A method for characterizing the position distribution of gate oxide traps in MOSFETs considering Coulomb scattering

Abstract

A characterization method for the distribution of gate oxide traps with position in MOSFETs considering Coulomb scattering, wherein the drain voltage is selected, the transfer characteristics of the MOSFET device to be measured and the drain current noise power spectrum under different gate voltages are measured; the channel carrier mobility determined by all scattering mechanisms other than the oxide trap Coulomb scattering is determined by an iterative method; the Coulomb scattering coefficient under different gate voltages and the change of the oxide trap density with position are determined by an iterative method. The present invention can be applied to different bias conditions including near the threshold voltage and the strong inversion region, can be applied to different trap position distributions, rather than being limited to uniform distribution or exponential distribution of traps with position, can reflect the differences between different individuals of the same type of samples, and has a stronger accuracy, and can be used to evaluate the performance and reliability of MOSFETs, and can be further used to guide the manufacture of MOSFETs.

Description

Technical Field

[0001] The present invention belongs to the technical field of semiconductor device characterization, and particularly relates to a method for characterizing the position-dependent distribution of gate oxide traps in MOSFETs considering Coulomb scattering. Background Art

[0002] The level and distribution of gate oxide traps are a key factor affecting the performance and reliability of MOSFETs. Gate oxide traps significantly affect the low-frequency noise of MOSFETs. Therefore, low-frequency noise can be used to reflect the trap distribution in the MOSFET oxide layer.

[0003] Traps in the MOSFET gate oxide capture and emit carriers, causing charge changes in the oxide layer. This change significantly affects the surface potential of the semiconductor, thereby causing fluctuations in the channel current. At the same time, the charge change in the oxide layer may modulate the change in the channel carrier mobility through Coulomb scattering, also triggering fluctuations in the channel current. The influence of this Coulomb scattering on the low-frequency noise of the channel current can sometimes be ignored and sometimes cannot be ignored. Therefore, it is very necessary to find a method for characterizing the position-dependent distribution of gate oxide traps in MOSFETs considering Coulomb scattering.

[0004] Existing methods for characterizing the modulation effect of Coulomb scattering of gate oxide traps on low-frequency noise all have the problem of limited applicable conditions.

[0005] For example, the method disclosed by Liang H et al. is only applicable near the threshold voltage and when the gate oxide traps are uniformly distributed or exponentially distributed with position, and cannot characterize the Coulomb scattering coefficient in the strong inversion region, nor is it applicable to other distributions of the gate oxide with position (Liang H, Zhao P, Jiahao L, et al. Gate Metal and Cap Layer Effects on Ge nMOSFETs Low-Frequency Noise Behavior. IEEE Transactions on Electron Devices, 2019, 66(2): 1050 - 1056.).

[0006] The method disclosed by Fang W et al. uses the formula (where C ox is the oxide capacitance per unit area, γ is the tunneling coefficient, and f is the frequency) to obtain the bulk density N Vg of oxide traps in the unit energy range from the noise S t , and uses the formula D ot = 4k B TN t T (where T is the oxide layer thickness) to obtain the surface density D of oxide trapsot , using the formula to fit the mobility μ and the surface density of oxide traps qD ot to obtain the Coulomb scattering coefficient α sc . It can characterize the threshold voltage and the Coulomb scattering effect near the strong inversion region. However, the premise for the formula to hold is that the oxide traps are uniformly distributed. Therefore, it is only suitable for the case where the oxide traps are uniformly distributed and still cannot be applied to other distributions of gate oxide traps other than uniform distribution or exponential distribution with position (Fang W, Simoen E, Arimura H, et al. Low-Frequency Noise Characterization of GeOx Passivated Germanium MOSFETs. IEEE Transactions on Electron Devices, 2015, 62(7): 2078 - 2083.).

