Abstract:
A method of simulating hot carrier deterioration of a P-MOS transistor uses the following formulas (A1), (A2), (A3) and (A4) or the following formulas (A1), (A5), (A3) and (A4) (A2), and coefficients A, n, B and m are determined by a preliminary measuring experiment, whereby a transistor lifetime r can be estimated: 
     
       Vth=Vfb+σ•Vd                                   (A1) 
     
     
       ΔVth=ΔVfb                                      (A2) 
     
     
       (ΔVfb).sub.f =A•σ.sup.n                  (A3) 
     
     
       τ=B•(Ig/W).sup.-m                                (A4) 
     
     
       ΔVth=ΔVfb+Δσ•Vd              (A5)

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a method of simulating hot carrier deterioration of an MOS transistor, and in particular to a method of simulating hot carrier deterioration of a P-MOS transistor during operation in the FWD and REV modes. 
     2. Description of the Background Art 
     Hot carrier deterioration of MOS transistors can be evaluated, for example, based on a rate (ΔId/Id) of a variation ΔId of a drain current to an initial drain current Id or a variation ΔVth of a threshold voltage with respect to an initial threshold voltage Vth. 
     FIG. 15 is an equivalent circuit diagram showing a concept of the hot carrier deterioration of the MOS transistor which can be found by a conventional simulation method. FIG. 15 shows at (A) a fact that a drain current Id flows in a fresh MOS tr. ansistor before application of a stress. FIG. 15 shows at (B) a fact that a drain current Id&#39; flows through the MOS transistor after hot carrier deterioration. Thus, the hot carrier deterioration changes the drain current flowing through the transistor by ΔId from the initial drain current Id. 
     A method of simulating hot carrier deterioration of a P-MOS transistor is described, for example, in IEEE Trans. Electron Devices, Vol. 37, pp 1658-1666 (1990) by Ong et al. 
     Under a static hot carrier stress condition by a DC (direct current), the hot carrier deterioration rate ΔId/Id can be expressed by the following formula (101): 
     
         ΔId/Id=A.sub.Id •t.sup.n                       ( 101) 
    
     where t represents a hot carrier stress time, characters &#34;A&#34; and &#34;n&#34; represent coefficients which depend on manufacturing process conditions of transistors and stress conditions. 
     Under the static hot carrier stress condition by DC, the hot carrier deterioration AVth can be expressed by the following formula (102): 
     
         ΔVth=A.sub.Vth •t.sup.n                        ( 102) 
    
     where t represents a hot carrier stress time, characters &#34;A&#34; and &#34;n&#34; represent coefficients which depend on the manufacturing process conditions of transistors and the stress conditions. 
     Assuming that a stress time which elapses until the variation rate of drain current attains to (ΔId/Id) f  is a lifetime τ of the transistor, the following formula (103) is obtained from the formula (101). For example, the time t at the relationship of (ΔId/Id) f  =10% is defined as the lifetime τ. 
     
         (ΔId/Id).sub.f =A.sub.Id •τ.sup.n          ( 103) 
    
     Assuming that the stress time which elapses until the variation rate of threshold voltage attains to (ΔVth) f  is a lifetime τ of the transistor, the following formula (104) is obtained from the formula (102). For example, the time t at the relationship of (ΔVth) f  =10 mV is defined as the lifetime τ. 
     
         (ΔVth).sub.f =A.sub.Vth •τ.sup.n           ( 104) 
    
     For performing a stress acceleration test of P-MOS transistor, the stress conditions applied to the transistor are determined such that the lifetime of transistor attains to the variation rate of (ΔId/Id) f  or (ΔVth) f  defined by the above formula (103) or (104) within a measurable time from about 1 to about 100000 seconds. Measurement is performed in an FWD mode, i.e., with the current flow of the same direction as that of current flow between source and drain in the stressed transistor, and also measurement is performed in an REV mode, i.e., with the current flow of the inverted direction. Thereby, the transistor lifetime related to ΔId/Id or ΔVth is obtained at a linear region and a saturated region. The stress voltage used in the acceleration test is determined to set the condition that the hot carrier deterioration quantity attains a maximum value in connection with a certain drain voltage Vd. Thus, in the P-MOS transistor, a gate voltage Vg which maximizes gate current Ig is used. 
     The above reference (Ong et al.) has proposed a simulation method in which the acceleration test method is expressed in formulas, and the formulas are used for the simulation. According to Ong et al., the lifetime τ of P-MOS transistor is expressed by the following experimental formula (105) using the gate current Ig. 
     
