Chapter 2 Self-assembly of PbSe quantum dot arrays
2.4 Results and discussion
2.4.2 Growth mechanisms
Several growth mechanisms play important roles to modulate the growth pattern. In the initial growth stage, the cluster occurrence correlates with a decrease of free energy in cluster formation. Moreover, the scaling function can be used to evaluate the critical nucleus size. As the coverage increases, a growth mechanism of the spinodal decomposition is involved. Moreover, as for our case of the QD growth in solution, a higher substrate temperature and the rapid evaporation of solvent shall induce a diffusion-limited aggregation.
Fig. 2.6(a)-(c) show the occurrence N of small clusters as a function of the cluster aggregation number n, calculated on the basis of fifty SEM images with an area of 2.8 × 2.1 μm2. The occurrence of single QD, 2-QD, 3-QD, and n-QD clusters are denoted as N1, N2, N3, and Nn, respectively. The data present with n in a range from 3 to 9. Because the formation of n-QD cluster from isolated QDs makes a decrease of energy (-En) and affects the occurrence (possible configurations) through a thermal equilibrium process, the occurrence shall follow the form [69]
Nn =N0(N1/N0)nexp(En/kT) (2.5) where N0 is the total number of QDs to cover the whole area of the SEM image. Using
the formation of n-QD cluster for n > 1 leads to a lowering of free energy (En = (2n - 3)αkT). This gives
Nn =N0(N1/N0)nexp((2n−3)α) (2.6) Usually the occurrence of a single QD is not easy to be counted possibly due to the limit of the image resolution. The other reason could be attributed to the fact that the stable nucleus size is larger than one, and therefore a single QD has a high tendency to coalesce on big islands. A nonlinear least-squares fitting was used to evaluate N1
and α. The fitting results are displayed as solid curves in Fig. 2.6(a)-(c). It is noted that Eq. 2 is valid only for n ≥ 2. A straight line is therefore given to simply connect the two calculated values of N1 and N2. The parameter α was determined to be 1.13, 0.99, and 0.85 for growth data at 50, 100, and 180°C, respectively. Thus the pair bond energies (αkT) were estimated to be 31, 32, and 33 meV for clusters grown at 50, 100, and 180°C. This gives an average value of about 32 meV, which is about one-half of the pair bond energy ε = 59 meV estimated from Fig. 2.5(a). In particular, the pair bond energy is very close to the calculated attraction energy of 40 meV (1.6 kT) and experimentally evaluated value of 88 meV (3.5 kT), as reported in ref 14.
On the other hand, the critical nucleus size can be determined by a scaled island size distribution. The island size distribution, which is normally Nn as a function of the QD number n, is scaled to Nnnavg2 /θ as a function of n/navg, where θ is the coverage and navg is the average size of the QD-formed islands. The scaled island size distributions with specified coverage of 5-15% and at different temperatures are displayed in Fig. 2.6(d)-(f). Because it is analyzed at low coverage, we can ignore the spinodal decomposition behavior. Thus only the temperature and solvent drying effects will be taken into account. In Fig. 2.6(e), a solid line of scaling function is displayed to indicate a critical nucleus of one QD, which means that island size with
two
Figure 2.6: Occurrence as a function of aggregation number (n, number of QDs) corresponding to small islands with 3-9 QDs at (a) 50, (b) 100, and (c) 180°C. The solid lines present the fitting results. Scaled island size distributions at (d) 70, (e) 100, and (f) 130°C. The coverage are (d) 4, (e) 13, and (f) 16%. The dashed lines are guides to eyes. The solid line in Figure (e) represents the scaling function
two more QDs are stable. Fig. 2.6(d) shows that, at 70°C, even a single QD can be stable and exist on the graphite surface. At a higher temperature of 100°C (Fig. 2.6(e)), the excess thermal energy makes single QD more mobile and islands having at least two QDs can stably exist in growth patterns. The critical nucleus size therefore increases and the smallest stable island must consist of at least two QDs. This result is in line with the temperature effect on critical nucleus reported for atomic growth behavior [76]. The critical nucleus size slightly changes back to one QD at an elevated temperature, 130°C (Fig. 2.6(f)). We believe that, at this high temperature, the solvent drying process should be taken into account. More interestingly, part e of Fig. 2.6 indicates a stable cluster of n ≥ 2. This result can also be confirmed in the bonding energy analysis displayed in Fig. 2.6(b). We found that the experimental value is much lower than the calculated value of occurrence of single adsorbed QD for growth at 100°C. On the contrary, the measured occurrence of single QD is more close to the calculated value in Fig. 2.6(a) and (c). It is noticed that the growth behavior varies with a different substrate temperature above the solvent boiling point (Fig. 2.6(c) and (f)).
