⬇ Download Markdown · ⬇ Download LaTeX · ⬇ original · view original · CSV: table 1
Spin-polarized electron transport for the altermagnet CrSb
V.D. Esin, 1 D.Yu. Kazmin, 1 A.V. Timonina, 1 N.N. Kolesnikov, 1 and E.V. Deviatov 1 , 2
arXiv:2607.07334v1 [cond-mat.mes-hall]
8
Jul 2026
1 Institute of Solid State Physics of the Russian Academy of Sciences, Chernogolovka, Moscow District, 2 Academician Ossipyan str., 142432 Russia 2 V.L. Ginzburg Research Centre for High-Temperature Superconductivity and Quantum Materials,
P.N. Lebedev Physical Institute of RAS, Moscow 119991, Russia
P.N. Lebedev Physical Institute of RAS, Moscow 119991, Russia (Dated: July 9, 2026) We experimentally investigate spin-polarized electron transport for the centrosymmetric altermag- net CrSb, which is known to reveal both altermagnetic and topological features. We demonstrate pronounced first-harmonic anomalous and second-harmonic non-linear Hall effects for a single-crystal CrSb flake with ferromagnetic nickel contacts, while both effects can not be seen for the reference samples with non-magnetic gold ones. For the anomalous Hall effect, we demonstrate bow-tie hys- teresis loop in Hall voltage, which is usually ascribed to surface spin textures in magnetic materials. The slope of the Hall curve changes a sign for two orientations of the Hall-bar contact configuration for the same sample, i.e. for the same sign of the charge carriers. We interpret the observed sign inversion and bow-tie hysteresis as the joint effect of the alternating bulk spin splitting and spin- polarized topological surface states in CrSb. The pronounced non-linear Hall effect with hysteresis in magnetic field confirms finite Berry curvature dipole under injection of spin-polarized electrons, i.e. the topological features for the altermagnetic candidate CrSb. PACS numbers: 73.40.Qv 71.30.+h
I. INTRODUCTION Recently, altermagnetic class of materials has been in- troduced 1 –3 as antiferromagnets with broken the com- bined spatial-inversion and time-reversal symmetry 4 . In this case, the spin-opposite sublattices cannot be related by translation or inversion 5 ,6 , so the up-polarized sub- band is obtained by π/2 rotation of the down-polarized one for the simplest example of the d-wave altermag- net 7 ,8 . As a result, the zero net magnetization is accom- panied by alternating spin splitting in the k-space 1 ,5,9 . For altermagnets, the anomalous Hall effect (AHE) as a hysteresis loop in Hall resistance has been regarded as the first experimental demonstration of their non-trivial spin splitting 1 0–14 . While weak ferromagnetism of anti- ferromagnets has already been known for materials with strong spin-orbit coupling 1 5 , the specifics of altermag- netism is the non-relativistic spin-momentum locking 1 . Usual spin-orbit induced spin-momentum locking is well known for topological materials, leading to spin po- larization of the topological surface states 1 6–19 . In mag- netic topological semimetals, the anomalous Hall effect is connected with dissipationless surface transport 1 8 . For non-magnetic systems, AHE is prohibited in the pres- ence of time-reversal symmetry, while the non-linear Hall effect (NLHE) 2 0 is predicted as a transverse second- harmonic voltage response due to the Berry curvature dipole in momentum space 2 0–24 . In the simplified pic- ture 2 0 , the longitudinal current generates the effective sample magnetization, which leads to the Hall effect even in zero external magnetic field. Hall voltage is therefore proportional to the square of the excitation current, so it can be detected as the second-harmonic V 2ω x y transverse voltage component. NLHE has been demonstrated for a wide range of materials 2 5–29 , being the direct manifesta- tion of finite Berry curvature in topological media 2 0 . Nowadays, it is a common agreement, that AHE still requires spin-orbit coupling in altermagnetic mate- rials 1 1,12,30–32 . For example, ARPES (angular-resolved photoemission spectroscopy) confirms the altermagnetic nature of spin-spliting in α-MnTe 3 3 . However, the prin- ciple origin of finite net magnetization 1 1,12,34,35 in MnTe is the spin-orbit coupling 3 1,32,36–38 . In contrast, spin- orbit coupling is negligible for centrosymmetric CrSb, while ARPES confirms high altermagnetic splitting for this material 3 9,40 . Moreover, signature of topological sur- face states 4 1,42 has been shown for CrSb 4 3 , so the room- temperature altermagnetic candidate CrSb reveals both altermagnetic and topological features 4 4 . AHE can not be expected for CrSb either from the topological surface states due to the time-reversal sym- metry 1 8 or from the bulk altermagnetic spectrum without spin-orbit coupling 3 1,32 . Similarly, NLHE is prohibited for centrosymmetric materials 2 0–24 . However, both AHE and NLHE becomes more intriguing in spin-injection ex- periments 4 5 due to the alternating bulk spin splitting 3 9,40 and strong spin polarization of the topological surface states 4 3,44 . Here, we experimentally investigate spin-polarized electron transport for the centrosymmetric altermag- net CrSb, which is known to reveal both altermagnetic and topological features. We demonstrate pronounced anomalous and second-harmonic non-linear Hall effects for a single-crystal CrSb flake with ferromagnetic nickel contacts, while both effects can not be seen for the refer- ence samples with non-magnetic gold ones.
