CN107203691A - The analytic method that a kind of prediction NBTI theoretical based on Reaction-diffusion terms recovers when long - Google Patents

Abstract

The invention discloses the analytic method that a kind of prediction NBTI theoretical based on Reaction-diffusion terms recovers when long, including：Step one：Obtain the device parameters of p MOSFET elements；Step 2：Based on basic reaction diffusion theory, the General Analytical model recovered when description NBTI is long is drawn；Step 3：Fast quick-recovery and H based on interface trap₂Locking-up effect, correct General Analytical model；Step 4：Based on D_H2It is the physical quantity changed over time, the expression formula that introducing parameter ξ is changed over time is modified, obtains the complete analytic modell analytical model for being used to describe to recover when NBTI is long；Step 5：According to analytic modell analytical model, predict and recover when the NBTI of p MOSFET elements is long.Analytic modell analytical model proposed by the present invention incorporates H₂Diffusion coefficient with recovery time decay and locking-up effect the two factors, and demonstrate effectiveness of the invention by being compared with R D model value solutions.

Description

Analytical method for predicting NBTI long-time recovery based on reaction-diffusion theory

Technical Field

The invention belongs to the technical field of semiconductors, and particularly relates to an analysis method for predicting NBTI long-time recovery based on a reaction-diffusion (R-D) model.

Background

As CMOS process dimensions continue to shrink, Negative Bias Temperature Instability (NBTI) has become one of the major factors affecting the performance of p-MOSFET devices. The NBTI effect causes device parameter degradation, such as threshold voltage (Δ V)_T) Linear and saturated drain currents, transconductance, and subthreshold slope, among others, thereby degrading performance of the circuit and system. Accurately describing and predicting the analytical model of NBTI degradation and recovery is a great problem to be perfected in the aspect of device reliability. The physical mechanism of NBTI has been studied extensively over the past decades and has led to different explanations. Generation of interface traps (Δ N) using the R-D theory_IT) Is believed to be the primary cause of NBTI degradation. On the other hand, the cavity is in the cavityProcess related gate insulator grown-in defects (Δ N)_HT) The trapping/releasing (T/D) mechanism in (a) is also considered to be not negligible. Studies show that_HTComplete saturation or recovery can be achieved in a few seconds, and this process can be explained by a two-energy-well model. It can be seen that only Δ N is of interest_ITOr Δ N_HTNone of the models of (a) provide a complete physical mechanism interpretation for NBTI. Except for delta N_ITAnd Δ N_HTIn addition, studies have found defects (Δ N) in the gate insulator that occur during stressing when a higher gate voltage is applied_OT) Also has a significant impact on NBTI degradation. Therefore, many studies have attempted to convert Δ N_IT，ΔN_HTAnd Δ N_OTA model is incorporated to give a better physical interpretation.

An important characteristic of NBTI is that damage to the device develops recovery immediately after the voltage applied to the gate of the p-MOSFET is removed. In general, NBTI recovery can be divided into three parts: 1) fast hole detachment from within the gate insulator layer defects originally present and created during stressing, 2) fast electron capture by interface defects, 3) slow recovery of remaining interface traps. Thus, long term NBTI recovery results only from slow repair of interface traps, which can be described by RD theory. Considering the slow recovery, H₂Reacts with defects at the interface, making the defects less and less, so that over the recovery time, H₂The probability of finding defects is constantly reduced, and this process can be represented by H₂Diffusion coefficient D of_H2The following relationship over time is characterized:

wherein D is_H20Finger H₂Constant diffusion coefficient in the pressed phase, t_recFinger device recovery time, t_strTo the device press time, B_DFor description of D_H2The degree of decay with time. In addition, due to H₂The locking effect of, i.e. part of, the device degradation processH₂Trapped defects are thus unable to participate in the recovery process and a portion of NBTI defects are unable to recover. During long-term recovery, the threshold voltage is degraded by Δ V_TThe variation relation with respect to time t can be given by a numerical solution of the R-D model.