With an increase of coverage, owing to a coexistence of two phases, the growth mechanism of spinodal decomposition engages to modulate sinusoidally the compositional (QD density) variation that generates a macroscopically uniform island size distribution with an interconnection labyrinthine structure. Two spinodal growth patterns assembled at 100 and 50°C are exhibited in Fig. 2.7(a) and (b), respectively.
Evidently, different wavelengths of the sinusoidal compositional modulation can be identified between the two SEM images. This wavelength variation has not been experimentally discovered in room-temperature QD assembly yet [13, 14, 62].
According to the spinodal phase separation theory [72], the wavelength is inversely proportional to the square root of the undercooling temperature, λ ∝ (TC - T)-1/2, where
TC is the critical-point temperature. To evaluate the wavelength of the sinusoidal composition modulation, the radial average intensity of fast-Fourier-transformed image was calculated (inserts of Fig. 2.7 as examples). The wavelengths at maximum intensity are 450 and 176 nm for growth patterns shown in Fig. 2.7(a) and (b), respectively. The temperature behavior of the wavelength at maximum intensity is thus summarized in Fig. 2.7(c). When the temperature is increased above room temperature, it was found that the wavelength increases up to a maximum value at 100°C. More excitingly, the spinodal growth patterns remain observable even if the substrate temperature is much higher than the critical-point temperature. This result is in contradiction to the gas or fluid phase above critical-point temperature that is predicted in the phase diagram (Fig. 2.5(b)). In addition, the wavelength at maximum intensity shown in Fig. 2.7(c) reveals a decline with an increase of the substrate temperature above the toluene boiling point. We therefore conclude that the elevated substrate temperature makes the solvent to evaporate and the growth mechanism of diffusion-limited aggregation is in turn employed to modify the growth pattern.
Because of the rapid evaporation of toluene solvent, the decrease of wavelength at elevated temperature shall be resulted from a reduced evolution time [77] required for long-wavelength spinodal patterns such as that displayed in Fig. 2.7(a). The interplay between spinodal decomposition and diffusion-limited aggregation could be studied and further controlled by using substrate temperature and QD coverage (concentration of QD suspension) so as to grow 2D islands with specified sizes and shapes.
Because the solvent evaporation introduces a diffusion-limited growth mechanism, the analysis method used in atomic growth of diffusion-limited aggregation can certainly be implemented here. Either the density-density correlation function or the radius of gyration as a function of cluster size (aggregation number n)
Figure 2.7: (a) SEM image of a spinodal pattern grown with coverage of ∼38% at 100°C. (b) SEM image of a spinodal pattern grown with coverage of 45% at 50°C.
The insets show fast-Fourier-transformed and radial averaged intensity as a function of inverse wavelength. (c) Wavelength at maximum intensity, estimated from the intensity distribution, like the inserts of parts (a) and (b), for QD growth with coverage of ∼42% and at different temperatures.
The analysis using radius of gyration, Rg, which is a mean square distance between the particle and the center of mass of the cluster, is adopted in this study. A fractal dimension D is introduced in the expression n ~ RgD [73]. Log-log plots of Rg as a function of aggregation number n are given in Fig. 2.8(a) and (b) for patterns (the inserts) grown at 100 and 180°C respectively, with coverage of ~20%. The fractal dimensions D’s are estimated to be 1.38 and 1.57 as indicated in the figure. This observable difference in D shall be originated from the substrate temperature. At the temperature of 100°C and below the toluene boiling point, the toluene solvent shall not be evaporated so quickly and the as-aggregated clusters may diffuse on surface and coalesce into larger ones. This experimental result of fractal dimension D = 1.38 is considerably close to the cluster diffusion models given by P. Meakin [78] (D = 1.47) and M. Kolb et al. [79] (D = 1.38). On the other hand, a high substrate temperature of 180°C leads to a random aggregate of cluster which by no means can diffuse on substrate anymore due to the solvent dry-up. This result of fractal dimension D = 1.57 is in line with the predicted value of 1.67 using the WS model [73]. The cluster diffusion at 100°C seems to be in connection with a long evolution time required to produce a long wavelength of spinodal growth patterns as discussed above. Here, the interplay between the spinodal decomposition and the diffusion-limited aggregation appears again. Two schemes to illustrate the cluster diffusion and single-particle diffusion models are given in Fig. 2.8(c) and (d). For the same aggregation number n, the cluster diffusion causes a longer radius of gyration and thus a smaller fractal dimension.
