Ballistic-like Space-charge-limited Currents in Halide Perovskites at Room Temperature

The emergence of halide perovskites in photovoltaics has diversified the research on this material family and extended their application towards several fields in the optoelectronics, such as photo- and ionizing-radiation-detectors. One of the most basic characterization protocols consist on measuring the dark current-voltage (J-V) curve of symmetrically contacted samples for identifying the different regimes of space-charge-limited current (SCLC). Customarily, J=C*V^n curves indicate the Mott-Gurney law when n=2, or the Child-Langmuir ballistic regime of SCLC when n=3/2. The latter can be often found in perovskite samples. In this work, we start by discussing the interpretation of currents proportional to V^(3/2) in relation to the masking effect of the dual electronic-ionic conductivity in halide perovskites. However, we do not discard the actual occurrence of SCLC transport with ballistic-like trends. For those cases, we introduce the models of: quasi-ballistic velocity-dependent dissipation (QvD) and the ballistic-like voltage-dependent mobility (BVM) regime of SCLC. The QvD model is shown to better describe electronic kinetics, whereas the BVM model is revealed as suitable for describing electronic or ionic kinetics in halide perovskites. The proposed formulations can be used as characterization tools for the evaluation of effective mobilities, charge carrier concentrations and times-of-flight from J-V curves and impedance spectroscopy spectra.


Introduction
Halide perovskites, e.g. methylammonium lead triiodide (MAPbI3), have emerged as one of the most attractive material families for optoelectronic applications, such as photovoltaics, [1] ionizing radiation detectors, [2] field-effect transistors, [3] memristors, [4] and energy storage. [5] Reason for that, is the optimal properties for photon absorption and charge transport, as well as the easy solution-based fabrication methods, which has motivated intensive research. At a material level, one of the most basic and commonly performed characterization techniques is the current density-voltage (J-V) measurement, which can inform on the transport mechanisms and properties. Notably, the study of the dark J-V curves of halide perovskites results particularly puzzling due to the dual ionicelectronic conductivity of these materials, resulting in many artefacts and anomalous behaviours, typically known as the J-V hysteresis effect.
The main transport mechanisms usually reported in symmetrically contacted samples (no built-in voltages Vbi at the electrodes) of hybrid perovskites are the ohmic and the space-charge-limited current (SCLC). In the ohmic regime, the current and the electrostatic potential ( ) are linear with the applied voltage and the position ( ), respectively (see FIG. 1(a,b)). Differently, in the SCLC regime the current may deviate from the linear behaviour due to significant modification in the charge density profile as the electric field increases with the externally applied voltage.
The most common form of SCLC is that of the mobility regime, where the current follows the Mott-Gurney law (J∝V 2 ). [6,7] Within this regime, one can often find the trapfilling (J∝V >2 ) sub-regime, [8] which allows the estimation of the concentration of deep level trap defects. [7] Subsequently, upon high enough injection of charge carriers, the velocity saturation regime (J∝V), [9,10] takes place. These regimes are well known and their identification in hybrid halide perovskites is generally not questioned with the classic distribution of FIG. 1(a).
Another fundamentally different SCLC regime is that of the ballistic transport, as the Child-Langmuir law, [11,12] where = 4 0 Here 0 is the vacuum permittivity; , the dielectric constant; , the distance between electrodes; and and are the charge and the effective mass of the charge carriers, respectively. In physical terms, unlike the mobility regime, in the ballistic regime no scattering is considered (the definition of mobility is not justified) and the maximum drift velocity results as a consequence of the conservative transformation of electrostatic energy into kinetic energy. This model was initially thought for vacuum conditions, and among semiconductors it has been considered for either low temperature conditions, [13] short enough distance [14] or time scale (near-ballistic regime). [15] Notably, in practice, with typical measurement conditions ( ≤100 V, room temperature ~300 K, electrode active area A>0.01 cm 2 , =23 [16] ) the ballistic currents of MAPbI3 samples would result significantly high for <10 μm (e.g. >10 A cm -2 at 10 V), as presented in Figure S1 in the supplementary material. In addition, if present, a bulk mechanism such as the SCLC is commonly overlapped by interface phenomena in thin film samples where the distance between electrodes is in the order of the diffusion and/or Debye lengths. Therefore, it makes sense to discard Child-Langmuir's ballistic SCLC transport as a major mechanism for operational currents in thin film samples.
