Spin Density wave instability in a ferromagnet

TL;DR

This paper reports the first observation of an incommensurate spin-density wave state emerging from a ferromagnetic ground state in Fe$_3$Ga$_4$, driven by Fermi surface nesting, combining neutron scattering and first principles calculations.

Contribution

It reveals a novel Fermi surface instability leading to ISDW formation in a ferromagnetic metal, supported by experimental and theoretical evidence.

Findings

ISDW state observed at intermediate temperatures

Re-emergence of ferromagnetism above 360 K

ISDW wavevector matches Fermi surface nesting condition

Abstract

Ferromagnetic (FM) and incommensurate spin-density wave (ISDW) states are an unusual set of competing magnetic orders that are seldom observed in the same material without application of a polarizing magnetic field. We report, for the first time, the discovery of an ISDW state that is derived from a FM ground state through a Fermi surface (FS) instability in Fe$_3$Ga$_4$. This was achieved by combining neutron scattering experiments with first principles simulations. Neutron diffraction demonstrates that Fe$_3$Ga$_4$ is in an ISDW state at intermediate temperatures and that there is a conspicuous re-emergence of ferromagnetism above 360 K. First principles calculations show that the ISDW ordering wavevector is in excellent agreement with a prominent nesting condition in the spin-majority FS demonstrating the discovery of a novel instability for FM metals; ISDW formation due to Fermi surface nesting in a spin-polarized Fermi surface.

Full text

Incommensurate spin density wave (ISDW) phases have long been known to be generated from paramagnetic (PM) metals either through a Fermi surface (FS) instability known as nesting or through the coupling of local magnetic moments via the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction[2, 1]. Interestingly, no ISDW phase stemming from an instability in the spin polarized FS of a ferromagnetic (FM) metal has been previously identified. We have discovered such a material in the transition metal binary compound Fe3Ga4 that hosts both FM and ISDW phases at zero applied field. Here, both of these phases are stable over wide temperature ($T$) ranges with the FM phase evident at temperatures above and below the $T$ range of stability of the ISDW phase. Furthermore, we have discovered the mechanism for the formation of the ISDW state in Fe3Ga4, which is distinct from any previously identified and that makes clear the connection between the FM and ISDW states. This mechanism, indicated by our data and simulations, is likely not limited to Fe3Ga4, suggesting that ISDW order or spin fluctuations stemming from spin-polarized FS instabilities should be investigated in a wide range of complex magnetic materials.

ISDW phases caused by FS instabilities are common in d-electron systems such as the prototypical ISDW system Cr[3, 4], while the local moment picture is usually associated with magnetism in rare earth materials where the magnetic moments derive from well localized f-electrons[5, 8, 6, 7, 9]. ISDW phases are commonly associated with antiferromagnetic (AFM) interactions, as opposed to incommensurate helical states which are more aptly described as being derived from FM states. In fact, transitions between ISDW and FM states in zero applied field are extraordinarily rare in contrast to helical magnets where transitions to FM ground states are common[10, 11]. There are only a handful of FM materials that display ISDW phases over narrow $T$ ranges with distinct, fairly restrictive mechanisms. These include: ferromagnets which seemingly avoid quantum criticality via a transition to a small ordered moment ISDW, Nb1-yFe2+y, YbRh2Si2, and PrPtAl[15, 14, 13, 12]; a frustrated magnetic insulator described as a Kagomé staircase system that undergoes transitions between several magnetic states, Co3V2O8[16]; and a very small number of Ce and U compounds which display ISDW states over a very narrow $T$ range just above their FM transitions and where the mechanism is unclear, CeRu2Al2B and UCu2Si2[17, 18]. The paucity of examples that we are able to find in the literature stemming from decades of intense investigation of magnetic materials is surprising.

