A model parameter extraction method under a BTI model framework

Abstract

The application discloses a model parameter extraction method under a BTI model framework, which comprises the following steps: a. extracting upper and lower limits of a time acceleration factor of a semiconductor device; b. optimizing the upper and lower limits of the time acceleration factor by using a PSO algorithm; and c. obtaining the optimized upper and lower limits of the time acceleration factor. The cross-optimization method is adopted to improve the parameter extraction and model decomposition precision of the new BTI model framework, and the method can also be downwardly compatible to extract the parameters of a traditional RD model and a TD model. Compared with the traditional method, the method has higher global search capability, and can extract the BTI model parameters without special and complex data preprocessing.

Description

TECHNICAL FIELD

[0001] The application specifically relates to a model parameter extraction method under a BTI model framework. BACKGROUND

[0002] With the shrinking of MOS device size, the BTI (Bias Temperature Instability) characteristics of semiconductor devices have an increasingly significant impact on device characteristics and circuit performance, and the physical mechanism of BTI is becoming more and more complex with the deepening of research. Under the new BTI model framework, the device reliability problem is no longer caused by a single physical mechanism, but is caused by the joint contribution of fast traps and slow traps. Moreover, most of the BTI device-level modeling is directly based on DC test results, which causes the dimension of the multi-trap model parameters to increase, leading to difficulty in parameter extraction and considerable error in the evaluation of the impact of BTI on circuits. Therefore, more accurate model parameter extraction is particularly important for the evaluation of BTI characteristics.

[0003] The method currently adopted is as follows: first, the model parameters are extracted one by one using sufficient data; and then, the parameter optimization is performed through upper and lower limits and error rate analysis in the model parameter extraction

[0004] The framework as shown in Figure 1 can be used to extract the RD model parameters, and the model parameters are roughly extracted for the first time and the upper and lower limits of each parameter are obtained, and the parameter optimization is performed within the range of each parameter for the second time.

[0005] Under the static stress of the MOS transistor, the nonlinear curve of the RD model is as follows:

[0006]

[0007] In the above RD model parameter extraction steps, first, the least square method is used to set the time index in the RD model to 1 / 6, and then the parameter values are fitted out respectively after linear conversion according to each group of degradation values. The first group of solutions of the model parameters is calculated by using the model parameter expression and the fitted parameter values.

[0008] The time index of the RD model is set to 1 / 4 again, and the second group of solutions is obtained by repeating the steps, and the range of the parameter solution is determined according to the two groups of solutions, and the range of each parameter solution is required to include the interval where the two solutions are located.

[0009] In the solving process, the expressions of the solutions related to the experimental data and the working conditions for different parameters are listed, such as the extraction of the temperature activation energy Ea, and the feasibility of parameter extraction and optimization can be obtained by equation derivation:

[0010]

[0011] The genetic algorithm is used for parameter optimization, a target function is determined, constraints are set by ranges of each model parameter, and finally the algorithm is executed to obtain accurate values of the model parameters.

[0012] However, the above method has the following disadvantages: 1. The method of extracting model parameters one by one using sufficient data requires that the data correspond to each model parameter one by one, which cannot be applied to parameter extraction between less data and more complex models. 2. The RD model parameter extraction and optimization method can make the extracted parameters more accurate, but cannot be applied to model decomposition of two types of traps, i.e. decoupling problem. 3. The solution obtained by the algorithm used in the parameter optimization process is still relatively rough. SUMMARY

[0013] The main purpose of the present application is to provide a method for extracting parameters of MOS transistor BTI effect fast trap and slow trap model based on particle swarm algorithm, which can accurately perform model decomposition.

[0014] To achieve the above-mentioned purposes, the technical scheme adopted by the present application comprises: a model parameter extraction method under a BTI model framework, which comprises the following steps:

[0015] a. Extracting the upper and lower limits of the time acceleration factor of the semiconductor device;

[0016] b. Optimizing the upper and lower limits of the time acceleration factor using the PSO algorithm;

[0017] c. Obtaining the optimized upper and lower limits of the time acceleration factor.

