Real-space observation of incommensurate spin density wave and coexisting charge density wave on Cr (001) surface

In itinerant magnetic systems, a spin density wave (SDW) state can be induced by Fermi surface nesting and electron-electron interaction. It may intertwine with other orders such as charge density wave (CDW), while their relation is still yet to be understood. Here via spin-polarized scanning tunneling microscopy, we directly observed long-range spin modulation on Cr(001) surface, which corresponds to the well-known incommensurate SDW of bulk Cr. It displays 6.0 nm in-plane period and anti-phase behavior between adjacent (001) planes. Meanwhile, we simultaneously observed the coexisting CDW with half the period of SDW. Such SDW/CDW have highly correlated domain structures and are in-phase. Surprisingly, the CDW displays a contrast inversion around a density-of-states dip at −22 meV, indicating an anomalous CDW gap opened below EF. These observations support that the CDW is a secondary order driven by SDW. Our work is not only a real-space characterization of incommensurate SDW, but also provides insights on how SDW and CDW coexist. Spin density waves are a spatial modulation of the spin, and can be either commensurate or incommensurate with the crystal lattice. Here, Hu et al. use spin-polarised scanning tunnelling microscopy to observe the incommensurate spin density wave on cleaned Chromium surface.

A spin density wave (SDW) state manifests itself as realspace spin modulations. It is usually formed in itinerant magnetic systems with Fermi surface nesting and electron-electron interactions 1 . The spatial period of SDW could be commensurate (C-SDW) or incommensurate (IC-SDW) to lattice constant. In the latter case, the spin modulation decouples from lattice, which is distinguished from local moment induced anti-ferromagnetic (AFM) order. Interestingly, SDW often coexists and sometimes intertwines with other orders in correlated systems, such as charge density wave (CDW) and superconductivity [1][2][3][4] . The interplay of these coexisting/intertwining orders has now become an important theme in condensed matter physics. To date, the commonly observed SDW states are commensurate SDW, such as the collinear/bicollinear SDW (AFM) state in iron-based superconductors 4,5 ; while incommensurate SDW is rarely seen, and particularly, its real space imaging is quite lacking.
Chromium (Cr) is one of the classic examples which shows itinerant magnetism with an IC-SDW ground state 6-8 below its Néel temperature (T N = 311 K). Such IC-SDW is stabilized by "imperfect" Fermi surface nesting condition 9 , as illustrated in Fig. 1a. Specifically, the Fermi surfaces of Cr (b.c.c. lattice) are composed of hole pockets at the corner and electron pocket at the center of the Brillouin zone 6 . The hole pocket is slightly larger than the electron pocket which yields two nesting vectors: Q ± = 2π/a (1 ± δ) (a = 2.9 Å being the lattice constant). Therefore, a long period IC-SDW with a wave vector Q SDW = 2πδ/a is generated which overlaps with the AFM coupling between Cr atoms (Fig. 1b). The wavelength of IC-SDW is reported to be 6.0 nm at T < 10 K (ref. 10 ), and Q SDW is along one of the <001> directions. The spin orientation of Cr atom is found to be perpendicular to Q SDW at T > T SF (123 K) but switched to be parallel at T < T SF (spin-flip transition 6 ).
In addition to IC-SDW, a charge density wave (CDW) with half period of the IC-SDW was also found in Cr [10][11][12][13] . Unlike the IC-SDW, the exact origin of such CDW is yet to be understood. It was often considered as the second-order harmonics of IC-SDW 12 , corresponding to a nesting vector Q CDW = 2Q SDW that connects the two folded bands at Γ (Fig. 1a); alternatively, it was suggested as a lattice strain wave induced by magneto-elastic coupling to the IC-SDW 6 . Therefore, being a pure element arranged in a simple structure, Cr is also a classical system to study the interplay of SDW/CDW orders.
However, after decades of research, the characterization of IC-SDW (and CDW) in Cr is still rather limited to spatially averaged method, such as neutron scattering 6,7 , x-ray diffraction 10,13 and photoemission spectroscopy 14,15 . In principle, SDW could also be detected by local probes at atomic scale, such as spin-polarized scanning tunneling microscopy (SP-STM) 16 . Although a few SP-STM studies have been performed on various Cr surfaces [17][18][19][20][21][22][23][24][25][26] , the real-space evidence of IC-SDW was rarely reported (some studies found CDW modulation on Cr (110) surface 23,27 , and argued the satellite FFT spots as an indication of IC-SDW 23 ). Most SP-STM studies on Cr (001) surface only observed in-plane ferromagnetism with AFM coupling between adjacent (001) planes [17][18][19][20][21][22] . To understand such a ferromagnetic spin arrangement on surface, it was suggested the magnetic moment is enhanced at the surface 28 and the IC-SDW antinodes are always pinned on the surface 8,19 , making it invisible to STM. However, we noticed that most previous STM studies on Cr (001) did not resolve clear atomic lattices, although mono-atomic terrace can be identified. This is likely due to local disorders induced by segregated impurities on the surface, which is a common problem in cleaning Cr single crystal. As the surface conditions could alter the surface magnetism 8 , it would be intriguing to search the IC-SDW in real space again on a well-ordered Cr surface.
