Generation of a pure ac-spin current

This section is closely based on Ref. [Hofer14a].

In this section, we discuss a proposal which makes use of two helical capacitors to create a pure ac-spin current. Since QPCs are not yet available in quantum spin Hall insulators, we further discuss a setup based on the quantum Hall regime where QPCs are readily available. This setup only requires building blocks that are experimentally available. Due to spin-orbit coupling, the spin is usually not a good quantum number in the considered systems. Therefore, we start our discussion by clarifying the role of spin and total angular momentum.

Spin vs. total angular momentum

We employ the BHZ Hamiltonian given in Eq. (2.43) to model the two-dimensional quantum spin Hall insulator. The basis states of the BHZ Hamiltonian are denoted by

|E,+1/2i, |H,+3/2i, |E,−1/2i, |H,−3/2i, (2.76) whereE(H) indicates that the state is derived from an electron-like (hole-like) band and the number denotes the projection of the total angular momentum on the z-axis mj. For a more detailed discussion on these basis states, we refer the reader to Refs. [Lunde13;Pfeuffer-Jeschke00]. Importantly,mj is a good quantum number for the basis states and does not depend on the specific material or sample geometry. We therefore discuss the proposed device as a source of total angular momentum current. The projection of the electron spin onto thez-axis,ms, is only a good quantum number for the states |H,±3/2iwhich havems=±1/2.

However, in the case of HgTe/CdTe, one can make an estimate of the spin polarization of the mj =±1/2 states based on Ref. [Pfeuffer-Jeschke00], which gives

hE,±1/2|Sˆz|E,±1/2i ≈ ±0.731

2, (2.77)

in units of ~= 1. Here, ˆSz is thez-component of the electron-spin operator. Although the last expression depends on the material and the design of the sample, we thus find that hSˆziand mj have equal signs for all basis states, which means that the currents associated with these two quantities are proportional to each other. This justifies the notion of spin current although we discuss the source in terms of total angular momentum current which is sample and material independent.

Solving the BHZ Hamiltonian for hard wall boundary conditions (see Refs. [K¨onig08; Zhou08]) yields states localized at the edges with a linear dispersion as solutions. These edge states have equal weight on the on the|E,±1/2iand|H,±3/2istate which yieldshJˆ_zi^α=±1, where ˆJ_zis thez-component of the total angular momentum operator andα=↑,↓labels the spin of the edge state. Tuning the Fermi energy into the bulk gap, only these edge modes contribute to transport. They constitute the scattering channels which are referred to below.

Time reversal invariant source

The ac-spin current source is sketched in Fig. 2.8 (a). It consists of two identical mesoscopic capacitors attached to the different edges of a strip of a two-dimensional quantum spin Hall insulator. Since time-reversal symmetry is preserved, both capacitors emit pairs of counter-propagating electrons (holes) whenever the energy levels of a Kramers pair are moved above (below) the Fermi energy. If both helical capacitors are operated with the same top gate potential (τ= 0 in Fig.2.8), they emit electrons and holes at the same times which leads to an ac charge current in both contacts LandR. Because the charge current is carried by two particles of opposite spin, the spin current vanishes in this case.

By operating the helical capacitors with a time-shiftτ in their top gate potentials, see Fig.2.8, electron emission of one capacitor can be synchronized with hole emission of the other capacitor. This cancels the

Figure 2.8: Proposed ac spin current source in the quantum spin Hall insulator regime. (a) Two mesoscopic capacitors are attached to a two-dimensional quantum spin Hall insulator. If they are operated in phase (τ = 0), they both emit electrons and holes at equal times resulting in a pure and equal ac chargecurrent in the two contacts (R/L). A delay,τ, can synchronize electron and hole emission on the different edges.

