This is a consequence of randomization:

some CNTs are less electrostatically screened causing them to surpass the emission of a perfect Quisinostat supplier array. Furthermore, most CNTs are screened, as can be seen in Figure 1d; so, only few CNTs are accounting for the total current [6]. Then, by increasing the external electric field, these few CNTs will become overloaded before most CNTs can start contributing to the current. Consequently, the maximum current density of non-uniform arrays is limited by the current that these few CNTs can support. We define I high as the highest CNT normalized current in the 3 × 3 array averaged over 100 runs. I high comprehends 1/9 or 11.1% of the most emissive CNTs. Figure 7 shows I high as a function of s for s > h and its standard deviation, σI high, shown in the figure as error bars. The σI high can be used to determine what part of the CNTs is expected to burn in the non-uniform array given their tolerance, as we shall indicate below. Figure 6 Normalized emission randomizing variables two at a time and all three variables simultaneously. Figure 7 Highest normalized emission I high and the standard deviation σI high as a function of the spacing. The σI high is shown as half error bars. These find more parameter can be used to estimate

the MK-8931 in vitro fraction of CNTs that will burn out at certain current given the degree of non-uniformity. The interpolating functions for the curves of Figure 6 are (8) (9) (10) (11) Equations (5) to (11) are valid for α = 1; however, our simulation results (not shown here) indicate that a quadratic function fits intermediate values 0 < α < 1 reasonably well. The following example gives a procedure to obtain the normalized current for any set (α p ,α r ,α h ), with normalized current I(α p ,α r ,α h ). In the simplest example, if only α p varies, then (12) where I p is given by Eq. (5). In another example, in which α p and α r are varying, then (13) where I pr is given in Eq. (9).

Finally, if all α parameters vary, we have (14) where I phr is given in Eq. (11). From the data shown in Figure 7, we derive Selleck Decitabine the following interpolating functions (15) where, α prh  = max(α p, α r, α h ) and (16) Equations (15) and (16) give an upper estimate of the maximum current carried by individual CNTs, as a function of our randomization parameter α prh . The fraction of CNTs expected to burn out can be evaluated from a Gaussian distribution as: (17) where erf(z) is the error function, I max is the normalized burn out current (or tolerance). The factor 11.1% is because Eqs. (15) and (16) account only for 1/9th of the CNTs in the 3 × 3 array. Let us give an example: consider a non-uniform array with α p  = 0.4, α r  = 0.5, α h =0.8 observed microscopically and s = 2 h yielding an average emission of 1 μA. From Eqs. (14), (15), and (16), we calculate a normalized current of I = 1.28, which corresponds to the 1 μA; I high = 4.94 (3.86 μA) and σI high = 1.