[001] 50 nm [100] [010]   In this work, by using low-temperature spin-polarized STM with vector magnetic field, we studied a thoroughly cleaned Cr (001) with a well-ordered surface. We observed clear spin modulation with a period of 6.0 nm, propagating along in-plane [100] or [010] directions, which well matches the projected bulk IC-SDW on (001) surface. Its SDW nature is confirmed by the contrast inversion upon switching tip's magnetization, and the anti-phase relation between adjacent terraces. Meanwhile, we also observed the coexisting CDW with a period 3.0 nm, and surprisingly found that it displays a π phase shift around gap structure about 22 meV below E F , which suggests its formation is beyond the intuitive Fermi surface nesting picture. Furthermore, as a local probe measurement, we directly observed the domain structure of SDW/CDW and revealed their in-phase relation. Our work not only gives a real-space investigation of IC-SDW, but also provide new insights on the general mechanism of coexisting SDW/CDW orders.

Results
STM characterization of Cr (001) surface and it's tunneling spectrum. The experiment was conducted in a cryogenic STM (UNISOKU) at T = 5.0 K. Details about the cleaning process of Cr (001) and STM measurement are described in the "Methods" section. Figure 1c shows a large scale STM image of the obtained Cr (001) surface. It displays atomically flat terraces with monoatomic height (≈0.14 nm), as indicated by the line profile in Fig. 1e. It is notable that the terrace edges prefer running along high symmetric directions such as [100], [010], and [110]. This is an indication of free surface atom diffusion during annealing 29 . Despite some randomly distributed defects, atomic lattice can be easily resolved in defect free area, as shown in Fig. 1d. It displays a centered 2 × 2 (or √2 × √2 R45°) lattice with respect to the pristine Cr 1 × 1 lattice. Some dislocation lines are observed where the atomic lattice on their two sides displays certain shift (black dashed lines in Fig. 1d). These dislocations do not show influence to SDW/CDW discussed below, more details are presented in Supplementary Fig. S1. We noticed some previous STM works on Cr (001) also observed c (2 × 2) structures 22,26,30 , but the electronic states of the present surface is quite different from those studies (shown below). Although the origin of this reconstruction is unclear at this stage, it is the first time to observe regular oriented terrace edges with well-ordered lattice on a sputtered/annealed Cr (001) surface.
The typical large energy scale dI/dV spectrum of the surface, measured by a normal W tip above defect-free area, is shown in Fig. 1f. There is a pronounced DOS peak located at −75 (±5) mV. We note although a DOS peak was widely observed on Cr surfaces [18][19][20][21][22][23][24][25][26]31,32 , the peak position varies significantly for different studies. The origin of such a peak was usually attributed to spin-polarized surface state 31,[33][34][35] or the orbital Kondo effect 32 . Our measurements shown below tend to support the former scenario. By zooming into a narrower energy range near E F (Fig. 1g), we found there is an additional DOS dip at E = −22(±1) meV, which has never been reported before. Such a DOS dip is repeatedly observed at different surface locations (see Supplementary Fig. S2 for more spectra). We will show later that it is likely an energy gap associated with the CDW order, but opens below the Fermi level.
Real-space imaging of incommensurate SDW. We then studied the surface with spin-polarized tips. Fig. 2a is a dI/dV map taken at V b = −150 meV with a tip coated with 40 nm thick Cr, which favors an in-plane spin polarization 16 . The mapping area is the same as that shown Fig. 1c. It is remarkable that stripe-like modulations can be observed, and there are two domains of such modulation which are perpendicular to each other. A zoomed-in dI/dV map around a domain wall is shown in Fig. 2b. The period of the stripe is 6.0 nm and the wave vector is either along [010] or [100] direction (as also seen in the FFT image in Fig. 2a inset). Such a period and propagating direction exactly match the projected bulk IC-SDW of Cr on a (001) surface. It is also seen that the domain walls in Fig. 2a (dashed curve) have no correlation with surface morphology (Fig. 1c), which indicates the stripes are not merely surface effects but of bulk origin. Figure 2c shows the typical dI/dV spectra taken on the stripe and between the stripes with the Cr-coated tip. There is observable difference on the intensity around the DOS peak, which is attributed to spin contrast as discussed below. More dI/dV maps taken at different energies and their FFT images can be found in Supplementary Figs. S3 and S4.
