Waveform measurement of charge- and spin-density wavepackets in a chiral Tomonaga–Luttinger liquid

Abstract

In contrast to a free-electron system, a Tomonaga–Luttinger (TL) liquid in a one-dimensional (1D) electron system hosts charge and spin excitations as independent entities^1,2,3,4. When an electron is injected into a TL liquid, it transforms into charge- and spin-density wavepackets that propagate at different group velocities and move away from each other. This process, known as spin–charge separation, is the hallmark of TL physics. While spin–charge separation has been probed in momentum- or frequency-domain measurements in various 1D systems^5,6,7,8,9, waveforms of separated excitations, which are a direct manifestation of the TL behaviour, have been long awaited to be measured. Here, we present a waveform measurement for the pseudospin–charge separation process in a chiral TL liquid comprising quantum Hall edge channels^{9,10,11,12,13}. The charge- and pseudospin-density waveforms are captured by utilizing a spin-resolved sampling scope that records the spin-up or -down component of the excitations. This experimental technique provides full information for time evolution of the 1D electron system, including not only propagation of TL eigenmodes but also their decay in a practical device¹⁴.

Main

Co-propagating spin-up and-down edge channels of the quantum Hall (QH) state at filling factor ν = 2 form a prototypical system for the study of TL physics. The TL eigenmodes have been identified by measuring interference of density waves⁹, shot noise generation^10,11,12, and charge-density correlation between two paths in a Hong–Ou–Mandel experiment¹³. Dynamics of the charge- and spin-density excitations, that is, their excitation, propagation, and attenuation, are elementary processes of non-equilibrium phenomena in the 1D system. For investigating these processes, a waveform measurement for each density excitation is highly desirable. In this study, we developed a spin-resolved sampling scope comprising a spin filter and a time-resolved charge detector^15,16, which enables one to perform the waveform measurement. We actually observe charge- and pseudospin-density excitations separated over a distance exceeding 200 μm. The TL parameters (group velocities of charge- (v_C) and pseudospin-density (v_S ≤ v_C) waves and mixing angle θ, which is introduced later) are directly read out from the waveforms; this is the first experiment in which all the TL parameters are estimated from a single measurement. Moreover, attenuation of TL wavepackets is evaluated from distortion of the waveforms. These results give quantitative information on various non-equilibrium phenomena in QH edge channels—for example, heat transport^17,18,19,20 and decoherence in interferometers^21,22.

The 1D electron dynamics in the co-propagating channels are formulated by the wave equation for the spin-up and-down charge densities ρ_{↑, ↓} (x, t)^23,24,25

where U_X is the inter-channel interaction and v_↑ (v_↓) is the group velocity renormalized by the intra-channel interaction in the spin-up (down) channel. While conventional TL theory assumes v_↑ = v_↓, we consider a more general case wherein they can differ, taking geometric asymmetry of the channels into account. By diagonalizing the matrix U, we obtain transport eigenmodes: the charge mode ρ_C = (ρ_C↑, ρ_C↓) = (cosθ, sinθ) and the pseudospin mode (sometimes called dipole or neutral mode; in the following we call it ‘spin mode’ for short) ρ_S = (ρ_S↑, ρ_S↓) = (sinθ, −cosθ)^9,10,13. The mixing angle θ(0 ≤ θ ≤ π/2) represents the asymmetry between the two channels. Pure spin–charge separation (θ = π/4) occurs for the symmetric case with v_↑ = v_↓.

A schematic of the measurement is shown in Fig. 1. We focus on the co-propagating channels labelled ‘target channels’ in the figure. First, an initial charge-mode wavepacket (i) is excited and fed to the spin filter SF_in placed at the entrance of the target channels (x = 0). A spin-up (Fig. 1a) or -down (Fig. 1b) charge packet (ii) is injected through SF_in by filtering out the other spin component. The spin-polarized charge packet splits into charge- and spin-mode packets (iii and iv), which are spatially separated after propagation. Each separated packet is decomposed by the second spin filter SF_det placed in front of the time-resolved detector (x = L). Only the spin-up charge components are measured to construct the objective waveforms.

