7.2 Co adatoms stabilized by Co nanowires
7.2.2 Theoretical modeling
The direct exchange interaction can be immediately excluded to be respon-sible for the finding because of the long range of the coupling behavior.
In order to evaluate the role of dipole-dipole interaction, the interaction energy between a Co adatom and a Co ML stripe is calculated according to the equation
Jdip−dip(d)= X
j
1
r3j [m~ ·m~j−3(m~ ·rˆj)(m~j·rˆj)], (7.1) where the summation is over all the atoms of the stripe,m~ and m~jare the magnetic moments of the adatom and of the atoms of the stripe, and ~rj
( ˆrj) is the relative distance (unit vector). Considering the perpendicular magnetic anisotropy, it can be reduced to
Jdip−dip(d)=m~ · X
j
1
r3j m~j. (7.2)
It is assumed that the Co stripe has a width of 10 nm and a homogeneous saturation magnetization of 1.3 × 106A/m so thatmjcan be deduced. The calculated result is shown with the black line in Figure 7.2. It is found thatJdip−dipalways favors AF and is negligible even at very short distance.
Therefore, it is concluded that the interaction is dominated by indirect exchange via the Pt conduction electrons.
F
AF
Distance from ML (nm)
E x c h a n g e e n e rg y J ( e V ) m
1.0 2.0 3.0 4.0 5.0
A B
C
-300 -200 -100 0 100 200 300
dipole-dipole 1D RKKY 2D RKKY 3D RKKY
Figure 7.2: RKKY coupling of adatoms to nanowires. Dots show the measured interaction energy as a function of distance from ML stripes as indicated by the arrow in Figure 7.1(d). The black line is the dipolar interaction calculated as described in the text. The red, blue, and green lines are fits to 1D, 2D, and 3D range functions for RKKY interaction.
Magenta points correspond to atom A, B, and C in Figure 7.1. Horizontal error bars are due to the roughness of the Co-ML-stripe edge, whereas the vertical ones are due to the uncertainty inBex. (Tunneling parameters:
Vstab= +0.3 V,Istab =0.8 nA,Vmod =20 mV (rms), andT=0.3 K.)
If only isotropic exchange is considered (neglect Dzyaloshinskii-Moriya interaction, see Chapter 7.3.2), the interaction can be described by the following Hamiltonian:
Hˆ = J(d~ij) ˆS~i ·~Sˆj. (7.3) Here, ˆ~Siand ˆ~Sjare the spin operators of the atoms at position~ri and~rj, and J(d~ij) is the exchange constant that depends on their distance d~ij = ~ri −~rj. J(d~ij) usually depends on the distance|d~ij|as well as on the direction ofd~ij. However, in an isotropic itinerant electron system, one can assumeJ(d~ij)= J(dij). If the exchange is dominated by three-dimensional (3D) electrons, J(d) is given by the following so called range function [117, 118, 119],
J(d)=6πZJ2N(EF)[sin(2kFd)
(2kFd)4 − cos(2kFd)
(2kFd)3 ], (7.4)
where Z is the number of conduction electrons per atom, J is the s-d ex-change constant, N(EF) is the density of states at EF, and kF = 2π/λF is the Fermi wave-vector. So it contains short- and long-range terms, and reduces to
J(d)= J0cos(2kFd+δ)
(2kFd)3 (7.5)
at large distance, where a phase factor δ is included to account for the scattering phase due to the charge difference between impurity and host and due to the former’s angular momentum [8]. J0depends on the density of the conduction electrons, thes-dexchange constant and the density of states at EF. The cos(2kFd) term determines the oscillating period as half the Fermi wavelength, while (2kFd)3 determines the decay rate, i.e., the larger kF the faster it decays. In order to evaluate the difference of the approximation Equation 7.5 from Equation 7.4, Figure 7.3(a) shows the curves calculated from these two equations assuming kF = 2π/(1.6 nm) which is a typical value for the Pt(1 1 1) surface-related electrons. It can be concluded from the curves that the short-range term sin(2kFd)/(2kFd)4is already negligible in this case whend > 1 nm while the long-range term cos(2kFd)/(2kFd)3 decays relatively slowly oscillating between FM and AF with distance.
Because RKKY interaction is mediated by conduction electrons, it de-pends on their dimensionality. In the case of isotropic D-dimensional electron systems which mediate the interaction, the long-range behavior
J3(a.u.) (a)
d(nm)
1 2 3 4 5
-8 -4 0 4
0 10 20 30 X106
(b)
d(nm)
1 2 3 4 5
JD(a.u.)
