Broadband Terahertz Probes of Anisotropic Magnetoresistance Disentangle Extrinsic and Intrinsic Contributions

Lukáš Nádvorník , Martin Borchert, Liane Brandt, Richard Schlitz, Koen A. de Mare, Karel Výborný, Ingrid Mertig, Gerhard Jakob , Matthias Kläui, Sebastian T. B. Goennenwein, Martin Wolf, Georg Woltersdorf, and Tobias Kampfrath 1,2 Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany Department of Physical Chemistry, Fritz Haber Institute of the Max Planck Society, 14195 Berlin, Germany Faculty of Mathematics and Physics, Charles University, 121 16 Prague, Czech Republic Institut für Physik, Martin-Luther-Universität, Halle, Germany Institut für Festkörperund Materialphysik, Technische Universität Dresden, 01062 Dresden, Germany Institute of Physics, Academy of Sciences of the Czech Republic, v.v.i., 162 00 Prague, Czech Republic Department of Applied Physics, Eindhoven University of Technology, Eindhoven 5612 AZ, Netherlands Institut für Physik, Johannes Gutenberg-Universität Mainz, 55128 Mainz, Germany


I. INTRODUCTION
The electrical resistance of a ferromagnet along the applied electric field is known to depend on the direction of the magnetization M [ Fig. 1(a)]. This anisotropic magnetoresistance (AMR) [1][2][3][4][5][6][7] is a well-studied magnetoresistive effect and a powerful tool to detect the magnetic order parameter of ferromagnets as well as ferrimagnets [8,9]. As AMR is even in the magnetic order parameter, it has received additional attention as a probe of the Néel vector of antiferromagnets [10]. Therefore, AMR has great potential for applications in future spintronic devices [11].
The canonic way to describe the origins of AMR relies on an extrinsic mechanism, that is, spin-dependent electron scattering due to crystal imperfections such as impurities and phonons. In transition metals, the M-dependent rate of electron scattering out of the current-carrying s-states is understood to arise from spin-orbit coupling, which reduces the symmetry of the target d-states [2,3,5,12]. Other extrinsic scenarios involve magnetic impurities acting on spin-orbit-coupled p-states (in, for instance, dilute magnetic semiconductors [13]) and nonmagnetic impurities acting on states with isotropic band dispersion but anisotropic wavefunctions [14].
Recently, theoretical works pointed out that AMR can already be significant in perfect crystals. An example of such an intrinsic (i.e., scattering-independent) mechanism is a change in the group velocity of Bloch states due to spinorbit coupling [15][16][17][18][19], which is a concept that extends to giant magnetoresistance [20]. First signatures of intrinsic contributions to dc AMR were reported [21,22] based on extensive electric transport measurements and ab initio theory. These highly promising results also show that more direct and versatile experimental methods are required to extract extrinsic and intrinsic AMR contributions.
To straightforwardly separate extrinsic (scatteringdependent) and intrinsic (scattering-independent) electron transport, we propose to probe the AMR dynamics on timescales that are both slower and faster than the timescale τ at which electron scattering takes place. To implement this idea, AMR needs to be measured over a wide frequency range from dc to several 10 THz. The lower frequencies ω=2π of this interval probe diffusive (i.e., scattering-dominated) transport in which an electron undergoes many collisions during one oscillation of the probing electric field (ωτ ≪ 1). In contrast, the frequencies at the higher end are more sensitive to electron motion in the ballistic limit (i.e., without scattering) because the probing electric field oscillates many times between subsequent electron collisions (ωτ ≫ 1). So far, measurements of AMR of common magnets were reported at either dc, at frequencies around 1 THz (Refs. [23][24][25]), or in the infrared, where AMR is usually referred to as magnetic linear birefringence [26,27].
In this work, we measure AMR of common ferromagnets in the regime of both ballistic and diffusive electron transport by means of low-noise broadband THz spectroscopy from dc to 28 THz. A frequency-resolved data analysis based on Boltzmann transport theory allows us to robustly separate the (A) extrinsic and (B) intrinsic components of AMR. We find that component (B) is significant and even dominates the total AMR for the case of polycrystalline Co. Using numerical estimates, we attribute this observation to the hexagonal structure of the Co crystallites. Owing to its instantaneous response up to at least 28 THz, the intrinsic AMR of Co is highly interesting for applications in future THz spintronic devices. Our results also highlight that broadband THz AMR is a powerful and versatile probe of ultrafast spin dynamics.

