Atomic Many-Body Selectivity in Cubic CsPbI₃ Solar Cell Memristor

Abstract

Using DFT+DMFT, we show the importance of spin-orbit coupling together with multi-orbital interactions in prescribing the reconstructed electronic state of the cubic CsPbI₃ crystal. Considering realistic Coulomb parameter values and Pb-spin-orbit interaction, we uncover relevant key features in the one-particle spectral functions for the Pb-6p and I-5p orbitals of semiconducting CsPbI₃ bulk crystal and the role played by p and n doping relevant for band-selective metallicity and current-voltage characteristics. The implication of our study for cubic CsPbI₃ is expected to be an important step to understanding the electronic structure of pure and doped broadband solar cell-based memristor materials for neuromorphic computing.

1. Introduction

Organic and inorganic perovskites as well as related hybrid compounds [1,2] have attracted considerable attention in recent years in view of their multifunctional properties, including, for example, magnetism [3], ferroelectricity [4], multiferroism [5,6], light harvesting [7], and photovoltaic [8,9] and thermoelectric responses [10], as well as resistive switching for perovskite-based memory devices [11,12]. Additionally, the fact that these and related properties in some of these compounds can be tuned by external perturbations like temperature, pressure, light, electric, and magnetic fields [13,14,15,16] has triggered a plethora of studies concerning their possible applications in switchable dielectric devices, sensors, cooling devices, and solar cells [1].

On general grounds, the halide perovskite $AB{X}_3$, where $A\left(B\right)$ is the organic or inorganic (metal) cation and X is a halogen anion (Cl, Br, or I) [17,18], has been considered in recent years for future resistive switching memory devices [19,20] and optoelectronic devices, including solar cells [21], photodetectors [22], and light-emitting diodes [23], due to their unusual $I-V$ hysteresis. The presence of organic components in these $AB{X}_3$ compounds is known to introduce functionalities usually not achieved in inorganic perovskites [2], providing new possibilities for finding multi-functional materials [17]. However, organometal halide perovskite (OHP) [11] materials are known to have unavoidable photo-thermal stability stints [24]. On the other hand, the stability of the OHPs can be improved by replacing the organic cations with Cs or Rb organic cations [11,25,26]. With this caveat, inorganic halide perovskites have been explored for various applications [27,28,29] and have been shown to perform similarly to the OHP materials.

Under ambient conditions, CsPbI₃ is known to have the most unstable crystal structure phase among the cesium lead halide perovskites CsPbX₃ ($X=$ Br and Cl) [11]. However, this instability might provide an opportunity to understand the role played by the distorted lattice structure [30,31,32,33] on resistive switching and solar cell behavior, which might be different from other lead trihalide perovskites. The $AB{X}_3$ [34] perovskite is made of divalent metals ($B=$ Pb, Sn) and halide X anions which are arranged in a octahedral framework hosting (A) cations in the center; see Figure 1. In these systems, hysteresis loops are observed in $I-V$ characteristics, which might be associated with ion migration processes [35,36] or other more fundamental mechanisms [8]. As pointed out by M. Loizos et al. [37], mixed ionic-electronic conductivity can be harmful to solar cell applications but it might be pertinent in resistance or resistive switching [38], adding the lead iodide perovskites as potential candidates for future memristive devices [8,29]. Finally, it should be also noted that cubic CsPbI₃ is expected to be a promising candidate for photovoltaic perovskite solar cells [39,40,41] due to its band gap of 1.73 eV [42], absorbing most of the visible-light spectrum up to 700 nm [43], ultrafast dynamics of charge carrier [44], long carrier diffusion length [39], and lifetimes exceeding 10 μs [45].

Nevertheless, before we turn to basic principles hidden in the changes of electronic structures induced by many-particle interactions and spin-orbit (SO) effects [9,46,47,48,49,50] in cubic CsPbI₃, it is worth noting that based on extant comparisons between computational studies and experimental results, it has become clear that density functional theory (DFT) [51,52] within the Generalized Gradient Approximation (GGA) is unable to accurately reproduce the experimental bandgap of $AB{X}_3$ halide perovskites. As pointed out by J. Brgoch et al. [53], based on GGA, the lead halide materials have a band gap that underestimates the experimental value by at least 0.5 eV. In view of this, screened hybrid functional [54] and the $GW$ method [9,55,56,57] were employed to improve the agreement with the experimental data. Regardless of the DFT functional, some studies also omit SO coupling, which is crucial in 6s, 6p systems [53] and it might be near to 1.0 eV in organometal halide perovskites [9].

