Disordered Mott-Hubbard Physics in Nanoparticle Solids: Persistent Gap Across the Disorder-localized-to-Mott-localized Transition

TL;DR

This paper introduces a new simulation tool and theoretical analysis to understand the persistent electronic gap in nanoparticle solids across disorder and Mott localization transitions.

Contribution

The study develops HINTS, a hierarchical transport simulator, and applies Dynamical Mean Field Theory to analyze disorder and correlation effects in nanoparticle solids.

Findings

HINTS accurately models nanoparticle solid transport properties.

The phase diagrams from HINTS and DMFT are consistent.

The persistent gap remains across the disorder-localized-to-Mott-localized transition.

Abstract

We show that Nanoparticle (NP) solids are an exciting platform to seek new insights into the disordered Mott-Hubbard physics. We developed a "Hierarchical Nanoparticle Transport Simulator" (HINTS), which builds from localized states to describe the Disorder-localized and Mott-localized phases, and the transitions out of these localized phases. We also studied the interplay between correlations and disorder in the corresponding multi-orbital Hubbard model at and away from integer filling by Dynamical Mean Field Theory. This approach is complementary to HINTS, as it builds from the metallic phase of the NP solid. The mobility scenarios and phase diagrams produced by the two methods are strikingly similar, and account for the mobilities measured in NP solids.

Figures

Full text

Nanoparticle (NP) solids are aggregates of nanometer scale particles with interesting and potentially useful emerging electronic functionalities. In NP solids, the wavefunctions are typically ”quantum confined” to the NPs, making their electronic properties tunable with the NP size Kovalenko et al. (2015). This tunability makes them attractive for a wide variety of opto-electronic applications Talapin et al. (2010), including photovoltaics Nozik (2002); Kamat (2008), light-emitting diodes Shirasaki et al. (2013), and field-effect transistors (FETs) Talapin  and Murray (2005); Hetsch et al. (2013). However, the same Quantum Confined localization also drives the NP solids insulating, hindering charge transport and thus their utility. Therefore, driving NP solids from their insulating phase across a Metal-Insulator Transition (MIT) into a conducting, metallic phase is a top priority to boost their utility. Recent experimental attempts to cross the MIT included atomic layer deposition (ALD) Liu et al. (2013), substitutional percolation Cargnello et al. (2015), chemical doping Chen et al. (2016); Choi et al. (2012), and photodoping Talgorn et al. (2011). ALD infilling with metal oxides already enhanced mobilities above 7 cm2/Vs Liu et al. (2013). Whether these enhanced-mobility NP solids support coherent metallic transport is still debated. Building on these advances, a few groups developed NP solar cells with 13-16% power conversion efficiencies Lan et al. (2016); Jiao et al. (2017); Sanehira et al. (2017), which is quite promising for their commercial viability.

There is a vast literature on the theory of the MIT in disordered systems Imada et al. (1998). For NP solids, Ref. Chen et al. (2016) developed insightful theories. Transport was described as nearest neighbor hopping at high T Yu et al. (2004); Liu et al. (2010a), and as Efros-Shklovskii (ES) variable range hopping at low T Efros  and Shklovskii (1976). The interplay of transport and Coulomb interactions was studied in Reich et al. (2014); Chandler et al. (2007). The concept of the ”Coulomb blockade/gap” is sometimes used to refer to the energy cost of the attraction between the hopping electron and the hole it leaves behind,Schmid (2006). This Coulomb blockade suppresses the transport at all fillings. Its dissolution across the MIT was analyzed in granular metals Efetov and Tschersich (2003); Beloborodov et al. (2005, 2007).

The Coulomb blockade is also used to describe an electron on an NP blocking the transport of additional electrons through that NP because of the Coulomb energy cost of double occupancy. This blockade occurs only at integer fillings. It was observed in experiments Stafford  and Das Sarma (1994); Leobandung et al. (1995); Romero  and Drndic (2005); Kang et al. (2011), and analyzed in a few theoretical papers Reich et al. (2014); Chandler et al. (2007), including ours on bilayer NP solids Qu et al. (2020). This blockade forces the finite temperature mobility to exhibit minima at integer fillings, exponentially suppressed by the Coulomb gap, and maxima between integer fillings.