2 group P 6 ′ 3 / m ′ m ′ c: two Cr sublattices with opposite spins are aligned along the c axis, being connected by C6z - a six-fold rotation combined with a 1/2 translation along c 4 6,47 . The N´eel temperature is about 705 K, which might be convenient for possible applications in spintronics.
(a)
Fig. 1 (b) shows a top-view image of the sample. Since CrSb is a three-dimensional altermagnet, we have to se- lect relatively thick (about 1 µm) single-crystal flakes. A thick flake also ensures sample homogeneity for correct determination of xx- and xy- voltage responses, however, the desired experimental geometry can not be defined by usual mesa etching for thick flakes. Instead, the contact geometry is formed by Ni leads pattern on the SiO 2 sur- face, as depicted in the inset to Fig. 1 (b). Ni leads are obtained by lift-off technique after thermal evaporation of 100 nm Ni film. As a reference, we use Au leads pat- tern for the samples without spin polarization of charge carriers.
(b)
FIG. 1. (Color online) (a) The X-ray powder diffraction pat- tern (Cu K α radiation), which is obtained for the crushed CrSb single crystal. The single-phase CrSb is confirmed with the space group P 6 3 / mmc (No. 194). (b) Optical image of the sample with a 1 µm thick CrSb flake on the ferromagnetic Ni leads with scheme of electrical connections for one of two Hall configurations. The leads pattern is shown in the inset: 100 nm thick, 5 µm separated leads allow to realize the cor- rect Hall (xy-) configuration for two orientations in respect to the CrSb flake. The depicted Hall configuration is specially designed to exclude possible admixture of the thermoelectric effects for reliable NLHE measurements 2 7 . The mechanically exfoliated ≈100 µm wide CrSb flake is transferred onto the metallic leads. After initial single- shot pressing by another oxidized silicon substrate, the flake is firmly connected to the leads, see Fig. 1 (b). This procedure provides high-quality contacts, the obtained samples are electrically and mechanically stable even in different cooling cycles, which has been verified before for a wide range of materials 4 8,49,51–53 . Also, the Ni-CrSb (or Au-CrSb) contacts are protected from any contam- ination (also oxygen and moisture) by SiO 2 substrate, as it has been demonstrated for sensitive materials like black phosphorus 5 3 . We investigate transverse (xy-) first- and second- harmonic voltage responses by standard four-point lock- in technique 2 7,29 . The ac current is applied between two current contacts in Fig. 1 (b), while the transverse (Hall) voltage is measured between the potential ones. The sig- nal is confirmed to be independent of the ac current fre- quency within 100 Hz – 10kHz range, which is defined by the applied filters. The measurement are performed at 30 mK temperature in a dilution refrigerator equipped with a superconducting solenoid. Magnetic field is directed normally to the sample plane to investigate the first- and second-harmonic Hall effects.
FIG. 1. (Color online) (a) The X-ray powder diffraction pat- tern (Cu K α radiation), which is obtained for the crushed CrSb single crystal. The single-phase CrSb is confirmed with the space group P 6 3 / mmc (No. 194). (b) Optical image of the sample with a 1 µm thick CrSb flake on the ferromagnetic Ni leads with scheme of electrical connections for one of two Hall configurations. The leads pattern is shown in the inset: 100 nm thick, 5 µm separated leads allow to realize the cor- rect Hall (xy-) configuration for two orientations in respect to the CrSb flake. The depicted Hall configuration is specially designed to exclude possible admixture of the thermoelectric effects for reliable NLHE measurements 2 7 . II. SAMPLES AND TECHNIQUE CrSb single crystals were synthesized by reaction of pure Cr (99.996%) and Sb (99.9999%), which were mixed in the stoichiometric ratio and then heated in an evacu- ated silica ampule up to 1000 ◦ C with the rate of 15 ◦ C/h in a gradient-free furnace. The load was held at 1000 ◦ C for 72 hours and then cooled down slowly (11 ◦ C/h) to the room temperature. The crystals grown are faceted single crystals with the space group P 6 3 / mmc (No. 194) and the stoichiometric composition, as confirmed by X- ray diffraction analysis, see Fig. 1 (a). CrSb is of hexagonal structure with the magnetic space We investigate transverse (xy-) first- and second- harmonic voltage responses by standard four-point lock- in technique 2 7,29 . The ac current is applied between two current contacts in Fig. 1 (b), while the transverse (Hall) voltage is measured between the potential ones. The sig- nal is confirmed to be independent of the ac current fre- quency within 100 Hz – 10kHz range, which is defined by the applied filters. The measurement are performed at 30 mK temperature in a dilution refrigerator equipped with a superconducting solenoid. Magnetic field is directed normally to the sample plane to investigate the first- and second-harmonic Hall effects. The leads pattern in the inset to Fig. 1 (b) allows to realize the correct Hall (xy-) configuration for two differ- ent Hall bar orientations in respect to the CrSb flake: one configuration 2 7 is shown in Fig. 1 (b), while the current line is 90 ◦ rotated for the other one 2 9 . In the contacts area, the leads are of 10 µm width, they are separated by 5 µm distance, so the voltage probe separation is 20- 80 µm for two different Hall configurations. The correct choice of the potential contacts is important for AHE and NLHE measurements. In particular, the contacts config- uration from Fig. 1 (b) is specially designed for reliable NLHE measurements, it has been verified to exclude pos- sible admixture of the thermoelectric effects, see Ref. 2 7 for details.
