Towards a table-top microscope for nanoscale magnetic imaging using picosecond thermal gradients

Research advancement in magnetoelectronics is challenged by the lack of a table-top magnetic measurement technique with the simultaneous temporal and spatial resolution necessary for characterizing magnetization dynamics in devices of interest, such as magnetic memory and spin torque oscillators. Although magneto-optical microscopy provides superb temporal resolution, its spatial resolution is fundamentally limited by optical diffraction. To address this challenge, we study heat rather than light as a vehicle to stroboscopically transduce a local magnetic moment into an electrical signal while retaining picosecond temporal resolution. Using this concept, we demonstrate spatiotemporal magnetic microscopy using the time-resolved anomalous Nernst effect (TRANE). Experimentally and with supporting numerical calculations, we find that TRANE microscopy has temporal resolution below 30 ps and spatial resolution determined by the area of thermal excitation. Based on these findings, we suggest a route to exceed the limits imposed by far-field optical diffraction.

shows the modeled, normalized -Lorentzian response function for the projected amplitude of the FMR precession for a resonant field, H r = 180 Oe and line-width of 80 Oe. The dashed, red curves show the corresponding signal line shape as detected by the lock-in when using a modulation amplitude of 20 Oe.

Supplementary Figure 10 | Collection Circuit Gain.
We plot the dependence of the collection circuit gain on the temporal width of a calibrating square pulse. The model used to fit the data estimates a transfer coefficient of 0.47 ± 0.04 for a 10 ps ANE pulse.

Supplementary Note 1 | Independence between TRANE Temporal Resolution and Circuit
Bandwidth.
This section shows how the temporal resolution of TRANE is only dependent on the lifetime of the thermal gradient, not on the bandwidth of the collection circuit. We discuss how the mixer is used to detect the pulsed signal and give relevant details pertaining to circuit bandwidth. An ideal frequency mixer outputs the voltage multiplication between two input ports.
Here, we label one input the sample voltage and the second input as the reference or local oscillator. If we set the local oscillator to a fixed frequency sine wave, then the mixer acts as a homodyne detector at the local oscillator frequency, producing a DC component at the output when the input is the same frequency as the local oscillator. Similarly, the mixer can be used to detect a pulse train. In this case, instead of a sine wave for the local oscillator, we use a reference pulse train with controllable duty cycle and relative delay.
To show the mixer output from a pulse train reference, we express the voltage multiplication in terms of a Fourier series expansion. Here, we define our Fourier expansion as where T is the period, t is the time, and are the Fourier coefficients defined as By applying the Fourier series expansion, the DC component of the mixer output is where and are the Fourier components of the sample voltage and reference voltage respectively. We have a multiplicative factor V 0 which accounts for the amplifier circuit gain and total insertion loss. K = f max /f 0 is the cutoff factor set by the bandwidth of the electrical components, f max , with f 0 as the laser repetition rate. We can see from Supplementary Equation 3 that we can maximize the output signal by setting the reference to have the same Fourier components as the pulsed signal. As expected, with a pulse train as the input signal, it is best to mix with a pulse train as a reference.
The bandwidth of the collection circuit does not affect the temporal resolution of TRANE. To show this, without loss of generality, we assume the temperature gradient is constant at the laser spot and 0 everywhere else. With this assumption, the time dependent ANE voltage V ANE from the resistor model is given by Where N is the Nernst coefficient, w is the voltage channel width, r is the thermal gradient lateral radius, and M s is the magnetic saturation. This is for measurement of the y-axis component of the magnetic moment, m y with a perpendicular-to-the-plane thermal gradient, . Applying this equation to Supplementary Equation 3, the measured voltage from the collection circuit can be expressed as The temperature gradient is non-zero only for a short time ( ≈ 10 ps according to numerical simulations). Using the fact that , which is true for the 1 GHz bandwidth circuit components, we can approximate the measured signal as This shows that TRANE measures the magnetic moment over the time period of . Therefore, the time resolution of TRANE is determined by the lifetime of the temperature gradient and it is not limited by the frequency bandwidth of the collection circuit.

