Correction: The Concept of Negative Capacitance in Ionically Conductive Van der Waals Ferroelectrics (Adv. Energy Mater., (2020), 10, (2001726), 10.1002/aenm.202001726)

Adv. Energy Mater. 2020, 10, 2001726 DOI: 10.1002/aenm.202001726 The polarization P(z) shown in Figure 1c in the original paper and reproduced here as Figure 1A was calculated by moving the Cu-atom sheets in a synchronous (concerted) way. The modern theory of polarization (MTP)^[1] however, yields the two curves shown in Figure 1B, defined by P(z) = P_calc(z) − P_ref, where P_ref are the polarizations for the Cu sheets at the mid-layer and mid-gap planes. In constructing Figure 1A, the red curve was mistakenly plotted as P(z) = P_ref − P_calc(z), which reversed its slope. Here it shown that a form of asynchronous Cu migration (ACM) is more appropriate to describe the experimental observations and results in the P(z) curve shown in Figure 1C. 1 Figure (Figure presented.) A. P(z) based on synchronous Cu migration from left to right as shown in Figure 1c in the original paper. B. The two MTP-calculated P(z) curves based on synchronous Cu migration and properly referenced to the mid-layer and mid-gap centrosymmetry planes, respectively. Periodic repetitions on the P axis (polarization quanta) are not shown. C. P(z) calculated using the asynchronous Cu-migration approach. The lines are obtained through Equation 1 and the points are the result of full DFT MTP calculations. Note that the results from the synchronous (panel B) and asynchronous (panel C) approaches coincide in the layer regions, but differ markedly in the gap regions. Synchronous ion motion has been used for common displacive ferroelectrics in which ion displacements are typically small. However, CuInP₂S₆ (CIPS) features large ionic displacements and is an order-disorder ferroelectric in which the paraelectric phase is described by individual Cu atoms occupying ± polarization states randomly.^[2,3] Under bias, according to quantum simulations,^[4,5] Cu migration occurs asynchronously, as individual Cu atoms cross energy barriers one at a time. Therefore, an asynchronous Cu-migration approach is implemented by considering a sequence of transitions between adjacent polarization states, P₁ and P₂ (P_1/2 = −LP/+LP, +LP/+HP, +HP/−HP, −HP/−LP, etc.). For each transition, all Cu atoms are initially in the P₁ polarization state and the Cu atoms are moved over the respective barrier one at a time until all Cu atoms are in P₂. Thus, snapshots during Cu migration between adjacent polarization states P₁ and P₂ would reveal all Cu atoms initially at P₁ and a gradual increase of the fraction f of Cu atoms that have transitioned to P₂. In the simplest approximation, when a fraction f of the Cu atoms has transitioned, the average position of all the transitioning Cu atoms defines a z coordinate between the P₁ and P₂ states and the net polarization is 1 (Formula presented.) (the z value for each f in Equation 1 coincides with hypothetical Cu sheets that transition synchronously). The net result is the piecewise straight line shown in Figure 1C. P(z) was also calculated using density functional theory (DFT) and the MTP for snapshots corresponding to sequences of fractions f for each P_1/2 segment. The result is the single-valued, continuous P(z) curve formed by the black squares in Figure 1C. The results within the layers are the same as obtained using synchronous Cu migration (black curve in both Figures 1A and 1B), whereby the four polarization states in the quadruple well^[4] remain unchanged. Notably, however, in the gap region, P(z) differs markedly from the red curve in Figure 1B; it decreases and goes through zero at the mid-gap plane, in agreement with the experimental data reported in the original manuscript. The markedly different behavior of the calculated synchronous Cu migration (SCM) and ACM P(z)'s can be traced to differences in the evolution of the Cu ions relative to the InP₂S₆ cages in the pertinent sequences of periodic unit cells. Midgap-to-midgap unit cells are adopted and the black curves in Figure 1B are compared with the ACM curve in Figure 1C. In the SCM, the Cu sheets always move from left to right relative to the stationary InP₂S₆ cages. Thus, P(z) rises monotonically in the entire unit cell and is then repeated periodically both on the z axis, as shown in Figure 1B, featuring a discontinuity from +25 to -25 μC/cm⁻², and on the P axis without discontinuities (the MTP's polarization quanta,^[1] not shown). In the ACM, as viewed in each unit cell, the average position of the Cu ions is identical to the actual positions of the SCM Cu sheets only while transiting within the layers from −HP to +HP. Thus, the SCM and ACM P(z)'s are identical only from −HP to +HP. From +HP on, however, the Cu ions transit across the vdW-gaps a fraction at a time, into the adjacent unit cells on the right. By periodicity, the same fractions of Cu ions enter each unit cell on the left. Thus, within each unit cell, the average position of the Cu ions appears to move from right to left, which leads to the calculated negative-slope P(z) in Figure 1C. Note that the negative-slope P(z) in effect replaces the discontinuity in the black curve in Figure 1B at the midgap planes with a gradual transition from +HP to −HP, in accord with the experimental data. Like the SCM curves, the ACM curve is also repeated periodically on the z and P axes. For a more detailed discussion of the asynchronous scheme and insights into the results, see Ref. [6]. The contents of this correction do not affect the analysis of the experimental data, and the conclusions in the original paper. The authors apologize for any inconvenience caused, they would also like to thank David Vanderbilt for inquiring about the slope of the red curve in Figure 1A and Raffaele Resta for valuable discussions.