Table 1 · 11 columns × 0 rows · p.3–3 · CSV I ac = 2 µA
-0.4 -0.2
I a c = 2 µA -10
0.0 B (T)
0.2 0.4 -0.4 -0.2 0.0 0.2 0.4
B (T)
FIG. 2. (Color online) Anomalous Hall effect for the CrSb sample with ferromagnetic Ni contacts. (a) AHE hysteresis is depicted as the magnetic field dependence of the transverse voltage V 1ω x y for two field sweep directions for the contacts configuration from Fig. 1 (b). Inset shows V 1ω x y (B) curves for the reference sample with normal gold contacts, no AHE can be seen in this case. (b) V 1ω x y (B) hysteresis for another Hall configuration, which is 90 ◦ rotated in respect to the one in (a). The AHE slope is inverted in this case for the same CrSb flake and for the same magnetic field orientation. For both Hall configurations, V 1ω x y (B) curves strongly resemble bow- tie magnetization hysteresis loops 5 5,56 , which is confirmed in the inset for another sample. The data are obtained at T = 30 mK temperature for the ac current amplitude I a c = 2µA.
III. EXPERIMENTAL RESULTS
Fig. 2 (a) shows typical AHE signal as the magnetic field dependence of the transverse voltage V 1ω x y for the sample with ferromagnetic Ni contacts. The data are obtained for the contacts configuration from Fig. 1 (b) at T = 30 mK temperature. The ac current amplitude I a c = 2µA guarantees the linear regime, e.g. it do not lead to the sample overheating for the low characteristic longitudinal (xx-) sample resistance ≈ 0.5Ω. The maxi- mum Hall resistance is 5 mΩ in Fig. 2 (a), it corresponds to Hall resistivity 0.5 µΩ·cm for 1 µm thick CrSb flake, which is at least one order of magnitude smaller than the usual values in magnetic materials 1 1,12,54 . To improve the signal to noise ratio for low V 1ω x y (B) values, we use multiple curves averaging, so the hysteresis in V 1ω x y (B) can be clearly seen for two magnetic field sweep direc- tions in Fig. 2 (a). This result is quite unusual for CrSb, since no AHE can be expected without spin-orbit coupling in altermagnetic materials 3 1,32,36,37 . We also do not observe any notice- able AHE signal for the reference samples with normal gold contacts, as depicted in the inset to Fig. 2 (a), so the observation of AHE should be a consequence of the spin-polarized current injection from the ferromagnetic Ni contacts. The leads pattern in the inset to Fig. 1 allows to re- alize another Hall (xy-) contact configuration, where the
0.0
0.1 0.2 0.3 0.4 0.5
I (mA)
FIG. 3. (Color online) Non-linear Hall effect 2 5–29 for the ferro- magnetic Ni contacts as a quadratic second-harmonic Hall-like response V 2ω x y ∼ I 2 to ac excitation current I. The curves are depicted for two 30 mK and 1.2 K temperatures as blue and red ones, respectively. V 2ω x y ∼ I 2 law is confirmed in the inset as strictly linear V 2ω x y (I 2 ) dependence for high ac excitation values. The second-harmonic Hall voltage V 2ω x y is one order of magnitude smaller for the reference sample with Au contacts for the same contact configuration, as depicted by the black curve. The data are obtained in zero magnetic field. current line is 90 ◦ rotated in respect to the previous case, the result is depicted in Fig. 2 (b). The V 1ω x y (B) hystere- sis is of the same ±0.3 T width in Fig. 2 (b), but of lower amplitude in comparison with one in Fig. 2 (a), the lat- ter can be ascribed to geometrical factor. However, the slope changes a sign in Fig. 2 (b), which can not be ex- pected for the same flake and the same magnetic field orientation 1 3,14,54 : in normal magnetic field, the angle between the magnetic field and the current is the same for two Hall bar configurations in Fig. 2. Since the slope of the V 1ω x y (B) Hall curve is determined by the sign of the charge carriers and the magnetic field direction, it should not depend on the current line orientation in respect to the CrSb flake. Also, the hysteresis loops strongly resemble a bow- tie magnetization hysteresis loop 5 5,56 in Fig. 2: the dis- tance between the curves for opposite sweep directions is smaller at zero field/zero voltage, which can be even bet- ter seen for another sample in the inset to Fig. 2 (b). The bow-tie hysteresis in magnetization is usually ascribed to surface spin textures in magnetic materials 5 5,56 . Fig. 3 shows the effect of spin-polarized current on the NLHE, i.e. on the second harmonic of the transverse (Hall) voltage V 2ω x y . For the CrSb sample with ferro- magnetic Ni contacts, typical behavior of the non-linear Hall effect 2 5–29 is demonstrated as a quadratic transverse Hall-like response V 2ω x y ∼ I 2 to ac excitation current I. In contrast, the second-harmonic Hall voltage is one or- der of magnitude smaller for the reference sample with Au contacts in Fig. 3, the data are obtained for the same contact configuration. For NLHE measurements, we only use the contact configuration depicted in Fig. 1 (b), to
4 0.1 0.2 0.3 0.4 0.5 0 2 4 B = 0 T V 2 ω xx (a) V
2
ω
x
y
(
µ
V
)
I (mA) -0.4 -0.2 0.0 0.2 0.4 4.76 4.80 4.84 (b) V
2
ω
x
y
(
µ
V
)
B (T) I a c = 0.5 mA -0.4 -0.2 0.0 0.2 0.4 0.02 0.03 V
2
ω
x
y
(
µ
V
)
