Method for deriving characteristic values of MOS transistor

Abstract

A method for deriving characteristic values of a MOS transistor is described. A set of ηk values is provided. A set of VBi values (i=1 to M, M≧3) is provided. A set of RSDi,j (i=1 to M−1, j=i+1 to M) values each under a pair of VBi and VBj, or a set of Vtq_q,j (q is one of 1 to M, j is 1 to M excluding q) values under VBq is derived for each ηk, with an iteration method. The ηk value making the set of RSDi,j values or Vtq_q,j values closest to each other is determined as an accurate ηk value. The mean value of RSDi,j at the accurate ηk value is calculated as an accurate RSD value.

Description

BACKGROUND OF THE INVENTION

Field of Invention

This invention relates to a method for probing a MOS transistor, and particularly relates to a method for deriving characteristic values of a MOS transistor.

Description of Related Art

The sum (RSD) of the drain resistance R_D and the source resistance R_S and the effective carrier mobility μ_eff are important characteristic values for a MOS transistor. Recently, RSD extraction by using a single device is reported in two papers including Da-Wen Lin et al., IEEE Electron Device Letters, Vol. 28, No. 12, pp. 1132-1134, December 2007, and Da-Wen Lin et al., IEEE Transactions on Electron Devices, Vol. 57, No. 4, April 2010. The μ_eff value is directly calculated from the extracted RSD value. The single device of MOS transistor is illustrated in FIG. 1, together with the designation of certain parameters thereof that will be used hereafter.

The V_t values at different bias configurations and the empirical factor η (˜3 for PFET and ˜2 for NFET) are only inputs in the RSD equations in the above papers. However, if the values of V_t and η are not accurate, the extracted RSD and the μ_eff derived therefrom are not accurate either.

This invention gives a self-constant iteration method to find the accurate V_t and η so that determine the accurate RSD and μ_eff.

SUMMARY OF THE INVENTION

In view of the foregoing, this invention provides a method for deriving correct characteristic values of a MOS transistor.

In the method for deriving correct characteristic values of a MOS transistor, the MOS transistor comprises a substrate, a gate, a channel region, a source and a drain, the threshold voltage of the channel region is V_t, the voltage applied to the substrate is V_B, the voltage applied to the gate is V_G, the voltage across the source and the drain is V_DS, the voltage across the gate and the source is V_GS, a V_GS making the MOS transistor operate in a constant-mobility region is V_{GS_c}, the current from the drain to the source is I_DS, and the sum of the resistance R_D of the drain and the resistance R_S of the source is RSD. When a correct η value has been given, a correct RSD is derived by steps comprising the following steps a) to g). Step a) is applying different V_B1 and V_B2, respectively, to the substrate. Step b) is measuring I_DS1 and I_DS2 that correspond to V_B1 and V_B2, respectively. Step c) is giving respective initial values of V_t1 and V_t2, wherein V_t1 correspond to V_B1 and V_t2 corresponds to V_B2. Step d) is calculating RSD with the correct η value, V_t1 and V_t2 by the equation of

$RSD=\left(\frac{V_{\mathrm{GS\_c}}+\left(\eta -1\right)V_{t1}-\eta V_{t2}-0.5V_{\mathrm{DS}}}{I_{\mathrm{DS}2}}-\frac{V_{\mathrm{GS\_c}}-V_{t1}-0.5V_{\mathrm{DS}}}{I_{\mathrm{DS}1}}\right)\times \frac{V_{\mathrm{DS}}}{\eta \times \left(V_{t1}-V_{t2}\right)}.$ Step e) is plotting a V_GS−I_{DS1_exclude_RSD} curve to derive V_{GS_a1} and plotting a V_GS−I_{DS2_exclude_RSD} curve to derive V_{GS_a2}, wherein I_{DS1_exclude_RSD} and I_{DS2_exclude_RSD} are calculated by the equations of

