4. Transport and Bound States on Topological Insulators with Induced Ferromag-
4.5. Finite Width Domain Wall - The z-y Configuration
4.5.1. The Exact Solution
At any rate, we can reuse the results from the previous section by noting that for the case ky = 0, the two configurations can be mapped onto each other by performing a rotation around they-axis by π2. The unitary spin rotation matrix is given by
U =e−iπ4σy ,
which performs UσxU† =−σz and UσzU† =σx. Thus, the transformed Hamiltonian which additionally has been brought into dimensionless form, reads
U HzyU†=iσy∂x−qσz−∆ sinθ σy−∆ cosθ σx=Hxy(q = 0)−qσz , (4.140) which is of the form of Hamiltonian (4.80), with the only difference that the transverse momentum q now couples to −σz instead of σx in Hxy. In the present case however, it is much easier to treat finiteq since, once we know the eigensystem forq = 0, i.e. the basis in whichHxy is diagonal, we find thatσz in this basis has also a very simple structure. We will now explicitly illustrate this by determining the bound states of this system.
We denote the eigenstates with eigenenergy±n,0 from result (4.107) as|±ni, so that Hxy(q = 0)|±ni=±n,0|±ni.
where entries not shown are zero. As inferred from the symmetry{Hxy, σz}= 0, we see that σz only couples pairs of positive and negative energy, with the only exception of the zero energy bound state. For n >0, the negative energy state corresponding to |+ni is given by
|−ni = σz|+ni, so that h+n|σz|−ni = 1 and h±n|σz|±ni = 0. Essentially, we now have to diagonalize the 2×2 sub-blocks
n,0 −q
k q
(a) Bulk spectrum
-4 -2 2 4
-4 -2 2 4
Continuum E(q)
q
n= 0 n= 1
n= 2 n= 3
(b) Spectrum in presence of a mass domain-wall
Figure 4.17.: The bulk spectrum is, contrary to the previous study of the x-y configuration, identical on both sides of the domain wall and is shown in (a). However, here we have an additional subtlety, since the mass term due to theMz component of the domain wall drives the system into the QAH state which is of opposite chirality on both sides of the domain wall due to opposite sign of Mz. As thoroughly discussed in chapter 3, the difference emerges for example in the Berry curvatures for quasi-particle transport in this system. The Berry curvatures do not coincide on both sides of the domain-wall, albeit the band structures are identical. Plot (b) shows the spectrum including the bound states at the domain-wall.
However, there is one exception to this namely, the zero energy state |0i which is invariant under the operation of σz and thus, we get −qh0|σz|0i = −q which gives us directly the linearly dispersing chiral state plotted as black straight line in Figure 4.17(b). The dispersion of the other bound states is also shown along with the region of continuum states which we treat in the next section. Like for the x-y wall configuration, we have a total of 2b∆c+ 1 bound states which are symmetric with respect to positive and negative energies, with the major difference being the n= 0 state which is chiral in the present configuration. Also, as opposed to before, here is no upper bound for the value ofq at which the bound states merge with the continuum. The bulk spectrum as it appears in absence of, or far away from the domain-wall is shown in Figure 4.17(a).
We obtain the spinor in the original frame by undoing the rotation, i.e. ψ±n,q(x) =U†|±n, qi. As in (4.113), we are interested in the spin and quasi-particle densities which can be related to the densities forq = 0 by the result
ρ0(n,q)=ψ†±n,qψ±n,q =ρ(n,0)(x)∓ q
n,qs(n,0)z (x) , (4.145) where the densities ρ(n,0)(x) and s(n,0)(x) were calculated in the previous section, c.f. Eqn (4.113). In order to derive this relation, we simply substituted (4.144) into the definition of
the densities. Furthermore
s0(n,q)x =ψ±n,q† σxψ±n,q =−s(n,0)z (x)± q n,q
ρ(n,0)(x) , (4.146)
s0(n,q)y =ψ†±n,qσyψ±n,q =±n,0
n,q s(n,0)y , (4.147)
s0(n,q)z =ψ±n,q† σzψ±n,q =±n,0
n,q s(n,0)x , (4.148)
where in the last two relations we made use of (4.138) and the fact that the perpendicular spin matricesσ⊥= (σx, σy) anti-commute with σz, i.e. {σ⊥, σz}= 0. The densities forq = 0 are identical to those shown in Figure 4.10, with the difference that s0x=ˆ −sz and s0z=sˆ x. In the semiclassical orbit shown in Figure 4.2(b), we can also see the asymmetry in s0x reduced by a finite value of q.
Using the results from (4.114)-(4.116), we can now readily give the expectation values of the spin-densities
hψ±n,q|σ|ψ±n,qi=
−eˆx n= 0
± q n,q
ˆ
ex n >0 , (4.149) so that the current flow along the domain wall becomes as expected,jy =vs0x =±v∂∂qn,q. It is also interesting to note that the spin-density of the zeroth bound state is compatible to the semiclassical expression (3.231),sxc(R, t)∝[ˆez(∇·M)−∇Mz]. For thex-ydomain wall configuration of the previous section, we have a non-vanishing∇·M which yields a finitesz component. In the presentz-yconfiguration on the other side, −∇Mz =−eˆx∂xMz correctly predicts finitesx with the same magnitude as before however, of opposite sign.