B (T) FIG. 4. (Color online) (a) Non-linear Hall effect for another sample with ferromagnetic Ni contacts in zero magnetic field. The magenta curve shows the longitudinal second-harmonic voltage V 2ω x x , which is one order of magnitude smaller. (b) Hysteresis in V 2ω x y for two magnetic field sweep directions at I a c = 0.5 mA for spin-polarized transport. Inset shows no V 2ω x y (B) field dependence for the reference sample with gold contacts. The data are obtained at T = 30 mK temperature. reliably exclude the thermoelectric effects, as it has been demonstrated in Ref. 2 7 . For NLHE, wide current interval is important to ob- serve non-linear effects. The NHLE curves V 2ω x y (I) still depend on temperature in 30 mK–1.2 K range, see red and blue curves, so the ac excitation do not overheat the sample. Moreover, sample overheating would lead to di- minishing of the measured voltage, while the V 2ω x y ∼ I 2 dependence is well confirmed for high ac excitations in the inset to Fig. 3. Qualitatively similar ∼ I 2 NLHE dependence is repro- duced for another sample with ferromagnetic Ni contacts, see Fig. 4 (a). The longitudinal second-harmonic voltage V 2ω x x is negligible in comparison with the Hall V 2ω x y one, which confirms well-defined experimental geometry and high sample homogeneity. As well as the first harmonic in Fig. 2, the second- harmonic Hall voltage shows hysteresis for two magnetic field sweep directions, see Fig. 4 (b). The hysteresis loops are of similar widths in Figs. 2 and. 4 (b). In contrast, there is no V 2ω x y (B) field dependence for the reference sample with gold contacts, as it is shown in the inset to Fig. 4 (b). We should conclude, that for the altermagnet CrSb both the first-harmonic anomalous and the second- harmonic non-linear Hall effects can only be observable for spin-polarized charge carriers. IV. DISCUSSION Both AHE and NLHE can not be expected for non- magnetic and centrosymmetric CrSb, which we confirm for the reference CrSb samples with gold contacts in Figs. 2,3,4. Also, the stoichiometric CrSb composition was verified by X-ray diffraction analysis, see Fig. 1 (a), so one can be sure in quality of our CrSb material. More- over, the AHE slope inversion in Fig. 2 can not be ex- pected even in magnetic materials for the same flake (i.e. for the same sign of the charge carriers) and the same magnetic field orientation. Thus, observation of AHE and NLHE should be connected with spin injection into the altermagnetic crystal CrSb. Let us start from the anomalous Hall effect in Fig. 2. In principle, CrSb exhibits low magnetocrys- talline anisotropy energy, enabling the manipulation of the N´eel vector in CrSb films through a suitable ferro- magnetic substrate, so the anomalous Hall and Nernst conductivities appear for the canted N´eel vector 5 7 . How- ever, this effect can only be observable within the bulk spin-relaxation length, i.e. near the contact regions in our experiment. Instead, we demonstrate the anomalous Hall effect for macroscopic (20-80 µm) contact separa- tion. On the other hand, Weyl topological surface states were shown for CrSb 4 3,44 . The states with opposite spin projections are counter-propagating for the vanishing net magnetization in CrSb, so there is no net AHE for the current injection from spin-unpolarized gold contacts 1 8 , as we corfirm for the reference samples. In contrast, spin injection from the ferromagnetic Ni contacts leads to the charge imbalance between counter-propagating sur- face states and, therefore, to the anomalous Hall effect in Fig. 2. The effect is observable for macroscopic dis- tances due to the topological protection of charge trans- port within Weyl surface states. The hysteresis width in Fig. 2 should be connected with Ni contact remagnetiza- tion, i.e., with the change of the spin polarization of the injected carriers. However, this picture is only valid for the directions of full spin compensation in bulk altermagnetic spec- trum. For other directions, AHE is affected by periodic bulk altermagnetic magnetization 5 8 , which leads to the direction-dependent sign of the AHE slope in Fig. 2. In this case, the bow-tie hysteresis loops in Fig. 2 is the re- sult of interaction 5 5,56 between the bulk alternating spin splitting and the surface spin polarization from the topo- logical surface states 4 3,44 . Thus, the AHE can be ex- plained as the joint effect of the k-dependent bulk alter- magnetic spin polarization and spin-polarized topological surface states for the spin-polarized charge carriers in the altermagnetic candidate CrSb. Similar mechanism of spin injection is valid for the non- linear Hall effect in Figs. 3 and 4. In general, NLHE 2 5–29 arises from the Berry curvature dipole in momentum space 2 0 . One can not expect non-zero Berry curvature dipole for centrosymmetric CrSb, which is confirmed by negligible V 2ω x y for the reference sample with gold contacts in Fig. 3. However, spin injection leads to the charge im- balance between the topological surface states, and, in order, to finite Berry curvature dipole due to the bulk-
