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Abstract

CdS is a kind of Ⅱ-Ⅵ type semiconductor which is well studied previously, that can guide our experiment to optimize the system efficient by modeling .In this part, we'll show some factors affecting the electron flux transformed into E.coli. and come up with some practical strategies to improve the efficiency of our system.

Basic theory

Carriers density:

According to quantum theory, states of electrons could be described by energy band:

Fig. 1 | The distribution of energy level and the process of creating free electrons

E_f is Fermi energy,which gives the average level of electrons' energy.

E_c is the minimum energy of conduction band.

E_v is the maximum energy of valence band.

Generally speaking, E_f and E_c,E_v have no direct relationship because E_c,E_v are derived from Schrodinger equation and E_f is determined by the real circumstances of electrons.

And the density of carriers is:

Among them, E represents the energy of each electron. g(E) represents the density of state(one energy line in the energy band image). f(E,E_f) shows the possibility whether the states will be occupied by one electron(Pauli exclusion principle tells that one state can only be occupied by one electron at most).

The concrete form of f(E,E_f) and g(E) is:

Expression of f(E,E_f) is called Fermi-Dirac distribution,which is derived from statistical mechanics,and g(E) is from the periodical boundary condition because we regard the CdS as the ideal crystal, that means, no dislocation, no impurity.

      ε:difference of E and Ec,represent the electrons' kinetic energy.

      m:valid mass of electron.

      h:Plank constant

Fig. 2 | A demonstrate of f(ε),g(ε) and f(ε)g(ε)

Substitute the expression in the integral ,we have:

That is the density of electrons in standard crystal.

Similarly, we can derive the concentration of holes in valence band as a function of E_f:

Let's consider about the situation that the semiconductor material keep electrical neutrality ,so we have :n=p

Substitute the expression for N_c and N_v ,we have:

Now E_f is called intrinsic Fermi level, it is proportion to the temperature where the semiconductor locates.

So we can derive the intrinsic carrier density：

We have to notice that the function is very simplified because we consider the CdS as the ideal crystal, neglect impurity and flaws, though inevitable in fact.

Band Gap:

Band Gap in quantum mechanics plays an important role to describe the circumstance when quantum leaping. Larger the gap is, harder the electron transit to the high energy band, making it be a carrier of electricity .It can be tested indirectly with Absorption spectrum.

Absorption spectrum can be measured in experiment and we can derive absorbance index α by:

T : material transmission of light with different wave length

d : the thickness of sample.

Thus we can get a curve of α as a function of wave length λ. To calculate the band gap of CdS we use, we can use the relationship between absorbance index and band gap:

We can get a diagram with axes (αhν)² and hν from the experiment, deriving intercept on x axis by plotting a straight line that is tangent to the raising part of the curve^[2]:

Fig. 3 | The curve of (αhν)² and hν

Now we know the band gap in CdS attached to Shewanella is 2.51eV

(Relation of α and hν are not sufficient consistently because under 2.51eV, electrons don't have enough energy to transit .Beyond 2.84eV, situation become more intrinsic because we have to consider about both direct absorption and indirect absorption.)