Introduction

The rapid increase in power conversion efficiency, from 3.8% (2009)1 to 26.1% (2023)2, reported in halide perovskites solar cells (PSCs), has started a new generation of photovoltaics and put these materials in the focus of optoelectronic research. Their unique properties (high absorption coefficient3,4,5, tunable bandgap6,7,8, long range carrier mobilities3,9,10,11,12, and low defect density13,14,15 opened a broad spectrum of possible applications such as solar cells16,17,18,19,20,21, photodetectors15,22,23,24,25,26, field-effect transistors27,28,29,30, light emitting diodes31,32,33,34, memristors35,36, and lasers37,38,39,40. Organometal halide perovskites typically come in the form of organic-inorganic crystalline compounds with the general formula of APbX341, where A is a monovalent cation (methylammonium, MA+, formamidinium FA+ or cesium Cs+), and X is a halide anion (chloride Cl–, bromide Br– and iodide I–).

Interestingly, their key optoelectronic properties, such as optical bandgap, can be easily tailored by chemical engineering before the synthesis of these solution-processable materials. For example, the bathochromic shift, that is the decrease of optical bandgap reported in MAPbX3, occurs when chloride (2.97 eV) is substituted with larger halide anions, bromide (2.22 eV) or iodide (1.50 eV)42. Other important factors in future practical applications are represented by the charge carrier transport and defect density, both of which are highly dependent on the type of the perovskite active layer. It has been widely accepted that perovskite single crystals exhibit significantly better charge transport properties compared to those of their polycrystalline thin film counterparts. The bulk perovskite single crystals show long periodicity of crystalline lattice and very low density of trap states (2–200 × 109 cm–3)43,44. On the contrary, multiple grain boundaries in polycrystalline thin films create deep trap states (trap density ≈1 × 1017 cm–3)45, and they also promote the electron-hole recombination; hence, they decrease the effectivity of charge carrier transport. Thus, single crystals make a good platform to explore the intrinsic properties of organometal halide perovskites.

Although extensive research has been performed, there is a lack of unanimous experimental and modeling procedures to determine both the charge carrier mobilities and concentration of trapped charge carriers in organometal halide perovskite. To determine the above parameters, the space-charge limited current (SCLC) method is commonly used. Essentially, this method is based on the complex analysis of current-voltage characteristics (also known as the J-V curves) of semiconductor materials with trap states. It identifies and distinguishes the three regions present in J-V curves: (1) ohmic region (i.e. the slope of current-voltage characteristic is equal to one); (2) trap filing region (the slope is greater than 2); and (3) quadratic region (the slope is equal to 2, i.e. the monoenergetic trap state or Mott–Gurney region).

However, this method has multiple shortcomings that may lead to underestimation and inaccurate interpretation of J-V characteristics. Firstly, commonly applied approaches use tangent fitting44,45,46,47,48,49 to identify the different regions of J-V characteristics and estimate the electric properties of the perovskite materials (mainly charge carrier mobility and concentration of the trapped states). Secondly, the halide perovskite materials are known for their electronic and ionic nature50,51,52, which affect the J-V characteristics and complicate their analysis and interpretation. Many efforts have been put toward understanding of ionic processes and their effects on J-V hysteresis and SCLC parameters. Duijnstee et al.53 and Le Corre et al. 54 developed complex experimental and modeling approach to describe the ionic effects on the measured J-V characteristics and quantify the concentrations of ions using drift diffusion model. In a recent SCLC analysis of MAPbI3-based devices Sajedi Alvar, et al.55 demonstrate the importance of temperature-dependent ion dynamics and temperature- and frequency-dependent dielectric constants. The authors suggest that only experimentally validated parameters should be used to build the SCLC model. Though the latest SCLC modeling approach enabled precise simulation of J-V curves and determination of ion influence on the shape of the characteristics, some of the fundamental properties describing the charge dynamics (e.g. energy and voltage dependance of modeled parameters) are still not described in detail.

