Figure 1

A schematic representation of the two-branch evolution.Reuse & Permissions

Figure 2

A second order diagram corresponding to the nuclear spin pair flip-flop processes.Reuse & Permissions

Figure 3

A set of third order diagrams.Reuse & Permissions

Figure 4

Some fourth order diagrams.Reuse & Permissions

Figure 5

The second order contribution to decoherence in LCE which includes a series of even orders of a conventional perturbation expansion.Reuse & Permissions

Figure 6

Renormalization of a Green’s function by the $I_i^z{I}_j^z$ interaction.Reuse & Permissions

Figure 7

Compensation of the spin excitations overlapping in time by a locked diagram. The time overlap is in one site (j).Reuse & Permissions

Figure 8

Compensation of the spin excitations overlapping by a locked diagram. The time overlap is in two sites [(i) and (j)].Reuse & Permissions

Figure 9

The total fourth order contribution of the nuclear spin excitations to the electron spin decoherence. The first diagram corresponds to the ring propagation of the spin excitation in the three-spin clusters ↑↓↓ and ↓↑↑. The second, single-site locked, diagram corrects the overlapping of the excitations. It contributes to the spin dynamics of three- and two-spin clusters as shown in Figs. 7, 8. The third, two-site locked, diagram corrects the overlapping of two spin excitations, Fig. 8. The last diagram describes the four-spin ring excitations in the spin clusters ↓↑↑↑, ↑↓↓↓, ↑↓↑↓, and ↑↑↓↓.Reuse & Permissions

Figure 10

Contributions of the linked clusters to the electron spin decoherence: (a) Crossover from the short-time evolution to the intermediate regime. (b) Development of the fourth order terms in the intermediate time regime. $V_2$ corresponds to the spin pair flip-flop process in Fig. 2; $V_{43}$, the three-spin excitation, the first term in Fig. 9; $V_{4l1}$, the single site spin locked diagram, the three-spin part of the second term in Fig. 9; $V_{4l2}$, correction of the overlapping of the two-spin excitations given within the brackets in Fig. 8a, and $V_4$, the four-spin ring diagram, the last term in Fig. 9.Reuse & Permissions

Figure 11

Time dependence of the second and fourth (inset) order terms in the exponent. FID vs spin echo decay.Reuse & Permissions

Figure 12

Calculated spin echo decay in comparison with the experimental data.[9] Orientation of the external magnetic field is along the (100) direction.Reuse & Permissions

Figure 13

Representation of a third order diagram using the cluster configurations.Reuse & Permissions