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Inverse Temperature Dependence of Charge Carrier Hopping in Quantum Dot Solids Rachel H. Gilmore,† Samuel W. Winslow,† Elizabeth M. Y. Lee,† Matthew Nickol Ashner,† Kevin G. Yager,§ Adam P. Willard,*,‡ and William A. Tisdale*,† Department of Chemical Engineering and ‡Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States § Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, United States ACS Nano 2018.12:7741-7749. Downloaded from pubs.acs.org by STOCKHOLM UNIV on 11/14/18. For personal use only.
†
S Supporting Information *
ABSTRACT: In semiconductors, increasing mobility with decreasing temperature is a signature of charge carrier transport through delocalized bands. Here, we show that this behavior can also occur in nanocrystal solids due to temperature-dependent structural transformations. Using a combination of broadband infrared transient absorption spectroscopy and numerical modeling, we investigate the temperature-dependent charge transport properties of well-ordered PbS quantum dot (QD) solids. Contrary to expectations, we observe that the QD-to-QD charge tunneling rate increases with decreasing temperature, while simultaneously exhibiting thermally activated nearest-neighbor hopping behavior. Using synchrotron grazing-incidence small-angle X-ray scattering, we show that this trend is driven by a temperature-dependent reduction in nearest-neighbor separation that is quantitatively consistent with the measured tunneling rate. KEYWORDS: nanocrystal, lead sulfide, temperature-dependent transport, transient absorption, thermal expansion, superlattice n inorganic and molecular semiconductors,1−16 charge transport typically proceeds through one of two mechanistic modes. Disordered or weakly coupled semiconducting materials, such as quantum dot (QD) solids, tend to exhibit an incoherent hopping mechanism, whereby transport proceeds through discrete site-to-site (e.g. QD-to-QD) transitions of localized charge carriers. Ordered or strongly coupled semiconducting materials, such as crystalline solids, tend to exhibit a band-like mechanism, whereby transport proceeds through ballistic motion interrupted by phonon scattering events. These two modes of charge transport (i.e., hopping or band-like) are often distinguished based on their implied temperature dependence: lowering temperature tends to enhance band-like transport (due to a decrease in the frequency of phonon scattering events)3−5,10 but impede hopping transport (due to a reduction in the available thermal energy).1,2,7 In QD solids, charge transport requires the tunneling of electrons between pairs of individual QDs.1−3,7,17−26 The rate for this microscopic tunneling process depends primarily on two components: an energetic component, which is determined by the overlap of electronic energy levels, and a distance component, which is determined by the overlap of the electronic wave functions.3,17,21,25,27 In this study, we examine how these two components combine to determine the temperature dependence of charge transport in ethanethiol-treated PbS QD solids. Specifically, we apply transient absorption (TA) spectroscopy, grazing-incidence small-angle X-ray scattering (GISAXS), and numerical modeling to show that these two components can have an opposing influence on the temper-
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© 2018 American Chemical Society
ature-dependent charge transport properties. The energetic component of the tunneling rate is thermally activated, driven by nuclear fluctuations, and is thus reduced upon lowering of temperature. At the same time, however, we show that temperature-induced variations of the QD superlattice (a reduction in QD nearest-neighbor separation of 1−2 Å as temperature is reduced from 300 to 150 K in ethanethiol-treated PbS) can have a significant effect on the distance component of the tunneling rate. We demonstrate that in energetically homogeneous PbS QD solids (site energy disorder ε o k exp o jj zz j i o k T B k(jiα) = o m k { o o o o o o k(α) ; εj ≤ εi o n
d (α) Pi (t ) = dt (1)
∑ kij(α)P(j α) − ∑ k(jiα)Pi(α) j≠i
j≠i
(2)
where P(α) i (t) denotes the probability to find an electron (α = “−”) or a hole (α = “+”) on QD i at time t. For the QD solids and temperature range (150−300 K) considered here, transport occurs via nearest-neighbor hopping. We do not observe a transition to variable range hopping, nor do we expect to; the transition from nearest neighbor to variable range hopping
(α)
where k is the phenomenological base hopping rate of a free charge carrier (α = “−” or “+” for an electron or hole, respectively), εi is the energy of the ith QD, kB is the Boltzmann constant, and T is the temperature. We choose to use the Miller−Abrahams hopping model, rather than Marcus rate 7743