II. AMR IN THE DRUDE-BOLTZMANN FRAMEWORK
The AMR contrast at frequency ω=2π is defined as where σ j ðωÞ is the conductivity for the magnetization M parallel (j ¼ k ) or perpendicular (j ¼ ⊥ ) to the applied electric-field amplitude EðωÞ. Typically, the dc AMR contrast (ω ¼ 0) is positive and reaches values of the order of 1% to 10% (Ref. [2]). Therefore, the difference Δσ ¼ σ k − σ ⊥ is relatively small, and σ k and σ ⊥ are very close to the mean conductivityσ ¼ ðσ k þ σ ⊥ Þ=2. We note that the σ j ðωÞ, like all frequency-domain quantities, are generally complex valued. In contrast to the anomalous Hall effect (which is of first order in M), there are significantly less theoretical studies of the microscopic mechanism of AMR (which is quadratic in the order parameter). A frequently used theoretical approach is based on the Boltzmann equation describing intraband transport [14,17,21,28,29]. Assuming state-independent relaxation rates, one can derive the Drude formula [30][31][32][33] where j ¼ k or ⊥ , ω=2π is the frequency of the driving field, σ j ð0Þ equals the dc conductivity, and τ j is the current relaxation time. The second part of Eq. (2) is a rewritten Drude formula, where Z 0 ≈ 377 Ω is the free-space impedance, c is the speed of light, and Ω plj =2π is the plasma frequency. This formulation allows us to identify (A) extrinsic contributions (due to electron scattering) and (B) intrinsic (scatteringindependent) contributions to the AMR contrast. Inequality σ k ≠ σ ⊥ of the k and ⊥ conductivities and, thus, AMR can arise from the M-direction dependence of (A) the current relaxation time (τ k ≠ τ ⊥ ) and (B) the plasma frequency (Ω pl k ≠ Ω pl ⊥ ). Because Ω 2 plj is given by a summation of the squared electron-group-velocity component j over the Fermi surface [34][35][36][37][38], it is a measure of the weight of intrinsic (scattering-independent) contributions to the conductivity. In contrast to Ω 2 plj , the velocity relaxation rate τ −1 j arises from electron-impurity and electron-phonon collisions and, thus, captures extrinsic (scattering-related) effects. Note that previous studies using the Boltzmann approach ascribed AMR to contribution (A) [2,28].
We expect that in the diffusive transport regime (ωτ j ≪ 1), both extrinsic and intrinsic effects contribute to AMR, whereas in the ballistic regime (ωτ j ≫ 1), intrinsic contributions dominate. To analyze this expectation quantitatively, we substitute the Drude formula [Eq. (2)] into the definition of the AMR contrast [see Eq. (1) and Appendix B]. The resulting relationship, has remarkable implications: First, the two terms on the right-hand side scale with A ¼ −Δτ=τ ⊥ and B ¼ −ΔðΩ 2 pl Þ=Ω 2 pl ⊥ , where Δ always refers to the difference of the k and ⊥ component-for instance, Δτ ¼ τ k − τ ⊥ . Therefore, the A and B terms, respectively, quantify the strength of the (A) extrinsic (scattering-based) and (B) intrinsic (scattering-independent) contribution to AMR. LUKÁŠ NÁDVORNÍK et al.
PHYS. REV. X 11, 021030 (2021) 021030-2 Second, as expected, components (A) and (B) exhibit a distinctly different frequency dependence. The extrinsic contribution (A) rolls off with frequency just as the conductivity σ ⊥ [Eq. (2)] does. The frequency scale of this decrease is set by the velocity relaxation rate τ −1 ⊥ =2π, which is typically of the order of 10 THz [39,40]. The intrinsic contribution (B), in contrast, is ω independent, thereby making it interesting for potential high-frequency applications in THz spintronics.
Finally, Eq. (3) shows us how to determine the weight of the two AMR contributions: We need to conduct a sufficiently broadband AMR measurement. Therefore, our goal is to measure the anisotropic conductivity of common ferromagnets over the broad range from ω=2π ∼ 0 to tens of THz.

A. Samples
As samples, we chose thin films of common ferromagnetic metals with in-plane magnetic anisotropy: Ni 81 Fe 19 (thickness of 8 nm), Ni 50 Fe 50 (10 nm), Ni (10 nm), and Co (10 nm). As detailed in Appendix A and Supplemental Material [41], they were grown by sputtering on isotropic Si substrates. Subsequently, the samples were cut in two pieces to enable the dc and THz experiments. For the THz experiments, part of the substrate was not covered by the metal layer to permit reference transmission measurements to extract the THz conductivity.
All thin films were prepared in the polycrystalline phase. As compiled in Table I, they consist of crystallites having cubic (Ni 81 Fe 19 , Ni 50 Fe 50 , Ni) or hexagonal symmetry (Co). Because the size of the randomly oriented crystals is orders of magnitude smaller than the wavelength of the probing THz radiation, all films are macroscopically isotropic in the sample plane in the absence of magnetic order (M ¼ 0). A symmetry analysis of our samples (see Appendix C) shows that the in-plane conductivity tensor is fully determined by the two conductivity values σ k and σ ⊥ parallel and perpendicular to the magnetization, independent of the sample azimuth. The difference σ k − σ ⊥ equals 2hGi xyxy M 2 , where hGi xyxy is the only relevant element of the rotationally averaged AMR tensor.
B. Conductivity and AMR measurements dc measurements.-In general, determination of the AMR contrast [Eq. (1)] of our samples relies on measuring the ratio of the conductivities for M k E and M ⊥ E. At dc frequency, this goal was achieved by a four-point approach [42]. A rectangular piece was cleaved from the sample and contacted in the corners. A constant current was applied along the longer side, and the voltage drop and, thus, resistance RðαÞ along this fixed direction were measured as a function of the angle α of the in-plane magnetization M [see Fig. 1(a)]. The AMR contrast [Eq. (1)] is given by −Δσ dc =σ dc ¼ ΔR=R, where ΔR andR are, respectively, the modulation depth and mean value of RðαÞ.
Note that the mean dc conductivityσ dc of the metal film is related toR through an unknown factor that is given by the current distribution. Consequently, we used the van der Pauw method [43,44] to measureσ dc .
THz measurements.-To determine the AMR contrast of our samples at THz frequencies, we did not use any electrical contacts. Instead, we measured the transmission of a broadband THz electromagnetic pulse through the specimen in a quasioptical manner [ Fig. 1(b)]. To this end, THz pulses were obtained by difference-frequency generation of femtosecond laser pulses (duration of 10 fs, center wavelength of 800 nm, energy of 1 nJ) from a Ti:sapphire laser oscillator (repetition rate of 80 MHz) in a suitable nonlinear-optical material. The THz pulses were linearly polarized along the x axis and normally incident onto the sample [see Fig. 1(b) and Fig. S1(a) in Ref. [41] for more details]. After transmission through the sample, a wire-grid polarizer projected the THz field E onto the x axis, that is, the polarization direction of the incident THz electric field. The THz pulses were detected by electro-optic sampling using a suitable electro-optic crystal [45]. The resulting THz signal S vs time t is related to the THz electric-field component E x ðtÞ directly behind the sample [ Fig. 1(d)] by a linear transfer function that cancels in the subsequent data analysis.
To ensure optimum frequency coverage and signal-tonoise ratio, we used various combinations of THz sources and detectors. For AMR measurements, we used a bias-free bimetallic emitter (TeraBlast, Protemics GmbH) and a 1-mm-thick ZnTe(110) crystal as the detector for the range 0. 2  -2 THz, while a 90 μm thick GaSe emitter and a 10 μm thin ZnTe detection crystal were employed for the range 8-28 THz. This combination delivers sufficient THz signal amplitude to resolve the small AMR-induced changes of the sample transmission upon rotation of the magnetization from 0°to 90°.
For measurement of the mean (diagonal) conductivityσ, where signal amplitudes are sufficiently large, we replaced the bimetallic emitter by a spintronic THz emitter [45] (TeraSpinTec GmbH) and used a 250 μm thick GaP crystal as a detector. This combination delivers an order-ofmagnitude less signal amplitude but covers the range 1-6 THz, which is useful for the precise determination of parameters of the Drude formula [Eq. (2)].
Typical examples of transmitted THz signal waveforms are shown in Fig. 1(d). As detailed in Sec. IV and Appendix A, the measured THz transmission signals can be used to determine the mean THz conductivityσ of a thin metallic layer. Similarly, by modulating the magnetization angle α between 90°and 0°, we can infer the THz AMR contrast.