         τ=B•W.sup.m •Ig.sup.-m                     ( 105) 
    
     where W represents a gate width of the transistor, B represents a coefficient depending on the manufacturing process condition of the transistor, and m represents an index which is deemed to be correlated to impact ionization by hot carriers. 
     From the formulas (103), (104) and (105), coefficients A Id  and A Vth  can be expresses by the following formulas (106) and (107): 
     
         A.sub.Id =(ΔId/Id).sub.f •(B•W.sup.-m •Ig.sup.-m).sup.-n                                  ( 106) 
    
     
         A.sub.Vth =(ΔVth).sub.f •(B•W.sup.m •Ig.sup.-m).sup.-n                                  ( 107) 
    
     Therefore, the following formulas (108) and (109) are obtained from the formulas (101), (102), (106) and (107): 
     
         A.sub.Id =(ΔId/Id).sub.f •B.sup.-n •W-.sup.mn •Ig.sup.mn •t.sup.n                           ( 108) 
    
     
         A.sub.Vth =(ΔVth).sub.f •B.sup.-n •W.sup.-mn •Ig.sup.mn •t.sup.n                           ( 109) 
    
     For the sake of illustration, the following formula (110) is defined, whereby the formulas (108) and (109) can be changed into the following formulas (111) and (112): 
     
         F(t)=B.sup.-n •W-.sup.mn •Ig.sup.mn •t.sup.n ( 110) 
    
     
         ΔId/Id=(ΔId/Id).sub.f •F(t)              (111) 
    
     
         ΔVth=(ΔVth).sub.f •F(t)                  (112) 
    
     Thus, F(t) represents a quantity of stress applied from start of application of the hot carrier stress to a time t. 
     FIG. 16 is a flow diagram showing steps in a method of simulating hot carrier deterioration of a P-MOS transistor utilizing the formula (111) or (112). In this flow diagram, a step S1 includes sub-steps S1a-S1e for extracting unknown parameters in the formula (111) or (112) by a preliminary measuring experiment. 
     In the sub-step S1a, which is executed for determining the gate current Ig in the formula (106) or (107), an experimental formula Ig=g (Vg, Vd) is determined so that it fits to data related to a plurality of measured points in the preliminary measuring experiment. A lucky electron model, which is an example of determining the gate current Ig, is described in IEEE Trans. Electron Device, Vol. ED-31, September 1984, pp. 1116-1125, by Tam et al. 
     In the sub-step S1b, transistor parameters such as a mobility μs (i.e., degree of movement) of carriers and a flat band voltage Vfb are extracted before application of the DC stress, for example, using a BSIM (Berkeley Short-Channel IGFET Model) method, which is specifically described by IEEE J. Solid-State Circuits, Vol. SC-22, August 1987, pp 558-566 by Sheu et al. In the subsequent sub-step S1c, the DC stress is applied to the transistor. In the sub-step S1d, the transistor parameters after application of the DC stress are extracted. 
     Extraction of the transistor parameters before and after application of the DC stress is required for coinciding characteristics of the transistor before application of the stress with characteristics of the transistor obtained by simulation, and is also required for estimating correlation between the actual hot carrier deterioration of the transistor after application of the stress and variation of the transistor parameters. 
     In the sub-step S1e, the coefficient B and index m are extracted based on comparison of the experimental formula (105) and data related to a plurality of measured points in the preliminary experiment. 
     In a step 2, the formula (111) or (112) is calculated using the parameters extracted in the step Si, whereby the hot carrier deterioration of P-MOS transistor is simulated. 
     In the simulation of the hot carrier deterioration of P-MOS transistor according to the prior art described above, the BSI method is used to obtain the transistor parameters after application of the DC stress. The transistor models according to the BSI method, however, do not include a model of lowering of the mobility of carriers, which is caused by the interface level or oxide film trap of electrons due to local hot carrier injection. For the coincidence of transistor characteristics after application of the stress, extraction of parameters is performed with various parameters of mobility μs and flat band voltage Vfb, and the simulation is performed with the parameters thus extracted. 
     In actual transistors, injection of hot carriers causes a difference between transistor characteristics in the FWD and REV modes as shown in FIGS. 17, 18 and 19. 
     FIG. 17 is a graph showing an example of Vd-Id characteristics in the FWD mode of the P-MOS transistor. In this figure, the solid curves represent the characteristics before application of the stress, and the broken curves show characteristics after application of the stress. 
     Likewise, FIG. 18 is a graph showing an example of Vd-Id characteristics in the REV mode of the P-MOS transistor. In this figure, the solid curves represent the characteristics before application of the stress, and the broken curves show characteristics after application of the stress. 
     FIG. 19 shows Vg-Id characteristics and Vg-gm characteristics in the FWD and REV modes of the P-MOS transistor, where gm represents a mutual conductance. In FIG. 19, the stress conditions were determined such that Ig attained a maximum value with Vd=-6.0V, and the stress was applied for 1000 seconds. Circular marks represent the transistor characteristics before application of the stress, triangular marks represent the transistor characteristics in the FWD mode after application of the stress, and square marks represent the transistor characteristics in the REV mode after application of the stress. The drain current Id was measured under two conditions of Vd=-1.5V and Vd=-0.2V. 
     A difference between the transistor characteristics in the FWD and REV modes is caused due to the fact that the hot carrier injection locally occurs at the vicinity of the drain in the transistor. Therefore, when using the BSI method which is based on a model that the transistor includes symmetrical source/drain, the transistor parameters after application of the stress must be extracted in both the FWD and REV modes. 
     According to the conventional simulation, it is impossible to simulate the hot carrier deterioration of transistors such as a pass-transistor in a circuit, which performs a bidirectional operation, i.e., which changes a direction of a current flow between source/drain. 
     SUMMARY OF THE INVENTION 
     In view of the above-mentioned disadvantages of the prior art, it is an object of the invention to provide a method by which hot carrier deterioration in a P-MOS transistor can be simulated in both the FWD and REV modes, and hot carrier deterioration in a transistor during bidirectional operation can be estimated by simulation with high accuracy. 
     According to a method of simulating hot carrier deterioration of a P-MOS transistor of an aspect of the invention, the following formulas (A1), (A2), (A3) and (A4) are used in an FWD mode that a direction of current flow between source/drain during application of a stress in a transistor is the same as that during measuring of transistor characteristics: 
     
