( x )] T .All the allowed energies of a quantum system are actu-ally determined by its boundary condition. For the ISW[see Fig. 1(a)], the boundary condition simply readsΨ( a ) = 0 . (5)For the FSW [see Fig. 1(b)], the boundary conditionreadsΨ( a + 0) = Ψ( a − , Ψ ′ ( a + 0) = Ψ ′ ( a − , (6)where Ψ ′ ( x ) is the first derivative of the eigenfunction.Note that the first equation is given by the continuouscondition of the wave function and the second equationis given by the integration lim ε → R a + εa − ε dx ( H − E )Ψ( x ) = 0in the vicinity of the site x = a .It should be noted that we do not need to considerthe boundary condition at the other site x = − a . Be-cause when the boundary condition [see Eq. (5) or (6)]at one site x = a is satisfied, the boundary condition at TABLE I. The parameters of a 1D InSb quantum dot used inour calculations ( m is the electron mass). m e /m [12] g [13] B (T) α (eV ˚A) a (nm) V (meV)0 . . . ∼ the other site x = − a is naturally satisfied due to the Z symmetry. It should be also noted that, in our followingcalculations, we have chosen InSb as our nanowire mate-rial. Unless otherwise stated, the model parameters aregiven in Table. I. III. THE BULK SPECTRUM AND THE BULKWAVE FUNCTIONS
Because of the specific form of the confining potential V ( x ) [see Fig. 1], the Hamiltonian H can be reduced toeither H b = p m + ασ z p +∆ σ x (inside the well) or H b + V (outside the well). In order to find the energy spectrumand the corresponding wave functions of our model, wefirst study the properties of the bulk Hamiltonian H b .The bulk spectrum and the corresponding bulk wavefunctions in the energy region E b ≥ − mα − ∆ mα canbe found elsewhere [35]. The bulk spectrum of plane-wave solution reads [35] E ± b = k m ± p α k + ∆ . (7)The bulk spectrum of exponential-function solutionreads [35] E ± b = − Γ m ± p − α Γ + ∆ . (8)Inside the well | x | < a , the eigenfunction Ψ( x ) of Hamil-tonian (1) can be expanded in terms of the four degener-ate bulk wave functions [35]. However, outside the well | x | > a for the FSW (classical forbidden region), the elec-tron must have a dissipative energy E b < − mα − ∆ mα ,otherwise, the bound state can not be formed. In thefollowing, we address the bulk spectrum and the corre-sponding bulk wave functions in the dissipative energyregion. The bulk wave function in this region can beassumed as Ψ b ( x ) = (cid:18) χ χ (cid:19) e ik ρ e iφ x , (9)where k ρ e iφ is a general complex number with amplitude k ρ and phase φ . This solution can also be considered as acombined plane-wave and exponential-function solution.Substituting the bulk wave function Ψ b ( x ) in Schr¨odingerequation ( H b − E b )Ψ b ( x ) = 0 with the above expression,we have k ρ e iφ m + α k ρ e iφ − E b ∆∆ k ρ e iφ m − α k ρ e iφ − E b ! · (cid:18) χ χ (cid:19) = 0 . (10)Letting the determinant of the matrix (the left 2 × k ρ cos 2 φ = 2 m ( E b + mα ) ,k ρ = 4 m ( E − ∆ ) . (11)Combining these two equations and eliminating the vari-able k ρ , we obtain the bulk spectrum E ± b mα = − ± q (1 − ∆ m α sin φ ) cos φ sin φ . (12)Once the bulk energy E b is obtained, we can obtain fourdegenerate bulk wave functions via Eq. (10)Ψ , ( x ) = (cid:18) R e ± i Φ (cid:19) e ik ρ x cos φ ∓ k ρ x sin φ , Ψ , ( x ) = (cid:18) R e ∓ i Φ (cid:19) e − ik ρ x cos φ ∓ k ρ x sin φ , (13)where R cos Φ = − mα + α k ρ cos φ ∆ ,R sin Φ = − k ρ sin 2 φ + 2 mα k ρ sin φ m ∆ . (14)Outside the well | x | > a (classical forbidden region), theeigenfunction Ψ( x ) of Hamiltonian (1) can be expandedin terms of the above four degenerate bulk wave func-tions.Here, taking the InSb nanowire quantum dot as anexample, we give the bulk spectrum of the Hamiltonianin