Magnetism in nanostructured materials provides an exciting field where technology development and fundamental research meet. Magnetic storage devices have shrunk towards dimensions where individual magnetic bits are just a few nanometers in size. At this scale quantum-magnetic effects can emerge.

An important figure of merit for magnetic clusters used in data storage devices is the uniaxial anisotropy energy per atom. Higher magnetic anisotropy per atom allows reduction of bit size without losing stability by perturbations such as thermal energy. The miniaturization of magnetic data storage has therefore been a hunt for materials with maximally high uniaxial anisotropy energy. Atoms adsorbed on metallic surfaces exhibit unusually large magnetic anisotropy For instance, Fe atoms on a monatomically thin copper nitride layer on copper (100) exhibit a magnetic anisotropy barrier of 6meV per atom [Cyrus 2007]. This suggests that an individual Fe atom could be a stable magnetic bit below 1.4K temperature and become room-temperature stable in a cluster of just 200 such Fe atoms; a factor of 5000 smaller than the current state.

Adatoms on surfaces often feature reduced crystal symmetry. In such environments the full magnetic anisotropy includes transverse anisotropy terms in addition to the uniaxial anisotropy. Transverse anisotropy can significantly alter the effective energy barrier for flipping of a magnetic moment by quantum spin tunneling. Hence, full characterization of the magnetic anisotropy in all spatial directions is crucial for predicting magnetic stability of individual atoms and few-atom clusters.

**Fig. 1:** Magnetic vector-field scanning tunneling microscope. (a) Schematic of the vector magnet coil assembly with one solenoid and two Helmholtz coil pairs. STM is located in the center of the coils where arbitrarily oriented magnetic fields can be generated. (b) Constant current topography of isolated Fe atoms on Cu_{2}N/Cu(100). Copper surface (yellow), copper nitride patches (black) and Fe atoms (red). Image size (12x12)nm^{2}, setpoint 10mV, 0.1nA.

For this purpose we have combined a low-temperature scanning tunneling microscope operating at 0.5K with a three axis vector magnet that offers arbitrary field rotation up to 2T magnitude. The magnet is assembled from nested superconducting coils. The inner coil is a solenoid that applies magnetic field parallel to the surface normal of the sample crystal. The two in-plane magnetic field directions are supplied by Helmholtz coil pairs that are rotated 90 degrees to each other, see.** Fig. 1(a)**. The coils can be energized independently resulting in an arbitrarily rotatable magnetic field at the position of the STM in the center of the coils.

Here we present the first measurements with this new vector-magnetic-field STM. We characterize the full magnetic anisotropy field of Fe atoms on Cu_{2}N/Cu(100). **Figure 1(b)** shows two Fe atoms located on separate patches of Cu_{2}N. An individual atom can be addressed by positioning the STM tip above it and recording the differential conductance (d*I*/d*V*) as a function of voltage. A magnetic field of 2 Tis applied and rotated stepwise while successive d*I*/d*V* spectra are recorded. At 0.5K, the thermal energy is smaller than the Zeeman energy and the atom’s spin primarily populates the spin ground state. Inelastic electron tunneling appears as steps in the dI/dV that indicate the energy of spin excitations out of the ground state. These steps are a direct measure of the magnetic anisotropy and it is therefore possible to obtain a full map of the magnetic anisotropy by tracking the variation of spin excitation energy as a function of magnetic field direction.

**Fig. 2:** Magneto-crystalline anisotropy of Fe on Cu_{2}N/Cu(100). (a) Schematics of the Cu_{2}N lattice with an Fe atom adsorbed on the Cu binding site and assignment of the x,y,z coordinate system. N (green), Cu (yellow) and Fe (Blue ball). Z: direction parallel to N atom row starting at the Fe atom, X: vacancy direction, Y: Surface normal. (b) Differential conductance spectra, d*I*/d*V*, recorded for 2T magnetic field applied in the high-symmetry directions x,y,z, compared to a d*I*/d*V* spectrum recorded without magnetic field, B=0T. Nominal junction impedance, 10MΩ, chosen at 10mV, 1nA. Successive spectra are vertically offset by 0.03nA/mV for clarity.

The Cu_{2}N lattice has a C_{2V} symmetry at the Cu binding sites where Fe atoms adsorb. Fe is therefore expected to feature two-fold symmetry with respect to three characteristic crystal symmetry directions (**Fig. 2(a)**): 1. the surface normal (*y* direction), 2. the direction parallel to the row of N atoms which is in contact with the Fe (*z* direction), 3. the direction parallel to the row of vacancy sites perpendicular to the row of nitrogen atoms (*x* direction).

