Ensemble-averaged Rabi oscillations in a ferromagnetic CoFeB film

Abstract

Rabi oscillations describe the process whereby electromagnetic radiation interacts coherently with spin states in a non-equilibrium interaction. To date, Rabi oscillations have not been studied in one of the most common spin ensembles in nature: spins in ferromagnets. Here, using a combination of femtosecond laser pulses and microwave excitations, we report the classical analogue of Rabi oscillations in ensemble-averaged spins of a ferromagnet. The microwave stimuli are shown to extend the coherence-time resulting in resonant spin amplification. The results we present in a dense magnetic system are qualitatively similar to those reported previously in semiconductors which have five orders of magnitude fewer spins and which require resonant optical excitations to spin-polarize the ensemble. Our study is a step towards connecting concepts used in quantum processing with spin-transport effects in ferromagnets. For example, coherent control may become possible without the complications of driving an electromagnetic field but rather by using spin-polarized currents.

Figure 1. Temporal responses in the non-adiabatic regime.

(a) Free induction decay responses of the TR-MOKE experiment for applied magnetic fields of 100 mT (blue), 300 mT (Red), 500 mT (yellow) and 700 mT (green). Traces are shifted along the y axis for clarity. (b) Schematic of the experimental set-up. A femtosecond pulse laser is phase-locked to a microwave oscillator. External magnetic field is applied in the sample plane causing precessions about the x axis while the out-of-plane component of the magnetization, m_z, is detected in a polar-MOKE configuration. The magnetic film was patterned into a square island and the microwave signal was transmitted by a shorted Au microwire. β represents the angle of polarization rotation. (c–e) Temporal responses of the pump-probe FMR measurement at 10 GHz and H₀ values of 422 mT (c), 434 mT (d), and 443 mT (e) and microwave field amplitude of∼0.8 mT. The pump pulse arrives at t=0 ps. The resonance field at this frequency is μ₀H_res∼450 mT. Each trace is normalized to the peak value. Arrows indicate the envelope minimum. (f) Measured temporal responses at 10 GHz for a complete range of applied fields. Each trace is normalized individually to the peak value. The solid red lines were plotted using the Rabi formula. Also the second Rabi oscillation is readily seen (black dashed line). Inset illustrates the trajectory of the magnetization vector, M, on the sphere of constant saturated magnetization. 10 GHz magnetization precessions take place about the axis while the slow variations in the , or component of M obey Rabi’s formulae.

Figure 2. Phase response.

(a) Dependence of the phase responses on the applied field at 10 GHz. Data set of Fig. 1f is presented in a two-dimensional contour plot to represent the phase information. Each temporal response was normalized individually. Blue curved guiding line indicates the location of the valley in Fig. 1f. (b) Phase response before the perturbation. The figure presents a close-up of the black dashed area of a for times between −300 and −100 ps. An overall phase shift of ∼0.75 π is measured across the resonance as indicated by the vertical black dashed lines representing the phase front at highest and lowest applied fields. Data are not normalized. (c) Phase response at long delays corresponding to black dashed area in a for times between 2,200 and 2,400 ps. Data are presented in normalized units. The measured net phase shift across the resonance is ∼2.75 π. Black dashed lines represent the phase front at highest and lowest applied fields. (d) Instantaneous frequency profiles at μ₀H₀ values of 424 mT (blue), 444 mT (red) and 468 mT (yellow) corresponding the blue, red and yellow dashed lines of a, respectively. Inset illustrates the decomposition of the total response to the steady state response of the system at ω_RF and the natural response at the angular frequency of γμ₀H₀. The same reasoning also accounts for the asymmetry seen in the responses of Fig. 1f.

Figure 3. Microwave power dependence of the Rabi frequency.

(a–c) Temporal responses for the CoFeB sample for microwave field amplitudes of 7.5 mT (a), 1.3 mT (b) and 0.25 mT (c). Responses were measured at a frequency of 10 GHz and μ₀H₀=446 mT. The envelopes (red dashed lines) exhibit a similar temporal form independent of the microwave amplitude. (d–f) Temporal responses of a 4 nm-thick molecular beam epitaxy (MBE) grown single-crystal Fe sample at microwave field amplitudes of 7.5 mT (d), 4.7 mT (e) and 2.3 mT (f). Responses were measured at a frequency of 12 GHz and μ₀H₀=143 mT. In contrast to the sputter deposited CoFeB film, the envelope exhibits a clear dependence on the microwave amplitude. Measurements in a–f were carried out at the resonance conditions. Guiding dashed lines indicate the envelopes.

Figure 4. Microwave spin-subgroup selection and resonant spin amplification.

(a) Schematic arrangement of the signals in time. (b–d) Dependence of the temporal response on the relative phase between the optical pump pulse and the microwave signal, Φ, for short intrinsic lifetime, τ_int. Data presented for a frequency of 10 GHz and μ₀H₀=450 mT and relative phases of 0° (b), 90° (c) and 180° (d). Variation of Φ has no effect on the envelope; the carrier signal shifts within the same envelope. (e) Dependence of temporal response on Φ for long intrinsic lifetime, τ_int. Data is presented for a frequency of 1 GHz and μ₀H₀=90 mT. For Φ=90°, constructive interference results in a sharp pulsation of the magnetization. Likewise, for additional 180°, at Φ=270°, pulsations of opposite polarity are generated. When the phase is tuned to Φ=0° and Φ=180°, destructive interference takes place and no pulsations are observed. (f) Illustration of the inhomogeneous broadening at 10 GHz corresponding to b–d. (g) Illustration of the inhomogeneous broadening at 1 GHz corresponding to e. In f,g, blue solid line indicates the total effective resonance line which includes contributions of the inhomogeneous broadening. Blue shaded resonances indicate the subgroups that are selected by the microwave. Red solid lines indicate the subgroups that are not interacting with the microwave signal. Data are shown for the CoFeB sample. The linewidths presented were determined from the experimental data.