the strong SOC regime ( mα > ∆). Figures 2(a),(b), and (c) respectively show the bulk spectrum of theplane-wave, the exponential-function, and the combinedplane-wave and exponential-function solutions. Also,from the detailed expressions of the bulk spectrum givenin Eqs. (7), (8), and (12), we have the following gen-eral results which are very useful for the following dis-cussions. For the plane-wave solution [see Fig. 2(a)], E +b ≥ ∆ and E − b ≥ − mα − ∆ mα . For the exponential-function solution [see Fig. 2(b)], − ∆ mα ≤ E +b ≤ ∆and − ∆ ≤ E − b ≤ − ∆ mα . For the combined solution[see Fig. 2(c)], − mα ≤ E +b ≤ − mα − ∆ mα and E − b ≤ − mα . IV. THE ENERGY SPECTRUM AND THEWAVE FUNCTIONS
Since the bulk spectrum and the corresponding bulkwave functions of our model are obtained, the calcula-tions for the energy spectrum are straightforward. Theeigenfunction Ψ( x ) of Hamiltonian (1) is expanded interms of the degenerate bulk wave functions [20, 22, 23].Imposing proper boundary condition [see Eq. (5) or (6)] -10 -5 0 5 10 k (in unit of 10 m -1 ) -1 0010203040 E+bE-b -0.4 -0.2 0 0.2 0.4 (in unit of 10 m -1 ) -1-0.500.51 E+bE-b sin -90-70-50-30-10 E+bE-b E b ( m e V ) (a) (b) (c) FIG. 2. The bulk spectrum of the quantum dot with strongSOC α = 2 . -10-8-6-4-2024 E ( m e V ) (eV ) Å (eV ) Å (a) (b) x P = 1 x P =-1
FIG. 3. The lowest two energy levels as a function of the SOCstrength α . (a) The results in the ISW. (b) The results in theFSW. on Ψ( x ), we analytically derive a series of transcendentalequations with respect to both the Z symmetry and theenergy region (for details see Appendix A and B). Thesolutions of these transcendental equations give us theexact energy spectrum.Figures 3(a) and (b) show the two lowest energy levelsas a function of the SOC α in the ISW and the FSW, re-spectively. First, with increasing the SOC, the effectiveZeeman splitting becomes smaller, similar results werealso obtained in a 2D quantum dot [21]. In the largeSOC limit mα ≫ ∆, i.e., ∆ →
0, Hamiltonian (1) istime reversal invariant, hence each level is 2-fold degen-erate due to Kramer’s degeneracy. Second, the effectiveZeeman splitting is much smaller (the spin-orbit effect ismuch stronger) in the FSW. The spin-orbit effect in thequantum dot can roughly be characterized by the relativeparameter h x i /x so [14, 27], where h x i is the width of thewave function and x so = ~ / ( mα ) is the spin-orbit length.Obviously, h x i F is larger than h x i I , hence the spin-orbiteffect is much stronger in the FSW.We also calculate the probability density distribution x (in unit of a) x (in unit of a) =1 eV Å=2 eV Å=3 eV Å x (in unit of a) V =82.8 meVV =13.8 meVV =4.14 meVV =1.38 meV
0 1 2 3 4 0 1 2 3 40 1 2 3 4 0 1 2 3 4 ( x )( x ) † ( x )( x ) † (a)(b) (c)(d) (e)(f) FIG. 4. (a-d) The probability density distribution in both theISW and the FSW with different SOC α (the height of theFSW is chosen as V = 1 .
38 meV). (a) For the ground statein the ISW. (b) For the first excited state in the ISW. (c) Forthe ground state in the FSW. (d) For the first excited statein the FSW. (e-f) The probability density distribution in theFSW with different potential height V (the SOC is chosenas α = 1 . in the quantum dot for both the ground state and thefirst excited state. It should be noted that the Zeemansublevels here are represented by the ground state andfirst excited state. Figures 4(a) and (b) show the prob-ability density distributions of the ground state and thefirst excited state in the ISW, respectively. Figures 4(c)and (d) show the probability density distributions of theground state and the first excited state in the FSW, re-spectively. When the well height of the FSW is small V = 1 .
