For a p-n junction, the essential electrical characteristic is that it constitutes a rectifier, which allows the easy flow of a charge in one direction but restrains the flow in the opposite direction. The voltage-current characteristic of such a device can be described by the Shockley equation, Equation 1, in which,I₀ is the reverse bias saturation current, q the charge of the electron, k is Boltzmann’s constant, and T is the temperature in Kelvin.

(1)

When the reverse bias is very large, the current I is saturated and equal to I₀. This saturation current is the sum of several different contributions. They are diffusion current, generation current inside the depletion zone, surface leakage effects and tunneling of carriers between states in the band gap. In a first approximation at a certain condition, I₀ can be interpreted as being solely due to minority carriers accelerated by the depletion zone field plus the applied potential difference. Therefore it can be shown that, Equation 2, where A is a constant, E_g the energy gap (slightly temperature dependent), and γ an integer depending on the temperature dependence of the carrier mobility µ.

(2)

We can show that γ is defined by the relation by a more advanced treatment, Equation 3.

(3)

After substituting the value of I₀ given by Equation 2 into Equation 1, we take the napierian logarithm of the two sides and multiply them by kT for large forward bias (qV > 3kT); thus, rearranging, we have Equation 4.

(4)

As InT can be considered as a slowly varying function in the 200 - 400 K interval, therefore for a constant current, I, flowing through the junction a plot of qV versus the temperature should approximate a straight line, and the intercept of this line with the qV axis is the required value of the band gap E_g extrapolated to 0 K. Through Equation 5 instead of qV, we can get a more precise value of Eg.

(5)

Equation 3 shows that the value of γ depends on the temperature and µ that is a very complex function of the particular materials, doping and processing. In the 200 - 400 K range, one can estimate that the variation ΔE_g produced by a change of Δγ in the value of γ is Equation 6. So a rough value of γ is sufficient for evaluating the correction. By taking the experimental data for the temperature dependence of the mobility µ, a mean value for γ can be found. Then the band gap energy qV can be determined.

(6)

The electrical circuit required for the measurement is very simple and the constant current can be provided by a voltage regulator mounted as a constant current source (see Figure 3). The potential difference across the junction can be measured with a voltmeter. Five temperature baths were used: around 90 °C with hot water, room temperature water, water-ice mixture, ice-salt-water mixture andmixture of dry ice and acetone. The result for GaAs is shown in Figure 4. The plot qV_corrected (qV_c) versus temperature gives E₁ = 1.56±0.02 eV for GaAs. This may be compared with literature value of 1.53 eV.

Figure 3: Schematic of the constant current source. (I_c = 5V/R). Adapted from Y. Canivez, Eur. J. Phys., 1983, 4, 42.

Figure 4: Plot of corrected voltage versus temperature for GaAs. Adapted from Y. Canivez, Eur. J. Phys., 1983, 4, 42.

Optical method can be described by using the measurement of a specific example, e.g., hexagonal boron nitride (h-BN, Figure 5). The UV-visible absorption spectrum was carried out for investigating the optical energy gap of the h-BN film based on its optically induced transition.

Figure 5: The structure of hexagonal boron nitride (h-BN).

For this study, a sample of h-BN was first transferred onto an optical quartz plate, and a blank quartz plate was used for the background as the reference substrate. The following Tauc’s equation was used to determine the optical band gap E_g , Equation 7, where ε is the optical absorbance, λ is the wavelength and ω = 2π/λ is the angular frequency of the incident radiation.

(7)

As Figure 6a shows, the absorption spectrum has one sharp absorption peak at 201 - 204 nm. On the basis of Tauc’s formulation, it is speculated that the plot of ε^1/2/λ versus 1/λ should be a straight line at the absorption range. Therefore, the intersection point with the x axis is 1/λ_g (λ_g is defined as the gap wavelength). The optical band gap can be calculated based on E_g) hc/λ_g. The plot in Figure 6b shows ε^1/2/λ versus 1/λ curve acquired from the thin h-BN film. For more than 10 layers sample, he calculated gap wavelength λ_g is about 223 nm, which corresponds to an optical band gap of 5.56 eV.

Figure 6: Ultraviolet-visible adsorption spectra of h-BN films of various thicknesses taken at room temperature. (a) UV adsorption spectra of 1L, 5L and thick (>10L) h-BN films. (b) Corresponding plots of ε ^1/2/λ versus 1/λ. (c) Calculated optical band gap for each h-BN films.

Previous theoretical calculations of a single layer of h-BN shows 6 eV band gap as the result. The thickness of h-BN film are 1 layer, 5 layers and thick (>10 layers) h-BN films, the measured gap is about 6.0, 5.8, 5.6 eV, respectively, which is consistent with the theoretical gap value. For thicker samples, the layer-layer interaction increases the dispersion of the electronic bands and tends to reduce the gap. From this example, we can see that the band gap is relative to the size of the materials, this is the most important feature of nano material.