Remarkably, in spite of all this progress, profound unmet needs persist in the field. (1) We are not aware of a theory of the MIT in NP solids that starts from the insulating phase. (2) In spite of inviting analogies, the connections between NP solids and the disordered Mott-Hubbard physics have been barely explored Sampaio et al. (2001); Byrnes et al. (2008); Stafford  and Das Sarma (1994); Stafford and Das Sarma (1994). For example, even though the mobility of NP solids has been observed to exhibit maxima and minima as a function of the filling, these are typically not attributed to Mott-Hubbard commensuration Kang et al. (2011).

Working on these challenges promises to pay additional synergistic benefits as well. (a) A theory of the MIT in NP solids that builds out from the insulating side can be very useful for the study of the disordered Hubbard model as well, since that is also lacking a description of the MIT from the insulating phase. And in reverse, the theoretical tools created for the Mott-Hubbard model building from the metallic phase can complement the theory of the MIT in NP solids. (b) There are accepted experimental platforms to study Mott-Hubbard models, such as strongly correlated electron materials, transition-metal oxides Dagotto (1994), especially high temperature superconductors, as well as cold atom systems in optical lattices Jaksch et al. (1998). However, in spite of decades of explorations, basic questions on the physics introduced by disorder effects remain open. Therefore, it can prove to be very helpful to advocate NP solids as a new class of experimental platforms to research the unanswered questions of disordered Mott-Hubbard physics.

To address the first need of a theoretical method to describe the MIT in NP solids building out from the localized phase, we developed the HIerarchical Nanoparticle Transport Simulator HINTS. HINTS is a Kinetic Monte Carlo transport simulator that is extended by an additional metallic transport channel. As such, it is capable of reaching the MIT from the insulating phase. In an initial report we demonstrated that HINTS can indeed reach the MIT Qu et al. (2017), which we interpreted as a Quantum Percolation Transition. However, we did not connect this MIT to Mott-Hubbard phenomena.

To address the second need, to explore connections between NP solids and Mott-Hubbard physics, we recall that the most widely-studied types of electron localization are the Mott and Anderson mechanisms. Mott localization emerges at $n$=1 electron/site filling, when the local Coulomb repulsion prevents electrons from jumping onto their occupied neighboring sites, thereby localizing electrons. Anderson, or disorder localization takes place in non-interacting systems independent of the filling, where the disorder of the site energies breaks up the site-to-site phase coherence of the originally extended wavefunctions, thereby localizing the electronsImada et al. (1998).

In systems with both site energy disorder and Coulomb interactions, the Mott and the Anderson localization scenarios interfere, as revealed by studies since the early 80s, most notably by scaling methods Finkel’shtein (1983); Altshuler et al. (1980); Castellani et al. (1987). In recent years, this research got re-energized by the adaptation of the Dynamical Mean Field Theory (DMFT)Aguiar et al. (2009); Bragança et al. (2015); Byczuk et al. (2005). An interesting and surprising prediction of DMFT was that at $n$=1, at intermediate repulsion, the Disorder-localized phase, formed at high disorder, transforms into a metal with decreasing disorder; and further decrease of the disorder transforms the metal into a Mott-insulator. In contrast, at strong repulsion decreasing disorder induces a transition from the Disorder-localized phase directly into the Mott-localized phase without an intervening metallic phase Bragança et al. (2015); Byczuk et al. (2005). Fillings $n\neq 1$ were not investigated. Notably, both the scaling and the DMFT techniques are built with extended wave functions, and thus indicate Anderson/disorder localization only as the boundary where their applicability breaks down. As such, they are ill-suited to describe the Disorder-localized phase itself.