This results in a pure ac spin current of opposite sign in the different contacts. The arrows indicate the expectation value of the spin. (b) Charge and spin current (|e|=~= 1) emitted by the source into the right contact. Black (solid) line shows the pure charge current for synchronous operation which is equal to the pure spin current forτ=T/2. The green (broken) line shows the charge current, the blue (dashed) line the spin current, for a finite delay. Changingτ moves the peaks and dips centered aroundτ andT/2 +τ, going from a pure charge (τ = 0) to a pure spin current (τ =T/2). Note that an equal (opposite) charge (spin) current enters the left contact. Figure reprinted from Ref. [Hofer14a].

charge current at all times. Since a spin up hole carries a negative spin current, the spin current of the emitted particles adds up yielding a pure ac spin current in the contacts.

For a quantitative analysis, we resort to the Floquet scattering theory. As discussed in Sec.2.3.3, each helical capacitor can be described as a pair of uncoupled chiral capacitors. Since there is no scattering between the different edges, our setup is thus equivalent to four uncoupled chiral capacitors and can be described by the Floquet scattering matrix which is given by the Fourier transform of Eq. (2.54). Here we consider the experimentally relevant step like top gate potential given in Eq. (2.52) which emits particles in wavepackets with exponential shape resulting in the current given in Eq. (2.55). Due to the linear dispersion, the spin current in a given channel α is proportional to the charge current with proportionality constant hJˆzi^α/e. The quantized emission of a single particle into channelαis therefore accompanied by the quantized emission of angular momentum in units ofhJˆzi^α. Operating the two capacitors with a time shiftτ leads to the currents in the right and left contacts

I_R^c(t) =I_L^c(t) =I^c(t) +I^c(t−τ),

I_R^s(t) =−I_L^s(t) = [I^c(t)−I^c(t−τ)]/e, (2.78) where the superscript labels spin (s) and charge (c) current, the subscript R/L denotes the contact, and I^c(t) is given by Eq. (2.55). To obtain the spin current, we made use ofhJˆ_zi^↑/↓=±1 for edge states.

The charge and spin currents measured at the right contact are plotted in Fig.2.8(b). Forτ = 0, each capacitor emits an electron at time t= 0 and a hole at timet=T/2. The charge current is thus twice the charge current of a single chiral capacitor and the spin current vanishes. For a finite τ, the upper capacitor still emits an electron at time t = 0 yielding a negative contribution to the charge current and a positive contribution to the spin current. The lower capacitor now emits an electron at timet=τ yielding a negative contribution to both the charge and the spin current. The opposite holds for the holes and for the spin current in the left contact. Finally, when τ=T/2, the electron of the upper capacitor is synchronized with the hole of the lower capacitor. The charge current vanishes and the spin current is maximized. Controlling the time shift thus allows one to tune from a pure charge to a pure spin current. Conveniently, mesoscopic capacitors are usually operated in the GHz range, where experimental detection of ac spin currents has recently been reported [Hahn13;Wei14;Hyde14;Weiler14].

Edge states in a magnetic field

Figure 2.9: Evolution of the band structure of InAs/GaSb in a magnetic field. The color scheme indicates the expectation value hJˆ_zi. a) For a weak magnetic field, spin up (down) states move up (down) in energy and the crossing of opposite spin states moves to negative (positive)k_yfor the upper (lower) edge. The crossings at zero energy are lifted by the inversion asymmetry terms as shown in the inset. b) In a strong magnetic field, Landau levels develop with a characteristic texture of hJˆ_zi. For fields stronger then B_c, the intrinsic band inversion is revoked and all the electron-like Landau levels are above the hole-like Landau levels. c) A split gate can move the band structure up (down) on the upper (lower) half of the sample by locally applying the potentialVu(Vl), with ∆V ≡Vu−Vl. The upper half then supports ahJˆzi^u= 3/2 and the lower half a hJˆzi^l=−1/2 edge channel. The inset shows the band structure including the Zeeman splitting for the same magnetic field and ∆V = 7 meV. For these calculations we used a tight-binding regularization of the BHZ model. Figure reprinted from Ref. [Hofer14a].

