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= Flat band potential =

In semiconductor physics, the flat band potential of a semiconductor defines the potential at which there is no depletion layer at the junction between a semiconductor and an electrolyte. This is a consequence of the condition that the redox Fermi level of the electrolyte must be equal to the Fermi level of the semiconductor and therefore preventing any band bending of the conduction and valence band. An application of the flat band potential can be found in the determining the width of the space charge region in a semiconductor-electrolyte junction. Furthermore, it is used in the Mott-Schottky equation to determine the capacitance of the semiconductor-electrolyte junction and plays a role in the photocurrent of a photoelectrochemical cell.

Background theory
In semiconductors, valence electrons are located in energy bands. According to band theory, the electrons are either located in the valence band (lower energy) or the conduction band (higher energy), which are separated by an energy gap. In general, electrons will occupy different energy levels following the Fermi-Dirac distribution; for energy levels higher than the Fermi energy $$E_f$$, the occupation will be minimal. Electrons in lower levels can be excited into the higher levels through thermal or photoelectric excitations, leaving a positively-charged hole in the band they left. Due to conservation of net charge, the concentration of electrons (n) and of protons or holes (p) in a (pure) semiconductor must always be equal. Semiconductors can be doped to increase these concentrations: n-doping increases the concentration of electrons while p-doping increases the concentration of holes. This also affects the Fermi energy of the electrons: n-doped means a higher Fermi energy, while p-doped means a lower energy. At the interface between a n-doped and p-doped region in a semiconductor, band bending will occur. Due to the different charge distributions in the regions, an electric field will be induced, creating a so-called depletion region at the interface. Similar interfaces also appear at junctions between (doped) semiconductors and other materials, such as metals/electrolytes. A way to counteract this band bending is by applying a potential to the system. This potential is the flat band potential and is defined to be the applied potential at which the conduction and valence bands become flat. The flat band potential of a semiconductor depends on many factors, such as the molecular structure of the material, but also its crystal structure, pH, presence of other materials and illumination level.

Gärtner model
The photocurrent is described by the Gärtner model. In the Gärtner model, the flat band potential is used to determine the width of the space charge region (w) in the semiconductor using :

$$ w = \left( \frac{2 \epsilon \epsilon_0}{qN_a} \right)^{0.5} (V - V_{fb})^{0.5} $$

Here, $$N_a$$ is the acceptor level in the p-doped semiconductor, q is the charge of an electron, $$\epsilon$$ is the dielectric constant of the material, $$\epsilon_0 $$ is the permittivity of air, V is the applied potential and $$V_{fb}$$ is the flat band potential. The width of the space charge region is then used to calculate the quantum efficiency φ of the system, according to:

$$   -ln(1-\phi) = ln(1+\alpha L_n) + \alpha w $$

Here $$L_n$$ is the outer end of the diffusion region (space between x = w and x = $$L_n$$). Using this method, the flat band potential can also be determined. If the left hand side is plotted against the second term on the right hand side the intersection with the horizontal axis would give the flat band potential of the semiconductor-electrolyte junction.

Mott-Schottky equation and plot
The Mott-Schottky equation relates the capacitance across a semiconductor-electrolyte junction to the voltage applied to the system  :

$$   \frac{1}{C^2} \ \propto \ V - V_{fb} - \frac{k_B T}{e} $$

Where Vfb is the flat band potential. If this is known, it can be used to determine capacitance, but also to measure various terms in the linearity factor, such as the volume of the depletion region or the dielectric constant of the material.

The Mott-Schottky equation can be used to determine the flat band potential, since the capacitance and applied potential can be measured, the flat band potential can be obtained. To do so, a Mott-Schottky plot is constructed. In this plot, a linear relation between the capacitance and applied voltage is easily visible, which follows the Mott-Schottky equation. The intersection with the horizontal axis of the plot corresponds with a potential of $$V = Vfb + KbT/e$$, from which the flat band potential can be determined.

Illumination of photoelectrochemical cell
A semiconductor-electrolyte junction can function as a photoelectrochemical cell. In dark conditions, the cell will have a certain potential. When illuminated, an electron may be excited to the conduction band from the valence band if the energy of the incoming light is higher than the energy of the band gap, as described by the photoelectric effect. Due to the present electric field in the depletion layer, excited electrons will move away from the semiconductor-electrolyte interface. On the other hand, the positively charged holes will be concentrated at the interface. The extra electrons-hole pairs generated through the photoelectric effect will induce a current in the circuit and thus causing the potential to shift with respect to the dark conditions. As such, the current through the circuit (and by extension the potential at the junction) is different in illuminated condition compared to dark conditions. This shift in current/potential will reach a maximum, corresponding with a flat valence band and a flat conduction band. Therefore, using high levels of illumination the flat band potential of the semiconductor can be determined.

Gärtner-Butler analysis
One can apply a voltage to the semiconductor-electrolyte junction and then measure the resulting photocurrent. Then the intersection of the square of photocurrent with the horizontal axis, is at the flat band potential. This follows from the Gärtner-Butler equation linking the photocurrent with the amount of band bending in the semiconductor :

$$   j_{photo} \approx I_{0,\lambda} \alpha_\lambda \left(\dfrac{2e\epsilon_0 \epsilon_r}{n_0}  \right)^{0.5} (\Delta \phi_{SC})^{0.5}$$

Where $$I_{\lambda,0}$$ is the intensity of the light for a certain wavelength, $$\alpha_\lambda$$ is the material absoprtion coefficient for a certain wavelength and the amount of band bending given as $$(\Delta \phi_{SC})^{0.5}$$. This equation holds only for monochromatic light. For multiple wavelengths the equation would need to be summed or integrated.

Optical spectroelectrochemistry
There is a clear connection between the photoelectric effect and band bending. By measuring absorption spectra of the semiconductor-electrolyte junction, the flat band potential can be measured. This would require for some of the parts to be transparent.

Factors influencing the flat band potential
When measuring the flat band potential for a semiconductor, it depends on the material, the doping of the semiconductor and the PH of the solution in which the semiconductor-electrolyte junction is present. Furthermore, it may depend on the crystal structure of the material, for example a different flat band potential exists for Rutile and Anatase which are two different crystal structures for a titanium dioxide molecule.