Coupling between Ion Drift and Kinetics of Electronic Current Transients in MAPbBr3 Single Crystals

The optoelectronic properties of halide perovskite materials have fostered their utilization in many applications. Unravelling their working mechanisms remains challenging because of their mixed ionic–electronic conductive nature. By registering, with high reproducibility, the long-time current transients of a set of single-crystal methylammonium lead tribromide samples, the ion migration process was proved. Sample biasing experiments (ionic drift), with characteristic times exhibiting voltage dependence as ∝ V–3/2, is interpreted with an ionic migration model obeying a ballistic-like voltage-dependent mobility (BVM) regime of space-charge-limited current. Ionic kinetics effectively modify the long-time electronic current, while the steady-state electronic currents’ behavior is nearly ohmic. Using the ionic dynamic doping model (IDD) for the recovering current at zero bias (ion diffusion), the ionic mobility is estimated to be ∼10–6 cm2 V–1 s–1. Our findings suggest that ionic currents are negligible in comparison to the electronic currents; however, they influence them via changes in the charge density profile.


III. UV-Visible spectroscopy & Tauc plot
For transmittance data, a Perkin Elmer Lambda 900 UV-Vis-NIR spectrometer was used with a wavelength range of 400-600 nm, a 0.1 nm step and equipped with a Tungsten lamp. Tauc plot were charted using equation 1, 2, 3 and 4, where is the absorption coefficient, is the Planck constant, is the photon frequency, is the band gap energy, ℎ is a constant, is a factor depending on the nature of the electron transition and equal to 1/2 or 2 for the direct and indirect transition band gaps (here =1/2), [11] is the speed of light, is the absorbance, is the transmittance, is the sample thickness and is the elementary charge. In the Tauc plot, was extracted at the intersection between the x axis and the linear fitting of the linear part in the onset region.   Figure S3. Comparation between the current-voltage characteristics (j-V) with scan rate of 90 mV/s and step: 1 V and the ionic drift currents J i immediately obtained after bias removal of 2 mm-thick MAPbBr 3 SC symmetrically contacted with Cr electrodes It is remarkable the ohmic character of the characteristics j-V curve, in agreement with previous analysis on Cr-contacted perovskite device. [9,12,13] Figure S5. Equivalent circuit used to fits the impedance spectra: with series resistance , geometric capacitance and , and the high-, and low-frequency resistances, respectively. The low-frequency capacitance usually labelled as surface capacitance .  Figure S6. Variation of the impedance after removing 10 V bias and evolution with time b)Variation of the resistance extracted from fitting impedance spectra after bias removal at t = 0. Note the log scale in the vertical axis that informs on the complex function (non-exponential) of the resistance variation c) Variation of resistance ratio as a function of time following the relationship in equation (5) Solid line represents a linear fitting.

