Magnetic field effect on the chiral magnetism of noncentrosymmetric UPtGe:
experiment and theory
The effect of differently oriented magnetic field on chiral incommensurate helimagnet UPtGe is studied both experimentally and theoretically. The magnetization measurements up to the field above the saturation have revealed an isotropic magnetic response below 20 T and a remarkable nonmonotonic anisotropy in high fields. Moreover, the two principally different phase transitions from the noncollinear incommensurate to the field-induced ferromagnetic state have been observed. These properties are successfully explained by density-functional theory calculations taking into account the noncollinearity of the magnetic structures, arbitrary directed magnetic field, and relativistic effects. We also estimate the strength of different competing magnetic interactions and discuss possible scenarios of the field-induced phase transformations.
Uranium intermetallic compounds show a wide variety of electronic properties, owing to the delicate balance of the hybridization between U 5 electrons and conduction electrons of ligand elements, and competing interaction energy scales Sechovsky1998 (). One of the intriguing phenomena observed in uranium based compounds is the coexistence of ferromagnetism (FM) and superconductivity Aoki2001 (); Huy2007 (); Aoki2012 (). Applying magnetic fields reveals new unexpected phenomena of these materials: e.g., the reentrance of the superconductivity is induced in URhGe when the magnetic field of 12 T is applied along the hard magnetization axis initiating spin-reorientation transition Levy2005 (). Obviously, the strength of the magnetic anisotropy (MA) plays an important role in the formation of such intriguing physical phenomena. In contrast to the Ising-like ferromagnet URhGe, UPtGe is a unique U system showing a chiral incommensurate helical magnetic ordering below 51 K Mannix2000 (). In the model of the helix suggested in Ref. Sandratskii2001 () the MA was considered to be negligibly small. The spin dynamics was recently studied via the NMR experiments, and the XY-type spin fluctuations were clarified Tokunaga2018 (). The chiral helical structures in various types of materials have been attracting strong research interest because of their importance in the physics of skyrmion lattices and chiral domains, topics of intense study in the field of spintronics Muhlbauer2009 (); Heinze2011 (); Bergmann2014 ().
A number of neutron diffraction studies lead to the same conclusion that the magnetic ground state of UPtGe is an incommensurate cycloid propagating along the axis with the wave vector in units of ; the U moments lie in the plane Szytula1992 (); Kawamata1992 (); Robinson1993 (); Mannix2000 (). Importantly, Mannix et al. Mannix2000 () clarified that the orthorhombic crystal structure of UPtGe is of the noncentrosymmetric EuAuGe type (, space group No. 44) (Fig. 1), which is different from the centrosymmetric TiNiSi type, such as of URhGe, or CeCu type assumed in previous studies Szytula1992 (); Kawamata1992 (); Robinson1993 (); Hoffmann2001 (); comment_lattice (). Another important property revealed in the experiment is a chiral character of the cycloid Mannix2000 (): All domains have the same wave vector whereas the domains with opposite wave vector are absent.
Previously proposed explanation Sandratskii2001 () of the origin of the incommensurate cycloid in UPtGe includes the following components: (i) a very small MA in the cycloid plane, which was treated as negligible, (ii) competing interatomic exchange interactions, and (iii) an active Dzyaloshinskii-Moriya interaction (DMI) due to the lack of the inversion symmetry. The first two features explain the formation of the incommensurate cycloid, and the third is responsible for the chiral character of the magnetic ground state. An important conclusion of Ref. Sandratskii2001 () is a critical sensitivity of the magnetic structure to the crystal lattice comment_lattice_sens ().
The study of the response of UPtGe to the magnetic field is expected to deepen the understanding of chiral incommensurate magnetism and of the field induced incommensurate-commensurate (I-C) phase transitions, which are fascinating phenomena of solid state physics. Despite the long history of the studies on this topic (see, e.g, early reviews Bak82 (); Izyumov1984 ()) the understanding of the I-C transitions is by far not complete. There are some exact statements, which are based on very simple theoretical models, like sine Gordon equation Dzyaloshinskii1964 (); Togawa2016 (). The applicability of these models to complex real materials is not self-evident. There are also more complex theoretical models solved numerically, e.g. atomistic spin Hamiltonians Jensen1996 (). However, such models are sensitive to the values of a large number of parameters whose choice is not unique.