[0018] Another optimization scheme, wherein the semiconductor device is a MOS transistor.

[0019] Another optimization scheme, in step a, the degradation values of the threshold voltage of the MOS transistor with respect to time under different gate voltage stress conditions and the degradation values of the threshold voltage with respect to time under different temperature stress conditions are obtained by establishing a nonlinear curve of the fast / slow trap model.

[0020] The nonlinearity of the fast / slow trap model is:

[0021] ΔV th =A*t B +C*log10(1+D*t) (1)

[0022] Wherein

[0023] A∝exp(d1 V g )exp(-E a1 / KT), C∝exp(d2 V g )exp(-E a2 / KT) (2).

[0024] Another optimization scheme, for the fast trap model, in the double logarithmic coordinates, using short-term characteristics, for formula (3)

[0025] ΔV th = a1 * t a2 + b1 * log10(1 + b2 * t) (3)

[0026] For each degradation curve, the particle swarm algorithm is used to fit the rough value a2, b2, and a plurality of a2, b2 values corresponding to different stress conditions are obtained. Due to the characteristics of the fast trap growth, the maximum a2 value is taken as the upper limit of the slow trap time acceleration factor B in formula (1), and the minimum b2 value is taken as the lower limit of the fast trap time acceleration factor D in formula (1);

[0027] For the slow trap model, in the bilinear coordinates, for formula (3), the particle swarm algorithm is used to fit the rough value a2, b2 for each degradation curve, and a plurality of different a2, b2 values are obtained. The minimum a2 value is taken as the lower limit of the slow trap time acceleration factor B in formula 3, and the maximum b2 value is taken as the upper limit of the fast trap time acceleration factor D in formula (1). Thus, the upper and lower limits of the slow trap time acceleration factor B and the fast trap time acceleration factor D are obtained. According to the gate voltage stress degradation curve family, the upper and lower limits of the slow trap time acceleration factor B and the fast trap time acceleration factor D are used to extract d1, d2 in formula (4);

[0028]

[0029] According to the temperature stress degradation curve family, the upper and lower limits of the slow trap time acceleration factor B and the fast trap time acceleration factor D are used to extract the temperature acceleration factor E in formula (4) a1 ,E a2 , to obtain the upper and lower limits of d1, d2, the slow trap term proportion coefficient c1, and the fast trap term proportion coefficient; wherein t: pressure time; Vg: gate voltage; T: temperature; K: Boltzmann constant; E a1 : slow trap temperature acceleration factor; E a2 : fast trap temperature acceleration factor.

[0030] Another optimization scheme, in step b, using the gate voltage stress degradation curve family, according to the range of the parameters d1, d2, B, D, and the term coefficient of formula (4) obtained, setting the constraint condition, using the particle swarm algorithm, using the traversal or the direction with small error rate of the upper and lower limits to optimize the model parameters, obtaining the optimized value of the parameters d1, d2, B, D; using the temperature stress degradation curve family, according to the model parameters E a1 ,E a2The range of B, D is set, constraints are used, the particle swarm algorithm is used, and the model parameters are optimized in the direction with small error rate of traversal or upper and lower limits, to obtain the optimized value of the parameters in formula (4) ; two different B, D values are obtained in two optimizations, and the value range of the parameters B and D is re-constructed by using the two groups of data, and the upper and lower limits are expanded outward by 15-30% with the difference as the standard; all data groups of curves are used, and the new B, D value constraints are used to accurately extract the parameters c1, c2, B, and D in formula (3).

[0031] Compared with the prior art, the application has the following advantages: the cross-optimization method is adopted, the extraction and model decomposition precision of the new BTI model framework parameters are improved, and the traditional RD model and TD model parameters can also be extracted downwardly compatible; compared with the traditional method, the global search ability is higher, and the BTI model parameters can be extracted without special complex data preprocessing. BRIEF DESCRIPTION OF DRAWINGS

[0032] Figure 1 RD model parameter extraction and optimization method for NBTI effect

[0033] Figure 2 PSO algorithm flow

[0034] Figure 3 Model parameter extraction and optimization flow DETAILED DESCRIPTION

[0035] The embodiments of the application will be described below in detail with reference to the accompanying drawings Figure 2 Figure 3 Flow, the embodiments of the application will be described in detail:

[0036] The application is used for model parameter extraction and model decomposition under the new model framework of MOS tube BTI (bias temperature instability) characteristics, including model fitting based on the particle swarm algorithm, model parameter constraint boundary acquisition and adjustment, and model parameter optimization.

