Photon Polarization Precession Spectroscopy for High-Resolution Studies of Spinwaves
A new type of spectroscopy for high-resolution studies of spin waves that relies on resonant scattering of hard x-rays is introduced. The energy transfer in the scattering process is encoded in the precession of the polarization vector of the scattered photons. Thus, the energy resolution of such a spectroscopy is independent of the bandwidth of the probing radiation. The measured quantity resembles the intermediate scattering function of the magnetic excitations in the sample. At pulsed x-ray sources, especially x-ray lasers, the proposed technique allows to take single-shot spectra of the magnetic dynamics. The method opens new avenues to study low-energy non-equilibrium magnetic processes in a pump-probe setup.
pacs:07.85.Nc, 78.70.Ck, 75.30.Ds, 75.25.-j
The enormous potential of fast spin manipulation for applications in information storage,
processing and retrieval stimulates a growing interest in the excited
states and non-equilibrium properties of magnetic structures.
The elementary quanta of excitations in an ordered ensemble of magnetic moments are magnons, also
known as spinwaves when described classically.
Magnetic excitations are of particular interest in magnetic
systems with competing interactions. For example, geometrically frustrated magnets
exhibit persistent magnetic excitations even at lowest temperatures
with most of their spectral weight shifted towards low energies
GDG*99 (); Ehl06 (). This has been shown for crystalline systems DAA*11 (); BSS12 () and
remains an interesting subject to be studied in artificially
structured systems MSL*10 (); MHR*11 ().
Moreover, the emerging field of spinwave engineering a.k.a magnonics NG09 (); KDG10 (); LUGM11 (); DS13 ()
relies on the understanding of low-energy magnetic excitations in nanostructured
For a precise measurement of the magnetic excitation spectrum, high-resolution spectroscopic techniques are required. In the optical regime, Brillouin light scattering allows to probe magnons with outstanding energy resolution SW79 (); HBG89 (); Hil00 (). The use of visible light, however, prevents the access to high momentum transfers and thus limits the range of accessible length scales. Dimensions down to interatomic distances can be reached via resonant inelastic x-ray scattering (RIXS) or inelastic neutron scattering (INS). While single-magnon spectroscopy with x-rays has been demonstrated just recently BBB*10 (), inelastic neutron scattering for magnon studies is an established technique since decades. In all these methods the energy resolution is determined (and limited) by the energy spread of the incoming particles (assuming a perfect analyzer), thus further bandwidth reduction to achieve better energy resolution goes at the expense of the signal to noise ratio. Due to limited instrumental resolution RIXS is restricted to energy transfers above 50 meV, so that the low-energy regime of magnetic excitations is still the domain of INS.
In fact, a very elegant decoupling of the energy resolution from the bandwidth of the probing particles has been achieved in the method of neutron spin echo (NSE) spectroscopy Mez72 (). Small energy transfers upon inelastic scattering are encoded as phase shift in the precessing polarization of the neutrons. In combination with momentum-resolved triple-axis spectrometry Pyn78 (), the dynamical structure factor of magnons can be effectively probed with eV resolution BKHK06 (); NKM*11 () although the energy bandwidth of the incident neutrons is much larger. As a time-of-flight method, the neutron spin echo technique relies on the finite rest mass of the neutron. Therefore, at first sight, this technique does not seem to be directly transferable to photons.
In this Letter we introduce a new type of inelastic x-ray spectroscopy
to probe magnetic excitations that exhibits
similarities to NSE, the basic principle of which is illustrated in Fig. 1.
The technique described here relies on resonant magnetic scattering of x-rays
in the presence of x-ray linear dichroism (XMLD). Under these
conditions a magnetic sample
with a spinwave of frequency exhibits exhibits properties of a half-wave
plate rotating with frequency . A half-wave plate
reverts the angular momentum of incident circularly
polarized light, thus it constitutes the angular variant of a mirror that reverses the
linear momentum of the light that is backreflected from it. In the
same fashion as photons reflected from a moving mirror experience a
linear Doppler shift of , the photons transmitted
through a rotating half-wave plate experience an angular Doppler
effect SW78 (); GA79 (); Gar81 (); BGS*94 (); BB97 (); CRD*98 () of , with and being linear and
angular momentum transfer, and and being linear and
angular velocity, respectively. Note, that a necessary
condition for the linear and angular Doppler effect is either a broken
translational invariance (interfaces, lattice planes) or a broken
rotational invariance (anisotropy, optical or magnetic axes),
respectively. While the linear Doppler effect forms the basis for
vibrational spectroscopies, we will exploit here the angular Doppler
effect for a new type of spectroscopy in the x-ray regime to probe
spinwaves within a frequency range reaching up to 100 Ghz.