$I_{\mathrm{DS}1\mathrm{\_exclude}\mathrm{\_RSD}}=\frac{I_{\mathrm{DS}1}\times V_{\mathrm{DS}}}{\left(V_{\mathrm{DS}}-I_{\mathrm{DS}1}\times RSD\right)}$ $\mathrm{and}$ $I_{\mathrm{DS}2\mathrm{\_exclude}\mathrm{\_RSD}}=\frac{I_{\mathrm{DS}2}\times V_{\mathrm{DS}}}{\left(V_{\mathrm{DS}}-I_{\mathrm{DS}2}\times RSD\right)},$ the V_GS at the cross point on the V_GS axis with a tangent line extrapolated from the point of the V_GS−I_{DS1_exclude_RSD} curve where dI_{DS1_exclude_RSD}/dV_GS is the maximal, and V_{GS_a2} is the V_GS at the cross point on the V_GS axis with a tangent line extrapolated from the point of the V_GS−I_{DS2_exclude_RSD} curve where dI_{DS2_exclude_RSD}/dV_GS is maximal. Step f) is calculating new V_t1 and V_t2 by using the equations of V_t1=V_{GS_a1}−V_DS/2 and V_t2=V_{Gs_a2}−V_DS/2. Step g) is repeating the above steps d), e) and f) until RSD, V_t1 and V_t2 approach constant values.

Because such obtained RSD, V_t1, and V_t2 are correct values, correct μ_eff1 and μ_eff2 values corresponding to V_B1 and V_B2 can then be calculated by using the correct RSD, V_t1, and V_t2:

${\mu }_{\mathrm{eff}1}=\frac{L_{\mathrm{eff}}}{W_{\mathrm{eff}}{C}_{\mathrm{ox}}}\times \frac{I_{\mathrm{DS}1}}{\left(V_{\mathrm{GS\_c}}-V_{t1}-V_{\mathrm{DS}}/2\right)\times \left(V_{\mathrm{DS}}-RSD\times I_{\mathrm{DS}1}\right)}$ $u_{\mathrm{eff}2}=\frac{L_{\mathrm{eff}}}{W_{\mathrm{eff}}{C}_{\mathrm{ox}}}\times \frac{I_{\mathrm{DS}2}}{\left(V_{\mathrm{GS\_c}}-V_{t2}-V_{\mathrm{DS}}/2\right)\times \left(V_{\mathrm{DS}}-RSD\times I_{\mathrm{DS}2}\right)}$ wherein L_eff is the channel length, W_eff is the channel width, and C_ox is the inversion gate-oxide capacitance.

When the correct η value is unknown, the method for deriving characteristic values of a MOS transistor of this invention includes the following steps to derive the correct η value together with a correct RSD value. A set of η_k values (k=1 to N) is provided. A set of V_Bi values (i=1 to M, M≧3) is provided. For each η_k, a set of RSD_i,j (i=1 to M−1, j=i+1 to M) values each under a pair of V_Bi and V_Bj, a set of μ_{effp_p,j} (p is one of 1 to M, j is 1 to M excluding p) values under a given V_Bp, and/or a set of V_{tq_q,j} (q is one of 1 to M, j is 1 to M excluding q) values under a given V_Bq are derived by the iteration method described in the steps a) to g). Next, an accurate η_k value, which makes the set of RSD_i,j values closest to each other, makes the set of μ_{effp_p,j} values closest to each other, and/or makes the set of V_{tq_q,j} closest to each other, is determined. The mean value of RSD_i,j is calculated as an accurate value for RSD.

With the method for deriving characteristic values of a MOS transistor of this invention, the RSD value can be determined more accurately by a more precise η value. Accurate η, RSD, μ_eff, and V_t could be obtained simultaneously and self-consistently.

In order to make the aforementioned and other objects, features and advantages of the present invention comprehensible, a preferred embodiment accompanied with figures is described in detail below.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a single device of MOS transistor whose characteristic values are to be investigated, together with the designation of certain parameters thereof.

FIG. 2 illustrates a method for deriving characteristic values of a MOS transistor according to an embodiment of this invention.

FIG. 3 illustrates the derivation of RSD_1,2, V_{t1_1,2}, V_{t2_1,2}, μ_{eff1_1,2}, and μ_{eff2_1,2} for each η_k value, as an example of the derivations of other RSD_i,j, V_{ti_i,j}, V_{tj_i,j}, μ_{eff_i,j}, and μ_{eff_i,j}, for the η_k value.