Figure 1: Principle of the experiment.

a, Spin-up charge injection into the target channels through SF_in. The spin-up charge components of separated charge and spin excitations are extracted through SF_det and are detected by the time-resolved charge detector. A pair of same-charge-polarity peaks should be detected. b, Spin-down charge injection. Peak and dip structure should be detected in this case.

Time evolution of a spin-polarized charge packet induced at x = 0, for example, spin-up charge (d(t), 0), is described as a superposition of the two modes

where d(t) is the real-time charge distribution function centred at t = 0. We measure the spin-up component ρ_↑(L, t). In an ideal TL liquid, ρ_C and ρ_S propagate without attenuation; thus, v_C, v_S and θ (and thus v_↑, v_↓ and U_X) are directly extracted from the waveforms. In an actual 1D electron system, attenuation of these eigenmodes results in distortion of the waveforms from which deviation from the TL behaviour can be evaluated.

Figure 2a shows the experimental set-up and optical micrograph of the device. The measurement was performed at 320 mK in a magnetic field B = 6.5 T (v = 2) applied perpendicular to the 2D electron system (2DES). Spin-up and -down edge channels (indicated by red and blue lines, respectively) are formed along the energized surface gates (green and yellow patterns). We focus on the 260-μm target channels along gate G_T, which are formed by applying a gate voltage V_T to completely deplete 2DES under the gate.

Figure 2: Experimental set-up.

a, Coloured optical micrograph of the sample fabricated on an AlGaAs/GaAs heterostructure. The 2DES under the yellow gates is completely depleted. The green gates are used to form SF_in and SF_det. An initial charge-mode packet is excited through capacitive coupling by a 4.2-mV step voltage V_ex1 or V_ex2 applied to the injector gate INJ₁ or INJ₂, respectively. (Inset, d.c. pinch-off trace of SF_in.) Spin filtering is confirmed in a wide e²/h plateau. To activate SF_in, the gate voltage is set at −0.32 V. b, Magnified view near the injection point (x = 0). c, Magnified view near the detection point (x = 260 μm). The detector QPC is opened temporarily by the voltage pulse V_det of amplitude 7.1 mV and width 100 ps.

An initial charge-mode packet is excited at the injector gates INJ₁ or INJ₂ by applying step voltages V_ex1 or V_ex2, respectively¹⁵. The spin filters SF_in and SF_det comprise local QH regions at ν = 1 formed under the gates, where only the spin-up channels travel through the gated regions (inset in Fig. 2a). When the initial packet is fed from INJ₁ (INJ₂), a spin-up (-down) charge packet is injected into the target channels (Fig. 2b). The spin-polarized charge packet splits into charge- and spin-mode packets, which are measured at different timings by the detector comprising SF_det and the quantum point contact (QPC) (Fig. 2c)^15,16.

The circles in Fig. 3a, b represent the central results obtained when spin-up and-down charge packets are injected, respectively. We observe the first peaks at the time of flight (approximately 0.5 ns) as well as an additional peak in Fig. 3a and a dip in Fig. 3b at approximately 4 ns. The distinct spin dependence of the current polarity allows us to unambiguously identify the first and second features as the charge and spin modes, respectively. This is reconfirmed by comparing the data with the results of control experiments, where SF_in is deactivated. As the injected spin-unpolarized charge packet is an eigenmode in this case, a single peak is observed, indicating no spin–charge separation (dashed lines). The peak positions coincide with those of the first peaks in the spin-polarized charge injection case.

Figure 3: Representative experimental results.

a,b, Representative waveforms measured at V_T = −2.0 V as a function of time of flight along the channels. Insets show the schematic of the measurements. Red circles in a: waveform obtained with spin-up charge (d(t), 0) injection. Blue circles in b: waveform measured with spin-down charge (0, d(t)) injection. Black dashed lines: waveforms obtained with unpolarized-charge injections. Solid lines: simulated curves using the parameters v_↑≅330 km s⁻¹, v_↓≅210 km s⁻¹ and U_X≅200 km s⁻¹. The data in a and b are measured with different cooldowns for rewiring radio frequency signal lines.

