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Long spin relaxation times in a transition metal atom in direct contact to a metal substrate Jan Hermenau, Markus Ternes, Manuel Steinbrecher, Roland Wiesendanger, and Jens Wiebe Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b05392 • Publication Date (Web): 22 Feb 2018 Downloaded from http://pubs.acs.org on February 23, 2018
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Long spin relaxation times in a transition metal atom in direct contact to a metal substrate Jan Hermenau1, Markus Ternes2, Manuel Steinbrecher1, Roland Wiesendanger1 and Jens Wiebe1,* 1
Department of Physics, Hamburg University, Jungiusstrasse 11, D-20355 Hamburg, Germany
2
Max-Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany *Corresponding author:
[email protected] Abstract Long spin relaxation times are a prerequisite for the use of spins in data storage or nanospintronics technologies. An atomic-scale solid-state realization of such a system is the spin of a transition metal atom adsorbed on a suitable substrate. For the case of a metallic substrate, which enables directly addressing the spin by conduction electrons, the experimentally measured lifetimes reported to date are on the order of only hundreds of femtoseconds. Here, we show that the spin states of iron atoms adsorbed directly on a conductive platinum substrate have a surprisingly long spin relaxation time in the nanosecond regime, which is comparable to that of a transition metal atom decoupled from the substrate electrons by a thin decoupling layer. The combination of long spin relaxation times and strong coupling to conduction electrons implies the possibility to use flexible coupling schemes in order to process the spin-information. Keywords: Spin relaxation, inelastic spin-resolved scanning tunneling spectroscopy, atomic spintronics, single atom bit, spin relaxation time, spin coherence time
The high sensitivity of magnetic quantum systems to their closest environment can be helpful in quantum sensors in order to improve their performance in comparison to classical detectors 1 or disturbing when large spin relaxation times (typically labeled with 𝑇𝑇1 ) and/or spin coherence
times (typically referred to as 𝑇𝑇2 ) are required. Robust magnetic properties are especially desired for quantum computing applications
2
, which are based on the quantum mechanical
superposition of states and thus require long 𝑇𝑇1 and 𝑇𝑇2 times, as well as for classical storage and spintronics applications, whose functionality particularly depends on long spin relaxation times 1 ACS Paragon Plus Environment
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𝑇𝑇1 . If the spin is coupled to a conductive substrate for the purpose of writing and reading the spin state, spin-flip scattering of the conduction electrons typically limits both time scales drastically.
Therefore, in the field of atomic spintronics and data storage, the search for long spin-lifetimes of atomic-sized magnets was mainly concentrated on systems which are decoupled from conduction electrons. In these systems, the magnetic moment bearing orbitals were protected against external perturbations by the use of insulating interlayers 3-5 and/or rare-earth elements 68
or molecular ligand fields 9. Also the effects of superconducting
10
or semiconducting
11
substrates were investigated. Due to the suppressed spin-flip scattering in such systems, long 𝑇𝑇1
times of transition metal atoms up to milliseconds have been realized, as revealed by pumpprobe measurements and real-time observations 3.
On the other hand, a strong decoupling of the magnetic moment hampers access of the contained magnetic information, which is a drawback relating to individual reading and writing
12
,
and also to controllably couple several spins, which is required for processing of the spin information. In this respect, adsorption of transition metal atoms directly on heavy substrate materials offers a high tunability of the couplings via distance dependent collinear or noncollinear Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions
13, 14
. However, for individual
transition metal atoms adsorbed on metal substrates, very short lifetimes in the regime of hundreds of femtoseconds were derived from the broadening of the excitation steps in inelastic scanning tunneling spectroscopy (ISTS) than 𝑇𝑇1
19
. Since 𝑇𝑇2 is typically orders of magnitudes shorter
15-18
, the lifetime broadening in ISTS is most probably dominated by the spin decoherence
effects. A direct measurement of 𝑇𝑇1 and 𝑇𝑇2 in these systems was missing so far. Here, we show that the spin relaxation times of the first excited states of an iron (Fe) atom in direct contact to the heavy substrate material platinum (Pt) are on the order of a nanosecond and thereby comparable to the spin relaxation times that have been found for transition metal atoms on thin insulating layers. All metal based spin processing schemes using non-collinear substrate electron mediated interactions between neighboring single transition metal atoms thereby get within reach 13, 14, 20.