In order to describe and research NBTI long-term recovery characteristics more easily, it is necessary to propose a more simplified and effective analytical model. In consideration of D_H2Decrease in sum H with time₂On the basis of the locking effect, the invention provides an analytic model for describing NBTI long-time recovery. The model has physical significance and simple form, and is consistent with the numerical solution of the R-D model. For the explanation of the physical mechanism of NBTI, reference is made in particular to the following paper [1 ]](N.Goel and S.Mahapatra, Modeling of DC and ACNBTI degradation and recovery for SiON and HKMG MOSFETs, "in fundamental of Bias temporal instrumentation in MOS Transistors, S.Mahapatra, Ed.,1 st. NewDelhi, India: Springer,2016, pp.209-263.). For general analytical model derivation for R-D theory, reference is made in particular to the following paper [2 ]](Sanjay V.Kumar,Chris H.Kim,Sachin S.Sapatnekar,A Finite‐OxideThickness‐Based Analytical Model for Negative Bias Temperature Instability,IEEE Transactions on Device and Materials Reliability,vol.9,no.4,December2009,pp.537‐556.)。

Disclosure of Invention

The invention provides an analysis method for predicting NBTI long-time recovery based on a reaction-diffusion theory, which comprises the following steps:

the method comprises the following steps: acquiring device parameters of a p-MOSFET device;

step two: obtaining a general analytical model for describing NBTI long-time recovery based on a basic reaction-diffusion theory;

step three: fast recovery and H based on interface traps₂Correcting the general analytical model;

step four: based on D_H2The parameter ξ is introduced to modify the physical quantity changing along with time to obtain a complete analytical model for describing NBTI long-time recovery;

step five: and predicting NBTI long-time recovery of the p-MOSFET device according to the analytical model.

In the analytic method for predicting NBTI long-time recovery based on the reaction-diffusion theory, the device parameters of the p-MOSFET device comprise: the amount of threshold voltage degradation.

In the analysis method for predicting NBTI long-time recovery based on the reaction-diffusion theory, the general analysis model comprises the relation of the interface trap concentration at the initial moment of the recovery stage along with time, and the relation is expressed by the following formula (i):

wherein,

in the formula (i), Δ N_IT(t) denotes the interface trap which is not restored at time t, Δ N_IT(t_str) Represents the interface trap concentration, Δ N, at the initial moment of the recovery phase_IT(t) represents the concentration of interface traps repaired at time t, t_strIndicating the device stress time, ξ is used to describe H₂Physical quantity of diffusion, Δ N_H2 ^*(x is 0, t) represents time H₂Concentration of diffusion front at interface, D_H2Represents H₂Diffusion coefficient of (d), t_recDenotes the device recovery time, and t denotes the sum of the compression time and the recovery time.

In the analytical method for predicting NBTI long-time recovery based on the reaction-diffusion theory, after the rapid recovery amount of the interface trap is introduced, a corrected general analytical model is represented by the following formula (ii):

in the formula (ii), Δ V_TRepresents the amount of threshold voltage degradation, Δ V, caused by interface traps that do not recover at time t_IT0Representing the amount of threshold voltage degradation introduced by the interface trap at the initial moment of recovery, F_FASTIndicating the specific gravity of the amount of rapid recovery of interface traps in the total interface defects, ξ is the function of t_rec/t_strParameter of variation, t_recDenotes the device recovery time, and t denotes the sum of the compression time and the recovery time.

In the analytical method for predicting NBTI long-time recovery based on the reaction-diffusion theory, H is further introduced₂After the lock-in effect, the modified general analytical model is represented by the following formula (iii):

in the formula (iii), Δ V_T(t) represents the amount of threshold voltage degradation by interface traps that do not recover at time t, Δ V_IT0Representing the amount of threshold voltage degradation introduced by the interface trap at the initial moment of recovery, F_FASTIndicating the proportion of the amount of rapid recovery of interface traps in the total interface defects, α indicating defect-locked H₂In a ratio of total, ξ is a function of t_rec/t_strThe parameter of variation, t, represents the total time.

