Principle of Operation

• A four-terminal device obtained from an extension of the two-terminal MIS structure by diffusing or implanting two n+ regions into the p-type substrate in order to form two ohmic contacts called the source and the drain.
• A thin layer separates the third contact (gate) from the channel region of the device, and a fourth contact (body or bulk or substrate) is connected to the substrate.
• When a positive voltage is applied to the gate, a thin channel of electrons is created near the Si- interface, which provides a conducting link between the source and the drain => on state of the device.
• In the absence of a conducting channel, no electrical continuity between the drain and the source exists => off state of the device.
• The depletion regions between the p-type substrate and n+ regions and n-channel provide the required isolation from other devices fabricated on the same substrate.
• In the on state of the device, an applied drain-to-source bias creates a drift field in the channel, and electrons move from the source to the drain => thus a current is established.
• The electron concentration in the channel (and, thus, the channel conductance and device current) can be modulated by a variation in the gate voltage.
  
  Note: the C-V characteristic of this device shows low-frequency behavior (of the two-terminal MIS structure) up to a fairly high frequency (of the order of the inverse transit time of the carriers across the channel), since the heavily doped source/drain regions provide an infinite reservoir, from which the carriers can move into the channel, or to which they can escape from the channel.
  
  

The I-V Characteristic

The Gradual Channel Approximation (GCA)

• The GCA, proposed by Shockley, is used in order to calculate the I-V characteristic of the device.
• This approximation states that the rate of variation of the lateral field within the channel is much smaller than the rate of variation of the vertical field, i.e., , and the channel potential is assumed to be a gradually changing function of position.
  
  Note: This approximation actually states that the channel potential varies very little along the channel over a distance of the order of the insulator thickness , i.e., this requires << L, where L is the channel length.
  
  
  The gradual channel approximation (GCA):
  Fig (a) schematic comparison of the parallel and perpendicular electric fields in the channel, and
  Fig (b) qualitative potential profile in the channel.
  

• However, modern MOSFETs have extremely short channel lengths, and this requirement is not often met; thus, the GCA fails for most of modern MOSFETs, nevertheless its discussion is important.
• According to GCA, the charge induced at any position along the channel can be determined from the formulas derived for the MIS structure, provided the constant surface potential for an MIS structure is replaced by a variable channel potential in the expression for the surface charge density per unit area in the semiconductor.
  
  Assumption: The device is operating in the above threshold regime, i.e., the gate voltage is sufficiently large to create strong inversion throughout the channel.
  
  
• The induced surface charge density is then given by
  
  
  
  where is the insulator capacitance per unit area, and the term within the square brackets is the voltage drop across the insulator.
  
  Band diagrams at
  
  Fig (a) the source side and
  Fig (b) the drain side of the channel for the direction perpendicular to the Si- interface.
  
  Note: here we are considering an n-channel device, however, all the results are also applicable for a p-channel device, provided appropriate sign changes are made.

• Assume that the source is grounded , the drain is connected to a potential , and the substrate is connected to the source
  
  
• The density of the free electrons in the channel can be found from the difference between the total surface charge density and the depletion charge density , i.e.,
  
  
  
  
  
  Qualitative two-dimensional plot of the conduction band edge for an n-channel MOSFET.

• Note: at the source side of the gate where = 0, is given by
  
  however, elsewhere in the channel, the total band bending between the substrate and the surface is , since the induced n-channel/p-substrate junction is reverse biased by the .
  
  
• The band bending increases in the channel as one moves from the source to the drain, which leads to an increase in the width of the depletion layer and of the depletion charge density, thus, the exact expression for can be given by
  
  
  
• Since the drain current is carried entirely due to drift, its expression can be given by
  
  
  where is the low-field electron mobility, and W is the channel width.
  
  In writing this equation, it is assumed that the electron drift velocity is proportional to the component of the electric field parallel to the Si- interface, i.e., .
  
  Note: for short channel devices, this electric field may be sufficiently high to cause velocity saturation in the channel.
  
  
  Thus, the drain current equation can be rewritten as
  
  
  Note: is a function of ; thus, substituting the expression for in the above equation, noting that is a constant throughout the channel, and integrating it from the source, i.e.,x = 0 ( = 0) to the drain, i.e., x = L ,
  
  the following I-V characteristic is obtained:
  
  
  
  
• This model is known as the Shockley model.
• This expression for is valid only if the inversion layer exists even at the drain side of the gate, i.e.,
• The condition is referred to as the pinch-off condition, and it occurs at the drain side of the gate when
  
  
  
  
• As (first-order approximation), where is the threshold voltage corresponding to the onset of strong inversion, and is given by
  
  
  
  
• In the presence of a substrate bias , the expression for gets modified to
  
  
  
  
  
  
  The band diagram of an n-channel MOSFET along the direction perpendicular to the Si- interface for a negative substrate bias.
  
  Note: is the voltage difference between the inversion layer at the source end and the substrate contact, and its sign should be such that it never forward biases the inversion layer-substrate junction (a small forward bias, much less than may be allowed in certain cases).