Commensuration effects in layered nanoparticle solids

We have developed a hierarchical nanoparticle transport simulator (hints) and adapted it to study commensuration effects in two classes of nanoparticle (NP) solids: (1) a bilayer NP solid (BNS) with an energy offset and (2) a BNS as part of a field effect transistor (FET). hints integrates the ab initio characterization of single NPs with the phonon-assisted tunneling transition model of the NP-NP transitions into a kinetic Monte Carlo based simulation of the charge transport in NP solids. First, we studied a BNS with an interlayer energy offset $\mathrm{\Delta }$, possibly caused by a fixed electric field. Our results include the following. First, in the independent energy-offset model, we observed the emergence of commensuration effects when scanning the electron filling factor (FF) across integer values. These commensuration effects were profound as they reduced the mobility by several orders of magnitude. We analyzed these commensuration effects in a five-dimensional parameter space, as a function of the on-site charging energy $E_C$, energy offset $\mathrm{\Delta }$, disorder $D$, the electron FF, and temperature $k_{B}T$. We demonstrated the complexity of our model by showing that at integer FFs commensuration effects are present in some regions of the parameter space, while they vanish in other regions, thus defining distinct dynamical phases of the model. We determined the phase boundaries between these dynamical phases. Second, using these results as a foundation, we shifted our focus to the experimentally much-studied NP-FETs. NP-FETs are also characterized by an interlayer energy offset $\mathrm{\Delta }$, which, in contrast to our first model, is set by the gate voltage $V_G$ and thereby related to the electron FF. We repeated many of our simulations and again demonstrated the emergence of commensuration effects and distinct dynamical phases in these NP-FETs. Notably, the commensuration effects in the NP-FETs showed many similarities to those in the independent energy-offset BNS.

Figure 1

Mobility in a single layer NP solid, exhibiting a clear commensuration-induced suppression. $E_C=120\mathrm{meV}$, $k_{B}T=80\text{K}$, $D(d=5.6\text{nm})=55\mathrm{meV}$, and $D(d=6.5\text{nm})=45\mathrm{meV}$.

Figure 2

The physical mechanism of the Coulomb blockade driving the commensuration-induced suppression of the mobility.

Figure 3

Illustration of a simulated bilayer nanoparticle solid.

Figure 4

Energy landscape of a BNS with an interlayer energy offset $\mathrm{\Delta }$.

Figure 5

Mobility as a function of the FF. $E_C=120\mathrm{meV}$, $k_{b}T=7\mathrm{meV}$, and $D=45\mathrm{meV}$. Energy offset $\mathrm{\Delta }$ is varied from 20 to 500 meV.

Figure 6

Energy diagrams to contextualize the various parameter regimes. (a) $\text{FF}=1$; $\mathrm{\Delta }=500$ and 100 meV. (b) $\text{FF}=1$; $\mathrm{\Delta }=40$ and 20 meV. (c) $\text{FF}=2$; $\mathrm{\Delta }=500\mathrm{meV}$. (d) $\text{FF}=2$; $\mathrm{\Delta }=100$, 40, and 20 meV. In all cases $E_C=120\mathrm{meV}$, $k_{B}T=7\mathrm{meV}$, and $D=45\mathrm{meV}$.

Figure 7

Fixed $\mathrm{\Delta }$ and different $E_C$: $\mathrm{\Delta }=500\mathrm{meV}$, $k_{B}T=7\mathrm{meV}$, and $D=45\mathrm{meV}$.

Figure 8

Dynamic phase diagram, capturing the dynamics of the BNS at the two filling factors $\text{FF}=1$ and 2 in terms of the DCM.

Figure 9

Mobility as a function of the electron filling factor FF. The four curves were generated by using four proportionality constants between the interlayer energy offset ${\mathrm{\Delta }}_{\text{FET}}$ and the FF such that ${\mathrm{\Delta }}_{\text{FET}}$ was equal to the four ${\mathrm{\Delta }}_{\text{BNS}}$ values used in Fig. 5 at $\text{FF}=1$.