         Vth=Vfb+σ•Vd                                   (A1) 
    
     
         ΔVth=ΔVfb                                      (A2) 
    
     
         (ΔVfb).sub.f =A•τ.sup.n                    (A 3) 
    
     
         τ=B•(Ig/W).sup.-m                                (A 4) 
    
     where Vth represents a threshold voltage, Vd represents a drain voltage, Vfb represents a threshold voltage at Vd=0V, Ig represents a gate current, W represents a gate width, ΔVth and ΔVfb represent variations of Vth and Vfd caused by hot carrier deterioration, σ represents a coefficient showing an effect of lowering of an oxide film barrier caused by Vd, and a lifetime τ of a transistor is defined by the formula (A3); and 
     coefficients A, n, B and m are determined by a preliminary measuring experiment, whereby the transistor lifetime τ can be estimated. 
     According to a method of simulating hot carrier deterioration of a P-MOS transistor of another aspect of the invention, the following formulas (A1), (A5), (A3) and (A4) are used in an REV mode that a direction of current flow between source/drain during application of a stress in a transistor is opposite to that during measuring of transistor characteristics: 
     
         Vth=Vfb+σ•Vd                                   (A1) 
    
     
         ΔVth=ΔVfb+Δσ•Vd              (A5) 
    
     
         (ΔVfb).sub.f =A•τ.sup.n                    (A 3) 
    
     
         (τ=B•(Ig/W).sup.-m                               (A 4) 
    
     where Vth represents a threshold voltage, Vd represents a drain voltage, Vfb represents a threshold voltage at Vd=0V, Ig represents a gate current, W represents a gate width, σ represents a coefficient showing an effect of lowering of an oxide film barrier caused by Vd, ΔVth, ΔVfb and Δσ represent variations of Vth, Vfd and σ caused by the hot carrier deterioration, and a lifetime τ of a transistor is defined by the formula (A3); and 
     coefficients A, n, B and m are determined by a preliminary measuring experiment, whereby the transistor lifetime τ can be estimated. 
     According to the method of simulating the hot carrier deterioration of the P-MOS transistor of the invention, the hot carrier deterioration can be simulated in both the FWD and REV modes. Also according to the invention, parameters which change due to the hot carrier deterioration are selected, a correlation between the parameters is extracted by the preliminary experiment, and the simulation is performed using the correlation, so that the hot carrier deterioration can be simulated with high accuracy. 
     The foregoing and other objects, features, aspects and advantages of the present invention will become more apparent from the following detailed description of the present invention when taken in conjunction with the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is an equivalent circuit diagram showing a concept of a simulation method of the invention; 
     FIG. 2 is a graph showing a relationship between a threshold voltage Vth and a drain voltage Vd in an FWD mode after a hot carrier stress in a P-MOS transistor; 
     FIG. 3 is a graph showing a relationship between a variation ΔVfb of a flat band voltage and a stress time in the FWD mode; 
     FIG. 4 is a graph showing a relationship between a lifetime τ affected by hot carrier deterioration and a gate current Ig in a P-MOS transistor; 
     FIG. 5 is a graph showing a relationship between a threshold voltage Vth and a drain voltage Vd in an REV mode after a hot carrier stress in the P-MOS transistor; 
     FIG. 6 is a graph showing a relationship between a variation ΔVfb of a flat band voltage and a variation Δσ of a DIBL effect in the REV mode; 
     FIG. 7 is a graph showing a relationship between an effective channel length Leff and a DIBL effect σ in the P-MOS transistor before application of the hot carrier stress; 
     FIG. 8 is a graph prepared by plotting 1/β in connection with differences of gate voltages Vg and threshold voltages Vth before and after application of the hot carrier stress in the P-MOS transistor; 
     FIG. 9 is a graph showing a relationship between a variation Δθ of dependency of a carrier mobility on a vertical field effect and a variation ΔVfb of a flat band voltage after the hot carrier deterioration in the P-MOS transistor; 
     FIG. 10 is a graph showing a relationship between variation ΔU0 of the mobility and variation ΔVfb of the flat band voltage with Vg=Vth after hot carrier deterioration in the P-MOS transistor; 
     FIG. 11 is a graph showing a relationship between variation rate ΔId/Id of the drain current and the stress time at a linear region in the FWD mode after the hot carrier stress in the P-MOS transistor; 
     FIG. 12 is a graph showing a relationship between variation rate ΔId/Id of the drain current and the stress time at the linear region in the REV mode after the hot carrier stress in the P-MOS transistor; 
     FIG. 13 is a graph showing a relationship between variation rate ΔId/Id of the drain current and the stress time at a saturation region in the FWD mode after the hot carrier stress in the P-MOS transistor; 
     FIG. 14 is a graph showing a relationship between variation rate ΔId/Id of the drain current and the stress time at the saturation region in the REV mode after the hot carrier stress in the P-MOS transistor; 
     FIG. 15 is an equivalent circuit diagram showing a concept of simulation of hot carrier deterioration of a P-MOS transistor in the prior art; 
     FIG. 16 is a flow diagram showing steps of simulation relating to the hot carrier deterioration of the P-MOS transistor in the prior art; 
     FIG. 17 is a graph showing Vd-Id characteristics measured in the FWD mode before and after application of the stress in the P-MOS transistor; 
     FIG. 18 is a graph showing Vd-Id characteristics measured in the REV mode before and after application of the stress in the P-MOS transistor; and 
     FIG. 19 is a graph showing Vg-Id characteristics at the linear region, Vg-Id characteristics at the saturation region and Vg-gm characteristics which were measured in the FWD and REV modes after and before application of the stress in the P-MOS transistor. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     FIG. 1 is an equivalent circuit diagram showing a concept of hot carrier deterioration of a P-MOS transistor according to simulation of the invention. In the conventional simulation, as shown in FIG. 15, injection of hot carriers causes change of transistor parameters in accordance with passage of time, and these transistor parameters are obtained based on a quantity of stress. According to the concept of simulation of the invention, however, the transistor parameters are maintained, and characteristics such as a drain current Id and a threshold voltage Vth which are changed by the hot carriers are represented by an equivalent circuit using a current source 1A of a voltage control type. 
     The concept of the invention is also characterized in that parameters characterizing the characteristics of voltage control type current source 1A are determined by a preliminary experiment with the hot carrier stress. Further, the concept is characterized in that the parameters which are changed by the hot carrier stress are selected and the hot carrier deterioration is simulated using a correlation between these parameters. 
     (Embodiment 1) 
     FIG. 2 shows a relationship between drain voltage Vd and threshold voltage Vth measured in an FWD mode after hot carrier deterioration caused by applying a stress voltage to a P-MOS transistor. In FIG. 2, a stress was applied for 0 second, 10 seconds, 100 seconds, 1000 seconds and 10000 seconds under the condition of Vd=-6.0V and the condition that achieves a maximum value of gate current Ig. From the graph of FIG. 2, it can be found that threshold voltage Vth is represented as a linear function of drain voltage Vd expressed by the following formula (1) 
     