The clearly resolved peak–peak (peak–dip) structure is a manifestation of the long-range ballistic transport of the TL eigenmodes. This implies that the co-propagating channels are well described by the chiral TL-liquid theory. From the arrival times, we obtain v_C≅480 km s⁻¹ and v_S≅65 km s⁻¹. The distance D between the excitations is estimated from these velocities. For example, D≅225 μm at the charge-mode detection timing (0.54 ns). The ratio of the charge- and spin-mode peak heights I_C and I_S, respectively, measured in case of the spin-up charge injection (Fig. 3a) provides the mixing angle θ≅0.2π. From these results, we obtain the TL parameters as v_↑≅330 km s⁻¹, v_↓≅210 km s⁻¹ and U_X≅200 km s⁻¹.

The solid lines in Fig. 3a, b represent the waveforms simulated using the obtained TL parameters without considering attenuation. In these calculations, d(t) is determined assuming d(t − L/v_C) ∝ ρ_{0, ↑}(L, t), where ρ_{0, ↑}(L, t) is the spin-up component of ρ(x, t) measured with the unpolarized-charge injections (dashed lines). The main features of the experiment and simulations agree fairly well. A small deviation in the shape of the spin-mode peaks indicates the presence of attenuation, particularly in spin-density waves, which is caused by inter-channel spin-flip tunnelling (Supplementary Information). Thus, an additional process that causes deviation from the TL model is identified from the obtained waveforms.

In the rest of this paper, we discuss tuning of the 1D electron dynamics. The transport properties of the TL eigenmodes depend on the surrounding environment, which screens the interactions in the channels^10,16. The screening strengths are represented by distributed electrochemical capacitances c_↑, c_↓ and c_X as shown in Fig. 4a^24,25 (see Methods). Figure 4b, c shows a schematic cross-section of the channels and the electron-density profile, respectively, at a filling factor slightly greater than v = 2. Compressible strips separated by the incompressible strip form the co-propagating channels²⁶. The widths and positions of these strips vary with experimental parameters, resulting in the tuning of c_↑, c_↓ and c_X, and thus the TL parameters.

Figure 4: Tuning transport properties.

a, Distributed-element circuit model of target channels. b, Schematic cross-section of the target channels. c, Schematic of the electron-density profile at the edge of the 2DES. d, Representative waveforms with spin-up charge injection measured at B = 6.5 T and at V_T = −2.0, −0.47 and −0.45 V. e, V_T dependences of v_C (filled) and v_S (open symbols). Solid black lines are visual guides. f, Magnetic field B dependences of v_C and v_S measured at V_T = −2.0 V. The velocity v_C varies following the 1/B curve (solid line), while v_S remains nearly constant at 64 km s⁻¹ (dotted line).

Figure 4d shows the waveforms for spin-up charge injection measured at several V_T values below the depletion voltage V_{T, dep} = −0.44 V. While the two-peak structures are observed in all traces, the shapes of the peaks vary with V_T. The group velocities v_C and v_S extracted from the peak positions are summarized in Fig. 4e. The velocities increase monotonically with decreasing V_T, reflecting the fact that the channels move apart from the gate metal, thereby reducing c_↑ and c_↓. The steep enhancement of the velocities at −0.55 V < V_T < −0.45 V indicates the edge configurations that change rapidly in this V_T range.

The TL parameters can also be tuned by B. Figure 4f shows the B dependences of v_C and v_S near v = 2. While v_C varies following the curve v ∝ 1/B (solid line) corresponding to the theory of edge magnetoplasmon transport²⁷, v_S remains nearly constant at 64 km s⁻¹. These behaviours are explained by considering asymmetric B dependences of the channel widths (Supplementary Information). In this manner, the TL behaviours are modified by the environmental parameters. The variety of such behaviours results in a wide range of non-equilibrium phenomena in QH systems.