We report a joint experimental and theoretical study of the magnetic field effect on UPtGe. The magnetization measurements are performed in fields up to 56 T that are above the saturation field for field directions in the cycloid plane. The isotropic behavior observed in low fields is replaced at higher fields by anisotropic field-induced phase transitions. This raises new important questions that we address on the basis of the density-functional theory (DFT) calculations. One of the focuses of our attention is the complex interplay of various interactions responsible for the unusual physical properties of UPtGe. Other focus is the field-induced phase transitions. Our experiment reveals the sequence of two very different phase transformations bringing the system from the chiral incommensurate helical state to the field-induced ferromagnetic state comment_field_induced_FM ().
Single crystals of UPtGe were prepared using the Czochralski pulling method in a tetra-arc furnace. Pulsed magnetic fields were generated using nondestructive magnets installed at the International MegaGauss Science Laboratory of the Institute for Solid State Physics at the University of Tokyo. The conventional induction method using coaxial pick-up coils was used for the magnetization measurements down to 1.4 K.
Iii Results of experiment
Magnetization curves for various field directions are presented in Fig. 2. The anisotropy between the plane and out-of plane axis is very large. For , the magnetization is linear without any anomalies and much smaller than for the field in the plane. In the following, we focus on the field directions in the cycloidal plane. Angle defining field direction is measured from the orthorhombic axis. Below 20 T, the magnetization is nearly isotropic. This isotropic magnetic response seems to be expected in connection with the assumed negligibly small in-plane MA Sandratskii2001 (). By contrast, two remarkable anisotropic increments of magnetization are seen at about 25 T and 42 T (Fig. 2) revealing the presence of considerable MA in the plane. The observed high-field anisotropy is nonmonotonic with respect to the field direction. Indeed, the saturated magnetic moment is maximal for the field parallel to the axis (). It strongly drops for , and then increases again for and ( axis) (Fig. 2). For , magnetization shows upturn near the maximum fields but does not saturate up to the maximum fields.
Iv Method of calculation
To understand the unusual angular dependence of the in-plane MA, we performed the calculation of the energy of the FM configuration for different directions of the atomic moments with respect to the crystallographic axes. The calculations were carried out with the augmented spherical waves code Williams1979 (); Eyert2012 () able to deal with noncollinear magnetism, spin-orbit coupling, and magnetic field along an arbitrary direction Sandratskii1998 (); Sandratskii2016 (). Since the localization of the 5 states can be underestimated in standard DFT calculations, we performed additional calculations introducing a scaling parameter to study the sensitivity of the results to the level of the 5-states localization. This parameter is used as a scaling factor for interatomic Hamiltonian and overlap integrals of U 5 wave functions entering the secular matrix. The exchange-correlation potential was used in the local density approximation (LDA) Barth1972 (). The k-vector sampling suggested by Monkhorst and Pack Monkhorst1976 () was employed in the integration over the Brillouin zone (BZ). The description of the crystal structure of the EuAuGe type and lattice parameters of UPtGe determined in the neutron diffraction experiment can be found in Ref. Mannix2000 (). In the calculations for orthorhombic unit cell containing four U atoms the number of the k points in the BZ varied between 8000 and 27000. For larger unit cells the number of the k points decreased in accordance with decreasing BZ volume. In very long-lasting calculations of self-consistent magnetic structures in large supercells the reduced numbers of the k points were used.
