pMOSFETs

The pMOSFETs are tested in the lab using ICS, an automated testing program. For more information, see Testing.

Theory

We can easily determine several characteristic parameters for the FETs, such as g_m (transconductance parameter), V_T (threshold voltage), L_eff (effective length), (channel length modulation factor), and (body-bias coefficient).

There are three ways to determine threshold voltage.

• The first method utilizes a plot of transconductance, g_m vs V_G for various V_D. By extrapolating the curves when the device is in saturation to the x-axis, we can determine the threshold voltage.
  
  In saturation:

  g_m = Δ I_D / Δ V_G ~ (W / L) μ C_i (V_g-V_t)
  
  g_m = Δ I_D / Δ V_G ~ k (V_g - V_t),

  where k is a constant including the channel length, L, width, W, capacitance per unit area, C_i, and the average mobility, μ.
  
  Therefore, (in saturation), at g_m=0, V_g=V_t.
  
  

• Another method is to plot the square root of I_D vs. V_g. By extrapolating the saturated portion of the curves to the x-axis, we can determine the threshold voltage.
  
  In saturation:

  I_D = (1/2) k (V_g- V_t)²,

  so sqrt(I_D) = sqrt(k/2) (V_g-V_t), and at sqrt(I_D) = 0, V_g = V_t.
  
  

• The third method to determine threshold voltage is based on the transistor operating in the linear region. By plotting ID vs. [V_g-V_D/2], we can read the threshold voltage off the x-intercept again.

  I_D = k [(V_g - V_t)V_DS - (1/2) V_DS²]
  
  I_D = k [(V_g-V_t)-V_DS/2] V_DS = k [(V_g - V_DS/2) - V_t] V_DS

  Then, when I_D = 0, (V_g-V_DS/2) = V_t

The body bias affects the threshold voltage, also. By determining the threshold voltages for several body bias conditions, we can fit the V_t and V_SB into the following:

V_t = V_t0 + γ[(V_SB + 2φF)^1/2 - (2φF)^1/2]

V_t ~ V_t0 + γ (V_SB)^1/2, when V_SB >> 2φF

From the traditional plot of I_D vs. V_D , we can determine the channel length modulation factor.

I_D ~ I_Dsat (1 + λ V_DS)

where the x-intercept of the extrapolation of the I-V curves in saturation is the reciprocal of the channel length modulation factor.

If two FETs are identical except for their channel lengths, then the effective channel lengths can be obtained from measurements of I_d vs. V_g in saturation (|V_d| > |V_g - V_t|).

As in the first method above for determining Vt:

in saturation:

g_m = d(I_d )/ d(V_g) ~ (W / L) μ C_i (V_g-V_t),

with the channel length, L, width, W, capacitance per unit area, C_i, and the average mobility, μ.

Then: g_m' = d(g_m) / d(V_g) ~ (W / L) μ C_i.

If the channel width, average mobility, and insulator capacitance per unit area are the same for the two devices, then the ratios of the values of g_m ' in saturation for the two devices is the ratio of the inverse of their channel lengths:

g_m1' / g_m2' = L_eff2 / L_eff1

where the effective channel length L_eff = L_design - ΔL.

If the causes of delta L are the same for both channel lengths (it is a reasonable assumption that they would be for process related changes in the channel length), then it is possible to find ΔL, since g_m1', g_m2', L_design1, and L_design2 are known.

Further Information

• Schroder, Dieter K. Semiconductor Material and Device Characterization, New York: John Wiley & Sons, 1990.
• Streetman, Ben G. Solid State Electronic Devices. Englewood Cliffs, NJ: Prentice- Hall, Inc. 1980.
• Martin G. Buehler, VLSI Electronics Microstructure Science, Norman G. Einspruch and Graydon B. Larrabee, Eds., (Academic Press, New York, 1983), Vol. 6, pp. 540-545.