In this letter, we focus on the investigation of the magnetic and electronic structure of Fe3Ga4 because previous magnetization, $M$, measurements indicated that there may be an interesting and unusual re-entrant FM phase surrounding an AFM phase at both low and high $T$[19]. Neither the magnetic structure, nor the character of the magnetic phases had been identified. The transitions between FM and AFM phases in Fe3Ga4 can be tuned via $T$, magnetic field, and pressure[19, 22, 20, 21]. We discover through neutron scattering measurements that Fe3Ga4 adopts an ISDW ordering between 60 and 360 K. This magnetic structure is nearly FM with magnetic moments being locally aligned but having a sinusoidal amplitude modulation occurring over several crystalline unit cells. In addition, we confirm the re-emergence of the FM phase above 360 K so that this material appears to display a thermodynamically odd re-entrant ferromagnetism[19]. A clue as to the cause for this unusual behavior comes from simulations of the electronic structure. In the non-magnetic case, our calculations do not indicate regions of FS nesting. However, we discovered significant nesting in the FM majority-spin FS with a nesting vector that is in excellent agreement with the measured ISDW wavevector, $q=$ (0 0 0.29). The FS instability along with anomalies in the charge transport support an itinerant mechanism for the magnetism in Fe3Ga4. This indicates that the ISDW emerges from an unstable FS in the FM phase so that a return to ferromagnetism is required prior to reaching the high $T$ PM phase. This mechanism, a FS instability in the majority band of a ferromagnet that drives a transition to an ISDW, has, to our knowledge, not been previously considered.

Fe3Ga4 forms in a monoclinic ($C2/m$) crystal structure with no evidence of structural change associated with the magnetic phase transitions[19, 20, 21, 24, 25]. The description of the crystal structure and an outline of our experimental methods can be found in [23]. A summary of our neutron scattering data is presented in Fig. 1a, which displays the intensity map for the scattering in the (2 0 $L$) reciprocal lattice direction. Three features of this plot represent our main experimental findings. First, the large increase in the scattering cross section at the (201) Bragg peak position below $\sim 60$ K demonstrates FM ordering consistent with the magnetic susceptibility, $\chi$ (Fig. 2a), previous measurements of $M$, and the M$\ddot{o}$ssbauer spectrum of Fe3Ga4[19, 21, 20, 27, 26]. Second, for $T$ just above 60 K, the reduced scattering at the Bragg peak position is accompanied by an increase in scattering centered at (2 0 0.71) indicating the existence of an incommensurate magnetic phase. The $T$ and $q$ dependence of this scattering contribution can be better viewed in Fig. 1b and in Fig. 2b and c where an abrupt loss of scattering is evident below 60 K while a more continuous change is seen above 300 K. At these higher temperatures, the scattering moves to somewhat larger $q$ before this signal is reduced below the background at $T>360$ K. Third, in this same $T$ range, the scattering at the (2 0 1) Bragg peak position increases (Fig. 1a, c and Fig. 2a) before decreasing at $T>420$ K. This is consistent with $\chi$ which indicates a PM state above 420 K[19]. Thus, our data indicate a FM ordered state below $T_{1}=60$ K transitioning to an incommensurate magnetic state for $T_{1}<T<T_{2}=360$ K along with a re-emergence of ferromagnetism between $T_{2}$ and $T_{3}=420$ K.

Surveys of a large number of reciprocal lattice positions have allowed a full refinement of the magnetic structure, the results of which are presented in Table S2 for the low $T$ FM state and Table S3 for the AFM order at 100 K in [23]. In Table S2[23], we report the size of the magnetic moment on each of the Fe sites refined with the magnetic moment constrained to lie along the crystallographic c-axis. These are comparable to the results of previously published M$\ddot{o}$ssbauer experiments[27], which are included in the table, and are in good agreement with the average magnetic moment determined from $M$[19, 21, 27]. When the constraint is removed a small contribution along the $a$ and $b$ directions is found that is smaller than the error in the refinement ($\sim 0.3$ $\mu_{B}$). In Fig. 2a we present a comparison of the scattering intensity, after subtraction of the Bragg scattering at 90 K, to $\chi$, measured on the same crystal. The two regions in $T$ where $\chi$ is large correspond well with the increased scattering at (2 0 1).