A typical area of the prepared Pt(111) surface with randomly distributed Fe atoms is shown in the constant-current scanning tunneling microscope (STM) image in Fig.1a (for experimental details, see Methods). On platinum the Fe atoms can be adsorbed on the two different hollow sites, face centered cubic (fcc) or hexagonally closed packed (hcp), of the (111) lattice. The type 2 ACS Paragon Plus Environment
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of adsorption site can be identified from the characteristic spin-excitation energies detectable in spin-averaging inelastic scanning tunneling spectroscopy (ISTS)
16
. ISTS on fcc Fe atoms
(Fig.1b, black curve) reveals a falling and a rising step at −0.75 mV and +0.75 mV, respectively,
with equal step heights, corresponding to the energy that it takes to excite the spin out of the ground state, with the spin oriented maximally perpendicular to the surface (up or down), into the first excited state
16
. The spin-averaging ISTS data could be reproduced using an effective spin
� = 𝑔𝑔µ𝐵𝐵 𝐁𝐁𝐒𝐒� + 𝐷𝐷𝑆𝑆̂𝑧𝑧2 with a spin of 𝑆𝑆 = 5, a uniaxial magnetic anisotropy constant of Hamiltonian 𝐻𝐻 2
𝐷𝐷 = −0.19 meV and a Landé 𝑔𝑔-factor of 𝑔𝑔 = 2.4 . In this model, the relatively large broadening 16
of the steps, which cannot be accounted for by modulation-voltage induced or thermal
broadening effects, was interpreted as due to lifetime broadening, revealing lifetimes of only 0.7 ps
. The easy-plane anisotropy of the hcp Fe atom with 𝐷𝐷 > 0 leads to zero energy Kondo
16
scattering between the two degenerate ground states with 𝑆𝑆𝑧𝑧 = ±
1 21 2
thereby limiting the ground
state lifetime. Therefore, we will focus on the out-of-plane system of fcc Fe here.
Figure 1 | Spin-resolved inelastic scanning tunneling spectroscopy of iron atoms on Pt(111). a, Constant-current image of the sample surface showing randomly distributed individual Fe atoms (image size 16 × 5 nm2, 𝑉𝑉 = 5 mV, 𝐼𝐼 = 500 pA). b, SP-ISTS spectra of the fcc atoms marked with arrows in the corresponding color in a (𝑉𝑉stab = 6 mV, 𝐼𝐼stab = 3 nA, 𝑉𝑉mod = 40 µV, 𝐵𝐵 = 0 T) in comparison to an ISTS spectrum recorded with a non-magnetic tip (black, atom not shown in a, 𝑉𝑉stab = 10 mV, 𝐼𝐼stab = 3 nA, 𝑉𝑉mod = 40 µV, 𝐵𝐵 = 0T). All spectra are normalized by dividing a substrate spectrum taken with the same parameters. A drastic change of the ISTS data is observed when the tip possesses a finite and stable magnetization with a significant out-of-plane component (Supplementary Note 1). As shown for three different fcc atoms marked in Fig.1a, the corresponding spin-polarized (SP-)ISTS spectra in Fig.1b are strongly asymmetric with respect to 𝑉𝑉 = 0 V, i.e. the heights of the steps at 3
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±0.75 mV and the conductance levels at larger bias are different on the positive and negative
bias side, with an inverted step slope at negative bias. Moreover, the steps reveal a peak-like overshooting. Additionally, without changing these main characteristics, the spectra recorded on different atoms differ significantly. This must be attributed to a high sensitivity of the underlying processes on the local environment of the particular atom. In an externally applied out-of-plane magnetic field 𝐵𝐵ext (Fig.2, see data with additional values of
𝐵𝐵ext and 𝐼𝐼s in Supplementary Fig.2), the step-like features in the SP-ISTS spectra shift towards higher bias according to the Zeeman interaction
