Transient Photovoltage Measurements in ... - ACS Publications

Article pubs.acs.org/JPCC

Transient Photovoltage Measurements in Nanocrystal-Based Solar Cells Weyde M. M. Lin, Deniz Bozyigit, Olesya Yarema, and Vanessa Wood* Laboratory for Nanoelectronics, Department of Information Technology and Electrical Engineering, ETH Zurich, Gloriastrasse 35, 8092 Zurich, Switzerland S Supporting Information *

ABSTRACT: Charge carrier lifetimes play an important role in determining the efficiency of a solar cell but remain poorly understood in nanocrystal-based devices. Carrier lifetimes are often determined using transient photovoltage measurements. Here, we perform transient photovoltage measurements on PbS nanocrystal-based solar cells and determine that the photovoltage decay time cannot be directly interpreted as the carrier lifetime. We show that the decay time can be modeled as the lifetime of an RC circuit, with the resistive component indicating the degree of trap-assisted Shockley−Read−Hall recombination and capacitive contribution coming from the space charge region. These results provide a model with which transient photovoltage data on nanocrystal devices can be analyzed and used to guide device design.

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INTRODUCTION Colloidal nanocrystals (NCs) are promising candidates for next generation photovoltaics, offering composition and size tunable band gaps as well as the possibility for low cost, solution-based manufacturing. Despite recent advances,1−3 NC-based solar cells still underperform traditional bulk semiconductor and other emerging thin film solar cells.4 One source of inefficiency is the recombination of photogenerated electrons and holes in the absorptive layer before they can be extracted, which decreases solar performance parameters including the short circuit current, the open circuit voltage, and the fill factor. Temperature-dependent current− voltage measurements on diode and heterojunction PbS NC solar cells identified that recombination occurs as Shockley− Read−Hall (SRH) trap-assisted recombination.5,6 Recombination is determined by diffusion of the charge carriers to trap states and is therefore sensitive to the number of trap states and the mobility of charge carriers in the NC film,6 which is related to the electronic overlap between NCs.7,8 While recombination in PbS NC-based solar cell has been decreased for example by modifying the surface properties of PbS NC,9,10 there is still room for improvement. Since the recombination of electrons (U(n)) or holes (U(p)) is inversely dU (n) related to the carrier lifetime (τn,p) by τn−1 = dn and τp−1 =

dU (p) , dp

ligand selection on the photoconductivity, carrier lifetime, and diffusion length in PbSe films.12 A benefit of these approaches is that they enable carrier lifetimes to be studied independently of the device; at the same time, complementary techniques to study carrier recombination within the device are of interest, since the semiconductor junctions and the band structure of the device can also play a role in defining the width of the recombination region. Here, we use transient photovoltage (TPV) measurements to characterize the decay times in NC-based solar cells containing PbS NCs as a function of illumination intensity (carrier generation rate) and temperature. TPV measurements have been previously applied to polymer solar cells, dye sensitized solar cells, and nanocrystal sensitized solar cells to determine carrier lifetimes,13,14 probe the steady-state charge densities,13 calculate activation energies for transport and recombination,15 measure charge transport from TPV rise times,16 and characterize the density of states.17 In dye sensitized solar cells (DSSCs), TPV measurements have been used to extract the density of trap states.18 The challenge of TPV measurements largely lies in the interpretation of the data, which requires modeling of the mechanisms leading to the observed carrier recombination.18,19 Typically in TPV measurements the perturbation of the short light pulse is small so that the response can always be described by a simple RC circuit. Both the resistive (R) and capacitive (C) components are readily determined from the height of the voltage transient (1/ΔV ∝ C) and the decay time (τ = RC) and are then related by a microscopic model to the properties of the semiconductor. In the case of bulk crystalline semiconductors such as silicon, R

being able to probe the carrier lifetime would

allow us to understand its origins and to systemically improve the device performance. A number of techniques have provided considerable insights into charge carrier dynamics in NC thin films. For example photoluminescence lifetime quenching using metallic nanoparticles dispersed in a PbS film was used to map exciton diffusion as a function of inter-NC coupling.11 Time-resolved terahertz spectroscopy was used to determine the effect of © 2016 American Chemical Society