38 meV, the probability density distribution inthe FSW is apparently distinct from that in the ISW,where the position of the maximal probability density ofthe first excited state is not at the center [see Fig. 4(d)].It is of interest to know how V affects the probabilitydensity distribution in the FSW. In Figs. 4(e) and (f), forvarious well heights V , we show the probability densitydistributions of the ground and the first excited statesrespectively. As can be seen from the figure, the positionof the maximal probability density of the ground state isalways at the center ( x = 0). When the well height islarge, e.g., V = 82 . x = 0). However, as we slowly lower V , thereexists a critical V c , below which the position of the max-imal probability density moves to x = 0 [see Fig. 4(f)].This will induce interesting phenomena in the followingdiscussion of the electric-dipole spin resonance.We also find that no matter how small the well height V is, there always exist at least two bound state in theFSW, one is labeled by the σ x P = − σ x P = 1 symmetry. V c0
1 2 3 41 1.5 2 2.5 0 20 40 60 80 R ( i n un it o f e E a ) R ( i n un it o f e E a ) R ( i n un it o f e E a ) (eV ) Å V (meV) (a)(b) (c) FIG. 5. (a) The Rabi frequency as a function of the SOC inthe ISW. (b) The Rabi frequency as a function of the SOCin the FSW. (c)The Rabi frequency as a function of the wellheight V in the FSW. The SOC is fixed at α = 1 . V. ELECTRIC-DIPOLE SPIN RESONANCE
In the presence of the SOC, the spin degree of freedomis mixed with the orbital degree of freedom, such thatthe spin in the quantum dot can respond to an exter-nal oscillating electric field eEx cos( ω t ), an effect calledelectric-dipole spin resonance [10–14, 36–44]. Becausethe wave functions in the quantum dot are obtained inthe previous section, we are able to calculate the Rabifrequency of the electric-dipole spin resonance.When the frequency ω of the electric field matchesthe level spacing of the Zeeman sublevels, the electricfield will induce an electric-dipole transition rate, i.e.,the Rabi frequency, between the Zeeman sublevelsΩ R = 2 eE (cid:12)(cid:12)(cid:12) Z Ξ0 dx Ψ † g ( x ) x Ψ e ( x ) (cid:12)(cid:12)(cid:12) , (15)where Ξ = a and ∞ represent the integration boundaryfor the FSW and the ISW respectively, and Ψ g,e ( x ) de-notes the ground (the first excited) state wave function.In Figs. 5(a) and (b), we show the Rabi frequency as afunction of the SOC α in the ISW and the FSW, respec-tively. The Rabi frequency in the FSW can be almostone order larger than that in the ISW. Why the spin-orbit effect is so large in the FSW? We trace back to thewave functions given in Fig. 4. In the FSW, the positionof the maximal probability density of the first excitedstate is not at x = 0, while the position of the maximalprobability density of the ground state is at x = 0, suchthat it is possible to produce a large Rabi frequency viaEq. (15). Also, in the large SOC limit α → ∞ , the Rabifrequency becomes zero [14, 27]. This is because in thelarge SOC limit mα ≫ ∆, i.e., ∆ →
0, the operator σ z would be a good quantum number [see Eq. (1)], hencethe Rabi frequency is zero.In Fig. 5(c), we show the dependence of the Rabi fre-quency on the well height V of the FSW. Obviously, inthe large V limit, e.g., V → ∞ , the Rabi frequencyin the FSW would coincide with that in the ISW (seeFigs. 5). Lower the well height can remarkably enhancethe SOC effect in the quantum dot. Interestingly, we findthere exists a critical well height V c , at which the Rabifrequency becomes maximal [see Fig. 5(c)], i.e., the spin-orbit effect is enhanced to maximal. Below the critical V c , if we continue to lower V , the Rabi frequency de-creases sharply. This result is reasonable, it is impossibleto infinitely enhance the spin-orbit effect. In the V → V c somewhere whenwe lower the well height. VI. SUMMARY
In the presence of both the strong SOC and the Zee-man field, we have obtained exactly the energy spectrumand the corresponding wave functions in both the ISWand the FSW. The spin-orbit effect is much stronger inthe FSW than that in the ISW. Moreover, the probabil-ity density distribution in the FSW can be very differentfrom that in the ISW. A strong enhancement of the SOCeffect is demonstrated by tuning the height of the con-fining potential. In particular, we show that there existsa critical well height, at which the spin-orbit effect isenhanced to maximal.
ACKNOWLEDGEMENTS
This work is supported by National Natural Sci-ence Foundation of China Grant No. 11404020 andPostdoctoral Science Foundation of China GrantNo. 2014M560039.
