Deep energy level transient spectrum data processing method based on dictionary learning

Abstract

The invention relates to a deep energy level transient spectrum data processing method based on dictionary learning. The method comprises the following steps: acquiring transient capacitance signals of a sample at different temperatures by using deep energy level transient spectrum equipment; establishing an Arrhenius equation that the defect emissivity changes along with the temperature; the defect emissivity under each group of values is calculated according to all possible values of the coefficient of the exhaustion Arrhenius equation; constructing a dictionary matrix based on the defect emissivity amplitude introducing Arrhenius constraint, and reconstructing a transient capacitance signal by using the dictionary matrix; and constructing a convex optimization problem, solving a dictionary index variable which enables a reconstruction signal of the transient capacitor to be closest to an acquisition signal, and extracting a non-zero solution in the dictionary index variable to obtain a corresponding defect emissivity amplitude spectrum. According to the method, the detection precision of the deep energy level defect can be improved by introducing the Arrhenius constraint relation.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of semiconductor material testing, in particular to a deep level transient spectroscopy data processing method based on dictionary learning. BACKGROUND

[0002] The main method for studying deep level defects in semiconductor materials is deep level transient spectroscopy (DLTS). DLTS is based on measuring the capacitance transient signal of a PN junction, Schottky junction or MOS structure at different temperatures to characterize the defects. The commonly used DLTS data processing method is the Boxcar method, which obtains the defect spectrum by selecting the capacitance signals at two time points and interpolating. However, for defects with two energy levels and close capture cross sections, this method cannot separate them.

[0003] The existing method for separating defects with two energy levels and close capture cross sections is inverse Laplace transform deep level transient spectroscopy (L-DLTS). The L-DLTS method performs inverse Laplace transform on the transient capacitance signal at a specific temperature to obtain the amplitude spectrum of different emission rate signals, thereby separating defects with different emission rates. By comparing the amplitude spectra at different temperatures, the energy levels and capture cross sections of the corresponding defects can be obtained. However, the L-DLTS method has the following shortcomings: In the L-DLTS method, the capacitance relaxation signals for different temperature intervals are processed independently. Although there is a theoretical correlation between the thermal emission rate of the defect energy level and the temperature based on the Arrhenius equation, the global constraint condition of the Arrhenius equation is not introduced in the algorithm architecture. This technical defect leads to a significant increase in the degree of freedom of the parameter solution space during the multi-temperature interval fitting process, and the convergence accuracy of the inversion result is systematically restricted due to the lack of temperature dimension correlation. SUMMARY

[0004] The technical problem to be solved by the present application is to provide a deep level transient spectroscopy data processing method based on dictionary learning, which can improve the detection accuracy of deep level defects by introducing the Arrhenius constraint relationship.

[0005] The technical solution adopted by the present application to solve its technical problem is: providing a deep level transient spectroscopy data processing method based on dictionary learning, comprising the following steps:

[0006] Collecting transient capacitance signals of the sample at different temperatures using a deep level transient spectroscopy device;

[0007] Establishing an Arrhenius equation for the change of defect emission rate with temperature;

[0008] Enumerating all possible values of the coefficients of the Arrhenius equation, and calculating the defect emission rate for each set of values;

[0009] The dictionary matrix is constructed based on the defect emissivity amplitude with the Arrhenius constraint introduced, and then the transient capacitance signal is reconstructed by using the dictionary matrix;

[0010] A convex optimization problem is constructed, the dictionary index variable that makes the reconstructed signal of the transient capacitance closest to the collected signal is solved, and the corresponding defect emissivity amplitude spectrum is obtained by extracting the non-zero solution of the dictionary index variable.

[0011] Further, each entry in the dictionary matrix corresponds to the defect emissivity amplitude under a set of values of the Arrhenius equation coefficient.