5 boundary correspondence, which appears as finite V 2ω x y transverse voltage in Figs. 3 and 4. Another possibility to have finite second-harmonic re- sponse is proposed in Ref. 5 9 . The flow of electric current in an altermagnet could result in the formation of homo- geneous electron spin orientation in the sample, which is quadratic in the current magnitude and does not re- quire broken inversion symmetry. In principle, the elec- tron spin orientation could be detected by ferromagnetic contacts, however, the effect should be observed primary for the xx- contact configuration. In contrast, the lon- gitudinal second-harmonic voltage V 2ω x x is negligible in comparison with the Hall V 2ω x y one in Fig. 4 (a). The second-order V 2ω x y (B) NLHE response is sensitive to the external magnetic field 6 0,61 . Being determined by spin injection from the ferromagnetic Ni contact, NLHE voltage shows hysteresis within the same field range in Fig. 4 (b), as the first-harmonic AHE response in Fig. 2. It is worth mentioning, that there is no field dependence for the reference sample, as it is shown in the inset to Fig. 4 (b). Also, NLHE should be affected by the bulk al- ternating spin splitting 5 8 similarly to AHE, which might be responsible for very different V 2ω x y values for samples in Figs. 3 and 4 (a). V. CONCLUSION As a conclusion, we experimentally investigate spin- polarized electron transport for the centrosymmetric al- termagnet CrSb, which is known to reveal both alter- magnetic and topological features. We demonstrate pro- nounced anomalous Hall and second-harmonic non-linear Hall effects for a thin single-crystal CrSb flake with fer- romagnetic nickel contacts, while they can not be seen for the reference samples with non-magnetic gold ones. For the anomalous Hall effect, we demonstrate bow-tie hysteresis in Hall voltage, which is usually ascribed to surface spin textures in magnetic materials. The slope of the Hall curve changes a sign for two Hall bar orientations for the same sample, i.e. for the same sign of the charge carriers. We interpret the observed sign inversion and bow-tie hysteresis as the joint effect of the k-dependent bulk altermagnetic spin polarization and spin-polarized topological surface states for the altermagnetic candidate CrSb. The pronounced non-linear Hall effect with hys- teresis in magnetic field confirms finite Berry curvature dipole under injection of spin-polarized electrons in al- termagnet CrSb. ACKNOWLEDGMENTS We wish to thank S.S Khasanov for X-ray sample characterization and Vladimir Zyuzin for valuable dis- cussions. We gratefully acknowledge financial support by the Russian Science Foundation, project RSF-26-12- 00001, https://rscf.ru/project/26-12-00001/ 1 Libor ˇ Smejkal, Jairo Sinova, and Tomas Jungwirth, ”Be- yond Conventional Ferromagnetism and Antiferromag- netism: A Phase with Nonrelativistic Spin and Crystal Ro- tation Symmetry”, Phys. Rev. X 12, 031042 (2022) DOI: 10.1103/PhysRevX.12.031042 2 Libor ˇ Smejkal, Jairo Sinova and Tomas Jungwirth, ”Emerging Research Landscape of Altermagnetism”, Phys. Rev. X 12, 040501 (2022). DOI: 10.1103/Phys- RevX.12.040501 3 Igor Mazin, ”Altermagnetism—A New Punch Line of Fun- damental Magnetism”, Phys. Rev. X 12, 040002 (2022); 10.1103/PhysRevX.12.040002 4 Sang-Wook Cheong, Fei-Ting Huang, ”Altermagnetism with non-collinear spins”, npj Quantum Mater. 9, 13 (2024) https://doi.org/10.1038/s41535-024-00626-6 5 J. Krempasky, L. ´ Smejkal, S.W. D’Souza, M. Hajlaoui, G. Springholz, K. Uhl´ıˇrov´a, F. Alarab, P. C. Constanti- nou, V. Strocov, D. Usanov, W. R. Pudelko, R. Gonz´alez- Hern´andez, A. Birk Hellenes, Z. Jansa, H. Reichlov´a, Z. ˇ Sob´aˇn, R. D. Gonzalez Betancourt, P. Wadley, J. Sinova, D. Kriegner, J. Min´ar, J. H. Dil and T. Jungwirth, ”Alter- magnetic lifting of Kramers spin degeneracy”, Nature 626, 517 (2024) https://doi.org/10.1038/s41586-023-06907-7 6 R. Jaeschke-Ubiergo, V. K. Bharadwaj, T. Jung- wirth, L. ˇ Smejkal, and J. Sinova, ”Supercell al- termagnets”, Phys. Rev. B 109, 094425 (2024) https://doi.org/10.1103/PhysRevB.109.094425 7 Igor I. Mazin ”Notes on altermagnetism and superconduc- tivity” arxiv:2203.05000 8 Sachchidanand Das, Dhavala Suri, Abhiram Soori, ”Trans- port across junctions of altermagnets with normal metals and ferromagnets” J. Phys. : Condens. Matter 35, 435302 (2023), https://doi.org/10.1088/1361-648X/acea12 9 Jabir Ali Ouassou, Arne Brataas, Jacob Lin- der, ”dc Josephson Effect in Altermagnets”, Physical Review Letters 131, 076003 (2023); https://doi.org/10.1103/PhysRevLett.131.076003 10 Z. Feng, X. Zhou, L. ´ Smejkal, L. Wu, Z. Zhu, H. Guo, R. Gonz´alez-Hern´andez, X. Wang, H. Yan, P. Qin, X. Zhang, H. Wu, H. Chen, Z. Xia, C. Jiang, M. Coey, J. Sinova, T. Jungwirth, and Z. Liu,”An anomalous Hall effect in altermagnetic ruthenium dioxide”, Nat. Electron. 5, 735 (2022). 11 R.D. Gonzalez Betancourt, J. Zubac, R. Gonzalez- Hernandez, K. Geishendorf, Z. Soban, G. Springholz, K. Olejnik, L. Smejkal, J. Sinova, T. Jungwirth, S.T.B. Goen- nenwein, A. Thomas, H. Reichlova, J. Zelezny, and D.