In the analytical method for predicting NBTI long-term recovery based on the reaction-diffusion theory, the expression of the time-varying parameter xi introduced in the fourth step is shown as the following formula (iv):

in formula (iv), ξ represents H₂Physical quantity of diffusion and with t_rec/t_strChange, t_strRepresents the device stress time, t_recRepresenting the device recovery time, a, ξ₀And η are fitting parameters.

In the analysis method for predicting NBTI long-term recovery based on the reaction-diffusion theory, an analysis model for describing the NBTI long-term recovery is shown as the following formula (v):

in the formula (V), Δ V_IT0Representing the amount of threshold voltage degradation introduced by the interface trap at the initial moment of recovery, F_FASTIndicating the proportion of the amount of rapid recovery of interface traps in the total interface defects, α indicating defect-locked H₂In a ratio of total, ξ is a function of t_rec/t_strThe parameter of variation, t, represents the total time.

In the analysis method for predicting NBTI long-term recovery based on the reaction-diffusion theory, the analysis model for describing NBTI long-term recovery is further verified before the fifth step, and the method comprises the following steps:

performing numerical simulation on NBTI recovery based on an RD model to obtain the NBTI unrecoverable quantity and the change relation of the recoverable quantity along with time;

and substituting the data obtained by simulation into the analytical model to obtain analytical model parameters, and obtaining the coincidence condition of the analytical model and the RD numerical solution.

The invention has the beneficial effects that: the analytical model provided by the invention is incorporated into H₂The diffusion coefficient of (2) decays along with the recovery time and the locking effect are two factors, and the effectiveness of the invention is verified by comparing with the numerical solution of the R-D model. The model is based on RD theory, and H is considered in the invention for the first time₂The diffusion coefficient of (1) decays with recovery time andon the basis of the latch-in effect, an analytical relationship that threshold voltage degradation caused by NBTI is recovered along with time in a device recovery phase is proposed. The prediction result of the model can describe the NBTI long-time recovery condition of the device more accurately and conveniently. The analytical model has the advantages of less required parameters and wide applicability, and provides simple and accurate prediction for the reliability of devices.

Drawings

FIG. 1 is an analytic method of predicting NBTI long-term recovery based on reaction-diffusion theory of the present invention;

FIG. 2 shows NBTI recovery phase H₂Approximate profile distribution in poly-Si;

FIG. 3 is the parameter ξ over t_rec/t_strThe variation relationship of (a);

FIG. 4 is a comparison of the RD model numerical solution for predicting NBTI long term recovery and the analytical model of the present invention.

Detailed Description

The present invention will be described in further detail with reference to the following specific examples and the accompanying drawings. The procedures, conditions, experimental methods and the like for carrying out the present invention are general knowledge and common general knowledge in the art except for the contents specifically mentioned below, and the present invention is not particularly limited.

The analysis method provided by the invention introduces an innovative NBTI analysis model, is based on the traditional RD theory and considers H₂The long-term recovery of NBTI degradation is accurately calculated by the diffusion coefficient of (A) fading along with the recovery time and the locking effect. The analysis method of the invention comprises the following steps:

the method comprises the following steps: and acquiring device parameters of the p-MOSFET device. The device parameters of the p-MOSFET device include: the amount of threshold voltage degradation.