To further verify these stripes are spin modulations, we performed measurement with a 16 nm-Fe-coated tip whose magnetization can be controlled by an external magnetic field 16 . Figure 3a, b are two dI/dV maps taken in the same region, but under opposite in-plane field of B X = ±1 T (X direction is perpendicular to stripes, as marked in figure). The 6.0 nm period stripes can be seen in both Fig. 3a, b, while they display a clear phase inversion, as further illustrated in their line profiles in Fig. 3e. Since the tip magnetization will follow such in-plane field, this gives a direct evidence that the stripes are SDW modulations, with opposite spin orientations on their peaks and troughs. We can also tune the tip magnetization along Y and Z directions (by applying B Y = 1 T and B Z = 1.5 T, respectively), the resulting dI/ dV maps are shown in Fig. 3c, d, respectively. In these two cases, the contrast of the stripes was significantly reduced and almost invisible. This confirmed that the spins are only polarized along X direction (the same direction of Q SDW ), which agrees with bulk measurements that the IC-SDW is longitudinal wave at T < T SF (ref. 6 ). We can extract the spin-polarization ratio (spin-contrast) of Fig. 3a, b by calculating their relative intensity difference, which is about 4% at the SDW peaks (Fig. 3f). We note here that for Cr-coated tip used in Fig. 2, its (in-plane) polarization direction is arbitrary 16 , that is why the two SDW domains in Fig. 2a, b can be simultaneously imaged but their contrasts are different.
We note that the above measurement under vector magnetic field also distinguished the SDW state from "spin-spiral" order which has been detected by SP-STM in other magnetic systems 36,37 . In a spin-spiral, the spins have a nearly constant magnitude but their orientations keep rotating with certain chirality, thus one would observe spin modulation in at least two of the X, Y, Z components (depends on the types of spin-spiral, e.g., helical or cycloidal 36 ). However here we only observed spin modulation along X direction.
Another signature of IC-SDW can be obtained near the atomic step edges. Figure 3g shows a topographic image of three adjacent mono-atomic height terraces, and Fig. 3h is the corresponding dI/dV map measured by a Cr-coated tip. As shown by dashed lines, there is a phase inversion of the modulations between adjacent terraces. This indicates the local spin of two adjacent (001) plane are still AFM coupled. Based on above observations, we now achieve a complete spin configuration of the present Cr (001) surface. As illustrated in Fig. 3i, the Cr have an out-of-plane AFM configuration with the spins lying in-plane, while a long wavelength, longitudinal IC-SDW (λ = 6.0 nm) is present in each (001) planes. Such a magnetic structure agrees with the neutron-scattering measurement for bulk Cr and thick Cr films 6-8 , but has not been visualized by a local probe before. Comparing with the commensurate SDW or AFM state 4,16 , the IC-SDW observed here are pure spin modulations that decoupled from the lattice. It can be considered as a "modulated" ferromagnetism for the top Cr plane.
Observation of the coexisting CDW. Moreover, in addition to the SDW modulation, we also observed another type of modulation with half the period of SDW (3.0 nm), as shown in the dI/ dV map in Fig. 2d (taken at V b = −10 mV). It displays the same domain structures with the SDW shown in Fig. 2b, however here the two domains have the same contrast. We further verified that such 3.0 nm modulation is also visible under a nonmagnetic PtIr tip, but the SDW is invisible (see Supplementary Fig. S5). Therefore, it is natural to assign such spin-unpolarized modulation to CDW with a Q CDW = 2Q SDW , as reported in X-ray studies of Cr 10,13 . We noticed a previous STM study on Cr (110) surface reported similar charge modulation that originated from bulk CDW 27 . Here, we are able to image the SDW and CDW simultaneously, enabling the study of their microscopic correlations. Figure 4a-h show a series of dI/dV maps taken in the same region, with the same Cr tip but at various V b . Their (averaged) line profiles are summarized in Fig. 4j, which display the evolvement of SDW/CDW modulations as the energy varies. Figure 4i shows the typical dI/dV spectrum of this mapping region, and the energy positions corresponding to line profiles in Fig. 4j are indicated by dashed lines. The SDW modulation is mainly observable in the energy range of −200 meV~−50 meV, which covers the large DOS peak in dI/dV. This suggests the peak is from certain spin-polarized state(s). As the mapping energy lowered to −50 meV~0, the 3.0 nm CDW modulation became pronounced. Interestingly, it displays an abrupt phase inversion between −30 and −10 meV (see also the dI/dV maps in Fig. 4f, g). Such π phase shift can also be seen in the phase of the Fourier transformations, as shown in Fig. 4k. We note this energy range right covers the DOS dip in the dI/dV spectrum (Fig. 4i), which suggests such a DOS dip is from a CDW gap, as the phase inversion of particle-hole states around the gap is a hallmark of CDW 38,39 . Our measurement at elevated temperatures (shown in Supplementary Figs. S6 and S7) also suggested the DOS dip and CDW are correlated, as they are still both visible at T = 78 K but disappeared together at T = 301 K, which is close to T N = 311 K.