C. Magnetization control
Slow modulation.-The magnetization angle α relative to the fixed direction of the applied dc or THz electric field [see Figs. 1(a) and 1(b)] was controlled by a suitable external magnetic field. For the dc measurements, we used a magnetic field of 1.1 T from a Halbach array of permanent magnets that was slowly rotated about the sample [see Fig. S1(b) in Ref. [41] ]. For the THz measurements as a function of all magnetization angles α between 0 and 360°, we employed a rotatable pair of permanent magnets with a field of approximately 40 mT at the sample position [ Fig. S1(c) in Ref. [41] ]. Fast modulation.-To drastically enhance the signal-tonoise ratio of the THz AMR measurements, we modulated the magnetization angle α at kilohertz rates by superimposing a sinusoidal ac magnetic field (frequency of 6 kHz) from an electromagnet and a perpendicular dc magnetic field from a permanent magnet [ Fig. S1(d) in Ref. [41] ]. As the two fields had an amplitude of approximately 30 mT at the sample position, the magnetization angle α was varied between α min ≈ 0°and α max ≈ 90°, that is, between approximately parallel and perpendicular to the polarization of the THz wave [see Fig. 1(b)]. Lock-in-type phase-sensitive demodulation of the THz signal allowed us to extract its magnetic-fieldinduced and, thus, AMR-induced signal variations.
The magnetic-field strength of the 6-kHz arrangement was sufficient to fully saturate the sample magnetization, as confirmed by measuring the magnetization direction by THz emission spectroscopy [45]. Note that the expected AMR signal is determined by the sample magnetization rather than the external magnetic field, which induces only isotropic and, thus, negligible conductivity changes [2,46,47]. Therefore, our various methods of magnetization modulation deliver conductivity modulations that can be directly compared to each other.

A. Impact of magnetization direction
To study the sample conductivity as a function of the magnetization angle α [ Fig. 1(b)], we vary the direction of the external magnetic field by the slowly rotating permanent magnets. Figure 1(c) shows the measured dc resistance of the Ni 81 Fe 19 thin film vs α. We observe the typical cos 2 α-like resistance modulation that is expected for samples described by two conductivities σ k and σ ⊥ (Ref. [2]). Indeed, a fit by R ⊥ þ ΔRcos 2 α yields excellent agreement with the experimental data. From the modulation depth and the average resistanceR ¼ ½Rð0°Þ þ Rð90°Þ=2, we estimate an AMR contrast ΔR=R ≈ −Δσ dc =σ dc of approximately 1%. The dc AMR data for the other samples are shown in Fig. S2 of Ref. [41], while the fit parameters are displayed in Table I.
We now turn to the THz measurements. Figure 1(d) displays the signals Sðt; αÞ of THz waveforms after traversal of the sample for α ¼ 0°and 90°. For these measurements, the emitter-detector pair covering the range 0.2-2 THz is used. While the two signals are nearly identical, a magnified plot around the signal minimum reveals that the signal for α ¼ 0°has larger amplitude than the signal for α ¼ 90°. This observation is consistent with the dc measurements [ Fig. 1(c)] and Eq. (A2): Changing the magnetization angle from α ¼ 0°to 90°yields a smaller sample resistance and, thus, larger conductivity, resulting in better screening of the incident THz field and, therefore, in a smaller THz field amplitude behind the sample.
To complete the picture, we determine the peak-to-peak amplitude of all THz signals Sðt; αÞ as indicated by the two black arrows in Fig. 1(d). The resulting THz peak-to-peak amplitude is displayed in Fig. 1(e) as a function of the magnetization angle α. It exhibits the same α dependence and comparable contrast as the dc resistance [ Fig. 1(c)]. Again, a cos 2 α fit yields excellent agreement with the experimental data [ Fig. 1(e)]. We explicitly confirm that the α-dependent signal component disappears when either (i) test samples without a magnetic layer are used, (ii) the strength of the magnetic field is lowered below a critical value, or (iii) the THz beam is blocked.
We conclude that the α-dependent THz signal arises from the anisotropic conductivity of the magnetic thin film under study. As the α dependence and relative magnitude of this signal [ Fig. 1(e)] coincide with that of the dc AMR signal [ Fig. 1(c)], we assign the α-dependent THz signal modulation to the AMR effect at THz frequencies.
B. THz AMR differential spectra To enable spectral analysis of the THz AMR with a strongly increased signal-to-noise ratio, we modulate the magnetization angle α at a frequency of 6 kHz between α min ≈ 0°and α max ≈ 90°[see Fig. 1(b)]. By demodulation with a lock-in-type technique, we obtain the difference signal while in a separate measurement, the mean signal  [39,40]. In contrast, the AMR contrast of Co remains constant up to 30 THz.

