- Supplementary for Narrow depletion
The Fermi level is the highest level occupied by conduction electrons in n-type semiconductors in thermal equilibrium. In thermal nonequilibrium, the number of conduction electrons temporarily increases or decreases because of current injection or current outflow, respectively. Assuming that the thermal equilibrium state is maintained temporarily and locally, the quasi-Fermi level become higher (or lower) when the conduction electron density increases (or decreases) compared with that of the equilibrium state [EImref,n = EC+kBT ln(n/ni)].
The Fermi level of the electron-doped (Ba,Sr)TiO3 inner layer (meaning bulk) containing oxygen vacancies is almost at the conduction band minimum. At the surface of (Ba,Sr)TiO3, adsorbrd oxygen forms a Schottky barrier with a height of approximately 0.7 eV. The electron affinity of oxygen molecule, which adsorbs onto (Ba,Sr)TiO3 surface, is not particularly high; however, the surface can become more stable when the adsorbed oxygen accepts electrons to become ionized because of the mirror potential resulting from mirror charges generated in SrTiO3. For SrTiO3 and (Ba,Sr)TiO3, which have many oxygen vacancies, a mid-gap state (MGS) having a peak at approximately 1.5 eV below the conduction band minimum is formed and can be observed in vacuum; however, it is not observed when there is a flow of oxygen since the adsorbed oxygen serves as an acceptor and attracts electrons not only from shallow-level donors but also from the MGSs around 1.5 eV. As a result, the quasi-Fermi level goes downward (not at 0.7 eV but around 1.5 eV). The same situation can take place at Pt-(Ba,Sr)TiO3 interface where Pt surface has adsorbed oxygen coming from the atomosphere.
Under a reverse bias the deep-level donors are ionized in a stepwise manner from shallower levels to deeper levels, causing a relaxation current to flow in a stepwise manner. Here, the strength of the electric field (potential gradient) increases as it approaches the surface. Therefore, donors are ionized sequentially from the surface, which leads to the downward bending of the quasi-Fermi level. In addition, the downward bending can also be explained by the fact that the number of deep-level donors increases as it approaches the interface with the electrode [T. Hara, "Electronic structures near surfaces of perovskite type oxides", Mater. Chem. Phys. 91 (2005) 243]. After reaching the equilibrium, the downward bending may be lost.
I adopted the diffusion model proposed by Wagner [C. Wagner, Phys. Z. 32 (1931) 641], Schottky and Spenk [W. Schottky, E. Spenk, Wiss. Veroff. Siemens Werke 18 (1939) 1], and Mott [N. F. Mott, Proc. R. Soc. A 171 (1939) 27]. In the diffusion model, the current is restricted by the diffusion current and drift current in the depletion layer. Therefore, this model is suitable for a semiconductor with a low carrier mobility.
In the thermal emission model proposed by Bethe [H. A. Bethe, MIT Radiat. Lab. Rep. No. 43 (1942)], which is generally used for the Schottky contact of conventional semiconductors (e.g., Si), the current is restricted by the current that flows over the Schottky barrier. This model is valid only when the mean free path of the conduction electrons is longer than the width of the depletion layer. Strictly speaking, the mean free path longer than the width from an end of the depletion layer to a point approximately kBT below the barrier top is acceptable. Therefore, the thermal emission model is not appropriate for (Ba,Sr)TiO3.
In the thermal emission model proposed by Bethe, the quasi-Fermi level of the semiconductor does not conform to that of the electrode even when a forward-bias voltage is applied. According to Gossick [B. R. Gossick, Solid State Electron. 6 (1963) 445], who supports Bethe’s thermal emission model, “In the diffusion model in which the Fermi level of the depletion layer matches that of the electrode when a forward-bias voltage is applied, injected electrons are considered to be in thermal equilibrium. The electrons flowing over the barrier, which are “hot” electrons having approximately 1 eV or higher energy, should be backscattered or they should be injected to the depletion layer after their energy decreases to reach thermal equilibrium (in the latter case, the heat release process is a rate-limiting process).” Thus, he denies the validity of the diffusion model. However, as mentioned earlier, the diffusion model is appropriate for semiconductors with a low mobility since the rates of drift and diffusion in the depletion layer are as low as the rate of thermal emission at the interface (the thermal emission that enables electrons to flow over the potential barrier and the heat exchange between electrons and lattices). Rhoderick [E. H. Rhoderick, J. Phys. D 5 (1972) 1920] states that the downward bending of the quasi-Fermi level under a forward bias is very small and negligible. In other words, theoretical and experimental results should be in reasonable agreement when the quasi-Fermi level is assumed almost flat in the depletion layer. Therefore, he concluded that the thermal emission model is valid for Si. In the case of Si, the model is appropriate since the following conditions are satisfied: the donor density can be assumed to be constant because of the high purity of Si [this assumption does not hold for defect-rich, sputter-grown (Ba,Sr)TiO3 films] and the dielectric constant is independent of the bias voltage [this does not hold for (Ba,Sr)TiO3 films]. Rhoderick attempted to raise the quasi-Fermi level of the depletion layer when a saturated current flowed stably under a forward-bias voltage so that it matched the Fermi level of the electrode. However, it is not necessary for the two levels to be equal when the relaxation current flows: relaxation current results from detrapping of deep-level donors; this is non-equilibrium.
- Debye length LD:
A charge q is screened by a surrounded electron cloud. When the distance r>LD, q is measured to be electrically neutral.
LD=((εε0kB)/(q2ND))1/2=(((kBT)/q)((εε0)/(qND)))1/2
For example, LD=(0.026×((400×8.854×10-12)/(1.602×10-19×1023)))1/2=7.58×10-8 m = 75.8 nm.
LD is expressed by another equation,
LD=(Dtd)1/2,
where D is the diffusion coefficient and td is the dielectric relaxation time.
- Depletion layer depth WD:
WD=εε0/CSC=2LD((eφSC/kBT)-1)1/2,
where CSC is the capacitance due to the space charges of the depletion layer and φSC is the drop in potential of the depletion layer.