         Vth=Vfb+σ•Vd                                   (1) 
    
     In the graph of FIG. 2, intersections of each line and Vth axis correspond to flat band voltages Vfb, and a gradient of each line corresponds to a DIBL (Drain Induced Barrier Lowering) effect u by the drain voltage. 
     It can be seen from FIG. 2 that flat band voltage Vfb changes depending on the stress time, but the DIBL effect σ is constant (gradient of each line is constant). Therefore, variation AVth of threshold voltage in the FWD mode is expressed by the following formula (2): 
     
         ΔVth=ΔVfb                                      (2) 
    
     In FIG. 3, a relationship between variation ΔVfb of the flat band voltage and the stress time in the FWD mode is plotted on a Log-Log scale. In FIG. 3V, -4.5V, -5V, -5.5V and -6V were applied as the drain voltage during a stress period. Gate voltage Vg was applied under the condition that gate current Ig attained a maximum value for maximizing the hot carrier variation. 
     It can be seen from the graph of FIG. 3 that variation ΔVfb of the flat band voltage can be expressed by the following formula (3) which is similar to the formula (102): 
     
         ΔVfb=A•t.sup.n                                 (3) 
    
     where A and n are coefficients depending on manufacturing process conditions of the transistor and stress conditions. 
     Therefore, by defining, for example, that lifetime τ of a transistor expires when (ΔVfb) r  attains to 10 mV, lifetime τ can be expressed by the following formula (4). 
     
         τ={(ΔVfb).sub.r /A}.sup.1/n                      (4) 
    
     FIG. 4 is a graph showing a relationship between lifetime τ and gate current Ig, and it can be seen that lifetime τ can be expressed by the following formula (5) similarly to the formula (105): 
     
         τ=B•(Ig/W).sup.-m                                (5) 
    
     Therefore, coefficient A in the formula (3) can be expressed by the following formula (6) similar to the formula (107): 
     
         A=A.sub.Vfb =(ΔVfb).sub.f •(B•W.sup.m •Ig.sup.-m).sup.-n                                  (6) 
    
     Therefore, variation AVfb of the flat band voltage can be expressed by the following formula (7) similar to the formula (109) 
     
         ΔVth=(ΔVth).sub.f •B.sup.-n •W.sup.mn •Ig.sup.mn •t.sup.n                           (7) 
    
     By extracting coefficients B, m and n in the formula (7) by a preliminary experiment, simulation can be performed to obtain flat band voltage Vfb after the hot carrier deterioration or threshold voltage Vth in the FWD mode of the P-MOS transistor. 
     More specifically, in the FWD mode of the embodiment 1, threshold voltage Vth after the application of stress can be simulated with high accuracy by changing flat band voltage Vfb by ΔVfb, because DIBL effect σ does not change. 
     (Embodiment 2) 
     FIG. 5 shows a relationship between threshold voltage Vth and drain voltage Vd which were measured in the REV mode when a stress voltage was applied in the P-MOS transistor to generate the hot carrier deterioration. The stress conditions in FIG. 5 are the same as those in FIG. 2. In the graph of FIG. 5, the gradients of straight lines increase as the stress time increases. Thus, it can be seen that the absolute value of DIBL effect σ in the formula (1) increases as the stress time increases. Therefore, variation ΔVth of threshold voltage in the REV mode is expressed not by the formula (2) but by the following formula (8). 
     
