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High-energy magnetic excitations and anomalous spin-wave damping in FeGe 2<br />
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2000 J. Phys.: Condens. Matter 12 8487<br />
(http://iopscience.iop.org/0953-8984/12/39/311)<br />
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J. Phys.: Condens. Matter 12 (2000) 8487–8493. Printed in the UK PII: S0953-8984(00)15871-0<br />
High-energy magnetic excitations and anomalous spin-wave<br />
damping in FeGe2<br />
1. Introduction<br />
C P Adams†, T E Mason† + , E Fawcett† ∗ , A Z Menshikov‡ ∗ , C D Frost§,<br />
J B Forsyth§, T G Perring§ and T M Holden #<br />
† Department of Physics, University of Toronto, Toronto, ON, Canada M5S 1A7<br />
‡ Institute for Metal Physics, Ekaterinburg, Russian Federation<br />
§ ISIS Facility, Rutherford-Appleton Laboratory, Chilton, Didcot OX11 0QX, UK<br />
<strong>National</strong> Research Council, Chalk River, ON, Canada K0J 1J0<br />
E-mail: carl.adams@nist.gov.<br />
Received 27 July 2000<br />
Abstract. Inelastic neutron scattering has been used to measure the high-energy, low-temperature<br />
magnetic excitations of the itinerant antiferromagnet FeGe2. The spin-wave excitations follow a<br />
Heisenberg dispersion with exchange constant along the c-axis SJFM = 68 ± 1 meV an order<br />
of magnitude higher than the basal-plane antiferromagnetic constant SJAF M =−4.4 ± 0.6 meV.<br />
The c-axis spin waves are highly damped, even at temperatures small compared to SJFM. The<br />
damping is roughly quadratic in energy (as in the hydrodynamic model) up to ∼250 meV, beyond<br />
which a continuum of excitations emerges.<br />
Over the last 10 years there has been growing evidence that a nearby spin-density-wave (SDW)<br />
instability is responsible for the anomalous normal-state transport properties of the high-Tc<br />
oxides [1] and possibly for the superconductivity itself [2–5]. Similar instabilities precede the<br />
SDW transitions in certain other Mott–Hubbard systems [6], such as V2−yO3 [7]. In both cases,<br />
strong electron–electron correlations exist and lead to a breakdown of conventional band theory<br />
and, as in the high-Tc cuprates, a metal–insulator transition. Despite some obvious differences<br />
these novel SDW systems have much in common with non-correlated, fully metallic, SDW<br />
systems, of which chromium [8] is the prototypical example. Notably in all SDW systems the<br />
periodic modulation of the conduction-electron spin density [9] develops as a result of Fermi<br />
surface nesting, i.e., electron exchange between two parallel pieces of Fermi surface [10].<br />
In the light of this commonality the recent interest in the high-Tc oxides has led to renewed<br />
studies of non-correlated SDW systems, where conventional band theory should be able to<br />
answer outstanding questions regarding transport and magnetic properties.<br />
One intriguing feature, common to both correlated and non-correlated SDW systems,<br />
is a complex spectrum of magnetic excitations in both ordered and non-ordered phases;<br />
Present address: NIST Center for <strong>Neutron</strong> Research, <strong>National</strong> Institute of Standards and Technology, Gaithersburg,<br />
MD 20899-8562, USA.<br />
+ Present address: <strong>Spallation</strong> <strong>Neutron</strong> <strong>Source</strong>, 701 Scarboro Road, <strong>Oak</strong> <strong>Ridge</strong> <strong>National</strong> Laboratory, <strong>Oak</strong> <strong>Ridge</strong>,<br />
TN 37830, USA.<br />
∗ Deceased.<br />
# Retired.<br />
0953-8984/00/398487+07$30.00 © 2000 IOP Publishing Ltd 8487
8488 C P Adams et al<br />
conventional oscillatory, propagating spin waves do not exist in these <strong>materials</strong> [7, 8, 11, 12].<br />
This has greatly hindered a complete description of SDW systems since it is the spin-wave<br />
dispersion that yields values for the microscopic interaction strengths. From this viewpoint<br />
FeGe2, the non-correlated SDW system that is the subject of this study, is an important<br />
