Anomalous helimagnetic domain shrinkage due to the weakening of Dzyaloshinskii-Moriya interaction in CrAs
B. Y. Pan, H. C. Xu, Y. Liu, R. Sutarto, F. He, Y. Shen, Y. Q. Hao, J. Zhao, Leland Harriger, D. L. Feng
AAnomalous helimagnetic domain shrinkage due to the weakening ofDzyaloshinskii-Moriya interaction in CrAs
B. Y. Pan,
1, 2, ∗ H. C. Xu, ∗ Y. Liu, R. Sutarto, F. He, Y. Shen, Y. Q. Hao, J. Zhao, Leland Harriger, and D. L. Feng
1, 6, 7, † State Key Laboratory of Surface Physics, Department of Physics,and Advanced Materials Laboratory, Fudan University, Shanghai 200433, China School of Physics and Optoelectronic Engineering,Ludong University, Yantai, Shandong 264025, China Center for Correlated Matter, Zhejiang University, Hangzhou, 310058, China Canadian Light Source, Saskatoon, Saskatchewan S7N 2V3, Canada NIST Center for Neutron Research, National Institute of Standards and Technology,100 Bureau Drive, Gaithersburg, Maryland 20899, USA Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China Hefei National Laboratory for Physical Science at Microscale and Department of Physics,University of Science and Technology of China, Hefei, Anhui 230026, China
CrAs is a well-known helimagnet with the double-helix structure originating from the competitionbetween the Dzyaloshinskii-Moriya interaction (DMI) and antiferromagnetic exchange interaction J .By resonant soft X-ray scattering (RSXS), we observe the magnetic peak (0 0 q m ) that emerges atthe helical transition with T S ≈ ∼
255 K, opposite to the conventional thermal effect. The weakening of DMI oncooling is found to play a critical role here. It causes the helical wave vector to vary, ordered spinsto rotate, and extra helimagnetic domain boundaries to form at local defects, thus leading to theanomalous shrinkage of helimagnetic domains. Our results indicate that the size of magnetic helicaldomains can be controlled by tuning DMI in certain helimagnets.
In correlated materials, multiple magnetic interac-tions, including the superexchange, Dzyaloshinskii-Moriya interaction (DMI), Kondo coupling,Ruderman − Kittel − Kasuya − Yosida interaction, mayco-exist, and they favor different ground states. Thecompetition between these magnetic interactions leadsto rich and novel phenomena such as the quantumcriticality in Kondo lattice[1], spin liquid states infrustrated systems[2], and the emergence of magneticskyrmions in helimagnets[3, 4]. Tuning the strength ofthese interactions would be a important way to engineerthe magnetic quantum states and properties. Take ahelimagnet system for example, its magnetic Hamitoniancan be generally written as: H = (cid:88) i,j (cid:126)D ij · ( (cid:126)S i × (cid:126)S j ) + J i,j (cid:126)S i · (cid:126)S j (1)in which (cid:126)D i,j and J i,j denote the anti-symmetric DMIand the symmetric exchange interactions betweeen (cid:126)S i and (cid:126)S j , respectively. By changing temperature, mag-netic field, material thickness, or pressure, the systemcan be continuously tuned into helical, conical, Skyrmecrystal or other quantum phases depending on the sub-tle balance of DMI, J , Zeeman coupling, and thermalfluctuations[5–8]. Especially, the nano-sized helimagneticdomains, a key ingredient for spintronics [9], can be del-icately manipulated by external fields. For example, in-situ Lorentz microscopy of Fe . Co . Si film showed thatmagnetic field can effectively deform, rotate and enlargethe helimagnetic domains by applying H along different directions[10]. Similar observation was recently reportedin Te-doped Cu OSeO [11]. Here, in the helimagnetCrAs, we observe an anomalous shrinkage of helimag-netic domains with decreasing temperature and find thedecrease of DMI in CrAs on cooling as its main drivingforce. This is a quantum effect opposite to conventionalthermal behavior and may be harnessed for domain en-gineering in spintronics.The helical transition temperature of CrAs is T S =265 K and its spin helix propagates along the c axis(Fig. 1(b))[5, 12, 13]. The local space-inversion sym-metry breaking at the Cr and As sites gives rise to DMI,which is essential in stabilizing the doule-helix spin struc-ture as revealed by group-theoretical approach[14]. Com-pared with other