Collective transport: MW model

Charge transport in an array which consists of small metallic dot exhibits a nonlinear behavior with a threshold. Middleton and Wingreen (MW) proposed a model to explain this collective charge transport behavior in one- and two-dimensional dot arrays [29]. Their model is based on the Coulomb blockade regime which means the electron transports among the dots in arrays by tunneling and the thermal fluctuation cannot exceed the charging energy. They predicted the current through a uniform

array behaves as

~( −1)^ς VT

I V (3.9)

with ζ = 1 and 5/3 for one- and two-dimension arrays, respectively.

Fig. 3.2(a) presents a schematic illustrating the build of charges in a one-dimensional array between two electrodes as the voltage is progressively increased to threshold. Each square represents a potential e/Cg for adding an electron on a dot, where Cg is the capacitance between dot and substrate. With randomly distributed offset charge on a one-dimensional array, electron can spontaneously tunnel to the next dot or be blocked in a down-ward or up-ward step, respectively. The threshold voltage VT means the potential difference VL - VR is enough to drive electron transport from left to right electrode. As the array size N (number of dot) is very large, statistically, an electron has to overcome N/2 up-ward step, and the thus we have

Cg

e N

V_T = 2 . (3.10) From Eq. (3.7), the tunneling rate at very low temperature is approximately ΔE/e²R. For the conduction state,V >V_T, since there are N dots in the array, the average energy difference EΔ is then e(V-VT)/N resulting a tunneling rate (V-VT)/eRN. With Eq.

(3.10), the current I = eΓis then given by Eq. (3.9) for one-dimensional array.

For a two-dimensional array with the size N × N, the extra charges driven by applied voltage penetrate the array and form an interface, at which the charges are blocked until the driving voltage is increased. Because of the disordered offset charge the interface is irregularly shaped and the voltage that makes the interface reach the other electrode is the threshold voltage VT. At V > VT, current-carrying channels are formed.

Similar to one-dimensional array, there will be (V −V_T)/(e/C_g) excess steps

(charge)

Figure 3.2: (a) A schematic illustrating the build of charges in a one-dimensional array as the voltage is progressively increased to threshold. (b) A scheme of branched channel in a N × N two-dimensional array.

(charges), on average, in a channel. Since the electrons can transport not only forward but also probably to lateral dots in the two-dimensional array, those excess steps can make a split in the channel. The average distance between each branch points is then

g

ξ . The Kardar-Parisi-Zhang (KPZ) equation [103], which

describes the growth of interfaces, is adopted to give that the transverse deviation ξ _⊥ will be ξ in a length _||^2/3 ξ_||. Fig. 3.2(b) shows a scheme of branched channel in an array. Thus the number of channel reaching right electrode will be

3 number of current channel is increased with the voltage. Before all the possible channels are turned on, the current will behave as I ~(V −V_T)^5/3. This region can be view as a transition from an isolated state to a conducted state, at which the current is linearly depended on the voltage.

3.3 Experiment

Several drops of the solution were put on a conducting and flat substrate. The chosen substrate was atomically flat terraces, with several hundred nanometers in size, on Au(111) surfaces on a ball prepared by melting a 2-mm gold wire in a gas flame. The as-deposited PbSe QDs formed islands with QD numbers from one to several tens and the assembling process can be controlled by the growth conditions of the substrate temperature and the solution concentration which we described in charter 2. The morphology of QD islands on Au(111) surfaces was inspected by field-emission

scanning electron microscope (JEOL, JSM-7000F) and atomic force microscope (Seiko Instruments Inc., SPA-300HV). The sample of as-assembled QD islands was loaded in a STM preparation chamber in an ultrahigh vacuum of 1 × 10-10 torr, and it was heated up to 100-150°C for more than 10 h. When annealing at 150°C, long time of thermal treatment helped to detach more organic ligands and to squeeze the distance between QDs. In addition, the PbSe QD islands liquefied and evaporated as the annealing temperature was increased up to 300°C. The thermal annealing converts the as-assembled QD islands to compact and ordering QD arrays as well. The sample of QD arrays on Au(111) surfaces was transferred to a STM main chamber and the STM analysis were carried out by Omicron LT STM. An electrochemically etched tungsten tip was used in our experiment. All STM images were taken in a constant current mode with a sample bias of 2.5 V and a tunneling current (setpoint) of 0.15 nA. After the PbSe QD array was specified in a STM topography image, current image tunneling spectroscopy (CITS) will be taken at the same place. The tunneling spectra were taken with voltages ramped from -1.5 to +1.5 V under a scanning condition of the sample bias 1.5 V and the tunneling current 0.15 nA. There were 100

× 100 points in a CITS image and every CITS image point contained an I-V curve having 200 steps. The STM images and CITS data were recorded at both room temperatures and 78 K. Since the slightly deviated separation distance between the STM tip and the sample could produce a large current fluctuation, the statistically averaged I-V curve from several tens or hundreds of CITS image points was evaluated for an individual PbSe QD array. All the data were analyzed using the software of scanning probe image processor (Image Metrology A/S, SPIP). The I-V was analyzed and fitted by the Levenberg-Marquardt nonlinear least squares method [78].

3.4 Results and discussion

The characterization of the PbSe QD size distribution is described in chapter 2 giving a uniform and average diameter of ~14.6 nm and a standard deviation of 17%. To investigate the electronic structure of these QD arrays using an STM, PbSe QDs are deposited on a conducting substrate. The QDs will form clusters as well as islands on the substrate with different sizes and shapes owing to different packing conditions.

Fig. 3.3(a) shows a typical STM image of PbSe QD arrays on a flat Au(111) surface.

Before annealing in an ultrahigh vacuum, PbSe QDs form a monolayer of two-dimensional islands demonstrating either disordered clusters or locally regular arrangements. The as-deposited QD islands seem to be mobile and cause a blurred image under the scanning measurements of STM. After annealing and removal of excess capping agents, the QD islands are organized as two-dimensional arrays and fixed on a flat gold surface to stabilize the STM scans. Fig. 3.3(b) shows a two-dimensional PbSe QD array with 229 QDs on the Au(111) surface. The arrays appear as a hexagonal or, sometimes, a square lattice in two dimensions and have a rounded shape. The average number of nearest neighbors of the PbSe QD in the arrays is about 6. The separation distance between the QDs is estimated: the STM topography image is transformed to a two-dimensional Fourier image like that shown in the inset of Fig. 3.3(c); the radially average intensity of the fast Fourier transform image is calculated and displayed in Fig. 3.3(c) to give a peak which indicates a special periodicity of ∼16.3 nm between the centers of two PbSe QDs. In particular, the fast Fourier transform image in the inset of Fig. 3.3(c) clearly exhibits 6 dots, symptomatic of a hexagonal lattice. Since the average diameter of the PbSe QDs is 14.6 nm, the separation distance between two PbSe QDs is evaluated to be 1.7 nm,

which

Figure 3.3: (a) STM image of PbSe QD arrays dispersing on a gold surface. (b) A two-dimensional PbSe QD array with 229 QDs on the Au(111) surface. (c) Radially averaged intensity of the two-dimensional Fourier transformed image displayed in the

which is about one layer of organic molecules of the TOP and oleic acid.