Appendix A: The transcendental equations in theISW quantum dot1. Energy region: − mα − ∆ mα ≤ E ≤ − ∆ In this energy region, as can be seen from the bulkspectrum [see Fig. 2(a)], one can find four k solutions ± k , from the ‘ − ’ dispersion relation given in Eq. (7) k , = √ mα s Emα ± r Emα + ∆ m α . (A1)Thus, the eigenfunction Ψ( x ) can been written as a lin-ear combination of these four degenerate bulk wave func-tions. Note that all of the four bulk wave functions belongto the ‘ − ’ branch. In the coordinate region | x | < a , theeigenfunction reads [35]Ψ( x ) = c (cid:18) sin θ − cos θ (cid:19) e ik x + c (cid:18) cos θ − sin θ (cid:19) e − ik x + c (cid:18) sin θ − cos θ (cid:19) e ik x + c (cid:18) cos θ − sin θ (cid:19) e − ik x , (A2) where θ , = arctan (cid:2) ∆ / ( α k , ) (cid:3) and c , , , are the co-efficients to be determined.As we emphasized before, we can specify the eigen-function Ψ( x ) with respect to the Z symmetry. Thissymmetry gives some constraints on the coefficients. Forthe σ x P = 1 symmetry, the relationship Ψ ( x ) = Ψ ( − x )gives rise to c = − c and c = − c . For the σ x P = − ( x ) = − Ψ ( − x ) gives riseto c = c and c = c . In other words, we only have twocoefficients c , to be determined. Using the hard-wallboundary condition Ψ , ( a ) = 0, we obtain the followingtranscendental equationsin[( k − k ) a ]sin[( k + k ) a ] = ∓ sin[( θ − θ ) / θ + θ ) / , (A3)where the minus sign ‘ − ’ and the plus sign ‘+’ correspondto the σ x P = 1 and σ x P = −
2. Energy region: ∆ ≤ E As also can be seen from the bulk spectrum [seeFig. 2(a)], one can find ± k and ± k solutions fromthe ‘+’ and ‘ − ’ dispersion relations given in Eq. (7), re-spectively. The eigenfunction Ψ( x ) can be written as alinear combination of these four degenerate bulk wavefunctions, i.e., two from the ‘+’ branch and two fromthe ‘ − ’ branch. In the coordinate region | x | < a , theeigenfunction reads [35]Ψ( x ) = c (cid:18) cos θ sin θ (cid:19) e ik x + c (cid:18) sin θ cos θ (cid:19) e − ik x + c (cid:18) sin θ − cos θ (cid:19) e ik x + c (cid:18) cos θ − sin θ (cid:19) e − ik x . (A4)The σ x P = 1 symmetry gives rise to c = c and c = − c , and the σ x P = − c = − c and c = c , such that only two coefficients c , are to be determined. The hard-wall boundary con-dition Ψ , ( a ) = 0 gives us the following transcendentalequation sin[( k + k ) a ]sin[( k − k ) a ] = ± sin[( θ + θ ) / θ − θ ) / , (A5)where the plus sign ‘+’ and the minus sign ‘ − ’ correspondto the σ x P = 1 and σ x P = −
3. Energy region: − ∆ ≤ E ≤ − ∆ mα In this energy region, one can find two solutions ± k from the ‘ − ’ branch dispersion relation given in Eq. (7) k = √ mα s Emα + r Emα + ∆ m α . (A6)One also can find two solutions ± Γ from the ‘ − ’ branchdispersion relation given in Eq. (8)Γ = √ mα s − − Emα + r Emα + ∆ m α . (A7)Thus, in the coordinate region | x | < a , the eigenfunctionΨ( x ) can be expanded as a linear combination of thefour degenerate bulk wave functions, i.e., two from the‘ − ’ branch of the plane-wave solution and two from the‘ − ’ branch of the exponential-function solution [35]Ψ( x ) = c e − Γ x (cid:18) − e − iϕ (cid:19) + c e Γ x (cid:18) − e iϕ (cid:19) + c e ikx (cid:18) sin θ − cos θ (cid:19) + c e − ikx (cid:18) cos θ − sin θ (cid:19) , (A8)where θ ≡ θ ( k ) = arctan [∆ / ( α k )] and ϕ ≡ ϕ (Γ) =arctan (cid:0) α Γ / √− α Γ + ∆ (cid:1) . For the σ x P = 1 sym-metry, the relationship Ψ ( x ) = Ψ ( − x ) gives rise to c = − c e − iϕ and c = − c . The boundary conditionΨ , ( a ) = 0 gives us the following transcendental equa-tioncos( ka − ϕ/ − e a cos( ka + ϕ/ ka + ϕ/ − e a cos( ka − ϕ/