Herein, we report a modeling approach to determine the energetic distribution of the charge carrier (electron and holes) mobilities, of trapped state density, and of Fermi level using advanced space charge limited current (A-SCLC) model. With this model, we can quantify the SCLC parameters in different regimes of the J-V curves (ohmic, trap filling, and quadratic regions) associated with the charge carrier injection properties of the perovskite-based devices and map the trap filling inside of the semiconductor bandgap. Moreover, we simulated the influence of Fermi level position, temperature, trap position and trap temperature on the energetic dependencies of the trapped charge concentration (Supplementary Figs. S8–S12). The capabilities of the proposed A-SCLC model were presented on two perspectives organometal halide perovskite single crystals MAPbBr3 and MAPbI3 (see Supplementary Fig. S1). The data for evaluation were collected by measuring the J-V characteristics of Carbon/Perovskite/Carbon devices in dark and under illumination. In addition, we discussed the effect of thermodynamic Fermi level shift and different charge injection properties of the studied materials.

Results and discussion

Analysis of J-V curves

In the early works56,57, A. Rose determined that the two requirements have to be fulfilled in order to observe space charge limited currents in low conducting materials: (i) at least one of the contact must be ohmic (i.e. the contact that provides an excess of carriers ready to enter the studied material) and (ii) the studied material has to be relatively free from trap states. The first requirement was fulfilled by contacting perovskites in sandwich-like (Supplementary Fig. S5) carbon contacts (e.g. C/MAPbBr3/C), and the monocrystalline form of perovskites were used to ensure the high periodicity of the crystalline lattice and low trap concentration needed to fulfill the second requirement. Structural and optical analysis of the prepared materials are presented in Supplementary Figs. S3, S4.

The current-voltage characteristics of a semiconductor material with trap states can be divided into three regions: ohmic, trap filling and quadratic region. The electric properties of these materials can be analyzed by two current equations that describe the J-V curves of SCLC. First is Ohm’s law describing the linear current dependence on applied bias voltage:

$$J=e{\mu }_{0}{p}_{{{\rm{f}}}}\,\frac{V}{L}$$
(1)

where e is an elementary charge (e = 1.602·10–19 C), µ0 is the microscopic mobility, pf (nf) is a concentration of free charge carriers (electrons or holes), L represents the thickness of the sample. The second equation is given by the combination of Poisson’s equation and Ohm’s law:

$$J={\varepsilon }_{0}{\varepsilon }_{{{\rm{r}}}}{\mu }_{0}{\varTheta} (1-\gamma ){(2-\gamma )}^{2}\frac{{V}^{2}}{{L}^{3}}$$
(2)

where ε0 is vacuum permittivity (ε0 = 8.854·10–12 F m–1), εr is relative permittivity (dielectric constant), Θ is the fraction of free (pf) and total (ps) charge carrier concentrations45,53 in the material (i.e. \({\varTheta }={p}_{{{\rm{f}}}}/{p}_{{{\rm{s}}}}={p}_{{{\rm{f}}}}/({p}_{{{\rm{f}}}}+{p}_{{{\rm{t}}}})\), where pt is a trapped charge concentration), and \(\gamma\) is inversely proportional to the logarithmic slope (m) of the current-voltage characteristic (i.e. \(\gamma =1/m={{\rm{d}}}\,{\mathrm{ln}}\,V/{{\rm{d}}}\,{\mathrm{ln}}\,J\))58,59. The parameter theta Θ and gamma \(\gamma\) can be used to identify different regions of the J-V curves. For instance, if the gamma equals one (\(m\,=\,1\,\to \,\gamma \,=\,1\)), ohmic region occurs. If the injected charge accumulates in the semiconductor (i.e. space charge region), the slope of the J-V curve increases and \(\gamma\) takes on values between zero and one (\(0\, < \,\gamma \, < \,1\,\)). In the space charge region, we can observe two different situations: (i) the slope m approaches infinity (\(m\,\to \,\infty \,\Rightarrow \,\gamma \,=\,1\)) for trap filled limit (TFL), and (ii) Mott–Gurney region for slope m equals two (\(m\,=\,2\,\Rightarrow \,\gamma \,=\,1\,/\,2\)). If the latter is the case, then Eq. (2) transforms to Mott–Gurney law

$$J=\frac{9}{8}{\varepsilon }_{0}{\varepsilon }_{{{\rm{r}}}}{\mu }_{0}{\varTheta} \frac{{V}^{2}}{{L}^{3}}$$
(3)