Supplementary Note 2 | Determination of Laser Induced Temperature Change
The finite element modeling of the thermal gradient evolution used for determining the temperature and thermal decay times was performed using the COMSOL Multiphysics Heat Transfer Module. We consider a single temperature diffusive model in which the laser is treated only as a heat source, rather than considering different phonon and electron temperatures. This is justified by the fact that the optically excited electrons are thermalized on time scales comparable to the laser pulse width of 3 ps 5 .
The spatiotemporal evolution of the thermal gradient in our system is calculated numerically with the Fourier diffusion equation using the material parameters given in Supplementary Table 1. The heat source ( ), is given by, where, δ x and δ y are the Gaussian widths in the x and y direction of the laser spot (311 nm), d is the skin depth (12 nm), Q o is the incident peak power of a single pulse (2.19 W), and τ is the pulse Gaussian temporal width of the 3 ps pulse.
The results of the simulations yield spatiotemporal profiles of the temperature and thermal gradient shown in Supplementary Figure 1. To apply the simulation for quantitative analysis we need a sample specific scaling factor determined experimentally (See below).
When the laser induces a temperature increase to create the TRANE signal, it also creates an increase in the local resistance. If there is electrical current in the sample, the resistance change creates an additional voltage contribution that is independent of the sample magnetization. In this section, we use the signal from the local resistance change to measure the temperature profile of the sample due to heating from the laser. We determine the local resistance change by measuring the signal dependence on an applied DC electrical current. The DC current is applied to the sample by introducing a bias-tee into the circuit as shown in Supplementary   Figure 2a. To show the relationship between the resistance change and the measured voltage, we begin with the sample voltage, which is given by where V ANE is the voltage from the anomalous Nernst effect and I(t) R(t) is the Ohmic voltage.
For the current scenario, we set a constant applied current, while the resistance and V ANE vary in time. From Supplementary Note 1, we find the voltage at the mixer output is given by By chopping laser power, the lock-in voltage from the mixer signal is given by is Fourier series component of the change in the resistance, ∆R, from room temperature. We show the results of these measurements in Supplementary Figure 2b, which displays the expected linear relationship between the collection signal and the applied DC current. We repeat the measurements at various laser powers and plot the slope as a function of laser power in Supplementary Figure 2c. These measurements were performed in the presence of a large saturating magnetic field, so that we can neglect current induced magnetization effects that may change the anomalous Nernst signal. Therefore, by relating the resistance change to a temperature increase, we can quantitatively determine the heating induced by the laser.
It is non-trivial to directly convert this data into a measure of resistance due to the nonlinearity of the circuit components. Instead, we compare this data to numerical simulations and calculations. We numerically simulate the temperature profile to calculate its corresponding resistance change and the resulting collection signal. Due to the unknown absorption coefficient of the sample, there is an uncertainty in the absolute value of the temperature change. Therefore there will be an overall factor which is determined by comparing the simulation results with the measured collection signal. By comparing the slopes of the calculated and measured signals as a function of DC current, we obtain the total temperature change and the temperature gradient.
