Effect of antiferromagnetic order on a propagating single-cycle THz pulse

Employing polarization sensitive terahertz (THz) transmission spectroscopy, we explored how the waveform of initially single-cycle linearly polarized THz pulses changes upon propagation through a thick antiferromagnetic crystal of CoF 2 . The changes upon propagation through CoF 2 are found to depend strongly on both the incoming polarization and temperature. In particular, the ellipticity and polarization rotation acquired by initially linearly polarized light are quantiﬁed and explained in terms of magnetic linear birefringence and dichroism. Although the magneto-optical effects are often considered to be relatively weak, our experiments reveal that the polarization of the THz pulse substantially changes along the pulse duration. The pulse shape is further complicated by features assigned to the formation of magnon-polaritons. The ﬁndings clearly show the importance of accounting for propagation effects in antiferromagnetic spintronics and magnonics.

On the quest to achieve ever faster and more energy-efficient magnetic writing, antiferromagnets are highly appealing for their outstanding properties and related functionalities.Opposite to ferromagnets which are commonly used in magnetic data storage devices but limited to the GHz frequency regime, antiferromagnets possess spin dynamics in the THz frequency range and, thus, have high potential for a faster and more energy efficient data storage.Although a theory has predicted ultrafast switching initiated by THz fields, 1 ultrashort pulses of light, 2 or spin currents, 3 the actual writing of antiferromagnetic bits at the picosecond timescale and at THz repetition rates has not been demonstrated experimentally yet.
A nearly single-cycle intense THz pulse is one of the most auspicious stimuli that was employed for driving coherent spin dynamics in antiferromagnets. 4In antiferromagnetic CoF 2 , the amplitudes of spin dynamics triggered by intense THz pulses were sufficiently high to push the dynamics into a nonlinear regime and opened a new channel of spin-lattice interaction, in particular.Aiming to optimize the polarization of the THz pulses and eventually achieve even higher amplitudes of spin dynamics sufficient for ultrafast switching of antiferromagnetic order parameter, one encounters a problem.Although it is quite well established that antiferromagnetic order can result in magnetic linear birefringence, [5][6][7][8][9][10][11] very often quantitative information about the birefringence in the THz spectral range is available only for a limited set of antiferromagnets.Thus, predictions of how a THz pulse is distorted upon propagation through antiferromagnetic CoF 2 , in particular, are not really possible.This lack of information hampers our understanding of how ultrashort THz pulses interact with the ensemble of spins in the antiferromagnet and eventually prevents us from the developing the scenario of THz control of antiferromagnetism.The goal of this paper is to solve this problem for antiferromagnetic CoF 2 .
Cobalt fluoride (CoF 2 ) recently attracted interest as a model antiferromagnet whose properties can be changed under ultrafast pumping with polarized light in the mid-infrared and THz spectral ranges. 12,13his single-crystal material is characterized by inherently strong piezomagnetic effects [13][14][15] and exhibits several magnonic and phononic states in the terahertz spectral region. 12,16,17This makes CoF 2 a promising candidate to study propagation effects of THz light in antiferromagnetic insulators.Here, we investigate these effects and show that a nearly single-cycle THz pulse acquires a giant rotation and significant ellipticity upon propagation through antiferromagnetic CoF 2 .