FIG. 4 illustrates the method of deriving V_{GS_a1} by plotting a V_GS−_{DS1_exclude_RSD} curve, as an example of deriving V_{GS_a1} (i=1, 2 . . . ).

FIG. 5 shows determination of the accurate η_k value that makes the set of RSD_i,j values, μ_{effp_p,j} values or V_{tq_q_j} values closest to each other in an example of this invention.

DESCRIPTION OF EMBODIMENTS

It is noted that the following embodiments are intended to further explain this invention but not to limit the scope thereof. For example, though a correct η value is unknown and is to be determined in the illustrated embodiment, the method of this invention can also be applied to cases where a correct η value has been known.

FIG. 2 illustrates a method for deriving characteristic values of a MOS transistor according to an embodiment of this invention.

Referring to FIG. 2, in the step 20, a set of η_k values (k=1 to N) is provided. The set of η_k values are any non-identical positive real numbers larger than 1. In an exemplary embodiment, the η_k values are chosen as 1.1, 1.2, 1.3 . . . 3.9 and 4.0, but not limit thereto.

In next step 22 shown in FIG. 2, a set of V_Bi values (i=1 to M, M≧3) is provided. In an embodiment, M=3, and the different substrate biases V_B1, V_B2 and V_B3 are, for example, 0 V, −0.5 V and −1V for NFET, and, 0 V, 0.5V, and 1V for PFET, respectively.

In next step 24, For each η_k, derive a set of RSD_i,j (i=1 to M−1, j=i+1 to M) values each under a pair of V_Bi and V_Bj, a set of μ_{effp_p,j} (p is one of 1 to M, j is 1 to M excluding p) values under a given V_Bp, and/or a set of V_{tq_q,j} (q is one of 1 to M, j is 1 to M excluding q) values under a given V_Bq corresponding to V_Bi and V_Bj by the iteration method of this invention that is exemplified later in FIG. 3.

In an embodiment of M=3 where three different substrate biases V_B1, V_B2 and V_B3 are applied respectively, the set of RSD_i,j values to be derived include: RSD_1,2 derived from V_B1 and V_B2, RSD_1,3 derived from V_B1 and V_B3, and RSD_2,3 derived from V_B2 and V_B3.

In an embodiment of M=3 where three different substrate biases V_B1, V_B2 and V_B3 are applied respectively and the μ_eff1 value under V_B1 is investigated, the set of μ_{eff1_1,j} values include: η_{eff1_1,2} derived from V_B1 and V_B2, and μ_{eff1_1,3} derived from V_B1 and V_B3.

In an embodiment of M=3 where three different substrate biases V_B1, V_B2 and V_B3 are applied respectively and the V_t2 value under V_B2 is investigated, the set of V_{t2_2,j} values include: V_{t2_1,2} derived from V_B1 and V_B2, and V_{t2_2,3} derived from V_B2 and V_B3.

In next step 26, the accurate η_k value, which makes the set of RSD_i,j values closest to each other, makes the μ_{eff_p, j} values closest to each other, and/or makes the V_{tq_q,j} values closest to each other, is determined.

In next step 28, the mean value of RSD_i,j at the accurate η_k value is calculated as an accurate RSD, the mean value of μ_{effp_p,j} at the accurate η_k value is calculated as an accurate μ_effp under the chosen V_Bp, and/or the mean value of V_{tq_q,j} at the accurate η_k value is calculated as an accurate V_tq under the chosen V_Bq.

The iterative derivation of RSD_i,j, V_{ti_i,j}, V_{tj_i,j}, μ_{effi_i,j}, and μ_{effj_i,j} for each η_k can be exemplified by the iterative derivation of RSD_1,2, V_{t1_1,2}, V_{t2_1,2}, μ_{eff1_1,2}, and μ_{eff2_1,2} for a given η_k value as illustrated in FIG. 3.