helimagnets in the MnP-type struc-ture, such as MnP and FeP, the DMI in CrAs is muchmore pronounced[14, 15]. In addition, recent studiesshow CrAs exhibits novel non-Fermi liquid behavior, un-conventional superconductivity, and quantum critical-ity under certain conditions [13, 16–20], and the helicalmagnetism is believed to be crucial on these fascinat-ing properties[21]. The strong DMI and rich quantumphases in CrAs make it an exciting playground to studythe behavior of magnetic domains under the competitionbetween DMI and J .In our experiment, we used soft X-ray absorption andresonant scattering to study the helical magnetism ofCrAs. The magnetic resonant peak (0 0 q m ) was observedat the chromium L -edge. Thanks to the high momentumresolution of resonant soft X-ray scattering (RSXS) tech-nique, we could reveal the average helimagnetic domain a r X i v : . [ c ond - m a t . s t r- e l ] J un (a)(b) (c) I n t en s i t y ( a r b . un i t ) ( a r b . un i t ) S ba q =(0 0 L ) k i k f π -pol. σ -pol. σ ’-pol. π ’-pol. CrAs
Scattering plane: ac S S S S S S S c FIG. 1. (a) X-ray absorption spectroscopy (XAS) in to-tal electron yield (TEY) mode of CrAs at the Cr L -edge(red solid line), in comparison with two reference compoundsCr O (blue dashed line) and CrO (Olive dashed line) fromliterature[22]. The bulk sensitive total fluorescence yield(TFY) spectra of CrAs (black dots) was simultaneously col-lected. (b) Illustration of the helical spin order in CrAs andthe RSXS scattering geometry. The gray and green dashedlines denote the double spin helix chains running along the c axis. The red dashed lines denote the nearest-neighboringspins in a unit cell. (c) L scans of the (0 0 q m ) magnetic peakwith the resonant energy E=578.7 eV (black dots) and non-resonant energy E=570 eV (orange filled triangles) at T =20 K. The incident photons are π -polarized. The blue dashedline is from neutron diffraction measurement on a CrAs singlecrystal, note its position has been shifted by +0.007 r.l.u. inorder to match the RSXS data. size ξ and its temperature-dependent evolution. Intrigu-ingly, unlike conventional magnets whose magnetic do-mains always grow larger on cooling, the domain size ofCrAs substantially decreases with lowering temperaturebelow ∼
255 K. The average domain size in the ac planeshrinks by ∼ J varies thehelimagnetic wave vector. As the helical magnetic chainspropagate across the crystal defects with decreasing DMIon cooling, extra helix domain boundaries form and theaverage domain size decreases, leading to the observeddomain shrinkage opposite to the conventional thermaleffect.CrAs single crystals were grown by the Sn-flux methoddescribed in previous report[23]. The obtained shinycrystals have needle-like shape with a typical size of 6 × × . The largest crystalline plane is (0 0 1).RSXS and X-ray absorption (XAS) experiments on aCrAs single crystal were performed using a four-circlediffractometer at the Resonant Elastic and Inelastic X-ray Scattering (REIXS) beamline of Canadian LightSource (CLS). The REIXS beamline is equipped withElliptical Polarized Undulators (EPU) and can provideboth σ and π polarized incident photons. The momen-tum resolution of the RSXS instrument is better than0.0005 ˚A − at 570 eV. For RSXS signal, a silicon photo-diode was used, while for the XAS in total electron yield(TEY) mode and total fluorescence yield (TFY) modewere collected using a drain current and micro-channelplate (MCP) detector, respectively.Figure 1(a) is the XAS of CrAs (red line) measuredby TEY at the Cr L -edge. In order to elucidate the Crvalance state, the XAS of Cr O and CrO were plot-ted together as the fingerprint of Cr and Cr valencestates[22], respectively. The bulk sensitive TFY spec-trum of CrAs was simultaneously collected. Both theTEY and TFY spectra of CrAs are consistent with thetypical Cr spectrum. These results show that the sam-ple surface is clean and the valance state is Cr with the3d electronic configuration. The spin-only magnetic mo-ment of Cr is 3.87 µ B , however, the observed value is1.7 µ B [24]. The reduction of magnetic moment in CrAsshould come from fluctuations and hybridization effect,similar to the case in MnP[25].In our RSXS