Inverse Temperature Dependence of Charge Carrier Hopping in

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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 ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.8b01643 • Publication Date (Web): 21 Jun 2018 Downloaded from http://pubs.acs.org on June 21, 2018

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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*‡, 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 * [email protected], [email protected] KEYWORDS: nanocrystal, lead sulfide, temperature-dependent transport, transient absorption, thermal expansion, superlattice

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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 (GISAXS), 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.

TOC figure:


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In inorganic and molecular semiconductors,1-16 charge transport typically proceeds through one of two mechanistic modes. Disordered or weakly coupled semiconducting materials, such as 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 quantum dot (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 wavefunctions.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 Xray scattering (GISAXS), and numerical modeling to show that these two components can have an opposing influence on the temperature-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 temperatureinduced variations of the QD superlattice (a reduction in QD nearest neighbor separation of 1–2 Å as temperature is reduced from 300 K to 150 K in ethanethiol-treated PbS) can have a


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significant effect on the distance component of the tunneling rate. We demonstrate that in energetically homogeneous PbS QD solids (site energy disorder = () ; ≤ .

(1)

where () is the phenomenological base hopping rate of a free charge carrier ( = “–” or “+”

for an electron or hole, respectively), is the energy of the ith QD, B is the Boltzmann constant, and is the temperature. We choose to use the Miller-Abrahams hopping model, rather

than Marcus rate expressions, because it requires fewer fitting parameters and was previously shown to sufficiently capture the salient features of nonequilibrium charge transport in this QD system.26 We define the hopping rate separately for electrons and holes to account for the possibility of unequal charge carrier mobilities.17 The factor of 2 in the denominator of the exponential term in Equation 1 is included based on the assumption that variation in the band gap energy is shared equally between the electron and hole levels, and that band gap inhomogeneity is the primary source of energetic disorder in the electron and hole transport levels.26 Because of


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the low excitation density used in these measurements (0.03–0.1 photons absorbed per QD per pulse), we assume that charge carriers do not interact. Thus, we explicitly neglect recombination, the difference between the trion and biexciton redshift, and other multi-carrier interaction effects. The consistent transient redshifts observed as a function of excitation power (Figure S3, Supplementary Note 4) confirms the assumption that we are not in a regime in which multicarrier effects have a significant impact on hopping dynamics. We express the time dependent population of charge carriers on the ith QD in terms of the master equation,31

where

() " (!)

!

"

()

() () (!) = # " − # " $

$

() ()

,

(2)

denotes the probability to find an electron ( = “–”) or a hole ( = “+”) 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 typically occurs at temperatures under 100K,1 colder than the conditions studied here. We simulate charge carrier dynamics by solving the master equation (Eq. 2), assuming a

uniform initial condition, i.e., " = 1/' , where ' 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, inh, is temperature independent (Supplementary Note 3).29 We solve for "

(+)

and "

(,)

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,

- .. = / exp 0−

12 4, 3

(3)


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where / is the attempt frequency and 12 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 Fig. 2a.18,32 The fraction of excitons that contribute to PL as a function of temperature can be expressed as33

5() =

6

6 + 86 + - ..

=

1 , / 1 1 + 86 + exp 9− 2 : 6 6 3

(4)

where 6 is the radiative recombination rate and 86 is the total recombination rate via

nonradiative pathways other than exciton dissociation. In this expression, 6 and 86 are assumed to be temperature-independent. Fitting Eqn. 4 to the normalized PL intensity as a function of temperature (Fig. 2a) gives an activation energy for exciton dissociation of 58 meV,

consistent with previously reported values.18,32 The function ;- .. () = 1 − (1 + 86 ⁄6 )5() determined by analysis of the temperature-dependent PL data (Fig. 2a) is subsequently used as an independent input to the charge transport model, described below.


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Figure 2. (a) Relative photoluminescence intensity is used to estimate the fraction of dissociated excitons as a function of temperature. (b) Master equation model fits compared to experimental data for the 5.0 nm QDs at selected temperatures. (c-d) Model results for the phenomenological base hopping rate constant (open circles) and the thermally-averaged per-pair total hopping rate (closed circles) for the 5.0 nm QDs at selected temperatures.

The exciton hopping time constant, as given by Förster resonant energy transfer (FRET), is estimated to be ~tens of nanoseconds in our PbS QD solids (Supplementary Note 5).34,35 Consequently, undissociated excitons contribute an essentially static bleach to the ensemble TA

spectrum at their initially excited energy, ̅, on the timescale of the TA measurement. The simulated ensemble TA bleach peak energy is then given by,

〈1(!)〉 = (1 − ;?@@ )̅ +

;?@@ A〈1 (+) (!)〉 + 〈1 (,) (!)〉B. 2

(5)

The quantities 〈1 + (!)〉 and 〈1 , (!)〉 represent the average energies of the electron and hole populations, respectively, and can be computed by evaluating the expression, 10 ACS Paragon Plus Environment