Received: April 11, 2016 Revised: May 13, 2016 Published: May 16, 2016 12900

DOI: 10.1021/acs.jpcc.6b03695 J. Phys. Chem. C 2016, 120, 12900−12908

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The Journal of Physical Chemistry C

Figure 1. Transient photovoltage (TPV) measurements as a function of generation rate and temperature. (a) Measurement setup for TPV and charge extraction measurement. By variation of illumination intensity of the LED, the generation rate can be changed. (b) (Top panel) Example of a TPV measurement, from which VOC(G,T), ΔVOC(G,T), and τ(G,T) can be extracted. (Middle panel) Example of a charge extraction measurement from which ΔQ(T) can be calculated by integrating the current with respect to time if the LED is turned off and only pulsed laser is active (lower panel). (c) Inverse decay time τ−1 as a function of VOC for temperatures from 180 K (blue) to 340 K (red). (d) Inverse decay time τ−1 as a function of generation rate G show for all temperatures (180−340 K).

1,2-ethanedithiol (EDT) cross-linked PbS NC film, and a LiF/ Al electrode, as previously described in refs 22 and 23. Here we use PbS NCs with an optical band gap of 1.4 eV. The protocols for NC synthesis, MSM diode fabrication, and device characterization can be found in the Supporting Information. The dark and light current characterization of the cells is shown in Supporting Information. We find a compensation voltage, where the photocurrent Jphoto is 0 (i.e., Jdark − Jlight = 0), of V0 = 0.59 V. Following common practice, we take the compensation voltage as an approximation for the built-in voltage (Vbi).24 A Vbi of about 0.6 V is smaller than the 0.8 V predicted by the work function difference between the ITO and Al electrodes, a fact that can be explained by Fermi level pinning by trap states.6 Transient Photovoltage Measurements. To perform the TPV measurements (shown schematically in Figure 1a), the devices are mounted in a cryostat (Janis ST-500) with an optical window and constantly illuminated with a 660 nm red LED (Thorlabs M660L3-C5). The intensity of the LED determines the baseline open circuit voltage (VOC(G,T)). This baseline VOC(G,T) is then perturbed using a pulsed laser (50 ps, 405 nm, Hamamatsu C1808-03). The resulting photovoltage transient is amplified (Femto DLPVA) and measured by a data acquisition card (National Instrument NI PCIe6361). An example transient is shown in the top panel of Figure

arises from Shockley−Read−Hall and band-to-band recombination of electrons or holes, while the capacitive contribution, C, comes from the chemical capacitance (i.e., the quasi Fermi level shifts in response to the changing population in conduction and valence band20). In the case of DSSCs, R and C are related to resistance and the chemical capacitance of trap states at the TiO2 surface.21 Here we show that, in contrast to silicon and DSSCs solar cells, in NC-based diodes, the R and C contributions do not necessarily represent intrinsic properties of the semiconducting NC film. While the R can be described by trap-assisted Shockley−Read−Hall (SRH) recombination, the C comes predominately from space-charge capacitance, the magnitude of which is determined by design of the device. This means that care must be taken when using TPV measurements to analyze NC diodes. Most importantly, the photovoltage decay time cannot be directly interpreted as the carrier lifetime and the density of mid-gap electronic trap states cannot be reliably quantified. This work presents a method for analyzing TPV measurements of NC-based solar cells.

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EXPERIMENTAL METHODS Device Fabrication and Characterization. We fabricate metal−semiconductor−metal (MSM) diodes consisting of a transparent indium tin oxide (ITO) electrode, a 100 nm thick 12901


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Figure 2. Interpreting photovoltage decay as an RC circuit. (a) Schematic band diagram showing the conduction (light gray) and valence (dark gray) bands as well as the quasi Fermi levels for electrons (dashed blue) and holes (dashed red). An equivalent RC circuit is depicted. (b) Inverse electrical resistance R−1(G,T) is calculated by taking the inverse of the derivative of VOC(G,T) with respect to G for temperatures from 180 (blue) to 340 K (red). (c) Inverse capacitance C−1(G,T) is calculated as ΔVOC divided by ΔQ. The inverse dielectric capacitance of the sample measured at room temperature at 10 kHz and 0 V is indicated as a dashed gray line. (d) The inverse decay time τ−1(G,T) calculated by multiplying R−1(G,T) with C−1(G,T) from (b) and (c). Note the agreement with τ −1(G,T) in Figure 1d, extracted directly from the TPV measurement.