Figure 2.8: Gyration radius, Rg, has a power law dependence on the aggregation number, n, in double logarithm scales for QD growth at (a) 100°C and (b) 180°C with coverage of 21% and 18%, respectively. Corresponding SEM images are displayed in the insets. Schemes of (c) cluster diffusion and d) single-particle diffusion models.
Chapter 3
Collective transport in PbSe quantum dot arrays
3.1 Introduction
Few electrons stored on small metal particles or semiconductor quantum dots (QDs) will build a charging energy to block additional electrons’ tunneling. This effect is called Coulomb blockage. It changes the conductivity of the current channel due to the small capacitance of the tiny-sized particles which give large blocking Coulomb energy [81, 82]. The few-electron charging effects have been applied not only to single-electron transistors but also to the fabrication of single-electron [83, 84] and nanocrystal based memories [6, 85] for operating at room temperatures. Moreover, another similar phenomenon of electrical bistability exhibits two conductivity states at the same voltage. This nonvolatile electrical bistability has been discovered and demonstrated in the system of nanoclusters embedded in organic or oxide layers [86, 87]. Recently, many organic bistable devices, containing different materials of metal particles or semiconductor QDs dispersed in organic layers, have displayed a large hysteresis in current- or capacitance-voltage curves that reveal bistable effects in
proposed to explore the device operation mechanism, investigation of the electrical coupling effect among particles from a microscopic viewpoint has not been intended yet.
In recent years, a high temperature organic solution-based wet-chemical strategy has been successfully developed in synthesizing high crystalline QDs in which the surface of particles is passivated by organic ligands [57]. These chemically generated QDs were introduced in a nanogap between metal electrodes for approaches to electrical studies of a single dot [90] and to a single-electron transistor [91].
Meanwhile, the QDs could self-assembled into quasi-one-dimensional chains and two- or three-dimensional arrays. The current-voltage (I-V) behavior at various temperatures has been measured for self-assembled chains of conducting carbon nanopartciles [30] and quasi-onedimensional arrays of gold nanocrystals [31]. In addition, the transport of electrons has been studied in the two-dimensional systems of cobalt-nanocrystal superlattices [2], Au nanocrystal arrays [92], CdSe QD arrays [93], and the topographically complex Au nanoparticle network [35]. These studies reveal a nonlinear I-V curve and a temperature-dependent threshold voltage Vth at lower temperatures. In particular, the I-V curves at different temperatures can be collapsed to a single power-law I(V-Vth) curve by translation on the voltage scale above Vth. This behavior is recently exposed in a system of Au nanoparticles self-assembling in organic thin films as well [94]. On the other hand, the scaling behavior has been observed in the early research of lithographically patterned GaAs QDs and Al island arrays [27, 28]. A collective transport model considered by Middleton and Wingreen (MW) can be used to explain the aforementioned experimental results [29].
In the very early studies, scanning tunneling microscope (STM) has been used to observe the Coulomb blockade characteristics of staircase features on I-V curves of an individual metal droplet or cluster at a very low temperature [95] as well as at room
temperatures [96]. Recently, not only the Coulomb effect but also the electronic structures of artificial-atom states in the colloidal and semiconducting QDs were investigated by several research groups [17, 20] through measurements of tunneling spectroscopy with the aid of low-temperature STM. In addition to the single-QD electronic states, the interparticle Coulomb interactions [23] and the level structure of InAs QDs in two-dimensional assemblies [24] draw attention for tunneling spectroscopy characterizations. However, the array size effects, which might exhibit during the assembling process from a single QD to a two-dimensional QD array, and the capacitive coupling among the QDs have not been explored to date. In this study, we employ STM to probe the size effects in the coupled PbSe QD arrays. The size-dependent tunneling spectra of the QD arrays may give a clue to understand the operation mechanism of memory devices from a microscopic viewpoint.