However, in some special cases, due to anomalies of charge transport (e.g. injection barrier), the slope m can be lower than one, in which case the current suddenly decreases in a forward biased scan. The parameter theta Θ equals one (Θ \(=1\)) in ohmic region when \({p}_{t}=0\) as well as in Mott-Gurney region, where \({p}_{{{\rm{s}}}}\approx {p}_{{{\rm{f}}}}\) (pt ≪ pf). In other regions of J-V curves theta takes on values between zero and one (\(0 < \) Θ \( < 1\)). The full explanation of the model including different regions of the curves and graphical presentation of theta are described in Supplementary Information (see Supplementary Figs. S13–S15). All described relations are derived for hole-only devices.

Figure 1 illustrates the band diagrams of two studied perovskite-based devices (C/MAPbBr3/C and C/MAPbI3/C) before contact, after contact at zero bias voltage and after applied bias voltage (forward scan). Moreover, the figure illustrates the band bending caused by illumination and applied bias voltage. As evident, the Fermi level equilibrates after contacting, and bands in MAPbBr3 and MAPbI3 bend for 0.172 eV and 0.126 eV, respectively. The thermodynamic Fermi level changes due to the illumination causing additional bending (0.260 eV for MAPbBr3 and 0.254 eV MAPbI3). The energy difference between the thermodynamic Fermi level positions under dark and illuminated conditions is reffered to as the Fermi level shift.

Fig. 1: Graphical illustration of band diagrams under various measurement conditions.
figure 1

Band diagrams of a Carbon/MAPbBr3/Carbon and b Carbon/MAPbI3/Carbon devices before and after contact (at 0 V and in forward regime). The diagrams consist of several energies: valence band energy (Ev), conduction band energy (Ec), energy of trap state (Et), thermodynamic Fermi level (EF0). The positions of valence band energy (Ev,dark and Ev,light), conduction band energy (Ec,dark and Ec,light) and Fermi level (EF0,dark and EF0,light) in dark and under illumination are clearly indicated.

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Figure 2 shows the J-V characteristics of MAPbBr3 and MAPbI3 single crystals measured under dark (yellow dots) and under white LED illumination of 34 W m–2 (orange dots) in forward scan regime only with low voltage scan rate of 1 mV s–1. Raw and average data plots are shown in Supplementary Information (see Supplementary Figs. S17, S18). Previous research suggests53,54,55 that ion migration plays an important role in the hysteresis effect and shape of J-V curves. Hence, we carried out an extensive analysis to determine the mobilities and ion concentrations in our halide perovskite single crystals (for further information see Supplementary Fig. S16). Based on the analysis and comparison with DD simulation model we decided to proceed using simplified Eq. (3) to model the measured J-V curves (see Fig. 2 and Supplementary Figs. S17, S18).

Fig. 2: J-V curves under various experimental conditions.
figure 2

Experimental data and estimated models of current-voltage characteristics for a MAPbBr3 and b MAPbI3 perovskite single crystals in the dark (Jdark) and under illumination (Jlight) fitted using injection and extraction model. Black dashed and solid lines represent linear and quadratic slopes (m) of J-V characteristics.

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To gain a better understanding of the current-voltage characteristics, we introduced the injection (solid line red) and, and the opposite, extraction model (dashed red line). The current density and voltage are plotted on a logarithmic scale. A detailed explanation of the applied models is presented in Supplementary Figs. S6, S7. The other two lines (i) dashed black line and (ii) solid black line correspond with the ohmic region (m = 1) and quadratic region (m = 2), respectively.

