Magnetic reversal of a quantum nanoferromagnet

When the external magnetic field applied to a ferromagnetically coupled atomic chain is reversed suddenly, the magnetization of the chain switches, due to the reversal of all the atomic magnetic moments in the chain. The quantum processes underlying the magnetization switching and the time required for the switching are analyzed for model magnetic chains adsorbed on a surface at 0 K. The sudden field reversal brings the chain into an excited state that relaxes towards the system ground state via interactions with the substrate electrons. Different mechanisms are outlined, ranging from the global stepwise rotation of the chain macrospin induced by spin-flip collisions with substrate electrons in the pure Heisenberg chain (Néel-Brown process) to a correlation-mediated direct switching process in the presence of strong magnetic anisotropies in short chains (the global spin of the chain reverses in a single electron interaction). The processes for magnetization switching induced by electrons tunneling from a scanning tunneling microscope tip are also analyzed.

Figure 1

Populations of the 13 sublevels of the $S_{\mathit{Tot}}$ $=$ 6 lowest manifold of a chain of 12 atoms as a function of time following the reversal of the direction of the applied B field (0.1 T). The system is initially in the highest state of the manifold ($M_{\mathit{Tot}}$ $=$ $+$6, full green line) and relaxes to the ground state ($M_{\mathit{Tot}}$ $=$ −6, full cyan line). The populations of the various $M_{\mathit{Tot}}$ states appear and disappear successively. The population of the central state of the manifold ($M_{\mathit{Tot}}$ $=$ 0) is shown in red.

Figure 2

Upper panel: variation of the magnetic moment of a chain of $N$ atoms as a function of the time following the reversal of the direction of the applied magnetic field (0.1 T). The magnetic moment is plotted relative to its maximum value. The number of atoms in the chain is given in the insert (the shorter the chain, the faster the magnetic switch). Lower panel: variation of the population of the ground state of a chain of $N$ atoms as a function of the time following the reversal of the direction of the applied magnetic field (0.1 T). Same color code as in the upper panel (the shorter the chain, the faster the magnetic switch).

Figure 3

Switching time of the magnetization of a chain of atoms when the direction of the applied B field (0.1 T) is reversed, as a function of the chain length (number of atoms in the chain). Two definitions of the switching time are shown: twice the time for the magnetic moment of the chain (projection on the field axis) to reach zero $\left(2T\right[\langle \mathrm{M}\rangle =0\left]\right)$ and the time for the population of the ground state to reach 90$\%$, $T$(90$\%$).

Figure 4

Population of the ground state of a chain of 12 atoms as a function of a scaled time following the reversal of the applied B field direction. The scaled time is equal to the product of the physical time in nanoseconds and the field strength expressed in Tesla. Various strengths of the B field are shown (see insert). At low fields, all the population curves are superimposed and represented by the dashed line, which shows the results for 0.1 T; as the B field is increased, the ground-state population increases earlier.

Figure 5

Populations of the 11 sublevels of the $S_{\mathit{Tot}}$ $=$ 5 lowest manifold of a chain of 10 atoms as a function of time following the reversal of the direction of the applied B field (0.1 T). The system is initially in the highest state of the manifold ($M_{\mathit{Tot}}$ $=$ $+$5) and relaxes to the ground state ($M_{\mathit{Tot}}$ $=$ −5). An electron from an STM tip with spin polarization down is injected into an end atom of the chain at intervals of 0.5 ns, creating the sharp structures in the populations. The populations of the various $M_{\mathit{Tot}}$ states (full lines) appear and disappear successively. The population of the initial upper state of the manifold ($M_{\mathit{Tot}}$ $=$ 5) is shown by a full green line, and that of the central state of the manifold ($M_{\mathit{Tot}}$ $=$ 0) is shown by a full red line. The dashed curves (green and blue) correspond to the case where no electron is injected (populations of the upper and lower states of the manifold, respectively).

Figure 6

Magnetization (relative mean magnetic moment of the chain) as a function of time following the reversal of the applied B field (B $=$ 0.1 T) for an $N$ $=$ 10 chain in which polarized electrons are injected into an end atom of the chain. The electrons are injected periodically with a constant $dt$ time interval. The different full lines correspond to different $dt$ intervals (see insert): the shorter the $dt$, the faster the magnetization change. The dashed line corresponds to the case where no electrons are injected, and the relaxation only proceeds via collisions with the substrate electrons.

Figure 7

Total number of injected electrons required to perform the magnetization switch as a function of the time interval, $dt$, between injected electrons. The chain contains 10 atoms, and the B field is 0.1 T. Full black circles and full black line: number of electrons for the ground-state population to reach 99$\%$. Full red diamonds: number of electrons for the ground-state population to reach 90$\%$.

Figure 8

Magnetization (relative mean magnetic moment of the chain) as a function of time following the reversal of the applied B field (B $=$ 0.1 T) for an $N$ $=$ 10 chain in which electrons are injected into an end atom of the chain. The electrons are injected periodically with a constant $dt$ $=$ 0.2 ns time interval. Full red line: the injected electrons are spin down. Full green line: the injected electrons are nonpolarized. Full black line: no injected electrons.

