• where are the longitudinal and transverse components of the electric field respectively, is the semiconductor dielectric permittivity, and is the charge density in the semiconductor
• The charge density consists of a mobile charge density and a depletion charge density is the substrate doping density.
  
• Integrating Eqn.(5.92) with respect to y from the semiconductor-insulator interface through the effective channel thickness , one obtains



where over the channel thickness and is the electron sheet density in the channel.


• At low substrate doping and with the device biased in strong inversion such that the vertical electric field at will be small compared to the vertical field at the interface, in which case can be neglected in Eqn.(5.93).
• Making the substitution where V is the average of the potential over the cross-section of the channel, Eqn.(5.93) can be written as

  

• The electric field at the interface is obtained by equating the electric displacement at the two sides of the semiconductor-insulator interface, leading to

  

• From the conditions of velocity saturation and current continuity, the electron sheet density should be a constant in the saturated region, and its value can therefore be determined at the boundary point where the GCA is still valid; thus,

  

  where is the threshold gate voltage, given by Eqn.(5.9).
  

• The combination of Eqns.(5.94) to (5.96) and (5.9) leads to the following second order differential equation for the channel potential in the saturated region:

  

  where is the characteristic length in the saturation region and is given by

  

• It should be noted that the solution of Eqn.(5.97) is very sensitive to the magnitude of the characteristic length for the saturated region.
• In comparisons with experimental data, it is therefore convenient to treat as a fitting parameter rather than using Eqn.(5.98), which itself is a result of rough estimates and approximations.
  
• The general solution of Eqn.(5.97) can be written in the following form:

  

• The coefficients A and B are determined from the boundary conditions, i.e., from the requirements that with the values respectively, leading to

• A relationship that links to the drain-source voltage is obtained by considering Eqn.(5.99) at the drain side of the channel:

  

  where with L being the gate length.

• Equation (5.100) can be solved with respect to resulting in

  

• Combining Eqns.(5.99) and (5.101), we find

  

• A self-consistent determination of is based on a model for the non-saturated part of the channel
• Owing to the complexity of Eqns.(5.99) to (5.101), it is extremely difficult to derive explicit, analytical expressions for important electrical properties, e.g., the I-V characteristics, using the present model for the saturation region.
• However, a numerical solution can readily be obtained which may serve as a physically based reference for simpler, more empirical models.
• Nonetheless, it is possible to simplify the equations somewhat in certain limiting cases.

• For i.e., just beyond the onset of saturation, it can be written to the first order in

  

• For > i.e., in deep saturation, we have

  

• From Eqn.(5.105), we obtain

  

• The solutions obtained represent only an approximation of the actual potential distribution in the saturation region, however, they clearly show that the potential rises exponentially with distance inside this region.
• Based on this result and on numerical simulations of the potential in the saturation region, a simplified empirical expression linking the drain-source voltage to the length of the saturation region has been proposed:

  

  where the constant is determined from the condition of continuity in the drain conductance.

Subthreshold Region

• Area of considerable research for the last few years due to low-voltage/low-power analog/digital circuit operation, where most of the devices operate very near the threshold region and some may even enter subthreshold operation.
• In the off state of the MOSFET, a finite drain current flows through the device, since the channel is weakly inverted, and also that there is a finite injection rate of carriers from the source into the channel.
  
• In the subthreshold regime in short channel devices, a drain voltage induces lowering of the energy barrier between the source and the channel, this effect is called the drain induced barrier lowering (DIBL) effect.
• DIBL causes excess injection of charge carriers from the source into the channel, and gives rise to an increased subthreshold current.
  
• This current is detrimental to both as well as digital operation.
  
• Figure 5.27 shows qualitatively the band diagram and the potential distribution at the interface in the channel,
  
• At the interface, the channel consists of three regions, the source-channel junction with length the drain-channel junction with length and the middle region of length
  
• At the interface potential in the middle of the channel can be taken to be approximately constant.
  
• A drain-source bias gives rise to a positive contribution V(x) to the channel potential => the minimum in the interface potential will be localized at the source side of the channel at
  
• Associated with the shift in the potential minimum, there will be a reduction in the interface energy barrier between the source and the channel by this is the so-called drain induced barrier lowering (DIBL) effect.



• DIBL is a short channel effect, which causes a drain voltage induced shift in the threshold voltage.
  
• The expression for the drain current in the drift-diffusion form can be given as

• where is the potential of the channel region referred to the potential of the source.

  

  
  Fig.5.27 Band diagram and potential profile at the semiconductor insulator interface of an n-channel MOSFET. The symmetrical profiles correspond to and the asymmetrical profiles to The figure indicates the origin of the Drain Induced Barrier Lowering (DIBL) effect.

• It is also assumed that the longitudinal electric field in the channel is sufficiently small (except for the junction region near the drain) such that velocity saturation can be neglected.
• Multiplying Eq.(5.108) by the integrating factor the right hand side of this equation can be made into an exact derivative, and a subsequent integration from source to drain yields (assuming that the current density remains independent of x):

  

  where n(L) = n(0) equals the drain and source contact doping density (neglecting degeneracy).

• With the source contact as the potential reference, at the source end, and at the drain end, where is the intrinsic drain-source voltage.
• When the device length is not too small, the channel potential can be taken to be independent of x over a portion of the channel length, i.e., and the integral in the denominator of Eq.(5.109) is determined by the contribution from this portion of the channel.
• Note: from Fig.5.27, the length of this section is approximately equal to and the current density can be expressed as

• For long channel devices, and the drain current can be obtained by integrating the current density over the cross-section of the conducting channel, thus,

  

  where is the effective channel thickness, and is the constant potential at the semiconductor-insulator interface, and is defined relative to the source electrode.
• Hence, although the interface potential relative to the interior of the p-type substrate is the built-in potential between the source contact and the substrate) is positive, will be negative for n-channel MOSFETs.
  
• At threshold, the interface potential in the channel relative to the source can be expressed as is the potential relative to the interior of the substrate at threshold
• For simplicity, it is assumed that the substrate is shorted to the source; the effects of a substrate-source bias are found simply by replacing of course, such a replacement is only valid for negative or small positive values of , a positive comparable to would lead to a large substrate leakage current.
• Below threshold, the interface potential can be written as
  
• All these equations predict that the subthreshold drain current decreases nearly exponentially with decreasing this current is practically independent of the drain-source voltage.
  
• The effective channel thickness is given by

  

• Note: this expression in only valid when i.e., in the depletion and weak inversion regions, and this condition is fulfilled for values of the drain current that are many orders of magnitude smaller than the threshold current.
  
• For short channel length devices, L should be replaced by as discussed earlier. 5.9.4 Drain Induced Barrier Lowering (DIBL)
• While dealing with short channel effects, the effective gate depletion charges were distributed evenly along the channel in order to estimate the threshold voltage shift.