Both characteristics exhibit noise for small voltage (ohmic region) and a clear change after illumination. In the case of MAPbBr3 the illumination causes 1 order of magnitude increase in photocurrent in the ohmic region, while the TFL regions exhibit little to no difference. The transition between ohmic and TFL region occurs at 0.5 V, followed by the transition from TFL to quadratic region at around 3 V. The dark and light current-voltage characteristics follow the proposed injection model, with no indication of the barrier behavior. However, both curves enter Mott–Gurney region at 10 V and 9 V, respectively. On the contrary, dark and light current voltage characteristics of MAPbI3 differ in the measured range. The dark J-V clearly shows the barrier behavior (i.e. follows the extraction model), while the light J-V follows the injection model with two transitions: (i) ohmic to TFL at 0.5 V and (ii) TFL to quadratic region at 3 V. This curve enters the Mott-Gurney region at ~10 V. Moreover, the current passing through the MAPbI3 single crystal increases by about 1.5 orders of magnitude after illumination. The conductivity (σ) of both materials was determined from Ohmic regions of dark J-V curves using the following formula \(\sigma =\frac{L}{A\,R}\), where L is the crystal thickness, A is the active area (Supplementary Table S1), R is the resistance. The conductivity of MAPbBr3 was calculated to be (3.9 ± 0.6) × 10–11 S cm–1, while that of MAPbI3 was found to be (2.6 ± 0.4) × 10–10 S cm–1.

To the best of our knowledge, the previously reported models44,45,46,47,48,49 apply the tangent fits to the defined regions (ohmic, TFL and Mott–Gurney) in dark current-voltage characteristics and determine the parameters such as VTFL, concentrations of trapped charge carriers (pt or nt), and their mobilities (μp and μn). However, the information about the energy distribution and voltage dependance of the determined parameters remains unclear. In addition, the localization of the trap state is usually missing, which might raise the question whether the simple fit by tangent is sufficient to describe these materials. Especially if we take into account that the trap state can cause the quadratic slope in J-V characteristics, same as the Mott–Gurney region (see Model Simulations in Supplementary Information). Hence, we developed the advanced space charge limited current model (A-SCLC) to investigate the electrical properties of halide perovskite materials. The model itself is a successor of temperature-modulated SCLC model60,61,62,63,64, previously published by our group and it can be applied not only to perovskites but to other semiconductor materials as well.

Determination of the charge carrier mobilities and concentrations

To determine the key parameters of semiconductor materials one must take into account the band diagram of the studied material and the contacts, as shown in Fig. 1 and Supplementary Fig. S6. With that being said, we developed a model consisting of two square root transport bands (conduction and valence band) and one monoenergetic trap state. Practically, the proposed model variates with five parameters: (i) microscopic mobility μ0, (ii) density of the trap states Nt, (iii) energy position of the trap state Et, (iv) trap temperature Tt, and (v) thermodynamic Fermi level position EF0 to fit the J-V characteristics and simultaneously fit the calculated parameters (effective mobility, concentration of free and trapped charge carriers, Fermi level shift, and parameter theta). A detailed description of the model used is described in Supplementary Note 1. Supplementary Table S2 presents all parameters used to build the model, while Supplementary Fig. S7 represents a scheme used to calculate and model the studied parameters (for better understanding, a guide to data analysis and modeling is also included and explained in Supplementary Note 2).

The first of the calculated parameters, the effective mobility58,62,65 of the charge carriers, can be derived from Eq. (2)

$${\mu }_{{{\rm{eff}}}}={\mu }_{0}{\varTheta}=\frac{{L}^{3}}{{\varepsilon }_{0}{\varepsilon }_{{{\rm{r}}}}(1-\gamma ){(2-\gamma )}^{2}}\frac{J}{{V}^{2}}\,$$
(4)

Essentially, the microscopic mobility (μ0) is the limiting case of effective mobility (μeff), which occurs when parameter Θ equals one (e.g. ohmic and Mott-Gurney regions). However, the trap states lower the effective mobility and thereby the magnitude of SCLC (Θ ≤ 1). Hence, to precisely determine the mobility, it is necessary to either measure J-V curves up to Mott-Gurney region, or to develop a model that can estimate at which voltage the device enters this region of current-voltage characteristics and then calculate the semiconductor’s properties.