-8 -4 0 4
0 10 20 30 X106
Figure 7.3: RKKY model within free electron approximation. (a) Com-parison of the range functions of Equation 7.4 (blue) and Equation 7.5 (red) for 3D RKKY interaction at large distance. Insetshows the short-distance behavior, from 0 to 0.5 nm. (b)Dimensionality dependent range functions.
Red: 3D; Blue: 2D; Green: 1D.Inset shows the short-distance behavior, from 0 to 0.5 nm.
correspondingly can be generalized to [130, 131, 132]
J(d)= J0·cos(2kFd+δ)
(2kFd)D . (7.6)
Figure 7.3(b) shows curves calculated for different dimensionalities with the samekFas in Figure 7.3(a). Obviously, the 1D curve decays most slowly and is still visible atd = 5 nm where the 2D and 3D curves are already very weak. Therefore, RKKY interaction becomes important for a low-dimensional system especially of the size comparable with the Fermi wave-length. This conclusion is understandable, because a low-dimensional electron gas is more unstable and therefore can be more easily perturbated by impurities.
In order to test whether an RKKY description is appropriate, the data points of Figure 7.2 are fitted using Equation 7.6 with different assumed dimensionalities,D. J0,kFandδare taken as variable parameters. Figure 7.2 shows the corresponding result. A good agreement is found forD=1 and a wavelength ofλF = 2π/kF ≈ 3±1 nm, corresponding to an oscillation period of the exchange energy of 1 to 2 nm.
For the RKKY interaction at the surface between adatoms and ML stripes a dimensionality below 2 is indeed expected. The interaction is dominated by surface related (2D) states due to their smallerkF as com-pared to those of the bulk. The superposition of the contributions from all Co atoms along the stripe edge attenuates the decay further resulting in a
island D
E
4 nm Distance from ML (nm)
J (eV)m
1 2 3 4 5
-300 -200 -100 0 100 200 300
1D RKKY
D E
a b
Figure 7.4: RKKY coupling of Co adatoms to a Co ML island. (a) STM topograph of an area with a Co ML island and Co adatoms nearby. Note that the island is located on the same terrace as the adatoms. (b) Same as Figure 7.2 but including the exchange energy of adatom D and E with the island (red points) whose topography is shown in (a). Due to the sharpness of the island-Pt edge, the horizontal error bars for points D and E are smaller. (Tunneling parameters: Vstab = +0.3 V, Istab = 0.8 nA, Vmod =20 mV (rms), andT=0.3 K.)
dimensionality close toD =1. This conclusion is analogous to the case of the exchange interaction between ferromagnetic layers separated by non-magnetic spacer layers where the dominating states are bulk (3D) states, while the summation over the atoms in the layer can result in a 2D asymp-totic behavior [133]. It can be concluded that the experimentally observed indirect exchange is due to a state with a strong localization in the surface.
The measured period of the RKKY interaction is consistent with the period extracted from the pairwise exchange (see Chapter 7.3) and from the KKR calculations of the spin polarization in the Pt(1 1 1) (see Chapter 8.3). It will be shown later by investigating the exchange interaction between adatom pairs, that the responsible state for the RKKY interaction is most probably the same surface resonance with an effective mass of 1.5me which is also responsible for the electron scattering at subsurface defects described in Chapter 4.1.2.
The RKKY interaction curve shown in Figure 7.2 is measured on ad-atoms residing on the Pt terrace which is adjacent to and thus one atomic layer higher than the one the nanowire is grown on. There are also ad-atoms found in the vicinity to the Co ML which are on the same Pt terrace than the ML. Because the hybridization of the nanowire to the Pt layer in these two cases is different leading to a different s−d exchange, one could expect a different strength of the RKKY interaction. Figure 7.4(a)
illustrates an atomic layer high Co island on the Pt(1 1 1) terrace. Several adatoms can be found nearby, of which single-atom magnetization curves are recorded. Similar to the previous case, AF and FM couplings are found.
For instance, the adatom D is coupled AF and the adatom E is coupled FM to the Co island. The coupling strength values are plotted with red dots in Figure 7.4(b). Obviously, the coupling strength is consistent with the previous case.
In conclusion, it has been demonstrated in this section that the RKKY interaction can be directly detected with atomic resolution at an energy scale of tens ofµeV by recording single-atom magnetization curves. The interaction with nearby Co nanowires can stabilize the magnetization of Co adatoms.