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ACS Nano typically occurs at temperatures under 100 K,1 colder than the conditions studied here. We simulate charge carrier dynamics by solving the master eq (eq 2), assuming a uniform initial condition, that is, Pi = 1/N, where N is the total number of QDs in the system. The hopping rate depends on temperature according to eq 1, however, we assume that the inhomogeneous broadening, σinh, is temperature independent (Supplementary Note 3).29 We solve for P(−) and i P(+) i separately and compute the TA spectrum by assuming that both species of charge carriers contribute equally to determining the band edge bleach. Mobile free charge carriers are the product of thermally activated exciton dissociation.18 The exciton dissociation rate is given by an Arrhenius expression: ji E zy kdiss = A expjjj− a zzz j kBT z k {
bleach energy contributed by a QD occupied by an electron− hole pair), which is expected to be a few meV in PbS.26,35 While this unknown parameter affects our model prediction of the equilibrium energy of charge carriers, it does not substantively affect our estimation of the charge carrier hopping rate, which is determined by the kinetics of the redshift. We compare our model to the results of the temperaturedependent TA data (plotted in Figure 1d) by fitting the values of σinh, k(−), and k(+) to maximize the overlap between simulated and experimental data. We note that our model cannot distinguish whether electron or hole transport is greater, and previous reports in the literature show that either case is possible depending on QD size and superlattice structure.17,21,36 We thus denote the two hopping rates k(1) and k(2) and take k(1) ≥ k(2). As illustrated in Figure 2b, our model results are in good agreement with experimental measurements. As Figure 2c,d illustrates, our numerical model indicates that the base hopping rates for electrons and holes (i.e., k(1) and k(2)) are both decreasing functions of temperature. Similarly, our model reveals that the mean equilibrium hopping rate, ⟨k(α) hop⟩, increases with decreasing temperature. These findings are contrary to expectations for a thermally activated site-to-site hopping process. Even for a perfectly homogeneous QD solid, we would still expect transport to be a thermally activated process due to the QD charging (i.e. solvation) energy.1,37 To understand our model results, we consider a general expression for a tunneling rate constant:
(3)
where A is the attempt frequency and Ea is the activation energy for exciton dissociation, which is related to the exciton binding energy and the relative energy of neighboring QDs. We can estimate the fraction of dissociated excitons in our sample from experimental measurements of the photoluminescence (PL) intensity, assuming dissociated charge carriers recombine nonradiatively, as illustrated in Figure 2a.18,32 The fraction of excitons that contribute to PL as a function of temperature can be expressed as18 η(T ) =
kr k r + k nr + kdiss
ij E yz k(α)(T ) = ko(α) exp(−β S)expjjj− C zzz j kBT z k {
1
= 1+
k nr kr
+
A kr
( ) E
exp − k Ta B
(4)
where kr is the radiative recombination rate and knr is the total recombination rate via nonradiative pathways other than exciton dissociation. In this expression, kr and knr are assumed to be temperature independent. Fitting eq 4 to the normalized PL intensity as a function of temperature (Figure 2a) gives an activation energy for exciton dissociation of 58 meV, consistent with previously reported values.18,32 The function fdiss(T) = 1 − (1 + knr/kr)η(T) determined by analysis of the temperaturedependent PL data (Figure 2a) is subsequently used as an independent input to the charge transport model, described below. The exciton hopping time constant, as given by Förster resonant energy transfer (FRET), is estimated to be approximately tens of nanoseconds in our PbS QD solids (Supplementary Note 5).33,34 Consequently, undissociated excitons contribute an essentially static bleach to the ensemble TA spectrum at their initially excited energy, ε,̅ on the time scale of the TA measurement. The simulated ensemble TA bleach peak energy is then given by ⟨E(t )⟩ = (1 − fdiss )ε ̅ + −
fdiss 2
(⟨E(−)(t )⟩ + ⟨E(+)(t )⟩)
where S is the tunneling distance, β is the effective tunneling barrier (β ∼ 0.9−1.1 Å−‑1 for alkanethiol ligands),17,18 and EC is the activation energy associated with QD charging. Typically, the first term in this expression is assumed to be independent of temperature, leading to an overall hopping rate expression that increases with temperature, contrary to our model predictions. Due to this discrepancy, we hypothesize that the distance component may exhibit temperature dependence.
