4. Transport and Bound States on Topological Insulators with Induced Ferromag-
4.4. Finite Width Domain Wall - The x-y Configuration
4.4.4. Bound States at the Domain Wall
We are now looking for bound states, i.e. solutions with imaginary wave-vector−ikR/L>0, so that the wave function becomes exponentially localized at the domain wall. We can easily investigate this with the help of the asymptotic expansion (4.103), and since Γ(−n)1 = 0 for n≥0, we find from definition (4.102) thatA(kL, kR) = 0, provided that
−i
2(kL+kR) = ∆−n , n= 0,1,2. . .b∆c . (4.105)
Using relation (4.87), we find for the energies of the bound states n,q =n,0
s
1− q2
(∆−n)2 , (4.106)
with the energy for zero transverse momentum q = 0, n,0 =p
(2∆−n)n . (4.107)
Therefore, we expect a total number of b∆c+ 1 bound states for energies ≥ 0. Due to {H, σz}= 0, the spectrum is symmetric with respect to positive and negative energies, so in total, we have 2b∆c+ 1 bound states at the domain wall, counting the zero energy staten= 0 only once. In fact, we see that the hypergeometric function in (4.100) and (4.101) terminates when the above condition is fulfilled and thus, we can directly obtain the wave function of the bound states in analytical form as a power series inz= 12(1−tanhx).
Once n,q = ∆− |q|, the bound states merge with the continuum (see Figure 4.8(b)) and become evanescent modes, i.e. a incoming wave that is totally reflected at the domain wall and decays exponentially on the other side of the wall. If the energy of an state is≥∆ +|q|, it becomes a scattering state which is doubly degenerate and exists as a plane wave on both sides of the wall. From the condition n,q = ∆− |q|, one obtains the minimal and maximal values for the transverse momentum,
qmax(n) = (∆−n)2
∆ , (4.108)
so that the bound state exists only for |q| ≤ q(n)max. For n= 0, we have a dispersionless (flat) band for values of the transverse momentum|q| ≤∆.
Restoring original units, we get for the dispersion relation En(ky) =p
(2M−nEw)nEw
s
1− ~2v2k2y
(M −nEw)2 , (4.109) and for the maximal transverse momentum
(ky)(n)max= (M −nEw)2
~vM , (4.110)
where we introduced the energy Ew ≡ ~wv which corresponds to the localization energy of a particle confined within the extensions of the domain wall of widthw, as it would be obtained in a semiclassical Bohr-Sommerfeld quantization scheme. The square root dependence of the energy is typical for Dirac fermions, and we have already seen an example thereof in the case of Landau levels for Dirac fermions in equation (4.33).
Actually, in the limit of wide walls and sufficiently low orbitn, the energyEn(0)→√
2M nEw, which is identical to the Landau-level spectrum of result (4.33), when we use the fact that in the center of the wall the effective field defined by (4.29) is constant for wide walls,Beff → vwM. Of course, when the quantum numbernbecomes comparable to ∆, we obtain corrections from the Landau level result as expressed by (4.109), since the orbits become large and extend into the outer regions of the domain wall where the assumption of constantBeff no longer is valid.
-3 -2 -1 1 2
Figure 4.10.: The 4 bound states at the domain wall for the parameters ∆ = M w
~v = 3.9 and q = 0. In this case, all the spin expectation values are zero, except for n= 0, i.e. hψn,0|σ|ψn,0i =δn,0 eˆz. The sx component essentially corresponds to the current along the domain wall, since jy(x) = vsx(x) can be interpreted as a current density (see continuity equation (4.5)). We see that on the left side of the domain wall (x < 0) the current flows in positive direction, and in the opposite direction on the other side (x > 0), thus constituting a persistent current circulating around the whole domain wall. We also note that wheneversx
attains a maximum,sz goes through zero (except forn= 0). This is nothing but a manifestation of relation (4.10), which for the present case reads ∂s∂xx = 2n,qsz. This implies that circulating currents will enclose regions of finite sz, with the sign ofszgiven by the chirality of current flow. The states forn≥1 correspond to the first type of semiclassical orbit shown in Figure 4.2(a). The sign ofsx is also consistent in both semiclassical and quantum picture.