         ΔVth=ΔVfb+Δσ•Vd              (8) 
    
     From comparison between FIGS. 5 and 4, it can be found that flat band voltage Vfb changes by the same amount of ΔVfb depending on the stress time in both the FWD and REV modes. Thus, it can be found that the change by ΔVfb of the flat band voltage is based on the oxide film trap of electrons caused by hot carrier injection. 
     Therefore, variation ΔVfb of the flat band voltage in the REV mode can be obtained similarly to the embodiment 1. By taking variation ΔVfb of the flat band voltage thus obtained as well as variation Δσ of the DIBL effect into consideration, it is possible to obtain variation ΔVth of the threshold voltage in the REV mode after the hot carrier deterioration based on the formula (8). 
     Thus, in the REV mode of the embodiment 2, Vth in the REV mode after the hot carrier deterioration can be simulated with high accuracy by taking variation ΔVfb of the flat band voltage depending on the stress time as well as variation Δσ of the DIBL effect σ depending on the stress time into consideration. 
     (Embodiment 3) 
     FIG. 6 shows a relationship between variation ΔVfb of the flat band voltage and variation Δσ of the DIBL effect in the REV mode which was found when the hot carrier deterioration was caused by application of the stress voltage in the P-MOS transistor. As the stress conditions, drain voltage Vd of -4.5V, -5.0V, -5.5V or -6.0V was applied. Gate current Vg was applied under the conditions that gate current Ig attained a maximum value. From FIG. 6, it can be seen that ΔVfb and Δσ are expressed by the following formula (9) regardless of a value of drain voltage Vd. 
     
         Δσ=C1•ΔVfb                         (9) 
    
     where coefficient C1 depends on manufacturing process conditions, a thickness of a gate oxide film and a gate length. 
     From a relationship between the formulas (8) and (9), variation ΔVth of the threshold voltage in the REV mode can be obtained from the following formula (10): 
     
         ΔVth=(1+C1•Vd)•ΔVfb                (10) 
    
     Therefore, simulation can be performed to obtain variation ΔVth of the threshold voltage by obtaining variation ΔVfb of the flat band voltage, which can be obtained by extracting coefficients B, m and n in the formula (1) described in the embodiment (1) by the preliminary experiment, and by determining coefficient C1 in the formula (9) by the preliminary experiment. 
     Thus, in the REV mode of the embodiment (3), threshold voltage Vth after application of the stress can be simulated with high accuracy only by obtaining ΔVfb, if the preliminary experiment is performed to determine coefficient C1 which determines a relationship between variation ΔVfb of the flat band voltage and variation do of the DIBL effect. 
     (Embodiment 4) 
     FIG. 7 shows a relationship between effective channel length Leff and DIBL effect o in the P-MOS transistor before application of the hot carrier stress. In the graph of FIG. 7, the relationship between Leff and σ is expressed by the following formula (11), where a coefficient C2 depends on the manufacturing process conditions and the thickness of the gate oxide film. 
     
         Leff∝σ.sup.1/C2                               (11) 
    
     Since the hot carrier stress may change DIBL effect σ as stated in the embodiment 2, variation Δσ of the DIBL effect and the formula (11) can be used to express, as the following formula (12), a shortening Let of effective channel length Leff which occurs due to the oxide film trap of electrons caused by the hot carrier stress: 
     
         (Leff-Let)/Leff={(σ+Δσ)/σ}.sup.1/C2 (12) 
    
     The formula (12) can be changed into the following formula (13): 
     
         Let=Leff•[{1-{(σ+Δσ)/σ}.sup.1/C2 ](13) 
    
     By determining coefficient C2 by the preliminary experiment, therefore, shortening Let of the effective channel length can be obtained. By incorporating this Let into the parameters of the P-MOS transistor subjected to the hot carrier stress as is done in an embodiment which will be described layer, the change of transistor characteristics caused by the hot carrier deterioration can be simulated more accurately. 
     Thus, in the embodiment 4, shortening Let of the effective channel length can be quantitatively obtained using DIBL effect σ and variation Δσ of the DIBL effect caused by the hot carrier stress. By utilizing this Let, the hot carrier deterioration of the P-MOS transistor can be simulated more accurately. 
     (Embodiment 5) 
     In FIG. 8, 1/β is plotted in connection with a difference between measured gate voltage Vg and threshold voltage Vth. Circular, square, triangular, diamond-shaped and solid circular marks represent the measured results after the stress time of 0 second, 10 seconds, 100 seconds, 1000 seconds and 10000 seconds. 
     β is defined in the linear region of the transistor by the following formula (14): 
     
         β=(∂Id/∂Vg)/Vd              (14) 
    
     β is used in the following model formula (15) of the drain current in the linear region, and is expressed by the following formula (16): ##EQU1## where Cox represents a capacitance of a gate oxide film, and Vmax represents a saturation speed. 
     It has been known that the carrier mobility μs can be expressed by the following formula: 
     
         μs=U0/{1+θ(Vg-Vth) }                              (17) 
    
     where U0 represents the mobility when Vg is equal to Vth, and θ represents dependency of the mobility on the vertical electric field. 
     Therefore, the formula (16) can be changed into the following formula (16a): 
     