3.2 Theoretical background
Based on the Coulomb blockade, a analytic form of tunneling current can be derived by a double tunnel junction model. As the junctions form an array, there will be a collective behavior of charge transport. We briefly introduce the double tunnel junction and collective transport model which are adopted to analyze our results.
3.2.1 Coulomb blockade and double tunnel junction model
The phenomenon “Coulomb blockade” was first observed in an assembly of double tunnel junction, which was made of small stannum particles embedded in an aluminum layer, by Giaever and Zeller [102]. In such a double tunnel junction system, the charged small island results in a large capacitance as well as electrostatic energy to block another electron from transporting into the island. Experimentally, there is a threshold voltage below which the current is suppressed. Under an appropriate bias, electron can tunnel through the junctions one by one. To observe Coulomb blockade, two conditions have to be met. Firstly, the charging energy of the small island Ec = e2/2C must exceed the thermal fluctuation kBT, where C is the capacitance of the small island. Secondly, the tunneling junction must have a resistance RT larger than the quantum resistance h/ e2 ≅ 25.8 kΩ to satisfy the requirement of Heisenberg uncertainty principle Δ tEΔ ≥h/2π (RC = Δt).
On the basis of Coulomb blockade, there is an orthodox theory of double tunnel junction that provides an analytic form for the scanning tunneling spectroscopy [98, 99]. Fig. 3.1 illustrates an STM geometry which is modeled as a double tunnel junction system. Consider a small island placed between an STM tip and a substrate, which is biased with a voltage V, the junction (junction 1) between the tip and the island possesses a capacitance C1 and a resistance R1. By analogy, C2 and R2 are the capacitance and resistance of the junction (junction 2) between the island and the substrate. With N extra electrons on the island, the voltage drops on junction 1 and 2 are , where Q0 is the residual charge. Physically, the Q0 ranges only between –e/2 to +e/2 and the excess part will be incorporated in N. For a steady-state condition that the
trans
Figure 3.1: Schematic diagram of an STM geometry which is modeled as a double tunnel junction system. The two junctions of capacitances C1 and C2, and resistances R1 and R2, are driven by a voltage source V.
transition rate from the state N to N+1 is equal to the reverse rate, we have
(r1(N,V)+l2(N,V))p(N,V)=(l1(N +1,V)+r2(N +1,V))p(N+1,V). (3.2) ri(N,V) and li(N,V) are the electron tunneling rate from right and left on the junction i, and p(N,V) is the probability that there are N electrons on the island with a bias V.
With the normalization condition Eq. (3.2) can be solved to obtain that
The average current is then
∑
+∞ To calculate quantitative tunneling current I, we have to know the expression of the four tunneling rates, i.e. r1, l1, r2, and l2. From Fermi’s golden rule, the tunneling rateHere E′ and E ′′ are the Fermi energies of different sides, with extra electrons, of the junction. f(E) is the Fermi-Dirac distribution f(E)=1/(1+eE/kBT) . Taking the transmission coefficientT(E)2, the density of statesD′(E), andD ′′(E) as constants, Eq. (3.6) is simplified to
, where R≡h/2πe2D0′D0′′T02 and ΔE=E′−E′′ . The EΔ means the energy that electrons gain after tunneling through the junction and can be considered as the sum of the change in electrostatic energy and the work done by the applied voltage. For tunneling rate
(Eq. (3.1a) and (3.1b) are used.) From Eq. (3.8), the voltage drops on each junction has to be the value larger than e/2(C1+C2) to have a non-zero EΔ for tunneling. Thus the V should exceed(e/2−Q0)/C>, where C> is the greater of C1 or C2, to have a current flow. The voltage with this value is called threshold voltage Vth, below which the current is suppressed.
3.2.2 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
VT = 2 . (3.10) From Eq. (3.7), the tunneling rate at very low temperature is approximately ΔE/e2R. For the conduction state,V >VT, 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 −VT)/(e/Cg) 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 −VT)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
Several drops of the solution were put on a conducting and flat substrate. The chosen