advances.sciencemag.org/cgi/content/full/3/5/e1603137/DC1

Supplementary Materials for

Nonlocally sensing the magnetic states of nanoscale antiferromagnets

with an atomic spin sensor

Shichao Yan, Luigi Malavolti, Jacob A. J. Burgess, Andrea Droghetti, Angel Rubio, Sebastian Loth

Published 26 May 2017, Sci. Adv. 3, e1603137 (2017)

DOI: 10.1126/sciadv.1603137

This PDF file includes:

section S1. Correspondence between nano-antiferromagnet Néel states and Fe

trimer lifetime T1

section S2. Response of T1 to different magnetic perturbations

section S3. Deducing magnetic interaction between nano-antiferromagnet and Fe

trimer spin sensor (Eq. 1)

section S4. Tuning the signal-to-noise ratio by external magnetic field and

interaction with magnetic tip

section S5. DFT calculations for the magnetic interaction

fig. S1. Spin-state diagram of the trimer.

fig. S2. Pump pulses induced switching of antiferromagnetic nanostructures.

fig. S3. Calibrating Fe trimer with external magnetic field.

fig. S4. Improving the signal-to-noise ratio by external magnetic field or local

magnetic field exerted by the magnetic STM tip.

fig. S5. Time trace of the pump-probe signal for 4000 s about the four-state

switching in Fig. 6.

fig. S6. Supercells used for the DFT calculations.

fig. S7. Calculated magnetic interaction energy for different spin chain and

supercell geometries.

References (42–44)

 

section S1. Correspondence between nano-antiferromagnet Néel states and Fe trimer

lifetime T1

The correspondence between the Néel states of the nano-antiferromagnet (nano-AF) and the two-

state fluctuations in T1 of Fe trimer were measured by an alternating sequence of spin polarized

STM images and pump-probe spectroscopies as following:

(1) first the state of the nano-AF was determined by spin-polarized imaging;

(2) immediately afterwards the tip was positioned on the trimer and T1 of the Fe trimer was

measured using pump-probe spectroscopy;

(3) finally the nearby nano-AF was re-imaged to verify its state.

When no switching was detected in the pump-probe trace of step (2), the nano-AF was observed

to be in the same Néel state in both step (1) and step (3). This verified that it remained in the

same state throughout the measurement. We repeated this procedure many times with the nano-

AF’s initial state being “1” or “0”. For all cases where a switch from longer T1 to shorter T1 in

step (2) is observed, the nano-AF is in Néel-state “0” in step (1) and Néel-state “1” in step (3),

and vice versa. By means of this procedure we were able to identify which state of the nano-AF

enhances the T1 of the trimer.

section S2. Response of T1 to different magnetic perturbations

We use an effective spin Hamiltonian model and electron-spin scattering up to second order in

the atomic spin to calculate the response of T1 of the Fe trimer to different magnetic

perturbations. First, we calculate the energy eigenstates of the Fe trimer based on the same spin

Hamiltonian, , as reported previously (42-44), but with an added term for the magnetic

perturbation, .

g

are the vector spin operators of the ith Fe atom and , , are the x,y,z components of it. This

model includes magneto-crystalline anisotropy of the Fe atoms, Di, Ei,, nearest-neighbour

 

Heisenberg exchange interaction Jh, and Zeeman energy, g (g is the Landé g-factor, B the

Bohr magneton and the magnetic bias field). The magnetic bias field is applied parallel to the

easy axis of the Fe atoms. The parameters Di, Ei, gi, and Jh can be determined from dI/dV spectra.

For the Fe trimer shown in Fig. 2 we find best agreement with the experiment for

D1,3 = 2.1 meV (for the corner atoms), D2 = 3.6 meV (for the center atom), E1,2,3 = 0.3 meV,

g1,2,3 = 2, Jh = 1.15 meV.

As perturbations we consider:

1. Longitudinal magnetic field, ∑ g || with B|| applied parallel to the easy axis

of Fe atoms in Fe trimer.

2. Perpendicular magnetic fields ∑ g ,, with applied in the hard

magnetic axis of Fe and in the remaining axis.