In thick halide perovskite samples the SCLC regimes with trends ∝ and 1 < < 2 have been identified [17] which may suggest the presence of some sort of ballisticlike transport. For example, FIG. 1(c) shows three different perovskite samples showing some section of the dark J-V curve with ≈ 3/2, characterized in our previous and simultaneous works and measured in room conditions with a Keithley Source-meter 2612B. [18,19] The chromium contacted 1-mm-thick polycrystalline (pc) pellet (Cr/pc-MAPbI3/Cr) [18] reports the lower currents and the 2 mm-thick single crystal (sc) of CH3NH3PbBr3 (Cr/sc-MAPbBr3/Cr) [20] sample presents the shorter voltage section with the ∝ 3/2 trend, whereas the platinum contacted 3 mm-thick CsPbBr3 sample (Pt/sc-CsPbBr3/Pt) [19] seems to behave the closest to the seemly ballistic comportment.   [11,12] of SCLC, (IIb) quasiballistic velocity-dependent dissipation (QvD) model of SCLC, (IIc) ballistic-like voltagedependent mobility (BVM) model of SCLC, (III) Mott-Gurney law [7] trap-empty SCLC, (IV) trap filling, [7] (V) Mott-Gurney law trap-filled [7] and (VI) velocity saturation. [7] (b) Electrostatic potential as a function of position between electrodes separated a distance upon application of voltage for several configurations from ohmic to one-sided abrupt (OSA) junction, as indicated. (c) Experimental curves from various perovskite samples (dots) reported in the supplementary materials of our simultaneous works, [19,21]  For all SCLC regimes, any trend from a dark J-V curve of halide perovskite samples should be taken with cautions, because of the hysteresis issues. For instance, Duijnstee et al. [22,23] recently suggested the use of temperature-dependent-pulsedvoltage-SCLC measurements as a validating technique for ensuring the identification of one or another transport regime. Also, Alvar et al. [24] showed how the frequency dependence of the permittivity of thin film perovskites, and their dependence on voltage scan rate and temperature, influence the analysis of the SCLC. Moreover, the temperature-modulated-SCLC spectroscopy study of Pospisil et al. [25] found multicomponent deep trap states in pure perovskite crystals, assumingly caused by the formation of nanodomains due to the presence of the mobile species in the perovskites.
These and more studies suggest that any claim of SCLC from J-V curves should be double checked for hysteresis or ion migration related artefacts, before extracting parameters. For instance, the J-V curves of FIG. 1(c) were not reproducible when long periods of pre-biasing preceded the voltage sweeps.
Regarding the classic ballistic regime of SCLC in halide perovskites, even if a J∝V 3/2 trend is validated as hysteresis-free, there is little likelihood for the mechanism to be present at room temperatures. On the one hand, the condition of negligible energy dissipation for charge carriers is arguably unrealistic for transport from one electrode to the other, regardless whether they are electronic or ionic charge carriers. Notably, even though there is evidence of ballistic transport lengths ~200 nm in MAPbI3 thin films, it has been reported in a time scale of 10-300 fs. [26][27][28] On the other hand, a fair assumption would be for the J∝V 3/2 tendency to more likely be an intermediate sub-regime of transition between the ohmic regime ( ∝ ) and the mobility regime ( ∝ ≥2 ), or between this latter and ohmic-seemly velocity saturation regime ( ∝ ). However, a J∝V 3/2 behaviour could occur in a way that resembles the SCLC ballistic mechanism at room temperature. For this to happen in the SCLC formalism, two main scenarios can be considered. First, a quasi-ballistic SCLC transport can occur in the case of an energy dissipation proportional to the velocity, resembling that due to Stoke's drag dissipation forces. Second, a ballistic-like SCLC transport takes place due to a behaviour of vd∝V 1/2 not related at all with energy conservation (ballistic regime) but with a bias-dependent energy dissipation (mobility regime).