experiment, the scattering crystallineplane is ac and the momentum transfer direction is along(0 0 L ) (Fig. 1(b)). In this configuration, the electricfield of horizontally (vertically) polarized incident pho-tons are perpendicular (parallel) to the b axis. In the he-lical state below T S , the Cr spin moments lie in the ab easy plane, and the magnetic propagation wavevector isabout (0 0 0.356) at T =4 K[13]. From sample alignment,we determined the lattice constants are a =5.412(9) ˚A, b =3.348(1) ˚A, c =6.007(9) ˚A at 20 K. Figure 1(c) presentsthe L scans around the magnetic wavevector k m =(0 0 q m )with the resonant (E = 578.7 eV, π polarization) andnon-resonant (E = 570 eV, π polarization) energies at T = 20 K. Strong resonant peak appears around (0 0 0.352),consistent with the previously reported helical propaga-tion wavevector[24]. The blue dashed line in Fig. 1(c)is from neutron diffraction measurement on CrAs singlecrystal which is much broader due to its relatively lowmomentum resolution. A detailed comparison of neu-tron diffraction and RSXS experiments on CrAs can befound in the Supplemental Material.To verify the magnetic nature of the (0 0 q m ) peak,we measured its resonant profiles as a function of X-ray energy and wavevector (0 0 L ) at T = 20 K (Fig.2). The incident X-ray is either vertically ( σ , Fig. 2(a))or horizontally ( π , Fig. 2(b)) polarized. No distinctionwas made on the polarization of scattered photons, sothe detected scattering intensities in our experiment are (a) (b) E ( e V ) I π q -integrated intensity (c) I σ I π I σ int int FIG. 2. Resonant profiles around (0 0 q m ) with (a) σ or (b) π polarized incident photons at T = 20 K. The resonance profilesare plotted as a function of photon energy covering the Cr L -edge and reciprocal lattice (0 0 L ). All data were measured bya photodiode detector and the fluorescence background has been subtracted. The color bar indicates scattering intensity inarbitrary unit. (c) Integrated intensity along the L direction of the resonances in (a,b) as a function of the incident photonenergy. The black and red lines represent the σ and π polarized incident photons, respectively. The dashed blue line is theXAS curve. I π = I ππ (cid:48) + I πσ (cid:48) and I σ = I σπ (cid:48) + I σσ (cid:48) . The observed resultsshow that I π is about 1.7 times stronger than I σ . Wefurther integrate the resonant intensity in Fig. 2(a) and2(b) along the L direction and get the q -integrated inten-sities I intσ and I intπ , as shown in Fig. 2(c). The lineshapeprofiles of I intσ (black line) and I intπ (red line) are similar,except the latter is apparently stronger. The polariza-tion dependence of resonant profiles is consistent withmagnetic scattering from a helimagnet, as evidenced bythe following theoretical analysis. The scattering inten-sity from a helimagnet can be expressed as I mag = | f mag | ,and | f mag | is the resonant magnetic scattering length[26–28]: f mag = (cid:18) f σσ (cid:48) f πσ (cid:48) f σπ (cid:48) f ππ (cid:48) (cid:19) = − iF (cid:18) M a cosθ + M c sinθM c sinθ − M a cosθ − M b sin θ (cid:19) (2)where σ (cid:48) and π (cid:48) denote the polarization of outgoing pho-tons, θ is the angle between the incident X-ray and thesample surface, and M a , M b , M c are the spin momentalong the three crystal axes. In our case, M a = M b , M c =0, so | f σπ (cid:48) | = | f πσ (cid:48) | , I σ = I σπ (cid:48) = I πσ (cid:48) . In this way, I π = I σπ (cid:48) + I ππ (cid:48) >I σ , so I π is always stronger than I σ withinthe Cr L -edge resonant energy range, this is consistentwith the experiment observation shown in Fig. 2.Detailed study on the temperature-dependent evolu-tion of (0 0 q m ) resonant peak was conducted by usingthe π -polarized incident photon with the resonant en-ergy 578.7 eV. Both L and H scans were performed atthe temperature range from 267.5 K to 20 K, as shownin Fig. 3(a) and 3(b). Along L scans, q m continuouslydecreases on cooling, consistent with our neutron diffrac-tion results shown in the Supplemental Material. The L and H scans take the Lorentzian and Gaussian line- -0.010 -0.005 0.000 0.005 0.0100.00.20.40.60.8 I n t en s i t y ( a r b . un i t ) H (r.l.u.)