6 Kriegner, ”Spontaneous Anomalous Hall Effect Arising from an Unconventional Compensated Magnetic Phase in a Semiconductor” Phys. Rev. Lett. 130, 036702 (2023). 12 K. P. Kluczyk, K. Gas, M. J. Grzybowski, P. Skupi´nski, M. A. Borysiewicz, T. Fas, J. Suffczy´nski, J. Z. Domagala, K. Grasza, A. Mycielski, M. Baj, K. H. Ahn, K. V´yborn´y, M. Sawicki, M. Gryglas-Borysiewicz, ”Coexistence of anoma- lous Hall effect and weak magnetization in a nominally collinear antiferromagnet MnTe” Physical Review B 110, 155201 (2024) 13 Miina Leivisk¨a, Javier Rial, Anton´ın Badura, Rafael Lopes Seeger, Ismaila Kounta, Sebastian Beckert, Do- minik Kriegner, Isabelle Joumard, Eva Schmoranzerov´a, Jairo Sinova, Olena Gomonay, Andy Thomas, Sebastian T.B. Goennenwein, Helena Reichlov´a, Libor ´ Smejkal, Lisa Michez, Tom´a´s Jungwirth, Vincent Baltz ”Anisotropy of the anomalous Hall effect in thin films of the altermagnet candidate Mn5Si3” Phys. Rev. B 109, 224430 (2024) 14 H. Reichlova, R. L. Seeger, R.Gonz´alez-Hern´andez, I. Kounta, R. Schlitz, D. Kriegner, M. Lammel, V. Petˇriˇcek, Ph. Ritzinger, M. Leivisk¨a, A. B. Hellenes, P. Doleˇzal, L. Horak, , K. Olejn´ık, E. Schmoranzerova, S. Bertaina, A. Thomas, A. Badura, L. Michez, J. Sinova, S. T. B. Goen- nenwein, V. Baltz, T. Jungwirth and L. ˇ Smejkal, ”Obser- vation of a spontaneous anomalous Hall response in the Mn 5 S i 3 d-wave altermagnet candidate”, Nature Commu- nications Vol 15, 4961 (2024) 15 I. Dzyaloshinsky, ”A thermodynamic theory of “weak” fer- romagnetism of antiferromagnetics” Journal of Physics and Chemistry of Solids 4, 241 (1958). 16 B.A. Volkov and O.A. Pankratov, ”Two-dimensional mass- less electrons in an inverted contact”, JETP Letters, 42, 178 (1985) 17 M. Z. Hasan and C. L. Kane, ”Colloquium: Topological insulators”, Reviews of Modern Physics, Vol. 82, pp. 3045– 3067 (2010) 18 N. P. Armitage, E. J. Mele, and Ashvin Vishwanath, ”Weyl and Dirac semimetals in three-dimensional solids” Rev. Mod. Phys. 90, 15001 (2018) 19 B. Ghosh, D. Mondal, C.-N. Kuo, C. S. Lue, J. Nayak, J. Fujii, I. Vobornik, A. Politano, and A. Agarwal, ”Observa- tion of bulk states and spin-polarized topological surface states in transition metal dichalcogenide Dirac semimetal candidate NiTe2”, Phys. Rev. B 100, 195134 (2019). 20 Inti Sodemann, Liang Fu., ”Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invari- ant materials” Phys. Rev. Lett. 115, 216806 (2015) 21 E. Deyo, L. E. Golub, E. L. Ivchenko, and B. Spivak, ”Semiclassical theory of the photogalvanic effect in non- centrosymmetric systems” arXiv:0904.1917 (2009). 22 L.E. Golub, E.L. Ivchenko, B.Z. Spivak, ”Photocurrent in gyrotropic Weyl semimetals” JETP Letters, 105, 782 (2017) 23 J. E. Moore and J. Orenstein, ”Confinement-induced Berry phase and helicity-dependent photocurrents” Phys. Rev. Lett., 105, 026805 (2010). 24 T. Low, Y. Jiang, and F. Guinea, ”Topological currents in black phosphorus with broken inversion symmetry” Phys- ical Review B 92, 235447 (2015). 25 Qiong Ma, Su-Yang Xu, Huitao Shen, David MacNeill, Valla Fatemi, Tay-Rong Chang, Andr´es M. Mier Val- divia, Sanfeng Wu, Zongzheng Du, Chuang-Han Hsu, Shi- ang Fang, Quinn D. Gibson, Kenji Watanabe, Takashi Taniguchi, Robert J. Cava, Efthimios Kaxiras, Hai-Zhou Lu, Hsin Lin, Liang Fu, Nuh Gedik and Pablo Jarillo- Herrero, ”Observation of the nonlinear Hall effect un- der time-reversal-symmetric conditions” Nature 565, 337 (2019). 26 K. Kang, T. Li, E. Sohn, J. Shan, and K. F. Mak, ”Non- linear anomalous Hall effect in few-layer WTe2” Nature Mater. 18, 324 (2019). 