Ni et al. (Research Articles, 20 March 2020, p. 1352) report bulk trap densities of 1011 cm–3 and an increase in interfacial trap densities by one to four orders of magnitude from drive-level capacitance profiling of lead halide perovskites. From electrostatic arguments, we show that the results are not trap densities but are a consequence of the geometrical capacitance and charge injection into the perovskite layer.
Despite the excellent optoelectronic properties of lead-halide perovskites, efforts to better understand the details of the remaining losses due to nonradiative recombination via defects are crucial to further improve the performance of photovoltaic or light-emitting devices. One method that can determine the energetic depth of a trap and its spatial position is the so-called drive-level capacitance profiling (DLCP) method. Ni et al. (1) recently applied this method to halide perovskite solar cells to resolve bulk trap densities as low as ~1011 cm–3 and interfacial trap densities that increase by one to four orders of magnitude from bulk values [see figure 3A of (1)]. However, a charge density can only be detected in capacitance measurements if it affects the electrostatic potential, which requires either sufficiently high charge densities, low permittivities, or sufficient thicknesses (2). Using basic electrostatic arguments, we show that capacitance-based methods cannot resolve the charge densities observed in (1), except for the measurement shown in their figure 1E. We show by numerical simulation that perovskite solar cells without any defects or dopant atoms yield a response that closely resembles the one in (1), indicating a universal threshold value below which the response cannot be considered to originate from a density of defects or dopants.
The inherent assumption required to obtain spatial information in capacitance profiling methods such as capacitance-voltage (CV) and DLCP measurements is the existence of a space-charge region of width w generated by a charge density Nd (dopant or trap densities), within the device of thickness d, that can be modified by the applied voltage V. Upon applying a perturbation, a response is obtained from the edge of the depletion region or from a density of emission-limited traps located at the junction transition region (3). Although DLCP is not a small-perturbation technique like a CV measurement, the electrostatic origin of the response is identical. Indeed, the two techniques often yield similar results, especially at low frequencies where the deep traps respond (3).
We use this property to illustrate the limitations of the DLCP technique in resolving charge densities, from numerical simulations of CV measurements of perovskite solar cells using SCAPS (4). A common representation is the doping density profile, which is a plot of Nd/a(w) = –2(dC–2/dV)–1/qεrε0 versus profiling distance w = εrε0/C(V), where C is the capacitance per unit area (F cm–2), εr and ε0 are the relative permittivity of the perovskite and permittivity of free space, respectively, and q is the elementary charge. A simulated doping profile for a dopant- and trap-free perovskite solar cell (Fig. 1A; parameters and band diagram in table S1 and fig. S1) is shown for the same thicknesses used in figure 3A of (1). The apparent doping profile is U-shaped and is nearly identical to the spatial trap density profile reported in figure 3A of (1). A similar effect is observed in Fig. 1B for an intrinsic dopant- and trap-free thin film solar cell, although the apparent doping densities are a few orders of magnitude higher, again in agreement with the values reported in (1).
(A) Simulated spatial doping profiles at 103 Hz of a p-i-n–type PTAA (10 nm)/perovskite/PCBM (25 nm) solar cell for the same thicknesses of the bulk perovskite layer as used in figure 3A of (1). Arrow indicates reduction of apparent bulk charge (dopant or trap) densities with increasing thickness. The profile is identical to figure 3A of (1) even in the absence of any dopant or trap densities in the model. (B) Different thicknesses between 300 and 800 nm representative of perovskite thin films. Arrow indicates apparent reduction of bulk charge densities with thickness. (C) Minimum charge densities (dopant or trap) that will be observed in a capacitance-voltage measurement (m = 2 is assumed) for different thicknesses and permittivities typical of perovskite (olive) and silicon or organic (cyan) solar cells, in comparison with measured minimum charge densities reported for bulk single crystal and polycrystalline thin films in (1). The green region represents charge densities that are experimentally accessible for the perovskite solar cell.
These doping profiles can be understood from the relation between a Mott-Schottky plot (C–2 versus V) and a doping profile (fig. S2). The increases in apparent dopant density at the interfaces are simply the plateaus at low and high forward bias of the Mott-Schottky plots (fig. S3), whereas the apparent doping density in the bulk corresponds to the linear apparent Mott-Schottky regime. Such a shape of the Mott-Schottky profile is actually a fundamental response caused by a geometrical electrode capacitance combined with charge injection. Charge injection at forward bias in a diode typically leads to an exponentially voltage-dependent capacitance (see supplementary materials for details). If we connect this capacitance in parallel to a geometric capacitance [i.e., C = Cg + C0 exp(qV/mkBT), where kBT is the thermal voltage and m is a factor that controls the slope of C versus V], the shape of the doping profiles can be analytically calculated (see supplementary materials).
If the doping and trap densities are too small to affect the electric field of the perovskite layer of thickness d, the condition w ≤ d is not satisfied. For example, for the lowest reported bulk trap densities of ~1011 cm–3 in ~39-μm-thick perovskite layers in (1), the theoretical space-charge layer width at the onset of the linear Mott-Schottky region would be w = 88.5 μm—that is, larger than the crystal thickness. In such situations, the geometric and injection capacitances dominate the response and yield a minimum charge density (derived in supplementary materials) given by
(1)This value (shown in Fig. 1C) sets the plateau region of the doping profile, and only measured charge densities greater than this limit (green shaded region) can be considered as a response from doping or from charged defects. Note that the condition Nd ≫ Nd,min holds for any measurement frequency (see supplementary materials). If the probed carrier or trap density does not comply with Nd ≫ Nd,min, the capacitance response must arise from charge injection likely combined with a capacitive response of the transport layers (see supplementary materials). Because the minimum charge density is inversely proportional to the square of the thickness of the device, intrinsic thin films will always show larger apparent doping and trap densities than bulk single crystal films, as was observed experimentally in figure S10B of (1).
As mentioned above, the apparent rise in interfacial charge densities is a direct consequence of charge in