V Results of calculations and discussion
A prominent feature of the theoretical dependence of the magnetic anisotropy energy, ) [Fig. 3(a)] is its nonmonotonic character, which is consistent with the curves (Fig. 2). An insight into the origin of the nonmonotonic ) is provided by the consideration of the dependence of the orbital moments of the two inequivalent U sublattices [see Fig. 3(b)], since there is deep physical connection between MA energy and orbital moments anisotropy (see, e.g., Refs. Bruno1989 (); Sandratskii2015 ()). The calculations gave the remarkable result that the orbital moments of the two U sublattices, though both monotonic functions of , have opposite character: decrease for the A sublattice and increase for the B sublattice. The competition of two opposite angular dependences explains unexpected properties of the MA of UPtGe: its unusual smallness for U compounds and the nonmonotonic behavior. The calculations with reduced overlap of the 5 functions scaled with parameter show that the nonmonotonic behavior of the ) is a robust property [Fig. 3(a)]. For the stronger scaling with parameter , the nonmonotonic features become weak.
After we have revealed the presence of sizable in-plane MA for the FM structures, we would like to reveal the origin of the isotropic response below 20 T (Fig.2). A direct DFT-calculation of the response of an incommensurate magnetic structure to differently oriented magnetic field is not feasible because of an infinite magnetic unit cell of the incommensurate structure. Therefore, we performed the following calculations to address this problem. We selected two parts of the helix with different directions of the atomic moments with respect to the crystal lattice [see Fig. 4(a)] and described these pieces with the supercells corresponding to commensurate helix with . In the first supercell, the initial directions of the atomic moments are collinear to the and axes [Fig. 4(b)]. In the second, the moments were rotated by 45 [Fig. 4(c)]. Next we calculate the magnetic response to the field of 23.5 T comment_23T () applied along different directions. We obtained two opposite monotonic dependencies of the induced moment on the field direction for two supercells [Fig. 4(d)]. This property explains the isotropy of the response of the cycloid to the applied magnetic field below 20 T as the result of the averaging of the anisotropic responses of different parts of the cycloid.
It is worth noting that the contributions of the orbital and spin moments to the induced moment shown in Fig. 4(d) have opposite signs: positive for the orbital moment and negative for the spin moment. The ratio of the magnitudes of the induced orbital and spin moments varies for the points of the ‘average’ line in the interval between 2.2 and 2.4. The fact that the induced spin moment is opposite to the direction of the magnetic field reveals stronger influence of the third Hund’s rule than the direct influence of the Zeeman coupling to the field (see also Ref. Sandratskii2016 ()).
It is important to compare the energy scales of different magnetic interactions, i.e., interatomic exchange interaction, DMI, and Zeeman energy. First, we estimate the interatomic exchange parameters. As a reference state of the system, we used the FM configuration with atomic moments parallel to the axis. To estimate the exchange interaction parameter between atoms and in Fig. 1(a), we evaluated the energies of the magnetic configurations with the moments of atoms and deviated in the plane by angle 10 in the same and opposite directions (see Fig. 5). The difference of these energies estimates the exchange energy corresponding to the angle 20 between moments of the atoms and . By dividing the energy by , we obtain exchange parameter . The values of the calculated exchange parameters are listed in Table 1.
|A1 , B1||0.275||0.821||1.373|
|A1 , A2||-0.013||-0.014||-0.014|
|A1 , B2||-0.076||0.009||0.139|
|A1 , A3||-0.202||-0.239||0.061|
|A1 , B3||-0.047||-0.105||-0.123|
|B1 , B2||-0.058||-0.017||-0.0257|
|B1 , B3||-0.014||0.061||0.082|
For unscaled calculation (), only exchange parameter is FM. It corresponds to the interaction between neighboring atoms of the zigzag chain (Fig. 1). This interaction is the strongest among all estimated interatomic exchange interactions. This result supports the picture of ferromagnetically ordered zigzag chains Kawamata1992 (); Robinson1993 (); Mannix2000 (). The decrease of the overlap given by , on one hand, increases the values of atomic moments and, on the other hand, diminishes the quantum-mechanical interaction integrals. Therefore, there is no way to relate the change in the electronic overlap to the character of the variation of interatomic exchange interaction. Indeed, the analysis of exchange parameters presented in Table 1 shows that there is no general trend in the variation of the exchange interactions with decreasing scaling parameter. For the calculation without scaling, all exchange parameters with the exception of the interchain one are antiferromagnetic (AFM) that leads to the frustration and canting of the atomic moments. The strongest AFM interaction is [Fig.1(a)], while is remarkably weak. For scaling factor the values of parameters are in good correlation with those for unscaled calculations. For stronger scaling with the exchange interactions become distinctly more FM.
