Observation of correlated spin-orbit order in a strongly anisotropic quantum wire system

Quantum wires with spin-orbit coupling provide a unique opportunity to simultaneously control the coupling strength and the screened Coulomb interactions where new exotic phases of matter can be explored. Here we report on the observation of an exotic spin-orbit density wave in Pb-atomic wires on Si(557) surfaces by mapping out the evolution of the modulated spin-texture at various conditions with spin- and angle-resolved photoelectron spectroscopy. The results are independently quantified by surface transport measurements. The spin polarization, coherence length, spin dephasing rate, and the associated quasiparticle gap decrease simultaneously as the screened Coulomb interaction decreases with increasing excess coverage, providing a new mechanism for generating and manipulating a spin-orbit entanglement effect via electronic interaction. Despite clear evidence of spontaneous spin-rotation symmetry breaking and modulation of spin-momentum structure as a function of excess coverage, the average spin-polarization over the Brillouin zone vanishes, indicating that time-reversal symmetry is intact as theoretically predicted.

of the (557) orientation to (223) oriented facets, thus introducing a new reciprocal lattice vector, g= 2πd −1 e, which fulfils the (spin-polarized) nesting condition 2k F = g along the [112] direction, which is one of the signatures for the formation of a SODW as discussed above (d is the interwire distance, k F the Fermi wave vector, e is the unit vector). This condition is superimposed on the local 2D character of the densely packed α − √ 3 × √ 3 phase of Pb on the mini-(111) terraces, which is reflected by closed Fermi circles rather than by open Fermi lines (Fig. 1c). The Fermi nesting in this system leads to the expected renormalization of the Fermi surface [24]. For the sake of simplicity and the fact that the gap is comparably small we show here the quasi-Fermi surfaces, i.e. the circularly shaped spectroscopic features seen at E F -50 meV. This representation takes also the replica bands into account which are seen by the MDCs taken necessarily also slightly below E F . A large degree of geometrical order along the wires, coupled with high 1D conductance, is evident by the 10-fold superperiodicity (s =10×a Si ) at the critical coverage of 1.31 ML, shown by high resolution scanning tunneling microscopy (STM) in Fig. 1a and also seen by spot splitting in low energy electron diffraction (LEED, see below). Strikingly, the system reveals giant Rashba-splitting of ∆k 0 =0.2Å −1 (Rashba parameter α R =1.9 eVÅ) [25] which is by far larger than the expected value for chemisorbed Pb monolayers on isotropic semiconducting surfaces, e.g. Pb/Ge(111) [26]. ∆k 0 corresponds to exactly g/2, i.e. the periodicity of spin ordering is twice than that of charge ordering at this critical Pb concentration, so that spins on adjacent terraces are coupled in an anti-parallel manner. This spin dependent splitting is marked in Figs. 1b,c by different colors. The different periodicities of spin and charge are a necessary prerequisite for non-trivial coupling between them.
Here we show that in this array of atomic wires bound to the (223) oriented facets on Si (557) we have the unique possibility to tailor the interwire coupling by adsorption of minute amounts of excess coverage δΘ. The screened Coulomb potential scales with the size of the excess coverage as U (δΘ) = U (0)·exp(−qδΘ), where q is the screening parameter [27]. As we will show, while the spin-orbit order is preserved (at least) up to a critical additional coverage of 0.1 ML, steps are decorated by this excess coverage not randomly, but by formation of superstructures whose periodicity depends on excess concentration of Pb [20,28]. This highlights that the mechanisms of spin and charge order are not the same. The coupling between charge and spin order is mediated by 1D in-gap states and is reduced with increasing excess coverage. This leads to spin depolarization as a function of excess coverage. Therefore this mechanism has much more fundamental consequences than the band shift associated with standard doping, an observation that has been made also in other metallic chain systems on surfaces [29,30], however with somewhat different physical scenarios. A collapse of the SODW phase is associated by an almost vanishing spin polarization close to δΘ ≈0.2 ML, where the Pb wires turn into an anisotropic 2D metal. Quite naturally, SODW formation and the reduced spin-charge coupling as a function of excess Pb concentration also influence other physical properties that are sensitive to both charge and spin, like magnetotransport.
As expected, the gradual spin depolarization seen in photoemission is closely related to the spin dephasing probed by magnetotransport measurements [31]. This relationship can be quantitatively described by SODW theory, as we will demonstrate below.