         1/β={(Leff/W)/Cox}•{1/U0+θ•(Vg-Vth)/U0}+{(Vd/Vmax)/W}/Cox                                                      (16a) 
    
     Since saturation speed Vmax is an invariable physical quantity which does not depend on the hot carrier stress, β in a region where .linevert split.Vg.linevert split. is larger than .linevert split.Vth.linevert split. by a relatively large amount can be approximately expressed as a linear function of Vg, whereby a proportionality constant al of the linear function can be expressed as the following formula (18a) based on the formula (16a): 
     
         a1=Leff•{(θ/W)/Cox}/U0                         (18a) 
    
     An intercept b1 of 1/β axis at Vg=Vth can be expressed by the following formula (18b) based on the formula (16a): 
     
         b1={(Leff/W)/Cox}/U0+{(Vd/Vmax)/W}/Cox                     (18b) 
    
     Based on the formulas (18a) and (18b), therefore, dependency θ of the mobility on the vertical electric field can be expressed by the following formula (19): 
     
         θ=a1/[b1-{(Vd/Vmax)/W}/Cox]                          (19) 
    
     In the graph of FIG. 8, it can be seen that the proportionality constant al in the relationship between 1/β and (Vg-Vth) changes depending on the stress time. The change of proportionality constant al depends on variation Δθ of dependency θ of the mobility on the vertical electric field in the formula (17). 
     In FIG. 9, there is shown a relationship between variation ΔVfb of the flat band voltage shown in the embodiments 1 and 2 and variation Δθ of dependency of the mobility on the vertical electric field. Circular, triangular and square marks show the stress with drain voltage Vd of -6.0V, -5.5V and -5.0V, respectively. Δθ can be calculated with the formula (19). As can be seen from the graph of FIG. 9, there is a linear relationship between Δθ and θVfb, which can be expressed by the following formula (20): 
     
         Δθ=C3•ΔVfb                         (20) 
    
     where C3 is a constant depending on the manufacturing process conditions, the thickness of gate oxide film and the gate length. 
     By determining the coefficient C3 by the preliminary experiment, therefore, it is possible to simulate variation Δθ of the dependency of the carrier mobility on the vertical electric field in the drain current after the hot carrier stress, using variation ΔVfb of the flat band voltage obtained in the embodiment 1 or 2. 
     Thus, in the embodiment 5, coefficient C3 is extracted by the preliminary experiment, and thereby it is possible to obtain variation Δθ of the dependency of the carrier mobility on the vertical electric field having a close relationship with the drain current characteristics after the hot carrier deterioration of the P-MOS transistor. Accordingly, mobility μs after the hot carrier stress can be accurately simulated based on the formula (17). 
     (Embodiment 6) 
     As already described in connection with FIG. 8, it is known that the formula (16a) can express 1/β in the region where .linevert split.Vg.linevert split. is relatively large with respect to .linevert split.Vth.linevert split.. Also, dependency θ of the mobility on the vertical electric field can be obtained from the formula (19). Therefore, mobility U0 at Vg=Vth can be expressed by the following formula (21) based on the formula (18a): 
     
         U0=Leff•{(θ/W)/Cox}/a1                         (21) 
    
     U0 can also be obtained from the following formula (22) using an intercept b1 of 1/β axis at Vg=Vth in FIG. 8. 
     
         U0=W•Cox•b1/Leff-(Vd/Vmax)/Leff                (22) 
    
     In FIG. 8, intercept b1 of 1/β axis at Vg=Vth changes depending on the hot carrier stress time. It can be seen from the formula (18b) that the change of intercept b1 is based on change (Leff-Let) of Leff and change ΔU0 of U1. Change ΔU0 of U0 can be obtained by incorporating shortening Let of Leff, which is obtained in the embodiment 4, into the formula (21) or (22). 
     FIG. 10 shows a relationship between change ΔU0 of the mobility at Vg-Vth obtained as described above and variation ΔVfb of the flat band voltage obtained in the embodiment 1 or 2. In the graph of FIG. 10, circular, triangular and square marks represent the stress with the drain voltage of -6.0V, -5.5V and -5.0V, respectively. It can be seen from the graph of FIG. 10 that ΔUO and ΔVfb have a linear relationship, and can be expressed by the following formula (23): 
     
         ΔU0=C4•ΔVfb                              (23) 
    
     where C4 is a constant depending on the manufacturing process conditions, the thickness of gate oxide film and the gate length. 
     By determining the coefficient C4 by the preliminary experiment, therefore, it is possible to simulate variation ΔU0 of the carrier mobility at Vg=Vth after the hot carrier stress based on ΔVfb obtained in the embodiment 1 or 2. 
     Thus, according to the embodiment 6, the coefficient C4 is determined by the preliminary experiment, and thereby it is possible to obtain the variation ΔU0 of the carrier mobility under the condition of Vg-Vth after the hot carrier stress, so that it is possible to simulate accurately the carrier mobility having a close relationship with the drain current of the P-MOS transistor. 
     (Embodiment 7) 
     In FIG. 11, a relationship between variation rate ΔId/Id of the drain current and the stress time during the FWD mode in the linear region of the P-MOS transistor is plotted on the Log-Log scale. For the stress conditions, drain voltage Vd of -6.0V and gate voltage Vg achieving the maximum gate current were applied. In the graph of FIG. 11, circular marks represent the results of measurement with Vd=-0.2V and Vg=-1.5V, and triangular marks represent the results of measurement with Vd=-0.2V and Vg=-2.0V. Solid curves represent the results obtained by the simulation. 
     It has been known that drain current Id in the linear region is expressed by the following formula (15): 
     
