Mott-Schottky relation

The potential dependence of the SCR capacity Csc on applied potential V is described by the Mott-Schottky relation ... 

The Relaxation Spectrum Analysis was carried out for a cell consisting of n-CdSe in a liquid junction configuration with NaOH/S=/S 1 1 1M as the electrolyte. Three parallel RC elements were identified for the equivalent circuit of this cell, and the fastest relaxing capacitive element obeys the Mott-Schottky relation. 

Due to the great extension of the space-charge region, almost all the potential drop occurs across it. So we can measure its capacity, Csc, and calculate from the Mott-Schottky relation... 

In the simplest case, as more fully discussed elsewhere [14, 15, 29], one obtains the Mott-Schottky relation (for the specific instance of an n-type semiconductor)... 

The general method for the determination of the flat band potential is based on the Mott-Schottky linear plot based on ca-pacitance/voltage relation. Starting from Eq. (9) the space charge distribution was calculated, and its potential dependence lead to the derivation of a model equivalent to a capacitance, given by ... 

For later discussions, we also define a potential Us, which is a potential at which the inverse square of the differential capacitance l/C tends to zero as determined from the 1/C vs potential plot (Mott-Schottky plot). It is related to E in the following way ... 

The Mott-Schottky regime spans about 1 V in applied bias potential for most semiconductor-electrolyte interfaces (i.e., in the region of depletion layer formation of the semiconductor space-charge layer, see above) [15]. The simple case considered here involves no mediator trap states or surface states at the interface such that the equivalent circuit of the interface essentially collapses to its most rudimentary form of Csc in series with the bulk resistance of the semiconductor. Further, in all the discussions above, it is reiterated that the redox electrolyte is sufficiently concentrated that the potential drop across the Gouy layer can be neglected. Specific adsorption and other processes at the semiconductor-electrolyte interface will influence Ffb these are discussed elsewhere [29, 30], as are anomalies related to the measurement process itself [31]. Figure 7 contains representative Mott-Schottky... 

Show that the capacity can be related to doping level (N — Na) and potential by the Mott-Schottky relationship... 

Equation (1.24) is the much-used Mott-Schottky equation, which relates the space charge capacity to the surface barrier potential Vs. Two important parameters can be determined by plotting versus Vapp the flatband potential Vn, at = 0 (where Vs = 0) and the density of charge in the space charge layer, that is, the doping concentration N. ... 

In the potential range where the hydroxyl surface was converted to a hydride surface, a high density of surface-states was found which was related to the radical or to a dangling bond formed as an intermediate (Memming and Neumann, 1968). Additional capacitance has also been observed in Mott-Schottky plots for doped semiconductors. In most cases, however, correlation to surface-states was not unambiguously possible. 

FRA systems are versatile, and they can be controlled to acquire and analyse the data required to construct Mott-Schottky plots, for example. Unfortunately, the ease of use of FRA-fitting software can lead to errors of interpretation that arise from a failure to relate fitting elements to the physical system. Several equivalent circuits may give the same frequency-dependent impedance response. No a priori distinction between degenerate circuits is possible, ft is necessary to study the system response as a function of additional experimental variables (DC voltage, concentration, mass transport conditions etc.) in order to establish whether the circuit elements are related in a predictable way to a model of the physical system. 

Finally, it should be noted that in many cases where < 0, is determined by the capacity method uncertainty arises, which is related to the frequency dependence of Mott-Schottky plots. (In particular, the frequency of the measuring current is increased in order to reduce the contribution of surface states to the capacity measured.) As the frequency varies, these plots, as well as the plots of the squared leakage resistance R vs. the potential (in the electrode equivalent circuit, R and C are connected in parallel), are deformed in either of two ways (see Figs. 6a and 6b). In most of the cases, only the slopes of these plots change but their intercepts on the potential axis remain unchanged and are the same for capacity and resistance plots (Fig. 6b). Sometimes, however, not only does the slope vary but the straight line shifts, as a whole, with respect to the potential axis, so that the intercept on this axis depends upon the frequency (Fig. 6a). 

The effect of CH on the Mott-Schottky plots obtained by using Eqs. (32)-(34), (37), (39), and (40) is shown in Fig. 11. The plots are linear when potential change is large but curved when (A Vsc + A VH) < ca. 0.3 V. The slope of the linear portion of the relation is almost identical whether A VH is neglected or not. Thus, the linearity of the Mott-Schottky plot does not necessarily mean the existence of band-edge pinning. 

The capacitance is a function of the electrode potential E. The relation between capacitance and potential is called the Mott-Schottky equation... 

The effect of bandwidth hmitations on a Mott- chottky measurement is illustrated in Fig. 3.17a. The data shown in this figure are simulated for a current range of 100 pA (Ri = 10 kO). The other parameters are the same as those used to simulate the 10 kQ curve of Fig. 3.16a. At 100 Hz and lOkHz, the latter figure shows a perfectly linear relation between Log(—Zim) and Log(f), with a slope of —1 that corresponds to an ideal capacitive response (—Zim = (mC) ). The resulting Mott-Schottky plots for these frequencies are therefore identical. At 50 kHz, however, the Mott-Schottky plot is significantly different, and at 100 kHz the data are entirely unreliable. This shows that one should stay weU away from the frequency region at which the slope of Log( Zi ) vs. Log(/) starts to deviate from -1. 

Two distinct differences can be seen in the relation for the space charge width in the Mott-Schottky compared to the Gouy-Chapman boundary conditions. When the majority defect carmot redistribute, the space charge width is dependent on the space charge potential, and the depletion width is greater in spatial extent due to a reduced charge screening ability. 

The capacity-potential relation, given by Eq. (10), is the so-called Mott-Schottky equation which is strictly valid only in the exhaustion region, i.e., for space charges in which the majority carrier density at the surface is smaller than the corresponding bulk concentration < o for n-type and Ps < po for p-type electrodes). 

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