The inverse decay time τ−1 measured from TPV can be described as the following differential equation:

1b. As indicated, the change in VOC due to the laser pulse, ΔVOC(G,T), can be extracted, and the photovoltage decay time, τ, is determined from a fit of the transient to a single exponential decay. Transients are measured for different LED intensities and from 180 to 340 K in 20 K increments, and in Figure 1c, the inverse decay time τ−1 is plotted versus baseline VOC(G,T). We see that for any temperature (T), increasing the background illumination (i.e., increasing the VOC(G,T)) results in a shorter decay time τ. To determine the charge ΔQ(T) generated by the laser pulse, we measure the current response of the device with the LED off using a current amplifier (Femto DHPCA-100) and an oscilloscope (Rhode & Schwarz RTM1054). Integrating the area under the current transient gives ΔQ(T) (see middle panel of Figure 1b). To relate the background illumination to the carrier generation rate (G), we use a power meter (Thorlabs PM100D) to measure the light intensity reaching the sample from the LED, which allows us to calculate the photon flux and carrier generation rate as described in the Supporting Information. We neglect the contribution to the generation rate from the laser pulse, since the charge injected per laser pulse (∼109 cm−2) is small compared to the light coming from the LED (1014−1017 cm−2 s−1). We now plot the inverse decay time versus the background carrier generation rate in Figure 1d and find a linear relationship between inverse decay time and generation rate that is invariant with temperature. In other words, the decay time is only dependent on the illumination intensity and ranges from milliseconds (τ−1 = 100 s−1) for low illumination intensities to microseconds (τ−1 = 106 s−1) for high illumination intensities.

δμ(t ) δμ(t ) dμ dδμ = =− =− τ dt dt RC

Here we have used the fact that the decay time can be associated with the time constant τ of an RC circuit: τ −1 = R−1C −1

(3)

with units of inverse resistance R−1 and capacitance C−1. Determining the R and C Components from TPV Measurements. To demonstrate that the photovoltage decay time is the decay constant of an RC circuit (τ(G,T) = RC), we consider how the inverse resistance R−1(G,T) and inverse capacitance C−1(G,T) can be separately and directly determined from our TPV measurements, independent of the direct measurement of decay time τ. As shown in the schematic equivalent circuit diagram in Figure 2a, we calculate the inverse resistance R−1(G,T) by taking the derivative of the recombination current JR with respect to the chemical potential μ (i.e, VOC). Since we vary VOC(G,T) by changing illumination (i.e., the generation rate, G) and, in steady state, G (cm2 s−1) times the elementary charge e is equal to the recombination current density JR (A cm−2) (i.e., eG = JR), we can write R−1 =

dJR dVOC

⎛ dG ⎞ ⎛ dV (G) ⎞−1 = e⎜ ⎟ = e⎜ OC ⎟ ⎝ dG ⎠ ⎝ dVOC ⎠

(4)

R−1(G,T) is plotted in Figure 2b. Our capacitance C−1(G,T) can be approximated as the change in VOC with the change in charge ΔQ (i.e., the change in the number of charge carriers times the elementary charge), which we approximate by

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RESULTS AND DISCUSSION Modeling Photovoltage Decay with an RC Circuit. As shown schematically in Figure 2a, TPV measurements are performed at open circuit conditions such that eVOC = μ, where the chemical potential μ is defined as difference in the quasi Fermi levels EFn and EFp: μ = EFn − EFp. The chemical potential changes in response to our laser pulse according to the following expression: ⎛ t⎞ μ(t ) = μOC + δμ(t ) = eVOC + eΔVOC exp⎜ − ⎟ ⎝ τ⎠

(2)

C −1 ≈

ΔVOC ΔQ

(5)

The results are plotted as a function of temperature and generation rate in Figure 2c. Multiplying eqs 4 and 5, we arrive at an expression for the inverse decay time given in terms of the values determined with in the TPV measurements ΔVOC(G,T), VOC(G,T), and ΔQ(T):

(1) 12902


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Figure 3. Schematic depiction of data analysis procedure and model. (Left side) From the TPV measurement, we extract four parameters: the photovoltage VOC(G,T), the change in photovoltage ΔVOC(G,T), the extracted charge ΔQ(T), and the decay time τ(G,T). From the VOC(G,T) we can calculate the resistance R(G,T), and from ΔVOC(G,T) and ΔQ(T), we can calculate the capacitance C(G,T). Their product RC matches the decay time τ(G,T) extracted directly from the transient measurements. (Right side) The recombination model allows calculation of the VOC(G,T). We determine the model parameters by fitting them to the measurement data. From VOC(G,T), we can calculate the resistance R(G,T) as well as the width of the space charge region wsc(G,T). This allows us to calculate the capacitance C(G,T).