GLOBAL FITTING MODEL PREDICTS LATTICE CONTRACTION As the first exponential term in eq 7 clearly indicates, small changes in the interparticle spacing can have a large impact on the charge carrier tunneling rate. The interparticle spacing in QD solids is determined in part by the layer of ligands that passivate the QD surface. Ligand configurations and their collective packing arrangements can depend sensitively on temperature. This dependence can lead to variation in superlattice structure as a function of temperature, as was recently observed in colloidal gold nanocrystal superlattices.38 If we assume that the phenomenological base hopping rates, k(1) and k(2), have the form given in eq 7, we can infer the temperature dependence of S by performing a global fit. Specifically, we assume a temperature-independent base tunneling rate of β = 1.1 Å−1,17 and a charging energy derived from the capacitance of a sphere, EC = 0.35e2/4πεε0d, where e is the elemental charge, ε is the dielectric constant of the QD solid, ε0 is the vacuum permittivity, and d is the QD diameter.1,37 Conceptually, the charging energy is similar to a solvation energy and represents the energy released by an electron polarizing its environment. Furthermore, we assume that the
(5)
The quantities ⟨E (t)⟩ and ⟨E (t)⟩ represent the average energies of the electron and hole populations, respectively, and can be computed by evaluating the expression: +
N
⟨E(α)(t )⟩ =
∑ εiPi(α)(t ) i=1
(7)
(6)
We note that our model neglects to account for the difference in energy between the trion (TA bleach energy contributed by a QD occupied by a single charge carrier) and biexciton (TA 7744
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Figure 3. Results of global fitting analysis. (a−c) Global model fits to the TA data. (d) Predicted temperature-dependent change in edge-to-edge spacing based on the model. (e−j) Calculated base hopping rate (open symbols) and the thermally averaged per-pair total hopping rate (closed symbols) for the 4.1 nm (e, f), 5.0 nm (g, h), and 5.8 nm (i, j) QDs.
Figure 4. Temperature-dependent lattice distortion of an ordered superlattice of 5.7 nm QDs with oleic acid ligands. (a) GISAXS patterns at 298 K showing a fcc lattice and (b) 133 K showing a bct lattice. (c) Change in neighbor center-to-center spacings with temperature for a and b axes (black, open circles), c axis (blue, open squares), d110 direction (red, open triangles), and nearest neighbors (purple crosses). (d, e) Schematics showing the superlattice distortion as a function of temperature with fcc unit cell (blue) and bct unit cell (red).
where S(300K) is the edge-to-edge spacing determined based on the known interparticle spacing at 300 K and c is a fitting parameter. The full, explicitly temperature-dependent hopping rate given in eq 1 is thus
inter-QD spacing in the superlattice depends linearly on temperature, following the form: S(T ) = S(300K) + c(T − 300K)
(8)
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Figure 5. Calculated mobilities μ(1) (a) and μ(2) (b) from KMC simulations based on the global hopping model fit. l o ij E yz ij (εj − εi)/2 yzz o o o zz ; εj > εi ko(α)exp( − β(S(300 K) + c(T − 300 K)))expjjjj− C zzzzexpjjjj− o o zz o j kBT o o k kBT { k { k(jiα) = o m o o o y i o jj EC zz o (α) o ; εj ≤ εi o o ko exp(− β(S(300 K) + c(T − 300 K)))expjjj− k T zzz o o k B { n
We use eq 9 with eqs 2 and 5 to compare our model to (2) experimental data, fitting the parameters σinh, k(1) 0 , k0 , and c. The results of the global fit indicate that the superlattice undergoes a slight contraction, with c ≈ 1 pm/K, over the range of temperature that we considered. This gives a linear thermal expansion coefficient of 1.2 × 10−4 K−1, which is an order-ofmagnitude larger than that of the PbS lattice (2 × 10−5 K−1),39 and similar to that of many polymers such as polyethylene (1−2 × 10−4 K−1) and polystyrene (0.7 × 10−4 K−1).40 A comparison of simulated and measured TA data for each QD solid is shown in Figure 3a−c, and the corresponding value of S(T ) is shown in Figure 3d. The fitted values of the phenomenological base and k(2) hopping rates, k(1) 0 0 , and the resulting equilibrium ⟩, are plotted in Figure 3e−j. In all cases the hopping rate, ⟨k(α) hop base hopping rates increase with decreasing temperature, and for the medium and large QDs, the mean equilibrium hopping rate does as well. In the smallest QDs with largest inhomogeneous line width (i.e., largest site energy disorder), the mean equilibrium hopping rate initially increases as temperature is reduced, but then begins to decrease as temperature is reduced further because there is insufficient thermal energy to overcome the inhomogeneous broadening in the ensemble. The global fit predicts a decrease of 0.9−1.7 Å in the edge-to-edge spacing when the temperature decreases from 300 to 150 K (Figure 3d). The lattice contraction at 150 K predicted by our global fitting analysis represents 10−15% of the edge-to-edge spacing at 300 K (Supplementary Note 6). There are two possible origins of this superlattice contraction: (1) contraction of the PbS nanocrystal and (2) contraction of the passivating ligand shell. The thermal expansion coefficient of bulk PbS is too small, by an order-of-magnitude, to account for the predicted superlattice contraction.39 We therefore hypothesize that the change in edge-to-edge spacing is due to temperature-dependent variations in the structure of the ligand shells. Grazing-incidence small-angle X-ray scattering (GISAXS) experiments confirm that lattice distortion upon cooling results in a reduction of nearest-neighbor separation (Figure 4). GISAXS patterns were collected cooling from 298 to 133 K using 5.7 nm diameter oleic acid capped QDs. Figure 4a,b show