In the following we evaluate the scattering of linearly polarized x-rays from magnetic samples that carry spinwave excitations. We first concentrate on small momentum transfers so that we can use the forward scattering amplitude to describe the scattering process that will be treated in the kinematical approximation. The spinwave is represented by a magnetization vector that precesses on a cone around the direction of the effective magnetic field, as illustrated in Fig. 1. The sample and optical elements like polarizers are described in the Jones matrix formalism by (22) matrices for a given polarization basis (e.g., linear or circular) with unit vectors . We assume the photon energy being tuned to a resonance (electronic or nuclear) that is sensitive to the sample magnetization which lifts the degeneracy of the magnetic sublevels due to a spin-orbit interaction of electronic levels or a magnetic hyperfine interaction of nuclear levels.
In an resonant inelastic scattering experiment at photon energy , the intensity scattered into an energy interval and solid angle is proportional to the double differential cross section :
where is the coherent atomic scattering amplitude for a given energy transfer , momentum transfer and change of polarization . is the dynamical structure factor of the spinwave with frequency and momentum . is derived from a matrix that accounts for the polarization dependence of the scattering process :
where the sum runs over all atomic species in the sample. For simplicity we drop the dependence in the following and assume that the scattering proceeds close to the forward direction. is then the 22 matrix of the coherent forward scattering length of the atomic species and is the number density of these atoms. It is convenient to decompose into a non-resonant part that describes electronic charge scattering (see supplementary material supplement ()) and a part that contains the contributions from resonant scattering :
With denoting the unit vector of the magnetization at the position of the atom, the resonant atomic scattering length for an electric dipole transition ( = 1) is typically written as :
with and being the energy dependent oscillator
strengths for resonant transitions between magnetic
sublevels with .
The three terms in Eq. (Photon Polarization Precession Spectroscopy for
High-Resolution Studies of Spinwaves) represent different
The first term is not sensitive to the sample magnetization.
The second term describes circular dichroism (XMCD) because it
depends on the difference between the resonant scattering
amplitudes and .
Since its polarization dependence is
, it describes orthogonal
scattering between the states in the polarization basis.
The third term is the important one here. It describes x-ray linear magnetic dichroism
(XMLD, see supplementary material supplement ()) and is responsible for the violation of rotational invariance that gives rise
to the angular Doppler effect on which the method relies.
It is convenient to express Eq. (Photon Polarization Precession Spectroscopy for High-Resolution Studies of Spinwaves) in terms of matrices within a circular polarization basis. With and we obtain for the scattering matrix (for derivation see supplementary material supplement ()) :
where is the unit matrix and
is the Jones matrix of a half-wave plate with the
fast axis oriented at an angle relative to the horizontal (see
In a spinwave, i.e., a collective motion of a large number of atomic magnetic moments in the sample, the magnetization performs precessional motion with angular frequency around the effective field as illustrated in Fig. 1. In order to describe the scattering process from such an ensemble of spins, we need to perform a transformation of the scattering matrix from the co-rotating frame into the fixed laboratory frame. This is accomplished via the transformation
is the expectation value of the operator that generates the precessional motion of the magnetization, given by (see supplementary material supplement ())
Applying this transformation to the scattering matrix in Eq. (Photon Polarization Precession Spectroscopy for High-Resolution Studies of Spinwaves) leaves the first two terms invariant, but changes and introduces a time dependence of the third one:
i.e. the spinwave acts like a half-wave plate that rotates with
angular velocity .
The scalar product accounts for the projection of the spin precession
cone on the incident wavevector. To simplify the following discussion,
we assume that
so that , and set .
In experiments with synchrotron radiation the incident field is usually linearly polarized in the horizontal plane, i.e., . To evaluate the amplitude of the scattered field , we write the incident horizontal polarization as superposition of left- and right-circular polarization, i.e., to obtain the following contribution from the third term in Eq. (Photon Polarization Precession Spectroscopy for High-Resolution Studies of Spinwaves) :
The left(right)-circular component of the incident linear polarization was converted into right(left)-circular polarization and shifted up(down) in frequency by . This means that the energy transfer from the spinwave to the scattered photon is encoded in the relative frequency shift of the two circular components. Note that this result is independent of the carrier frequency , so that the effective energy resolution of the method is decoupled from the frequency bandwidth of the incident radiation. Writing Eq. (9) in the linear basis yields
which shows that the polarization performs a precessional motion in space, illustrated in Fig. 2a.
Since two orthogonal polarizations do not interfere with each other this frequency shift is not directly observable. The interference can be induced, however, if the radiation is analyzed by a linear polarizer which projects parallel polarization components. The scattered field behind a vertical analyzer with Jones matrix is given by . Inserting given by Eq. (Photon Polarization Precession Spectroscopy for High-Resolution Studies of Spinwaves) with we obtain (see supplementary material supplement ()) :
Thus, the intense, horizontally polarized direct beam that was scattered by the sample
without interaction with the spinwave is completely blocked by the vertical linear
polarizer. The linearly polarized wave that is transmitted by the vertical
analyzer is modulated with a frequency of .