C. From signals to conductivities and AMR
Mean conductivities.-To better understand these observations, we also determine the mean conductivityσ of our samples at 1-6 THz and 8-28 THz (see Sec. III B). For this purpose, we measure the signalsSðtÞ [mean signal of Eq. (5) with respect to the full sample] and the reference signal S ref ðtÞ corresponding to transmission through the plain substrate in sample regions without a metal film. Using the Tinkham formula [48] (see Appendix A), we obtain the mean conductivity of the metal layer.
The real and imaginary parts of the mean conductivityσ vs frequency ω=2π are displayed in Figs. 3(a)-3(d) (top panels) along with the dc conductivity σ dc for all four samples studied. We note that the measured dc conductivity agrees well with the THz mean conductivity between 1 and 4 THz.
To gain access to microscopic parameters, we fit the measured conductivities using the Drude formula [Eq. (2)]. As shown by the solid lines of Figs. 3(a)-3(d), the Drude-Boltzmann framework provides a very good description of our experimental data over more than two frequency decades. Broadband Drude-like behavior of metals is quite common and was previously observed also for other magnetic thin films [49][50][51][52], magnetic multilayers [33], and nonmagnetic metals [30,53]. The best-fit parameters of our data, the mean zero-frequency conductivityσð0Þ and the mean scattering rateτ, are summarized in Table I. Again, we obtain a good match between the measured dc conductivity σ dc and the zero-frequency extrapolation σð0Þ. The current relaxation timesτ are found to be of THz signal (arb. units)
AMR contrast.-To infer the AMR contrast [Eq. (1)], we use theσðωÞ as determined by the fits above and the relationship (see Appendix A) Here, n S and n A are the frequency-dependent refractive indices of air and substrate, and d is the thickness of the metal layer. The modulus jAMRðωÞj is displayed in Fig. 3 (bottom panels) vs frequency for all materials investigated. We see that the AMR contrast is approximately frequency independent for ω=2π < 2 THz with magnitudes ranging from 0.3% (Ni) up to 1.6% (Ni 81 Fe 19 ). These values are compatible with the dc quantities obtained by contact-based measurements within the uncertainties of our methodology. The various error sources are discussed in Appendix A. We do not attempt to determine the phase of the AMR contrast because the signals ΔS andS are taken at different times. Therefore, the complex-valued ratio ΔSðωÞ=SðωÞ may be subject to an unknown phase shift, which does not allow us to determine the phase of AMRðωÞ through Eq. (6). We emphasize that this lack of information is, however, no issue because the modulus of AMRðωÞ is fully sufficient to determine the ratio B=A of intrinsic and extrinsic AMR contributions as shown in the following.  3)]. For ω=2π > 2 THz, we find that the AMR contrast decreases by about 50% from 10 to 20 THz for both Ni 81 Fe 19 and Ni 50 Fe 50 . The slope of this decrease is similar to that of the conductivity Reσ. This observation and the discussion following Eq. (3) suggest that the AMR contribution (A) is dominant for these films. In contrast, for Co, we find an AMR decrease of less than 10% from 10 to 20 THz, although the conductivity rolls off by more than 50% in this range. This finding is consistent with Fig. 2

(f). Along with Eq. (3), it indicates that the AMR of Co has a significant frequency-independent contribution (B).
To address the last point quantitatively, we determine the weights A and B of the two AMR contributions (A) and (B) by fitting Eq. (3) to the measured jAMRðωÞj (Fig. 3). Here, A, B are the only fit parameters, whereas the value of the scattering time τ ⊥ ≈τ is fixed by our analysis of the mean conductivity. Fitting is performed over both the full frequency range 0.2-28 THz and the high frequency range 8-28 THz (see Supplemental Material [41]). With both procedures, we obtain excellent and consistent agreement of measured data and fits for all four investigated materials (Fig. 3).
The relevant parameters are summarized in Table I. We find very small ratios B=A of the order of 10 −3 for Ni 81 Fe 19 , Ni 50 Fe 50 , and Ni. According to Eq. We witness a strongly contrasting behavior for our Co thin film. A fit without the presence of an intrinsic contribution [B ¼ 0 in Eq. (3)] yields a curve with significantly larger slope above 8 THz [grey dotted line in Fig. 3(d)], which agrees poorly with experimental data. A fit without this constraint results in very good agreement of Eq. (3) with the measured modulus of the AMR contrast for B=A ¼ 2.5 AE 1.5. Therefore, the intrinsic contribution to the AMR contrast [red dashed horizontal line in Fig. 3(d)] is a factor of about 2 larger than the extrinsic component. At the same time, Co exhibits a THz AMR of 0.7%, only 50% smaller than that of Ni 81 Fe 19 , which turns out to have the largest THz AMR of the four materials studied here. Thus, we have found direct experimental evidence for intrinsic contributions to AMR in a common ferromagnet.