(9)
the initial and final scattering patterns. Indexing the peaks41 reveals a lattice distortion in which an initial face-centered cubic (fcc) lattice expands along the a and b axes while contracting along the c axis. The resulting superlattice is a body-centered tetragonal (bct) lattice with a reduction of 5.2 Å in nearestneighbor spacing overall (Figure 4c). Oleic acid ligands are longer than the ethanethiol ligands used in the TA experiments. We expect the reduction in neighbor spacing upon distortion to be smaller in these superlattices, which is consistent with the model prediction that only a reduction of 0.9−1.7 Å is needed to explain the TA data in these samples. Attempts at capturing a similar distortion using ethanethiol ligand-exchanged QDs were unable to differentiate between lattice types due to lack of higher-order scattering peaks (see Supplementary Note 7 and Figure S5).
CHARGE CARRIER MOBILITY INCREASES WITH DECREASING TEMPERATURE With our parametrized model of charge transport in PbS QD solids, we can simulate the dynamics of charge carriers in these systems. Specifically, we compute electron and hole mobility based on the diffusion constant for each species, which we extract from the simulated mean-squared displacement of individual excited charge carriers following thermalization to a Boltzmann distribution. The mobilities of the two charge carrier types are plotted in Figure 5. We observe that the charge carrier mobility increases with decreasing temperature, indicating that the hopping enhancement due to decreased interparticle spacing outweighs any decrease due to Arrhenius thermally activated processes. Increasing mobility with decreasing temperature is expected for band-like transport, as occurs in bulk semiconductors. However, whether demonstrating dμ/dT < 0 is sufficient evidence to claim the observation of band-like transport is the subject of significant debate.3,7 It has previously been suggested that hopping transport with low activation energy, Ea, can give dμ/dT < 0 at higher temperatures because of the temperature dependence of the prefactor in the hopping mobility expression 7746
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inversely proportional to the excitation density.6 Epitaxially connected QDs retain defined absorption peaks that redshift by a few to a few tens of meV relative to unattached QDs,6,43−45 which is similar to the optical properties of other QD solids exhibiting possible band-like transport. However, epitaxially connected superlattices have also been shown to have electron and hole localization lengths on the order of a few QD diameters,21 and local mobilities measured by terahertz spectroscopy that are an order of magnitude greater than the DC mobilities.44 Thus, while some delocalization exists in these materials, long-range delocalization is probably limited by structural disorder and missing connections in the epitaxially connected solid.21,44
derived from Einstein’s relations between diffusivity and mobility: μ hop =
ij ed 2Ea E yz expjjj−βl − a zzz j 3ℏkBT kBT z{ k
(10)
where e is the unit charge, d is the center-to-center distance between QDs, and ℏ is Planck’s constant.7 Here, we show that the nearest-neighbor edge-to-edge spacing, l, may also be a function of temperature, so the observation of dμ/dT < 0 for hopping transport may extend to lower temperatures, depending on the lattice transformations present. To the best of our knowledge, previous studies of charge transport in similar PbS and PbSe QD solids with short organic ligands have not shown negative temperature dependence under single exciton excitation conditions. There are two key aspects to our work that allowed us to make this observation. First, our QD solids are energetically homogeneous. A measured site energy disorder of only ∼17 meV (Figure 3) was required to observe increasing total hopping rate and mobility with decreasing temperature over the entire 150−300 K temperature range. QD solids with just 49 meV site energy disorder showed a maximum hopping rate and mobility at 240 K, even though the phenomenological base hopping rate still increases at lower temperatures. If all site energy disorder is attributed to variation in QD diameter, then these energetic disorder numbers correspond to estimated QD size dispersity of