The result of the refinement of the magnetic scattering at 100 K (AFM phase) is significantly more complex with the best fit to our data consisting of a structure that closely resembles that of an ISDW having a propagation vector of $q=$ (0 0 0.29). After establishing the ISDW character of the magnetic state, we have further constrained the model such that crystallographically equivalent sites were required to have the same magnetic moment amplitude. The results of this refinement are displayed in Fig. 1d. In Table S3[23] the magnitude of the magnetic moments in one particular unit cell is presented, but we note that the incommensurate nature of the ordering results in moments that vary from cell-to-cell accordingly. The refinement places the magnetic moments mostly along the a-axis with a maximum magnitude of 2.31(6) $\mu_{B}$. However, there is also a considerable contribution along the b- and c-axis of nearly equal magnitude ($m_{max}$=0.58(6) $\mu_{B}$). Thus, there is a significant non-collinear and non-coplanar magnetic moment indicated by our data. Previously, we had speculated about such a non-coplanar magnetic moment based upon a large topological Hall effect in the range of temperatures and fields where the AFM is stable[19] and the data presented here confirm this interpretation.

The magnetic state represented in Fig. 1d retains a FM orientation of neighboring magnetic moments with an amplitude that is modulated by the wave vector $\delta=0.29$ in reciprocal lattice units. This is distinct from the simpler ISDW materials such as Cr[3] where neighboring sites are AFM aligned. Thus, the ISDW state appears to be a long wavelength modulation of a FM state. In fact, the average size of the magnetic moments in the ISDW state are within error of those in the FM state indicating a conservation of total moment magnitude at $T_{1}$. However, there is a significant change in the direction of the magnetic moments in the FM and AFM structures. The complexity of the ISDW phase and the significant differences in the two competing magnetic phases highlight the question of the mechanism driving the ISDW ordering and its instability at low $T$.

The evolution of $\delta$ and the correlation length for the magnetic scattering, $\xi$, with $T$ can lend insight into the character of the magnetic state and the competing interactions which are of clear importance in Fe3Ga4[23]. In Figs. 1b, 2b and 2d the $T$ evolution of $\delta$ is displayed. This $T$ dependence is usually associated with the competition between the electronic degrees of freedom responsible for the incommensurate nature of the wavevector and the lattice degrees of freedom which prefer a commensurate density wave[29]. Here, we observe a decrease in $\delta$ as the FM phases are approached either by warming or cooling with a large decrease in $\delta$ apparent above 250 K. Preceding this decrease in $\delta$ is a precipitous reduction in $\xi$ as determined from the widths of the (2 0 1-$\delta$) scattering peak for $T>200$ K (Fig. 2a and d). Above 300 K $\xi$ has decreased such that it is equivalent to several wavelengths of the ISDW phase ($2\pi/\delta$). Thus, there is an extended $T$ range where only short range ordering exists.

Independent of the mechanism responsible, most ISDW phases are accompanied by a discontinuous decrease in conductivity, $\sigma$, and an increase in the Hall constant as the result of a partial FS gapping. This is a consequence of either FS nesting[1] or simply the additional periodicity associated with the ISDW phase. Fe3Ga4 does not disappoint in this regard as we identify both a decreased $\sigma(T)$ at $T_{1}$ and a somewhat smaller increase at $T_{2}$ (Fig. 3) indicating a change to the FS[21, 19]. Furthermore, the ordinary Hall coefficient is observed to undergo large changes at both $T_{1}$ and $T_{2}$[19].

To establish the itinerant nature of the magnetism, the electronic structure of Fe3Ga4 was calculated in the nonmagnetic and spin polarized phases employing the full potential linearized augmented plane wave method[30]. The resulting Fermi Surfaces were examined for possible nesting along the $c$-axis. Although we find no such nesting condition in the non-magnetic FS, the majority spin-band FS in the FM state contains a FS sheet with a substantial region that is flat and perpendicular to the $c^{*}$ direction (see Fig. 4). Here, the nesting across the Brillouin zone boundary is demonstrated indicating the likely formation of density wave ordering. The nesting wavevector shown in this figure, $q_{nest}=$(0 0 0.276), is in very good agreement with the experimentally observed ISDW providing compelling evidence that the FM state is unstable to the formation of a density wave.

