Temperature-dependent resistance measurement of a thin film on different substrates.

The aim of this measurement was to determine the current-voltage characteristics of a InN-thin film that was deposited on both, a Si- and a glass substrate with plasma-induced Atomic Layer Deposition (ALD). The measurement was performed with the four-wire method which is thoroughly described here. A picture of the samples can be seen below:

Fig. 1: Samples on Si- and glass substrate. The circle marks the two samples that were used for further measurements.

Modelling of the temperature-dependent resistance

Drift regime: A thin film can be considered to be a 2-dimensional system. It is common to use the square sheet resistance $R_S$ to describe the electrical properties of such a system,

\begin{equation} R_S=\frac{\pi\cdot U}{\ln(2)\cdot I}, \end{equation}

with the current I and the voltage U. It is closely related to the resistivity $\rho$ and the sample thickness $t$ (20 nm):

\begin{equation} R_S=\frac{\rho}{t} \end{equation}

Furthemore, the temperature-dependent resistance needs to be taken into account. The behaviour can be obtained by looking at the electron ($n$) and hole concentration ($p$) of an intrinsic semiconductor, which are equivalent to the square-root of the intrinsic carrier concentration $n_i$ according to the law of mass action:

\begin{equation} n=p=n_i=\sqrt{N_c N_v}\cdot e^{-\frac{E_g}{2 k_B T}} \end{equation}

This equation links the band gap $E_g$ to the carrier densities. The effective electron ($N_c$) and hole concentrations ($N_v$) are also weakly temperature-dependent but this can be neglected in the measured temperature range. The final step is the relationship between the electrical conductance $\sigma$ and the carrier densities:

\begin{equation} \sigma=n e_c \mu_n +p e_c \mu_p=(\mu_n+\mu_p) e_c e^{-\frac{E_g}{2 k_B T}}, \end{equation}

with the elementary charge $e_c$. This means that the sheet resistance must show an exponential temperature-dependence acording to:

\begin{equation} R \propto e^{\frac{E_g}{2 k_B T}} \end{equation}

This means that a fit of $ln(R_S)$ vs $\frac{1}{T}$ should give an estimate for the band gap of the measured semiconductor.

Variable-range-hopping (VRH): However, the conclusions above were drawn under the assumption that diffusion is the main transport mechanism in the measured semiconductor. VRH is another model which describes the behaviour of semiconductors at lower temperatures:

\begin{equation} R=R_0 \cdot \exp\left(\left(\frac{T_0}{T}\right)^{\frac{1}{d+1}}\right), \end{equation}

with a typical temperature $T_0$ and the dimension $d$. In our case, a sample thickness of 20 nm satisfies the conditions for a 3d-system in this model.