⎛ dV (G) ⎞−1 ΔVOC τ −1 = e⎜ OC ⎟ ⎝ dG ⎠ ΔQ

Instead, we attribute the capacitance to the space charge (or depletion) region. Space charge capacitance is given by

(6)

Csc =

In Figure 2d, we plot eq 6, i.e., the multiple of the R−1(G,T) (Figure 2b) and C−1(G,T) (Figure 2c), and find excellent agreement with the decay time extracted directly from the transients that is plotted in Figure 1d. This demonstrates that for our devices, the three parameters extracted from a TPV measurement (VOC, ΔVOC, τ) can be described by only two components, which are the resistive and the capacitive element in eqs 4 and 5. Origins of the Resistive and Capacitive Contributions to the Decay Time. To model the decay time in our devices, we must attribute the resistive and capacitive terms to specific effects. From temperature-dependent current−voltage characterization of PbS NC solids,6,25 we expect the resistive term to come from trap-assisted SRH recombination. Qualitatively, this is consistent with the measurement data in Figure 2b showing that for all temperatures, R(G,T) decreases with increasing generation rate. In the SRH model, generating more carriers will result in more recombination (i.e., a smaller resistance). We then consider the origins of the capacitive contribution to the decay time. Figure 2c shows that the capacitance C(G,T) extracted from TPV measurements (see eq 5) is larger than the dielectric capacitance of our device CD = 4.21 nF (or 0.24 μF cm−2 given the device area of 0.0175 cm2) (gray line) measured in the dark at room temperature with an impedance analyzer (Solartron Analytical ModuLab MTS) at 0 V bias using an amplitude modulation of 10 mV at 10 kHz. We model the chemical capacitance from free charge carriers and from trap states (see Supporting Information) and determine that, unlike silicon solar cells26 and DSSCs,20 the capacitive contribution shown in Figure 2c does not come primarily from chemical capacitance. The chemical capacitance from free charge carriers increases with temperature, while the C(G,T) we measure decreases with temperature, and both the chemical capacitances from free charge carriers and from trap states are of different orders of magnitude than our measured C(G,T).

Aεrε0 wsc

(7)

where A is the device area, ε0 is the electric constant, εr is the relative permittivity of the absorber layer, and wsc is the width of the space charge layer. Using a previously reported dielectric constant of εr = 14 (ref 27), we estimate the width of the space charge region as ∼50 nm. This means that in the dark, the absorptive NC layer is approximately half depleted. The width of the space charge layer wsc decreases with increasing device voltage VD according to the approximation:28 wsc =

⎛ 2εrε0 ⎞1/2 ⎜ ⎟ (Vbi − VD)1/2 ⎝ eN ⎠

(8)

−3

N (cm ) is related to the donor and acceptor concentrations, and here we determine N by inserting eq 8 in eq 7, solving for N using the device capacitance CD (measured in the dark at VD = 0) for the space charge capacitance Csc. We get N = 3.38 × 1017 cm−3, which is comparable to literature value of (1−7) × 1016 cm−3.22,29 If we bias the device with light (i.e., increasing VOC = VD), the width of the space charge region will decrease and the capacitance will increase. This is consistent with the trend in measured capacitance shown in Figure 2c, where for all temperatures, increasing the illumination increases the capacitance. Model for the Resistive and Capacitive Components. As outlined in Figure 3, now that we have proposed a picture for decay time as being determined by a resistive contribution coming from SRH recombination and capacitive contribution coming from the space charge region, we implement a model that describes these effects and check that the experimentally determined R(G,T) and C(G,T) from the TPV measurements match those predicted by the model. Starting with the resistance, we note that since 1 dV (G)