[0012] Further, the defect emissivity amplitude with the Arrhenius constraint introduced is calculated by the following method:

[0013] Based on the sampling accuracy of the deep level transient spectrum device, an emissivity sequence is created;

[0014] The defect emissivity calculated by the Arrhenius equation is the target frequency, the linear interpolation estimation of the amplitude of the target frequency at the same temperature is performed by using the two emissivities adjacent to the target frequency in the created emissivity sequence, and the defect emissivity amplitude with the Arrhenius constraint introduced is obtained.

[0015] Further, the Arrhenius equation is expressed as

[0016]

[0017] Where, α, β, p are set parameters, s m (T) is the defect emissivity with the Arrhenius constraint introduced at temperature T.

[0018] Further, the entry is expressed as

[0019]

[0020] Where, w {α,β,p} (s(i), T(j)) is the defect emissivity amplitude with the Arrhenius constraint introduced corresponding to the entry α, β, p, and s(i) is the emissivity sequence.

[0021] Further, the transient capacitance signal is reconstructed by using the dictionary matrix, comprising:

[0022] The dictionary matrix and its index variable are multiplied to obtain the defect emissivity amplitude spectrum;

[0023] The defect emissivity amplitude spectrum is converted into the transient capacitance signal in the time domain.

[0024] Further, the defect emissivity amplitude spectrum is converted into the transient capacitance signal in the time domain by Laplace transform.

[0025] Further, the convex optimization problem is expressed as

[0026]

[0027] s.t.F=D·X

[0028] Wherein, C is the collection signal matrix of transient capacitance, F is the amplitude variable matrix, D is the dictionary matrix, X is the dictionary index variable, and A is the Laplace transform kernel matrix.

[0029] Further, the Laplace transform kernel matrix is expressed as

[0030]

[0031] Wherein, s(i), i=1, 2, …, N s represents the emissivity sequence, t(k), k=1, 2, …, N t represents the measurement time.

[0032] Advantages

[0033] Compared with the prior art, the present application has the following advantages and positive effects: based on dictionary learning, the present application establishes a related dictionary matrix according to the Arrhenius equation of defect emissivity in DLTS related theory, realizes amplitude constraint in multiple temperature intervals, reduces the degree of freedom of amplitude variable, and improves fitting accuracy and reliability. BRIEF DESCRIPTION OF DRAWINGS

[0034] Figure 1 is a flowchart of the embodiment of the present application;

[0035] Figure 2 is an Arrhenius curve diagram obtained by a traditional L-DLTS method (L1 regularization + L-curve);

[0036] Figure 3 is an Arrhenius curve diagram obtained by the embodiment of the present application. DETAILED DESCRIPTION

[0037] The present application will be further described below in conjunction with specific embodiments. It should be understood that these embodiments are only used to illustrate the present application and not used to limit the scope of the present application. In addition, it should be understood that after reading the content taught by the present application, those skilled in the art can make various modifications or changes to the present application, and these equivalent forms also fall within the scope defined by the appended claims of the present application.

[0038] The embodiment of the present application relates to a deep level transient spectrum data processing method based on dictionary learning, as shown in Figure 1 , comprising the following steps:

[0039] The transient capacitance signals of the sample at different temperatures are collected by a deep level transient spectroscopy device;

[0040] An Arrhenius equation of the defect emission rate changing with temperature is established;

[0041] All possible values of the coefficients of the Arrhenius equation are exhausted, and the defect emission rate under each set of values is calculated;

[0042] A dictionary matrix is constructed based on the defect emission rate with the Arrhenius constraint, and then the transient capacitance signal is reconstructed by using the dictionary matrix;

[0043] A convex optimization problem is constructed, the dictionary index variable that makes the reconstructed signal of the transient capacitance closest to the collected signal is solved, and the corresponding defect emission rate amplitude spectrum is obtained by extracting the non-zero solution in the dictionary index variable.

[0044] Each entry in the dictionary matrix corresponds to the defect emission rate under a set of values of the coefficients of the Arrhenius equation. The defect emission rate with the Arrhenius constraint can be calculated by the following method:

[0045] The emission rate sequence is extracted based on the collected transient capacitance signal;

[0046] The defect emission rate calculated by the Arrhenius equation is used as the target frequency, and the two emission rates adjacent to the target frequency in the extracted emission rate sequence are used for linear interpolation estimation of the target frequency at the same temperature.