3. Magnetic exchange interaction with another magnetic nano-object. For simplicity we assume

a net Heisenberg interaction which results from the interaction of all atoms of the nano-object

with all atoms of the trimer, ∑ , where JnAF is the net exchange interaction

strength.

Having computed the energy eigenstates of the perturbed Hamiltonian we proceed to calculate

the spin relaxation time of the Fe trimer. It is determined by electron-scattering from the first

excited spin state, |, to the spin ground state, |, and given by

1 Δ ∙ ∞

∞

where f() is the Fermi function, E the energy splitting between and GS is the effective

conductance of the scattering channel which is determined by the density of states of the Cu

substrate and the coupling strength of the Fe atom spin to them (42-44). Throughout this work, it

is maintained as a constant, GS = 4.0 µS. M is the transition matrix element between | and |

which up to second order in the impurity spin is given by

, ′ ,, ,

 

The scattering operator is chosen analogous to previous reports as the sum of a spin-independent

term, u, and an exchange scattering term, , between Fe atom spin and substrate electron. We

maintained u = 0.7 for all calculations because this value fits dI/dV spectra of individual Fe

atoms on Cu2N (44). The scattering operator acts on the states | and |’ which are product

states of the electron spin and the Fe trimer spin.

We note that at 0.5 K and at finite E the Fermi functions can be approximated by step functions

and the integral simplifies. Then, the spin relaxation time is Δ . For simplicity

we use this approximation in the main text but the calculations shown in Fig. 3b use the full

solution.

section S3. Deducing magnetic interaction between nano-antiferromagnet and Fe trimer

spin sensor (Eq. 1)

As shown in the calculation in Fig. 3b and the experimental measurements in fig. S2, T1 of Fe

trimer changes linearly with the perturbation energy induced by external magnetic bias field,

Δ || . The slope of the linear dependence can be expressed as:

Δ || ||.

||

Δ || ||.

1

g ∑1

4g || (S1)

where g is the Landé g-factor for Fe atoms on Cu2N surface, and the Bohr magneton. and

are the expectation values of the z-component of the spin operator of the i-th Fe atom for the

first excited state and the ground state of Fe trimer. For all magnetic field and perturbation

strengths that do not move the trimer into the avoided level crossing where dT1/dE becomes

nonlinear, these expectation values are equal to +2 or 2 to sufficient accuracy. Hence,

∑ , , 2 and ∑ , , 2.

Analogously, T1 of the Fe trimer varies linearly as the perturbation energy induced by the

exchange interaction with the nearby antiferromanget (∆ ). This linear dependence is:

∆

∆ (S2)

 

T1 is the difference in T1 as the nearby nano-AF switches between its Néel states. When the

nano-AF is in Néel-state “0”, the magnetic interaction with Fe trimer will increase the energy

difference between the two low-energy states of Fe trimer by E/2, whereas it will decrease by

E/2 for Néel-state “1”, Fig. 3a. So, the energy difference induced by the Néel-state switching

of the nano-AF is E.

Since the linear dependence of T1 with B|| and J, ∆ ||

and ∆

, are almost identical (as

shown in Fig.2b), we can combine Eq. (S1) and Eq. (S2) to obtain the relation between T1 and

∆E(J) that gives the energy perturbation to the Fe trimer induced by the nearby nano-AF that is

shown as Eq. (1) in the main text

∆4g

1||

Δ 1

section S4. Tuning the signal-to-noise by external magnetic field and interaction with

magnetic tip

The change in T1 upon nano-AF switching (T1) is constant for each separation between Fe

trimer and nano-AF. The signal-to-noise ratio in our non-local sensing measurement scales with

T1 / T1 (fig. S4), and can therefore be improved by decreasing T1. This can be achieved either

by reducing the external magnetic bias field (fig. S4 c) or by adding a local exchange bias field

with the magnetic tip (fig. S4 d). The local exchange bias field acts on the trimer just like an

externally applied magnetic field. Since different magnetic perturbations sum each other (see

Fig. 3c) either possibility or a combination of the two can be used. Fig. S4 shows that, in order to

clearly detect the difference between the two spin relaxation curves recorded for the nano-AF in

its two Néel states, T1 / T1 has to be tuned to be larger than 0.