, our SRH recombination model should R(G ,T ) = e OC dG provide the correct dependence of Voc as a function of G. In 12903


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The Journal of Physical Chemistry C Table 1. Parameters Used To Calculate Carrier Lifetimes name

variable

optical band gap valence band level conduction band level effective density of states in the conduction band effective density of states in the valence band trap level trap sate density recombination coefficient capture rate coefficient electrons capture rate coefficient holes fermi level

Eg EV EC NC NV ET NT Rec φn φp EF0

built in voltage charger carrier density absorber thickness temperature dielectric constant

Vbi N d T εR

value 1.4 0 1.4 1019 1019 1.1 1017 10−10 1012 1012 intrinsic 0.7 n-type 1.2 p-type 0.2 0.59 3.38 × 1017 100 180−340 14

units eV eV eV cm−3 cm−3 eV cm−3 cm−3 s−1 cm3 s−1 cm3 s−1 eV

V cm−3 nm K

source optical band gap of material used in this study =EV + Eg See text. See text. 27 27 35 See text. See text.

compensation voltage See text. thickness of films used in this study temperatures used in this study 27

Figure 4. Microscopic origins of resistive and capacitive components of decay time. Schematic (a) band diagram for an illuminated device and (b) depiction of the recombination rate USRH(x) as a function of position in the film x with the estimated recombination region width wR and the maximal recombination rate Umax. (c) Measured VOC(G,T) as a function of generation rate and for temperatures from 180 (blue) to 340 K (red). (d) Modeled VOC(G,T). (e) Resistance derived from the measured VOC in (c). (f) Resistance derived from the modeled VOC in (d). (g) Capacitance extracted from the TPV measurements as a function of generation rate and temperature. (h) Capacitance modeled based on the space charge capacitance. (i) Recombination width wR(G,T) and (j) the space charge region width wsc(G,T) calculated from the model.

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To calculate Umax or wR, we need to know the average Fermi level EF0, which depends on how the space charge redistributes within the device. The average Fermi level EF0 is related to the quasi Fermi levels EFn and EFp: μ E Fn = E F0 + 2 μ E Fp = E F0 − (16) 2

SRH-type recombination, carriers recombine via trap states by the recombination rate: USRH(n ,p) =

φn−1φp−1NT(np − ni 2) φn−1(n + n1) − φp−1(p + p1 )

(9)

where φn and φp are the capture rate coefficients of electrons and holes (cm3 s−1) and NT is the trap state density (cm−3). n1 and p1 are defined as

We assume that charge will redistribute to minimize the chemical energy μ. We use this assumption to determine EF0 by solving eq 12 for μ(EF0) and then minimizing μ(EF0) to find EF0 for which μ is smallest. Note that this analysis is restricted to the case of μ < Vbi. On the basis of the data (see Figure 4c) showing that the VOC plateaus at Vbi, we assume that for higher generation rates G, we have μ = Vbi and G = Umax × d. After observing the correct trend for VOC(G,T) (Supporting Information), we fit the model to our VOC(G,T) data, leaving the mobility band gap Eμ, which characterizes the energy difference between the free charge carriers, the energy depth of trap states ET, the number density of trap states NT, the capture rate coefficients of electrons and holes φn, φp, and Vbi as variable parameters (see values Table 2).

⎛ E − ET ⎞ n1 = NC exp⎜ − C ⎟ kBT ⎠ ⎝ ⎛ E − EV ⎞ p1 = NV exp⎜ − T ⎟ kBT ⎠ ⎝

(10)

where ET is the trap state energy level (eV) and NV and NC are the effective density of states in the conduction and valence band (cm−3), respectively. The values used are summarized in Table 1. We estimate NV and NC to be on the order of the NC density of 5.2 × 1019 cm−3. As described in previous work, the carrier capture coefficients for diffusion-controlled recombination can be calculated using6,7 φn−1 = φp−1 = 4πR *

kBT μn , p e

Table 2. Parameters Determined by Fitting

(11)