exception, since the first inelastic neutron scattering measurements showed that its lowtemperature<br />
magnetic excitations are propagating spin waves or magnons [13, 14]. A simple<br />
Heisenberg model successfully describes the low-energy magnetic dispersion. However, there<br />
appears to be an intrinsic damping of the spin waves even at low temperatures. Damping of this<br />
magnitude is impossible in a purely localized system where magnon–magnon and magnon–<br />
phonon scattering processes dominate, but in metallic systems, such as FeGe2, scattering<br />
can simply arise from the itinerant electrons even at low T . This dual behaviour (spin-wave<br />
dispersion of local moments, significant damping from itinerant electrons) is also found in<br />
the ferromagnetic (FM) metals iron [15] and nickel [16] and La0.85Sr0.15MnO3 [17], a FM<br />
transition metal oxide near the Mott transition.<br />
A further novel feature of FeGe2 is that the dispersion is so anisotropic that it is well<br />
described as quasi-one-dimensional (quasi-1D), unique among metallic magnetic systems [14].<br />
While the entire a-axis magnon dispersion was measured using thermal neutron scattering at<br />
the Chalk River reactor, the much larger energy scale associated with the c-axis magnons<br />
prevented the observation of resolved spin-wave modes in constant-Q scans. A quantitative<br />
determination of the damping (seen as the intrinsic linewidth) and the high-Q dispersion was<br />
not possible. This prompted a continuation of these measurements up to higher energies at the<br />
ISIS spallation source.<br />
FeGe2 has a body-centred tetragonal crystal structure, as in CuAl2, with space group<br />
I4/mcm and two zero-field magnetic phase transitions [18, 19]. One is a second-order Néel<br />
transition from a paramagnetic phase to an incommensurate (IC) SDW state at TN = 289 K<br />
and the other at TC−IC = 263 K is a typical first-order transition from an IC to a commensurate<br />
phase [19]. The ordering wavevector of the commensurate phase, (2π/a)(1, 0, 0), changes to<br />
(2π/a)(1+δ, 0, 0) in the IC phase where δ varies from 0 to ∼0.05. Nearest-neighbour (NN)<br />
iron atoms are 2.478 Å apart along the c-axis and their magnetic moments have FM alignment<br />
in both phases. The next-NN iron atoms are separated by 4.178 Å along the [110] direction with<br />
antiferromagnetic (AFM) alignment of the moments below TC−IC. The higher-temperature IC<br />
phase appears as a long-wavelength modulation of this structure in the basal plane. AFM IC<br />
structures in metals are a strong indication of an interaction that is mediated by the conduction<br />
electrons (indirect exchange). Superexchange through the germanium ions probably also plays<br />
a role in the net AFM basal-plane interaction. In contrast, the NN spacing along the c-axis is so<br />
close to that of elemental iron (2.50 Å) that the primary magnetic interaction in this direction<br />
is probably a direct FM interaction.<br />
2. Experimental results and discussion<br />
To measure the high-energy magnetic excitations and the damping of spin waves in FeGe2, a<br />
new single crystal sample was prepared. It is roughly semi-cylindrical, 15 mm in radius and<br />
40 mm in length, with a mass of 270 g. The measurements were performed at 11 K in the<br />
low-T AFM phase (effectively at T = 0 compared to TN) using HET, a direct-geometry chopper<br />
spectrometer at the ISIS spallation source. The sample was oriented with [010] vertical so the<br />
horizontal banks of detectors were sensitive to momentum transfers Q with (h0l) components,<br />
designated by Qa and Qc. Since HET is a time-of-flight instrument, energy transfer E is<br />
coupled to Q in a way that in general prevents extraction of constant-E and constant-Q scans<br />
(conventionally employed when using a triple-axis spectrometer). Despite this complication,
High-energy magnetic excitations in FeGe2<br />
8489<br />
a substantial part of the magnetic dispersion can often be measured at one time provided that<br />