Figure 3 shows the dependence of effective mobility on the applied voltage under dark and light measurement conditions (experimental data), their models (solid red curves) and estimated microscopic mobility values (dashed lines). Interestingly, both dark and light effective mobility of MAPbBr3 approaches the M-G region (i.e. effective mobility approaches the microscopic mobility), while in the case of MAPbI3 only the light μeff follows the same trend. The dark μeff decreases with increasing voltage, reflecting the barrier behavior previously seen in the current-voltage characteristics. In our case, the microscopic mobility of MAPbI3 under dark can be determined only from ohmic region. We note that such behavior was previously measured in MAPbI3-based hole-only devices48. Estimated microscopic mobilities for both single crystals are presented in Table 1. The voltage dependence of Θ is shown in Supplementary Fig. S19.

Fig. 3: Tracking the change of effective mobility under applied bias voltage.
figure 3

Effective mobility (μeff) of a MAPbBr3 and b MAPbI3 perovskite single crystals, in the dark (μeff,dark) and under illumination (μeff,light). Red and black dashed lines represent estimated microscopic mobilities (μ0,dark and μ0,light), while red solid lines represent models of effective mobility (μeff,m).

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Table 1 Summarized parameters for MAPbBr3 and MAPbI3 single crystals
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Once microscopic mobility is estimated, we can calculate the concentration of free and trapped charge carriers (in our case holes), pf and pt using following relations

$${p}_{{{\rm{f}}}}=\frac{L\,J}{e{\mu }_{0}\,(2{-}\gamma )\,V}$$
(5)
$${p}_{{{\rm{t}}}}=\frac{{\varepsilon }_{0}{\varepsilon }_{{{\rm{r}}}}(1-\gamma )(2-\gamma )\,V}{e\,{L}^{2}}$$
(6)

Figure 4a, b shows voltage dependences of experimental pf and pt and their models for two studied crystals MAPbBr3 and MAPbI3, respectively. As evident, the illuminated MAPbBr3 single crystal exhibits an increase of free carriers in ohmic region, while MAPbI3 exhibits an increase in free charge carriers over the entire measurement range. These findings correlate with the J-V curves (Fig. 1). Moreover, the concentration of trapped charge carriers can also be plotted as a function of free charge carrier concentration and Fermi level (top and bottom axis in Fig. 4c, d). The model concentrations of free and trapped charge carriers were plotted for holes (solid ptm and dashed pfm red lines indicating positive charge) and for electrons (solid ntm and dashed nfm blue lines indicating negative charge). This representation of the modeled and calculated concentration gives a deeper understanding of the charge carrier transport mechanisms. Firstly, note that in the case of MAPbBr3, the concentration of trapped charge follows the model of hole carriers (red line), while in MAPbI3 only the points calculated from light J-V characteristics do so. This trend responds with injection behavior (i.e. ohmic and TFL region transition). A detailed explanation regarding the different regions of the J-V curves and their correlation with the pt (pf) and nt (nf) functions is shown in the model simulation (see Supplementary Fig. S15). In addition, the saturated value of pt (see Fig. 4c, d), that is the total concentration of trapped charge carriers, increases from 7.58 × 109 cm–3 (under dark) to 8.84 × 109 cm–3 (after illumination) for MAPbBr3. In the case of MAPbI3 we notice a different behavior under dark. Trapped charge carriers follow the trend of nt instead of pt. This could be related to a blocking mechanism at the injection contact causing extraction dominance and sudden decrease of free charge carriers (see Fig. 4d and Supplementary Fig. S22b, c). However, there is a change in concentration of trapped charge carriers from 1.15 × 1010 cm–3 under dark to 8.63 × 109 cm–3 after illumination.

Fig. 4: Change of carrier concentrations and their mapping using A-SCLC model.
figure 4

Voltage dependence of free (pf) and trapped (pt) charge carriers for a MAPbBr3 and b MAPbI3 and energy dependances of pf and pt for c MAPbBr3 and d MAPbI3. The solid blue and red lines represent model curves (ntm and ptm) of trapped charge carriers. Red dashed line represents model curve of free charge carriers (pfm).