where R* is the radius of the solid sphere from which a free carrier can recombine (cm), μn,p is the mobility (cm2 V−1 s−1), and e is the elementary charge (C). From experimental and theoretical work on the size dependent mobility in PbS NC solids,6,8 we know that μn,p is between 102 and 104 cm2 V−1 s−1. Assuming an R* of 1 nm, we get reasonable carrier capture coefficients φn, φp in the range from 109 to 1012 cm3 s−1, in agreement with previously reported values.6 We choose our initial φn, φp to be 1012 cm3 s−1. As previously mentioned, TPV measurements are performed at open circuit conditions (i.e., eVOC = μ) as shown in Figure 4a and in steady state, which means that the generation rate G (cm−2 s−1) is equal to the total recombination rate Utot: G = Utot. The total recombination rate is thus found by integrating the position-dependent recombination rate U(x) over the film thickness x. As shown schematically in Figure 4b, we approximate the total recombination rate as the maximum recombination Umax (cm−2 s−1) multiplied by the width of the recombination region wR (cm): Utot = Umax(E F0 ,μ) × wR (μ)

πkBT eE

Eμ ET NT φn, φp Vbi

eV eV cm−3 cm−3 s−1 V

1.10 Eμ/2 1.8 × 1018 1 × 109 0.57

1 dV (G)

sc

−1 ⎛ ⎞−1 πk T + d −1⎟ . width wR (G ,T ) = ⎜ (V − VB ) / w ⎝ bi OC sc ⎠ Comparing the experimentally obtained capacitance ΔQ C(G ,T ) = ΔV from eq 5 (Figure 4g) to the capacitance

(12)

(

)

OC

from the model (Figure 4h), we again find the correct trends and magnitude. We note that the shape and magnitude of our data are comparable to previous reports in refs 9 and 31. Examining the wR(G,T) (Figure 4i) and the wsc(G,T) (Figure 4j) calculated from our model, we find that with increasing generation rate G, the width of the space charge region decreases from about 50 to 0 nm while the width of recombination region increases from several nanometers to 100 nm. In Figure 5, we sketch this scenario. Since we know that there is an energy difference between the two electrodes ITO and aluminum but we do not know where in the film the

(13)

(14)

The built-in voltage Vbi = 0.59 V is set based on the compensation voltage, and the width of the space charge region wsc is defined in eq 8. Since wR cannot be larger than device thickness d (wR < d), we write wR as wR = (wR,est(μ)−1 + d −1)−1

value

calculated from VOC(G,T) using R(G ,T ) = e OC from the dG experiment (Figure 4e) and the model (Figure 4f) show the same trend of decreasing resistance with increase generation rate for all temperatures and higher resistance at higher temperatures. As indicated in Figure 3, using VOC(G,T) from our model, we can calculate the temperature- and generation-dependent width of the space charge layer wsc(G,T) = w0(Vbi − VOC)1/2, the Aε ε associated capacitance C(G ,T ) = wr 0 , and the recombination

where E is the electric field across the space charge region: E = (Vbi − μ)/wsc

unit

Comparing the experimentally measured VOC(G,T) (Figure 4c) to the calculated VOC(G,T) from the SRH model (Figure 4d) shows good agreement. Likewise, the resistances R(G,T)

The estimated recombination width wR,est as first described by Sah et al.30 and later applied by us for NC diodes6 is wR,est(μ) =

parameter

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photovoltage decay time to the carrier lifetime, one would assume that increasing the capacitive contribution, which increases the measured decay time, is a sign of improved carrier lifetime. However, in the case of the NC solar cell studies here, increasing the capacitance means decreasing the space charge region width, which is known to have a negative impact on the solar cell performance.32 If the recombination region width is known, the carrier lifetimes τn,p of the NC film material can be estimated from the SRH model. At generation rates exceeding 1017 cm−2 s−1, where we assume that recombination is occurring across the device thickness, we find carrier lifetimes of ∼10−6 s, which are consistent with previous findings for carrier lifetimes. However, it is important to remember that the recombination rate and the width of the recombination region change depending on design of the device, illumination, and temperature. Our findings of SRH trap-assisted recombination imply that longer carrier lifetimes τn,p and higher performing solar cells can be achieved by reducing recombination by either eliminating recombination centers (i.e., trap states) or decreasing the coupling of charge carriers to the trap states.6 While TPV is used as a tool to determine trap state densities (NT) in DSSCs,18 our results strongly suggest that TPV cannot be used to evaluate NT in NC-based diodes. First, using TPV to determine NT relies on the capacitive term coming from a chemical capacitance (see Supporting Information). However, we have shown that for NC-based solar cells, the capacitive term is due to the space charge region. Second, although the trap state density NT is present in the SRH model for recombination, which we use to explain the resistive component of the decay time, NT cannot be reliably determined from R. Determining NT from R would require detailed knowledge of the capture coefficients for electrons and holes (φn, φp), which is currently not available. In the model above, we use previously reported values for (φn, φp) (see Supporting Information), which yields an estimation of NT on the order of 1 per every 10 NCs. This is in good agreement with independent measurements of trap state densities.6 Nonetheless, measurements of the trap state density NT in order to evaluate a NC solid are best performed by capacitive27 or transient techniques33 designed to isolate this parameter.