a suitable orientation of the sample is chosen.<br />
An example of an HET scan is shown in figure 1. The curved surface in (Q,E) space<br />
scanned by the instrument is projected onto the (Qa,Qc) plane. The contours of constant-E<br />
reflect this curvature and approach the incident energy Ei = 120 meV at the apex. Absolute<br />
scattering intensity, resulting from vanadium normalization, is indicated by spectral colour.<br />
In terms of actual neutron counts, 1 mb meV −1 sr −1 is equal to 20 counts over a 12-hour<br />
scan. No correction was made for sample shape or absorption, which could lead to an error of<br />
15% in the absolute cross-section values. Several features are evident in figure 1: incoherent<br />
elastic scattering along the E = 0 contour, complicated phonon and magnon scattering for<br />
E
8490 C P Adams et al<br />
for various values of Ei are shown in figures 2(a) and 2(b) and resemble the more familiar<br />
constant-E and constant-Q triple-axis scans. Peaks are resolved to high energies (>200 meV)<br />
in both types of cut. The dispersion, which is quite steep in comparison to the Qc-resolution, is<br />
measured with a resolution corrected fit to the peaks in constant-E cuts as shown in figure 2(a).<br />
The constant-Qc cuts show damped magnons with a significant linewidth that increases with<br />
energy. Along the a-axis, similar spin-wave behaviour and damping have been observed with<br />
triple-axis measurements [13] but the zone boundary energy is only 30 meV, much less than<br />
the c-axis spin-wave energies in figures 2(a) and 2(b).<br />
Intensity (mb meV −1 sr −1<br />
)<br />
(a)<br />
0.4<br />
0.2<br />
0.0<br />
x10<br />
x 3<br />
h¯ ω=250 meV<br />
h¯ ω=155 meV<br />
h¯ ω=<br />
77.5 meV<br />
1.2 1.6 2.0 2.4 2.8<br />
Q c (c*)<br />
(b)<br />
0.6<br />
0.4<br />
0.2<br />
0.0<br />
Q c =1.6 c*<br />
Q c =1.74 c*<br />
Q c =1.87 c*<br />
50 100 150 200<br />
Energy (meV)<br />
Figure 2. (a) Constant-E and (b) constant-Qc cuts showing the c-axis magnons. Cuts are taken<br />
from data reduced to the (Qc,E) plane and have been offset vertically (the thick dashed lines<br />
represent zero intensity above the background). Two sets have been rescaled as indicated. The<br />
data in (b) have had the background removed by subtraction of a nearby non-magnetic cut. The<br />
solid lines are the resolution-corrected scattering cross-sections (described in more detail in a later<br />
section).<br />
The c-axis spin-wave dispersion, as measured by Gaussian fits to constant-E cuts, is shown<br />
in figure 3.<br />
The 3D classical, NN Heisenberg model has a Hamiltonian of the form<br />
H =− Jij Si · Sj .<br />
Combining this with the known low-T AFM structure of FeGe2 gives the spin-wave dispersion<br />
[20]:<br />
¯hωQ = 4S({JFM[1 − cos(πl)] − 2JAF M[1 − cos(πh) cos(πk)]}<br />
×{JFM[1 − cos(πl)] − 2JAF M[1 + cos(πh) cos(πk)]}) 1/2<br />
(1)<br />
where the spin-wave energy is ¯hωQ, S is the on-site spin, JFM is the c-axis exchange constant,<br />
JAF M is the [110] exchange constant, and h, k, and l express Q in terms of reciprocal-lattice<br />
coordinates. Triple-axis measurements [13] have shown a gapless (
High-energy magnetic excitations in FeGe2<br />
Energy (meV)<br />
300<br />
200<br />
100<br />
SJ =−4.4 meV<br />
AFM<br />
SJFM =68 meV<br />
0<br />
0.0 0.1 0.2 0.3<br />
Qc (c*)<br />
0.4 0.5 0.6<br />
8491<br />
Figure 3. The c-axis magnon dispersion and the prediction of a NN Heisenberg model. The data<br />
have been corrected for the Qa-energy contribution. Excitations beyond the plotted E-range are<br />
highly damped and not well defined.<br />
gap in the 1D planar model). However, using equation (1) with SJAF M =−4.4 meV (from<br />
the data of [13]) significantly improves the fit to the low-energy dispersion and the overall<br />
scattered intensity (see the next section). The 3D behaviour can be seen in figure 2(b) where<br />
the variation in Qa along a ‘constant-Q’ cut leads to peaks at 45 meV and at 75 meV for<br />
Qc = 1.87 c∗ ; an effect that is correctly predicted by the simulation. The small Qa-dependent<br />
