Controlled creation and annihilation of isolated robust emergent magnetic monopole like charged vertices in square artificial spin ice

Magnetic analogue of an isolated free electric charge, i.e., a magnet with a single north or south pole, is a long sought-after particle which remains elusive so far. In magnetically frustrated pyrochlore solids, a classical analogue of monopole was observed as a result of excitation of spin ice vertices. Direct visualization of such excitations were proposed and later confirmed in analogous artificial spin ice (ASI) systems of square as well as Kagome geometries. However, such magnetically charged vertices are randomly created as they are thermally driven and are always associated with corresponding equal and opposite emergent charges, often termed as monopole–antimonopole pairs, connected by observable strings. Here, we demonstrate a controlled stabilisation of a robust isolated emergent monopole-like magnetically charged vertices in individual square ASI systems by application of an external magnetic field. The excitation conserves the magnetic charge without the involvement of a corresponding excitation of opposite charge. Well supported by Monte Carlo simulations our experimental results enable, in absence of a true elemental magnetic monopole, creation of electron vortices and studying electrodynamics in presence of a monopole-like field in a solid state environment.


Measurements for average magnetization of individual spin ice vertices
. For our measurements, multiple (more than a million) isolated single vertices for both sample types were patterned using EBL so that the measurements could be performed with the sensitivity of SQUID or vibrating sample magnetometer. For convenience of fabrication, we lithographically patterned four separate grids of 180 × 180 vertices with closed edges. Since each of these individual closed-edge vertex structures has 12 nanomagnets, total no. of nanomagnets patterned is 180 × 180 × 4 × 12, i.e. 1555200. Fig. S1(a) shows the SEM image of four grids and Fig. S1(b,c) show a part of one of the four grids for stained window sample. The deformed stained glass window sample was patterned in the same way ( Fig. S1(d)) . The measurements of net magnetic moment for these nanomagnets were carried out at room temperature using a vibrating sample magnetometer (VSM) equipped in a physical properties measurement system (PPMS, make: Quantum Design). The magnetic field was applied in-plane. Fig. S1(e) shows the magnetization data observed at room temperature for the two types of samples. From the magnetization measurements, average magnetic moment of the individual nanomagnet was determined by dividing the net magnetic moment at saturation by the total no. of nanomagnets, i.e., 1555200. The average magnetic moment of the individual nanomagnets was estimated to be 1.5×10 −5 1555200 = 9.65 × 10 −12 emu. Table-1 shows the parameters obtained from the global magnetic measurements.

MFM measurements:
Magnetic imaging of the stained-glass window samples were carried out using a commercial magnetic force microscope (make: Asylum Research, model: MFP-3D). The MFM system is fitted with a permanent magnet fixed on a rotation module which allowed the in-plane field to vary in the range of ±250 mT. Data were collected using Co-Cr coated tip (ASYMFM, Asylum Research) commercially procured from Asylum research. The tip's average magnetic moment is 1 × 10 −13 emu.

Monte Carlo simulations:
For Monte Carlo simulations, the same field configuration as used in the experiments, i.e., the external field for stained window glass sample was applied at an angle of 10 • with respect to the easy axis of the vertical nanoislands (except that here the field is applied upward, which does not affect for a square lattice). For the deformed stained glass sample, likewise, the external field was applied at an angle of 7 • with respect to the easy axis of the vertical nanoislands. The exact misalignement angle of 30 • was considered for the respective nanomagnet. All the simulations were performed for room temperature and varying magnetic field. We calculated the energy and probability for all 2 12 possible configurations by using a canonical ensemble at varying B (see supplementry material). From these calculations we constructed the energy histograms for a range of B and determined the most probable magnetic configurations obtained at different B values.

Calculations of magnetic field lines:
The magnetic field lines for the system excitations are obtained by subtracting the magnetic field of the fundamental state ( B fund ) from that of the excited state ( B exc ) i.e., ∆ B = B exc − B fund . B fund is the field for magnetic configuration (a) in Fig. 3 for both samples. From this vector field, we draw the tangent lines that follow the same vector field direction. The length scales used are in the units of lattice constant of the artificial lattice.

Field-evolution of magnetic charges in the Monte Carlo calculations
The magnetic charges for the most probable magnetic configurations at different external magnetic fields as obtained by Monte Carlo calculations (Fig. 3 in main text) are analysed according to the the dumbbell model of a magnetic dipole. According to the charge description shown in Fig. 1(c-e) in the main text, we find that both deformed and undeformed samples remain magnetically neutral for all configurations before the system saturates. Fig. S2 shows that the net zero magnetic charge for each magnetic configurations remains conserved during the creation and annihilation of isolated emergent monopole state.

Video of energy histograms
The videos show the probability of all configurations and their energies, within a canonical ensemble. Individual videos (file names: i) Undeformed-sample.mov and ii) Deformed-sample.mov ) for the corresponding two different samples are constructed as follows: first, for a given applied external magnetic field B ext , the energies for all 2 12 possible configurations is calculated. All the energies are shown in the turquoise graph (in video), normalized by the maximum energy of the system for that applied field. It is important to note that each configuration is previously labeled from 1 to 2 12 . Then, the partition function Z = 4096 i=1 e βEi , for this finite case is obtained. Next, the probability for each configuration, P i = e βEi /Z, is calculated and plotted on the graph (red curve). This process is repeated for each value of the applied field and the graph-video is then updated. It is also important to note that the most likely states within a given canonical ensemble provide us with an estimate of the possible states which are observed in the actual experiment. In our case, there is an agreement between the most likely states and the experiment for temperatures near room temperature. The two degenerate most probable states for initial fields are seen as two configurations of probabilities ∼ 0.5. The final configuration with probability 1 is the state when the system saturates at high external fields. We note here that although the videos show the most probable states (different magnetic configurations) which are most likely to be observed in the experiments, a discrepency in the corresponding exact B ext values in the videos and calculations of the energy per spin versus B ext (Fig. 3 in main text) arises due to the use of finite partition functions used for calculations for preparing the videos. The finite partition functions were used to calculate the probability of accessing a state within the ensemble. For the calculations of energy vs B ext (Fig. 3 in main text), Boltzmanntype weight was used to calculate the probability which in this case is more realistic. The videos are presented only to show the field-dependent most propbable accessible states (i.e., magnetic configurations) for the two samples. The vidoes were generated using GNUPlot-x11 and FFmpeg.

Monte Carlo simulation for a broken window glass
As a further test of the precondition of the emergent monopole behavior, Monte Carlo calculations were performed for another type of defective vertex where 50% of the edge-islands were removed (thereby forming a kind of a broken stained glass window) so that there is a charge imbalance in the border charges (see Fig. S3). In the broken glass window, we again observe a type-I ground state (state a). The calculations show an excited state for a very small magnetic field for B ext ∼ 2.5D/gµ (state (b) In Fig. S3). Even in the presence of the charge imbalance of border charges for this case of broken glass window, the magnetizations of respective nanoislands are found to orient such that the net zero magnetic charge of the whole system is maintained at the monopole state. Thus, this provides an additional test of the condition for creation of emergent monopoles in individual vertices.