The effects of SO coupling in halide perovskite have been experimentally observed—for example, in Refs. [58,59]—and explored in a series of theory studies over the last few years [9,53,57,60]. Acting on the 6p orbitals of lead, the SO coupling increases the band dispersion [53]. This in turn significantly reduces the one-particle bandgap by inducing a large splitting of the conduction band states [60], leading to additional difficulties in describing the electronic structure of $AB{X}_3$ halide perovskite analogs based on DFT calculations. However, accounting for the effect of the SO coupling along with correlated many-body interactions through DFT combined with dynamical mean-field theory (DFT+DMFT) approximation [61] has been shown to work reliably well for elemental bismuth [46] and Bi chalcogenide topological insulators [47,48], but to the best of our knowledge, an investigation based on multi-orbital (MO) DFT+DMFT calculations has been largely overlooked for the $AB{X}_3$ halide perovskite compounds, and this is our focus in this work.

Notwithstanding, before going into the multiband, MO problem of cubic CsPbI₃, we shall also mention here that understanding the role played by electron–electron interactions in Mott insulators [62,63] and correlated semiconductors [64,65,66,67] remains a problem of fundamental and applied importance in condensed matter and materials physics phenomena. In recent years, notable progress has been achieved in studies of metal-insulator-metal ($m-i-m$) structures displaying different types of voltage-dependent resistance switching voltages in two-terminal in random access memories (RAMs) [68]. The insulating compound of the $m-i-m$ structure is either a transition-metal oxide [69,70,71,72] or an organic material [73,74]. The switching behavior is associated with the hysteresis observed in the current-voltage ($I-V$) characteristics. These $m-i-m$ systems are now referred to as resistance random access memories or resistance-switching random access memories (RRAMs) [20] and the most fundamental issue within these systems is the hysteretic response observed in the $I-V$ curves and their link to volatility [75].

Exploring and using functional electronic materials might greatly improve energy efficiency and the scalability of electronic future devices [76,77]. By using materials in which a metal-insulator transition (MIT) can be triggered electrically makes it possible to mimic the behavior of biological neurons in real circuits made of few memristor components [78,79]. Therefore, an understanding of memristors’ physical properties and their responses to external perturbations is important for designing the applications by using advanced electronic systems [29]. Although different switching mechanisms for the $m-i-m$ structures have been reported in recent years [20,80], a generic view for the electric switching from a Mott insulator to a metal (good or not) in materials such as VO₂ [81,82] and NbO₂ [83,84] is by now established [71]. An applied external electric field can induce phase transition, due to Joule heating and/or field-induced carrier doping [85,86], which might result in the formation of metallic filaments inside the insulating system serving as channels for the electric current to flow. The filament formation causes strong nonlinear V-dependence in the $I-V$ characteristics, including negative differential resistance [87]. A different type of resistive switching, where an electrical stimulus drives the material from a semiconducting (correlated or not) into a metallic phase, is a rare phenomenon and understanding microscopically the emergent metallic state which might show reversible $p-i-n$ structures [8] is still lacking. In this work, we demonstrate that resistive switching in air stable CsPbI₃ inorganic perovskite [11,12] from a semiconductor into a metallic phase might be linked to anisotropic, atom-selective p and n doping the parent compound.

2. Theory and Results

The DFT calculations for the cubic CsPbI₃ bulk crystal were performed using the SIESTA simulation package [88]. GGA within the PBE framework [89] was applied as the exchange-correlation functional. Norm-conserving scalar relativistic [90] pseudopotentials in the nonlocal form [91] were used to represent the ionic core potentials. For the Pb atoms, the 5d semicore states were included in the DFT calculations. Moreover, a linear combination of atomic orbitals was also used to expand the Kohn–Sham orbitals [52] in a finite range determined by an energy shift of 0.01 Ry [92]. The real-space grid integration was made using an energy cutoff of 200 Ry [93]. A $10 \times 10 \times 10$ Monkhorst–Pack grid [94] was used for geometry optimization calculations and, finally, a $40 \times 40 \times 40$ grid was used to compute the density of states (DOS) of the CsPbI₃ bulk crystal. In this work, we have considered a cubic primitive cell of CsPbI₃, whose space group is $Pm\overline{3}m\left[221\right]$ and the Pearson symbol is $cP5$. Moreover, the lattice parameters were taken from Ref. [30], while the atomic positions were optimized until all the forces were found to be smaller than 0.03 eV/through the conjugated-gradient algorithm.