[0007] Vandamme E P et al. and Pacelli A et al. respectively proposed an empirical model. Both of these empirical models require the use of fitting parameters, and the fitting parameters are only applicable to a specific type of sample. Therefore, they also do not have universality (Vandamme E P, Vandamme L. Critical discussion on unified 1 / f noise models for MOSFETs. IEEE Transactions on Electron Devices, 2000, 47(11): 2146 - 2152., Pacelli A, Villa S. Quantum effects on the extraction of MOS oxide traps by 1 / f noise measurements. IEEE Transactions on Electron Devices, 1999, 46(5): 1029 - 1035.). Summary of the Invention

[0008] In order to overcome the above - mentioned drawbacks of the prior art, the object of the present invention is to provide a method for characterizing the distribution of gate oxide traps of MOSFETs with position considering Coulomb scattering, so as to avoid the uniform distribution or exponential distribution of oxide traps with position, obtain the Coulomb scattering coefficient under different gate voltages and the variation of oxide trap density with position, thereby being able to evaluate the performance and reliability of MOSFETs and further being used to guide the manufacture of MOSFETs.

[0009] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0010] A method for characterizing the position-dependent distribution of oxide traps in MOSFETs considering Coulomb scattering, comprising the following steps:

[0011] Step 1: Select the drain voltage V D , and measure the transfer characteristic I D -V g and the drain current noise power spectrum S g -f of the MOSFET device to be measured at different gate voltages V Id . The drain current noise power spectrum S Id -f is a function of the power spectrum amplitude S Id and the frequency f, and I D represents the drain current;

[0012] Step 2: Determine the channel carrier mobility μ 0 determined by all other scattering mechanisms except Coulomb scattering of oxide traps through an iterative method in the discrete form of the low-frequency noise model;

[0013] Step 3: Determine the Coulomb scattering coefficient α g and the variation of the oxide trap density with position N sc (z t ) at different gate voltages V i through an iterative method.

[0014] In one embodiment, in Step 1, the range of the drain voltage V D is from 0.05 V to 0.1 V to ensure that the MOSFET device to be measured operates in the linear mode during low-frequency noise measurement.

[0015] In one embodiment, Step 2 includes:

[0016] 2.1. From the transfer characteristic I D -V g , use to obtain the curve g m -V g , substitute it into to obtain the variation of μ with V g , expressed as μ-V g ; where g m is the transconductance, μ is the effective channel carrier mobility, W and L respectively represent the width and length of the channel, and C ox is the oxide capacitance per unit area;

[0017] 2.2. At the gate voltage V g(1) Under the condition, numerically solve the discrete form of the low-frequency noise model to obtain the variation of the oxide layer trap density with position N t (z i ). At this time, both S Id and f are m-dimensional vectors. Substitute S Id - f into the discrete form of the low-frequency noise model to obtain a linear system of equations with n unknowns and m equations. Solve this linear system of equations by non-negative least squares method to obtain N t (z i ); where q is the electron charge, k is the Boltzmann constant, T is the absolute temperature, n is the total number of layers divided in the oxide layer, τ = τ 0 exp(γ·z), τ is the time constant, γ is the tunneling coefficient, z is the position coordinate of the oxide layer, and the typical value of τ 0 is 10 -10 s, z i is the coordinate of a certain position in the i-th layer of the oxide layer, and Δz i is the thickness of the divided unit oxide layer;

[0018] 2.3. Substitute the solution result N t (z i ) obtained in 2.2 into to obtain the areal density D ot of the oxide layer trap number;

[0019] 2.4. Repeat 2.2 and 2.3 for all gate voltages V g to obtain qD g under different gate voltages V ot . Combine μ-V g in 2.1 to obtain and the one-to-one correspondence of qD ot . Use the formula to linearly fit the Coulomb scattering coefficient α sc ;

[0020] 2.5. At the gate voltage V g (1), substitute S Id - f and α sc into the discrete form of the low-frequency noise model including the Coulomb scattering coefficient α sc of the oxide layer trap to obtain a linear system of equations with n unknowns and m equations. Solve this linear system of equations by methods such as non-negative least squares method to obtain N (z t (z i );

[0021] 2.6. Substitute the solution result N t (z i ) obtained in 2.5 into Obtain the surface density D of oxide layer traps ot ;

