Scattering States and Ballistic Transport

We consider the experimental setup for ballistic transport across the domain wall illustrated in Figure 4.12 and which we study using the formalism of scattering theory. The general scattering problem can be treated by finding the asymptotics of our exact wave-functions, which generally is of the form (see [122] for the general scattering formalism and Landauer formula)

ψ=









 aL

e^ikLx

√vL|χ_Li+bL

e^−ikLx

√vL |χ^∗

Li x→ −∞

aR

e^−ikRx

√vR

|χ^∗_Ri+bR

e^ikRx

√vR

|χ_Ri x→+∞ .

(4.118)

The coefficientsaL/R describe the amplitudes of the incoming waves, while bL/R are the am-plitudes of the outgoing scattered wave. The scattering region in this particular case is the domain wall described by the magnetization profile (4.78). For finite transverse momentum

q, the longitudinal (along the transport direction) group velocities on the left and the right side of the domain wall differ and explicitly read (see result (4.92))

vL/R =vkL/R

, (4.119)

and have been incorporated into the definition of the scattering states, so that the coeffi-cients a,b correspond to transmission and reflection amplitudes. The ingoing and outgoing coefficients are related through the scattering matrix

bL

The reflection and transmission coefficients are readily extracted from the asymptotic expan-sion (4.103) and (4.104),

r = −A(−kL, kR)

A(kL, kR) , (4.121)

t= 1

A(kL, kR) , (4.122)

and our solutions indeed show the symmetry that is required for the scattering matrix to be unitary, S^†S = 1₂. Current conservation is also incorporated into this condition of unitar-ity, which can be seen by explicitly consideringψ₁ from result (4.103) which in the present language can be written as

The stateψ₁ yields the current contribution in the transport direction

jx∝1− |r|²=|t|² , (4.124) as is directly obtained using result (4.93) and taking the real part thereof. Thus,R_q() =|r|² and Tq() = |t|² can indeed be given a physically meaningful definition as reflection and transmission probabilities, respectively. These probabilities explicitly depend on energyand transverse momentumq.

Using our results (4.102) and (4.122) and some properties of the Gamma function (E.8)-(E.12), we eventually arrive at the transmission probability

Tq() = sinh(πkL) sinh(πkR)

sin²(π∆) + sinh^{2 1}₂(kL+kR)π , (4.125) where kL/R is given in (4.87).

A central result in quantum transport is the Landauer formula for the linear conductance G G= e²

2π~ X

q

T_q(_F) , (4.126)

which states that the current is essentially given by the sum over all transverse channels, each with a weight given by the transmission probability. The fundamental conductance quantum GQ = _2π^e2

~ is the numerical contribution of each channel to the conductance [122]. In linear response, the current is then simply given by jxW = I = GV, where V is the externally applied voltage and W is the transverse dimension of the ballistic contact, while j_x is the current density (per unit length) along the x-direction. In the linear transport regime, all quantities are evaluated at the Fermi level_F.

The transverse momentum q runs over all values that lie inside the shaded yellow area in Figure 4.8(b), i.e. |q| ≤_F−∆, and beyond this range, we either have total reflection or no states at all, so there is no contribution. Therefore,

G=GQ

W 2π

Z _F−∆

−(F−∆)

dq Tq(_F) , (4.127)

where we rewrote the sum as an integral and _2π^W is the density ink_y-space. In absence of the domain wall, the conductance simply becomes

G₀ =G_Q W 2π

Z F−∆

−_F−∆

dq =G_Q W _F

π . (4.128)

We are now interested in change of the conductance due to the presence of the domain wall, so we define a domain wall resistance as

δG=−G_DW−G₀

G₀ =δG_T+δG_DW , (4.129)

which we split into a topological contribution δG_T and the contribution δG_DW that is de-pendent on the specific domain wall profile. In particular, the former depends only on the magnetization M viz. ∆, and does not depend on the details of the domain wall, while the latter depends for example on the domain wall width. The topological contribution is due to the mere reduction of states available for transport across the domain wall, since some of them become totally reflecting. This can be seen in Figure 4.8(b) for example, since in the presence of the domain wall, the contributing transport channels get reduced to the yellow area, while without the domain wall the whole area within the red cone would be available.

We remark that the appearance of the two Dirac cones on both sides that are shifted with respect to each other, is due to the opposite magnetization direction on both sides. These observations lead to

δGT= ∆ _F = M

E_F >0 , (4.130)

and is the fraction of the contribution from the red shaded area of totally reflecting states to the total contribution from the Dirac cone (yellow + red shaded areas in fig. 4.8(b)). In the end, δG_T incorporates the differences in the bulk spectrum on both sides of the wall, as far as it concerns transport at the Fermi level.

The remaining contribution from the domain wall is then related to the reflection from the wall profile and is explicitly given by

δG_DW = 1 2_F

Z _F−∆

−(_F−∆)

dq (1−T_q(_F)) = 1 2_F

Z _F−∆

−(_F−∆)

dq R_q(_F) . (4.131)

2 3 4

Figure 4.13.: 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 is already pretty close to the asymptotic curve for the limit of sharp walls, ∆→0 (see Figure 4.15(a)).