[0037] Referring to Figure 2 , the PSO (particle swarm optimization algorithm) algorithm flow is as follows:

[0038] A. Initialize a group of particles (population size m), including random position and speed;

[0039] B. Define the fitness function to evaluate the fitness of each particle;

[0040] C. For each particle, compare its fitness value with the individual extreme value Pbest it has experienced, and if it is better, take it as the current optimal individual extreme value;

[0041] D. For each particle, compare its fitness value with the global experienced optimal individual extreme value, if better, reset the index number of the global optimal value Gbest;

[0042] E. Update the velocity and position of the particle;

[0043] F. If the end condition (usually a good enough fitness value or reaching a preset maximum iteration number) is not reached, return to B.

[0044] Referring to Figure 3 , the clear model parameter extraction optimization process is as follows:

[0045] (I). The time acceleration factor and its constraint condition are obtained by using the data in the double logarithmic coordinates and linear coordinates;

[0046] (II). The voltage and temperature acceleration factor constraint conditions are obtained by using the degradation data of the gate voltage and temperature change and the upper and lower limits of the time acceleration factor, respectively;

[0047] (III). The gate voltage and temperature acceleration factors are optimized by using the degradation data of the gate voltage and temperature change, respectively;

[0048] (IV). The function proportional coefficient and the time acceleration factor are optimized and extracted by using the degradation data in the linear coordinates.

[0049] The present application gives the following examples.

[0050] Example 1: BTI model decomposition under static stress is realized by using an optimization algorithm

[0051] The BTI model (formula 1) based on fast / slow traps contains the following parameters:

[0052] t: stress time, unit: s

[0053] Vg: gate voltage, unit: V

[0054] T: temperature, unit: K

[0055] K: Boltzmann constant, 8.6173e-5eV / K

[0056] c1: slow trap item proportional coefficient, constant, process related

[0057] c2: fast trap item proportional coefficient, constant, process related

[0058] d1: slow trap gate voltage acceleration factor, constant, process related, 1 / V

[0059] d2: fast trap gate voltage acceleration factor, constant, process related, 1 / V

[0060] E a1: slow trap temperature acceleration factor, constant, process dependent, eV

[0061] E a2 : fast trap temperature acceleration factor, constant, process dependent, eV

[0062] B: slow trap time acceleration factor, exponential term

[0063] D: fast trap time acceleration factor

[0064] Step 1, data preparation.

[0065] 1a) Perform BTI effect test on MOS transistor under different gate voltage, get multiple sets of degradation data of gate voltage change.

[0066] 1b) Perform BTI effect test on MOS transistor under different temperature, get multiple sets of degradation data of temperature change.

[0067] Step 2, prepare PSO algorithm m file.

[0068] 2a) Parameter setting:

[0069] Set self and global learning factor: c1 = 2, c2 = 2.1;

[0070] Set particle size: size = 500;

[0071] Set parameter dimension: 4;

[0072] Set maximum number of iterations: 500;

[0073] Set maximum particle velocity;

[0074] Set parameter interval range;

[0075] Set iteration inertia weight: 0.05;

[0076] If there is no special description later, the above default values are used.

[0077] 2b) Particle swarm initialization

[0078] Initialize particle position and velocity;

[0079] Generate reasonable random position within parameter constraint range;

[0080] 2c) Set current best position before initial iteration and historical optimal position

[0081] 2d) Set fitness function:

[0082] V = W * V + C1 * R1 * (X best -X) + C2 * R2 * (G best- X) + R3*(G best - X best ) (5)

[0083] Step 3, obtain the upper and lower limits of time acceleration factor.

[0084] 3a) directly use the simplified model form, as shown in the following formula.

