CN111505399A - Method for measuring space charge distribution of polymer dielectric film - Google Patents
Method for measuring space charge distribution of polymer dielectric film Download PDFInfo
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- CN111505399A CN111505399A CN202010351989.6A CN202010351989A CN111505399A CN 111505399 A CN111505399 A CN 111505399A CN 202010351989 A CN202010351989 A CN 202010351989A CN 111505399 A CN111505399 A CN 111505399A
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Abstract
The invention relates to the field of polymer dielectric films, in particular to a method for measuring space charge distribution of a polymer dielectric film, which comprises the following steps: carrying out metallization treatment on two sides of the polymer dielectric film, and applying direct-current voltage; pulse laser is hit to a metallized electrode on one side of the polymer dielectric film, the generated response current is measured, and a time domain response current expression is constructed; carrying out Fourier transform on the time domain response current expression by combining a Poisson equation and a one-dimensional heat conduction equation to obtain a frequency domain expression of response current related to space charge distribution; performing high-frequency approximation on the frequency domain expression to obtain an approximate frequency domain expression; and solving the approximate frequency domain expression to obtain space charge distribution. Compared with the prior art, the efficiency of data analysis is improved, the sensitivity of the data analysis to noise is reduced, and the precision of the measurement result is improved.
Description
Technical Field
The invention relates to the field of polymer dielectric films, in particular to a method for measuring space charge distribution of a polymer dielectric film.
Background
The space charge distribution in a polymer dielectric film has a crucial influence on its performance. In the existing measurement method for space charge, the heat pulse method has the advantages of short measurement period, high resolution and the like, and is particularly suitable for measuring micron-sized polymer films. The method generates thermal disturbance by applying a thermal pulse to a sample, and the thermal pulse drives charges in the process of propagating in a medium to trigger a measurable thermal response displacement current. The data analysis of the displacement current relates to a first Fredholm integral equation, and a proper numerical calculation method is adopted for solving. Traditionally in data analysis of non-polar dielectrics, the solution to the equation is aimed at the electric field distribution in the medium. From the poisson equation, space charge distribution information can be obtained from the calculation result by simple differentiation.
The first class of Fredholm integral equation is mathematically ill-conditioned, and very small data errors can lead to completely different results, i.e., ill-characterization. In actual measurement, a displacement current signal generated by a film sample is a nanoamp-level weak signal, so that the displacement current signal is easily interfered by an experimental circuit and the environment, and a noise signal is inevitably superposed on the current signal, thereby bringing certain difficulty to mathematical processing. At present, several numerical methods widely used in experiments for solving the displacement current equation, such as a scale transformation method, a Tikhonov regularization method, a Monte Carlo method, and the like, have certain sensitivity to noise. Therefore, in the data analysis of the measured signal, the electric field distribution curve obtained may have a certain jitter due to the existence of noise in the current signal and the characteristics of the algorithm. In this case, the spatial charge distribution curve obtained by performing differential calculation on the electric field may have unreasonable distributions such as partial regional distortion and frequent polarity inversion, and has a large deviation from the actual distribution. Therefore, when data processing is performed, the electric field calculation result is required to be smooth at a later stage, but the true information of partial charge distribution can be masked.
Disclosure of Invention
The present invention is directed to a method for measuring space charge distribution of a polymer dielectric thin film, which overcomes the above-mentioned drawbacks of the prior art.
The purpose of the invention can be realized by the following technical scheme:
a method for measuring space charge distribution of a polymer dielectric film, the method comprising the steps of:
step S1: carrying out metallization treatment on two sides of the polymer dielectric film, and applying direct-current voltage;
step S2: pulse laser is hit to a metallized electrode on one side of the polymer dielectric film, the generated response current is measured, and a time domain response current expression is constructed;
step S3: carrying out Fourier transform on the time domain response current expression by combining a Poisson equation and a one-dimensional heat conduction equation to obtain a frequency domain expression of response current related to space charge distribution;
step S4: performing high-frequency approximation on the frequency domain expression to obtain an approximate frequency domain expression;
step S5: and solving the approximate frequency domain expression to obtain space charge distribution.
The time domain response current expression is a Fredholm integral equation, and the time domain response current expression is as follows:
wherein i (T) is a response current time domain representation, x is a spatial position, T is time, Δ T (x, T) is a temperature change of the polymer dielectric film, A is an irradiation area of the polymer dielectric film hit by the pulsed laser, d is a thickness of the polymer dielectric film, and g (x) is a distribution function.