energy contribution arising from equation (1) has been subtracted from the data in figure 3<br />
to leave only the Qc-dependence. The system becomes more 1D at higher energies as this<br />
correction decreases in comparison to ¯hωQ. The fit shown in figure 3 has SJFM = 68±1 meV,<br />
roughly a factor of 15 larger in strength than SJAF M, a uniquely large anisotropy among<br />
3D metals.<br />
Although the Gaussian fits give the dispersion of well defined magnons, they provide no<br />
clear information about the damping since the intrinsic linewidth remains convoluted with the<br />
experimental resolution. This linewidth is determined by using a simulation of the resolution<br />
and a full model for the scattering cross-section per iron atom [20]:<br />
d2σ kf<br />
∝ |F (Q)|<br />
d dE ′<br />
ki<br />
2<br />
1/2 JFM[1 − cos(πl)] − 2JAF M[1 − cos(πh) cos(πk)]<br />
JFM[1 − cos(πl)] − 2JAF M[1 + cos(πh) cos(πk)]<br />
<br />
× 1 − exp − E<br />
−1 4 ¯hωQEƔ<br />
kBT π [E2 − (¯hωQ) 2 ] 2 +4(EƔ) 2<br />
<br />
. (2)<br />
ki and kf are the lengths of the incident and final neutron wavevectors. F (Q) is a magnetic<br />
form factor appropriate for Fe3+ [22]. The next term, the dynamical structure factor, predicts<br />
that the intensity of pure FM spin waves is uniform throughout the magnetic Brillouin zone<br />
and AFM spin waves have a diverging intensity at magnetic zone centres and zero intensity at<br />
nuclear zone centres. The principle of detailed balance is satisfied by inclusion of the thermal
8492 C P Adams et al<br />
population of magnons. The final term is the appropriate susceptibility for a damped simple<br />
harmonic oscillator (DSHO) with linewidth Ɣ and free-mode energy ¯hωQ. This cross-section<br />
correctly approaches δ-functions in the Ɣ ≪ ¯hωQ limit.<br />
The simulation, using equation (2) and the previously mentioned SJ-values, successfully<br />
describes the spin-wave intensity and dispersion through several magnetic Brillouin zones.<br />
Subsequent fixing of the proportionality in equation (2) gives the results for Ɣ shown in<br />
figure 4. This confirms that the peak broadening in figure 2 is due to increasing Ɣ rather than the<br />
coarsening resolution at higher Ei. The damping increases rapidly beyond ¯hωQ = 220 meV,<br />
wiping out the well defined magnon peaks; uncertainties in peak position and linewidth increase<br />
accordingly. Such large values of Ɣ for T ≪ TN are not expected on the basis of local<br />
models where magnon–magnon scattering dominates. Hence, this damping is anomalous and<br />
must be due to itinerant electrons, a conclusion previously reached in studies of several FM<br />
systems [15–17]. In fact, the itinerant effects eventually predominate over the local behaviour<br />
and the spin waves merge into a continuum of excitations (Ɣ ∼ E). Calculation of Ɣ in<br />
itinerant systems often requires detailed knowledge of the electronic band structure [23] but<br />
in FeGe2 the reduced dimensionality and the lack of electron correlations may simplify such<br />
a prediction, providing important insight into several classes of itinerant magnetic systems.<br />
One result that appears to be preserved from a local-moment model is the quadratic<br />
variation of Ɣ with E predicted by the hydrodynamic model [24]. A range of appropriate<br />
quadratic parametrizations is shown in figure 4 with the best description being given for<br />
Ɣ = 2.7 × 10 −3 E 2 (meV units). The reciprocal of the prefactor is 370 ± 80 meV, comparable<br />
to 4SJFM = 272 meV, half of the zone boundary energy. The hydrodynamic model includes<br />
SJ in a similar way but also includes a T -dependent term, (T /TN) 3 , which would eliminate<br />
low-T damping. This is clearly in disagreement with figure 4.<br />
Figure 4. The magnon linewidth Ɣ versus E. The solid and dashed lines represent the function and<br />
the uncertainty given in the figure. Note the rapid increase of Ɣ beyond 220 meV and the gradual<br />
merging into a continuum of excitations.