V. DISCUSSION
To summarize, we successfully measure AMR of thin films of the standard ferromagnets Ni 81 Fe 19 , Ni 50 Fe 50 , Ni, and Co from dc to 28 THz. Our data can be excellently described by the Drude formula for the conductivity parallel and perpendicular to the sample magnetization. We identify two distinctly different contributions to AMR: (A) a frequency-dependent extrinsic component due to magnetization-dependent electron scattering and (B) a frequency-independent intrinsic component arising from magnetization-dependent electronic group velocities. While contribution (B) is usually neglected in Boltzmanntype models of AMR [2,3,5,12], it can be significant already at dc and even dominate the AMR above 20 THz in Co.

A. Origin of the intrinsic AMR of Co
The question arises as to why contribution (B) to AMR is much larger in Co than in Ni, Ni 50 Fe 50 , and Ni 81 Fe 19 . We ascribe this distinctly different behavior to the crystal symmetry of the materials studied here. While crystalline Ni, Ni 50 Fe 50 , and Ni 81 Fe 19 are cubic (fcc, point group m3m), Co has hexagonal symmetry (hcp, point group 6/mmm). The lower symmetry of Co allows for different values of observables for directions parallel and perpendicular to the c axis. Examples include the refractive index (making Co optically anisotropic already for M ¼ 0), the electron orbital angular momentum, and spin-orbit coupling energies [54].
The strongly anisotropic spin-orbit coupling strength implies that the electronic band structure changes substantially when the magnetization M is parallel or perpendicular to the c axis. Therefore, the squared plasma frequency Ω 2 plj , which is a summation of the squared electron-group-velocity component j over the Fermi surface [34,35,38], should change strongly as well.
We put this expectation to the test by numerically estimating the weight B of the scattering-independent component (B) of the conductivity, that is, the Mdependent variation of the squared plasma frequency Ω 2 plj of Co [see Eq.
(3) and Appendix C]. Preliminary results indicate that when M is tilted out of the basal plane into a direction parallel to the c axis (local z axis) of Co, the plasma frequency Ω plz decreases by a value of the order of 4%. In contrast, the calculations for Ni indicate that the plasma frequency varies significantly less than 1% as a function of the magnetization direction. Our numerical estimates, thus, confirm the expected variation of the plasma frequency Ω plz when M is rotated out of the basal plane of Co.

B. Impact of polycrystallinity
We note that the samples of our experiment are polycrystalline. In a simplified picture, one can imagine this situation as an ensemble of three subsets of Co crystallites whose c axes point along either the x, y, or z axis with the same probability of 1=3. For simplicity, we assume that only the magnetization component along the c axis will modify the conductivity. When the driving THz field E is applied along the z direction and the resulting current density j is measured along E, the relevant conductivity σ zz changes only due to those crystallites whose c axis is parallel to the z axis. Therefore, the current density along E will change when M is rotated from M k E to M ⊥ E, and at least part of the AMR of the crystallites is inherited by the polycrystalline sample.
In a more rigorous way, the polycrystallinity of the sample can be taken into account by averaging the conductivity tensor over all crystal orientations while keeping the magnetization M fixed. Equivalently, one can perform a rotational average of the AMR tensor G jklm ¼ ð1=2Þ∂ 2 σ jk =∂M l ∂M m (see Appendix C). The elements G jjll are proportional to the change in Ω 2 pll with respect to M 2 l . While the refractive index of polycrystalline Co in the absence of magnetic order (M ¼ 0) becomes completely isotropic, the AMR, to a large extent, survives the rotational averaging process. For polycrystalline Co, we estimate the scattering-independent AMR contrast by a linear combination of the numerically estimated tensor elements G jjll . As detailed in Appendix C, we obtain an AMR contrast of ð0.8 AE 0.5Þ% for Co and 0% for Ni, which is in excellent agreement with the measured scattering-independent contribution of B ¼ ð0.5 AE 0.1Þ% and ð0 AE 0.07Þ% (Table I), respectively.