We measure the temperature dependence of resistivity to map the simulated temperature profile to a total resistance change. We consider the linear response regime of the resistivity dependence on temperature, such that where ρ 0 is the resistivity at the base temperature T 0 , which we set as room temperature at 293 K, and α is the temperature coefficient of resistivity. To determine the temperature coefficient of resistivity, we measure the 4-point resistance as a function of temperature with a physical property measurement system (PPMS), with the results shown in Supplementary Figure 3. With the 4-point resistance measurement, we remove contributions due to contact resistance, therefore the resistivity is related to the resistance by , where A is the cross-sectional area and L is the length between the measurement probes. By fitting the data in Supplemental Figure 3, we find the temperature coefficient of resistance in the 30 nm permalloy to be α = 0.0025 Ω K -1 .
We calculate the total time-varying resistance induced from laser heating by mapping the numerically simulated temperature profile to a resistivity profile using the measured resistivity versus temperature. The total resistance of the sample in terms of the spatially dependent resistivity is given by where ( ( )) is the temperature dependent resistivity. Applying the linear temperature dependence of the resistivity gives From the measurements of the resistivity temperature dependence, it is safe to assume , where is the resistance at T 0 . Therefore, we find the total resistance change by performing a series expansion and taking the first order term to be This shows that the total resistance change is proportional to the mean temperature change through the length of the wire. Supplementary Figure 4 shows the calculated total resistance change as a function of time for the simulated temporal profile from Supplemental Figure 1.
We note from Supplementary Figure 4 that the decay lifetime for resistance change is much larger than the pulsed anomalous Nernst voltage. This is because the anomalous Nernst voltage is dependent on the vertical component of the temperature gradient while the resistance change is dependent on the overall mean temperature. Therefore, lateral thermal diffusion from the heat source into other regions of the ferromagnet will reduce the vertical thermal gradient, causing the decay in the anomalous Nernst voltage. Conversely, lateral thermal diffusion has less influence on the overall mean temperature, and thus the total resistance changes more slowly.
We determine the temperature profile by scaling the simulated temperature profile in By comparing the measured hard-axis TRANE hysteresis with the resistor model, we measure an anomalous Nernst coefficient of V K -1 T -1 . This value within an order of magnitude as reported in the literature [6][7][8] . There is no consensus value because the anomalous Hall coefficient for permalloy, which is related to the anomalous Nernst effect through the Seebeck coefficient, is highly dependent on the thickness and resistivity 9 . This suggests that TRANE is a viable technique to measure the anomalous Nernst coefficient in materials without specialized thermal measurement apparatus. The error for the anomalous Nernst coefficient accounts for the experimental error in the transfer coefficient and the timing of the mixing pulse. It has ignored the uncertainty of values used in the numerical simulation, which include the laser pulse temporal profile and the material parameters. These errors would change the overall scaling factor used to predict the temperature and anomalous Nernst coefficients, but it does not influence the technique to measure possible variations within the sample.