We studyδGDW as a function of the two relevant parameters ˜EF =EF/M, the Fermi level in units of the magnetic exchange fieldM, and ∆ =w/lM, i.e. the domain wall width in units of the magnetic length l_M =~v/M. We introduce the new integration variable ζ = q/∆ in order to bring the above expression into the more convenient form

δG_DW = 1

When ∆ is an odd multiple of 1/2, we can see slightly enhanced peaks when E_F is close toM (see for example the green curve for ∆ = 5.5 in diagram 4.13(b)). This behavior can be traced back to the term sin²(π∆) in the denominator of the transmission probability (4.125) which is zero for ∆ being an integer number and thus enhances transmission. For ∆ being an odd multiple of 1/2, this term is maximal and reduces transmission so that reflection increases and so doesδGDW. When the Fermi level EF approachesM and eventually reaches the crossing point of the left and right Dirac cones (crossing of red and green line in Figure 4.8(b)), the number of states available for transport drastically reduces and one reaches the point of minimal conductance which is not well described in our theory. At this point, transport is due to evanescent modes [127, 178], which is beyond the present study.

The total change in conductanceδG=δG_T+δG_DW is shown in Figure 4.14, and we find that the topological contributionδG_T strongly dominates, except for short walls for which ∆.1.

We also study the behavior ofδG_DW with the wall width, which is shown in Figure 4.15. As before, the change in conductance is largest for sharp walls and monotonically decreases as the wall width becomes larger. As the fermi level moves deeper into the metallic regime, we also see an overall decrease of the change in conductance.

2 3 4 0

20 40 60 80

EF

M

δGin % _{∆ = 0.01}

∆ = 0.5

∆ = 1

∆ = 1.5

Figure 4.14.: A plot of the total reduction of the conductance due to the domain wall,δG= δG_T+δG_DW. The parameters are the same as in Figure 4.13(a), and for values of

∆ significantly larger than 1, the contribution ofδGTdominates and thus, in the above plot the red and cyan lines already lie very close together. Experimentally, one could subtract the topological contribution (determined by measuring the bulk properties) in order to obtain the contribution from the domain wall profile.

Let us now consider the two limits of sharp (abrupt) walls and very wide walls for which we will derive analytical expressions.

Sharp wall

In this regime,wis very small compared to the magnetic lengthl_M, so that ∆1 and we can expand sin(π∆)≈π∆ and sinh(π∆k)≈π∆kin the result for the transmission probability (4.125), and find an expression that only depends on ˜E_F ≡_F/∆ =E_F/M

δG_DW ^∆1→ 1 E˜F

Z E^˜F−1 0

dζ

1− 4k₊k−

4 +k++k−

. (4.133)

For ˜E_F 1, we can approximately solve the integral and find the asymptotic result δG_DW

∆1˜ E_F1

→ 1 3 ˜EF

= M

3E_F , (4.134)

which equals to one third of the topological contribution δG_T. Therefore, the domain wall contribution δG_DW is at best 1/3 of the topological contribution δG_T, since for wider walls δG_DW decreases further, while δG_T does not depend on the wall width.

Wide wall

Here, we investigate the opposite regime of wide walls where ∆ 1, so that we can replace the hyperbolic sines by its exponential representation,

δG_DW = 1 E˜_F

Z E˜F−1 0

dζ 1− (1−e^−2π∆k+) (1−e^{−2π∆k−})

4 sin²(π∆)e^{−π∆(k++k−)}+ 1−e^{−π∆(k++k−)}2

!

0.5 1.0 1.5

(a) Change in conductance as a function of width

10 15

(b) Limit of a short domain wall

Figure 4.15.: The plot on the left shows the dependence of the domain wall resistance on the wall widthw in units of the magnetic lengthlM =~v/M, and we can see a monotonic decrease with wall width. For widths much larger than the magnetic lengthl_M, we find the analytic dependence ∝1/w². The right plot shows the situation for a sharp wallw→0 together with the approximate solution (4.134) (green dashed line). The maximum is at ≈ 1.68 so that the highest possible value achievable is δG_DW .18.4%.

and note that due to the largeness of ∆, only exponentials whose argument vanishes at some point during integration can contribute. This is the case for e^−2π∆k+, since k₊ → 0 for ζ→E˜_F−1, whereask− ≥2p

E˜_F−1 (we assume that ˜E_F is sufficiently larger than 1) in the whole region of integration. Therefore, almost all the terms play no role during integration, and we can evaluate the much simpler integral

δG_DW^∆1→ where we rescaled the integration variable and set the lower integration bound to 0, since relevant contributions arise only for values of ζ close to 1. In a second step, we once more changed the integration variable to z = p

1−ζ², expand the prefactor around z = 0 and set the upper integration limit to∞ since now, contributions to the integral only come from valuesz1.³

We find that the ballistic domain wall resistance δG_DW decays with the inverse square of the domain wall width, but the topological contribution δG_T = _E^M

F is also present in limit and is the dominating one. We note that the diffusive version of the conductance change found in chapter 3.4.4 is of the same type as δG_T, which can be seen by integrating the local contribution (3.234) over the contact of length L, and for the one-dimensional domain-wall geometry considered in this section,

We can see that this result depends only on the total change of the magnetizationMx(+∞)− Mx(−∞), as it is the case forδG_T.

3This is essentially the method of the steepest descent or the saddle point method [179].

Topological Insulator Ferromagnet

x y z

Figure 4.16.: The system of topological insulator with an insulating ferromagnetic layer on top which contains a N´eel domain wall, and contrary to the previous system of Figure 4.7(a), corresponds to a mass domain wall.

Im Dokument Quantum Transport in Non-Collinear Magnetic Nanostructures (Seite 166-172)