Results
The effect of excess coverage. Adsorption of excess coverage on the perfect Pb-chain system, as shown in Fig. 1a, changes primarily the periodicities of the Si(557) surface perpendicular to the wires. We demonstrate by the results collected in Fig. 2 that spin and geometric order indeed do not have the same periodicities, although they are connected and commensurate to each other by the underlying terrace structure. Moreover, while the charge order follows the geometry, the spin order turns out to be robust and unchanged by excess coverage.
The respective charge and spin order can be seen in Fig which is shifted by 0.2Å −1 and has the opposite spin helicity [25]. In Fig. 2a we marked the extrema of the x-component of the spin polarization by red and blue colored circles. This equidistant splitting is a spectroscopic hallmark for a SODW in its ground state [19]. Note that we present throughout the paper spin-integrated MDCs measured with high resolution.
The color-coded spin components shown are deduced from additional MDCs recorded with the Mott detector (see Supplementary Note 1 and Supplementary Figure 1). In order to reveal good signal/noise ratios in reasonable time the spin distribution has been measured 50 meV below the Fermi level. However, the excellent agreement with results from DC-and magnetotransport demonstrate that this limitation does not affect our conclusions at all.
With increasing δΘ, we see that the intensities in the MDCs are slightly altered due to changing photoelectron scattering conditions related to the geometrical superperiodicities (see below). However, and more importantly, the positions of the maxima of spin polarization remain unchanged in k-space. In contrast to the geometric order, the original spin periodicity found at δΘ=0 ML of 0.2 and 0.4Å −1 , respectively, prevails until we reach the limit of second Pb layer growth at an excess coverage of 0.2 ML. At this excess coverage also the original SODW phase collapses, as seen by negligible spin polarization of all three components. Since the enhanced spin-texture along the Fermi surface (FS) nesting direction only appears below 0.2 ML excess coverage and below 78 K, it strongly indicates to be an interaction effect, rather than a single particle SOC effect. We therefore conclude that the periodicity of spin order up to δΘ=0.2 ML is exclusively determined by the periodicity of the terrace structure of (223) facets. This terrace induced spin order is reached with maximum spin polarization at the critical concentration of Θ=1.31 ML, and is interpreted to represent the ground state of the SODW.
Looking at the geometric order, the diffraction pattern of the perfect phase (δΘ=0 ML, Upon deposition of excess coverage, superstructure spots appear within the initial diffraction pattern of the 1.31 ML phase whose periodicity is reduced by integer multiples of the terrace lattice constant as a function of δΘ (see diffraction line profiles perpendicular to the steps plotted in Fig. 2b). As an example, for δΘ=0.1 ML the periodicity along the [112] direction is doubled (see red arrows) by decoration of every second terrace [28].
This variation of geometric order induced by excess coverage is coupled with the reduction of the band gap, as measured previously [27]. The results, reproduced in Fig. 2e show that subbands are formed within the band gap (see schematic of Fig. 2d), which gradually fill the gap as a function of δΘ. The gradual decrease of the band gap follows ∆ = 20 meV · exp(−δΘ/0.038 ML), which corresponds to q=26.3 ML −1 . Regarding the influence of geometric periodicities on the spin order observed, there is obviously no direct impact of the modified chain structures, but the (223) structure is still observed in LEED at all excess coverages and at low temperature (cf. Fig. 2b). This is the only geometric feature remaining constant upon adsorption of excess coverage. Therefore coupling between spins on adjacent terraces can only be mediated via the band gap. Consequently, the reduction of the band gap also reduces the effective coupling between spin order on different terraces.
In other words, decoration of the step edges with various periodicities and excess coverages increases the screened Coulomb interaction, which couples to spin order via spin-orbit interaction. This is exactly the SODW scenario. further favors the SODW scenario [19].
In order to quantify the spin depolarization we have plotted the peak-to-peak values of the most intense oscillations for all three spin components (Fig. 3b). The S x -component decreases exponentially with increasing excess coverage and can be well described by S x (δΘ) = 0.56 · exp(−δΘ/0.036 ML) + 0.12, i.e. q=27.8 ML −1 . The (solid) line is a fit according to the SODW theory (see below). We want to emphasize that the gradual decrease of the band gap ∆ deduced independently from DC-transport measurements reveals an almost identical q-value. As shown in Fig. 3b the S y -and S z -components are considerably weaker (see also Supplementary Figure 2). For an ideal Rashba system these components should vanish, while in presence of excess coverage this rule is lifted and might be indicative of spin spirals [11]. After the band gap has vanished a fast dephasing of the spin-polarized photoholes within the SODW state mimics almost unpolarized surface bands. As expected, the effect of depolarization comes along with a reduction of the coherence length ξ. Fig. 3c shows ξ of the photoelectrons deduced from the full width of half maximum FWHM=2ξ −1 for two MDC peaks as a function of excess coverage which follow a similar trend as the spin polarization.
Moreover, the spin texture along the wires for the quasi-perfect structure has also been measured. As obvious from the MDC shown in Fig. 4b, the most intense features stem from behavior seen in angle resolved photoemission spectroscopy (ARPES). We want to point out that this is per se not self-evident because ARPES and transport probe electrons at different energies. Regardless, it is remarkable that the spin polarization follows the same trend which in turn demonstrates the close entanglement of both quantities and is a hallmark of strong electronic correlation. As we will show below, both can be consistently explained in terms of the SODW formalism.

