Lecture-45 MOS Field-Effect-Transistors 7.4. Threshold voltage In this section we summarize the calculation of the threshold voltage and discuss the dependence of the threshold voltage on the bias applied to the substrate, called the substrate bias effect. 7.4.1. Threshold voltage calculation The threshold voltage equals the sum of the flatband voltage, twice the bulk potential and the voltage across the oxide due to the depletion layer charge, or: (7.4.1) where the flatband voltage, VFB, is given by: (7.4.2) With (7.4.3) and (7.4.4) The threshold voltage of a p-type MOSFET with an n-type substrate is obtained using the following equations: (7.4.5) where the flatband voltage, VFB, is given by: (7.4.6)
With (7.4.7) and (7.4.8) The threshold voltage dependence on the doping density is illustrated with Figure 7.4.1 for both n-type and p-type MOSFETs with an aluminum gate metal. Figure 7.4.1 Threshold voltage of n-type (upper curve) and p-type (lower curve) MOSFETs versus substrate doping density. : The threshold of both types of devices is slightly negative at low doping densities and differs by 4 times the absolute value of the bulk potential. The threshold of nmosfets increases with doping while the threshold of pmosfets decreases with doping in the same way. A variation of the flatband voltage due to oxide charge will cause a reduction of both threshold voltages if the charge is positive and an increase if the charge is negative. 7.4.2. The substrate bias effect The voltage applied to the back contact affects the threshold voltage of a MOSFET. The voltage difference between the source and the bulk, VBS changes the width of the depletion layer and therefore also the voltage across the oxide due to the change of the charge in the depletion region. This results in a modified expression for the threshold voltage, as given by:
(7.4.9) The threshold difference due to an applied source-bulk voltage can therefore be expressed by: (7.4.10) Where is the body effect parameter given by: (7.4.11) The variation of the threshold voltage with the applied bulk-to-source voltage can be observed by plotting the transfer curve for different bulk-to-source voltages. The expected characteristics, as calculated using the quadratic model and the variable depletion layer model, are shown in Figure 7.4.2. Figure 7.4.2 Square root of ID versus the gate-source voltage as calculated using the : quadratic model (upper curves) and the variable depletion layer model (lower curves). First, we observe that the threshold shift is the same for both models. For a device biased at the threshold voltage, drain saturation is obtained at zero drain-to-source voltage so that the depletion layer width is constant along the channel. As the drain-source voltage at saturation
is increased, there is an increasing difference between the drain current as calculated with each model. The difference however reduces as a more negative bulk-source voltage is applied. This is due to the larger depletion layer width, which reduces the relative variation of the depletion layer charge along the channel. Example 7.3 Calculate the threshold voltage of a silicon nmosfet when applying a substrate voltage, VBS = 0, -2.5, -5, -7.5 and -10 V. The capacitor has a substrate doping Na = 1017 cm-3, a 20 nm thick oxide ( ox = 3.9 0) and an aluminum gate ( M = 4.1 V). Assume there is no fixed charge in the oxide or at the oxide-silicon interface. Solution The threshold voltage at VBS = -2.5 V equals: Where the flatband voltage without substrate bias, VT0, was already calculated in example 6.2. The body effect parameter was obtained from: The threshold voltages for the different substrate voltages are listed in the table below. Saturation or active mode[23][24] When VGS > Vth and VDS ( VGS Vth ) The switch is turned on, and a channel has been created, which allows current to flow between the drain and source. Since the drain voltage is higher than the source voltage, the electrons spread out, and conduction is not through a narrow channel but
through a broader, two- or three-dimensional current distribution extending away from the interface and deeper in the substrate. The onset of this region is also known as pinch-off to indicate the lack of channel region near the drain. Although the channel does not extend the full length of the device, the electric field between the drain and the channel is very high, and conduction continues. The drain current is now weakly dependent upon drain voltage and controlled primarily by the gate source voltage, and modeled approximately as: The additional factor involving λ, the channel-length modulation parameter, models current dependence on drain voltage due to the Early effect, or channel length modulation. According to this equation, a key design parameter, the MOSFET transconductance is: where the combination V ov = V GS V th is called the overdrive voltage, [25] and where V DSsat = V GS V th (which Sedra neglects) accounts for a small discontinuity in which would otherwise appear at the transition between the triode and saturation regions. Another key design parameter is the MOSFET output resistance r out given by:. r out is the inverse of g DS where. I D is the expression in saturation region. If λ is taken as zero, an infinite output resistance of the device results that leads to unrealistic circuit predictions, particularly in analog circuits. As the channel length becomes very short, these equations become quite inaccurate. New physical effects arise. For example, carrier transport in the active mode may become limited by velocity saturation. When velocity saturation dominates, the
saturation drain current is more nearly linear than quadratic in V GS. At even shorter lengths, carriers transport with near zero scattering, known as quasi-ballistic transport. In the ballistic regime, the carriers travel at an injection velocity that may exceed the saturation velocity and approaches the Fermi velocity at high inversion charge density. In addition, drain-induced barrier lowering increases off-state (cutoff) current and requires an increase in threshold voltage to compensate, which in turn reduces the saturation current. Body effect[edit] Band diagram showing body effect. V SB splits Fermi levels F n for electrons and F p for holes, requiring larger V GB to populate the conduction band in an nmos MOSFET The occupancy of the energy bands in a semiconductor is set by the position of the Fermi level relative to the semiconductor energy-band edges. Application of a source-to-substrate reverse bias of the source-body pn-junction introduces a split between the Fermi levels for electrons and holes, moving the Fermi level for the channel further from the band edge, lowering the occupancy of the channel. The effect is to increase the gate voltage necessary to establish the channel, as seen in the figure. This change in channel strength by application of reverse bias is called the 'body effect'. Simply put, using an nmos example, the gate-to-body bias V GB positions the conductionband energy levels, while the source-to-body bias V SB positions the electron Fermi level near the interface, deciding occupancy of these levels near the interface, and hence the strength of the inversion layer or channel.
The body effect upon the channel can be described using a modification of the threshold voltage, approximated by the following equation: where V TB is the threshold voltage with substrate bias present, and V T0 is the zero-v SB value of threshold voltage, is the body effect parameter, and 2φ B is the approximate potential drop between surface and bulk across the depletion layer when V SB = 0 and gate bias is sufficient to insure that a channel is present. [26] As this equation shows, a reverse bias V SB > 0 causes an increase in threshold voltage V TB and therefore demands a larger gate voltage before the channel populates. The body can be operated as a second gate, and is sometimes referred to as the "back gate"; the body effect is sometimes called the "back-gate effect". [27] Application Digital integrated circuits such as microprocessors and memory devices contain thousands to millions of integrated MOSFET transistors on each device, providing the basic switching functions required to implement logic gates and data storage. Discrete devices are widely used in applications such as switch mode power supplies, variable frequency drives and other power electronics applications where each device may be switching hundreds or thousands of watts. Radio-frequency amplifiers up to the UHF spectrum use MOSFET transistors as analog signal and power amplifiers. Radio systems also use MOSFETs as oscillators, or mixers to convert frequencies. MOSFET devices are also applied in audiofrequency power amplifiers for public address systems, sound reinforcement and home and automobile sound systems. The basic principle of this kind of transistor