Here are then the main messages of our paper in response of the above needs. (1) NP solids are an exciting experimental and theoretical platform to seek new insights into the disordered Mott-Hubbard physics. (2) HINTS is a powerful method to describe the various transitions starting out from the disorder-localized phase, providing a complementary approach to existing methods building from the metallic-delocalized phase. (3) For $n$=1 filling and large interactions, HINTS showed that decreasing disorder drives a disorder-localized–to–Mott-localized transition without an intervening metallic phase, i.e. a direct transition, with a persistent gap. (4) For $n\neq 1$, HINTS showed that decreasing disorder drives a Disorder-localized–to–Metal transition. (5) We modeled NP solids in their metallic phase as a multi-orbital Hubbard model, which we analyzed with a disordered DMFT technique, complementary to HINTS. This DMFT work at and away from $n=1$ determined how the mobility and the Mott gap were affected by the disorder.

We developed the HINTS code to simulate NP solids by the below-described hierarchical approach. This allowed us to tackle the computational challenges associated with the huge number of degrees of freedom. We computed the electronic energy levels by using a parameterized band structure of individual PbSe NPs with diameters in the 3-8 nm range Kang  and Wise (1997). We then generated a superlattice of NPs with diameters selected from a Gaussian distribution. The evaluation of the electron configuration energies further includes the determination of the on-site Coulomb charging energy $E_{c}$ with a hybrid empirical-perturbative method Delerue  and Lannoo (2004). The long range part of the Coulomb interaction could be optionally included as the experimental conditions warrant. We modeled the NP-NP transitions as either activated hopping or non-activated metallic transitions, depending on whether the energy difference between the initial and final states was larger or smaller than a hybridization energy. And finally, we computed the mobility of the NP solid by an Extended Kinetic Monte Carlo (KMC) algorithm with a sufficiently small voltage bias. Details of HINTS are provided in the Supplementary Material. We move to presenting the results of the HINTS simulations.

We first consider the Disorder-localized–to–Mott-localized transition as a function of the filling $n$. Two experimental groups measured the dependence of the mobility on the NP diameter in NP-FETs, and reported an initial rise followed by a maximum Liu et al. (2010b); Kang et al. (2011). In previous work we simulated NP solids with increasing NP diameters and found that their mobility also exhibited a maximum Carbone et al. (2013), consistent with the experiments Liu et al. (2010b). However, we did not develop an explanation of the results.

We now extend our previous work by using HINTS to compute the conductivity of NP solids as a function of NP diameter well past the maximum. Fig. 1 shows the HINTS-computed mobilities with increasing NP diameter, overlaid on the experimental data of Kang et al. on PbSe NP FETs Kang et al. (2011). The HINTS mobilities and the Kang et al. data exhibit remarkable agreement over the extended diameter range. Similar mobilities were reported by the Law group Liu et al. (2010b).

Notably, the $n$ filling/NP increases with the NP diameter at a fixed volumetric electron density. Conspicuously, the conductivity maximum at a NP diameter of 6 nm is observed to be the beginning of a maximum-minimum-increase pattern, centered on the $n=1$ filling. Interestingly, this is also the behavior expected when scanning across $n=1$ in the repulsive Hubbard model, as the Coulomb repulsion opens a Mott gap at $n=1$, suppressing the transport through the occupied NPs.

In our simulations the temperature dependence of the conductivity is activated for all fillings, having a smaller, disorder-induced gap away from $n=1$, boosted to a larger gap by the Coulomb blockade at $n=1$. Thus, scanning with $n$ through $n=1$ crosses from a Disorder-localized phase through a Mott-localized phase back to the Disorder-localized phase, again as expected in the disordered Hubbard model. These remarkable correspondences between experimental data and our simulations are compelling evidence that NP solids are well-controlled and tunable experimental realizations of the disordered Hubbard model.

Changing the NP diameter also varies other parameters. This convolutes crossing the Mott-localized phase with other trends, such as changing site energies and hopping transition rates. Therefore, next we isolate the mobility’s dependence on the filling without impacting the other parameters by analyzing how the mobility of a fixed diameter NP solid varies when only the filling is tuned. Experimentally this can be achieved by varying the gate voltage applied to a FET formed from an NP solid.

Fig. 2 shows the HINTS-simulated mobility of the NP solid as the filling $n$ was varied. As before, the mobility exhibits pronounced minima at integer fillings, accompanied by maxima close to half-integer fillings. This is a re-confirmation of the Disorder-localized-to-Mott-localized transition in the NP solids as the filling $n$ is tuned.