S2. Ionic Dynamic Doping (IDD) model
The IDD model, [6,9] relies on the way the redistribution of ionic species along the layer bulk modulate the electronic carrier density. Mobile ions acts as dopants and locally alter the doping density by a dynamic doping process. Therefore, the resistance variation can be explained in terms of the increase in electronic carrier density. A detailed view of the high-frequency resistance decrese under 0-V bias can be observed in Figure S6a. By extracting the fitting parameters in Figure S6b the resistance shows a large initial value 1 at shorter relaxing times and a steady-state value (background state) after a long 0 equilibration time. A simple expression based in the IDD model [6] is thereby derived where and are the limiting resistances which can be normalized by the resistance where and stands for the sample thickness and the width of the high doping region respectively. The normalized resistance is formulated as a linear function of giving rise to the expression (S5), that again predicts a linear dependence of the sample resistance with the width of the high doping zone .
Let us also assume that ions initially accumulate following a narrow distribution near the contact with standing for the distribution width. The accumulation occurs by effect of the applied electrical field. This ionic accumulation is expected by formation of a diffusion ion layer in the vicinity of the contacts and by very limited interfacial chemical interaction within the experimental time framework. Mobile ions depleted from the layer bulk induce a sort of electronic de-doping, reducing as a consequence the doping level and producing higher resistance. When bias is removed, ions are released and tend to diffuse back to their equilibrium position in such a way that the mean square displacement of the ion distribution enlarges with time. The simple outlined model would entail diffusion takes place in 1D ( ), but the real situation might be more complex with = 1 diffusion dimensionality approaching larger values because of the polycrystalline structure. For the sake of simplicity, let's assume that the distribution width spreads by diffusion as with accounts for the ion diffusion coefficient, for 1D diffusion, and is the time. = 1 Equation S6 informs on how the high doping region width extends within the perovskite layer bulk as the ion distribution spreads to attain initial homogeneity.
An additional component of the model concerns the relation between the width of the high doping zone and the spead of the ion distribution . For a Gaussian distribution, one can observe that nearly 99% of the ions concentrates within as to assume that 3 . We recognize that this is in fact an oversimplification of a complex problem that = 3 should include concrete ion distributions and how they influence the local doping density. In any case, one can derive a simple linear expresion for the variation of the resistance that suffices for our purposes.
By combining Equation (S5) and (S6), an expression for and = 1 the time dependence of the normalized resistance step results as -0 which predicts a linear dependence as with slope determined by the diffusion ∝ coefficient. One can also obtain a value for the ion diffusion coefficient by assuming the Einstein's relation, equation where is the elementary charge, stands for the Boltzmann constant, and is the absolute temperature. It's important to mentioned that due to the simplicity related to the model, a deviation occurs at shorter times, presumably unable to capture second order depolarization effects for greater moving ion concentrations.

S3. Ballistic-like voltage-dependent mobility (BVM) model of space-charge-limited current (SCLC)
In the classic mobility regime of SCLC, the drift velocity is a function of the electric field and the bias-and space-independent constant mobility as = 0 (S9) Equation (S9) results in the Mott-Gurney law [14] where the current density is quadratic with the external applied voltage ( ), the electrostatic potential behaves ∝ 2 with the position as and the time of flight can be approximated to .
In the BVM model, [15] the mobility is taken as function of the field, and thus the bias, as (S10) where is the distance between electrodes, is the onset voltage for the BVM regime, 0 is the threshold mobility for the transition between ohmic and SCLC regime and the absolute field value is considered as at each position in between the | | = | / | elecctrodes and at the active electrode. A deduction with an expression for | | ≈ / 0 is presented in the original publication [15] under two main assumptions: (i) the larger the larger , where is Frenkel's equation [16] for the distance between the ions and their local potential maxima upon application of an external field; and (ii) the smaller the larger , where is a Debye length for the accumulation of mobile ions towards the electrodes.
By subsituting Equation (S10) in (S9), the BVM drift velocity results as = 0 (S11) Equation (S10) can also be approximated as a particular case of Poole-Frenkel [16][17][18] ionized-trap-mediated transport when field dependent charge carrier density and ∝ for narrow ranges of . [15] Furthermore, Equation (S10) can be substituted in the expression for the associated total current density = (S12) where charge. Taking from equations (S12) and (S10), the Poisson equation can be solved for trend such as that of Equation (S10), which results in a ∝ ( / ) 1/2 potential (S13) The possible effect of a potential as Equation (S13) is presented in Figure S11. Moreover, after evaluating Equation (S13) at where , and since , the = = 500/243 ≈ 2 current density results (S15) one can substitute (S14) and (S12) in (S15) to obtain the BVM approximation of the time of flight as (S16) where is now an effective homogeneous charge carrier density. Typical values of the BVM model are presented in Figure 3a of the main manuscript. Figure S11. Proposed energy diagram of the evolution of transport for electron (top) and hole (bottom) charge carriers. From an initially ohmic behaviour, the mobile ions modify the space charge to create BVM SCLC within characteristic times ks and s ubsequently pile up towards the electrodes resulting in a bulkily ohmic transport eased by interface ionic dynamic doping.