Referring to FIG. 3, in step 30, the different V_B1 and V_B2 are respectively applied to the substrate, with the other voltages shown in FIG. 1 unchanged. In next step 32, the transistor current I_DS1 and I_DS2 that correspond to V_B1 and V_B2, respectively, are measured. In next step 34, respective initial values of V_t1 and V_t2 are given, wherein V_t1 correspond to V_B1, and V_t2 corresponds to V_B2. V_{GS_c} is a given V_GS making the MOS transistor operate in a constant-mobility region, and the range of V_GS corresponding to the constant-mobility region is determined by the well-known method described in the papers mentioned in [0002].

In next step 36, RSD is calculated with the η_k value, V_{t1_1,2} and V_{t2_1,2} by the equation of

$RS{D}_{1,2}=\left(\frac{V_{\mathrm{GS\_c}}+\left({\eta }_k-1\right)V_{t1\_1,2}-{\eta }_k{V}_{t2\_1,2}-0.5V_{\mathrm{DS}}}{I_{\mathrm{DS}2}}-\frac{V_{\mathrm{GS\_c}}-V_{t1\_1,2}-0.5V_{\mathrm{DS}}}{I_{\mathrm{DS}1}}\right)\times \frac{V_{\mathrm{DS}}}{{\eta }_k\times \left(V_{t1\_1,2}-V_{t2\_1,2}\right)}$

In next step 38, a V_GS−I_{DS1_exclude_RSD} curve is plotted to derive V_{GS_a1}, and a V_GS−I_{DS2_exclude_RSD} curve is plotted to derive V_{GS_a2}, wherein I_{DS1_exclude_RSD} and I_{DS2_exclude_RSD} are calculated by the equations of

$I_{\mathrm{DS}1\mathrm{\_exclude}\mathrm{\_RSD}}=\frac{I_{\mathrm{DS}1}\times V_{\mathrm{DS}}}{\left(V_{\mathrm{DS}}-I_{\mathrm{DS}1}\times RS{D}_{1,2}\right)}$ $\mathrm{and}$ $I_{\mathrm{DS}2\mathrm{\_exclude}\mathrm{\_RSD}}=\frac{I_{\mathrm{DS}2}\times V_{\mathrm{DS}}}{\left(V_{\mathrm{DS}}-I_{\mathrm{DS}2}\times RS{D}_{1,2}\right)},$ is the V_GS at the cross point on the V_GS axis with a tangent line extrapolated from the point of the V_GS−I_{DS1_exclude_RSD} curve where dI_{DS1_exclude_RSD}/dV_GS is maximal, and V_{GS_a2} is the V_GS at the cross point on the V_GS axis with a tangent line extrapolated from the point of the V_GS−I_{DS2_exclude_RSD} curve where dI_{DS2_exclude_RSD} dV_GS is maximal.

The method of deriving V_{GS_a1} from the plotted V_GS−I_{DS1_exclude_RSD} curve is taken as an example of deriving V_{GS_ai} from the plotted V_GS−I_{DS1_exclude_RSD} as shown in FIG. 4. Referring to FIGS. 1 and 4, to plot the V_GS−I_{DS1_exclude_RSD} curve, a series of V_G's are respectively applied to the gate of the MOS transistor while the substrate bias is fixed at V_B1, the corresponding series of V_GS's are calculated from the series of V_G's, the corresponding series of I_DS1's are measured, the corresponding series of I_{DS1_exclude_RSD}'S are calculated by the equation of

$I_{\mathrm{DS}1\mathrm{\_exclude}\mathrm{\_RSD}}=\frac{I_{\mathrm{DS}1}\times V_{\mathrm{DS}}}{\left(V_{\mathrm{DS}}-I_{\mathrm{DS}1}\times RS{D}_{1,2}\right)},$ and the I_{DS1_exclude_RSD} curve values are plotted with respect to the V_GS. The first-order differential of I_{DS1_exclude_RSD} with respect to V_GS is plotted to determine the point of the curve with the maximal slope, a tangent line is plotted at the maximal-slope point of the curve, and the intercept of the tangent line with the x-axis is determined as V_{GS_a1}. V_{GS_a2} can be derived from the V_GS−I_{DS2_exclude_RSD} curve in a similar manner.