Discussion
Based on our findings and the fact that within first order the spin lifetime τ s is given by the spin-orbit scattering time τ so in a strongly spin-orbit coupled system, the spin dynamics can be described by a kinetic equation [35]: where S k is the spin polarization vector, P is any external spin source, and η is the scattering time. Ω k is the effective Larmor frequency defined in our case by where Ω R = 2α R k F / , and Ω SODW = 2∆/ are the frequencies corresponding to Rashba-type SOC and SODW, respectively. In our case, P=0, and the initial condition for the spin is Inserting the exponential dependence of the SODW gap on the excess coverage and keeping all other parameters constant, we get a good fit of the spin dephasing time τ so to the experimental value for a reasonable parameter set of SODW gap ∆=20 meV, η =1 ×10 −12 s and Ω R =2.6 ×10 12 Hz which is small compared to Ω SODW up to δΘ=0.1 ML. Using the value of α R =1.9 eVÅ as the Rashba parameter found in our previous study [25], this refers to an extremely small Fermi-wave vector component along the wires (k F,x ≈10 −4Å−1 ). This in turn explains the insensitivity of the propagating electrons along the wires against atomic sized defects [23].
Focusing on the direction along the wires, the emergent modulated spin texture in real space can be described by dynamic spin spirals which are antiferromagnetically coupled between the wires as illustrated in Fig. 4e. The antiferromagnetic coupling along the [112] direction is a consequence of the helical nesting found in ARPES [25]. The spin texture along the direction of wires represents the dynamic analogue to static one-dimensional skyrmions as observed by STM in magnetic structures [36,37]. Our results are representative for systems where SOC and electron correlations are of comparable strength and with a negative exchange coupling. While the SODW is robust and insensitive to these couplings the spin polarization is strongly affected and only in the pure SODW phase highly spin-polarized transport can be expected. In this respect, the continuous improvement in the field of ultrahigh spin-resolved spectroscopy will allow us to investigate this new quantum phase and the associated band gap at soonest in more detail [38,39]. Our investigations have further shown that the electrons within the SODW phase are strongly correlated with electrons residing as gap states. With a large screened interaction by reducing the excess coverage, the electronic states also become insulating, opening up a possibility for the realization of SODW order induced Luttinger liquid in quantum wires.

Methods
Sample preparation and measurements. Spin-and angle-resolved photoemission (SARPES) measurements have been performed at the COPHEE end station at the SIS beamline of the Swiss Light Source [40]. The data was analysed as described in Ref. [41].
Atomic wire structures were grown by evaporation of Pb on Si (557)

Additional information
Competing financial interests: The authors declare no competing financial interests. The two colors denote the spin orientation in each of the subbands. The distance between bands with the same spin helicity is g = 2π/d=0.4Å −1 . The Fermi nesting driven energy gap ∆=20 meV has been determined by ARPES and transport measurements [23,24,33]. Right: the equidistant sequence of both spin bands is nicely reflected by the MDC (spin-integrated) taken at the valence band maximum along the [112] direction. The spin signature of the subbands has been deduced from spin-resolved measurements [25].