JPH03152483A - Extracting method for characteristic parameter of field effect transistor element - Google Patents

Abstract

PURPOSE:To speedily extract a parameter easy to calculate while maintaining accuracy by first determining a true value of an unfixed constant from the quantitative evaluating value of a deviation between a temporary linear model including a temporary value of the unfixed constant and an actually-measured value. CONSTITUTION:Based on a nonlinear mathematic model of a field effect transistor, a value of an unfixed constant in a formula is obtained by mathematic and statistical means from a measuring value of a current, voltage or the like, thereby to extract a characteristic parameter of an element such as a threshold voltage, conductance or the like. At this time, a temporary value is given to a part of the unfixed constants in the model formula, so that the nonlinear formula is turned to be linear, and the remaining unfixed constants are temporarily determined. A true value of the unfixed constant to which the temporary value is first given is determined from the quantitative evaluating value of a deviation between the temporary linear model including the temporary value and the actually-measuring value, and then the nonlinear formula is made truly linear. Accordingly, true values of the remaining unfixed constants can be determined without using the general iteration method.

Description

Detailed Description of the Invention [Industrial Application Field] The present invention relates to device characteristic parameters for quantitatively evaluating the electrical characteristics of a semiconductor device manufactured singly or a field effect transistor constituting an integrated circuit. Regarding the extraction method.

[Conventional technology]

The basic electrical characteristics of field effect transistors are generally modeled by several mathematical equations. Device characteristic parameters such as threshold voltage and conductance appear as undetermined constants in these formulas, so in order to quantitatively evaluate the electrical characteristics of a field effect transistor, current,
Mathematics and statistical operations are required to determine these undetermined constants by fitting the model formula to actual measured values such as voltage using a parameter fitting method.The above model formula is nonlinear with respect to the undetermined constants. Therefore, the steepest descent method or Gaussian method is used for parameter fitting.
Various iterative calculation methods such as the s-Newton method or combinations thereof are widely used.

[Problem to be solved by the invention] However, the iterative calculation method using conventional technology has drawbacks such as requiring a long calculation time and making it difficult to set the initial value because the answer may not be obtained due to divergence depending on the initial value. For example, it is almost impossible to apply it to the daily management of mass-produced products.

For this reason, parameter extraction using conventional techniques is limited to purposes such as research and development, and for daily management of mass-produced products, the model itself is simplified and pseudo-linearized. Methods such as the so-called square characteristic method and the maximum slope method, which are simple but have inferior model accuracy and whose parameters have a small amount of information, have been used.

The present invention solves the problems associated with the conventional field effect transistor element characteristic parameter extraction method.
The purpose is to provide a parameter extraction algorithm that is easy to calculate and fast, without sacrificing the inherent accuracy of the field effect transistor characteristic model.

[Means for Solving the Problem] The field effect transistor device characteristic parameter extraction method of the present invention is based on a nonlinear mathematical model of a field effect transistor, and uses mathematical and statistical means from actual measured values such as current and voltage. In this method, device characteristic parameters such as threshold voltage and conductance are extracted by determining the values of undetermined constants in the formula of the mathematical model. By linearizing the mathematical expression, the values of the remaining undetermined constants are tentatively determined, and the relationship between the linear tentative model containing the "tentative values" and the actual measured values is calculated.
First, determine the true value of the undetermined constant to which a "temporary value" is initially given based on the quantitative evaluation value of "deviation", that is, the minimum evaluation error condition, and use this to truly linearize the nonlinear formula. It is characterized by determining the true value of the remaining undetermined constants without using the general iterative calculation method.

[Operation] According to the above method of the present invention, an appropriate one is selected from among the undetermined constants in the formula of the mathematical model, and a "temporary value" is given to it to linearize the nonlinear formula. This allows the values of the remaining undetermined constants to be determined by simple linear parameter fitting such as the least squares method. Since the evaluation error in linear parameter fitting is expressed as the sum of squared deviations, the relationship between the "tentative value" and the evaluation error is a simple function, and the minimum evaluation error is calculated from the "tentative value" of several points. can be determined by simple calculations. Also, as a value corresponding to this minimum value, first a "temporary value"
Once the true value of the undetermined constant given is determined, the true values of the remaining undetermined constants can also be determined by the same operation as linearization using [temporary values].

[Example] The present invention will be described below based on an example of extraction of threshold voltage, transconductance, and field drop coefficient based on a bipolar pre-operating region model of a field effect transistor.

Generally, the bipolar pre-operating region model of a field effect transistor is expressed by the following formula.