16
. Most importantly, they strongly change their
appearance and asymmetries with respect to 𝑉𝑉 = 0 V when the field polarity is reversed. The spectra at negative 𝐵𝐵ext show steps with asymmetric heights and an overshooting
characteristics, while the spectra at positive 𝐵𝐵ext reveal steps with inverted slope. Note, that the data was taken using the same magnetic tip with a sensitivity to the out-of-plane component of
the magnetization, which is not affected by 𝐵𝐵ext . Similar shapes of spectra and asymmetries in the step heights with respect to 𝑉𝑉 = 0 V are well-known characteristics of non-equilibrium effects in SP-ISTS of an atomic spin in the presence of a magnetic field
22, 23
. Our observation of a
strongly asymmetric appearance of the SP-ISTS spectra in zero magnetic field (Fig.1b) further points towards an additional stabilization of the atom’s magnetization by the magnetic tip as we will show below.
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Figure 2 | Magnetic field and stabilization current dependence of SP-ISTS. SP-ISTS spectra recorded on the fcc atom shown in the inset taken at the indicated magnetic fields using stabilization currents of 𝐼𝐼stab = 1 nA (blue), 𝐼𝐼stab = 6 nA (orange) and 𝐼𝐼stab = 16 nA (purple). All spectra are normalized by dividing a substrate spectrum taken at the same parameters (𝑉𝑉stab = 5 mV, 𝑉𝑉mod = 80 µV). For better visibility the spectra recorded at different magnetic fields are shifted by an artificial vertical offset. The peak-like overshooting of the excitation steps can principally arise from higher order Kondo scattering effects excited states
24, 25
or from pumping processes enabled by long spin relaxation times of the
4, 23, 26
. In fact, the characteristic features in the spectra show a systematic
dependence on the stabilization current 𝐼𝐼stab , in particular for small negative 𝐵𝐵ext (Fig.2). This
strongly indicates that dynamic processes are the origin of the complex line shapes
5, 27
. In
general, dynamic processes are only relevant in ISTS if the spin relaxation times of the excited states are at least on the order of the average time between two tunneling electrons 𝜏𝜏𝐼𝐼 (𝑉𝑉) =
𝑒𝑒 , 𝐼𝐼(𝑉𝑉)
where 𝑒𝑒 is the elementary charge and 𝐼𝐼(𝑉𝑉) the measured tunneling current. Even at the lowest
stabilization current 𝐼𝐼stab = 500 pA used, the line shapes still indicate dynamic processes. A
lower limit of the spin relaxation times of the excited states can hence be estimated, considering that only about 10% of these tunneling electrons excite the spin 5 ACS Paragon Plus Environment
, resulting in about 𝜏𝜏min (𝑉𝑉 =
16
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0) ≈ 3 ns. This value is three to four orders of magnitude larger than the time that has been estimated from the lifetime broadening of the excitation steps 16.
In order to extract the spin relaxation times from the SP-ISTS spectra quantitatively, the experimental data is simulated within a third order perturbation theory model
24
(see Fig.3, data
and simulations for further values of 𝐼𝐼stab are given in Supplementary Figs.3 and 4). It contains
the coupling of substrate and tunneling electrons to the atom’s spin, which is described by an effective spin Hamiltonian (see Methods) that takes into account the Zeeman energy and the complete set of anisotropy constants for the 𝑆𝑆 =
5 2
system in a crystal field of C3v-symmetry, i.e.