[0085] ΔV th = A * t B + C * log10(1 + D * t) (1)

[0086] Wherein, A ∝ exp(d1 V g )exp(-E a1 / KT), C ∝ exp(d2 V g )exp(-E a2 / KT) (2);

[0087] First extraction, set A, C, B, D parameters are all greater than 0, parameter dimension 4, according to each group of parameter degradation data directly using PSO algorithm fitting A, C, B, D, the obtained multiple sets of parameter values are different from each other.

[0088] In order to accurately find the range of B, D, the parameter values B, D obtained by fitting in logarithmic coordinates and bilinear are removed, and the maximum and minimum values are found. The maximum and minimum values of B, D are set as the upper and lower limits of the model parameters of B, D.

[0089] Step 4, respectively obtain the upper and lower limits of voltage acceleration factor and temperature acceleration factor

[0090] The value range of parameters B, D obtained in the last step, in this iteration, obtain the voltage acceleration factor or temperature acceleration factor.

[0091] ΔV th = a1 * exp(d1 V g ) * t B + a2 * exp(d2 V g ) * log10(1 + D * t) (7)

[0092] ΔV th = a1 * exp(-E a2 / KT) * t B + a2 * exp(-E a2 / KT) * log10(1 + D * t) (8)

[0093] Using the upper and lower limits of B, D parameters, input parameter dimension 4, parameter constraint condition a1>0, a2>0; d1<0, d2<0; Ea1 >0,E a2 >0

[0094] 4a) Directly using the simplified model shown in formula 7, input the lower limit and upper limit of B and D respectively, and extract the gate voltage acceleration factor according to the gate voltage change degradation curve family, to obtain two different combinations of d1 and d2, and use the two sets of data to regenerate the upper and lower limits of the gate voltage acceleration factor;

[0095] 4b) The same step, using the simplified model shown in formula 8 and the lower limit and upper limit of B and D, obtaining the gate voltage acceleration factor E according to the temperature change degradation curve family. a1 ,E a2 The upper and lower limits of the gate voltage acceleration factor.

[0096] In order to weaken the defect that the range is actually smaller, the gate voltage acceleration factor and the temperature acceleration factor are respectively expanded outward by 25% with the difference between the upper and lower limits as the standard.

[0097] So far, the value range of time, temperature, gate voltage acceleration factor, and the value range of the proportional coefficient of the term in the simplified model of formula 7 and formula 8 are obtained.

[0098] Step 5, optimization of temperature acceleration factor and gate voltage acceleration factor

[0099] The value range of time, temperature, gate voltage acceleration factor is determined to form a parameter constraint condition. Input the constraint condition and input the parameter dimension 4.

[0100] 5a) In the formula 7 model, three groups of parameter value ranges are obtained, and the value range of the term coefficients a1 and a2 is divided into 20 equal parts using the gate voltage change degradation curve family. The optimal value is searched by using the traversal method combined with particle swarm algorithm to obtain the optimized model parameter values d1, d2, B and D.

[0101] 5b) Synchronously with step 5a), the temperature change degradation curve family is used to perform traversal search using the term coefficient to obtain the optimized model parameter values E a1 ,E a2 ,B,D.

[0102] Step 6, optimization of time acceleration factor and extraction of term coefficient parameters.

[0103]

[0104] This step uses the equation shown in formula 4, and step 5 obtains two different time acceleration factors B and D from the gate voltage and temperature change data respectively. The value range of the new parameters B and D is formed with the difference between the upper and lower limits as the standard, and then it is expanded outward by 25%. Input the constraint condition of parameters B and D, c1>0, c2>0. Input the parameter dimension 4, input d1, d2, E a1 ,Ea2 .

[0105] 6a) Using all the degradation data, fit the time acceleration factor to the term coefficients;

[0106] At this point, all the parameters are extracted.

[0107] The above examples are only for illustrating the technical concept and characteristics of the present application, and the purpose is to enable those skilled in the art to understand the content of the present application and to implement it, and cannot limit the protection scope of the present application. Any equivalent changes or modifications made in accordance with the spirit and essence of the present application shall be covered within the protection scope of the present application.

