Charge transport in PbSe nanocrystal arrays

We report electrical transport measurements of arrays of PbSe nanocrystals forming the channels of field-effect transistors. We measure the current in these devices as a function of source-drain voltage, gate voltage, and temperature. Annealing is necessary to observe measurable current, after which a simple model of hopping between intrinsic localized states describes the transport properties of the nanocrystal solid. We find that the majority carriers are holes, which are thermally released from acceptor states. At low source-drain voltages, the activation energy for the conductivity is given by the energy required to generate holes plus the activation over barriers resulting from site disorder. At high source-drain voltages, the activation energy is given by the former only. The thermal activation energy of the zero-bias conductance indicates that the Fermi energy is close to the highest-occupied valence level, the $1S_h$ state, and this is confirmed by field-effect measurements, which give a density of states of approximately 8 per nanocrystal as expected from the degeneracy of the $1S_h$ state.

Figure 1

(Color online) Schematic of the inverted FET and the circuit used in the experiments. Silicon oxide $330\mathrm{nm}$ thick is grown on a $n$-doped silicon wafer. Gold electrodes $200\times 800\times 0.1\mu {\mathrm{m}}^3$, separated by a distance of $1\mu \mathrm{m}$, are microfabricated on the silicon oxide surface. NCs $6.2\mathrm{nm}$ in diameter are drop cast from solution on the structure, resulting in a film approximately $300\mathrm{nm}$ thick.

Figure 2

(a) Optical absorption spectrum of NCs in solution. The first absorption peak occurs at $695\mathrm{meV}$. The insets are TEM images of a NC film on a carbon grid before annealing (top inset) and after annealing (bottom inset). The NCs have a mean diameter of $6.2\pm 0.4\mathrm{nm}$. (b) GISAXS diffraction pattern of a NC film as deposited on a silicon substrate (solid line) and after annealing at $400\mathrm{K}$ for $35\min$ (dotted line). The thickness of the capping layer decreases from $\approx 2\mathrm{nm}\text{to}\approx 1\mathrm{nm}$ upon annealing.

Figure 3

(Color online) Current versus drain-source voltage with $V_{\mathrm{g}}=0\mathrm{V}$ for various values of temperature. The lines serve as guides for the eye. The triangle marker indicates the value of $V_{\mathrm{ds}}$ at which a crossover from Ohmic to non-Ohmic behavior is predicted as explained in the text. The top axis gives the corresponding field. The inset is a plot of current versus $V_{\mathrm{ds}}$ at $4\mathrm{K}$ and $V_{\mathrm{g}}=0\mathrm{V}$.

Figure 4

(Color online) Temperature dependence of the zero-bias conductance in the range $30<T<295\mathrm{K}$ with $V_{\mathrm{g}}=0$. The solid line is an Arrhenius fit to the data with an activation energy of $42\mathrm{meV}$.

Figure 5

(Color online) (a) Temperature dependence of the current at different values of $V_{\mathrm{ds}}$ with $V_{\mathrm{g}}=0$ (filled circle, $V_{\mathrm{ds}}=10\mathrm{V}$; open circle, $V_{\mathrm{ds}}=7.5\mathrm{V}$; filled square, $V_{\mathrm{ds}}=3\mathrm{V}$; open square, $V_{\mathrm{ds}}=1\mathrm{V}$; filled triangle, $V_{\mathrm{ds}}=200\mathrm{mV}$; open triangle, $V_{\mathrm{ds}}=10\mathrm{mV}$). The solid lines are fits to Arrhenius equations in the range $30\mathrm{K}<T<200\mathrm{K}$. (b) Activation energy $E_A$ versus the field $F$ established by the drain-source voltage. The solid line is a linear fit of $E_A$ versus $F$ for ${10}^2\mathrm{V}∕\mathrm{cm}<F<{10}^4\mathrm{V}∕\mathrm{cm}$. The dotted line is a linear fit for $3\times {10}^4\mathrm{V}∕\mathrm{cm}<F<{10}^5\mathrm{V}∕\mathrm{cm}$.

Figure 6

(Color online) Current versus $V_{\mathrm{ds}}$ for various values of $V_{\mathrm{g}}$. (a) $T=4\mathrm{K}$ and (b) $T=77\mathrm{K}$. The lines serve as guides for the eye.

Figure 7

Differential conductance as a function of gate voltage at $V_{\mathrm{ds}}=0\mathrm{V}$. $V_{\mathrm{g}}$ is varied in increments of $10\mathrm{V}$ from $0\text{to}+60\mathrm{V}$, from $+60\text{to}-60\mathrm{V}$, and from $-60\mathrm{V}$ back to $0\mathrm{V}$. For each value of $V_{\mathrm{g}}$, the voltage is applied for $5\mathrm{s}$ and then set to zero for a recovery period of $5\mathrm{s}$. The lines serve as guides for the eye.

Figure 8

(Color online) (a) Temperature dependence of the current at different values of $V_{\mathrm{g}}$ with $V_{\mathrm{ds}}=6\mathrm{V}$. The solid lines are fits to an Arrhenius equation. (b) The activation energy $E_A$ versus $V_{\mathrm{g}}$. The solid line is a guide for the eye.

Figure 9

Sketch of the density of states versus energy. In this model, impurity states in the band gap are broadly distributed in energy. $\{{\epsilon }_F-{\epsilon }_{1S_h}\}$ is the energy required to generate a hole in the $1S_h$ state. ${\epsilon }^{*}$ is the width of the disorder-broadened band of $1S_h$ states. (a) $V_g=0$ and (b) $V_g<0$.

Figure 10

(Color online) (a) $1S_h$ states are distributed in energy versus position with the variation given by ${\epsilon }^{*}$. At $F=0$, the acceptor states right above the Fermi energy are vacant and the energy to excite a hole into the $1S_h$ state is $\{{\epsilon }_F-{\epsilon }_{1S_h}\}$. The subsequent energy required to hop through the array is given by ${\epsilon }^{*}$. (b) When $F\ne 0$, the energy of the drain electrode is raised by $eFL$, where $L$ is the spacing between the source and drain electrodes. The energy to excite a hole is reduced by $eFx$ (where $x$ is the distance between the core and the surface of the NC.) For $F>F_c$ (where $F_c$ is described in the text), holes always hop to NCs with higher site energies and, thus, no longer require the absorption of a phonon. The current is limited by the energy required to excite a hole, which is the difference between the energy of the acceptor state on the surface and the energy of the $1S_h$ state in the NC and is given by $\{{\epsilon }_F-{\epsilon }_{1S_h}\}-eFx$.