two axial anisotropy constants 𝐵𝐵20 and 𝐵𝐵40 as well as a transverse anisotropy constant 𝐵𝐵43
28
. The
latter anisotropy leads to a mixing of three subsets of the 𝑆𝑆̂𝑧𝑧 eigenstates (Fig.3e) into the six
eigenstates 𝛹𝛹𝑖𝑖 of the Hamiltonian, whose energy levels in a positive magnetic field are illustrated
in Fig.3f for the present case of out-of-plane anisotropy (𝐵𝐵20 < 0)
16
. The model consistently
reproduces almost every detail of the discussed line shapes for the complete set of measured SP-ISTS spectra using different magnetic fields and stabilization currents (Fig.3a and Supplementary Figs.3, 4). To this end, the model parameters have been slightly varied within
reasonable constraints (see Table 1, Supplementary Tables 1 and 2, and Supplementary Fig.5), and we will later discuss the resulting offset magnetic field and a systematic dependence of the electron-to-local-spin coupling constants which are caused by tip-related effects. As a side note, we would like to mention that the SP-ISTS spectra taken at 𝐵𝐵ext = +0.1T closely resemble the ab-initio calculated renormalized local vacuum density of states from Ref. 29.
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Figure 3 | Spin relaxation times and occupations from the simulation of SP-ISTS. a, Simulations (magenta) of the experimental SP-ISTS spectra (gray) recorded using a stabilization current of 𝐼𝐼stab = 1 nA for the indicated magnetic fields (𝑉𝑉stab = 5 mV, 𝑉𝑉mod = 80 µV). For better visibility all spectra (except the 𝐵𝐵ext = 0 T spectra) are shifted vertically by an artificial offset indicated by the shift in the conductance at 𝑉𝑉stab = 5 mV. The experimental data was normalized by dividing a substrate spectrum taken with the same parameters before it was rescaled to real units by the stabilization. The differential conductance is given in units of the quantum of conductance (qoc). b, c, d Bias voltage dependence of (b) the spin relaxation times of the two ground states (𝜏𝜏1 and 𝜏𝜏2 ), (c) the higher spin-states, and (d) the occupations of all spin-states (see f for the assignment between line colors and spin states) calculated from the simulation of the experimental data taken at 𝐼𝐼stab = 1 nA and 𝐵𝐵exp = 0 T. The average time between two tunneling electrons 𝜏𝜏𝐼𝐼 =
e 𝐼𝐼(𝑉𝑉)
is given in black. The vertical dashed lines mark the voltage
corresponding to the first excitation energy determined from ISTS measurements. e, f, Illustrations of the mixing of 𝑆𝑆̂𝑧𝑧 eigenstates with the same line style (solid, dashed, wiggly lines) due to transversal magnetic anisotropy (e) and of the resulting energy levels of the spin states (f, 𝐵𝐵 > 0) with colors used for the corresponding lines in (b-d).
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𝐵𝐵20 (meV)
−0.084 ± 0.003
𝐵𝐵40 (meV)
0.0014 ± 0.0002
𝐵𝐵43 (meV)
0.0014 ± 0.0015
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𝑈𝑈
3.62 ± 0.55
𝑇𝑇eff (K) 0.65 ± 0.06
𝜂𝜂t
0.17 ± 0.02
𝐵𝐵𝑦𝑦 (T)
0.28 ± 0.14
Table 1 | Simulation parameters. Averaged values of the anisotropy constants 𝐵𝐵20 , 𝐵𝐵40 , 𝐵𝐵43 , of the potential scattering parameter 𝑈𝑈, of the effective temperature 𝑇𝑇eff, of the tip’s spinpolarization 𝜂𝜂t , and of the transverse magnetic field 𝐵𝐵𝑦𝑦 . The values have been calculated by averaging of the parameters given in Supplementary Tables 1 and 2. The error is given by the standard deviation.