[0022] 2.7. For all V g Repeat steps 2.5 and 2.6 to obtain qD at different gate voltages ot , combined with μ-V in step 2.1 g , to obtain and qD ot 's one-to-one correspondence, and use the formula to linearly fit and α sc,new , where α sc,new is the Coulomb scattering coefficient updated by the iterative method;

[0023] 2.8. Judge whether the absolute value error between α sc adopted in step 2.5 and α sc,new output in step 2.7 is less than the set threshold c (c is less than 100 and can be set smaller according to the accuracy requirement). If it holds, output μ 0 in step 2.7, and the iteration stops; if it does not hold, let α sc =α sc,new , and repeat steps 2.5 to 2.8.

[0024] In one embodiment, step 3 includes:

[0025] 3.1. At the gate voltage V g (1), substitute the measurement result S Id -f in step 1 and α sc in step 2.8 or step 3.4 into the discrete form of the low-frequency noise model sc including the oxide layer trap Coulomb scattering coefficient α to obtain a linear system of equations with n unknowns and m equations. Solve this linear system of equations by methods such as non-negative least squares method to obtain N t (z i );

[0026] 3.2. Substitute the solution result N t (z i ) in step 3.1 into to obtain the surface density D of oxide layer traps ot ;

[0027] 3.3. According to μ-V in 2.1 g , find the g (1) at the gate voltage V and substitute it together with D ot obtained in step 3.2 into the formula to solve the equation to obtain α sc,new ;

[0028] 3.4. Determine the α used in step 3.1 sc and the α obtained in step 3.3 sc,new Whether the absolute value error is less than the set threshold c (c is less than 100 and can be set smaller according to the accuracy requirement). If it holds, output α sc and N in 3.1 t (z i ), and the iteration stops; if not, let α sc = α sc,new , and repeat 3.1 - 3.4;

[0029] 3.5. Repeat 3.1 - 3.4 for all voltages to obtain α sc and N t (z i ) at all voltages.

[0030] Compared with the prior art, based on the discretized low - frequency noise model, the present invention considers the scattering effect of gate oxide traps on carrier mobility. By using non - negative least squares method and iterative method to solve the discretized low - frequency noise model considering Coulomb scattering effect, the distribution of oxide traps with position and the Coulomb scattering coefficient can be obtained simultaneously. The present invention can be applied to different bias conditions including near the threshold voltage, sub - threshold region, and strong inversion region, etc., and can be applied to different trap position distributions, rather than being limited to uniform or exponential distributions of traps with position. Moreover, it can reflect the differences between different individuals of the same type of samples and has higher precision. Furthermore, it can be used to evaluate the performance and reliability of MOSFETs and can be further used to guide the manufacturing of MOSFETs. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 is a schematic flow chart of the present invention.

[0032] Figure 2 is the measurement result of an embodiment of the present invention, where (a) is the measurement result of the transfer characteristics and (b) is the measurement result of the low - frequency noise characteristics.

[0033] Figure 3 is a schematic diagram of the relationship with qD ot .

[0034] Figure 4 is the characterization result under different gate voltages of an embodiment of the present invention, where (a) is the variation of α sc with the gate voltage V g , and (b) is the variation of N t (z) with the gate voltage V g . DETAILED DESCRIPTION OF THE INVENTION

[0035] The embodiments of the present invention will be described in detail below in conjunction with the accompanying drawings and examples.

[0036] The present invention relates to a method for characterizing the position-dependent distribution of traps in the gate oxide layer of MOSFETs. Different from the prior art, in order to avoid the uniform or exponential distribution of oxide traps with position, the present invention takes into account the Coulomb scattering effect, divides the oxide layer into n layers, and proposes a discrete form of the low-frequency noise model:

[0037]

[0038] From this, a linear system of equations with n unknowns and m equations can be obtained. Solving this linear system of equations gives N t (z i ).