ratio

 

section S5. DFT calculations for the magnetic interaction

The Cu2N/Cu(100) system was modelled as a slab containing four layers of Cu and one layer of

Cu2N. 20 Å of vacuum were introduced to separate the slab from its periodic image. The typical

supercells employed in the calculations of the magnetic interaction are shown in fig. S6, where

the red (blue) spheres represents Fe atoms with spin up (down). The fig. S6, a and b contain two

Fe atom chains and two Fe atom ladders. In particular, fig. S6a and S6b correspond to the Néel

state “0” and “1” respectively. The inclusion of two ladders and two chains is required to avoid

frustration induced by the periodic boundary conditions. Furthermore, the presence of several

magnetic atoms makes the computed energy differences per supercell much larger than the

numerical accuracy (that is 1 meV per supercell as discussed in the Methods section). Figs. S6, c

and d show supercells, which contain two parallel Fe atom chains and that were used to extract

the magnetic interaction energy for distances up to 2.6 nm. Finally, in order to obtain insight into

the importance of finite size effects, we also considered supercells with single ladder-chain and

chain-chain pairs separated by approximately 3 nm from their periodic images (fig. S6e). Since

at such distance the magnetic interaction energy drops below numerical accuracy, we expect the

result to be closer to that for isolated chain-chain and chain-ladder pairs. The computed magnetic

interaction energies for the different supercell geometries are presented in fig. S7.

Overall the results all agree in predicting an antiferromagnetic interaction and on the general

decay. Still, we note that finite size effects play an important role for the smallest distances. For

both the ladder-chain and the chain-chain case, the interaction energy tends to increase when the

coupling of the nanostructures with their periodic images becomes negligible (compare the

corresponding filled and open symbols in fig. S7). In the limit of completely isolated chain-chain

pairs one would likely recover the exponential decay found experimentally.

 

 

fig. S1. Spin-state diagram of the trimer. The eigenstates computed for the Fe trimer are plotted as a function of the projection of the state’s total spin in the z direction and the state energy. A field of 0.5 T applied along the easy axis of the trimer (z direction) is used in this

calculation. The two low-lying states are indicated with blue (ground state |) and red (first

excited state |) points. Inset diagrams show the z spin projections of each atom of the trimer for

| (at left) and | (at right) with red indicating negative spin, and blue indicating positive spin. Each state is close to a pure antiferromagnetic Neel state; each atom has a z spin projection |Sz| = 1.99 compared to the Fe spin of 2. Nevertheless finite overlap exists between these two states, and permit transitions. Magnetic perturbations change the magnitude of this overlap and

consequently change the relaxation rate between | and |.

 

fig. S2. Pump pulses induced switching of antiferromagnetic nanostructures. Time traces of the pump-probe signal measured on the Fe trimer with different pump voltages. The time trace monitors the switching of a 12-atom antiferromagnet that was separated from the Fe trimer by 2.2 nm (see main text Fig. 2 second panel from left for STM topograph of the configuration). Pulse amplitude and duration: pump pulse: variable amplitude and fixed duration, 80 ns; probe pulse: 3 mV, 100 ns; time delay between pump and probe pulse fixed at 200 ns. Magnetic field 0.5 T. Large pump pulse amplitudes induced fast switching of the antiferromagnets between its two Néel states. Decreasing the pump pulse amplitude reduced the switching rate quickly and it reached once per tens of minutes for pump pulses of 10 mV amplitude or less. This indicates that the spin sensing method is invasive. But its influence can be made sufficiently weak to allow measurements that are non-perturbative on the time-scale of our experiment (main text Fig. 1b). Since the switching rate increases with pulse amplitude we speculate that it is induced by hot carriers injected from the tip into the sample which propagate to the nanomagnets. Similar behavior has been observed for adsorbed molecules on surfaces.

 

fig. S3. Calibrating Fe trimer with external magnetic field. Spin relaxation time, T1, of an Fe trimer as a function of external magnetic field. a, For the configuration shown in main text Fig. 2a top panel with 3.0 nm separation between Fe trimer and nanomagnet. b, For 2.2 nm separation as shown in main text Fig. 2a second panel. c, For 1.5 nm separation as shown in main text Fig. 2a third panel. d, For 3.0 nm separation as shown in main text Fig. 2a bottom panel. Red points are measured values of T1 as the nanomagnet stays in Néel state “0” and blue points are measured for Néel state “1”. Solid lines are linear fits and the fitted slope is indicated in each plot.