High-energy magnetic excitations in FeGe2<br />
8493<br />
The observation of well defined spin-wave modes in an itinerant system such as FeGe2<br />
is particularly noteworthy given the fact that this sort of response has not been seen in Cr,<br />
the prototypical SDW system. In Cr, complex behaviour at low frequencies is observed with<br />
a suggestion of spin-wave modes with a high velocity [8, 25]. However, measurements to<br />
higher frequencies carried out using the same time-of-flight techniques as are employed in the<br />
present study have failed to reveal resolved spin-wave modes such as those shown in figure 2.<br />
The behaviour of FeGe2 is similar to that seen in itinerant FM <strong>materials</strong> where the magnon<br />
modes have non-zero damping, even in the low-T ground state. The reduced dimensionality<br />
of FeGe2 may hasten a theoretical description of the transition from local to fully itinerant<br />
magnetism.<br />
Acknowledgments<br />
We wish to thank V G Guk from the Ural Technical University for growing the FeGe2 single<br />
crystal and S M Hayden who provided the computer code for the resolution-corrected fits.<br />
Work at Toronto was supported by the Natural Sciences and Engineering Research Council of<br />
Canada and the Canadian Institute for Advanced Research. TEM acknowledges the support<br />
of the A P Sloan Foundation. AZM recognizes the support of the programme ‘<strong>Neutron</strong> Study<br />
of Matter’ and RFFI Grant No 98-02-16165.<br />
References<br />
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[2] Scalapino D J, Loh E Jr and Hirsch J E 1987 Phys. Rev. B 35 6694<br />
[3] Monthoux P, Balatsky A V and Pines D 1992 Phys. Rev. B 46 14 803<br />
[4] Levin K, Zha Y, Radtke R J, Si Q, Norman M R and Schüttler H-B 1994 J. Supercond. 7 563<br />
[5] Ruvalds J, Rieck C T, Tewari S, Thoma J and Virosztek A 1995 Phys. Rev. B 51 3797<br />
[6] Mott N F 1990 Metal–Insulator Transitions 2nd edn (New York: Taylor and Francis) pp 123–44, 171–97<br />
[7] Bao W, Broholm C, Carter S A, Rosenbaum T F and Aeppli G 1993 Phys. Rev. Lett. 71 766<br />
[8] Fawcett E 1988 Rev. Mod. Phys. 60 209<br />
[9] Herring C 1966 Magnetism vol 4, ed G T Rado and H Suhl (New York: Academic) pp 85–119<br />
[10] Overhauser A W 1962 Phys. Rev. 128 1437<br />
[11] Noakes D R, Holden T M, Fawcett E and deCamargo P C 1990 Phys. Rev. Lett. 65 369<br />
[12] Hayden S M, Doubble R, Aeppli G, Perring T G, Fawcett E, Lowden J and Mitchell P W 1997 Physica B<br />
237+238 421<br />
[13] Holden T M, Menshikov A Z and Fawcett E 1996 J. Phys.: Condens. Matter 8 L291<br />
[14] Mason T E, Adams C P, Mentink SAM,Fawcett E, Menshikov A Z, Frost C D, Forsyth J B, Perring TGand<br />
Holden T M 1997 Physica B 237+238 449<br />
[15] Paul D M c K, Mitchell P W, Mook H A and Steigenberger U 1988 Phys. Rev. B 38 580<br />
[16] Mook HAandPaulDM c K 1985 Phys. Rev. Lett. 54 227<br />
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14 913<br />
[18] Corliss L M, Hastings J M, Kunnmann W, Thomas R, Zhuang J, Butera R and Mukamel D 1985 Phys. Rev. B<br />
31 4337<br />
[19] Menshikov A Z, Dorofeev Yu A, Budrina G L and Syromyatnikov V M 1988 J. Magn. Magn. Mater. 73 211<br />
[20] Lovesey S M 1984 Theory of Inelastic <strong>Neutron</strong> Scattering vol 2 (New York: Oxford University Press) pp 109–19<br />
[21] Villain J 1973 J. Phys. C: Solid State Phys. 6 L97<br />
[22] Brown P J 1995 International Tables for Crystallography volC,edAJCWilson (Boston, MA: Kluwer Academic)<br />
p 392<br />
[23] Cooke J F, Lynn J W and Davis H L 1980 Phys. Rev. B 21 4118<br />
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[25] Fukuda T and Endoh Y 1998 Physica B 241–243 619
arXiv:cond-mat/9610009v1 [cond-mat.str-el] 1 Oct 1996<br />
Itinerant Antiferromagnetism in FeGe2<br />
T.E. Mason, C.P. Adams, S.A.M. Mentink, E. Fawcett<br />
Department of Physics, University of Toronto, Toronto, ON M5S 1A7, Canada<br />