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C

〈1 () (!)〉 = # " () (!). DE

(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 bleach energy contributed by a QD occupied by an electron-hole pair), which is expected to be a few meV in PbS.26,36 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 temperature-dependent TA data (plotted in Fig. 1d)

by fitting the values of inh, (+) , and (,) 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.17,21,37 We thus denote the two hopping rates (E) and (F) and

take (E) ≥ (F) . As illustrated in Fig. 2b, our model results are in good agreement with

experimental measurements. As Fig. 2c and d illustrate, our numerical model indicates that the

base hopping rates for electrons and holes (i.e., (E) and (F) ) are both decreasing functions of

() temperature. Similarly, our model reveals that the mean equilibrium hopping rate, 〈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.1,38 To understand our model results, we consider a general expression for a tunneling rate constant,


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() () = I exp(−Jℓ) exp 0− ()

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1L 4

(7)

where ℓ is the tunneling distance, J is the effective tunneling barrier (β ~ 0.9–1.1 Å–-1 for

alkanethiol ligands),17,18 and 1L 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. (77 clearly indicates, small changes in the inter-particle spacing can have a large impact on the charge carrier tunneling rate. The inter-particle 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 super-lattice structure as a function of temperature, as was recently observed in colloidal gold nanocrystal superlattices.39

If we assume that the phenomenological base hopping rates, (E) and (F) , have the form given

in Eq. 7, we can infer the temperature dependence of ℓ by performing a global fit. Specifically,

we assume a temperature-independent base tunneling rate of J = 1.1 Å-1,17 and a charging

energy derived from the capacitance of a sphere, 1L = 0.35P F ⁄4RS , where e is the elemental

charge, ε is the dielectric constant of the QD solid, S is the vacuum permittivity, and d is the QD diameter.1,38 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 inter-QD spacing in the super-lattice depends linearly on temperature, following the form, 12 ACS Paragon Plus Environment

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ℓ() = ℓ(300T) + U( − 300T),

(8)

where ℓ(300T) is the edge-to-edge spacing determined based on the known inter-particle

spacing at 300 K and U is a fitting parameter. The full, explicitly temperature-dependent hopping rate given in Eqn. 1 is thus = ()

Y () exp 9−JAℓ(300T) + U( − 300T)B: exp 0− 1L 4 exp − ( − )/2 ; W I

1L X () 4 ; W I exp 9−JAℓ(300T) + U( − 300T)B: exp 0− V

>

≤ .

(9)

We use Eqn. 9 with Eqns. 2 and 5 to compare our model to experimental data, fitting the parameters inh , S , S , and c. (E)

(F)

The results of the global fit indicate that the super-lattice undergoes a slight contraction,

with U ≈ −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-of-magnitude larger than that of the PbS lattice (2×10–5 K–1),40 and similar to that of many polymers such as polyethylene (1–2×10–4 K–1) and polystyrene (0.7×10–4 K–1).41 A comparison of simulated and measured TA data for each QD

solid is shown in Fig. 3 a-c, and the corresponding value of ℓ() is shown in Fig. 3d. The fitted

values of the phenomenological base hopping rates, S and S , and the resulting equilibrium (E)

(F)

() hopping rate, 〈hop 〉 are plotted in Figure 3e-j. In all cases the 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 linewidth (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


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decrease of 0.9–1.7 Å in the edge-to-edge spacing when the temperature decreases from 300 K 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 super-lattice 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 super-lattice contraction.40 We therefore hypothesize that the change in edge-to-edge spacing is due to temperature-dependent variations in the structure of the ligand shells.

Figure 3. Results of global fitting analysis. (a-c) Global model fits to the transient absorption 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.


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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 K to 133 K using 5.7 nm diameter oleic acid capped QDs. Figure 4a and 4b show the initial and final scattering patterns. Indexing the peaks42 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 nearest neighbor 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 ligandexchanged QDs were unable to differentiate between lattice types due to lack of higher-order scattering peaks (see Supplementary Note 7 and Figure S5).


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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 face-centered cubic lattice and (b) 133 K showing a body-centered tetragonal 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).

Charge Carrier Mobility Increases with Decreasing Temperature With our parameterized 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


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that the hopping enhancement due to decreased inter-particle spacing outweighs any decrease due to Arrhenius thermally activated processes.

Figure 5. Calculated mobilities [ (E) (a) and [ (F) (b) from KMC simulations based on the global hopping model fit.

Increasing mobility with decreasing temperature is expected for band-like transport, as occurs

in bulk semiconductors. However, whether demonstrating [⁄ \ 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

[⁄ \ 0 at higher temperatures because of the temperature dependence of the prefactor in the

hopping mobility expression derived from Einstein’s relations between diffusivity and mobility,

[]I^ =

P F 12 12 exp 0−J` − 4 3_3 3

(10)

where e is the unit charge, d is the center-to-center distance between QDs, and h 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 [⁄ \ 0 for hopping transport may extend to lower temperatures, depending on the lattice transformations present.


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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 10 meV (Fig. 3) was required to observe increasing total hopping rate and mobility with decreasing temperature over the entire 150K-300K temperature range. QD solids with just ~17 meV site energy disorder showed a maximum hopping rate and mobility at 240K, even though the phenomenological base hopping rate still increases at lower temperatures. If all site energy disorder is attributed to variation in QD diameter, these energetic disorder numbers correspond to estimated QD size dispersity of

Inverse Temperature Dependence of Charge Carrier Hopping in

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