Terahertz (THz) time domain spectroscopy (THz-TDS) of a CoCr2O4 single crystal has been performed under magnetic fields up to 8 Tesla. The magnetic field dependences of inter-sublattice exchange resonance at different temperatures have been investigated. Benefiting from the phase and polarization sensitive detection technique in THz-TDS, the circular absorption dichroism and Faraday ellipticity in the THz frequency region are observed and are found to be tunable by the external magnetic field. The complex indices of refraction are obtained under different magnetic field, which present distinct rotatory dispersions arising from the exchange magnetic resonance.
© 2015 Optical Society of America
The magnetic dynamics theory shows that there are two resonance modes with different frequency and opposite precession chirality in the ferrimagnet containing two non-equivalent spin sublattices [1, 2 ]. The lower frequency mode corresponds to the ordinary ferromagnetic resonance dominated by total magnetization and effective magnetic field, which normally fall in the microwave range and is observable by electron spin resonance (ESR) spectroscopy. In addition, the exchange interaction between sublattice spins give rise to an upper branch resonance,, which is determined by the sublattice magnetization ,, as well as the coupling constant .This exchange magnon can be described by the following equation [1, 3 ],3, 4 ].The CoCr2O4 cobalt chromite is a cubic normal spinel ferrimagnet with Tc = 97 K. It is composed with two magnetic sublattices, in which the magnetic Co2+ ions occupy the A sites and magnetic Cr3+ ions occupy the B sites. The Magnetic order in CoCr2O4 consists of long-range ferrimagnetic component below TC, and the short-range spiral component appears below 50 K [5, 6 ].An interest in the CoCr2O4 compound was renewed after the discovery of multiferroic effects in this system in recent years [7–10 ]. Very recently, D. Kamenskyi et al. observed an intersublattice-exchange magnetic resonance in THz spectra region in CoCr2O4 powder samples by backward-wave oscillators and Fourier transform infrared spectrometer .
The development of terahertz (THz) spectroscopy stimulated the research interests on the THz spin dynamics [11–14 ]. This is mainly attributed to the great advantage on the coherently generation and detection of the broadband THz electromagnetic pulse via time domain spectroscopy (TDS) [15–17 ]. Especially, recently developed terahertz (THz) magneto-optical time domain spectroscopy (MO-TDS) allows for the coherently detection of the field amplitude, phase and polarization of broadband THz pulse [11, 18–21 ]. Such techniques have been successfully used to study the THz optical chiralities such as circular dichroism and Faraday rotation induced by the ferromagnetic resonance, electron cyclotron resonance, magneto-plasmon resonance in magnetic materials and semiconductors [11, 22–26 ], which have significant importance to understand the spin dynamics and develop the tunable polarization optics in THz frequency range. However, such magneto-optic effects related to the intersublattice exchange magnon in ferrimagnets in THz frequency region remain unexplored so far. In this work, we investigated the THz transmission spectroscopy of CoCr2O4 single crystal under low temperature and magnetic field up to 8 Tesla by using THz MO-TDS. We observed distinct resonance absorption induced by the magnetic dipole transition from inter-sublattice exchange mode. The THz Faraday ellipsometry spectroscopy revealed the exchange-magnon-induced magneto-optic phenomena including optical rotatory dispersion, circular absorption dichroism, as well as the Faraday ellipticity in THz spectra range under different magnetic fields.
2. Samples and experiments
The single crystals of CoCr2O4 with <111> face were grown by the chemical vapor transport method . The measured sample has trapezoid shape (2mm x 3mm size) with 0.8 mm thickness and is optically polished. For THz time domain spectroscopy experiment, a mode-locked Ti:sapphire laser is used to deliver femtosecond pulses with duration of 150 fs, center wavelength of 800 nm, and repetition rate of 76 MHz, which are divided into pump and probe beams. The pumping laser pulses are focused onto a LT-GaAs photoconductive antenna, which generates the broadband THz pulses. Free-space electro-optic sampling in a ZnTe single crystal is employed to detect the p-polarized electric field amplitude of the THz waveforms in the time domain. The superconducting magnet (Oxford Spectromag SM4000) is used to exert the magnetic field and tune the temperature for the sample. During the measurement for the transmission, the Faraday geometry was used, i.e., the direction of the magnetic field was parallel or antiparallel to the propagation direction of the THz beam. The incident THz pulse has an electric field with p polarization (polarized in the x-direction). The sample is fixed on a copper cold finger with 2 mm-diameter hole and is orientated arbitrarily around the <111> crystal axis. We used the THz MO-TDS method as reported in reference [19, 26 ] to measure the THz polarization change. One wire-grid THz polarizer (WGP) was placed to resolve the polarization of the transmitted THz pulse into one of two orthorhombic components defined by the rotation of its transmission axis to either + 45° or −45°. The transmitted pulse is further projected on a second WGP with horizontal orientation. Finally, a coordinate transformation is applied to obtain the THz electric field vector in two dimensional (x-y) space. The THz spectroscopy measurements were taken under a dry nitrogen purge.