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Another interesting situation occurs at the point where concentration of trapped holes equals the concentration of free holes, \({p}_{{{\rm{t}}}}\approx {p}_{{{\rm{f}}}}\,\). Given that in this situation \({\varTheta }={p}_{{{\rm{f}}}}/{p}_{{{\rm{s}}}}={p}_{{{\rm{f}}}}/({p}_{{{\rm{t}}}}+{p}_{{{\rm{f}}}})\), we can deduce that \({\mu }_{e{{\rm{ff}}}}={\mu }_{0}{\varTheta} ={\mu }_{0}/2\). Using the proposed model one can precisely determine where this situation occurs and test the modeling of the microscopic mobility.

Determination of quasi-Fermi level position

Fermi level position is another parameter that could be fitted using the A-SCLC model. To determine the position of the Fermi level, we must use the following equation

$${p}_{{{\rm{f}}}}={N}_{{{\rm{v}}}}{e}^{\left(-\frac{\varDelta {E}_{{{\rm{F}}}}}{{k}_{{{\rm{B}}}\,}T}\right)}$$
(7)

where Nv is the concentration of delocalized states in the valence band and ΔEF is the Fermi level shift (\(\varDelta E_{{\rm{F}}} = E_{{\rm{F}}} - E_{{\rm{v}}}\)) relative to the position of valence band (as shown in Fig. 1). Figure 5a, b shows calculated dark and light Fermi level positions as well as the corresponding model curves for MAPbBr3 and MAPbI3. Again, we notice different trends in two-measured single crystals. Light and dark quasi-Fermi level of MAPbBr3 moves closer to transport band with increasing voltage, which agrees with proposed injection mechanism and increase of the pf. On the contrary, dark quasi-Fermi level of MAPbI3 single crystal moves farther away from the transport band up to approx. 2 V, after which the Fermi level shows the opposite trend. This supports our earlier statement that the presence of injection barrier leads to dominance of extraction mechanisms and decrease of free charge carriers (see Fig. 4b).

Fig. 5: Effect of applied bias voltage on Fermi level position.
figure 5

Voltage dependence of Fermi level of a MAPbBr3 and b MAPbI3, in the dark (EF,dark) and under illumination (EF,light). The solid red line represents model of Fermi level energy (EF,m). The thermodynamic Fermi level positions in the dark (EF0,dark) and under illumination (EF0,light) are indicated by black arrows.

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However, the injection barrier is removed either with higher voltage (>2 V) or by illumination, as shown in Fig. 4b and Supplementary Fig. S23. Expectedly, we noticed a shift of thermodynamic Fermi level EF0 (EF at 0 V) after illumination in both measured single crystals: from –4.828 eV to –4.874 eV (ΔEF0 = 0.046 eV) in MAPbBr3 and from –4.740 eV to –4.746 eV (ΔEF0 = 0.006 eV) in MAPbI3.

Energy distributions of the studied parameters

Energy distributions of calculated pt, pf and model curves ptm, pfm (ntm, nfm) give in-depth information about the charge transport processes inside of the semiconductor’s bandgap. Firstly, about the tendency of the carriers to move closer or further away from the transport band, and secondly about the quantity of the trapped (free) charge carrier concentrations. The model itself consists of two square root (approximately monoenergetic) transport bands (valence and conduction band) and one bi-exponential (approximately monoenergetic) trap states. All states are described with function g(E), presented with a green dashed line in Fig. 6. Previous reports suggest that halide perovskite materials have significant defect tolerance10,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28. Nevertheless, the trap states influence the charge transport and thus they cannot be neglected in the SCLC model. The effect of trap position and trap temperature on the ptm and nfm is presented in the Model Simulation section of Supplementary Information. As can be seen from this section, trap temperature does not significantly affect ptm and pfm for temperatures lower than the temperature of the experiment T (K), where EF0 ≈ Et. However, the position of thermodynamic Fermi level has the highest impact on the shape of ptm and ntm. Supplementary Fig. S8 shows three possible positions of the thermodynamic Fermi level relative to the trap position: (i) EF0 closer to the conduction band, (ii) EF0 in the position of the trap state, and (iii) EF0 closer to the valence band. Furthermore, the proposed A-SCLC model can distinguish minor changes in the concentration of trapped charge carriers and free carriers in the bandgap of semiconductor materials. The traditional SCLC models (tangent fit of J-V regions) usually do not interpret these changes.