In Figure 2, we show our results for the band structure and atom-resolved DOS along the Brillouin zone high symmetric $R,\Gamma ,X,M$ path. The one-particle band structure of cubic CsPbI₃, from −6.0 eV to 6.0 eV with the Fermi level $\left(E_F\right)$ being the origin of energy, is shown in the left panel of Figure 2. This figure shows that cubic CsPbI₃ crystal is a direct R-point bandgap semiconductor. This intrinsic bandgap behavior is known to be relevant for solar cell materials, because the system transmits the absorbed light more efficiently than other indirect band gap materials, enabling higher optical absorption or optical conductivity [10]. Based on the SIESTA code, the calculated bandgap for cubic CsPbI₃ is approximately 1.45 eV. Although this value is underestimated compared to the estimated experimental value of 1.73 eV, our bandgap is found to be consistent with the 1.35 eV value found by Z. Wang et al. [33] for cubic CsPbI₃. Moreover, our results for the partial density of states (DOS) are also consistent with that reported by Ahmadi et al. [10], showing that the bottom of the conduction band minimum is composed mostly of the $6p$ orbitals of lead and the valence band maximum is originated by $5p$ orbitals of iodine atom, with clear Pb-I hybridization which gives rise to bonding-antibonding bands. Also interesting is the intensity of partial DOS of the I atom, showing a highest valence band edge peak as compared to the one seen in the conduction p band DOS of Pb.

From DFT, the one-electron Hamiltonian ($H_{\alpha }^0$) for the relevant atomic channels $\alpha$, where $\alpha =(1,2)$ corresponds, respectively, to (I, Pb) of cubic CsPbI₃ is

$$\begin{array}{c}\hfill H_{\alpha }^0=\sum _{\mathbf{k},a,\sigma }{ϵ}_{\alpha ,a}\left(\mathbf{k}\right)c_{\alpha ,\mathbf{k},a,\sigma }^{†}{c}_{\alpha ,\mathbf{k},a,\sigma }-\mathsf{\mu }\sum _{i,\alpha ,a,\sigma }{n}_{i,\alpha ,a,\sigma },\end{array}$$

where $a=x,y,x$ are the diagonalized (in orbital basis) p-band DOS as in Figure 2. In the usual notation, within the two-band and three-orbital problem of CsPbI₃, ${ϵ}_{\alpha ,a}\left(\mathbf{k}\right)$ is the one-electron band dispersion and $\mathsf{\mu }$ is the chemical potential. Local two-particle interactions in CsPbI₃ bulk crystal are included in $H_{\alpha }^{int}$ [47], which contains, respectively, the intra-orbital (U) and inter-orbital ($U^{\prime }$) Coulomb repulsion and the Hund’s rule coupling $\left(J_H\right)$. To treat dynamical effects (induced by $U,U^{\prime },J_H$) in the many-body Hamiltonian $H^{MO}={\sum }_{\alpha }{H}_{\alpha }^0+H_{\alpha }^{int}$ of cubic CsPbI₃, we use the DMFT approximation [95]. Finally, to investigate the role of SO interaction [26,53,60] within our formulation, let us consider, for the sake of simplicity, only the local component of this quantum interaction [96,97], which can be treated exactly within DMFT [98,99,100]. Similar to Refs. [46,47,48], on atomic Pb, the local SO Hamiltonian reads $H_2^{SO}=\lambda {\sum }_{i,a}(c_{2,i,a\uparrow }^{†}{c}_{2,i,a\downarrow }+H.C.)$, with $\lambda$ being the SO coupling, which is taken as model parameter in our description. Physically, $H_2^{SO}$ acts as a transverse, spin-flip term [98,99,100], and locally mixes the $p_a$ spin states of lead. As pointed out in Refs. [46,47,48], in order to have strong SO interactions, one needs materials with heavy atoms, as the SO coupling scales as $Z^4$ [101], which is predicted to be close to one in Pb [9]. However, the role played by the SO interaction in iodine has received little attention in the extant literature of halide perovskite systems. Thus, since the SO coupling in I is expected to be small compared to the one in Pb, as well as to the bare bandwidth $(W_{DFT}\approx 10$ eV, see Figure 2) of cubic CsPbI₃, this contribution is not considered in our theory. Importantly, our choice for $\lambda$ is consistent with a study by Xia et al. [102], where similar value for the SO coupling was used in atomic Bi of Bi₂Se₃ topological insulator.