The quasi-ballistic velocity-dependent dissipation regime
The core of the deduction of the classic SCLC resides in neglecting the diffusion currents (div =0) and assuming that the total current density is mostly the drift component with a given relation between and the electrostatic potential or the absolute value of the electric field | | = | / |. Mathematically, for a sample with distance between electrodes at an external voltage and ∝ , one can always find a solution of the Poisson equation with a position-independent current such as = +1 −2 , where is a constant (see Equation S10 in Section S1 of the supplementary material). Thus, ∝ 1/2 results in the Child-Langmuir ( ∝ 3/2 ) [11,12] law with a potential ∝ 4/3 (see FIG. 1).
Unlike the classic ballistic approach, where all the electrostatic energy is converted into kinetic energy, a dissipation term is included as a correction in the QvD model. The energy dissipated during the transport of the charge carrier from one electrode to the other is considered in the form of = , similarly to a Stoke's drag dissipation mechanism. Here, the momentum relaxation rate as = 0 ( / 0 ) 1/2 is such that it agrees with Teitel and Wilkins' conditions for the typical time overshoot, as previously considered for near-ballistic transport in one-valley semiconductors. [29] The characteristic dissipation frequency 0 is in the order of the black-body radiation and increases with the potential, and thus , over the threshold value 0 , attaining a maximum at the biased electrode. Moreover, 0 is also expected to relate to balancing the dissipation energy in a form that only depends on the position through and . Subsequently, the energy conservation results as where the dimensionless QvD coefficient is and the effective mobility is = 0 2 / 0 . Notably, the use of effective mobilities has been earlier introduced in quasi-ballistic transport in high electron mobility transistors. [30] In FIG. 2(a), the typical values for as function of 0 and for a charge carrier with the elementary charge and the electron mass . The exact solution of Equation (3) is only different to (2) by a factor ((2 + ) 1/2 ± 1/2 ) which is 2 1/2 in the limit of → 0. Note that the solution with the plus must be discarded because the dissipation cannot increase the velocity, For ≪ 2 (e.g., low temperatures, low mobilities, and/or low effective mass charge carriers), the dissipation does not significantly affects the Child-Langmuir law [11,12] as in Equations (1) and (2). Thus, the use of the ballistic SCLC model would be justified with typical current values as in FIG. 1. On the other hand, the case could be a large dissipation for ≫ 2 (e.g. high temperatures, high mobilities, and/or high effective mass charge carriers). In this situation the ballistic regime would no longer be present.
In halide perovskites, the QvD-SCLC would be more appropriate for describing electronic kinetics due to their significantly higher mobility (~1 cm -2 V -1 s -1 ), in comparison with that of the mobile ion charge carriers (<10 -6 cm -2 V -1 s -1 ). The low mobility of the ions lowers , which makes the currents to follow the Child-Langmuir law, that could attain unrealistically large ionic currents.

The ballistic-like voltage-dependent mobility regime
The collapse of the QvD-SCLC model above presented occurs when the large dissipation ( ≫ 2) produces current values out of the rage of those found in the experiment. In this case, the presence of SCLC with a ∝ 3/2 trend could still be justified with a modified SCLC mobility regime. In the classic mobility regime, instead of the relation (2) For to be other than linear with , the mobility should be field-and thus voltagedependent, which is in agreement with earlier experimental reports for halide perovskites. [19,31] Importantly, if is no longer linear with , the definition of mobility is no longer satisfied. Nevertheless, one can propose an effective threshold mobility 0 for the transition from the ohmic to the BVM regime of SCLC above an onset voltage Subsequently, one can substitute Equation (9) in (8) to obtain the BVM drift velocity The conceptual ideas behind the expressions (9) and (10)  , where is Frenkel's equation [32] 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. In addition, Equation (10) can be deduced as a particular case of Poole-Frenkel's [9,32,33] ionized-trap-assisted transport where the charge carrier concentration is proportional to the field in a narrow bias range (see also Section S2.1). Summing up, it is the dual ionic-electronic conductivity of perovskites that enables the BVM regime of SCLC. The introduction of ionic space charges and mechanisms of field-dependent ionization is also suggested from the potential distribution, as presented in FIG. 1(b).