Fig. 6: A-SCLC bandgap maps.
figure 6

Energy distribution of density of states g(E) model (green dashed lines), modeled ptm (pfm) and ntm (nfm) curves (red and blue lines), the concentration increment dptm/dE (dashed pink lines) and experimental data points pt and pf plotted for light (a, b) and dark (c, d) measurements performed on MAPbBr3 and MAPbI3, respectively. The trap state (Et) position (green dashed line) and the Fermi level (EF0) position (dashed black line) are clearly indicated.

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It our case, both MAPbBr3 and MAPbI3-based devices have thermodynamic Fermi level position closer to the valence band, EF0 < Et (i.e. deep trap state for holes, see Table 1). Figure 6 shows the discussed trends of hole concentrations in the wider energy range. In addition, the increment of charge carrier concentration (dpt/dE) is plotted to better demonstrate the increase/decrease of the trapped holes. The effect of temperature on ptm (ntm) and dptm/dE is simulated in Supplementary Figs. S10, S11.

Figure 6 also reveals the change of the free and trapped holes in the bandgap of the two studied halide perovskite materials. As mentioned above, the trapped holes in MAPbBr3 follow the ptm, which corresponds with injection mechanism in C/MAPbBr3/C (Figs. 4a, c and 6a, c) both under dark and after illumination (i.e. pt increases with increasing voltage and ptm fits pt). On the contrary, the hole injection mechanism is negligible in C/MAPbI3/C device under the dark condition, and the hole extraction dominates, leading the device into barrier-like mode (Fig. 2b). As a result of that, we notice a low concentration of free charge carriers under dark (7 × 106 cm–3), which stays almost constant up to 2.5 V (Fig. 4b, d). Under higher voltage, the hole injection surpasses the extraction, and the pf concentration increases (the injection barrier is overcome). Figures 4b and 6d suggest that barrier can be overcome with illumination as well. Thus, after illumination we notice that the pt follows ptm and that concentration of pt significantly increases after illumination. To get a better understanding of illumination effect on MAPbBr3 and MAPbI3 we compared all parameters determined by the A-SCLC model in Table 1.

The described model is applicable not only to halide perovskites but to other semiconductor materials as well. Nevertheless, there are certain criteria that must be considered before evaluating the measured current-voltage characteristics. Firstly, the mapping of trap state is possible only for lower temperatures (see Supplementary Fig. S10). If temperatures surpass the critical temperature, the trap is not visible in the ptm and ntm. Secondly, in materials such as halide perovskites, where diffusion of ions, polarization effects, reactions between ions and electrodes, influence the J-V curves, the measurements have to be carried out in steady state regime. In addition, it is noteworthy to mention that 2-electrode configuration can suffer from surface current. Thus, we recommend others to use ring-guard electrode configuration, described by Almora et al.66 prior to application of our A-SCLC model.

Conclusions

Herein, we report a comprehensive analysis of the current-voltage characteristics of two commonly studied perovskite materials: methylammonium lead bromide (MAPbBr3) and methylammonium lead iodide (MAPbI3) single crystals. Using advanced space charge limited current (A-SCLC) model we determined the key semiconductor properties (microscopic mobility, thermodynamic Fermi level, trap position and concentration of majority charge carriers) in two carbon/perovskite/carbon devices under dark and after white LED illumination (all results are summarized in Table 1). Two different behaviors can be seen from the measured and modeled parameters. The C/MAPbBr3/C device showed clear injection properties, where the concentration of free and trapped holes (i.e. majority charge carriers) increased after applied bias voltage, both under dark and after illumination. On the contrary, C/MAPbI3/C showed presence of injection barrier that altered the shape of the J-V curve under dark measurement conditions. This could be caused by interface defects or by self-doping of MAPbI3 due to the presence of intrinsic defect state (i.e. ion vacancies, interstitials etc.)67. As shown in our results, the injection barrier can be overcome with higher voltage (>2 V) and after illumination. The proposed advanced- SCLC model presented in this work is a guide for detailed analysis of current-voltage characteristics, applicable to a broad range of semiconductor materials.