Inserting the above expression for given by Eq. (11) into Eq. (1) and integrating over all frequencies of energy loss and energy gain in the scattering process, we obtain for the intensity observed behind the analyser :
with and ,
where we have used that and
that being the static structure
factor for given .
The first term in Eq. (12) is
independent of and thus contributes a constant background to the
measured intensity. The second term contains information about the
spinwave dynamics. It resembles the intermediate scattering function
as it is obtained in neutron spin echo spectroscopy Mez72 ().
This quantity can also be obtained via time-domain
interferometry based on nuclear resonant scattering BFM*97 () but the
resolution and signal strength of this method
is governed by the intrinsic bandwidth of the nuclear resonant
For symmetric functions , however, the second term in Eq. (12) vanishes. This is typically the case in the limit under the condition of detailed balance at thermal equilibrium, i.e., , where the Stokes () and anti-Stokes () contributions in the spectrum are almost equal, i.e., for small energy transfers like in quasielastic scattering or at high temperatures. In magnetic systems, however, the condition of detailed balance is violated because it requires the system to possess time-reversal invariance. This is not the case in the presence of a magnetic field SRD*09 (): The equation of motion of the magnetization in the effective field enforces only a right-handed precession. The time-reverted state of a left-handed precession is not supported. Therefore one can expect a significant asymmetry in the dynamical structure factor of magnetic systems, leading to a non-zero value for the integral in Eq. (12). This asymmetry is further enhanced with decreasing temperature. Moreover, detailed balance is significantly violated for systems that are strongly driven out of thermal equilibrium as it is the case, e.g., for spinwaves that are excited by a microwave field. Since this scattering geometry with a vertical analyser enables polarization rejection ratios up to in a multiple reflection geometry TAS*95 (); AST00 (); MUH*11 (), a very strong suppression of the non-resonant and non-orthogonal scattering can be achieved so that even weak signals can be detected with good signal-to-noise ratio. This technique appears to be very attractive for studies at the L-edges of the rare earths that are constituents of materials with complex and unconventional magnetic properties and, due to their crystalline structure, should exhibit a substantial XMLD. For these energies in the range of 6 - 9 keV one finds Bragg reflections of Si or Ge with a Bragg angle close to 45 to ensure sufficiently high polarization rejection (see supplementary material supplement ()).
A case that will be frequently encountered in experiments is a spinwave with a wavevector that lies on the dispersion surface of the excitation with frequency , and Lorentzian lineshape with half-width . We assume that the Stokes/anti-Stokes asymmetry for that excitation can be expressed as . Inserting this into Eq. (12), we obtain
The exponential results from the fanning out of the photon polarization vectors
with increasing travel distance from the sample due to the distribution of spinwave frequencies, as illustrated in Fig. 2a.
Experimentally, the time is translated into the travel distance
of the photons after the scattering process, i.e., , so
that can be measured via a position
sensitive analyzer behind the sample, consisting of a strongly
asymmetrically cut crystal with a Bragg angle equal or close to
45 (the Brewster angle for hard x-rays), as illustrated in
Fig. 2b (see also supplementary material
supplement ()). Thus, one period of the spinwave is
mapped to a distance of . Assuming that the spatial
point spread function introduced by analyzer and detector has a width of m, and that about 10 sampling points are required to resolve one modulation
period, one finds that spinwaves up to a frequency of GHz can be detected. This
covers a wide range of magnetic dynamics that can be excited, e.g.,
via microwave fields.
It should be noted that the formalism laid out in this paper is based on a classical description. This approach is valid either when a large number of magnons is excited in the system, e.g., via pumping by an external stimulus or when (with being the exchange interaction constant) but . Since , this can be valid only when , i.e., for large spins. This applies for many of the rare earth (RE) elements that exhibit large magnetic moments close to that of a free ion. At the L-edges of RE compounds and transition metal oxides like cuprates and maganites one often finds a relatively large XMLD contribution where this spectroscopy relies on. Another very interesting realization of the classical limit are collective spinwave modes in nanoparticles in which at sufficiently low temperatures only the mode is populated where all spins precess in unison HLL93 (). These modes (that can also be excited by microwave fields GFJ*06 ()) lead to peculiar magnetic properties of antiferromagnetic nanoparticles FM04 ().
The scheme proposed here offers the possibility to record for a given in a single shot at x-ray free-electron lasers in combination with pump-probe schemes with unprecedented energy resolution. This allows to reveal magnetic microstates as they are populated, for example, during magnetic switching and reversal processes. If phase locked to a periodic excitation process, similar studies can be done in a stroboscopic fashion already at conventional synchrotron radiation sources. The combination with efficient micro- and nanofocusing of high-brilliance x-rays allows to uniquely access magnetic dynamics in low-dimensional systems as they are relevant in the field of spintronics and magnonics.
Acknowledgements.I acknowledge fruitful discussions with Guido Meier, Lars Bocklage and Liudmila Dzemiantsova.
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