27 O. O. Shvetsov, V. D. Esin, A. V. Timonina, N. N. Kolesnikov, and E. V. Deviatov, ”Nonlinear Hall effect in three-dimensional Weyl and Dirac semimetals” JETP Let- ters, 109, 715 (2019). DOI: 10.1134/S0021364019110018 28 A. Tiwari, F. Chen, Sh. Zhong, E. Drueke, J. Koo, A. Kaczmarek, C. Xiao, J. Gao, X. Luo, Q. Niu, Y. Sun, B. Yan, L. Zhao and A. W. Tsen, ”Giant c-axis nonlinear anomalous Hall effect in Td-MoTe2 and WTe2” Nat. Com- mun. 12, 2049 (2021). https://doi.org/10.1038/s41467-021- 22343-5 29 N. N. Orlova, A. V. Timonina, N. N. Kolesnikov, and E. V. Deviatov, ”Gate-Dependent Nonlinear Hall Ef- fect at Room Temperature in Topological Semimetal GeTe”, Chinese Physics Letters 40, 077302 (2023) https://doi.org/10.1088/0256-307X/40/7/077302 30 Merce Roig, Yue Yu, Rune C. Ekman, Andreas Kreisel, Brian M. Andersen, Daniel F. Agterberg, ”Quasi- symmetry Constrained Spin Ferromagnetism in Altermag- nets” Phys. Rev. Lett. 135, 016703 (2025) 31 Chao Chen Ye, Karma Tenzin, Jagoda Sl awi´nska, Carmine Autieri, ”Dominant orbital magnetization in the prototyp- ical altermagnet MnTe”, Phys. Rev. B 113, 014413 (2026) 32 Yufei Zhao, Yiyang Jiang, Kamal Das, Chao-Xing Liu, Binghai Yan, ”Residual orbital magnetization governs the anomalous Hall effect in altermagnets”, arXiv:2606.25999 33 T. Osumi, S. Souma, T. Aoyama, K. Yamauchi, A. Honma, K. Nakayama, T. Takahashi, K. Ohgushi, and T. Sato,”Observation of Giant Band Splitting in Altermag- netic MnTe”, Phys. Rev. B 109, 115102 (2024) 34 N.N. Orlova, A.A. Avakyants, A.V. Timonina, N.N. Kolesnikov, and E.V. Deviatov, ”Crossover from rela- tivistic to non-relativistic net magnetization for MnTe al- termagnet candidate”, JETP Letters, 120, 360 (2024). https://doi.org/10.1134/S0021364024602926 35 N. N. Orlova, V. D. Esin, A. V. Timonina , N. N. Kolesnikov and E. V. Deviatov, ”Magnetization symmetry for the altermagnetic candidate MnTe”, Phys. Rev. B 111, 224414 (2025) DOI: https://doi.org/10.1103/br1r-bjzk 36 Satoru Hayami and Hiroaki Kusunose, ”Essential role of the anisotropic magnetic dipole in the anomalous Hall ef- fect” Phys. Rev. B 103, L180407 (2021). 37 M. Hajlaoui, S.W. D’Souza, L. Smejkal, D. Kriegner, G. Krizman, T. Zakusylo, N. Olszowska, O. Caha, J. Michaliˇcka, A. Marmodoro, K. V´yborn´y, A. Ernst, M. Cinchetti, J. Minar, T. Jungwirth, G. Springholz, ”Tem- perature Dependence of Relativistic Valence Band Split- ting Induced by an Altermagnetic Phase Transition” Adv. Mater. 36, 2314076 (2024) 38 A. Hariki, A. Dal Din, O. J. Amin, T. Yamaguchi, A. Badura, D. Kriegner, K. W. Edmonds, R. P. Campion, P. Wadley, D. Backes, L. S. I. Veiga, S. S. Dhesi, G. Springholz, L. Smejkal, K. Vyborny, T. Jungwirth, J. Kunes, ”X-Ray Magnetic Circular Dichroism in Altermag- netic α-MnTe” Phys. Rev. Lett. 132, 176701 (2024) 39 Sonka Reimers, Lukas Odenbreit, Libor Smejkal, Vladimir N. Strocov, Procopios Constantinou, Anna Birk Hellenes, Rodrigo Jaeschke Ubiergo, Warlley H. Campos, Venkata
7 K. Bharadwaj, Atasi Chakraborty, Thiboud Denneulin, Wen Shi, Rafal E. Dunin-Borkowski, Suvadip Das, Mathias Kl¨aui, Jairo Sinova, Martin Jourdan, ”Direct observation of altermagnetic band splitting in CrSb thin films”, Nat Commun 15, 2116 (2024). https://doi.org/10.1038/s41467- 024-46476-5 40 G. Yang, Zh. Li, S. Yang, J. Li, H. Zheng, W. Zhu, Z. Pan, Y. Xu, S. Cao, W. Zhao, A. Jana, J. Zhang, M. Ye, Yu Song, L.-H. Hu, L. Yang, J. Fujii, I. Vobornik, M. Shi, H. Yuan, Y. Zhang, Y. Xu and Y. Liu, ”Three-dimensional mapping of the altermagnetic spin splitting in CrSb”, Na- ture Communications Vol. 16, 1442, pp. 1 (2025) 41 Daniil S. Antonenko, Rafael M. Fernandes, Jorn W. F. Venderbos, ”Mirror Chern Bands and Weyl Nodal Loops in Altermagnets”, Phys. Rev. Lett. 134, 096703 (2025) 42 Rafael M. Fernandes, Vanuildo S. de Carvalho, Turan Birol, Rodrigo G. Pereira, ”Topological transition from nodal to nodeless Zeeman splitting in altermagnets”, Phys. Rev. B 109, 024404 (2024). 