4.5.2. Scattering States and Ballistic Transport
For the scattering states, it is of course also possible to take advantage of the results obtained in the previous section. First, we note that kL = kR = k, since the bulk band structure is identical on both sides of the domain wall. We determine the scattering states for finite q exactly as we did in the case of the bound states. It boils down to transforming the two orthogonal solutions from (4.103) and (4.104) independently with a relation like (4.144). Since this transformation only affects spin-space, it acts only on the spinors|χL/Ridefined in (4.88), so we essentially need to consider
|χk,qi= ∓qσz+ (k,0+k,q)σ0
p2k,q(k,q+k,0) |χk,0i , (4.150) where the full energy dispersion becomes
±k,q =±q
2k,0+q2 =±p
∆2+k2+q2 , (4.151) just as in (4.143).
R
∆ k
Figure 4.18.: A plot of the reflection probability (4.152) as a function of ∆ = w/lM and the momentum kin units of the inverse wall width. We see that the reflection probability is strongly suppressed whenever the wall width is an integer multiple of the magnetic length lM.
In the end, the spinors |χk,qidirectly relate to the group velocity by virtue of hχk,q|jx|χk,qi=−vhχk,q|σy|χk,qi=∓vk,0
k,qhχk,0|σy|χk,0i=±v k k,q , where we first used (4.147) and then (4.89). Using (4.146), we can readily see that
hχk,q|jy|χk,qi=±v q k,q
,
and thus is consistent with the definition of the group velocity, or current as j=∇k,qk,q. We see that finite q only modifies the spinor structure to yield the expected group velocity and therefore, the transmission coefficients remain independent of q. Thus, the transmission probability can be directly obtained from (4.125) and noting thatkL=kR=k,
R(∆, k) = sin2(π∆)
sin2(π∆) + sinh2(πk) , (4.152) T(∆, k) = sinh2(πk)
sin2(π∆) + sinh2(πk) , (4.153) which obviously satisfies the conservation lawR+T = 1. The reflection probability is plotted in Figure 4.18.
We now calculate the ballistic conductivity using the Landauer formula (for definitions see (4.126))
G=GQ
W 2π
Z kF
−kF
dq T
∆, q
k2F−q2
, (4.154)
where we introduced the magnitude of the wave vector at the Fermi energykF=q
2F−∆2. Since we evaluate the transmission probability on-shell (i.e. on states with energy equal to the
2 3 4
Figure 4.19.: Relative decrease in the conductance due to the presence of the domain wall.
We see that the values are rather large for sharp walls and decrease significantly when the wall width becomes of the order of the magnetic length ~v/M. The blue curve in (a) for ∆ = 0.01 is already very close to the asymptotic curve for the limit ∆→0, and coincides with the analytical result for an abrupt wall given in (4.156). Note that we considered only values of ∆ which are an odd multiple of 1/2, since the wall becomes transparent when ∆ is an integer number.
Fermi level), the additional constraintF =p
∆2+k2+q2 leads to an implicit dependency onq through the momentumkin the above integration.
In absence of the domain wall, the conductance simply becomesG0 =GQ 2πW 2kF, so with the same definition as in (4.129), we need to solve the following integral in order to obtain the change in conductance, Note that contrary to the previous domain wall configuration, there is no topological contri-butionδGT, since the bulk spectrum is identical on both sides of the wall. The value ofδGis shown in Figure 4.19 as a function of the Fermi levelEF for various values of the parameter
∆, and in Figure 4.20 as a function of the wall widthw.
The change in conductance, and thus the reflection at the domain wall is strongly suppressed when the wall widthw equals the magnetic lengthlM, or equivalently, when the localization energyEw matches the magnetic energyM. This is a direct consequence of the transmission probability (4.153), since the wall becomes transparent whenever ∆ is an integer number, i.e.
then T = 1 for all momenta k and q. This is contrary to the x-y wall configuration, where this transparency occurs only at the singular pointq= 0 viz. kL=kR, so that the signatures are far less pronounced than in the present situation. In particular, only the first minimum is really noticeable in the plots (for example, note the difference between the two curves for
∆ = 1 and ∆ = 1.5 in Figure 4.13(a)).
0.5 1.0 1.5 2.0
Figure 4.20.: The plot in (a) shows the dependence of the domain wall resistance on the wall width w in units of the magnetic length lM = ~v/M, and we can see a monotonous decrease with wall width along with a periodic modulation due to the factor from eq. (4.158). Whenever the domain wall width is an integer multiple of the magnetic length, the domain becomes transparent for the Dirac Fermions andδGdrops to zero. Figure (b) shows the same diagram as in (a) on a logarithmic scale, together with the approximate formula (4.157) (black thin lines). We can see a very good agreement, except for very sharp domain walls.
To further discuss these results, we analytically investigate the two opposite limits of short and wide walls.
Abrupt wall
When the wall is much shorter than any other relevant transport length scales of the system, i.e. ∆, kF 1, we can expand the sines and perform the integration analytically,
δGwl→M
F , while in the opposite limiting case when the Fermi level comes close to the band edgeEF ≈M, we have δG→1, i.e. the abrupt domain wall blocks all transport channels. However, in the latter case, we should be careful, since transport at the Dirac point or at band edges is not well described in the present theory (see remark following equation (4.132)).
Wide wall
In order to obtain an analytical result for the limit of wide walls ∆, kF 1, we use very similar tricks as before in the derivation of (4.135). We find
δG∆1→
with the modulation function
f(∆) = 2 sin2(π∆)
1 + sin2(π∆) . (4.158)
We see that the envelope decreases with 1/w2 as it was the case for the in-plane x-y configu-ration of the domain wall (see result (4.135)). There however, the reduction of the reflection probability for w an integer multiple oflM plays no role in the wide wall limit, which is con-trary to the present configuration, where the modulation function (4.158) still goes to zero for
∆ an integer number. We find that the approximate formula fits well already for ∆&0.5, and in Figure 4.20(b), we show a comparison between numerical integration and the approximate solution (4.157).