For non-polar media, g (x) is expressed as:
g(x)＝0 r(α-αx)E(x)
wherein, αIs temperature coefficient of dielectric constant, αxWhich is the coefficient of thermal expansion of the polymer dielectric film,0in order to have a dielectric constant in a vacuum,re (x) is the electric field distribution of the polymer dielectric film.
The frequency domain expression is as follows:
wherein D is the thermal diffusion coefficient, p, of the polymer dielectric film(x) Is the space charge distribution of the polymer dielectric film, omega is the angular frequency, x is the spatial position, A is the irradiated area of the polymer dielectric film struck by the pulsed laser, d is the thickness of the polymer dielectric film, αIs temperature coefficient of dielectric constant, αxΔ T (x, ω) is a frequency domain representation of the temperature change of the polymer dielectric film, and I (ω) is a frequency domain representation of the response current, which is the coefficient of thermal expansion of the polymer dielectric film.
In the frequency domain expression, let:
an approximate frequency domain expression is obtained, where q is the energy density of the pulsed laser at the metal electrode, η is the absorption rate of the pulsed laser at the metal electrode, k is the thermal conductivity of the polymer dielectric thin film,
the approximate frequency domain expression is as follows:
wherein M isρ＝cρod/[ηq(α-αx)]And c is the specific heat capacity, rho, of the polymer dielectric film0Is the density of the polymer dielectric film.
The step S5 is solved by a scaling method.
The space charge distribution obtained by the scale transformation method is as follows:
wherein the content of the first and second substances,
denotes the difference between the real and imaginary parts of I (ω) multiplied by γ.
The metallization treatment is to evaporate electrodes on two sides of the polymer dielectric film.
Compared with the prior art, the invention has the following advantages:
(1) the solution target is directly converted into space charge distribution, the space charge distribution can be obtained only by solving the numerical value of a first Fredholm integral equation, namely, the space charge distribution in the tested polymer dielectric film can be directly obtained only by solving a ill-condition equation, and the efficiency of data analysis is improved.
(2) The sensitivity of data analysis to noise is improved, the differential operation process is avoided, the influence of the noise on electric field distribution calculation is prevented from being further amplified in a space charge distribution curve, and the sensitivity to the noise in actual measurement is lower.
(3) The unreasonable oscillation of space charge data is greatly reduced, and the precision of the measurement result is improved.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a graph showing the results of the present invention.
Detailed Description
The invention is described in detail below with reference to the figures and specific embodiments. The present embodiment is implemented on the premise of the technical solution of the present invention, and a detailed implementation manner and a specific operation process are given, but the scope of the present invention is not limited to the following embodiments.
Examples
In this embodiment, the space charge distribution of the thermal pulse data processing is directly calculated by taking a scale transform method in the frequency domain as an example. The first Fredholm integral equation (response current expression) involved in the analysis of the thermal pulse method measurement space charge data is as follows:
wherein i (T) is a response current time domain representation, x is a spatial position, T is time, Δ T (x, T) is a temperature change of the polymer dielectric film, A is an irradiation area of the polymer dielectric film hit by the pulsed laser, d is a thickness of the polymer dielectric film, g (x) is a distribution function, and g (x) is expressed as:
g(x)＝0 r(α-αx)E(x) (2)
wherein, αIs temperature coefficient of dielectric constant, αxWhich is the coefficient of thermal expansion of the polymer dielectric film,0in order to have a dielectric constant in a vacuum,re (x) is the electric field distribution of the polymer dielectric film.
The basic steps are as follows:
performing metallization treatment on the polymer dielectric film (evaporating electrodes on two sides), applying direct-current voltage, striking pulse laser to the metal electrode on one side of the polymer dielectric film, measuring response current generated by the pulse laser, and constructing a response current expression;
according to one-dimensional heat conduction equation
Where D is the thermal diffusivity of the polymer dielectric film, equation (1) translates to:
and Fourier transform is carried out on the response current expression, and the formula (1) is converted into a first Fredholm integral equation in a frequency domain form:
where Δ T (x, ω) is a frequency domain representation of the temperature change of the polymer dielectric film.