Supplementary Note 3 | Temporal Convolution Using an Electrical Mixer
For TRANE microscopy to be a truly stroboscopic, time-domain method, the voltage induced by the thermal gradient has to decay faster than the probed dynamical behavior because the thermal gradient lifetime defines the interaction time between the magnetization and the probe. Direct measurement of the voltage pulse using an oscilloscope is difficult due to the short temporal duration (ps scale) and the small voltage amplitude (nV scale). As an alternative, we measure the convolution between the pulse and a reference electrical pulse of known width. In this scheme, we amplify the pulse with two 10 kHz-15 GHz, 15 dB amplifiers We note that when used for TRANE measurements, the delay is fixed and the pulse is multiplied by a pulse from a pulse pattern generator (pulse width ~1.5 ns) after amplification by two 0.1-1 GHz bandwidth 20 dB gain amplifiers (Mini-Circuits model ZFL-1000LN+). This form of homodyne detection is used to convert the short-lived TRANE pulse into a low-frequency signal that can be measured by the lock-in. Lower frequency detection circuitry does not reduce the temporal resolution (see section 1) and allows us to filter gigahertz frequency noise induced in the magnetic channel by the AC driving field. Additionally, using a wider decreases the measurement's sensitivity to variations in the delay between the and .

Supplementary Note 4 | Sensitivity
The sensitivity is calculated using the field-dependent magnetization measurements shown in Fig. 1c in the main text and Supplementary Figure 7. This measurement is done in the transverse geometrythe saturated moment is perpendicular to the voltage pick-upsso that

Supplementary Note 6 | Modification of the Resonant Line-shapes due to Field Modulation
To measure the FMR of the permalloy wires we detect the projected magnetic moment perpendicular to the wire. The magnetic moment of the wire precesses about the externally applied magnetic field when driven by a microwave field generated in a microwave antenna patterned parallel to the magnetic wire. The FMR precession angle of a ferromagnet in the linear response regime can be modeled as a driven damped oscillator. The projection amplitude of this motion is the linear combination of even and odd Lorentzian functions given by where is the phase between the even and odd Lorentzian functions, H is the applied field and is the resonant field, and is the linewidth of the Lorentzian. When measuring magnetic dynamics with TRANE, a time-varying magnetic field is applied across the ferromagnet. This induces an electrical current in the ferromagnetic wire, which creates a large background voltage across the wire which is removed with a lowpass filter.
In addition to the magnetic signal due to FMR, we also detect an induced electrical response from coupling between the microwave antenna and the magnetic channel that is detected because of the temperature induced resistance change. However, since the resistance change contribution is independent of magnetization, we are able to remove it by using a cascaded lock-in technique. We detect the signal by using two lock in amplifiers connected in series, the first demodulation was referenced to a square modulated 9.7 kHz signal from an optical chopper and the second demodulation was referenced to a 14 Oe sinusoidal field, H mod , modulated at =10 Hz (5 Hz for FMR frequencies above 10 GHz). The TRANE signal detected by the second lock-in can be modeled by The resulting analytical equation is then used to fit the resonance data obtained with TRANE to quantify the values of the linewidth, amplitude, phase, and center frequency. We note that the modification to the Lorentzian shape does not add free parameters to the fitting function because the modulation amplitude is a known value. The modulation does impact the uncertainty and it reduces the overall signal amplitude, but at the benefit of increased angular sensitivity.

Supplementary Note 7 | Collection Circuit Transfer Coefficient
To determine the transfer coefficient of the collection circuit depicted in Fig. 1b and Fig.   3a in the main text, we measure the collection voltage from a calibration pulse. Numerical simulations indicate that has a width of 10 ps which for our magnetic system translates into a V ANE pulse that also has a width of 10 ps. With the electronics available, we cannot create a 10 ps pulse to directly measure the transfer coefficient. Instead, we extrapolate it through measuring the gain of square pulses of wider widths. Supplementary Figure 10 shows the total gain in the collection circuit as a function of the square pulse width and the fit with our model.
We use Supplementary Equation 3 to model the gain where V 0 is the free parameter to fit the model. The pulse pattern generator signal into the mixer is treated as a periodic triangular function such that where is the peak voltage, δ is the rise and fall time, f 0 = 1/T is the laser repetition rate and is a triangular function given by For all measurements, we set the pulse pattern generator to have a peak voltage of 800 mV and a rise and fall time of 800 ps. We can express this in terms of a Fourier series as Similarly, we can express the calibrating square pulse train generated by the AWG as where is the peak voltage, τ is the square wave width and is a square pulse function given by The square pulse train can be expressed in terms of the Fourier series as The square pulse voltage is measured with a sampling oscilloscope to be 2.22 mV and the pulse width is varied from 300 ps to 3 ns.
With the two input signals, the DC component of the mixer output voltage is The bandwidth of the amplifiers and mixer set the maximum frequency of the sum to K = f 0 /f max where f max is the maximum frequency bandwidth. For the measurements, the collection circuit bandwidth is limited to a maximum of f max = 1 GHz and the laser repetition rate is f 0 = 25.3 MHz.
By fitting Supplementary Equation 24 to the calibration measurement with V 0 as the only free parameter, we obtain a best fit of V 0 = 0.41 ± 0.04 mV.
It is desirable to describe the total measured voltage in terms of the peak anomalous Nernst voltage in terms of a collection circuit transfer coefficient. We define the transfer coefficient γ as We can determine γ by using the numerically simulated anomalous Nernst voltage in Supplementary Note 2 and applying it to calibration fit. Using the voltage multiplier coefficient from the fit in Supplementary Figure 10, our model estimates a transfer coefficient of 0.47 ± 0.04 for the numerically simulated pulse of approximately 10 ps in width.