Let us now discuss the wave functions of these bound states by plugging the energy dispersion (4.106) into definition (4.87), so that we obtain the inverse localization lengths for the left and right side of the domain wall,
−ikL/R = ∆−n± ∆ q
∆−n . (4.111)
Of course, for highern (i.e. higher excited states), the Dirac fermions become less localized and spread farther out.
Furthermore, due to the symmetry [K,H] = 0, we can choose our solutions ϕ↑ and ϕ↓ to be real, which leads to a vanishing y-component of the spin-expectation value. This in turn implies that there is no current flow in transport direction, sincej=veˆz×σ.
Let us first focus on the zero energy bound staten= 0, which we find by plugging−ikL/R=
-3 -2 -1 1 2 0.2
0.4 0.6 0.8
x ρ
sx
sz
n= 0
-3 -2 -1 1 2
-0.4 -0.2 0.2 0.4
x
ρ sx
sz
n= 1
Figure 4.11.: The bound states for finite transverse momentumq are asymmetric. We chose the parameters ∆ = 3.9 andq = 1.5, so that there are in total 2 bound states at the domain wall. We see that the density distribution becomes asymmetric, which is a feature also exhibited by the semiclassical orbits. The state n = 1 corresponds to the third type of semiclassical orbit shown in Figure 4.2(a), with a group velocity pointing along the negativey-axis, since ∂qn,q<0. On the right side of the domain wall, sx <0. Note that the state withn= 0 is not captured by semiclassical descriptions since it involves a coherent excitation between the upper and lower band, i.e. it sits at the Dirac point. As in Figure 4.10, the identity ∂s∂xx = 2n,qsz applies here too.
∆(1±q) into (4.100) and (4.101),
ϕ(0,q)↑ (x) =N0
eqx (coshx)∆ ,
ϕ(0,q)↓ (x) = 0 , (4.112)
which has been properly normalized to R+∞
−∞ dx |ϕ↑|2+|ϕ↓|2
= 1 by setting the normal-ization factor toN02 = 22∆∆Γ(∆−q)Γ(∆+q)Γ(1+2∆) . The higher excited states can be straightforwardly obtained by substituting (4.111) into (4.100) and (4.101), and this step terminates the hyper-geometric series, turning the solution into a polynomial of degree n.
The spin densities are defined as
s(x) = (sx, sy, sz) =ψ†σψ= 2Reϕ∗↑ϕ↓,2Imϕ∗↑ϕ↓, ϕ∗↑ϕ↑−ϕ∗↓ϕ↓
, (4.113)
and the probability density of an excitation in the bound state is simplyρ(x) =ϕ∗↑ϕ↑+ϕ∗↓ϕ↓. As noted above, sy vanishes identically, and the other densities for the bound states are shown in figures 4.10 and 4.11. We see thatsx(−x) =sx is antisymmetric whilesz andρ are symmetric in the case q = 0. For finite transverse momentum q, the bound states become asymmetric and attain a higher weight on one side of the domain wall, which is clear, since according to (4.87), one side is energetically favorable.
The spin-expectation values in these states are obtained by integrating s(x) over all x ∈R,
Topological Insulator Ferromagnet
x y z
V
Figure 4.12.: The experimental setup for studying ballistic transport through a domain wall.
Two leads with external voltage bias V are attached to the structure of topo-logical insulator coated with a thin ferromagnetic layer.
and can be summarized as
hψn,q|σx|ψn,qi=− q 2n,0
(∆−n)2n,q = ∂n,q
∂q , (4.114)
hψn,q|σy|ψn,qi= 0 , (4.115)
hψn,q|σz|ψn,qi=δn,0 , (4.116)
where the expectation value ofσz is a direct result of the symmetry{H, σz}= 0, sinceσz|ψn,qi is the state belonging to the negative energy eigenvalue −n,q and which is orthogonal to its positive energy counterpart. Of course, the zero energy bound state is its own counterpart σz|ψn,qi = |ψn,qi so for n = 0, Eq. (4.116) constitutes just the normalization of the state.
Finally, the expectation value of σx exhibits the asymmetry induced by the transverse mo-mentumq and vanishes for q = 0. In fact, from relation (4.91) we see that the expectation value ofσxis directly related to the current in the transverse direction, i.e. along the domain wall, and it simply yields the usual definition of the group velocity derived from the dispersion (4.109),
jy =vhψn,q|σx|ψn,qi=v∂n,q
∂q . (4.117)