Step two: based on the basic RD theory, a general analytic model for describing NBTI long-time recovery is obtained

In the recovery phase, a portion of H is generated by the reaction near the original interface₂Continue to diffuse into poly-Si with a portion of H near the interface₂And reacting with an interface trap to repair the defect and realize the recovery of NBTI. H₂The diffusion profile in poly-Si can be approximated as a triangle, as shown in fig. 2. Let Δ N_IT(t_str) The interface trap concentration at the initial time of the recovery phase can be expressed by the area of the triangle below the solid line in fig. 2. Delta N_IT ^*(t) represents the concentration of interface traps repaired at time t, and can be represented by the area of the shaded portion in FIG. 2. Thus, at time t, an interface trap that is not restored may be expressed as

According to FIG. 2, the interface trap repaired at time t can be expressed as

Where parameter ξ is a time-varying physical quantity the area of the triangle under the solid line in fig. 2 is equal to the area under the dashed line, and therefore the interface trap concentration deltan at the initial moment of the recovery phase_IT(t_str) Can be expressed as

It should be noted that one H₂Corresponding to two defects, the area of the triangle representing the defect concentration is therefore calculated without being multiplied by 1/2. From the expressions (1) to (3), the relationship of the interface trap concentration with time in the recovery stage is as follows

In the above formulas 1 to 4,. DELTA.N_IT(t) denotes the interface trap which is not restored at time t, Δ N_IT(t_str) Represents the interface trap concentration, Δ N, at the initial moment of the recovery phase_IT ^*(t) represents the concentration of interface traps repaired at time t, t_strIndicating the device stress time, ξ is used to describe H₂Physical quantity of diffusion, Δ N_H2 ^*(x is 0, t) represents time H₂Concentration of diffusion front at interface, D_H2Represents H₂Diffusion coefficient of (d), t_recDenotes the device recovery time, t denotes the total time equal to (═ t)_rec+t_str)。

Equation (4) above is consistent with conventional RD theory and can be used to blur the description of the slow recovery of NBTI. However, equation (4) ignores a part of the interface traps recovered due to the fast capture of electrons. Combined formulae (4) and Δ V_IT＝q*(ΔN_IT)/C_oxThe amount of threshold voltage degradation caused by NBTI during the recovery phase varies with respect to time as follows (4'):

wherein, is Δ V_IT0Indicating NBTI degradation caused by the original interface traps. Equation (4') can briefly describe NBTI recovery behavior, but theoretically has serious limitations: 1) fast recovery is not considered; 2) do not consider H₂The amount of irrecoverability due to lock-in effects; 3) is not taking into account H₂Based on the time decay of the diffusion coefficient of (c), the time dependence of the parameter ξ is obtained.

Step three: consider the fast recovery of a portion of interface traps and a portion of H₂The locking effect of (2), the model is corrected.

Step 3a: introduction parameter F_FAST. If using Δ V_IT0Representing NBTI degradation due to the original interface trap, then Δ V_IT1＝ΔV_IT0*F_FASTThe amount of rapid recovery that can be used to represent NBTI degradation. It should be noted that after some of the interface traps are neutralized by electrons, they can still react with H₂The defect is reacted and repaired, but the Si-H bond recovery at this time does not contribute to the overall NBTI recovery.

According to the formula (4), based on H₂The amount of interfacial repair can be expressed as Δ N_IT0*(ξ*t_rec/t)^1/2But wherein the ratio is F_FASTHas been neutralized by electrons and no Δ V can be recalculated_TSo that the total recovery is Δ N_IT0*F_FAST+ΔN_IT0*(1-F_FAST)*(ξ*t_rec/t)^1/2. Bound Δ V_IT＝q*(ΔN_IT)/C_oxNBTI long term recovery can be expressed as

And step 3 b: during the device compression phase, a portion H₂Locked in traps, causing a portion of NBTI degeneration to fail recovery, so a parameter α needs to be introduced to describe the amount of unrecoverable H of FIG. 2₂Does not contain locked H in the distribution₂Accordingly, the combination of formula (4) is based on H₂Should correct the amount of interfacial repair to Δ N_IT0*(1-α)*(ξ*t_rec/t)^1/2. The total recovery amount can be expressed as Δ N in view of electron capture_IT0*F_FAST+ΔN_IT0*(1-F_FAST)*(1-α)*(ξ*t_rec/t)^1/2. Equation (5) can be modified to

In formulae (5) and (6), Δ V_T(t) denotes the boundary that did not recover at time tAmount of threshold voltage degradation, Δ V, due to surface traps_IT0Representing the amount of threshold voltage degradation introduced by the interface trap at the initial moment of recovery, F_FASTIndicating the proportion of the amount of rapid recovery of interface traps in the total interface defects, α indicating defect-locked H₂In the proportion of the total amount, t_strRepresents the device stress time, t_recDenotes the device recovery time, t denotes the sum of the compression time and the recovery time, ξ is description H₂Physical amount of diffusion.

