Enter the thermal voltage, the depletion layer capacitance, and the gate-oxide capacitance into the Subthreshold Swing Calculator. The calculator will evaluate the Subthreshold Swing.
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Subthreshold Swing Formula
Subthreshold swing describes how much gate voltage is needed to change the drain current by one decade while a MOSFET is operating in the subthreshold region. It is typically reported in mV/dec. A lower subthreshold swing means the transistor switches more sharply and requires less gate-voltage change for the same current reduction.
S = \ln(10) \cdot V_T \cdot \left(1 + \frac{C_d}{C_{ox}}\right)If the thermal voltage is written explicitly, the same equation becomes:
S = \ln(10) \cdot \frac{kT}{q} \cdot \left(1 + \frac{C_d}{C_{ox}}\right)This relationship shows that subthreshold swing depends on both temperature and device electrostatics. Increasing depletion capacitance raises the swing, while increasing gate-oxide capacitance lowers it.
Variable Definitions
| Variable | Meaning | Typical Unit |
|---|---|---|
| S | Subthreshold swing | mV/dec or V/dec |
| VT | Thermal voltage | mV or V |
| Cd | Depletion layer capacitance | F, mF, µF, nF |
| Cox | Gate-oxide capacitance | F, mF, µF, nF |
| k | Boltzmann constant | Constant |
| T | Absolute temperature | K |
| q | Electron charge | Constant |
Useful Rearrangements
Because the calculator can solve for any missing value, the formula can be rearranged in several ways:
V_T = \frac{S}{\ln(10)\left(1 + \frac{C_d}{C_{ox}}\right)}C_d = C_{ox}\left(\frac{S}{\ln(10)\,V_T} - 1\right)C_{ox} = \frac{C_d}{\frac{S}{\ln(10)\,V_T} - 1}How to Calculate Subthreshold Swing
- Determine the thermal voltage.
- Measure or estimate the depletion layer capacitance.
- Measure or estimate the gate-oxide capacitance.
- Compute the capacitance ratio Cd/Cox.
- Add 1 to that ratio.
- Multiply the result by ln(10) and the thermal voltage.
For the capacitance ratio to be valid, Cd and Cox must be entered in the same unit. The ratio itself is dimensionless, so the output unit of the swing follows the unit used for thermal voltage.
Example Calculation
If the thermal voltage is 30 mV, the depletion layer capacitance is 14 nF, and the gate-oxide capacitance is 30 nF, then:
S = \ln(10) \cdot 30 \cdot \left(1 + \frac{14}{30}\right)S \approx 101.3 \text{ mV/dec}This means the gate voltage must change by about 101.3 mV to change the drain current by one decade in the subthreshold region.
How to Interpret the Result
- Lower S: steeper turn-on behavior and better switching efficiency.
- Higher S: more gate-voltage change is needed for the same current change.
- Higher Cd: increases the body effect and worsens swing.
- Higher Cox: improves gate control and reduces swing.
- Higher temperature: increases thermal voltage and usually increases swing.
Body Factor Form
Many semiconductor texts group the capacitance term into a body factor:
n = 1 + \frac{C_d}{C_{ox}}S = \ln(10) \cdot n \cdot V_T
In this form, n captures how strongly the depletion region weakens gate control. The closer n is to 1, the closer the device is to ideal subthreshold behavior.
Ideal Room-Temperature Reference
At approximately room temperature, the thermal voltage is about 25.85 mV. If the depletion effect is negligible, the minimum long-channel MOSFET swing is near:
S_{min} \approx \ln(10) \cdot 25.85 \text{ mV} \approx 59.6 \text{ mV/dec}This reference point is useful when comparing real devices to ideal switching behavior.
Common Input Mistakes
- Using different capacitance units for Cd and Cox.
- Entering temperature directly when the calculator expects thermal voltage.
- Forgetting that subthreshold swing is expressed per decade.
- Assuming a larger gate-oxide capacitance increases swing, when it actually reduces it.