[0047] More specifically, the above convex optimization problem can be represented as:

[0048]

[0049] s.t.F=D·X

[0050] Where C is the collected signal matrix of the transient capacitance, F is the amplitude variable matrix, D is the dictionary matrix, X is the dictionary index variable, and A is the Laplace transform kernel matrix; is the objective function; F, X are the variables to be optimized; is the square of the two-norm of C-A·F; ||X||1 is the one-norm of X; F=D·X is the relevant constraint, in which the elements of F and D·X are arranged in rows one by one and correspond to each other.

[0051] The convex optimization problem can be effectively solved by using some convex optimization solvers to estimate the values of F and X, thereby obtaining the relevant defect information.

[0052] The collected signal matrix C includes the transient capacitance signal, the time signal and the temperature signal collected, and can be represented as:

[0053]

[0054] where C is the transient capacitance data matrix; t(i), i = 1, 2, …, N t denotes the measured time signals arranged in ascending order; T(j), j = 1, 2, …, N T denotes the measured temperature signals arranged in ascending order; c(t(i), T(j)) denotes the measured capacitance signal at time point t(i) under temperature T(j), N t denotes the number of time points, N T denotes the number of temperature points; the dimension of the matrix C is (N t , N T ).

[0055] The Laplace transform kernel matrix A can be expressed as:

[0056]

[0057] where A is the Laplace transform kernel matrix; s(i), i = 1, 2, …, N s denotes the constructed emissivity sequence, whose value is greater than or equal to zero, arranged in ascending order; exp denotes the exponential function; the dimension of the matrix A is (N t , N s ).

[0058] The amplitude variable matrix F is:

[0059]

[0060] where F is the constructed amplitude variable matrix; f(s(i), T(j)) denotes the amplitude of the defect emissivity under temperature T(j) at emissivity s(i), which is the variable to be optimized; the dimension of the matrix F is (N s , N T ).

[0061] The dictionary matrix D is:

[0062]

[0063] where D is the constructed dictionary matrix, w n (s(i), T(j)) denotes the defect emissivity amplitude under s(i), T(j) corresponding to the word n with the introduction of the Arrhenius constraint; the dimension of the matrix D is (N s × N T , N w ).

[0064] More specifically, the word w i can be realized by the following formula:

[0065]

[0066]

[0067] where, a, b, p are the attributes of the word itself, selected according to the Arrhenius equation for the defect emissivity of formula (5) and the actual material parameter range; w n The subscript depends on the value of a, b, p, and each set of values corresponds to a word. The permutation combination of different a, b, p values makes the word subscript increase.

[0068] In formula (6), the word function w {α,β,p} According to the value of s(i), T(j), the corresponding target emissivity position s m (T(j)) is interpolated near the corresponding frequency according to the discretized s(i) sequence, and is normalized at the same temperature, that is, ∑ {α,β,p} w {α,β,p} (s(i), T(j)) = 1.

[0069] The index variable can be represented as:

[0070]

[0071] where X is the constructed index variable; is the index value of the dictionary, which is the variable to be optimized; the dimension of the X matrix is (N w ,1). Using the solved index variable, the values of a, b, and p can be read from the dictionary matrix, and the defect emissivity amplitude spectrum can be reproduced. Among them, the non-zero solution of the index variable corresponds to the defect of the capture cross section and the energy level, and the non-zero solution in the corresponding amplitude matrix F is the amplitude of the defect.

[0072] The following uses the fitting results of two simulated point defect caused capacitance transient signals as a specific embodiment to verify the effectiveness of the present embodiment.

[0073] Consider two point defects in N-type doped silicon, with energy levels of 0.200 eV and 0.210 eV from the conduction band, and capture cross sections of 3×10 -14 cm 2 and 1×10 -14 cm 2 , respectively. The varying amplitude is 1 pF and 2 pF, the base capacitance value is 200 pF, and the normal noise standard deviation of the capacitance signal is 0.04 pF. The sampling time sequence is 8×10 -5 s, 9×10 -5 s, …, 1×10 -2 s, and the sampling temperature sequence is 90 K, 92 K, …, 126 K.