Step four: consider D_H2Is a time-varying physical quantity, and introduces a time-varying expression of the parameter ξ

With restoration of interface traps, H₂The time required to find a defect is longer and longer, so D can be equivalently regarded as_H2Decays over time. The above formula (4') is deduced to be H₂The diffusion coefficient is uniformly constant in the compression phase and the recovery phase, but theoretically, in the recovery phase D_H2Is a physical quantity that changes with time.

H in poly-Si during device recovery₂There are two possibilities of diffusion when diffusing: (1) diffusing to the interface and repairing the defect; (2) continue to diffuse into the poly-Si. If it is symmetric diffusion (symmetric diffusion means H in the system)₂The same degree of diffusion into the interface or into the poly-Si), ξ should be equal to 0.5, but at the beginning of recovery, near the interface H₂Higher concentration of H near the interior of poly-Si₂Low concentration and H in the system₂The concentration is constantly changing, therefore H₂Not likely to be symmetric diffusion, ξ is a time-varying parameter if H₂Are all flared, ξ equals 1 if H₂All extending toward the interface, ξ should be equal to 0. during the initial phase of recovery, H is near the interface₂Is consumed rapidly to cause H diffusion to the vicinity of the interface₂The concentration gradient is much greater than the diffusion of H to the deep poly₂A concentration gradient. Thus, H₂Diffusion rate near the interface is much higher than ambient, making ξ less than 0.5 with H₂Consumption of H₂The concentration gradient decreases continuously, but close to the interfaceThe rate of decrease of the nearby concentration gradient must be faster than the rate of decrease at the poly depth, so ξ increases with time, and as recovery progresses, the interfacial defects gradually decrease, H₂The longer and longer the time required to find the interface defect, the H will be caused finally₂Build-up at the interface, thereby reducing the diffusion of H to the interface₂Concentration gradient such that ξ increases with time, known paper [2 ]]The general RD model numerical solution and analytic model are used to obtain the relationship of the parameter ξ with continuously increasing time, but no specific relationship of ξ with time is determined.

Thus, the present invention introduces an expression of the time-varying parameter ξ for characterizing H₂The effect of diffusion coefficient decay over time on NBTI recovery. In consideration of D_H2Decay with time and H₂On the basis of the locking effect, the change relation of the parameter ξ with respect to time is obtained by using an RD model numerical solution and an analytic model, and the details are as follows:

determining parameter F using RD model_FASTAnd α, combining the numerical solution of RD model and the formula (6), calculating ξ value and obtaining ξ about t_rec/t_strAs indicated by the symbol in fig. 3, it can be seen that ξ relates to t_rec/t_strExhibits an approximately logarithmic relationship, but ξ is related to log (t)_rec/t_str) Is not perfectly proportional, so the correction term (t) continues to be introduced_rec/t_str)^ηξ are obtained for t_rec/t_strIs expressed as follows

Wherein a, ξ₀And η are fitting parameters.

From analytical formula (7), the solid line in FIG. 3 is obtained, and it can be seen that formula (7) can accurately express ξ with respect to t_rec/t_strThe analytical relationship of (1).

In the above formula (7), ξ is the description H₂Diffused articleRational and with t_rec/t_strChange, t_strRepresents the device stress time, t_recRepresenting the device recovery time, a, ξ₀And η are fitting parameters.