2) = tan( θ/ . (A9)For the σ x P = − ( x ) = − Ψ ( − x ) gives rise to c = c e − iϕ and c = c . Theboundary condition Ψ , ( a ) = 0 gives us the followingtranscendental equationsin( ka − ϕ/
2) + e a sin( ka + ϕ/ ka + ϕ/
2) + e a sin( ka − ϕ/
2) = tan( θ/ . (A10)
4. Energy region: − ∆ mα ≤ E ≤ ∆ In this energy region, one can find two solutions ± k from the ‘ − ’ branch dispersion relation given in Eq. (7).One also can find two solutions ± Γ from the ‘+’ branchdispersion relation of given in Eq. (8). Thus, in the co-ordinate region | x | < a , the eigenfunction Ψ( x ) can beexpanded as a linear combination of these four degener-ate bulk wave functions, i.e., two from the ‘ − ’ branch ofthe plane-wave solution and two from the ‘+’ branch ofthe exponential-function solution [35]Ψ( x ) = c e − Γ x (cid:18) e iϕ (cid:19) + c e Γ x (cid:18) e − iϕ (cid:19) + c e ikx (cid:18) sin θ − cos θ (cid:19) + c e − ikx (cid:18) cos θ − sin θ (cid:19) . (A11)For the σ x P = 1 symmetry, the relationship Ψ ( x ) =Ψ ( − x ) gives rise to c = c e iϕ and c = − c . The boundary condition Ψ , ( a ) = 0 gives us the followingtranscendental equationsin( ka + ϕ/
2) + e a sin( ka − ϕ/ ka − ϕ/
2) + e a sin( ka + ϕ/
2) = − tan( θ/ . (A12)For the σ x P = − ( x ) = − Ψ ( − x ) gives rise to c = − c e iϕ and c = c . Theboundary condition Ψ , ( a ) = 0 gives us the followingtranscendental equationcos( ka + ϕ/ − e a cos( ka − ϕ/ ka − ϕ/ − e a cos( ka + ϕ/
2) = − tan( θ/ . (A13) Appendix B: The transcendental equations in theFSW quantum dot
Outside the square well x > a , because of the con-straint lim x →∞ Ψ( x ) = 0, the eigenfunction can only bewritten asΨ( x ) = c (cid:18) R e i Φ (cid:19) e ik ρ x cos φ − k ρ x sin φ + c (cid:18) R e − i Φ (cid:19) e − ik ρ x cos φ − k ρ x sin φ , (B1)where c , are the coefficients to be determined and k ρ cos φ = mα s E − V mα + r ( E − V ) − ∆ m α ,k ρ sin φ = mα s − − E − V mα + r ( E − V ) − ∆ m α . (B2)The other two bulk wave functions Ψ , b ( x ) are divergentin the limit x → ∞ . Inside the square well | x | < a ,the eigenfunction can still be written as those given inEqs. (A2), (A4), (A8), and (A11) with respect to theenergy region.The eigenfunctions can still be specified with respectto the Z symmetry. For eigenfunction inside the well,we have two coefficients c , to be determined. Also, foreigenfunction outside the well, we have the other twocoefficients c , to be determined [see Eq. (B1)]. Theboundary condition, given by Eq. (6), give us a matrixequation M · C = 0 . (B3)where M is a 4 × C = ( c , c , c , c ) T . Letthe determinant of the matrix M equal to 0, we obtain atranscendental equation which is an implicit equation ofthe energy E det( M ) = 