 

fig. S4. Improving the signal-to-noise ratio by external magnetic field or local magnetic field exerted by the magnetic STM tip. a, Constant current topograph of a Fe trimer and a nanomagnet recorded with a spin-polarized tip. Image size, (5.8 × 5.8) nm2. The separation between Fe trimer and nanomagnet is 2.2 nm. b, Red and blue dots show pump-probe spectra measured on the Fe trimer as the nanomagnet is in different Néel states. Solid lines are exponential fits. External magnetic field is 0.75 T and tunnel junction setpoint is 5 mV, 200 pA. c, Same measurement as (b) but measured at 0.25 T external magnetic field. The difference between red and blue pump-probe curves is significantly more pronounced than in b, because of

an increased T1/T1. d, Same measurement as b but measured with 5 mV, 1.2 nA tunnel condition which will move the magnetic tip closer to Fe trimer. The magnetic tip exerts a local

exchange bias field on Fe trimer that acts analogous to a magnetic field and also increases T1/T1.

 

fig. S5. Time trace of the pump-probe signal for 4000 seconds about the four-state switching in Fig.6. The delay time between pump and probe pulses is 180 ns. External magnetic field, 1.5 T, tunnel junction setpoint, 5 mV, 500 pA. Pulse amplitude and pulse duration: pump pulse, 9mV, 50ns; probe pulse: 3mV, 100ns.

 

fig. S6 Supercells used for the DFT calculations. The distance d between the two chains or the chain and the ladder is varied in the calculations. Color code: Cu atom - brown sphere, N atom – grey sphere, Fe atom with spin up – red sphere, Fe atom with spin down – blue sphere. a Supercell with two Fe atom chains and two Fe atom ladders in the magnetic configuration that corresponds to the chain-ladder antiferromagnetic alignment that was found to be lower in

energy (/2, see Fig. 4C inset). b, Same supercell as in a, but showing the chain-ladder

ferromagnetic alignment that was found to be higher in energy (+/2, see Fig. 4C inset). c

Supercell with two Fe atom chains for the calculation of the magnetic interaction energy between the chains for all distances up to 2.6 nm. The plotted supercell is for 1.44 nm distance and shows the antiferromagnetic alignment of the two chains that was found to be the magnetic ground state

(/2). d, Same supercell as in c but showing the ferromagnetic alignment of the two chains

(+/2). e, Supercell with a Fe chain pair at 1.44 nm distance with an additional section of free Cu2N, placing the periodic images of the Fe chains created by the periodic boundary conditions at about 3 nm distance.

 

fig. S7. Calculated magnetic interaction energy for different spin chain and supercell

geometries. Purple squares: DFT results obtained for a Fe ladder coupled to a Fe chain

calculated as the energy difference between the supercells in Figs. S6a and 6b. Since these

supercells contain two ladders and two chains, the interaction energy between only one chain and

one ladders is ¼ of the total energy difference per supercell. Blue filled circles: DFT results

obtained for two coupled Fe chains calculated as the energy difference between the supercells in

Figs. S6c and S6d. The interaction energy between the two chains is ½ of the total energy

difference per supercell because the chain at the left supercell edge couples equally to the second

chain within the supercell and its periodic image created by the periodic boundary conditions.

Blue open circles: DFT results obtained for two Fe chains in a supercell as shown in Fig. S6e.

The supercell contains additional Cu2N surface to place the periodic image at 3 nm distance.

Therefore, the magnetic interaction energy between two Fe chains is directly given by the total

energy difference per supercell (refer to Figs. S6c and S6d). Black diamonds: DFT results

obtained for a Fe ladder and a Fe chain for a supercell with additional Cu2N surface analogous to

Fig. 6e. The magnetic interaction energy is directly given by the total energy difference per

supercell. Note that the supercells contain two Fe atoms along the chain so that the energy

difference represents the energy per 2-atom segment of an infinite chain.