[0039] Combined with the equation:

[0040]

[0041]

[0042] By iteratively solving the above equation, the Coulomb scattering coefficient α g at different gate voltages V sc and the variation of the oxide trap density with position N t (z i ) that meet the preset solution accuracy can be obtained simultaneously.

[0043] The specific steps of the present invention and the meaning and parameter definitions of the above model will be described in detail through the following steps.

[0044] Referring to Figure 1 , the specific steps of the present invention are as follows:

[0045] Step 1: Measure the transfer characteristics and low-frequency noise characteristics of the MOSFETs device to be tested.

[0046] Specifically, in this step, the drain voltage V D is selected. Under this test condition, the transfer characteristics I D -V g and the drain current noise power spectrum S g -f at different gate voltages V Id of the MOSFETs device to be tested are measured. Among them, I D represents the drain current, and the measured drain current noise power spectrum S Id -f is the power spectrum amplitude S IdFunction of frequency f. To ensure that the MOSFETs under test operate in the linear mode during low-frequency noise measurement, the drain voltage V is selected in the range of 0.05V to 0.1V D . Transfer characteristic I D -V g The measurement results are shown in Figure 2 (a). The drain current noise power spectrum S g -f under different gate voltages V Id The measurement results are shown in Figure 2 (b).

[0047] Step 2: Discrete form of the low-frequency noise model. The channel carrier mobility μ determined by all scattering mechanisms other than the Coulomb scattering of oxide traps is determined by an iterative method 0 .

[0048] Specifically, the implementation process of this step can be described as follows:

[0049] 2.1, Calculate μ-V D -V g from the transfer characteristic I g .

[0050] In the linear region, from the transfer characteristic I D -V g Using (g m is the transconductance), the curve g m -V g is obtained and substituted into (where W and L represent the width and length of the channel respectively, and C ox is the oxide capacitance per unit area), and the variation of μ with V g can be obtained. μ is the effective mobility of channel carriers.

[0051] 2.2, At the gate voltage V g (1), numerically solve the discrete form of the low-frequency noise model to obtain the variation of the oxide trap density with position N t (z i ).

[0052] At this time, both S Id and f are m-dimensional vectors. Substitute the measurement results S Id -f in Step 1 into the discrete form of the low-frequency noise model (where τ is the time constant, τ = τ 0 exp(γ·z), the typical value of τ 0 is 10 -10 s, γ is the tunneling coefficient, z is the position coordinate of the oxide layer, f is the frequency, and z iis the coordinate of a certain position in the i-th layer of the oxide layer, Δz i is the divided unit oxide layer thickness, n is the total number of divided layers, I D is the drain current, k is the Boltzmann constant, T is the absolute temperature, q is the electron charge), a linear equation system with n unknowns and m equations can be obtained, and by solving this linear equation system through non-negative least squares method, N t (z i ) can be obtained. Exemplarily, this linear equation system can also be solved by other methods.

[0053] 2.3, substitute the solution result N t (z i ) obtained in 2.2 into to obtain the surface density D of the oxide layer trap number ot .

[0054] 2.4, linearly fit the Coulomb scattering coefficient α sc .

[0055] For all gate voltages V g Repeat 2.2 and 2.3 to obtain qD g at different gate voltages V ot . Combining μ-V g in 2.1, and qD ot a one-to-one correspondence can be obtained, and using the formula to linearly fit the Coulomb scattering coefficient α sc .

[0056] 2.5, at the gate voltage V g (1), numerically solve the discrete form low-frequency noise model containing the Coulomb scattering coefficient α sc to obtain the variation of the oxide layer trap density with position N t (z i ).

