Ocaya-Yakuphanoglu method

The Ocaya-Yakuphanoğlu method (OYM) is a generalized method proposed in 2021 for analyzing the Schottky diode equation within the thermionic emission (TE) model in the presence of appreciable series resistance, $R_s$, with the express purpose of extracting the parameters of the diode. The method is applicable for metal-semiconductor (MS), metal-semiconductor-metal (MSM), metal-insulator-semiconductor (MIS) diodes. It relies on a group theory approach to address this complex problem, yielding an accessible algorithm for empirical data analysis. The OYM posits the existence of an inherent symmetry in semiconductor current-voltage (IV) curves that leads to a reduction of the Schottky diode equation.

In group theory, Emmy Noether's first theorem establishes a connection between conservative forces and symmetries in systems. On the outset, the search for symmetry is a non-trivial, and typically mathematically abstract, endeavor. This fact alone has traditionally led many to avoid it, although the final results can reveal an underlying, and basically simple symmetry, and yet powerful in its applications.

OYM sets out to identify this symmetry, which is suggested by the visual similarity in the I-V characteristic series of a given device. Then, it demonstrates that the symmetry exists and depends strongly on the instantaneous value of $R_s$. The Schottky equation is then simplified to a form that does not contain $R_s$ by adjusting the applied bias term. Consequently, more accurate calculations of barrier height, $\mathrm{\Phi }$ and diode ideality factor, $n$, are then realized through the TE model. Symmetry analysis, though traditionally associated with abstract mathematics, offers a practical means of solving ordinary differential equations by systematically varying infinitesimal parameters.

OYM hypothesis

The diode current $I$ in a Schottky diode in the presence of appreciable diode series resistance $R_s$ is cyclically dependent on itself, the applied voltage, $V$, the absolute diode temperature $T$, effective barrier height, $\mathrm{\Phi }$, the ideality factor $n$ i.e.,$$I=A{A}^{*}{T}^2\exp (-\frac{q\mathrm{\Phi }}{kT})(\exp (\frac{qV-I{R}_s}{(nkT)})-1).$$where $q$ is electronic charge, $k$ is the Boltzmann constant, $A$ is the diode area, $A^{*}$is the Richardson constant.

Eq. \ref{eqn:schottky} is difficult to solve explicitly because of $R_s$. Existing methods estimate $\mathrm{\Phi }$ and $n$ by using simplifying approximations. A commonly used assumption is that $qV>3kT$, such that $R_s$ and $n$ are estimated through the natural logarithm functions $ln(I)$ and $ln(V)$. This limits the method to the forward low-bias region. The barrier height $\mathrm{\Phi }$ is known to depend on applied bias and many simplifying assumptions and treatments of this variation are also described in the literature.

Postulates

The OYM approach is supported by two main theorems.

• Existence

For any Schottky diode that can be described in terms of the TE model, there exists a translational, $R_s$-dependent symmetry that reversibly maps the $I-V$ characteristics of the device, of the form:

$\hat{I}=I+\delta \epsilon \text{<mrow data-mjx-texclass="ORD">and</mrow>}\hat{V}=V+\mu \epsilon ,$

where $\epsilon$ is some infinitesimal variational parameter, and

• Uniqueness

There exists one and only one symmetry i.e. the symmetry is unique such that $\mu =\delta R_s.$

Proof outline

To outline the proof of existence of symmetry, an ordinary differential equation (ODE) is constructed from the TE equation and then shown to have a translational symmetry. The TE equation can be written in the more brief form:

$y=(1/s)e^{[-b\varphi (x)+c(x-yr)]}\{1-e^{-b(x-yr)}\},$

where

where $x=V,y=I,b=q/kT,c=b/n,r=R_s,\text{<mrow data-mjx-texclass="ORD">and</mrow>}(1/s)=A{A}^{*}{T}^2.$

This form can be expressed as an ODE by differentiating w.r.t $x$:

$y^{\prime }=\frac{\beta (x,y)}{p+r\beta (x,y)}=\omega (x,y),$

where

$\beta =\beta (x,y)=[(b-c)e^{-b(x-yr)}+c]e^{c(x-yr)}.$

References

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