When Va, >> Vos, Ion = [no/(1+e (Vas-VyH
l)](]Van-Vvh-Vos/21Vo*H
H+ (1)Co “uocoyVI*tt/L,*
tt...(2) However, VDn is the drain voltage and is a known constant, Vow is the gate voltage, Ins is the current between the drain and source, and the undetermined constants Vd and θ are the threshold voltage and electric field drop coefficient, respectively. Transconductance is β. (2) Since it is expressed as o1, in order to determine the above three parameters, it is sufficient to find three undetermined constants, VTH1θ and β0. Note that in equation (2), μ.

is the mobility, Cox is the oxide film capacitance+Lft is the effective channel width, and L and ff are the effective channel lengths.

Transforming equation (1), (Van-Vy, I-Vos/21/Ls=(θ/βo
Vosl (Van-Vyo) + (1/BoV
osl (3) Kokote, X = Vas-VvH, Y = fVa*-
By performing variable conversion with VyH-Vos/2)/Ls, it can be seen that equation (3) becomes a linear equation of X and Y*V
Fix ow to a constant value and change VaS appropriately to obtain I. Measure the value of 3 and obtain the triode region data V.

, (1), Obtain IomNl *If the value of VTll is temporarily given, X +++ and Ycl+ corresponding to each data can be calculated, and the undetermined coefficient (θ/β) of equation (3) can be calculated by the -order least squares method. ■., ) and (1/β.vo,) can be temporarily obtained. At this time, the sum of squared deviations from regression S
is expressed as S=Σ(YLlji(θ/β. Voml This is a quadratic equation of VTH, and the relationship between the two is expressed by the following equation.

5=aV7H" + bVtn + c...
(5) Therefore, perform the above linear regression calculation for each provisional VvHIJI set at several points, and calculate the corresponding S Nl
is calculated, and the coefficient a of equation (5) is calculated using the second-order least squares method.
, b, c, and when S takes the minimum value, VTM (7
) value, i.e. VT, =-b/2a is true (7) VTHtft! .

By substituting this value into Equation (3) and executing the above linear regression calculation again, (θ/β.va, l and (1/β.Vos
The true value of l has been found, that is, the values of β0 and θ have been determined.

In order to demonstrate the simplicity of calculation, the above operations will be explained along the flowchart shown in FIG. First, ■ Number of measurement points n
and ``determine the number m of temporary VTHJ 0 Usually, n is 20~
30. About 5 points is sufficient for m. Next, measure the data at n points in the diode operating region and calculate VO@lIl+l6s.
Let it be H1. However, a V where i=1.2-..., n
The relationship between as+u and log+u is as shown in Figure 21
Assuming that the "temporary VTHJ of m points is v7□," perform the linear regression calculation based on equation (3) for each "temporary VTHJ", and calculate the corresponding S IJI (j=1+2,..., m
). In other words, the linear regression calculation based on equation (3) is performed m times.

The relationship between VTIIIJI and 5IJI is as shown in FIG.

Next, (2) perform a quadratic regression calculation based on equation (5) for the m sets of Vto IJI and SIJ+ obtained, and obtain the true VTH from the minimum condition of S; By feeding this back into equation (3) and performing linear regression calculation,
β. and θ are obtained.

As described above, the method according to the present invention requires only a few times of -order and quadratic regression calculations, and is much more efficient than the conventional iterative method of repeating the calculation of the 0-by-m partial differential coefficient matrix at least ten times. It is simple.

[Effects of the Invention] As described above, according to the present invention, it is not only necessary to calculate a complex partial differential coefficient matrix (Jacobian) for each iteration, but also to adjust the initial value by trial and error while considering convergence. extremely fast compared to the iterative calculation method that requires
It is possible to simply and stably extract the element characteristic variations of field effect transistors, and since the model itself is not simplified, it has the excellent effect of being able to obtain the same level of accuracy as the iterative calculation method described above. has.

Flowchart showing an embodiment of the device characteristic parameter extraction method. FIG. 2 is a current-voltage characteristic diagram in the triode operation region of the field effect transistor in the embodiment. FIG. A diagram showing a relationship with a temporary VTHJ.

that's all

Claims (1)

[Claims] Based on a nonlinear mathematical model of a field-effect transistor, device characteristic parameters such as threshold voltage and conductance can be determined by calculating the values of undetermined constants in the formula from actual measured values such as current and voltage using mathematical and statistical means. In this method, the values of the remaining undetermined constants are tentatively determined by linearizing the nonlinear formula by giving "temporary values" to some of the undetermined constants in the formula of the mathematical model, and ``Difference'' between the linear temporary model containing the ``value'' and the actual measured value.
From the quantitative evaluation value of , a method for extracting characteristic parameters of a field effect transistor, characterized in that the true value of a remaining undetermined constant is determined without using a general iterative calculation method.