Justified by the excellent agreement between simulation and data, we use the model to extract the spin relaxation times 𝜏𝜏𝑖𝑖 and occupation probabilities 𝑝𝑝𝑖𝑖 of the six spin-states 𝛹𝛹𝑖𝑖 in 1 2
dependence of the bias voltage (Fig.3b-d). Note, that, with a spin larger than 𝑆𝑆 = , an individual
spin relaxation time 𝜏𝜏𝑖𝑖 (𝑖𝑖 > 2) is assigned to each excited state 𝛹𝛹𝑖𝑖
30
, which is the average time
before any scattering event transfers the system into a different eigenstate. In Fig.3b,c the bias voltage dependence of 𝜏𝜏𝑖𝑖 , calculated from the simulations of the 𝐵𝐵ext = 0 T data recorded with
𝐼𝐼stab = 1 nA, are plotted together with the average time between two tunneling electrons 𝜏𝜏𝐼𝐼 (𝑉𝑉) =
𝑒𝑒 . 𝐼𝐼(𝑉𝑉)
Above the voltage of the first spin-excitation (see the vertical lines at ±0.75 mV), all
six states have a spin relaxation time considerably longer (𝛹𝛹1 , 𝛹𝛹2 , 𝛹𝛹3 , 𝛹𝛹4 ) than, or on the same
order (𝛹𝛹5 , 𝛹𝛹6 ) of 𝜏𝜏𝐼𝐼 (𝑉𝑉), corroborating the spin-pumping effect of the tunneling electrons in this
system. In particular, the spin relaxation times of the first excited states 𝛹𝛹3 and 𝛹𝛹4 for a bias
voltage close to the excitation threshold are longer than 1 ns, which is three orders of magnitude
larger than the lifetime determined from the analysis of the broadening in ISTS measurements . Surprisingly long spin relaxation times are found for the two ground states (𝛹𝛹1 , 𝛹𝛹2 ) as shown
16
in Fig.3b. For voltages above the tunneling electron excitation threshold the lifetimes of both states decrease due to the interaction with an increasing number of tunneling electrons, as shown by the nearly constant ratio
τ1/2
𝜏𝜏𝐼𝐼 (𝑉𝑉)
~10. This means that on average every tenth tunneling
electron excites the spin out of the ground state. Below this threshold the extracted ground state lifetimes increase up to values ranging between micro- and milliseconds. Resulting from the comparatively long spin relaxation times of the spin states, there is a significant non-thermal occupation of the higher energy spin states due to the interplay of the spin-pumping effect and a non-vanishing spin transfer torque caused by the spin-polarization of the tunneling current
. Fig.3d shows the occupation probabilities 𝑝𝑝𝑖𝑖 of the states 𝛹𝛹𝑖𝑖 for the
27
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situation of a small positive magnetic field, where 𝛹𝛹1 on the left side of the anisotropy barrier is the non-degenerate ground state (cf. Fig.3f). Note, that there is an effective magnetic field
induced by the magnetic tip (see below). Consequently, for bias voltages of the tunneling electrons below the excitation threshold (vertical lines in Fig.3d) the occupation probability 𝑝𝑝1 of
𝛹𝛹1 , is highest. However, if the spin is driven by tunneling electrons at a negative bias above the excitation threshold, the situation gets reversed. Now, the occupation of the energetically higher
lying states 𝛹𝛹2 , 𝛹𝛹4 , and 𝛹𝛹6 on the right side of the anisotropy barrier is larger as compared to
their energetically lower partners 𝛹𝛹1 , 𝛹𝛹3 , and 𝛹𝛹5 on the left side. Obviously, for this choice of the
bias polarity, subsequent excitations induced by the spin-polarized tunneling electrons pump the spin from the energetically favored ground state across the anisotropy barrier, i.e. the spin is reversed
27
. This spin-pumping effect enables the atom’s spin to be driven into the desired