C. Role of interband transitions
The Drude-Boltzmann theory of AMR and other electronic transport phenomena relies on intraband transitions: The probing THz field, possibly in conjunction with a phonon or an impurity, causes an electron to scatter from one Bloch state into another in the same band of the electronic band structure. Above a certain probing frequency, however, interband transitions-that is, transitions between different bands-become operative.
Intraband transitions can often be well described by the Drude formula [Eq. (2)], and their contribution to the conductivity decays with 1=ω for large enough frequencies.
For interband transitions, we expect a different frequency dependence of the conductivity than for intraband transitions. Such a crossover from intraband to vertical (i.e., wavevector-conserving) interband transitions was, for example, observed for the semimetal graphite already at frequencies between 10 and 20 THz (Ref. [55]).
In our conductivity spectra (Fig. 3), however, we do not observe indications of interband transitions because we are able to describe all measured curves well by the simple Drude formula [Eq. (2)] over the full frequency range 0-28 THz. For Ni, this notion is consistent with earlier work [49] in which the onset of interband transitions was found at a photon energy of 0.15 eV (corresponding to 36 THz), which is outside the frequency range considered here. Similarly, for Co and Fe, previous studies report that the lowest interband transitions are at 0.18 eV (44 THz) [56,57] and 0.20 eV (48 THz) [50,51]. We conclude that in the materials studied here, intraband transitions dominate the response at least up to 30 THz. Therefore, our insights into intrinsic and extrinsic AMR contributions at THz frequencies can directly be transferred to the dc AMR.

VI. CONCLUSIONS
In conclusion, low-noise broadband THz spectroscopy enables one to measure AMR from about 0.2 to tens of THz. The wide bandwidth provides access to important transport parameters. Our measurements reveal extrinsic and sizable intrinsic contributions to the AMR contrast, thereby providing new and surprising insights into a mature effect.
Polycrystalline Co exhibits a sizable intrinsic contribution, which can consistently be ascribed to the crystalline anisotropy of the hexagonal (hcp) structure of Co crystallites. Our interpretation is supported by rotational averaging of the AMR tensor and numerical estimates. It highlights a strategy to identify materials with a large intrinsic AMR contribution, which is relevant for potential broadband THz spintronic applications.
Probing of the intrinsic AMR component is also highly interesting from a spectroscopic viewpoint because it reports on magnetic-order-induced variations of the electronic band structure. We anticipate that broadband THz AMR will be a highly useful, versatile, and ultrafast probe of all flavors of magnetic order and transport parameters of spintronic materials. It can be applied to standard thin films, both crystalline [22] or polycrystalline, and under ambient conditions without the need for microstructuring and contacting. In particular, THz AMR should also be applicable to metallic antiferromagnets such as CuMnAs and Mn 2 Au, which have recently moved into the focus of spintronics research [11,58].
As our THz radiation is pulsed, THz AMR can be measured with a time resolution down to 100 fs. This feature opens up the exciting possibility to monitor material-relevant parameters on the timescales of spin, electron, and lattice dynamics [59]. In this way, THz AMR complements other recently developed ultrafast spintronic techniques such as the THz anomalous Hall effect [52,60,61], THz tunnel magnetoresistance [25], THz giant magnetoresistance [33], and magnetization-dependent THz emission [45,[62][63][64][65], which have provided new insights into the dynamics of spin transport and spin-to-chargecurrent conversion. Finally, because the THz range coincides with a variety of excitations (such as phonons and magnons), the method presented here allows us to study the impact of such resonances on magnetotransport at their natural frequencies.

ACKNOWLEDGMENTS
We acknowledge funding by the German Research Foundation through collaborative research centers SFB TRR 227 "Ultrafast spin dynamics" (projects A05, B02, and B04) and SFB TRR 173 "Spin+X"/Project No. 268565370/TRR173 (projects A01 and B02). We thank the ERC for support through the Horizon2020 projects CoG TERAMAG/

Sample growth and characterization
All samples were deposited by sputtering or thermal evaporation techniques [66] on thermally oxidized SijSiO 2 ð100 nmÞ substrates. Details on growth and characterization can be found in the Supplemental Material [41]. In brief, the Ni 81 Fe 19 thin film (thickness of 8 nm) was grown by dc sputtering (sputter power of 800 W, Ar pressure of 0.5 Pa). X-ray diffraction (θ-2θ scans) reveals a very weak (111) reflection, indicating a crystallite size of about 3 nm. A weak (220) reflection confirms the polycrystalline growth of the sample. The layer thicknesses were inferred from the X-ray reflectometry measurements.
The Ni 50 Fe 50 (10 nm) layer was deposited by dc magnetron sputtering (sputter power 30 W, Ar pressure 0.4 Pa). After the sputtering process, the sample was capped with an MgO (7 nm) layer grown by in situ molecular beam epitaxy (MBE) and electron-beam evaporation and by an Al 2 O 3 (5 nm) layer by ex situ atomic-layer deposition. The Co (10 nm) film was grown by thermal evaporation in ultrahigh vacuum (MBE) and capped by MgO (5 nm) and Al 2 O 3 (5 nm) using the techniques described above. The Ni (10 nm) film was prepared by thermal evaporation in a vacuum chamber and capped by Al (3 nm), which fully oxidizes under ambient conditions [67]. The crystal structure of the Co film was monitored during growth using reflection high-energy electron diffraction. We find a polycrystalline hcp structure with random crystal orientation.