And after the fourth step, further verifying the analytical model. The specific verification steps are as follows:

step 4 a: and performing numerical simulation on the NBTI recovery based on the RD model to obtain the change relation of the NBTI recoverable quantity and the unrecoverable quantity along with time.

And 4 b: and substituting the data into the analytical model to obtain analytical model parameters, and obtaining the coincidence condition of the analytical model and the RD numerical solution.

NBTI Long term recovery is shown as (6), equivalent to Δ V_IT2Is recovered and can be rewritten as

Due to the locking effect of H2, a part of the interface is broken (N)_IT0α) is unrecoverable, but a portion thereof is neutralized by electrons, so the total unrecoverable amount should be

P_TG＝ΔV_IT0(1-F_FAST)α (9)

From (8) and (9), Δ V_IT2Can be divided into unrecoverable amount and recoverable amount, the latter can be expressed as

Fig. 4 shows that the proposed analytical model can be well matched with the numerical solution of the RD model, thereby verifying the validity of the analytical model.

In the above formulae 8-10,. DELTA.V_IT2(t) represents the amount of unrecovered threshold voltage degradation introduced by the interfacial defect at time t, Δ V_IT2,SDRepresenting the amount of threshold voltage degradation, P, that can be recovered at time t but has not yet been recovered_TGDenotes the amount of unrecoverability,. DELTA.V_IT0Representing the amount of threshold voltage degradation introduced by the interface trap at the initial moment of recovery, F_FASTIndicating the proportion of the amount of rapid recovery of interface traps in the total interface defects, α indicating defect-locked H₂In the proportion of the total amount, t_strRepresents the device stress time, t_recIndicating device recovery time, ξ is description H₂Physical amount of diffusion.

Step five: and predicting NBTI long-time recovery of the p-MOSFET device according to a complete analytical model. The method comprises the following specific steps:

step 5a, placing the p-MOSFETs device under an NBTI stressed condition, and testing the relation of the change of the threshold voltage of the device along with time;

step 5b, after the device is stressed for a long time, removing the grid voltage to enable the device to enter a recovery state, and continuously testing the time-varying relation of the threshold voltage of the device;

and 5c, obtaining the threshold voltage degradation quantity delta V caused by the interface trap of the device at the initial recovery moment according to the RD model by using the data_IT0Fitting to obtain related parameters in an analytical model by combining the change data of the threshold voltage degradation quantity of the device in the recovery stage with respect to time;

and 5d, predicting NBTI long-time recovery by the analytical model by using the parameters obtained by fitting.

The model considers H for the first time according to RD theory₂On the basis of the diffusion coefficient attenuation along with the recovery time and the locking effect, the analytical relation that the threshold voltage degradation caused by NBTI is recovered along with the time in the device recovery phase is provided. The model can predict the NBTI long-time recovery condition of the device more accurately.

The protection of the present invention is not limited to the above embodiments. Variations and advantages that may occur to those skilled in the art may be incorporated into the invention without departing from the spirit and scope of the inventive concept, and the scope of the appended claims is intended to be protected.

Claims (8)

1. An analytical method for predicting NBTI long-term recovery based on a reaction-diffusion theory is characterized by comprising the following steps:

the method comprises the following steps: acquiring device parameters of a p-MOSFET device;

step two: obtaining a general analytical model for describing NBTI long-time recovery based on a basic reaction-diffusion theory;

step three: fast recovery and H based on interface traps₂Correcting the general analytical model;

step four: based on D_H2Is varied with timeIntroducing an expression of the parameter ξ changing along with time to correct the physical quantity to obtain a complete analytical model for describing NBTI long-time recovery;

step five: and predicting NBTI long-time recovery of the p-MOSFET device according to the analytical model.

2. The analytical method for predicting NBTI long-term recovery based on reaction-diffusion theory as claimed in claim 1, wherein the device parameters of the p-MOSFET device comprise: the amount of threshold voltage degradation.