CoF 2 possesses a tetragonal rutile crystal structure (P4 2 =mnm space group) as illustrated in Fig. S1(a) in the supplementary material.The lattice parameters at room temperature are a ¼ b ¼ 4:695 and c ¼ 3.1817 A ˚. 18 The sample used in our experiment was a ac plane single crystal plate with the b axis along the normal to the surface and thickness d % 900 lm.The spins of Co 2þ ions are aligned along the c axis with opposite directions at the center and corners of the unit cell below the N eel temperature T N ¼ 39 K 19,20 [see Fig. S1(a) in the supplementary material].The simplest representation decomposes antiferromagnetic spin structure of CoF 2 into two antiparallel sublattices with their net magnetizations represented by M 1 and M 2 , respectively.Exchange coupling of these sublattices and relatively strong anisotropy lift the frequency of spin resonances in antiferromagnetic fluorides to the THz regime.Spin dynamics in antiferromagnets are described in terms of two vectors, the net magnetization M ¼ M 1 þ M 2 and the antiferromagnetic N eel vector L ¼ M 1 À M 2 .Conventionally, it is assumed that in antiferromagnets magnetic linear birefringence is proportional to L 2 .In real materials, however, the situation can be complicated due to mechanical deformations induced by antiferromagnetic order.As a result, the temperature dependence of the linear magnetic birefringence of CoF 2 in the visible spectral range, in particular, does not follow the temperature dependence of L 2 . 21 mode-locked Ti:sapphire regenerative amplifier laser with 50 fs pulse duration at a center wavelength of 800 nm and a repetition rate of 1 kHz was used in the experiment.Employing optical rectification in ZnTe, we generated nearly single-cycle pulses of THz radiation which are focused tightly onto the surface of our CoF 2 sample via offaxis parabolic mirrors.After passing through the sample, the THz light was directed and focused on a second ZnTe crystal where a small, time-delayed part of the infrared radiation was used to probe the waveform of the THz pulse.Hence, the THz electric field induced birefringence leading to ellipticity of the linearly polarized probe pulse at a central wavelength of 800 nm.The diameter of the THz beam is at the order of 1-2 mm and, thus, significantly larger than the one of the probe pulse which is at the order of hundreds of micrometer.The ellipticity of the probe pulse was further measured using a quarter wave plate, a Wollastone prism, and a balanced detector as explained elsewhere. 22The sample of CoF 2 was placed in an optical helium cryostat, which allowed us to carry out experiments in the temperature range from 5 to 300 K.
For polarization sensitive THz spectroscopy, we employed a set of wire-grid polarizers (WGPs).Directly after the generating ZnTe crystal, we placed the first wire-grid polarizer WGP 1 that allows only THz light polarized vertically, i.e., at 45 between a and c axes to pass through the CoF 2 crystal.The second wire-grid polarizer WGP 2 was set right after the crystal and could be oriented either along the a or the c axis, respectively.The third polarizer WGP 3 was positioned right before the detecting electro-optical crystal and its orientation was again fixed at 45 between the a and c axes.We illustrate this scheme of THz polarimetry in Fig. S1 changes, we employed a Lorentzian fit function and deduced the center frequency x center from the fit (see the inset in Fig. 2).The peak has a frequency of 1.12 THz at 10 K, undergoes a red-shift to 0.99 THz at 24 K, and broadens by more than 0.2 THz.We identify this peak as the antiferromagnetic resonance in CoF 2 and emphasize excellent agreement of the resonance frequency temperature dependence with previous reports in Refs.21, 24, and 25  Since spins are aligned along the c axis, an electromagnetic wave with the electric field along the a axis and the magnetic field along the c axis does not exert any torque on the spins.This confirms the fact that the antiferromagnetic resonance is seen exclusively for the case of detection of the electric field E THz along the crystallographic c axis.Remarkably, we also revealed a previously unreported feature evidenced by the beating-like behavior in the time domain as well as the double-peak feature in the frequency domain.
A similar feature was previously observed for THz time-domain spectroscopy of TmFeO 3 and was assigned to the formation of magnon polaritons. 26In the case of CoF 2 , we observe similar linewidths and comparable central frequencies of the antiferromagnetic resonance.Comparing the oscillator strengths and sample thicknesses, we do not expect a large difference between the Co 2þ -and Fe 3þ -spins, respectively.Thus, we assign the experimentally observed double-peak feature in CoF 2 to a magnon-polariton.We also emphasize the agreement of the antiferromagnetic resonance frequency dependence on temperature in our data with previously reported values from Ref. 27.