         Id=•W•(Cox/Leff)•[μs/{1+μs•(Vd/Vmax)/Leff}].cndot.Vd•(Vg-Vth-Vd/2)                                (15) 
    
     Threshold voltage Vth&#39; in the FWD mode after the hot carrier stress can be expressed by the following formula (24) based on the relationship with the formula (2) in the embodiment 1: 
     
         Vth&#39;=Vth-ΔVth=Vth-ΔVfb                         (24) 
    
     Dependency θ&#39; of the mobility on the vertical electric field in the FWD mode after the hot carrier stress can be expressed by the following formula (25) based on the embodiment 5, and the mobility U0&#39; at Vg=Vth can be expressed by the following formula (26) based on the embodiment 6. Therefore, carrier mobility μs&#39; after the hot carrier deterioration can be expressed by the following formula (27). 
     
         θ&#39;=θ+Δθ                            (25) 
    
     
         U0&#39;=θ+Δθ                                 (26) 
    
     
         μs&#39;=U0&#39;/{1+θ&#39;•(Vg-Vth&#39;) }                   (27) 
    
     Effective channel length Leff&#39; after the hot carrier stress is expressed by the following formula (27.5) based on the embodiment 4: 
     
         Leff&#39;=Leff-Let=Leff•{(σ+Δσ)/σ}.sup.1/C2 (27.5) 
    
     Therefore, drain current Id&#39;.sub.(FWD, LIN) at the linear region in the FWD mode after the hot carrier stress is expressed by the following formula (28) using Vth&#39; in the formula (24), μs&#39; in the formula (27) and Leff in the formula (27.5): ##EQU2## 
     Accordingly, the drain current variation rate ΔId&#39;/Id.sub.(FWD,LIN) at the linear region in the FWD mode after the hot carrier stress can be expressed by the following formula (29): 
     
         ΔId&#39;.sub.(FWD, LIN) =(Id&#39;-Id)/Id 
    
     
         =μs&#39;/μs•{(μs•Vd+Vmax•Leff) 
    
     
         /(μs&#39;•Vd+Vmax•Leff&#39;)}•{(Vg-Vth&#39;-Vd/2) /(Vg-Vth-Vd/2)}-1                                         (29) 
    
     The solid curves in the graph of FIG. 11 represent the results of simulation using the formula (29), and it can be seen that the results are accurately coincident with the results of actual measurement. 
     Thus, in the embodiment 7, the drain current Id at the linear region in the FWD mode after the hot carrier stress can be accurately simulated by using ΔVth, Δθ, ΔU0 and Leff&#39; obtained by the formula (29) and the foregoing embodiments. 
     (Embodiment 8) 
     In FIG. 12, a relationship between variation rate ΔId/Id of the drain current and the stress time at the linear region of the P-MOS transistor in the REV mode after the hot carrier stress is plotted on the Log-Log scale. In the graph of FIG. 12, the stress condition was that drain voltage Vd of -6.0V and gate voltage Vg achieving the maximum gate current were applied. Circular marks represent the results of measurement with Vd=-0.2V and Vg=-1.5V, and triangular marks represent the results of measurement with Vd=-0.2V and Vg=-2.0V. Solid curves represent the results obtained by the simulation. In connection with the REV mode after the hot carrier stress, formulas (15), (25), (26), (27) and (27.5) can also be utilized. In addition to these formulas, the threshold voltage Vth&#39; after the hot carrier stress in the REV mode can be expressed by the following formula (30) based on the relationship between the formula (8) in the embodiment 2 and the formula (10) in the embodiment 3. 
     
         Vth&#39;=Vth-ΔVth=Vth-ΔVfb-Δσ•Vd =Vth-(1+C1•Vd)•ΔVfb                     (30) 
    
     Therefore, drain current Id&#39;.sub.(REV, LIN) at the linear region in the REV mode after the hot carrier stress can be expressed by the following formula (31) using Vth&#39; in the formula (30), μs&#39; in the formula (27) and Leff&#39; in the formula (27.5). ##EQU3## 
     As a result, variation rate ΔId/Id.sub.(REV, LIN) of the drain current at the linear region in the REV mode after the hot carrier stress is expressed by the following formula (32): ##EQU4## 
     Solid curves in the graph of FIG. 12 show the results of simulation using this formula (32), and it can be seen that the results are coincident with the results of actual measurement with high accuracy. 
     Thus, in the embodiment 8, the drain current at the linear region in the REV mode after the hot carrier stress can be accurately simulated by using ΔVth, Δθ, ΔU0 and Leff&#39; obtained in the formula (32) and the foregoing embodiments. 
     (Embodiment 9) 
     In FIG. 13, a relationship between variation rate ΔId/Id of the drain current and the stress time at the saturation region in the FWD mode of the P-MOS transistor is plotted on the Log-Log scale. In the graph of FIG. 13, the stress condition was that drain voltage Vd of -6.0V and gate voltage Vg achieving the maximum gate current were applied. Square marks represent the results of measurement with Vd=-1.5V and Vg=-1.5V, and diamond-like marks represent the results of measurement with Vd=-2.0V and Vg=-2.0V. Solid curves represent the results obtained by the simulation. 
     It has been known that drain current Id in the saturation region is expressed by the following formula (33): ##EQU5## 
     Saturation drain voltage Vdsat in the formula (33) is expressed by the following formula (34), and saturation speed region length ΔL is expressed by the following formula (35): 
     