3. The analytical method for predicting NBTI long-term recovery according to claim 1, wherein the general analytical model includes a time-dependent relationship between the concentration of interface traps at the initial time of the recovery stage, which is expressed by the following formula (i):

<mrow> <msub> <mi>&amp;Delta;N</mi> <mrow> <mi>I</mi> <mi>T</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>&amp;Delta;N</mi> <mrow> <mi>I</mi> <mi>T</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>s</mi> <mi>t</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>&amp;Delta;N</mi> <mrow> <mi>I</mi> <mi>T</mi> </mrow> <mo>*</mo> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>&amp;Delta;N</mi> <mrow> <mi>I</mi> <mi>T</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>s</mi> <mi>t</mi> <mi>r</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msqrt> <mfrac> <mrow> <msub> <mi>&amp;xi;t</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> </mrow> <mi>t</mi> </mfrac> </msqrt> <mo>)</mo> </mrow> <mo>,</mo> </mrow>

wherein,

in the formula (i), Δ N_IT(t) denotes the interface trap which is not restored at time t, Δ N_IT(t_str) Represents the interface trap concentration, Δ N, at the initial moment of the recovery phase_IT ^*(t) represents the concentration of interface traps repaired at time t, t_strIndicating the device stress time, ξ is used to describe H₂Physical quantity of diffusion, Δ N_H2(x is 0, t) represents time H₂Concentration of diffusion front at interface, D_H2Represents H₂Diffusion coefficient of (d), t_recDenotes the device recovery time, and t denotes the sum of the compression time and the recovery time.

4. The analytical method for predicting NBTI long-term recovery based on reaction-diffusion theory as claimed in claim 1, wherein after introducing the fast recovery amount of interface traps, the modified general analytical model is represented by the following formula (ii):

<mrow> <msub> <mi>&amp;Delta;V</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>&amp;Delta;V</mi> <mrow> <mi>I</mi> <mi>T</mi> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>F</mi> <mrow> <mi>F</mi> <mi>A</mi> <mi>S</mi> <mi>T</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msqrt> <mfrac> <mrow> <msub> <mi>&amp;xi;t</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> </mrow> <mi>t</mi> </mfrac> </msqrt> <mo>)</mo> </mrow> <mo>;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mi>i</mi> <mi>i</mi> <mo>)</mo> </mrow> </mrow>

in the formula (ii), Δ V_TRepresents the amount of threshold voltage degradation, Δ V, caused by interface traps that do not recover at time t_IT0Representing the amount of threshold voltage degradation introduced by the interface trap at the initial moment of recovery, F_FASTIndicating the specific gravity of the amount of rapid recovery of interface traps in the total interface defects, ξ is the function of t_rec/t_strParameter of variation, t_recDenotes the device recovery time, and t denotes the sum of the compression time and the recovery time.

5. The analytical method for predicting NBTI long-term recovery based on the reaction-diffusion theory as claimed in claim 4, wherein H is further introduced₂After the lock-in effect, the modified general analytical model is represented by the following formula (iii):

<mrow> <msub> <mi>&amp;Delta;V</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>&amp;Delta;V</mi> <mrow> <mi>I</mi> <mi>T</mi> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>F</mi> <mrow> <mi>F</mi> <mi>A</mi> <mi>S</mi> <mi>T</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msqrt> <mfrac> <mrow> <msub> <mi>&amp;xi;t</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> </mrow> <mi>t</mi> </mfrac> </msqrt> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&amp;Delta;V</mi> <mrow> <mi>I</mi> <mi>T</mi> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>F</mi> <mrow> <mi>F</mi> <mi>A</mi> <mi>S</mi> <mi>T</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>&amp;alpha;</mi> <mo>;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mi>i</mi> <mi>i</mi> <mi>i</mi> <mo>)</mo> </mrow> </mrow>

in the formula (iii), Δ V_T(t) represents the amount of threshold voltage degradation by interface traps that do not recover at time t, Δ V_IT0Representing the amount of threshold voltage degradation introduced by the interface trap at the initial moment of recovery, F_FASTIndicating the proportion of the amount of rapid recovery of interface traps in the total interface defects, α indicating defect-locked H₂In a ratio of total, ξ is a function of t_rec/t_strThe parameter of variation, t, represents the total time.