Since the fabrication of QPCs in quantum spin Hall insulators remains an open challenge, we extend our proposal to the quantum Hall regime, where QPCs can be implemented using gates. To this end, we first describe the behavior of the helical edge states in a perpendicular magnetic field. The transition from the quantum spin Hall to the quantum Hall regime in a honeycomb lattice is discussed in Refs. [Shevtsov12], the effect of a magnetic field in HgTe/CdTe quantum wells in Refs. [Scharf12; Chen12]. In the light of recent experiments [Du15;Spanton14], we consider an InAs/GaSb quantum well with the material parameters given in Ref. [Wang14] and shown in Table 2.1. The band structure evolution in a magnetic field is qualitatively similar for all materials described by the BHZ model. However, the required magnetic field depends on the material parameters. It is quantitatively similar in HgTe/CdTe while in Ge/GaAs it is two orders of magnitude higher. Since we are interested in the spin texture, we take into account the spin coupling terms that arise in this material due to bulk (BIA) and structural inversion asymmetry (SIA) [Liu08]

H_BIA(k) =

wherek_±=k_x±ik_y andδ,δ_e/h, andRare material dependent constants. We verified numerically that the Zeeman splitting only quantitatively modifies those features required for the generation of spin currents [see the inset in Fig. 2.9 (c)] and thus neglect this effect in the general discussion, returning to it below. The Hamiltonian we consider is thus the sum of Eqs. (2.43) and (2.79), where the magnetic field is introduced using the minimal coupling in the Landau gauge kx→ −i∂x,ky→ky−eBx.

The behavior of the edge states in a magnetic field is illustrated in Fig. 2.9 (a) and (b). For small magnetic fields, the up (down) spin states are pushed up (down) in energy. This shifts the crossing of the opposite spin states at zero energy to negativeky values for the upper edge and to positiveky values for the lower edge. Since these crossings are no longer protected by time reversal invariance and since the spins are coupled by the inversion asymmetry terms, a gap opens up [see inset of Fig. 2.9(a)].

Further increasing the magnetic field moves the crossings between the states of equal spin polarization into the bulk states. Coupling of the different edges via the bulk states lifts these degeneracies and Landau levels are formed. The spin up states give rise to a hole-like Landau level which is above the electron-like spin-down Landau level due to the intrinsic band inversion (not shown). Increasing the magnetic field further shifts the hole-like spin-up level down and the electron-like spin-down level up in energy, until they

eventually undergo an avoided crossing at B_c =M/(|e|B)≈8.23 T [K¨onig08]. For higher magnetic fields, all electron-like Landau levels are above the hole-like Landau levels and they alternate in the sign of hJˆ_zi (see Fig. 2.9(b). Since at high magnetic fields,k_y is directly proportional to thex-component of the center of cyclotron motion, we can read off the localization of the electrons directly from the band structure.

A[eV·˚A] B[eV·˚A²] D[eV·˚A²] M[eV] δ[eV] δe[eV·˚A] δh [eV·˚A] R[eV·˚A]

0.3 −40 −30 −5·10⁻³ 2·10⁻⁴ 6.6·10⁻⁴ 6·10⁻⁴ −8·10⁻⁴

Table 2.1: Parameters of the BHZ Hamiltonian for InAs/GaSb/AlSb quantum wells taken from [Wang14].

Quantum Hall source

In this section, we extend our proposal to the quantum Hall regime, where it consists exclusively of experi-mentally available building blocks.