ground state only by choosing the proper bias voltage polarity 5. In the following we focus on effects of the tip on the Fe atom in addition to the spin-pumping of the tunneling electrons. First, there is a strong asymmetry with respect to zero bias even at zero magnetic field (Fig.1b, Fig.2, Fig.3a), which hints towards an additional inherent magnetic field. Indeed, as shown in Fig.4, the magnetic field 𝐵𝐵sim resulting in the correct simulation of the data
has a small offset of about +0.1T to the applied magnetic field 𝐵𝐵ext , which is compensated at
𝐵𝐵ext ≈ −0.1 T (see inset of Fig.4), dependent on 𝐼𝐼stab. Obviously, there is an additional effective
magnetic field caused by the tip 𝐵𝐵tip , whose magnitude depends on the experimentally given tip𝐼𝐼
sample conductance 𝐺𝐺ts = stab , where 𝑉𝑉stab = 5 mV is the stabilization voltage, and thereby on 𝑉𝑉 stab
the tip-atom separation adjusted by 𝐼𝐼stab. 𝐵𝐵tip most probably originates from a stray field and/or
an exchange interaction between tip apex and atom
31, 32
. Interestingly, as visible in the inset of
Fig.4, the resulting field at the location of the atom even reverts its orientation from negative to positive polarity when 𝐼𝐼stab is increased from 1 nA to 16 nA. In the data, the effect of the 𝐼𝐼stab
dependent reversal of the magnetic field polarity can be seen for the 𝐵𝐵ext = − 0.1 T data in Fig.2,
where the spectra at low 𝐼𝐼stab resemble those taken at large negative 𝐵𝐵ext and the spectra at
larger currents those taken at positive 𝐵𝐵ext . Moreover, we can show (see Supplementary Note and Figure 6) that the spin relaxation times of the Fe atoms on Pt(111) are mainly limited by the
spin-flip scattering of tip electrons, even for the largest tip-sample separations. Consequently, with further decreasing 𝐼𝐼stab, we expect even larger zero voltage spin-lifetimes until they finally approach their intrinsic values limited by substrate electron scattering.
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Figure 4 | STM tip effective field. Out-of-plane magnetic field used for the simulations (𝐵𝐵sim ) in dependence of the external field applied in the experiment (𝐵𝐵ext ) for the indicated stabilization currents. The inset shows a zoom-in of the area marked with the dashed box.
We finally discuss the reliability of and possible reasons for the observed surprisingly long relaxation times of the spin-states of Fe atoms on Pt(111). The spin relaxation times of the excited states extracted from the simulations (Fig.3c) are comparable to the lower limit 𝜏𝜏min (𝑉𝑉) ≈ 3 ns which was roughly estimated from 𝐼𝐼stab. Therefore, we believe that the simulation
results for the spin relaxation times of the excited states are quite reliable. 𝜏𝜏3 /𝜏𝜏4 , which are the longest 𝑇𝑇1 times of all excited states, are thus three orders of magnitude longer than the lifetime of 0.7 − 1.1 ps extracted from the broadening analysis of spin-averaging ISTS
16
or of SP-ISTS
via the fitted effective temperature (see Table 1 and Methods). We therefore propose, that the experimentally observed broadening of the ISTS spectra is dominated by an effective spin coherence time 𝑇𝑇2∗ of the excitations, which effectively takes into account all the decoherence
effects of all states involved in the process. 𝑇𝑇2∗ is typically several orders of magnitude shorter
than the 𝑇𝑇1 times (see Supplementary Note 3). Regarding the extremely long lifetimes of the ground states, the comparison of the outcomes of Ref.