The chiral magnetism of UPtGe essentially arises from the DMI Sandratskii2001 (). The strength of DMI is calculated as the difference of the energies of the cycloids with = and = and has the following values: 0.11 mRy/U for unscaled calculation, 0.21 mRy/U for and 0.22 mRy/U for . This estimation shows that the energy scale of the DMI is of the same order of magnitude as the exchange interaction. We also obtained considerable dependence of the DMI strength on the localization parameter.
Finally, we estimate the scale of the Zeeman energy. The orientation of atomic moments parallel to the magnetic field gives the energy gain of where is the value of the atomic magnetic moment per U atom. At 40 T and we obtain the value of 0.21 mRy/U, which is close to the values of the AFM exchange interactions (see Table 1).
As shown above the calculations give a complex balance of several interactions. The chiral magnetic ground state of UPtGe comes from competing exchange interactions and contributions of the DMI and MA. In applied magnetic fields, the Zeeman energy overcomes these interactions, leading to the FM transformation.
Now we turn to the discussion of the two field-induced phase transitions (Fig. 2). The applicability of the direct DFT calculations to the description of these phase transitions is rather limited comment_limitation (). Nevertheless, it is instructive to perform the following calculations. We consider supercells of different moderate sizes and start the iterational process with various accidentally chosen magnetic configurations. The magnetic moments are allowed to relax to a self-consistent state. On the next step, we apply the magnetic field and consider the change of the magnetic configurations. The calculations with self-consistently determined directions of the magnetic moments were performed as follows. In the first step, the calculations were carried out with a relatively small number of points in the BZ that varied from about 200 for the supercell containing eight U atoms to about 50 for the supercell with 16 U atoms. When the convergence of the directions of all U moments reached 0.1 the number of points was approximately doubled and the calculations continued until the convergence of the directions is better than 0.01 comment_involvedCalc ().
The analysis of performed calculations shows that starting from different initial magnetic configurations we generally obtain different self-consistent magnetic states. Among the self-consistent states there are both magnetically compensated [see examples in Figs. 6(a), 6(b), and 6(c)] and uncompensated states [Figs. 6(d) and 6(e)]. The fact that the calculations give multiple self-consistent states indicates the presence of numerous local minima in the complex high-dimensional energy landscape describing the energy of the system as a function of the directions of atomic moments.
Figures 6(f), 6(g), and 6(h) show the result of the calculations with external field of 100 T. In Figs. 6(f) and 6(g), we present the transformation of the compensated magnetic structure (c) in the fields directed along the and axes, respectively. For both field directions there is the modification of the magnetic structure resulting in an induced magnetic moment. The response to the magnetic field is anisotropic. The induced net spin, orbital and total magnetic moments along field direction for structure (f) are, respectively, , 7.60, and 3.92 per supercell. For structure (g) these values are larger: , 12.37, and 5.45 . The energies of the in-field structures (f) and (g) are lower than the energy of the compensated structure (c) by 6.2 and 3.1 mRy, respectively. The decreases in the Zeeman energy are, however, only 2.0 and 2.7 mRy per supercell. This reveals that self-consistent response of the electron system to the applied field is considerably more complex than a rotation of the rigid atomic moments.
Figure 6(h) shows the transformation of the uncompensated magnetic structure (e) with the field along the axis. Interestingly, in this case the modification of the magnetic structure is especially strong and the in-field state is rather close to the FM state. The numerical results give the following values: the differences of the net spin, orbital, and total moments of the structures (e) and (h) are , 16.3, and 7.0 per supercell. The energy of structure (h) is lower than that of structure (e) by 5.9 mRy, whereas the difference of the Zeeman energies of the two structures is only 3.5 mRy per supercell.
