**2. Materials and Methods**

Copper single crystals of 99.999% purity oriented in [001], were cyclically deformed in tension/compression at room temperature to 157 cycles at a strain amplitude of 4.0 <sup>×</sup> 10−<sup>3</sup> and a strain rate of 2 <sup>×</sup> 10−<sup>3</sup> s<sup>−</sup>1. This single crystal copper was cyclically deformed in multiple slip with 8 active slip systems of identical Schmid factor of 0.408. Figure 1a illustrates the stress versus strain behavior of the cyclically deformed specimen. Figure 1b illustrates that the copper single crystal was fatigued to saturation to an axial stress of 275 MPa (a resolved shear stress of 112 MPa in the <110> direction on a {111} plane). A deceleration in the cyclic hardening rate up to the maximum peak stress at the 108th cycle followed by a very slow softening until the 157th cycle was observed. The specimens were stored in a liquid nitrogen container to suppress recovery and recrystallization subsequent to deformation, which has been observed in other high purity metals [28,29]. TEM disks from the (100), (001), and (010) planes were prepared using conventional jet electropolishing with a Fischione twin jet (Fischione Instruments Inc., Export, PA, USA). Foil preparation details can be found in earlier publication by the authors [1]. Figure 1a shows that the macroscopic back stress is about one half of the maximum stress, independent from the number of cycles (a few cycles: no labyrinth structure versus about 80 cycles close to the quasi-saturation stage, considered as the labyrinth domain).

CBED studies were done on both the cyclically deformed copper and a 99.999% pure unstrained copper using the JEOL JEM-2100F transmission electron microscope (TEM) (JEOL Inc., Peabody, MA, USA) at the University of Southern California at an accelerating voltage of 200 kV and a beam diameter of about 40 nm. In order to obtain the lattice parameter from CBED patterns, comparisons between the experimental CBED patterns with simulated ones were made [27]. In this study the EMSOFT codes that consider a dynamical behavior of the electrons within the specimen were used for CBED patterns simulations [30].

**Figure 1.** (**a**) The cyclic deformation of [001]-oriented copper single crystal at 298 K. Strains are plastic and elastic. (**b**) The evolution of the maximum and minimum peak stress versus the number of cycles confirms saturation is reached.

Figure 2 illustrates the 3-dimensional microstructure based on TEM images taken from the (100), (010), and (001) planes of the cyclically deformed copper [1]. The stress axis is parallel to the vertical [001] direction. The heterogeneous labyrinth dislocation microstructure consisting of orthogonal high dislocation density walls and low dislocation density channels is observed. Dislocation density in the walls and channels of the labyrinth structure was 8.6 <sup>×</sup> <sup>10</sup><sup>14</sup> <sup>m</sup>/m3 and 1.55 <sup>×</sup> <sup>10</sup><sup>13</sup> <sup>m</sup>/m<sup>3</sup> respectively [1].

Figure 3 illustrates TEM micrographs of the labyrinth structure viewing from the [010] direction. All the dipole height measurements [1] and CBED analysis were performed on the labyrinth structure on the (010) planes.

**Figure 2.** A transmission electron microscope "cube" based on images taken from the (100), (010), and (001) planes of a copper specimen cyclically deformed to saturation as shown in Figure 1a. The stress axis is parallel to the [001] direction (Taken from Ref. [1]).

**Figure 3.** Transition electron microscope (TEM) micrographs of the labyrinth structure from the {010} planes of the cyclically deformed copper.
