Terahertz Spin-Conductance Spectroscopy: Probing Coherent and Incoherent Ultrafast Spin Tunneling

Thin-film stacks | consisting of a ferromagnetic-metal layer and a heavy-metal layer are spintronic model systems. Here, we present a method to measure the ultrabroadband spin conductance across a layer between and at terahertz frequencies, which are the natural frequencies of spin-transport dynamics. We apply our approach to MgO tunneling barriers with thickness d = 0-6 Å. In the time domain, the spin conductance Gs has two components. An instantaneous feature arises from processes like coherent spin tunneling. Remarkably, a longer-lived component is a hallmark of incoherent resonant spin tunneling mediated by MgO defect states, because its relaxation time grows monotonically with d to as much as 270 fs at d = 6.0 Å. Our results are in full agreement with an analytical model. They indicate that terahertz spin-conductance spectroscopy will yield new and relevant insights into ultrafast spin transport in a wide range of spintronic nanostructures.

Here,  ̃ () is the setup transfer function that captures THz propagation to the detector and the electrooptic sampling process.Note that the case  = coincides with the ℱ|ℋ reference sample.We determine  ̃ () by using a well-understood emitter 1 , GaP(110), with a thickness of 50 µm.Eqs.(S1) and (S2) imply that the spin current can be extracted by The impedance for ℱ||ℋ is measured by THz transmission spectroscopy 2 , and we find  ̃() =  ̃ref () = 8 Ω for all  (see Fig.

Supporting Note 2. THz spin conductance of layer 𝓧
As seen in Supporting Note S1, the extraction of the setup transfer function  ̃ and, thus, the spin current from the THz-emission signal is not straightforward.However, the determination of the THz spin conductance  does not require knowledge of  ̃ and is, therefore, much more easily to implement.

(S4)
Here, Δ s =  s ℱ −  s ℋ is the difference of the transient spin voltage  s ℱ and  s ℋ between layer ℱ and ℋ, respectively, and  ̃s() is the spin conductance of layer .The connection of ̃s() to the actually measured signal  ̃() is given by Eqs.(S1) and (S2).
To access the spin conductance  ̃s() without knowledge of the setup response function  ̃ () [Eq.(S2)], we consider ℱ|ℋ as suitable reference sample, which is identical to all the ℱ||ℋ samples apart from a lacking layer  ( = ).If the ℱ||ℋ sample and the ℱ|ℋ reference sample are placed at the same position on the optical axis, the two resulting THz electric fields directly behind the metal stack are at the same position on the optical axis and have the same relevant lateral spatial distribution 2 .Therefore, the connection between signal and field is described by the same instrument response function  ̃ (), even though the pump-beam diameter of about 30 µm is comparable or even smaller than the relevant THz wavelengths 2 .
Likewise,  ̃SH () and  ̃rel () are the same for both ℱ|ℋ and ℱ||ℋ.As shown previously 4 and in Supporting Note S3, the dynamics Δ ̃s() of the spin voltage is strongly dominated by the spin-voltage dynamics in ℱ, i.e., To summarize, the unknown  ̃SH (),  ̃rel () and  ̃ () all cancel in Eq. (S7), and one can measure the THz spin conductance of a layer  without knowledge of the setup transfer function and the difficultto-measure  ̃SH () and  ̃rel ().

Supporting Note 3. Spin conductance of the CoFeB/Pt interface and Pt spin voltage
To determine the conductance of the CoFeB/Pt interface of the CoFeB|Pt reference stack, we directly interrogate the spin-voltage dynamics  s ℱ () of ℱ by measuring the rate of change    = / of the magnetization of a single film ℱ = CoFeB, which fulfills 4   () ∝   ℱ ().The emitted THz field symmetric with respect to sample turning by 180° about the magnetization vector is dominated by magnetic-dipole radiation due to 4   .In the frequency domain, the THz field amplitude directly behind the sample is where  ̃() is the refractive index of the substrate,  ℱ is the ℱ thickness,  ̃ℱ() is the impedance of the ℱ sample and  is the speed of light 4 .On the other hand, the rate of magnetization change i ̃() is given by 4 i ̃() =  sf ̃s ℱ (), (S9) where the coefficient  sf quantifies the spin-flip strength of ℱ.
Note that  ̃() and  ̃ref () are measured under identical experimental conditions.Therefore, the measured THz signal  ̃() is given by  ̃() =  ̃ () ̃() with an identical setup transfer function  ̃ () as in Eq. (S2  S2b shows their Fourier amplitude.We observe that the signals have the same dynamics as found previously 4 .Further, Fig. S2c,d displays the amplitude and phase of the ratio  ̃() =  ̃ref ()/ ̃(), which is constant to very good approximation in the relevant frequency range.
S4 in Ref. 3).The  ̃SH and  ̃rel are assumed to be frequencyindependent.Typical signals are shown in Figs.2a and S1, and extracted spin currents are shown in Fig. 2c.