Methods

Chemicals and reagents

Methylammonium bromide (6876-37-5, ≥99%, CH3NH3Br, MABr), methylammonium iodide (14965-49-2, ≥99%, CH3NH3I, MAI), lead (II) bromide (10031-22-8, 99% PbBr2), lead (II) iodide (10101-63-0, 99% PbI2), N,N-dimethylformamide (68-12-2, 99% DMF), γ-butyrolactone (96-48-0, 99% GBL), and propylene carbonate (108-32-7, 99% PC) were purchased from Merck (Sigma-Aldrich). All chemicals were used as received without further purification.

Preparation of the MAPbBr3 and MAPbI3 single crystals

Organometal halide perovskite single crystals (MAPbBr3 and MAPbI3) were synthesized using inverse temperature crystallization (ITC) method. Firstly, the equimolar ratios of perovskite precursors MABr (MAI) and PbBr2 (PbI2) were dissolved in polar aprotic solvents: bromides in DMF and iodides in GBL/PC (65%:35% v/v). As prepared, 1 ml perovskite solutions were then stirred for 3 h at 25 °C; subsequently, they were filtered using 0.45 μm polytetrafluoroethylene (PTFE) filters. Those solutions were later poured in 4 ml vials and placed into silicon oil bath to ensure the uniform temperature distribution. The whole set-up was then placed onto the SuperNuova+ (Thermo Scientific™) hotplate with PT1000 temperature probe and heated from 25 °C to 80 °C. As a result, the desired millimeter sized, MAPbBr3 and MAPbI3 single crystals were obtained (see Supplementary Figs. S1, S2).

Structural characterization

The crystalline structure of the prepared samples was determined using an X-ray diffractometer (XRD) equipped with the Cu Kα X-ray tubes. The single crystal X-ray diffraction (SC-XRD) measurements were performed on an XtaLAB Synergy-I diffractometer with a HyPix3000 hybrid pixel array detector and microfocused PhotonJet-I X-ray source (Cu Kα). The data integration was done using CrysAlisPro 1.171.40.82a software. The XRD patterns for MAPbBr3 and MAPbI3 single crystals are shown in Supplementary Fig. S3.

Optical characterization

Microscopic photos of the prepared single crystals were captured using a Hirox RH-2000 digital microscope. The UV-VIS absorbance spectra were recorded at room temperature using a Varian Cary 50 UV-VIS Spectrophotometer operating in the 200–800 nm region with a step of 1 nm. Samples were excited by 380 nm nanosecond LED source from the front side at an angle of 45°. Absorption and photoluminescence spectra of the prepared materials are shown in Supplementary Fig. S4.

Device fabrication

C/MAPbBr3/C and C/MAPbI3/C devices were prepared by depositing the perovskite grade carbon paste (DM-CAP-4703S, Dycotec Materials) at the two opposite facets of the prepared bulk single crystals as shown in Supplementary Fig. S5. Subsequently, the samples were encapsulated with Ossila epoxy resin to minimize the potential degradation due to the exposure to oxygen and moisture. This encapsulation was performed in a glovebox under an inert atmosphere (<1 ppm O2).

A-SCLC measurements

Advanced space charge limited current spectroscopy was performed using the Keithley Electrometer/High-resistance meter (model 6571B) at constant temperature (300 K) in vacuum/isolation chamber. The dark and light (34 W m–2 white LED reflector LR4605.5dDG-WFL) current-voltage (J-V) characteristics were measured in forward scan only in sandwich configuration Carbon/Perovskite/Carbon (see Supplementary Fig. S5), with a voltage scan rate of 1 mV s–1. The temperature was regulated and measured using a Lauda ECO Silver RE 415 and a Digit NanoVolt/MicroOhm meter (Agilent 34420 A thermocouple type K).

Impedance spectroscopy

Analysis of ion mobilities and concentrations was performed using Solartron SI 1260 Impedance/Gain-Phase Analyzer with Solartron Dielectric Interface 1296. Measurements were carried out at ambient temperature under dark (at 0 V voltage bias and 3V-biased), see Supplementary Fig. S16.