In next step 40, V_{t1_1,2} is calculated with V_{GS_a1} and V_{t2_1,2} is calculated with V_{GS_a2}, by the equations of and V_{t1_1,2}=V_{GS_a1}−V_DS/2 and V_{t2_1,2}=V_{GS_a2}−V_DS/2. The above steps 36, 38 and 40 are repeated until RSD_1,2, V_{t1_1,2} and V_{t2_1,2} approach constant values without change after several times iterations.

In step 42, μ_{eff1_1,2} and μ_{eff2_1,2} values can be calculated from the above derived RSD_1,2 V_{t1_1,2}, and V_{t2_1,2} values, by the equation of

${\mu }_{\mathrm{eff}1\_1,2}=\frac{L_{\mathrm{eff}}}{W_{\mathrm{eff}}{C}_{\mathrm{ox}}}\times \frac{I_{\mathrm{DS}1}}{\left(V_{\mathrm{GS\_c}}-V_{t1\_1,2}-V_{\mathrm{DS}}/2\right)\times \left(V_{\mathrm{DS}}-RS{D}_{1,2}\times I_{\mathrm{DS}1}\right)},{\mu }_{\mathrm{eff}2\_1,2}=\frac{L_{\mathrm{eff}}}{W_{\mathrm{eff}}{C}_{\mathrm{ox}}}\times \frac{I_{\mathrm{DS}2}}{\left(V_{\mathrm{GS\_c}}-V_{t2\_1,2}-V_{\mathrm{DS}}/2\right)\times \left(V_{\mathrm{DS}}-RS{D}_{1,2}\times I_{\mathrm{DS}2}\right)}$ wherein L_eff is the channel length, W_eff is the channel width, and C_ox is the inversion gate-oxide capacitance.

FIG. 5 shows determination of the η_k value that makes the set of RSD_i,j values, μ_{effp_p,j} values or V_{tq_q,j} values closest to each other in an example of this invention. In the example, the set of η_k values used were 1.1, 1.2, 1.3 . . . 3.9 and 4.0 (N=30), and three different biases (M=3), V_B1=0 V, V_B2=−0.5 V and V_B3=−1V, were applied to the substrate of an NFET, respectively. For each η_k value, RSD_1,2, RSD_1,3 RSD_2,3, μ_{eff1_1,2}, μ_{eff_1,3}, V_{t1_1,2} and V_{t1_1,3} were derived based on the method shown in FIGS. 3-4.

Referring to FIG. 5, for each η_k value, how the set of RSD_1,2, RSD_1,3 and RSD_2,3, the set of μ_{eff1_1,2} and μ_{eff_1,3}, or the set of V_{t1_1,2} and V_{t1_1,3} are close to each other was determined by the ratio of the difference between the maximal value and the minimal value in the set of values to the average of the set of values. Because “3.1” was the η value that made the set of RSD, μ_eff1 or V_t1 values closest to each other, it was the accurate η value, and the corresponding RSD, μ_eff1, or V_t1 calculated as the mean value of RSD_1,2, RSD_1,3 and RSD_2,3, the mean value of μ_{eff1_1,2} and μ_{eff_1,3}, or the mean value of V_{t1_1,2} and V_{t1_1,3} was also the accurate value.

With the method for deriving characteristic values of a MOS transistor of this invention, the RSD value and the μ_eff value can be determined more accurately, and a more precise η value can be obtained as well.

This invention has been disclosed above in the preferred embodiments, but is not limited to those. It is known to persons skilled in the art that some modifications and innovations may be made without departing from the spirit and scope of this invention. Hence, the scope of this invention should be defined by the following claims.