         Vdsat={(Vg=Vth)•Esat•Leff}/{Esat•Leff+(Vg-Vth) }(34) 
    
     
         ΔL=k•In[{(Vd-Vdsat)/k+Em}/Esat ]               (35) 
    
     Internal electric field Em in the formula (35) is expressed by the following formula (36), and k is expressed by the following formula (37) including a junction depth Xj and a gate oxide film thickness tox: 
     
         Em={(Vd-Vdsat).sup.2 /k.sup.2 +Esat.sup.2 }.sup.1/2        (36) 
    
     
         k=0.2•Kj.sup.1/2 •tox .sup.1/3                 (37) 
    
     Threshold voltage Vth&#39; in the FWD mode after the hot carrier stress is expressed by the formula (24), and mobility μs&#39; is expressed by the formula (27). Saturation drain voltage Vdsat&#39; in the FWD mode after the hot carrier stress is expressed by the following formula (38): 
     
         Vdsat&#39;={(Vg-Vth-ΔVth)•Esat•Leff}/{Esat•Leff+(Vg-Vth+ΔVth)}                                              (38) 
    
     Therefore, drain current Id&#39; FWD , SAT) at the saturation region in the FWD mode after the hot carrier stress is expressed by the following formula (39), using Vth&#39; in the formula (24), μs&#39; in the formula (27) and Vdsat&#39; in the formula (38 ##EQU6## 
     As a result, variation rate ΔId/Id.sub.(FWD, SAT) of the drain current at the saturation region in the FWD mode after the hot carrier stress is expressed by the following formula (40): ##EQU7## 
     Solid curves in the graph of FIG. 13 represent the results of simulation using the formula (40), and it can be seen that the results are accurately coincident with the results of actual measurement. Thus, according to the embodiment 9, the drain current at the saturation mode in the FWD mode after the hot carrier stress can be accurately simulated by using ΔVth, Δθ and ΔU0 obtained in the foregoing embodiments and Vdsat&#39; obtained in the formula (38) as well as the formula (40). 
     (Embodiment 10) 
     In FIG. 14, a relationship between variation rate ΔId/Id of the drain current and the stress time at the saturation region in the REV mode after the hot carrier stress of the P-MOS transistor is plotted on the Log-Log scale. In the graph of FIG. 14, the stress condition was that drain voltage Vd of -6.0V and gate voltage Vg achieving the maximum gate current were applied. Square marks represent the results of measurement with Vd=-1.5V and Vg=-1.5V, and diamond-like marks represent the results of measurement with Vd=-2.0V and Vg=-2.0V. Solid curves represent the results obtained by the simulation. Before application of the stress, as described before, saturation drain voltage Vdsat is expressed by the formula (34), and saturation measurement region length ΔL is expressed by the formula (35). At the same time, internal electric field Em is expressed by the formula (36), and k is expressed by the formula (37). Threshold voltage Vth&#39; in the REV mode after the hot carrier stress is expressed by the formula (30), and mobility μs&#39; is expressed by the formula (27). 
     Effective channel length Leff&#39; in the REV mode after the hot carrier stress is expressed by the formula (27.5), and saturation drain voltage Vdsat&#39; is expressed by the following formula (41). 
     
         Vdsat&#39;={(Vg-Vth-ΔVth)•Esat•Leff&#39;}/{Esat•Leff&#39;+(Vg-Vth+ΔVth)}                                            (41) 
    
     Therefore, drain current Id&#39;.sub.(REV, SAT) at the saturation region in the REV mode after the hot carrier stress is expressed by the following formula (42) using Vth&#39; in the formula (30), μs&#39; in the formula (27), Leff&#39; in the formula (27.5) and Vdsat&#39; in the formula (41). ##EQU8## 
     Consequently, the variation rate ΔId/Id.sub.(REV, SAT) of the drain current at the saturation region in the REV mode after the hot carrier stress is expressed by the following formula (43): ##EQU9## 
     Solid curves in the graph of FIG. 14 represent the results of simulation using the formula (43), and it can be seen that the results are accurately coincident with the results of actual measurement. 
     Thus, according to the embodiment 10, the drain current at the saturation region in the REV mode after the hot carrier stress can be accurately simulated by using ΔVth, Δθ, ΔUO and Leff&#39; obtained in the foregoing embodiments, and Vdsat&#39; obtained in the formula (41) as well as the formula (43). 
     As described hereinbefore, the invention can provide the method by which it is possible to simulate accurately the hot carrier deterioration of various transistor characteristics in the FWD and REV modes after the hot carrier stress of the P-MOS transistor. 
     Although the present invention has been described and illustrated in detail, it is clearly understood that the same is by way of illustration and example only and is not to be taken by way of limitation, the spirit and scope of the present invention being limited only by the terms of the appended claims.