[0074] In this embodiment, the Arrhenius curve obtained by using the conventional L-DLTS method (L1 regularization + L-curve) is as shown in Figure 2 The obtained defect energy level, capture cross section and amplitude are 0.170 eV and 0.201 eV, 1.35 x 10 - 15 cm 2 and 4.09 x 10 -15 cm 2 , 0.767 pF and 1.425 pF respectively.

[0075] In this embodiment, the Arrhenius curve obtained by using the proposed deep level transient spectroscopy method based on dictionary learning is as shown in Figure 3 The obtained defect energy level, capture cross section and amplitude are 0.200 eV and 0.209 eV, 3.09 x 10 - 14 cm 2 and 8.56 x 10 -15 cm 2 , 0.987 pF and 1.988 pF respectively. The present application is superior to the conventional L-DLTS method in the characterization results of all defect parameters.

Claims

1. A method for processing deep-level transient spectral data based on dictionary learning, characterized in that, Includes the following steps: The transient capacitance signals of the sample at different temperatures were acquired using a deep-level transient spectroscopy device; Establish the Arrhenius equation for the change of defect emissivity with temperature; By exhaustively listing all possible values ​​of the coefficients of the Arrhenius equation, the defect emissivity under each set of values ​​is calculated; A dictionary matrix is ​​constructed based on the defect emissivity amplitude with Arrhenius constraint, and then the transient capacitance signal is reconstructed using the dictionary matrix. A convex optimization problem is constructed, and the dictionary index variable that makes the reconstructed signal of the transient capacitance closest to the acquired signal is found. The non-zero solutions in the dictionary index variable are extracted to obtain the corresponding defect emissivity amplitude spectrum.

2. The method according to claim 1, characterized in that, Each entry in the dictionary matrix corresponds to a set of values ​​for the coefficients of the Arrhenius equation, representing the amplitude of the defect emissivity.

3. The method according to claim 1, characterized in that, The defect emissivity amplitude with Arrhenius constraint introduced is calculated using the following method: Emissivity sequences were created based on the sampling accuracy of deep-level transient spectral devices; The defect emissivity calculated using the Arrhenius equation is the target frequency. The amplitude of the target frequency is estimated by linear interpolation at the same temperature using two emissivity values ​​adjacent to the target frequency in the created emissivity sequence, thus obtaining the defect emissivity amplitude with Arrhenius constraint.

4. The method according to claim 3, characterized in that, The Arrhenius equation is expressed as: Where α, β, p are set parameters, s m (T) represents the defect emissivity at temperature T with Arrhenius constraint introduced.

5. The method according to claim 4, characterized in that, The term is represented as Among them, w {α,β,p} (s(i),T(j)) represents the emission rate amplitude of the defect corresponding to the terms α, β, p with the introduction of Arrhenius constraints, and s(i) represents the emission rate sequence.

6. The method according to claim 1, characterized in that, The process of reconstructing the transient capacitance signal using a dictionary matrix includes: Multiplying the dictionary matrix and its index variables yields the defect emissivity amplitude spectrum; The defect emissivity amplitude spectrum is converted into a transient capacitance signal in the time domain.

7. The method according to claim 6, characterized in that, The conversion of the defect emissivity amplitude spectrum into a transient capacitance signal in the time domain is achieved through Laplace transform.

8. The method according to claim 1, characterized in that, The convex optimization problem is expressed as: stF=D·X Where C is the transient capacitance acquisition signal matrix, F is the amplitude variable matrix, D is the dictionary matrix, X is the dictionary index variable, and A is the Laplace transform kernel matrix.

9. The method according to claim 1, characterized in that, The Laplace transform kernel matrix is ​​represented as follows: Where s(t), i=1,2,…,N s Let t(k) represent the emissivity sequence, k = 1, 2, ..., N. t Indicates the measurement time.

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