According to the Poisson equation
e0erdE(x)/dx＝ρ(x) (6)
Converting equation (5) into a frequency domain expression of the response current with respect to the space charge distribution, which is targeted for solving the space charge distribution ρ (x) of the polymer dielectric thin film:
where D is the thermal diffusion coefficient of the polymer dielectric thin film, and I (ω) is the frequency domain representation of the response current.
The polymer dielectric film can be considered adiabatic on both sides during the measurement time, approximating the spatial position derivative for temperature change:
where q is the energy density of the pulsed laser at the metal electrode, η is the absorption rate of the pulsed laser at the metal electrode, k is the thermal conductivity of the polymer dielectric thin film,
from equation (8), an approximate expression of equation (7) is obtained:
wherein M isρ＝cρ0d/[ηq(α-αx)]And c is the specific heat capacity, rho, of the polymer dielectric film0Is the density of the polymer dielectric film.
Solving the charge density distribution solution in the polymer dielectric film according to a scale conversion method:
wherein the content of the first and second substances,
is the real part and imaginary part after multiplying I (omega) by gammaThe difference of the sections.
The following is a specific example:
in the case where a certain level of noise is superimposed on the response current, the result of comparing the method of the present embodiment with the conventional method is shown in fig. 2, and it can be seen that the method of the present embodiment obtains a more accurate result.
Claims (9)
1. A method for measuring space charge distribution of a polymer dielectric film, comprising the steps of:
step S1: carrying out metallization treatment on two sides of the polymer dielectric film, and applying direct-current voltage;
step S2: pulse laser is hit to a metallized electrode on one side of the polymer dielectric film, the generated response current is measured, and a time domain response current expression is constructed;
step S3: carrying out Fourier transform on the time domain response current expression by combining a Poisson equation and a one-dimensional heat conduction equation to obtain a frequency domain expression of response current related to space charge distribution;
step S4: performing high-frequency approximation on the frequency domain expression to obtain an approximate frequency domain expression;
step S5: and solving the approximate frequency domain expression to obtain space charge distribution.
2. The method as claimed in claim 1, wherein the time-domain response current expression is Fredholm integral equation, and the time-domain response current expression is:
wherein i (T) is a response current time domain representation, x is a spatial position, T is time, Δ T (x, T) is a temperature change of the polymer dielectric film, A is an irradiation area of the polymer dielectric film hit by the pulsed laser, d is a thickness of the polymer dielectric film, and g (x) is a distribution function.
3. The method of claim 2, wherein g (x) is expressed as:
g(x)＝0 r(α-αx)E(x)
wherein, αIs temperature coefficient of dielectric constant, αxWhich is the coefficient of thermal expansion of the polymer dielectric film,0in order to have a dielectric constant in a vacuum,re (x) is the electric field distribution of the polymer dielectric film.
4. The method as claimed in claim 1, wherein the frequency domain expression is:
wherein D is the thermal diffusion coefficient of the polymer dielectric film, ρ (x) is the space charge distribution of the polymer dielectric film, ω is the angular frequency, x is the spatial position, A is the irradiation area of the polymer dielectric film struck by the pulsed laser, D is the thickness of the polymer dielectric film, αIs temperature coefficient of dielectric constant, αxΔ T (x, ω) is a frequency domain representation of the temperature change of the polymer dielectric film, and I (ω) is a frequency domain representation of the response current, which is the coefficient of thermal expansion of the polymer dielectric film.
5. The method as claimed in claim 4, wherein in the frequency domain expression, the following steps are performed:
obtaining an approximate frequency domain expression, wherein q is the energy density of the pulse laser at the metal electrode, and η is the pulseThe absorption of the laser light at the metal electrode, k is the thermal conductivity of the polymer dielectric film,
6. the method as claimed in claim 5, wherein the approximate frequency domain expression is:
wherein M isρ＝cρ0d/[ηq(α-αx)]And c is the specific heat capacity, rho, of the polymer dielectric film0Is the density of the polymer dielectric film.
7. The method as claimed in claim 6, wherein the step S5 is solved by a scale transformation method.
8. The method of claim 7, wherein the space charge distribution obtained by the scaling method is:
wherein the content of the first and second substances,
denotes the difference between the real and imaginary parts of I (ω) multiplied by γ.
9. The method as claimed in claim 1, wherein the metallization process is evaporation plating of electrodes on both sides of the polymer dielectric film.
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| CN113419120A (en) * | 2021-05-08 | 2021-09-21 | 同济大学 | Method and system for measuring thermal resistance of dielectric film and metal interface |
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