Einstein relation for electron diffusion on arrays of weakly coupled quantum dots

The transport of electrons on arrays of weakly coupled quantum dots is studied using computer simulations. Distinct minima in the conductivity of the arrays are found for conditions that correspond to filled shells and to integer charging of the quantum dots when the size distribution of the dots is sufficiently small. It is found that the Fickian diffusion coefficient for electrons deviates strongly from that predicted using the Einstein relation, especially in regions where the conductivity spectrum of the array shows minima due to a Coulomb blockade or shell filling.

Figure 1

Normalized mobility (open circles) and Fickian diffusion coefficient (closed circles) as a function of average electron loading (right pane) and electrochemical potential (left pane) for an array of 30 Å diameter CdSe particles with a 5% wide (lognormal) size dispersion $({\sigma }_R=0.025)$. $\alpha$ was taken equal to 2 Å. The interparticle distance was, on average, 10 Å. Particles of 30 Å diameter have a $J=70\mathrm{meV},{ϵ}_{\mathit{\text{out}}}$ was 12, ${ϵ}_{\mathit{\text{in}}}$ was assumed to be 6. The Debye length was assumed to be 15 Å.

Figure 2

(Color online) Open circles: Thermodynamic factor $T_n[T_n=Dq∕\left(\mu kT\right)]$ as a function of the electrochemical potential for the quantum dot array described in Fig. 1. The (blue) continuous line indicates the result of calculating $T_n$ from Eq. (10). The (red) dashed lines indicate the potentials at which $n=2$ and $n=8$.

Figure 3

(Color online) Electron mobility as a function of the size dispersion parameter for an array of CdSe particles. The legend indicates the value of ${\sigma }_R$ used. $R_0$ was 15 Å; $\alpha$ was taken equal to 2 Å. The interparticle distance was 10 Å. Particles of 30 Å diameter have a $J=0.23\mathrm{eV}$ (Ref. 32), ${ϵ}_{\mathit{\text{out}}}$ was 6, ${ϵ}_{\mathit{\text{in}}}$ was assumed to be 6. The Debye length was assumed to be 15 Å.

Figure 4

(Color online) Thermodynamic factor for the data in Fig. 3. The vertical (red/dashed) lines indicate the electrochemical potentials at which $n$ is an integer.