We compute the correlated Green’s functions

$$G_{\alpha ,a,\sigma }(\omega ,\mathbf{k})={\left[{\xi }_{\alpha ,a,\sigma }\left(\omega \right)-{ϵ}_{\alpha ,a}\left(\mathbf{k}\right)-{\delta }_{\alpha ,2}{\displaystyle \frac{{\lambda }^2}{{\xi }_{\alpha ,a,\sigma }\left(\omega \right)}}\right]}^{-1},$$

where ${\xi }_{\alpha ,a,\sigma }\left(\omega \right)\equiv \omega +\mathsf{\mu }+i\eta -{\Sigma }_{\alpha ,a,\sigma }(\omega +i\eta )$, of the model Hamiltonian $H=H^{MO}+H_{Pb}^{SO}$ for bulk CsPbI₃ within DFT+DMFT [61]. To evaluate self-consistently the DMFT equations, we employ the MO iterated-perturbation theory (MO-IPT) as impurity solver [103]. The MO-IPT is numerically very efficient and self-energies $\left[{\Sigma }_{\alpha ,a,\sigma }\left(\omega \right)\right]$ of the two atomic channels can be numerically assessed at small and large frequencies. Given the complexity in cubic CsPbI₃ bulk crystal with three ($a=x,y,z$) p bands on the Pb and I atomic channels, these are relevant features to describe electron interactions and SO coupling in correlated semiconductors [64,65,66].

In Figure 3, we show the role played by electron interactions in the many-body local spectral function (DOS) for fixed $J_H=0.5$ eV and $\lambda =0.0$. A clear electronic reconstruction from the bare semiconductor to a correlated semiconducting state occurs upon increasing U. Looking at Figure 3, we see that ${\Sigma }_{\alpha ,a,\sigma }\left(\omega \right)$ vanishes in the gap region. This behavior is a fingerprint of a correlated Kondo insulator, where the band-gap size is linked to the combined effects of electronic correlations and one-particle, interband hybridization. Thus, the semiconducting state of CsPbI₃ is, within our DFT+DMFT approach, that of a Kondo-like insulator [104]. In our theory, the bare semiconducting state is renormalized due to Kondoness, which in contrast to the usual bare band semiconductors, correlation effects together with relevant band structure details play a fundamental microscopic role. As is apparent from the atom-resolved self-energies, correlation effects manifest themselves immediately above the bandgap as sharp peak-like structures, revealing the underlying Kondoness in the system. Thus, since the CsPbI₃ is Kondo-correlated, spectral weight redistribution in response to other perturbations induced, for example, by the SO interaction, can drive the system into a channel-selective metallic phase, as shown below.

Figure 4 shows the changes on the two-atom DOS upon consideration of the SO interaction. Consistent with the ab initio results by E. Even et al. [60], the SO coupling splits the conduction band of the Pb-channel into two branches, resulting in the emergence of a bound state at the conduction band edge similar to that reported for Bi-chalcogenide topological insulators [47,48]. As seen in Figure 4, this bound state broadens and it is dynamically shifted towards $E_F$ with increasing $\lambda$, giving rise to metallicity within the weakly correlated regime of cubic CsPbI₃. Moreover, consistent with SO inducing bulk mass enhancement due to the formation of bound states and spectral weight transfer [47,48], at finite $\lambda$, the DMFT self-energies develop a pole below $E_F$ (see the insets of the Figure 4 lower panel). This suggests that an exotic metallic state can be tuned by appropriate compound engineering or structural variations enforced by mutual external perturbations. Bound states in the Pb DOS suggest hidden quantum criticality [105] and, thus, the emergence of novel states of quantum matter which could be seen in future studies.