0 . (B4)Similar to the discussions in the ISW, here we also canobtain a series of transcendental equations with respectto both the Z symmetry and the energy region. Thedetailed expression of the matrix M is given as followes. In the energy region − mα − ∆ mα ≤ E ≤ − ∆, thematrix M reads M ± = e ik a sin θ ∓ e − ik a cos θ e ik a sin θ ∓ e − ik a cos θ − e ik x a − k y a − R × e − i ( k x a +Φ) − k y a − e ik a cos θ ± e − ik a sin θ − e ik a cos θ ± e − ik a sin θ − R × − e − ik x a − k y a e i ( k x a +Φ) − k y a ik (cid:0) e ik a sin θ ± e − ik a cos θ (cid:1) ik (cid:0) e ik a sin θ ± e − ik a cos θ (cid:1) − ( ik x − k y ) × R ( ik x + k y ) × e ik x a − k y a e − i ( k x a +Φ) − k y a − ik (cid:0) e ik a cos θ ± e − ik a sin θ (cid:1) − ik (cid:0) e ik a cos θ ± e − ik a sin θ (cid:1) − R ( ik x − k y ) × ( ik x + k y ) × e i ( k x a +Φ) − k y a e − ik x a − k y a . (B5)In the energy region E ≥ ∆, the matrix M reads M ± = e ik a cos θ ± e − ik a sin θ e ik a sin θ ∓ e − ik a cos θ − e ik x a − k y a − R × e − i ( k x a +Φ) − k y a e ik a sin θ ± e − ik a cos θ − e ik a cos θ ± e − ik a sin θ − R × − e − ik x a − k y a e i ( k x a +Φ) − k y a ik (cid:0) e ik a cos θ ∓ e − ik a sin θ (cid:1) ik (cid:0) e ik a sin θ ± e − ik a cos θ (cid:1) − ( ik x − k y ) × R ( ik x + k y ) × e ik x a − k y a e − i ( k x a +Φ) − k y a ik (cid:0) e ik a sin θ ∓ e − ik a cos θ (cid:1) − ik (cid:0) e ik a cos θ ± e − ik a sin θ (cid:1) − R ( ik x − k y ) × ( ik x + k y ) × e i ( k x a +Φ) − k y a e − ik x a − k y a . (B6)In the energy region − ∆ ≤ E ≤ − ∆ mα , the matrix M reads M ± = − e − Γ a − iϕ ± e Γ a e ika sin θ ∓ e − ika cos θ − e ik x a − k y a − R × e − i ( k x a +Φ) − k y a e − Γ a ∓ e Γ a − iϕ − e ika cos θ ± e − ika sin θ − R × − e − ik x a − k y a e i ( k x a +Φ) − k y a Γ (cid:0) e − Γ a − iϕ ± e Γ a (cid:1) ik (cid:0) e ika sin θ ± e − ika cos θ (cid:1) − ( ik x − k y ) × R ( ik x + k y ) × e ik x a − k y a e − i ( k x a + ϕ ) − k y a − Γ (cid:0) e − Γ a ± e Γ a − iϕ (cid:1) − ik (cid:0) e ika cos θ ± e − ika sin θ (cid:1) − R ( ik x − k y ) × ( ik x + k y ) × e i ( k x a +Φ) − k y a e − ik x a − k y a . (B7)In the energy region − ∆ mα ≤ E ≤ ∆, the matrix M reads M ± = e − Γ a + iϕ ± e Γ a e ika sin θ ∓ e − ika cos θ − e ik x a − k y a − R × e − i ( k x a +Φ) − k y a e − Γ a ± e Γ a + iϕ − e ika cos θ ± e − ika sin θ − R × − e − ik x a − k y a e i ( k x a +Φ) − k y a − Γ (cid:0) e − Γ a + iϕ ∓ e Γ a (cid:1) ik (cid:0) e ika sin θ ± e − ika cos θ (cid:1) − ( ik x − k y ) × R ( ik x + k y ) × e ik x a − k y a e − i ( k x a + ϕ ) − k y a − Γ (cid:0) e − Γ a ∓ e Γ a + iϕ (cid:1) − ik (cid:0) e ika cos θ ± e − ika sin θ (cid:1) − R ( ik x − k y ) × ( ik x + k y ) × e i ( k x a +Φ) − k y a e − ik x a − k y a . (B8)Here k x = k ρ cos φ , k y = k ρ sin φ , and M ± means M σ x P = ± . [1] Winkler, R. Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems. Springer (2003). [2] Hasan, M. Z. & Kane, C. L. Colloquium : Topological insulators. Rev. Mod. Phys. , 3045 (2010).[3] Konig, M., Wiedmann, S., Brune, C., Roth, A., Buh-mann, H., Molenkamp, L. W., Qi, X. L. & Zhang, S.C. 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