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Band Gap Measurement

Module by: Yongji Gong, Andrew R. Barron. E-mail the authorsEdited By: Andrew R. Barron

Introduction

In solid state physics a band gap also called an energy gap, is an energy range in an ideal solid where no electron states can exist. As shown in Figure 1 for an insulator or semiconductor the band gap generally refers to the energy difference between the top of the valence band and the bottom of the conduction band. This is equivalent to the energy required to free an outer shell electron from its orbit about the nucleus to become a mobile charge carrier, able to move freely within the solid material.

Figure 1: Schematic explanation of band gap.
Figure 1 (graphics1.jpg)

The band gap is a major factor determining the electrical conductivity of a solid. Substances with large band gaps are generally insulators (i.e., dielectric), those with smaller band gaps are semiconductors, while conductors either have very small band gaps or no band gap (because the valence and conduction bands overlap as shown in Figure 2).

Figure 2: Schematic representation of the band gap difference in a metal, a semiconductor and an insulator.
Figure 2 (graphics2.jpg)

The theory of bands in solids is one of the most important steps in the comprehension of the properties of solid matter. The existence of a forbidden energy gap in semiconductors is an essential concept in order to be able to explain the physics of semiconductor devices. For example, the magnitude of the bad gap of solid determines the frequency or wavelength of the light, which will be adsorbed. Such a value is useful for photocatalysts and for the performance of a dye sensitized solar cell.

Nanocomposites materials are of interest to researchers the world over for various reasons. One driver for such research is the potential application in next-generation electronic and photonic devices. Particles of a nanometer size exhibit unique properties such as quantum effects, short interface migration distances (and times) for photoinduced holes and electrons in photochemical and photocatalytic systems, and increased sensitivity in thin film sensors.

Measurement methods

Electrical measurement method

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.

Eq41.jpg
(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, Eg the energy gap (slightly temperature dependent), and γ an integer depending on the temperature dependence of the carrier mobility µ.

Eq42.jpg
(2)

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

Eq43.jpg
(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.

Eq44.jpg
(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 Eg extrapolated to 0 K. Through Equation 5 instead of qV, we can get a more precise value of Eg.

Eq45.jpg
(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 ΔEg 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.

Eq46.jpg
(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 qVcorrected (qVc) 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. (Ic = 5V/R). Adapted from Y. Canivez, Eur. J. Phys., 1983, 4, 42.
Figure 3 (Fig23.jpg)
Figure 4: Plot of corrected voltage versus temperature for GaAs. Adapted from Y. Canivez, Eur. J. Phys., 1983, 4, 42.
Figure 4 (bandgap.jpg)

Optical measurement method

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).
Figure 5 (HBN.png)

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 Eg , Equation 7, where ε is the optical absorbance, λ is the wavelength and ω = 2π/λ is the angular frequency of the incident radiation.

Eq47.jpg
(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 ε1/2/λ versus 1/λ should be a straight line at the absorption range. Therefore, the intersection point with the x axis is 1/λg (λg is defined as the gap wavelength). The optical band gap can be calculated based on Eg) hc/λg. The plot in Figure 6b shows ε1/2/λ versus 1/λ curve acquired from the thin h-BN film. For more than 10 layers sample, he calculated gap wavelength λg 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.
Figure 6 (graphics5.jpg)

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.

Bibliography:

  • Y. Canivez, Eur. J. Phys. 1983, 4, 42.
  • S. U. M. Khan, M. Al-Shahry, and W. B. Ingler Jr., Science, 2002, 297, 2243.
  • B. O’Regan and M. Grätzel, Nature, 1991, 353, 737.
  • S. Uchida, Y. Yamamoto, Y. Fujishiro, A. Watanabe, O. Ito, and T. Sato, J. Chem. Soc., Farady Trans., 1997, 93, 3229.
  • T. Sato, Y. Yamamoto, Y. Fujishiro, and S. Uchida, J. Chem. Soc., Farady Trans., 1996, 92, 5089.
  • T. Sato, K. Masaki, K. Sato, Y. Fujishiro, and A. Okuwaki, J. Chem. Tech. Biotechnol., 1996, 67, 339.
  • L. Song, L. Ci, and P. M. Ajayan, Nano Lett. 2010, 10, 3209.
  • J. Tauc, R. Grigorovici, and A. Vancu, Phys. Status Solidi. 1996, 15, 627.
  • X. Blase, A. Rubio, S. G. Louie, and M. L. Cohen, Phys. Rev. B, 1995, 51, 6868.
  • D. M. Hoffman, G. L. Doll, and P. C. Eklund, Phys. Rev. B, 1984, 30, 6051.

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