3. Results and discussions
3.1 Exchange magnetic resonance
Figure 1 shows the amplitude of the complex transmission, , in the range from 0.2 to 1.2 THz as measured in different external magnetic fields and temperatures. Here and correspond to the fast Fourier (FFT) spectra for the THz waveforms transmitted through sample and empty sample holder, respectively. These spectra were taken without applying the THz WGPs and only present the p-polarized THz electric field transmission. It can be seen that the two absorption dips can be well distinguished in the spectrum as indicated by the dot lines. The low energy absorption around 0.3 THz is independent of applied magnetic field and temperature, which may be related to the birefringence in single crystal . The high energy absorption peaks as guided by blue dot lines, which are originated from the inter-sublattice exchange resonance , moves toward high frequency side with increasing external magnetic field. In Fig. 2 , we plotted the minimum values of absorption dips as the function of magnetic field at different temperature. For comparison, the experimental data at 30 K taken from  are also displayed as open circles in this figure. The magnetic-field dependences of the magnetic resonances show nearly linear dependences in the studied magnetic field range at each temperature, which agree well with the experimental results measured by backward-wave oscillators and Fourier transform infrared spectrometer in . The measurements were performed on powder samples in , where the anisotropic field can be counteracted. Therefore, the consistency with  indicates that the anisotropic-field contribution to the exchange mode can be neglected in our measurement configuration. Actually, it is reasonable considering that the anisotropic field in CoCr2O4 was estimated to be only in the order of 0.1 T by the earlier ESR experiment . As the temperature goes down, the resonance frequency is increased and arrive maximum at 25 K, however, shows a small drop at 15 K. This anomaly is consistent with the structural transition occurs at Ts = 26 K, below which an incommensurate conical magnetic structure sets in .
3.2 Circular dichroic absorption
To further reveal the THz transmission chirality arising from the exchange resonance, we measured the electric-field-vector temporal profiles of transmitted THz pulses in k (wave vector) plane by using the same experimental method as in reference [19, 26 ]. The obtained x- (Ex) and y-components (Ey) of THz electric field waveforms under different magnetic field and 25 K temperature are displayed in Fig. 3 . In the following, the symbol + and - represent the magnetic fields parallel and antiparallel to the propagation direction of THz beam, respectively. As can be seen, the Ey is not zero without the application of magnetic field although the incident THz beam is x-direction polarized. The main feature revealed by Fig. 3 is that the sign of Ey is changed when reversing the external magnetic field, however, the Ex is independent of the magnetic field direction. Such polarity reversion of the transmitted THz waveforms indicates that the circular dichroic absorption take place, which can be revealed more clearly in frequency domain. The left-handed circularly polarized (LCP) or right-handed circularly polarized (RCP) THz radiation can be regarded as the superposition of two linearly polarized components orthogonal to each other with the phase delay of or [19, 26 ], i.e.,Fig. 3, respectively. Figure 4 shows the amplitude of the complex transmission for LCP (-) and RCP ( + ) light,, under different external magnetic fields and at 25 K temperature. Here and correspond to the complex circular FFT spectrum transmitted through the sample and through the empty sample holder, respectively. It should be mentioned that there is an absorption dip around 0.3 THz for the RCP transmission, which is independent of applied magnetic field and disappear in the LCP transmission spectra. This anomalous absorption may be related to the THz waveform distortion due to the birefringence in single crystal as mentioned above. In the case of zero magnetic field, a absorption dip from the exchange resonance at 0.5 THz shows up in the RCP THz transmission spectrum, while is absent for the LCP transmission. When the external magnetic field parallel with the wave vector of THz radiation, i.e., positive field, is applied, the LCP exchange magnon absorption is active while the RCP absorption is suppressed obviously. In contrast, the negative magnetic field tends to enhance RCP exchange resonance absorption but weaken the LCP absorption. It is clear that the exchange-resonance absorption dip moves toward high frequency side with increasing external magnetic field as guided by dot lines. The magnetic field dependence of resonance frequency accords with the result measured without using WGP as shown in Fig. 1.