Figure 5 displays the role played by the SO interaction within the intermediate correlated electronic state of cubic CsPbI₃. As seen, the metallic state, which has emerged in the weakly correlated regime in Figure 4, is no longer present at $U=8.0$ eV and $\lambda =0.8$ eV, where the spectral function and self-energies of Pb show characteristics akin to correlated semiconductors [64,65,66]. Although this emergent insulating state is due to the influence of one-electron hybridization and is adiabatically connected to a band insulator, it preserves specific features associated with electron correlations just above the bandgap. Thus, the insulating features seen in the lower panel of Figure 5 are partially related to the $\omega$-dependence of $Re{\Sigma }_{2,a}$ (see inset of Figure 5) in a channel where $Im{\Sigma }_{2,a}$ vanishes in the gap region. Although some correlated semiconductors might show mixed Mott and Kondo insulating states, the insulating state in the Pb-channel of cubic CsPbI₃ is an interesting demonstration of a Kondo insulator, since the hybridization gap is intrinsically associated with vanishing $Im{\Sigma }_{2,a}\left(\omega \right)$, rather than its divergence as found in Mott insulators.

To give additional insights into the electronic structure reconstruction of cubic CsPbI₃, in Figure 6 and Figure 7, we reveal the effect of, respectively, increasing and decreasing its total band filling n. This is motivated by the fact that within DFT+DMFT electron/hole ($p/n$) doping is not described by the rigid band shifts of the bare DFT spectrum, but they are self-consistently computed using the DFT+DMFT formalism. Therefore, understanding the effect of electron band filling is crucial. Despite existing transport and current-voltage $(I-V)$ data, the generic appearance of novel electronic states and the instabilities of such states to metallicity (Fermi liquid or not) in a variety of other correlated semiconducting and semi-metallic (Dirac fermion) systems highlights the importance of this question. Our aim here is to model and analyze the effect of changing the total band filling in cubic CsPbI₃. In particular, we shall introduce a set of predictions which might be corroborated in future tunneling and spectroscopy measurements by controlling the energy position of the Fermi level [106], or the chemical potential μ.

In Figure 6 and Figure 7, we show the modifications in the correlated DOS and self-energies (computed using $U=$10.0 eV and $J_H=0.5$ eV) upon slightly increasing/decreasing the total band filling ($n=n_{DFT}\pm \delta$) of $p/n$-doped CsPbI₃. An intriguing observation in our results is that atomic selectivity occurs starting at small $\delta$ (i.e., $\delta \le \pm 0.125$), suggesting the relevance of electron addition/removal in broadband systems. Moreover, as $\delta$ increases to about $0.25$, a more visible atom-selective behavior develops in the electronic structure of electron-doped CsPbI₃, see Figure 6. According to our results, channel selectivity at $\delta =+0.25$ is characterized by the presence of a bound state with Pb-channel character and correlated semiconducting iodine spectral functions. A reverse effect is obtained via hole doping the CsPbI₃ parent compound, where the correlated Pb-channel remains semiconducting while the I-channel goes towards a metallic state. The origin of these atom-selective features can be attributed to scattering between different carriers in channel states due to specific crystal field and interaction effects on the bare DFT lineshapes. The latter has two implications: channel-dependent shifts of the $6p$ and $5p$ states relative to each other due to Hartree contributions from the static part of the self-energies, and dynamical effects emerging from sizable $U,U^{\prime }$, and $\lambda$, which induces spectral weight transfer over considerable large energies upon electron/hole doping. This latter response leads to atom-selective changes in the DOS and self-energies, as shown in Figure 6 and Figure 7. Thus, for electron/hole-doped CsPbI₃, coexisting semiconducting, normal metal, and narrow bound states are expected to be seen near $E_F$. Intrinsic atom selectivity, together with the large transfer of spectral weight are relevant fingerprints arising from the correlated electronic state of cubic CsPbI₃ broadband semiconductor and future observation is called to place it based on different experimental grounds.

The central message of our approach in Figure 6 and Figure 7 is thus that a correlated electron scenario of cubic CsPbI₃ leads to a microscopic description for the atom-selective electronic reconstruction from a Kondo insulator to a coherent Kondo metal [107] or to a weakly correlated FL metal with electron/hole doping, respectively. Being derived based on a DFT+DMFT framework for the multichannel and MO electronic structure of bulk CsPbI₃, our proposal is a primary many-body contribution to extant theory works on correlated halide perovskite system [26,42,60]. In fact, the intermediate coupling regime underpins the physical behavior of the system to be seen in future spectroscopy studies.