Claims

1. A method for deriving characteristic values of a MOS transistor, wherein the MOS transistor comprises a substrate, a gate, a channel region, a source and a drain, a threshold voltage of the channel region is Vt, a voltage applied to the substrate is VB, a voltage applied to the gate is VG, a voltage across the source and the drain is VDS, a voltage across the gate and the source is VGS, a VGS making the MOS transistor operate in a constant-mobility region is VGS_c, a current from the drain to the source is IDS, and a sum of a resistance RD of the drain and a resistance RS of the source is RSD, the method comprising: a) applying different VB1 and VB2, respectively, to the substrate;b) measuring IDS1 and IDS2 that correspond to VB1 and VB2, respectively;c) giving respective initial values of Vt1 and Vt2, wherein Vt1 corresponds to VB1 and Vt2 corresponds to VB2;d) calculating RSD with a given η value, Vt1 and Vt2 by an equation of

2. The method of claim 1, further comprising deriving effective carrier mobility μeff1 and μeff2 under VB1 and VB2, respectively, by equations of

3. A method for deriving characteristic values of a MOS transistor, wherein the MOS transistor comprises a substrate, a gate, a channel region, a source and a drain, a threshold voltage of the channel region is Vt, a voltage applied to the substrate is VB, a voltage applied to the gate is VG, a voltage across the source and the drain is VDS, a voltage across the gate and the source is VGS, a VGS making the MOS transistor operate in a constant-mobility region is VGS_c, a current from the drain to the source is IDS, and a sum of a resistance RD of the drain and a resistance RS of the source is RSD, the method comprising: providing a set of ηk values (k=1 to N);providing a set of VBi values (i=1 to M, M≧3);for each ηk, derive a set of RSDi,j (i=1 to M−1, j=i+1 to M) values each under a pair of VBi and VBj, a set of μeffp_p,j (p is one of 1 to M, j is 1 to M excluding p) values each under a VBp, and/or a set of Vtq_q,j (q is one of 1 to M, j is 1 to M excluding q) values each under a VBq by the method of claim 2;determine an accurate ηk value that makes the set of RSDi,j values closest to each other, makes the set of μeffp_p,j values closest to each other, and/or makes the set of Vtq_q,j values closest to each other; andcalculating a mean value of RSDi,j at the accurate ηk value as an accurate value of RSD.

4. The method of claim 3, wherein how the set of RSDi,j values, μeffp_p,j values or Vtq_q,j values are close to each other is determined by a ratio of a difference between a maximal value and a minimal value in the set of values to an average of the set of values.

5. The method of claim 3, further comprising: calculating a mean value of μeffp_p,j at the accurate ηk value as an accurate value of μeffp under VBp; and/orcalculating a mean value of Vtq_q,j at the accurate ηk value as an accurate value of Vtq under VBq.

6. The method of claim 3, wherein M is equal to 3, and three different biases VB1, VB2 and VB3 are respectively applied to the substrate.

7. A method for deriving characteristic values of a MOS transistor, wherein the MOS transistor comprises a substrate, a gate, a channel region, a source and a drain, a threshold voltage of the channel region is Vt, a voltage applied to the substrate is VB, a voltage applied to the gate is VG, a voltage across the source and the drain is VDS, a voltage across the gate and the source is VGS, a VGS making the MOS transistor operate in a constant-mobility region is VGS_c, a current from the drain to the source is IDS, and a sum of a resistance RD of the drain and a resistance RS of the source is RSD, the method comprising: providing a set of ηk values (k=1 to N);providing a set of VBi values (i=1 to M, M≧3);for each ηk, derive a set of RSDi,j (i=1 to M−1, j=i+1 to M) values each under a pair of VBi and VBj, and/or a set of Vtq_q,j (q is one of 1 to M, j is 1 to M excluding q) values each under a VBq by the iteration method described in the steps a) to g) of claim 1;determine an accurate ηk value that makes the set of RSDi,j values closest to each other, and/or makes the set of Vtq_q,j values closest to each other; andcalculating a mean value of RSDi,j at the accurate ηk value as an accurate value of RSD.

8. The method of claim 7, wherein how the set of RSDi,j values or Vtq_q,j values are close to each other is determined by a ratio of a difference between a maximal value and a minimal value in the set of values to an average of the set of values.

9. The method of claim 7, further comprising: calculating a mean value of Vtq_q,j at the accurate ηk value as an accurate value of Vtq under VBq.

10. The method of claim 7, wherein M is equal to 3, and three different biases VB1, VB2 and VB3 are respectively applied to the substrate.