THz conductivity measurements
Our sample system is a stack SjFjA consisting of a metal thin film F (thickness d) between substrate S (refractive index n S ) and air A (refractive index n A ). To determine the conductance of F, we conduct transmission measurements [see Fig. 1(b)]. In a first measurement, we characterize the THz field E directly behind the F layer. As the field E inc incident on the sample is unknown, we conduct a second measurement on a reference sample SjRjA, where the sample film F is replaced by a reference film R with known refractive index. In practice, the reference measurement is performed in sample regions where no metal film is deposited. Thus, our reference material is air (R ¼ A), whose refractive index equals 1 to a very good approximation.
In our setup, the field E inc ¼ E inc u is normally incident on the sample and linearly polarized parallel to the vertical unit vector u [ Fig. 1(b)]. The field E behind the sample is projected onto the same direction u by means of a polarizer. Thus, we measure a signal S which is related to the projection u · E through the transfer function of our setup. Likewise, the signal S ref from the reference sample is obtained. By dividing the signals SðωÞ and S ref ðωÞ in the frequency domain, the setup transfer function is canceled. As derived in Appendix B, the ratio SðωÞ=S ref ðωÞ is related to sample-intrinsic parameters by where Z ref ¼ Z 0 =½n S ðωÞ þ n A ðωÞ is the impedance of the reference sample.
As the AMR contrast AMRðωÞ amounts to only a few percent, the second term in the square brackets of Eq. (A1) is much smaller than the first one. Consequently, we determine the first term by a simple transmission measurement through the sample averaged over all magnetization directions and through the reference without metal film. We obtain the familiar Tinkham formula [48] SðωÞ which implies that To measure the AMR-related term in Eq. (A1), we modulate α between 0 and 90°. We obtain ΔSðωÞ ¼ Sðω; 0°Þ − Sðω; 90°Þ, and thus, which is equivalent to Eq. (6) of the main text.

Error considerations
THz measurements.-The uncertainties of the fit param-etersσð0Þ,τ [Eq. (2)] and A, B [Eq. (3)] are given by the uncertainties of the signals ΔS andS and the statistics of the fit procedure. They are summarized in Table I and in  Table S1 of Ref. [41].
The precision ofS is estimated by the standard error of repeated measurements of this signal. To estimate the uncertainties of ΔS, two contributions are considered. The first one is the statistical error of ΔS that arises from the shot noise of our measurement. It is estimated by the constant noise floor outside the signal bandwidth of the THz-emitter-detector configuration used. An example of the noise floor for Ni 81 Fe 19 is shown in Fig. S6 in Ref. [41].
The second error contribution to ΔS arises from the finite precision with which the magnets for rapid modulation of the angle α of the external magnetic field could be positioned. As a consequence, the minimum angle α min and the maximum angle α max deviate from the target angles 0°and 90°, respectively. This systematic error only results in an overall rescaling of ΔS by an estimated upper limit of 30%. Importantly, it does not affect the frequency dependence of ΔS. It may, however, differ between the measurements in the ranges 0.2-2 THz and 8-28 THz, where the permanent magnet and the electromagnet were repositioned. This issue was tackled by fitting Eq. (3) over the full frequency range 0.2-28 THz and the higher range 8-28 THz only. We obtain consistent results, as summarized in the Supplemental Material [41], including Table S1.
dc measurements.-For the electrical measurements of σ dc by the van der Pauw method, the measurement error is of the order of 5% and predominantly arises from the nonvanishing size of the contacts and their positioning within the sample perimeter [42][43][44]. The error of the electrical measurement of the AMR contrast by our fourpoint approach is governed by the uncertainty of the direction of the current flow between electrical contacts, the directional homogeneity of the external magnetic field of the Halbach array, and the fit statistics of the raw data (see Fig. S1 of Ref. [41]).
Comparison of dc and THz.-From Fig. 3, we observe that the values of the THz AMR contrast below 2 THz are smaller than the dc AMR contrast for Ni 50 Fe 50 , Ni, and Co. This behavior can be explained by a deviation of α max − α min from 90°, which leads to a reduction of the measured THz AMR contrast. For Ni 81 Fe 19 , we observe the opposite behavior, which we believe arises from the different aspect ratios of the rectangular samples used for the dc AMR measurements. While the aspect ratio of the Ni 81 Fe 19 sample is close to 1∶1, it is roughly 4∶1 for the other samples. As a consequence, the current flow in the Ni 81 Fe 19 sample is less homogeneous, resulting in an apparently smaller measured dc AMR contrast. The following derivations refer to complex-valued quantities in the frequency domain. For the sake of simplicity, the argument ω is omitted.