20 K 50 K 100 K 150 K 180 K 250 K 255 K 260 K 262.5 K 265 K
H scan L scan I n t en s i t y ( × - a r b . un i t ) T (K) q m (r . l . u . )
20 K 50 K 100 K 150 K 180 K 250 K 255 K 260 K 262.5 K 265 K 267.5 K I n t en s i t y ( a r b . un i t ) L (r.l.u.) ξ ( Å ) ξ c ξ a (a)(b) (c)(d)(e) FIG. 3. Temperature-dependent evolution of the (0 0 q m )resonant peak along (a) L and (b) H directions in the tem-perature range from 267.5 K to 20 K. The incident photon en-ergy is 578.7 eV with π polarization. The sample was warmedup from 20 K to 267.5 K. The solid lines in (a) and (b) areLorentzian and Gaussian function fittings, respectively. (c)Integrated intensity of resonant peak, (d) wavevector q m , and(e) the average helimagnetic domain size ξ versus tempera-ture for the L (red dots) and H (black dots) scans. The solidlines are guides to the eye. shapes, respectively [Fig. 3(a) and 3(b)], indicating dif-fernt domain size distribution along the c and a axes.This anisotropy may come from the unique propagatingdirection of the helimagnetic wavevector or the elongatedneedle-like crystal shape, both of which are along c axisand could cause anisotropic grain and strain distributioninside the sample.The temperature-dependent evolutions of peak inten-sity, propagation wavevector q m , and the average heli-magnetic domain size ξ ( ξ =1/FWHM, FWHM is for full-width-at-half-maximum) in L and H scans are presentedin Fig. 3(c)-3(e). In Fig. 3(c), the magnetic peak inten-sity rapidly saturates below T S , consistent with the firstorder transition character[13]. The change of q m withdecreasing temperature (Fig. 3(d)) indicates that thebalance of competition between different magnetic inter-actions in CrAs varies with temperature. The averagehelimagnetic domain size ξ c and ξ a rapidly grow from T S to 255 K (Fig. 3(e)), consistent with the typical criti-cal behavior near a transition point[29]. However, below ∼
255 K both ξ c and ξ a abnormally decrease on cooling(Fig. 3(e)). The broadening of the magnetic peak alsoleads to a slight decreasing of peak intensity along H (Fig. 3(c)). Usually, the average helimagnetic domainsize of a magnetic order should monotonically increase below T S because thermal fluctuations are weakened andspins become more correlated on cooling. The anomalous ξ versus T in CrAs indicates the average helimagneticdomain size actually shrinks with lowering temperature.Cooling and warming sequences (see Supplemental Mate-rial) show little thermal history effect in the temperature-dependent evolution of ξ and q m .The anomalous shrinkage of magnetic domains on cool-ing can be interpreted by the weakening of DMI in CrAs.In ref. 13, the authors gave a detailed description on themagnetic interactions in CrAs. Its magnetic Hamitonianwas represented by Eq. 1, in which (cid:126)D i,j and J i,j arethe DMI and antiferromagnetic interactions between thenearest neighbors, respectively. The nearest-neighboringspins in a single unit cell are illustrated by the red dashedlines in Fig. 1(b). q m can be expressed as[13]: q m = β + β π (3)where β is the angle between (cid:126)S and (cid:126)S , β is the anglebetween (cid:126)S and (cid:126)S . In the temperature range of ourstudy, β barely changes [13]. Therefore, the decreaseof q m on cooling is mainly attributed to the variation of β , which is determined by: β = tg − ( D c /J ) (4)in which D c is the (cid:126)D component along the c axis.There are antiferromagnetic interactions between allnearest spins, in contrast, DMI exists between (cid:126)S and (cid:126)S but is absent between (cid:126)S and (cid:126)S [13]. Moreover, theDMI in CrAs is exceptionally larger than the antiferro-magnetic interaction ( | D | > | J | ) [14]. Therefore, the in-duced change of β and the decreasing of q m should bedominated by the variation of D c :∆( q m ) ∝ ∆( D c ) (5) a ξ c = 293.2 Å, ξ a = 275.5 Å ξ c = 180.5 Å, ξ a = 250.0 Å c DMI dereases a c
