4.1.1 Local Threshold and Flatband Voltages

**Figure 4.2:** Definition of (a) local threshold voltage V_th(x) and (b) local flatband voltage V_fb(x) for n-MOSFET. The positive direction of voltage pointsdownwards.
(a) (b)

Foremost, it is necessary to introduce the notion of local threshold and local flatband voltages [173]. These quantities describe the dependencies of CP characteristics on the gate pulse amplitude V_A, the frequency f, the reverse bias on source and drain, etc. Moreover, the mentioned values are the key parameters for the determination of lateral defect distributions. The local threshold voltage V_th(x) can be defined as the gate voltage required to induce a certain density of minority carriers (electrons in the case of n-type MOSFET) and the local flatband voltage V_fb(x) is the gate voltage required to induce a desired density of majority carriers (holes) at each position x along the Si/SiO₂ interface. The values of these densities should be defined such that the interface states can capture holes and electrons during the inversion and accumulation periods of the gate pulse [173]. In other words, V_th(x) and V_fb(x) depend on the dynamics of the capture and emission process at the interface. The distribution of V_th(x) and V_fb(x) can be calculated with the device simulator MiniMOS-NT using a widely-adopted routine [173,174,22,175,176]. Namely, as it was explaned above, the local threshold (flatband) voltage of a point at the interface of an n-type MOSFET is defined as the gate voltage required to accumulate the electron (hole) concentration n_e/h [173]

Consequently, the dependence of the local threshold and flatband voltages on the coordinate along the interface x is determined by the oxide thickness and the amount of charges in the oxide and/or at the device interface. Additionally, these voltages significantly change with the position around the drain/source junctions as a result of variations in impurity concentration and two-dimensional effects in the potential. As shown in Figure 4.2a, an interface of an n-type MOSFET with a local threshold lower than V_g is flooded with electrons supplied from the source or drain. Similarly, as shown in Figure 4.2b, an interface whose local flatband voltage is higher than V_g is filled with holes from the substrate. Furthermore, it is necessary to consider the dynamic response of free carriers around the drain contact when periodic pulses of high level V_gh and low level V_gl are applied to the gate electrode. When V_g equals V_gh, the interface between the drain and x₁ is flooded with electrons from the drain (see, Figure 4.3). Simultaneously, the holes are rapidly swept into the substrate. As a consequence, interface states in this region are emptied of holes. However, when V_g falls to V_gl, free electrons go back to the drain and holes flood back from the substrate again and reach x₂. This tendency is reflected in Figure 4.3. Therefore, the interface traps up to x₂ are filled with holes.

**Figure 4.3:** The response of the free carriers to the applied periodic gate pulse. At V_g=V_gh electrons flood into x₁ and holes return to the substrate. At V_g=V_gl electrons return to the drain and holes flood into x₂.

v_th	- thermal carrier velocity ( = 1× 10⁷cm/s),
τ_e/h	- time constant for electron (hole) trapping ( = 1/(2f)),
σ_e/h	- capture cross section for electrons (holes).