[0057] At this time, S Id and f are both m-dimensional vectors. Substitute the measurement result S Id -f in step 1 and the α sc obtained in step 2.4 or step 2.8 (when this step is first executed for each gate voltage, use the α sc in step 2.4, and use the α sc in step 2.8 for the rest) into the discrete form low-frequency noise model containing the Coulomb scattering coefficient α sc of the oxide layer trap (divide the oxide layer into n layers), a linear equation system with n unknowns and m equations is obtained, and by solving this linear equation system through non-negative least squares method or other algorithms, N (z t (z i)。

[0058] 2.6. Substitute the solution result N of 2.5 into t (z i ) to obtain the surface density D of the oxide layer trap number. ot 。

[0059] 2.7. For all V g Repeat 2.5 and 2.6 to obtain qD under different gate voltages. ot Combined with μ-V in step 2.1 g , obtain and the one-to-one correspondence relationship between qD ot . Use the formula to linearly fit and α sc,new , where α sc,new is the Coulomb scattering coefficient updated by the iterative method.

[0060] 2.8. Determine whether the absolute value error between α sc adopted in step 2.5 and α sc,new output in step 2.7 is less than 10 -9 (generally less than 100 is sufficient). If it holds, output μ in step 2.7 0 , and the iteration stops; if it does not hold, let α sc = α sc,new , and repeat 2.5, 2.6, 2.7, 2.8. The extraction result of Figure 3 is as shown.

[0061] Step 3: Determine the Coulomb scattering coefficient α g at different gate voltages V sc and the variation of the oxide layer trap density with position N t (z i ) by the iterative method.

[0062] Specifically, the implementation process of this step can be described as follows:

[0063] 3.1. At the gate voltage V g (1), numerically solve the discrete form low-frequency noise model including the Coulomb scattering coefficient α sc to obtain the variation of the oxide layer trap density with position N t (z i ).

[0064] At this time, S id and f are both m-dimensional vectors. Substitute the measurement result S Id -f of step 1 and α in step 2.8 or step 3.4 (when this step is first executed for each gate voltage, use α in step 2.8)sc , and the rest all use α in step 3.4 sc ) of α sc Substitute into the low-frequency noise model in discrete form that includes the Coulomb scattering coefficient α of the oxide layer traps sc (Divide the oxide layer into n layers), a linear equation system with n unknowns and m equations can be obtained. By solving this linear equation system using methods such as non-negative least squares, N can be obtained t (z i );

[0065] 3.2, Substitute the solution result N t (z i ) obtained in step 3.1 into to obtain the surface density D of the oxide layer traps ot .

[0066] 3.3, According to μ-V in 2.1 g , find the gate voltage V g (1) under the and D obtained in step 3.2 ot are substituted into the formula to solve the equation to obtain α sc,new .

[0067] 3.4, Judge whether the absolute value error of α sc used in step 3.1 and α sc,new obtained in step 3.3 is less than 10 -9 (Generally, less than 100 is fine). If it holds, output α sc and N in 3.1 t (z i ), and the iteration stops; if it does not hold, let α sc =α sc,new , and repeat 3.1, 3.2, 3.3, 3.4

[0068] 3.5, Repeat 3.1, 3.2, 3.3, 3.4 for all voltages to obtain α sc and N t (z i ) under all voltages; The calculation results are as shown in (a) and (b) in Figure 4 . Obviously, according to this result, the overall oxide layer trap level of this transistor is relatively high, and when the gate voltage is from 4.0V to 4.2V, the oxide layer trap density near 2.5nm from the substrate-oxide layer interface is the highest, reaching the order of 10 21 / (cm 3 ·eV).​

Claims

1. A method for characterizing the position distribution of gate oxide traps in MOSFETs considering Coulomb scattering, characterized in that: The steps include: Step 1: Select the drain voltage V D , the transfer characteristics of the MOSFETs device to be tested I D -V g and different gate voltage V g The drain current noise power spectrum S Id -f is measured, the drain current noise power spectrum S Id -f is the power spectrum amplitude S Id As a function of frequency f, I D represents the drain current; Step 2: Establish a low-frequency noise model in a discrete form and determine the channel carrier mobility μ0 determined by all scattering mechanisms except the oxide layer trap Coulomb scattering through an iterative method; include: 2.

1. Transfer Characteristics I D -V g use Get curve g m -V g , substitute Get μ with V g The change in μ-V g ; Among them, g m is the transconductance, μ is the effective mobility of channel carriers, W and L represent the width and length of the channel respectively, C ox is the oxide layer capacitance per unit area; 2.