3.3 Faraday ellipticity
The circular dichroic absorption is expected to induce the polarization state change of transmitted THz wave with the external magnetic field, which can be visualized in time domain. In Fig. 5 , we plotted the E x and E y with the parameters of time delays that correspond to the projections of the electric-field vector tips of THz pulse on the x-y plane and describe the rotation trajectory of THz electric-field vector. For comparisons, the electric field amplitude of the THz pulse (reference) passing through empty sample holder is scaled down by 0.33. The reference. i.e., incident THz pulse, shows almost linear polarization state along x direction as displayed in Fig. 5(a). However, the passing through the sample makes it elliptical with the counterclockwise sense at zero magnetic field. As shown in Figs. 5(c) and 5(d), the distinct polarization ellipticity persists by applying the magnetic field. Meanwhile, the polarization rotation sense with time is reversed as reversing the magnetic field, but the ellipticity under −3 T is larger than that in + 3 T. In the case of positive magnetic field, the exchange resonance tends to selectively absorb the LCP THz photon due to the precession motion of sublattice magnetization. As a result, the transmitted THz radiation exhibits the RCP ellipticity from the point of receiver. The local magnetic moment is reversed with reversing the magnetic field direction, which induce the opposite optical chirality. The coupling between AC magnetic field component of electromagnetic wave and precession of sublatice magnetizations in ferrimagnet can produce two resonance modes, i.e., ferromagnetic resonanc and exchange resonance. In principle, the exchange resonance mode can be excited by the circular polarization electromagnetic wave with the sense opposite to that excites the ferromagnetic resonance in microwave range . As for CoCr2O4, the total magnetic moment is a sum of a long-range order of the ferrimagnetic component and a short-range order of a spiral incommensurate component at low temperature . This adds the complexity to understand its optical chirality quantitatively. CoCr2O4 single crystal has easy axis along  direction for the ferrimagnetic component . For our measurement, the wave vector of THz beam is along  direction. At low temperature and zero external magnetic field, the spontaneous magnetization is aligned in the  direction and the spiral component of spins rotate in its plane [3, 6 ]. Such spin structure induced magnetization precession may be associated with the THz ellipticity at 0 T.
The Faraday ellipticity in frequency domain is defined as [8, 12 ]Fig. 6 . As can be seen, the Faraday ellipticity spectrum has two extreme values. The low-frequency valley with negative ellipticity is independent of the applied magnetic field strength and direction, which emphasizes the birefringence related to the anisotropic dielectric constant in CoCr2O4 single crystal. The high-frequency ellipticity takes place in the opposite sign when the magnetic field is reversed, which coincide with typical magnetic resonance behavior . With increasing external magnetic field strength, the absolute value of the high-frequency Faraday ellipticity is increased and the profile is moved to higher frequency side. The peak position,, as the function of external magnetic field is plotted in the inset, which shows nearly linear magnetic field dependence same as the result in Fig. 2.
3.4 THz rotatory dispersion
The observed magnetic resonance is due to the strong interaction between the THz electric-magnetic pulse and the electron spin occurs through the magnetic field component, which can be described by the circular magnetic susceptibility. The complex index of refraction that determined by both magnetic susceptibility and dielectric permeability, i.e., ,can be derived from the circular polarized THz transmission according to Fresnel laws,Fig. 7 . As shown in Fig. 7(a), the imaginary part of refractive index for RCP light is peaked around 0.5THz in the case without external magnetic field, which corresponds to the exchange magnon resonance absorption. When the positive magnetic fields are applied, no resonance absorption peaks can be distinguished, however, a negative magnetic field can strengthen the RCP exchange resonance. Oppositely, the LCP resonance peaks in imaginary parts become more robust when the applied magnetic field has positive direction as illustrated in Fig. 7(b). The resonance peak shifts toward the high frequency side with increasing magnetic field. The same THz propagation chirality can be revealed by the real part of refractive index in Figs. 7(c) and 7(d), i.e., the positive magnetic field is able to boost the LCP THz dispersion and induce the blueshift of the dispersion profile, but the negative magnetic field lead to the suppression of the exchange-resonance dispersion.
In summary, we have investigated the THz transmission through CoCr2O4 single crystal with THz-TDS at different temperatures and under the magnetic fields up to 8 T. The magnetic resonance originating from the sublattice exchange magnon in CoCr2O4 ferrimagnet are revealed. We observed the circular absorption dichroism in THz spectra range by THz MO-TDS. When the applied magnetic field is along THz wave vector, the LCP absorption is active and RCP absorption is suppressed. The pronouncedly Faraday ellipticity can be found both in time and frequency domain. The complex indices of refraction for LCP and RCP THz are obtained under different magnetic field. The dispersions show remarkable chirality determined by the exchange magnon resonance.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11004199, 11374304, 11304323, 11304317) and Natural Science Foundation of Anhui (1208085MA10). X. L. was supported by the Joint Funds of the National Natural Science Foundation of China and the Chinese Academy of Sciences' Large-Scale Scientific Facility (Grant No. U1432139).
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