Finally, in order to clarify the interplay between channel-selective metallicity and electron/hole-doping effects into the evolution in the $I-V$ characteristic of CsPbI₃ memristor in Figure 8, we display our DFT+DMFT results obtained for three different $\delta (=0.0,$$\pm 0.25$ values considered in Figure 6 and Figure 7. Firstly, within the wideband limit of the electrodes, the current formula reads $I={\displaystyle \frac{2e}{\hslash }}{\sum }_{\sigma }\int d\omega \tilde{\Gamma }\left(\omega \right)\left\{f_L\left(\omega \right)-f_R\left(\omega \right)\right\}{\rho }_{\alpha ,a,\sigma }\left(\omega \right)$ [108,109], where $\tilde{\Gamma }\left(\omega \right)={\Gamma }_L\left(\omega \right){\Gamma }_R\left(\omega \right)/\Gamma \left(\omega \right)$, with ${\Gamma }_{\gamma }\left(\omega \right)=\pi {\sum }_k{\left|t_k^{\gamma }\right|}^2\delta (\omega -{\epsilon }_{k\gamma })$ being the coupling strength between the right (R) and left (L) electrode ($\gamma$ = L,R) and the central region [110,111], and $\Gamma \left(\omega \right)={\Gamma }_L\left(\omega \right)+{\Gamma }_R\left(\omega \right)$. $f_{\gamma }\left(\omega \right)=1/(e^{\beta (\omega -{\mathsf{\mu }}_{\gamma })}+1)$ and ${\rho }_{\alpha ,a,\sigma }\left(\omega \right)=-{\displaystyle \frac{1}{\pi }}Im{G}_{\alpha ,a,\sigma }\left(\omega \right)$ are, respectively, the Fermi function of the electrode $\gamma$ and the total DOS of the $5p$ and $6p$ channels with spin-$\sigma$ of CsPbI₃ memristor. Secondly, here we suppose a symmetric voltage drop, ${\mathsf{\mu }}_L=-{\mathsf{\mu }}_R=eV$, with constant DOS for the leads. As shown in Ref. [112], these assumptions allow for a microscopic description of the $I-V$ characteristic of cubic CsPbI₃.

In Figure 8, we show the semi-logarithmic current-voltage $(I-V)$ characteristic curves obtained using the DFT+DMFT spectral functions for $\delta =0.0,\pm 0.25$, shown in Figure 6 and Figure 7. As seen, cubic CsPbI₃ exhibits memristive behavior, as reported in extant experimental studies [11,12], where the resistance steady-state value exhibits an ON/OFF ratio [8] as high as 10⁴ in our theory assisted by modulation $p/n$-type doping. This behavior suggests that doped CsPbI₃ memristor exhibits analog resistive memory with unipolar (or symmetric), low-resistance, and high-resistance states. However, unlike the switching behavior derived within our theoretical framework, which is irrespective of the voltage polarity and current signal, in the experiment, the $I-V$ characteristic of CsPbI₃ and analogs are usually bipolar (or antisymmetric) since the set (to an ON state) occurs at one voltage polarity and the reset (to an OFF) state takes place on a reversed polarity [20]. This is due to the fact that the memristive circuitry system has some intrinsic asymmetry arising from different electrode materials or the polarity during the initial electroforming step [20], resulting in a bipolar switching response. Based on our theory results, the unipolar switching is linked to channel-selective $p/n$ doping, resulting in an analog memristive functionality of cubic CsPbI₃ for future neuromorphic computing.

3. Conclusions

In summary, we have studied the effect of electron–electron and spin-orbit interactions as well as $p/n$-type doping in a two-channel, three-orbital Hubbard model for the Pb- and I-electronic states of cubic CsPbI₃ perovskite solar cell memristor. Our intermediate correlated DFT+DMFT results provide a microscopic description of the electronic reconstruction induced by dynamical spectral weight transfer, a characteristic akin to correlated electrons in solids. Moreover, based on DFT+DMFT calculations, we provide evidence of doping-induced channel-selective metallicity and resistive phase switching relevant to future neuromorphic devices based on solar cell materials. Our distinct current-voltage characteristics at finite $p/n$ doping confirm that cubic CsPbI₃ can be tuned into an electronic state, which gives unipolar memristive responses [20]. This proposal could be verified in tip-assisted investigations [106,113] implemented in electronic systems exploiting the Kondo insulating state [114,115] in correlated semiconductors.