Figure 2.10: Proposed ac spin current source in the quantum Hall regime. (a) A strong magnetic field is used to produce Landau levels. Using a split gate (VuandVl, blue and red shading), the upper half is tuned into a regime where one edge state withhJˆzi^u = 3/2 (blue, solid) propagates clockwise, while in the lower half an edge state with hJˆzi^l = −1/2 (red, dashed) propagates counter clockwise. The leftmoving states are localized in the middle of the sample, ensuring the absence of backscattering. Figure reprinted from Ref. [Hofer14a]. (b) Charge and spin current (|e|=~= 1) emitted by the source into the right contact. The black (solid) line shows the charge current and the red (dashed-dotted) line the spin current for synchronous operation. The green (broken) line shows the charge current and the blue (dashed) line the spin current for a finite delay. While a pure spin current can still be obtained, a pure charge current can no longer be obtained in contrast to the time reversal symmetric case.

A bottom gate can be used to shift the band structure relative toEF. As experimentally demonstrated in Ref. [Br¨une12], a split gate can be used to shift the band structure up in energy in one half, and down in energy in the other half of the sample. We model this by including a potentialV(x) which is positive in the upper half and negative in the lower half of the sample [see Fig.2.10(a)]. This leads to the band structure shown in Fig. 2.9 (c) with four states atEF. The states at the edges of the sample both propagate to the right and have spin polarizations of opposite signs but different magnitudes. The state at the upper edge has hJˆzi^u = 3/2, the state at the lower edgehJˆzi^l =−1/2. Since the leftmoving states are located in the middle of the sample, there is no backscattering.

To achieve a source of ac spin current, a mesoscopic capacitor is attached to each side of the sample as sketched in Fig.2.10(a). Note that in this setup, the channels connected to the mesoscopic capacitors both propagate to the right, meaning that I_L^c =I_L^s = 0. The charge current in the right contact is still given by Eq. (2.78) and the spin current reads

I_R^s(t) = 1 e

3

2I^c(t)−1

2I^c(t−τ)

. (2.80)

Even atτ = 0, this source thus emits a spin current ofI^c(t); a pure charge current can no longer be created.

However, a pure spin current of 2I^c(t) can still be generated by an offsetτ =T/2. The emitted charge and spin currents are plotted in Fig.2.10(b).

We end this section with some remarks on the feasibility of our proposal. As shown in Fig. 2.9(c), the spin-down state that is coupled to the lower capacitor has a spin-up state in its vicinity. The gap between these states is approximately 1 meV. However, even if the Fermi energy lies above the spin-up Landau level, the QPC of the capacitor can be operated to be transmitting only for the outer, spin-down edge channel.

In the inset of Fig. 2.9 (c), we include the Zeeman splitting with estimates for the g-factor of −8 for electrons and−3 for holes [Nilsson06]. The Zeeman splitting has no observable effect on the spin polarization Although the band inversion is no longer revoked at 9.5 T, the available states atEF, and thus our results, remain unchanged.

Recent work indicates that our proposal is robust against disorder and deformations of the quantum dots [Xing14]. Although we focused on InAs/GaSb quantum wells, our results are quantitatively similar for HgTe/CdTe quantum wells making our proposal relevant for both experimentally available quantum spin Hall insulator materials [K¨onig07;Roth09; Du15;Spanton14].

Summary

Using two mesoscopic capacitors attached to the sides of a quantum spin Hall insulator, we propose an ac current source that can be tuned from a pure charge to a pure spincurrent source by varying an offset in the driving potentials of the capacitors. So far, QPCs have not been implemented in quantum spin Hall insulators. We therefore extend our proposal to the quantum Hall regime, where QPCs are provided by gates. To this end, we discuss the behavior of the helical edge states in a perpendicular magnetic field. The emerging Landau levels have a characteristic angular momentum structure due to the intrinsic spin-orbit coupling of quantum spin Hall insulators. Using a split gate [Br¨une12], we produce a situation where two co-propagating states with angular momentum expectation value of opposite directions are localized at the opposite edges of the sample. Analogous to the source in the time-reversal invariant regime, the charge and spin currents can be manipulated by varying the offset in the driving potentials. Since the magnitude of the angular momentum expectation value on the two edges is different, a pure charge current cannot be produced any more. However, a pure ac spin current can still be generated.