43 Cong Li, Mengli Hu, Zhilin Li, Balasubramanian Thia- garajan, Yang Wang, Wanyu Chen, Mats Leandersson, Craig Polley, Cosma Fulga, Maia G. Vergniory, Oleg Jan- son, Timur Kim, Oscar Tjernberg, Jeroen van den Brink, ”Topological Weyl altermagnetism in CrSb”, Communica- tions Physics 8, 311, pp. 1 (2025) 44 Wenlong Lu, Shiyu Feng, Yuzhi Wang, Dong Chen, Zihan Lin, Xin Liang, Siyuan Liu, Wanxiang Feng, Kohei Ya- magami, Junwei Liu, Claudia Felser, Quansheng Wu and Junzhang Ma, ”Signature of Topological Surface Bands in Altermagnetic Weyl Semimetal CrSb”, Nano Letters, Vol 25, 18, pp. 7343, (2025) 45 Koki Mizuno ”High-harmonic spin-current signatures of al- termagnetic spin-group symmetry” arXiv:2606.31573 46 A. I. Snow, ”Neutron diffraction investigation of the atomic magnetic moment orientation in the antiferromagnetic compound CrSb”, Phys. Rev. 85, 365 (1952) 47 W. J. Takei, D. E. Cox, and G. Shirane, ”Magnetic Struc- tures in the MnSb-CrSb System”, Physical Review, 129, No. 5, pp. 2008–2018 (1963) 48 V.D. Esin, D.Yu. Kazmin, Yu.S. Barash, A.V. Tim- onina, N.N. Kolesnikov, and E.V. Deviatov, ”Joseph- son diode and spin-valve effects on the surface of al- termagnet CrSb”, JETP Letters, 123, 556 (2025) DOI: 10.1134/S0021364026600667 49 D.Yu. Kazmin, V.D. Esin, Yu.S. Barash, A.V. Ti- monina, N.N. Kolesnikov, E.V. Deviatov, ”An- dreev reflection for MnTe altermagnet candidate”, Physica B: Condensed Matter, 696, 416602 (2025) https://doi.org/10.1016/j.physb.2024.416602 50 O. O. Shvetsov, V. D. Esin, A. V. Timonina, N. N. Kolesnikov, and E. V. Deviatov, ”Surface superconductiv- ity in a three-dimensional Cd3As2 semimetal at the inter- face with a gold contact” Phys. Rev. B 99, 125305 (2019) 51 O. O. Shvetsov, V. D. Esin, A. V. Timonina, N. N. Kolesnikov, E. V. Deviatov, ”Multiple magnon modes in the Co3Sn2S2 Weyl semimetal candidate”, EPL, 127, 57002 (2019) 52 V. D. Esin, D. N. Borisenko, A. V. Timonina, N. N. Kolesnikov, and E. V. Deviatov, ”Spin-dependent trans- port through a Weyl semimetal surface” Phys. Rev. B 101, 155309 (2020) 53 N. N. Orlova, N. S. Ryshkov, A. A. Zagitova, V. I. Kulakov, A. V. Timonina, D. N. Borisenko, N. N. Kolesnikov, and E. V. Deviatov, ”Band gap reconstruction at the interface between black phosphorus and a gold electrode” Phys. Rev. B 101, 235316 (2020) 54 Guoxin Zheng, Arjyama Bordoloi, Mingjun Fan, Shun- suke Kitou, Hiraku Saito, Taro Nakajima, Sobhit Singh, Takashi Kurumaji, and Linda Ye, ”Dominant in-plane anomalous Hall effect in a monoclinic room-temperature ferromagnet”, arxiv:2606.10063 55 Felipe Tejo, Denilson Toneto, Sim´on Oyarz´un, Jos´e Her- mosilla, Caroline S. Danna, Juan L. Palma, Ricardo B. da Silva, Lucio S. Dorneles, and Juliano C. Denardin, ”Stabi- lization of magnetic skyrmions on arrays of self-assembled hexagonal nanodomes for magnetic recording applications” ACS Appl. Mater. Interfaces, 12, 47, 53454 (2020). 56 A.A. Avakyants, N.N. Orlova, A.V. Timonina, N.N. Kolesnikov, E.V. Deviatov, ”Evidence for surface spin structures from first order reversal curves in Co3Sn2S2 and Fe3GeTe2 magnetic topological semimetals”, Journal of Magnetism and Magnetic Materials, 573, 170668 (2023), https://doi.org/10.1016/j.jmmm.2023.170668. 57 Tianye Yu, Ijaz Shahid, Peitao Liu, Ding-Fu Shao, Xing- Qiu Chen and Yan Sun, ”N´eel vector-dependent anomalous transport in altermagnetic metal CrSb” npj Quantum Ma- terials 10, 47 (2025). https://doi.org/10.1038/s41535-025- 00766-3 58 N.N. Orlova, A.A. Avakyants, V.D. Esin, A.V. Timonina, N.N. Kolesnikov, E.V. Deviatov, ”Altermagnetic bulk and topological surface magnetizations for CrSb single crys- tals”, arXiv:2512.11344 59 L. E. Golub, L. ˇ Smejkal ”Spin orientation by electric cur- rent in altermagnets”, arXiv:2503.12203 60 Debottam Mandal, Kamal Das, Amit Agarwal, ”Chiral anomaly and nonlinear magnetotransport in time rever- sal symmetric Weyl semimetals” Phys. Rev. B 106, 035423 (2022). 61 A. A. Zyuzin and A. Yu. Zyuzin, ”Chiral anomaly and second-harmonic generation in Weyl semimetals” Phys. Rev. B 95, 085127 (2017). DOI: 10.1103/Phys- RevB.95.085127