Derivation of Eq. (A1)
Wave equation.-In our setup, the incident THz pulse propagates along the z axis, which is perpendicular to the sample plane [see Fig. 1(b)]. Therefore, z is the only relevant spatial coordinate, and we choose its origin such that the metal film F is located between z ¼ 0 and d. We assume that the substrate S and air A are optically isotropic and homogeneous and can thus be described by scalar refractive indices n S and n A , respectively. The metal thin film F, in contrast, is allowed to be inhomogeneous along z and is optically anisotropic. It is adequately described by the conductivity tensor (matrix) σðzÞ.
In frequency space, the THz field EðzÞ is determined by the wave equation [68] Here, Q ext quantifies the sample-external source of the incident THz wave, and the squared wave-number matrix β 2 ðzÞ captures the linear-optical properties of the system. Its difference from the reference system fulfills where Z 0 ≈ 377Ω is the free-space impedance. The reference system is the sample without the metal film, that is, just the substrate and air half-spaces. We rewrite Eq. (B3) as where the term QðzÞ quantifies the source of the field component that arises from the response of the metal film with conductivity σ. By inverting the operator ∂ 2 z þ β 2 ref in Eq. (B5), one obtains the integral equation [68] EðzÞ where G ref ðz; z 0 Þ is the optical Green's function of the reference sample [45]. Equation (B6) has a clear physical interpretation: The total THz field EðzÞ is the sum of the field E ref ðzÞ of the reference sample (no metal film) plus the field generated by the field-induced currents in the metal.
Thin-film approximation.-To solve Eq. (B6), we apply the so-called thin-film approximation and assume that the field is constant throughout the thickness of the metal film, that is, in the vicinity of z ¼ 0. This assumption is fulfilled if the thickness of the metal film is much smaller than the wavelength and the attenuation length of the THz wave inside the metal. Likewise, for z ≈ 0, the Green's function of the reference sample becomes a ðz; z 0 Þ-independent scalar, that is [45], where β j ¼ n j ω=c is the wave number of the substrate (j ¼ S) and air (j ¼ A). By combining Eqs. (B5), (B7), and (B8) with Eq. (B6), we finally obtain the total field where respectively, are the impedance of the reference system close to the S/A interface and the (anisotropic) conductance of the metal film F. For a homogeneous film with z-independent conductivity σ, we have G F ¼ σd.
Application to our sample.-For our magnetic thin film F, the conductivity can be split according to σ ¼ σ 0 þ Δσ, where σ 0 is the isotropic conductivity in the absence of magnetization (M ¼ 0), while the anisotropic part Δσ captures all magnetoresistive effects. By linearizing Eq. (B9) with respect to Δσ, we find where G F0 ¼ R dz 0 σ 0 ðz 0 Þ is the conductance for M ¼ 0 and ΔG F ¼ R dz 0 Δσðz 0 Þ is the magnetoresistive contribution. In our experiment, the incident field and, thus, reference are linearly polarized parallel to the x axis [see Fig. 1(b)]. Therefore, they can be written as E inc ¼ E inc u and of the x axis. In addition, the field E behind the sample is projected onto the same direction u by a polarizer, resulting in E ¼ u · E. We multiply Eq. (B11) by u from the left side and arrive at where ΔG Fuu ¼ u · ΔG F u is the magnetization-induced change in the conductance projected onto u. For our polycrystalline, homogeneous, and ferromagnetic F layer, we have G F0 ¼ σ 0 d and ΔG F ¼ Δσd. According to Eq. (C2), the magnetoresistive part Δσ of the conductivity tensor fulfills The first term is the anomalous Hall effect [52], and the second term is the AMR. The contribution of isotropic magnetoresistance can be added to σ 0 , but it was neglected here because it is much smaller than σ 0 . The constants a and b are material-specific, and b is directly related to the AMR tensor. Equations (B13) and (C3) imply that u · Δσu ¼ bM 2 cos 2 α ¼ ðσ k − σ ⊥ Þcos 2 α, where α is the angle between M and u [ Fig. 1(b)]. Therefore, projection of the transmitted field onto u does not contain any contribution of the anomalous Hall effect since these field changes are perpendicular to u. By substituting this result into Eq. (B12), we obtain Here, we replaced σ 0 byσ ¼ ðσ k þ σ ⊥ Þ=2 in the denominators of Eq. (B14) with negligible error because jσ k − σ ⊥ j ≪ σ 0 ≈σ. Along with the AMR contrast AMR ¼ −ðσ k − σ ⊥ Þ=σ, Eq. (B14) turns into the desired relationship APPENDIX C: AMR OF POLYCRYSTALLINE SAMPLES

Symmetry analysis
The AMR tensor of an arbitrary ferromagnetic material is defined by G jklm ¼ ð1=2Þ∂ 2 σ jk =∂M l ∂M m . Depending on the point-symmetry group of the material, a substantial number of tensor elements are strictly zero or depend on each other, thereby resulting in a relatively small number of independent tensor elements. For the hexagonal crystal structure of Co (hcp, point group 6/mmm), we have [69,70] 0 where the z axis is oriented along the c axis of Co.
For the cubic crystal structure of Ni, Ni 50 Fe 50 , and Ni 81 Fe 19 (fcc, point group m3m), one has the additional constraints G xxyy ¼ G xxzz ¼ G zzxx , G xxxx ¼ G zzzz , and G xxxx − G xxyy ¼ 2G yzyz . Therefore, two independent elements such as G xxyy and G yzyz ¼ G zxzx ¼ G xyxy completely determine the AMR tensor in this case.

Rotational averaging
The AMR tensor of a polycrystalline material is obtained by rotational averaging [71,72] of the tensor hGi jklm of the crystalline material. The resulting tensor hGi jklm fulfills the same symmetry constraints as the AMR tensor of a cubic crystal with point group m3m. Thus, knowledge of the two independent elements hGi xyxy and hGi xxyy is sufficient. For a ferromagnetic material of this symmetry class, the current density j induced by an electric field E, up to second order in the magnetization M, can be compactly written as [69,70] Here, the first term on the right-hand side is the current in the absence of magnetic order, the second term with constant a is the anomalous Hall effect, the third term is the AMR with b ¼ 2hGi xyxy , and the last term is an isotropic magnetoresistance with c ¼ hGi xxyy . For a thin film of this material with M and E in the film plane, the conductivity is For polycrystalline Co, we perform rotational averaging [71,72] of the AMR tensor and obtain whereas for polycrystalline Ni, we find 3. AMR estimate of polycrystalline samples We conducted ab initio calculations of the plasma frequency of crystalline Ni and Co as detailed in Supplemental Material [41]. These values can be compared to the weight B of the intrinsic AMR contribution we measured on polycrystalline samples of Ni and Co. For Ni, the calculated plasma frequencies are independent of the magnetization direction (see Table S3 in Ref. [41]), consistent with B ¼ ð0 AE 0.07Þ% as inferred from our measurements.