Figure 5. Schematic band diagrams of the device. EC is the edge of the conduction band, Ev is the edge of the valence, Eg is the band gap, Vbi is the built potential, and x is an arbitrary space coordinate. (a) The device is in the dark, and the space charge region is large compared to the recombination region. In (b) the device is under illumination, which results in a Fermi level splitting (EFn − EFp = qVOC). The width of the space charge region approaches zero, while the recombination region is approaching the device thickness.

band bending occurs and what shape the band bending assumes, we therefore depict the bands as a function of an arbitrary space coordinate x, which can be transformed to real space by a transformation function. The position of the space charge region is irrelevant for our model. In Figure 5a, the device is in the dark, so there is no Fermi level splitting and the recombination width is very small while the space charge region is large. This changes when the device is under high illumination (Figure 5b). In this case, the Fermi level is split, the recombination width has increased dramatically, while the space charger region has shrunk. Increase in generations rates G leads to higher VOC(G,T); however, the model is only defined for VOC(G,T) < Vbi. For higher generation rates G, the VOC(G,T) is limited by Vbi, leading to the plateauing of VOC(G,T) for high generation rates in Figures 4c,d. With higher temperatures, the VOC(G,T) plateau shifts to higher generation rates. This also means that for a given generation rate G (VOC < Vbi), we find smaller wR(G,T) for larger temperatures. Since the measurements are performed in steady state such that G = Umax × wR, a smaller wR(G,T) means that for a given generation rate, we have a higher recombination rate Umax (i.e., Umax gets larger for higher temperatures). This is expected, since recombination depends strongly on the charge carrier density, which itself is temperature activated. Increasing the generation rate increases the recombination current, which results in a decreased resistance and decay time. In conclusion, our model of SRH-type trap-assisted recombination and a capacitance stemming from the spacecharge region in the device fully captures the dependences R(G,T) and C(G,T) measured experimentally. Since we showed that the photovoltage decay time is τ(G,T) = RC, we can now correctly interpret TPV data on NC-based solar cells. Implications for Solar Cells Characterization and Engineering. We have found that the photovoltage decay time measured in a NC solar cell is determined not only by the properties of the semiconducting NC layer but also by the widths of space charge and recombination regions, which are dependent on device design and illumination intensity. The photovoltage decay time τ cannot directly be taken as a measure of the carrier lifetime τn,p. If one naively equates the

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CONCLUSIONS

We have investigated what kind of information can be obtained by transient photovoltage (TPV) measurements in NC solar cells as a function of illumination (G) and temperature (T). The decay time τ(G,T) is fully described by independent resistive R(G,T) and capacitive C(G,T) components (τ = RC). We determined that in our diodes, the resistive contribution comes from the trap-assisted recombination current and the capacitive contribution comes predominately from the capacitance of the space charge (depletion) layer. This means that the decay time measured with TPV cannot be directly associated with a carrier lifetime τn,p and implies that TPV measurements cannot generally be used to determine trap state densities in NC-based devices. To increase the carrier lifetime and thereby achieve improved NC device performance, one must decrease recombination (i.e., increase R), which can be achieved by decreasing trap state density and/or the coupling rate of charge carriers to the trap states.6,34 12906


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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b03695. TEM images and optical absorption data of the PbS nanocrystals; NC solar cell fabrication protocol and characterization; calculations of generation rate; models for recombination; calculation of chemical capacitance in NC solar cells (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +41-44-632 66 54. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge funding from an ETH research grant and the Swiss National Science Foundation through the NCCR Quantum Science and Technology. TEM microscopy was performed at ScopeM, ETH Zurich.

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