**Figure 2(b)** shows d*I*/d*V* spectra recorded with magnetic field applied in these high symmetry directions. The *z*-direction is the uniaxial anisotropy direction indicated by a strong modification compared to a spectrum recorded at *B*=0T. The spectra for magnetic field in *x*- and *y*- direction show a small difference which is an indication of finite transverse anisotropy. Previously, the magnetic field dependence along these three directions was fitted with a model Spin Hamiltonian that inputs C_{2v} symmetry of the Fe atom. It yields uniaxial anisotropy, *D*=-1.55meV, and transverse anisotropy, *E*=+0.3meV [1].

With our current set-up, the full three-dimensional characterization of the magnetic anisotropy provides model-free information about the Fe atom’s symmetry. We record three field rotations: one in the plane of the sample (*z*,*x* plane), one in the plane spanned by the vacancy row direction and the surface normal (*x*,*y* direction), and one in the plane spanned by N row direction and the surface normal (*z*,*y* direction). **Figure 3(a)−(c)** show the corresponding d*I*/d*V* spectra. To visualize the rotation-dependence we plot the absolute value of the second derivative of the tunnel current |d^{2}*I*/d*V*^{2}|. Steps in the d*I*/d*V* appear as peaks in the d^{2}*I*/d*V*^{2} making shifts in energy and amplitude of the steps easier to spot.

**Fig. 3:** Evolution of spin excitations in three-dimensional magnetic field rotations. (a)−(c) differential conductance spectra recorded as a function of orientation of a 2T magnetic field rotating in (a) x−z plane with angle *θ*, (b) y−z plane with angle *Φ* and (c) z−y plane with angle *γ*. To highlight the evolution of the spin excitations we plot |d^{2}*I*/d*V*^{2}| color coded as a function of bias voltage and field angle (details see text). All spectra in (a) and (b) were measured on the same Fe atom and (c) on another Fe atom. (d)−(f) Calculated conductance spectra using the quadratic Spin Hamiltonian plotted in the same manner as (a)−(c). Fit parameters are *g*=2.11, *D*=-1.62meV, *E*=0.31meV for (d) and (e); *g*=2.11, *D*=-1.55meV, *E*=0.31meV for (f).

To reduce the measurement time we chose field angles that are incommensurate with the expected symmetry of the Fe atom. The real measurements are then replicated according to the symmetry rules of C_{2V}. Deviations from the C_{2V} symmetry would show up as discontinuities in the evolution of the spin excitations as the field is rotated. No such discontinuities appear in any of the three magnetic field rotations. The measured anisotropy shows a prominent 180 deg. rotation symmetry and mirror symmetry with respect to the high-symmetry axes (*x*,*y*,*z*). This confirms the C_{2V} symmetry of the Fe atom

To obtain a quantitative measure of the full anisotropy we fit the measured magnetic field dependence with numerically derived d*I*/d*V* spectra. The spin states as a function of magnetic field are calculated using a quadratic Spin Hamiltonian [2]

*H = -gμ*_{B}**BS** + DS_{z}^{2}* +E(S*_{x}^{2}* – S*_{y}^{2}*)*

The first term represents the Zeeman energy resulting from the external magnetic field *B*, where *g* is the Landé g-factor, *μ*_{B} the Bohr magneton and *S* the spin operator representing the Fe spin. The spin magnitude of Fe on this surface is *S*=2. Note that the three-dimensional vector operators must be used because the magnetic field no longer coincides with the high-symmetry directions. The remaining terms represent the quadratic magneto-crystalline anisotropies experienced by the Fe spin, quantified by the uniaxial anisotropy parameter *D*, and the transverse anisotropy parameter E. To obtain a simulated dI/dV spectrum we solve for the bias-voltage dependent intensity for inelastic tunneling [2]. The simulated spectra (dotted black lines in Fig. 3(a)−(c) show remarkable agreement with the measured vector-field dependence data. The positions of the spin excitations as well as the relative intensities are well reproduced for *D*=-1.62meV, and *E*=0.31meV.

It is worth noting that small deviations in excitation energy of order 0.1meV occur at the angles 0 deg and 180 deg that cannot be accounted for by the above model. In addition, the intensity of the spin excitation at ≈1meV appears to diminish faster than predicted when the field is rotated away from 90 deg and 270 deg. in the sample plane. The quadratic anisotropy constrains the possible shape of the vector-field dependence. Consequently, these deviations may indicate the presence of higher orders in the spin operators [3] and will be the subject of further studies.

In summary we have commissioned a new scanning tunneling microscope capable of mapping the full three-dimensional magnetic anisotropy field of individual atoms. In a first application of this technique we verify the C_{2V} symmetry of Fe atoms on Cu_{2}N/Cu(100). Our results show that a quadratic Spin Hamiltonian provides an accurate description of the Fe atom’s anisotropy. In the future the ability to apply arbitrary magnetic fields will be used to characterize the magnetic anisotropy of complex spin systems such as three dimensional single molecule magnets or nanostructures, and surfaces with non-collinear magnetism.