However, it is unusual that the gap is not opened at E F here, but about 22 meV below, which is rather unexpected for conventional CDW 1 . This gap is also unlikely associated with the SDW as its size (≈10 meV) is too small with comparing to the Néel temperature of Cr (311 K). We note in previous ARPES study on Cr (110), a SDW gap is found to be about 200 meV and located above E F (ref. 14 ), which can be understood through AFM coupling induced band folding (see Supplementary Fig. S8, the hole pocket of Cr is slightly larger than electron pocket, their crossing point upon folding is above E F ). However, here we did not observe an obvious SDW gap in tunneling spectrum, although the tunneling conductance at positive energy is indeed low in Fig. 1f (whether it is related to SDW gap needs further investigation). Assuming the Cr sample here has similar band structure to that reported in ref. 14 , a CDW gap below E F cannot be induced by the band folding scenario either. It therefore suggests the formation of CDW is beyond the intuitive Fermi surface nesting picture. We note a recent STM study on TiSe 2 also b d 30 nm [010] [100] indicated a CDW gap opened away from E F (ref. 40 ), which was attributed to strong electron correlations.
Further information can be extracted from the real space imaging of SDW and CDW is their phase relation. As discussed before, the SDW modulations in dI/dV are directly induced by spin contrast (Fig. 3). Their maximum and minimum positions are where the absolute spin density reach maximum. Meanwhile, the CDW modulation in dI/dV is the local DOS variation induced by periodic charge distribution 38 . Figure 4j shows that the locations with maximum spin density (tracked by solid and dashed lines) always have minimum LDOS at −30 mV and maximum LDOS at −10 meV. As usually the charge density is proportional to the LDOS of occupied state near E F , our data suggests the SDW and CDW in Cr are in-phase, i.e., the positions with maximum spin density also have maximum charge density (sketched in Fig. 1b). We note that previously such relation can only be obtained through combined X-ray and neutron diffraction measurement after extensive data analysis 41,42 , while here we provide a rather direct evidence on the same system. The in-phase relation appears to consist with the theory which treat CDW as a second harmonics of SDW 12,43,44 .
At last, beside the static SDW/CDW modulations, we also observed dispersive quasi-particle interference (QPI) on the surface. As shown in Fig. 5a for example, clear short wavelength interference patterns are visible around the defects. Figure 5b is the FFT image of Fig. 5a which displays a square shaped scattering ring (more QPI data is shown in Supplementary Fig. S9). By summarizing the FFT line profile taken at various energies (Fig. 5c), an electron-like dispersion with q F ≈ 1.1 Å −1 is visualized. We note previous DFT calculations had predicted multiple spin-polarized surface states on Cr(001) 33 , the observed QPI could be originated from one of the surface states, as bulk bands usually do not generate strong QPI; and the DOS peak at E = −75 meV in dI/dV (Fig. 1f) could be from the onset of this band. The clear observation of QPI here (which is absent in previous STM studies) is also an indication of improved surface condition. We expect it will help to elucidate the surface electronic states of Cr with the help of further theoretical calculations.  The mechanism of coexisting spin/charge orders has long been an important issue in condensed matter physics, particularly for correlated materials such as cuprates 3,45 and iron-based superconductors 4,46 , our new spectroscopic and microscopic information provide insights on the comprehensive understanding SDW/ CDW in Cr and other correlated materials. Our work is one of the few cases in which simultaneous imaging of spin/charge order with high resolution is achieved (ref. 5 is another example). We expected similar SP-STM measurement shall also be applied to other systems and would inspire more studies on the coexisting quantum orders.

Methods
Cr (001) sample preparation and STM measurements. Cr (001) single crystal (Mateck, purity: 99.999%) was intensively cleaned by repeated cycles of Ar sputtering at 750°C (for 15 min) and annealing at 800°C (for 20 min), until a well-ordered surface is obtained. Spin-resolved tunneling spectroscopy and conductance mapping were performed by Cr-coated and Fe-coated STM tips, which are prepared by depositing 40 nm Cr or 16 nm Fe layers on W tip. The W tip was electrochemically etched and flashed up to ≈2000 K for cleaning before coating. The tunneling conductance (dI/dV) was collected by standard lock-in method and the bias voltage (V b ) is applied to the sample.

Data availability
The main data supporting the findings of this study are available within the article and its Supplementary Information files. All the raw data generated in this study are available from the corresponding author upon reasonable request.

Code availability
All the data analysis codes related to this study are available from the corresponding author upon reasonable request.