T=255 K T=20 K (a) (b) S S S S β β
12 12 =-100.9 o =-107.9 o FIG. 4. Cartoon illustration of the angle ( β ) between (cid:126)S and (cid:126)S and helimagnetic domains in the ac plane at (a) 255 Kand (b) 20 K. The values of β are from ref. 13. As the DMIterm in the magnetic Hamiltonian has the form (cid:126)D · ( (cid:126)S × (cid:126)S ),thus (cid:126)D favours non-collinear spin alignment. The tendencyto antiferromagnetic spin alignment between (cid:126)S and (cid:126)S from255 K to 20 K indicates the DMI gets weaker on cooling. According to this equation, the 6.70% decrease of prop-agation wavevector, from q m =0.3773(5) at 255 K to q m =0.3520(6) at 20 K, indicates that D c becomeweaker. Since D c favors non-collinear spin alignmentand J favors antiparallel spin alignment in CrAs [13], (cid:126)S and (cid:126)S tends to be more antiparallel[13], as illustrated by β in Fig. 4. This again evidences the weakening of DMIwith decreasing temperature. As DMI is the dominantforce determining the spin rotation along the helix chain,its weakening will make the helimagnetic domains easierto break up at the defect sites. As q m varies with tem-perature, the neighboring spins continuously modulatetheir relative spin angles on cooling, which would gen-erate additional domain boundaries at defect sites giventhe weakening DMI, in other words, the helimagnetic do-mains shrink. A straightforward cartoon illustration forthe DMI controlled spin angle β and the accompanieddomain shrinkage is presented in Fig. 4.The helimagnetic domain shrinkage is anisotropic andmainly takes place along the c direction. As shown inFig. 3(e), from 255 K to 20 K the percentage drop of ξ c and ξ a are 38.44 % and 9.26 %, respectively. Here we de-fine the spatial anisotropic ratio of domain shrinkage as γ ca = ∆ ξ c /ξ c (255K)∆ ξ a /ξ a (255K) =4.15. This is consistent with the factthat the helimagnetic order is propagating along the c di-rection. Meanwhile, it is intriguing to note that the DMIof CrAs is (cid:126)D ≈ D (-0.17, -0.5, 0.85)[13], so the ratio ofDMI components along c and a is κ ca = | D c /D a | =5.The similar size of γ ca and κ ca implies possible role playedby DMI in the anisotropy of domain shrinkage, as certaininteractions exist in a and b directions as well.By contrast, in our previous RSXS investigation onMnP, a helimagnet similar to CrAs in lattice and mag-netic structures but its propagation wavevector increaseson cooling[5], the domain shrinkage behavior was notobserved[25]. This implies that the decrease of q m orDMI on cooling is the key to the formation of new domainboundaries at defect sites inside the sample. It shouldbe noted that in most 3 d -transition metal pnictides thestrength of DMI is much smaller than J , however, CrAsis an exception in which | D | > | J | [14]. The strong DMI ofCrAs even drives the spin reorientation transition and de-crease of magnetic wavevector under pressure[13]. There-fore, we conclude that the pronounced DMI strengthcombined with its decrease on cooling are essential in-gredients for the anomalous helimagnetic domain shrink-age behavior in CrAs. Broadening of magnetic peak atlow temperature was also observed in Ca Co O withferromagnetic chains, while it is attributed to the devel-opment of a short range order [30], which is distinct fromthe single-component magnetic peak in CrAs and doesnot involve DMI.In summary, we find the Cr valance state in CrAsand our RSXS experiment reveals its helimagnetic do-mains shrink on cooling below ∼