A.Z. Menshikov<br />
Institute for Metal Physics, Ekaterinburg, Russia<br />
C.D. Frost, J.B. Forsyth, T.G. Perring<br />
ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX,<br />
United Kingdom<br />
Abstract<br />
T.M. Holden<br />
AECL, Chalk River Laboratories, Chalk River, ON K0J 1J0, Canada<br />
FeGe2, and lightly doped compounds based on it, have a Fermi surface driven instability<br />
which drive them into an incommensurate spin density wave state. Studies<br />
of the temperature and magnetic field dependence of the resistivity have been used<br />
to determine the magnetic phase diagram of the pure material which displays an<br />
incommensurate phase at high temperatures and a commensurate structure below<br />
263 K in zero field. Application of a magnetic field in the tetragonal basal plane<br />
decreases the range of temperatures over which the incommensurate phase is stable.<br />
We have used inelastic neutron scattering to measure the spin dynamics of FeGe2.<br />
Despite the relatively isotropic transport the magnetic dynamics is quasi-one dimensional<br />
in nature. Measurements carried out on HET at ISIS have been used to<br />
map out the spin wave dispersion along the c-axis up the 400 meV, more than an<br />
order of magnitude higher than the zone boundary magnon for wavevectors in the<br />
basal plane.<br />
Preprint submitted to Elsevier Preprint 1 February 2008
Studies of the spin dynamics of the high Tc superconductor, La2−xSrxCuO4,<br />
have shown that this material has incommensurate magnetic fluctuations characteristic<br />
of a metal close to a spin density wave instability[1]. This has led to<br />
a renewed interest in systems which exhibit similar instabilities since there are<br />
many unanswered questions regarding the transport and magnetic properties<br />
of spin density wave systems. The prototypical SDW system is the itinerant<br />
antiferromagnet Cr and dilute alloys of Cr with other transition metals[2].<br />
SDW ordering also occurs in Mott-Hubbard systems where strong correlations<br />
lead to a break down of conventional band theory and, as in the high Tc<br />
cuprates, a metal insulator transition[3]. FeGe2 and lightly doped compounds<br />
based on it have a similar (Fermi surface) spin density wave instability which<br />
drives the system to order and, in its undoped form, exhibits both commensurate<br />
and incommensurate magnetic ordering as temperature is varied. We are<br />
studying the transport and spin dynamics of this system in order to obtain a<br />
better microscopic understanding of how this instability is manifested in the<br />
spin fluctuations.<br />
The properties of FeGe2, a tetragonal material with the CuAl2 structure, have<br />
been studied since 1943[4] and many of the details of the crystal and magnetic<br />
structure have been determined [5–8]. FeGe2 is characterised by two magnetic<br />
phase transitions, the first of which is the Néel transition from a paramagnetic<br />
phase to an incommensurate spin density wave state at TN = 289 K in zero<br />
field. The second one is a commensurate-incommensurate (C-IC) transition<br />
into a collinear antiferromagnetic structure. This transition occurs at Tc =<br />
263 K in zero field. The propagation vector Q of the SDW is parallel to [100]<br />
and varies from 1 to roughly 1.05 a ∗ between the two transitions which are<br />
seen in resistivity[9,10], ultrasonic attenuation[11], heat capacity[5,12], and<br />
AC susceptibility[13] studies. The spins lie in the basal plane and are ferromagnetically<br />
aligned along the c-axis. Although this compound has been<br />
well characterised in zero field there has been very little work done in magnetic<br />
fields and not much is known about the details of the magnetic phase<br />
diagram. The magnon and phonon dispersion were measured (primarily at<br />