24
and Ref. 5 lead to the conclusion that a
large number of scattering processes do not leave their fingerprints in SP-ISTS spectra. Such hidden scattering processes might cause an overestimation of the intrinsic spin-lifetimes of the 10 ACS Paragon Plus Environment
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ground states calculated by the model. Therefore, the ground state lifetimes need to be confirmed by a more direct technique as, e.g., a pump-probe experiment 4. Ground state lifetimes in the range of micro- to milliseconds are comparable to that of Fe atoms which are decoupled from substrate conduction electrons by thin insulating layers 3. From the simulation we conclude that the reasons for our finding of long living spin-states in Fe atoms directly coupled to the conductive Pt substrate are i) the beneficially large ratio of axial and transverse anisotropy and ii) the small coupling to the substrate electron bath 𝐺𝐺s . Indeed, the simulated values for 𝐺𝐺s are on the same order of magnitude as values found for manganese
atoms on the insulating Cu2N substrate 24. Note, that Pt is a large-susceptibility substrate leading to a large cluster of non-collinearly magnetized Pt atoms beneath the Fe
14, 16
, which will have a
drastic effect on 𝐺𝐺s . It is, therefore, an interesting question, whether the long living spin-states
observed here in an all-metallic system are a general property of adatom/substrate systems using substrates that almost fulfill the Stoner criterion for ferromagnetism, e.g. Rh 33 or Pd 34.
Methods Experimental procedures The measurements have been performed in a low-temperature scanning tunneling microscope facility with an external magnetic field 𝐵𝐵ext applied perpendicular to the sample surface
35
using
the same preparation as well as the same measurement conditions as described in Ref.14. The cleaned Pt(111) surface was coated with a fraction of a monolayer Co resulting in monolayer islands and stripes (see Supplementary Note 1) before single Fe atoms were deposited on the pre-cooled sample leading to a statistical distribution of Fe atoms on the Pt terraces. The spinpolarization of the tip was achieved by using a Cr-coated tungsten tip whose spin-contrast was optimized by dipping into Co or picking up Fe atoms until a strong and magnetically stable outof-plane contrast was observed in the differential conductance channel on the Co islands or stripes. SP-ISTS measurements were performed by stabilizing the tip at the bias voltage 𝑉𝑉stab (applied to
the sample) and current 𝐼𝐼stab on top of the fcc Fe atoms. During the voltage sweep, the feedback loop was swiched off in order to keep the tip-sample separation constant.
d𝐼𝐼 d𝑉𝑉
was recorded as
the response to a small modulation voltage 𝑉𝑉mod (𝑓𝑓mod = 4142 Hz) added to the bias using lockin technique. All shown spectra were normalized via dividing by a substrate spectrum taken with
the same measurement parameters in order to suppress spectral features resulting from the tip. 11 ACS Paragon Plus Environment
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Third order perturbation theory model The experimental SP-ISTS data were modeled by employing a third order perturbation theory 24, which is based on the effective spin Hamiltonian �𝐶𝐶 = 𝑔𝑔𝜇𝜇B 𝐁𝐁 ⋅ 𝐒𝐒� + (3𝐵𝐵20 − (30𝑆𝑆(𝑆𝑆 + 1) − 25)𝐵𝐵40 ) ⋅ 𝑆𝑆̂𝑧𝑧 2 + 35𝐵𝐵40 ⋅ 𝑆𝑆̂𝑧𝑧 4 𝐻𝐻 3𝑣𝑣 1 3 3 3 3 + 𝐵𝐵43 �𝑆𝑆̂𝑧𝑧 �𝑆𝑆̂+ + 𝑆𝑆̂− � + �𝑆𝑆̂+ + 𝑆𝑆̂− � 𝑆𝑆̂𝑧𝑧 � 2
(see Supplementary Note 3), which is consistent with a spin of quantum number 𝑆𝑆 = 5�2
crystal field of 𝐶𝐶3𝑣𝑣 symmetry
factor (𝑔𝑔 = 2.4)
16
in a
28
. Here, the first term is the Zeeman energy with the Landé g-
, the Bohr magneton 𝜇𝜇B , the magnetic field 𝐁𝐁, and the vector of the spin
16
operators 𝐒𝐒�. The second and third terms contain the axial magnetic anisotropy with the operator