We have carried out an extensive neutron diffraction investigation of Fe3Ga4 solving, for the first time, the magnetic structure of the AFM state finding an ISDW phase. The ground state is confirmed as a robust FM state and the re-emergence of this ferromagnetism is also confirmed for $360\leq T\leq 420$ K. Although the magnetic contribution to the neutron scattering signal is small and data too sparse above 360 K to make a convincing comparison between the low and high $T$ FM states, our $\chi(T)$ and $M(H,T)$ data display identical anisotropy as well as similar moment magnitudes and field dependencies indicating that these FM states are likely identical[19]. Furthermore, electronic structure calculations reveal a FS sheet in the majority band of the FM state that is unstable towards density wave formation with a nesting vector that matches the neutron data both in direction and magnitude. Interestingly, the ISDW state likely results from a FS instability in only one of the spin-polarized bands of Fe3Ga4, a mechanism that, thus far, has not been considered or identified in other materials. The details of how the transition between itinerant FM and ISDW states takes place are not yet clear since a residual magnetic moment may be expected for a simple FS reconstruction in conflict with our $M(H)$[19] and neutron diffraction data.

These conclusions place Fe3Ga4 as a unique compound among the large number of FM materials reported in the literature such that identifying materials that are more than tangentially related is difficult. Extending the conversation to examples of FM materials with consequential FS nesting in their PM states broadens the number of comparisons slightly as this is only somewhat rare. Most of these are associated with competing orders and none, that we are aware of, report ISDW states resulting from nesting. Competing orders occur, for example, in SmNiC2 which has a charge density wave (CDW) phase that exists only above its Curie $T$[31]. In addition, nesting of the PM FS is common in the heavy rare earth elemental metals which tend to have FM ground states[10, 11]. This nesting leads to transitions from PM into helimagnetic, rather than ISDW, states. More interesting is the idea that triplet superconductivity in FM UGe2 is related to coupled CDW-ISDW fluctuations that emerge with pressure within the FM phase[32].

We are left with two important unanswered questions about the mechanism we discovered for the transitions between FM and ISDW states in Fe3Ga4. The first is the character of the magnetism and its relationship to the conducting charge carriers. Somewhat in contrast to an itinerant magnetism in Fe3Ga4 are the large magnetic moments evident in $M$, M$\ddot{o}$ssbauer spectra, simulation, and the neutron scattering cross-sections suggesting magnetic moments may be more localized to the Fe sites. This would indicate an important coupling mechanism between the charge carriers and the more localized electrons responsible for the magnetic moments. A similar cooperative mechanism has been proposed for GdSi where both RKKY and FS nesting are thought to play a role in creating an ISDW state[33] and for the Fe-based superconducting families where this coupling is responsible for the various AFM structures[34, 35, 36]. The second related question is the cause of the FM ground state. We speculate that the re-emergence of FM ground state at low $T$ may be a result of a loss of RKKY coupling with the establishment of partial energy gaps in the FS of the ISDW state, as has been suggested for the helimagnetic rare earth elemental metals such as Dy and Tb[10, 38, 37, 11]. In these materials an abrupt transition from FM-to-helimagnetic ordering results from superzone gaps in the FS of the incommensurate phase along with considerations of spin-orbit coupling[10]. As such, the incommensurate phase appears to be self limiting as the ordering creates a partial energy gap that removes carriers responsible for the RKKY coupling. Additionally, there is a complex and interesting variation of the magnetic wavevector and metamagnetic fields that is in many ways similar to what we observe in Fe3Ga4. Despite these remaining issues, our data and computational results indicate that Fe3Ga4 is the first material that has been discovered to evolve from a FM FS to an ISDW, a discovery that suggests that there is likely a rich and mostly unrecognized competition between magnetic states caused by FS instabilities of spin polarized bands.