We remark that the magnetic field used in the calculations presented in Figs. 6(f), (g) and (h) is about two times larger than the maximal experimental field. This large field was selected to make the influence of the field clearly visible in the figures. The property that this field does not lead in all cases to the magnetic structure close to ferromagnet is explained as follows. The energy of the system as a function of the directions of the atomic moments is a very complex unknown function with many local minima and barriers separating these minima. It is to be expected that these barriers are often higher than the characteristic Zeeman energies even for large magnetic fields. In the DFT calculations we deal with electron interactions that are of larger scale. Since in the calculations there are no fluctuations that can be responsible for the tunneling of the system from a local minimum to a deeper minimum the system remains near one of the quasistable states. It is also important that we can perform calculations only for relatively small supercells whereas in the I-C transitions the intermediate structures with larger period can be important. This makes the problem of the quantitative theoretical description of the system very challenging. Our paper makes a step towards this description but remains on the level revealing qualitatively new properties.
Coming back to our experiment, we remark that the interpretation of the lower-field phase transition at around 25-30 T is rather straightforward. The system transforms discontinuously in a magnetically uncompensated fan-type structure. The examples of such a structure are shown in Figs. 6(d) and 6(e) comment_fan_structure ().
The nature of the higher-field phase transition to the ferromagnetic state around 40 T is principally different, as clearly seen in Fig. 2. First, it does not have a noticeable hysteresis. Second, the curves just below the transition have a strong convex curvature. There are two possible scenarios of this transition. One possibility is a special property of the energy landscape of UPtGe consisting in (i) almost equal energies of the fan structure just before the transition and the ferromagnetic structure after the transition and (ii) the existence of a barrier-free path between these points of the energy landscape. The process resembling this type of transformation is obtained in our numerical experiments [Figs. 6(e),6(h), and 6(i)]. This calculation shows that the system can, in principle, relatively easily move from the fan configuration to the state close to collinear ferromagnetism.
As an alternative scenario, we mention the possibility of soliton-lattice formation. Here, with increasing fields the regions of noncollinear magnetic moments are separated by increasing ferromagnetically aligned domains (see, e.g., figures in Refs. Zheludev1998 (); Togawa2016 () illustrating this kind of transition). The convex curve and hysteresis-free transitions are characteristic features of the soliton-lattice type transition Togawa2016 () which is consistent with the in-field behavior obtained as the solution of the sine-Gordon equation.
It is possible that both scenarios contribute to the transition. The neutron diffraction or resonant x-ray scattering studies of the transitions would be very useful but are challenging in such high-field regions. On the theoretical side, one can study the energetics of the system using a lattice spin-model. However, a very large crystal domain must be used to describe the structures of different periodicity, and a large number of parameters must be employed to reflect the complexity of the system. Though further progress appears rather laborious, we hope that our work will stimulate new deep studies on the I-C transition of the chiral helical magnetic structures.
We have reported the magnetic properties of chiral incommensurate magnet UPtGe in the fields of varying directions and up to above the saturation. We have revealed that magnetic response, isotropic for fields below 20 T, becomes strongly anisotropic for higher fields where two principally different phase transitions are observed. Remarkably, this anisotropy possesses an unusual nonmonotonic field-orientation dependence. Our DFT calculations successfully explain the apparently contradicting properties obtained in the low- and high-field experiments with the identification of competing contributions into magnetic interactions, magnetic anisotropy, and magnetization process. We suggest an interpretation of the nature of the two phase transitions that bring the system from the incommensurate to the field-induced ferromagnetic state. Our work deepens the understanding of the physical origin of the wide variety of the properties of the U intermetallics.
Acknowledgements.The authors are grateful to A. Pourret and Y. Tokunaga for fruitful discussions. This research was carried out (in part) at the International Research Center for Nuclear Materials Science, Institute for Materials Research, Tohoku University. This work was partially supported by the MEXT of Japan Grants-in-Aid for Scientific Research (JP15K17700, JP15K05156, JP15H05882, JP15K05884, JP15K21732, JP15KK0149, and JP16H04006).
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