low temperatures) and revealed a significant anisotropy of the magnetic interactions<br />
with the spin wave velocity along the c-axis much higher than that<br />
observed for modes propagating in the basal plane [14]. For those thermal<br />
neutron triple-axis measurements it was not possible to fully characterise the<br />
spin wave dispersion along c due to the unsuitability of that technique for<br />
energy transfers beyond about 50 meV.<br />
Incommensurate ordering of the type seen in FeGe2 gives rise to four magnetic<br />
satellites around (100) and related positions occurring at (1±q, 0, 0) and<br />
(1,±q, 0). Elastic neutron scattering can be used to determine the wavevector<br />
of this magnetic ordering, a scan through the incommensurate peaks at 275 K<br />
is shown in the upper panel of Figure 1 compared to that of Fe(Ge0.96Ga0.04)2<br />
at the same temperature. Doping has the effect of increasing the magnitude<br />
2
Intensity (arb. units)<br />
4000<br />
3000<br />
2000<br />
1000<br />
0<br />
600<br />
400<br />
200<br />
(ζ,0,2)<br />
(ζ,0,0)<br />
FeGe 2<br />
T=275 K<br />
Fe(Ge 0.96 Ga 0.04 ) 2<br />
0<br />
0.7 0.8 0.9 1.0<br />
ζ<br />
1.1 1.2 1.3<br />
Fig. 1. Scans through the incommensurate magnetic Bragg peaks for pure (upper<br />
panel) and Ga doped (lower panel) FeGe2 at 275 K obtained on the E3 and C5<br />
triple axis spectrometers at Chalk River. The peak in the centre arises from the two<br />
peaks displaced from the commensurate position along the b-axis, seen due to the<br />
coarse vertical collimation employed in these measurements.<br />
Resistivity (µΩ-cm)<br />
Resistivity (µΩ-cm) Resistivity (µΩ-cm)<br />
100<br />
50<br />
Resistivity (µΩ-cm)<br />
0<br />
70<br />
60<br />
50<br />
40<br />
82<br />
77<br />
250 300<br />
Temperature Tc = 263 K (K)<br />
H=0, i||[110]<br />
T c = 182 K<br />
T N = 289 K<br />
undoped<br />
T N = 285 K<br />
Ga doped<br />
0 50 100 150 200 250 300<br />
Temperature (K)<br />
Fig. 2. Temperature dependence of the resistivity for pure and Ga doped FeGe2<br />
(measured on pieces of the same samples used for Figure 1).<br />
of the incommensurate wavevector (similar to what is seen in lightly doped<br />
alloys of Cr[2]). In the 4 at% Ga doped sample the ordering wavevector varies<br />
between q = 0.12 just below TN = 285K and q = 0.1 below a first order lockin<br />
transition at Tc = 182K. In the pure material the transition to the low<br />
temperature commensurate state (with q = 0) is continuous. The zero field<br />
resistivity for these two compounds is shown in Figure 2. In the doped sample<br />
(bottom panel) the two phase transitions are clearly visible and are denoted by<br />
arrows. In the pure material the anomalies are more subtle but the transitions<br />
3
Magnetic Field (T)<br />
30<br />
25<br />
20<br />
15<br />
10<br />
5<br />
H // [110]<br />
commensurate SDW<br />
incommensurate<br />
SDW<br />
FeGe 2<br />
paramagnetic<br />
0<br />
250 260 270<br />
theory<br />
280 290 300<br />
Temperature (K)<br />
Fig. 3. Magnetic phase diagram for FeGe2 for a magnetic field applied along the [110]<br />
direction determined by resistivity measurements and (for H=0) neutron diffraction.<br />
The line denoted theory corresponds to the approximate phase boundary expected<br />
based on the Ginzburg-Landau theory of Reference [13]<br />
Intensity (mb meV -1 sr -1 )<br />
0.25<br />
0.20<br />
0.15<br />
0.10<br />
0.05<br />
E i =120 meV<br />
∆E=77.5 meV<br />
0.00<br />
1.5 2.0<br />
Q (c*) c<br />
2.5<br />
Fig. 4. Constant energy cut at 77.5 meV through the two counter-propagating spin<br />
wave modes obtained on HET for FeGe2 at 11 K using an incident neutron energy<br />
of 120 meV.<br />
are seen in the inset. The transition temperatures deduced from the resistive<br />
anomalies agree well with those determined by neutron scattering. We have<br />
used similar measurements, together with constant temperature, field scans<br />