The BVM model corresponds to an electrostatic potential situation somehow in between the Mott-Gurney law of SCLC (electronic bulk effect) and the quadratic case of the rectifying junctions (mobile ions depletion effect towards interfaces). Moreover, the idea of gradients in the transport properties such as in (9) has also been proposed in the ionic dynamic doping model, [18,20,34] where the biasing produce accumulation of mobile ions towards one electrode with slow kinetics. Consequently, one can substitute Equation (10) in (5) to solve the Poisson equation whose solution between = 0 and = , with (0) = 0 and ( ) = , [11] results in the BVM current density The current values for Equation (12) Experimentally, Equation (14) relates to the resistance from impedance spectroscopy (IS) characterizations. Notably, the low frequency resistance from IS spectra have been found to behave as ∝ −1 2 ⁄ in our simultaneous work on 3 mmthick CsPbBr3 single crystals. [19] Importantly, discerning whether ∝ 3 2 ⁄ implies a QvD or a BVM regime of SCLC as Equations (7) and (12)

Conclusions
In summary, the use of the Child-Langmuir formalism of ballistic SCLC in halide perovskite has been reviewed and discussed. Our analyses suggest that, even though the where the constant is In the ballistic regime, the Child-Langmuir law [1,2] of space-charge-limited current (SCLC) uses = (2 / ) 1/2 and = 1/2, then the electrostatic potential results as   Figure S4: Current density as a function of the external applied voltage for the ballistic Child-Langmuir law [1,2] of SCLC for a MAPbI3 sample with =23 [3] and Q and M being the electron charge and mass, respectively, in Equation (S27).
In the quasi-ballistic velocity-dependent dissipation (QvD) [4]  where the constant is In the Mott-Gurney law [5] of SCLC, = and = 1, therefore, the electrostatic potential results as and the current density is The current as in Equation (S40)(S27) is illustrated in Figure S5 for typical values.  Figure S5: Current density as a function of the external applied voltage and mobility for the Mott-Gurney law [5] of mobility regime of SCLC for a MAPbI3 sample with =23 [3] and L=1.0 mm in Equation (S40)(S27).

S2.1. The onset voltage of the BVM regime of SCLC
In the SCLC formalism, the ∝ ( / ) 1/2 can explain a ∝ 3/2 , as above demonstrated. Typically, in the mobility regime the absolute value of the drift velocity is considered as The use of Equation (S43) leads to the Mott-Gurney law. [5] However, assuming a transition from Ohmic to the BVM regime around an onset voltage 0 , the conjunction of both field-dependent ionization and accumulation of mobile ions towards the interface can be producing a voltage-dependent mobility as where 0 is an effective mobility independent of field and position, is Frenkel's equation [6] for the distance between the ions and their local potential maxima upon application of an external field The values for 0 are presented in Figure S6 for a MAPI sample at room temperature. Note that one may expect values in the range 1-10 V for a 1.0 mm thickness sample, meaning that the concentration of mobile ions towards the interface is around 10 14 cm -3 .  Figure S6: Onset voltage as a function of the distance between electrodes and the concentration of mobile ions towards the electrodes in the BVM regime of SCLC for a MAPbI3 sample with =23, [3] = 300 K and as the elementary charge in Equation (S48).
The BVM drift velocity can then be rewritten by substituting (S47) in (S43) as Alternatively, one can assume the BVM model as an approximation to a particular case of the Poole-Frenkel [6][7][8] ionized-trap-mediated transport when the field and the charge carrier profile meet certain specific criteria. For a start, we consider the Poole-