6. The analytical method for predicting NBTI long-term recovery based on the reaction-diffusion theory as claimed in claim 1, wherein the expression of the time-dependent variation of the parameter xi introduced in step four is shown as the following formula (iv):

<mrow> <mi>&amp;xi;</mi> <mo>=</mo> <mi>a</mi> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>t</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> <msub> <mi>t</mi> <mrow> <mi>s</mi> <mi>t</mi> <mi>r</mi> </mrow> </msub> </mfrac> <mo>)</mo> </mrow> <mi>&amp;eta;</mi> </msup> <mo>&amp;CenterDot;</mo> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>t</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> <msub> <mi>t</mi> <mrow> <mi>s</mi> <mi>t</mi> <mi>r</mi> </mrow> </msub> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&amp;xi;</mi> <mn>0</mn> </msub> <mo>;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mi>i</mi> <mi>v</mi> <mo>)</mo> </mrow> </mrow>

in formula (iv), ξ represents H₂Physical quantity of diffusion and with t_rec/t_strChange, t_strRepresents the device stress time, t_recRepresenting the device recovery time, a, ξ₀And η are fitting parameters.

7. The analytic method for predicting NBTI long-term recovery based on reaction-diffusion theory as claimed in claim 1, wherein the analytic model for describing NBTI long-term recovery is as shown in the following formula (v):

<mrow> <msub> <mi>&amp;Delta;V</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>&amp;Delta;V</mi> <mrow> <mi>I</mi> <mi>T</mi> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>F</mi> <mrow> <mi>F</mi> <mi>A</mi> <mi>S</mi> <mi>T</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msqrt> <mfrac> <mrow> <msub> <mi>&amp;xi;t</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> </mrow> <mi>t</mi> </mfrac> </msqrt> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&amp;Delta;V</mi> <mrow> <mi>I</mi> <mi>T</mi> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>F</mi> <mrow> <mi>F</mi> <mi>A</mi> <mi>S</mi> <mi>T</mi> </mrow> </msub> <mo>)</mo> </mrow> <mi>&amp;alpha;</mi> <mo>,</mo> <mi>&amp;xi;</mi> <mo>=</mo> <mi>a</mi> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>t</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> <msub> <mi>t</mi> <mrow> <mi>s</mi> <mi>t</mi> <mi>r</mi> </mrow> </msub> </mfrac> <mo>)</mo> </mrow> <mi>&amp;eta;</mi> </msup> <mo>&amp;CenterDot;</mo> <mi>log</mi> <mo>(</mo> <mfrac> <msub> <mi>t</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> <msub> <mi>t</mi> <mrow> <mi>s</mi> <mi>t</mi> <mi>r</mi> </mrow> </msub> </mfrac> <mo>)</mo> <mo>+</mo> <msub> <mi>&amp;xi;</mi> <mn>0</mn> </msub> <mo>;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow>

in the formula (V), Δ V_IT0Representing the amount of threshold voltage degradation introduced by the interface trap at the initial moment of recovery, F_FASTIndicating the proportion of the amount of rapid recovery of interface traps in the total interface defects, α indicating defect-locked H₂The proportion of the total amount of the composition,ξ is with t_rec/t_strThe parameter of variation, t, represents the total time.

8. The analytical method for predicting NBTI long-term recovery based on reaction-diffusion theory as claimed in claim 1, wherein the analytical model for describing NBTI long-term recovery is further verified before step five, comprising the following steps:

performing numerical simulation on NBTI recovery based on an RD model to obtain the NBTI unrecoverable quantity and the change relation of the recoverable quantity along with time;

and substituting the data obtained by simulation into the analytical model to obtain analytical model parameters, and obtaining the coincidence condition of the analytical model and the RD numerical solution.