which clearly show the lower phase transition, with a magnetic field applied<br />
in the basal plane to determine the magnetic phase diagram for FeGe2, shown<br />
in Figure 3, the behaviour is similar for a field applied along [100].[15]<br />
In order to fully characterise the ground state of FeGe2 we have measured the<br />
spin dynamics at low temperatures (11 K) using the direct geometry timeof-flight<br />
spectrometer HET at ISIS. This instrument allows access to much<br />
higher incident neutron energies than the thermal triple-axis measurements of<br />
Holden et. al.[14] making it possible to fully map out the c-axis dispersion.<br />
Figure 4 shows a constant energy cut through the time-of-flight trajectories of<br />
the 2.5 m detector bank on HET for an energy transfer of 77.5 meV. This was<br />
obtained for an integrated current of 2200 µA-hrs using an incident energy of<br />
120 meV. The two spin wave modes emanating from Qc = 2 are clearly visible,<br />
4
Energy Transfer (meV)<br />
400<br />
300<br />
200<br />
100<br />
0<br />
0.00 0.20 0.40 0.60 0.80 1.00<br />
Momentum Transfer (c*)<br />
Fig. 5. Spin wave dispersion relation obtained from the triple axis data (lowest<br />
energies) and incident energies of 80, 120, 300, 500, and 700 meV on HET. The line<br />
is a fit to a model for a planar one-dimensional ferromagnet with a zone boundary<br />
magnon energy of 400 meV.<br />
for energies higher than 25 meV FeGe2 behaves as a quasi-one-dimensional<br />
ferromagnet so it is permissible to ignore the variation of Qa with energy<br />
transfer, a fact confirmed by more detailed analysis of these results[16]. By<br />
carrying out measurements with incident energies of 80, 120, 300, 500, and<br />
700 meV we have fully mapped out the c-axis dispersion relation which is<br />
shown in Figure 5. The line is the result of a fit to a planar one-dimensional<br />
ferromagnet model in appropriate in the zero temperature classical limit. The<br />
zone boundary magnon energy is 400 meV, more an order of magnitude larger<br />
than the 25 meV found for the a-axis[14]. This substantial anisotropy reflects<br />
the structure of FeGe2, the Fe-Fe distance in the c-axis chains of Fe spins is<br />
similar to that in pure Fe, whereas in the ab-plane the distance between the<br />
Fe ions is 4.2 ˚A. Along the c-axis the interaction is ferromagnetic and mainly<br />
due to d-orbital overlap while in the basal plane the weaker antiferromagnetic<br />
coupling has its origins in the conduction electrons.<br />
FeGe2 is a unique example of quasi-one-dimensional ferromagnetism in an<br />
itinerant electron system. Studies of the magnetic dynamics at higher temperatures[16]<br />
reveal the presence of well resolved magnon-like excitations even<br />
in the paramagnetic state. The much weaker basal plane coupling drives the<br />
three dimensional ordering, the doping dependence of the ordering wavevector<br />
suggests that this is Fermi surface driven, hole doping by substituting Ga for<br />
Ge increases the incommensurate wavevector similar to the effect of increased<br />
Sr doping in La2−xSrxCuO4 [1] or V doping in CrxV1−x[2]. which are quasi-twodimensional<br />
and three-dimensional realizations of systems which exhibit Fermi<br />
surface driven spin density wave instability. The Fe(Ge1−xGax)2 alloy system<br />
represents another such system which is predominantly one-dimensional magnetically<br />
although the relatively small anisotropy in the resistivity (a factor<br />
of two with higher resistivity along c) indicates the charge carriers are not<br />
restricted to the c-axis.<br />
5
We would like to thank the technical staff of ISIS and Chalk River Laboratories<br />
for expert assistance with the neutron scattering measurements described in<br />
this paper. This work was financially supported by NSERC and the CIAR.<br />
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6