Claims

1. A method for extracting model parameters under a BTI model framework, characterized in that, It comprises the following steps: a. extracting upper and lower limits of time acceleration factor of semiconductor device; b. optimizing the upper and lower limits of time acceleration factor by using PSO algorithm; c. obtaining the optimized upper and lower limits of time acceleration factor; In step a, the degradation values of the threshold voltage of the mos tube with respect to time under different gate voltage stress conditions and the degradation values of the threshold voltage with respect to time under different temperature stresses are obtained by establishing a nonlinear curve of the fast / slow trap model; The nonlinearity of the fast / slow trap model is: ΔV th = A * t B + C * log10(1 + D * t) (1) Wherein A α exp(d1 V g ) exp(-E a1 / KT), C α exp(d2 Vg) exp(-E a2 / KT) (2); In the formula, d1 is the slow trap gate voltage acceleration factor, d2 is the fast trap gate voltage acceleration factor, B is the slow trap time acceleration factor; D is the fast trap time acceleration factor; For the fast trap model, in the double logarithmic coordinates, the short-term characteristics are used to express, and formula (3) is used to express the degradation curve of the threshold voltage of the mos tube with respect to time under different gate voltage stress conditions ΔV th = a1*t a2 + b1*log10(1 + b2*t) (3); For each degradation curve, the particle swarm algorithm is used to fit the rough values a2 and b2, and a plurality of groups of a2 and b2 values corresponding to different stress conditions are obtained. Due to the characteristics of the fast trap that the short-term characteristics contribute greatly, the maximum a2 value is taken as the upper limit of the slow trap time acceleration factor B in formula (1), and the minimum b2 value is taken as the lower limit of the fast trap time acceleration factor D in formula (1). For the slow trap model, in the double linear coordinates, formula (3) is used to express the degradation curve of the threshold voltage of the mos tube with respect to time under different gate voltage stress conditions According to the temperature stress degradation curve family, using the upper and lower limits of the slow trap time acceleration factor B and the fast trap time acceleration factor D, the temperature acceleration factor E in formula (4) is extracted a1 , a2 , to obtain the upper and lower limits of d1, d2, and the slow trap term proportionality coefficient c1 and the fast trap term proportionality coefficient c2; wherein t: pressure application time; V g : gate voltage; T: temperature; K: Boltzmann constant; E a1 : slow trap temperature acceleration factor; E a2 : fast trap temperature acceleration factor; In step b, the constraint condition is set according to the range of the coefficients of the parameters d1, d2, B and D in formula (4) obtained by using the gate voltage stress degradation curve family, the particle swarm algorithm is used, the model parameters are optimized in the direction of small error rate of traversal or upper and lower limits, and the optimized values of the parameters d1, d2, B and D are obtained; the constraint condition is set according to the range of the model parameters E a1 ,E a2 ,B and D, the particle swarm algorithm is used, the model parameters are optimized in the direction of small error rate of traversal or upper and lower limits, and the optimized values of the parameters in formula (4) are obtained; two groups of different B and D values are obtained in the two optimizations, the value range of the parameters B and D is reconstituted by using the two groups of data, and the upper and lower limits are both expanded outward by 15-30% with the difference as the standard; the constraint condition of the new B and D values is used, and the parameters c1, c2, B and D in formula (4) are accurately extracted by using all the data groups of curves.

2. The method of claim 1, wherein: For each degradation curve, the particle swarm algorithm is used to fit the rough values a2 and b2, and a plurality of groups of a2 and b2 values are obtained. The minimum a2 value is taken as the lower limit of the slow trap time acceleration factor B in formula (3), and the maximum b2 value is taken as the upper limit of the fast trap time acceleration factor D in formula (1). Thus, the upper and lower limits of the power index law time index B and the logarithmic law time acceleration factor D are obtained. According to the gate voltage stress degradation curve family, the upper and lower limits of the slow trap time acceleration factor B and the fast trap time acceleration factor D are used to extract d1 and d2 in formula (4) The semiconductor device is a mos tube.

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