255 K. The domainshrinkage has similar temperature-dependent evolutionwith that of DMI, indicating DMI is the main drivingforce in this anomalous behavior. Our results reveal aquantum effect that is opposite to conventional thermaleffect, and suggest that DMI may be tuned to manipulatethe domain boundaries in helimagnets which may haveapplication in future spintronics.The authors are grateful for the helpful discussionswith Prof. Jiang Xiao, Prof. Yi-Zheng Wu, and Prof.Yan Chen of Fudan University. This work is supported bythe National Natural Science Foundation of China (GrantNos. 11888101, 11790312, 11804137, 11704074), the Na-tional Key Research and Development Program of China(Grant No. 2016YFA0300200 and No. 2017YFA0303104),the Science and Technology Commission of Shanghai Mu-nicipality (Grant No. 15ZR1402900), and the NaturalScience Foundation of Shandong Province (Grant No.ZR2018BA026). 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We used Physical Properties Measurement System (PPMS) to characterize theelectronic resistivity of the experimental sample from 300 K to 2 K, as shown in Fig.S1. The sample’s resistivity shows a transition into the helical magnetic phase at 263K, in consistence with previous reports. The inset panel shows the CrAs crystal withneedle-like shape.Figure S1: Resistivity of CrAs single crystal. The helical transition appears at T S =263K.
2. Comparison of the results from resonance soft X-ray scattering and neutrondiffraction
We have performed both RSXS and neutron diffraction (ND) on CrAs singlecrystal for a comparative study. The RSXS experiment was carried out at the REIXSbeamline of Canadian Light Source. Single crystal ND was measured at the SPINScold triple-axis spectrometer of the NIST Center for Neutron Research. Themomentum transfer q is along the (0 0 L ) direction in both experiments. The measured(0 0 q m ) magnetic peak at various temperatures by RSXS (E=578.7 eV, polarization)and ND are shown in Fig. S2(a) and (b), respectively. It is obvious that the peak widthin RSXS data is much narrower, illustrating the instrumental resolution of RSXS ishigher than that of ND. Actually, the RSXS’s instrumental resolution in ourexperiment is less than 0.0005 Å -1 at 570 eV , much smaller than the resonance peakwidth of CrAs. This unique ability of RSXS directly gives us intrinsic magnetic peak FIG. 5. width or correlation length.The magnetic peak intensity, propagation wavevector, and FWHM at varioustemperatures are shown in Fig. S2(c-e). It can be seen that RSXS and ND agree witheach other at the temperature dependence of peak intensity and propagationwavevector. The FWHM of ND data are 7 times broader than that of RSXS data dueto the relatively poor momentum resolution of ND.Figure S2: Temperature dependence of the magnetic peak in CrAs single crystalsprobed by (a) RSXS and (b) neutron diffraction. The neutron diffraction peaks aremuch broader than the ones from RSXS. Comparison of the (c) peak intensity, (d)propagation wavevector, and (e) FWHM of RSXS (red dots) and neutron diffraction(black dots) as a function of temperature.
3. Evolution of the (0 0 q m ) resonant peak on the cooling and warm up processes Detailed study on the temperature evolution of the (0 0 q m ) resonant peak wasconducted by using π polarized incident photon with the energy E=578.7 eV. Thesample was first cooled down from above T S to 20 K (step 1), and then warmed upback to above T S (step 2). The raw data of L scans at several temperatures are shownin Fig. S3(a,b). The solid lines are fittings by the Lorentzian function. The peakintensity and correlation length are shown in Fig. S3(c,d). There is no obviousdifference between the L scans in the cooling and warm up processes, suggesting thetemperature evolution of magnetic domains has little dependence on thermal history. FIG. 6.
Figure S3: (a) Temperature evolution of the (0 0 q m ) resonant peak along the L direction with incident photon E = 578.7 eV, π-polarization. The sample was first (a)cooled down from above T S to 20 K (step 1), and then (b) warmed up back to 267.5K (step 2). The solid lines are Lorentzian fittings. Temperature dependence of (c)resonant peak intensity and (d) correlation length ξ c in the cooling (blue circles) andwarm up (red dots) processes.in the cooling (blue circles) andwarm up (red dots) processes.