Figure 9

Energy diagram of an $N$ $=$ 5 chain of $S$ $=$ 2 spins as a function of the applied B field (in Tesla). The local spins are coupled by a ferromagnetic Heisenberg coupling and interact with the environment via an anisotropy $D$ term (easy axis kind). The lowest-lying state energies are shown as full black lines; the energies of the initial and final states of the magnetic switching process ($M$ $=$ $+$10 and $M$ $=$ −10 states) are highlighted in full green and dashed red, respectively.

Figure 10

Magnetic switching induced by tunneling electrons in a ferromagnetic chain of 5 local spins S $=$ 2, with a longitudinal anisotropy. The full symbols present the excitation probability from the initial $M_{\mathit{Tot}}$ $=$ $+$10 state to an excited state $j$ as a function of the excited state energy for various strengths of the applied B field (see insert). The open symbols present the probability for the magnetic switch of the chain via the excited state $j$ (transition from the $M_{\mathit{Tot}}$ $=$ $+$10 state through the intermediate excited $j$ state to the final $M_{\mathit{Tot}}$ $=$ −10 state). Note that for the highest field (40 T), open and full symbols are superimposed. For the sake of clarity, the energies of the intermediate states $j$ are plotted relative to the energy of the ground state for a vanishing field.

Figure 11

Energy diagram of an $N$ $=$ 5 chain of $S$ $=$ 2 spins as a function of the applied B field (in Tesla). The local spins are coupled by a ferromagnetic Heisenberg coupling and interact with the environment via anisotropy $D$ and $E$ terms (easy axis kind). The lowest-lying state energies are shown as full black lines. The full green and dashed red curves that overrun the avoided crossings highlight the energy of the initial and final states of the magnetic switching process ($M$ ≈ $+$10 and $M$ ≈ −10 states).

Figure 12

Magnetization switching time for the direct transitions induced by substrate electrons: The transition between the two extreme magnetization states occurs in a single electron collision. The magnetization switching time (inverse of the switching rate) is shown in seconds for three different chain lengths ($N$ $=$ 4, 5, and 6 atoms; see insert) as a function of the transverse magnetic anisotropy of the chain, $E$. The applied B field is 1 T.

Figure 13

Lifetime of the upper state ($M_{\mathit{Tot}}$ ≈ $+$10 state) of the magnetization switch induced by substrate electrons in a single collision event for a length chain $N$ $=$ 5. The transverse magnetic anisotropy, $E$, is equal to 0.5 meV. The state lifetime (inverse of the state decay rate) is shown in seconds as a function of the applied B field in tesla.

Figure 14

Probability for the electron-induced indirect magnetization switch process in a chain of $N$ $=$ 5 atoms as a function of the intermediate state excitation energy; the anisotropy terms are $D$ $=$ 1.5 meV and $E$ $=$ 0.3 meV, and the B field is equal to 1 T. Black full circles joined by a dashed line: excitation probability $P_{\mathit{Exc}}(\mathit{init}\to \mathit{j})$ from the initial state (init : $M_{\mathit{Tot}}$ ≈ $+$10) to the excited state ($j$). Red full diamonds: probability, $P_{\mathit{Ind}}(\mathit{init}\to \mathit{j}\to \mathit{final})$, for the indirect process (init → j → final), where ‘final’ is the ground state ($M_{\mathit{Tot}}$ ≈ −10). The intermediate state energies are referred to that of the ground state of the system for a vanishing field, and the largest $P_{\mathit{Exc}}(\mathit{init}\to \mathit{j})$ probability around 2 meV corresponds to the elastic probability in the initial channel.

Figure 15

Probability for the electron-induced indirect magnetization switch process in a chain of $N$ $=$ 5 atoms; the longitudinal anisotropy term is $D$ $=$ 1.5 meV, and the B field is equal to 1 T. The anisotropy term, $E$, is variable (see insert). The probability, $P_{\mathit{Ind}}(\mathit{init}\to \mathit{j}\to \mathit{final})$, of the indirect process (init → j → final) is shown as a function of the energy of the intermediate state, $j$, referred to that of the ground state for a vanishing field. The magnetization switch occurs between the ‘init’ ($M_{\mathit{Tot}}$ ≈ $+$10) state and the ‘final’ state ($M_{\mathit{Tot}}$ ≈ −10; ground state).

Figure 16

Total probability for the electron-induced indirect magnetization switch process in a chain of $N$ $=$ 5 atoms; the longitudinal anisotropy term is $D$ $=$ 1.5 meV, and the B field is equal to 1 T. The magnetization switch occurs between the ‘init’ ($M_{\mathit{Tot}}$ ≈ $+$10) state and the ‘final’ state ($M_{\mathit{Tot}}$ ≈ −10; ground state). The total probability, ${\sum }_j{P}_{\mathit{Ind}}(\mathit{init}\to \mathit{j}\to \mathit{final})$, of the indirect process for a tunneling electron with energy larger than all excitation thresholds is shown (full black line with open circles) as a function of the transverse magnetic anisotropy term, $E$. The dashed red line shows the total excitation probability ${\sum }_j{P}_{\mathit{Exc}}(\mathit{init}\to \mathit{j})$ for a tunneling electron with energy larger than all excitation thresholds.