Fig. 3 provides a schematic guide for the various scans performed across the phase diagram. In scans 1 and 2, the filling was tuned across $n=1$, respectively shown in Figs. 1 and 2. Scan 1 is tilted to indicate the simultaneous change of the other parameters.

Next, we analyze the Disorder-localized–to–Mott-localized transition as a function of disorder $W$ around $n$=1. Fig. 4 shows the $W$ dependent gap $\Delta(W)$ at fillings $n$=1 (scan 4a), and $n=1+\delta$, $\delta=0.001$ off the integer $n$=1 (scan 4b). At $n$=1 and small disorder the gap $\Delta$ is Mott-like, with a value set by the charging energy $E_{c}$ that suppresses transport, thus creating a Mott-localized phase. With increasing disorder, the gap $\Delta$ gets renormalized to a lower value by the disorder. The physics of this lower gap region is clarified by scan 4b of $\Delta(W)$ at a filling $n=1+\delta$. The gap $\Delta(W)$ is the same for $n=1$ and $n=1+\delta$ above a critical value $W_{c}(Mott)/2t\approx 3.5$. The gap’s insensitivity to the charging energy and to commensuration identifies this phase as a Disorder-localized phase, into which the Mott-localized phase of $n$=1 transitions for $W>W_{c}(Mott)$. The relative constancy of $\Delta(W)$ in this phase is explained by the charging energy $E_{c}$ screening the disorder $W$. Clearly, the transition at $n=1$ across $W_{c}(Mott)$ is a direct one, having a persistent gap and no intervening gapless metallic phase.

A qualitatively similar direct Mott-to-Anderson transition was found by DMFT with a disorder scan in the Hubbard model at $n$=1 for $U/t>10$ Bragança et al. (2015). Thus, our work and Ref.Bragança et al. (2015) both report a direct Mott-localized–to–Disorder-localized transition with a persistent gap as $W$ crosses a $W_{c}(Mott)$ in the disordered Hubbard model at $n$=1. Next, we go beyond previous work by exploring the filling dependence away from $n=1$.

Scan 4b at $n=1+\delta$ shows that as the disorder $W$ decreases, the gap shrinks to zero at $W_{c}(MIT)/2t\approx 1.5$. HINTS does not track the phases of the electron wavefunctions, and thus describes the transport with decreasing disorder only to $W=W_{c}(MIT)$. We identify $W_{c}(MIT)$ as the Metal-Insulator Transition of the $incommensurate$ NP solid from the Disorder-localized phase to a metal. This MIT at $n=1+\delta$ is disorder-driven, as opposed to the interaction-driven Mott transition at $W=W_{c}(Mott)$ at $n=1$.

Our results also mean that a filling scan across $n=1$ at a fixed disorder $W$ between $W_{c}(MIT)$ and $W_{c}(Mott)$ induces a filling-driven Disorder-localized–to–Mott-localized transition with a persistent gap, another new result beyond the scope of previous works Bragança et al. (2015).

We now turn to the Disordered-metal–to–Mott-localized transition as function of filling $n$. We already established that the physics of the NP solids in the localized phases is analogous to the disordered Hubbard model. Now we complete the analogy by describing the metallic phase of the NP solid as a Hubbard model as well whose sites represent the individual NPs. We put the NP solid on a regular lattice to start from a bona fide metallic state and explore the combined effect of Coulomb repulsion and site energy disorder. This approach is complementary to HINTS, which starts from localized states. We use similar parameters as before by associating the on-site Hubbard repulsion $U$ with the NP charging energy to take $U=E_{c}=100meV$, and the kinetic term with the NP-NP hopping amplitude of $t=7meV$, making $U/2t=7$. We take into account the degeneracy of the electronic states of the PbSe NPs An et al. (2006) by adopting a 8-fold degenerate multi-orbital Hubbard model Rozenberg (1997).