of the 𝑧𝑧-component of the spin 𝑆𝑆̂𝑧𝑧 and the axial anisotropy constants 𝐵𝐵20 and 𝐵𝐵40 . The fourth term
is the transverse magnetic anisotropy with the raising (𝑆𝑆̂+ ) and lowering (𝑆𝑆̂− ) ladder operators
and the corresponding transverse anisotropy constant 𝐵𝐵43 . The model is appropriate for an accurate determination of spin relaxation times of excited states, as shown by a comparison of
the lifetime determined by fitting recent SP-ISTS data using the rate equation approach with the one directly measured by pump-probe techniques
23,31
. Besides the anisotropy constants, the
following model parameters have been varied during the fitting of the simulation to the experimental data (see Supplementary Fig.6c): (i) the potential scattering parameter 𝑈𝑈 =
𝑈𝑈pot �𝐽𝐽 , which determines the ratio between potential and Kondo scattering; (ii) the Kondo K
scattering strength of the substrate electrons on the local spin 𝐽𝐽K ⋅ 𝜌𝜌s ; (iii) the coupling between
tunneling electrons and the local spin 𝑇𝑇02 , which is determined by the tunneling current; (iv) the 𝜌𝜌 −𝜌𝜌
tip’s spin polarization 𝜂𝜂t = 𝜌𝜌↑+𝜌𝜌↓, whose strength is given by the imbalance in the spin-resolved ↑
↓
densities of states 𝜌𝜌↑/↓ ; (v) the out-of-plane magnetic field 𝐵𝐵sim ; (vi) an in-plane magnetic field
component 𝐵𝐵𝑦𝑦 ; and (vii) the effective temperature 𝑇𝑇eff. The parameters which have been used in order to simulate the spectra are given in Table 1 and Supplementary Tables 1 and 2. Note, that the extracted 𝐵𝐵20 is consistent with the value data
𝐷𝐷 3
extracted from the simulation of the spin-averaged
. The large values 𝑈𝑈 ≫ 0 display the high efficiency of elastic tunneling contributions
16
through the atom as compared to inelastic transitions. The values for 𝜂𝜂t are in the typical range
of spin-polarizations that were found for magnetically coated tips in SP-STS measurements. The
effective temperature 𝑇𝑇eff = (0.65 ± 0.06) K (see Table 1), which is the only parameter within our 12
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model that considers the additional broadening due to spin decoherence effects (Supplementary Note 3), is considerably larger than the thermodynamic temperature in the experiments (𝑇𝑇 =
0.3 K). We can use it in order to estimate the effective lifetime due to spin decoherence effects via 𝑇𝑇2∗ =
ℏ 2⋅(5.4⋅𝑘𝑘B 𝑇𝑇eff )
≈ 1.1 ps which is consistent with the value determined from the broadening
of the spin-averaged ISTS data
. The small in-plane magnetic fields 𝐵𝐵𝑦𝑦 are probably due to a
16
transverse stray field of the tip. See a description of additional impacts of the simulation parameters on the simulated spectra in the Supplementary Materials. Data availability The authors declare that the main data supporting the findings of this study are available within the article and its Supplementary Information files. Extra data are available from the corresponding author upon reasonable request. Supporting Information SP-ISTS on fcc Fe atoms RKKY-coupled to a Co stripe, additional spectra and simulations, description of the simulation procedure, tables with all fitting parameters.
Notes The authors declare no competing financial interest.
Acknowledgements We would like to thank Samir Lounis, Manuel dos Santos Dias, and Sebastian Loth for fruitful discussions. J.H., M.S., R.W., and J.W. acknowledge funding from the SFB668-A1 and from the GrK1286 of the DFG. M.T. acknowledges funding via SFB767.
Author contributions J.H., M.S. and J.W. designed the experiments. J.H. carried out the measurements and did the analysis of the experimental data together with M.T.. J.H. and J.W. wrote the manuscript, to which all authors contributed via discussions and corrections.
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