Thus, we have presented a waveform measurement for the pseudospin–charge separation process. While this study focuses on the integer QH edge channels, the concept of decomposing TL eigenmodes can be used to investigate electron dynamics in various 1D systems; for example, neutral-mode transport in reconstructed edges in fractional QH systems²⁸.

Methods

Sample fabrication.

We fabricated the sample in a standard AlGaAs/GaAs heterostructure with a two-dimensional electron gas located 95 nm below the surface having electron density of 3.0 × 10¹¹ cm⁻² and mobility of 1.0 × 10⁶ cm² V⁻¹ s⁻¹. The sample was patterned using photolithography for chemical etching and coarse metallized structures, as well as electron-beam lithography for fine gate structures. The ohmic contacts shown by the white squares with cross marks in Fig. 2a were formed by alloying Au–Ge–Ni onto the surface.

Time-resolved charge detection.

This technique was performed by applying two radio frequency (RF) voltages. One is the square-wave voltage V_ex1 or V_ex2 (amplitude: 4.2 mV peak to peak, period: 40 ns) applied to the gate INJ₁ or INJ₂ to excite an initial charge packet. The other is the square pulse voltage V_det (amplitude: 7.1 mV peak to peak, width: 100 ps) for temporary opening of the quantum point contact (QPC). The initial charge packets are excited at the edge of the square wave. In this study, we focused on the positive charge packets excited at the rising edge. Note that the main results are reproduced with the opposite charge polarity on the negative charge packets excited at the falling edge. These RF voltages generate a charge flow through the QPC when the detection pulse synchronizes with the arrival of a charge excitation at the QPC. We measured the current I_QPC as a function of the time delay τ of the detection pulse from the excitation voltage, under successive applications of these voltages at a repetition frequency of 25 MHz.

We assume that the initial charge packet is spin unpolarized. While this is not the case when θ ≠ π/4 or when the capacitive coupling of the injection gate with the inner and outer channels are unequal, in the present experiment the deviation from this assumption is small and does not affect the main results. This can be confirmed in the waveforms without activating SF_in (dashed lines in Fig. 3a, b), where only single peaks are observed.

In this experiment, the time origin of the waveform measurement is defined as the moment of the spin-polarized charge excitation at x = 0. We estimated the moment in the I_QPC trace by the following procedure. First, we defined a straight channel from INJ₁ to the detector by applying −2.0 V to the gates (total length of the channels is L_total = 560 μm). Second, an I_QPC trace was measured with the charge-mode packet injection at INJ₁ without activating SF_in. The arrival time τ_C of the packet at the detector was evaluated from the peak position. Then, the excitation timing τ₀ with charge injection at INJ₁ was calculated as τ₀ = τ_C × (L_total − L)/L_total, where L = 260 μm is the length of the target channels. With charge injection at INJ₂, we performed similar measurement to estimate arrival time τ_C′ and calculated the excitation timing as τ₀′ = τ_C′ − L/v_C. In this way, we determined the time origins of the waveform measurements as τ₀ and τ₀′.

Distributed-element circuit model of co-propagating edge channels.

Intra- and inter-channel Coulomb interaction in co-propagating edge channels can be modelled by distributed electrochemical capacitances^24,25. Assuming the corresponding circuit model shown in Fig. 4a, the wave equation for charge excitations ρ_↑ and ρ_↓ described in the main text can be substituted by

where σ_xy = e²/h is the Hall conductivity, c_↑ (c_↓) is the spin-up (-down) channel capacitance, and c_X is the inter-channel capacitance. The TL parameters v_↑, v_↓, and U_X can be expressed by using these capacitances as

By substituting the experimental results v_↑≅330 km s⁻¹, v_↓≅210 km s⁻¹, and U_X≅200 km s⁻¹, we obtain c_↑≅15 pF m⁻¹, c_↓≅170 pF m⁻¹, and c_X≅240 pF m⁻¹, which are comparable to the values in the case of counter-propagating edge channels^24,25.

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.