From the experimental data, we can estimate the complex refractive index, where the real part Re½ñ ¼ c vac Dt=Dd is obtained from the speed of light in vacuum c vac , the difference of the arrival times Dt of the reference, and the THz signal and the sample thickness d.The difference in arrival time Dt between the signal and the reference pulse is % 5 ps and, thus, the real part of the refractive index Re½ñ % 2:4.Based on the approach introduced in Ref. 28, the spectral dependence of Re½nðxÞ and the absorption coefficient aðxÞ have the following forms: Here, /ðxÞ ¼ /ðxÞ À / ref ðxÞ is the phase difference of the transmitted THz pulse through the sample /ðxÞ and the reference pulse (freespace propagation) / ref ðxÞ.The corresponding spectra for aðxÞ are shown in Fig. 2. Finally, using the time-traces for the two projections of the THz electric field transmitted through the sample E THz k a and E THz k c, we reconstructed a three-dimensional waveform of the corresponding pulses.The THz electric field E THz sampled for free space propagation is linearly polarized (see the blue line in Fig. 3), while the THz electric field after interaction with the CoF 2 sample exhibits considerable changes of polarization.The most pronounced deviation from the linear polarization appears as an elliptical feature between 6 and 8 ps (10 K, see Fig. 3) and occurs during the temporal overlap of THz and probe pulses.
From this simple three-dimensional illustration, we identified the region of the first 10 ps as hosting the most significant changes.To characterize the evolution of the THz polarization state which we express as ellipticity and rotation, we employed a direct (i.e., noniterative) and ellipse specific in terms of occlusion and noise sensitivity least squares fitting method proposed for generic applications in Ref. 29.Starting from the general expression of a conic as which an ellipse can be modeled and referring to Ref. 29 for detailed steps, we arrive at a generalized eigenvalue problem.Solving the same, we compute fitting parameters for an ellipse, namely, the major a and minor b axes as well as the rotation h with respect to the center coordinates.An exemplary illustration for such a fit of the a and c axis projection is indicated by the red ellipse in Fig. 3.
We can easily deduce (see Fig. 3) the relative change in ellipticity e rel and rotation h rel of the THz pulse by rel ¼ s À ref and  temperature in Fig. 4. Our data are in qualitative agreement with the temperature dependence of birefringence reported in the visible spectral range (632 nm) in Ref. 21.We believe that the ellipticity and the rotation are due to linear birefringence in the antiferromagnetic CoF 2 crystal.Maximum amplitudes of ordinary and extraordinary refractive indices were determined in Ref. 21 as n o À n e % 2 Â 10 À4 (see the normalized data in Fig. 4).Despite a huge difference in the photon energy practically by a factor of 10 3 , the THz rotation and ellipticity and the birefringence in the visible spectral range closely follow a similar law.It was shown that the latter is determined by the interplay of different contributions, including the pure magnetic linear birefringence and the magnetostriction, 21 and, thus, is anomalous and does not follow the conventional dependence of L 2 .
In conclusion, by performing THz-TDS, we showed that (i) antiferromagnetic CoF 2 and, in particular, the antiferromagnetic mode impose significant rotation and ellipticity on the polarization state of a propagating nearly single-cycle THz pulse; (ii) rotation and ellipticity can be well quantified by employing elliptical fits; (iii) both the THz ellipticity and polarization rotation are due to magnetic linear birefringence and pronounced in the spectral range of the antiferromagnetic resonance.Their temperature dependencies follow a law that is qualitatively similar to the linear birefringence (magnetic and natural) measured in the visible spectral range; and (iv) our findings clearly show the importance of accounting for propagation effect in THz spintronics and magnonics.
See the supplementary material for an extended discussion on the spin structure of CoF 2 , the experimental setup, and the calculation of Fourier transformation and loss function.
FIG. 1.Time domain traces of THz pulses transmitted through antiferromagnetic CoF 2 and detected using the polarizer WGP 2 to select the projection of the THz electric field E THz along the (left) a and (right) c axes, respectively.
FIG. 2. Absorption coefficient for THz transmission along the crystallographic a axis (dotted lines) and c axis (solid lines) as a function of temperature.Inset: Temperature dependence of the magnon frequency.

FIG. 3 .
FIG.3.Three-dimensional representation of the THz pulse shape after propagating through CoF 2 at T ¼ 10 K (red).Elliptical fitting to the two-dimensional projection along the crystallographic a and c axes is indicated in the exposed plot (red).The THz pulse propagating in free space is linearly polarized and shown as reference (blue).

FIG. 4 .
FIG. 4. Ellipticity (blue triangles) and rotation (red squares) extracted from ellipse fits to the projection of THz pulse propagation in the time range of 0-10 ps as a function of temperature.The solid lines represent birefringence as measured in Ref. 21 normalized to ellipticity (blue) and rotation (red), respectively.