2. At the gate voltage V g (1) is numerically solved to obtain the variation of the oxide layer trap density with position N. t (z i ), at this time, S Id and f are both m-dimensional vectors, S Id -f Substitute the low-frequency noise model in discrete form We get a linear system of equations with n unknowns and m equations. By solving this linear system, we can get N t (z i ), where q is the electron charge, k is the Boltzmann constant, T is the absolute temperature, n is the total number of layers divided by the oxide layer, τ = τ0exp(γ·z), τ is the time constant, γ is the tunneling coefficient, z is the position coordinate of the oxide layer, and the typical value of τ0 is 10 -10 s, z i is the coordinate of a position in the i-th oxide layer, Δz i is the unit oxide layer thickness divided; 2.

3. Substitute the solution N of 2.2 t (z i ) The surface density D of the oxide layer trap number is obtained ot ; 2.

4. For all gate voltages V g Repeat 2.2 and 2.3 to get different gate voltages V g qD ot , combined with the μ-V in 2.1 g ,get and qD ot One-to-one correspondence, using the formula Linear fitting of Coulomb scattering coefficient α sc ; 2.

5. At the gate voltage V g (1) Next, change S Id -f and α sc Substitute the Coulomb scattering coefficient α including the oxide layer trap sc The low-frequency noise model in discrete form is We get a linear system of equations with n unknowns and m equations. By solving this linear system, we can get N t (z i ); 2.

6. Substitute the solution N of 2.5 t (z i ) The surface density D of the oxide layer trap number is obtained ot ; 2.

7. For all V g Repeat 2.5 and 2.6 to obtain qD at different gate voltages ot , combined with the μ-V in step 2.1 g ,get and qD ot One-to-one correspondence, using the formula Linear fitting and α sc,new , α sc,new is the Coulomb scattering coefficient updated by iterative method; 2.

8. Determine the α used in step 2.5 sc and the α output from step 2.7 sc,new Is the absolute value error less than the set threshold c? If so, output μ0 in step 2.7 and the iteration stops; if not, set α sc =α sc,new , repeat 2.5-2.8; Step 3: Determine different gate voltages V by iterative method g The Coulomb scattering coefficient α under sc And the variation of oxide layer trap density with position N t (z i ),include: 3.

1. At the gate voltage V g (1) Next, the measurement result S of step 1 is Id -f and α from step 2.8 or step 3.4 sc Substitute the Coulomb scattering coefficient α including the oxide layer trap sc The low-frequency noise model in discrete form is We get a linear system of equations with n unknowns and m equations. By solving this linear system, we can get N t (z i ); 3.

2. The solution N of step 3.1 t (z i ) The surface density D of the oxide layer trap number is obtained ot ; 3.

3. According to μ-V in 2.1 g , find the gate voltage V g (1) D obtained in step 3.2 ot Substitute into the formula together Solving the equation yields α sc,new ; 3.

4. Determine the α used in step 3.1 sc and α obtained in step 3.3 sc,new Is the absolute value error less than the set threshold c? If so, output α sc and N of 3.1 t (z i ), the iteration stops; if it does not hold, let α sc =α sc,new , repeat 3.1 to 3.4; 3.

5. Repeat 3.1 to 3.4 for all gate voltages to obtain α under all gate voltages. sc and N t (z i ).

2. The method for characterizing the distribution of gate oxide traps in MOSFETs with Coulomb scattering according to claim 1, characterized in that: In step 1, the drain voltage V D The range is 0.05V to 0.1V to ensure that the MOSFETs under test operate in linear mode when making low frequency noise measurements.

3. The method for characterizing the distribution of MOSFET gate oxide traps with position considering Coulomb scattering according to claim 1, characterized in that: The system of linear equations is solved by the non-negative least squares method.

4. The method for characterizing the distribution of gate oxide traps in MOSFETs with Coulomb scattering according to claim 1, characterized in that: The set threshold c is less than 100.

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