Since $t/U<<1$, the Hubbard model is in strong coupling. Therefore, we adopt the Dynamical Mean Field Theory (DMFT)Georges et al. (1996). DMFT proved very successful for the study of the interplay between correlation effects and the Mott-localization of an electron band, producing many useful insights into the Mott metal-insulator transition. Crucially, DMFT can also be extended to include disorder effects Aguiar et al. (2009); Bragança et al. (2015); Byczuk et al. (2005) (see Sup. Mat. for details).

The DMFT approach maps the original lattice model onto an auxiliary quantum impurity model supplemented with a self-consistency condition. The quantum impurity problem is then tackled by numerical simulations that we performed using the continuous-time quantum Monte Carlo (CTQMC) methodGull et al. (2011), described in Ref. Werner et al. (2009). That method samples a diagrammatic expansion of the partition function in powers of the impurity-bath hybridization. For simplicity, for the non-interacting electrons we adopt a semi-circular density of states of bandwidth $4t$. The disorder is introduced through the site energy random variable (diagonal disorder) with a uniform distribution in the energy interval $[-W,W]$. (see Sup. Mat. for details.)

Fig. 5 shows the electron mobility and the chemical potential $\mu$ of this Hubbard model as function of the filling $n$ for the clean and disordered cases (scans 5a/5b in Fig.3). In the clean case, $\mu$ shows a jump at $n=1$: this indicates the emergence of a Mott gap that localizes the electronsRozenberg (1997). The Mott gap makes the mobility exhibit a minimum as the filling crosses $n$=1 (i.e. band filling=1/8, well below half filling). Clearly, DMFT established that the Mott gap/Coulomb blockade also suppresses the mobility in the metallic phase when the filling is scanned across $n=1$, just like in the insulating phase (cf. Fig. 2). This confirms and completes the phase diagram obtained by HINTS that was built from the localized phase.

A main result of the DMFT study is that the clean trends persist even for a substantial disorder $W/2t=1$. Indeed, the Mott gap (jump of $\mu$) at $n=1$ only decreases by a small amount and remains robust. This is reasonable since $W/2t=1<<W_{c}(Mott)/2t=3.5$. At the same time, the mobility away from $n=1$ is reduced notably because the $W/2t=1$ disorder is relatively close to the off-commensuration MIT at $W_{c}(MIT)/2t=1.5$. Finally, Fig. 5 shows that the relative reduction of the mobility by the disorder is strongest at small $n$. This makes physical sense, since at small $n$ the Fermi energy becomes comparable to the disorder, and thus the relative importance of the disorder grows. To our knowledge, the present study of disorder-DMFT is the first one done for a multi-orbital model as a function of electron filling.

One of the most appealing results of our study is that despite HINTS and disorder-DMFT describing the MIT from dual opposite starting points, they produce qualitatively the same scenarios. This is well illustrated by the strikingly similar Figs.2 and 5. The discrepancy between the magnitudes of the insulating and metallic mobilities is due to the insulating phase having a gap that exponentially suppresses the mobility, whereas the metallic phase lacking this suppression factor. Reassuringly, experiments often report orders of magnitude changes when the mobility crosses the MITImada et al. (1998).

To conclude, we argued that NP solids are an exciting platform to seek new insights into the disordered Mott-Hubbard physics. We presented HINTS as a new powerful method to describe the various transitions from the Disorder-localized phase; and showed how the DMFT method can complement these studies from the delocalized phase. In particular, at $n=1$ filling, HINTS showed that at large interactions the Disorder-localized–to–Mott-localized transition occurs with a persistent gap. These results articulate an important message for how to optimize the mobility of NP-based optoelectronic and solar devices: tune the disorder below the Metal-Insulator transition, and tune the electron filling far from commensurate values. In FETs this can be done by tuning the gate voltage. In solar cells, by cobaltocene dopingKoh et al. (2013); Vörös et al. (2015).

Acknowledgments: We acknowledge helpful discussions with L. Qu, M. Voros and M. Law. This work was supported by the UC Office of the President under the UC Laboratory Fees Research Program Collaborative Research and Training Award LFR-17-477148. A.C. and J.B. gratefully acknowledge support from CONICET and UBACyT.